1178 lines
68 KiB
JSON
1178 lines
68 KiB
JSON
[
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{
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"id": "badc82fe-f6be-4340-9164-cc25529a9bed",
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"section": "General",
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"subject": "Physics",
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"topic": "Kinematics",
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"subtopic": "Projectile motion",
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"difficulty": "Easy",
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"type": "MCQ_Single",
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"language": "English",
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"question_text": "A ball is projected from level ground with a speed of $20\\ \\mathrm{m\\,s^{-1}}$ at an angle of $45^\\circ$ above the horizontal. Taking $g = 10\\ \\mathrm{m\\,s^{-2}}$, the time of flight of the ball is:",
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"options": {
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"A": "$2.0\\ \\mathrm{s}$",
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"B": "$2\\sqrt{2}\\ \\mathrm{s}$",
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"C": "$2\\sqrt{3}\\ \\mathrm{s}$",
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"D": "$4.0\\ \\mathrm{s}$"
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},
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"correct_answer": {
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"value": "B"
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},
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"solution_text": "Only the vertical component governs the time of flight.\nStep 1: $u_y = u\\sin\\theta = 20\\sin 45^\\circ = 20 \\times \\dfrac{1}{\\sqrt{2}} = 10\\sqrt{2}\\ \\mathrm{m\\,s^{-1}}$.\nStep 2: For symmetric flight returning to the same level, $T = \\dfrac{2u_y}{g} = \\dfrac{2 \\times 10\\sqrt{2}}{10} = 2\\sqrt{2}\\ \\mathrm{s} \\approx 2.83\\ \\mathrm{s}$.\nStep 3 (check): Range $= u_x T = (10\\sqrt{2})(2\\sqrt{2}) = 40\\ \\mathrm{m} = \\dfrac{u^2 \\sin 90^\\circ}{g} = \\dfrac{400}{10}$, consistent.\nHence, the correct option is (B)."
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},
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{
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"id": "03b40792-a94a-403a-84da-058cf9eff7d8",
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"section": "General",
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"subject": "Physics",
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"topic": "Work, Energy and Power",
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"subtopic": "Vertical circular motion",
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"difficulty": "Medium",
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"type": "MCQ_Single",
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"language": "English",
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"question_text": "A small particle tied to a string of length $5\\ \\mathrm{m}$ is given the minimum speed at the lowest point so that it just completes a vertical circle ($g = 10\\ \\mathrm{m\\,s^{-2}}$). Its speed at the highest point is:",
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"options": {
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"A": "$10\\ \\mathrm{m\\,s^{-1}}$",
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"B": "$5\\sqrt{2}\\ \\mathrm{m\\,s^{-1}}$",
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"C": "$5\\sqrt{10}\\ \\mathrm{m\\,s^{-1}}$",
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"D": "$10\\sqrt{2}\\ \\mathrm{m\\,s^{-1}}$"
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},
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"correct_answer": {
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"value": "B"
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},
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"solution_text": "Step 1: To just complete the circle, the string must remain taut at the top. In the limiting case the tension at the top is zero, so gravity alone supplies the centripetal force:\n$mg = \\dfrac{mv_{top}^2}{R} \\Rightarrow v_{top}^2 = gR$.\nStep 2: Energy conservation between the lowest point (speed $u$) and the highest point:\n$\\dfrac{1}{2}mu^2 = \\dfrac{1}{2}mv_{top}^2 + mg(2R) \\Rightarrow u^2 = gR + 4gR = 5gR = 250 \\Rightarrow u = 5\\sqrt{10}\\ \\mathrm{m\\,s^{-1}}$.\nStep 3: Speed at the highest point $v_{top} = \\sqrt{gR} = \\sqrt{50} = 5\\sqrt{2}\\ \\mathrm{m\\,s^{-1}} \\approx 7.07\\ \\mathrm{m\\,s^{-1}}$.\nHence, the correct option is (B)."
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},
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{
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"id": "2f3bf2bf-1773-4b76-88b6-e5c52d548eb5",
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"section": "General",
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"subject": "Physics",
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"topic": "Oscillations",
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"subtopic": "Spring-mass simple harmonic motion",
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"difficulty": "Easy",
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"type": "MCQ_Single",
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"language": "English",
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"question_text": "A block of mass $0.5\\ \\mathrm{kg}$ attached to a horizontal spring of force constant $200\\ \\mathrm{N\\,m^{-1}}$ performs SHM on a frictionless surface with amplitude $10\\ \\mathrm{cm}$. Its maximum speed is:",
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"options": {
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"A": "$1\\ \\mathrm{m\\,s^{-1}}$",
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"B": "$2\\ \\mathrm{m\\,s^{-1}}$",
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"C": "$2.5\\ \\mathrm{m\\,s^{-1}}$",
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"D": "$4\\ \\mathrm{m\\,s^{-1}}$"
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},
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"correct_answer": {
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"value": "B"
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},
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"solution_text": "Step 1: Angular frequency $\\omega = \\sqrt{\\dfrac{k}{m}} = \\sqrt{\\dfrac{200}{0.5}} = \\sqrt{400} = 20\\ \\mathrm{rad\\,s^{-1}}$.\nStep 2: Maximum speed $v_{max} = A\\omega = 0.10 \\times 20 = 2.0\\ \\mathrm{m\\,s^{-1}}$.\nStep 3 (energy check): $E = \\dfrac{1}{2}kA^2 = \\dfrac{1}{2}(200)(0.01) = 1\\ \\mathrm{J}$, so $v_{max} = \\sqrt{\\dfrac{2E}{m}} = \\sqrt{\\dfrac{2}{0.5}} = 2\\ \\mathrm{m\\,s^{-1}}$, consistent.\nHence, the correct option is (B)."
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},
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{
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"id": "c33ccf66-1dac-4bc5-ae8d-04cff93c94d1",
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"section": "General",
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"subject": "Physics",
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"topic": "Electrostatics",
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"subtopic": "Electric field due to a dipole",
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"difficulty": "Medium",
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"type": "MCQ_Single",
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"language": "English",
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"question_text": "Two point charges $+2\\ \\mu\\mathrm{C}$ and $-2\\ \\mu\\mathrm{C}$ are fixed $10\\ \\mathrm{cm}$ apart in air. The magnitude of the resultant electric field at the midpoint of the line joining the charges is ($k = 9\\times10^{9}\\ \\mathrm{N\\,m^{2}\\,C^{-2}}$):",
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"options": {
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"A": "$3.6\\times10^{6}\\ \\mathrm{N\\,C^{-1}}$",
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"B": "$7.2\\times10^{6}\\ \\mathrm{N\\,C^{-1}}$",
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"C": "$1.44\\times10^{7}\\ \\mathrm{N\\,C^{-1}}$",
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"D": "$2.88\\times10^{7}\\ \\mathrm{N\\,C^{-1}}$"
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},
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"correct_answer": {
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"value": "C"
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},
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"solution_text": "Step 1: The midpoint is at distance $r = 5\\ \\mathrm{cm} = 0.05\\ \\mathrm{m}$ from each charge.\nStep 2: Field due to one charge: $E_1 = \\dfrac{kq}{r^2} = \\dfrac{(9\\times10^{9})(2\\times10^{-6})}{(0.05)^2} = \\dfrac{1.8\\times10^{4}}{2.5\\times10^{-3}} = 7.2\\times10^{6}\\ \\mathrm{N\\,C^{-1}}$.\nStep 3: At the midpoint on the axial line of the dipole, both fields point from $+$ to $-$, i.e. in the same direction, so they add:\n$E = 2E_1 = 1.44\\times10^{7}\\ \\mathrm{N\\,C^{-1}}$.\nHence, the correct option is (C)."
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},
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{
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"id": "4d2d15b5-1ec0-4e53-a60d-c024711326a0",
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"section": "General",
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"subject": "Physics",
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"topic": "Current Electricity",
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"subtopic": "EMF, internal resistance and terminal voltage",
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"difficulty": "Easy",
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"type": "MCQ_Single",
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"language": "English",
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"question_text": "A cell of emf $6\\ \\mathrm{V}$ and internal resistance $0.5\\ \\Omega$ is connected across an external resistance of $2.5\\ \\Omega$. The terminal potential difference of the cell is:",
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"options": {
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"A": "$4.5\\ \\mathrm{V}$",
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"B": "$5.0\\ \\mathrm{V}$",
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"C": "$5.5\\ \\mathrm{V}$",
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"D": "$6.0\\ \\mathrm{V}$"
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},
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"correct_answer": {
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"value": "B"
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},
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"solution_text": "Step 1: Current in the circuit $I = \\dfrac{\\varepsilon}{R + r} = \\dfrac{6}{2.5 + 0.5} = \\dfrac{6}{3} = 2\\ \\mathrm{A}$.\nStep 2: Terminal voltage $V = \\varepsilon - Ir = 6 - (2)(0.5) = 5.0\\ \\mathrm{V}$.\nStep 3 (check): $V = IR = 2 \\times 2.5 = 5\\ \\mathrm{V}$, consistent.\nHence, the correct option is (B)."
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},
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{
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"id": "2ded5abe-b546-40a9-953a-0554849452af",
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"section": "General",
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"subject": "Physics",
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"topic": "Dual Nature of Matter and Radiation",
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"subtopic": "Photoelectric effect and Einstein's equation",
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"difficulty": "Medium",
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"type": "MCQ_Single",
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"language": "English",
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"question_text": "Light of wavelength $400\\ \\mathrm{nm}$ falls on a metal surface of work function $2.0\\ \\mathrm{eV}$. The maximum kinetic energy of the emitted photoelectrons is ($hc = 1240\\ \\mathrm{eV\\,nm}$):",
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"options": {
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"A": "$1.1\\ \\mathrm{eV}$",
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"B": "$2.0\\ \\mathrm{eV}$",
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"C": "$3.1\\ \\mathrm{eV}$",
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"D": "$5.1\\ \\mathrm{eV}$"
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},
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"correct_answer": {
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"value": "A"
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},
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"solution_text": "Step 1: Photon energy $E = \\dfrac{hc}{\\lambda} = \\dfrac{1240}{400} = 3.1\\ \\mathrm{eV}$.\nStep 2: Einstein's photoelectric equation: $K_{max} = E - \\phi = 3.1 - 2.0 = 1.1\\ \\mathrm{eV}$.\nSince $K_{max} > 0$, emission occurs.\nHence, the correct option is (A)."
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},
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{
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"id": "227ee1d1-b81e-4721-bfad-3eaf6f839998",
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"section": "General",
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"subject": "Physics",
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"topic": "Atoms",
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"subtopic": "Bohr model - hydrogen spectrum (Balmer series)",
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"difficulty": "Medium",
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"type": "MCQ_Single",
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"language": "English",
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"question_text": "A hydrogen atom de-excites from $n = 3$ to $n = 2$. The wavelength of the emitted photon is closest to ($hc = 1240\\ \\mathrm{eV\\,nm}$; energy of the ground state $= -13.6\\ \\mathrm{eV}$):",
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"options": {
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"A": "$121.6\\ \\mathrm{nm}$",
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"B": "$486\\ \\mathrm{nm}$",
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"C": "$656\\ \\mathrm{nm}$",
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"D": "$1875\\ \\mathrm{nm}$"
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},
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"correct_answer": {
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"value": "C"
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},
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"solution_text": "Step 1: Energy of the emitted photon:\n$\\Delta E = 13.6\\left(\\dfrac{1}{2^2} - \\dfrac{1}{3^2}\\right) = 13.6 \\times \\dfrac{9-4}{36} = 13.6 \\times \\dfrac{5}{36} \\approx 1.89\\ \\mathrm{eV}$.\nStep 2: Wavelength $\\lambda = \\dfrac{hc}{\\Delta E} = \\dfrac{1240}{1.89} \\approx 656\\ \\mathrm{nm}$ — this is the $H_\\alpha$ line of the Balmer series (visible).\nCross-check: $486\\ \\mathrm{nm}$ corresponds to $n = 4 \\to 2$; $121.6\\ \\mathrm{nm}$ is Lyman-$\\alpha$ ($2 \\to 1$); $1875\\ \\mathrm{nm}$ is Paschen ($4 \\to 3$).\nHence, the correct option is (C)."
