question: A particle of mass m is initially at rest at the origin. It is subjected to a force and starts moving along the x-axis. Its kinetic energy K changes with time as dK/dt = γt, where γ is a positive constant of appropriate dimensions. Which of the following statements is (are) true?

response_choices:
(A) The force applied on the particle is constant
(B) The speed of the particle is proportional to time
(C) The distance of the particle from the origin increases linearly with time
(D) The force is conservative

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question_subject: physics


question: A wire is bent in the shape of a right angled triangle and is placed in front of a concave mirror of focal length f, as shown in the figure. Which of the figures shown in the four options qualitatively represent(s) the shape of the image of the bent wire? (These figures are not to scale.)

response_choices: A, B, C, D

answer: B

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question_subject: physics


question: In a radioactive decay chain, 232Th nucleus decays to 212Pb nucleus. Let N_α and N_β be the number of α and β^− particles, respectively, emitted in this decay process. Which of the following statements is (are) true?

answer: (C) N_β = 2

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question_subject: physics


question: In an experiment to measure the speed of sound by a resonating air column, a tuning fork of frequency 500 Hz is used. The length of the air column is varied by changing the level of water in the resonance tube. Two successive resonances are heard at air columns of length 50.7 cm and 83.9 cm. Which of the following statements is (are) true?

response_choices: (A) The speed of sound determined from this experiment is 332 m s^-1 (B) The end correction in this experiment is 0.9 cm (C) The wavelength of the sound wave is 66.4 cm (D) The resonance at 50.7 cm corresponds to the fundamental harmonic

answer: [Answer not provided]

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question_subject: physics


question: A solid horizontal surface is covered with a thin layer of oil. A rectangular block of mass m = 0.4 kg is at rest on this surface. An impulse of 1.0 N s is applied to the block at time t = 0 so that it starts moving along the x-axis with a velocity v(t) = v0e^(-t/τ), where v0 is a constant and τ = 4 s. The displacement of the block, in metres, at t = τ is _____________.

response_choices: No response choices provided

answer: Numerical answer required

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question_subject: physics


question: A ball is projected from the ground at an angle of 45° with the horizontal surface. It reaches a maximum height of 120 m and returns to the ground. Upon hitting the ground for the first time, it loses half of its kinetic energy. Immediately after the bounce, the velocity of the ball makes an angle of 30° with the horizontal surface. The maximum height it reaches after the bounce, in metres, is _____________.

response_choices: No response choices provided

answer: Numerical answer required

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question_subject: physics


question: A particle, of mass 10^(-3)kg and charge 1.0 C, is initially at rest. At time t = 0, the particle comes under the influence of an electric field E(t) = E0 sin ωt i, where E0 = 1.0 N C^(-1)and ω = 10³ rad s^(-1). Consider the effect of only the electrical force on the particle. Then the maximum speed, in m s^(-1), attained by the particle at subsequent times is _____________.

response_choices: No response choices provided

answer: Numerical answer required

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question_subject: physics


question: A moving coil galvanometer has 50 turns and each turn has an area 2 × 10^(-4) m². The magnetic field produced by the magnet inside the galvanometer is 0.02 T. The torsional constant of the suspension wire is 10^(-4) N m rad^(-1). When a current flows through the galvanometer, a full scale deflection occurs if the coil rotates by 0.2 rad. The resistance of the coil of the galvanometer is 50 Ω. This galvanometer is to be converted into an ammeter capable of measuring current in the range 0 → 1.0 A. For this purpose, a shunt resistance is to be added in parallel to the galvanometer. The value of this shunt resistance, in ohms, is _____________.

response_choices: No response choices provided

answer: Numerical answer required

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question_subject: physics


question: A steel wire of diameter 0.5 mm and Young's modulus 2 × 10¹¹ N m⁻² carries a load of mass M. The length of the wire with the load is 1.0 m. A vernier scale with 10 divisions is attached to the end of this wire. Next to the steel wire is a reference wire to which a main scale, of least count 1.0 mm, is attached. The 10 divisions of the vernier scale correspond to 9 divisions of the main scale. Initially, the zero of vernier scale coincides with the zero of main scale. If the load on the steel wire is increased by 1.2 kg, the vernier scale division which coincides with a main scale division is _________. Take g = 10 m s⁻² and π = 3.2.

