question: A flat plate is moving normal to its plane through a gas under the action of a constant force F. The gas is kept at a very low pressure. The speed of the plate v is much less than the average speed u of the gas molecules. Which of the following options is/are true?

response_choices: [A] The pressure difference between the leading and trailing faces of the plate is proportional to uv, [B] The resistive force experienced by the plate is proportional to v, [C] The plate will continue to move with constant non-zero acceleration, at all times, [D] At a later time the external force F balances the resistive force

answer: (A), (B), and (D)

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question_subject: physics


question: A block of mass M has a circular cut with a frictionless surface as shown. The block rests on the horizontal frictionless surface of a fixed table. Initially the right edge of the block is at x = 0, in a co-ordinate system fixed to the table. A point mass m is released from rest at the topmost point of the path as shown and it slides down. When the mass loses contact with the block, its position is x and the velocity is v. At that instant, which of the following options is/are correct?

response_choices:
[A] The position of the point mass m is: x = -√(2*(mR)/(M+m))
[B] The velocity of the point mass m is: v = √((2gR)/(1+(M/m)))
[C] The x component of displacement of the center of mass of the block M is: -(mR)/(M+m)
[D] The velocity of the block M is: V = (m/M)*√(2gR)

answer: B and C

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question_subject: physics


question: A block M hangs vertically at the bottom end of a uniform rope of constant mass per unit length. The top end of the rope is attached to a fixed rigid support at O. A transverse wave pulse (Pulse 1) of wavelength λo is produced at point O on the rope. The pulse takes time ToA to reach point A. If the wave pulse of wavelength λo is produced at point A (Pulse 2) without disturbing the position of M it takes time TAO to reach point O. Which of the following options is/are correct?

response_choices:
[A] The time TAO = TOA

[B] The velocities of the two pulses (Pulse 1 and Pulse 2) are the same at the midpoint of rope

[C] The wavelength of Pulse 1 becomes longer when it reaches point A

[D] The velocity of any pulse along the rope is independent of its frequency and wavelength

answer: (A) and (D)

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question_subject: physics


question: A human body has a surface area of approximately 1 m². The normal body temperature is 10 K above the surrounding room temperature T₀. Take the room temperature to be T₀ = 300 K. For T₀ = 300 K, the value of σT₀⁴ = 460 Wm⁻² (where σ is the Stefan-Boltzmann constant). Which of the following options is/are correct?

response_choices:
A] The amount of energy radiated by the body in 1 second is close to 60 Joules

B] If the surrounding temperature reduces by a small amount ΔT₀ << T₀, then to maintain the same body temperature the same (living) human being needs to radiate ΔW = 4σT₀³ΔT₀ more energy per unit time

C] Reducing the exposed surface area of the body (e.g. by curling up) allows humans to maintain the same body temperature while reducing the energy lost by radiation

D] If the body temperature rises significantly then the peak in the spectrum of electromagnetic radiation emitted by the body would shift to longer wavelengths

answer: C

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question_subject: physics


question: A circular insulated copper wire loop is twisted to form two loops of area A and 2A as shown in the figure. At the point of crossing the wires remain electrically insulated from each other. The entire loop lies in the plane (of the paper). A uniform magnetic field B points into the plane of the paper. At t = 0, the loop starts rotating about the common diameter as axis with a constant angular velocity ω in the magnetic field. Which of the following options is/are correct?

response_choices:
[A] The emf induced in the loop is proportional to the sum of the areas of the two loops

[B] The amplitude of the maximum net emf induced due to both the loops is equal to the amplitude of maximum emf induced in the smaller loop alone

[C] The net emf induced due to both the loops is proportional to cos ωt

[D] The rate of change of the flux is maximum when the plane of the loops is perpendicular to plane of the paper

answer: B) and D)

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question_subject: physics


question: In the circuit shown, L = 1 μH, C = 1 μF and R = 1 kΩ. They are connected in series with an a.c. source V = V0 sin ωt as shown. Which of the following options is/are correct?

response_choices:
[A] The current will be in phase with the voltage if ω = 10⁴ rad.s⁻¹
[B] The frequency at which the current will be in phase with the voltage is independent of R
[C] At ω∼0 the current flowing through the circuit becomes nearly zero
[D] At ω ≫ 10⁶ rad.s⁻¹, the circuit behaves like a capacitor

answer: B) and C)

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question_subject: physics


question: For an isosceles prism of angle A and refractive index μ, it is found that the angle of minimum deviation δm = A. Which of the following options is/are correct?

response_choices:
[A] For the angle of incidence i1 = A, the ray inside the prism is parallel to the base of the prism
[B] For this prism, the refractive index μ and the angle of prism A are related as A = 2cos⁻¹(μ/2)
[C] At minimum deviation, the incident angle i1 and the refracting angle r1 at the first refracting surface are related by r1 = (i1/2)
[D] For this prism, the emergent ray at the second surface will be tangential to the surface when the angle of incidence at the first surface is i1 = sin⁻¹[sinA√(A cos²(A/2) - 1 - cos A)]

answer: A), C) and D)