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},
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{
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"id": "15d650bc-7f53-4591-94fd-7fb01f68b81b",
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"section": "General",
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"subject": "Physics",
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"topic": "Ray Optics",
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"subtopic": "Thin lenses - image formation",
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"difficulty": "Easy",
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"type": "MCQ_Single",
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"language": "English",
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"question_text": "An object is placed $30\\ \\mathrm{cm}$ in front of a thin converging lens of focal length $10\\ \\mathrm{cm}$. The image formed is:",
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"options": {
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"A": "at $15\\ \\mathrm{cm}$ from the lens, real and inverted",
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"B": "at $15\\ \\mathrm{cm}$ from the lens, virtual and erect",
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"C": "at $20\\ \\mathrm{cm}$ from the lens, real and inverted",
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"D": "at $7.5\\ \\mathrm{cm}$ from the lens, virtual and erect"
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},
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"correct_answer": {
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"value": "A"
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},
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"solution_text": "Step 1: Using the lens equation $\\dfrac{1}{v} - \\dfrac{1}{u} = \\dfrac{1}{f}$ with $u = -30\\ \\mathrm{cm}$ and $f = +10\\ \\mathrm{cm}$:\n$\\dfrac{1}{v} = \\dfrac{1}{10} - \\dfrac{1}{30} = \\dfrac{3-1}{30} = \\dfrac{2}{30} \\Rightarrow v = +15\\ \\mathrm{cm}$.\nStep 2: Positive $v$ means the image is formed $15\\ \\mathrm{cm}$ behind the lens — real.\nStep 3: Magnification $m = \\dfrac{v}{u} = \\dfrac{15}{-30} = -0.5$ — inverted and diminished (half the object size).\nHence, the correct option is (A)."
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},
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{
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"id": "7acaff9c-6123-423d-ba22-6f0ddccd1517",
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"section": "General",
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"subject": "Physics",
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"topic": "Kinematics",
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"subtopic": "Projectile motion - trajectory and energy",
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"difficulty": "Medium",
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"type": "MCQ_Multiple",
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"language": "English",
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"question_text": "A particle is projected from level ground with speed $u$ at an angle $\\theta$ ($0^\\circ < \\theta < 90^\\circ$) with the horizontal. Air resistance is negligible. Which of the following statements are TRUE?",
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"options": {
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"A": "The trajectory of the particle is a parabola.",
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"B": "At the highest point of the trajectory, the velocity is purely horizontal.",
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"C": "The horizontal range is maximum when $\\theta = 45^\\circ$.",
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"D": "The kinetic energy of the particle is maximum at the highest point."
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},
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"correct_answer": {
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"value": [
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"A",
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"B",
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"C"
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]
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},
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"solution_text": "(A) Eliminating $t$ from $x = u\\cos\\theta\\,t$ and $y = u\\sin\\theta\\,t - \\dfrac{1}{2}gt^2$ gives $y = x\\tan\\theta - \\dfrac{gx^2}{2u^2\\cos^2\\theta}$, a parabola. TRUE.\n(B) At the highest point $v_y = 0$ while $v_x = u\\cos\\theta$ is unchanged, so the velocity is purely horizontal. TRUE.\n(C) $R = \\dfrac{u^2\\sin 2\\theta}{g}$ is maximum when $\\sin 2\\theta = 1 \\Rightarrow 2\\theta = 90^\\circ \\Rightarrow \\theta = 45^\\circ$. TRUE.\n(D) At the highest point the speed is minimum ($v = u\\cos\\theta$), so the kinetic energy $\\dfrac{1}{2}mu^2\\cos^2\\theta$ is minimum there; it is maximum at the launch and landing points. FALSE.\nHence, the correct options are (A), (B) and (C)."
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},
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{
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"id": "d947a6fa-2057-406c-ae1e-637501f89ddc",
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"section": "General",
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"subject": "Physics",
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"topic": "Electrostatics",
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"subtopic": "Capacitor with dielectric (isolated capacitor)",
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"difficulty": "Hard",
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"type": "MCQ_Multiple",
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"language": "English",
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"question_text": "An ideal parallel-plate air capacitor is charged by a battery and then disconnected from it. A dielectric slab of dielectric constant $K > 1$ is now inserted so as to completely fill the space between the plates. Which of the following statements are TRUE?",
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"options": {
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"A": "The charge on the plates remains unchanged.",
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"B": "The potential difference between the plates increases.",
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"C": "The capacitance increases by a factor $K$.",
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"D": "The electrostatic energy stored in the capacitor decreases."
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},
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"correct_answer": {
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"value": [
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"A",
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"C",
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"D"
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]
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},
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"solution_text": "After disconnection, the charge $Q$ on the plates is trapped (no path for charge flow).\n(A) Charge stays $Q$. TRUE.\n(C) With the slab filling the gap, $C' = KC$. TRUE.\n(B) New potential difference $V' = \\dfrac{Q}{C'} = \\dfrac{V}{K} < V$ — the p.d. decreases, it does not increase. FALSE.\n(D) $U' = \\dfrac{Q^2}{2C'} = \\dfrac{U}{K} < U$ — energy decreases (work is done by the field in pulling the slab in). TRUE.\nHence, the correct options are (A), (C) and (D)."
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},
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{
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"id": "bd330625-7851-49d8-a280-ed569c57a4b2",
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"section": "General",
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"subject": "Physics",
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"topic": "Rotational Mechanics",
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"subtopic": "Rolling motion on an inclined plane",
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"difficulty": "Hard",
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"type": "MCQ_Multiple",
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"language": "English",
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"question_text": "A uniform solid sphere rolls without slipping down a fixed inclined plane of inclination $\\theta$. Which of the following statements are TRUE?",
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"options": {
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"A": "The friction force on the sphere acts up the incline.",
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"B": "The acceleration of the centre of mass is $\\dfrac{5}{7}g\\sin\\theta$.",
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"C": "The net work done by static friction on the sphere during the rolling is zero.",
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"D": "The ratio of translational kinetic energy to total kinetic energy is $5:7$."
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},
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"correct_answer": {
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"value": [
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"A",
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"B",
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"C",
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"D"
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]
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},
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"solution_text": "Write $a$ = acceleration of centre of mass (down the incline) and $f$ = static friction (up the incline).\nTranslation: $mg\\sin\\theta - f = ma$.\nRotation about centre: $fR = I\\alpha = \\dfrac{2}{5}mR^2\\alpha$, with rolling constraint $a = R\\alpha$.\nStep 1: $f = \\dfrac{2}{5}ma$ (up the incline, providing the spin-up torque). TRUE for (A).\nStep 2: $mg\\sin\\theta = ma + \\dfrac{2}{5}ma = \\dfrac{7}{5}ma \\Rightarrow a = \\dfrac{5}{7}g\\sin\\theta$. TRUE for (B).\nStep 3: The contact point of a rolling body is instantaneously at rest, so static friction does no work; the loss in gravitational PE appears entirely as kinetic energy. TRUE for (C).\nStep 4: $K_{trans} = \\dfrac{1}{2}mv^2$, $K_{rot} = \\dfrac{1}{2}I\\omega^2 = \\dfrac{1}{2}\\cdot\\dfrac{2}{5}mR^2\\cdot\\dfrac{v^2}{R^2} = \\dfrac{1}{5}mv^2$.\n$\\dfrac{K_{trans}}{K_{total}} = \\dfrac{\\frac{1}{2}}{\\frac{1}{2}+\\frac{1}{5}} = \\dfrac{1/2}{7/10} = \\dfrac{5}{7}$. TRUE for (D).\nHence, all of (A), (B), (C), (D) are correct."
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},
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{
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"id": "013ca5d3-a04f-4fa5-a426-bfb3ffa60e92",
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"section": "General",
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"subject": "Physics",
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"topic": "Thermodynamics",
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"subtopic": "First law - free (Joule) expansion of an ideal gas",
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"difficulty": "Medium",
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"type": "MCQ_Multiple",
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"language": "English",
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"question_text": "An ideal gas kept in an insulated rigid vessel is allowed to rush into a second, initially evacuated chamber rigidly connected to it (Joule free expansion). Which of the following statements are TRUE?",
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"options": {
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"A": "The heat absorbed by the gas is zero.",
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"B": "The work done by the gas is zero.",
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"C": "The temperature of the gas increases.",
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"D": "The internal energy of the gas remains unchanged."
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},
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"correct_answer": {
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"value": [
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"A",
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"B",
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"D"
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]
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},
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"solution_text": "(A) The walls are insulated, so $Q = 0$. TRUE.\n(B) The gas expands into vacuum; the external opposing pressure is zero, so $W = \\int p_{ext}\\,dV = 0$. TRUE.\n(D) First law: $\\Delta U = Q - W = 0 - 0 = 0$. TRUE.\n(C) For an ideal gas, $U = U(T)$, so $\\Delta U = 0 \\Rightarrow \\Delta T = 0$; the temperature does not increase. FALSE.\nHence, the correct options are (A), (B) and (D)."
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},
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{
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"id": "5c7f8ba3-d172-4b6e-9d57-af81fffd8953",
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"section": "General",
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"subject": "Physics",
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"topic": "Magnetic Effects of Current and Magnetism",
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"subtopic": "Motion of a charged particle in a uniform magnetic field",
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"difficulty": "Hard",
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"type": "MCQ_Multiple",
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"language": "English",
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"question_text": "A charged particle of mass $m$ and charge $q$ enters a uniform magnetic field $\\vec{B}$ with velocity $\\vec{v}$ perpendicular to $\\vec{B}$. Gravity and all other fields are absent. Which of the following statements are TRUE?",
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"options": {
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"A": "The kinetic energy of the particle remains constant.",
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"B": "The linear momentum of the particle remains constant.",
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"C": "The particle moves in a circle of radius $\\dfrac{mv}{qB}$.",
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"D": "The time period of revolution is independent of the speed $v$."
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},
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"correct_answer": {
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"value": [
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"A",
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"C",
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"D"
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]
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},
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"solution_text": "(A) The magnetic force $\\vec{F} = q\\vec{v}\\times\\vec{B}$ is always perpendicular to $\\vec{v}$, so it does no work and the kinetic energy stays constant. TRUE.\n(B) A nonzero force continuously changes the direction of momentum; only its magnitude $mv$ is constant. FALSE.\n(C) The magnetic force supplies the centripetal force: $qvB = \\dfrac{mv^2}{r} \\Rightarrow r = \\dfrac{mv}{qB}$. TRUE.\n(D) $T = \\dfrac{2\\pi m}{qB}$ contains no $v$; faster particles travel larger circles in the same period. TRUE.\nHence, the correct options are (A), (C) and (D)."
