question: Consider an expanding sphere of instantaneous radius R whose total mass remains constant. The expansion is such that the instantaneous density ρ remains uniform throughout the volume. The rate of fractional change in density (1/ρ dρ/dt) is constant. The velocity v of any point on the surface of the expanding sphere is proportional to

answer: A

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question_subject: physics


question: Consider regular polygons with number of sides n = 3, 4, 5, ..... as shown in the figure. The center of mass of all the polygons is at height h from the ground. They roll on a horizontal surface about the leading vertex without slipping and sliding as depicted. The maximum increase in height of the locus of the center of mass for each polygon is Δ. Then Δ depends on n and h as

answer: B

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question_subject: mathematics


question: A photoelectric material having work-function ϕ0 is illuminated with light of wavelength λ (λ < hc/ϕ0). The fastest photoelectron has a de Broglie wavelength λd. A change in wavelength of the incident light by Δλ results in a change Δλd in λd. Then the ratio Δλd/Δλ is proportional to

answer: D

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question_subject: physics


question: A symmetric star shaped conducting wire loop is carrying a steady state current I as shown in the figure. The distance between the diametrically opposite vertices of the star is 4a. The magnitude of the magnetic field at the center of the loop is

answer: A

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question_subject: physics


question: Three vectors P, Q and R are shown in the figure. Let S be any point on the vector R. The distance between the points P and S is b|R|. The general relation among vectors P, Q and S is

answer: A

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question_subject: physics


question: A rocket is launched normal to the surface of the Earth, away from the Sun, along the line joining the Sun and the Earth. The Sun is 3×10^5 times heavier than the Earth and is at a distance 2.5×10^8 times larger than the radius of the Earth. The escape velocity from Earth's gravitational field is ve = 11.2 km s^-1. The minimum initial velocity (vs) required for the rocket to be able to leave the Sun-Earth system is closest to (Ignore the rotation and revolution of the Earth and the presence of any other planet)

answer: B

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question_subject: physics


question: A person measures the depth of a well by measuring the time interval between dropping a stone and receiving the sound of impact with the bottom of the well. The error in his measurement of time is δT = 0.01 seconds and he measures the depth of the well to be L = 20 meters. Take the acceleration due to gravity g = 10 ms^-2 and the velocity of sound is 300 ms^-1. Then the fractional error in the measurement, δL/L, is closest to

answer: B

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question_subject: physics


question: A uniform magnetic field B exists in the region between x = 0 and x = 3R (region 2 in the figure) pointing normally into the plane of the paper. A particle with charge +Q and momentum p directed along x-axis enters region 2 from region 1 at point P1 (y = -R). Which of the following option(s) is/are correct?

response_choices:
[A] For B > 2 p/(3 QR), the particle will re-enter region 1

[B] For B = 8 p/(13 QR), the particle will enter region 3 through the point P2 on x-axis

[C] When the particle re-enters region 1 through the longest possible path in region 2, the magnitude of the change in its linear momentum between point P1 and the farthest point from y-axis is p/√2

[D] For a fixed B, particles of same charge Q and same velocity v, the distance between the point P1 and the point of re-entry into region 1 is inversely proportional to the mass of the particle

answer: (A) and (B)

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question_subject: physics


question: The instantaneous voltages at three terminals marked X, Y and Z are given by Vx = V0 sin ωt, Vy = V0 sin (ωt + 2π/3) and Vz = V0 sin (ωt + 4π/3). An ideal voltmeter is configured to read rms value of the potential difference between its terminals. It is connected between points X and Y and then between Y and Z. The reading(s) of the voltmeter will be

response_choices: [A] VXYrms = V0√(3/2), [B] VYZrms = V0√(1/2), [C] VXYrms = V0, [D] independent of the choice of the two terminals

answer: A and D

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question_subject: physics


question: A point charge +Q is placed just outside an imaginary hemispherical surface of radius R as shown in the figure. Which of the following statements is/are correct?

response_choices: [A] The electric flux passing through the curved surface of the hemisphere is (Q/(2ε0)) * (1 - 1/2), [B] Total flux through the curved and the flat surfaces is Q/ε0, [C] The component of the electric field normal to the flat surface is constant over the surface, [D] The circumference of the flat surface is an equipotential

answer: A and D

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question_subject: physics


question: Two coherent monochromatic point sources S1 and S2 of wavelength λ = 600 nm are placed symmetrically on either side of the center of the circle as shown. The sources are separated by a distance d = 1.8 mm. This arrangement produces interference fringes visible as alternate bright and dark spots on the circumference of the circle. The angular separation between two consecutive bright spots is Δθ. Which of the following options is/are correct?

