diff --git a/backend/main.py b/backend/main.py index 7addfb7..4f96c04 100644 --- a/backend/main.py +++ b/backend/main.py @@ -628,8 +628,6 @@ def create_manual_test( created_by=data.admin_id ) session.add(test_obj) - session.commit() - session.refresh(test_obj) # 2. Parse and create questions sections_dict = {} @@ -649,8 +647,6 @@ def create_manual_test( created_by=data.admin_id ) session.add(q_obj) - session.commit() - session.refresh(q_obj) sec_name = q_data.section if sec_name not in sections_dict: @@ -678,4 +674,69 @@ def toggle_test_status(test_id: UUID, session: Session = Depends(get_session)): session.add(test_obj) session.commit() - return {"status": "success", "is_active": test_obj.is_active} \ No newline at end of file + return {"status": "success", "is_active": test_obj.is_active} + +@app.delete("/api/admin/tests/{test_id}", tags=["Admin"]) +def delete_test(test_id: UUID, session: Session = Depends(get_session)): + from sqlmodel import select + test_obj = session.get(models.Test, test_id) + if not test_obj: + raise HTTPException(status_code=404, detail="Test not found") + + # 1. Identify all question IDs associated with the test + question_ids = [] + if test_obj.sections: + for section_name, section_data in test_obj.sections.items(): + question_ids.extend(section_data.get("questions", [])) + + # Convert question IDs to UUIDs + import uuid + q_uuids = [] + for q_id in question_ids: + try: + q_uuids.append(uuid.UUID(q_id) if isinstance(q_id, str) else q_id) + except ValueError: + continue + + # 2. Identify all attempts associated with this test + attempts_query = select(models.Attempt).where( + models.Attempt.source_id == test_id + ) + attempts = session.exec(attempts_query).all() + attempt_ids = [att.id for att in attempts] + + # 3. Delete Mistake records referencing these attempts + if attempt_ids: + mistakes_by_attempt_query = select(models.Mistake).where( + models.Mistake.attempt_id.in_(attempt_ids) + ) + mistakes_by_attempt = session.exec(mistakes_by_attempt_query).all() + for mistake in mistakes_by_attempt: + session.delete(mistake) + + # 4. Delete Mistake records referencing these questions + if q_uuids: + mistakes_by_q_query = select(models.Mistake).where( + models.Mistake.question_id.in_(q_uuids) + ) + mistakes_by_q = session.exec(mistakes_by_q_query).all() + for mistake in mistakes_by_q: + try: + session.delete(mistake) + except Exception: + pass + + # 5. Delete the Attempt records + for att in attempts: + session.delete(att) + + # 6. Delete the Question records + for q_uuid in q_uuids: + q_obj = session.get(models.Question, q_uuid) + if q_obj: + session.delete(q_obj) + + # 7. Delete the Test itself + session.delete(test_obj) + session.commit() + return {"status": "success", "message": "Test and all associated questions, attempts, and mistakes deleted successfully"} \ No newline at end of file diff --git a/jee_advanced_full_90q.json b/jee_advanced_full_90q.json new file mode 100644 index 0000000..605c953 --- /dev/null +++ b/jee_advanced_full_90q.json @@ -0,0 +1,1832 @@ +[ + { + "id": "101b3483-2514-48b4-b86b-630264373e6f", + "section": "General", + "subject": "Physics", + "topic": "Physics Topic 1", + "subtopic": "Physics Subtopic 1", + "difficulty": "Hard", + "type": "MCQ_Single", + "language": "English", + "question_text": "A sample Single Choice question for Physics #1.", + "options": { + "A": "Option 1", + "B": "Option 2", + "C": "Option 3", + "D": "Option 4" + }, + "correct_answer": { + "value": "A" + }, + "solution_text": "Explanation for this single choice question." + }, + { + "id": "17f0c361-04e9-423a-9a9c-c2f2b34fcbc8", + "section": "General", + "subject": "Physics", + "topic": "Physics Topic 1", + "subtopic": "Physics Subtopic 2", + "difficulty": "Medium", + "type": "MCQ_Multiple", + "language": "English", + "question_text": "Which of the following are correct for Physics #2?", + "options": { + "A": "Statement P is correct", + "B": "Statement Q is incorrect", + "C": "Statement R is correct", + "D": "Statement S is incorrect" + }, + "correct_answer": { + "value": [ + "A", + "C" + ] + }, + "solution_text": "A and C are correct because..." + }, + { + "id": "c59422e5-1459-483b-bd99-c1563d30f7bf", + "section": "General", + "subject": "Physics", + "topic": "Physics Topic 1", + "subtopic": "Physics Subtopic 3", + "difficulty": "Hard", + "type": "Numerical", + "language": "English", + "question_text": "Calculate the required value for Physics #3.", + "options": null, + "correct_answer": { + "value": "42.5" + }, + "solution_text": "Applying the formula yields 42.5." + }, + { + "id": "51feadbd-ee79-42c3-9beb-e7ced3b5ca98", + "section": "General", + "subject": "Physics", + "topic": "Physics Topic 1", + "subtopic": "Physics Subtopic 4", + "difficulty": "Hard", + "type": "MCQ_Single", + "language": "English", + "question_text": "A sample Single Choice question for Physics #4.", + "options": { + "A": "Option 1", + "B": "Option 2", + "C": "Option 3", + "D": "Option 4" + }, + "correct_answer": { + "value": "A" + }, + "solution_text": "Explanation for this single choice question." + }, + { + "id": "ef4ce35b-30ee-4dce-a177-92bd2b5f8501", + "section": "General", + "subject": "Physics", + "topic": "Physics Topic 1", + "subtopic": "Physics Subtopic 5", + "difficulty": "Medium", + "type": "MCQ_Multiple", + "language": "English", + "question_text": "Which of the following are correct for Physics #5?", + "options": { + "A": "Statement P is correct", + "B": "Statement Q is incorrect", + "C": "Statement R is correct", + "D": "Statement S is incorrect" + }, + "correct_answer": { + "value": [ + "A", + "C" + ] + }, + "solution_text": "A and C are correct because..." + }, + { + "id": "a3560537-2027-47b4-8476-65892ffdb9b2", + "section": "General", + "subject": "Physics", + "topic": "Physics Topic 2", + "subtopic": "Physics Subtopic 1", + "difficulty": "Medium", + "type": "Numerical", + "language": "English", + "question_text": "Calculate the required value for Physics #6.", + "options": null, + "correct_answer": { + "value": "42.5" + }, + "solution_text": "Applying the formula yields 42.5." + }, + { + "id": "703eaa26-5e3b-4955-80b6-95cc9bd24cb5", + "section": "General", + "subject": "Physics", + "topic": "Physics Topic 2", + "subtopic": "Physics Subtopic 2", + "difficulty": "Hard", + "type": "MCQ_Single", + "language": "English", + "question_text": "A sample Single Choice question for Physics #7.", + "options": { + "A": "Option 1", + "B": "Option 2", + "C": "Option 3", + "D": "Option 4" + }, + "correct_answer": { + "value": "A" + }, + "solution_text": "Explanation for this single choice question." + }, + { + "id": "6a41f0cc-eeb0-4785-ac1d-0f771fbafbbb", + "section": "General", + "subject": "Physics", + "topic": "Physics Topic 2", + "subtopic": "Physics Subtopic 3", + "difficulty": "Hard", + "type": "MCQ_Multiple", + "language": "English", + "question_text": "Which of the following are correct for Physics #8?", + "options": { + "A": "Statement P is correct", + "B": "Statement Q is incorrect", + "C": "Statement R is correct", + "D": "Statement S is incorrect" + }, + "correct_answer": { + "value": [ + "A", + "C" + ] + }, + "solution_text": "A and C are correct because..." + }, + { + "id": "13332aa0-532b-4126-a60e-ae50cbd515a7", + "section": "General", + "subject": "Physics", + "topic": "Physics Topic 2", + "subtopic": "Physics Subtopic 4", + "difficulty": "Hard", + "type": "Numerical", + "language": "English", + "question_text": "Calculate the required value for Physics #9.", + "options": null, + "correct_answer": { + "value": "42.5" + }, + "solution_text": "Applying the formula yields 42.5." + }, + { + "id": "7e33107e-8587-48d0-a8a4-bd816353ddb9", + "section": "General", + "subject": "Physics", + "topic": "Physics Topic 2", + "subtopic": "Physics Subtopic 5", + "difficulty": "Hard", + "type": "MCQ_Single", + "language": "English", + "question_text": "A sample Single Choice question for Physics #10.", + "options": { + "A": "Option 1", + "B": "Option 2", + "C": "Option 3", + "D": "Option 4" + }, + "correct_answer": { + "value": "A" + }, + "solution_text": "Explanation for this single choice question." + }, + { + "id": "abdb40b9-9c6e-4ad9-ac5e-5954d55b023b", + "section": "General", + "subject": "Physics", + "topic": "Physics Topic 3", + "subtopic": "Physics Subtopic 1", + "difficulty": "Medium", + "type": "MCQ_Multiple", + "language": "English", + "question_text": "Which of the following are correct for Physics #11?", + "options": { + "A": "Statement P is correct", + "B": "Statement Q is incorrect", + "C": "Statement R is correct", + "D": "Statement S is incorrect" + }, + "correct_answer": { + "value": [ + "A", + "C" + ] + }, + "solution_text": "A and C are correct because..." + }, + { + "id": "636f33a4-d72d-4eb1-a454-15ccf1e313f4", + "section": "General", + "subject": "Physics", + "topic": "Physics Topic 3", + "subtopic": "Physics Subtopic 2", + "difficulty": "Hard", + "type": "Numerical", + "language": "English", + "question_text": "Calculate the required value for Physics #12.", + "options": null, + "correct_answer": { + "value": "42.5" + }, + "solution_text": "Applying the formula yields 42.5." + }, + { + "id": "79f449ef-5b45-4f55-b5c1-be0b2ff0b8b4", + "section": "General", + "subject": "Physics", + "topic": "Physics Topic 3", + "subtopic": "Physics Subtopic 3", + "difficulty": "Hard", + "type": "MCQ_Single", + "language": "English", + "question_text": "A sample Single Choice question for Physics #13.", + "options": { + "A": "Option 1", + "B": "Option 2", + "C": "Option 3", + "D": "Option 4" + }, + "correct_answer": { + "value": "A" + }, + "solution_text": "Explanation for this single choice question." + }, + { + "id": "c9a9efd0-e4d7-414a-825f-c9862ff02397", + "section": "General", + "subject": "Physics", + "topic": "Physics Topic 3", + "subtopic": "Physics Subtopic 4", + "difficulty": "Medium", + "type": "MCQ_Multiple", + "language": "English", + "question_text": "Which of the following are correct for Physics #14?", + "options": { + "A": "Statement P is correct", + "B": "Statement Q is incorrect", + "C": "Statement R is correct", + "D": "Statement S is incorrect" + }, + "correct_answer": { + "value": [ + "A", + "C" + ] + }, + "solution_text": "A and C are correct because..." + }, + { + "id": "1949f4b6-65a9-45e8-9040-bdf838dfde1c", + "section": "General", + "subject": "Physics", + "topic": "Physics Topic 3", + "subtopic": "Physics Subtopic 5", + "difficulty": "Hard", + "type": "Numerical", + "language": "English", + "question_text": "Calculate the required value for Physics #15.", + "options": null, + "correct_answer": { + "value": "42.5" + }, + "solution_text": "Applying the formula yields 42.5." + }, + { + "id": "ba68a73d-c7f8-49dc-9b35-24be29d44db3", + "section": "General", + "subject": "Physics", + "topic": "Physics Topic 4", + "subtopic": "Physics Subtopic 1", + "difficulty": "Hard", + "type": "MCQ_Single", + "language": "English", + "question_text": "A sample Single Choice question for Physics #16.", + "options": { + "A": "Option 1", + "B": "Option 2", + "C": "Option 3", + "D": "Option 4" + }, + "correct_answer": { + "value": "A" + }, + "solution_text": "Explanation for this single choice question." + }, + { + "id": "6f53e1e7-2b64-4509-add9-435f79415971", + "section": "General", + "subject": "Physics", + "topic": "Physics Topic 4", + "subtopic": "Physics Subtopic 2", + "difficulty": "Hard", + "type": "MCQ_Multiple", + "language": "English", + "question_text": "Which of the following are correct for Physics #17?", + "options": { + "A": "Statement P is correct", + "B": "Statement Q is incorrect", + "C": "Statement R is correct", + "D": "Statement S is incorrect" + }, + "correct_answer": { + "value": [ + "A", + "C" + ] + }, + "solution_text": "A and C are correct because..." + }, + { + "id": "0535bed9-e2f9-4348-95cf-ba085ab6f06d", + "section": "General", + "subject": "Physics", + "topic": "Physics Topic 4", + "subtopic": "Physics Subtopic 3", + "difficulty": "Medium", + "type": "Numerical", + "language": "English", + "question_text": "Calculate the required value for Physics #18.", + "options": null, + "correct_answer": { + "value": "42.5" + }, + "solution_text": "Applying the formula yields 42.5." + }, + { + "id": "edbb3595-7e87-45c7-be98-f5b45934a017", + "section": "General", + "subject": "Physics", + "topic": "Physics Topic 4", + "subtopic": "Physics Subtopic 4", + "difficulty": "Medium", + "type": "MCQ_Single", + "language": "English", + "question_text": "A sample Single Choice question for Physics #19.", + "options": { + "A": "Option 1", + "B": "Option 2", + "C": "Option 3", + "D": "Option 4" + }, + "correct_answer": { + "value": "A" + }, + "solution_text": "Explanation for this single choice question." + }, + { + "id": "636a4f58-7401-408b-bc2e-3718580a42ff", + "section": "General", + "subject": "Physics", + "topic": "Physics Topic 4", + "subtopic": "Physics Subtopic 5", + "difficulty": "Hard", + "type": "MCQ_Multiple", + "language": "English", + "question_text": "Which of the following are correct for Physics #20?", + "options": { + "A": "Statement P is correct", + "B": "Statement Q is incorrect", + "C": "Statement R is correct", + "D": "Statement S is incorrect" + }, + "correct_answer": { + "value": [ + "A", + "C" + ] + }, + "solution_text": "A and C are correct because..." + }, + { + "id": "d7881877-08fe-4037-8c30-adec70bd6ae0", + "section": "General", + "subject": "Physics", + "topic": "Physics Topic 5", + "subtopic": "Physics Subtopic 1", + "difficulty": "Hard", + "type": "Numerical", + "language": "English", + "question_text": "Calculate the required value for Physics #21.", + "options": null, + "correct_answer": { + "value": "42.5" + }, + "solution_text": "Applying the formula yields 42.5." + }, + { + "id": "f3a937a3-c2f9-43d2-8213-f4295398e5a2", + "section": "General", + "subject": "Physics", + "topic": "Physics Topic 5", + "subtopic": "Physics Subtopic 2", + "difficulty": "Hard", + "type": "MCQ_Single", + "language": "English", + "question_text": "A sample Single Choice question for Physics #22.", + "options": { + "A": "Option 1", + "B": "Option 2", + "C": "Option 3", + "D": "Option 4" + }, + "correct_answer": { + "value": "A" + }, + "solution_text": "Explanation for this single choice question." + }, + { + "id": "bbdaf043-e749-4592-9ec5-12a8753b2c16", + "section": "General", + "subject": "Physics", + "topic": "Physics Topic 5", + "subtopic": "Physics Subtopic 3", + "difficulty": "Medium", + "type": "MCQ_Multiple", + "language": "English", + "question_text": "Which of the following are correct for Physics #23?", + "options": { + "A": "Statement P is correct", + "B": "Statement Q is incorrect", + "C": "Statement R is correct", + "D": "Statement S is incorrect" + }, + "correct_answer": { + "value": [ + "A", + "C" + ] + }, + "solution_text": "A and C are correct because..." + }, + { + "id": "dfe35647-6337-4957-bdfb-a6c178b42c48", + "section": "General", + "subject": "Physics", + "topic": "Physics Topic 5", + "subtopic": "Physics Subtopic 4", + "difficulty": "Hard", + "type": "Numerical", + "language": "English", + "question_text": "Calculate the required value for Physics #24.", + "options": null, + "correct_answer": { + "value": "42.5" + }, + "solution_text": "Applying the formula yields 42.5." + }, + { + "id": "4682f89a-ce27-4b56-a060-23e04fa816b3", + "section": "General", + "subject": "Physics", + "topic": "Physics Topic 5", + "subtopic": "Physics Subtopic 5", + "difficulty": "Hard", + "type": "MCQ_Single", + "language": "English", + "question_text": "A sample Single Choice question for Physics #25.", + "options": { + "A": "Option 1", + "B": "Option 2", + "C": "Option 3", + "D": "Option 4" + }, + "correct_answer": { + "value": "A" + }, + "solution_text": "Explanation for this single choice question." + }, + { + "id": "0063c154-e362-43db-9558-5e4b669c1d03", + "section": "General", + "subject": "Physics", + "topic": "Physics Topic 6", + "subtopic": "Physics Subtopic 1", + "difficulty": "Hard", + "type": "MCQ_Multiple", + "language": "English", + "question_text": "Which of the following are correct for Physics #26?", + "options": { + "A": "Statement P is correct", + "B": "Statement Q is incorrect", + "C": "Statement R is correct", + "D": "Statement S is incorrect" + }, + "correct_answer": { + "value": [ + "A", + "C" + ] + }, + "solution_text": "A and C are correct because..." + }, + { + "id": "cd0c58a8-059a-4b56-8e26-8eb28aaf4002", + "section": "General", + "subject": "Physics", + "topic": "Physics Topic 6", + "subtopic": "Physics Subtopic 2", + "difficulty": "Hard", + "type": "Numerical", + "language": "English", + "question_text": "Calculate the required value for Physics #27.", + "options": null, + "correct_answer": { + "value": "42.5" + }, + "solution_text": "Applying the formula yields 42.5." + }, + { + "id": "c120ae76-4a2c-4b62-a577-a1442842e490", + "section": "General", + "subject": "Physics", + "topic": "Physics Topic 6", + "subtopic": "Physics Subtopic 3", + "difficulty": "Hard", + "type": "MCQ_Single", + "language": "English", + "question_text": "A sample Single Choice question for Physics #28.", + "options": { + "A": "Option 1", + "B": "Option 2", + "C": "Option 3", + "D": "Option 4" + }, + "correct_answer": { + "value": "A" + }, + "solution_text": "Explanation for this single choice question." + }, + { + "id": "ca85f595-6ae3-475c-baa6-07932433de22", + "section": "General", + "subject": "Physics", + "topic": "Physics Topic 6", + "subtopic": "Physics Subtopic 4", + "difficulty": "Hard", + "type": "MCQ_Multiple", + "language": "English", + "question_text": "Which of the following are correct for Physics #29?", + "options": { + "A": "Statement P is correct", + "B": "Statement Q is incorrect", + "C": "Statement R is correct", + "D": "Statement S is incorrect" + }, + "correct_answer": { + "value": [ + "A", + "C" + ] + }, + "solution_text": "A and C are correct because..." + }, + { + "id": "83b9ab3c-a752-4378-8d80-6ac6a82e59b4", + "section": "General", + "subject": "Physics", + "topic": "Physics Topic 6", + "subtopic": "Physics Subtopic 5", + "difficulty": "Medium", + "type": "Numerical", + "language": "English", + "question_text": "Calculate the required value for Physics #30.", + "options": null, + "correct_answer": { + "value": "42.5" + }, + "solution_text": "Applying the formula yields 42.5." + }, + { + "id": "d601a599-6e41-4fac-a057-7113b5693f25", + "section": "General", + "subject": "Chemistry", + "topic": "Chemistry Topic 1", + "subtopic": "Chemistry Subtopic 1", + "difficulty": "Medium", + "type": "MCQ_Single", + "language": "English", + "question_text": "A sample Single Choice question for Chemistry #1.", + "options": { + "A": "Option 1", + "B": "Option 2", + "C": "Option 3", + "D": "Option 4" + }, + "correct_answer": { + "value": "A" + }, + "solution_text": "Explanation for this single choice question." + }, + { + "id": "79de929c-d77b-438b-a863-5de333f6c704", + "section": "General", + "subject": "Chemistry", + "topic": "Chemistry Topic 1", + "subtopic": "Chemistry Subtopic 2", + "difficulty": "Medium", + "type": "MCQ_Multiple", + "language": "English", + "question_text": "Which of the following are correct for Chemistry #2?", + "options": { + "A": "Statement P is correct", + "B": "Statement Q is incorrect", + "C": "Statement R is correct", + "D": "Statement S is incorrect" + }, + "correct_answer": { + "value": [ + "A", + "C" + ] + }, + "solution_text": "A and C are correct because..." + }, + { + "id": "9b08b698-6f60-4d92-b91d-2b7f86a31ab1", + "section": "General", + "subject": "Chemistry", + "topic": "Chemistry Topic 1", + "subtopic": "Chemistry Subtopic 3", + "difficulty": "Hard", + "type": "Numerical", + "language": "English", + "question_text": "Calculate the required value for Chemistry #3.", + "options": null, + "correct_answer": { + "value": "42.5" + }, + "solution_text": "Applying the formula yields 42.5." + }, + { + "id": "acea9034-a499-42c8-a83a-459334f74f12", + "section": "General", + "subject": "Chemistry", + "topic": "Chemistry Topic 1", + "subtopic": "Chemistry Subtopic 4", + "difficulty": "Hard", + "type": "MCQ_Single", + "language": "English", + "question_text": "A sample Single Choice question for Chemistry #4.", + "options": { + "A": "Option 1", + "B": "Option 2", + "C": "Option 3", + "D": "Option 4" + }, + "correct_answer": { + "value": "A" + }, + "solution_text": "Explanation for this single choice question." + }, + { + "id": "ca417255-9dcd-47e5-893a-33d1309bfb5a", + "section": "General", + "subject": "Chemistry", + "topic": "Chemistry Topic 1", + "subtopic": "Chemistry Subtopic 5", + "difficulty": "Hard", + "type": "MCQ_Multiple", + "language": "English", + "question_text": "Which of the following are correct for Chemistry #5?", + "options": { + "A": "Statement P is correct", + "B": "Statement Q is incorrect", + "C": "Statement R is correct", + "D": "Statement S is incorrect" + }, + "correct_answer": { + "value": [ + "A", + "C" + ] + }, + "solution_text": "A and C are correct because..." + }, + { + "id": "5813fa9f-be5c-4114-9a45-000da895bbd8", + "section": "General", + "subject": "Chemistry", + "topic": "Chemistry Topic 2", + "subtopic": "Chemistry Subtopic 1", + "difficulty": "Hard", + "type": "Numerical", + "language": "English", + "question_text": "Calculate the required value for Chemistry #6.", + "options": null, + "correct_answer": { + "value": "42.5" + }, + "solution_text": "Applying the formula yields 42.5." + }, + { + "id": "fb4a737a-66f8-4dd4-9c1d-77142d784093", + "section": "General", + "subject": "Chemistry", + "topic": "Chemistry Topic 2", + "subtopic": "Chemistry Subtopic 2", + "difficulty": "Medium", + "type": "MCQ_Single", + "language": "English", + "question_text": "A sample Single Choice question for Chemistry #7.", + "options": { + "A": "Option 1", + "B": "Option 2", + "C": "Option 3", + "D": "Option 4" + }, + "correct_answer": { + "value": "A" + }, + "solution_text": "Explanation for this single choice question." + }, + { + "id": "623ad0ff-8a05-48be-afe9-6ee0822d54f5", + "section": "General", + "subject": "Chemistry", + "topic": "Chemistry Topic 2", + "subtopic": "Chemistry Subtopic 3", + "difficulty": "Hard", + "type": "MCQ_Multiple", + "language": "English", + "question_text": "Which of the following are correct for Chemistry #8?", + "options": { + "A": "Statement P is correct", + "B": "Statement Q is incorrect", + "C": "Statement R is correct", + "D": "Statement S is incorrect" + }, + "correct_answer": { + "value": [ + "A", + "C" + ] + }, + "solution_text": "A and C are correct because..." + }, + { + "id": "101770cb-429e-43ba-968b-87749e71cadd", + "section": "General", + "subject": "Chemistry", + "topic": "Chemistry Topic 2", + "subtopic": "Chemistry Subtopic 4", + "difficulty": "Medium", + "type": "Numerical", + "language": "English", + "question_text": "Calculate the required value for Chemistry #9.", + "options": null, + "correct_answer": { + "value": "42.5" + }, + "solution_text": "Applying the formula yields 42.5." + }, + { + "id": "a01ed7b6-f52d-493a-ae7d-c0e1aafb2200", + "section": "General", + "subject": "Chemistry", + "topic": "Chemistry Topic 2", + "subtopic": "Chemistry Subtopic 5", + "difficulty": "Medium", + "type": "MCQ_Single", + "language": "English", + "question_text": "A sample Single Choice question for Chemistry #10.", + "options": { + "A": "Option 1", + "B": "Option 2", + "C": "Option 3", + "D": "Option 4" + }, + "correct_answer": { + "value": "A" + }, + "solution_text": "Explanation for this single choice question." + }, + { + "id": "72df0d56-84fa-4da5-84ee-207d9e67f888", + "section": "General", + "subject": "Chemistry", + "topic": "Chemistry Topic 3", + "subtopic": "Chemistry Subtopic 1", + "difficulty": "Hard", + "type": "MCQ_Multiple", + "language": "English", + "question_text": "Which of the following are correct for Chemistry #11?", + "options": { + "A": "Statement P is correct", + "B": "Statement Q is incorrect", + "C": "Statement R is correct", + "D": "Statement S is incorrect" + }, + "correct_answer": { + "value": [ + "A", + "C" + ] + }, + "solution_text": "A and C are correct because..." + }, + { + "id": "5c3aafcf-0012-4fee-aa87-1e932f96e5e7", + "section": "General", + "subject": "Chemistry", + "topic": "Chemistry Topic 3", + "subtopic": "Chemistry Subtopic 2", + "difficulty": "Hard", + "type": "Numerical", + "language": "English", + "question_text": "Calculate the required value for Chemistry #12.", + "options": null, + "correct_answer": { + "value": "42.5" + }, + "solution_text": "Applying the formula yields 42.5." + }, + { + "id": "57a12afc-f50a-4bfb-acab-c3ebc440a0a6", + "section": "General", + "subject": "Chemistry", + "topic": "Chemistry Topic 3", + "subtopic": "Chemistry Subtopic 3", + "difficulty": "Medium", + "type": "MCQ_Single", + "language": "English", + "question_text": "A sample Single Choice question for Chemistry #13.", + "options": { + "A": "Option 1", + "B": "Option 2", + "C": "Option 3", + "D": "Option 4" + }, + "correct_answer": { + "value": "A" + }, + "solution_text": "Explanation for this single choice question." + }, + { + "id": "5447d8b2-0c00-43a7-b488-7a420677d37c", + "section": "General", + "subject": "Chemistry", + "topic": "Chemistry Topic 3", + "subtopic": "Chemistry Subtopic 4", + "difficulty": "Hard", + "type": "MCQ_Multiple", + "language": "English", + "question_text": "Which of the following are correct for Chemistry #14?", + "options": { + "A": "Statement P is correct", + "B": "Statement Q is incorrect", + "C": "Statement R is correct", + "D": "Statement S is incorrect" + }, + "correct_answer": { + "value": [ + "A", + "C" + ] + }, + "solution_text": "A and C are correct because..." + }, + { + "id": "f805baad-8747-4ba2-b18e-dbdf96fd4c82", + "section": "General", + "subject": "Chemistry", + "topic": "Chemistry Topic 3", + "subtopic": "Chemistry Subtopic 5", + "difficulty": "Hard", + "type": "Numerical", + "language": "English", + "question_text": "Calculate the required value for Chemistry #15.", + "options": null, + "correct_answer": { + "value": "42.5" + }, + "solution_text": "Applying the formula yields 42.5." + }, + { + "id": "f4d08d8e-3958-495d-a30d-557b88a7e003", + "section": "General", + "subject": "Chemistry", + "topic": "Chemistry Topic 4", + "subtopic": "Chemistry Subtopic 1", + "difficulty": "Medium", + "type": "MCQ_Single", + "language": "English", + "question_text": "A sample Single Choice question for Chemistry #16.", + "options": { + "A": "Option 1", + "B": "Option 2", + "C": "Option 3", + "D": "Option 4" + }, + "correct_answer": { + "value": "A" + }, + "solution_text": "Explanation for this single choice question." + }, + { + "id": "bf39f2c6-f6f1-470a-8afa-7e0bd15a9198", + "section": "General", + "subject": "Chemistry", + "topic": "Chemistry Topic 4", + "subtopic": "Chemistry Subtopic 2", + "difficulty": "Hard", + "type": "MCQ_Multiple", + "language": "English", + "question_text": "Which of the following are correct for Chemistry #17?", + "options": { + "A": "Statement P is correct", + "B": "Statement Q is incorrect", + "C": "Statement R is correct", + "D": "Statement S is incorrect" + }, + "correct_answer": { + "value": [ + "A", + "C" + ] + }, + "solution_text": "A and C are correct because..." + }, + { + "id": "6876fbf5-6dc4-4d48-baaf-7354b0a749a6", + "section": "General", + "subject": "Chemistry", + "topic": "Chemistry Topic 4", + "subtopic": "Chemistry Subtopic 3", + "difficulty": "Hard", + "type": "Numerical", + "language": "English", + "question_text": "Calculate the required value for Chemistry #18.", + "options": null, + "correct_answer": { + "value": "42.5" + }, + "solution_text": "Applying the formula yields 42.5." + }, + { + "id": "ab0fc4ad-111e-4d63-aaf0-aaf187ab9e18", + "section": "General", + "subject": "Chemistry", + "topic": "Chemistry Topic 4", + "subtopic": "Chemistry Subtopic 4", + "difficulty": "Hard", + "type": "MCQ_Single", + "language": "English", + "question_text": "A sample Single Choice question for Chemistry #19.", + "options": { + "A": "Option 1", + "B": "Option 2", + "C": "Option 3", + "D": "Option 4" + }, + "correct_answer": { + "value": "A" + }, + "solution_text": "Explanation for this single choice question." + }, + { + "id": "ba9c184e-2f69-4619-94e9-229e250cf903", + "section": "General", + "subject": "Chemistry", + "topic": "Chemistry Topic 4", + "subtopic": "Chemistry Subtopic 5", + "difficulty": "Medium", + "type": "MCQ_Multiple", + "language": "English", + "question_text": "Which of the following are correct for Chemistry #20?", + "options": { + "A": "Statement P is correct", + "B": "Statement Q is incorrect", + "C": "Statement R is correct", + "D": "Statement S is incorrect" + }, + "correct_answer": { + "value": [ + "A", + "C" + ] + }, + "solution_text": "A and C are correct because..." + }, + { + "id": "7e849d03-a387-4134-a5d2-4d6daee2539b", + "section": "General", + "subject": "Chemistry", + "topic": "Chemistry Topic 5", + "subtopic": "Chemistry Subtopic 1", + "difficulty": "Medium", + "type": "Numerical", + "language": "English", + "question_text": "Calculate the required value for Chemistry #21.", + "options": null, + "correct_answer": { + "value": "42.5" + }, + "solution_text": "Applying the formula yields 42.5." + }, + { + "id": "dc815332-4f86-471a-b378-fd981b6aca0a", + "section": "General", + "subject": "Chemistry", + "topic": "Chemistry Topic 5", + "subtopic": "Chemistry Subtopic 2", + "difficulty": "Hard", + "type": "MCQ_Single", + "language": "English", + "question_text": "A sample Single Choice question for Chemistry #22.", + "options": { + "A": "Option 1", + "B": "Option 2", + "C": "Option 3", + "D": "Option 4" + }, + "correct_answer": { + "value": "A" + }, + "solution_text": "Explanation for this single choice question." + }, + { + "id": "8d38c199-df59-4704-986e-e27420a899df", + "section": "General", + "subject": "Chemistry", + "topic": "Chemistry Topic 5", + "subtopic": "Chemistry Subtopic 3", + "difficulty": "Medium", + "type": "MCQ_Multiple", + "language": "English", + "question_text": "Which of the following are correct for Chemistry #23?", + "options": { + "A": "Statement P is correct", + "B": "Statement Q is incorrect", + "C": "Statement R is correct", + "D": "Statement S is incorrect" + }, + "correct_answer": { + "value": [ + "A", + "C" + ] + }, + "solution_text": "A and C are correct because..." + }, + { + "id": "9e6983ae-7a18-4d49-aa40-d064b809db76", + "section": "General", + "subject": "Chemistry", + "topic": "Chemistry Topic 5", + "subtopic": "Chemistry Subtopic 4", + "difficulty": "Medium", + "type": "Numerical", + "language": "English", + "question_text": "Calculate the required value for Chemistry #24.", + "options": null, + "correct_answer": { + "value": "42.5" + }, + "solution_text": "Applying the formula yields 42.5." + }, + { + "id": "062c9856-39b8-45ce-bb71-6d81616c24bb", + "section": "General", + "subject": "Chemistry", + "topic": "Chemistry Topic 5", + "subtopic": "Chemistry Subtopic 5", + "difficulty": "Medium", + "type": "MCQ_Single", + "language": "English", + "question_text": "A sample Single Choice question for Chemistry #25.", + "options": { + "A": "Option 1", + "B": "Option 2", + "C": "Option 3", + "D": "Option 4" + }, + "correct_answer": { + "value": "A" + }, + "solution_text": "Explanation for this single choice question." + }, + { + "id": "0a5a77ef-5053-468c-be39-8a771f3d12e1", + "section": "General", + "subject": "Chemistry", + "topic": "Chemistry Topic 6", + "subtopic": "Chemistry Subtopic 1", + "difficulty": "Medium", + "type": "MCQ_Multiple", + "language": "English", + "question_text": "Which of the following are correct for Chemistry #26?", + "options": { + "A": "Statement P is correct", + "B": "Statement Q is incorrect", + "C": "Statement R is correct", + "D": "Statement S is incorrect" + }, + "correct_answer": { + "value": [ + "A", + "C" + ] + }, + "solution_text": "A and C are correct because..." + }, + { + "id": "d389a400-1172-46a7-8ea0-9a8b36ff57cf", + "section": "General", + "subject": "Chemistry", + "topic": "Chemistry Topic 6", + "subtopic": "Chemistry Subtopic 2", + "difficulty": "Medium", + "type": "Numerical", + "language": "English", + "question_text": "Calculate the required value for Chemistry #27.", + "options": null, + "correct_answer": { + "value": "42.5" + }, + "solution_text": "Applying the formula yields 42.5." + }, + { + "id": "36c062e0-39a7-4007-8109-b3d054d8f8b9", + "section": "General", + "subject": "Chemistry", + "topic": "Chemistry Topic 6", + "subtopic": "Chemistry Subtopic 3", + "difficulty": "Hard", + "type": "MCQ_Single", + "language": "English", + "question_text": "A sample Single Choice question for Chemistry #28.", + "options": { + "A": "Option 1", + "B": "Option 2", + "C": "Option 3", + "D": "Option 4" + }, + "correct_answer": { + "value": "A" + }, + "solution_text": "Explanation for this single choice question." + }, + { + "id": "35a71a95-4ad7-4d13-b749-8ff310cfb1df", + "section": "General", + "subject": "Chemistry", + "topic": "Chemistry Topic 6", + "subtopic": "Chemistry Subtopic 4", + "difficulty": "Hard", + "type": "MCQ_Multiple", + "language": "English", + "question_text": "Which of the following are correct for Chemistry #29?", + "options": { + "A": "Statement P is correct", + "B": "Statement Q is incorrect", + "C": "Statement R is correct", + "D": "Statement S is incorrect" + }, + "correct_answer": { + "value": [ + "A", + "C" + ] + }, + "solution_text": "A and C are correct because..." + }, + { + "id": "b8465837-7cd8-408d-9ecb-54d8552ea977", + "section": "General", + "subject": "Chemistry", + "topic": "Chemistry Topic 6", + "subtopic": "Chemistry Subtopic 5", + "difficulty": "Hard", + "type": "Numerical", + "language": "English", + "question_text": "Calculate the required value for Chemistry #30.", + "options": null, + "correct_answer": { + "value": "42.5" + }, + "solution_text": "Applying the formula yields 42.5." + }, + { + "id": "2b0b4d4b-7109-408f-b092-83e39a7242de", + "section": "General", + "subject": "Mathematics", + "topic": "Mathematics Topic 1", + "subtopic": "Mathematics Subtopic 1", + "difficulty": "Hard", + "type": "MCQ_Single", + "language": "English", + "question_text": "A sample Single Choice question for Mathematics #1.", + "options": { + "A": "Option 1", + "B": "Option 2", + "C": "Option 3", + "D": "Option 4" + }, + "correct_answer": { + "value": "A" + }, + "solution_text": "Explanation for this single choice question." + }, + { + "id": "20a99e4c-5267-4cce-9a57-cbdafd36fb01", + "section": "General", + "subject": "Mathematics", + "topic": "Mathematics Topic 1", + "subtopic": "Mathematics Subtopic 2", + "difficulty": "Medium", + "type": "MCQ_Multiple", + "language": "English", + "question_text": "Which of the following are correct for Mathematics #2?", + "options": { + "A": "Statement P is correct", + "B": "Statement Q is incorrect", + "C": "Statement R is correct", + "D": "Statement S is incorrect" + }, + "correct_answer": { + "value": [ + "A", + "C" + ] + }, + "solution_text": "A and C are correct because..." + }, + { + "id": "c2e1387b-3ddf-497a-a694-8a2ce1eccd69", + "section": "General", + "subject": "Mathematics", + "topic": "Mathematics Topic 1", + "subtopic": "Mathematics Subtopic 3", + "difficulty": "Hard", + "type": "Numerical", + "language": "English", + "question_text": "Calculate the required value for Mathematics #3.", + "options": null, + "correct_answer": { + "value": "42.5" + }, + "solution_text": "Applying the formula yields 42.5." + }, + { + "id": "b5487dff-617d-44dd-96e9-766b28a00ac0", + "section": "General", + "subject": "Mathematics", + "topic": "Mathematics Topic 1", + "subtopic": "Mathematics Subtopic 4", + "difficulty": "Hard", + "type": "MCQ_Single", + "language": "English", + "question_text": "A sample Single Choice question for Mathematics #4.", + "options": { + "A": "Option 1", + "B": "Option 2", + "C": "Option 3", + "D": "Option 4" + }, + "correct_answer": { + "value": "A" + }, + "solution_text": "Explanation for this single choice question." + }, + { + "id": "2f8845ab-4f98-40ab-b230-e66c46cbb4c2", + "section": "General", + "subject": "Mathematics", + "topic": "Mathematics Topic 1", + "subtopic": "Mathematics Subtopic 5", + "difficulty": "Medium", + "type": "MCQ_Multiple", + "language": "English", + "question_text": "Which of the following are correct for Mathematics #5?", + "options": { + "A": "Statement P is correct", + "B": "Statement Q is incorrect", + "C": "Statement R is correct", + "D": "Statement S is incorrect" + }, + "correct_answer": { + "value": [ + "A", + "C" + ] + }, + "solution_text": "A and C are correct because..." + }, + { + "id": "000bcdbf-5c70-4188-8980-33754190d593", + "section": "General", + "subject": "Mathematics", + "topic": "Mathematics Topic 2", + "subtopic": "Mathematics Subtopic 1", + "difficulty": "Hard", + "type": "Numerical", + "language": "English", + "question_text": "Calculate the required value for Mathematics #6.", + "options": null, + "correct_answer": { + "value": "42.5" + }, + "solution_text": "Applying the formula yields 42.5." + }, + { + "id": "137b6fa6-23d8-4d10-a1ae-60697dd77b25", + "section": "General", + "subject": "Mathematics", + "topic": "Mathematics Topic 2", + "subtopic": "Mathematics Subtopic 2", + "difficulty": "Medium", + "type": "MCQ_Single", + "language": "English", + "question_text": "A sample Single Choice question for Mathematics #7.", + "options": { + "A": "Option 1", + "B": "Option 2", + "C": "Option 3", + "D": "Option 4" + }, + "correct_answer": { + "value": "A" + }, + "solution_text": "Explanation for this single choice question." + }, + { + "id": "9bc7d11a-37f4-4e67-a566-676d60d98333", + "section": "General", + "subject": "Mathematics", + "topic": "Mathematics Topic 2", + "subtopic": "Mathematics Subtopic 3", + "difficulty": "Medium", + "type": "MCQ_Multiple", + "language": "English", + "question_text": "Which of the following are correct for Mathematics #8?", + "options": { + "A": "Statement P is correct", + "B": "Statement Q is incorrect", + "C": "Statement R is correct", + "D": "Statement S is incorrect" + }, + "correct_answer": { + "value": [ + "A", + "C" + ] + }, + "solution_text": "A and C are correct because..." + }, + { + "id": "afbf0dcc-8927-4685-8b7c-6d05c5afc90a", + "section": "General", + "subject": "Mathematics", + "topic": "Mathematics Topic 2", + "subtopic": "Mathematics Subtopic 4", + "difficulty": "Hard", + "type": "Numerical", + "language": "English", + "question_text": "Calculate the required value for Mathematics #9.", + "options": null, + "correct_answer": { + "value": "42.5" + }, + "solution_text": "Applying the formula yields 42.5." + }, + { + "id": "593706ee-9e39-44fc-bd60-efdbca51868b", + "section": "General", + "subject": "Mathematics", + "topic": "Mathematics Topic 2", + "subtopic": "Mathematics Subtopic 5", + "difficulty": "Medium", + "type": "MCQ_Single", + "language": "English", + "question_text": "A sample Single Choice question for Mathematics #10.", + "options": { + "A": "Option 1", + "B": "Option 2", + "C": "Option 3", + "D": "Option 4" + }, + "correct_answer": { + "value": "A" + }, + "solution_text": "Explanation for this single choice question." + }, + { + "id": "e73de16c-a16b-4213-8bf4-f8820414bb72", + "section": "General", + "subject": "Mathematics", + "topic": "Mathematics Topic 3", + "subtopic": "Mathematics Subtopic 1", + "difficulty": "Hard", + "type": "MCQ_Multiple", + "language": "English", + "question_text": "Which of the following are correct for Mathematics #11?", + "options": { + "A": "Statement P is correct", + "B": "Statement Q is incorrect", + "C": "Statement R is correct", + "D": "Statement S is incorrect" + }, + "correct_answer": { + "value": [ + "A", + "C" + ] + }, + "solution_text": "A and C are correct because..." + }, + { + "id": "450c48a4-9982-401a-b0d9-5a09adaeef95", + "section": "General", + "subject": "Mathematics", + "topic": "Mathematics Topic 3", + "subtopic": "Mathematics Subtopic 2", + "difficulty": "Hard", + "type": "Numerical", + "language": "English", + "question_text": "Calculate the required value for Mathematics #12.", + "options": null, + "correct_answer": { + "value": "42.5" + }, + "solution_text": "Applying the formula yields 42.5." + }, + { + "id": "50d88f8f-3997-4e1f-9b30-16c222b5bc3a", + "section": "General", + "subject": "Mathematics", + "topic": "Mathematics Topic 3", + "subtopic": "Mathematics Subtopic 3", + "difficulty": "Hard", + "type": "MCQ_Single", + "language": "English", + "question_text": "A sample Single Choice question for Mathematics #13.", + "options": { + "A": "Option 1", + "B": "Option 2", + "C": "Option 3", + "D": "Option 4" + }, + "correct_answer": { + "value": "A" + }, + "solution_text": "Explanation for this single choice question." + }, + { + "id": "655bf082-4822-437e-b3cc-47fb7997f3a7", + "section": "General", + "subject": "Mathematics", + "topic": "Mathematics Topic 3", + "subtopic": "Mathematics Subtopic 4", + "difficulty": "Medium", + "type": "MCQ_Multiple", + "language": "English", + "question_text": "Which of the following are correct for Mathematics #14?", + "options": { + "A": "Statement P is correct", + "B": "Statement Q is incorrect", + "C": "Statement R is correct", + "D": "Statement S is incorrect" + }, + "correct_answer": { + "value": [ + "A", + "C" + ] + }, + "solution_text": "A and C are correct because..." + }, + { + "id": "dfa762be-19f9-4c85-aa07-6fdb53217ab8", + "section": "General", + "subject": "Mathematics", + "topic": "Mathematics Topic 3", + "subtopic": "Mathematics Subtopic 5", + "difficulty": "Medium", + "type": "Numerical", + "language": "English", + "question_text": "Calculate the required value for Mathematics #15.", + "options": null, + "correct_answer": { + "value": "42.5" + }, + "solution_text": "Applying the formula yields 42.5." + }, + { + "id": "e4983846-4964-4dae-8a57-696f5658ee38", + "section": "General", + "subject": "Mathematics", + "topic": "Mathematics Topic 4", + "subtopic": "Mathematics Subtopic 1", + "difficulty": "Hard", + "type": "MCQ_Single", + "language": "English", + "question_text": "A sample Single Choice question for Mathematics #16.", + "options": { + "A": "Option 1", + "B": "Option 2", + "C": "Option 3", + "D": "Option 4" + }, + "correct_answer": { + "value": "A" + }, + "solution_text": "Explanation for this single choice question." + }, + { + "id": "7c9f8759-eebd-4d9c-8ad5-0f1814256838", + "section": "General", + "subject": "Mathematics", + "topic": "Mathematics Topic 4", + "subtopic": "Mathematics Subtopic 2", + "difficulty": "Medium", + "type": "MCQ_Multiple", + "language": "English", + "question_text": "Which of the following are correct for Mathematics #17?", + "options": { + "A": "Statement P is correct", + "B": "Statement Q is incorrect", + "C": "Statement R is correct", + "D": "Statement S is incorrect" + }, + "correct_answer": { + "value": [ + "A", + "C" + ] + }, + "solution_text": "A and C are correct because..." + }, + { + "id": "89abd7fb-6e44-4001-a889-e55679474fc0", + "section": "General", + "subject": "Mathematics", + "topic": "Mathematics Topic 4", + "subtopic": "Mathematics Subtopic 3", + "difficulty": "Medium", + "type": "Numerical", + "language": "English", + "question_text": "Calculate the required value for Mathematics #18.", + "options": null, + "correct_answer": { + "value": "42.5" + }, + "solution_text": "Applying the formula yields 42.5." + }, + { + "id": "9ff01fbd-77e5-403d-bd1a-213b7173e580", + "section": "General", + "subject": "Mathematics", + "topic": "Mathematics Topic 4", + "subtopic": "Mathematics Subtopic 4", + "difficulty": "Medium", + "type": "MCQ_Single", + "language": "English", + "question_text": "A sample Single Choice question for Mathematics #19.", + "options": { + "A": "Option 1", + "B": "Option 2", + "C": "Option 3", + "D": "Option 4" + }, + "correct_answer": { + "value": "A" + }, + "solution_text": "Explanation for this