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question_subject: physics


question: A drop of liquid of radius R = 10^-2 m having surface tension S = 0.1 Nm^-1 divides itself into K identical drops. In this process the total change in the surface energy ΔU = 10^-3 J. If K = 10^6 then the value of κ is

response_choices: Multiple choice options not provided

answer: 6

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question_subject: physics


question: An electron in a hydrogen atom undergoes a transition from an orbit with quantum number ni to another with quantum number nf. Vi and Vf are respectively the initial and final potential energies of the electron. If Vf = 6.25, then the smallest possible ni is

response_choices: Multiple choice options not provided

answer: 5

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question_subject: physics


question: A monochromatic light is travelling in a medium of refractive index n = 1.6. It enters a stack of glass layers from the bottom side at an angle θ = 30°. The interfaces of the glass layers are parallel to each other. The refractive indices of different glass layers are monotonically decreasing as nm = n - mΔn, where nm is the refractive index of the mth slab and Δn = 0.1 (see the figure). The ray is refracted out parallel to the interface between the (m - 1)th and mth slabs from the right side of the stack. What is the value of m?

response_choices: No response choices provided

answer: (8)

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question_subject: physics


question: A stationary source emits sound of frequency f0 = 492 Hz. The sound is reflected by a large car approaching the source with a speed of 2 ms^-1. The reflected signal is received by the source and superposed with the original. What will be the beat frequency of the resulting signal in Hz? (Given that the speed of sound in air is 330 ms^-1 and the car reflects the sound at the frequency it has received).

answer: (6)

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question_subject: physics


question: 131I is an isotope of Iodine that β decays to an isotope of Xenon with a half-life of 8 days. A small amount of a serum labelled with 131I is injected into the blood of a person. The activity of the amount of 131I injected was 2.4 ×105 Becquerel (Bq). It is known that the injected serum will get distributed uniformly in the blood stream in less than half an hour. After 11.5 hours, 2.5 ml of blood is drawn from the person's body, and gives an activity of 115 Bq. The total volume of blood in the person's body, in liters is approximately (you may use e^ = 1 + x for |x| ≪ 1 and ln 2 ≈ 0.7).

answer: (5)

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question_subject: chemistry


question: In which case will the particle move in a straight line with constant velocity?

answer: D

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question_subject: physics


question: In which case will the particle describe a helical path with axis along the positive z direction?

answer: A

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question_subject: physics


question: In which case would the particle move in a straight line along the negative direction of y-axis (i.e., move along - y)?

answer: B

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question_subject: physics


question: Answer Q.16, Q.17 and Q.18 by appropriately matching the information given in the three columns of the following table.

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question_subject: physics


question: An ideal gas is undergoing a cyclic thermodynamic process in different ways as shown in the corresponding P – V diagrams in column 3 of the table. Consider only the path from state 1 to state 2. W denotes the corresponding work done on the system. The equations and plots in the table have standard notations as used in thermodynamic processes. Here γ is the ratio of heat capacities at constant pressure and constant volume. The number of moles in the gas is n.

response_choices: (I) Isothermal, (II) Isochoric, (III) Isobaric, (IV) Adiabatic

answer: (I) Path 2->1, (II) Path 2->1, (III) Path 1->2, (IV) Path 1->2

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question_subject: physics


question: Match the equations in Column 1 and processes in Column 2 with the corresponding P-V plots in Column 3.

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question_subject: physics


question: Which of the following options is the only correct representation of a process in which ΔU = ΔQ - PΔV?

answer: D

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question_subject: physics


question: Which one of the following options is the correct combination?

answer: B

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question_subject: physics


question: Which one of the following options correctly represents a thermodynamic process that is used as a correction in the determination of the speed of sound in an ideal gas?

answer: D

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question_subject: physics


question: An ideal gas is expanded from (p1, V1, T1) to (p2, V2, T2) under different conditions. The correct statement(s) among the following is(are)

response_choices:
[A] The work done on the gas is maximum when it is compressed irreversibly from (p2, V2) to (p1, V1) against constant pressure p1

[B] If the expansion is carried out freely, it is simultaneously both isothermal as well as adiabatic

[C] The work done by the gas is less when it is expanded reversibly from V1 to V2 under adiabatic conditions as compared to that when expanded reversibly from V1 to V2 under isothermal conditions

[D] The change in internal energy of the gas is (i) zero, if it is expanded reversibly with T1 = T2, and (ii) positive, if it is expanded reversibly under adiabatic conditions with T1 ≠ T2

answer: (A), (B), and (C)