response_choices:
[A] A dark spot will be formed at the point P2

[B] At P2 the order of the fringe will be maximum

[C] The total number of fringes produced between P1 and P2 in the first quadrant is close to 3000

[D] The angular separation between two consecutive bright spots decreases as we move from P1 to P2 along the first quadrant

answer: B) and (C)

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question_subject: physics


question: A source of constant voltage V is connected to a resistance R and two ideal inductors L1 and L2 through a switch S as shown. There is no mutual inductance between the two inductors. The switch S is initially open. At t = 0, the switch is closed and current begins to flow. Which of the following options is/are correct?

response_choices:
[A] After a long time, the current through L1 will be V * L2 / (R L1 + L2)

[B] After a long time, the current through L2 will be V * L1 / (R L1 + L2)

[C] The ratio of the currents through L1 and L2 is fixed at all times (t > 0)

[D] At t = 0, the current through the resistance R is V/R

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

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question_subject: physics


question: A rigid uniform bar AB of length L is slipping from its vertical position on a frictionless floor (as shown in the figure). At some instant of time, the angle made by the bar with the vertical is θ. Which of the following statements about its motion is/are correct?

response_choices:
[A] The midpoint of the bar will fall vertically downward

[B] The trajectory of the point A is a parabola

[C] Instantaneous torque about the point in contact with the floor is proportional to sin θ

[D] When the bar makes an angle θ with the vertical, the displacement of its midpoint from the initial position is proportional to (1 − cos θ)

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

image_available: yes

question_subject: physics


question: A wheel of radius R and mass M is placed at the bottom of a fixed step of height R as shown in the figure. A constant force is continuously applied on the surface of the wheel so that it just climbs the step without slipping. Consider the torque τ about an axis normal to the plane of the paper passing through the point Q. Which of the following options is/are correct?

response_choices:
[A] If the force is applied at point P tangentially then τ decreases continuously as the wheel climbs

[B] If the force is applied normal to the circumference at point X then τ is constant

[C] If the force is applied normal to the circumference at point P then τ is zero

[D] If the force is applied tangentially at point S then τ ≠ 0 but the wheel never climbs the step

answer: {{C}} or {{C} and {D}}

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question_subject: physics


question: In Process 1, the energy stored in the capacitor EC and heat dissipated across resistance ED are related by:

response_choices: [A] EC = ED, [B] EC = ED ln 2, [C] EC = 1/2 ED, [D] EC = 2ED

answer: A

image_available: yes

question_subject: physics


question: In Process 2, total energy dissipated across the resistance ED is:

answer: C

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question_subject: physics


question: The total kinetic energy of the ring is

response_choices:
[A] Mω₀²R²
[B] ½ Mω₀²(R − r)²
[C] Mω₀²(R − r)²
[D] ²⁄₃ Mω₀²(R − r)²

answer: Due to internal review, all candidates are awarded +3 marks.

image_available: yes

question_subject: physics


question: The minimum value of ω₀ below which the ring will drop down is

answer: A

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question_subject: physics


question: Pure water freezes at 273 K and 1 bar. The addition of 34.5 g of ethanol to 500 g of water changes the freezing point of the solution. Use the freezing point depression constant of water as 2 K kg mol^-1. The figures shown below represent plots of vapour pressure (V.P.) versus temperature (T). [molecular weight of ethanol is 46 g mol^-1] Among the following, the option representing change in the freezing point is

answer: C

image_available: yes

question_subject: chemistry


question: For the following cell, Zn(s) | ZnSO4(aq) || CuSO4(aq) | Cu(s) when the concentration of Zn^2+ is 10 times the concentration of Cu^2+, the expression for ΔG (in J mol^-1) is [F is Faraday constant; R is gas constant; T is temperature; E^0(cell) = 1.1 V]

answer: B

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question_subject: chemistry


question:
The standard state Gibbs free energies of formation of C(graphite) and C(diamond) at T = 298 K are
 Δf G°[C(graphite)] = 0 kJ mol^-1
 Δf G°[C(diamond)] = 2.9 kJ mol^-1.
The standard state means that the pressure should be 1 bar, and substance should be pure at a given temperature. The conversion of graphite [C(graphite)] to diamond [C(diamond)] reduces its volume by 2×10^-6 m³ mol^-1. If C(graphite) is converted to C(diamond) isothermally at T = 298 K, the pressure at which C(graphite) is in equilibrium with C(diamond), is
[Useful information: 1 J = 1 kg m² s^-2; 1 Pa = 1 kg m^-1 s^-2; 1 bar = 10⁵ Pa]

answer: A

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question_subject: chemistry


question: Which of the following combination will produce H2 gas?