single choice question." + }, + { + "id": "65d3fd65-1a76-42b9-8ce2-c63ff4d31b94", + "section": "General", + "subject": "Mathematics", + "topic": "Mathematics Topic 4", + "subtopic": "Mathematics Subtopic 5", + "difficulty": "Hard", + "type": "MCQ_Multiple", + "language": "English", + "question_text": "Which of the following are correct for Mathematics #20?", + "options": { + "A": "Statement P is correct", + "B": "Statement Q is incorrect", + "C": "Statement R is correct", + "D": "Statement S is incorrect" + }, + "correct_answer": { + "value": [ + "A", + "C" + ] + }, + "solution_text": "A and C are correct because..." + }, + { + "id": "cea876ec-3c06-4e7d-b05b-dcfbc31fa2d3", + "section": "General", + "subject": "Mathematics", + "topic": "Mathematics Topic 5", + "subtopic": "Mathematics Subtopic 1", + "difficulty": "Medium", + "type": "Numerical", + "language": "English", + "question_text": "Calculate the required value for Mathematics #21.", + "options": null, + "correct_answer": { + "value": "42.5" + }, + "solution_text": "Applying the formula yields 42.5." + }, + { + "id": "df41e3a2-d048-4efc-bd62-6725590bfedc", + "section": "General", + "subject": "Mathematics", + "topic": "Mathematics Topic 5", + "subtopic": "Mathematics Subtopic 2", + "difficulty": "Medium", + "type": "MCQ_Single", + "language": "English", + "question_text": "A sample Single Choice question for Mathematics #22.", + "options": { + "A": "Option 1", + "B": "Option 2", + "C": "Option 3", + "D": "Option 4" + }, + "correct_answer": { + "value": "A" + }, + "solution_text": "Explanation for this single choice question." + }, + { + "id": "cb19dab2-8a1a-4b8e-a564-15b6b4d7252b", + "section": "General", + "subject": "Mathematics", + "topic": "Mathematics Topic 5", + "subtopic": "Mathematics Subtopic 3", + "difficulty": "Medium", + "type": "MCQ_Multiple", + "language": "English", + "question_text": "Which of the following are correct for Mathematics #23?", + "options": { + "A": "Statement P is correct", + "B": "Statement Q is incorrect", + "C": "Statement R is correct", + "D": "Statement S is incorrect" + }, + "correct_answer": { + "value": [ + "A", + "C" + ] + }, + "solution_text": "A and C are correct because..." + }, + { + "id": "aa08e317-b9f6-409c-b2b3-dcc6b4408d45", + "section": "General", + "subject": "Mathematics", + "topic": "Mathematics Topic 5", + "subtopic": "Mathematics Subtopic 4", + "difficulty": "Medium", + "type": "Numerical", + "language": "English", + "question_text": "Calculate the required value for Mathematics #24.", + "options": null, + "correct_answer": { + "value": "42.5" + }, + "solution_text": "Applying the formula yields 42.5." + }, + { + "id": "5282dc5a-de89-4f33-9a1f-ee21ae2c00f7", + "section": "General", + "subject": "Mathematics", + "topic": "Mathematics Topic 5", + "subtopic": "Mathematics Subtopic 5", + "difficulty": "Medium", + "type": "MCQ_Single", + "language": "English", + "question_text": "A sample Single Choice question for Mathematics #25.", + "options": { + "A": "Option 1", + "B": "Option 2", + "C": "Option 3", + "D": "Option 4" + }, + "correct_answer": { + "value": "A" + }, + "solution_text": "Explanation for this single choice question." + }, + { + "id": "c12be55a-cf2e-4ae7-b9b1-b0fcb70a26ed", + "section": "General", + "subject": "Mathematics", + "topic": "Mathematics Topic 6", + "subtopic": "Mathematics Subtopic 1", + "difficulty": "Hard", + "type": "MCQ_Multiple", + "language": "English", + "question_text": "Which of the following are correct for Mathematics #26?", + "options": { + "A": "Statement P is correct", + "B": "Statement Q is incorrect", + "C": "Statement R is correct", + "D": "Statement S is incorrect" + }, + "correct_answer": { + "value": [ + "A", + "C" + ] + }, + "solution_text": "A and C are correct because..." + }, + { + "id": "250c8530-c9fb-4ad9-b91c-c623cc4759ef", + "section": "General", + "subject": "Mathematics", + "topic": "Mathematics Topic 6", + "subtopic": "Mathematics Subtopic 2", + "difficulty": "Hard", + "type": "Numerical", + "language": "English", + "question_text": "Calculate the required value for Mathematics #27.", + "options": null, + "correct_answer": { + "value": "42.5" + }, + "solution_text": "Applying the formula yields 42.5." + }, + { + "id": "354d3a5d-7622-4857-9512-6b2a1476645e", + "section": "General", + "subject": "Mathematics", + "topic": "Mathematics Topic 6", + "subtopic": "Mathematics Subtopic 3", + "difficulty": "Hard", + "type": "MCQ_Single", + "language": "English", + "question_text": "A sample Single Choice question for Mathematics #28.", + "options": { + "A": "Option 1", + "B": "Option 2", + "C": "Option 3", + "D": "Option 4" + }, + "correct_answer": { + "value": "A" + }, + "solution_text": "Explanation for this single choice question." + }, + { + "id": "25c299b0-b0f7-4360-b22c-c3b5219126b1", + "section": "General", + "subject": "Mathematics", + "topic": "Mathematics Topic 6", + "subtopic": "Mathematics Subtopic 4", + "difficulty": "Medium", + "type": "MCQ_Multiple", + "language": "English", + "question_text": "Which of the following are correct for Mathematics #29?", + "options": { + "A": "Statement P is correct", + "B": "Statement Q is incorrect", + "C": "Statement R is correct", + "D": "Statement S is incorrect" + }, + "correct_answer": { + "value": [ + "A", + "C" + ] + }, + "solution_text": "A and C are correct because..." + }, + { + "id": "38524542-ed8c-43c0-97d9-1219dc330b7f", + "section": "General", + "subject": "Mathematics", + "topic": "Mathematics Topic 6", + "subtopic": "Mathematics Subtopic 5", + "difficulty": "Hard", + "type": "Numerical", + "language": "English", + "question_text": "Calculate the required value for Mathematics #30.", + "options": null, + "correct_answer": { + "value": "42.5" + }, + "solution_text": "Applying the formula yields 42.5." + } +] diff --git a/neet_180_questions.json b/neet_180_questions.json new file mode 100644 index 0000000..ceb5539 --- /dev/null +++ b/neet_180_questions.json @@ -0,0 +1,3782 @@ +[ + { + "section": "General", + "subject": "Physics", + "topic": "Units and Measurements (Set 1)", + "subtopic": "Dimensions", + "difficulty": "Medium", + "type": "MCQ_Single", + "question_text": "The dimension of $\\frac{1}{\\sqrt{\\mu_0 \\varepsilon_0}}$ is that of:", + "options": { + "A": "Velocity", + "B": "Time", + "C": "Capacitance", + "D": "Distance" + }, + "correct_answer": { + "value": "A" + }, + "solution_text": "We know that $c = \\frac{1}{\\sqrt{\\mu_0 \\varepsilon_0}}$, where $c$ is the speed of light in vacuum. Therefore, the dimension of $\\frac{1}{\\sqrt{\\mu_0 \\varepsilon_0}}$ is that of velocity, which is $[L T^{-1}]$.", + "language": "English", + "id": "858a5001-97c3-4d03-873c-b83247223ed7" + }, + { + "section": "General", + "subject": "Physics", + "topic": "Kinematics (Set 1)", + "subtopic": "Projectile Motion", + "difficulty": "Easy", + "type": "MCQ_Single", + "question_text": "Two projectiles are thrown with the same initial velocity at angles of $30^\\circ$ and $60^\\circ$ with the horizontal. The ratio of their maximum heights is:", + "options": { + "A": "1:3", + "B": "3:1", + "C": "1:\\sqrt{3}", + "D": "1:2" + }, + "correct_answer": { + "value": "A" + }, + "solution_text": "Maximum height $H = \\frac{u^2 \\sin^2 \\theta}{2g}$.\nFor $\\theta_1 = 30^\\circ$, $H_1 = \\frac{u^2 (1/2)^2}{2g} = \\frac{u^2}{8g}$.\nFor $\\theta_2 = 60^\\circ$, $H_2 = \\frac{u^2 (\\sqrt{3}/2)^2}{2g} = \\frac{3u^2}{8g}$.\nRatio $H_1 : H_2 = 1 : 3$.", + "language": "English", + "id": "b6e37b17-d6bb-4e79-b3d4-b96a80a060b4" + }, + { + "section": "General", + "subject": "Physics", + "topic": "Thermodynamics (Set 1)", + "subtopic": "Carnot Engine", + "difficulty": "Medium", + "type": "MCQ_Single", + "question_text": "A Carnot engine takes heat from a reservoir at $627^\\circ\\text{C}$ and rejects heat to a sink at $27^\\circ\\text{C}$. Its efficiency is:", + "options": { + "A": "2/3", + "B": "1/3", + "C": "3/4", + "D": "1/2" + }, + "correct_answer": { + "value": "A" + }, + "solution_text": "Temperature of source, $T_1 = 627 + 273 = 900\\text{ K}$.\nTemperature of sink, $T_2 = 27 + 273 = 300\\text{ K}$.\nEfficiency $\\eta = 1 - \\frac{T_2}{T_1} = 1 - \\frac{300}{900} = 1 - \\frac{1}{3} = \\frac{2}{3}$.", + "language": "English", + "id": "c01d4590-27ad-47e9-9f07-217b6dcfa0b9" + }, + { + "section": "General", + "subject": "Physics", + "topic": "Electrostatics (Set 1)", + "subtopic": "Capacitors", + "difficulty": "Hard", + "type": "MCQ_Single", + "question_text": "A parallel plate capacitor with air between the plates has a capacitance of $8\\text{ pF}$. What will be the capacitance if the distance between the plates is reduced by half, and the space between them is filled with a substance of dielectric constant $6$?", + "options": { + "A": "96 pF", + "B": "8 pF", + "C": "48 pF", + "D": "12 pF" + }, + "correct_answer": { + "value": "A" + }, + "solution_text": "Initial capacitance $C_0 = \\frac{\\varepsilon_0 A}{d} = 8\\text{ pF}$.\nNew distance $d' = d/2$ and new dielectric constant $K = 6$.\nNew capacitance $C' = \\frac{K \\varepsilon_0 A}{d'} = \\frac{6 \\varepsilon_0 A}{d/2} = 12 \\left(\\frac{\\varepsilon_0 A}{d}\\right) = 12 \\times 8\\text{ pF} = 96\\text{ pF}$.", + "language": "English", + "id": "4698d5d3-0bbf-42f5-88b6-06cd17d09f75" + }, + { + "section": "General", + "subject": "Physics", + "topic": "Current Electricity (Set 1)", + "subtopic": "Meter Bridge", + "difficulty": "Medium", + "type": "MCQ_Single", + "question_text": "In a meter bridge experiment, the null point is obtained at $20\\text{ cm}$ from one end of the wire when resistance $X$ is balanced against another resistance $Y$. If $X < Y$, then what will be the new position of the null point from the same end, if one decides to balance a resistance of $4X$ against $Y$?", + "options": { + "A": "50 cm", + "B": "40 cm", + "C": "80 cm", + "D": "20 cm" + }, + "correct_answer": { + "value": "A" + }, + "solution_text": "Initially, $\\frac{X}{Y} = \\frac{20}{100-20} = \\frac{20}{80} = \\frac{1}{4}$. So $Y = 4X$.\nFor new resistance $4X$ against $Y$, ratio is $\\frac{4X}{Y} = \\frac{4X}{4X} = 1$.\nLet new null point be $l$, then $\\frac{l}{100-l} = 1 \\implies l = 100 - l \\implies 2l = 100 \\implies l = 50\\text{ cm}$.", + "language": "English", + "id": "b845ae53-3ffe-4ade-9c9a-1d7c96cad1ea" + }, + { + "section": "General", + "subject": "Physics", + "topic": "Magnetism (Set 1)", + "subtopic": "Biot-Savart Law", + "difficulty": "Easy", + "type": "MCQ_Single", + "question_text": "The magnetic field at the centre of a circular current carrying coil of radius $R$ is $B_0$. If the radius is doubled keeping the current same, the new magnetic field at the centre will be:", + "options": { + "A": "$B_0/2$", + "B": "$B_0/4$", + "C": "$2B_0$", + "D": "$B_0$" + }, + "correct_answer": { + "value": "A" + }, + "solution_text": "Magnetic field at the centre of a circular coil is $B = \\frac{\\mu_0 I}{2R}$.\nSince $B \\propto \\frac{1}{R}$, doubling the radius will make the magnetic field half. Thus, new field is $B_0/2$.", + "language": "English", + "id": "6c96124e-c093-4070-a60b-937451db625b" + }, + { + "section": "General", + "subject": "Physics", + "topic": "Alternating Current (Set 1)", + "subtopic": "LCR Circuit", + "difficulty": "Medium", + "type": "MCQ_Single", + "question_text": "In a series LCR circuit, the voltage across the resistance, capacitance, and inductance is $10\\text{ V}$ each. If the capacitance is short-circuited, the voltage across the inductance will be:", + "options": { + "A": "10 V", + "B": "$10\\sqrt{2}\\text{ V}$", + "C": "$10/\\sqrt{2}\\text{ V}$", + "D": "Zero" + }, + "correct_answer": { + "value": "C" + }, + "solution_text": "Initially, $V_R = 10\\text{ V}$, $V_L = 10\\text{ V}$, $V_C = 10\\text{ V}$. Since $V_L = V_C$, the circuit is in resonance. The applied voltage $V = \\sqrt{V_R^2 + (V_L - V_C)^2} = V_R = 10\\text{ V}$.\nIf capacitance is shorted, it becomes an LR circuit. Impedance $Z = \\sqrt{R^2 + X_L^2}$. Since $V_R = V_L$ initially, $R = X_L$, so $Z = R\\sqrt{2}$.\nNew current $I' = \\frac{V}{Z} = \\frac{10}{R\\sqrt{2}}$.\nNew voltage across inductance $V_L' = I' X_L = \\frac{10}{R\\sqrt{2}} \\times R = \\frac{10}{\\sqrt{2}}\\text{ V}$.", + "language": "English", + "id": "71dd39a3-1a78-4589-893a-1bbd030006cf" + }, + { + "section": "General", + "subject": "Physics", + "topic": "Optics (Set 1)", + "subtopic": "Wave Optics", + "difficulty": "Medium", + "type": "MCQ_Single", + "question_text": "In Young's double slit experiment, if the separation between coherent sources is halved and the distance of the screen from the coherent sources is doubled, then the fringe width becomes:", + "options": { + "A": "Half", + "B": "Four times", + "C": "One-fourth", + "D": "Double" + }, + "correct_answer": { + "value": "B" + }, + "solution_text": "Fringe width $\\beta = \\frac{\\lambda D}{d}$.\nNew slit separation $d' = d/2$.\nNew screen distance $D' = 2D$.\nNew fringe width $\\beta' = \\frac{\\lambda (2D)}{(d/2)} = 4 \\frac{\\lambda D}{d} = 4\\beta$.", + "language": "English", + "id": "21936288-06c6-4710-9fd5-1d3e085e42b0" + }, + { + "section": "General", + "subject": "Physics", + "topic": "Modern Physics (Set 1)", + "subtopic": "Photoelectric Effect", + "difficulty": "Hard", + "type": "MCQ_Single", + "question_text": "When the light of frequency $2\\nu_0$ (where $\\nu_0$ is threshold frequency) is incident on a metal plate, the maximum velocity of electrons emitted is $v_1$. When the frequency of the incident radiation is increased to $5\\nu_0$, the maximum velocity of electrons emitted from the same plate is $v_2$. The ratio of $v_1$ to $v_2$ is:", + "options": { + "A": "1:2", + "B": "1:4", + "C": "4:1", + "D": "2:1" + }, + "correct_answer": { + "value": "A" + }, + "solution_text": "From Einstein's photoelectric equation: $K_{\\max} = h\\nu - h\\nu_0 = \\frac{1}{2} m v^2$.\nCase 1: $\\frac{1}{2} m v_1^2 = h(2\\nu_0) - h\\nu_0 = h\\nu_0$.\nCase 2: $\\frac{1}{2} m v_2^2 = h(5\\nu_0) - h\\nu_0 = 4h\\nu_0$.\nDividing the two equations: $\\frac{v_1^2}{v_2^2} = \\frac{1}{4} \\implies \\frac{v_1}{v_2} = \\frac{1}{2}$.", + "language": "English", + "id": "658541df-4e92-49f1-8861-e1011ad39233" + }, + { + "section": "General", + "subject": "Physics", + "topic": "Semiconductors (Set 1)", + "subtopic": "Logic Gates", + "difficulty": "Easy", + "type": "MCQ_Single", + "question_text": "Which logic gate is represented by the boolean expression $Y = \\overline{A \\cdot B}$?", + "options": { + "A": "NOR", + "B": "OR", + "C": "NAND", + "D": "AND" + }, + "correct_answer": { + "value": "C" + }, + "solution_text": "The boolean expression $Y = \\overline{A \\cdot B}$ represents the NAND gate. It gives an output of 0 only when both inputs A and B are 1, and 1 otherwise.", + "language": "English", + "id": "1c1ade08-e7dd-45b8-94fe-373d320b359e" + }, + { + "section": "General", + "subject": "Chemistry", + "topic": "Physical Chemistry (Set 1)", + "subtopic": "Mole Concept", + "difficulty": "Easy", + "type": "MCQ_Single", + "question_text": "The number of water molecules is maximum in:", + "options": { + "A": "18 grams of water", + "B": "18 moles of water", + "C": "18 molecules of water", + "D": "1.8 grams of water" + }, + "correct_answer": { + "value": "B" + }, + "solution_text": "18 grams of water = 1 mole = $N_A$ molecules.\n18 moles of water = $18 N_A$ molecules.\n1.8 grams of water = 0.1 mole = $0.1 N_A$ molecules.\nThus, 18 moles of water contains the maximum number of molecules.", + "language": "English", + "id": "dd5d5b06-f0a7-4688-9f44-1581d9a2eae6" + }, + { + "section": "General", + "subject": "Chemistry", + "topic": "Physical Chemistry (Set 1)", + "subtopic": "Atomic Structure", + "difficulty": "Medium", + "type": "MCQ_Single", + "question_text": "The maximum number of electrons in an atom that can have the quantum numbers $n = 4$ and $m_l = +1$ is:", + "options": { + "A": "2", + "B": "4", + "C": "6", + "D": "8" + }, + "correct_answer": { + "value": "C" + }, + "solution_text": "For $n = 4$, the possible values of $l$ are 0, 1, 2, 3.\nThe condition $m_l = +1$ is possible for $l = 1$, $l = 2$, and $l = 3$.\nEach subshell ($l=1, 2, 3$) will have one orbital with $m_l = +1$.\nSince each orbital can hold 2 electrons, the total number of electrons is $3 \\times 2 = 6$.", + "language": "English", + "id": "0913c5cf-e9b2-4319-a51a-3ffaf1fdf5fd" + }, + { + "section": "General", + "subject": "Chemistry", + "topic": "Inorganic Chemistry (Set 1)", + "subtopic": "Chemical Bonding", + "difficulty": "Medium", + "type": "MCQ_Single", + "question_text": "The species having pyramidal shape is:", + "options": { + "A": "$\\text{SO}_3$", + "B": "$\\text{BrF}_3$", + "C": "$\\text{SiO}_3^{2-}$", + "D": "$\\text{OSF}_2$" + }, + "correct_answer": { + "value": "D" + }, + "solution_text": "In $\\text{OSF}_2$, the central sulfur atom has 4 electron domains (1 double bond to O, 2 single bonds to F, and 1 lone pair). This corresponds to an $sp^3$ hybridization with a trigonal pyramidal shape. $\\text{SO}_3$ is trigonal planar, $\\text{BrF}_3$ is T-shaped.", + "language": "English", + "id": "b0fd946f-02c7-45d2-af59-c2c490019ba6" + }, + { + "section": "General", + "subject": "Chemistry", + "topic": "Physical Chemistry (Set 1)", + "subtopic": "Equilibrium", + "difficulty": "Medium", + "type": "MCQ_Single", + "question_text": "For the reversible reaction $\\text{N}_2(\\text{g}) + 3\\text{H}_2(\\text{g}) \\rightleftharpoons 2\\text{NH}_3(\\text{g}) + \\text{Heat}$, the equilibrium shifts in forward direction:", + "options": { + "A": "By increasing the concentration of $\\text{NH}_3(\\text{g})$", + "B": "By decreasing the pressure", + "C": "By decreasing the concentrations of $\\text{N}_2(\\text{g})$ and $\\text{H}_2(\\text{g})$", + "D": "By increasing pressure and decreasing temperature" + }, + "correct_answer": { + "value": "D" + }, + "solution_text": "According to Le Chatelier's principle, an exothermic reaction ($\\Delta H < 0$) is favored by low temperatures. A reaction involving a decrease in the number of gaseous moles ($4 \\rightarrow 2$) is favored by high pressures. Therefore, increasing pressure and decreasing temperature will shift the equilibrium in the forward direction.", + "language": "English", + "id": "43658efb-4da6-48c1-b6cd-afc56706986f" + }, + { + "section": "General", + "subject": "Chemistry", + "topic": "Physical Chemistry (Set 1)", + "subtopic": "Electrochemistry", + "difficulty": "Hard", + "type": "MCQ_Single", + "question_text": "Given $E^\\circ_{\\text{Cr}^{3+}/\\text{Cr}} = -0.74\\text{ V}$; $E^\\circ_{\\text{MnO}_4^-/\\text{Mn}^{2+}} = 1.51\\text{ V}$; $E^\\circ_{\\text{Cr}_2\\text{O}_7^{2-}/\\text{Cr}^{3+}} = 1.33\\text{ V}$; $E^\\circ_{\\text{Cl}/\\text{Cl}^-} = 1.36\\text{ V}$.\nBased on the data given above, strongest oxidizing agent will be:", + "options": { + "A": "$\\text{Cl}$", + "B": "$\\text{Cr}^{3+}$", + "C": "$\\text{Mn}^{2+}$", + "D": "$\\text{MnO}_4^-$" + }, + "correct_answer": { + "value": "D" + }, + "solution_text": "Higher the standard reduction potential ($E^\\circ$), stronger is the oxidizing power. The $E^\\circ$ value for $\\text{MnO}_4^-/\\text{Mn}^{2+}$ is the highest ($+1.51\\text{ V}$). Therefore, $\\text{MnO}_4^-$ is the strongest oxidizing agent among the given species.", + "language": "English", + "id": "46384cb2-62ea-47cb-88ea-4d8b800f7fd2" + }, + { + "section": "General", + "subject": "Chemistry", + "topic": "Organic Chemistry (Set 1)", + "subtopic": "General Organic Chemistry", + "difficulty": "Medium", + "type": "MCQ_Single", + "question_text": "Which of the following carbocations is most stable?", + "options": { + "A": "Allyl carbocation", + "B": "Primary alkyl carbocation", + "C": "Tert-butyl carbocation", + "D": "Triphenylmethyl carbocation" + }, + "correct_answer": { + "value": "D" + }, + "solution_text": "Stability of carbocations is determined by delocalization of charge. The triphenylmethyl carbocation, $(\\text{C}_6\\text{H}_5)_3\\text{C}^+$, has 3 phenyl rings providing extensive resonance stabilization, making it exceptionally stable compared to the others.", + "language": "English", + "id": "e2d87153-02ff-4342-88e0-8e496e408532" + }, + { + "section": "General", + "subject": "Chemistry", + "topic": "Organic Chemistry (Set 1)", + "subtopic": "Hydrocarbons", + "difficulty": "Hard", + "type": "MCQ_Single", + "question_text": "Ozonolysis of an organic compound gives formaldehyde as one of the products. This confirms the presence of:", + "options": { + "A": "Two isomeric double bonds", + "B": "A vinyl group", + "C": "An isopropyl group", + "D": "An acetylenic triple bond" + }, + "correct_answer": { + "value": "B" + }, + "solution_text": "Ozonolysis cleaves a carbon-carbon double bond and adds oxygen. Formaldehyde ($\\text{HCHO}$) is produced if the original double bond was at the end of a chain, specifically involving a terminal $=\\text{CH}_2$ group, which is characteristic of a vinyl group ($-\\text{CH}=\\text{CH}_2$).", + "language": "English", + "id": "b2da608f-c808-4fde-a3cf-45fbd0b4f4ff" + }, + { + "section": "General", + "subject": "Chemistry", + "topic": "Inorganic Chemistry (Set 1)", + "subtopic": "Coordination Compounds", + "difficulty": "Medium", + "type": "MCQ_Single", + "question_text": "The IUPAC name of $[\\text{Pt}(\\text{NH}_3)_2\\text{Cl}(\\text{NO}_2)]$ is:", + "options": { + "A": "Platinum diaminechloronitrite", + "B": "Chloronitrito-N-ammineplatinum(II)", + "C": "Diamminechloridonitrito-N-platinum(II)", + "D": "Diamminechloronitrito-O-platinate(II)" + }, + "correct_answer": { + "value": "C" + }, + "solution_text": "Ligands are named in alphabetical order: ammine, chlorido, nitrito-N (since it coordinates through nitrogen). The oxidation state of Pt is $x + 0 + (-1) + (-1) = 0 \\implies x = +2$. Thus, the correct name is Diamminechloridonitrito-N-platinum(II).", + "language": "English", + "id": "e4e0e913-c5a9-4abc-bd86-b2caf406e03f" + }, + { + "section": "General", + "subject": "Chemistry", + "topic": "Physical Chemistry (Set 1)", + "subtopic": "Chemical Kinetics", + "difficulty": "Easy", + "type": "MCQ_Single", + "question_text": "The unit of rate constant for a zero order reaction is:", + "options": { + "A": "$\\text{s}^{-1}$", + "B": "$\\text{mol}\\cdot\\text{L}^{-1}\\cdot\\text{s}^{-1}$", + "C": "$\\text{L}\\cdot\\text{mol}^{-1}\\cdot\\text{s}^{-1}$", + "D": "$\\text{L}^2\\cdot\\text{mol}^{-2}\\cdot\\text{s}^{-1}$" + }, + "correct_answer": { + "value": "B" + }, + "solution_text": "For a zero-order reaction, Rate = $k[A]^0 = k$. The unit of rate is concentration per unit time ($\\text{mol}\\cdot\\text{L}^{-1}\\cdot\\text{s}^{-1}$). Thus, the unit of the rate constant $k$ is also $\\text{mol}\\cdot\\text{L}^{-1}\\cdot\\text{s}^{-1}$.", + "language": "English", + "id": "48f8e5ea-8b89-433f-9aef-4252a70fece4" + }, + { + "section": "General", + "subject": "Chemistry", + "topic": "Organic Chemistry (Set 1)", + "subtopic": "Biomolecules", + "difficulty": "Medium", + "type": "MCQ_Single", + "question_text": "Which of the following hormones is produced under the condition of stress which stimulates glycogenolysis in the liver of human beings?", + "options": { + "A": "Thyroxin", + "B": "Insulin", + "C": "Adrenaline", + "D": "Estradiol" + }, + "correct_answer": { + "value": "C" + }, + "solution_text": "Adrenaline (epinephrine) is the 'fight or flight' hormone released during stress. It promotes the breakdown of glycogen into glucose (glycogenolysis) in the liver to increase blood glucose levels for energy.", + "language": "English", + "id": "2d40b28f-5af9-4b15-9fc1-5ada60dd098a" + }, + { + "section": "General", + "subject": "Biology", + "topic": "Cell Structure and Function (Set 1)", + "subtopic": "Cell Cycle and Cell Division", + "difficulty": "Medium", + "type": "MCQ_Single", + "question_text": "Crossing over occurs during which stage of meiosis?", + "options": { + "A": "Leptotene", + "B": "Pachytene", + "C": "Diplotene", + "D": "Zygotene" + }, + "correct_answer": { + "value": "B" + }, + "solution_text": "Crossing over, the exchange of genetic material between non-sister chromatids of homologous chromosomes, occurs during the pachytene stage of prophase I of meiosis.", + "language": "English", + "id": "4c3a00cd-776b-4336-a8ce-56dfdf13a834" + }, + { + "section": "General", + "subject": "Biology", + "topic": "Genetics and Evolution (Set 1)", + "subtopic": "Principles of Inheritance and Variation", + "difficulty": "Easy", + "type": "MCQ_Single", + "question_text": "A cross between two tall plants resulted in offspring having few dwarf plants. What would be the genotypes of both the parents?", + "options": { + "A": "TT and Tt", + "B": "Tt and Tt", + "C": "TT and TT", + "D": "Tt and tt" + }, + "correct_answer": { + "value": "B" + }, + "solution_text": "Since both parents are tall but they produce some dwarf offspring, both parents must carry the recessive allele for dwarfness. Thus, their genotype must be heterozygous (Tt). A cross of Tt x Tt yields a 3:1 ratio of tall to dwarf.", + "language": "English", + "id": "6f3fa081-48bc-43f6-a814-005acb92f91b" + }, + { + "section": "General", + "subject": "Biology", + "topic": "Genetics and Evolution (Set 1)", + "subtopic": "Molecular Basis of Inheritance", + "difficulty": "Medium", + "type": "MCQ_Single", + "question_text": "Which of the following RNAs should be most abundant in an animal cell?", + "options": { + "A": "tRNA", + "B": "mRNA", + "C": "miRNA", + "D": "rRNA" + }, + "correct_answer": { + "value": "D" + }, + "solution_text": "Ribosomal RNA (rRNA) constitutes about 80% of the total RNA in an actively growing cell. It forms the structural and catalytic core of ribosomes.", + "language": "English", + "id": "5466289f-920a-4ab7-9b17-38914bf67664" + }, + { + "section": "General", + "subject": "Biology", + "topic": "Human Physiology (Set 1)", + "subtopic": "Digestion and Absorption", + "difficulty": "Hard", + "type": "MCQ_Single", + "question_text": "Which of the following terms describes human dentition?", + "options": { + "A": "Thecodont, Diphyodont, Homodont", + "B": "Thecodont, Diphyodont, Heterodont", + "C": "Pleurodont, Monophyodont, Homodont", + "D": "Pleurodont, Diphyodont, Heterodont" + }, + "correct_answer": { + "value": "B" + }, + "solution_text": "Human dentition is Thecodont (teeth embedded in jaw sockets), Diphyodont (two sets of teeth: deciduous and permanent), and Heterodont (different types of teeth: incisors, canines, premolars, molars).", + "language": "English", + "id": "eb80dcf1-5f97-4527-a393-79cf9646a285" + }, + { + "section": "General", + "subject": "Biology", + "topic": "Reproduction (Set 1)", + "subtopic": "Human Reproduction", + "difficulty": "Medium", + "type": "MCQ_Single", + "question_text": "Which hormone is responsible for the ovulation in human females?", + "options": { + "A": "FSH (Follicle Stimulating Hormone)", + "B": "Progesterone", + "C": "LH (Luteinizing Hormone)", + "D": "Estrogen" + }, + "correct_answer": { + "value": "C" + }, + "solution_text": "A surge in LH (Luteinizing Hormone) mid-cycle triggers the rupture of the Graafian follicle and the subsequent release of the ovum, a process known as ovulation.", + "language": "English", + "id": "90c4e3a8-0736-47f9-8aca-8bd942cdc2e2" + }, + { + "section": "General", + "subject": "Biology", + "topic": "Plant Physiology (Set 1)", + "subtopic": "Photosynthesis in Higher Plants", + "difficulty": "Medium", + "type": "MCQ_Single", + "question_text": "The first stable product of CO2 fixation in C3 plants is:", + "options": { + "A": "Phosphoglyceraldehyde (PGAL)", + "B": "Malic acid", + "C": "Oxaloacetic acid (OAA)", + "D": "3-Phosphoglyceric acid (3-PGA)" + }, + "correct_answer": { + "value": "D" + }, + "solution_text": "In the Calvin cycle of C3 plants, CO2 combines with RuBP to form a 6-carbon unstable intermediate that immediately splits into two molecules of 3-phosphoglyceric acid (3-PGA), the first stable product.", + "language": "English", + "id": "a9b47475-4f9b-4758-a5e1-2972379862e8" + }, + { + "section": "General", + "subject": "Biology", + "topic": "Ecology (Set 1)", + "subtopic": "Ecosystem", + "difficulty": "Easy", + "type": "MCQ_Single", + "question_text": "The pyramid of energy is always:", + "options": { + "A": "Upright", + "B": "Inverted", + "C": "Spindle shaped", + "D": "Variable" + }, + "correct_answer": { + "value": "A" + }, + "solution_text": "The pyramid of energy is always upright because energy is lost as heat at each trophic level (10% law), so energy always decreases as it flows from producers to higher trophic levels.", + "language": "English", + "id": "4fd0bba0-8227-4af6-995a-4d536730b45d" + }, + { + "section": "General", + "subject": "Biology", + "topic": "Biotechnology (Set 1)", + "subtopic": "Principles and Processes", + "difficulty": "Medium", + "type": "MCQ_Single", + "question_text": "The cutting of DNA at specific locations became possible with the discovery of:", + "options": { + "A": "Selectable markers", + "B": "Ligases", + "C": "Restriction enzymes", + "D": "Probes" + }, + "correct_answer": { + "value": "C" + }, + "solution_text": "Restriction endonucleases (or restriction enzymes) cut DNA at specific recognition sequences, making recombinant DNA technology possible.", + "language": "English", + "id": "6a84494e-15cb-4c75-86a3-98aeb1c02eba" + }, + { + "section": "General", + "subject": "Biology", + "topic": "Human Physiology (Set 1)", + "subtopic": "Neural Control and Coordination", + "difficulty": "Hard", + "type": "MCQ_Single", + "question_text": "Receptor sites for neurotransmitters are present on:", + "options": { + "A": "Pre-synaptic membrane", + "B": "Tips of axons", + "C": "Post-synaptic membrane", + "D": "Membrane of synaptic vesicles" + }, + "correct_answer": { + "value": "C" + }, + "solution_text": "Neurotransmitters are released from synaptic vesicles in the pre-synaptic neuron and diffuse across the synaptic cleft to bind with specific receptor sites located on the post-synaptic membrane.", + "language": "English", + "id": "2427b46d-139b-4526-9eb4-341d192b7124" + }, + { + "section": "General", + "subject": "Biology", + "topic": "Diversity in Living World (Set 1)", + "subtopic": "Biological Classification", + "difficulty": "Medium", + "type": "MCQ_Single", + "question_text": "Which of the following are found in extreme saline conditions?", + "options": { + "A": "Eubacteria", + "B": "Cyanobacteria", + "C": "Mycobacteria", + "D": "Archaebacteria" + }, + "correct_answer": { + "value": "D" + }, + "solution_text": "Archaebacteria are known to live in some of the most harsh habitats. Specifically, halophiles live in extreme saline conditions (e.g., Dead Sea).", + "language": "English", + "id": "6a691f62-8492-4494-8954-b3089e772ae3" + }, + { + "section": "General", + "subject": "Physics", + "topic": "Units and Measurements (Set 2)", + "subtopic": "Dimensions", + "difficulty": "Medium", + "type": "MCQ_Single", + "question_text": "The dimension of $\\frac{1}{\\sqrt{\\mu_0 \\varepsilon_0}}$ is that of:", + "options": { + "A": "Velocity", + "B": "Time", + "C": "Capacitance", + "D": "Distance" + }, + "correct_answer": { + "value": "A" + }, + "solution_text": "We know that $c = \\frac{1}{\\sqrt{\\mu_0 \\varepsilon_0}}$, where $c$ is the speed of light in vacuum. Therefore, the dimension of $\\frac{1}{\\sqrt{\\mu_0 \\varepsilon_0}}$ is that of velocity, which is $[L T^{-1}]$.", + "language": "English", + "id": "7b762697-6eec-42af-88d2-89633265013f" + }, + { + "section": "General", + "subject": "Physics", + "topic": "Kinematics (Set 2)", + "subtopic": "Projectile Motion", + "difficulty": "Easy", + "type": "MCQ_Single", + "question_text": "Two projectiles are thrown with the same initial velocity at angles of $30^\\circ$ and $60^\\circ$ with the horizontal. The ratio of their maximum heights is:", + "options": { + "A": "1:3", + "B": "3:1", + "C": "1:\\sqrt{3}", + "D": "1:2" + }, + "correct_answer": { + "value": "A" + }, + "solution_text": "Maximum height $H = \\frac{u^2 \\sin^2 \\theta}{2g}$.\nFor $\\theta_1 = 30^\\circ$, $H_1 = \\frac{u^2 (1/2)^2}{2g} = \\frac{u^2}{8g}$.\nFor $\\theta_2 = 60^\\circ$, $H_2 = \\frac{u^2 (\\sqrt{3}/2)^2}{2g} = \\frac{3u^2}{8g}$.\nRatio $H_1 : H_2 = 1 : 3$.", + "language": "English", + "id": "360c8296-f0b1-486a-9c9b-5cbf0edeae7e" + }, + { + "section": "General", + "subject": "Physics", + "topic": "Thermodynamics (Set 2)", + "subtopic": "Carnot Engine", + "difficulty": "Medium", + "type": "MCQ_Single", + "question_text": "A Carnot engine takes heat from a reservoir at $627^\\circ\\text{C}$ and rejects heat to a sink at $27^\\circ\\text{C}$. Its efficiency is:", + "options": { + "A": "2/3", + "B": "1/3", + "C": "3/4", + "D": "1/2" + }, + "correct_answer": { + "value": "A" + }, + "solution_text": "Temperature of source, $T_1 = 627 + 273 = 900\\text{ K}$.\nTemperature of sink, $T_2 = 27 + 273 = 300\\text{ K}$.\nEfficiency $\\eta = 1 - \\frac{T_2}{T_1} = 1 - \\frac{300}{900} = 1 - \\frac{1}{3} = \\frac{2}{3}$.", + "language": "English", + "id": "03d674bb-9da5-436f-be5a-44164bb2b169" + }, + { + "section": "General", + "subject": "Physics", + "topic": "Electrostatics (Set 2)", + "subtopic": "Capacitors", + "difficulty": "Hard", + "type": "MCQ_Single", + "question_text": "A parallel plate capacitor with air between the plates has a capacitance of $8\\text{ pF}$. What will be the capacitance if the distance between the plates is reduced by half, and the space between them is filled with a substance of dielectric constant $6$?", + "options": { + "A": "96 pF", + "B": "8 pF", + "C": "48 pF", + "D": "12 pF" + }, + "correct_answer": { + "value": "A" + }, + "solution_text": "Initial capacitance $C_0 = \\frac{\\varepsilon_0 A}{d} = 8\\text{ pF}$.\nNew distance $d' = d/2$ and new dielectric constant $K = 6$.\nNew capacitance $C' = \\frac{K \\varepsilon_0 A}{d'} = \\frac{6 \\varepsilon_0 A}{d/2} = 12 \\left(\\frac{\\varepsilon_0 A}{d}\\right) = 12 \\times 8\\text{ pF} = 96\\text{ pF}$.", + "language": "English", + "id": "ca859cf2-0784-4c06-8298-3252a5e9ede0" + }, + { + "section": "General", + "subject": "Physics", + "topic": "Current Electricity (Set 2)", + "subtopic": "Meter Bridge", + "difficulty": "Medium", + "type": "MCQ_Single", + "question_text": "In a meter bridge experiment, the null point is obtained at $20\\text{ cm}$ from one end of the wire when resistance $X$ is balanced against another resistance $Y$. If $X < Y$, then what will be the new position of the null point from the same end, if one decides to balance a resistance of $4X$ against $Y$?", + "options": { + "A": "50 cm", + "B": "40 cm", + "C": "80 cm", + "D": "20 cm" + }, + "correct_answer": { + "value": "A" + }, + "solution_text": "Initially, $\\frac{X}{Y} = \\frac{20}{100-20} = \\frac{20}{80} = \\frac{1}{4}$. So $Y = 4X$.\nFor new resistance $4X$ against $Y$, ratio is $\\frac{4X}{Y} = \\frac{4X}{4X} = 1$.\nLet new null point be $l$, then $\\frac{l}{100-l} = 1 \\implies l = 100 - l \\implies 2l = 100 \\implies l = 50\\text{ cm}$.", + "language": "English", + "id": "c3048971-d687-4b55-a166-43df4ab61b2e" + }, + { + "section": "General", + "subject": "Physics", + "topic": "Magnetism (Set 2)", + "subtopic": "Biot-Savart Law", + "difficulty": "Easy", + "type": "MCQ_Single", + "question_text": "The magnetic field at the centre of a circular current carrying coil of radius $R$ is $B_0$. If the radius is doubled keeping the current same, the new magnetic field at the centre will be:", + "options": { + "A": "$B_0/2$", + "B": "$B_0/4$", + "C": "$2B_0$", + "D": "$B_0$" + }, + "correct_answer": { + "value": "A" + }, + "solution_text": "Magnetic field at the centre of a circular coil is $B = \\frac{\\mu_0 I}{2R}$.\nSince $B \\propto \\frac{1}{R}$, doubling the radius will make the magnetic field half. Thus, new field is $B_0/2$.", + "language": "English", + "id": "ef4bc469-b252-43c9-a8ce-ef0888a44b03" + }, + { + "section": "General", + "subject": "Physics", + "topic": "Alternating Current (Set 2)", + "subtopic": "LCR Circuit", + "difficulty": "Medium", + "type": "MCQ_Single", + "question_text": "In a series LCR circuit, the voltage across the resistance, capacitance, and inductance is $10\\text{ V}$ each. If the capacitance is short-circuited, the voltage across the inductance will be:", + "options": { + "A": "10 V", + "B": "$10\\sqrt{2}\\text{ V}$", + "C": "$10/\\sqrt{2}\\text{ V}$", + "D": "Zero" + }, + "correct_answer": { + "value": "C" + }, + "solution_text": "Initially, $V_R = 10\\text{ V}$, $V_L = 10\\text{ V}$, $V_C = 10\\text{ V}$. Since $V_L = V_C$, the circuit is in resonance. The applied voltage $V = \\sqrt{V_R^2 + (V_L - V_C)^2} = V_R = 10\\text{ V}$.\nIf capacitance is shorted, it becomes an LR circuit. Impedance $Z = \\sqrt{R^2 + X_L^2}$. Since $V_R = V_L$ initially, $R = X_L$, so $Z = R\\sqrt{2}$.\nNew current $I' = \\frac{V}{Z} = \\frac{10}{R\\sqrt{2}}$.\nNew voltage across inductance $V_L' = I' X_L = \\frac{10}{R\\sqrt{2}} \\times R = \\frac{10}{\\sqrt{2}}\\text{ V}$.", + "language": "English", + "id": "76fe2e5f-d75e-4a67-91e9-1e4e6f214a8b" + }, + { + "section": "General", + "subject": "Physics", + "topic": "Optics (Set 2)", + "subtopic": "Wave Optics", + "difficulty": "Medium", + "type": "MCQ_Single", + "question_text": "In Young's double slit experiment, if the separation between coherent sources is halved and the distance of the screen from the coherent sources is doubled, then the fringe width becomes:", + "options": { + "A": "Half", + "B": "Four times", + "C": "One-fourth", + "D": "Double" + }, + "correct_answer": { + "value": "B" + }, + "solution_text": "Fringe width $\\beta = \\frac{\\lambda D}{d}$.\nNew slit separation $d' = d/2$.\nNew screen distance $D' = 2D$.\nNew fringe width $\\beta' = \\frac{\\lambda (2D)}{(d/2)} = 4 \\frac{\\lambda D}{d} = 4\\beta$.", + "language": "English", + "id": "9bb0969b-efef-4ef1-8790-e0bc31d4cc67" + }, + { + "section": "General", + "subject": "Physics", + "topic": "Modern Physics (Set 2)", + "subtopic": "Photoelectric Effect", + "difficulty": "Hard", + "type": "MCQ_Single", + "question_text": "When the light of frequency $2\\nu_0$ (where $\\nu_0$ is threshold frequency) is incident on a metal plate, the maximum velocity of electrons emitted is $v_1$. When the frequency of the incident radiation is increased to $5\\nu_0$, the maximum velocity of electrons emitted from the same plate is $v_2$. The ratio of $v_1$ to $v_2$ is:", + "options": { + "A": "1:2", + "B": "1:4", + "C": "4:1", + "D": "2:1" + }, + "correct_answer": { + "value": "A" + }, + "solution_text": "From Einstein's photoelectric equation: $K_{\\max} = h\\nu - h\\nu_0 = \\frac{1}{2} m v^2$.\nCase 1: $\\frac{1}{2} m v_1^2 = h(2\\nu_0) - h\\nu_0 = h\\nu_0$.\nCase 2: $\\frac{1}{2} m v_2^2 = h(5\\nu_0) - h\\nu_0 = 4h\\nu_0$.\nDividing the two equations: $\\frac{v_1^2}{v_2^2} = \\frac{1}{4} \\implies \\frac{v_1}{v_2} = \\frac{1}{2}$.", + "language": "English", + "id": "8247e89c-e0a3-429a-ae55-aed2cb229a4b" + }, + { + "section": "General", + "subject": "Physics", + "topic": "Semiconductors (Set 2)", + "subtopic": "Logic Gates", + "difficulty": "Easy", + "type": "MCQ_Single", + "question_text": "Which logic gate is represented by the boolean expression $Y = \\overline{A \\cdot B}$?", + "options": { + "A": "NOR", + "B": "OR", + "C": "NAND", + "D": "AND" + }, + "correct_answer": { + "value": "C" + }, + "solution_text": "The boolean expression $Y = \\overline{A \\cdot B}$ represents the NAND gate. It gives an output of 0 only when both inputs A and B are 1, and 1 otherwise.", + "language": "English", + "id": "2dcf2400-ab08-47f6-b5d7-acb58d4eba16" + }, + { + "section": "General", + "subject": "Chemistry", + "topic": "Physical Chemistry (Set 2)", + "subtopic": "Mole Concept", + "difficulty": "Easy", + "type": "MCQ_Single", + "question_text": "The number of water molecules is maximum in:", + "options": { + "A": "18 grams of water", + "B": "18 moles of water", + "C": "18 molecules of water", + "D": "1.8 grams of water" + }, + "correct_answer": { + "value": "B" + }, + "solution_text": "18 grams of water = 1 mole = $N_A$ molecules.\n18 moles of water = $18 N_A$ molecules.\n1.8 grams of water = 0.1 mole = $0.1 N_A$ molecules.\nThus, 18 moles of water contains the maximum number of molecules.", + "language": "English", + "id": "8500561b-4357-467a-b112-6f208ed4478f" + }, + { + "section": "General", + "subject": "Chemistry", + "topic": "Physical Chemistry (Set 2)", + "subtopic": "Atomic Structure", + "difficulty": "Medium", + "type": "MCQ_Single", + "question_text": "The maximum number of electrons in an atom that can have the quantum numbers $n = 4$ and $m_l = +1$ is:", + "options": { + "A": "2", + "B": "4", + "C": "6", + "D": "8" + }, + "correct_answer": { + "value": "C" + }, + "solution_text": "For $n = 4$, the possible values of $l$ are 0, 1, 2, 3.\nThe condition $m_l = +1$ is possible for $l = 1$, $l = 2$, and $l = 3$.\nEach subshell ($l=1, 2, 3$) will have one orbital with $m_l = +1$.\nSince each orbital can hold 2 electrons, the total number of electrons is $3 \\times 2 = 6$.", + "language": "English", + "id": "486fe0d3-26c6-4e64-8a6f-8bd78d82953e" + }, + { + "section": "General", + "subject": "Chemistry", + "topic": "Inorganic Chemistry (Set 2)", + "subtopic": "Chemical Bonding", + "difficulty": "Medium", + "type": "MCQ_Single", + "question_text": "The species having pyramidal shape is:", + "options": { + "A": "$\\text{SO}_3$", + "B": "$\\text{BrF}_3$", + "C": "$\\text{SiO}_3^{2-}$", + "D": "$\\text{OSF}_2$" + }, + "correct_answer": { + "value": "D" + }, + "solution_text": "In $\\text{OSF}_2$, the central sulfur atom has 4 electron domains (1 double bond to O, 2 single bonds to F, and 1 lone pair). This corresponds to an $sp^3$ hybridization with a trigonal pyramidal shape. $\\text{SO}_3$ is trigonal planar, $\\text{BrF}_3$ is T-shaped.", + "language": "English", + "id": "de84de18-b5fe-44c1-807c-a7da3e06033d" + }, + { + "section": "General", + "subject": "Chemistry", + "topic": "Physical Chemistry (Set 2)", + "subtopic": "Equilibrium", + "difficulty": "Medium", + "type": "MCQ_Single", + "question_text": "For the reversible reaction $\\text{N}_2(\\text{g}) + 3\\text{H}_2(\\text{g}) \\rightleftharpoons 2\\text{NH}_3(\\text{g}) + \\text{Heat}$, the equilibrium shifts in forward direction:", + "options": { + "A": "By increasing the concentration of $\\text{NH}_3(\\text{g})$", + "B": "By decreasing the pressure", + "C": "By decreasing the concentrations of $\\text{N}_2(\\text{g})$ and $\\text{H}_2(\\text{g})$", + "D": "By increasing pressure and decreasing temperature" + }, + "correct_answer": { + "value": "D" + }, + "solution_text": "According to Le Chatelier's principle, an exothermic reaction ($\\Delta H < 0$) is favored by low temperatures. A reaction involving a decrease in the number of gaseous moles ($4 \\rightarrow 2$) is favored by high pressures. Therefore, increasing pressure and decreasing temperature will shift the equilibrium in the forward direction.", + "language": "English", + "id": "8eb9b9c7-d262-4c0e-9366-e76fc9e3311f" + }, + { + "section": "General", + "subject": "Chemistry", + "topic": "Physical Chemistry (Set 2)", + "subtopic": "Electrochemistry", + "difficulty": "Hard", + "type": "MCQ_Single", + "question_text": "Given $E^\\circ_{\\text{Cr}^{3+}/\\text{Cr}} = -0.74\\text{ V}$; $E^\\circ_{\\text{MnO}_4^-/\\text{Mn}^{2+}} = 1.51\\text{ V}$; $E^\\circ_{\\text{Cr}_2\\text{O}_7^{2-}/\\text{Cr}^{3+}} = 1.33\\text{ V}$; $E^\\circ_{\\text{Cl}/\\text{Cl}^-} = 1.36\\text{ V}$.