question: The electrostatic energy of Z protons uniformly distributed throughout a spherical nucleus of radius R is given by E=3Z(Z-1)e^2/5(4πϵ₀R). The measured masses of the neutron, ¹H, ¹⁵N and ¹⁷O are 1.008665 u, 1.007825 u, 15.000109 u and 15.003065 u, respectively. Given that the radii of both the ¹⁵N and ¹⁷O nuclei are same, 1 u = 931.5 MeV/c^2 (c is the speed of light) and e²/(4πϵ₀)=1.44 MeV fm. Assuming that the difference between the binding energies of ¹⁵N and ¹⁷O is purely due to the electrostatic energy, the radius of either of the nuclei is (1 fm = 10^-15 m)

answer: C

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question_subject: physics


question: An accident in a nuclear laboratory resulted in deposition of a certain amount of radioactive material of half-life 18 days inside the laboratory. Tests revealed that the radiation was 64 times more than the permissible level required for safe operation of the laboratory. What is the minimum number of days after which the laboratory can be considered safe for use?

answer: C

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question_subject: physics


question: A gas is enclosed in a cylinder with a movable frictionless piston. Its initial thermodynamic state at pressure Pi=10^6 Pa and volume Vi=10^(-3) m^3 changes to a final state at Pf=(1/32)×10^6 Pa and Vf=8×10^(-3) m^3 in an adiabatic quasi-static process, such that PV^(gamma)= constant. Consider another thermodynamic process that brings the system from the same initial state to the same final state in two steps: an isobaric expansion at Pi followed by an isochoric (isovolumetric) process at volume Vf. The amount of heat supplied to the system in the two-step process is approximately

answer: C

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question_subject: physics


question: The ends Q and R of two thin wires, PQ and RS, are soldered (joined) together. Initially each of the wires has a length of 1 m at 10 °C. Now the end P is maintained at 10 °C, while the end S is heated and maintained at 400 °C. The system is thermally insulated from its surroundings. If the thermal conductivity of wire PQ is twice that of the wire RS and the coefficient of linear thermal expansion of PQ is 1.2 × 10^(-5) K^(-1), the change in length of the wire PQ is

answer: A

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question_subject: physics


question: A small object is placed 50 cm to the left of a thin convex lens of focal length 30 cm. A convex spherical mirror of radius of curvature 100 cm is placed to the right of the lens at a distance of 50 cm. The mirror is tilted such that the axis of the mirror is at an angle θ = 30° to the axis of the lens, as shown in the figure. If the origin of the coordinate system is taken to be at the centre of the lens, the coordinates (in cm) of the point (x, y) at which the image is formed are

answer: (C)

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question_subject: physics


question: There are two Vernier calipers both of which have 1 cm divided into 10 equal divisions on the main scale. The Vernier scale of one of the calipers (C1) has 10 equal divisions that correspond to 9 main scale divisions. The Vernier scale of the other caliper (C2) has 10 equal divisions that correspond to 11 main scale divisions. The readings of the two calipers are shown in the figure. The measured values (in cm) by calipers C1 and C2, respectively, are

answer: B

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question_subject: physics


question: Two thin circular discs of mass m and 4m, having radii of a and 2a, respectively, are rigidly fixed by a massless, rigid rod of length l = √24a through their centers. This assembly is laid on a firm and flat surface, and set rolling without slipping on the surface so that the angular speed about the axis of the rod is ω. The angular momentum of the entire assembly about the point 'O' is L (see the figure). Which of the following statement(s) is(are) true?

response_choices: (A) The center of mass of the assembly rotates about the z-axis with an angular speed of ω/5, (B) The magnitude of angular momentum of center of mass of the assembly about the point O is 81ma²ω, (C) The magnitude of angular momentum of the assembly about its center of mass is 17ma²ω/2, (D) The magnitude of the z-component of L is 55ma²ω

answer: (A) OR (A) and (C)

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question_subject: physics


question: Light of wavelength λph falls on a cathode plate inside a vacuum tube as shown in the figure. The work function of the cathode surface is ϕ and the anode is a wire mesh of conducting material kept at a distance d from the cathode. A potential difference V is maintained between the electrodes. If the minimum de Broglie wavelength of the electrons passing through the anode is λc, which of the following statement(s) is(are) true?

response_choices:
(A) λc decreases with increase in ϕ and λph
(B) λc is approximately halved, if d is doubled
(C) For large potential difference (V \gg ϕe), λc is approximately halved if V is made four times
(D) λc increases at the same rate as λph for λph < hc/ϕ

answer: C

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question_subject: physics


question: In an experiment to determine the acceleration due to gravity g, the formula used for the time period of a periodic motion is T' = 2π√((7(R - r) / 5g). The values of R and r are measured to be (60 ± 1) mm and (10 ± 1) mm, respectively. In five successive measurements, the time period is found to be 0.52 s, 0.56 s, 0.57 s, 0.54 s and 0.59 s. The least count of the watch used for the measurement of time period is 0.01 s. Which of the following statement(s) is(are) true?

response_choices: (A) The error in the measurement of r is 10% (B) The error in the measurement of T' is 3.57% (C) The error in the measurement of T' is 2% (D) The error in the determined value of g is 11%

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

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question_subject: physics


question: Consider two identical galvanometers and two identical resistors with resistance R. If the internal resistance of the galvanometers Rc < R/2, which of the following statement(s) about any one of the galvanometers is(are) true?

