[
  {
    "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"
  }
]