\nBased on the data given above, strongest oxidizing agent will be:", + "options": { + "A": "$\\text{Cl}$", + "B": "$\\text{Cr}^{3+}$", + "C": "$\\text{Mn}^{2+}$", + "D": "$\\text{MnO}_4^-$" + }, + "correct_answer": { + "value": "D" + }, + "solution_text": "Higher the standard reduction potential ($E^\\circ$), stronger is the oxidizing power. The $E^\\circ$ value for $\\text{MnO}_4^-/\\text{Mn}^{2+}$ is the highest ($+1.51\\text{ V}$). Therefore, $\\text{MnO}_4^-$ is the strongest oxidizing agent among the given species.", + "language": "English", + "id": "3c83e93c-1f88-4c55-8e4f-8dbbee38e61e" + }, + { + "section": "General", + "subject": "Chemistry", + "topic": "Organic Chemistry (Set 2)", + "subtopic": "General Organic Chemistry", + "difficulty": "Medium", + "type": "MCQ_Single", + "question_text": "Which of the following carbocations is most stable?", + "options": { + "A": "Allyl carbocation", + "B": "Primary alkyl carbocation", + "C": "Tert-butyl carbocation", + "D": "Triphenylmethyl carbocation" + }, + "correct_answer": { + "value": "D" + }, + "solution_text": "Stability of carbocations is determined by delocalization of charge. The triphenylmethyl carbocation, $(\\text{C}_6\\text{H}_5)_3\\text{C}^+$, has 3 phenyl rings providing extensive resonance stabilization, making it exceptionally stable compared to the others.", + "language": "English", + "id": "7904b9d7-9e66-40cc-8380-b4cf3bd1cb31" + }, + { + "section": "General", + "subject": "Chemistry", + "topic": "Organic Chemistry (Set 2)", + "subtopic": "Hydrocarbons", + "difficulty": "Hard", + "type": "MCQ_Single", + "question_text": "Ozonolysis of an organic compound gives formaldehyde as one of the products. This confirms the presence of:", + "options": { + "A": "Two isomeric double bonds", + "B": "A vinyl group", + "C": "An isopropyl group", + "D": "An acetylenic triple bond" + }, + "correct_answer": { + "value": "B" + }, + "solution_text": "Ozonolysis cleaves a carbon-carbon double bond and adds oxygen. Formaldehyde ($\\text{HCHO}$) is produced if the original double bond was at the end of a chain, specifically involving a terminal $=\\text{CH}_2$ group, which is characteristic of a vinyl group ($-\\text{CH}=\\text{CH}_2$).", + "language": "English", + "id": "e97d91f2-e924-48d0-aa02-2541f12b3f85" + }, + { + "section": "General", + "subject": "Chemistry", + "topic": "Inorganic Chemistry (Set 2)", + "subtopic": "Coordination Compounds", + "difficulty": "Medium", + "type": "MCQ_Single", + "question_text": "The IUPAC name of $[\\text{Pt}(\\text{NH}_3)_2\\text{Cl}(\\text{NO}_2)]$ is:", + "options": { + "A": "Platinum diaminechloronitrite", + "B": "Chloronitrito-N-ammineplatinum(II)", + "C": "Diamminechloridonitrito-N-platinum(II)", + "D": "Diamminechloronitrito-O-platinate(II)" + }, + "correct_answer": { + "value": "C" + }, + "solution_text": "Ligands are named in alphabetical order: ammine, chlorido, nitrito-N (since it coordinates through nitrogen). The oxidation state of Pt is $x + 0 + (-1) + (-1) = 0 \\implies x = +2$. Thus, the correct name is Diamminechloridonitrito-N-platinum(II).", + "language": "English", + "id": "57f1959d-5345-4286-abf1-bb2d2c747a96" + }, + { + "section": "General", + "subject": "Chemistry", + "topic": "Physical Chemistry (Set 2)", + "subtopic": "Chemical Kinetics", + "difficulty": "Easy", + "type": "MCQ_Single", + "question_text": "The unit of rate constant for a zero order reaction is:", + "options": { + "A": "$\\text{s}^{-1}$", + "B": "$\\text{mol}\\cdot\\text{L}^{-1}\\cdot\\text{s}^{-1}$", + "C": "$\\text{L}\\cdot\\text{mol}^{-1}\\cdot\\text{s}^{-1}$", + "D": "$\\text{L}^2\\cdot\\text{mol}^{-2}\\cdot\\text{s}^{-1}$" + }, + "correct_answer": { + "value": "B" + }, + "solution_text": "For a zero-order reaction, Rate = $k[A]^0 = k$. The unit of rate is concentration per unit time ($\\text{mol}\\cdot\\text{L}^{-1}\\cdot\\text{s}^{-1}$). Thus, the unit of the rate constant $k$ is also $\\text{mol}\\cdot\\text{L}^{-1}\\cdot\\text{s}^{-1}$.", + "language": "English", + "id": "90bd6f4a-954b-4e71-930e-e34a41bdc1f4" + }, + { + "section": "General", + "subject": "Chemistry", + "topic": "Organic Chemistry (Set 2)", + "subtopic": "Biomolecules", + "difficulty": "Medium", + "type": "MCQ_Single", + "question_text": "Which of the following hormones is produced under the condition of stress which stimulates glycogenolysis in the liver of human beings?", + "options": { + "A": "Thyroxin", + "B": "Insulin", + "C": "Adrenaline", + "D": "Estradiol" + }, + "correct_answer": { + "value": "C" + }, + "solution_text": "Adrenaline (epinephrine) is the 'fight or flight' hormone released during stress. It promotes the breakdown of glycogen into glucose (glycogenolysis) in the liver to increase blood glucose levels for energy.", + "language": "English", + "id": "2e0d89c0-6321-45f4-9508-6ea46c503193" + }, + { + "section": "General", + "subject": "Biology", + "topic": "Cell Structure and Function (Set 2)", + "subtopic": "Cell Cycle and Cell Division", + "difficulty": "Medium", + "type": "MCQ_Single", + "question_text": "Crossing over occurs during which stage of meiosis?", + "options": { + "A": "Leptotene", + "B": "Pachytene", + "C": "Diplotene", + "D": "Zygotene" + }, + "correct_answer": { + "value": "B" + }, + "solution_text": "Crossing over, the exchange of genetic material between non-sister chromatids of homologous chromosomes, occurs during the pachytene stage of prophase I of meiosis.", + "language": "English", + "id": "f1cb705f-1484-4481-9e51-9dea8f6267ba" + }, + { + "section": "General", + "subject": "Biology", + "topic": "Genetics and Evolution (Set 2)", + "subtopic": "Principles of Inheritance and Variation", + "difficulty": "Easy", + "type": "MCQ_Single", + "question_text": "A cross between two tall plants resulted in offspring having few dwarf plants. What would be the genotypes of both the parents?", + "options": { + "A": "TT and Tt", + "B": "Tt and Tt", + "C": "TT and TT", + "D": "Tt and tt" + }, + "correct_answer": { + "value": "B" + }, + "solution_text": "Since both parents are tall but they produce some dwarf offspring, both parents must carry the recessive allele for dwarfness. Thus, their genotype must be heterozygous (Tt). A cross of Tt x Tt yields a 3:1 ratio of tall to dwarf.", + "language": "English", + "id": "1f046583-5f89-40bc-9391-3c1ab7cffd62" + }, + { + "section": "General", + "subject": "Biology", + "topic": "Genetics and Evolution (Set 2)", + "subtopic": "Molecular Basis of Inheritance", + "difficulty": "Medium", + "type": "MCQ_Single", + "question_text": "Which of the following RNAs should be most abundant in an animal cell?", + "options": { + "A": "tRNA", + "B": "mRNA", + "C": "miRNA", + "D": "rRNA" + }, + "correct_answer": { + "value": "D" + }, + "solution_text": "Ribosomal RNA (rRNA) constitutes about 80% of the total RNA in an actively growing cell. It forms the structural and catalytic core of ribosomes.", + "language": "English", + "id": "3feb4d38-73a0-4d6b-90db-de47a0d690f7" + }, + { + "section": "General", + "subject": "Biology", + "topic": "Human Physiology (Set 2)", + "subtopic": "Digestion and Absorption", + "difficulty": "Hard", + "type": "MCQ_Single", + "question_text": "Which of the following terms describes human dentition?", + "options": { + "A": "Thecodont, Diphyodont, Homodont", + "B": "Thecodont, Diphyodont, Heterodont", + "C": "Pleurodont, Monophyodont, Homodont", + "D": "Pleurodont, Diphyodont, Heterodont" + }, + "correct_answer": { + "value": "B" + }, + "solution_text": "Human dentition is Thecodont (teeth embedded in jaw sockets), Diphyodont (two sets of teeth: deciduous and permanent), and Heterodont (different types of teeth: incisors, canines, premolars, molars).", + "language": "English", + "id": "84049894-bf1b-4b34-bf54-188d81ddb524" + }, + { + "section": "General", + "subject": "Biology", + "topic": "Reproduction (Set 2)", + "subtopic": "Human Reproduction", + "difficulty": "Medium", + "type": "MCQ_Single", + "question_text": "Which hormone is responsible for the ovulation in human females?", + "options": { + "A": "FSH (Follicle Stimulating Hormone)", + "B": "Progesterone", + "C": "LH (Luteinizing Hormone)", + "D": "Estrogen" + }, + "correct_answer": { + "value": "C" + }, + "solution_text": "A surge in LH (Luteinizing Hormone) mid-cycle triggers the rupture of the Graafian follicle and the subsequent release of the ovum, a process known as ovulation.", + "language": "English", + "id": "decff3ac-b594-4fe4-b876-408926384c47" + }, + { + "section": "General", + "subject": "Biology", + "topic": "Plant Physiology (Set 2)", + "subtopic": "Photosynthesis in Higher Plants", + "difficulty": "Medium", + "type": "MCQ_Single", + "question_text": "The first stable product of CO2 fixation in C3 plants is:", + "options": { + "A": "Phosphoglyceraldehyde (PGAL)", + "B": "Malic acid", + "C": "Oxaloacetic acid (OAA)", + "D": "3-Phosphoglyceric acid (3-PGA)" + }, + "correct_answer": { + "value": "D" + }, + "solution_text": "In the Calvin cycle of C3 plants, CO2 combines with RuBP to form a 6-carbon unstable intermediate that immediately splits into two molecules of 3-phosphoglyceric acid (3-PGA), the first stable product.", + "language": "English", + "id": "ce615ee2-8ae5-41fe-9cb8-df2d0974fa70" + }, + { + "section": "General", + "subject": "Biology", + "topic": "Ecology (Set 2)", + "subtopic": "Ecosystem", + "difficulty": "Easy", + "type": "MCQ_Single", + "question_text": "The pyramid of energy is always:", + "options": { + "A": "Upright", + "B": "Inverted", + "C": "Spindle shaped", + "D": "Variable" + }, + "correct_answer": { + "value": "A" + }, + "solution_text": "The pyramid of energy is always upright because energy is lost as heat at each trophic level (10% law), so energy always decreases as it flows from producers to higher trophic levels.", + "language": "English", + "id": "b8931a19-6f8b-4bc0-8ef3-19c7bd151780" + }, + { + "section": "General", + "subject": "Biology", + "topic": "Biotechnology (Set 2)", + "subtopic": "Principles and Processes", + "difficulty": "Medium", + "type": "MCQ_Single", + "question_text": "The cutting of DNA at specific locations became possible with the discovery of:", + "options": { + "A": "Selectable markers", + "B": "Ligases", + "C": "Restriction enzymes", + "D": "Probes" + }, + "correct_answer": { + "value": "C" + }, + "solution_text": "Restriction endonucleases (or restriction enzymes) cut DNA at specific recognition sequences, making recombinant DNA technology possible.", + "language": "English", + "id": "a26ad786-743a-4669-82cb-adcc472ae82a" + }, + { + "section": "General", + "subject": "Biology", + "topic": "Human Physiology (Set 2)", + "subtopic": "Neural Control and Coordination", + "difficulty": "Hard", + "type": "MCQ_Single", + "question_text": "Receptor sites for neurotransmitters are present on:", + "options": { + "A": "Pre-synaptic membrane", + "B": "Tips of axons", + "C": "Post-synaptic membrane", + "D": "Membrane of synaptic vesicles" + }, + "correct_answer": { + "value": "C" + }, + "solution_text": "Neurotransmitters are released from synaptic vesicles in the pre-synaptic neuron and diffuse across the synaptic cleft to bind with specific receptor sites located on the post-synaptic membrane.", + "language": "English", + "id": "cf1d3d36-d823-4045-8574-4670011b4da1" + }, + { + "section": "General", + "subject": "Biology", + "topic": "Diversity in Living World (Set 2)", + "subtopic": "Biological Classification", + "difficulty": "Medium", + "type": "MCQ_Single", + "question_text": "Which of the following are found in extreme saline conditions?", + "options": { + "A": "Eubacteria", + "B": "Cyanobacteria", + "C": "Mycobacteria", + "D": "Archaebacteria" + }, + "correct_answer": { + "value": "D" + }, + "solution_text": "Archaebacteria are known to live in some of the most harsh habitats. Specifically, halophiles live in extreme saline conditions (e.g., Dead Sea).", + "language": "English", + "id": "acb7d418-b090-4fa9-9dbe-d11d80dc769d" + }, + { + "section": "General", + "subject": "Physics", + "topic": "Units and Measurements (Set 3)", + "subtopic": "Dimensions", + "difficulty": "Medium", + "type": "MCQ_Single", + "question_text": "The dimension of $\\frac{1}{\\sqrt{\\mu_0 \\varepsilon_0}}$ is that of:", + "options": { + "A": "Velocity", + "B": "Time", + "C": "Capacitance", + "D": "Distance" + }, + "correct_answer": { + "value": "A" + }, + "solution_text": "We know that $c = \\frac{1}{\\sqrt{\\mu_0 \\varepsilon_0}}$, where $c$ is the speed of light in vacuum. Therefore, the dimension of $\\frac{1}{\\sqrt{\\mu_0 \\varepsilon_0}}$ is that of velocity, which is $[L T^{-1}]$.", + "language": "English", + "id": "4f6acacf-edec-4889-afa0-fccbb163fd14" + }, + { + "section": "General", + "subject": "Physics", + "topic": "Kinematics (Set 3)", + "subtopic": "Projectile Motion", + "difficulty": "Easy", + "type": "MCQ_Single", + "question_text": "Two projectiles are thrown with the same initial velocity at angles of $30^\\circ$ and $60^\\circ$ with the horizontal. The ratio of their maximum heights is:", + "options": { + "A": "1:3", + "B": "3:1", + "C": "1:\\sqrt{3}", + "D": "1:2" + }, + "correct_answer": { + "value": "A" + }, + "solution_text": "Maximum height $H = \\frac{u^2 \\sin^2 \\theta}{2g}$.\nFor $\\theta_1 = 30^\\circ$, $H_1 = \\frac{u^2 (1/2)^2}{2g} = \\frac{u^2}{8g}$.\nFor $\\theta_2 = 60^\\circ$, $H_2 = \\frac{u^2 (\\sqrt{3}/2)^2}{2g} = \\frac{3u^2}{8g}$.\nRatio $H_1 : H_2 = 1 : 3$.", + "language": "English", + "id": "b99d4c8f-591b-4f11-9109-852f1f14be6a" + }, + { + "section": "General", + "subject": "Physics", + "topic": "Thermodynamics (Set 3)", + "subtopic": "Carnot Engine", + "difficulty": "Medium", + "type": "MCQ_Single", + "question_text": "A Carnot engine takes heat from a reservoir at $627^\\circ\\text{C}$ and rejects heat to a sink at $27^\\circ\\text{C}$. Its efficiency is:", + "options": { + "A": "2/3", + "B": "1/3", + "C": "3/4", + "D": "1/2" + }, + "correct_answer": { + "value": "A" + }, + "solution_text": "Temperature of source, $T_1 = 627 + 273 = 900\\text{ K}$.\nTemperature of sink, $T_2 = 27 + 273 = 300\\text{ K}$.\nEfficiency $\\eta = 1 - \\frac{T_2}{T_1} = 1 - \\frac{300}{900} = 1 - \\frac{1}{3} = \\frac{2}{3}$.", + "language": "English", + "id": "88ae576d-44d1-42bc-aeac-295d0c841a97" + }, + { + "section": "General", + "subject": "Physics", + "topic": "Electrostatics (Set 3)", + "subtopic": "Capacitors", + "difficulty": "Hard", + "type": "MCQ_Single", + "question_text": "A parallel plate capacitor with air between the plates has a capacitance of $8\\text{ pF}$. What will be the capacitance if the distance between the plates is reduced by half, and the space between them is filled with a substance of dielectric constant $6$?", + "options": { + "A": "96 pF", + "B": "8 pF", + "C": "48 pF", + "D": "12 pF" + }, + "correct_answer": { + "value": "A" + }, + "solution_text": "Initial capacitance $C_0 = \\frac{\\varepsilon_0 A}{d} = 8\\text{ pF}$.\nNew distance $d' = d/2$ and new dielectric constant $K = 6$.\nNew capacitance $C' = \\frac{K \\varepsilon_0 A}{d'} = \\frac{6 \\varepsilon_0 A}{d/2} = 12 \\left(\\frac{\\varepsilon_0 A}{d}\\right) = 12 \\times 8\\text{ pF} = 96\\text{ pF}$.", + "language": "English", + "id": "823af790-369b-4ad6-ba99-6cfc54295d63" + }, + { + "section": "General", + "subject": "Physics", + "topic": "Current Electricity (Set 3)", + "subtopic": "Meter Bridge", + "difficulty": "Medium", + "type": "MCQ_Single", + "question_text": "In a meter bridge experiment, the null point is obtained at $20\\text{ cm}$ from one end of the wire when resistance $X$ is balanced against another resistance $Y$. If $X < Y$, then what will be the new position of the null point from the same end, if one decides to balance a resistance of $4X$ against $Y$?", + "options": { + "A": "50 cm", + "B": "40 cm", + "C": "80 cm", + "D": "20 cm" + }, + "correct_answer": { + "value": "A" + }, + "solution_text": "Initially, $\\frac{X}{Y} = \\frac{20}{100-20} = \\frac{20}{80} = \\frac{1}{4}$. So $Y = 4X$.\nFor new resistance $4X$ against $Y$, ratio is $\\frac{4X}{Y} = \\frac{4X}{4X} = 1$.\nLet new null point be $l$, then $\\frac{l}{100-l} = 1 \\implies l = 100 - l \\implies 2l = 100 \\implies l = 50\\text{ cm}$.", + "language": "English", + "id": "c067d0b8-b2f4-4d86-add5-b4f8a711235c" + }, + { + "section": "General", + "subject": "Physics", + "topic": "Magnetism (Set 3)", + "subtopic": "Biot-Savart Law", + "difficulty": "Easy", + "type": "MCQ_Single", + "question_text": "The magnetic field at the centre of a circular current carrying coil of radius $R$ is $B_0$. If the radius is doubled keeping the current same, the new magnetic field at the centre will be:", + "options": { + "A": "$B_0/2$", + "B": "$B_0/4$", + "C": "$2B_0$", + "D": "$B_0$" + }, + "correct_answer": { + "value": "A" + }, + "solution_text": "Magnetic field at the centre of a circular coil is $B = \\frac{\\mu_0 I}{2R}$.\nSince $B \\propto \\frac{1}{R}$, doubling the radius will make the magnetic field half. Thus, new field is $B_0/2$.", + "language": "English", + "id": "b4101faa-9c9d-4c51-815e-306614afbdb0" + }, + { + "section": "General", + "subject": "Physics", + "topic": "Alternating Current (Set 3)", + "subtopic": "LCR Circuit", + "difficulty": "Medium", + "type": "MCQ_Single", + "question_text": "In a series LCR circuit, the voltage across the resistance, capacitance, and inductance is $10\\text{ V}$ each. If the capacitance is short-circuited, the voltage across the inductance will be:", + "options": { + "A": "10 V", + "B": "$10\\sqrt{2}\\text{ V}$", + "C": "$10/\\sqrt{2}\\text{ V}$", + "D": "Zero" + }, + "correct_answer": { + "value": "C" + }, + "solution_text": "Initially, $V_R = 10\\text{ V}$, $V_L = 10\\text{ V}$, $V_C = 10\\text{ V}$. Since $V_L = V_C$, the circuit is in resonance. The applied voltage $V = \\sqrt{V_R^2 + (V_L - V_C)^2} = V_R = 10\\text{ V}$.\nIf capacitance is shorted, it becomes an LR circuit. Impedance $Z = \\sqrt{R^2 + X_L^2}$. Since $V_R = V_L$ initially, $R = X_L$, so $Z = R\\sqrt{2}$.\nNew current $I' = \\frac{V}{Z} = \\frac{10}{R\\sqrt{2}}$.\nNew voltage across inductance $V_L' = I' X_L = \\frac{10}{R\\sqrt{2}} \\times R = \\frac{10}{\\sqrt{2}}\\text{ V}$.", + "language": "English", + "id": "7869286a-9de6-4a20-8589-602bf552ed8d" + }, + { + "section": "General", + "subject": "Physics", + "topic": "Optics (Set 3)", + "subtopic": "Wave Optics", + "difficulty": "Medium", + "type": "MCQ_Single", + "question_text": "In Young's double slit experiment, if the separation between coherent sources is halved and the distance of the screen from the coherent sources is doubled, then the fringe width becomes:", + "options": { + "A": "Half", + "B": "Four times", + "C": "One-fourth", + "D": "Double" + }, + "correct_answer": { + "value": "B" + }, + "solution_text": "Fringe width $\\beta = \\frac{\\lambda D}{d}$.\nNew slit separation $d' = d/2$.\nNew screen distance $D' = 2D$.\nNew fringe width $\\beta' = \\frac{\\lambda (2D)}{(d/2)} = 4 \\frac{\\lambda D}{d} = 4\\beta$.", + "language": "English", + "id": "b53b002e-9b01-41e6-b43c-105894b69fd9" + }, + { + "section": "General", + "subject": "Physics", + "topic": "Modern Physics (Set 3)", + "subtopic": "Photoelectric Effect", + "difficulty": "Hard", + "type": "MCQ_Single", + "question_text": "When the light of frequency $2\\nu_0$ (where $\\nu_0$ is threshold frequency) is incident on a metal plate, the maximum velocity of electrons emitted is $v_1$. When the frequency of the incident radiation is increased to $5\\nu_0$, the maximum velocity of electrons emitted from the same plate is $v_2$. The ratio of $v_1$ to $v_2$ is:", + "options": { + "A": "1:2", + "B": "1:4", + "C": "4:1", + "D": "2:1" + }, + "correct_answer": { + "value": "A" + }, + "solution_text": "From Einstein's photoelectric equation: $K_{\\max} = h\\nu - h\\nu_0 = \\frac{1}{2} m v^2$.\nCase 1: $\\frac{1}{2} m v_1^2 = h(2\\nu_0) - h\\nu_0 = h\\nu_0$.\nCase 2: $\\frac{1}{2} m v_2^2 = h(5\\nu_0) - h\\nu_0 = 4h\\nu_0$.\nDividing the two equations: $\\frac{v_1^2}{v_2^2} = \\frac{1}{4} \\implies \\frac{v_1}{v_2} = \\frac{1}{2}$.", + "language": "English", + "id": "bc8b21fa-04f2-4fd6-901e-303b2f1a8c60" + }, + { + "section": "General", + "subject": "Physics", + "topic": "Semiconductors (Set 3)", + "subtopic": "Logic Gates", + "difficulty": "Easy", + "type": "MCQ_Single", + "question_text": "Which logic gate is represented by the boolean expression $Y = \\overline{A \\cdot B}$?", + "options": { + "A": "NOR", + "B": "OR", + "C": "NAND", + "D": "AND" + }, + "correct_answer": { + "value": "C" + }, + "solution_text": "The boolean expression $Y = \\overline{A \\cdot B}$ represents the NAND gate. It gives an output of 0 only when both inputs A and B are 1, and 1 otherwise.", + "language": "English", + "id": "90148afe-d488-4e11-b3b1-38fd0733f024" + }, + { + "section": "General", + "subject": "Chemistry", + "topic": "Physical Chemistry (Set 3)", + "subtopic": "Mole Concept", + "difficulty": "Easy", + "type": "MCQ_Single", + "question_text": "The number of water molecules is maximum in:", + "options": { + "A": "18 grams of water", + "B": "18 moles of water", + "C": "18 molecules of water", + "D": "1.8 grams of water" + }, + "correct_answer": { + "value": "B" + }, + "solution_text": "18 grams of water = 1 mole = $N_A$ molecules.\n18 moles of water = $18 N_A$ molecules.\n1.8 grams of water = 0.1 mole = $0.1 N_A$ molecules.\nThus, 18 moles of water contains the maximum number of molecules.", + "language": "English", + "id": "8e30b99f-bc31-4edc-bfe7-e1ff8d1d8062" + }, + { + "section": "General", + "subject": "Chemistry", + "topic": "Physical Chemistry (Set 3)", + "subtopic": "Atomic Structure", + "difficulty": "Medium", + "type": "MCQ_Single", + "question_text": "The maximum number of electrons in an atom that can have the quantum numbers $n = 4$ and $m_l = +1$ is:", + "options": { + "A": "2", + "B": "4", + "C": "6", + "D": "8" + }, + "correct_answer": { + "value": "C" + }, + "solution_text": "For $n = 4$, the possible values of $l$ are 0, 1, 2, 3.\nThe condition $m_l = +1$ is possible for $l = 1$, $l = 2$, and $l = 3$.\nEach subshell ($l=1, 2, 3$) will have one orbital with $m_l = +1$.\nSince each orbital can hold 2 electrons, the total number of electrons is $3 \\times 2 = 6$.", + "language": "English", + "id": "b956e495-8e61-4cd2-9b90-d36a3a235946" + }, + { + "section": "General", + "subject": "Chemistry", + "topic": "Inorganic Chemistry (Set 3)", + "subtopic": "Chemical Bonding", + "difficulty": "Medium", + "type": "MCQ_Single", + "question_text": "The species having pyramidal shape is:", + "options": { + "A": "$\\text{SO}_3$", + "B": "$\\text{BrF}_3$", + "C": "$\\text{SiO}_3^{2-}$", + "D": "$\\text{OSF}_2$" + }, + "correct_answer": { + "value": "D" + }, + "solution_text": "In $\\text{OSF}_2$, the central sulfur atom has 4 electron domains (1 double bond to O, 2 single bonds to F, and 1 lone pair). This corresponds to an $sp^3$ hybridization with a trigonal pyramidal shape. $\\text{SO}_3$ is trigonal planar, $\\text{BrF}_3$ is T-shaped.", + "language": "English", + "id": "d629e2eb-96f3-4c6b-9a8d-097965bd330c" + }, + { + "section": "General", + "subject": "Chemistry", + "topic": "Physical Chemistry (Set 3)", + "subtopic": "Equilibrium", + "difficulty": "Medium", + "type": "MCQ_Single", + "question_text": "For the reversible reaction $\\text{N}_2(\\text{g}) + 3\\text{H}_2(\\text{g}) \\rightleftharpoons 2\\text{NH}_3(\\text{g}) + \\text{Heat}$, the equilibrium shifts in forward direction:", + "options": { + "A": "By increasing the concentration of $\\text{NH}_3(\\text{g})$", + "B": "By decreasing the pressure", + "C": "By decreasing the concentrations of $\\text{N}_2(\\text{g})$ and $\\text{H}_2(\\text{g})$", + "D": "By increasing pressure and decreasing temperature" + }, + "correct_answer": { + "value": "D" + }, + "solution_text": "According to Le Chatelier's principle, an exothermic reaction ($\\Delta H < 0$) is favored by low temperatures. A reaction involving a decrease in the number of gaseous moles ($4 \\rightarrow 2$) is favored by high pressures. Therefore, increasing pressure and decreasing temperature will shift the equilibrium in the forward direction.", + "language": "English", + "id": "0b78c170-738d-4d7b-b02d-cac8c9dc2191" + }, + { + "section": "General", + "subject": "Chemistry", + "topic": "Physical Chemistry (Set 3)", + "subtopic": "Electrochemistry", + "difficulty": "Hard", + "type": "MCQ_Single", + "question_text": "Given $E^\\circ_{\\text{Cr}^{3+}/\\text{Cr}} = -0.74\\text{ V}$; $E^\\circ_{\\text{MnO}_4^-/\\text{Mn}^{2+}} = 1.51\\text{ V}$; $E^\\circ_{\\text{Cr}_2\\text{O}_7^{2-}/\\text{Cr}^{3+}} = 1.33\\text{ V}$; $E^\\circ_{\\text{Cl}/\\text{Cl}^-} = 1.36\\text{ V}$.\nBased on the data given above, strongest oxidizing agent will be:", + "options": { + "A": "$\\text{Cl}$", + "B": "$\\text{Cr}^{3+}$", + "C": "$\\text{Mn}^{2+}$", + "D": "$\\text{MnO}_4^-$" + }, + "correct_answer": { + "value": "D" + }, + "solution_text": "Higher the standard reduction potential ($E^\\circ$), stronger is the oxidizing power. The $E^\\circ$ value for $\\text{MnO}_4^-/\\text{Mn}^{2+}$ is the highest ($+1.51\\text{ V}$). Therefore, $\\text{MnO}_4^-$ is the strongest oxidizing agent among the given species.", + "language": "English", + "id": "4e54fdf6-c2c0-4cf0-9181-112b46da2bd7" + }, + { + "section": "General", + "subject": "Chemistry", + "topic": "Organic Chemistry (Set 3)", + "subtopic": "General Organic Chemistry", + "difficulty": "Medium", + "type": "MCQ_Single", + "question_text": "Which of the following carbocations is most stable?", + "options": { + "A": "Allyl carbocation", + "B": "Primary alkyl carbocation", + "C": "Tert-butyl carbocation", + "D": "Triphenylmethyl carbocation" + }, + "correct_answer": { + "value": "D" + }, + "solution_text": "Stability of carbocations is determined by delocalization of charge. The triphenylmethyl carbocation, $(\\text{C}_6\\text{H}_5)_3\\text{C}^+$, has 3 phenyl rings providing extensive resonance stabilization, making it exceptionally stable compared to the others.", + "language": "English", + "id": "61cf97ef-68f5-4aae-9916-a6f28bf2777f" + }, + { + "section": "General", + "subject": "Chemistry", + "topic": "Organic Chemistry (Set 3)", + "subtopic": "Hydrocarbons", + "difficulty": "Hard", + "type": "MCQ_Single", + "question_text": "Ozonolysis of an organic compound gives formaldehyde as one of the products. This confirms the presence of:", + "options": { + "A": "Two isomeric double bonds", + "B": "A vinyl group", + "C": "An isopropyl group", + "D": "An acetylenic triple bond" + }, + "correct_answer": { + "value": "B" + }, + "solution_text": "Ozonolysis cleaves a carbon-carbon double bond and adds oxygen. Formaldehyde ($\\text{HCHO}$) is produced if the original double bond was at the end of a chain, specifically involving a terminal $=\\text{CH}_2$ group, which is characteristic of a vinyl group ($-\\text{CH}=\\text{CH}_2$).", + "language": "English", + "id": "fe32e132-b359-4db0-a078-84942d8738cc" + }, + { + "section": "General", + "subject": "Chemistry", + "topic": "Inorganic Chemistry (Set 3)", + "subtopic": "Coordination Compounds", + "difficulty": "Medium", + "type": "MCQ_Single", + "question_text": "The IUPAC name of $[\\text{Pt}(\\text{NH}_3)_2\\text{Cl}(\\text{NO}_2)]$ is:", + "options": { + "A": "Platinum diaminechloronitrite", + "B": "Chloronitrito-N-ammineplatinum(II)", + "C": "Diamminechloridonitrito-N-platinum(II)", + "D": "Diamminechloronitrito-O-platinate(II)" + }, + "correct_answer": { + "value": "C" + }, + "solution_text": "Ligands are named in alphabetical order: ammine, chlorido, nitrito-N (since it coordinates through nitrogen). The oxidation state of Pt is $x + 0 + (-1) + (-1) = 0 \\implies x = +2$. Thus, the correct name is Diamminechloridonitrito-N-platinum(II).", + "language": "English", + "id": "d7a0f59f-0e20-450f-96af-e46e066bff59" + }, + { + "section": "General", + "subject": "Chemistry", + "topic": "Physical Chemistry (Set 3)", + "subtopic": "Chemical Kinetics", + "difficulty": "Easy", + "type": "MCQ_Single", + "question_text": "The unit of rate constant for a zero order reaction is:", + "options": { + "A": "$\\text{s}^{-1}$", + "B": "$\\text{mol}\\cdot\\text{L}^{-1}\\cdot\\text{s}^{-1}$", + "C": "$\\text{L}\\cdot\\text{mol}^{-1}\\cdot\\text{s}^{-1}$", + "D": "$\\text{L}^2\\cdot\\text{mol}^{-2}\\cdot\\text{s}^{-1}$" + }, + "correct_answer": { + "value": "B" + }, + "solution_text": "For a zero-order reaction, Rate = $k[A]^0 = k$. The unit of rate is concentration per unit time ($\\text{mol}\\cdot\\text{L}^{-1}\\cdot\\text{s}^{-1}$). Thus, the unit of the rate constant $k$ is also $\\text{mol}\\cdot\\text{L}^{-1}\\cdot\\text{s}^{-1}$.", + "language": "English", + "id": "07db4654-f99d-4d08-ba02-fd29d7848f61" + }, + { + "section": "General", + "subject": "Chemistry", + "topic": "Organic Chemistry (Set 3)", + "subtopic": "Biomolecules", + "difficulty": "Medium", + "type": "MCQ_Single", + "question_text": "Which of the following hormones is produced under the condition of stress which stimulates glycogenolysis in the liver of human beings?", + "options": { + "A": "Thyroxin", + "B": "Insulin", + "C": "Adrenaline", + "D": "Estradiol" + }, + "correct_answer": { + "value": "C" + }, + "solution_text": "Adrenaline (epinephrine) is the 'fight or flight' hormone released during stress. It promotes the breakdown of glycogen into glucose (glycogenolysis) in the liver to increase blood glucose levels for energy.", + "language": "English", + "id": "aaf5a7d6-8337-4efd-920b-400820bcfa07" + }, + { + "section": "General", + "subject": "Biology", + "topic": "Cell Structure and Function (Set 3)", + "subtopic": "Cell Cycle and Cell Division", + "difficulty": "Medium", + "type": "MCQ_Single", + "question_text": "Crossing over occurs during which stage of meiosis?", + "options": { + "A": "Leptotene", + "B": "Pachytene", + "C": "Diplotene", + "D": "Zygotene" + }, + "correct_answer": { + "value": "B" + }, + "solution_text": "Crossing over, the exchange of genetic material between non-sister chromatids of homologous chromosomes, occurs during the pachytene stage of prophase I of meiosis.", + "language": "English", + "id": "b3899a79-4864-4482-a1c1-7d049124a643" + }, + { + "section": "General", + "subject": "Biology", + "topic": "Genetics and Evolution (Set 3)", + "subtopic": "Principles of Inheritance and Variation", + "difficulty": "Easy", + "type": "MCQ_Single", + "question_text": "A cross between two tall plants resulted in offspring having few dwarf plants. What would be the genotypes of both the parents?", + "options": { + "A": "TT and Tt", + "B": "Tt and Tt", + "C": "TT and TT", + "D": "Tt and tt" + }, + "correct_answer": { + "value": "B" + }, + "solution_text": "Since both parents are tall but they produce some dwarf offspring, both parents must carry the recessive allele for dwarfness. Thus, their genotype must be heterozygous (Tt). A cross of Tt x Tt yields a 3:1 ratio of tall to dwarf.", + "language": "English", + "id": "94f205a1-1e11-4253-8ee6-77b3b61e392c" + }, + { + "section": "General", + "subject": "Biology", + "topic": "Genetics and Evolution (Set 3)", + "subtopic": "Molecular Basis of Inheritance", + "difficulty": "Medium", + "type": "MCQ_Single", + "question_text": "Which of the following RNAs should be most abundant in an animal cell?", + "options": { + "A": "tRNA", + "B": "mRNA", + "C": "miRNA", + "D": "rRNA" + }, + "correct_answer": { + "value": "D" + }, + "solution_text": "Ribosomal RNA (rRNA) constitutes about 80% of the total RNA in an actively growing cell. It forms the structural and catalytic core of ribosomes.", + "language": "English", + "id": "d5114d8a-9d77-4d3f-b2bb-36d82ad3bd26" + }, + { + "section": "General", + "subject": "Biology", + "topic": "Human Physiology (Set 3)", + "subtopic": "Digestion and Absorption", + "difficulty": "Hard", + "type": "MCQ_Single", + "question_text": "Which of the following terms describes human dentition?", + "options": { + "A": "Thecodont, Diphyodont, Homodont", + "B": "Thecodont, Diphyodont, Heterodont", + "C": "Pleurodont, Monophyodont, Homodont", + "D": "Pleurodont, Diphyodont, Heterodont" + }, + "correct_answer": { + "value": "B" + }, + "solution_text": "Human dentition is Thecodont (teeth embedded in jaw sockets), Diphyodont (two sets of teeth: deciduous and permanent), and Heterodont (different types of teeth: incisors, canines, premolars, molars).", + "language": "English", + "id": "50c3586b-339f-4430-8f73-63e1b6780f14" + }, + { + "section": "General", + "subject": "Biology", + "topic": "Reproduction (Set 3)", + "subtopic": "Human Reproduction", + "difficulty": "Medium", + "type": "MCQ_Single", + "question_text": "Which hormone is responsible for the ovulation in human females?", + "options": { + "A": "FSH (Follicle Stimulating Hormone)", + "B": "Progesterone", + "C": "LH (Luteinizing Hormone)", + "D": "Estrogen" + }, + "correct_answer": { + "value": "C" + }, + "solution_text": "A surge in LH (Luteinizing Hormone) mid-cycle triggers the rupture of the Graafian follicle and the subsequent release of the ovum, a process known as ovulation.", + "language": "English", + "id": "d1a663d5-10dd-4885-9655-71060103c3bd" + }, + { + "section": "General", + "subject": "Biology", + "topic": "Plant Physiology (Set 3)", + "subtopic": "Photosynthesis in Higher Plants", + "difficulty": "Medium", + "type": "MCQ_Single", + "question_text": "The first stable product of CO2 fixation in C3 plants is:", + "options": { + "A": "Phosphoglyceraldehyde (PGAL)", + "B": "Malic acid", + "C": "Oxaloacetic acid (OAA)", + "D": "3-Phosphoglyceric acid (3-PGA)" + }, + "correct_answer": { + "value": "D" + }, + "solution_text": "In the Calvin cycle of C3 plants, CO2 combines with RuBP to form a 6-carbon unstable intermediate that immediately splits into two molecules of 3-phosphoglyceric acid (3-PGA), the first stable product.", + "language": "English", + "id": "1abfefc9-f273-4cd4-add4-df7ad5ec1c3e" + }, + { + "section": "General", + "subject": "Biology", + "topic": "Ecology (Set 3)", + "subtopic": "Ecosystem", + "difficulty": "Easy", + "type": "MCQ_Single", + "question_text": "The pyramid of energy is always:", + "options": { + "A": "Upright", + "B": "Inverted", + "C": "Spindle shaped", + "D": "Variable" + }, + "correct_answer": { + "value": "A" + }, + "solution_text": "The pyramid of energy is always upright because energy is lost as heat at each trophic level (10% law), so energy always decreases as it flows from producers to higher trophic levels.", + "language": "English", + "id": "2b9d9743-53fc-46d6-a9f6-55d4fcab0d4b" + }, + { + "section": "General", + "subject": "Biology", + "topic": "Biotechnology (Set 3)", + "subtopic": "Principles and Processes", + "difficulty": "Medium", + "type": "MCQ_Single", + "question_text": "The cutting of DNA at specific locations became possible with the discovery of:", + "options": { + "A": "Selectable markers", + "B": "Ligases", + "C": "Restriction enzymes", + "D": "Probes" + }, + "correct_answer": { + "value": "C" + }, + "solution_text": "Restriction endonucleases (or restriction enzymes) cut DNA at specific recognition sequences, making recombinant DNA technology possible.", + "language": "English", + "id": "03ea2344-2db9-4eb9-a231-9a9fb00d2d0c" + }, + { + "section": "General", + "subject": "Biology", + "topic": "Human Physiology (Set 3)", + "subtopic": "Neural Control and Coordination", + "difficulty": "Hard", + "type": "MCQ_Single", + "question_text": "Receptor sites for neurotransmitters are present on:", + "options": { + "A": "Pre-synaptic membrane", + "B": "Tips of axons", + "C": "Post-synaptic membrane", + "D": "Membrane of synaptic vesicles" + }, + "correct_answer": { + "value": "C" + }, + "solution_text": "Neurotransmitters are released from synaptic vesicles in the pre-synaptic neuron and diffuse across the synaptic cleft to bind with specific receptor sites located on the post-synaptic membrane.", + "language": "English", + "id": "ade2064c-c01e-4257-89cf-57f3db564e05" + }, + { + "section": "General", + "subject": "Biology", + "topic": "Diversity in Living World (Set 3)", + "subtopic": "Biological Classification", + "difficulty": "Medium", + "type": "MCQ_Single", + "question_text": "Which of the following are found in extreme saline conditions?", + "options": { + "A": "Eubacteria", + "B": "Cyanobacteria", + "C": "Mycobacteria", + "D": "Archaebacteria" + }, + "correct_answer": { + "value": "D" + }, + "solution_text": "Archaebacteria are known to live in some of the most harsh habitats. Specifically, halophiles live in extreme saline conditions (e.g., Dead Sea).", + "language": "English", + "id": "71dd9778-d209-4c67-aa11-e8da44d9cc0d" + }, + { + "section": "General", + "subject": "Physics", + "topic": "Units and Measurements (Set 4)", + "subtopic": "Dimensions", + "difficulty": "Medium", + "type": "MCQ_Single", + "question_text": "The dimension of $\\frac{1}{\\sqrt{\\mu_0 \\varepsilon_0}}$ is that of:", + "options": { + "A": "Velocity", + "B": "Time", + "C": "Capacitance", + "D": "Distance" + }, + "correct_answer": { + "value": "A" + }, + "solution_text": "We know that $c = \\frac{1}{\\sqrt{\\mu_0 \\varepsilon_0}}$, where $c$ is the speed of light in vacuum. Therefore, the dimension of $\\frac{1}{\\sqrt{\\mu_0 \\varepsilon_0}}$ is that of velocity, which is $[L T^{-1}]$.", + "language": "English", + "id": "125dc892-386f-4774-93de-1577084cea04" + }, + { + "section": "General", + "subject": "Physics", + "topic": "Kinematics (Set 4)", + "subtopic": "Projectile Motion", + "difficulty": "Easy", + "type": "MCQ_Single", + "question_text": "Two projectiles are thrown with the same initial velocity at angles of $30^\\circ$ and $60^\\circ$ with the horizontal. The ratio of their maximum heights is:", + "options": { + "A": "1:3", + "B": "3:1", + "C": "1:\\sqrt{3}", + "D": "1:2" + }, + "correct_answer": { + "value": "A" + }, + "solution_text": "Maximum height $H = \\frac{u^2 \\sin^2 \\theta}{2g}$.\nFor $\\theta_1 = 30^\\circ$, $H_1 = \\frac{u^2 (1/2)^2}{2g} = \\frac{u^2}{8g}$.\nFor $\\theta_2 = 60^\\circ$, $H_2 = \\frac{u^2 (\\sqrt{3}/2)^2}{2g} = \\frac{3u^2}{8g}$.\nRatio $H_1 : H_2 = 1 : 3$.", + "language": "English", + "id": "00f4623c-1416-4d3a-b58a-a5222d810e54" + }, + { + "section": "General", + "subject": "Physics", + "topic": "Thermodynamics (Set 4)", + "subtopic": "Carnot Engine", + "difficulty": "Medium", + "type": "MCQ_Single", + "question_text": "A Carnot engine takes heat from a reservoir at $627^\\circ\\text{C}$ and rejects heat to a sink at $27^\\circ\\text{C}$. Its efficiency is:", + "options": { + "A": "2/3", + "B": "1/3", + "C": "3/4", + "D": "1/2" + }, + "correct_answer": { + "value": "A" + }, + "solution_text": "Temperature of source, $T_1 = 627 + 273 = 900\\text{ K}$.\nTemperature of sink, $T_2 = 27 + 273 = 300\\text{ K}$.\nEfficiency $\\eta = 1 - \\frac{T_2}{T_1} = 1 - \\frac{300}{900} = 1 - \\frac{1}{3} = \\frac{2}{3}$.", + "language": "English", + "id": "69d7036d-c152-41f0-9edb-a008d62a5b0f" + }, + { + "section": "General", + "subject": "Physics", + "topic": "Electrostatics (Set 4)", + "subtopic": "Capacitors", + "difficulty": "Hard", + "type": "MCQ_Single", + "question_text": "A parallel plate capacitor with air between the plates has a capacitance of $8\\text{ pF}$. What will be the capacitance if the distance between the plates is reduced by half, and the space between them is filled with a substance of dielectric constant $6$?", + "options": { + "A": "96 pF", + "B": "8 pF", + "C": "48 pF", + "D": "12 pF" + }, + "correct_answer": { + "value": "A" + }, + "solution_text": "Initial capacitance $C_0 = \\frac{\\varepsilon_0 A}{d} = 8\\text{ pF}$.\nNew distance $d' = d/2$ and new dielectric constant $K = 6$.\nNew capacitance $C' = \\frac{K \\varepsilon_0 A}{d'} = \\frac{6 \\varepsilon_0 A}{d/2} = 12 \\left(\\frac{\\varepsilon_0 A}{d}\\right) = 12 \\times 8\\text{ pF} = 96\\text{ pF}$.", + "language": "English", + "id": "8461826b-c62a-4494-9bb0-681d1bbfe967" + }, + { + "section": "General", + "subject": "Physics", + "topic": "Current Electricity (Set 4)", + "subtopic": "Meter Bridge", + "difficulty": "Medium", + "type": "MCQ_Single", + "question_text": "In a meter bridge experiment, the null point is obtained at $20\\text{ cm}$ from one end of the wire when resistance $X$ is balanced against another resistance $Y$. If $X < Y$, then what will be the new position of the null point from the same end, if one decides to balance a resistance of $4X$ against $Y$?", + "options": { + "A": "50 cm", + "B": "40 cm", + "C": "80 cm", + "D": "20 cm" + }, + "correct_answer": { + "value": "A" + }, + "solution_text": "Initially, $\\frac{X}{Y} = \\frac{20}{100-20} = \\frac{20}{80} = \\frac{1}{4}$. So $Y = 4X$.\nFor new resistance $4X$ against $Y$, ratio is $\\frac{4X}{Y} = \\frac{4X}{4X} = 1$.\nLet new null point be $l$, then $\\frac{l}{100-l} = 1 \\implies l = 100 - l \\implies 2l = 100 \\implies l = 50\\text{ cm}$.", + "language": "English", + "id": "e16f6881-5dd0-42c2-b91d-59bfa3a8b307" + }, + { + "section": "General", + "subject": "Physics", + "topic": "Magnetism (Set 4)", + "subtopic": "Biot-Savart Law", + "difficulty": "Easy", + "type": "MCQ_Single", + "question_text": "The magnetic field at the centre of a circular current carrying coil of radius $R$ is $B_0$. If the radius is doubled keeping the current same, the new magnetic field at the centre will be:", + "options": { + "A": "$B_0/2$", + "B": "$B_0/4$", + "C": "$2B_0$", + "D": "$B_0$" + }, + "correct_answer": { + "value": "A" + }, + "solution_text": "Magnetic field at the centre of a circular coil is $B = \\frac{\\mu_0 I}{2R}$.\nSince $B \\propto \\frac{1}{R}$, doubling the radius will make the magnetic field half. Thus, new field is $B_0/2$.", + "language": "English", + "id": "f4a6d234-002b-41ad-a4f2-69fb2fd1cd7d" + }, + { + "section": "General", + "subject": "Physics", + "topic": "Alternating Current (Set 4)", + "subtopic": "LCR Circuit", + "difficulty": "Medium", + "type": "MCQ_Single", + "question_text": "In a series LCR circuit, the voltage across the resistance, capacitance, and inductance is $10\\text{ V}$ each. If the capacitance is short-circuited, the voltage across the inductance will be:", + "options": { + "A": "10 V", + "B": "$10\\sqrt{2}\\text{ V}$", + "C": "$10/\\sqrt{2}\\text{ V}$", + "D": "Zero" + }, + "correct_answer": { + "value": "C" + }, + "solution_text": "Initially, $V_R = 10\\text{ V}$, $V_L = 10\\text{ V}$, $V_C = 10\\text{ V}$. Since $V_L = V_C$, the circuit is in resonance. The applied voltage $V = \\sqrt{V_R^2 + (V_L - V_C)^2} = V_R = 10\\text{ V}$.\nIf capacitance is shorted, it becomes an LR circuit. Impedance $Z = \\sqrt{R^2 + X_L^2}$. Since $V_R = V_L$ initially, $R = X_L$, so $Z = R\\sqrt{2}$.\nNew current $I' = \\frac{V}{Z} = \\frac{10}{R\\sqrt{2}}$.\nNew voltage across inductance $V_L' = I' X_L = \\frac{10}{R\\sqrt{2}} \\times R = \\frac{10}{\\sqrt{2}}\\text{ V}$.", + "language": "English", + "id": "f6f9fedc-a555-41f5-b24d-09ff0a4e3af9" + }, + { + "section": "General", + "subject": "Physics", + "topic": "Optics (Set 4)", + "subtopic": "Wave Optics", + "difficulty": "Medium", + "type": "MCQ_Single", + "question_text": "In Young's double slit experiment, if the separation between coherent sources is halved and the distance of the screen from the coherent sources is doubled, then the fringe width becomes:", + "options": { + "A": "Half", + "B": "Four times", + "C": "One-fourth", + "D": "Double" + }, + "correct_answer": { + "value": "B" + }, + "solution_text": "Fringe width $\\beta = \\frac{\\lambda D}{d}$.\nNew slit separation $d' = d/2$.\nNew screen distance $D' = 2D$.\nNew fringe width $\\beta' = \\frac{\\lambda (2D)}{(d/2)} = 4 \\frac{\\lambda D}{d} = 4\\beta$.", + "language": "English", + "id": "47133d9d-a7f2-4b1a-8913-0920e21006a4" + }, + { + "section": "General", + "subject": "Physics", + "topic": "Modern Physics (Set 4)", + "subtopic": "Photoelectric Effect", + "difficulty": "Hard", + "type": "MCQ_Single", + "question_text": "When the light of frequency $2\\nu_0$ (where $\\nu_0$ is threshold frequency) is incident on a metal plate, the maximum velocity of electrons emitted is $v_1$. When the frequency of the incident radiation is increased to $5\\nu_0$, the maximum velocity of electrons emitted from the same plate is $v_2$. The ratio of $v_1$ to $v_2$ is:", + "options": { + "A": "1:2", + "B": "1:4", + "C": "4:1", + "D": "2:1" + }, + "correct_answer": { + "value": "A" + }, + "solution_text": "From Einstein's photoelectric equation: $K_{\\max} = h\\nu - h\\nu_0 = \\frac{1}{2} m v^2$.