response_choices: No response choices provided

answer: No answer provided

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question_subject: physics


question: One mole of a monatomic ideal gas undergoes an adiabatic expansion in which its volume becomes eight times its initial value. If the initial temperature of the gas is 100 K and the universal gas constant R = 8.0 J mol⁻¹K⁻¹, the decrease in its internal energy, in Joule, is________.

response_choices: No response choices provided

answer: No answer provided

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question_subject: physics


question: In a photoelectric experiment a parallel beam of monochromatic light with power of 200 W is incident on a perfectly absorbing cathode of work function 6.25 eV. The frequency of light is just above the threshold frequency so that the photoelectrons are emitted with negligible kinetic energy. Assume that the photoelectron emission efficiency is 100%. A potential difference of 500 V is applied between the cathode and the anode. All the emitted electrons are incident normally on the anode and are absorbed. The anode experiences a force F = n x 10⁻⁴ N due to the impact of the electrons. The value of n is _________. Mass of the electron mₑ = 9 × 10⁻³¹kg and 1.0 eV = 1.6 × 10⁻¹⁹ J.

response_choices: No response choices provided

answer: No answer provided

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question_subject: physics


question: Consider a hydrogen-like ionized atom with atomic number Z with a single electron. In the emission spectrum of this atom, the photon emitted in the n = 2 to n = 1 transition has energy 74.8 eV higher than the photon emitted in the n = 3 to n = 2 transition. The ionization energy of the hydrogen atom is 13.6 eV. The value of Z is _________.

response_choices: No response choices provided

answer: No answer provided

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question_subject: physics


question: The electric field E is measured at a point P(0, 0, d) generated due to various charge distributions and the dependence of E on d is found to be different for different charge distributions. List-I contains different relations between E and d. List-II describes different electric charge distributions, along with their locations. Match the functions in List-I with the related charge distributions in List-II.

answer: C) P → 5; Q → 3; R → 1, 2; S → 4

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question_subject: physics


question: A planet of mass M has two natural satellites with masses m1 and m2. The radii of their circular orbits are R1 and R2 respectively. Ignore the gravitational force between the satellites. Define v1, L1, K1 and T1 to be, respectively, the orbital speed, angular momentum, kinetic energy and Time period of revolution of satellite 1; and v2, L2, K2 and T2 to be the corresponding quantities of satellite 2. Given m1/m2 = 2 and R1/R2 = 1/4, match the ratios in List-I to the numbers in List-II.

response_choices:
A) P → 4; Q → 2; R → 1; S → 3
B) P → 3; Q → 2; R → 4; S → 1
C) P → 2; Q → 3; R → 1; S → 4
D) P → 2; Q → 3; R → 4; S → 1

answer: D

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question_subject: physics


question: One mole of a monatomic ideal gas undergoes four thermodynamic processes as shown schematically in the PV-diagram below. Among these four processes, one is isobaric, one is isochoric, one is isothermal and one is adiabatic. Match the processes mentioned in List-I with the corresponding statements in List-II.

response_choices:
A. P → 4; Q → 3; R → 1; S → 2
B. P → 1; Q → 3; R → 2; S → 4
C. P → 3; Q → 4; R → 1; S → 2
D. P → 3; Q → 4; R → 2; S → 1

answer: C

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question_subject: physics


question: In the List-I below, four different paths of a particle are given as functions of time. In these functions, α and β are positive constants of appropriate dimensions and α ≠ β. In each case, the force acting on the particle is either zero or conservative. In List-II, five physical quantities of the particle are mentioned: p̂ is the linear momentum, L̂ is the angular momentum about the origin, K is the kinetic energy, U is the potential energy and E is the total energy. Match each path in List-I with those quantities in List-II, which are conserved for that path.

response_choices: P. r⃗(t) = αt î + βt ĵ, Q. r⃗(t) = α cos ωt î + β sin ωt ĵ, R. r⃗(t) = α (cos ωt î + sin ωt ĵ), S. r⃗(t) = αt î + 1⁄2 t^2 ĵ

answer: (A) P → 1, 2, 3, 4, 5;  Q → 2, 5;  R → 2, 3, 4, 5;  S → 5 (B) P → 1, 2, 3, 4, 5;  Q → 3, 5;  R → 2, 3, 4, 5;  S → 2, 5 (C) P → 2, 3, 4;  Q → 5;  R → 1, 2, 4;  S → 2, 5 (D) P → 1, 2, 3, 5;  Q → 2, 5;  R → 2, 3, 4, 5;  S → 2, 5