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},
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{
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|
"id": "f574d7cf-eabb-4ffd-927d-4073b8456276",
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|
"section": "General",
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"subject": "Physics",
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"topic": "Atoms",
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"subtopic": "Bohr model - quantization and scaling laws",
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"difficulty": "Medium",
|
|
"type": "MCQ_Multiple",
|
|
"language": "English",
|
|
"question_text": "According to Bohr's model of the hydrogen atom, which of the following statements are TRUE?",
|
|
"options": {
|
|
"A": "The angular momentum of the electron in the $n$-th orbit is $\\dfrac{nh}{2\\pi}$.",
|
|
"B": "The radius of the $n$-th orbit is proportional to $n$.",
|
|
"C": "The total energy of the electron in the $n$-th orbit is $-\\dfrac{13.6}{n^2}\\ \\mathrm{eV}$.",
|
|
"D": "The orbital speed of the electron in the $n$-th orbit is inversely proportional to $n$."
|
|
},
|
|
"correct_answer": {
|
|
"value": [
|
|
"A",
|
|
"C",
|
|
"D"
|
|
]
|
|
},
|
|
"solution_text": "(A) Bohr's quantization condition: $mvr_n = \\dfrac{nh}{2\\pi}$. TRUE.\n(B) $r_n = 0.529\\,n^2\\ \\text{\\AA}$, i.e. $r_n \\propto n^2$, not $n$. FALSE.\n(C) $E_n = -\\dfrac{13.6}{n^2}\\ \\mathrm{eV}$. TRUE.\n(D) From $v_n = \\dfrac{2\\pi k e^2}{nh}\\left(\\dfrac{1}{4\\pi\\varepsilon_0}\\ \\text{form}\\right)$, $v_n \\propto \\dfrac{1}{n}$. TRUE.\nHence, the correct options are (A), (C) and (D)."
|
|
},
|
|
{
|
|
"id": "34bce33a-55ca-425f-acd6-82957a704127",
|
|
"section": "General",
|
|
"subject": "Physics",
|
|
"topic": "Work, Energy and Power",
|
|
"subtopic": "Coefficient of restitution",
|
|
"difficulty": "Easy",
|
|
"type": "Numerical",
|
|
"language": "English",
|
|
"question_text": "A ball is dropped from a height of $20\\ \\mathrm{m}$ onto a hard horizontal floor and rebounds to a height of $5\\ \\mathrm{m}$. The coefficient of restitution between the ball and the floor is ____ (round off to two decimal places).",
|
|
"options": null,
|
|
"correct_answer": {
|
|
"value": "0.50"
|
|
},
|
|
"solution_text": "Step 1: Speed just before impact (downward): $v_1 = \\sqrt{2gh_1} = \\sqrt{2 \\times 10 \\times 20} = 20\\ \\mathrm{m\\,s^{-1}}$.\nStep 2: Speed just after rebound (upward): $v_2 = \\sqrt{2gh_2} = \\sqrt{2 \\times 10 \\times 5} = 10\\ \\mathrm{m\\,s^{-1}}$.\nStep 3: Coefficient of restitution $e = \\dfrac{v_{after}}{v_{before}} = \\dfrac{10}{20} = 0.50$."
|
|
},
|
|
{
|
|
"id": "e5ee9d3e-3036-43ce-94b9-ad8804078a80",
|
|
"section": "General",
|
|
"subject": "Physics",
|
|
"topic": "Laws of Motion",
|
|
"subtopic": "Newton's second law in vector form",
|
|
"difficulty": "Medium",
|
|
"type": "Numerical",
|
|
"language": "English",
|
|
"question_text": "A block of mass $5\\ \\mathrm{kg}$ moving on a frictionless horizontal surface is acted upon by a constant force $\\vec{F} = (6\\hat{i} + 8\\hat{j})\\ \\mathrm{N}$. The magnitude of its acceleration is ____ $\\mathrm{m\\,s^{-2}}$ (round off to two decimal places).",
|
|
"options": null,
|
|
"correct_answer": {
|
|
"value": "2.00"
|
|
},
|
|
"solution_text": "Step 1: Magnitude of the force $|\\vec{F}| = \\sqrt{6^2 + 8^2} = \\sqrt{36 + 64} = \\sqrt{100} = 10\\ \\mathrm{N}$.\nStep 2: Newton's second law: $a = \\dfrac{|\\vec{F}|}{m} = \\dfrac{10}{5} = 2.00\\ \\mathrm{m\\,s^{-2}}$."
|
|
},
|
|
{
|
|
"id": "4a148ad8-8fce-41de-8e53-ef2c2507bee8",
|
|
"section": "General",
|
|
"subject": "Physics",
|
|
"topic": "Work, Energy and Power",
|
|
"subtopic": "One-dimensional elastic collision",
|
|
"difficulty": "Medium",
|
|
"type": "Numerical",
|
|
"language": "English",
|
|
"question_text": "A block of mass $2\\ \\mathrm{kg}$ moving with speed $6\\ \\mathrm{m\\,s^{-1}}$ on a smooth horizontal surface collides head-on and elastically with a stationary block of mass $1\\ \\mathrm{kg}$. The final speed of the $1\\ \\mathrm{kg}$ block is ____ $\\mathrm{m\\,s^{-1}}$ (round off to two decimal places).",
|
|
"options": null,
|
|
"correct_answer": {
|
|
"value": "8.00"
|
|
},
|
|
"solution_text": "For a 1-D elastic collision with the target at rest:\nStep 1: $v_2' = \\dfrac{2m_1}{m_1 + m_2}\\,u_1 = \\dfrac{2\\times2}{2+1}\\times6 = \\dfrac{4}{3}\\times6 = 8.00\\ \\mathrm{m\\,s^{-1}}$.\nStep 2 (conservation checks): Momentum: $2(2) + 1(8) = 12 = 2(6)$ ✓. Kinetic energy: $\\dfrac{1}{2}(2)(2)^2 + \\dfrac{1}{2}(1)(8)^2 = 4 + 32 = 36\\ \\mathrm{J} = \\dfrac{1}{2}(2)(6)^2$ ✓."
|
|
},
|
|
{
|
|
"id": "db34cbbe-8fa1-49ba-acb8-cb881d4b5402",
|
|
"section": "General",
|
|
"subject": "Physics",
|
|
"topic": "Oscillations, Gravitation and Kinematics",
|
|
"subtopic": "Mixed formula matching (List-I / List-II)",
|
|
"difficulty": "Hard",
|
|
"type": "MCQ_Multiple",
|
|
"language": "English",
|
|
"question_text": "Match the physical situation in List-I with the correct expression in List-II.\n\nList-I:\n(P) Time period of a simple pendulum of length $L$\n(Q) Escape speed from the surface of a planet of mass $M$ and radius $R$\n(R) Maximum height above the ground attained by a projectile launched with speed $u$ at $45^\\circ$\n(S) Angular frequency of a block of mass $m$ attached to a spring of force constant $k$\n\nList-II:\n(1) $2\\pi\\sqrt{\\dfrac{L}{g}}$\n(2) $\\sqrt{\\dfrac{2GM}{R}}$\n(3) $\\dfrac{u^{2}}{4g}$\n(4) $\\sqrt{\\dfrac{k}{m}}$\n(5) $\\sqrt{\\dfrac{mg}{k}}$\n\nWhich of the following matchings are CORRECT?",
|
|
"options": {
|
|
"A": "P $\\to$ 1",
|
|
"B": "Q $\\to$ 2",
|
|
"C": "R $\\to$ 3",
|
|
"D": "S $\\to$ 5"
|
|
},
|
|
"correct_answer": {
|
|
"value": [
|
|
"A",
|
|
"B",
|
|
"C"
|
|
]
|
|
},
|
|
"solution_text": "(P) Simple pendulum: $T = 2\\pi\\sqrt{L/g}$, so P-1 is correct. TRUE.\n(Q) Escape speed: $v_e = \\sqrt{\\dfrac{2GM}{R}}$ (from $\\dfrac{1}{2}mv_e^2 = \\dfrac{GMm}{R}$), so Q-2 is correct. TRUE.\n(R) $H = \\dfrac{u^2\\sin^2 45^\\circ}{2g} = \\dfrac{u^2/2}{2g} = \\dfrac{u^2}{4g}$, so R-3 is correct. TRUE.\n(S) $\\omega = \\sqrt{k/m}$, so the correct match is S-4, not S-5. FALSE.\nHence, the correct options are (A), (B) and (C)."
|
|
},
|
|
{
|
|
"id": "c482e3b7-8c67-42ad-b669-020ede4a8217",
|
|
"section": "General",
|
|
"subject": "Chemistry",
|
|
"topic": "Chemical Bonding and Molecular Structure",
|
|
"subtopic": "Hybridisation and shape of molecules",
|
|
"difficulty": "Easy",
|
|
"type": "MCQ_Single",
|
|
"language": "English",
|
|
"question_text": "The hybridisation of the central iodine atom in the $I_3^-$ ion and its shape are respectively:",
|
|
"options": {
|
|
"A": "$sp^3$ and linear",
|
|
"B": "$sp^3d$ and linear",
|
|
"C": "$sp^3d$ and bent",
|
|
"D": "$sp^2$ and linear"
|
|
},
|
|
"correct_answer": {
|
|
"value": "B"
|
|
},
|
|
"solution_text": "Step 1: Lewis structure of $I_3^-$ (total valence electrons $= 3\\times7 + 1 = 22$): the central iodine carries 3 lone pairs and is bonded to two terminal iodines, with 3 lone pairs on each terminal I.\nStep 2: Steric number of central I $=$ (bonds) $+$ (lone pairs) $= 2 + 3 = 5 \\Rightarrow sp^3d$ hybridisation (trigonal-bipyramidal electron-pair geometry; lone pairs occupy equatorial positions).\nStep 3: With the three lone pairs equatorial, the two I--I bonds occupy the axial positions $\\Rightarrow$ linear shape ($\\angle I-I-I = 180^\\circ$).\nHence, the correct option is (B)."
|
|
},
|
|
{
|
|
"id": "0a791914-f938-4062-a7d6-a0a310bdc0e3",
|
|
"section": "General",
|
|
"subject": "Chemistry",
|
|
"topic": "Solutions",
|
|
"subtopic": "Molarity-molality relation using density",
|
|
"difficulty": "Medium",
|
|
"type": "MCQ_Single",
|
|
"language": "English",
|
|
"question_text": "The density (in $\\mathrm{g\\,mL^{-1}}$) of a $3.60\\ \\mathrm{M}$ sulphuric acid solution that is $23\\%$ $H_2SO_4$ by mass is closest to (molar mass of $H_2SO_4 = 98\\ \\mathrm{g\\,mol^{-1}}$):",
|
|
"options": {
|
|
"A": "1.45",
|
|
"B": "1.55",
|
|
"C": "1.65",
|
|
"D": "1.35"
|
|
},
|
|
"correct_answer": {
|
|
"value": "B"
|
|
},
|
|
"solution_text": "Take $100\\ \\mathrm{g}$ of solution:\nStep 1: Mass of $H_2SO_4 = 23\\ \\mathrm{g} \\Rightarrow n = \\dfrac{23}{98} = 0.2347\\ \\mathrm{mol}$.\nStep 2: Molarity is $3.60\\ \\mathrm{M}$, so the solution volume $V = \\dfrac{n}{M} = \\dfrac{0.2347}{3.60} = 0.06519\\ \\mathrm{L} = 65.19\\ \\mathrm{mL}$.\nStep 3: Density $= \\dfrac{\\text{mass}}{\\text{volume}} = \\dfrac{100}{65.19} = 1.53\\ \\mathrm{g\\,mL^{-1}}$, closest to $1.55$.\nHence, the correct option is (B)."
|
|
},
|
|
{
|
|
"id": "fee2042f-c39e-425c-b573-c735f796aa65",
|
|
"section": "General",
|
|
"subject": "Chemistry",
|
|
"topic": "Coordination Compounds",
|
|
"subtopic": "Hybridisation, shape and magnetic moment",
|
|
"difficulty": "Medium",
|
|
"type": "MCQ_Single",
|
|
"language": "English",
|
|
"question_text": "The magnetic moment (spin-only value) and the number of unpaired electrons in $[Fe(CN)_6]^{3-}$ are respectively:",
|
|
"options": {
|
|
"A": "$5.92\\ \\mathrm{BM}$ and 5",
|
|
"B": "$1.73\\ \\mathrm{BM}$ and 1",
|
|
"C": "$2.83\\ \\mathrm{BM}$ and 2",
|
|
"D": "$0\\ \\mathrm{BM}$ and 0"
|
|
},
|
|
"correct_answer": {
|
|
"value": "B"
|
|
},
|
|
"solution_text": "Step 1: In $[Fe(CN)_6]^{3-}$, Fe is in the $+3$ state: $d^5$ configuration ($3d^5$).\nStep 2: $CN^-$ is a strong-field ligand $\\Rightarrow$ low-spin octahedral complex, i.e. $t_{2g}^5 e_g^0$.\nStep 3: Number of unpaired electrons $= 5 - 4 = 1$ (pairing of four of the five $t_{2g}$ electrons).\nStep 4: Spin-only moment $\\mu = \\sqrt{n(n+2)} = \\sqrt{1\\times3} = 1.73\\ \\mathrm{BM}$.\nHence, the correct option is (B)."