response_choices:
[A] Fe metal and conc. HNO3
[B] Cu metal and conc. HNO3
[C] Zn metal and NaOH(aq)
[D] Au metal and NaCN(aq) in the presence of air

answer: C

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question_subject: chemistry


question: The order of the oxidation state of the phosphorus atom in H3PO2, H3PO3, H3PO4, and H4P2O6 is

answer: C

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question_subject: chemistry


question: The major product of the following reaction is

answer: C

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question_subject: chemistry


question: The order of basicity among the following compounds is

response_choices: [A] II > I > IV > III, [B] IV > II > III > I, [C] IV > I > II > III, [D] I > IV > III > II

answer: C

image_available: yes

question_subject: chemistry


question: The correct statement(s) about surface properties is(are)

response_choices: [A] Adsorption is accompanied by decrease in enthalpy and decrease in entropy of the system, [B] The critical temperatures of ethane and nitrogen are 563 K and 126 K, respectively. The adsorption of ethane will be more than that of nitrogen on same amount of activated charcoal at a given temperature, [C] Cloud is an emulsion type of colloid in which liquid is dispersed phase and gas is dispersion medium, [D] Brownian motion of colloidal particles does not depend on the size of the particles but depends on viscosity of the solution

answer: A, B

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question_subject: chemistry


question: For a reaction taking place in a container in equilibrium with its surroundings, the effect of temperature on its equilibrium constant K in terms of change in entropy is described by

response_choices:
[A] With increase in temperature, the value of K for exothermic reaction decreases because the entropy change of the system is positive

[B] With increase in temperature, the value of K for endothermic reaction increases because unfavourable change in entropy of the surroundings decreases

[C] With increase in temperature, the value of K for endothermic reaction increases because the entropy change of the system is negative

[D] With increase in temperature, the value of K for exothermic reaction decreases because favourable change in entropy of the surroundings decreases

answer: B and D

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question_subject: chemistry


question: In a bimolecular reaction, the steric factor P was experimentally determined to be 4.5. The correct option(s) among the following is(are)

response_choices:
[A] The activation energy of the reaction is unaffected by the value of the steric factor

[B] Experimentally determined value of frequency factor is higher than that predicted by Arrhenius equation

[C] Since P = 4.5, the reaction will not proceed unless an effective catalyst is used

[D] The value of frequency factor predicted by Arrhenius equation is higher than that determined experimentally

answer: A and B

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question_subject: chemistry


question: For the following compounds, the correct statement(s) with respect to nucleophilic substitution reactions is(are)

response_choices:
[A] I and III follow SN1 mechanism

[B] I and II follow SN2 mechanism

[C] Compound IV undergoes inversion of configuration

[D] The order of reactivity for I, III and IV is: IV > I > III

answer: {{(A), (B), and (C)}} or {{(A), (B), (C), and (D)}}

image_available: yes

question_subject: chemistry


question: Among the following, the correct statement(s) is(are)

response_choices:
[A] Al(CH3)3 has the three-centre two-electron bonds in its dimeric structure

[B] BH3 has the three-centre two-electron bonds in its dimeric structure

[C] AlCl3 has the three-centre two-electron bonds in its dimeric structure

[D] The Lewis acidity of BCl3 is greater than that of AlCl3

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

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question_subject: chemistry


question: The option(s) with only amphoteric oxides is(are)

response_choices:
[A] Cr2O3, BeO, SnO, SnO2

[B] Cr2O3, CrO, SnO, PbO

[C] NO, B2O3, PbO, SnO2

[D] ZnO, Al2O3, PbO, PbO2

answer: (A) and (D)

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question_subject: chemistry


question: Compounds P and R upon ozonolysis produce Q and S, respectively. The molecular formula of Q and S is C8H8O. Q undergoes Cannizzaro reaction but not haloform reaction, whereas S undergoes haloform reaction but not Cannizzaro reaction.

answer: (A) and (B)

image_available: yes

question_subject: chemistry


question: W and X are, respectively

response_choices:
[A] O3 and P4O6
[B] O2 and P4O6
[C] O2 and P4O10
[D] O3 and P4O10

answer: C

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question_subject: chemistry


question: Y and Z are, respectively

response_choices:
[A] N2O3 and H3PO4
[B] N2O5 and HPO3
[C] N2O4 and HPO3
[D] N2O4 and H3PO4

answer: B

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question_subject: chemistry


question: The product S is

response_choices: [A] (Structure A) [B] (Structure B) [C] (Structure C) [D] (Structure D)