\nCase 1: $\\frac{1}{2} m v_1^2 = h(2\\nu_0) - h\\nu_0 = h\\nu_0$.\nCase 2: $\\frac{1}{2} m v_2^2 = h(5\\nu_0) - h\\nu_0 = 4h\\nu_0$.\nDividing the two equations: $\\frac{v_1^2}{v_2^2} = \\frac{1}{4} \\implies \\frac{v_1}{v_2} = \\frac{1}{2}$.", + "language": "English", + "id": "c26de02a-bede-4dd8-8e08-2d0e8a9e1170" + }, + { + "section": "General", + "subject": "Physics", + "topic": "Semiconductors (Set 4)", + "subtopic": "Logic Gates", + "difficulty": "Easy", + "type": "MCQ_Single", + "question_text": "Which logic gate is represented by the boolean expression $Y = \\overline{A \\cdot B}$?", + "options": { + "A": "NOR", + "B": "OR", + "C": "NAND", + "D": "AND" + }, + "correct_answer": { + "value": "C" + }, + "solution_text": "The boolean expression $Y = \\overline{A \\cdot B}$ represents the NAND gate. It gives an output of 0 only when both inputs A and B are 1, and 1 otherwise.", + "language": "English", + "id": "a0a61994-1783-4988-bba8-697192131e2a" + }, + { + "section": "General", + "subject": "Chemistry", + "topic": "Physical Chemistry (Set 4)", + "subtopic": "Mole Concept", + "difficulty": "Easy", + "type": "MCQ_Single", + "question_text": "The number of water molecules is maximum in:", + "options": { + "A": "18 grams of water", + "B": "18 moles of water", + "C": "18 molecules of water", + "D": "1.8 grams of water" + }, + "correct_answer": { + "value": "B" + }, + "solution_text": "18 grams of water = 1 mole = $N_A$ molecules.\n18 moles of water = $18 N_A$ molecules.\n1.8 grams of water = 0.1 mole = $0.1 N_A$ molecules.\nThus, 18 moles of water contains the maximum number of molecules.", + "language": "English", + "id": "a485be71-df65-433d-b4b1-0e28e7425c9f" + }, + { + "section": "General", + "subject": "Chemistry", + "topic": "Physical Chemistry (Set 4)", + "subtopic": "Atomic Structure", + "difficulty": "Medium", + "type": "MCQ_Single", + "question_text": "The maximum number of electrons in an atom that can have the quantum numbers $n = 4$ and $m_l = +1$ is:", + "options": { + "A": "2", + "B": "4", + "C": "6", + "D": "8" + }, + "correct_answer": { + "value": "C" + }, + "solution_text": "For $n = 4$, the possible values of $l$ are 0, 1, 2, 3.\nThe condition $m_l = +1$ is possible for $l = 1$, $l = 2$, and $l = 3$.\nEach subshell ($l=1, 2, 3$) will have one orbital with $m_l = +1$.\nSince each orbital can hold 2 electrons, the total number of electrons is $3 \\times 2 = 6$.", + "language": "English", + "id": "a45008df-c27b-4b28-8aa7-381aac5a0f05" + }, + { + "section": "General", + "subject": "Chemistry", + "topic": "Inorganic Chemistry (Set 4)", + "subtopic": "Chemical Bonding", + "difficulty": "Medium", + "type": "MCQ_Single", + "question_text": "The species having pyramidal shape is:", + "options": { + "A": "$\\text{SO}_3$", + "B": "$\\text{BrF}_3$", + "C": "$\\text{SiO}_3^{2-}$", + "D": "$\\text{OSF}_2$" + }, + "correct_answer": { + "value": "D" + }, + "solution_text": "In $\\text{OSF}_2$, the central sulfur atom has 4 electron domains (1 double bond to O, 2 single bonds to F, and 1 lone pair). This corresponds to an $sp^3$ hybridization with a trigonal pyramidal shape. $\\text{SO}_3$ is trigonal planar, $\\text{BrF}_3$ is T-shaped.", + "language": "English", + "id": "12111a4a-c754-4c2f-8a09-e5e62832fa6a" + }, + { + "section": "General", + "subject": "Chemistry", + "topic": "Physical Chemistry (Set 4)", + "subtopic": "Equilibrium", + "difficulty": "Medium", + "type": "MCQ_Single", + "question_text": "For the reversible reaction $\\text{N}_2(\\text{g}) + 3\\text{H}_2(\\text{g}) \\rightleftharpoons 2\\text{NH}_3(\\text{g}) + \\text{Heat}$, the equilibrium shifts in forward direction:", + "options": { + "A": "By increasing the concentration of $\\text{NH}_3(\\text{g})$", + "B": "By decreasing the pressure", + "C": "By decreasing the concentrations of $\\text{N}_2(\\text{g})$ and $\\text{H}_2(\\text{g})$", + "D": "By increasing pressure and decreasing temperature" + }, + "correct_answer": { + "value": "D" + }, + "solution_text": "According to Le Chatelier's principle, an exothermic reaction ($\\Delta H < 0$) is favored by low temperatures. A reaction involving a decrease in the number of gaseous moles ($4 \\rightarrow 2$) is favored by high pressures. Therefore, increasing pressure and decreasing temperature will shift the equilibrium in the forward direction.", + "language": "English", + "id": "11a28113-cb35-4715-8fea-7025b97d98f6" + }, + { + "section": "General", + "subject": "Chemistry", + "topic": "Physical Chemistry (Set 4)", + "subtopic": "Electrochemistry", + "difficulty": "Hard", + "type": "MCQ_Single", + "question_text": "Given $E^\\circ_{\\text{Cr}^{3+}/\\text{Cr}} = -0.74\\text{ V}$; $E^\\circ_{\\text{MnO}_4^-/\\text{Mn}^{2+}} = 1.51\\text{ V}$; $E^\\circ_{\\text{Cr}_2\\text{O}_7^{2-}/\\text{Cr}^{3+}} = 1.33\\text{ V}$; $E^\\circ_{\\text{Cl}/\\text{Cl}^-} = 1.36\\text{ V}$.\nBased on the data given above, strongest oxidizing agent will be:", + "options": { + "A": "$\\text{Cl}$", + "B": "$\\text{Cr}^{3+}$", + "C": "$\\text{Mn}^{2+}$", + "D": "$\\text{MnO}_4^-$" + }, + "correct_answer": { + "value": "D" + }, + "solution_text": "Higher the standard reduction potential ($E^\\circ$), stronger is the oxidizing power. The $E^\\circ$ value for $\\text{MnO}_4^-/\\text{Mn}^{2+}$ is the highest ($+1.51\\text{ V}$). Therefore, $\\text{MnO}_4^-$ is the strongest oxidizing agent among the given species.", + "language": "English", + "id": "bfa2a8aa-fa96-431a-896c-9b969280068e" + }, + { + "section": "General", + "subject": "Chemistry", + "topic": "Organic Chemistry (Set 4)", + "subtopic": "General Organic Chemistry", + "difficulty": "Medium", + "type": "MCQ_Single", + "question_text": "Which of the following carbocations is most stable?", + "options": { + "A": "Allyl carbocation", + "B": "Primary alkyl carbocation", + "C": "Tert-butyl carbocation", + "D": "Triphenylmethyl carbocation" + }, + "correct_answer": { + "value": "D" + }, + "solution_text": "Stability of carbocations is determined by delocalization of charge. The triphenylmethyl carbocation, $(\\text{C}_6\\text{H}_5)_3\\text{C}^+$, has 3 phenyl rings providing extensive resonance stabilization, making it exceptionally stable compared to the others.", + "language": "English", + "id": "c12d1985-e70b-4ba8-a8c9-967e001dbf46" + }, + { + "section": "General", + "subject": "Chemistry", + "topic": "Organic Chemistry (Set 4)", + "subtopic": "Hydrocarbons", + "difficulty": "Hard", + "type": "MCQ_Single", + "question_text": "Ozonolysis of an organic compound gives formaldehyde as one of the products. This confirms the presence of:", + "options": { + "A": "Two isomeric double bonds", + "B": "A vinyl group", + "C": "An isopropyl group", + "D": "An acetylenic triple bond" + }, + "correct_answer": { + "value": "B" + }, + "solution_text": "Ozonolysis cleaves a carbon-carbon double bond and adds oxygen. Formaldehyde ($\\text{HCHO}$) is produced if the original double bond was at the end of a chain, specifically involving a terminal $=\\text{CH}_2$ group, which is characteristic of a vinyl group ($-\\text{CH}=\\text{CH}_2$).", + "language": "English", + "id": "b6759d7f-dff9-4a3a-86ee-d054b701aff8" + }, + { + "section": "General", + "subject": "Chemistry", + "topic": "Inorganic Chemistry (Set 4)", + "subtopic": "Coordination Compounds", + "difficulty": "Medium", + "type": "MCQ_Single", + "question_text": "The IUPAC name of $[\\text{Pt}(\\text{NH}_3)_2\\text{Cl}(\\text{NO}_2)]$ is:", + "options": { + "A": "Platinum diaminechloronitrite", + "B": "Chloronitrito-N-ammineplatinum(II)", + "C": "Diamminechloridonitrito-N-platinum(II)", + "D": "Diamminechloronitrito-O-platinate(II)" + }, + "correct_answer": { + "value": "C" + }, + "solution_text": "Ligands are named in alphabetical order: ammine, chlorido, nitrito-N (since it coordinates through nitrogen). The oxidation state of Pt is $x + 0 + (-1) + (-1) = 0 \\implies x = +2$. Thus, the correct name is Diamminechloridonitrito-N-platinum(II).", + "language": "English", + "id": "7d335b70-4352-4ee5-b17b-281edbbabd30" + }, + { + "section": "General", + "subject": "Chemistry", + "topic": "Physical Chemistry (Set 4)", + "subtopic": "Chemical Kinetics", + "difficulty": "Easy", + "type": "MCQ_Single", + "question_text": "The unit of rate constant for a zero order reaction is:", + "options": { + "A": "$\\text{s}^{-1}$", + "B": "$\\text{mol}\\cdot\\text{L}^{-1}\\cdot\\text{s}^{-1}$", + "C": "$\\text{L}\\cdot\\text{mol}^{-1}\\cdot\\text{s}^{-1}$", + "D": "$\\text{L}^2\\cdot\\text{mol}^{-2}\\cdot\\text{s}^{-1}$" + }, + "correct_answer": { + "value": "B" + }, + "solution_text": "For a zero-order reaction, Rate = $k[A]^0 = k$. The unit of rate is concentration per unit time ($\\text{mol}\\cdot\\text{L}^{-1}\\cdot\\text{s}^{-1}$). Thus, the unit of the rate constant $k$ is also $\\text{mol}\\cdot\\text{L}^{-1}\\cdot\\text{s}^{-1}$.", + "language": "English", + "id": "5ca190aa-4050-423c-93d8-a739a54d9dc6" + }, + { + "section": "General", + "subject": "Chemistry", + "topic": "Organic Chemistry (Set 4)", + "subtopic": "Biomolecules", + "difficulty": "Medium", + "type": "MCQ_Single", + "question_text": "Which of the following hormones is produced under the condition of stress which stimulates glycogenolysis in the liver of human beings?", + "options": { + "A": "Thyroxin", + "B": "Insulin", + "C": "Adrenaline", + "D": "Estradiol" + }, + "correct_answer": { + "value": "C" + }, + "solution_text": "Adrenaline (epinephrine) is the 'fight or flight' hormone released during stress. It promotes the breakdown of glycogen into glucose (glycogenolysis) in the liver to increase blood glucose levels for energy.", + "language": "English", + "id": "0a0fc9fd-6a7e-4b8f-90c8-aeaf0bbc6619" + }, + { + "section": "General", + "subject": "Biology", + "topic": "Cell Structure and Function (Set 4)", + "subtopic": "Cell Cycle and Cell Division", + "difficulty": "Medium", + "type": "MCQ_Single", + "question_text": "Crossing over occurs during which stage of meiosis?", + "options": { + "A": "Leptotene", + "B": "Pachytene", + "C": "Diplotene", + "D": "Zygotene" + }, + "correct_answer": { + "value": "B" + }, + "solution_text": "Crossing over, the exchange of genetic material between non-sister chromatids of homologous chromosomes, occurs during the pachytene stage of prophase I of meiosis.", + "language": "English", + "id": "746b509c-3d31-41e2-82a0-dbc633bfd99e" + }, + { + "section": "General", + "subject": "Biology", + "topic": "Genetics and Evolution (Set 4)", + "subtopic": "Principles of Inheritance and Variation", + "difficulty": "Easy", + "type": "MCQ_Single", + "question_text": "A cross between two tall plants resulted in offspring having few dwarf plants. What would be the genotypes of both the parents?", + "options": { + "A": "TT and Tt", + "B": "Tt and Tt", + "C": "TT and TT", + "D": "Tt and tt" + }, + "correct_answer": { + "value": "B" + }, + "solution_text": "Since both parents are tall but they produce some dwarf offspring, both parents must carry the recessive allele for dwarfness. Thus, their genotype must be heterozygous (Tt). A cross of Tt x Tt yields a 3:1 ratio of tall to dwarf.", + "language": "English", + "id": "68389613-4924-4439-893d-cbcabfe50a4d" + }, + { + "section": "General", + "subject": "Biology", + "topic": "Genetics and Evolution (Set 4)", + "subtopic": "Molecular Basis of Inheritance", + "difficulty": "Medium", + "type": "MCQ_Single", + "question_text": "Which of the following RNAs should be most abundant in an animal cell?", + "options": { + "A": "tRNA", + "B": "mRNA", + "C": "miRNA", + "D": "rRNA" + }, + "correct_answer": { + "value": "D" + }, + "solution_text": "Ribosomal RNA (rRNA) constitutes about 80% of the total RNA in an actively growing cell. It forms the structural and catalytic core of ribosomes.", + "language": "English", + "id": "153c10b8-304e-457e-b753-4be0dfda153d" + }, + { + "section": "General", + "subject": "Biology", + "topic": "Human Physiology (Set 4)", + "subtopic": "Digestion and Absorption", + "difficulty": "Hard", + "type": "MCQ_Single", + "question_text": "Which of the following terms describes human dentition?", + "options": { + "A": "Thecodont, Diphyodont, Homodont", + "B": "Thecodont, Diphyodont, Heterodont", + "C": "Pleurodont, Monophyodont, Homodont", + "D": "Pleurodont, Diphyodont, Heterodont" + }, + "correct_answer": { + "value": "B" + }, + "solution_text": "Human dentition is Thecodont (teeth embedded in jaw sockets), Diphyodont (two sets of teeth: deciduous and permanent), and Heterodont (different types of teeth: incisors, canines, premolars, molars).", + "language": "English", + "id": "e5a4e179-10c6-4a03-8483-9a7320a211ec" + }, + { + "section": "General", + "subject": "Biology", + "topic": "Reproduction (Set 4)", + "subtopic": "Human Reproduction", + "difficulty": "Medium", + "type": "MCQ_Single", + "question_text": "Which hormone is responsible for the ovulation in human females?", + "options": { + "A": "FSH (Follicle Stimulating Hormone)", + "B": "Progesterone", + "C": "LH (Luteinizing Hormone)", + "D": "Estrogen" + }, + "correct_answer": { + "value": "C" + }, + "solution_text": "A surge in LH (Luteinizing Hormone) mid-cycle triggers the rupture of the Graafian follicle and the subsequent release of the ovum, a process known as ovulation.", + "language": "English", + "id": "039a4deb-fbd5-4b17-bbab-a48cc325a2fd" + }, + { + "section": "General", + "subject": "Biology", + "topic": "Plant Physiology (Set 4)", + "subtopic": "Photosynthesis in Higher Plants", + "difficulty": "Medium", + "type": "MCQ_Single", + "question_text": "The first stable product of CO2 fixation in C3 plants is:", + "options": { + "A": "Phosphoglyceraldehyde (PGAL)", + "B": "Malic acid", + "C": "Oxaloacetic acid (OAA)", + "D": "3-Phosphoglyceric acid (3-PGA)" + }, + "correct_answer": { + "value": "D" + }, + "solution_text": "In the Calvin cycle of C3 plants, CO2 combines with RuBP to form a 6-carbon unstable intermediate that immediately splits into two molecules of 3-phosphoglyceric acid (3-PGA), the first stable product.", + "language": "English", + "id": "b3dcd78a-0342-4036-9102-5b517706649b" + }, + { + "section": "General", + "subject": "Biology", + "topic": "Ecology (Set 4)", + "subtopic": "Ecosystem", + "difficulty": "Easy", + "type": "MCQ_Single", + "question_text": "The pyramid of energy is always:", + "options": { + "A": "Upright", + "B": "Inverted", + "C": "Spindle shaped", + "D": "Variable" + }, + "correct_answer": { + "value": "A" + }, + "solution_text": "The pyramid of energy is always upright because energy is lost as heat at each trophic level (10% law), so energy always decreases as it flows from producers to higher trophic levels.", + "language": "English", + "id": "ee00348e-d154-4bee-9fb6-1dd3875b05e0" + }, + { + "section": "General", + "subject": "Biology", + "topic": "Biotechnology (Set 4)", + "subtopic": "Principles and Processes", + "difficulty": "Medium", + "type": "MCQ_Single", + "question_text": "The cutting of DNA at specific locations became possible with the discovery of:", + "options": { + "A": "Selectable markers", + "B": "Ligases", + "C": "Restriction enzymes", + "D": "Probes" + }, + "correct_answer": { + "value": "C" + }, + "solution_text": "Restriction endonucleases (or restriction enzymes) cut DNA at specific recognition sequences, making recombinant DNA technology possible.", + "language": "English", + "id": "c90a3d1d-a327-4090-a4a4-2478622c8c74" + }, + { + "section": "General", + "subject": "Biology", + "topic": "Human Physiology (Set 4)", + "subtopic": "Neural Control and Coordination", + "difficulty": "Hard", + "type": "MCQ_Single", + "question_text": "Receptor sites for neurotransmitters are present on:", + "options": { + "A": "Pre-synaptic membrane", + "B": "Tips of axons", + "C": "Post-synaptic membrane", + "D": "Membrane of synaptic vesicles" + }, + "correct_answer": { + "value": "C" + }, + "solution_text": "Neurotransmitters are released from synaptic vesicles in the pre-synaptic neuron and diffuse across the synaptic cleft to bind with specific receptor sites located on the post-synaptic membrane.", + "language": "English", + "id": "400b1d45-f375-47b6-aeb1-39ba6c53b56b" + }, + { + "section": "General", + "subject": "Biology", + "topic": "Diversity in Living World (Set 4)", + "subtopic": "Biological Classification", + "difficulty": "Medium", + "type": "MCQ_Single", + "question_text": "Which of the following are found in extreme saline conditions?", + "options": { + "A": "Eubacteria", + "B": "Cyanobacteria", + "C": "Mycobacteria", + "D": "Archaebacteria" + }, + "correct_answer": { + "value": "D" + }, + "solution_text": "Archaebacteria are known to live in some of the most harsh habitats. Specifically, halophiles live in extreme saline conditions (e.g., Dead Sea).", + "language": "English", + "id": "cd4b967a-d552-49ed-ba86-bccdac81e439" + }, + { + "section": "General", + "subject": "Physics", + "topic": "Units and Measurements (Set 5)", + "subtopic": "Dimensions", + "difficulty": "Medium", + "type": "MCQ_Single", + "question_text": "The dimension of $\\frac{1}{\\sqrt{\\mu_0 \\varepsilon_0}}$ is that of:", + "options": { + "A": "Velocity", + "B": "Time", + "C": "Capacitance", + "D": "Distance" + }, + "correct_answer": { + "value": "A" + }, + "solution_text": "We know that $c = \\frac{1}{\\sqrt{\\mu_0 \\varepsilon_0}}$, where $c$ is the speed of light in vacuum. Therefore, the dimension of $\\frac{1}{\\sqrt{\\mu_0 \\varepsilon_0}}$ is that of velocity, which is $[L T^{-1}]$.", + "language": "English", + "id": "b2f04447-2710-46bc-a2af-c7e918701550" + }, + { + "section": "General", + "subject": "Physics", + "topic": "Kinematics (Set 5)", + "subtopic": "Projectile Motion", + "difficulty": "Easy", + "type": "MCQ_Single", + "question_text": "Two projectiles are thrown with the same initial velocity at angles of $30^\\circ$ and $60^\\circ$ with the horizontal. The ratio of their maximum heights is:", + "options": { + "A": "1:3", + "B": "3:1", + "C": "1:\\sqrt{3}", + "D": "1:2" + }, + "correct_answer": { + "value": "A" + }, + "solution_text": "Maximum height $H = \\frac{u^2 \\sin^2 \\theta}{2g}$.\nFor $\\theta_1 = 30^\\circ$, $H_1 = \\frac{u^2 (1/2)^2}{2g} = \\frac{u^2}{8g}$.\nFor $\\theta_2 = 60^\\circ$, $H_2 = \\frac{u^2 (\\sqrt{3}/2)^2}{2g} = \\frac{3u^2}{8g}$.\nRatio $H_1 : H_2 = 1 : 3$.", + "language": "English", + "id": "2b66c506-0b95-413f-925c-0d5e83c4d9d8" + }, + { + "section": "General", + "subject": "Physics", + "topic": "Thermodynamics (Set 5)", + "subtopic": "Carnot Engine", + "difficulty": "Medium", + "type": "MCQ_Single", + "question_text": "A Carnot engine takes heat from a reservoir at $627^\\circ\\text{C}$ and rejects heat to a sink at $27^\\circ\\text{C}$. Its efficiency is:", + "options": { + "A": "2/3", + "B": "1/3", + "C": "3/4", + "D": "1/2" + }, + "correct_answer": { + "value": "A" + }, + "solution_text": "Temperature of source, $T_1 = 627 + 273 = 900\\text{ K}$.\nTemperature of sink, $T_2 = 27 + 273 = 300\\text{ K}$.\nEfficiency $\\eta = 1 - \\frac{T_2}{T_1} = 1 - \\frac{300}{900} = 1 - \\frac{1}{3} = \\frac{2}{3}$.", + "language": "English", + "id": "348fa2cc-1306-46da-9a1e-3723beed6dd7" + }, + { + "section": "General", + "subject": "Physics", + "topic": "Electrostatics (Set 5)", + "subtopic": "Capacitors", + "difficulty": "Hard", + "type": "MCQ_Single", + "question_text": "A parallel plate capacitor with air between the plates has a capacitance of $8\\text{ pF}$. What will be the capacitance if the distance between the plates is reduced by half, and the space between them is filled with a substance of dielectric constant $6$?", + "options": { + "A": "96 pF", + "B": "8 pF", + "C": "48 pF", + "D": "12 pF" + }, + "correct_answer": { + "value": "A" + }, + "solution_text": "Initial capacitance $C_0 = \\frac{\\varepsilon_0 A}{d} = 8\\text{ pF}$.\nNew distance $d' = d/2$ and new dielectric constant $K = 6$.\nNew capacitance $C' = \\frac{K \\varepsilon_0 A}{d'} = \\frac{6 \\varepsilon_0 A}{d/2} = 12 \\left(\\frac{\\varepsilon_0 A}{d}\\right) = 12 \\times 8\\text{ pF} = 96\\text{ pF}$.", + "language": "English", + "id": "133d4adf-5cba-47bc-a919-8983d8bf6c99" + }, + { + "section": "General", + "subject": "Physics", + "topic": "Current Electricity (Set 5)", + "subtopic": "Meter Bridge", + "difficulty": "Medium", + "type": "MCQ_Single", + "question_text": "In a meter bridge experiment, the null point is obtained at $20\\text{ cm}$ from one end of the wire when resistance $X$ is balanced against another resistance $Y$. If $X < Y$, then what will be the new position of the null point from the same end, if one decides to balance a resistance of $4X$ against $Y$?", + "options": { + "A": "50 cm", + "B": "40 cm", + "C": "80 cm", + "D": "20 cm" + }, + "correct_answer": { + "value": "A" + }, + "solution_text": "Initially, $\\frac{X}{Y} = \\frac{20}{100-20} = \\frac{20}{80} = \\frac{1}{4}$. So $Y = 4X$.\nFor new resistance $4X$ against $Y$, ratio is $\\frac{4X}{Y} = \\frac{4X}{4X} = 1$.\nLet new null point be $l$, then $\\frac{l}{100-l} = 1 \\implies l = 100 - l \\implies 2l = 100 \\implies l = 50\\text{ cm}$.", + "language": "English", + "id": "fc788d01-fddc-4615-82d4-e0761143edf6" + }, + { + "section": "General", + "subject": "Physics", + "topic": "Magnetism (Set 5)", + "subtopic": "Biot-Savart Law", + "difficulty": "Easy", + "type": "MCQ_Single", + "question_text": "The magnetic field at the centre of a circular current carrying coil of radius $R$ is $B_0$. If the radius is doubled keeping the current same, the new magnetic field at the centre will be:", + "options": { + "A": "$B_0/2$", + "B": "$B_0/4$", + "C": "$2B_0$", + "D": "$B_0$" + }, + "correct_answer": { + "value": "A" + }, + "solution_text": "Magnetic field at the centre of a circular coil is $B = \\frac{\\mu_0 I}{2R}$.\nSince $B \\propto \\frac{1}{R}$, doubling the radius will make the magnetic field half. Thus, new field is $B_0/2$.", + "language": "English", + "id": "9be2d264-5f15-4205-acf6-971f7df348e3" + }, + { + "section": "General", + "subject": "Physics", + "topic": "Alternating Current (Set 5)", + "subtopic": "LCR Circuit", + "difficulty": "Medium", + "type": "MCQ_Single", + "question_text": "In a series LCR circuit, the voltage across the resistance, capacitance, and inductance is $10\\text{ V}$ each. If the capacitance is short-circuited, the voltage across the inductance will be:", + "options": { + "A": "10 V", + "B": "$10\\sqrt{2}\\text{ V}$", + "C": "$10/\\sqrt{2}\\text{ V}$", + "D": "Zero" + }, + "correct_answer": { + "value": "C" + }, + "solution_text": "Initially, $V_R = 10\\text{ V}$, $V_L = 10\\text{ V}$, $V_C = 10\\text{ V}$. Since $V_L = V_C$, the circuit is in resonance. The applied voltage $V = \\sqrt{V_R^2 + (V_L - V_C)^2} = V_R = 10\\text{ V}$.\nIf capacitance is shorted, it becomes an LR circuit. Impedance $Z = \\sqrt{R^2 + X_L^2}$. Since $V_R = V_L$ initially, $R = X_L$, so $Z = R\\sqrt{2}$.\nNew current $I' = \\frac{V}{Z} = \\frac{10}{R\\sqrt{2}}$.\nNew voltage across inductance $V_L' = I' X_L = \\frac{10}{R\\sqrt{2}} \\times R = \\frac{10}{\\sqrt{2}}\\text{ V}$.", + "language": "English", + "id": "6ed6639d-484a-4cd7-b1bb-c2b70dfc2f3a" + }, + { + "section": "General", + "subject": "Physics", + "topic": "Optics (Set 5)", + "subtopic": "Wave Optics", + "difficulty": "Medium", + "type": "MCQ_Single", + "question_text": "In Young's double slit experiment, if the separation between coherent sources is halved and the distance of the screen from the coherent sources is doubled, then the fringe width becomes:", + "options": { + "A": "Half", + "B": "Four times", + "C": "One-fourth", + "D": "Double" + }, + "correct_answer": { + "value": "B" + }, + "solution_text": "Fringe width $\\beta = \\frac{\\lambda D}{d}$.\nNew slit separation $d' = d/2$.\nNew screen distance $D' = 2D$.\nNew fringe width $\\beta' = \\frac{\\lambda (2D)}{(d/2)} = 4 \\frac{\\lambda D}{d} = 4\\beta$.", + "language": "English", + "id": "1005e9e4-0e0d-4ed7-a9d2-e7b57f011f0d" + }, + { + "section": "General", + "subject": "Physics", + "topic": "Modern Physics (Set 5)", + "subtopic": "Photoelectric Effect", + "difficulty": "Hard", + "type": "MCQ_Single", + "question_text": "When the light of frequency $2\\nu_0$ (where $\\nu_0$ is threshold frequency) is incident on a metal plate, the maximum velocity of electrons emitted is $v_1$. When the frequency of the incident radiation is increased to $5\\nu_0$, the maximum velocity of electrons emitted from the same plate is $v_2$. The ratio of $v_1$ to $v_2$ is:", + "options": { + "A": "1:2", + "B": "1:4", + "C": "4:1", + "D": "2:1" + }, + "correct_answer": { + "value": "A" + }, + "solution_text": "From Einstein's photoelectric equation: $K_{\\max} = h\\nu - h\\nu_0 = \\frac{1}{2} m v^2$.\nCase 1: $\\frac{1}{2} m v_1^2 = h(2\\nu_0) - h\\nu_0 = h\\nu_0$.\nCase 2: $\\frac{1}{2} m v_2^2 = h(5\\nu_0) - h\\nu_0 = 4h\\nu_0$.\nDividing the two equations: $\\frac{v_1^2}{v_2^2} = \\frac{1}{4} \\implies \\frac{v_1}{v_2} = \\frac{1}{2}$.", + "language": "English", + "id": "59778b7a-fe97-417b-ac19-739e04688ccf" + }, + { + "section": "General", + "subject": "Physics", + "topic": "Semiconductors (Set 5)", + "subtopic": "Logic Gates", + "difficulty": "Easy", + "type": "MCQ_Single", + "question_text": "Which logic gate is represented by the boolean expression $Y = \\overline{A \\cdot B}$?", + "options": { + "A": "NOR", + "B": "OR", + "C": "NAND", + "D": "AND" + }, + "correct_answer": { + "value": "C" + }, + "solution_text": "The boolean expression $Y = \\overline{A \\cdot B}$ represents the NAND gate. It gives an output of 0 only when both inputs A and B are 1, and 1 otherwise.", + "language": "English", + "id": "adac2d1f-cc43-4634-a49b-8bb47e0139e7" + }, + { + "section": "General", + "subject": "Chemistry", + "topic": "Physical Chemistry (Set 5)", + "subtopic": "Mole Concept", + "difficulty": "Easy", + "type": "MCQ_Single", + "question_text": "The number of water molecules is maximum in:", + "options": { + "A": "18 grams of water", + "B": "18 moles of water", + "C": "18 molecules of water", + "D": "1.8 grams of water" + }, + "correct_answer": { + "value": "B" + }, + "solution_text": "18 grams of water = 1 mole = $N_A$ molecules.\n18 moles of water = $18 N_A$ molecules.\n1.8 grams of water = 0.1 mole = $0.1 N_A$ molecules.\nThus, 18 moles of water contains the maximum number of molecules.", + "language": "English", + "id": "ff604fd9-3cd8-4622-b8e5-953e99100e6a" + }, + { + "section": "General", + "subject": "Chemistry", + "topic": "Physical Chemistry (Set 5)", + "subtopic": "Atomic Structure", + "difficulty": "Medium", + "type": "MCQ_Single", + "question_text": "The maximum number of electrons in an atom that can have the quantum numbers $n = 4$ and $m_l = +1$ is:", + "options": { + "A": "2", + "B": "4", + "C": "6", + "D": "8" + }, + "correct_answer": { + "value": "C" + }, + "solution_text": "For $n = 4$, the possible values of $l$ are 0, 1, 2, 3.\nThe condition $m_l = +1$ is possible for $l = 1$, $l = 2$, and $l = 3$.\nEach subshell ($l=1, 2, 3$) will have one orbital with $m_l = +1$.\nSince each orbital can hold 2 electrons, the total number of electrons is $3 \\times 2 = 6$.", + "language": "English", + "id": "5d975bc3-87b3-4715-90f6-6f33cf559566" + }, + { + "section": "General", + "subject": "Chemistry", + "topic": "Inorganic Chemistry (Set 5)", + "subtopic": "Chemical Bonding", + "difficulty": "Medium", + "type": "MCQ_Single", + "question_text": "The species having pyramidal shape is:", + "options": { + "A": "$\\text{SO}_3$", + "B": "$\\text{BrF}_3$", + "C": "$\\text{SiO}_3^{2-}$", + "D": "$\\text{OSF}_2$" + }, + "correct_answer": { + "value": "D" + }, + "solution_text": "In $\\text{OSF}_2$, the central sulfur atom has 4 electron domains (1 double bond to O, 2 single bonds to F, and 1 lone pair). This corresponds to an $sp^3$ hybridization with a trigonal pyramidal shape. $\\text{SO}_3$ is trigonal planar, $\\text{BrF}_3$ is T-shaped.", + "language": "English", + "id": "64a9af48-1868-42db-a1d4-b2186f81bb11" + }, + { + "section": "General", + "subject": "Chemistry", + "topic": "Physical Chemistry (Set 5)", + "subtopic": "Equilibrium", + "difficulty": "Medium", + "type": "MCQ_Single", + "question_text": "For the reversible reaction $\\text{N}_2(\\text{g}) + 3\\text{H}_2(\\text{g}) \\rightleftharpoons 2\\text{NH}_3(\\text{g}) + \\text{Heat}$, the equilibrium shifts in forward direction:", + "options": { + "A": "By increasing the concentration of $\\text{NH}_3(\\text{g})$", + "B": "By decreasing the pressure", + "C": "By decreasing the concentrations of $\\text{N}_2(\\text{g})$ and $\\text{H}_2(\\text{g})$", + "D": "By increasing pressure and decreasing temperature" + }, + "correct_answer": { + "value": "D" + }, + "solution_text": "According to Le Chatelier's principle, an exothermic reaction ($\\Delta H < 0$) is favored by low temperatures. A reaction involving a decrease in the number of gaseous moles ($4 \\rightarrow 2$) is favored by high pressures. Therefore, increasing pressure and decreasing temperature will shift the equilibrium in the forward direction.", + "language": "English", + "id": "6d191cad-6a77-4dcd-8f19-ba3134024517" + }, + { + "section": "General", + "subject": "Chemistry", + "topic": "Physical Chemistry (Set 5)", + "subtopic": "Electrochemistry", + "difficulty": "Hard", + "type": "MCQ_Single", + "question_text": "Given $E^\\circ_{\\text{Cr}^{3+}/\\text{Cr}} = -0.74\\text{ V}$; $E^\\circ_{\\text{MnO}_4^-/\\text{Mn}^{2+}} = 1.51\\text{ V}$; $E^\\circ_{\\text{Cr}_2\\text{O}_7^{2-}/\\text{Cr}^{3+}} = 1.33\\text{ V}$; $E^\\circ_{\\text{Cl}/\\text{Cl}^-} = 1.36\\text{ V}$.\nBased on the data given above, strongest oxidizing agent will be:", + "options": { + "A": "$\\text{Cl}$", + "B": "$\\text{Cr}^{3+}$", + "C": "$\\text{Mn}^{2+}$", + "D": "$\\text{MnO}_4^-$" + }, + "correct_answer": { + "value": "D" + }, + "solution_text": "Higher the standard reduction potential ($E^\\circ$), stronger is the oxidizing power. The $E^\\circ$ value for $\\text{MnO}_4^-/\\text{Mn}^{2+}$ is the highest ($+1.51\\text{ V}$). Therefore, $\\text{MnO}_4^-$ is the strongest oxidizing agent among the given species.", + "language": "English", + "id": "fc75431f-f3ab-4f54-9296-7e53de4c4531" + }, + { + "section": "General", + "subject": "Chemistry", + "topic": "Organic Chemistry (Set 5)", + "subtopic": "General Organic Chemistry", + "difficulty": "Medium", + "type": "MCQ_Single", + "question_text": "Which of the following carbocations is most stable?", + "options": { + "A": "Allyl carbocation", + "B": "Primary alkyl carbocation", + "C": "Tert-butyl carbocation", + "D": "Triphenylmethyl carbocation" + }, + "correct_answer": { + "value": "D" + }, + "solution_text": "Stability of carbocations is determined by delocalization of charge. The triphenylmethyl carbocation, $(\\text{C}_6\\text{H}_5)_3\\text{C}^+$, has 3 phenyl rings providing extensive resonance stabilization, making it exceptionally stable compared to the others.", + "language": "English", + "id": "315893aa-46ad-4b6e-9d65-fda1187eba80" + }, + { + "section": "General", + "subject": "Chemistry", + "topic": "Organic Chemistry (Set 5)", + "subtopic": "Hydrocarbons", + "difficulty": "Hard", + "type": "MCQ_Single", + "question_text": "Ozonolysis of an organic compound gives formaldehyde as one of the products. This confirms the presence of:", + "options": { + "A": "Two isomeric double bonds", + "B": "A vinyl group", + "C": "An isopropyl group", + "D": "An acetylenic triple bond" + }, + "correct_answer": { + "value": "B" + }, + "solution_text": "Ozonolysis cleaves a carbon-carbon double bond and adds oxygen. Formaldehyde ($\\text{HCHO}$) is produced if the original double bond was at the end of a chain, specifically involving a terminal $=\\text{CH}_2$ group, which is characteristic of a vinyl group ($-\\text{CH}=\\text{CH}_2$).", + "language": "English", + "id": "5592a3c2-e4e3-493b-8ec7-d36f366a6dad" + }, + { + "section": "General", + "subject": "Chemistry", + "topic": "Inorganic Chemistry (Set 5)", + "subtopic": "Coordination Compounds", + "difficulty": "Medium", + "type": "MCQ_Single", + "question_text": "The IUPAC name of $[\\text{Pt}(\\text{NH}_3)_2\\text{Cl}(\\text{NO}_2)]$ is:", + "options": { + "A": "Platinum diaminechloronitrite", + "B": "Chloronitrito-N-ammineplatinum(II)", + "C": "Diamminechloridonitrito-N-platinum(II)", + "D": "Diamminechloronitrito-O-platinate(II)" + }, + "correct_answer": { + "value": "C" + }, + "solution_text": "Ligands are named in alphabetical order: ammine, chlorido, nitrito-N (since it coordinates through nitrogen). The oxidation state of Pt is $x + 0 + (-1) + (-1) = 0 \\implies x = +2$. Thus, the correct name is Diamminechloridonitrito-N-platinum(II).", + "language": "English", + "id": "dde0d306-3a56-44fa-a01d-8eec040f4ba7" + }, + { + "section": "General", + "subject": "Chemistry", + "topic": "Physical Chemistry (Set 5)", + "subtopic": "Chemical Kinetics", + "difficulty": "Easy", + "type": "MCQ_Single", + "question_text": "The unit of rate constant for a zero order reaction is:", + "options": { + "A": "$\\text{s}^{-1}$", + "B": "$\\text{mol}\\cdot\\text{L}^{-1}\\cdot\\text{s}^{-1}$", + "C": "$\\text{L}\\cdot\\text{mol}^{-1}\\cdot\\text{s}^{-1}$", + "D": "$\\text{L}^2\\cdot\\text{mol}^{-2}\\cdot\\text{s}^{-1}$" + }, + "correct_answer": { + "value": "B" + }, + "solution_text": "For a zero-order reaction, Rate = $k[A]^0 = k$. The unit of rate is concentration per unit time ($\\text{mol}\\cdot\\text{L}^{-1}\\cdot\\text{s}^{-1}$). Thus, the unit of the rate constant $k$ is also $\\text{mol}\\cdot\\text{L}^{-1}\\cdot\\text{s}^{-1}$.", + "language": "English", + "id": "242645d1-3f78-4a8d-b6e9-4fe1dfec026d" + }, + { + "section": "General", + "subject": "Chemistry", + "topic": "Organic Chemistry (Set 5)", + "subtopic": "Biomolecules", + "difficulty": "Medium", + "type": "MCQ_Single", + "question_text": "Which of the following hormones is produced under the condition of stress which stimulates glycogenolysis in the liver of human beings?", + "options": { + "A": "Thyroxin", + "B": "Insulin", + "C": "Adrenaline", + "D": "Estradiol" + }, + "correct_answer": { + "value": "C" + }, + "solution_text": "Adrenaline (epinephrine) is the 'fight or flight' hormone released during stress. It promotes the breakdown of glycogen into glucose (glycogenolysis) in the liver to increase blood glucose levels for energy.", + "language": "English", + "id": "2d74deba-b576-413c-bdc9-b38ebc4133ff" + }, + { + "section": "General", + "subject": "Biology", + "topic": "Cell Structure and Function (Set 5)", + "subtopic": "Cell Cycle and Cell Division", + "difficulty": "Medium", + "type": "MCQ_Single", + "question_text": "Crossing over occurs during which stage of meiosis?", + "options": { + "A": "Leptotene", + "B": "Pachytene", + "C": "Diplotene", + "D": "Zygotene" + }, + "correct_answer": { + "value": "B" + }, + "solution_text": "Crossing over, the exchange of genetic material between non-sister chromatids of homologous chromosomes, occurs during the pachytene stage of prophase I of meiosis.", + "language": "English", + "id": "d35a85de-ae37-43c4-aefb-08a8634e97d9" + }, + { + "section": "General", + "subject": "Biology", + "topic": "Genetics and Evolution (Set 5)", + "subtopic": "Principles of Inheritance and Variation", + "difficulty": "Easy", + "type": "MCQ_Single", + "question_text": "A cross between two tall plants resulted in offspring having few dwarf plants. What would be the genotypes of both the parents?", + "options": { + "A": "TT and Tt", + "B": "Tt and Tt", + "C": "TT and TT", + "D": "Tt and tt" + }, + "correct_answer": { + "value": "B" + }, + "solution_text": "Since both parents are tall but they produce some dwarf offspring, both parents must carry the recessive allele for dwarfness. Thus, their genotype must be heterozygous (Tt). A cross of Tt x Tt yields a 3:1 ratio of tall to dwarf.", + "language": "English", + "id": "74e69024-34e1-436b-91c1-05795ddc8991" + }, + { + "section": "General", + "subject": "Biology", + "topic": "Genetics and Evolution (Set 5)", + "subtopic": "Molecular Basis of Inheritance", + "difficulty": "Medium", + "type": "MCQ_Single", + "question_text": "Which of the following RNAs should be most abundant in an animal cell?", + "options": { + "A": "tRNA", + "B": "mRNA", + "C": "miRNA", + "D": "rRNA" + }, + "correct_answer": { + "value": "D" + }, + "solution_text": "Ribosomal RNA (rRNA) constitutes about 80% of the total RNA in an actively growing cell. It forms the structural and catalytic core of ribosomes.", + "language": "English", + "id": "428e0b29-ea6a-4373-98c5-3496a5efd50a" + }, + { + "section": "General", + "subject": "Biology", + "topic": "Human Physiology (Set 5)", + "subtopic": "Digestion and Absorption", + "difficulty": "Hard", + "type": "MCQ_Single", + "question_text": "Which of the following terms describes human dentition?", + "options": { + "A": "Thecodont, Diphyodont, Homodont", + "B": "Thecodont, Diphyodont, Heterodont", + "C": "Pleurodont, Monophyodont, Homodont", + "D": "Pleurodont, Diphyodont, Heterodont" + }, + "correct_answer": { + "value": "B" + }, + "solution_text": "Human dentition is Thecodont (teeth embedded in jaw sockets), Diphyodont (two sets of teeth: deciduous and permanent), and Heterodont (different types of teeth: incisors, canines, premolars, molars).", + "language": "English", + "id": "7902ca11-3492-42bb-ad58-61bda135d377" + }, + { + "section": "General", + "subject": "Biology", + "topic": "Reproduction (Set 5)", + "subtopic": "Human Reproduction", + "difficulty": "Medium", + "type": "MCQ_Single", + "question_text": "Which hormone is responsible for the ovulation in human females?", + "options": { + "A": "FSH (Follicle Stimulating Hormone)", + "B": "Progesterone", + "C": "LH (Luteinizing Hormone)", + "D": "Estrogen" + }, + "correct_answer": { + "value": "C" + }, + "solution_text": "A surge in LH (Luteinizing Hormone) mid-cycle triggers the rupture of the Graafian follicle and the subsequent release of the ovum, a process known as ovulation.", + "language": "English", + "id": "dfe577bd-71d7-4e0f-9321-defb739fcaa5" + }, + { + "section": "General", + "subject": "Biology", + "topic": "Plant Physiology (Set 5)", + "subtopic": "Photosynthesis in Higher Plants", + "difficulty": "Medium", + "type": "MCQ_Single", + "question_text": "The first stable product of CO2 fixation in C3 plants is:", + "options": { + "A": "Phosphoglyceraldehyde (PGAL)", + "B": "Malic acid", + "C": "Oxaloacetic acid (OAA)", + "D": "3-Phosphoglyceric acid (3-PGA)" + }, + "correct_answer": { + "value": "D" + }, + "solution_text": "In the Calvin cycle of C3 plants, CO2 combines with RuBP to form a 6-carbon unstable intermediate that immediately splits into two molecules of 3-phosphoglyceric acid (3-PGA), the first stable product.", + "language": "English", + "id": "656e0b14-518f-4145-a244-20fc9ab3b7f0" + }, + { + "section": "General", + "subject": "Biology", + "topic": "Ecology (Set 5)", + "subtopic": "Ecosystem", + "difficulty": "Easy", + "type": "MCQ_Single", + "question_text": "The pyramid of energy is always:", + "options": { + "A": "Upright", + "B": "Inverted", + "C": "Spindle shaped", + "D": "Variable" + }, + "correct_answer": { + "value": "A" + }, + "solution_text": "The pyramid of energy is always upright because energy is lost as heat at each trophic level (10% law), so energy always decreases as it flows from producers to higher trophic levels.", + "language": "English", + "id": "2e290702-ad9e-4f58-9ae4-c8a2fc10df09" + }, + { + "section": "General", + "subject": "Biology", + "topic": "Biotechnology (Set 5)", + "subtopic": "Principles and Processes", + "difficulty": "Medium", + "type": "MCQ_Single", + "question_text": "The cutting of DNA at specific locations became possible with the discovery of:", + "options": { + "A": "Selectable markers", + "B": "Ligases", + "C": "Restriction enzymes", + "D": "Probes" + }, + "correct_answer": { + "value": "C" + }, + "solution_text": "Restriction endonucleases (or restriction enzymes) cut DNA at specific recognition sequences, making recombinant DNA technology possible.", + "language": "English", + "id": "b471b34c-bb0e-4c19-a08d-0bfd44dd970b" + }, + { + "section": "General", + "subject": "Biology", + "topic": "Human Physiology (Set 5)", + "subtopic": "Neural Control and Coordination", + "difficulty": "Hard", + "type": "MCQ_Single", + "question_text": "Receptor sites for neurotransmitters are present on:", + "options": { + "A": "Pre-synaptic membrane", + "B": "Tips of axons", + "C": "Post-synaptic membrane", + "D": "Membrane of synaptic vesicles" + }, + "correct_answer": { + "value": "C" + }, + "solution_text": "Neurotransmitters are released from synaptic vesicles in the pre-synaptic neuron and diffuse across the synaptic cleft to bind with specific receptor sites located on the post-synaptic membrane.", + "language": "English", + "id": "d132f31d-b66c-4873-885a-8d76aad4eb59" + }, + { + "section": "General", + "subject": "Biology", + "topic": "Diversity in Living World (Set 5)", + "subtopic": "Biological Classification", + "difficulty": "Medium", + "type": "MCQ_Single", + "question_text": "Which of the following are found in extreme saline conditions?", + "options": { + "A": "Eubacteria", + "B": "Cyanobacteria", + "C": "Mycobacteria", + "D": "Archaebacteria" + }, + "correct_answer": { + "value": "D" + }, + "solution_text": "Archaebacteria are known to live in some of the most harsh habitats. Specifically, halophiles live in extreme saline conditions (e.g., Dead Sea).", + "language": "English", + "id": "cd148387-e310-4c96-9378-fad03647b30d" + }, + { + "section": "General", + "subject": "Physics", + "topic": "Units and Measurements (Set 6)", + "subtopic": "Dimensions", + "difficulty": "Medium", + "type": "MCQ_Single", + "question_text": "The dimension of $\\frac{1}{\\sqrt{\\mu_0 \\varepsilon_0}}$ is that of:", + "options": { + "A": "Velocity", + "B": "Time", + "C": "Capacitance", + "D": "Distance" + }, + "correct_answer": { + "value": "A" + }, + "solution_text": "We know that $c = \\frac{1}{\\sqrt{\\mu_0 \\varepsilon_0}}$, where $c$ is the speed of light in vacuum. Therefore, the dimension of $\\frac{1}{\\sqrt{\\mu_0 \\varepsilon_0}}$ is that of velocity, which is $[L T^{-1}]$.", + "language": "English", + "id": "ff3df799-7390-43f9-82a8-da1891d371bb" + }, + { + "section": "General", + "subject": "Physics", + "topic": "Kinematics (Set 6)", + "subtopic": "Projectile Motion", + "difficulty": "Easy", + "type": "MCQ_Single", + "question_text": "Two projectiles are thrown with the same initial velocity at angles of $30^\\circ$ and $60^\\circ$ with the horizontal. The ratio of their maximum heights is:", + "options": { + "A": "1:3", + "B": "3:1", + "C": "1:\\sqrt{3}", + "D": "1:2" + }, + "correct_answer": { + "value": "A" + }, + "solution_text": "Maximum height $H = \\frac{u^2 \\sin^2 \\theta}{2g}$.\nFor $\\theta_1 = 30^\\circ$, $H_1 = \\frac{u^2 (1/2)^2}{2g} = \\frac{u^2}{8g}$.\nFor $\\theta_2 = 60^\\circ$, $H_2 = \\frac{u^2 (\\sqrt{3}/2)^2}{2g} = \\frac{3u^2}{8g}$.\nRatio $H_1 : H_2 = 1 : 3$.", + "language": "English", + "id": "2a49c38b-9c45-4ed3-93c7-1b98c13c7418" + }, + { + "section": "General", + "subject": "Physics", + "topic": "Thermodynamics (Set 6)", + "subtopic": "Carnot Engine", + "difficulty": "Medium", + "type": "MCQ_Single", + "question_text": "A Carnot engine takes heat from a reservoir at $627^\\circ\\text{C}$ and rejects heat to a sink at $27^\\circ\\text{C}$. Its efficiency is:", + "options": { + "A": "2/3", + "B": "1/3", + "C": "3/4", + "D": "1/2" + }, + "correct_answer": { + "value": "A" + }, + "solution_text": "Temperature of source, $T_1 = 627 + 273 = 900\\text{ K}$.