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question_subject: chemistry


question: For a solution formed by mixing liquids L and M, the vapour pressure of L plotted against the mole fraction of M in solution is shown in the following figure. Here xL and xM represent mole fractions of L and M, respectively, in the solution. The correct statement(s) applicable to this system is(are)

response_choices: A) The point Z represents vapour pressure of pure liquid M and Raoult's law is obeyed from xL = 0 to xL = 1; B) The point Z represents vapour pressure of pure liquid L and Raoult's law is obeyed when xL > 1; C) The point Z represents vapour pressure of pure liquid M and Raoult's law is obeyed when xL > 0; D) Attractive intermolecular interactions between L-L in pure liquid L and M-M in pure liquid M are stronger than those between L-M when mixed in solution

answer: B) and D)

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question_subject: chemistry


question: The correct statement(s) about the oxoacids, HClO4 and HClO3 is(are)

response_choices:
[A] The central atom in both HClO4 and HClO3 is sp3 hybridized
[B] HClO4 is more acidic than HClO3 because of the resonance stabilization of its anion
[C] HClO4 is formed in the reaction between Cl2 and H2O
[D] The conjugate base of HClO4 is weaker base than H2O

answer: (A), (B), and (D)

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question_subject: chemistry


question: The colour of the X2 molecules of group 17 elements changes gradually from yellow to violet down the group. This is due to

response_choices:
[A] the physical state of X2 at room temperature changes from gas to solid down the group
[B] decrease in ionization energy down the group
[C] decrease in π*-σ* gap down the group
[D] decrease in HOMO-LUMO gap down the group

answer: (C) and (D)

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question_subject: chemistry


question: Addition of excess aqueous ammonia to a pink coloured aqueous solution of MCl2·6H2O (X) and NH4Cl gives an octahedral complex Y in the presence of air. In aqueous solution, complex Y behaves as 1:3 electrolyte. The reaction of X with excess HCl at room temperature results in the formation of a blue coloured complex Z. The calculated spin only magnetic moment of X and Z is 3.87 B.M., whereas it is zero for complex Y. Among the following options, which statement(s) is(are) correct?

response_choices:
[A] Addition of silver nitrate to Y gives only two equivalents of silver chloride
[B] The hybridization of the central metal ion in Y is d sp3
[C] Z is a tetrahedral complex
[D] When X and Z are in equilibrium at 0°C, the colour of the solution is pink

answer: (B), (C), and (D)

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question_subject: chemistry


question: The IUPAC name(s) of the following compound is(are)

response_choices:
[A] 1-chloro-4-methylbenzene
[B] 4-chlorotoluene
[C] 4-methylchlorobenzene
[D] 1-methyl-4-chlorobenzene

answer: (A) and (B)

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question_subject: chemistry


question: The correct statement(s) for the following addition reactions is(are)

response_choices:
[A] Q and P are identical molecules

[B] (M and O) and (N and P) are two pairs of diastereomers

[C] (M and O) and (N and P) are two pairs of enantiomers

[D] Bromination proceeds through trans-addition in both the reactions

answer: (B) and (D)

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question_subject: chemistry


question: A crystalline solid of a pure substance has a face-centred cubic structure with a cell edge of 400 pm. If the density of the substance in the crystal is 8 g cm^-3, then the number of atoms present in 256 g of the crystal is N × 10^24. The value of N is

response_choices: There are no multiple choice options provided for this question.

answer: 2

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question_subject: chemistry


question: The conductance of a 0.0015 M aqueous solution of a weak monobasic acid was determined by using a conductivity cell consisting of platinized Pt electrodes. The distance between the electrodes is 120 cm with an area of cross section of 1 cm^2. The conductance of this solution was found to be 5 × 10^-7 S. The pH of the solution is 4. The value of limiting molar conductivity (λ_m) of this weak monobasic acid in aqueous solution is Z × 10^-5 S cm^-1 mol^-1. The value of Z is

response_choices: There are no multiple choice options provided for this question.

answer: 6

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question_subject: chemistry


question: The sum of the number of lone pairs of electrons on each central atom in the following species is [TeBr6]2-, [BrF5]2-, SNF3, and [XeF5]+ (Atomic numbers: N = 7, F = 9, S = 16, Br = 35, Te = 52, Xe = 54)

answer: 6

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question_subject: chemistry


question: Among H2, He2+, Li2, Be2, B2-, C2, N2, O2-, and F2, the number of diamagnetic species is (Atomic numbers: H = 1, He = 2, Li = 3, Be = 4, B = 5, C = 6, N = 7, O = 8, F = 9)