answer: A

image_available: yes

question_subject: chemistry


question: The reactions, Q to R and R to S, are

response_choices: [A] Dehydration and Friedel-Crafts acylation [B] Aromatic sulfonation and Friedel-Crafts acylation [C] Friedel-Crafts alkylation, dehydration and Friedel-Crafts acylation [D] Friedel-Crafts alkylation and Friedel-Crafts acylation

answer: D

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question_subject: chemistry


question: The equation of the plane passing through the point (1, 1, 1) and perpendicular to the planes 2x + y - 2z = 5 and 3x - 6y - 2z = 7, is

answer: C

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question_subject: mathematics


question:
Let O be the origin and let PQR be an arbitrary triangle. The point S is such that OP · OQ + OR · OS = OR · OP + OQ · OS = OQ · OR + OP · OS
Then the triangle PQR has S as its

response_choices:
A) centroid
B) circumcentre
C) incentre
D) orthocenter

answer: D

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question_subject: mathematics


question: If y = y(x) satisfies the differential equation (8/sqrt(9 + sqrt(x)) dy = ((4 + sqrt(9 + sqrt(x)))^-1) dx, x > 0) and y(0) = sqrt(7), then y(256) =

answer: A

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question_subject: mathematics


question: If f: R → R is a twice differentiable function such that f''(x) > 0 for all x ∈ R, and f(1/2) = 1/2 f'(1) = 1, then

answer: D

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question_subject: mathematics


question: How many 3×3 matrices M with entries from {0,1,2} are there, for which the sum of the diagonal entries of M^TM is 5?

answer: B

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question_subject: mathematics


question: Let S = {1,2,3,...,9}. For k = 1,2,...,5, let Nk be the number of subsets of S, each containing five elements out of which exactly k are odd. Then N1 + N2 + N3 + N4 + N5 =

answer: D

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question_subject: mathematics


question: Three randomly chosen nonnegative integers x, y and z are found to satisfy the equation x + y + z = 10. Then the probability that z is even, is

answer: B

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question_subject: mathematics


question: If g(x) = (∫sin(2x)/sinx)*1(t) dt, then

answer: [A], [C] and [D]

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question_subject: mathematics


question: Let α and β be nonzero real numbers such that 2(cos β - cos α) + cos α cos β = 1. Then which of the following is/are true?

answer: All candidates are awarded +4 marks.

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question_subject: mathematics


question: If f: R → R is a differentiable function such that f'(x) > 2f(x) for all x ∈ R, and f(0) = 1, then

response_choices:
[A] f(x) is increasing in (0, ∞)
[B] f(x) is decreasing in (0, ∞)
[C] f(x) > e^2x in (0, ∞)
[D] f'(x) < e^2x in (0, ∞)

answer: (A) and (C)

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question_subject: mathematics


question: Let f(x) = (1-x^(1+|1-x|)) / |1-x| * cos(1 / 1-x) for x ≠ 1. Then

response_choices:
[A] lim(x→1-) f(x) = 0
[B] lim(x→1+) f(x) does not exist
[C] lim(x→1-) f(x) = 0
[D] lim(x→1+) f(x) does not exist

answer: (A) and (D)

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question_subject: mathematics


question:
If f(x) = [cos(2x)  cos(2x)  sin(2x)]  [- cos x    cos x    - sin x], then
                  [ sin x     sin x     cos x ]

response_choices:
[A] f'(x) = 0 at exactly three points in (-π, π)
[B] f'(x) = 0 at more than three points in (-π, π)
[C] f(x) attains its maximum at x = 0
[D] f(x) attains its minimum at x = 0

answer: (B) and (C)

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question_subject: mathematics


question: If the line x = α divides the area of region R = {(x, y) ∈ R^2 : x^3 ≤ y ≤ x, 0 ≤ x ≤ 1} into two equal parts, then

answer: B and C

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question_subject: mathematics


question: If I = ∑(98 k=1) ∫(k k+1) (k+1 x(x+1)) dx, then

answer: B and D

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question_subject: mathematics


question: | OX × OY |=

response_choices: [A] sin(P + Q) [B] sin 2R [C] sin(P + R) [D] sin(Q + R)

answer: A

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question_subject: mathematics


question: If the triangle PQR varies, then the minimum value of cos(P + Q) + cos(Q + R) + cos(R + P) is

answer: B

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question_subject: mathematics


question: Let p, q be integers and let α, β be the roots of the equation, x^2 - x - 1 = 0, where α ≠ β. For n = 0, 1, 2, ... let an = pα^n + qβ^n. FACT: If a and b are rational numbers and a + b√5 = 0, then a = 0 = b. Q.53 a12 =

answer: B

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question_subject: mathematics


question: If a4 = 28, then p + 2q =

answer: D

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question_subject: mathematics