\nTemperature of sink, $T_2 = 27 + 273 = 300\\text{ K}$.\nEfficiency $\\eta = 1 - \\frac{T_2}{T_1} = 1 - \\frac{300}{900} = 1 - \\frac{1}{3} = \\frac{2}{3}$.", + "language": "English", + "id": "d096dcda-f450-44d2-93a4-ba122e8f59f3" + }, + { + "section": "General", + "subject": "Physics", + "topic": "Electrostatics (Set 6)", + "subtopic": "Capacitors", + "difficulty": "Hard", + "type": "MCQ_Single", + "question_text": "A parallel plate capacitor with air between the plates has a capacitance of $8\\text{ pF}$. What will be the capacitance if the distance between the plates is reduced by half, and the space between them is filled with a substance of dielectric constant $6$?", + "options": { + "A": "96 pF", + "B": "8 pF", + "C": "48 pF", + "D": "12 pF" + }, + "correct_answer": { + "value": "A" + }, + "solution_text": "Initial capacitance $C_0 = \\frac{\\varepsilon_0 A}{d} = 8\\text{ pF}$.\nNew distance $d' = d/2$ and new dielectric constant $K = 6$.\nNew capacitance $C' = \\frac{K \\varepsilon_0 A}{d'} = \\frac{6 \\varepsilon_0 A}{d/2} = 12 \\left(\\frac{\\varepsilon_0 A}{d}\\right) = 12 \\times 8\\text{ pF} = 96\\text{ pF}$.", + "language": "English", + "id": "a26c476c-25ae-4da3-ae3f-072830afda60" + }, + { + "section": "General", + "subject": "Physics", + "topic": "Current Electricity (Set 6)", + "subtopic": "Meter Bridge", + "difficulty": "Medium", + "type": "MCQ_Single", + "question_text": "In a meter bridge experiment, the null point is obtained at $20\\text{ cm}$ from one end of the wire when resistance $X$ is balanced against another resistance $Y$. If $X < Y$, then what will be the new position of the null point from the same end, if one decides to balance a resistance of $4X$ against $Y$?", + "options": { + "A": "50 cm", + "B": "40 cm", + "C": "80 cm", + "D": "20 cm" + }, + "correct_answer": { + "value": "A" + }, + "solution_text": "Initially, $\\frac{X}{Y} = \\frac{20}{100-20} = \\frac{20}{80} = \\frac{1}{4}$. So $Y = 4X$.\nFor new resistance $4X$ against $Y$, ratio is $\\frac{4X}{Y} = \\frac{4X}{4X} = 1$.\nLet new null point be $l$, then $\\frac{l}{100-l} = 1 \\implies l = 100 - l \\implies 2l = 100 \\implies l = 50\\text{ cm}$.", + "language": "English", + "id": "027fff15-6c63-4b2b-b472-b4a853e6f707" + }, + { + "section": "General", + "subject": "Physics", + "topic": "Magnetism (Set 6)", + "subtopic": "Biot-Savart Law", + "difficulty": "Easy", + "type": "MCQ_Single", + "question_text": "The magnetic field at the centre of a circular current carrying coil of radius $R$ is $B_0$. If the radius is doubled keeping the current same, the new magnetic field at the centre will be:", + "options": { + "A": "$B_0/2$", + "B": "$B_0/4$", + "C": "$2B_0$", + "D": "$B_0$" + }, + "correct_answer": { + "value": "A" + }, + "solution_text": "Magnetic field at the centre of a circular coil is $B = \\frac{\\mu_0 I}{2R}$.\nSince $B \\propto \\frac{1}{R}$, doubling the radius will make the magnetic field half. Thus, new field is $B_0/2$.", + "language": "English", + "id": "657cc163-4800-4c4c-8c84-d0bd96eb0535" + }, + { + "section": "General", + "subject": "Physics", + "topic": "Alternating Current (Set 6)", + "subtopic": "LCR Circuit", + "difficulty": "Medium", + "type": "MCQ_Single", + "question_text": "In a series LCR circuit, the voltage across the resistance, capacitance, and inductance is $10\\text{ V}$ each. If the capacitance is short-circuited, the voltage across the inductance will be:", + "options": { + "A": "10 V", + "B": "$10\\sqrt{2}\\text{ V}$", + "C": "$10/\\sqrt{2}\\text{ V}$", + "D": "Zero" + }, + "correct_answer": { + "value": "C" + }, + "solution_text": "Initially, $V_R = 10\\text{ V}$, $V_L = 10\\text{ V}$, $V_C = 10\\text{ V}$. Since $V_L = V_C$, the circuit is in resonance. The applied voltage $V = \\sqrt{V_R^2 + (V_L - V_C)^2} = V_R = 10\\text{ V}$.\nIf capacitance is shorted, it becomes an LR circuit. Impedance $Z = \\sqrt{R^2 + X_L^2}$. Since $V_R = V_L$ initially, $R = X_L$, so $Z = R\\sqrt{2}$.\nNew current $I' = \\frac{V}{Z} = \\frac{10}{R\\sqrt{2}}$.\nNew voltage across inductance $V_L' = I' X_L = \\frac{10}{R\\sqrt{2}} \\times R = \\frac{10}{\\sqrt{2}}\\text{ V}$.", + "language": "English", + "id": "95c1c552-22e9-4859-946e-e5ea233c427d" + }, + { + "section": "General", + "subject": "Physics", + "topic": "Optics (Set 6)", + "subtopic": "Wave Optics", + "difficulty": "Medium", + "type": "MCQ_Single", + "question_text": "In Young's double slit experiment, if the separation between coherent sources is halved and the distance of the screen from the coherent sources is doubled, then the fringe width becomes:", + "options": { + "A": "Half", + "B": "Four times", + "C": "One-fourth", + "D": "Double" + }, + "correct_answer": { + "value": "B" + }, + "solution_text": "Fringe width $\\beta = \\frac{\\lambda D}{d}$.\nNew slit separation $d' = d/2$.\nNew screen distance $D' = 2D$.\nNew fringe width $\\beta' = \\frac{\\lambda (2D)}{(d/2)} = 4 \\frac{\\lambda D}{d} = 4\\beta$.", + "language": "English", + "id": "eb6c484a-b031-428f-be49-8ae3880c0ee5" + }, + { + "section": "General", + "subject": "Physics", + "topic": "Modern Physics (Set 6)", + "subtopic": "Photoelectric Effect", + "difficulty": "Hard", + "type": "MCQ_Single", + "question_text": "When the light of frequency $2\\nu_0$ (where $\\nu_0$ is threshold frequency) is incident on a metal plate, the maximum velocity of electrons emitted is $v_1$. When the frequency of the incident radiation is increased to $5\\nu_0$, the maximum velocity of electrons emitted from the same plate is $v_2$. The ratio of $v_1$ to $v_2$ is:", + "options": { + "A": "1:2", + "B": "1:4", + "C": "4:1", + "D": "2:1" + }, + "correct_answer": { + "value": "A" + }, + "solution_text": "From Einstein's photoelectric equation: $K_{\\max} = h\\nu - h\\nu_0 = \\frac{1}{2} m v^2$.\nCase 1: $\\frac{1}{2} m v_1^2 = h(2\\nu_0) - h\\nu_0 = h\\nu_0$.\nCase 2: $\\frac{1}{2} m v_2^2 = h(5\\nu_0) - h\\nu_0 = 4h\\nu_0$.\nDividing the two equations: $\\frac{v_1^2}{v_2^2} = \\frac{1}{4} \\implies \\frac{v_1}{v_2} = \\frac{1}{2}$.", + "language": "English", + "id": "b6164e29-a4e2-4782-94a4-28763cf3878b" + }, + { + "section": "General", + "subject": "Physics", + "topic": "Semiconductors (Set 6)", + "subtopic": "Logic Gates", + "difficulty": "Easy", + "type": "MCQ_Single", + "question_text": "Which logic gate is represented by the boolean expression $Y = \\overline{A \\cdot B}$?", + "options": { + "A": "NOR", + "B": "OR", + "C": "NAND", + "D": "AND" + }, + "correct_answer": { + "value": "C" + }, + "solution_text": "The boolean expression $Y = \\overline{A \\cdot B}$ represents the NAND gate. It gives an output of 0 only when both inputs A and B are 1, and 1 otherwise.", + "language": "English", + "id": "490441d3-ebee-4c9a-8398-07d009223c22" + }, + { + "section": "General", + "subject": "Chemistry", + "topic": "Physical Chemistry (Set 6)", + "subtopic": "Mole Concept", + "difficulty": "Easy", + "type": "MCQ_Single", + "question_text": "The number of water molecules is maximum in:", + "options": { + "A": "18 grams of water", + "B": "18 moles of water", + "C": "18 molecules of water", + "D": "1.8 grams of water" + }, + "correct_answer": { + "value": "B" + }, + "solution_text": "18 grams of water = 1 mole = $N_A$ molecules.\n18 moles of water = $18 N_A$ molecules.\n1.8 grams of water = 0.1 mole = $0.1 N_A$ molecules.\nThus, 18 moles of water contains the maximum number of molecules.", + "language": "English", + "id": "de008807-5fc1-461f-9139-d6a1771c5525" + }, + { + "section": "General", + "subject": "Chemistry", + "topic": "Physical Chemistry (Set 6)", + "subtopic": "Atomic Structure", + "difficulty": "Medium", + "type": "MCQ_Single", + "question_text": "The maximum number of electrons in an atom that can have the quantum numbers $n = 4$ and $m_l = +1$ is:", + "options": { + "A": "2", + "B": "4", + "C": "6", + "D": "8" + }, + "correct_answer": { + "value": "C" + }, + "solution_text": "For $n = 4$, the possible values of $l$ are 0, 1, 2, 3.\nThe condition $m_l = +1$ is possible for $l = 1$, $l = 2$, and $l = 3$.\nEach subshell ($l=1, 2, 3$) will have one orbital with $m_l = +1$.\nSince each orbital can hold 2 electrons, the total number of electrons is $3 \\times 2 = 6$.", + "language": "English", + "id": "f9e4a4ec-b2a0-49f3-bbc1-cc050ed33432" + }, + { + "section": "General", + "subject": "Chemistry", + "topic": "Inorganic Chemistry (Set 6)", + "subtopic": "Chemical Bonding", + "difficulty": "Medium", + "type": "MCQ_Single", + "question_text": "The species having pyramidal shape is:", + "options": { + "A": "$\\text{SO}_3$", + "B": "$\\text{BrF}_3$", + "C": "$\\text{SiO}_3^{2-}$", + "D": "$\\text{OSF}_2$" + }, + "correct_answer": { + "value": "D" + }, + "solution_text": "In $\\text{OSF}_2$, the central sulfur atom has 4 electron domains (1 double bond to O, 2 single bonds to F, and 1 lone pair). This corresponds to an $sp^3$ hybridization with a trigonal pyramidal shape. $\\text{SO}_3$ is trigonal planar, $\\text{BrF}_3$ is T-shaped.", + "language": "English", + "id": "2d265d5d-17f2-4aed-8c06-c2c99dce4515" + }, + { + "section": "General", + "subject": "Chemistry", + "topic": "Physical Chemistry (Set 6)", + "subtopic": "Equilibrium", + "difficulty": "Medium", + "type": "MCQ_Single", + "question_text": "For the reversible reaction $\\text{N}_2(\\text{g}) + 3\\text{H}_2(\\text{g}) \\rightleftharpoons 2\\text{NH}_3(\\text{g}) + \\text{Heat}$, the equilibrium shifts in forward direction:", + "options": { + "A": "By increasing the concentration of $\\text{NH}_3(\\text{g})$", + "B": "By decreasing the pressure", + "C": "By decreasing the concentrations of $\\text{N}_2(\\text{g})$ and $\\text{H}_2(\\text{g})$", + "D": "By increasing pressure and decreasing temperature" + }, + "correct_answer": { + "value": "D" + }, + "solution_text": "According to Le Chatelier's principle, an exothermic reaction ($\\Delta H < 0$) is favored by low temperatures. A reaction involving a decrease in the number of gaseous moles ($4 \\rightarrow 2$) is favored by high pressures. Therefore, increasing pressure and decreasing temperature will shift the equilibrium in the forward direction.", + "language": "English", + "id": "c1663444-4775-48ad-a006-a801c5bb4366" + }, + { + "section": "General", + "subject": "Chemistry", + "topic": "Physical Chemistry (Set 6)", + "subtopic": "Electrochemistry", + "difficulty": "Hard", + "type": "MCQ_Single", + "question_text": "Given $E^\\circ_{\\text{Cr}^{3+}/\\text{Cr}} = -0.74\\text{ V}$; $E^\\circ_{\\text{MnO}_4^-/\\text{Mn}^{2+}} = 1.51\\text{ V}$; $E^\\circ_{\\text{Cr}_2\\text{O}_7^{2-}/\\text{Cr}^{3+}} = 1.33\\text{ V}$; $E^\\circ_{\\text{Cl}/\\text{Cl}^-} = 1.36\\text{ V}$.\nBased on the data given above, strongest oxidizing agent will be:", + "options": { + "A": "$\\text{Cl}$", + "B": "$\\text{Cr}^{3+}$", + "C": "$\\text{Mn}^{2+}$", + "D": "$\\text{MnO}_4^-$" + }, + "correct_answer": { + "value": "D" + }, + "solution_text": "Higher the standard reduction potential ($E^\\circ$), stronger is the oxidizing power. The $E^\\circ$ value for $\\text{MnO}_4^-/\\text{Mn}^{2+}$ is the highest ($+1.51\\text{ V}$). Therefore, $\\text{MnO}_4^-$ is the strongest oxidizing agent among the given species.", + "language": "English", + "id": "9ac3c12a-3bb9-4233-bcc3-7eb8f5f0e060" + }, + { + "section": "General", + "subject": "Chemistry", + "topic": "Organic Chemistry (Set 6)", + "subtopic": "General Organic Chemistry", + "difficulty": "Medium", + "type": "MCQ_Single", + "question_text": "Which of the following carbocations is most stable?", + "options": { + "A": "Allyl carbocation", + "B": "Primary alkyl carbocation", + "C": "Tert-butyl carbocation", + "D": "Triphenylmethyl carbocation" + }, + "correct_answer": { + "value": "D" + }, + "solution_text": "Stability of carbocations is determined by delocalization of charge. The triphenylmethyl carbocation, $(\\text{C}_6\\text{H}_5)_3\\text{C}^+$, has 3 phenyl rings providing extensive resonance stabilization, making it exceptionally stable compared to the others.", + "language": "English", + "id": "533ebd7e-f374-471f-8163-7b4f858db478" + }, + { + "section": "General", + "subject": "Chemistry", + "topic": "Organic Chemistry (Set 6)", + "subtopic": "Hydrocarbons", + "difficulty": "Hard", + "type": "MCQ_Single", + "question_text": "Ozonolysis of an organic compound gives formaldehyde as one of the products. This confirms the presence of:", + "options": { + "A": "Two isomeric double bonds", + "B": "A vinyl group", + "C": "An isopropyl group", + "D": "An acetylenic triple bond" + }, + "correct_answer": { + "value": "B" + }, + "solution_text": "Ozonolysis cleaves a carbon-carbon double bond and adds oxygen. Formaldehyde ($\\text{HCHO}$) is produced if the original double bond was at the end of a chain, specifically involving a terminal $=\\text{CH}_2$ group, which is characteristic of a vinyl group ($-\\text{CH}=\\text{CH}_2$).", + "language": "English", + "id": "5ca357c9-3d85-466f-82a0-73352cee31b2" + }, + { + "section": "General", + "subject": "Chemistry", + "topic": "Inorganic Chemistry (Set 6)", + "subtopic": "Coordination Compounds", + "difficulty": "Medium", + "type": "MCQ_Single", + "question_text": "The IUPAC name of $[\\text{Pt}(\\text{NH}_3)_2\\text{Cl}(\\text{NO}_2)]$ is:", + "options": { + "A": "Platinum diaminechloronitrite", + "B": "Chloronitrito-N-ammineplatinum(II)", + "C": "Diamminechloridonitrito-N-platinum(II)", + "D": "Diamminechloronitrito-O-platinate(II)" + }, + "correct_answer": { + "value": "C" + }, + "solution_text": "Ligands are named in alphabetical order: ammine, chlorido, nitrito-N (since it coordinates through nitrogen). The oxidation state of Pt is $x + 0 + (-1) + (-1) = 0 \\implies x = +2$. Thus, the correct name is Diamminechloridonitrito-N-platinum(II).", + "language": "English", + "id": "3fcb25d5-2b6e-420b-86cb-1cc004e59d14" + }, + { + "section": "General", + "subject": "Chemistry", + "topic": "Physical Chemistry (Set 6)", + "subtopic": "Chemical Kinetics", + "difficulty": "Easy", + "type": "MCQ_Single", + "question_text": "The unit of rate constant for a zero order reaction is:", + "options": { + "A": "$\\text{s}^{-1}$", + "B": "$\\text{mol}\\cdot\\text{L}^{-1}\\cdot\\text{s}^{-1}$", + "C": "$\\text{L}\\cdot\\text{mol}^{-1}\\cdot\\text{s}^{-1}$", + "D": "$\\text{L}^2\\cdot\\text{mol}^{-2}\\cdot\\text{s}^{-1}$" + }, + "correct_answer": { + "value": "B" + }, + "solution_text": "For a zero-order reaction, Rate = $k[A]^0 = k$. The unit of rate is concentration per unit time ($\\text{mol}\\cdot\\text{L}^{-1}\\cdot\\text{s}^{-1}$). Thus, the unit of the rate constant $k$ is also $\\text{mol}\\cdot\\text{L}^{-1}\\cdot\\text{s}^{-1}$.", + "language": "English", + "id": "2174d7a8-3b23-4ca0-91f8-975aa8e34c04" + }, + { + "section": "General", + "subject": "Chemistry", + "topic": "Organic Chemistry (Set 6)", + "subtopic": "Biomolecules", + "difficulty": "Medium", + "type": "MCQ_Single", + "question_text": "Which of the following hormones is produced under the condition of stress which stimulates glycogenolysis in the liver of human beings?", + "options": { + "A": "Thyroxin", + "B": "Insulin", + "C": "Adrenaline", + "D": "Estradiol" + }, + "correct_answer": { + "value": "C" + }, + "solution_text": "Adrenaline (epinephrine) is the 'fight or flight' hormone released during stress. It promotes the breakdown of glycogen into glucose (glycogenolysis) in the liver to increase blood glucose levels for energy.", + "language": "English", + "id": "88c7ec81-dc43-4eda-b6d4-60d14d49fa85" + }, + { + "section": "General", + "subject": "Biology", + "topic": "Cell Structure and Function (Set 6)", + "subtopic": "Cell Cycle and Cell Division", + "difficulty": "Medium", + "type": "MCQ_Single", + "question_text": "Crossing over occurs during which stage of meiosis?", + "options": { + "A": "Leptotene", + "B": "Pachytene", + "C": "Diplotene", + "D": "Zygotene" + }, + "correct_answer": { + "value": "B" + }, + "solution_text": "Crossing over, the exchange of genetic material between non-sister chromatids of homologous chromosomes, occurs during the pachytene stage of prophase I of meiosis.", + "language": "English", + "id": "2a0e2fd6-5c37-43bc-92d6-766ef87cf765" + }, + { + "section": "General", + "subject": "Biology", + "topic": "Genetics and Evolution (Set 6)", + "subtopic": "Principles of Inheritance and Variation", + "difficulty": "Easy", + "type": "MCQ_Single", + "question_text": "A cross between two tall plants resulted in offspring having few dwarf plants. What would be the genotypes of both the parents?", + "options": { + "A": "TT and Tt", + "B": "Tt and Tt", + "C": "TT and TT", + "D": "Tt and tt" + }, + "correct_answer": { + "value": "B" + }, + "solution_text": "Since both parents are tall but they produce some dwarf offspring, both parents must carry the recessive allele for dwarfness. Thus, their genotype must be heterozygous (Tt). A cross of Tt x Tt yields a 3:1 ratio of tall to dwarf.", + "language": "English", + "id": "2d687e42-b22d-4b2b-9551-3d9afdab6e0f" + }, + { + "section": "General", + "subject": "Biology", + "topic": "Genetics and Evolution (Set 6)", + "subtopic": "Molecular Basis of Inheritance", + "difficulty": "Medium", + "type": "MCQ_Single", + "question_text": "Which of the following RNAs should be most abundant in an animal cell?", + "options": { + "A": "tRNA", + "B": "mRNA", + "C": "miRNA", + "D": "rRNA" + }, + "correct_answer": { + "value": "D" + }, + "solution_text": "Ribosomal RNA (rRNA) constitutes about 80% of the total RNA in an actively growing cell. It forms the structural and catalytic core of ribosomes.", + "language": "English", + "id": "21bf4e9b-fb36-404e-bf18-b362ac089093" + }, + { + "section": "General", + "subject": "Biology", + "topic": "Human Physiology (Set 6)", + "subtopic": "Digestion and Absorption", + "difficulty": "Hard", + "type": "MCQ_Single", + "question_text": "Which of the following terms describes human dentition?", + "options": { + "A": "Thecodont, Diphyodont, Homodont", + "B": "Thecodont, Diphyodont, Heterodont", + "C": "Pleurodont, Monophyodont, Homodont", + "D": "Pleurodont, Diphyodont, Heterodont" + }, + "correct_answer": { + "value": "B" + }, + "solution_text": "Human dentition is Thecodont (teeth embedded in jaw sockets), Diphyodont (two sets of teeth: deciduous and permanent), and Heterodont (different types of teeth: incisors, canines, premolars, molars).", + "language": "English", + "id": "69b32d02-bce7-46ce-89d8-835699c1f509" + }, + { + "section": "General", + "subject": "Biology", + "topic": "Reproduction (Set 6)", + "subtopic": "Human Reproduction", + "difficulty": "Medium", + "type": "MCQ_Single", + "question_text": "Which hormone is responsible for the ovulation in human females?", + "options": { + "A": "FSH (Follicle Stimulating Hormone)", + "B": "Progesterone", + "C": "LH (Luteinizing Hormone)", + "D": "Estrogen" + }, + "correct_answer": { + "value": "C" + }, + "solution_text": "A surge in LH (Luteinizing Hormone) mid-cycle triggers the rupture of the Graafian follicle and the subsequent release of the ovum, a process known as ovulation.", + "language": "English", + "id": "92e6ed7d-649b-4aec-b393-939dbeb78eee" + }, + { + "section": "General", + "subject": "Biology", + "topic": "Plant Physiology (Set 6)", + "subtopic": "Photosynthesis in Higher Plants", + "difficulty": "Medium", + "type": "MCQ_Single", + "question_text": "The first stable product of CO2 fixation in C3 plants is:", + "options": { + "A": "Phosphoglyceraldehyde (PGAL)", + "B": "Malic acid", + "C": "Oxaloacetic acid (OAA)", + "D": "3-Phosphoglyceric acid (3-PGA)" + }, + "correct_answer": { + "value": "D" + }, + "solution_text": "In the Calvin cycle of C3 plants, CO2 combines with RuBP to form a 6-carbon unstable intermediate that immediately splits into two molecules of 3-phosphoglyceric acid (3-PGA), the first stable product.", + "language": "English", + "id": "0956708f-eff5-4517-9d40-90ca12f90b93" + }, + { + "section": "General", + "subject": "Biology", + "topic": "Ecology (Set 6)", + "subtopic": "Ecosystem", + "difficulty": "Easy", + "type": "MCQ_Single", + "question_text": "The pyramid of energy is always:", + "options": { + "A": "Upright", + "B": "Inverted", + "C": "Spindle shaped", + "D": "Variable" + }, + "correct_answer": { + "value": "A" + }, + "solution_text": "The pyramid of energy is always upright because energy is lost as heat at each trophic level (10% law), so energy always decreases as it flows from producers to higher trophic levels.", + "language": "English", + "id": "b64efac0-7a70-487d-8c6d-7ee3c218b704" + }, + { + "section": "General", + "subject": "Biology", + "topic": "Biotechnology (Set 6)", + "subtopic": "Principles and Processes", + "difficulty": "Medium", + "type": "MCQ_Single", + "question_text": "The cutting of DNA at specific locations became possible with the discovery of:", + "options": { + "A": "Selectable markers", + "B": "Ligases", + "C": "Restriction enzymes", + "D": "Probes" + }, + "correct_answer": { + "value": "C" + }, + "solution_text": "Restriction endonucleases (or restriction enzymes) cut DNA at specific recognition sequences, making recombinant DNA technology possible.", + "language": "English", + "id": "bee1707b-4169-418c-8e09-c3e37feb355d" + }, + { + "section": "General", + "subject": "Biology", + "topic": "Human Physiology (Set 6)", + "subtopic": "Neural Control and Coordination", + "difficulty": "Hard", + "type": "MCQ_Single", + "question_text": "Receptor sites for neurotransmitters are present on:", + "options": { + "A": "Pre-synaptic membrane", + "B": "Tips of axons", + "C": "Post-synaptic membrane", + "D": "Membrane of synaptic vesicles" + }, + "correct_answer": { + "value": "C" + }, + "solution_text": "Neurotransmitters are released from synaptic vesicles in the pre-synaptic neuron and diffuse across the synaptic cleft to bind with specific receptor sites located on the post-synaptic membrane.", + "language": "English", + "id": "a53f69e9-44f8-4b63-a0b9-4db7c5a175df" + }, + { + "section": "General", + "subject": "Biology", + "topic": "Diversity in Living World (Set 6)", + "subtopic": "Biological Classification", + "difficulty": "Medium", + "type": "MCQ_Single", + "question_text": "Which of the following are found in extreme saline conditions?", + "options": { + "A": "Eubacteria", + "B": "Cyanobacteria", + "C": "Mycobacteria", + "D": "Archaebacteria" + }, + "correct_answer": { + "value": "D" + }, + "solution_text": "Archaebacteria are known to live in some of the most harsh habitats. Specifically, halophiles live in extreme saline conditions (e.g., Dead Sea).", + "language": "English", + "id": "5978c4fe-94ec-4e92-a6a0-c5a1af6d960a" + } +] \ No newline at end of file diff --git a/neet_sample_questions.md b/neet_sample_questions.md new file mode 100644 index 0000000..b46a9dd --- /dev/null +++ b/neet_sample_questions.md @@ -0,0 +1,479 @@ +# NEET Sample Questions + +Here are some realistic sample questions for a NEET exam. + +### Q1. The dimension of $\frac{1}{\sqrt{\mu_0 \varepsilon_0}}$ is that of: +**Subject**: Physics | **Topic**: Units and Measurements | **Difficulty**: Medium + +- **A**: Velocity +- **B**: Time +- **C**: Capacitance +- **D**: Distance + +**Correct Answer**: A + +**Solution**: +We know that $c = \frac{1}{\sqrt{\mu_0 \varepsilon_0}}$, where $c$ is the speed of light in vacuum. Therefore, the dimension of $\frac{1}{\sqrt{\mu_0 \varepsilon_0}}$ is that of velocity, which is $[L T^{-1}]$. + +--- + +### Q2. Two projectiles are thrown with the same initial velocity at angles of $30^\circ$ and $60^\circ$ with the horizontal. The ratio of their maximum heights is: +**Subject**: Physics | **Topic**: Kinematics | **Difficulty**: Easy + +- **A**: 1:3 +- **B**: 3:1 +- **C**: 1:\sqrt{3} +- **D**: 1:2 + +**Correct Answer**: A + +**Solution**: +Maximum height $H = \frac{u^2 \sin^2 \theta}{2g}$. +For $\theta_1 = 30^\circ$, $H_1 = \frac{u^2 (1/2)^2}{2g} = \frac{u^2}{8g}$. +For $\theta_2 = 60^\circ$, $H_2 = \frac{u^2 (\sqrt{3}/2)^2}{2g} = \frac{3u^2}{8g}$. +Ratio $H_1 : H_2 = 1 : 3$. + +--- + +### Q3. A Carnot engine takes heat from a reservoir at $627^\circ\text{C}$ and rejects heat to a sink at $27^\circ\text{C}$. Its efficiency is: +**Subject**: Physics | **Topic**: Thermodynamics | **Difficulty**: Medium + +- **A**: 2/3 +- **B**: 1/3 +- **C**: 3/4 +- **D**: 1/2 + +**Correct Answer**: A + +**Solution**: +Temperature of source, $T_1 = 627 + 273 = 900\text{ K}$. +Temperature of sink, $T_2 = 27 + 273 = 300\text{ K}$. +Efficiency $\eta = 1 - \frac{T_2}{T_1} = 1 - \frac{300}{900} = 1 - \frac{1}{3} = \frac{2}{3}$. + +--- + +### Q4. A parallel plate capacitor with air between the plates has a capacitance of $8\text{ pF}$. What will be the capacitance if the distance between the plates is reduced by half, and the space between them is filled with a substance of dielectric constant $6$? +**Subject**: Physics | **Topic**: Electrostatics | **Difficulty**: Hard + +- **A**: 96 pF +- **B**: 8 pF +- **C**: 48 pF +- **D**: 12 pF + +**Correct Answer**: A + +**Solution**: +Initial capacitance $C_0 = \frac{\varepsilon_0 A}{d} = 8\text{ pF}$. +New distance $d' = d/2$ and new dielectric constant $K = 6$. +New capacitance $C' = \frac{K \varepsilon_0 A}{d'} = \frac{6 \varepsilon_0 A}{d/2} = 12 \left(\frac{\varepsilon_0 A}{d}\right) = 12 \times 8\text{ pF} = 96\text{ pF}$. + +--- + +### Q5. In a meter bridge experiment, the null point is obtained at $20\text{ cm}$ from one end of the wire when resistance $X$ is balanced against another resistance $Y$. If $X < Y$, then what will be the new position of the null point from the same end, if one decides to balance a resistance of $4X$ against $Y$? +**Subject**: Physics | **Topic**: Current Electricity | **Difficulty**: Medium + +- **A**: 50 cm +- **B**: 40 cm +- **C**: 80 cm +- **D**: 20 cm + +**Correct Answer**: A + +**Solution**: +Initially, $\frac{X}{Y} = \frac{20}{100-20} = \frac{20}{80} = \frac{1}{4}$. So $Y = 4X$. +For new resistance $4X$ against $Y$, ratio is $\frac{4X}{Y} = \frac{4X}{4X} = 1$. +Let new null point be $l$, then $\frac{l}{100-l} = 1 \implies l = 100 - l \implies 2l = 100 \implies l = 50\text{ cm}$. + +--- + +### Q6. The magnetic field at the centre of a circular current carrying coil of radius $R$ is $B_0$. If the radius is doubled keeping the current same, the new magnetic field at the centre will be: +**Subject**: Physics | **Topic**: Magnetism | **Difficulty**: Easy + +- **A**: $B_0/2$ +- **B**: $B_0/4$ +- **C**: $2B_0$ +- **D**: $B_0$ + +**Correct Answer**: A + +**Solution**: +Magnetic field at the centre of a circular coil is $B = \frac{\mu_0 I}{2R}$. +Since $B \propto \frac{1}{R}$, doubling the radius will make the magnetic field half. Thus, new field is $B_0/2$. + +--- + +### Q7. In a series LCR circuit, the voltage across the resistance, capacitance, and inductance is $10\text{ V}$ each. If the capacitance is short-circuited, the voltage across the inductance will be: +**Subject**: Physics | **Topic**: Alternating Current | **Difficulty**: Medium + +- **A**: 10 V +- **B**: $10\sqrt{2}\text{ V}$ +- **C**: $10/\sqrt{2}\text{ V}$ +- **D**: Zero + +**Correct Answer**: C + +**Solution**: +Initially, $V_R = 10\text{ V}$, $V_L = 10\text{ V}$, $V_C = 10\text{ V}$. Since $V_L = V_C$, the circuit is in resonance. The applied voltage $V = \sqrt{V_R^2 + (V_L - V_C)^2} = V_R = 10\text{ V}$. +If capacitance is shorted, it becomes an LR circuit. Impedance $Z = \sqrt{R^2 + X_L^2}$. Since $V_R = V_L$ initially, $R = X_L$, so $Z = R\sqrt{2}$. +New current $I' = \frac{V}{Z} = \frac{10}{R\sqrt{2}}$. +New voltage across inductance $V_L' = I' X_L = \frac{10}{R\sqrt{2}} \times R = \frac{10}{\sqrt{2}}\text{ V}$. + +--- + +### Q8. In Young's double slit experiment, if the separation between coherent sources is halved and the distance of the screen from the coherent sources is doubled, then the fringe width becomes: +**Subject**: Physics | **Topic**: Optics | **Difficulty**: Medium + +- **A**: Half +- **B**: Four times +- **C**: One-fourth +- **D**: Double + +**Correct Answer**: B + +**Solution**: +Fringe width $\beta = \frac{\lambda D}{d}$. +New slit separation $d' = d/2$. +New screen distance $D' = 2D$. +New fringe width $\beta' = \frac{\lambda (2D)}{(d/2)} = 4 \frac{\lambda D}{d} = 4\beta$. + +--- + +### Q9. When the light of frequency $2\nu_0$ (where $\nu_0$ is threshold frequency) is incident on a metal plate, the maximum velocity of electrons emitted is $v_1$. When the frequency of the incident radiation is increased to $5\nu_0$, the maximum velocity of electrons emitted from the same plate is $v_2$. The ratio of $v_1$ to $v_2$ is: +**Subject**: Physics | **Topic**: Modern Physics | **Difficulty**: Hard + +- **A**: 1:2 +- **B**: 1:4 +- **C**: 4:1 +- **D**: 2:1 + +**Correct Answer**: A + +**Solution**: +From Einstein's photoelectric equation: $K_{\max} = h\nu - h\nu_0 = \frac{1}{2} m v^2$. +Case 1: $\frac{1}{2} m v_1^2 = h(2\nu_0) - h\nu_0 = h\nu_0$. +Case 2: $\frac{1}{2} m v_2^2 = h(5\nu_0) - h\nu_0 = 4h\nu_0$. +Dividing the two equations: $\frac{v_1^2}{v_2^2} = \frac{1}{4} \implies \frac{v_1}{v_2} = \frac{1}{2}$. + +--- + +### Q10. Which logic gate is represented by the boolean expression $Y = \overline{A \cdot B}$? +**Subject**: Physics | **Topic**: Semiconductors | **Difficulty**: Easy + +- **A**: NOR +- **B**: OR +- **C**: NAND +- **D**: AND + +**Correct Answer**: C + +**Solution**: +The boolean expression $Y = \overline{A \cdot B}$ represents the NAND gate. It gives an output of 0 only when both inputs A and B are 1, and 1 otherwise. + +--- + +### Q11. The number of water molecules is maximum in: +**Subject**: Chemistry | **Topic**: Physical Chemistry | **Difficulty**: Easy + +- **A**: 18 grams of water +- **B**: 18 moles of water +- **C**: 18 molecules of water +- **D**: 1.8 grams of water + +**Correct Answer**: B + +**Solution**: +18 grams of water = 1 mole = $N_A$ molecules. +18 moles of water = $18 N_A$ molecules. +1.8 grams of water = 0.1 mole = $0.1 N_A$ molecules. +Thus, 18 moles of water contains the maximum number of molecules. + +--- + +### Q12. The maximum number of electrons in an atom that can have the quantum numbers $n = 4$ and $m_l = +1$ is: +**Subject**: Chemistry | **Topic**: Physical Chemistry | **Difficulty**: Medium + +- **A**: 2 +- **B**: 4 +- **C**: 6 +- **D**: 8 + +**Correct Answer**: C + +**Solution**: +For $n = 4$, the possible values of $l$ are 0, 1, 2, 3. +The condition $m_l = +1$ is possible for $l = 1$, $l = 2$, and $l = 3$. +Each subshell ($l=1, 2, 3$) will have one orbital with $m_l = +1$. +Since each orbital can hold 2 electrons, the total number of electrons is $3 \times 2 = 6$. + +--- + +### Q13. The species having pyramidal shape is: +**Subject**: Chemistry | **Topic**: Inorganic Chemistry | **Difficulty**: Medium + +- **A**: $\text{SO}_3$ +- **B**: $\text{BrF}_3$ +- **C**: $\text{SiO}_3^{2-}$ +- **D**: $\text{OSF}_2$ + +**Correct Answer**: D + +**Solution**: +In $\text{OSF}_2$, the central sulfur atom has 4 electron domains (1 double bond to O, 2 single bonds to F, and 1 lone pair). This corresponds to an $sp^3$ hybridization with a trigonal pyramidal shape. $\text{SO}_3$ is trigonal planar, $\text{BrF}_3$ is T-shaped. + +--- + +### Q14. For the reversible reaction $\text{N}_2(\text{g}) + 3\text{H}_2(\text{g}) \rightleftharpoons 2\text{NH}_3(\text{g}) + \text{Heat}$, the equilibrium shifts in forward direction: +**Subject**: Chemistry | **Topic**: Physical Chemistry | **Difficulty**: Medium + +- **A**: By increasing the concentration of $\text{NH}_3(\text{g})$ +- **B**: By decreasing the pressure +- **C**: By decreasing the concentrations of $\text{N}_2(\text{g})$ and $\text{H}_2(\text{g})$ +- **D**: By increasing pressure and decreasing temperature + +**Correct Answer**: D + +**Solution**: +According to Le Chatelier's principle, an exothermic reaction ($\Delta H < 0$) is favored by low temperatures. A reaction involving a decrease in the number of gaseous moles ($4 \rightarrow 2$) is favored by high pressures. Therefore, increasing pressure and decreasing temperature will shift the equilibrium in the forward direction. + +--- + +### Q15. Given $E^\circ_{\text{Cr}^{3+}/\text{Cr}} = -0.74\text{ V}$; $E^\circ_{\text{MnO}_4^-/\text{Mn}^{2+}} = 1.51\text{ V}$; $E^\circ_{\text{Cr}_2\text{O}_7^{2-}/\text{Cr}^{3+}} = 1.33\text{ V}$; $E^\circ_{\text{Cl}/\text{Cl}^-} = 1.36\text{ V}$. +Based on the data given above, strongest oxidizing agent will be: +**Subject**: Chemistry | **Topic**: Physical Chemistry | **Difficulty**: Hard + +- **A**: $\text{Cl}$ +- **B**: $\text{Cr}^{3+}$ +- **C**: $\text{Mn}^{2+}$ +- **D**: $\text{MnO}_4^-$ + +**Correct Answer**: D + +**Solution**: +Higher the standard reduction potential ($E^\circ$), stronger is the oxidizing power. The $E^\circ$ value for $\text{MnO}_4^-/\text{Mn}^{2+}$ is the highest ($+1.51\text{ V}$). Therefore, $\text{MnO}_4^-$ is the strongest oxidizing agent among the given species. + +--- + +### Q16. Which of the following carbocations is most stable? +**Subject**: Chemistry | **Topic**: Organic Chemistry | **Difficulty**: Medium + +- **A**: Allyl carbocation +- **B**: Primary alkyl carbocation +- **C**: Tert-butyl carbocation +- **D**: Triphenylmethyl carbocation + +**Correct Answer**: D + +**Solution**: +Stability of carbocations is determined by delocalization of charge. The triphenylmethyl carbocation, $(\text{C}_6\text{H}_5)_3\text{C}^+$, has 3 phenyl rings providing extensive resonance stabilization, making it exceptionally stable compared to the others. + +--- + +### Q17. Ozonolysis of an organic compound gives formaldehyde as one of the products. This confirms the presence of: +**Subject**: Chemistry | **Topic**: Organic Chemistry | **Difficulty**: Hard + +- **A**: Two isomeric double bonds +- **B**: A vinyl group +- **C**: An isopropyl group +- **D**: An acetylenic triple bond + +**Correct Answer**: B + +**Solution**: +Ozonolysis cleaves a carbon-carbon double bond and adds oxygen. Formaldehyde ($\text{HCHO}$) is produced if the original double bond was at the end of a chain, specifically involving a terminal $=\text{CH}_2$ group, which is characteristic of a vinyl group ($-\text{CH}=\text{CH}_2$). + +--- + +### Q18. The IUPAC name of $[\text{Pt}(\text{NH}_3)_2\text{Cl}(\text{NO}_2)]$ is: +**Subject**: Chemistry | **Topic**: Inorganic Chemistry | **Difficulty**: Medium + +- **A**: Platinum diaminechloronitrite +- **B**: Chloronitrito-N-ammineplatinum(II) +- **C**: Diamminechloridonitrito-N-platinum(II) +- **D**: Diamminechloronitrito-O-platinate(II) + +**Correct Answer**: C + +**Solution**: +Ligands are named in alphabetical order: ammine, chlorido, nitrito-N (since it coordinates through nitrogen). The oxidation state of Pt is $x + 0 + (-1) + (-1) = 0 \implies x = +2$. Thus, the correct name is Diamminechloridonitrito-N-platinum(II). + +--- + +### Q19. The unit of rate constant for a zero order reaction is: +**Subject**: Chemistry | **Topic**: Physical Chemistry | **Difficulty**: Easy + +- **A**: $\text{s}^{-1}$ +- **B**: $\text{mol}\cdot\text{L}^{-1}\cdot\text{s}^{-1}$ +- **C**: $\text{L}\cdot\text{mol}^{-1}\cdot\text{s}^{-1}$ +- **D**: $\text{L}^2\cdot\text{mol}^{-2}\cdot\text{s}^{-1}$ + +**Correct Answer**: B + +**Solution**: +For a zero-order reaction, Rate = $k[A]^0 = k$. The unit of rate is concentration per unit time ($\text{mol}\cdot\text{L}^{-1}\cdot\text{s}^{-1}$). Thus, the unit of the rate constant $k$ is also $\text{mol}\cdot\text{L}^{-1}\cdot\text{s}^{-1}$. + +--- + +### Q20. Which of the following hormones is produced under the condition of stress which stimulates glycogenolysis in the liver of human beings? +**Subject**: Chemistry | **Topic**: Organic Chemistry | **Difficulty**: Medium + +- **A**: Thyroxin +- **B**: Insulin +- **C**: Adrenaline +- **D**: Estradiol + +**Correct Answer**: C + +**Solution**: +Adrenaline (epinephrine) is the 'fight or flight' hormone released during stress. It promotes the breakdown of glycogen into glucose (glycogenolysis) in the liver to increase blood glucose levels for energy. + +--- + +### Q21. Crossing over occurs during which stage of meiosis? +**Subject**: Biology | **Topic**: Cell Structure and Function | **Difficulty**: Medium + +- **A**: Leptotene +- **B**: Pachytene +- **C**: Diplotene +- **D**: Zygotene + +**Correct Answer**: B + +**Solution**: +Crossing over, the exchange of genetic material between non-sister chromatids of homologous chromosomes, occurs during the pachytene stage of prophase I of meiosis. + +--- + +### Q22. A cross between two tall plants resulted in offspring having few dwarf plants. What would be the genotypes of both the parents? +**Subject**: Biology | **Topic**: Genetics and Evolution | **Difficulty**: Easy + +- **A**: TT and Tt +- **B**: Tt and Tt +- **C**: TT and TT +- **D**: Tt and tt + +**Correct Answer**: B + +**Solution**: +Since both parents are tall but they produce some dwarf offspring, both parents must carry the recessive allele for dwarfness. Thus, their genotype must be heterozygous (Tt). A cross of Tt x Tt yields a 3:1 ratio of tall to dwarf. + +--- + +### Q23. Which of the following RNAs should be most abundant in an animal cell? +**Subject**: Biology | **Topic**: Genetics and Evolution | **Difficulty**: Medium + +- **A**: tRNA +- **B**: mRNA +- **C**: miRNA +- **D**: rRNA + +**Correct Answer**: D + +**Solution**: +Ribosomal RNA (rRNA) constitutes about 80% of the total RNA in an actively growing cell. It forms the structural and catalytic core of ribosomes. + +--- + +### Q24. Which of the following terms describes human dentition? +**Subject**: Biology | **Topic**: Human Physiology | **Difficulty**: Hard + +- **A**: Thecodont, Diphyodont, Homodont +- **B**: Thecodont, Diphyodont, Heterodont +- **C**: Pleurodont, Monophyodont, Homodont +- **D**: Pleurodont, Diphyodont, Heterodont + +**Correct Answer**: B + +**Solution**: +Human dentition is Thecodont (teeth embedded in jaw sockets), Diphyodont (two sets of teeth: deciduous and permanent), and Heterodont (different types of teeth: incisors, canines, premolars, molars). + +--- + +### Q25. Which hormone is responsible for the ovulation in human females? +**Subject**: Biology | **Topic**: Reproduction | **Difficulty**: Medium + +- **A**: FSH (Follicle Stimulating Hormone) +- **B**: Progesterone +- **C**: LH (Luteinizing Hormone) +- **D**: Estrogen + +**Correct Answer**: C + +**Solution**: +A surge in LH (Luteinizing Hormone) mid-cycle triggers the rupture of the Graafian follicle and the subsequent release of the ovum, a process known as ovulation. + +--- + +### Q26. The first stable product of CO2 fixation in C3 plants is: +**Subject**: Biology | **Topic**: Plant Physiology | **Difficulty**: Medium + +- **A**: Phosphoglyceraldehyde (PGAL) +- **B**: Malic acid +- **C**: Oxaloacetic acid (OAA) +- **D**: 3-Phosphoglyceric acid (3-PGA) + +**Correct Answer**: D + +**Solution**: +In the Calvin cycle of C3 plants, CO2 combines with RuBP to form a 6-carbon unstable intermediate that immediately splits into two molecules of 3-phosphoglyceric acid (3-PGA), the first stable product. + +--- + +### Q27. The pyramid of energy is always: +**Subject**: Biology | **Topic**: Ecology | **Difficulty**: Easy + +- **A**: Upright +- **B**: Inverted +- **C**: Spindle shaped +- **D**: Variable + +**Correct Answer**: A + +**Solution**: +The pyramid of energy is always upright because energy is lost as heat at each trophic level (10% law), so energy always decreases as it flows from producers to higher trophic levels. + +--- + +### Q28. The cutting of DNA at specific locations became possible with the discovery of: +**Subject**: Biology | **Topic**: Biotechnology | **Difficulty**: Medium + +- **A**: Selectable markers +- **B**: Ligases +- **C**: Restriction enzymes +- **D**: Probes + +**Correct Answer**: C + +**Solution**: +Restriction endonucleases (or restriction enzymes) cut DNA at specific recognition sequences, making recombinant DNA technology possible. + +--- + +### Q29. Receptor sites for neurotransmitters are present on: +**Subject**: Biology | **Topic**: Human Physiology | **Difficulty**: Hard + +- **A**: Pre-synaptic membrane +- **B**: Tips of axons +- **C**: Post-synaptic membrane +- **D**: Membrane of synaptic vesicles + +**Correct Answer**: C + +**Solution**: +Neurotransmitters are released from synaptic vesicles in the pre-synaptic neuron and diffuse across the synaptic cleft to bind with specific receptor sites located on the post-synaptic membrane. + +--- + +### Q30. Which of the following are found in extreme saline conditions? +**Subject**: Biology | **Topic**: Diversity in Living World | **Difficulty**: Medium + +- **A**: Eubacteria +- **B**: Cyanobacteria +- **C**: Mycobacteria +- **D**: Archaebacteria + +**Correct Answer**: D + +**Solution**: +Archaebacteria are known to live in some of the most harsh habitats. Specifically, halophiles live in extreme saline conditions (e.g., Dead Sea). + +--- diff --git a/package-lock.json b/package-lock.json index 9ef1a30..af48019 100644 --- a/package-lock.json +++ b/package-lock.json @@ -17,6 +17,7 @@ "katex": "^0.17.0", "lucide-react": "^1.21.0", "next": "16.2.9", + "next-themes": "^0.4.6", "react": "19.2.4", "react-datepicker": "^9.1.0", "react-dom": "19.2.4", @@ -27,6 +28,7 @@ "rehype-raw": "^7.0.0", "remark-math": "^6.0.0", "tailwind-merge": "^3.6.0", + "zod": "^4.4.3", "zustand": "^5.0.14" }, "devDependencies": { @@ -6754,6 +6756,16 @@ } } }, + "node_modules/next-themes": { + "version": "0.4.6", + "resolved": "https://registry.npmjs.org/next-themes/-/next-themes-0.4.6.tgz", + "integrity": "sha512-pZvgD5L0IEvX5/9GWyHMf3m8BKiVQwsCMHfoFosXtXBMnaS0ZnIJ9ST4b4NqLVKDEm8QBxoNNGNaBv2JNF6XNA==", + "license": "MIT", + "peerDependencies": { + "react": "^16.8 || ^17 || ^18 || ^19 || ^19.0.0-rc", + "react-dom": "^16.8 || ^17 || ^18 || ^19 || ^19.0.0-rc" + } + }, "node_modules/next/node_modules/postcss": { "version": "8.4.31", "resolved": "https://registry.npmjs.org/postcss/-/postcss-8.4.31.tgz", @@ -8888,7 +8900,6 @@ "version": "4.4.3", "resolved": "https://registry.npmjs.org/zod/-/zod-4.4.3.tgz", "integrity": "sha512-ytENFjIJFl2UwYglde2jchW2Hwm4GJFLDiSXWdTrJQBIN9Fcyp7n4DhxJEiWNAJMV1/BqWfW/kkg71UDcHJyTQ==", - "dev": true, "license": "MIT", "funding": { "url": "https://github.com/sponsors/colinhacks" diff --git a/package.json b/package.json index 743bf3b..09c092e 100644 --- a/package.json +++ b/package.json @@ -18,6 +18,7 @@ "katex": "^0.17.0", "lucide-react": "^1.21.0", "next": "16.2.9", + "next-themes": "^0.4.6", "react": "19.2.4", "react-datepicker": "^9.1.0", "react-dom": "19.2.4", @@ -28,6 +29,7 @@ "rehype-raw": "^7.0.0", "remark-math": "^6.0.0", "tailwind-merge": "^3.6.0", + "zod": "^4.4.3", "zustand": "^5.0.14" }, "devDependencies": { diff --git a/real_jee_questions_rendering_test.json b/real_jee_questions_rendering_test.json new file mode 100644 index 0000000..fedbf1e --- /dev/null +++ b/real_jee_questions_rendering_test.json @@ -0,0 +1,1872 @@ +[ + { + "section": "General", + "subject": "Physics", + "topic": "Rotational Mechanics Set 1", + "subtopic": "Angular Momentum & Collisions", + "difficulty": "Hard", + "type": "MCQ_Single", + "question_text": "A thin uniform rod of mass $M$ and length $L$ is pivoted at its upper end and hangs vertically. A small particle of mass $m$ ($m \\ll M$) moving horizontally with velocity $v_0$ strikes the rod at a distance $x$ from the pivot and sticks to it. If the rod deflects by an angle $\\theta$ such that $\\cos\\theta = 1 - \\frac{6mv_0^2}{MgL}$, the collision point $x$ is:", + "options": { + "A": "$\\frac{L}{\\sqrt{3}}$", + "B": "$\\frac{L}{2}$", + "C": "$\\frac{2L}{3}$", + "D": "$\\frac{L}{\\sqrt{2}}$" + }, + "correct_answer": { + "value": "B" + }, + "solution_text": "By conservation of angular momentum about the pivot:\n\n$$m v_0 x = I \\omega = \\left(\\frac{1}{3}ML^2 + m x^2\\right) \\omega \\approx \\frac{1}{3}ML^2 \\omega \\implies \\omega = \\frac{3mv_0x}{ML^2}$$\n\nFrom conservation of energy after collision:\n\n$$\\frac{1}{2} I \\omega^2 = M g h_{\\text{cm}} \\implies \\frac{1}{2} \\left(\\frac{1}{3}ML^2\\right) \\left(\\frac{3mv_0x}{ML^2}\\right)^2 = M g \\left(\\frac{L}{2}\\right) (1 - \\cos\\theta)$$\n\n$$\\frac{3}{2}\\frac{m^2 v_0^2 x^2}{ML^2} = M g \\frac{L}{2} \\left(\\frac{6mv_0^2}{MgL}\\right) = 3mv_0^2 \\implies x = \\frac{L}{2}$$", + "language": "English", + "id": "1811a71f-fb8d-4f31-a541-c4fcffffb523" + }, + { + "section": "General", + "subject": "Physics", + "topic": "Electromagnetism Set 1", + "subtopic": "Electromagnetic Induction", + "difficulty": "Hard", + "type": "MCQ_Multiple", + "question_text": "A rectangular loop of dimensions $a \\times b$ with resistance $R$ is placed near an infinitely long straight wire carrying a time-dependent current $I(t) = I_0 \\sin(\\omega t)$. The loop is at a distance $r_0$ from the wire. Which of the following statements is/are correct?", + "options": { + "A": "The maximum mutual inductance $M$ between the wire and the loop is $\\frac{\\mu_0 b}{2\\pi} \\ln\\left(1 + \\frac{a}{r_0}\\right)$.", + "B": "The amplitude of the induced current in the loop is $\\frac{\\mu_0 b I_0 \\omega}{2\\pi R} \\ln\\left(1 + \\frac{a}{r_0}\\right)$.", + "C": "The net force on the loop is zero at all times.", + "D": "The induced EMF in the loop is in phase with the current in the straight wire." + }, + "correct_answer": { + "value": [ + "A", + "B" + ] + }, + "solution_text": "Magnetic flux $\\Phi(t)$ through the loop is:\n\n$$\\Phi(t) = \\int_{r_0}^{r_0+a} \\frac{\\mu_0 I(t)}{2\\pi r} b \\, dr = \\frac{\\mu_0 b I(t)}{2\\pi} \\ln\\left(\\frac{r_0+a}{r_0}\\right)$$\n\nMutual inductance $M = \\frac{\\Phi(t)}{I(t)} = \\frac{\\mu_0 b}{2\\pi} \\ln\\left(1 + \\frac{a}{r_0}\\right).