|
|
},
|
|
{
|
|
"id": "8ac6227a-5a53-4d95-aa69-608782363cf8",
|
|
"section": "General",
|
|
"subject": "Chemistry",
|
|
"topic": "Chemical Kinetics",
|
|
"subtopic": "First order kinetics - half-life and time",
|
|
"difficulty": "Easy",
|
|
"type": "MCQ_Single",
|
|
"language": "English",
|
|
"question_text": "A first-order reaction has a half-life of $10\\ \\mathrm{min}$. The time required for the concentration of the reactant to fall to $6.25\\%$ of its initial value is:",
|
|
"options": {
|
|
"A": "$30\\ \\mathrm{min}$",
|
|
"B": "$40\\ \\mathrm{min}$",
|
|
"C": "$50\\ \\mathrm{min}$",
|
|
"D": "$60\\ \\mathrm{min}$"
|
|
},
|
|
"correct_answer": {
|
|
"value": "B"
|
|
},
|
|
"solution_text": "For a first-order reaction the half-life is concentration-independent.\nStep 1: Each half-life reduces the concentration by half: after $t$ minutes, $\\dfrac{[A]}{[A]_0} = \\left(\\dfrac{1}{2}\\right)^{t/t_{1/2}}$.\nStep 2: $6.25\\% = 0.0625 = \\dfrac{1}{16} = \\left(\\dfrac{1}{2}\\right)^4 \\Rightarrow$ four half-lives.\nStep 3: $t = 4 \\times 10 = 40\\ \\mathrm{min}$.\nHence, the correct option is (B)."
|
|
},
|
|
{
|
|
"id": "c57a69a5-9a40-4962-92a4-bddcbf65bef4",
|
|
"section": "General",
|
|
"subject": "Chemistry",
|
|
"topic": "Equilibrium",
|
|
"subtopic": "pH of buffer solutions - Henderson-Hasselbalch",
|
|
"difficulty": "Medium",
|
|
"type": "MCQ_Single",
|
|
"language": "English",
|
|
"question_text": "A buffer solution contains $0.1\\ \\mathrm{M}$ acetic acid and $0.1\\ \\mathrm{M}$ sodium acetate. If $0.01\\ \\mathrm{mol}$ of NaOH is added to $1\\ \\mathrm{L}$ of this buffer (volume change neglected), the pH of the resulting solution is ($pK_a$ of acetic acid $= 4.74$):",
|
|
"options": {
|
|
"A": "4.74",
|
|
"B": "4.83",
|
|
"C": "4.65",
|
|
"D": "4.56"
|
|
},
|
|
"correct_answer": {
|
|
"value": "B"
|
|
},
|
|
"solution_text": "Step 1: NaOH consumes the acid: $CH_3COOH + OH^- \\to CH_3COO^- + H_2O$.\nAfter reaction: acid $= 0.10 - 0.01 = 0.09\\ \\mathrm{mol}$, salt $= 0.10 + 0.01 = 0.11\\ \\mathrm{mol}$ (same total volume, so use mole ratio).\nStep 2: Henderson-Hasselbalch: $pH = pK_a + \\log\\dfrac{[\\text{salt}]}{[\\text{acid}]} = 4.74 + \\log\\dfrac{0.11}{0.09} = 4.74 + \\log(1.222)$.\nStep 3: $pH = 4.74 + 0.087 = 4.83$.\nHence, the correct option is (B)."
|
|
},
|
|
{
|
|
"id": "e1d32832-baba-4de5-a808-ef71d74b0f84",
|
|
"section": "General",
|
|
"subject": "Chemistry",
|
|
"topic": "Organic Chemistry - Some Basic Principles and Techniques",
|
|
"subtopic": "Inductive, resonance and stability of carbocations",
|
|
"difficulty": "Medium",
|
|
"type": "MCQ_Single",
|
|
"language": "English",
|
|
"question_text": "Which one of the following carbocations is the MOST stable?",
|
|
"options": {
|
|
"A": "$(CH_3)_3C^+$",
|
|
"B": "$CH_3CH_2^+$",
|
|
"C": "$C_6H_5CH_2^+$ (benzyl)",
|
|
"D": "$CH_2=CHCH_2^+$ (allyl)"
|
|
},
|
|
"correct_answer": {
|
|
"value": "C"
|
|
},
|
|
"solution_text": "Compare stabilisation mechanisms:\nStep 1: $(CH_3)_3C^+$ is stabilised by the $+I$ effect and hyperconjugation of nine $\\beta$ C--H bonds (tertiary, $\\sim$ order 3 on the aliphatic scale).\nStep 2: Allyl $CH_2=CHCH_2^+$ is resonance-stabilised over two carbons (order $\\sim$ 2-3).\nStep 3: Benzyl $C_6H_5CH_2^+$ is stabilised by resonance delocalisation of the positive charge into the aromatic ring; the empty $p$ orbital conjugates with the ring $\\pi$ system, spreading the charge over the ortho and para positions — five canonical structures in total. This delocalisation over the extended aromatic $\\pi$ system makes it the most stable of the four.\nStep 4: Primary ethyl carbocation is the least stable.\nStability order: benzyl $>$ $t$-alkyl $\\approx$ allyl $>$ primary.\nHence, the correct option is (C)."
|
|
},
|
|
{
|
|
"id": "303782f2-1585-417b-b6db-3031a92eda8a",
|
|
"section": "General",
|
|
"subject": "Chemistry",
|
|
"topic": "d- and f- Block Elements",
|
|
"subtopic": "Lanthanoid contraction and oxidation states",
|
|
"difficulty": "Easy",
|
|
"type": "MCQ_Single",
|
|
"language": "English",
|
|
"question_text": "Which of the following statements about the lanthanoids is INCORRECT?",
|
|
"options": {
|
|
"A": "The general oxidation state exhibited by lanthanoids is +3.",
|
|
"B": "The ionic radii of the lanthanoid $M^{3+}$ ions decrease regularly with increase in atomic number (lanthanoid contraction).",
|
|
"C": "Eu$^{2+}$ is a strong reducing agent while Yb$^{2+}$ is a mild oxidising agent.",
|
|
"D": "Ln$^{3+}$ compounds of early lanthanoids (La$^{3+}$, Pr$^{3+}$) are generally colourless."
|
|
},
|
|
"correct_answer": {
|
|
"value": "C"
|
|
},
|
|
"solution_text": "(A) $+3$ is the general/most stable oxidation state of the lanthanoids. CORRECT.\n(B) Poor shielding by the $4f$ electrons causes a regular decrease in $M^{3+}$ radii across the series — the lanthanoid contraction. CORRECT.\n(C) FALSE. The $+2$ state is unusually stable at Eu$^{2+}$ ($4f^7$) and Yb$^{2+}$ ($4f^{14}$), but Eu$^{2+}$ readily loses an electron to attain the common $+3$ state, so it is a strong REDUCING agent; Yb$^{2+}$ ($4f^{14}$) is also a (mild) reductant, NOT an oxidising agent. (Tb$^{4+}$, with its $4f^7$ configuration, is the species that acts as an oxidant.)\n(D) Early lanthanoid $Ln^{3+}$ ions (e.g. $f^0$ La$^{3+}$) absorb little in the visible region and their compounds are generally colourless. CORRECT.\nThe INCORRECT statement is (C), which reverses the redox character of Yb$^{2+}$.\nHence, the correct option is (C)."
|
|
},
|
|
{
|
|
"id": "02a1f516-6470-4ed4-90de-e9293508bf44",
|
|
"section": "General",
|
|
"subject": "Chemistry",
|
|
"topic": "Redox Reactions and Electrochemistry",
|
|
"subtopic": "Nernst equation - cell EMF",
|
|
"difficulty": "Medium",
|
|
"type": "MCQ_Single",
|
|
"language": "English",
|
|
"question_text": "For the cell reaction at 298 K:\n$Zn(s) + Cu^{2+}(aq) \\to Zn^{2+}(aq) + Cu(s)$\nwith $[Cu^{2+}] = 0.1\\ \\mathrm{M}$ and $[Zn^{2+}] = 1.0\\ \\mathrm{M}$, the cell EMF is ($E^\\circ_{cell} = 1.10\\ \\mathrm{V}$, $\\dfrac{2.303RT}{F} = 0.059\\ \\mathrm{V}$):",
|
|
"options": {
|
|
"A": "1.10 V",
|
|
"B": "1.07 V",
|
|
"C": "1.13 V",
|
|
"D": "1.16 V"
|
|
},
|
|
"correct_answer": {
|
|
"value": "B"
|
|
},
|
|
"solution_text": "Step 1: Two electrons are transferred ($n = 2$). Nernst equation:\n$E = E^\\circ - \\dfrac{0.059}{2}\\log\\dfrac{[Zn^{2+}]}{[Cu^{2+}]}$.\nStep 2: $\\log\\dfrac{1.0}{0.1} = \\log 10 = 1$.\nStep 3: $E = 1.10 - \\dfrac{0.059}{2}(1) = 1.10 - 0.0295 = 1.07\\ \\mathrm{V}$.\nHence, the correct option is (B)."
|
|
},
|
|
{
|
|
"id": "2c20f4ff-5472-4082-bb2f-50525945c832",
|
|
"section": "General",
|
|
"subject": "Chemistry",
|
|
"topic": "Chemical Bonding and Molecular Structure",
|
|
"subtopic": "Bond parameters and dipole moments",
|
|
"difficulty": "Hard",
|
|
"type": "MCQ_Multiple",
|
|
"language": "English",
|
|
"question_text": "Which of the following statements are TRUE?",
|
|
"options": {
|
|
"A": "The bond order of the $O_2$ molecule is 2.",
|
|
"B": "The $CO_2$ molecule has a net dipole moment of zero.",
|
|
"C": "The bond angle in $H_2O$ is larger than that in $H_2S$.",
|
|
"D": "The $N_2$ molecule is diamagnetic."
|
|
},
|
|
"correct_answer": {
|
|
"value": [
|
|
"A",
|
|
"B",
|
|
"C",
|
|
"D"
|
|
]
|
|
},
|
|
"solution_text": "(A) MO configuration of $O_2$: $\\sigma 2s^2 \\sigma^*2s^2 \\sigma 2p_z^2 \\pi 2p_x^2 \\pi 2p_y^2 \\pi^*2p_x^1 \\pi^*2p_y^1$.\nBond order $= \\dfrac{8 - 4}{2} = 2$. TRUE.\n(B) $CO_2$ is linear ($O=C=O$); the two equal, opposite bond dipoles cancel, so $\\mu = 0$. TRUE.\n(C) Bond angle $H_2O \\approx 104.5^\\circ$; in $H_2S$ the larger S atom and more diffuse $3p$ orbitals give $\\approx 92.1^\\circ$. So $H_2O > H_2S$. TRUE.\n(D) $N_2$: all electrons are paired ($\\sigma 2s^2\\sigma^{*}2s^2\\pi 2p_x^2\\pi 2p_y^2\\sigma 2p_z^2$), bond order 3, no unpaired electrons $\\Rightarrow$ diamagnetic. TRUE.\nHence, all of (A), (B), (C), (D) are correct."