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question_subject: chemistry


question: Among the following, the number of aromatic compound(s) is

answer: 5

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question_subject: chemistry


question: For the given orbital in Column 1, the only CORRECT combination for any hydrogen-like species is

answer: C

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question_subject: physics


question: For hydrogen atom, the only CORRECT combination is

answer: A

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question_subject: physics


question: For He+ ion, the only INCORRECT combination is

answer: C

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question_subject: physics


question: For the synthesis of benzoic acid, the only CORRECT combination is

answer: A

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question_subject: chemistry


question: The only CORRECT combination that gives two different carboxylic acids is

answer: C

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question_subject: chemistry


question: The only CORRECT combination in which the reaction proceeds through radical mechanism is

answer: D

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question_subject: chemistry


question: If 2x − y + 1 = 0 is a tangent to the hyperbola x^2/a^2 - y^2/16 = 1, then which of the following CANNOT be sides of a right angled triangle?

answer: (A), (B), and (C)

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question_subject: mathematics


question: If a chord, which is not a tangent, of the parabola y^2 = 16x has the equation 2x + y = p, and midpoint (h, k), then which of the following is(are) possible value(s) of p, h and k?

answer: (C)

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question_subject: mathematics


question: Let [x] be the greatest integer less than or equals to x. Then, at which of the following point(s) the function f(x) = x cos(π(x + [x])) is discontinuous?

answer: Due to printing inconsistency in Q.39 in Code1 of Paper1 (and the corresponding question in all other codes), +4 marks will be awarded to all candidates for this question.

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question_subject: mathematics


question: Let f: ℝ → (0,1) be a continuous function. Then, which of the following function(s) has(have) the value zero at some point in the interval (0,1)?

response_choices: [A] x^9 - f(x), [B] x - ∫(0 to x) f(t) cos t dt, [C] ex - ∫(0 to x) f(t) sin t dt, [D] f(x) + ∫(0 to x) f(t) sin t dt

answer: Ans for Q.40: (A) and (B)

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question_subject: mathematics


question: Which of the following is(are) NOT the square of a 3×3 matrix with real entries?

answer: Ans for Q.41: (B) and (D)

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question_subject: mathematics


question: Let a, b, x and y be real numbers such that a – b = 1 and y ≠ 0. If the complex number z = x + iy satisfies Im((a+b)/(x+y)) = y, then which of the following is(are) possible value(s) of x?

answer: (A) and (B)

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question_subject: mathematics


question: Let X and Y be two events such that P(X) = 1/3, P(X|Y) = 1/2 and P(Y|X) = 2/5. Then

answer: (A) and (B)

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question_subject: mathematics


question: For how many values of p, the circle x^2 + y^2 + 2x + 4y - p = 0 and the coordinate axes have exactly three common points?

response_choices: A, B, C, D, E, F, G, H, I, J (0 to 9 digits)

answer: 2

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question_subject: mathematics


question: Let f: R → R be a differentiable function such that f(0) = 0, f(π/2) = 3 and f'(0) = 1. If g(x) = ∫[x to π/2] [f'(t) coset - cot t coset f(t)] dt for x ∈ (0, π/2], then lim(x→0) g(x) =

response_choices: A, B, C, D, E, F, G, H, I, J (0 to 9 digits)

answer: 2

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question_subject: mathematics


question: For a real number α, if the system [1 α α^2] [X] = [1] [α 1 α] [Y] = [-1] [α^2 α 1] [Z] [-1] of linear equations, has infinitely many solutions, then 1 + α + α^2 =

response_choices: No multiple choice options provided.

answer: (1)

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question_subject: mathematics


question: Words of length 10 are formed using the letters A,B,C,D,E,F,G,H,I,J. Let x be the number of such words where no letter is repeated; and let y be the number of such words where exactly one letter is repeated twice and no other letter is repeated. Then, y/x =

response_choices: No multiple choice options provided.

answer: (5)

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question_subject: mathematics


question: The sides of a right angled triangle are in arithmetic progression. If the triangle has area 24, then what is the length of its smallest side?

response_choices: No multiple choice options provided.

answer: (6)

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question_subject: mathematics


question: For a = √2, if a tangent is drawn to a suitable conic (Column 1) at the point of contact (-1, 1), then which of the following options is the only CORRECT combination for obtaining its equation?

answer: B

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question_subject: mathematics


question: If a tangent to a suitable conic (Column 1) is found to be y = x + 8 and its point of contact is (8, 16), then which of the following options is the only CORRECT combination?

answer: C

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question_subject: mathematics


question: The tangent to a suitable conic (Column 1) at (√3, 1/√3) is found to be √3x + 2y = 4, then which of the following options is the only CORRECT combination?

answer: D

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question_subject: mathematics


question: Which of the following options is the only CORRECT combination?

answer: B

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question_subject: mathematics


question: Which of the following options is the only CORRECT combination?

answer: B

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question_subject: mathematics


question: Which of the following options is the only INCORRECT combination?

answer: C

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question_subject: mathematics