\n\nInduced EMF $\\mathcal{E}(t) = -\\frac{d\\Phi}{dt} = -M I_0 \\omega \\cos(\\omega t)$.\n\nInduced current $i(t) = \\frac{\\mathcal{E}(t)}{R} = -\\frac{M I_0 \\omega}{R} \\cos(\\omega t)$.\n\nTherefore, the amplitude of the induced current is $\\frac{\\mu_0 b I_0 \\omega}{2\\pi R} \\ln\\left(1 + \\frac{a}{r_0}\\right)$. Since EMF varies as $\\cos(\\omega t)$ and current in the wire is $\\sin(\\omega t)$, they are out of phase by $\\pi/2$. Net force is non-zero because the magnetic field varies across the loop width.", + "language": "English", + "id": "7aac5446-8c86-4582-befc-32de35913317" + }, + { + "section": "General", + "subject": "Physics", + "topic": "Modern Physics Set 1", + "subtopic": "Photoelectric Effect", + "difficulty": "Medium", + "type": "Numerical", + "question_text": "In a photoelectric effect experiment, light of wavelength $\\lambda = 300\\text{ nm}$ is incident on a metal plate. The stopping potential is found to be $V_1 = 1.85\\text{ V}$. When light of wavelength $\\lambda' = 200\\text{ nm}$ is incident on the same metal plate, the stopping potential becomes $V_2$ (in Volts). Calculate the value of $V_2$.\n\n(Take $hc = 1240\\text{ eV}\\cdot\\text{nm}$ and electron charge $e = 1.6 \\times 10^{-19}\\text{ C}$)", + "options": null, + "correct_answer": { + "value": "3.92" + }, + "solution_text": "From Einstein's Photoelectric Equation:\n\n$$eV_1 = \\frac{hc}{\\lambda} - \\Phi_0 \\implies \\Phi_0 = \\frac{1240}{300} - 1.85 = 4.13 - 1.85 = 2.28\\text{ eV}$$\n\nFor the second wavelength:\n\n$$eV_2 = \\frac{hc}{\\lambda'} - \\Phi_0 \\implies V_2 = \\frac{1240}{200} - 2.28 = 6.20 - 2.28 = 3.92\\text{ V}$$", + "language": "English", + "id": "a174daee-81b7-4df4-9d1f-76e37f16fd27" + }, + { + "section": "General", + "subject": "Chemistry", + "topic": "Coordination Chemistry Set 1", + "subtopic": "Crystal Field Theory", + "difficulty": "Hard", + "type": "MCQ_Single", + "question_text": "For the octahedral complex ion $[Fe(H_2O)_6]^{3+}$, the crystal field splitting energy is denoted by $\\Delta_0$ and the pairing energy is $P$. Which of the following options correctly represents the magnetic properties and d-electron configuration of the metal ion?", + "options": { + "A": "Low spin, $t_{2g}^5 e_g^0$, paramagnetic with $\\mu_{\\text{eff}} = \\sqrt{35}\\text{ B.M.}$", + "B": "High spin, $t_{2g}^3 e_g^2$, paramagnetic with $\\mu_{\\text{eff}} = \\sqrt{35}\\text{ B.M.}$", + "C": "High spin, $t_{2g}^3 e_g^2$, paramagnetic with $\\mu_{\\text{eff}} = \\sqrt{15}\\text{ B.M.}$", + "D": "Low spin, $t_{2g}^5 e_g^0$, paramagnetic with $\\mu_{\\text{eff}} = \\sqrt{3}\\text{ B.M.}$" + }, + "correct_answer": { + "value": "B" + }, + "solution_text": "$Fe^{3+}$ has a $d^5$ electron configuration. Since $H_2O$ is a weak field ligand, $\\Delta_0 < P$, resulting in a high-spin complex.\n\nConfiguration: $t_{2g}^3 e_g^2$.\n\nNumber of unpaired electrons $n = 5$.\n\nSpin-only magnetic moment $\\mu_{\\text{eff}} = \\sqrt{n(n+2)} = \\sqrt{5(7)} = \\sqrt{35}\\text{ B.M.}$", + "language": "English", + "id": "aa5735b2-91ef-4862-8e42-eb0492d61ae7" + }, + { + "section": "General", + "subject": "Chemistry", + "topic": "Physical Chemistry Set 1", + "subtopic": "Chemical Kinetics", + "difficulty": "Hard", + "type": "MCQ_Multiple", + "question_text": "Consider the reaction network showing first-order parallel decomposition of a reactant $A$:\n\n$$A \\xrightarrow{k_1} B \\quad \\text{and} \\quad A \\xrightarrow{k_2} C$$\n\nAt a temperature $T_1$, the rate constants are $k_1 = 2.0 \\times 10^{-4}\\text{ s}^{-1}$ and $k_2 = 1.0 \\times 10^{-4}\\text{ s}^{-1}$. Let the activation energies for the two paths be $E_{a1} = 40\\text{ kJ/mol}$ and $E_{a2} = 80\\text{ kJ/mol}$ respectively. Which of the following statements is/are correct?", + "options": { + "A": "The fractional yield of $B$ at temperature $T_1$ is $\\frac{2}{3}$.", + "B": "If the temperature is increased to $T_2 > T_1$, the ratio $[B]/[C]$ decreases.", + "C": "The overall rate constant for the disappearance of $A$ is $k_{\\text{eff}} = 3.0 \\times 10^{-4}\\text{ s}^{-1}$ at $T_1$.", + "D": "The effective activation energy for the disappearance of $A$ is $E_{\\text{eff}} \\approx 53.3\\text{ kJ/mol}$ at $T_1$." + }, + "correct_answer": { + "value": [ + "A", + "B", + "C", + "D" + ] + }, + "solution_text": "Let us check each option:\n\n1. Yield of $B = \\frac{k_1}{k_1+k_2} = \\frac{2.0}{2.0+1.0} = \\frac{2}{3}$. (A is correct)\n\n2. Since $E_{a2} > E_{a1}$, the rate constant $k_2$ increases more rapidly with temperature than $k_1$. Hence, the ratio $\\frac{[B]}{[C]} = \\frac{k_1}{k_2}$ decreases as temperature increases. (B is correct)\n\n3. For parallel first-order reactions, $k_{\\text{eff}} = k_1 + k_2 = 3.0 \\times 10^{-4}\\text{ s}^{-1}$. (C is correct)\n\n4. The effective activation energy is $E_{\\text{eff}} = \\frac{k_1 E_{a1} + k_2 E_{a2}}{k_1 + k_2} = \\frac{2(40) + 1(80)}{3} = 53.33\\text{ kJ/mol}$. (D is correct)", + "language": "English", + "id": "ab8fcfea-0219-4e42-b4e1-c104288339f7" + }, + { + "section": "General", + "subject": "Chemistry", + "topic": "Physical Chemistry Set 1", + "subtopic": "Thermodynamics", + "difficulty": "Medium", + "type": "Numerical", + "question_text": "Calculate the change in entropy $\\Delta S$ (in $\\text{J}\\cdot\\text{K}^{-1}\\cdot\\text{mol}^{-1}$) when $2\\text{ moles}$ of an ideal gas ($C_{V,m} = \\frac{3}{2}R$) are heated reversibly from $300\\text{ K}$ to $600\\text{ K}$ at constant pressure.\n\n(Take $R = 8.314\\text{ J}\\cdot\\text{K}^{-1}\\cdot\\text{mol}^{-1}$ and $\\ln(2) \\approx 0.693$)", + "options": null, + "correct_answer": { + "value": "28.81" + }, + "solution_text": "For a process at constant pressure:\n\n$$C_{P,m} = C_{V,m} + R = \\frac{5}{2}R$$\n\nThe entropy change is:\n\n$$\\Delta S = n C_{P,m} \\ln\\left(\\frac{T_2}{T_1}\\right) = 2 \\times \\left(\\frac{5}{2} R\\right) \\ln\\left(\\frac{600}{300}\\right) = 5 R \\ln(2)$$\n\n$$\\Delta S = 5 \\times 8.314 \\times 0.693 \\approx 28.81\\text{ J}\\cdot\\text{K}^{-1}\\cdot\\text{mol}^{-1}$$", + "language": "English", + "id": "4893b467-8896-42ed-bb93-3e70f7ce5495" + }, + { + "section": "General", + "subject": "Mathematics", + "topic": "Calculus Set 1", + "subtopic": "Definite Integration", + "difficulty": "Hard", + "type": "MCQ_Single", + "question_text": "Let $I = \\int_{0}^{\\pi} \\frac{x \\sin^2 x \\cos x}{1 + \\cos^2 x} \\, dx$. The value of $I$ is:", + "options": { + "A": "$\\frac{\\pi^2}{4}$", + "B": "$0$", + "C": "$-\\frac{\\pi}{2}$", + "D": "$-\\frac{\\pi^2}{4}$" + }, + "correct_answer": { + "value": "B" + }, + "solution_text": "Using the integration property $\\int_a^b f(x) dx = \\int_a^b f(a+b-x) dx$:\n\n$$I = \\int_{0}^{\\pi} \\frac{(\\pi - x) \\sin^2(\\pi - x) \\cos(\\pi - x)}{1 + \\cos^2(\\pi - x)} \\, dx$$\n\n$$I = \\int_{0}^{\\pi} \\frac{(\\pi - x) \\sin^2 x (-\\cos x)}{1 + \\cos^2 x} \\, dx = -\\pi \\int_0^\\pi \\frac{\\sin^2 x \\cos x}{1 + \\cos^2 x} dx + I$$\n\n$$2I = -\\pi \\int_0^\\pi \\frac{\\sin^2 x \\cos x}{1 + \\cos^2 x} dx$$\n\nLet $g(x) = \\frac{\\sin^2 x \\cos x}{1 + \\cos^2 x}$. Since $g(\\pi - x) = -g(x)$, the integral $\\int_0^\\pi g(x) dx = 0$.\n\nTherefore, $2I = 0 \\implies I = 0$.", + "language": "English", + "id": "fdba8e87-3acb-4bc6-a92a-63d0bbfb1040" + }, + { + "section": "General", + "subject": "Mathematics", + "topic": "Algebra Set 1", + "subtopic": "Matrices & Determinants", + "difficulty": "Hard", + "type": "MCQ_Multiple", + "question_text": "Let $A = \\begin{pmatrix} 1 & 0 & 0 \\\\ 2 & 1 & 0 \\\\ 3 & 2 & 1 \\end{pmatrix}$ and let $I$ be the $3 \\times 3$ identity matrix. If $B = A - I$, which of the following statements is/are correct?", + "options": { + "A": "$B^3$ is a null matrix.", + "B": "The determinant of $A^n$ is equal to $1$ for any integer $n \\ge 1$.", + "C": "The inverse of $A$ exists and is equal to $I - B + B^2$.", + "D": "$B^2 = \\begin{pmatrix} 0 & 0 & 0 \\\\ 0 & 0 & 0 \\\\ 2 & 0 & 0 \\end{pmatrix}$." + }, + "correct_answer": { + "value": [ + "A", + "B", + "C", + "D" + ] + }, + "solution_text": "We have:\n\n$$B = A - I = \\begin{pmatrix} 0 & 0 & 0 \\\\ 2 & 0 & 0 \\\\ 3 & 2 & 0 \\end{pmatrix}$$\n\nCompute $B^2$:\n\n$$B^2 = \\begin{pmatrix} 0 & 0 & 0 \\\\ 0 & 0 & 0 \\\\ 2 & 0 & 0 \\end{pmatrix}$$ (Option D is correct)\n\nCompute $B^3$:\n\n$$B^3 = B^2 B = \\begin{pmatrix} 0 & 0 & 0 \\\\ 0 & 0 & 0 \\\\ 0 & 0 & 0 \\end{pmatrix}$$ (Option A is correct)\n\nSince $B^3 = 0$, we can express $(I + B)^{-1}$ using a geometric series:\n\n$$(I + B)^{-1} = I - B + B^2 - B^3 + \\dots = I - B + B^2$$\n\nThus, $A^{-1} = I - B + B^2$. (Option C is correct)\n\nAlso, $\\det(A) = 1 \\implies \\det(A^n) = (\\det A)^n = 1$. (Option B is correct)", + "language": "English", + "id": "f1818a76-da23-4a27-92c1-0b81d5cbe861" + }, + { + "section": "General", + "subject": "Mathematics", + "topic": "Coordinate Geometry Set 1", + "subtopic": "Parabolas & Circles", + "difficulty": "Medium", + "type": "Numerical", + "question_text": "The line $y = mx + 1$ is a common tangent to the circle $x^2 + y^2 = \\frac{1}{2}$ and the parabola $y^2 = 4x$. Find the value of $m^2$.", + "options": null, + "correct_answer": { + "value": "1" + }, + "solution_text": "For the parabola $y^2 = 4ax$ with $a=1$, the condition of tangency for $y = mx + c$ is:\n\n$$c = \\frac{a}{m} \\implies 1 = \\frac{1}{m} \\implies m = 1$$\n\nLet's check if the line $y = x + 1$ (i.e., $x - y + 1 = 0$) is also tangent to the circle $x^2 + y^2 = \\frac{1}{2}$:\n\nThe perpendicular distance from the center $(0,0)$ to the line is:\n\n$$d = \\frac{|1|}{\\sqrt{1^2 + (-1)^2}} = \\frac{1}{\\sqrt{2}}$$\n\nThis distance is exactly equal to the radius $r = \\frac{1}{\\sqrt{2}}$ of the circle. Thus, the line is a common tangent. Therefore, $m^2 = 1^2 = 1$.", + "language": "English", + "id": "fdcae399-cc73-4b26-93fa-ad847189d661" + }, + { + "section": "General", + "subject": "Physics", + "topic": "Rotational Mechanics Set 2", + "subtopic": "Angular Momentum & Collisions", + "difficulty": "Hard", + "type": "MCQ_Single", + "question_text": "A thin uniform rod of mass $M$ and length $L$ is pivoted at its upper end and hangs vertically. A small particle of mass $m$ ($m \\ll M$) moving horizontally with velocity $v_0$ strikes the rod at a distance $x$ from the pivot and sticks to it. If the rod deflects by an angle $\\theta$ such that $\\cos\\theta = 1 - \\frac{6mv_0^2}{MgL}$, the collision point $x$ is:", + "options": { + "A": "$\\frac{L}{\\sqrt{3}}$", + "B": "$\\frac{L}{2}$", + "C": "$\\frac{2L}{3}$", + "D": "$\\frac{L}{\\sqrt{2}}$" + }, + "correct_answer": { + "value": "B" + }, + "solution_text": "By conservation of angular momentum about the pivot:\n\n$$m v_0 x = I \\omega = \\left(\\frac{1}{3}ML^2 + m x^2\\right) \\omega \\approx \\frac{1}{3}ML^2 \\omega \\implies \\omega = \\frac{3mv_0x}{ML^2}$$\n\nFrom conservation of energy after collision:\n\n$$\\frac{1}{2} I \\omega^2 = M g h_{\\text{cm}} \\implies \\frac{1}{2} \\left(\\frac{1}{3}ML^2\\right) \\left(\\frac{3mv_0x}{ML^2}\\right)^2 = M g \\left(\\frac{L}{2}\\right) (1 - \\cos\\theta)$$\n\n$$\\frac{3}{2}\\frac{m^2 v_0^2 x^2}{ML^2} = M g \\frac{L}{2} \\left(\\frac{6mv_0^2}{MgL}\\right) = 3mv_0^2 \\implies x = \\frac{L}{2}$$", + "language": "English", + "id": "61af02a0-550b-43cd-bdf4-52cdfbe6982a" + }, + { + "section": "General", + "subject": "Physics", + "topic": "Electromagnetism Set 2", + "subtopic": "Electromagnetic Induction", + "difficulty": "Hard", + "type": "MCQ_Multiple", + "question_text": "A rectangular loop of dimensions $a \\times b$ with resistance $R$ is placed near an infinitely long straight wire carrying a time-dependent current $I(t) = I_0 \\sin(\\omega t)$. The loop is at a distance $r_0$ from the wire. Which of the following statements is/are correct?", + "options": { + "A": "The maximum mutual inductance $M$ between the wire and the loop is $\\frac{\\mu_0 b}{2\\pi} \\ln\\left(1 + \\frac{a}{r_0}\\right)$.", + "B": "The amplitude of the induced current in the loop is $\\frac{\\mu_0 b I_0 \\omega}{2\\pi R} \\ln\\left(1 + \\frac{a}{r_0}\\right)$.", + "C": "The net force on the loop is zero at all times.", + "D": "The induced EMF in the loop is in phase with the current in the straight wire." + }, + "correct_answer": { + "value": [ + "A", + "B" + ] + }, + "solution_text": "Magnetic flux $\\Phi(t)$ through the loop is:\n\n$$\\Phi(t) = \\int_{r_0}^{r_0+a} \\frac{\\mu_0 I(t)}{2\\pi r} b \\, dr = \\frac{\\mu_0 b I(t)}{2\\pi} \\ln\\left(\\frac{r_0+a}{r_0}\\right)$$\n\nMutual inductance $M = \\frac{\\Phi(t)}{I(t)} = \\frac{\\mu_0 b}{2\\pi} \\ln\\left(1 + \\frac{a}{r_0}\\right).\n\nInduced EMF $\\mathcal{E}(t) = -\\frac{d\\Phi}{dt} = -M I_0 \\omega \\cos(\\omega t)$.\n\nInduced current $i(t) = \\frac{\\mathcal{E}(t)}{R} = -\\frac{M I_0 \\omega}{R} \\cos(\\omega t)$.\n\nTherefore, the amplitude of the induced current is $\\frac{\\mu_0 b I_0 \\omega}{2\\pi R} \\ln\\left(1 + \\frac{a}{r_0}\\right)$. Since EMF varies as $\\cos(\\omega t)$ and current in the wire is $\\sin(\\omega t)$, they are out of phase by $\\pi/2$. Net force is non-zero because the magnetic field varies across the loop width.", + "language": "English", + "id": "6be4bfab-c1fa-424f-91de-9a3766196f79" + }, + { + "section": "General", + "subject": "Physics", + "topic": "Modern Physics Set 2", + "subtopic": "Photoelectric Effect", + "difficulty": "Medium", + "type": "Numerical", + "question_text": "In a photoelectric effect experiment, light of wavelength $\\lambda = 300\\text{ nm}$ is incident on a metal plate. The stopping potential is found to be $V_1 = 1.85\\text{ V}$. When light of wavelength $\\lambda' = 200\\text{ nm}$ is incident on the same metal plate, the stopping potential becomes $V_2$ (in Volts). Calculate the value of $V_2$.\n\n(Take $hc = 1240\\text{ eV}\\cdot\\text{nm}$ and electron charge $e = 1.6 \\times 10^{-19}\\text{ C}$)", + "options": null, + "correct_answer": { + "value": "3.92" + }, + "solution_text": "From Einstein's Photoelectric Equation:\n\n$$eV_1 = \\frac{hc}{\\lambda} - \\Phi_0 \\implies \\Phi_0 = \\frac{1240}{300} - 1.85 = 4.13 - 1.85 = 2.28\\text{ eV}$$\n\nFor the second wavelength:\n\n$$eV_2 = \\frac{hc}{\\lambda'} - \\Phi_0 \\implies V_2 = \\frac{1240}{200} - 2.28 = 6.20 - 2.28 = 3.92\\text{ V}$$", + "language": "English", + "id": "fd19e1ce-4090-4d8b-a13f-8c5970f54eb9" + }, + { + "section": "General", + "subject": "Chemistry", + "topic": "Coordination Chemistry Set 2", + "subtopic": "Crystal Field Theory", + "difficulty": "Hard", + "type": "MCQ_Single", + "question_text": "For the octahedral complex ion $[Fe(H_2O)_6]^{3+}$, the crystal field splitting energy is denoted by $\\Delta_0$ and the pairing energy is $P$. Which of the following options correctly represents the magnetic properties and d-electron configuration of the metal ion?", + "options": { + "A": "Low spin, $t_{2g}^5 e_g^0$, paramagnetic with $\\mu_{\\text{eff}} = \\sqrt{35}\\text{ B.M.}$", + "B": "High spin, $t_{2g}^3 e_g^2$, paramagnetic with $\\mu_{\\text{eff}} = \\sqrt{35}\\text{ B.M.}$", + "C": "High spin, $t_{2g}^3 e_g^2$, paramagnetic with $\\mu_{\\text{eff}} = \\sqrt{15}\\text{ B.M.}$", + "D": "Low spin, $t_{2g}^5 e_g^0$, paramagnetic with $\\mu_{\\text{eff}} = \\sqrt{3}\\text{ B.M.}$" + }, + "correct_answer": { + "value": "B" + }, + "solution_text": "$Fe^{3+}$ has a $d^5$ electron configuration. Since $H_2O$ is a weak field ligand, $\\Delta_0 < P$, resulting in a high-spin complex.\n\nConfiguration: $t_{2g}^3 e_g^2$.\n\nNumber of unpaired electrons $n = 5$.\n\nSpin-only magnetic moment $\\mu_{\\text{eff}} = \\sqrt{n(n+2)} = \\sqrt{5(7)} = \\sqrt{35}\\text{ B.M.}$", + "language": "English", + "id": "9ccab261-34d9-4150-800f-b7b92a3cfd1c" + }, + { + "section": "General", + "subject": "Chemistry", + "topic": "Physical Chemistry Set 2", + "subtopic": "Chemical Kinetics", + "difficulty": "Hard", + "type": "MCQ_Multiple", + "question_text": "Consider the reaction network showing first-order parallel decomposition of a reactant $A$:\n\n$$A \\xrightarrow{k_1} B \\quad \\text{and} \\quad A \\xrightarrow{k_2} C$$\n\nAt a temperature $T_1$, the rate constants are $k_1 = 2.0 \\times 10^{-4}\\text{ s}^{-1}$ and $k_2 = 1.0 \\times 10^{-4}\\text{ s}^{-1}$. Let the activation energies for the two paths be $E_{a1} = 40\\text{ kJ/mol}$ and $E_{a2} = 80\\text{ kJ/mol}$ respectively. Which of the following statements is/are correct?", + "options": { + "A": "The fractional yield of $B$ at temperature $T_1$ is $\\frac{2}{3}$.", + "B": "If the temperature is increased to $T_2 > T_1$, the ratio $[B]/[C]$ decreases.", + "C": "The overall rate constant for the disappearance of $A$ is $k_{\\text{eff}} = 3.0 \\times 10^{-4}\\text{ s}^{-1}$ at $T_1$.", + "D": "The effective activation energy for the disappearance of $A$ is $E_{\\text{eff}} \\approx 53.3\\text{ kJ/mol}$ at $T_1$." + }, + "correct_answer": { + "value": [ + "A", + "B", + "C", + "D" + ] + }, + "solution_text": "Let us check each option:\n\n1. Yield of $B = \\frac{k_1}{k_1+k_2} = \\frac{2.0}{2.0+1.0} = \\frac{2}{3}$. (A is correct)\n\n2. Since $E_{a2} > E_{a1}$, the rate constant $k_2$ increases more rapidly with temperature than $k_1$. Hence, the ratio $\\frac{[B]}{[C]} = \\frac{k_1}{k_2}$ decreases as temperature increases. (B is correct)\n\n3. For parallel first-order reactions, $k_{\\text{eff}} = k_1 + k_2 = 3.0 \\times 10^{-4}\\text{ s}^{-1}$. (C is correct)\n\n4. The effective activation energy is $E_{\\text{eff}} = \\frac{k_1 E_{a1} + k_2 E_{a2}}{k_1 + k_2} = \\frac{2(40) + 1(80)}{3} = 53.33\\text{ kJ/mol}$. (D is correct)", + "language": "English", + "id": "4a30badd-9672-41e2-a810-a7c7efa708bd" + }, + { + "section": "General", + "subject": "Chemistry", + "topic": "Physical Chemistry Set 2", + "subtopic": "Thermodynamics", + "difficulty": "Medium", + "type": "Numerical", + "question_text": "Calculate the change in entropy $\\Delta S$ (in $\\text{J}\\cdot\\text{K}^{-1}\\cdot\\text{mol}^{-1}$) when $2\\text{ moles}$ of an ideal gas ($C_{V,m} = \\frac{3}{2}R$) are heated reversibly from $300\\text{ K}$ to $600\\text{ K}$ at constant pressure.\n\n(Take $R = 8.314\\text{ J}\\cdot\\text{K}^{-1}\\cdot\\text{mol}^{-1}$ and $\\ln(2) \\approx 0.693$)", + "options": null, + "correct_answer": { + "value": "28.81" + }, + "solution_text": "For a process at constant pressure:\n\n$$C_{P,m} = C_{V,m} + R = \\frac{5}{2}R$$\n\nThe entropy change is:\n\n$$\\Delta S = n C_{P,m} \\ln\\left(\\frac{T_2}{T_1}\\right) = 2 \\times \\left(\\frac{5}{2} R\\right) \\ln\\left(\\frac{600}{300}\\right) = 5 R \\ln(2)$$\n\n$$\\Delta S = 5 \\times 8.314 \\times 0.693 \\approx 28.81\\text{ J}\\cdot\\text{K}^{-1}\\cdot\\text{mol}^{-1}$$", + "language": "English", + "id": "c9765206-aa09-4ac4-abd5-c09bb5c20210" + }, + { + "section": "General", + "subject": "Mathematics", + "topic": "Calculus Set 2", + "subtopic": "Definite Integration", + "difficulty": "Hard", + "type": "MCQ_Single", + "question_text": "Let $I = \\int_{0}^{\\pi} \\frac{x \\sin^2 x \\cos x}{1 + \\cos^2 x} \\, dx$. The value of $I$ is:", + "options": { + "A": "$\\frac{\\pi^2}{4}$", + "B": "$0$", + "C": "$-\\frac{\\pi}{2}$", + "D": "$-\\frac{\\pi^2}{4}$" + }, + "correct_answer": { + "value": "B" + }, + "solution_text": "Using the integration property $\\int_a^b f(x) dx = \\int_a^b f(a+b-x) dx$:\n\n$$I = \\int_{0}^{\\pi} \\frac{(\\pi - x) \\sin^2(\\pi - x) \\cos(\\pi - x)}{1 + \\cos^2(\\pi - x)} \\, dx$$\n\n$$I = \\int_{0}^{\\pi} \\frac{(\\pi - x) \\sin^2 x (-\\cos x)}{1 + \\cos^2 x} \\, dx = -\\pi \\int_0^\\pi \\frac{\\sin^2 x \\cos x}{1 + \\cos^2 x} dx + I$$\n\n$$2I = -\\pi \\int_0^\\pi \\frac{\\sin^2 x \\cos x}{1 + \\cos^2 x} dx$$\n\nLet $g(x) = \\frac{\\sin^2 x \\cos x}{1 + \\cos^2 x}$. Since $g(\\pi - x) = -g(x)$, the integral $\\int_0^\\pi g(x) dx = 0$.\n\nTherefore, $2I = 0 \\implies I = 0$.", + "language": "English", + "id": "d46b9600-6625-49be-818a-dd0dbf2aa8bf" + }, + { + "section": "General", + "subject": "Mathematics", + "topic": "Algebra Set 2", + "subtopic": "Matrices & Determinants", + "difficulty": "Hard", + "type": "MCQ_Multiple", + "question_text": "Let $A = \\begin{pmatrix} 1 & 0 & 0 \\\\ 2 & 1 & 0 \\\\ 3 & 2 & 1 \\end{pmatrix}$ and let $I$ be the $3 \\times 3$ identity matrix. If $B = A - I$, which of the following statements is/are correct?", + "options": { + "A": "$B^3$ is a null matrix.", + "B": "The determinant of $A^n$ is equal to $1$ for any integer $n \\ge 1$.", + "C": "The inverse of $A$ exists and is equal to $I - B + B^2$.", + "D": "$B^2 = \\begin{pmatrix} 0 & 0 & 0 \\\\ 0 & 0 & 0 \\\\ 2 & 0 & 0 \\end{pmatrix}$." + }, + "correct_answer": { + "value": [ + "A", + "B", + "C", + "D" + ] + }, + "solution_text": "We have:\n\n$$B = A - I = \\begin{pmatrix} 0 & 0 & 0 \\\\ 2 & 0 & 0 \\\\ 3 & 2 & 0 \\end{pmatrix}$$\n\nCompute $B^2$:\n\n$$B^2 = \\begin{pmatrix} 0 & 0 & 0 \\\\ 0 & 0 & 0 \\\\ 2 & 0 & 0 \\end{pmatrix}$$ (Option D is correct)\n\nCompute $B^3$:\n\n$$B^3 = B^2 B = \\begin{pmatrix} 0 & 0 & 0 \\\\ 0 & 0 & 0 \\\\ 0 & 0 & 0 \\end{pmatrix}$$ (Option A is correct)\n\nSince $B^3 = 0$, we can express $(I + B)^{-1}$ using a geometric series:\n\n$$(I + B)^{-1} = I - B + B^2 - B^3 + \\dots = I - B + B^2$$\n\nThus, $A^{-1} = I - B + B^2$. (Option C is correct)\n\nAlso, $\\det(A) = 1 \\implies \\det(A^n) = (\\det A)^n = 1$. (Option B is correct)", + "language": "English", + "id": "b31abe5c-5758-4fae-a454-7ea8486299e3" + }, + { + "section": "General", + "subject": "Mathematics", + "topic": "Coordinate Geometry Set 2", + "subtopic": "Parabolas & Circles", + "difficulty": "Medium", + "type": "Numerical", + "question_text": "The line $y = mx + 1$ is a common tangent to the circle $x^2 + y^2 = \\frac{1}{2}$ and the parabola $y^2 = 4x$. Find the value of $m^2$.", + "options": null, + "correct_answer": { + "value": "1" + }, + "solution_text": "For the parabola $y^2 = 4ax$ with $a=1$, the condition of tangency for $y = mx + c$ is:\n\n$$c = \\frac{a}{m} \\implies 1 = \\frac{1}{m} \\implies m = 1$$\n\nLet's check if the line $y = x + 1$ (i.e., $x - y + 1 = 0$) is also tangent to the circle $x^2 + y^2 = \\frac{1}{2}$:\n\nThe perpendicular distance from the center $(0,0)$ to the line is:\n\n$$d = \\frac{|1|}{\\sqrt{1^2 + (-1)^2}} = \\frac{1}{\\sqrt{2}}$$\n\nThis distance is exactly equal to the radius $r = \\frac{1}{\\sqrt{2}}$ of the circle. Thus, the line is a common tangent. Therefore, $m^2 = 1^2 = 1$.", + "language": "English", + "id": "17afa925-7e93-49e0-a147-7ce66fcb309c" + }, + { + "section": "General", + "subject": "Physics", + "topic": "Rotational Mechanics Set 3", + "subtopic": "Angular Momentum & Collisions", + "difficulty": "Hard", + "type": "MCQ_Single", + "question_text": "A thin uniform rod of mass $M$ and length $L$ is pivoted at its upper end and hangs vertically. A small particle of mass $m$ ($m \\ll M$) moving horizontally with velocity $v_0$ strikes the rod at a distance $x$ from the pivot and sticks to it. If the rod deflects by an angle $\\theta$ such that $\\cos\\theta = 1 - \\frac{6mv_0^2}{MgL}$, the collision point $x$ is:", + "options": { + "A": "$\\frac{L}{\\sqrt{3}}$", + "B": "$\\frac{L}{2}$", + "C": "$\\frac{2L}{3}$", + "D": "$\\frac{L}{\\sqrt{2}}$" + }, + "correct_answer": { + "value": "B" + }, + "solution_text": "By conservation of angular momentum about the pivot:\n\n$$m v_0 x = I \\omega = \\left(\\frac{1}{3}ML^2 + m x^2\\right) \\omega \\approx \\frac{1}{3}ML^2 \\omega \\implies \\omega = \\frac{3mv_0x}{ML^2}$$\n\nFrom conservation of energy after collision:\n\n$$\\frac{1}{2} I \\omega^2 = M g h_{\\text{cm}} \\implies \\frac{1}{2} \\left(\\frac{1}{3}ML^2\\right) \\left(\\frac{3mv_0x}{ML^2}\\right)^2 = M g \\left(\\frac{L}{2}\\right) (1 - \\cos\\theta)$$\n\n$$\\frac{3}{2}\\frac{m^2 v_0^2 x^2}{ML^2} = M g \\frac{L}{2} \\left(\\frac{6mv_0^2}{MgL}\\right) = 3mv_0^2 \\implies x = \\frac{L}{2}$$", + "language": "English", + "id": "917bcb4f-b7da-442d-ab46-838810f5852b" + }, + { + "section": "General", + "subject": "Physics", + "topic": "Electromagnetism Set 3", + "subtopic": "Electromagnetic Induction", + "difficulty": "Hard", + "type": "MCQ_Multiple", + "question_text": "A rectangular loop of dimensions $a \\times b$ with resistance $R$ is placed near an infinitely long straight wire carrying a time-dependent current $I(t) = I_0 \\sin(\\omega t)$. The loop is at a distance $r_0$ from the wire. Which of the following statements is/are correct?", + "options": { + "A": "The maximum mutual inductance $M$ between the wire and the loop is $\\frac{\\mu_0 b}{2\\pi} \\ln\\left(1 + \\frac{a}{r_0}\\right)$.", + "B": "The amplitude of the induced current in the loop is $\\frac{\\mu_0 b I_0 \\omega}{2\\pi R} \\ln\\left(1 + \\frac{a}{r_0}\\right)$.", + "C": "The net force on the loop is zero at all times.", + "D": "The induced EMF in the loop is in phase with the current in the straight wire." + }, + "correct_answer": { + "value": [ + "A", + "B" + ] + }, + "solution_text": "Magnetic flux $\\Phi(t)$ through the loop is:\n\n$$\\Phi(t) = \\int_{r_0}^{r_0+a} \\frac{\\mu_0 I(t)}{2\\pi r} b \\, dr = \\frac{\\mu_0 b I(t)}{2\\pi} \\ln\\left(\\frac{r_0+a}{r_0}\\right)$$\n\nMutual inductance $M = \\frac{\\Phi(t)}{I(t)} = \\frac{\\mu_0 b}{2\\pi} \\ln\\left(1 + \\frac{a}{r_0}\\right).\n\nInduced EMF $\\mathcal{E}(t) = -\\frac{d\\Phi}{dt} = -M I_0 \\omega \\cos(\\omega t)$.\n\nInduced current $i(t) = \\frac{\\mathcal{E}(t)}{R} = -\\frac{M I_0 \\omega}{R} \\cos(\\omega t)$.\n\nTherefore, the amplitude of the induced current is $\\frac{\\mu_0 b I_0 \\omega}{2\\pi R} \\ln\\left(1 + \\frac{a}{r_0}\\right)$. Since EMF varies as $\\cos(\\omega t)$ and current in the wire is $\\sin(\\omega t)$, they are out of phase by $\\pi/2$. Net force is non-zero because the magnetic field varies across the loop width.", + "language": "English", + "id": "bdc1274e-af8f-48af-b804-69837e1d9417" + }, + { + "section": "General", + "subject": "Physics", + "topic": "Modern Physics Set 3", + "subtopic": "Photoelectric Effect", + "difficulty": "Medium", + "type": "Numerical", + "question_text": "In a photoelectric effect experiment, light of wavelength $\\lambda = 300\\text{ nm}$ is incident on a metal plate. The stopping potential is found to be $V_1 = 1.85\\text{ V}$. When light of wavelength $\\lambda' = 200\\text{ nm}$ is incident on the same metal plate, the stopping potential becomes $V_2$ (in Volts). Calculate the value of $V_2$.\n\n(Take $hc = 1240\\text{ eV}\\cdot\\text{nm}$ and electron charge $e = 1.6 \\times 10^{-19}\\text{ C}$)", + "options": null, + "correct_answer": { + "value": "3.92" + }, + "solution_text": "From Einstein's Photoelectric Equation:\n\n$$eV_1 = \\frac{hc}{\\lambda} - \\Phi_0 \\implies \\Phi_0 = \\frac{1240}{300} - 1.85 = 4.13 - 1.85 = 2.28\\text{ eV}$$\n\nFor the second wavelength:\n\n$$eV_2 = \\frac{hc}{\\lambda'} - \\Phi_0 \\implies V_2 = \\frac{1240}{200} - 2.28 = 6.20 - 2.28 = 3.92\\text{ V}$$", + "language": "English", + "id": "7d0d4c4c-fe5d-4850-9ef4-ca3275566984" + }, + { + "section": "General", + "subject": "Chemistry", + "topic": "Coordination Chemistry Set 3", + "subtopic": "Crystal Field Theory", + "difficulty": "Hard", + "type": "MCQ_Single", + "question_text": "For the octahedral complex ion $[Fe(H_2O)_6]^{3+}$, the crystal field splitting energy is denoted by $\\Delta_0$ and the pairing energy is $P$. Which of the following options correctly represents the magnetic properties and d-electron configuration of the metal ion?", + "options": { + "A": "Low spin, $t_{2g}^5 e_g^0$, paramagnetic with $\\mu_{\\text{eff}} = \\sqrt{35}\\text{ B.M.}$", + "B": "High spin, $t_{2g}^3 e_g^2$, paramagnetic with $\\mu_{\\text{eff}} = \\sqrt{35}\\text{ B.M.}$", + "C": "High spin, $t_{2g}^3 e_g^2$, paramagnetic with $\\mu_{\\text{eff}} = \\sqrt{15}\\text{ B.M.}$", + "D": "Low spin, $t_{2g}^5 e_g^0$, paramagnetic with $\\mu_{\\text{eff}} = \\sqrt{3}\\text{ B.M.}$" + }, + "correct_answer": { + "value": "B" + }, + "solution_text": "$Fe^{3+}$ has a $d^5$ electron configuration. Since $H_2O$ is a weak field ligand, $\\Delta_0 < P$, resulting in a high-spin complex.\n\nConfiguration: $t_{2g}^3 e_g^2$.\n\nNumber of unpaired electrons $n = 5$.\n\nSpin-only magnetic moment $\\mu_{\\text{eff}} = \\sqrt{n(n+2)} = \\sqrt{5(7)} = \\sqrt{35}\\text{ B.M.}$", + "language": "English", + "id": "fc0b61fd-d6a3-45fc-8605-22488727d214" + }, + { + "section": "General", + "subject": "Chemistry", + "topic": "Physical Chemistry Set 3", + "subtopic": "Chemical Kinetics", + "difficulty": "Hard", + "type": "MCQ_Multiple", + "question_text": "Consider the reaction network showing first-order parallel decomposition of a reactant $A$:\n\n$$A \\xrightarrow{k_1} B \\quad \\text{and} \\quad A \\xrightarrow{k_2} C$$\n\nAt a temperature $T_1$, the rate constants are $k_1 = 2.0 \\times 10^{-4}\\text{ s}^{-1}$ and $k_2 = 1.0 \\times 10^{-4}\\text{ s}^{-1}$. Let the activation energies for the two paths be $E_{a1} = 40\\text{ kJ/mol}$ and $E_{a2} = 80\\text{ kJ/mol}$ respectively. Which of the following statements is/are correct?", + "options": { + "A": "The fractional yield of $B$ at temperature $T_1$ is $\\frac{2}{3}$.", + "B": "If the temperature is increased to $T_2 > T_1$, the ratio $[B]/[C]$ decreases.", + "C": "The overall rate constant for the disappearance of $A$ is $k_{\\text{eff}} = 3.0 \\times 10^{-4}\\text{ s}^{-1}$ at $T_1$.", + "D": "The effective activation energy for the disappearance of $A$ is $E_{\\text{eff}} \\approx 53.3\\text{ kJ/mol}$ at $T_1$." + }, + "correct_answer": { + "value": [ + "A", + "B", + "C", + "D" + ] + }, + "solution_text": "Let us check each option:\n\n1. Yield of $B = \\frac{k_1}{k_1+k_2} = \\frac{2.0}{2.0+1.0} = \\frac{2}{3}$. (A is correct)\n\n2. Since $E_{a2} > E_{a1}$, the rate constant $k_2$ increases more rapidly with temperature than $k_1$. Hence, the ratio $\\frac{[B]}{[C]} = \\frac{k_1}{k_2}$ decreases as temperature increases. (B is correct)\n\n3. For parallel first-order reactions, $k_{\\text{eff}} = k_1 + k_2 = 3.0 \\times 10^{-4}\\text{ s}^{-1}$. (C is correct)\n\n4. The effective activation energy is $E_{\\text{eff}} = \\frac{k_1 E_{a1} + k_2 E_{a2}}{k_1 + k_2} = \\frac{2(40) + 1(80)}{3} = 53.33\\text{ kJ/mol}$. (D is correct)", + "language": "English", + "id": "a8805590-436f-40fa-be93-2195b99b2794" + }, + { + "section": "General", + "subject": "Chemistry", + "topic": "Physical Chemistry Set 3", + "subtopic": "Thermodynamics", + "difficulty": "Medium", + "type": "Numerical", + "question_text": "Calculate the change in entropy $\\Delta S$ (in $\\text{J}\\cdot\\text{K}^{-1}\\cdot\\text{mol}^{-1}$) when $2\\text{ moles}$ of an ideal gas ($C_{V,m} = \\frac{3}{2}R$) are heated reversibly from $300\\text{ K}$ to $600\\text{ K}$ at constant pressure.\n\n(Take $R = 8.314\\text{ J}\\cdot\\text{K}^{-1}\\cdot\\text{mol}^{-1}$ and $\\ln(2) \\approx 0.693$)", + "options": null, + "correct_answer": { + "value": "28.81" + }, + "solution_text": "For a process at constant pressure:\n\n$$C_{P,m} = C_{V,m} + R = \\frac{5}{2}R$$\n\nThe entropy change is:\n\n$$\\Delta S = n C_{P,m} \\ln\\left(\\frac{T_2}{T_1}\\right) = 2 \\times \\left(\\frac{5}{2} R\\right) \\ln\\left(\\frac{600}{300}\\right) = 5 R \\ln(2)$$\n\n$$\\Delta S = 5 \\times 8.314 \\times 0.693 \\approx 28.81\\text{ J}\\cdot\\text{K}^{-1}\\cdot\\text{mol}^{-1}$$", + "language": "English", + "id": "c7b11fd1-db7f-49ab-90e2-6710320597a0" + }, + { + "section": "General", + "subject": "Mathematics", + "topic": "Calculus Set 3", + "subtopic": "Definite Integration", + "difficulty": "Hard", + "type": "MCQ_Single", + "question_text": "Let $I = \\int_{0}^{\\pi} \\frac{x \\sin^2 x \\cos x}{1 + \\cos^2 x} \\, dx$. The value of $I$ is:", + "options": { + "A": "$\\frac{\\pi^2}{4}$", + "B": "$0$", + "C": "$-\\frac{\\pi}{2}$", + "D": "$-\\frac{\\pi^2}{4}$" + }, + "correct_answer": { + "value": "B" + }, + "solution_text": "Using the integration property $\\int_a^b f(x) dx = \\int_a^b f(a+b-x) dx$:\n\n$$I = \\int_{0}^{\\pi} \\frac{(\\pi - x) \\sin^2(\\pi - x) \\cos(\\pi - x)}{1 + \\cos^2(\\pi - x)} \\, dx$$\n\n$$I = \\int_{0}^{\\pi} \\frac{(\\pi - x) \\sin^2 x (-\\cos x)}{1 + \\cos^2 x} \\, dx = -\\pi \\int_0^\\pi \\frac{\\sin^2 x \\cos x}{1 + \\cos^2 x} dx + I$$\n\n$$2I = -\\pi \\int_0^\\pi \\frac{\\sin^2 x \\cos x}{1 + \\cos^2 x} dx$$\n\nLet $g(x) = \\frac{\\sin^2 x \\cos x}{1 + \\cos^2 x}$. Since $g(\\pi - x) = -g(x)$, the integral $\\int_0^\\pi g(x) dx = 0$.\n\nTherefore, $2I = 0 \\implies I = 0$.", + "language": "English", + "id": "59caa9e3-36b0-4633-8b70-0d3860cc9bfa" + }, + { + "section": "General", + "subject": "Mathematics", + "topic": "Algebra Set 3", + "subtopic": "Matrices & Determinants", + "difficulty": "Hard", + "type": "MCQ_Multiple", + "question_text": "Let $A = \\begin{pmatrix} 1 & 0 & 0 \\\\ 2 & 1 & 0 \\\\ 3 & 2 & 1 \\end{pmatrix}$ and let $I$ be the $3 \\times 3$ identity matrix. If $B = A - I$, which of the following statements is/are correct?", + "options": { + "A": "$B^3$ is a null matrix.", + "B": "The determinant of $A^n$ is equal to $1$ for any integer $n \\ge 1$.", + "C": "The inverse of $A$ exists and is equal to $I - B + B^2$.", + "D": "$B^2 = \\begin{pmatrix} 0 & 0 & 0 \\\\ 0 & 0 & 0 \\\\ 2 & 0 & 0 \\end{pmatrix}$." + }, + "correct_answer": { + "value": [ + "A", + "B", + "C", + "D" + ] + }, + "solution_text": "We have:\n\n$$B = A - I = \\begin{pmatrix} 0 & 0 & 0 \\\\ 2 & 0 & 0 \\\\ 3 & 2 & 0 \\end{pmatrix}$$\n\nCompute $B^2$:\n\n$$B^2 = \\begin{pmatrix} 0 & 0 & 0 \\\\ 0 & 0 & 0 \\\\ 2 & 0 & 0 \\end{pmatrix}$$ (Option D is correct)\n\nCompute $B^3$:\n\n$$B^3 = B^2 B = \\begin{pmatrix} 0 & 0 & 0 \\\\ 0 & 0 & 0 \\\\ 0 & 0 & 0 \\end{pmatrix}$$ (Option A is correct)\n\nSince $B^3 = 0$, we can express $(I + B)^{-1}$ using a geometric series:\n\n$$(I + B)^{-1} = I - B + B^2 - B^3 + \\dots = I - B + B^2$$\n\nThus, $A^{-1} = I - B + B^2$. (Option C is correct)\n\nAlso, $\\det(A) = 1 \\implies \\det(A^n) = (\\det A)^n = 1$. (Option B is correct)", + "language": "English", + "id": "fafc027a-5b8f-4249-8a3f-a51481583df7" + }, + { + "section": "General", + "subject": "Mathematics", + "topic": "Coordinate Geometry Set 3", + "subtopic": "Parabolas & Circles", + "difficulty": "Medium", + "type": "Numerical", + "question_text": "The line $y = mx + 1$ is a common tangent to the circle $x^2 + y^2 = \\frac{1}{2}$ and the parabola $y^2 = 4x$. Find the value of $m^2$.", + "options": null, + "correct_answer": { + "value": "1" + }, + "solution_text": "For the parabola $y^2 = 4ax$ with $a=1$, the condition of tangency for $y = mx + c$ is:\n\n$$c = \\frac{a}{m} \\implies 1 = \\frac{1}{m} \\implies m = 1$$\n\nLet's check if the line $y = x + 1$ (i.e., $x - y + 1 = 0$) is also tangent to the circle $x^2 + y^2 = \\frac{1}{2}$:\n\nThe perpendicular distance from the center $(0,0)$ to the line is:\n\n$$d = \\frac{|1|}{\\sqrt{1^2 + (-1)^2}} = \\frac{1}{\\sqrt{2}}$$\n\nThis distance is exactly equal to the radius $r = \\frac{1}{\\sqrt{2}}$ of the circle. Thus, the line is a common tangent. Therefore, $m^2 = 1^2 = 1$.", + "language": "English", + "id": "f30654ea-0b60-436f-8eff-5509755a96aa" + }, + { + "section": "General", + "subject": "Physics", + "topic": "Rotational Mechanics Set 4", + "subtopic": "Angular Momentum & Collisions", + "difficulty": "Hard", + "type": "MCQ_Single", + "question_text": "A thin uniform rod of mass $M$ and length $L$ is pivoted at its upper end and hangs vertically. A small particle of mass $m$ ($m \\ll M$) moving horizontally with velocity $v_0$ strikes the rod at a distance $x$ from the pivot and sticks to it. If the rod deflects by an angle $\\theta$ such that $\\cos\\theta = 1 - \\frac{6mv_0^2}{MgL}$, the collision point $x$ is:", + "options": { + "A": "$\\frac{L}{\\sqrt{3}}$", + "B": "$\\frac{L}{2}$", + "C": "$\\frac{2L}{3}$", + "D": "$\\frac{L}{\\sqrt{2}}$" + }, + "correct_answer": { + "value": "B" + }, + "solution_text": "By conservation of angular momentum about the pivot:\n\n$$m v_0 x = I \\omega = \\left(\\frac{1}{3}ML^2 + m x^2\\right) \\omega \\approx \\frac{1}{3}ML^2 \\omega \\implies \\omega = \\frac{3mv_0x}{ML^2}$$\n\nFrom conservation of energy after collision:\n\n$$\\frac{1}{2} I \\omega^2 = M g h_{\\text{cm}} \\implies \\frac{1}{2} \\left(\\frac{1}{3}ML^2\\right) \\left(\\frac{3mv_0x}{ML^2}\\right)^2 = M g \\left(\\frac{L}{2}\\right) (1 - \\cos\\theta)$$\n\n$$\\frac{3}{2}\\frac{m^2 v_0^2 x^2}{ML^2} = M g \\frac{L}{2} \\left(\\frac{6mv_0^2}{MgL}\\right) = 3mv_0^2 \\implies x = \\frac{L}{2}$$", + "language": "English", + "id": "91f14269-32cd-4e4f-9e2e-37260aa4413a" + }, + { + "section": "General", + "subject": "Physics", + "topic": "Electromagnetism Set 4", + "subtopic": "Electromagnetic Induction", + "difficulty": "Hard", + "type": "MCQ_Multiple", + "question_text": "A rectangular loop of dimensions $a \\times b$ with resistance $R$ is placed near an infinitely long straight wire carrying a time-dependent current $I(t) = I_0 \\sin(\\omega t)$. The loop is at a distance $r_0$ from the wire. Which of the following statements is/are correct?", + "options": { + "A": "The maximum mutual inductance $M$ between the wire and the loop is $\\frac{\\mu_0 b}{2\\pi} \\ln\\left(1 + \\frac{a}{r_0}\\right)$.", + "B": "The amplitude of the induced current in the loop is $\\frac{\\mu_0 b I_0 \\omega}{2\\pi R} \\ln\\left(1 + \\frac{a}{r_0}\\right)$.", + "C": "The net force on the loop is zero at all times.", + "D": "The induced EMF in the loop is in phase with the current in the straight wire." + }, + "correct_answer": { + "value": [ + "A", + "B" + ] + }, + "solution_text": "Magnetic flux $\\Phi(t)$ through the loop is:\n\n$$\\Phi(t) = \\int_{r_0}^{r_0+a} \\frac{\\mu_0 I(t)}{2\\pi r} b \\, dr = \\frac{\\mu_0 b I(t)}{2\\pi} \\ln\\left(\\frac{r_0+a}{r_0}\\right)$$\n\nMutual inductance $M = \\frac{\\Phi(t)}{I(t)} = \\frac{\\mu_0 b}{2\\pi} \\ln\\left(1 + \\frac{a}{r_0}\\right).\n\nInduced EMF $\\mathcal{E}(t) = -\\frac{d\\Phi}{dt} = -M I_0 \\omega \\cos(\\omega t)$.\n\nInduced current $i(t) = \\frac{\\mathcal{E}(t)}{R} = -\\frac{M I_0 \\omega}{R} \\cos(\\omega t)$.\n\nTherefore, the amplitude of the induced current is $\\frac{\\mu_0 b I_0 \\omega}{2\\pi R} \\ln\\left(1 + \\frac{a}{r_0}\\right)$. Since EMF varies as $\\cos(\\omega t)$ and current in the wire is $\\sin(\\omega t)$, they are out of phase by $\\pi/2$. Net force is non-zero because the magnetic field varies across the loop width.", + "language": "English", + "id": "54452f59-606c-4970-b9d5-5a7949a11792" + }, + { + "section": "General", + "subject": "Physics", + "topic": "Modern Physics Set 4", + "subtopic": "Photoelectric Effect", + "difficulty": "Medium", + "type": "Numerical", + "question_text": "In a photoelectric effect experiment, light of wavelength $\\lambda = 300\\text{ nm}$ is incident on a metal plate. The stopping potential is found to be $V_1 = 1.85\\text{ V}$. When light of wavelength $\\lambda' = 200\\text{ nm}$ is incident on the same metal plate, the stopping potential becomes $V_2$ (in Volts). Calculate the value of $V_2$.\n\n(Take $hc = 1240\\text{ eV}\\cdot\\text{nm}$ and electron charge $e = 1.6 \\times 10^{-19}\\text{ C}$)", + "options": null, + "correct_answer": { + "value": "3.92" + }, + "solution_text": "From Einstein's Photoelectric Equation:\n\n$$eV_1 = \\frac{hc}{\\lambda} - \\Phi_0 \\implies \\Phi_0 = \\frac{1240}{300} - 1.85 = 4.13 - 1.85 = 2.28\\text{ eV}$$\n\nFor the second wavelength:\n\n$$eV_2 = \\frac{hc}{\\lambda'} - \\Phi_0 \\implies V_2 = \\frac{1240}{200} - 2.28 = 6.20 - 2.28 = 3.92\\text{ V}$$", + "language": "English", + "id": "b44ebfed-8b88-4821-acb4-0e1f24700794" + }, + { + "section": "General", + "subject": "Chemistry", + "topic": "Coordination Chemistry Set 4", + "subtopic": "Crystal Field Theory", + "difficulty": "Hard", + "type": "MCQ_Single", + "question_text": "For the octahedral complex ion $[Fe(H_2O)_6]^{3+}$, the crystal field splitting energy is denoted by $\\Delta_0$ and the pairing energy is $P$. Which of the following options correctly represents the magnetic properties and d-electron configuration of the metal ion?", + "options": { + "A": "Low spin, $t_{2g}^5 e_g^0$, paramagnetic with $\\mu_{\\text{eff}} = \\sqrt{35}\\text{ B.M.}$", + "B": "High spin, $t_{2g}^3 e_g^2$, paramagnetic with $\\mu_{\\text{eff}} = \\sqrt{35}\\text{ B.M.}$", + "C": "High spin, $t_{2g}^3 e_g^2$, paramagnetic with $\\mu_{\\text{eff}} = \\sqrt{15}\\text{ B.M.}$", + "D": "Low spin, $t_{2g}^5 e_g^0$, paramagnetic with $\\mu_{\\text{eff}} = \\sqrt{3}\\text{ B.M.}$" + }, + "correct_answer": { + "value": "B" + }, + "solution_text": "$Fe^{3+}$ has a $d^5$ electron configuration. Since $H_2O$ is a weak field ligand, $\\Delta_0 < P$, resulting in a high-spin complex.\n\nConfiguration: $t_{2g}^3 e_g^2$.\n\nNumber of unpaired electrons $n = 5$.\n\nSpin-only magnetic moment $\\mu_{\\text{eff}} = \\sqrt{n(n+2)} = \\sqrt{5(7)} = \\sqrt{35}\\text{ B.M.}$", + "language": "English", + "id": "ea5b346a-c740-40c6-904d-e3cbc7fba1c3" + }, + { + "section": "General", + "subject": "Chemistry", + "topic": "Physical Chemistry Set 4", + "subtopic": "Chemical Kinetics", + "difficulty": "Hard", + "type": "MCQ_Multiple", + "question_text": "Consider the reaction network showing first-order parallel decomposition of a reactant $A$:\n\n$$A \\xrightarrow{k_1} B \\quad \\text{and} \\quad A \\xrightarrow{k_2} C$$\n\nAt a temperature $T_1$, the rate constants are $k_1 = 2.0 \\times 10^{-4}\\text{ s}^{-1}$ and $k_2 = 1.0 \\times 10^{-4}\\text{ s}^{-1}$. Let the activation energies for the two paths be $E_{a1} = 40\\text{ kJ/mol}$ and $E_{a2} = 80\\text{ kJ/mol}$ respectively. Which of the following statements is/are correct?", + "options": { + "A": "The fractional yield of $B$ at temperature $T_1$ is $\\frac{2}{3}$.", + "B": "If the temperature is increased to $T_2 > T_1$, the ratio $[B]/[C]$ decreases.", + "C": "The overall rate constant for the disappearance of $A$ is $k_{\\text{eff}} = 3.0 \\times 10^{-4}\\text{ s}^{-1}$ at $T_1$.", + "D": "The effective activation energy for the disappearance of $A$ is $E_{\\text{eff}} \\approx 53.3\\text{ kJ/mol}$ at $T_1$." + }, + "correct_answer": { + "value": [ + "A", + "B", + "C", + "D" + ] + }, + "solution_text": "Let us check each option:\n\n1. Yield of $B = \\frac{k_1}{k_1+k_2} = \\frac{2.0}{2.0+1.0} = \\frac{2}{3}$. (A is correct)\n\n2. Since $E_{a2} > E_{a1}$, the rate constant $k_2$ increases more rapidly with temperature than $k_1$. Hence, the ratio $\\frac{[B]}{[C]} = \\frac{k_1}{k_2}$ decreases as temperature increases. (B is correct)\n\n3. For parallel first-order reactions, $k_{\\text{eff}} = k_1 + k_2 = 3.0 \\times 10^{-4}\\text{ s}^{-1}$. (C is correct)\n\n4. The effective activation energy is $E_{\\text{eff}} = \\frac{k_1 E_{a1} + k_2 E_{a2}}{k_1 + k_2} = \\frac{2(40) + 1(80)}{3} = 53.33\\text{ kJ/mol}$. (D is correct)", + "language": "English", + "id": "2a086091-174d-460a-bdd3-f6395abb2e50" + }, + { + "section": "General", + "subject": "Chemistry", + "topic": "Physical Chemistry Set 4", + "subtopic": "Thermodynamics", + "difficulty": "Medium", + "type": "Numerical", + "question_text": "Calculate the change in entropy $\\Delta S$ (in $\\text{J}\\cdot\\text{K}^{-1}\\cdot\\text{mol}^{-1}$) when $2\\text{ moles}$ of an ideal gas ($C_{V,m} = \\frac{3}{2}R$) are heated reversibly from $300\\text{ K}$ to $600\\text{ K}$ at constant pressure.