|
|
},
|
|
{
|
|
"id": "09ea6ab7-c1e4-49eb-82b8-114e2ceb0169",
|
|
"section": "General",
|
|
"subject": "Chemistry",
|
|
"topic": "Equilibrium",
|
|
"subtopic": "Factors affecting chemical equilibrium (Le Chatelier)",
|
|
"difficulty": "Medium",
|
|
"type": "MCQ_Multiple",
|
|
"language": "English",
|
|
"question_text": "For the exothermic gas-phase reaction $N_2(g) + 3H_2(g) \\rightleftharpoons 2NH_3(g)$ at equilibrium in a rigid vessel, which of the following actions will shift the equilibrium towards the product side?",
|
|
"options": {
|
|
"A": "Increasing the total pressure by decreasing the volume at constant temperature.",
|
|
"B": "Increasing the temperature at constant volume.",
|
|
"C": "Removing some $NH_3$ from the vessel at constant temperature and volume.",
|
|
"D": "Adding an inert gas (He) at constant volume and constant temperature."
|
|
},
|
|
"correct_answer": {
|
|
"value": [
|
|
"A",
|
|
"C"
|
|
]
|
|
},
|
|
"solution_text": "(A) Decreasing the volume raises partial pressures; the system shifts toward the side with fewer gaseous moles ($2 < 1 + 3 = 4$), i.e. toward $NH_3$. Shifts to products. CORRECT.\n(B) For an exothermic reaction, raising $T$ shifts equilibrium towards reactants (endothermic direction) and lowers $K$. Does NOT favour products. INCORRECT.\n(C) Removing $NH_3$ lowers $Q$ below $K$; the system responds by forming more $NH_3$ until $Q = K$ again. CORRECT.\n(D) At constant volume, an inert gas changes the total pressure but not the partial pressures of the reacting gases or $Q$; no shift occurs. INCORRECT.\nHence, the correct options are (A) and (C)."
|
|
},
|
|
{
|
|
"id": "7bc8a39a-eb7f-4245-93cd-f45d238f77d2",
|
|
"section": "General",
|
|
"subject": "Chemistry",
|
|
"topic": "Biomolecules",
|
|
"subtopic": "Carbohydrates - structure and reactions of glucose",
|
|
"difficulty": "Hard",
|
|
"type": "MCQ_Multiple",
|
|
"language": "English",
|
|
"question_text": "D-(+)-Glucose is subjected to the following tests/reagents. Which of the following observations/conclusions are CORRECT?",
|
|
"options": {
|
|
"A": "With bromine water, glucose is oxidised to gluconic acid, proving the presence of an aldehyde group.",
|
|
"B": "Glucose does not react with sodium hydrogensulphite addition reagent, which supports the presence of a cyclic hemiacetal structure.",
|
|
"C": "Glucose forms a pentaacetate, indicating five free hydroxyl groups.",
|
|
"D": "Glucose is oxidised by nitric acid to a dicarboxylic acid (saccharic acid), indicating both primary and secondary alcohol groups."
|
|
},
|
|
"correct_answer": {
|
|
"value": [
|
|
"A",
|
|
"B",
|
|
"C",
|
|
"D"
|
|
]
|
|
},
|
|
"solution_text": "(A) Bromine water is a mild oxidant: $-CHO \\to -COOH$; glucose gives gluconic acid. TRUE.\n(B) Glucose fails the NaHSO$_3$ addition test (typical of free aldehydes) and does not form an oxime with hydroxylamine under ordinary conditions; this is explained by the cyclic hemiacetal (pyranose) form predominating in solution. TRUE (standard textbook evidence).\n(C) Acetylation with acetic anhydride gives glucose pentaacetate $\\Rightarrow$ five $-\\mathrm{OH}$ groups (one primary, four secondary). TRUE.\n(D) Oxidation with conc. $HNO_3$ oxidises both $-CHO$ and $-CH_2OH$ ends to give saccharic (glucaric) acid, a dicarboxylic acid. TRUE.\nHence, all of (A), (B), (C), (D) are correct."
|
|
},
|
|
{
|
|
"id": "15ba4def-2a9c-408d-a5f2-0297208a6ee5",
|
|
"section": "General",
|
|
"subject": "Chemistry",
|
|
"topic": "General Principles and Processes of Isolation of Elements",
|
|
"subtopic": "Extraction principles - Ellingham diagram, roasting and reduction",
|
|
"difficulty": "Medium",
|
|
"type": "MCQ_Multiple",
|
|
"language": "English",
|
|
"question_text": "Which of the following statements about the principles of metallurgical extraction are TRUE?",
|
|
"options": {
|
|
"A": "In the Ellingham diagram, a metal can reduce the oxide of another metal lying above its own oxide line at a given temperature.",
|
|
"B": "Sulphide ores are generally converted to oxides by roasting before reduction.",
|
|
"C": "In the extraction of copper from its sulphide ore, the copper matte is converted to blister copper by oxidation in a Bessemer converter.",
|
|
"D": "Carbon cannot be used to reduce $Al_2O_3$ at temperatures below about 2000 K."
|
|
},
|
|
"correct_answer": {
|
|
"value": [
|
|
"A",
|
|
"B",
|
|
"C",
|
|
"D"
|
|
]
|
|
},
|
|
"solution_text": "(A) In an Ellingham diagram, the metal whose oxide line lies LOWER (more negative $\\Delta G$) reduces the oxide whose line lies above it at that temperature. TRUE.\n(B) Roasting heats the sulphide ore in excess air: $2ZnS + 3O_2 \\to 2ZnO + 2SO_2$; oxide formation is the standard prerequisite for reduction. TRUE.\n(C) Matte ($Cu_2S + FeS$) in the Bessemer converter: $2FeS + 3O_2 \\to 2FeO + 2SO_2$, FeO is slagged with silica, then self-reduction $2Cu_2S + 3O_2 \\to 2Cu_2O + 2SO_2$ and $Cu_2O + Cu_2S \\to 6Cu + SO_2$; the SO$_2$ bubbles give blister copper. TRUE.\n(D) The Ellingham lines of Al$_2$O$_3$ and CO lie close only above $\\sim$2000 K, and carbon reduction of alumina is not practical below about 2000 K (Al is extracted electrolytically). TRUE.\nHence, all of (A), (B), (C), (D) are correct."
|
|
},
|
|
{
|
|
"id": "3bb7c68a-99da-4739-8ae4-f8f85b3644fb",
|
|
"section": "General",
|
|
"subject": "Chemistry",
|
|
"topic": "Aldehydes, Ketones and Carboxylic Acids",
|
|
"subtopic": "Named reactions and nucleophilic addition (aldehydes vs ketones)",
|
|
"difficulty": "Hard",
|
|
"type": "MCQ_Multiple",
|
|
"language": "English",
|
|
"question_text": "Which of the following statements about aldehydes and ketones are TRUE?",
|
|
"options": {
|
|
"A": "Aldehydes are generally more reactive than ketones towards nucleophilic addition reactions.",
|
|
"B": "Acetaldehyde gives a positive iodoform test.",
|
|
"C": "Benzaldehyde gives a positive Fehling test.",
|
|
"D": "Acetophenone ($C_6H_5COCH_3$) can be prepared by the Friedel-Crafts acylation of benzene with acetyl chloride."
|
|
},
|
|
"correct_answer": {
|
|
"value": [
|
|
"A",
|
|
"B",
|
|
"D"
|
|
]
|
|
},
|
|
"solution_text": "(A) Less steric hindrance and fewer $+I$ electron-donating groups in aldehydes make the carbonyl carbon more electrophilic than in ketones. TRUE.\n(B) Acetaldehyde ($CH_3CHO$) contains the $CH_3CO-$ unit required for the iodoform test: yellow $CHI_3$ is formed. TRUE.\n(C) Benzaldehyde has no $\\alpha$-hydrogen and gives a negative Fehling test (no red $Cu_2O$ precipitate); aromatic aldehydes generally fail Fehling. FALSE.\n(D) $C_6H_6 + CH_3COCl \\xrightarrow{AlCl_3} C_6H_5COCH_3$ (acetophenone) + HCl — the standard Friedel-Crafts acylation. TRUE.\nHence, the correct options are (A), (B) and (D)."
|
|
},
|
|
{
|
|
"id": "554fb98f-3646-4727-ad54-86dbbe260555",
|
|
"section": "General",
|
|
"subject": "Chemistry",
|
|
"topic": "Polymers, Biomolecules and Chemistry in Everyday Life",
|
|
"subtopic": "Polymer types and examples",
|
|
"difficulty": "Medium",
|
|
"type": "MCQ_Multiple",
|
|
"language": "English",
|
|
"question_text": "Which of the following statements about polymers are TRUE?",
|
|
"options": {
|
|
"A": "Nylon 6,6 is formed by condensation polymerisation of hexamethylenediamine and adipic acid.",
|
|
"B": "Natural rubber is a polyisoprene (addition polymer of isoprene units).",
|
|
"C": "Teflon is a branched-chain polymer of tetrafluoroethylene.",
|
|
"D": "Buna-S is a copolymer of butadiene and styrene."
|
|
},
|
|
"correct_answer": {
|
|
"value": [
|
|
"A",
|
|
"B",
|
|
"D"
|
|
]
|
|
},
|
|
"solution_text": "(A) $n\\,H_2N(CH_2)_6NH_2 + n\\,HOOC(CH_2)_4COOH \\to [-NH(CH_2)_6NHCO(CH_2)_4CO-]_n + 2n\\,H_2O$ — condensation (polyamide). TRUE.\n(B) Natural rubber is cis-1,4-polyisoprene, an addition polymer of 2-methyl-1,3-butadiene (isoprene). TRUE.\n(C) Teflon $[-CF_2-CF_2-]_n$ is a LINEAR addition polymer of tetrafluoroethylene, not branched. FALSE.\n(D) Buna-S: buta(diene) + S(tyrene), a copolymer made by free-radical copolymerisation. TRUE.\nHence, the correct options are (A), (B) and (D)."
|
|
},
|
|
{
|
|
"id": "b9e89b65-1a6a-452c-9197-6c891eb2e29e",
|
|
"section": "General",
|
|
"subject": "Chemistry",
|
|
"topic": "Solid State",
|
|
"subtopic": "Density of a unit cell",
|
|
"difficulty": "Easy",
|
|
"type": "Numerical",
|
|
"language": "English",
|
|
"question_text": "Sodium metal crystallises in a body-centred cubic lattice. The edge length of the unit cell is $4.29\\ \\text{\\AA}$. The density of sodium is ____ $\\mathrm{g\\,cm^{-3}}$ (molar mass of Na $= 23\\ \\mathrm{g\\,mol^{-1}}$, $N_A = 6.0\\times10^{23}\\ \\mathrm{mol^{-1}}$; round off to two decimal places).",
|
|
"options": null,
|
|
"correct_answer": {
|
|
"value": "0.97"
|
|
},
|
|
"solution_text": "Step 1: For bcc, $Z = 2$ atoms per unit cell.\nStep 2: Density $\\rho = \\dfrac{Z\\,M}{N_A\\,a^3}$.\nStep 3: $a^3 = (4.29\\times10^{-8}\\ \\mathrm{cm})^3 = 7.89\\times10^{-23}\\ \\mathrm{cm^3}$.\nStep 4: $\\rho = \\dfrac{2\\times23}{(6.0\\times10^{23})(7.89\\times10^{-23})} = \\dfrac{46}{47.34} = 0.97\\ \\mathrm{g\\,cm^{-3}}$."