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question_subject: physics


question: The correct option(s) regarding the complex [Co(en)(NH3)4(H2O)]3+ (en = H2NCH2CH2NH2) is (are)

response_choices: (A) It has two geometrical isomers (B) It will have three geometrical isomers if bidentate 'en' is replaced by two cyanide ligands (C) It is paramagnetic (D) It absorbs light at longer wavelength as compared to [Co(en)(NH3)6]3+

answer: C

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question_subject: chemistry


question: The correct option(s) to distinguish nitrate salts of Mn2+ and Cu2+ taken separately is (are)

response_choices: (A) Mn2+ shows the characteristic green colour in the flame test (B) Only Cu2+ shows the formation of precipitate by passing H2S in acidic medium (C) Only Mn2+ shows the formation of precipitate by passing H2S in faintly basic medium (D) Cu2+/Cu has higher reduction potential than Mn2+/Mn (measured under similar conditions)

answer: A,B,D

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question_subject: chemistry


question: Aniline reacts with mixed acid (conc. HNO3 and conc. H2SO4) at 288 K to give P (51 %), Q (47%) and R (2%). The major product(s) of the following reaction sequence is (are)

answer: (D)

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question_subject: chemistry


question: The Fischer presentation of D-glucose is given below. The correct structure(s) of β-L-glucopyranose is (are)

answer: (B)

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question_subject: chemistry


question: For a first order reaction A(g) → 2B(g) + C(g) at constant volume and 300 K, the total pressure at the beginning (t = 0) and at time t are P0 and Pt, respectively. Initially, only A is present with concentration [A]0, and t1/3 is the time required for the partial pressure of A to reach 1/3rd of its initial value. The correct option(s) is (are) (Assume that all these gases behave as ideal gases)

answer: Graphical response choices, no selected answer marked

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question_subject: chemistry


question: For a reaction, A ⇌ P, the plots of [A] and [P] with time at temperatures T1 and T2 are given below. If T2 > T1, the correct statement(s) is (are) (Assume ΔHᶱ and ΔSᶱ are independent of temperature and ratio of lnK at T1 to lnK at T2 is greater than T2/T1. Here H, S, G and K are enthalpy, entropy, Gibbs energy and equilibrium constant, respectively.)

response_choices:
(A) ΔHᶱ < 0, ΔSᶱ < 0
(B) ΔGᶱ < 0, ΔHᶱ > 0
(C) ΔGᶱ < 0, ΔSᶱ < 0
(D) ΔGᶱ < 0, ΔSᶱ > 0

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question_subject: chemistry


question: The total number of compounds having at least one bridging oxo group among the molecules given below is ____. N2O5, N2O4, P4O6, P4O7, H4P2O5, H3PO10, H2S2O5, H2S2O8

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question_subject: chemistry


question: Galena (an ore) is partially oxidized by passing air through it at high temperature. After some time, the passage of air is stopped, but the heating is continued in a closed furnace such that the contents undergo self-reduction. The weight (in kg) of Pb produced per kg of O2 consumed is ____. (Atomic weights in g mol^-1: O = 16, S = 32, Pb = 207)

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question_subject: chemistry


question: To measure the quantity of MnCl2 dissolved in an aqueous solution, it was completely converted to KMnO4 using the reaction, MnCl2 + K2S2O8 + H2O -> KMnO4 + H2SO4 + HCl (equation not balanced). Few drops of concentrated HCl were added to this solution and gently warmed. Further, oxalic acid (225 mg) was added in portions till the colour of the permanganate ion disappeared. The quantity of MnCl2 (in mg) present in the initial solution is ____. (Atomic weights in g mol^-1: Mn = 55, Cl = 35.5)

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question_subject: chemistry


question: For the given compound X, the total number of optically active stereoisomers is ____.

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question_subject: chemistry


question: In the following reaction sequence, the amount of D (in g) formed from 10 moles of acetophenone is ____.