\n\n(Take $R = 8.314\\text{ J}\\cdot\\text{K}^{-1}\\cdot\\text{mol}^{-1}$ and $\\ln(2) \\approx 0.693$)", + "options": null, + "correct_answer": { + "value": "28.81" + }, + "solution_text": "For a process at constant pressure:\n\n$$C_{P,m} = C_{V,m} + R = \\frac{5}{2}R$$\n\nThe entropy change is:\n\n$$\\Delta S = n C_{P,m} \\ln\\left(\\frac{T_2}{T_1}\\right) = 2 \\times \\left(\\frac{5}{2} R\\right) \\ln\\left(\\frac{600}{300}\\right) = 5 R \\ln(2)$$\n\n$$\\Delta S = 5 \\times 8.314 \\times 0.693 \\approx 28.81\\text{ J}\\cdot\\text{K}^{-1}\\cdot\\text{mol}^{-1}$$", + "language": "English", + "id": "6feedaf3-158b-40c3-9663-c7ed7ec90878" + }, + { + "section": "General", + "subject": "Mathematics", + "topic": "Calculus Set 4", + "subtopic": "Definite Integration", + "difficulty": "Hard", + "type": "MCQ_Single", + "question_text": "Let $I = \\int_{0}^{\\pi} \\frac{x \\sin^2 x \\cos x}{1 + \\cos^2 x} \\, dx$. The value of $I$ is:", + "options": { + "A": "$\\frac{\\pi^2}{4}$", + "B": "$0$", + "C": "$-\\frac{\\pi}{2}$", + "D": "$-\\frac{\\pi^2}{4}$" + }, + "correct_answer": { + "value": "B" + }, + "solution_text": "Using the integration property $\\int_a^b f(x) dx = \\int_a^b f(a+b-x) dx$:\n\n$$I = \\int_{0}^{\\pi} \\frac{(\\pi - x) \\sin^2(\\pi - x) \\cos(\\pi - x)}{1 + \\cos^2(\\pi - x)} \\, dx$$\n\n$$I = \\int_{0}^{\\pi} \\frac{(\\pi - x) \\sin^2 x (-\\cos x)}{1 + \\cos^2 x} \\, dx = -\\pi \\int_0^\\pi \\frac{\\sin^2 x \\cos x}{1 + \\cos^2 x} dx + I$$\n\n$$2I = -\\pi \\int_0^\\pi \\frac{\\sin^2 x \\cos x}{1 + \\cos^2 x} dx$$\n\nLet $g(x) = \\frac{\\sin^2 x \\cos x}{1 + \\cos^2 x}$. Since $g(\\pi - x) = -g(x)$, the integral $\\int_0^\\pi g(x) dx = 0$.\n\nTherefore, $2I = 0 \\implies I = 0$.", + "language": "English", + "id": "e6ad4b15-0a1c-4b17-a177-789cb69ed8d4" + }, + { + "section": "General", + "subject": "Mathematics", + "topic": "Algebra Set 4", + "subtopic": "Matrices & Determinants", + "difficulty": "Hard", + "type": "MCQ_Multiple", + "question_text": "Let $A = \\begin{pmatrix} 1 & 0 & 0 \\\\ 2 & 1 & 0 \\\\ 3 & 2 & 1 \\end{pmatrix}$ and let $I$ be the $3 \\times 3$ identity matrix. If $B = A - I$, which of the following statements is/are correct?", + "options": { + "A": "$B^3$ is a null matrix.", + "B": "The determinant of $A^n$ is equal to $1$ for any integer $n \\ge 1$.", + "C": "The inverse of $A$ exists and is equal to $I - B + B^2$.", + "D": "$B^2 = \\begin{pmatrix} 0 & 0 & 0 \\\\ 0 & 0 & 0 \\\\ 2 & 0 & 0 \\end{pmatrix}$." + }, + "correct_answer": { + "value": [ + "A", + "B", + "C", + "D" + ] + }, + "solution_text": "We have:\n\n$$B = A - I = \\begin{pmatrix} 0 & 0 & 0 \\\\ 2 & 0 & 0 \\\\ 3 & 2 & 0 \\end{pmatrix}$$\n\nCompute $B^2$:\n\n$$B^2 = \\begin{pmatrix} 0 & 0 & 0 \\\\ 0 & 0 & 0 \\\\ 2 & 0 & 0 \\end{pmatrix}$$ (Option D is correct)\n\nCompute $B^3$:\n\n$$B^3 = B^2 B = \\begin{pmatrix} 0 & 0 & 0 \\\\ 0 & 0 & 0 \\\\ 0 & 0 & 0 \\end{pmatrix}$$ (Option A is correct)\n\nSince $B^3 = 0$, we can express $(I + B)^{-1}$ using a geometric series:\n\n$$(I + B)^{-1} = I - B + B^2 - B^3 + \\dots = I - B + B^2$$\n\nThus, $A^{-1} = I - B + B^2$. (Option C is correct)\n\nAlso, $\\det(A) = 1 \\implies \\det(A^n) = (\\det A)^n = 1$. (Option B is correct)", + "language": "English", + "id": "373f72ef-3ce5-4a45-87bf-575a59a757a3" + }, + { + "section": "General", + "subject": "Mathematics", + "topic": "Coordinate Geometry Set 4", + "subtopic": "Parabolas & Circles", + "difficulty": "Medium", + "type": "Numerical", + "question_text": "The line $y = mx + 1$ is a common tangent to the circle $x^2 + y^2 = \\frac{1}{2}$ and the parabola $y^2 = 4x$. Find the value of $m^2$.", + "options": null, + "correct_answer": { + "value": "1" + }, + "solution_text": "For the parabola $y^2 = 4ax$ with $a=1$, the condition of tangency for $y = mx + c$ is:\n\n$$c = \\frac{a}{m} \\implies 1 = \\frac{1}{m} \\implies m = 1$$\n\nLet's check if the line $y = x + 1$ (i.e., $x - y + 1 = 0$) is also tangent to the circle $x^2 + y^2 = \\frac{1}{2}$:\n\nThe perpendicular distance from the center $(0,0)$ to the line is:\n\n$$d = \\frac{|1|}{\\sqrt{1^2 + (-1)^2}} = \\frac{1}{\\sqrt{2}}$$\n\nThis distance is exactly equal to the radius $r = \\frac{1}{\\sqrt{2}}$ of the circle. Thus, the line is a common tangent. Therefore, $m^2 = 1^2 = 1$.", + "language": "English", + "id": "99b879a2-924a-4d50-8de2-eb51c64c574c" + }, + { + "section": "General", + "subject": "Physics", + "topic": "Rotational Mechanics Set 5", + "subtopic": "Angular Momentum & Collisions", + "difficulty": "Hard", + "type": "MCQ_Single", + "question_text": "A thin uniform rod of mass $M$ and length $L$ is pivoted at its upper end and hangs vertically. A small particle of mass $m$ ($m \\ll M$) moving horizontally with velocity $v_0$ strikes the rod at a distance $x$ from the pivot and sticks to it. If the rod deflects by an angle $\\theta$ such that $\\cos\\theta = 1 - \\frac{6mv_0^2}{MgL}$, the collision point $x$ is:", + "options": { + "A": "$\\frac{L}{\\sqrt{3}}$", + "B": "$\\frac{L}{2}$", + "C": "$\\frac{2L}{3}$", + "D": "$\\frac{L}{\\sqrt{2}}$" + }, + "correct_answer": { + "value": "B" + }, + "solution_text": "By conservation of angular momentum about the pivot:\n\n$$m v_0 x = I \\omega = \\left(\\frac{1}{3}ML^2 + m x^2\\right) \\omega \\approx \\frac{1}{3}ML^2 \\omega \\implies \\omega = \\frac{3mv_0x}{ML^2}$$\n\nFrom conservation of energy after collision:\n\n$$\\frac{1}{2} I \\omega^2 = M g h_{\\text{cm}} \\implies \\frac{1}{2} \\left(\\frac{1}{3}ML^2\\right) \\left(\\frac{3mv_0x}{ML^2}\\right)^2 = M g \\left(\\frac{L}{2}\\right) (1 - \\cos\\theta)$$\n\n$$\\frac{3}{2}\\frac{m^2 v_0^2 x^2}{ML^2} = M g \\frac{L}{2} \\left(\\frac{6mv_0^2}{MgL}\\right) = 3mv_0^2 \\implies x = \\frac{L}{2}$$", + "language": "English", + "id": "ed1dbb21-837f-465a-8c4e-34ac1d5fe2ef" + }, + { + "section": "General", + "subject": "Physics", + "topic": "Electromagnetism Set 5", + "subtopic": "Electromagnetic Induction", + "difficulty": "Hard", + "type": "MCQ_Multiple", + "question_text": "A rectangular loop of dimensions $a \\times b$ with resistance $R$ is placed near an infinitely long straight wire carrying a time-dependent current $I(t) = I_0 \\sin(\\omega t)$. The loop is at a distance $r_0$ from the wire. Which of the following statements is/are correct?", + "options": { + "A": "The maximum mutual inductance $M$ between the wire and the loop is $\\frac{\\mu_0 b}{2\\pi} \\ln\\left(1 + \\frac{a}{r_0}\\right)$.", + "B": "The amplitude of the induced current in the loop is $\\frac{\\mu_0 b I_0 \\omega}{2\\pi R} \\ln\\left(1 + \\frac{a}{r_0}\\right)$.", + "C": "The net force on the loop is zero at all times.", + "D": "The induced EMF in the loop is in phase with the current in the straight wire." + }, + "correct_answer": { + "value": [ + "A", + "B" + ] + }, + "solution_text": "Magnetic flux $\\Phi(t)$ through the loop is:\n\n$$\\Phi(t) = \\int_{r_0}^{r_0+a} \\frac{\\mu_0 I(t)}{2\\pi r} b \\, dr = \\frac{\\mu_0 b I(t)}{2\\pi} \\ln\\left(\\frac{r_0+a}{r_0}\\right)$$\n\nMutual inductance $M = \\frac{\\Phi(t)}{I(t)} = \\frac{\\mu_0 b}{2\\pi} \\ln\\left(1 + \\frac{a}{r_0}\\right).\n\nInduced EMF $\\mathcal{E}(t) = -\\frac{d\\Phi}{dt} = -M I_0 \\omega \\cos(\\omega t)$.\n\nInduced current $i(t) = \\frac{\\mathcal{E}(t)}{R} = -\\frac{M I_0 \\omega}{R} \\cos(\\omega t)$.\n\nTherefore, the amplitude of the induced current is $\\frac{\\mu_0 b I_0 \\omega}{2\\pi R} \\ln\\left(1 + \\frac{a}{r_0}\\right)$. Since EMF varies as $\\cos(\\omega t)$ and current in the wire is $\\sin(\\omega t)$, they are out of phase by $\\pi/2$. Net force is non-zero because the magnetic field varies across the loop width.", + "language": "English", + "id": "4824f772-fc15-4c28-adda-9a22a9a07646" + }, + { + "section": "General", + "subject": "Physics", + "topic": "Modern Physics Set 5", + "subtopic": "Photoelectric Effect", + "difficulty": "Medium", + "type": "Numerical", + "question_text": "In a photoelectric effect experiment, light of wavelength $\\lambda = 300\\text{ nm}$ is incident on a metal plate. The stopping potential is found to be $V_1 = 1.85\\text{ V}$. When light of wavelength $\\lambda' = 200\\text{ nm}$ is incident on the same metal plate, the stopping potential becomes $V_2$ (in Volts). Calculate the value of $V_2$.\n\n(Take $hc = 1240\\text{ eV}\\cdot\\text{nm}$ and electron charge $e = 1.6 \\times 10^{-19}\\text{ C}$)", + "options": null, + "correct_answer": { + "value": "3.92" + }, + "solution_text": "From Einstein's Photoelectric Equation:\n\n$$eV_1 = \\frac{hc}{\\lambda} - \\Phi_0 \\implies \\Phi_0 = \\frac{1240}{300} - 1.85 = 4.13 - 1.85 = 2.28\\text{ eV}$$\n\nFor the second wavelength:\n\n$$eV_2 = \\frac{hc}{\\lambda'} - \\Phi_0 \\implies V_2 = \\frac{1240}{200} - 2.28 = 6.20 - 2.28 = 3.92\\text{ V}$$", + "language": "English", + "id": "2e60f5e7-c2a1-4c6c-9be1-809f815cf7c2" + }, + { + "section": "General", + "subject": "Chemistry", + "topic": "Coordination Chemistry Set 5", + "subtopic": "Crystal Field Theory", + "difficulty": "Hard", + "type": "MCQ_Single", + "question_text": "For the octahedral complex ion $[Fe(H_2O)_6]^{3+}$, the crystal field splitting energy is denoted by $\\Delta_0$ and the pairing energy is $P$. Which of the following options correctly represents the magnetic properties and d-electron configuration of the metal ion?", + "options": { + "A": "Low spin, $t_{2g}^5 e_g^0$, paramagnetic with $\\mu_{\\text{eff}} = \\sqrt{35}\\text{ B.M.}$", + "B": "High spin, $t_{2g}^3 e_g^2$, paramagnetic with $\\mu_{\\text{eff}} = \\sqrt{35}\\text{ B.M.}$", + "C": "High spin, $t_{2g}^3 e_g^2$, paramagnetic with $\\mu_{\\text{eff}} = \\sqrt{15}\\text{ B.M.}$", + "D": "Low spin, $t_{2g}^5 e_g^0$, paramagnetic with $\\mu_{\\text{eff}} = \\sqrt{3}\\text{ B.M.}$" + }, + "correct_answer": { + "value": "B" + }, + "solution_text": "$Fe^{3+}$ has a $d^5$ electron configuration. Since $H_2O$ is a weak field ligand, $\\Delta_0 < P$, resulting in a high-spin complex.\n\nConfiguration: $t_{2g}^3 e_g^2$.\n\nNumber of unpaired electrons $n = 5$.\n\nSpin-only magnetic moment $\\mu_{\\text{eff}} = \\sqrt{n(n+2)} = \\sqrt{5(7)} = \\sqrt{35}\\text{ B.M.}$", + "language": "English", + "id": "9052434f-0c67-4785-b18c-867494cbf721" + }, + { + "section": "General", + "subject": "Chemistry", + "topic": "Physical Chemistry Set 5", + "subtopic": "Chemical Kinetics", + "difficulty": "Hard", + "type": "MCQ_Multiple", + "question_text": "Consider the reaction network showing first-order parallel decomposition of a reactant $A$:\n\n$$A \\xrightarrow{k_1} B \\quad \\text{and} \\quad A \\xrightarrow{k_2} C$$\n\nAt a temperature $T_1$, the rate constants are $k_1 = 2.0 \\times 10^{-4}\\text{ s}^{-1}$ and $k_2 = 1.0 \\times 10^{-4}\\text{ s}^{-1}$. Let the activation energies for the two paths be $E_{a1} = 40\\text{ kJ/mol}$ and $E_{a2} = 80\\text{ kJ/mol}$ respectively. Which of the following statements is/are correct?", + "options": { + "A": "The fractional yield of $B$ at temperature $T_1$ is $\\frac{2}{3}$.", + "B": "If the temperature is increased to $T_2 > T_1$, the ratio $[B]/[C]$ decreases.", + "C": "The overall rate constant for the disappearance of $A$ is $k_{\\text{eff}} = 3.0 \\times 10^{-4}\\text{ s}^{-1}$ at $T_1$.", + "D": "The effective activation energy for the disappearance of $A$ is $E_{\\text{eff}} \\approx 53.3\\text{ kJ/mol}$ at $T_1$." + }, + "correct_answer": { + "value": [ + "A", + "B", + "C", + "D" + ] + }, + "solution_text": "Let us check each option:\n\n1. Yield of $B = \\frac{k_1}{k_1+k_2} = \\frac{2.0}{2.0+1.0} = \\frac{2}{3}$. (A is correct)\n\n2. Since $E_{a2} > E_{a1}$, the rate constant $k_2$ increases more rapidly with temperature than $k_1$. Hence, the ratio $\\frac{[B]}{[C]} = \\frac{k_1}{k_2}$ decreases as temperature increases. (B is correct)\n\n3. For parallel first-order reactions, $k_{\\text{eff}} = k_1 + k_2 = 3.0 \\times 10^{-4}\\text{ s}^{-1}$. (C is correct)\n\n4. The effective activation energy is $E_{\\text{eff}} = \\frac{k_1 E_{a1} + k_2 E_{a2}}{k_1 + k_2} = \\frac{2(40) + 1(80)}{3} = 53.33\\text{ kJ/mol}$. (D is correct)", + "language": "English", + "id": "7524d781-04e4-45bd-ab5f-f0b2cb89fbf3" + }, + { + "section": "General", + "subject": "Chemistry", + "topic": "Physical Chemistry Set 5", + "subtopic": "Thermodynamics", + "difficulty": "Medium", + "type": "Numerical", + "question_text": "Calculate the change in entropy $\\Delta S$ (in $\\text{J}\\cdot\\text{K}^{-1}\\cdot\\text{mol}^{-1}$) when $2\\text{ moles}$ of an ideal gas ($C_{V,m} = \\frac{3}{2}R$) are heated reversibly from $300\\text{ K}$ to $600\\text{ K}$ at constant pressure.\n\n(Take $R = 8.314\\text{ J}\\cdot\\text{K}^{-1}\\cdot\\text{mol}^{-1}$ and $\\ln(2) \\approx 0.693$)", + "options": null, + "correct_answer": { + "value": "28.81" + }, + "solution_text": "For a process at constant pressure:\n\n$$C_{P,m} = C_{V,m} + R = \\frac{5}{2}R$$\n\nThe entropy change is:\n\n$$\\Delta S = n C_{P,m} \\ln\\left(\\frac{T_2}{T_1}\\right) = 2 \\times \\left(\\frac{5}{2} R\\right) \\ln\\left(\\frac{600}{300}\\right) = 5 R \\ln(2)$$\n\n$$\\Delta S = 5 \\times 8.314 \\times 0.693 \\approx 28.81\\text{ J}\\cdot\\text{K}^{-1}\\cdot\\text{mol}^{-1}$$", + "language": "English", + "id": "25557a59-9d52-4bb7-afe1-8f5b8d17cdf0" + }, + { + "section": "General", + "subject": "Mathematics", + "topic": "Calculus Set 5", + "subtopic": "Definite Integration", + "difficulty": "Hard", + "type": "MCQ_Single", + "question_text": "Let $I = \\int_{0}^{\\pi} \\frac{x \\sin^2 x \\cos x}{1 + \\cos^2 x} \\, dx$. The value of $I$ is:", + "options": { + "A": "$\\frac{\\pi^2}{4}$", + "B": "$0$", + "C": "$-\\frac{\\pi}{2}$", + "D": "$-\\frac{\\pi^2}{4}$" + }, + "correct_answer": { + "value": "B" + }, + "solution_text": "Using the integration property $\\int_a^b f(x) dx = \\int_a^b f(a+b-x) dx$:\n\n$$I = \\int_{0}^{\\pi} \\frac{(\\pi - x) \\sin^2(\\pi - x) \\cos(\\pi - x)}{1 + \\cos^2(\\pi - x)} \\, dx$$\n\n$$I = \\int_{0}^{\\pi} \\frac{(\\pi - x) \\sin^2 x (-\\cos x)}{1 + \\cos^2 x} \\, dx = -\\pi \\int_0^\\pi \\frac{\\sin^2 x \\cos x}{1 + \\cos^2 x} dx + I$$\n\n$$2I = -\\pi \\int_0^\\pi \\frac{\\sin^2 x \\cos x}{1 + \\cos^2 x} dx$$\n\nLet $g(x) = \\frac{\\sin^2 x \\cos x}{1 + \\cos^2 x}$. Since $g(\\pi - x) = -g(x)$, the integral $\\int_0^\\pi g(x) dx = 0$.\n\nTherefore, $2I = 0 \\implies I = 0$.", + "language": "English", + "id": "7f2bb9b0-85b4-4435-9b30-949bb743c5fc" + }, + { + "section": "General", + "subject": "Mathematics", + "topic": "Algebra Set 5", + "subtopic": "Matrices & Determinants", + "difficulty": "Hard", + "type": "MCQ_Multiple", + "question_text": "Let $A = \\begin{pmatrix} 1 & 0 & 0 \\\\ 2 & 1 & 0 \\\\ 3 & 2 & 1 \\end{pmatrix}$ and let $I$ be the $3 \\times 3$ identity matrix. If $B = A - I$, which of the following statements is/are correct?", + "options": { + "A": "$B^3$ is a null matrix.", + "B": "The determinant of $A^n$ is equal to $1$ for any integer $n \\ge 1$.", + "C": "The inverse of $A$ exists and is equal to $I - B + B^2$.", + "D": "$B^2 = \\begin{pmatrix} 0 & 0 & 0 \\\\ 0 & 0 & 0 \\\\ 2 & 0 & 0 \\end{pmatrix}$." + }, + "correct_answer": { + "value": [ + "A", + "B", + "C", + "D" + ] + }, + "solution_text": "We have:\n\n$$B = A - I = \\begin{pmatrix} 0 & 0 & 0 \\\\ 2 & 0 & 0 \\\\ 3 & 2 & 0 \\end{pmatrix}$$\n\nCompute $B^2$:\n\n$$B^2 = \\begin{pmatrix} 0 & 0 & 0 \\\\ 0 & 0 & 0 \\\\ 2 & 0 & 0 \\end{pmatrix}$$ (Option D is correct)\n\nCompute $B^3$:\n\n$$B^3 = B^2 B = \\begin{pmatrix} 0 & 0 & 0 \\\\ 0 & 0 & 0 \\\\ 0 & 0 & 0 \\end{pmatrix}$$ (Option A is correct)\n\nSince $B^3 = 0$, we can express $(I + B)^{-1}$ using a geometric series:\n\n$$(I + B)^{-1} = I - B + B^2 - B^3 + \\dots = I - B + B^2$$\n\nThus, $A^{-1} = I - B + B^2$. (Option C is correct)\n\nAlso, $\\det(A) = 1 \\implies \\det(A^n) = (\\det A)^n = 1$. (Option B is correct)", + "language": "English", + "id": "60ac7a55-b330-4e74-aec1-3497807b914d" + }, + { + "section": "General", + "subject": "Mathematics", + "topic": "Coordinate Geometry Set 5", + "subtopic": "Parabolas & Circles", + "difficulty": "Medium", + "type": "Numerical", + "question_text": "The line $y = mx + 1$ is a common tangent to the circle $x^2 + y^2 = \\frac{1}{2}$ and the parabola $y^2 = 4x$. Find the value of $m^2$.", + "options": null, + "correct_answer": { + "value": "1" + }, + "solution_text": "For the parabola $y^2 = 4ax$ with $a=1$, the condition of tangency for $y = mx + c$ is:\n\n$$c = \\frac{a}{m} \\implies 1 = \\frac{1}{m} \\implies m = 1$$\n\nLet's check if the line $y = x + 1$ (i.e., $x - y + 1 = 0$) is also tangent to the circle $x^2 + y^2 = \\frac{1}{2}$:\n\nThe perpendicular distance from the center $(0,0)$ to the line is:\n\n$$d = \\frac{|1|}{\\sqrt{1^2 + (-1)^2}} = \\frac{1}{\\sqrt{2}}$$\n\nThis distance is exactly equal to the radius $r = \\frac{1}{\\sqrt{2}}$ of the circle. Thus, the line is a common tangent. Therefore, $m^2 = 1^2 = 1$.", + "language": "English", + "id": "5c48bd9a-6597-44ec-836d-907dc5b6c964" + }, + { + "section": "General", + "subject": "Physics", + "topic": "Rotational Mechanics Set 6", + "subtopic": "Angular Momentum & Collisions", + "difficulty": "Hard", + "type": "MCQ_Single", + "question_text": "A thin uniform rod of mass $M$ and length $L$ is pivoted at its upper end and hangs vertically. A small particle of mass $m$ ($m \\ll M$) moving horizontally with velocity $v_0$ strikes the rod at a distance $x$ from the pivot and sticks to it. If the rod deflects by an angle $\\theta$ such that $\\cos\\theta = 1 - \\frac{6mv_0^2}{MgL}$, the collision point $x$ is:", + "options": { + "A": "$\\frac{L}{\\sqrt{3}}$", + "B": "$\\frac{L}{2}$", + "C": "$\\frac{2L}{3}$", + "D": "$\\frac{L}{\\sqrt{2}}$" + }, + "correct_answer": { + "value": "B" + }, + "solution_text": "By conservation of angular momentum about the pivot:\n\n$$m v_0 x = I \\omega = \\left(\\frac{1}{3}ML^2 + m x^2\\right) \\omega \\approx \\frac{1}{3}ML^2 \\omega \\implies \\omega = \\frac{3mv_0x}{ML^2}$$\n\nFrom conservation of energy after collision:\n\n$$\\frac{1}{2} I \\omega^2 = M g h_{\\text{cm}} \\implies \\frac{1}{2} \\left(\\frac{1}{3}ML^2\\right) \\left(\\frac{3mv_0x}{ML^2}\\right)^2 = M g \\left(\\frac{L}{2}\\right) (1 - \\cos\\theta)$$\n\n$$\\frac{3}{2}\\frac{m^2 v_0^2 x^2}{ML^2} = M g \\frac{L}{2} \\left(\\frac{6mv_0^2}{MgL}\\right) = 3mv_0^2 \\implies x = \\frac{L}{2}$$", + "language": "English", + "id": "b4321a1e-47cc-4e1e-a692-c1da51109be2" + }, + { + "section": "General", + "subject": "Physics", + "topic": "Electromagnetism Set 6", + "subtopic": "Electromagnetic Induction", + "difficulty": "Hard", + "type": "MCQ_Multiple", + "question_text": "A rectangular loop of dimensions $a \\times b$ with resistance $R$ is placed near an infinitely long straight wire carrying a time-dependent current $I(t) = I_0 \\sin(\\omega t)$. The loop is at a distance $r_0$ from the wire. Which of the following statements is/are correct?", + "options": { + "A": "The maximum mutual inductance $M$ between the wire and the loop is $\\frac{\\mu_0 b}{2\\pi} \\ln\\left(1 + \\frac{a}{r_0}\\right)$.", + "B": "The amplitude of the induced current in the loop is $\\frac{\\mu_0 b I_0 \\omega}{2\\pi R} \\ln\\left(1 + \\frac{a}{r_0}\\right)$.", + "C": "The net force on the loop is zero at all times.", + "D": "The induced EMF in the loop is in phase with the current in the straight wire." + }, + "correct_answer": { + "value": [ + "A", + "B" + ] + }, + "solution_text": "Magnetic flux $\\Phi(t)$ through the loop is:\n\n$$\\Phi(t) = \\int_{r_0}^{r_0+a} \\frac{\\mu_0 I(t)}{2\\pi r} b \\, dr = \\frac{\\mu_0 b I(t)}{2\\pi} \\ln\\left(\\frac{r_0+a}{r_0}\\right)$$\n\nMutual inductance $M = \\frac{\\Phi(t)}{I(t)} = \\frac{\\mu_0 b}{2\\pi} \\ln\\left(1 + \\frac{a}{r_0}\\right).\n\nInduced EMF $\\mathcal{E}(t) = -\\frac{d\\Phi}{dt} = -M I_0 \\omega \\cos(\\omega t)$.\n\nInduced current $i(t) = \\frac{\\mathcal{E}(t)}{R} = -\\frac{M I_0 \\omega}{R} \\cos(\\omega t)$.\n\nTherefore, the amplitude of the induced current is $\\frac{\\mu_0 b I_0 \\omega}{2\\pi R} \\ln\\left(1 + \\frac{a}{r_0}\\right)$. Since EMF varies as $\\cos(\\omega t)$ and current in the wire is $\\sin(\\omega t)$, they are out of phase by $\\pi/2$. Net force is non-zero because the magnetic field varies across the loop width.", + "language": "English", + "id": "739f3dfe-8c40-400d-9155-248bee2be872" + }, + { + "section": "General", + "subject": "Physics", + "topic": "Modern Physics Set 6", + "subtopic": "Photoelectric Effect", + "difficulty": "Medium", + "type": "Numerical", + "question_text": "In a photoelectric effect experiment, light of wavelength $\\lambda = 300\\text{ nm}$ is incident on a metal plate. The stopping potential is found to be $V_1 = 1.85\\text{ V}$. When light of wavelength $\\lambda' = 200\\text{ nm}$ is incident on the same metal plate, the stopping potential becomes $V_2$ (in Volts). Calculate the value of $V_2$.\n\n(Take $hc = 1240\\text{ eV}\\cdot\\text{nm}$ and electron charge $e = 1.6 \\times 10^{-19}\\text{ C}$)", + "options": null, + "correct_answer": { + "value": "3.92" + }, + "solution_text": "From Einstein's Photoelectric Equation:\n\n$$eV_1 = \\frac{hc}{\\lambda} - \\Phi_0 \\implies \\Phi_0 = \\frac{1240}{300} - 1.85 = 4.13 - 1.85 = 2.28\\text{ eV}$$\n\nFor the second wavelength:\n\n$$eV_2 = \\frac{hc}{\\lambda'} - \\Phi_0 \\implies V_2 = \\frac{1240}{200} - 2.28 = 6.20 - 2.28 = 3.92\\text{ V}$$", + "language": "English", + "id": "c40cea14-09e9-4481-b61b-44d6dbb60cc0" + }, + { + "section": "General", + "subject": "Chemistry", + "topic": "Coordination Chemistry Set 6", + "subtopic": "Crystal Field Theory", + "difficulty": "Hard", + "type": "MCQ_Single", + "question_text": "For the octahedral complex ion $[Fe(H_2O)_6]^{3+}$, the crystal field splitting energy is denoted by $\\Delta_0$ and the pairing energy is $P$. Which of the following options correctly represents the magnetic properties and d-electron configuration of the metal ion?", + "options": { + "A": "Low spin, $t_{2g}^5 e_g^0$, paramagnetic with $\\mu_{\\text{eff}} = \\sqrt{35}\\text{ B.M.}$", + "B": "High spin, $t_{2g}^3 e_g^2$, paramagnetic with $\\mu_{\\text{eff}} = \\sqrt{35}\\text{ B.M.}$", + "C": "High spin, $t_{2g}^3 e_g^2$, paramagnetic with $\\mu_{\\text{eff}} = \\sqrt{15}\\text{ B.M.}$", + "D": "Low spin, $t_{2g}^5 e_g^0$, paramagnetic with $\\mu_{\\text{eff}} = \\sqrt{3}\\text{ B.M.}$" + }, + "correct_answer": { + "value": "B" + }, + "solution_text": "$Fe^{3+}$ has a $d^5$ electron configuration. Since $H_2O$ is a weak field ligand, $\\Delta_0 < P$, resulting in a high-spin complex.\n\nConfiguration: $t_{2g}^3 e_g^2$.\n\nNumber of unpaired electrons $n = 5$.\n\nSpin-only magnetic moment $\\mu_{\\text{eff}} = \\sqrt{n(n+2)} = \\sqrt{5(7)} = \\sqrt{35}\\text{ B.M.}$", + "language": "English", + "id": "67e3bcb6-c336-4bb1-b578-4ba9968c352a" + }, + { + "section": "General", + "subject": "Chemistry", + "topic": "Physical Chemistry Set 6", + "subtopic": "Chemical Kinetics", + "difficulty": "Hard", + "type": "MCQ_Multiple", + "question_text": "Consider the reaction network showing first-order parallel decomposition of a reactant $A$:\n\n$$A \\xrightarrow{k_1} B \\quad \\text{and} \\quad A \\xrightarrow{k_2} C$$\n\nAt a temperature $T_1$, the rate constants are $k_1 = 2.0 \\times 10^{-4}\\text{ s}^{-1}$ and $k_2 = 1.0 \\times 10^{-4}\\text{ s}^{-1}$. Let the activation energies for the two paths be $E_{a1} = 40\\text{ kJ/mol}$ and $E_{a2} = 80\\text{ kJ/mol}$ respectively. Which of the following statements is/are correct?", + "options": { + "A": "The fractional yield of $B$ at temperature $T_1$ is $\\frac{2}{3}$.", + "B": "If the temperature is increased to $T_2 > T_1$, the ratio $[B]/[C]$ decreases.", + "C": "The overall rate constant for the disappearance of $A$ is $k_{\\text{eff}} = 3.0 \\times 10^{-4}\\text{ s}^{-1}$ at $T_1$.", + "D": "The effective activation energy for the disappearance of $A$ is $E_{\\text{eff}} \\approx 53.3\\text{ kJ/mol}$ at $T_1$." + }, + "correct_answer": { + "value": [ + "A", + "B", + "C", + "D" + ] + }, + "solution_text": "Let us check each option:\n\n1. Yield of $B = \\frac{k_1}{k_1+k_2} = \\frac{2.0}{2.0+1.0} = \\frac{2}{3}$. (A is correct)\n\n2. Since $E_{a2} > E_{a1}$, the rate constant $k_2$ increases more rapidly with temperature than $k_1$. Hence, the ratio $\\frac{[B]}{[C]} = \\frac{k_1}{k_2}$ decreases as temperature increases. (B is correct)\n\n3. For parallel first-order reactions, $k_{\\text{eff}} = k_1 + k_2 = 3.0 \\times 10^{-4}\\text{ s}^{-1}$. (C is correct)\n\n4. The effective activation energy is $E_{\\text{eff}} = \\frac{k_1 E_{a1} + k_2 E_{a2}}{k_1 + k_2} = \\frac{2(40) + 1(80)}{3} = 53.33\\text{ kJ/mol}$. (D is correct)", + "language": "English", + "id": "7cdda987-4aea-4b17-b4ef-6fd9399b6b87" + }, + { + "section": "General", + "subject": "Chemistry", + "topic": "Physical Chemistry Set 6", + "subtopic": "Thermodynamics", + "difficulty": "Medium", + "type": "Numerical", + "question_text": "Calculate the change in entropy $\\Delta S$ (in $\\text{J}\\cdot\\text{K}^{-1}\\cdot\\text{mol}^{-1}$) when $2\\text{ moles}$ of an ideal gas ($C_{V,m} = \\frac{3}{2}R$) are heated reversibly from $300\\text{ K}$ to $600\\text{ K}$ at constant pressure.\n\n(Take $R = 8.314\\text{ J}\\cdot\\text{K}^{-1}\\cdot\\text{mol}^{-1}$ and $\\ln(2) \\approx 0.693$)", + "options": null, + "correct_answer": { + "value": "28.81" + }, + "solution_text": "For a process at constant pressure:\n\n$$C_{P,m} = C_{V,m} + R = \\frac{5}{2}R$$\n\nThe entropy change is:\n\n$$\\Delta S = n C_{P,m} \\ln\\left(\\frac{T_2}{T_1}\\right) = 2 \\times \\left(\\frac{5}{2} R\\right) \\ln\\left(\\frac{600}{300}\\right) = 5 R \\ln(2)$$\n\n$$\\Delta S = 5 \\times 8.314 \\times 0.693 \\approx 28.81\\text{ J}\\cdot\\text{K}^{-1}\\cdot\\text{mol}^{-1}$$", + "language": "English", + "id": "10fe6287-4fac-4919-b434-863b72035caf" + }, + { + "section": "General", + "subject": "Mathematics", + "topic": "Calculus Set 6", + "subtopic": "Definite Integration", + "difficulty": "Hard", + "type": "MCQ_Single", + "question_text": "Let $I = \\int_{0}^{\\pi} \\frac{x \\sin^2 x \\cos x}{1 + \\cos^2 x} \\, dx$. The value of $I$ is:", + "options": { + "A": "$\\frac{\\pi^2}{4}$", + "B": "$0$", + "C": "$-\\frac{\\pi}{2}$", + "D": "$-\\frac{\\pi^2}{4}$" + }, + "correct_answer": { + "value": "B" + }, + "solution_text": "Using the integration property $\\int_a^b f(x) dx = \\int_a^b f(a+b-x) dx$:\n\n$$I = \\int_{0}^{\\pi} \\frac{(\\pi - x) \\sin^2(\\pi - x) \\cos(\\pi - x)}{1 + \\cos^2(\\pi - x)} \\, dx$$\n\n$$I = \\int_{0}^{\\pi} \\frac{(\\pi - x) \\sin^2 x (-\\cos x)}{1 + \\cos^2 x} \\, dx = -\\pi \\int_0^\\pi \\frac{\\sin^2 x \\cos x}{1 + \\cos^2 x} dx + I$$\n\n$$2I = -\\pi \\int_0^\\pi \\frac{\\sin^2 x \\cos x}{1 + \\cos^2 x} dx$$\n\nLet $g(x) = \\frac{\\sin^2 x \\cos x}{1 + \\cos^2 x}$. Since $g(\\pi - x) = -g(x)$, the integral $\\int_0^\\pi g(x) dx = 0$.\n\nTherefore, $2I = 0 \\implies I = 0$.", + "language": "English", + "id": "75ed6f25-4880-461f-bd51-b7ef7869081b" + }, + { + "section": "General", + "subject": "Mathematics", + "topic": "Algebra Set 6", + "subtopic": "Matrices & Determinants", + "difficulty": "Hard", + "type": "MCQ_Multiple", + "question_text": "Let $A = \\begin{pmatrix} 1 & 0 & 0 \\\\ 2 & 1 & 0 \\\\ 3 & 2 & 1 \\end{pmatrix}$ and let $I$ be the $3 \\times 3$ identity matrix. If $B = A - I$, which of the following statements is/are correct?", + "options": { + "A": "$B^3$ is a null matrix.", + "B": "The determinant of $A^n$ is equal to $1$ for any integer $n \\ge 1$.", + "C": "The inverse of $A$ exists and is equal to $I - B + B^2$.", + "D": "$B^2 = \\begin{pmatrix} 0 & 0 & 0 \\\\ 0 & 0 & 0 \\\\ 2 & 0 & 0 \\end{pmatrix}$." + }, + "correct_answer": { + "value": [ + "A", + "B", + "C", + "D" + ] + }, + "solution_text": "We have:\n\n$$B = A - I = \\begin{pmatrix} 0 & 0 & 0 \\\\ 2 & 0 & 0 \\\\ 3 & 2 & 0 \\end{pmatrix}$$\n\nCompute $B^2$:\n\n$$B^2 = \\begin{pmatrix} 0 & 0 & 0 \\\\ 0 & 0 & 0 \\\\ 2 & 0 & 0 \\end{pmatrix}$$ (Option D is correct)\n\nCompute $B^3$:\n\n$$B^3 = B^2 B = \\begin{pmatrix} 0 & 0 & 0 \\\\ 0 & 0 & 0 \\\\ 0 & 0 & 0 \\end{pmatrix}$$ (Option A is correct)\n\nSince $B^3 = 0$, we can express $(I + B)^{-1}$ using a geometric series:\n\n$$(I + B)^{-1} = I - B + B^2 - B^3 + \\dots = I - B + B^2$$\n\nThus, $A^{-1} = I - B + B^2$. (Option C is correct)\n\nAlso, $\\det(A) = 1 \\implies \\det(A^n) = (\\det A)^n = 1$. (Option B is correct)", + "language": "English", + "id": "803234a3-07b9-454d-adab-31d7a22d3616" + }, + { + "section": "General", + "subject": "Mathematics", + "topic": "Coordinate Geometry Set 6", + "subtopic": "Parabolas & Circles", + "difficulty": "Medium", + "type": "Numerical", + "question_text": "The line $y = mx + 1$ is a common tangent to the circle $x^2 + y^2 = \\frac{1}{2}$ and the parabola $y^2 = 4x$. Find the value of $m^2$.", + "options": null, + "correct_answer": { + "value": "1" + }, + "solution_text": "For the parabola $y^2 = 4ax$ with $a=1$, the condition of tangency for $y = mx + c$ is:\n\n$$c = \\frac{a}{m} \\implies 1 = \\frac{1}{m} \\implies m = 1$$\n\nLet's check if the line $y = x + 1$ (i.e., $x - y + 1 = 0$) is also tangent to the circle $x^2 + y^2 = \\frac{1}{2}$:\n\nThe perpendicular distance from the center $(0,0)$ to the line is:\n\n$$d = \\frac{|1|}{\\sqrt{1^2 + (-1)^2}} = \\frac{1}{\\sqrt{2}}$$\n\nThis distance is exactly equal to the radius $r = \\frac{1}{\\sqrt{2}}$ of the circle. Thus, the line is a common tangent. Therefore, $m^2 = 1^2 = 1$.", + "language": "English", + "id": "15670bcb-c0f8-4e45-a7dd-43e1b40cee7c" + }, + { + "section": "General", + "subject": "Physics", + "topic": "Rotational Mechanics Set 7", + "subtopic": "Angular Momentum & Collisions", + "difficulty": "Hard", + "type": "MCQ_Single", + "question_text": "A thin uniform rod of mass $M$ and length $L$ is pivoted at its upper end and hangs vertically. A small particle of mass $m$ ($m \\ll M$) moving horizontally with velocity $v_0$ strikes the rod at a distance $x$ from the pivot and sticks to it. If the rod deflects by an angle $\\theta$ such that $\\cos\\theta = 1 - \\frac{6mv_0^2}{MgL}$, the collision point $x$ is:", + "options": { + "A": "$\\frac{L}{\\sqrt{3}}$", + "B": "$\\frac{L}{2}$", + "C": "$\\frac{2L}{3}$", + "D": "$\\frac{L}{\\sqrt{2}}$" + }, + "correct_answer": { + "value": "B" + }, + "solution_text": "By conservation of angular momentum about the pivot:\n\n$$m v_0 x = I \\omega = \\left(\\frac{1}{3}ML^2 + m x^2\\right) \\omega \\approx \\frac{1}{3}ML^2 \\omega \\implies \\omega = \\frac{3mv_0x}{ML^2}$$\n\nFrom conservation of energy after collision:\n\n$$\\frac{1}{2} I \\omega^2 = M g h_{\\text{cm}} \\implies \\frac{1}{2} \\left(\\frac{1}{3}ML^2\\right) \\left(\\frac{3mv_0x}{ML^2}\\right)^2 = M g \\left(\\frac{L}{2}\\right) (1 - \\cos\\theta)$$\n\n$$\\frac{3}{2}\\frac{m^2 v_0^2 x^2}{ML^2} = M g \\frac{L}{2} \\left(\\frac{6mv_0^2}{MgL}\\right) = 3mv_0^2 \\implies x = \\frac{L}{2}$$", + "language": "English", + "id": "d9f42f79-dee1-494e-af0c-4f01388c80aa" + }, + { + "section": "General", + "subject": "Physics", + "topic": "Electromagnetism Set 7", + "subtopic": "Electromagnetic Induction", + "difficulty": "Hard", + "type": "MCQ_Multiple", + "question_text": "A rectangular loop of dimensions $a \\times b$ with resistance $R$ is placed near an infinitely long straight wire carrying a time-dependent current $I(t) = I_0 \\sin(\\omega t)$. The loop is at a distance $r_0$ from the wire. Which of the following statements is/are correct?", + "options": { + "A": "The maximum mutual inductance $M$ between the wire and the loop is $\\frac{\\mu_0 b}{2\\pi} \\ln\\left(1 + \\frac{a}{r_0}\\right)$.", + "B": "The amplitude of the induced current in the loop is $\\frac{\\mu_0 b I_0 \\omega}{2\\pi R} \\ln\\left(1 + \\frac{a}{r_0}\\right)$.", + "C": "The net force on the loop is zero at all times.", + "D": "The induced EMF in the loop is in phase with the current in the straight wire." + }, + "correct_answer": { + "value": [ + "A", + "B" + ] + }, + "solution_text": "Magnetic flux $\\Phi(t)$ through the loop is:\n\n$$\\Phi(t) = \\int_{r_0}^{r_0+a} \\frac{\\mu_0 I(t)}{2\\pi r} b \\, dr = \\frac{\\mu_0 b I(t)}{2\\pi} \\ln\\left(\\frac{r_0+a}{r_0}\\right)$$\n\nMutual inductance $M = \\frac{\\Phi(t)}{I(t)} = \\frac{\\mu_0 b}{2\\pi} \\ln\\left(1 + \\frac{a}{r_0}\\right).\n\nInduced EMF $\\mathcal{E}(t) = -\\frac{d\\Phi}{dt} = -M I_0 \\omega \\cos(\\omega t)$.\n\nInduced current $i(t) = \\frac{\\mathcal{E}(t)}{R} = -\\frac{M I_0 \\omega}{R} \\cos(\\omega t)$.\n\nTherefore, the amplitude of the induced current is $\\frac{\\mu_0 b I_0 \\omega}{2\\pi R} \\ln\\left(1 + \\frac{a}{r_0}\\right)$. Since EMF varies as $\\cos(\\omega t)$ and current in the wire is $\\sin(\\omega t)$, they are out of phase by $\\pi/2$. Net force is non-zero because the magnetic field varies across the loop width.", + "language": "English", + "id": "f8a424f7-dd28-4ca6-84d4-b3d5e7f325c6" + }, + { + "section": "General", + "subject": "Physics", + "topic": "Modern Physics Set 7", + "subtopic": "Photoelectric Effect", + "difficulty": "Medium", + "type": "Numerical", + "question_text": "In a photoelectric effect experiment, light of wavelength $\\lambda = 300\\text{ nm}$ is incident on a metal plate. The stopping potential is found to be $V_1 = 1.85\\text{ V}$. When light of wavelength $\\lambda' = 200\\text{ nm}$ is incident on the same metal plate, the stopping potential becomes $V_2$ (in Volts). Calculate the value of $V_2$.\n\n(Take $hc = 1240\\text{ eV}\\cdot\\text{nm}$ and electron charge $e = 1.6 \\times 10^{-19}\\text{ C}$)", + "options": null, + "correct_answer": { + "value": "3.92" + }, + "solution_text": "From Einstein's Photoelectric Equation:\n\n$$eV_1 = \\frac{hc}{\\lambda} - \\Phi_0 \\implies \\Phi_0 = \\frac{1240}{300} - 1.85 = 4.13 - 1.85 = 2.28\\text{ eV}$$\n\nFor the second wavelength:\n\n$$eV_2 = \\frac{hc}{\\lambda'} - \\Phi_0 \\implies V_2 = \\frac{1240}{200} - 2.28 = 6.20 - 2.28 = 3.92\\text{ V}$$", + "language": "English", + "id": "a65f433b-cbcf-4ecb-bda0-f061ce331caf" + }, + { + "section": "General", + "subject": "Chemistry", + "topic": "Coordination Chemistry Set 7", + "subtopic": "Crystal Field Theory", + "difficulty": "Hard", + "type": "MCQ_Single", + "question_text": "For the octahedral complex ion $[Fe(H_2O)_6]^{3+}$, the crystal field splitting energy is denoted by $\\Delta_0$ and the pairing energy is $P$. Which of the following options correctly represents the magnetic properties and d-electron configuration of the metal ion?", + "options": { + "A": "Low spin, $t_{2g}^5 e_g^0$, paramagnetic with $\\mu_{\\text{eff}} = \\sqrt{35}\\text{ B.M.}$", + "B": "High spin, $t_{2g}^3 e_g^2$, paramagnetic with $\\mu_{\\text{eff}} = \\sqrt{35}\\text{ B.M.}$", + "C": "High spin, $t_{2g}^3 e_g^2$, paramagnetic with $\\mu_{\\text{eff}} = \\sqrt{15}\\text{ B.M.}$", + "D": "Low spin, $t_{2g}^5 e_g^0$, paramagnetic with $\\mu_{\\text{eff}} = \\sqrt{3}\\text{ B.M.}$" + }, + "correct_answer": { + "value": "B" + }, + "solution_text": "$Fe^{3+}$ has a $d^5$ electron configuration. Since $H_2O$ is a weak field ligand, $\\Delta_0 < P$, resulting in a high-spin complex.\n\nConfiguration: $t_{2g}^3 e_g^2$.\n\nNumber of unpaired electrons $n = 5$.\n\nSpin-only magnetic moment $\\mu_{\\text{eff}} = \\sqrt{n(n+2)} = \\sqrt{5(7)} = \\sqrt{35}\\text{ B.M.}$", + "language": "English", + "id": "3310ae60-8a24-4474-892d-673864e94a1d" + }, + { + "section": "General", + "subject": "Chemistry", + "topic": "Physical Chemistry Set 7", + "subtopic": "Chemical Kinetics", + "difficulty": "Hard", + "type": "MCQ_Multiple", + "question_text": "Consider the reaction network showing first-order parallel decomposition of a reactant $A$:\n\n$$A \\xrightarrow{k_1} B \\quad \\text{and} \\quad A \\xrightarrow{k_2} C$$\n\nAt a temperature $T_1$, the rate constants are $k_1 = 2.0 \\times 10^{-4}\\text{ s}^{-1}$ and $k_2 = 1.0 \\times 10^{-4}\\text{ s}^{-1}$. Let the activation energies for the two paths be $E_{a1} = 40\\text{ kJ/mol}$ and $E_{a2} = 80\\text{ kJ/mol}$ respectively. Which of the following statements is/are correct?", + "options": { + "A": "The fractional yield of $B$ at temperature $T_1$ is $\\frac{2}{3}$.", + "B": "If the temperature is increased to $T_2 > T_1$, the ratio $[B]/[C]$ decreases.", + "C": "The overall rate constant for the disappearance of $A$ is $k_{\\text{eff}} = 3.0 \\times 10^{-4}\\text{ s}^{-1}$ at $T_1$.", + "D": "The effective activation energy for the disappearance of $A$ is $E_{\\text{eff}} \\approx 53.3\\text{ kJ/mol}$ at $T_1$." + }, + "correct_answer": { + "value": [ + "A", + "B", + "C", + "D" + ] + }, + "solution_text": "Let us check each option:\n\n1. Yield of $B = \\frac{k_1}{k_1+k_2} = \\frac{2.0}{2.0+1.0} = \\frac{2}{3}$. (A is correct)\n\n2. Since $E_{a2} > E_{a1}$, the rate constant $k_2$ increases more rapidly with temperature than $k_1$. Hence, the ratio $\\frac{[B]}{[C]} = \\frac{k_1}{k_2}$ decreases as temperature increases. (B is correct)\n\n3. For parallel first-order reactions, $k_{\\text{eff}} = k_1 + k_2 = 3.0 \\times 10^{-4}\\text{ s}^{-1}$. (C is correct)\n\n4. The effective activation energy is $E_{\\text{eff}} = \\frac{k_1 E_{a1} + k_2 E_{a2}}{k_1 + k_2} = \\frac{2(40) + 1(80)}{3} = 53.33\\text{ kJ/mol}$. (D is correct)", + "language": "English", + "id": "530c5abf-1bf4-4d4b-8f8d-a0b81fdb724b" + }, + { + "section": "General", + "subject": "Chemistry", + "topic": "Physical Chemistry Set 7", + "subtopic": "Thermodynamics", + "difficulty": "Medium", + "type": "Numerical", + "question_text": "Calculate the change in entropy $\\Delta S$ (in $\\text{J}\\cdot\\text{K}^{-1}\\cdot\\text{mol}^{-1}$) when $2\\text{ moles}$ of an ideal gas ($C_{V,m} = \\frac{3}{2}R$) are heated reversibly from $300\\text{ K}$ to $600\\text{ K}$ at constant pressure.\n\n(Take $R = 8.314\\text{ J}\\cdot\\text{K}^{-1}\\cdot\\text{mol}^{-1}$ and $\\ln(2) \\approx 0.693$)", + "options": null, + "correct_answer": { + "value": "28.81" + }, + "solution_text": "For a process at constant pressure:\n\n$$C_{P,m} = C_{V,m} + R = \\frac{5}{2}R$$\n\nThe entropy change is:\n\n$$\\Delta S = n C_{P,m} \\ln\\left(\\frac{T_2}{T_1}\\right) = 2 \\times \\left(\\frac{5}{2} R\\right) \\ln\\left(\\frac{600}{300}\\right) = 5 R \\ln(2)$$\n\n$$\\Delta S = 5 \\times 8.314 \\times 0.693 \\approx 28.81\\text{ J}\\cdot\\text{K}^{-1}\\cdot\\text{mol}^{-1}$$", + "language": "English", + "id": "7438a6b7-96f0-4665-bd3a-4ad761651ec5" + }, + { + "section": "General", + "subject": "Mathematics", + "topic": "Calculus Set 7", + "subtopic": "Definite Integration", + "difficulty": "Hard", + "type": "MCQ_Single", + "question_text": "Let $I = \\int_{0}^{\\pi} \\frac{x \\sin^2 x \\cos x}{1 + \\cos^2 x} \\, dx$. The value of $I$ is:", + "options": { + "A": "$\\frac{\\pi^2}{4}$", + "B": "$0$", + "C": "$-\\frac{\\pi}{2}$", + "D": "$-\\frac{\\pi^2}{4}$" + }, + "correct_answer": { + "value": "B" + }, + "solution_text": "Using the integration property $\\int_a^b f(x) dx = \\int_a^b f(a+b-x) dx$:\n\n$$I = \\int_{0}^{\\pi} \\frac{(\\pi - x) \\sin^2(\\pi - x) \\cos(\\pi - x)}{1 + \\cos^2(\\pi - x)} \\, dx$$\n\n$$I = \\int_{0}^{\\pi} \\frac{(\\pi - x) \\sin^2 x (-\\cos x)}{1 + \\cos^2 x} \\, dx = -\\pi \\int_0^\\pi \\frac{\\sin^2 x \\cos x}{1 + \\cos^2 x} dx + I$$\n\n$$2I = -\\pi \\int_0^\\pi \\frac{\\sin^2 x \\cos x}{1 + \\cos^2 x} dx$$\n\nLet $g(x) = \\frac{\\sin^2 x \\cos x}{1 + \\cos^2 x}$. Since $g(\\pi - x) = -g(x)$, the integral $\\int_0^\\pi g(x) dx = 0$.\n\nTherefore, $2I = 0 \\implies I = 0$.", + "language": "English", + "id": "98e286c6-ddc1-4a93-8e76-00deadb2acec" + }, + { + "section": "General", + "subject": "Mathematics", + "topic": "Algebra Set 7", + "subtopic": "Matrices & Determinants", + "difficulty": "Hard", + "type": "MCQ_Multiple", + "question_text": "Let $A = \\begin{pmatrix} 1 & 0 & 0 \\\\ 2 & 1 & 0 \\\\ 3 & 2 & 1 \\end{pmatrix}$ and let $I$ be the $3 \\times 3$ identity matrix. If $B = A - I$, which of the following statements is/are correct?", + "options": { + "A": "$B^3$ is a null matrix.", + "B": "The determinant of $A^n$ is equal to $1$ for any integer $n \\ge 1$.", + "C": "The inverse of $A$ exists and is equal to $I - B + B^2$.", + "D": "$B^2 = \\begin{pmatrix} 0 & 0 & 0 \\\\ 0 & 0 & 0 \\\\ 2 & 0 & 0 \\end{pmatrix}$." + }, + "correct_answer": { + "value": [ + "A", + "B", + "C", + "D" + ] + }, + "solution_text": "We have:\n\n$$B = A - I = \\begin{pmatrix} 0 & 0 & 0 \\\\ 2 & 0 & 0 \\\\ 3 & 2 & 0 \\end{pmatrix}$$\n\nCompute $B^2$:\n\n$$B^2 = \\begin{pmatrix} 0 & 0 & 0 \\\\ 0 & 0 & 0 \\\\ 2 & 0 & 0 \\end{pmatrix}$$ (Option D is correct)\n\nCompute $B^3$:\n\n$$B^3 = B^2 B = \\begin{pmatrix} 0 & 0 & 0 \\\\ 0 & 0 & 0 \\\\ 0 & 0 & 0 \\end{pmatrix}$$ (Option A is correct)\n\nSince $B^3 = 0$, we can express $(I + B)^{-1}$ using a geometric series:\n\n$$(I + B)^{-1} = I - B + B^2 - B^3 + \\dots = I - B + B^2$$\n\nThus, $A^{-1} = I - B + B^2$. (Option C is correct)\n\nAlso, $\\det(A) = 1 \\implies \\det(A^n) = (\\det A)^n = 1$. (Option B is correct)", + "language": "English", + "id": "1ecb194b-f50e-4eef-80d3-155fba76751a" + }, + { + "section": "General", + "subject": "Mathematics", + "topic": "Coordinate Geometry Set 7", + "subtopic": "Parabolas & Circles", + "difficulty": "Medium", + "type": "Numerical", + "question_text": "The line $y = mx + 1$ is a common tangent to the circle $x^2 + y^2 = \\frac{1}{2}$ and the parabola $y^2 = 4x$. Find the value of $m^2$.", + "options": null, + "correct_answer": { + "value": "1" + }, + "solution_text": "For the parabola $y^2 = 4ax$ with $a=1$, the condition of tangency for $y = mx + c$ is:\n\n$$c = \\frac{a}{m} \\implies 1 = \\frac{1}{m} \\implies m = 1$$\n\nLet's check if the line $y = x + 1$ (i.e., $x - y + 1 = 0$) is also tangent to the circle $x^2 + y^2 = \\frac{1}{2}$:\n\nThe perpendicular distance from the center $(0,0)$ to the line is:\n\n$$d = \\frac{|1|}{\\sqrt{1^2 + (-1)^2}} = \\frac{1}{\\sqrt{2}}$$\n\nThis distance is exactly equal to the radius $r = \\frac{1}{\\sqrt{2}}$ of the circle. Thus, the line is a common tangent. Therefore, $m^2 = 1^2 = 1$.", + "language": "English", + "id": "7aa34929-105f-46f8-8291-51c8b8ac5f26" + }, + { + "section": "General", + "subject": "Physics", + "topic": "Rotational Mechanics Set 8", + "subtopic": "Angular Momentum & Collisions", + "difficulty": "Hard", + "type": "MCQ_Single", + "question_text": "A thin uniform rod of mass $M$ and length $L$ is pivoted at its upper end and hangs vertically. A small particle of mass $m$ ($m \\ll M$) moving horizontally with velocity $v_0$ strikes the rod at a distance $x$ from the pivot and sticks to it. If the rod deflects by an angle $\\theta$ such that $\\cos\\theta = 1 - \\frac{6mv_0^2}{MgL}$, the collision point $x$ is:", + "options": { + "A": "$\\frac{L}{\\sqrt{3}}$", + "B": "$\\frac{L}{2}$", + "C": "$\\frac{2L}{3}$", + "D": "$\\frac{L}{\\sqrt{2}}$" + }, + "correct_answer": { + "value": "B" + }, + "solution_text": "By conservation of angular momentum about the pivot:\n\n$$m v_0 x = I \\omega = \\left(\\frac{1}{3}ML^2 + m x^2\\right) \\omega \\approx \\frac{1}{3}ML^2 \\omega \\implies \\omega = \\frac{3mv_0x}{ML^2}$$\n\nFrom conservation of energy after collision:\n\n$$\\frac{1}{2} I \\omega^2 = M g h_{\\text{cm}} \\implies \\frac{1}{2} \\left(\\frac{1}{3}ML^2\\right) \\left(\\frac{3mv_0x}{ML^2}\\right)^2 = M g \\left(\\frac{L}{2}\\right) (1 - \\cos\\theta)$$\n\n$$\\frac{3}{2}\\frac{m^2 v_0^2 x^2}{ML^2} = M g \\frac{L}{2} \\left(\\frac{6mv_0^2}{MgL}\\right) = 3mv_0^2 \\implies x = \\frac{L}{2}$$", + "language": "English", + "id": "d3703f78-17ef-4ce0-9cc7-3342102231a9" + }, + { + "section": "General", + "subject": "Physics", + "topic": "Electromagnetism Set 8", + "subtopic": "Electromagnetic Induction", + "difficulty": "Hard", + "type": "MCQ_Multiple", + "question_text": "A rectangular loop of dimensions $a \\times b$ with resistance $R$ is placed near an infinitely long straight wire carrying a time-dependent current $I(t) = I_0 \\sin(\\omega t)$. The loop is at a distance $r_0$ from the wire. Which of the following statements is/are correct?", + "options": { + "A": "The maximum mutual inductance $M$ between the wire and the loop is $\\frac{\\mu_0 b}{2\\pi} \\ln\\left(1 + \\frac{a}{r_0}\\right)$.", + "B": "The amplitude of the induced current in the loop is $\\frac{\\mu_0 b I_0 \\omega}{2\\pi R} \\ln\\left(1 + \\frac{a}{r_0}\\right)$.", + "C": "The net force on the loop is zero at all times.", + "D": "The induced EMF in the loop is in phase with the current in the straight wire." + }, + "correct_answer": { + "value": [ + "A", + "B" + ] + }, + "solution_text": "Magnetic flux $\\Phi(t)$ through the loop is:\n\n$$\\Phi(t) = \\int_{r_0}^{r_0+a} \\frac{\\mu_0 I(t)}{2\\pi r} b \\, dr = \\frac{\\mu_0 b I(t)}{2\\pi} \\ln\\left(\\frac{r_0+a}{r_0}\\right)$$\n\nMutual inductance $M = \\frac{\\Phi(t)}{I(t)} = \\frac{\\mu_0 b}{2\\pi} \\ln\\left(1 + \\frac{a}{r_0}\\right).