|
|
},
|
|
{
|
|
"id": "bedc7ef3-046f-4819-83d1-d4698c0d303c",
|
|
"section": "General",
|
|
"subject": "Chemistry",
|
|
"topic": "Thermodynamics",
|
|
"subtopic": "Enthalpy of formation from bond enthalpies",
|
|
"difficulty": "Medium",
|
|
"type": "Numerical",
|
|
"language": "English",
|
|
"question_text": "For the reaction\n$H_2(g) + Cl_2(g) \\to 2HCl(g)$\nthe bond enthalpies are $H-H = 435\\ \\mathrm{kJ\\,mol^{-1}}$, $Cl-Cl = 240\\ \\mathrm{kJ\\,mol^{-1}}$ and $H-Cl = 430\\ \\mathrm{kJ\\,mol^{-1}}$. The magnitude of $\\Delta_r H$ for the reaction is ____ $\\mathrm{kJ\\,mol^{-1}}$ (round off to two decimal places).",
|
|
"options": null,
|
|
"correct_answer": {
|
|
"value": "185.00"
|
|
},
|
|
"solution_text": "Step 1: Bonds broken: one $H-H$ and one $Cl-Cl$: $435 + 240 = 675\\ \\mathrm{kJ\\,mol^{-1}}$.\nStep 2: Bonds formed: two $H-Cl$: $2 \\times 430 = 860\\ \\mathrm{kJ\\,mol^{-1}}$ released.\nStep 3: $\\Delta_r H = \\Sigma H_{broken} - \\Sigma H_{formed} = 675 - 860 = -185\\ \\mathrm{kJ\\,mol^{-1}}$.\nThe magnitude $|\\Delta_r H| = 185.00\\ \\mathrm{kJ\\,mol^{-1}}$ (the reaction is exothermic)."
|
|
},
|
|
{
|
|
"id": "ac3f9cd2-69b5-436c-b6b5-79c2a4fdb17b",
|
|
"section": "General",
|
|
"subject": "Chemistry",
|
|
"topic": "Solutions",
|
|
"subtopic": "Colligative properties - freezing point depression",
|
|
"difficulty": "Medium",
|
|
"type": "Numerical",
|
|
"language": "English",
|
|
"question_text": "The depression in freezing point of a $0.10\\ \\mathrm{m}$ aqueous solution of a non-electrolyte solute is ($K_f$ of water $= 1.86\\ \\mathrm{K\\,kg\\,mol^{-1}}$) ____ K. (Enter the magnitude, round off to two decimal places.)",
|
|
"options": null,
|
|
"correct_answer": {
|
|
"value": "0.19"
|
|
},
|
|
"solution_text": "Step 1: For a non-electrolyte, van't Hoff factor $i = 1$.\nStep 2: $\\Delta T_f = i\\,K_f\\,m = 1 \\times 1.86 \\times 0.10 = 0.186\\ \\mathrm{K}$.\nStep 3: Rounded to two decimal places: $0.19\\ \\mathrm{K}$."
|
|
},
|
|
{
|
|
"id": "582c0d66-0bee-4b22-adfb-c306f215ff90",
|
|
"section": "General",
|
|
"subject": "Chemistry",
|
|
"topic": "p-Block and Coordination Chemistry",
|
|
"subtopic": "Mixed matching (List-I / List-II)",
|
|
"difficulty": "Hard",
|
|
"type": "MCQ_Multiple",
|
|
"language": "English",
|
|
"question_text": "Match the species in List-I with the property in List-II.\n\nList-I:\n(P) $XeF_4$\n(Q) $SF_4$\n(R) $PCl_5$\n(S) $BF_3$\n\nList-II:\n(1) Square planar shape\n(2) See-saw shape\n(3) Trigonal bipyramidal shape\n(4) Trigonal planar shape\n(5) Pyramidal shape\n\nWhich of the following matchings are CORRECT?",
|
|
"options": {
|
|
"A": "P $\\to$ 1",
|
|
"B": "Q $\\to$ 2",
|
|
"C": "R $\\to$ 3",
|
|
"D": "S $\\to$ 5"
|
|
},
|
|
"correct_answer": {
|
|
"value": [
|
|
"A",
|
|
"B",
|
|
"C"
|
|
]
|
|
},
|
|
"solution_text": "(P) $XeF_4$: Xe has $sp^3d^2$ electron geometry (octahedral) with 2 lone pairs trans to each other $\\Rightarrow$ square planar; P-1 correct. TRUE.\n(Q) $SF_4$: S has $sp^3d$ electron geometry with one equatorial lone pair $\\Rightarrow$ see-saw; Q-2 correct. TRUE.\n(R) $PCl_5$: P has $sp^3d$ with no lone pairs $\\Rightarrow$ trigonal bipyramidal; R-3 correct. TRUE.\n(S) $BF_3$: B is $sp^2$, electron-deficient $\\Rightarrow$ trigonal planar, so the correct match is S-4, not S-5. FALSE.\nHence, the correct options are (A), (B) and (C)."
|
|
},
|
|
{
|
|
"id": "908efdb6-792b-4aa4-a468-c4f3899e1d4b",
|
|
"section": "General",
|
|
"subject": "Mathematics",
|
|
"topic": "Complex Numbers and Quadratic Equations",
|
|
"subtopic": "Modulus and conjugate of complex numbers",
|
|
"difficulty": "Easy",
|
|
"type": "MCQ_Single",
|
|
"language": "English",
|
|
"question_text": "If $z$ is a complex number satisfying $z\\bar{z} + z + \\bar{z} = 8$, then $|z + 1|$ is equal to:",
|
|
"options": {
|
|
"A": "$2$",
|
|
"B": "$3$",
|
|
"C": "$4$",
|
|
"D": "$9$"
|
|
},
|
|
"correct_answer": {
|
|
"value": "B"
|
|
},
|
|
"solution_text": "Step 1: Write $z = x + iy$, so $z\\bar z = x^2 + y^2$ and $z + \\bar z = 2x$.\nThe condition becomes $x^2 + y^2 + 2x = 8$.\nStep 2: Completing the square: $(x+1)^2 + y^2 = 9$, i.e. $|z+1|^2 = 9$.\nStep 3: Hence $|z+1| = 3$.\nHence, the correct option is (B)."
|
|
},
|
|
{
|
|
"id": "4465e1e4-16eb-44c8-8c8e-9836eadcd67e",
|
|
"section": "General",
|
|
"subject": "Mathematics",
|
|
"topic": "Sequences and Series",
|
|
"subtopic": "Arithmetic progression - sum of terms",
|
|
"difficulty": "Easy",
|
|
"type": "MCQ_Single",
|
|
"language": "English",
|
|
"question_text": "The sum of all natural numbers between 1 and 100 that are NOT divisible by 3 is:",
|
|
"options": {
|
|
"A": "3267",
|
|
"B": "3367",
|
|
"C": "3467",
|
|
"D": "3567"
|
|
},
|
|
"correct_answer": {
|
|
"value": "B"
|
|
},
|
|
"solution_text": "Step 1: Sum of all natural numbers from 1 to 100: $S_1 = \\dfrac{100 \\times 101}{2} = 5050$.\nStep 2: Sum of multiples of 3 up to 99: $\\dfrac{33 \\times (3 + 99)}{2} = 33 \\times 51 = 1683$.\nStep 3: Required sum $= 5050 - 1683 = 3367$.\nHence, the correct option is (B)."
|
|
},
|
|
{
|
|
"id": "5266c185-750d-4cd5-87c3-06b19265fcdb",
|
|
"section": "General",
|
|
"subject": "Mathematics",
|
|
"topic": "Trigonometry and Inverse Trigonometry",
|
|
"subtopic": "Trigonometric equations",
|
|
"difficulty": "Medium",
|
|
"type": "MCQ_Single",
|
|
"language": "English",
|
|
"question_text": "The number of solutions of the equation $|\\cos x| = \\sin x$ in the interval $[0, 2\\pi]$ is:",
|
|
"options": {
|
|
"A": "1",
|
|
"B": "2",
|
|
"C": "3",
|
|
"D": "4"
|
|
},
|
|
"correct_answer": {
|
|
"value": "B"
|
|
},
|
|
"solution_text": "Step 1: Since $|\\cos x| \\ge 0$, we need $\\sin x \\ge 0$, so $x \\in [0, \\pi]$.\nStep 2: For $x \\in [0, \\tfrac{\\pi}{2}]$, $|\\cos x| = \\cos x$, so the equation is $\\cos x = \\sin x \\Rightarrow \\tan x = 1 \\Rightarrow x = \\dfrac{\\pi}{4}$.\nStep 3: For $x \\in (\\tfrac{\\pi}{2}, \\pi]$, $|\\cos x| = -\\cos x$, so $-\\cos x = \\sin x \\Rightarrow \\tan x = -1 \\Rightarrow x = \\dfrac{3\\pi}{4}$.\nStep 4: Check the boundary $x = \\pi$: $|\\cos \\pi| = 1 \\neq \\sin \\pi = 0$; not a solution.\nSo the solutions are $x = \\dfrac{\\pi}{4}$ and $x = \\dfrac{3\\pi}{4}$ — exactly 2 solutions.\nHence, the correct option is (B)."
|
|
},
|
|
{
|
|
"id": "154214c9-4711-4f2c-a417-53e2f2c2466e",
|
|
"section": "General",
|
|
"subject": "Mathematics",
|
|
"topic": "Straight Lines and Circles",
|
|
"subtopic": "Intercept form of a line",
|
|
"difficulty": "Easy",
|
|
"type": "MCQ_Single",
|
|
"language": "English",
|
|
"question_text": "The length of the intercept made by the line $3x + 4y = 12$ between the coordinate axes is:",
|
|
"options": {
|
|
"A": "5",
|
|
"B": "7",
|
|
"C": "9",
|
|
"D": "12"
|
|
},
|
|
"correct_answer": {
|
|
"value": "A"
|
|
},
|
|
"solution_text": "Step 1: The line meets the $x$-axis at $y = 0$: $x = 4 \\Rightarrow A(4, 0)$; and the $y$-axis at $x = 0$: $y = 3 \\Rightarrow B(0, 3)$.\nStep 2: Intercept length $AB = \\sqrt{(4-0)^2 + (0-3)^2} = \\sqrt{16 + 9} = \\sqrt{25} = 5$.\nHence, the correct option is (A)."
|
|
},
|
|
{
|
|
"id": "bab14720-d2f2-4c86-b126-1027b1466278",
|
|
"section": "General",
|
|
"subject": "Mathematics",
|
|
"topic": "Limits, Continuity and Differentiability",
|
|
"subtopic": "Limit of the form $1^\\infty$",
|
|
"difficulty": "Medium",
|
|
"type": "MCQ_Single",
|
|
"language": "English",
|
|
"question_text": "$\\displaystyle\\lim_{x \\to \\infty}\\left(1 + \\frac{3}{x}\\right)^{x+2}$ is equal to:",
|
|
"options": {
|
|
"A": "$e^2$",
|
|
"B": "$e^3$",
|
|
"C": "$e^5$",
|
|
"D": "$e^6$"
|
|
},
|
|
"correct_answer": {
|
|
"value": "B"
|
|
},
|
|
"solution_text": "Step 1: Split the exponent: $\\left(1+\\dfrac{3}{x}\\right)^{x+2} = \\left(1+\\dfrac{3}{x}\\right)^{x} \\cdot \\left(1+\\dfrac{3}{x}\\right)^{2}$.\nStep 2: $\\displaystyle\\lim_{x\\to\\infty}\\left(1+\\dfrac{3}{x}\\right)^{x} = e^{3}$ (standard $1^\\infty$ limit).\nStep 3: $\\displaystyle\\lim_{x\\to\\infty}\\left(1+\\dfrac{3}{x}\\right)^{2} = 1$.\nStep 4: Hence the limit is $e^3 \\times 1 = e^3$.\nHence, the correct option is (B)."