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question_subject: chemistry


question: The surface of copper gets tarnished by the formation of copper oxide. N2 gas was passed to prevent the oxide formation during heating of copper at 1250 K. However, the N2 gas contains 1 mole % of water vapour as impurity. The water vapour oxidises copper as per the reaction given below: 2Cu(s) + H2O(g) → Cu2O(s) + H2(g) pH2 is the minimum partial pressure of H2 (in bar) needed to prevent the oxidation at 1250 K. The value of ln(pH2) is _____. (Given: total pressure = 1 bar, R (universal gas constant) = 8 J K-1 mol-1, ln(10) = 2.3, Cu(s) and Cu2O(s) are mutually immiscible. At 1250 K: 2Cu(s) + ½ O2(g) → Cu2O(s); ΔG^0 = - 78,000 J mol-1 H2(g) + ½ O2(g) → H2O(g); ΔG^0 = - 1,78,000 J mol-1; G is the Gibbs energy)

response_choices: No response choices provided

answer: No answer provided

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question_subject: chemistry


question: Consider the following reversible reaction, A(g) + B(g) ⇌ AB(g). The activation energy of the backward reaction exceeds that of the forward reaction by 2RT (in J mol-1). If the pre-exponential factor of the forward reaction is 4 times that of the reverse reaction, the absolute value of ΔG^0 (in J mol-1) for the reaction at 300 K is _____. (Given: ln(2) = 0.7, RT = 2500 J mol-1 at 300 K and G is the Gibbs energy)

response_choices: No response choices provided

answer: No answer provided

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question_subject: chemistry


question: Consider an electrochemical cell: A(s) | Am+ (aq, 2 M) || B2m+ (aq, 1 M) | B(s). The value of ΔH^0 for the cell reaction is twice that of ΔG^0 at 300 K. If the emf of the cell is zero, the ΔS^0 (in J K-1 mol-1) of the cell reaction per mole of B formed at 300 K is _____. (Given: ln(2) = 0.7, R (universal gas constant) = 8.3 J K-1 mol-1, H, S and G are enthalpy, entropy and Gibbs energy, respectively.)

response_choices: No response choices provided

answer: No answer provided

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question_subject: chemistry


question: Match each set of hybrid orbitals from LIST-I with complex(es) given in LIST-II.

answer: (D) P → 4,6; Q → 5,6; R → 1,2; S → 3

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question_subject: chemistry


question: The desired product X can be prepared by reacting the major product of the reactions in LIST-I with one or more appropriate reagents in LIST-II. (given, order of migratory aptitude: aryl > alkyl > hydrogen)

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question_subject: chemistry


question: LIST-I contains reactions and LIST-II contains major products. Match each reaction in LIST-I with one or more products in LIST-II and choose the correct option.

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question_subject: chemistry


question: Dilution processes of different aqueous solutions with water, are given in LIST-I. The effects of dilution of the solutions on [H+] are given in LIST-II. (Note: Degree of dissociation (α) of weak acid and weak base is << 1; degree of hydrolysis of salt <<1; [H+] represents the concentration of H+ ions) Match each process given in LIST-I with one or more effect(s) in LIST-II.

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question_subject: chemistry


question: For any positive integer n, define fn: (0, ∞) → ℝ as fn(x) = ∑^n_(j=1) tan^-1 ((1)/((1+(x+j)(x+j-1))) for all x ∈ (0, ∞). (Here, the inverse trigonometric function tan^-1 x assumes values in (-π/2, π/2).) Then, which of the following statement(s) is (are) TRUE?

response_choices: (A) ∑^5_(j=1) tan^2(fj(0)) = 55 (B) ∑^10_(j=1) (1 + fj'(0)) sec^2(fj(0)) = 10 (C) For any fixed positive integer n, lim_(x→∞) tan (fn(x)) = 1/n (D) For any fixed positive integer n, lim_(x→∞) sec^2(fn(x)) = 1

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question_subject: mathematics


question: Let T be the line passing through the points P(-2, 7) and Q(2, -5). Let F1 be the set of all pairs of circles (S1, S2) such that T is tangent to S1 at P and tangent to S2 at Q, and also such that S1 and S2 touch each other at a point, say, M. Let E1 be the set representing the locus of M as the pair (S1, S2) varies in F1. Let the set of all straight line segments joining a pair of distinct points of E1 and passing through the point R(1, 1) be F2. Let E2 be the set of the mid-points of the line segments in the set F2. Then, which of the following statement(s) is (are) TRUE?