\n\nInduced EMF $\\mathcal{E}(t) = -\\frac{d\\Phi}{dt} = -M I_0 \\omega \\cos(\\omega t)$.\n\nInduced current $i(t) = \\frac{\\mathcal{E}(t)}{R} = -\\frac{M I_0 \\omega}{R} \\cos(\\omega t)$.\n\nTherefore, the amplitude of the induced current is $\\frac{\\mu_0 b I_0 \\omega}{2\\pi R} \\ln\\left(1 + \\frac{a}{r_0}\\right)$. Since EMF varies as $\\cos(\\omega t)$ and current in the wire is $\\sin(\\omega t)$, they are out of phase by $\\pi/2$. Net force is non-zero because the magnetic field varies across the loop width.", + "language": "English", + "id": "ca276921-6f5e-423f-9b4d-3f9a3f8d32f6" + }, + { + "section": "General", + "subject": "Physics", + "topic": "Modern Physics Set 8", + "subtopic": "Photoelectric Effect", + "difficulty": "Medium", + "type": "Numerical", + "question_text": "In a photoelectric effect experiment, light of wavelength $\\lambda = 300\\text{ nm}$ is incident on a metal plate. The stopping potential is found to be $V_1 = 1.85\\text{ V}$. When light of wavelength $\\lambda' = 200\\text{ nm}$ is incident on the same metal plate, the stopping potential becomes $V_2$ (in Volts). Calculate the value of $V_2$.\n\n(Take $hc = 1240\\text{ eV}\\cdot\\text{nm}$ and electron charge $e = 1.6 \\times 10^{-19}\\text{ C}$)", + "options": null, + "correct_answer": { + "value": "3.92" + }, + "solution_text": "From Einstein's Photoelectric Equation:\n\n$$eV_1 = \\frac{hc}{\\lambda} - \\Phi_0 \\implies \\Phi_0 = \\frac{1240}{300} - 1.85 = 4.13 - 1.85 = 2.28\\text{ eV}$$\n\nFor the second wavelength:\n\n$$eV_2 = \\frac{hc}{\\lambda'} - \\Phi_0 \\implies V_2 = \\frac{1240}{200} - 2.28 = 6.20 - 2.28 = 3.92\\text{ V}$$", + "language": "English", + "id": "edc92315-4004-4cbf-9a23-0bf34139bf50" + }, + { + "section": "General", + "subject": "Chemistry", + "topic": "Coordination Chemistry Set 8", + "subtopic": "Crystal Field Theory", + "difficulty": "Hard", + "type": "MCQ_Single", + "question_text": "For the octahedral complex ion $[Fe(H_2O)_6]^{3+}$, the crystal field splitting energy is denoted by $\\Delta_0$ and the pairing energy is $P$. Which of the following options correctly represents the magnetic properties and d-electron configuration of the metal ion?", + "options": { + "A": "Low spin, $t_{2g}^5 e_g^0$, paramagnetic with $\\mu_{\\text{eff}} = \\sqrt{35}\\text{ B.M.}$", + "B": "High spin, $t_{2g}^3 e_g^2$, paramagnetic with $\\mu_{\\text{eff}} = \\sqrt{35}\\text{ B.M.}$", + "C": "High spin, $t_{2g}^3 e_g^2$, paramagnetic with $\\mu_{\\text{eff}} = \\sqrt{15}\\text{ B.M.}$", + "D": "Low spin, $t_{2g}^5 e_g^0$, paramagnetic with $\\mu_{\\text{eff}} = \\sqrt{3}\\text{ B.M.}$" + }, + "correct_answer": { + "value": "B" + }, + "solution_text": "$Fe^{3+}$ has a $d^5$ electron configuration. Since $H_2O$ is a weak field ligand, $\\Delta_0 < P$, resulting in a high-spin complex.\n\nConfiguration: $t_{2g}^3 e_g^2$.\n\nNumber of unpaired electrons $n = 5$.\n\nSpin-only magnetic moment $\\mu_{\\text{eff}} = \\sqrt{n(n+2)} = \\sqrt{5(7)} = \\sqrt{35}\\text{ B.M.}$", + "language": "English", + "id": "1d98fbaa-4087-4762-a63b-f190c5f7579e" + }, + { + "section": "General", + "subject": "Chemistry", + "topic": "Physical Chemistry Set 8", + "subtopic": "Chemical Kinetics", + "difficulty": "Hard", + "type": "MCQ_Multiple", + "question_text": "Consider the reaction network showing first-order parallel decomposition of a reactant $A$:\n\n$$A \\xrightarrow{k_1} B \\quad \\text{and} \\quad A \\xrightarrow{k_2} C$$\n\nAt a temperature $T_1$, the rate constants are $k_1 = 2.0 \\times 10^{-4}\\text{ s}^{-1}$ and $k_2 = 1.0 \\times 10^{-4}\\text{ s}^{-1}$. Let the activation energies for the two paths be $E_{a1} = 40\\text{ kJ/mol}$ and $E_{a2} = 80\\text{ kJ/mol}$ respectively. Which of the following statements is/are correct?", + "options": { + "A": "The fractional yield of $B$ at temperature $T_1$ is $\\frac{2}{3}$.", + "B": "If the temperature is increased to $T_2 > T_1$, the ratio $[B]/[C]$ decreases.", + "C": "The overall rate constant for the disappearance of $A$ is $k_{\\text{eff}} = 3.0 \\times 10^{-4}\\text{ s}^{-1}$ at $T_1$.", + "D": "The effective activation energy for the disappearance of $A$ is $E_{\\text{eff}} \\approx 53.3\\text{ kJ/mol}$ at $T_1$." + }, + "correct_answer": { + "value": [ + "A", + "B", + "C", + "D" + ] + }, + "solution_text": "Let us check each option:\n\n1. Yield of $B = \\frac{k_1}{k_1+k_2} = \\frac{2.0}{2.0+1.0} = \\frac{2}{3}$. (A is correct)\n\n2. Since $E_{a2} > E_{a1}$, the rate constant $k_2$ increases more rapidly with temperature than $k_1$. Hence, the ratio $\\frac{[B]}{[C]} = \\frac{k_1}{k_2}$ decreases as temperature increases. (B is correct)\n\n3. For parallel first-order reactions, $k_{\\text{eff}} = k_1 + k_2 = 3.0 \\times 10^{-4}\\text{ s}^{-1}$. (C is correct)\n\n4. The effective activation energy is $E_{\\text{eff}} = \\frac{k_1 E_{a1} + k_2 E_{a2}}{k_1 + k_2} = \\frac{2(40) + 1(80)}{3} = 53.33\\text{ kJ/mol}$. (D is correct)", + "language": "English", + "id": "c2032eb9-c7fc-44a7-bf77-621bec626fe6" + }, + { + "section": "General", + "subject": "Chemistry", + "topic": "Physical Chemistry Set 8", + "subtopic": "Thermodynamics", + "difficulty": "Medium", + "type": "Numerical", + "question_text": "Calculate the change in entropy $\\Delta S$ (in $\\text{J}\\cdot\\text{K}^{-1}\\cdot\\text{mol}^{-1}$) when $2\\text{ moles}$ of an ideal gas ($C_{V,m} = \\frac{3}{2}R$) are heated reversibly from $300\\text{ K}$ to $600\\text{ K}$ at constant pressure.\n\n(Take $R = 8.314\\text{ J}\\cdot\\text{K}^{-1}\\cdot\\text{mol}^{-1}$ and $\\ln(2) \\approx 0.693$)", + "options": null, + "correct_answer": { + "value": "28.81" + }, + "solution_text": "For a process at constant pressure:\n\n$$C_{P,m} = C_{V,m} + R = \\frac{5}{2}R$$\n\nThe entropy change is:\n\n$$\\Delta S = n C_{P,m} \\ln\\left(\\frac{T_2}{T_1}\\right) = 2 \\times \\left(\\frac{5}{2} R\\right) \\ln\\left(\\frac{600}{300}\\right) = 5 R \\ln(2)$$\n\n$$\\Delta S = 5 \\times 8.314 \\times 0.693 \\approx 28.81\\text{ J}\\cdot\\text{K}^{-1}\\cdot\\text{mol}^{-1}$$", + "language": "English", + "id": "fdf69a10-bce9-494b-b267-fc130f84394d" + }, + { + "section": "General", + "subject": "Mathematics", + "topic": "Calculus Set 8", + "subtopic": "Definite Integration", + "difficulty": "Hard", + "type": "MCQ_Single", + "question_text": "Let $I = \\int_{0}^{\\pi} \\frac{x \\sin^2 x \\cos x}{1 + \\cos^2 x} \\, dx$. The value of $I$ is:", + "options": { + "A": "$\\frac{\\pi^2}{4}$", + "B": "$0$", + "C": "$-\\frac{\\pi}{2}$", + "D": "$-\\frac{\\pi^2}{4}$" + }, + "correct_answer": { + "value": "B" + }, + "solution_text": "Using the integration property $\\int_a^b f(x) dx = \\int_a^b f(a+b-x) dx$:\n\n$$I = \\int_{0}^{\\pi} \\frac{(\\pi - x) \\sin^2(\\pi - x) \\cos(\\pi - x)}{1 + \\cos^2(\\pi - x)} \\, dx$$\n\n$$I = \\int_{0}^{\\pi} \\frac{(\\pi - x) \\sin^2 x (-\\cos x)}{1 + \\cos^2 x} \\, dx = -\\pi \\int_0^\\pi \\frac{\\sin^2 x \\cos x}{1 + \\cos^2 x} dx + I$$\n\n$$2I = -\\pi \\int_0^\\pi \\frac{\\sin^2 x \\cos x}{1 + \\cos^2 x} dx$$\n\nLet $g(x) = \\frac{\\sin^2 x \\cos x}{1 + \\cos^2 x}$. Since $g(\\pi - x) = -g(x)$, the integral $\\int_0^\\pi g(x) dx = 0$.\n\nTherefore, $2I = 0 \\implies I = 0$.", + "language": "English", + "id": "1032bec1-3a49-44b7-82a6-c09c6c049101" + }, + { + "section": "General", + "subject": "Mathematics", + "topic": "Algebra Set 8", + "subtopic": "Matrices & Determinants", + "difficulty": "Hard", + "type": "MCQ_Multiple", + "question_text": "Let $A = \\begin{pmatrix} 1 & 0 & 0 \\\\ 2 & 1 & 0 \\\\ 3 & 2 & 1 \\end{pmatrix}$ and let $I$ be the $3 \\times 3$ identity matrix. If $B = A - I$, which of the following statements is/are correct?", + "options": { + "A": "$B^3$ is a null matrix.", + "B": "The determinant of $A^n$ is equal to $1$ for any integer $n \\ge 1$.", + "C": "The inverse of $A$ exists and is equal to $I - B + B^2$.", + "D": "$B^2 = \\begin{pmatrix} 0 & 0 & 0 \\\\ 0 & 0 & 0 \\\\ 2 & 0 & 0 \\end{pmatrix}$." + }, + "correct_answer": { + "value": [ + "A", + "B", + "C", + "D" + ] + }, + "solution_text": "We have:\n\n$$B = A - I = \\begin{pmatrix} 0 & 0 & 0 \\\\ 2 & 0 & 0 \\\\ 3 & 2 & 0 \\end{pmatrix}$$\n\nCompute $B^2$:\n\n$$B^2 = \\begin{pmatrix} 0 & 0 & 0 \\\\ 0 & 0 & 0 \\\\ 2 & 0 & 0 \\end{pmatrix}$$ (Option D is correct)\n\nCompute $B^3$:\n\n$$B^3 = B^2 B = \\begin{pmatrix} 0 & 0 & 0 \\\\ 0 & 0 & 0 \\\\ 0 & 0 & 0 \\end{pmatrix}$$ (Option A is correct)\n\nSince $B^3 = 0$, we can express $(I + B)^{-1}$ using a geometric series:\n\n$$(I + B)^{-1} = I - B + B^2 - B^3 + \\dots = I - B + B^2$$\n\nThus, $A^{-1} = I - B + B^2$. (Option C is correct)\n\nAlso, $\\det(A) = 1 \\implies \\det(A^n) = (\\det A)^n = 1$. (Option B is correct)", + "language": "English", + "id": "13a7fd63-6f95-4d24-87eb-5204b9c37e59" + }, + { + "section": "General", + "subject": "Mathematics", + "topic": "Coordinate Geometry Set 8", + "subtopic": "Parabolas & Circles", + "difficulty": "Medium", + "type": "Numerical", + "question_text": "The line $y = mx + 1$ is a common tangent to the circle $x^2 + y^2 = \\frac{1}{2}$ and the parabola $y^2 = 4x$. Find the value of $m^2$.", + "options": null, + "correct_answer": { + "value": "1" + }, + "solution_text": "For the parabola $y^2 = 4ax$ with $a=1$, the condition of tangency for $y = mx + c$ is:\n\n$$c = \\frac{a}{m} \\implies 1 = \\frac{1}{m} \\implies m = 1$$\n\nLet's check if the line $y = x + 1$ (i.e., $x - y + 1 = 0$) is also tangent to the circle $x^2 + y^2 = \\frac{1}{2}$:\n\nThe perpendicular distance from the center $(0,0)$ to the line is:\n\n$$d = \\frac{|1|}{\\sqrt{1^2 + (-1)^2}} = \\frac{1}{\\sqrt{2}}$$\n\nThis distance is exactly equal to the radius $r = \\frac{1}{\\sqrt{2}}$ of the circle. Thus, the line is a common tangent. Therefore, $m^2 = 1^2 = 1$.", + "language": "English", + "id": "2e237253-e2c7-4582-a0b7-ff34ccf3bde5" + }, + { + "section": "General", + "subject": "Physics", + "topic": "Rotational Mechanics Set 9", + "subtopic": "Angular Momentum & Collisions", + "difficulty": "Hard", + "type": "MCQ_Single", + "question_text": "A thin uniform rod of mass $M$ and length $L$ is pivoted at its upper end and hangs vertically. A small particle of mass $m$ ($m \\ll M$) moving horizontally with velocity $v_0$ strikes the rod at a distance $x$ from the pivot and sticks to it. If the rod deflects by an angle $\\theta$ such that $\\cos\\theta = 1 - \\frac{6mv_0^2}{MgL}$, the collision point $x$ is:", + "options": { + "A": "$\\frac{L}{\\sqrt{3}}$", + "B": "$\\frac{L}{2}$", + "C": "$\\frac{2L}{3}$", + "D": "$\\frac{L}{\\sqrt{2}}$" + }, + "correct_answer": { + "value": "B" + }, + "solution_text": "By conservation of angular momentum about the pivot:\n\n$$m v_0 x = I \\omega = \\left(\\frac{1}{3}ML^2 + m x^2\\right) \\omega \\approx \\frac{1}{3}ML^2 \\omega \\implies \\omega = \\frac{3mv_0x}{ML^2}$$\n\nFrom conservation of energy after collision:\n\n$$\\frac{1}{2} I \\omega^2 = M g h_{\\text{cm}} \\implies \\frac{1}{2} \\left(\\frac{1}{3}ML^2\\right) \\left(\\frac{3mv_0x}{ML^2}\\right)^2 = M g \\left(\\frac{L}{2}\\right) (1 - \\cos\\theta)$$\n\n$$\\frac{3}{2}\\frac{m^2 v_0^2 x^2}{ML^2} = M g \\frac{L}{2} \\left(\\frac{6mv_0^2}{MgL}\\right) = 3mv_0^2 \\implies x = \\frac{L}{2}$$", + "language": "English", + "id": "d12b093a-9ea9-42fe-ac34-0e415484e6e1" + }, + { + "section": "General", + "subject": "Physics", + "topic": "Electromagnetism Set 9", + "subtopic": "Electromagnetic Induction", + "difficulty": "Hard", + "type": "MCQ_Multiple", + "question_text": "A rectangular loop of dimensions $a \\times b$ with resistance $R$ is placed near an infinitely long straight wire carrying a time-dependent current $I(t) = I_0 \\sin(\\omega t)$. The loop is at a distance $r_0$ from the wire. Which of the following statements is/are correct?", + "options": { + "A": "The maximum mutual inductance $M$ between the wire and the loop is $\\frac{\\mu_0 b}{2\\pi} \\ln\\left(1 + \\frac{a}{r_0}\\right)$.", + "B": "The amplitude of the induced current in the loop is $\\frac{\\mu_0 b I_0 \\omega}{2\\pi R} \\ln\\left(1 + \\frac{a}{r_0}\\right)$.", + "C": "The net force on the loop is zero at all times.", + "D": "The induced EMF in the loop is in phase with the current in the straight wire." + }, + "correct_answer": { + "value": [ + "A", + "B" + ] + }, + "solution_text": "Magnetic flux $\\Phi(t)$ through the loop is:\n\n$$\\Phi(t) = \\int_{r_0}^{r_0+a} \\frac{\\mu_0 I(t)}{2\\pi r} b \\, dr = \\frac{\\mu_0 b I(t)}{2\\pi} \\ln\\left(\\frac{r_0+a}{r_0}\\right)$$\n\nMutual inductance $M = \\frac{\\Phi(t)}{I(t)} = \\frac{\\mu_0 b}{2\\pi} \\ln\\left(1 + \\frac{a}{r_0}\\right).\n\nInduced EMF $\\mathcal{E}(t) = -\\frac{d\\Phi}{dt} = -M I_0 \\omega \\cos(\\omega t)$.\n\nInduced current $i(t) = \\frac{\\mathcal{E}(t)}{R} = -\\frac{M I_0 \\omega}{R} \\cos(\\omega t)$.\n\nTherefore, the amplitude of the induced current is $\\frac{\\mu_0 b I_0 \\omega}{2\\pi R} \\ln\\left(1 + \\frac{a}{r_0}\\right)$. Since EMF varies as $\\cos(\\omega t)$ and current in the wire is $\\sin(\\omega t)$, they are out of phase by $\\pi/2$. Net force is non-zero because the magnetic field varies across the loop width.", + "language": "English", + "id": "9b524a6b-286c-4161-97ae-f60a9d0d8f8e" + }, + { + "section": "General", + "subject": "Physics", + "topic": "Modern Physics Set 9", + "subtopic": "Photoelectric Effect", + "difficulty": "Medium", + "type": "Numerical", + "question_text": "In a photoelectric effect experiment, light of wavelength $\\lambda = 300\\text{ nm}$ is incident on a metal plate. The stopping potential is found to be $V_1 = 1.85\\text{ V}$. When light of wavelength $\\lambda' = 200\\text{ nm}$ is incident on the same metal plate, the stopping potential becomes $V_2$ (in Volts). Calculate the value of $V_2$.\n\n(Take $hc = 1240\\text{ eV}\\cdot\\text{nm}$ and electron charge $e = 1.6 \\times 10^{-19}\\text{ C}$)", + "options": null, + "correct_answer": { + "value": "3.92" + }, + "solution_text": "From Einstein's Photoelectric Equation:\n\n$$eV_1 = \\frac{hc}{\\lambda} - \\Phi_0 \\implies \\Phi_0 = \\frac{1240}{300} - 1.85 = 4.13 - 1.85 = 2.28\\text{ eV}$$\n\nFor the second wavelength:\n\n$$eV_2 = \\frac{hc}{\\lambda'} - \\Phi_0 \\implies V_2 = \\frac{1240}{200} - 2.28 = 6.20 - 2.28 = 3.92\\text{ V}$$", + "language": "English", + "id": "6e5e8cd3-be05-4508-ab32-35a5d1d142c6" + }, + { + "section": "General", + "subject": "Chemistry", + "topic": "Coordination Chemistry Set 9", + "subtopic": "Crystal Field Theory", + "difficulty": "Hard", + "type": "MCQ_Single", + "question_text": "For the octahedral complex ion $[Fe(H_2O)_6]^{3+}$, the crystal field splitting energy is denoted by $\\Delta_0$ and the pairing energy is $P$. Which of the following options correctly represents the magnetic properties and d-electron configuration of the metal ion?", + "options": { + "A": "Low spin, $t_{2g}^5 e_g^0$, paramagnetic with $\\mu_{\\text{eff}} = \\sqrt{35}\\text{ B.M.}$", + "B": "High spin, $t_{2g}^3 e_g^2$, paramagnetic with $\\mu_{\\text{eff}} = \\sqrt{35}\\text{ B.M.}$", + "C": "High spin, $t_{2g}^3 e_g^2$, paramagnetic with $\\mu_{\\text{eff}} = \\sqrt{15}\\text{ B.M.}$", + "D": "Low spin, $t_{2g}^5 e_g^0$, paramagnetic with $\\mu_{\\text{eff}} = \\sqrt{3}\\text{ B.M.}$" + }, + "correct_answer": { + "value": "B" + }, + "solution_text": "$Fe^{3+}$ has a $d^5$ electron configuration. Since $H_2O$ is a weak field ligand, $\\Delta_0 < P$, resulting in a high-spin complex.\n\nConfiguration: $t_{2g}^3 e_g^2$.\n\nNumber of unpaired electrons $n = 5$.\n\nSpin-only magnetic moment $\\mu_{\\text{eff}} = \\sqrt{n(n+2)} = \\sqrt{5(7)} = \\sqrt{35}\\text{ B.M.}$", + "language": "English", + "id": "21cc1080-caad-41b3-8131-53407d368738" + }, + { + "section": "General", + "subject": "Chemistry", + "topic": "Physical Chemistry Set 9", + "subtopic": "Chemical Kinetics", + "difficulty": "Hard", + "type": "MCQ_Multiple", + "question_text": "Consider the reaction network showing first-order parallel decomposition of a reactant $A$:\n\n$$A \\xrightarrow{k_1} B \\quad \\text{and} \\quad A \\xrightarrow{k_2} C$$\n\nAt a temperature $T_1$, the rate constants are $k_1 = 2.0 \\times 10^{-4}\\text{ s}^{-1}$ and $k_2 = 1.0 \\times 10^{-4}\\text{ s}^{-1}$. Let the activation energies for the two paths be $E_{a1} = 40\\text{ kJ/mol}$ and $E_{a2} = 80\\text{ kJ/mol}$ respectively. Which of the following statements is/are correct?", + "options": { + "A": "The fractional yield of $B$ at temperature $T_1$ is $\\frac{2}{3}$.", + "B": "If the temperature is increased to $T_2 > T_1$, the ratio $[B]/[C]$ decreases.", + "C": "The overall rate constant for the disappearance of $A$ is $k_{\\text{eff}} = 3.0 \\times 10^{-4}\\text{ s}^{-1}$ at $T_1$.", + "D": "The effective activation energy for the disappearance of $A$ is $E_{\\text{eff}} \\approx 53.3\\text{ kJ/mol}$ at $T_1$." + }, + "correct_answer": { + "value": [ + "A", + "B", + "C", + "D" + ] + }, + "solution_text": "Let us check each option:\n\n1. Yield of $B = \\frac{k_1}{k_1+k_2} = \\frac{2.0}{2.0+1.0} = \\frac{2}{3}$. (A is correct)\n\n2. Since $E_{a2} > E_{a1}$, the rate constant $k_2$ increases more rapidly with temperature than $k_1$. Hence, the ratio $\\frac{[B]}{[C]} = \\frac{k_1}{k_2}$ decreases as temperature increases. (B is correct)\n\n3. For parallel first-order reactions, $k_{\\text{eff}} = k_1 + k_2 = 3.0 \\times 10^{-4}\\text{ s}^{-1}$. (C is correct)\n\n4. The effective activation energy is $E_{\\text{eff}} = \\frac{k_1 E_{a1} + k_2 E_{a2}}{k_1 + k_2} = \\frac{2(40) + 1(80)}{3} = 53.33\\text{ kJ/mol}$. (D is correct)", + "language": "English", + "id": "96e9ae8a-6f55-4154-8d84-2f739fda1ce9" + }, + { + "section": "General", + "subject": "Chemistry", + "topic": "Physical Chemistry Set 9", + "subtopic": "Thermodynamics", + "difficulty": "Medium", + "type": "Numerical", + "question_text": "Calculate the change in entropy $\\Delta S$ (in $\\text{J}\\cdot\\text{K}^{-1}\\cdot\\text{mol}^{-1}$) when $2\\text{ moles}$ of an ideal gas ($C_{V,m} = \\frac{3}{2}R$) are heated reversibly from $300\\text{ K}$ to $600\\text{ K}$ at constant pressure.\n\n(Take $R = 8.314\\text{ J}\\cdot\\text{K}^{-1}\\cdot\\text{mol}^{-1}$ and $\\ln(2) \\approx 0.693$)", + "options": null, + "correct_answer": { + "value": "28.81" + }, + "solution_text": "For a process at constant pressure:\n\n$$C_{P,m} = C_{V,m} + R = \\frac{5}{2}R$$\n\nThe entropy change is:\n\n$$\\Delta S = n C_{P,m} \\ln\\left(\\frac{T_2}{T_1}\\right) = 2 \\times \\left(\\frac{5}{2} R\\right) \\ln\\left(\\frac{600}{300}\\right) = 5 R \\ln(2)$$\n\n$$\\Delta S = 5 \\times 8.314 \\times 0.693 \\approx 28.81\\text{ J}\\cdot\\text{K}^{-1}\\cdot\\text{mol}^{-1}$$", + "language": "English", + "id": "fe355e15-7bc6-479f-b6cf-a0c857bf8f77" + }, + { + "section": "General", + "subject": "Mathematics", + "topic": "Calculus Set 9", + "subtopic": "Definite Integration", + "difficulty": "Hard", + "type": "MCQ_Single", + "question_text": "Let $I = \\int_{0}^{\\pi} \\frac{x \\sin^2 x \\cos x}{1 + \\cos^2 x} \\, dx$. The value of $I$ is:", + "options": { + "A": "$\\frac{\\pi^2}{4}$", + "B": "$0$", + "C": "$-\\frac{\\pi}{2}$", + "D": "$-\\frac{\\pi^2}{4}$" + }, + "correct_answer": { + "value": "B" + }, + "solution_text": "Using the integration property $\\int_a^b f(x) dx = \\int_a^b f(a+b-x) dx$:\n\n$$I = \\int_{0}^{\\pi} \\frac{(\\pi - x) \\sin^2(\\pi - x) \\cos(\\pi - x)}{1 + \\cos^2(\\pi - x)} \\, dx$$\n\n$$I = \\int_{0}^{\\pi} \\frac{(\\pi - x) \\sin^2 x (-\\cos x)}{1 + \\cos^2 x} \\, dx = -\\pi \\int_0^\\pi \\frac{\\sin^2 x \\cos x}{1 + \\cos^2 x} dx + I$$\n\n$$2I = -\\pi \\int_0^\\pi \\frac{\\sin^2 x \\cos x}{1 + \\cos^2 x} dx$$\n\nLet $g(x) = \\frac{\\sin^2 x \\cos x}{1 + \\cos^2 x}$. Since $g(\\pi - x) = -g(x)$, the integral $\\int_0^\\pi g(x) dx = 0$.\n\nTherefore, $2I = 0 \\implies I = 0$.", + "language": "English", + "id": "511ba3f4-50c7-4014-aac9-274f6041b60b" + }, + { + "section": "General", + "subject": "Mathematics", + "topic": "Algebra Set 9", + "subtopic": "Matrices & Determinants", + "difficulty": "Hard", + "type": "MCQ_Multiple", + "question_text": "Let $A = \\begin{pmatrix} 1 & 0 & 0 \\\\ 2 & 1 & 0 \\\\ 3 & 2 & 1 \\end{pmatrix}$ and let $I$ be the $3 \\times 3$ identity matrix. If $B = A - I$, which of the following statements is/are correct?", + "options": { + "A": "$B^3$ is a null matrix.", + "B": "The determinant of $A^n$ is equal to $1$ for any integer $n \\ge 1$.", + "C": "The inverse of $A$ exists and is equal to $I - B + B^2$.", + "D": "$B^2 = \\begin{pmatrix} 0 & 0 & 0 \\\\ 0 & 0 & 0 \\\\ 2 & 0 & 0 \\end{pmatrix}$." + }, + "correct_answer": { + "value": [ + "A", + "B", + "C", + "D" + ] + }, + "solution_text": "We have:\n\n$$B = A - I = \\begin{pmatrix} 0 & 0 & 0 \\\\ 2 & 0 & 0 \\\\ 3 & 2 & 0 \\end{pmatrix}$$\n\nCompute $B^2$:\n\n$$B^2 = \\begin{pmatrix} 0 & 0 & 0 \\\\ 0 & 0 & 0 \\\\ 2 & 0 & 0 \\end{pmatrix}$$ (Option D is correct)\n\nCompute $B^3$:\n\n$$B^3 = B^2 B = \\begin{pmatrix} 0 & 0 & 0 \\\\ 0 & 0 & 0 \\\\ 0 & 0 & 0 \\end{pmatrix}$$ (Option A is correct)\n\nSince $B^3 = 0$, we can express $(I + B)^{-1}$ using a geometric series:\n\n$$(I + B)^{-1} = I - B + B^2 - B^3 + \\dots = I - B + B^2$$\n\nThus, $A^{-1} = I - B + B^2$. (Option C is correct)\n\nAlso, $\\det(A) = 1 \\implies \\det(A^n) = (\\det A)^n = 1$. (Option B is correct)", + "language": "English", + "id": "ec110ef2-d5f2-43dc-b214-be06221626a9" + }, + { + "section": "General", + "subject": "Mathematics", + "topic": "Coordinate Geometry Set 9", + "subtopic": "Parabolas & Circles", + "difficulty": "Medium", + "type": "Numerical", + "question_text": "The line $y = mx + 1$ is a common tangent to the circle $x^2 + y^2 = \\frac{1}{2}$ and the parabola $y^2 = 4x$. Find the value of $m^2$.", + "options": null, + "correct_answer": { + "value": "1" + }, + "solution_text": "For the parabola $y^2 = 4ax$ with $a=1$, the condition of tangency for $y = mx + c$ is:\n\n$$c = \\frac{a}{m} \\implies 1 = \\frac{1}{m} \\implies m = 1$$\n\nLet's check if the line $y = x + 1$ (i.e., $x - y + 1 = 0$) is also tangent to the circle $x^2 + y^2 = \\frac{1}{2}$:\n\nThe perpendicular distance from the center $(0,0)$ to the line is:\n\n$$d = \\frac{|1|}{\\sqrt{1^2 + (-1)^2}} = \\frac{1}{\\sqrt{2}}$$\n\nThis distance is exactly equal to the radius $r = \\frac{1}{\\sqrt{2}}$ of the circle. Thus, the line is a common tangent. Therefore, $m^2 = 1^2 = 1$.", + "language": "English", + "id": "f8f9850c-ed28-416c-864b-f4aa851ee324" + }, + { + "section": "General", + "subject": "Physics", + "topic": "Rotational Mechanics Set 10", + "subtopic": "Angular Momentum & Collisions", + "difficulty": "Hard", + "type": "MCQ_Single", + "question_text": "A thin uniform rod of mass $M$ and length $L$ is pivoted at its upper end and hangs vertically. A small particle of mass $m$ ($m \\ll M$) moving horizontally with velocity $v_0$ strikes the rod at a distance $x$ from the pivot and sticks to it. If the rod deflects by an angle $\\theta$ such that $\\cos\\theta = 1 - \\frac{6mv_0^2}{MgL}$, the collision point $x$ is:", + "options": { + "A": "$\\frac{L}{\\sqrt{3}}$", + "B": "$\\frac{L}{2}$", + "C": "$\\frac{2L}{3}$", + "D": "$\\frac{L}{\\sqrt{2}}$" + }, + "correct_answer": { + "value": "B" + }, + "solution_text": "By conservation of angular momentum about the pivot:\n\n$$m v_0 x = I \\omega = \\left(\\frac{1}{3}ML^2 + m x^2\\right) \\omega \\approx \\frac{1}{3}ML^2 \\omega \\implies \\omega = \\frac{3mv_0x}{ML^2}$$\n\nFrom conservation of energy after collision:\n\n$$\\frac{1}{2} I \\omega^2 = M g h_{\\text{cm}} \\implies \\frac{1}{2} \\left(\\frac{1}{3}ML^2\\right) \\left(\\frac{3mv_0x}{ML^2}\\right)^2 = M g \\left(\\frac{L}{2}\\right) (1 - \\cos\\theta)$$\n\n$$\\frac{3}{2}\\frac{m^2 v_0^2 x^2}{ML^2} = M g \\frac{L}{2} \\left(\\frac{6mv_0^2}{MgL}\\right) = 3mv_0^2 \\implies x = \\frac{L}{2}$$", + "language": "English", + "id": "533787b4-ff1d-4b43-b4b8-7c95a2e1f07a" + }, + { + "section": "General", + "subject": "Physics", + "topic": "Electromagnetism Set 10", + "subtopic": "Electromagnetic Induction", + "difficulty": "Hard", + "type": "MCQ_Multiple", + "question_text": "A rectangular loop of dimensions $a \\times b$ with resistance $R$ is placed near an infinitely long straight wire carrying a time-dependent current $I(t) = I_0 \\sin(\\omega t)$. The loop is at a distance $r_0$ from the wire. Which of the following statements is/are correct?", + "options": { + "A": "The maximum mutual inductance $M$ between the wire and the loop is $\\frac{\\mu_0 b}{2\\pi} \\ln\\left(1 + \\frac{a}{r_0}\\right)$.", + "B": "The amplitude of the induced current in the loop is $\\frac{\\mu_0 b I_0 \\omega}{2\\pi R} \\ln\\left(1 + \\frac{a}{r_0}\\right)$.", + "C": "The net force on the loop is zero at all times.", + "D": "The induced EMF in the loop is in phase with the current in the straight wire." + }, + "correct_answer": { + "value": [ + "A", + "B" + ] + }, + "solution_text": "Magnetic flux $\\Phi(t)$ through the loop is:\n\n$$\\Phi(t) = \\int_{r_0}^{r_0+a} \\frac{\\mu_0 I(t)}{2\\pi r} b \\, dr = \\frac{\\mu_0 b I(t)}{2\\pi} \\ln\\left(\\frac{r_0+a}{r_0}\\right)$$\n\nMutual inductance $M = \\frac{\\Phi(t)}{I(t)} = \\frac{\\mu_0 b}{2\\pi} \\ln\\left(1 + \\frac{a}{r_0}\\right).\n\nInduced EMF $\\mathcal{E}(t) = -\\frac{d\\Phi}{dt} = -M I_0 \\omega \\cos(\\omega t)$.\n\nInduced current $i(t) = \\frac{\\mathcal{E}(t)}{R} = -\\frac{M I_0 \\omega}{R} \\cos(\\omega t)$.\n\nTherefore, the amplitude of the induced current is $\\frac{\\mu_0 b I_0 \\omega}{2\\pi R} \\ln\\left(1 + \\frac{a}{r_0}\\right)$. Since EMF varies as $\\cos(\\omega t)$ and current in the wire is $\\sin(\\omega t)$, they are out of phase by $\\pi/2$. Net force is non-zero because the magnetic field varies across the loop width.", + "language": "English", + "id": "195dcb08-b6e6-4870-896f-ba7f17085117" + }, + { + "section": "General", + "subject": "Physics", + "topic": "Modern Physics Set 10", + "subtopic": "Photoelectric Effect", + "difficulty": "Medium", + "type": "Numerical", + "question_text": "In a photoelectric effect experiment, light of wavelength $\\lambda = 300\\text{ nm}$ is incident on a metal plate. The stopping potential is found to be $V_1 = 1.85\\text{ V}$. When light of wavelength $\\lambda' = 200\\text{ nm}$ is incident on the same metal plate, the stopping potential becomes $V_2$ (in Volts). Calculate the value of $V_2$.\n\n(Take $hc = 1240\\text{ eV}\\cdot\\text{nm}$ and electron charge $e = 1.6 \\times 10^{-19}\\text{ C}$)", + "options": null, + "correct_answer": { + "value": "3.92" + }, + "solution_text": "From Einstein's Photoelectric Equation:\n\n$$eV_1 = \\frac{hc}{\\lambda} - \\Phi_0 \\implies \\Phi_0 = \\frac{1240}{300} - 1.85 = 4.13 - 1.85 = 2.28\\text{ eV}$$\n\nFor the second wavelength:\n\n$$eV_2 = \\frac{hc}{\\lambda'} - \\Phi_0 \\implies V_2 = \\frac{1240}{200} - 2.28 = 6.20 - 2.28 = 3.92\\text{ V}$$", + "language": "English", + "id": "722da39b-72fe-490d-8c69-640466e9377d" + }, + { + "section": "General", + "subject": "Chemistry", + "topic": "Coordination Chemistry Set 10", + "subtopic": "Crystal Field Theory", + "difficulty": "Hard", + "type": "MCQ_Single", + "question_text": "For the octahedral complex ion $[Fe(H_2O)_6]^{3+}$, the crystal field splitting energy is denoted by $\\Delta_0$ and the pairing energy is $P$. Which of the following options correctly represents the magnetic properties and d-electron configuration of the metal ion?", + "options": { + "A": "Low spin, $t_{2g}^5 e_g^0$, paramagnetic with $\\mu_{\\text{eff}} = \\sqrt{35}\\text{ B.M.}$", + "B": "High spin, $t_{2g}^3 e_g^2$, paramagnetic with $\\mu_{\\text{eff}} = \\sqrt{35}\\text{ B.M.}$", + "C": "High spin, $t_{2g}^3 e_g^2$, paramagnetic with $\\mu_{\\text{eff}} = \\sqrt{15}\\text{ B.M.}$", + "D": "Low spin, $t_{2g}^5 e_g^0$, paramagnetic with $\\mu_{\\text{eff}} = \\sqrt{3}\\text{ B.M.}$" + }, + "correct_answer": { + "value": "B" + }, + "solution_text": "$Fe^{3+}$ has a $d^5$ electron configuration. Since $H_2O$ is a weak field ligand, $\\Delta_0 < P$, resulting in a high-spin complex.\n\nConfiguration: $t_{2g}^3 e_g^2$.\n\nNumber of unpaired electrons $n = 5$.\n\nSpin-only magnetic moment $\\mu_{\\text{eff}} = \\sqrt{n(n+2)} = \\sqrt{5(7)} = \\sqrt{35}\\text{ B.M.}$", + "language": "English", + "id": "876ec1a7-1b2b-4c89-b3bb-ac21ed558989" + }, + { + "section": "General", + "subject": "Chemistry", + "topic": "Physical Chemistry Set 10", + "subtopic": "Chemical Kinetics", + "difficulty": "Hard", + "type": "MCQ_Multiple", + "question_text": "Consider the reaction network showing first-order parallel decomposition of a reactant $A$:\n\n$$A \\xrightarrow{k_1} B \\quad \\text{and} \\quad A \\xrightarrow{k_2} C$$\n\nAt a temperature $T_1$, the rate constants are $k_1 = 2.0 \\times 10^{-4}\\text{ s}^{-1}$ and $k_2 = 1.0 \\times 10^{-4}\\text{ s}^{-1}$. Let the activation energies for the two paths be $E_{a1} = 40\\text{ kJ/mol}$ and $E_{a2} = 80\\text{ kJ/mol}$ respectively. Which of the following statements is/are correct?", + "options": { + "A": "The fractional yield of $B$ at temperature $T_1$ is $\\frac{2}{3}$.", + "B": "If the temperature is increased to $T_2 > T_1$, the ratio $[B]/[C]$ decreases.", + "C": "The overall rate constant for the disappearance of $A$ is $k_{\\text{eff}} = 3.0 \\times 10^{-4}\\text{ s}^{-1}$ at $T_1$.", + "D": "The effective activation energy for the disappearance of $A$ is $E_{\\text{eff}} \\approx 53.3\\text{ kJ/mol}$ at $T_1$." + }, + "correct_answer": { + "value": [ + "A", + "B", + "C", + "D" + ] + }, + "solution_text": "Let us check each option:\n\n1. Yield of $B = \\frac{k_1}{k_1+k_2} = \\frac{2.0}{2.0+1.0} = \\frac{2}{3}$. (A is correct)\n\n2. Since $E_{a2} > E_{a1}$, the rate constant $k_2$ increases more rapidly with temperature than $k_1$. Hence, the ratio $\\frac{[B]}{[C]} = \\frac{k_1}{k_2}$ decreases as temperature increases. (B is correct)\n\n3. For parallel first-order reactions, $k_{\\text{eff}} = k_1 + k_2 = 3.0 \\times 10^{-4}\\text{ s}^{-1}$. (C is correct)\n\n4. The effective activation energy is $E_{\\text{eff}} = \\frac{k_1 E_{a1} + k_2 E_{a2}}{k_1 + k_2} = \\frac{2(40) + 1(80)}{3} = 53.33\\text{ kJ/mol}$. (D is correct)", + "language": "English", + "id": "a28e6819-a227-4c8a-99b5-103defe6af3c" + }, + { + "section": "General", + "subject": "Chemistry", + "topic": "Physical Chemistry Set 10", + "subtopic": "Thermodynamics", + "difficulty": "Medium", + "type": "Numerical", + "question_text": "Calculate the change in entropy $\\Delta S$ (in $\\text{J}\\cdot\\text{K}^{-1}\\cdot\\text{mol}^{-1}$) when $2\\text{ moles}$ of an ideal gas ($C_{V,m} = \\frac{3}{2}R$) are heated reversibly from $300\\text{ K}$ to $600\\text{ K}$ at constant pressure.\n\n(Take $R = 8.314\\text{ J}\\cdot\\text{K}^{-1}\\cdot\\text{mol}^{-1}$ and $\\ln(2) \\approx 0.693$)", + "options": null, + "correct_answer": { + "value": "28.81" + }, + "solution_text": "For a process at constant pressure:\n\n$$C_{P,m} = C_{V,m} + R = \\frac{5}{2}R$$\n\nThe entropy change is:\n\n$$\\Delta S = n C_{P,m} \\ln\\left(\\frac{T_2}{T_1}\\right) = 2 \\times \\left(\\frac{5}{2} R\\right) \\ln\\left(\\frac{600}{300}\\right) = 5 R \\ln(2)$$\n\n$$\\Delta S = 5 \\times 8.314 \\times 0.693 \\approx 28.81\\text{ J}\\cdot\\text{K}^{-1}\\cdot\\text{mol}^{-1}$$", + "language": "English", + "id": "a851cb6a-3e3e-4e8e-8386-5bc9312dfa77" + }, + { + "section": "General", + "subject": "Mathematics", + "topic": "Calculus Set 10", + "subtopic": "Definite Integration", + "difficulty": "Hard", + "type": "MCQ_Single", + "question_text": "Let $I = \\int_{0}^{\\pi} \\frac{x \\sin^2 x \\cos x}{1 + \\cos^2 x} \\, dx$. The value of $I$ is:", + "options": { + "A": "$\\frac{\\pi^2}{4}$", + "B": "$0$", + "C": "$-\\frac{\\pi}{2}$", + "D": "$-\\frac{\\pi^2}{4}$" + }, + "correct_answer": { + "value": "B" + }, + "solution_text": "Using the integration property $\\int_a^b f(x) dx = \\int_a^b f(a+b-x) dx$:\n\n$$I = \\int_{0}^{\\pi} \\frac{(\\pi - x) \\sin^2(\\pi - x) \\cos(\\pi - x)}{1 + \\cos^2(\\pi - x)} \\, dx$$\n\n$$I = \\int_{0}^{\\pi} \\frac{(\\pi - x) \\sin^2 x (-\\cos x)}{1 + \\cos^2 x} \\, dx = -\\pi \\int_0^\\pi \\frac{\\sin^2 x \\cos x}{1 + \\cos^2 x} dx + I$$\n\n$$2I = -\\pi \\int_0^\\pi \\frac{\\sin^2 x \\cos x}{1 + \\cos^2 x} dx$$\n\nLet $g(x) = \\frac{\\sin^2 x \\cos x}{1 + \\cos^2 x}$. Since $g(\\pi - x) = -g(x)$, the integral $\\int_0^\\pi g(x) dx = 0$.\n\nTherefore, $2I = 0 \\implies I = 0$.", + "language": "English", + "id": "9c777bd5-c406-4f8b-8cab-56f9bba0eee7" + }, + { + "section": "General", + "subject": "Mathematics", + "topic": "Algebra Set 10", + "subtopic": "Matrices & Determinants", + "difficulty": "Hard", + "type": "MCQ_Multiple", + "question_text": "Let $A = \\begin{pmatrix} 1 & 0 & 0 \\\\ 2 & 1 & 0 \\\\ 3 & 2 & 1 \\end{pmatrix}$ and let $I$ be the $3 \\times 3$ identity matrix. If $B = A - I$, which of the following statements is/are correct?", + "options": { + "A": "$B^3$ is a null matrix.", + "B": "The determinant of $A^n$ is equal to $1$ for any integer $n \\ge 1$.", + "C": "The inverse of $A$ exists and is equal to $I - B + B^2$.", + "D": "$B^2 = \\begin{pmatrix} 0 & 0 & 0 \\\\ 0 & 0 & 0 \\\\ 2 & 0 & 0 \\end{pmatrix}$." + }, + "correct_answer": { + "value": [ + "A", + "B", + "C", + "D" + ] + }, + "solution_text": "We have:\n\n$$B = A - I = \\begin{pmatrix} 0 & 0 & 0 \\\\ 2 & 0 & 0 \\\\ 3 & 2 & 0 \\end{pmatrix}$$\n\nCompute $B^2$:\n\n$$B^2 = \\begin{pmatrix} 0 & 0 & 0 \\\\ 0 & 0 & 0 \\\\ 2 & 0 & 0 \\end{pmatrix}$$ (Option D is correct)\n\nCompute $B^3$:\n\n$$B^3 = B^2 B = \\begin{pmatrix} 0 & 0 & 0 \\\\ 0 & 0 & 0 \\\\ 0 & 0 & 0 \\end{pmatrix}$$ (Option A is correct)\n\nSince $B^3 = 0$, we can express $(I + B)^{-1}$ using a geometric series:\n\n$$(I + B)^{-1} = I - B + B^2 - B^3 + \\dots = I - B + B^2$$\n\nThus, $A^{-1} = I - B + B^2$. (Option C is correct)\n\nAlso, $\\det(A) = 1 \\implies \\det(A^n) = (\\det A)^n = 1$. (Option B is correct)", + "language": "English", + "id": "f3ce0149-90d3-4536-9427-81d3c9a22dec" + }, + { + "section": "General", + "subject": "Mathematics", + "topic": "Coordinate Geometry Set 10", + "subtopic": "Parabolas & Circles", + "difficulty": "Medium", + "type": "Numerical", + "question_text": "The line $y = mx + 1$ is a common tangent to the circle $x^2 + y^2 = \\frac{1}{2}$ and the parabola $y^2 = 4x$. Find the value of $m^2$.", + "options": null, + "correct_answer": { + "value": "1" + }, + "solution_text": "For the parabola $y^2 = 4ax$ with $a=1$, the condition of tangency for $y = mx + c$ is:\n\n$$c = \\frac{a}{m} \\implies 1 = \\frac{1}{m} \\implies m = 1$$\n\nLet's check if the line $y = x + 1$ (i.e., $x - y + 1 = 0$) is also tangent to the circle $x^2 + y^2 = \\frac{1}{2}$:\n\nThe perpendicular distance from the center $(0,0)$ to the line is:\n\n$$d = \\frac{|1|}{\\sqrt{1^2 + (-1)^2}} = \\frac{1}{\\sqrt{2}}$$\n\nThis distance is exactly equal to the radius $r = \\frac{1}{\\sqrt{2}}$ of the circle. Thus, the line is a common tangent. Therefore, $m^2 = 1^2 = 1$.", + "language": "English", + "id": "9c7898d1-ba7b-41d3-ba4f-4603394dd3f2" + } +] \ No newline at end of file diff --git a/src/app/admin/tests/page.tsx b/src/app/admin/tests/page.tsx index b1e6b1b..221e622 100644 --- a/src/app/admin/tests/page.tsx +++ b/src/app/admin/tests/page.tsx @@ -6,24 +6,51 @@ import { Button } from "@/components/ui/button" import { Badge } from "@/components/ui/badge" import { Switch } from "@/components/ui/switch" import { Input } from "@/components/ui/input" -import { Loader2, Upload, FileText, Clock, CalendarIcon } from "lucide-react" +import { Loader2, Upload, FileText, Clock, CalendarIcon, CheckCircle2, AlertCircle, X, Trash2 } from "lucide-react" import DatePicker from "react-datepicker" import "react-datepicker/dist/react-datepicker.css" +import { z } from "zod" import { useActiveUser } from "@/hooks/useActiveUser" +// Strict Zod schema for client-side validation +const QuestionSchema = z.object({ + id: z.string().optional(), + section: z.string().default("General"), + subject: z.string(), + topic: z.string(), + subtopic: z.string(), + difficulty: z.string(), + type: z.enum(["MCQ_Single", "MCQ_Multiple", "Numerical"]), + question_text: z.string(), + options: z.record(z.string(), z.any()).nullable().optional(), + correct_answer: z.object({ + value: z.any() + }), + solution_text: z.string(), + language: z.enum(["English", "Hindi", "Bilingual"]) +}).passthrough(); + +const TestSchema = z.array(QuestionSchema); + export default function AdminTestsPage() { const [tests, setTests] = useState([]) const [loading, setLoading] = useState(true) const [uploading, setUploading] = useState(false) const { userId } = useActiveUser() + // Upload Form State const [title, setTitle] = useState("") const [examPattern, setExamPattern] = useState("JEE_Main") const [timeLimit, setTimeLimit] = useState("180") const [scheduledDate, setScheduledDate] = useState(null) const [endDate, setEndDate] = useState(null) - const [questionsJson, setQuestionsJson] = useState("") + + // Drag and Drop State + const [isDragging, setIsDragging] = useState(false) + const [validQuestions, setValidQuestions] = useState[] | null>(null) + const [validationErrors, setValidationErrors] = useState<{ path: string, message: string }[]>([]) + const fileInputRef = useRef(null) const fetchTests = async () => { try { @@ -40,6 +67,7 @@ export default function AdminTestsPage() { } useEffect(() => { + // eslint-disable-next-line react-hooks/set-state-in-effect fetchTests() }, []) @@ -56,28 +84,106 @@ export default function AdminTestsPage() { } } + const handleDelete = async (testId: string) => { + if (!confirm("Are you sure you want to delete this test and all its questions? This action cannot be undone.")) return + try { + const res = await fetch(`/api/admin/tests/${testId}`, { + method: "DELETE" + }) + if (res.ok) { + alert("Test deleted successfully!") + fetchTests() + } else { + alert("Failed to delete test.") + } + } catch (e) { + console.error(e) + alert("Error deleting test") + } + } + + const handleDragOver = (e: React.DragEvent) => { + e.preventDefault() + setIsDragging(true) + } + + const handleDragLeave = (e: React.DragEvent) => { + e.preventDefault() + setIsDragging(false) + } + + const handleDrop = (e: React.DragEvent) => { + e.preventDefault() + setIsDragging(false) + + if (e.dataTransfer.files && e.dataTransfer.files.length > 0) { + processFile(e.dataTransfer.files[0]) + } + } + + const handleFileChange = (e: React.ChangeEvent) => { + if (e.target.files && e.target.files.length > 0) { + processFile(e.target.files[0]) + } + } + + const processFile = (file: File) => { + if (!file.name.endsWith('.json')) { + setValidationErrors([{ path: "File", message: "Only .json files are allowed." }]) + setValidQuestions(null) + return + } + + const reader = new FileReader() + reader.onload = (e) => { + try { + const json = JSON.parse(e.target?.result as string) + + // Zod validation + const result = TestSchema.safeParse(json) + if (result.success) { + setValidQuestions(result.data) + setValidationErrors([]) + } else { + setValidQuestions(null) + const errors = (result as any).error.errors.map((err: any) => ({ + path: err.path.join('.'), + message: err.message + })) + setValidationErrors(errors) + } + } catch { + setValidQuestions(null) + setValidationErrors([{ path: "JSON Parsing", message: "Invalid JSON format. Check for syntax errors." }]) + } + } + reader.readAsText(file) + } + + const removeFile = () => { + setValidQuestions(null) + setValidationErrors([]) + if (fileInputRef.current) { + fileInputRef.current.value = "" + } + } + const handleUpload = async (e: React.FormEvent) => { e.preventDefault() - if (!questionsJson) return + if (!validQuestions || validQuestions.length === 0) { + alert("Please upload a valid JSON file with questions first.") + return + } if (!userId) return setUploading(true) - let parsedQuestions = []; - try { - parsedQuestions = JSON.parse(questionsJson); - } catch(err) { - alert("Invalid JSON format. Please paste a valid JSON array."); - setUploading(false); - return; - } - const payload = { title, exam_pattern: examPattern, time_limit: parseInt(timeLimit), admin_id: userId, - questions_json: parsedQuestions, + questions_json: validQuestions, scheduled_date: scheduledDate ? scheduledDate.toISOString() : null, end_date: endDate ? endDate.toISOString() : null, }; @@ -92,10 +198,11 @@ export default function AdminTestsPage() { if (res.ok) { alert("Test created successfully!") setTitle("") - setQuestionsJson("") + removeFile() fetchTests() } else { - alert("Creation failed") + const err = await res.json() + alert(`Creation failed: ${JSON.stringify(err)}`) } } catch (error) { console.error(error) @@ -115,13 +222,13 @@ export default function AdminTestsPage() {
{/* Upload Form */} - + Create Manual Test - Paste the structured JSON block of questions in LaTeX format to create a deterministic test. + Upload a structured JSON array of questions in LaTeX format to create a deterministic test.
@@ -150,7 +257,7 @@ export default function AdminTestsPage() {
- + setScheduledDate(date)} @@ -160,11 +267,11 @@ export default function AdminTestsPage() { timeCaption="time" dateFormat="MMMM d, yyyy h:mm aa" className="flex h-10 w-full rounded-md border border-input bg-background px-3 py-2 text-sm ring-offset-background placeholder:text-muted-foreground focus:outline-none focus:ring-2 focus:ring-ring focus:ring-offset-2" - placeholderText="Select start date & time" + placeholderText="Optional" />
- + setEndDate(date)} @@ -174,24 +281,66 @@ export default function AdminTestsPage() { timeCaption="time" dateFormat="MMMM d, yyyy h:mm aa" className="flex h-10 w-full rounded-md border border-input bg-background px-3 py-2 text-sm ring-offset-background placeholder:text-muted-foreground focus:outline-none focus:ring-2 focus:ring-ring focus:ring-offset-2" - placeholderText="Select end date & time" + placeholderText="Optional" />
-
- -