|
|
},
|
|
{
|
|
"id": "bbd0eb6c-e69d-466c-880a-e6c08818831f",
|
|
"section": "General",
|
|
"subject": "Mathematics",
|
|
"topic": "Definite Integration",
|
|
"subtopic": "Definite integral using symmetry (King property)",
|
|
"difficulty": "Medium",
|
|
"type": "MCQ_Single",
|
|
"language": "English",
|
|
"question_text": "$\\displaystyle\\int_0^{\\pi/2} \\frac{1}{1 + \\sqrt{\\tan x}}\\,dx$ is equal to:",
|
|
"options": {
|
|
"A": "$\\dfrac{\\pi}{4}$",
|
|
"B": "$\\dfrac{\\pi}{2}$",
|
|
"C": "$\\dfrac{\\pi}{8}$",
|
|
"D": "$\\pi$"
|
|
},
|
|
"correct_answer": {
|
|
"value": "A"
|
|
},
|
|
"solution_text": "Let $I = \\displaystyle\\int_0^{\\pi/2} \\dfrac{dx}{1+\\sqrt{\\tan x}}$.\nStep 1: Apply the substitution $x \\to \\dfrac{\\pi}{2} - x$:\n$I = \\displaystyle\\int_0^{\\pi/2} \\dfrac{dx}{1+\\sqrt{\\cot x}} = \\int_0^{\\pi/2} \\dfrac{\\sqrt{\\tan x}}{1+\\sqrt{\\tan x}}\\,dx$.\nStep 2: Adding the two expressions: $2I = \\displaystyle\\int_0^{\\pi/2} \\left(\\dfrac{1}{1+\\sqrt{\\tan x}} + \\dfrac{\\sqrt{\\tan x}}{1+\\sqrt{\\tan x}}\\right) dx = \\int_0^{\\pi/2} dx = \\dfrac{\\pi}{2}$.\nStep 3: $I = \\dfrac{\\pi}{4}$.\nHence, the correct option is (A)."
|
|
},
|
|
{
|
|
"id": "f14cd525-53f0-4ee4-af49-8f82ee9c501e",
|
|
"section": "General",
|
|
"subject": "Mathematics",
|
|
"topic": "Three Dimensional Geometry",
|
|
"subtopic": "Angle between a line and a plane",
|
|
"difficulty": "Medium",
|
|
"type": "MCQ_Single",
|
|
"language": "English",
|
|
"question_text": "The angle between the line $\\dfrac{x-1}{2} = \\dfrac{y}{3} = \\dfrac{z+1}{6}$ and the plane $2x - 2y + z = 5$ is:",
|
|
"options": {
|
|
"A": "$\\sin^{-1}\\dfrac{8}{21}$",
|
|
"B": "$\\cos^{-1}\\dfrac{8}{21}$",
|
|
"C": "$\\sin^{-1}\\dfrac{4}{21}$",
|
|
"D": "$\\sin^{-1}\\dfrac{2}{7}$"
|
|
},
|
|
"correct_answer": {
|
|
"value": "C"
|
|
},
|
|
"solution_text": "Step 1: The line has direction vector $\\vec{d} = (2, 3, 6)$ and the plane has normal $\\vec{n} = (2, -2, 1)$.\nStep 2: $|\\vec{d}| = \\sqrt{4+9+36} = 7$; $|\\vec{n}| = \\sqrt{4+4+1} = 3$.\nStep 3: $\\vec{d}\\cdot\\vec{n} = (2)(2) + (3)(-2) + (6)(1) = 4 - 6 + 6 = 4$.\nStep 4: For the angle $\\theta$ between a line and a plane, $\\sin\\theta = \\dfrac{|\\vec{d}\\cdot\\vec{n}|}{|\\vec{d}|\\,|\\vec{n}|} = \\dfrac{4}{21}$, so $\\theta = \\sin^{-1}\\dfrac{4}{21}$.\nHence, the correct option is (C)."
|
|
},
|
|
{
|
|
"id": "3def3940-8a8f-4041-ab3b-8cbaefcb3e2e",
|
|
"section": "General",
|
|
"subject": "Mathematics",
|
|
"topic": "Matrices and Determinants",
|
|
"subtopic": "Adjoint and determinant properties",
|
|
"difficulty": "Medium",
|
|
"type": "MCQ_Single",
|
|
"language": "English",
|
|
"question_text": "If $A$ is a $3\\times3$ matrix with $|A| = 4$, then $|2\\,\\mathrm{adj}(A)|$ is equal to:",
|
|
"options": {
|
|
"A": "32",
|
|
"B": "64",
|
|
"C": "128",
|
|
"D": "256"
|
|
},
|
|
"correct_answer": {
|
|
"value": "C"
|
|
},
|
|
"solution_text": "Step 1: For an $n \\times n$ matrix, $|\\mathrm{adj}(A)| = |A|^{n-1}$. Here $n = 3$, so $|\\mathrm{adj}(A)| = |A|^2 = 16$.\nStep 2: For a $3\\times3$ matrix $M$ and scalar $k$, $|kM| = k^3|M|$, so $|2\\,\\mathrm{adj}(A)| = 2^3 \\times 16 = 128$.\nHence, the correct option is (C)."
|
|
},
|
|
{
|
|
"id": "9191e9a1-2425-4a20-8ae0-17bd0889f35f",
|
|
"section": "General",
|
|
"subject": "Mathematics",
|
|
"topic": "Sets, Relations and Functions",
|
|
"subtopic": "Cubic function - injectivity, surjectivity and range",
|
|
"difficulty": "Hard",
|
|
"type": "MCQ_Multiple",
|
|
"language": "English",
|
|
"question_text": "Let $f: \\mathbb{R} \\to \\mathbb{R}$ be defined by $f(x) = x^3 - 3x$. Which of the following statements are TRUE?",
|
|
"options": {
|
|
"A": "$f$ is not one-one.",
|
|
"B": "$f$ is onto.",
|
|
"C": "$f$ is a periodic function.",
|
|
"D": "The range of $f$ restricted to the interval $[-1, 1]$ is $[-2, 2]$."
|
|
},
|
|
"correct_answer": {
|
|
"value": [
|
|
"A",
|
|
"B",
|
|
"D"
|
|
]
|
|
},
|
|
"solution_text": "(A) $f(0) = f(\\sqrt{3}) = f(-\\sqrt{3}) = 0$: three distinct inputs give the same value, so $f$ is not injective. TRUE.\n(B) $f$ is a continuous odd-degree polynomial with $f(x) \\to -\\infty$ as $x \\to -\\infty$ and $f(x) \\to +\\infty$ as $x \\to +\\infty$; by the intermediate value theorem it attains every real value. TRUE.\n(C) A non-constant polynomial cannot be periodic (a periodic continuous function on $\\mathbb{R}$ is bounded, while $|f(x)| \\to \\infty$). FALSE.\n(D) On $[-1, 1]$: $f'(x) = 3x^2 - 3 < 0$ for $-1 < x < 1$, so $f$ is strictly decreasing there; $f(-1) = 2$ and $f(1) = -2$, giving the range $[-2, 2]$. TRUE.\nHence, the correct options are (A), (B) and (D)."
|
|
},
|
|
{
|
|
"id": "d6558b31-30af-43b0-94b2-7240808dfd7d",
|
|
"section": "General",
|
|
"subject": "Mathematics",
|
|
"topic": "Probability",
|
|
"subtopic": "Classical probability and independence of events",
|
|
"difficulty": "Hard",
|
|
"type": "MCQ_Multiple",
|
|
"language": "English",
|
|
"question_text": "Two fair dice are rolled. Let $A$ be the event \"the sum of the numbers shown is 8\" and $B$ be the event \"at least one die shows 6\". Which of the following statements are TRUE?",
|
|
"options": {
|
|
"A": "$P(A) = \\dfrac{5}{36}$.",
|
|
"B": "$P(A \\cap B) = \\dfrac{1}{18}$.",
|
|
"C": "$A$ and $B$ are independent events.",
|
|
"D": "$P(B) = \\dfrac{11}{36}$."
|
|
},
|
|
"correct_answer": {
|
|
"value": [
|
|
"A",
|
|
"B",
|
|
"D"
|
|
]
|
|
},
|
|
"solution_text": "(A) Favourable outcomes for sum 8: $(2,6), (3,5), (4,4), (5,3), (6,2)$ — 5 of 36 outcomes, so $P(A) = \\dfrac{5}{36}$. TRUE.\n(B) Among these, those with at least one six are $(2,6)$ and $(6,2)$, so $P(A\\cap B) = \\dfrac{2}{36} = \\dfrac{1}{18}$. TRUE.\n(D) $P(B) = 1 - P(\\text{no six}) = 1 - \\dfrac{5\\times5}{36} = \\dfrac{11}{36}$. TRUE.\n(C) $P(A)\\,P(B) = \\dfrac{5}{36}\\times\\dfrac{11}{36} = \\dfrac{55}{1296} \\approx 0.042$, while $P(A\\cap B) = \\dfrac{4}{1296} \\approx 0.003$. Since $P(A\\cap B) \\neq P(A)P(B)$, the events are dependent. FALSE.\nHence, the correct options are (A), (B) and (D)."
|
|
},
|
|
{
|
|
"id": "e9df289c-e8eb-4b1f-a9d7-c2cbc1599871",
|
|
"section": "General",
|
|
"subject": "Mathematics",
|
|
"topic": "Vector Algebra",
|
|
"subtopic": "Scalar triple product, coplanarity and cross product",
|
|
"difficulty": "Medium",
|
|
"type": "MCQ_Multiple",
|
|
"language": "English",
|
|
"question_text": "Let $\\vec{a} = \\hat{i} + \\hat{j} + \\hat{k}$, $\\vec{b} = 2\\hat{i} - \\hat{j} + \\hat{k}$ and $\\vec{c} = \\hat{i} - 2\\hat{j} + 3\\hat{k}$. Which of the following statements are TRUE?",
|
|
"options": {
|
|
"A": "The volume of the parallelepiped with coterminous edges $\\vec{a}, \\vec{b}, \\vec{c}$ is 11 cubic units.",
|
|
"B": "The vectors $\\vec{a}, \\vec{b}, \\vec{c}$ are non-coplanar.",
|
|
"C": "A unit vector perpendicular to both $\\vec{a}$ and $\\vec{b}$ is $\\dfrac{2\\hat{i} - \\hat{j} - 3\\hat{k}}{\\sqrt{14}}$.",
|
|
"D": "$\\vec{a} \\times \\vec{c}$ is parallel to $\\hat{j} + \\hat{k}$."
|
|
},
|
|
"correct_answer": {
|
|
"value": [
|
|
"B",
|
|
"C"
|
|
]
|
|
},
|
|
"solution_text": "Step 1: Scalar triple product:\n$[\\vec{a}\\,\\vec{b}\\,\\vec{c}] = \\begin{vmatrix} 1 & 1 & 1\\\\ 2 & -1 & 1\\\\ 1 & -2 & 3\\end{vmatrix} = 1(-3+2) - 1(6-1) + 1(-4+1) = -1 - 5 - 3 = -9$.\nVolume of the parallelepiped $= |[\\vec{a}\\,\\vec{b}\\,\\vec{c}]| = 9$, not 11. FALSE for (A).\nStep 2: Since $[\\vec{a}\\vec{b}\\vec{c}] = -9 \\neq 0$, the three vectors are non-coplanar. TRUE for (B).\nStep 3: $\\vec{a}\\times\\vec{b} = \\begin{vmatrix} \\hat i & \\hat j & \\hat k\\\\ 1&1&1\\\\ 2&-1&1\\end{vmatrix} = (2)\\hat i - (-1)\\hat j + (-3)\\hat k = (2, 1, -3)$, with magnitude $\\sqrt{4+1+9} = \\sqrt{14}$.\nA unit vector perpendicular to both $\\vec{a}$ and $\\vec{b}$ is $\\dfrac{(2,-1,-3)}{\\sqrt{14}} = \\dfrac{2\\hat{i} - \\hat{j} - 3\\hat{k}}{\\sqrt{14}}$. TRUE for (C).\nStep 4: $\\vec{a}\\times\\vec{c} = \\begin{vmatrix} \\hat i & \\hat j & \\hat k\\\\ 1&1&1\\\\ 1&-2&3\\end{vmatrix} = (5, -2, -3)$.\n$(5,-2,-3)\\cdot(0,1,1) = -5 \\neq 0$, so $\\vec{a}\\times\\vec{c}$ is not parallel to $\\hat{j} + \\hat{k}$. FALSE for (D).\nHence, the correct options are (B) and (C)."