response_choices: (A) The point (-2, 7) lies in E1, (B) The point (4, 7/5) does NOT lie in E2, (C) The point (1/2, 1) lies in E2, (D) The point (0, 3/2) does NOT lie in E1

answer: No selected answer visible

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question_subject: mathematics


question: Let S be the set of all column matrices [b1, b2, b3] such that b1, b2, b3 ∈ R and the system of equations (in real variables) -x + 2y + 5z = b1, 2x - 4y + 3z = b2, x - 2y + 2z = b3 has at least one solution. Then, which of the following system(s) (in real variables) has (have) at least one solution for each [b1, b2, b3] ∈ S?

answer: No selected answer visible

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question_subject: mathematics


question: Consider two straight lines, each of which is tangent to both the circle x^2 + y^2 = 1/2 and the parabola y^2 = 4x. Let these lines intersect at the point Q. Consider the ellipse whose center is at the origin O(0,0) and whose semi-major axis is OQ. If the length of the minor axis of this ellipse is √2 , then which of the following statement(s) is (are) TRUE?

response_choices: (A) For the ellipse, the eccentricity is 1/√2 and the length of the latus rectum is 1 (B) For the ellipse, the eccentricity is 1/2 and the length of the latus rectum is 1/2 (C) The area of the region bounded by the ellipse between the lines x = 1/√2 and x = 1 is 1/4√2 (π − 2) (D) The area of the region bounded by the ellipse between the lines x = 1/√2 and x = 1 is 1/16 (π − 2)

answer: B

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question_subject: mathematics


question: Let s, t, r be non-zero complex numbers and L be the set of solutions z = x + iy (x, y ∈ ℝ, i = √−1) of the equation sz + tz + r = 0, where z = x − iy. Then, which of the following statement(s) is (are) TRUE?

response_choices: (A) If L has exactly one element, then |s| ≠ |t| (B) If |s| = |t|, then L has infinitely many elements (C) The number of elements in L ∩ {z : |z − 1 + i| = 5} is at most 2 (D) If L has more than one element, then L has infinitely many elements

answer: B

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question_subject: mathematics


question: Let f: (0, π) → ℝ be a twice differentiable function such that lim [(f(x) sin(t-x)/(t-x) - f(t) sin x/(t-x)) / sin^2 x] = 0 for all x ∈ (0, π). If f'(π/2) = - π/12, then which of the following statement(s) is (are) TRUE?

response_choices:
A) f(π/4) = π/√12
B) f(x) ≤ x^4/6 - x^2 for all x ∈ (0,π)
C) There exists α ∈ (0, π) such that f''(α) = 0
D) f''(π/2) + f'(π/2) = 0

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question_subject: mathematics


question: The value of the integral ∫(0 to (1 + √3)/2) (((x + 1)^2(1 - x))^(-1)) dx

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question_subject: mathematics


question: Let P be a matrix of order 3 × 3 such that all the entries in P are from the set {-1, 0, 1}. Then, the maximum possible value of the determinant of P is _____.

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question_subject: mathematics


question: Let X be a set with exactly 5 elements and Y be a set with exactly 7 elements. If α is the number of one-one functions from X to Y and β is the number of onto functions from Y to X, then the value of (1/5)(β - α) is _____.

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question_subject: mathematics


question: Let f: ℝ → ℝ be a differentiable function with f(0) = 0. If y = f(x) satisfies the differential equation dy/dx = (2 + 5y)(5y - 2), then the value of lim(x→0) f(x) is _____.

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question_subject: mathematics


question: Let f: ℝ → ℝ be a differentiable function with f(0) = 1 and satisfying the equation f(x + y) = f(x)f'(y) + f'(x)f(y) for all x, y ∈ ℝ. Then, the value of loge(f(4)) is _____.

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question_subject: mathematics


question: Let P be a point in the first octant, whose image Q in the plane x + y = 3 (that is, the line segment PQ is perpendicular to the plane x + y = 3 and the mid-point of PQ lies in the plane x + y = 3) lies on the z-axis. Let the distance of P from the x-axis be 5. If R is the image of P in the xy-plane, then the length of PR is _____.

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question_subject: mathematics


question: Consider the cube in the first octant with sides OP, OQ and OR of length 1, along the x-axis, y-axis and z-axis, respectively, where O(0,0,0) is the origin. Let S(1/2, 1/2, 1/2) be the centre of the cube and T be the vertex of the cube opposite to the origin O such that S lies on the diagonal OT. If p⃗ = 5i⃗, q⃗ = 5j⃗, r⃗ = 5k⃗ and t⃗ = 5T⃗, then the value of |(p⃗×q⃗)×(r⃗×t⃗)| is _____.

response_choices: No multiple choice options provided

answer: No answer provided

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question_subject: mathematics


question: Let X = (^10C_1 )^2 + 2(^10C_2 )^2 + 3(^10C_3 )^2 + ... + 10(^10C_10 )^2, where ^nC_r, r ∈ {1, 2, ..., 10} denote binomial coefficients. Then, the value of 1/1430 X is _____.

response_choices: No multiple choice options provided

answer: No answer provided

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question_subject: mathematics


question: Let E1 = {x ∈ ℝ : x ≠ 1 and x^-1 > 0} and E2 = {x ∈ E1 : sin^-1 (log_e (x^-1)) is a real number}. (Here, the inverse trigonometric function sin^-1 x assumes values in [-π/2, π/2]). Let f: E1 → ℝ be the function defined by f(x) = log_e (x^-1) and g: E2 → ℝ be the function defined by g(x) = sin^-1 (log_e (x^-1)).

response_choices:
P. The range of f is
Q. The range of g contains
R. The domain of f contains
S. The domain of g is

answer: A

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question_subject: mathematics


question: In a high school, a committee has to be formed from a group of 6 boys M1, M2, M3, M4, M5, M6 and 5 girls G1, G2, G3, G4, G5. (i) Let α1 be the total number of ways in which the committee can be formed such that the committee has 5 members, having exactly 3 boys and 2 girls. (ii) Let α2 be the total number of ways in which the committee can be formed such that the committee has at least 2 members, and having an equal number of boys and girls. (iii) Let α3 be the total number of ways in which the committee can be formed such that the committee has 5 members, at least 2 of them being girls. (iv) Let α4 be the total number of ways in which the committee can be formed such that the committee has 4 members, having at least 2 girls and such that both M1 and G1 are NOT in the committee together.

response_choices: LIST-I: P. The value of α1 is, Q. The value of α2 is, R. The value of α3 is, S. The value of α4 is; LIST-II: 1. 136, 2. 189, 3. 192, 4. 200, 5. 381, 6. 461

answer: (A) P → 4; Q → 6; R → 2; S → 1

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question_subject: mathematics


question: Let H: x^2/a^2 - y^2/b^2 = 1, where a > b > 0, be a hyperbola in the xy-plane whose conjugate axis LM subtends an angle of 60° at one of its vertices N. Let the area of the triangle LMN be 4√3.

response_choices:
(A) P → 4; Q → 2; R → 1; S → 3
(B) P → 4; Q → 3; R → 1; S → 2
(C) P → 4; Q → 1; R → 3; S → 2
(D) P → 3; Q → 4; R → 2; S → 1

answer: (A) P → 4; Q → 2; R → 1; S → 3

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question_subject: mathematics


question: Let f1: R -> R, f2: (π/2, π/2) -> R, f3: (-1, e^2 - 2) -> R and f4: R -> R be functions defined by (i) f1(x) = sin(sqrt(1 - e^(-x^2))), (ii) f2(x) = { sin x / tan^-1 x  if x ≠ 0, 1 if x = 0 }, where the inverse trigonometric function tan^-1 x assumes values in (- π/2, π/2), (iii) f3(x) = [ sin(log_e(x + 2))], where, for t ∈ R, [t] denotes the greatest integer less than or equal to t, (iv) f4(x) = { x^2 sin (1/x)  if x ≠ 0, 0 if x = 0 }

response_choices: LIST-I: P. The function f1 is, Q. The function f2 is, R. The function f3 is, S. The function f4 is. LIST-II: 1. NOT continuous at x = 0, 2. continuous at x = 0 and NOT differentiable at x = 0, 3. differentiable at x = 0 and its derivative is NOT continuous at x = 0, 4. differentiable at x = 0 and its derivative is continuous at x = 0. The correct option is: (A) P -> 2; Q -> 3; R -> 1; S -> 4, (B) P -> 4; Q -> 1; R -> 2; S -> 3, (C) P -> 4; Q -> 2; R -> 1; S -> 3, (D) P -> 2; Q -> 1; R -> 4; S -> 3

answer: (C) P -> 4; Q -> 2; R -> 1; S -> 3

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question_subject: mathematics