|
|
},
|
|
{
|
|
"id": "690c0ab0-71f0-4589-b74c-8a951cba371e",
|
|
"section": "General",
|
|
"subject": "Mathematics",
|
|
"topic": "Differential Equations",
|
|
"subtopic": "Linear first order ODE with integrating factor",
|
|
"difficulty": "Medium",
|
|
"type": "MCQ_Multiple",
|
|
"language": "English",
|
|
"question_text": "The differential equation $\\dfrac{dy}{dx} + y\\tan x = \\sec x$ with $y(0) = 0$ is considered. Which of the following statements are TRUE?",
|
|
"options": {
|
|
"A": "The integrating factor of the equation is $\\sec x$.",
|
|
"B": "The solution of the equation is $y = \\sin x$.",
|
|
"C": "The solution satisfies $y\\left(\\dfrac{\\pi}{4}\\right) = \\dfrac{1}{\\sqrt{2}}$.",
|
|
"D": "The solution is periodic with period $2\\pi$."
|
|
},
|
|
"correct_answer": {
|
|
"value": [
|
|
"A",
|
|
"B",
|
|
"C",
|
|
"D"
|
|
]
|
|
},
|
|
"solution_text": "Step 1: This is a linear first-order ODE $\\dfrac{dy}{dx} + P(x)y = Q(x)$ with $P(x) = \\tan x$.\nIntegrating factor $= e^{\\int \\tan x\\,dx} = e^{\\ln(\\sec x)} = \\sec x$. TRUE for (A).\nStep 2: The solution is $y\\,\\sec x = \\displaystyle\\int \\sec x\\cdot\\sec x\\,dx = \\int \\sec^2 x\\,dx = \\tan x + C$, i.e. $y = \\sin x + C\\cos x$.\nApplying $y(0) = 0$: $0 = 0 + C \\Rightarrow C = 0$, so $y = \\sin x$. TRUE for (B).\nStep 3: $y\\left(\\dfrac{\\pi}{4}\\right) = \\sin\\dfrac{\\pi}{4} = \\dfrac{1}{\\sqrt{2}}$. TRUE for (C).\nStep 4: $\\sin(x + 2\\pi) = \\sin x$ for all $x$, so the solution has period $2\\pi$. TRUE for (D).\nHence, all of (A), (B), (C) and (D) are correct."
|
|
},
|
|
{
|
|
"id": "ff371961-91bd-4486-954a-d6f7f9362741",
|
|
"section": "General",
|
|
"subject": "Mathematics",
|
|
"topic": "Straight Lines and Circles",
|
|
"subtopic": "Circle - position of points and tangents",
|
|
"difficulty": "Hard",
|
|
"type": "MCQ_Multiple",
|
|
"language": "English",
|
|
"question_text": "The circle $x^2 + y^2 - 4x - 6y + 4 = 0$ is considered. Which of the following statements are TRUE?",
|
|
"options": {
|
|
"A": "The centre of the circle is $(2, 3)$ and its radius is 3.",
|
|
"B": "The point $(2, 2)$ lies inside the circle.",
|
|
"C": "The point $(5, 6)$ lies inside the circle.",
|
|
"D": "The line $x = 5$ is a tangent to the circle."
|
|
},
|
|
"correct_answer": {
|
|
"value": [
|
|
"A",
|
|
"B",
|
|
"D"
|
|
]
|
|
},
|
|
"solution_text": "Step 1: Completing the square: $(x-2)^2 + (y-3)^2 = 4 + 9 - 4 = 9$, so the centre is $(2, 3)$ and the radius is $3$. TRUE for (A).\nStep 2: Distance of $(2, 2)$ from the centre: $\\sqrt{(2-2)^2 + (2-3)^2} = 1 < 3$, so the point lies inside the circle. TRUE for (B).\nStep 3: Distance of $(5, 6)$ from the centre: $\\sqrt{(5-2)^2 + (6-3)^2} = \\sqrt{18} \\approx 4.24 > 3$, so the point lies outside the circle. FALSE for (C).\nStep 4: Distance of the line $x = 5$ from the centre $(2, 3)$: $|2 - 5| = 3 = $ radius, so the line is a tangent to the circle. TRUE for (D).\nHence, the correct options are (A), (B) and (D)."
|
|
},
|
|
{
|
|
"id": "3c3b8976-f086-4500-a9bb-822affdf9049",
|
|
"section": "General",
|
|
"subject": "Mathematics",
|
|
"topic": "Permutations and Combinations, Binomial Theorem",
|
|
"subtopic": "Binomial theorem - coefficients and divisibility",
|
|
"difficulty": "Medium",
|
|
"type": "MCQ_Multiple",
|
|
"language": "English",
|
|
"question_text": "Which of the following statements are TRUE?",
|
|
"options": {
|
|
"A": "The coefficient of $x^2$ in the expansion of $(1+x)^6$ is 15.",
|
|
"B": "The number of terms in the expansion of $(1+x)^{10}$ is 11.",
|
|
"C": "The greatest coefficient in the expansion of $(1+x)^{10}$ is $\\binom{10}{5}$.",
|
|
"D": "$2^{10} - 1$ is divisible by 7."
|
|
},
|
|
"correct_answer": {
|
|
"value": [
|
|
"A",
|
|
"B",
|
|
"C"
|
|
]
|
|
},
|
|
"solution_text": "(A) $\\binom{6}{2} = \\dfrac{6\\times5}{2} = 15$. TRUE.\n(B) The expansion of $(1+x)^n$ has $n+1$ terms, so $(1+x)^{10}$ has 11 terms. TRUE.\n(C) For even $n$, the binomial coefficients strictly increase up to the middle term; the greatest coefficient in $(1+x)^{10}$ is $\\binom{10}{5} = 252$. TRUE.\n(D) $2^{10} - 1 = 1023 = 7 \\times 146 + 1$, which is not divisible by 7 (in fact $1023 = 3 \\times 11 \\times 31$). FALSE.\nHence, the correct options are (A), (B) and (C)."
|
|
},
|
|
{
|
|
"id": "b23f5408-f103-4278-8838-50a8062a666a",
|
|
"section": "General",
|
|
"subject": "Mathematics",
|
|
"topic": "Definite Integration",
|
|
"subtopic": "Definite integral of an even function",
|
|
"difficulty": "Easy",
|
|
"type": "Numerical",
|
|
"language": "English",
|
|
"question_text": "The value of $\\displaystyle\\int_{-1}^{1} (x^2 + 2x + 5)\\,dx$ is ____ (round off to two decimal places).",
|
|
"options": null,
|
|
"correct_answer": {
|
|
"value": "10.67"
|
|
},
|
|
"solution_text": "Step 1: Split the integral: $I = \\displaystyle\\int_{-1}^{1} x^2\\,dx + 2\\int_{-1}^{1} x\\,dx + \\int_{-1}^{1} 5\\,dx$.\nStep 2: The middle integral vanishes because $x$ is an odd function integrated over a symmetric interval: $\\int_{-1}^{1} x\\,dx = 0$.\nStep 3: $\\int_{-1}^{1} x^2\\,dx = \\left[\\dfrac{x^3}{3}\\right]_{-1}^{1} = \\dfrac{2}{3}$ and $\\int_{-1}^{1} 5\\,dx = 10$.\nStep 4: $I = \\dfrac{2}{3} + 0 + 10 = \\dfrac{32}{3} \\approx 10.67$."
|
|
},
|
|
{
|
|
"id": "7cc9d8f9-114f-4cfb-9e3a-0412d6eb8a62",
|
|
"section": "General",
|
|
"subject": "Mathematics",
|
|
"topic": "Sequences and Series",
|
|
"subtopic": "Infinite geometric series",
|
|
"difficulty": "Easy",
|
|
"type": "Numerical",
|
|
"language": "English",
|
|
"question_text": "The sum of the infinite series $1 + \\dfrac{1}{3} + \\dfrac{1}{9} + \\dfrac{1}{27} + \\cdots$ is ____ (round off to two decimal places).",
|
|
"options": null,
|
|
"correct_answer": {
|
|
"value": "1.50"
|
|
},
|
|
"solution_text": "Step 1: This is a geometric series with first term $a = 1$ and common ratio $r = \\dfrac{1}{3}$, and $|r| < 1$.\nStep 2: $S_\\infty = \\dfrac{a}{1-r} = \\dfrac{1}{1 - \\frac{1}{3}} = \\dfrac{3}{2} = 1.50$."
|
|
},
|
|
{
|
|
"id": "c1eba270-f644-45ff-b1dc-501f2621c28c",
|
|
"section": "General",
|
|
"subject": "Mathematics",
|
|
"topic": "Probability",
|
|
"subtopic": "Binomial distribution - probability of exactly r successes",
|
|
"difficulty": "Medium",
|
|
"type": "Numerical",
|
|
"language": "English",
|
|
"question_text": "A fair coin is tossed 5 times. The probability of obtaining exactly 3 heads is ____ (round off to two decimal places).",
|
|
"options": null,
|
|
"correct_answer": {
|
|
"value": "0.31"
|
|
},
|
|
"solution_text": "Step 1: The tosses form a binomial distribution with $n = 5$ and $p = q = \\dfrac{1}{2}$.\nStep 2: $P(X = 3) = \\dbinom{5}{3}\\left(\\dfrac{1}{2}\\right)^3\\left(\\dfrac{1}{2}\\right)^2 = 10 \\times \\dfrac{1}{32} = \\dfrac{10}{32} = 0.3125$.\nStep 3: Rounded to two decimal places: $0.31$."
|
|
},
|
|
{
|
|
"id": "67c5cae0-68ce-4449-bb72-3ab9e29bd1f6",
|
|
"section": "General",
|
|
"subject": "Mathematics",
|
|
"topic": "Calculus - Limits, Integrals and Differential Equations",
|
|
"subtopic": "Mixed matching (List-I / List-II)",
|
|
"difficulty": "Hard",
|
|
"type": "MCQ_Multiple",
|
|
"language": "English",
|
|
"question_text": "Match the expression in List-I with its value in List-II.\n\nList-I:\n(P) $\\displaystyle\\lim_{x \\to 0} \\frac{\\sin 3x}{x}$\n(Q) $\\displaystyle\\int_0^1 x\\,e^{x}\\,dx$\n(R) $\\displaystyle\\lim_{n \\to \\infty}\\left(1 + \\frac{1}{n}\\right)^{n}$\n(S) $\\dfrac{d}{dx}\\left[x^2 e^x\\right]$ at $x = 1$\n\nList-II:\n(1) $3$\n(2) $1$\n(3) $e$\n(4) $2e$\n(5) $e - 1$\n\nWhich of the following matchings are CORRECT?",
|
|
"options": {
|
|
"A": "P $\\to$ 1",
|
|
"B": "Q $\\to$ 2",
|
|
"C": "R $\\to$ 3",
|
|
"D": "S $\\to$ 4"
|
|
},
|
|
"correct_answer": {
|
|
"value": [
|
|
"A",
|
|
"B",
|
|
"C"
|
|
]
|
|
},
|
|
"solution_text": "(P) $\\displaystyle\\lim_{x\\to0}\\dfrac{\\sin 3x}{x} = 3\\lim_{u\\to0}\\dfrac{\\sin u}{u}$ (with $u = 3x$) $= 3$; so P-1 is correct. TRUE.\n(Q) By integration by parts: $\\displaystyle\\int_0^1 xe^x\\,dx = \\left[xe^x\\right]_0^1 - \\int_0^1 e^x\\,dx = e - (e - 1) = 1$; so Q-2 is correct. TRUE.\n(R) $\\displaystyle\\lim_{n\\to\\infty}\\left(1+\\dfrac{1}{n}\\right)^{n} = e$; so R-3 is correct. TRUE.\n(S) $\\dfrac{d}{dx}\\left(x^2e^x\\right) = (2x + x^2)e^x$; at $x = 1$ the value is $3e \\approx 8.15$, not $2e \\approx 5.44$; so S-4 is incorrect. FALSE.\nHence, the correct options are (A), (B) and (C)."
|
|
}
|
|
] |