response_choices: (A) The maximum voltage range is obtained when all the components are connected in series (B) The maximum voltage range is obtained when the two resistors and one galvanometer are connected in series, and the second galvanometer is connected in parallel to the first galvanometer (C) The maximum current range is obtained when all the components are connected in parallel (D) The maximum current range is obtained when the two galvanometers are connected in series and the combination is connected in parallel with both the resistors

answer: (B) and (C)

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question_subject: physics


question: In the circuit shown below, the key is pressed at time t = 0. Which of the following statement(s) is(are) true?

response_choices:
(A) The voltmeter displays - 5 V as soon as the key is pressed, and displays + 5 V after a long time
(B) The voltmeter will display 0 V at time t = ln 2 seconds
(C) The current in the ammeter becomes 1/e of the initial value after 1 second
(D) The current in the ammeter becomes zero after a long time

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

image_available: yes

question_subject: physics


question: A block with mass M is connected by a massless spring with stiffness constant k to a rigid wall and moves without friction on a horizontal surface. The block oscillates with small amplitude A about an equilibrium position x0. Consider two cases: (i) when the block is at x0; and (ii) when the block is at x = x0 + A. In both the cases, a particle with mass m(< M) is softly placed on the block after which they stick to each other. Which of the following statement(s) is(are) true about the motion after the mass m is placed on the mass M?

response_choices:
(A) The amplitude of oscillation in the first case changes by a factor of √(M/(m+M)), whereas in the second case it remains unchanged
(B) The final time period of oscillation in both the cases is same
(C) The total energy decreases in both the cases
(D) The instantaneous speed at x0 of the combined masses decreases in both the cases

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

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question_subject: physics


question: While conducting the Young's double slit experiment, a student replaced the two slits with a large opaque plate in the x-y plane containing two small holes that act as two coherent point sources (S1, S2) emitting light of wavelength 600 nm. The student mistakenly placed the screen parallel to the x-z plane (for z > 0) at a distance D = 3 m from the mid-point of S1S2, as shown schematically in the figure. The distance between the sources d = 0.6003 mm. The origin O is at the intersection of the screen and the line joining S1S2. Which of the following is(are) true of the intensity pattern on the screen?

response_choices:
(A) Straight bright and dark bands parallel to the x-axis
(B) The region very close to the point O will be dark
(C) Hyperbolic bright and dark bands with foci symmetrically placed about O in the x-direction
(D) Semi circular bright and dark bands centered at point O

answer: (B) and (D)

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question_subject: physics


question: A rigid wire loop of square shape having side of length L and resistance R is moving along the x-axis with a constant velocity v0 in the plane of the paper. At t = 0, the right edge of the loop enters a region of length 3L where there is a uniform magnetic field B0 into the plane of the paper, as shown in the figure. For sufficiently large v0, the loop eventually crosses the region. Let x be the location of the right edge of the loop. Let v(x), I(x) and F(x) represent the velocity of the loop, current in the loop, and force on the loop, respectively, as a function of x. Counter-clockwise current is taken as positive. Which of the following schematic plot(s) is(are) correct? (Ignore gravity)

response_choices: A, B, C, D

answer: A and B

image_available: yes

question_subject: physics


question: This section contains instructions about the format and scoring of the following questions. No specific question text is provided.

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question_subject: mathematics


question: The distance r of the block at time t is

response_choices: (A) R/4 (eωt + e-ωt) (B) R/2 cos ωt (C) R/4 (e2ωt + e-2ωt) (D) R/2 cos 2ωt

answer: A

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question_subject: physics


question: The net reaction of the disc on the block is

response_choices: (A) 1/2 mω2R(e2ωt - e-2ωt) j + mgk (B) 1/2 mω2R(eωt - e-ωt) j + mgk (C) - mω2R cos ωt j - mgk (D) mω2R sin ωt j - mgk

answer: B

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question_subject: physics


question: Which one of the following statements is correct?

response_choices:
(A) The balls will stick to the top plate and remain there
(B) The balls will bounce back to the bottom plate carrying the same charge they went up with
(C) The balls will bounce back to the bottom plate carrying the opposite charge they went up with
(D) The balls will execute simple harmonic motion between the two plates

answer: C

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question_subject: physics


question: The average current in the steady state registered by the ammeter in the circuit will be

response_choices:
(A) zero
(B) proportional to the potential V₀
(C) proportional to V₀¹⁄²
(D) proportional to V₀²

answer: D

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question_subject: physics


question: For the following electrochemical cell at 298 K, Pt(s)|H2(g,1 bar)| H⁺(aq,1 M)| M⁺⁺(aq),M⁺⁺(aq)| Pt(s) E_cell = 0.092 V when [M^(+2)]/[M^(+4)] = 10^4. Given : E^0_(M^(+4)/M^(+2)) = 0.151 V;2.303RT/F = 0.059 V The value of x is

answer: D

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question_subject: chemistry


question: The qualitative sketches I, II and III given below show the variation of surface tension with molar concentration of three different aqueous solutions of KCl, CH3OH and CH3(CH2)11OSO3-Na+ at room temperature. The correct assignment of the sketches is

response_choices:
(A) I : KCl

II : CH3OH

III : CH3(CH2)11OSO3-Na+

(B) I : CH3(CH2)11OSO3-Na+ II : CH3OH

III : KCl

(C) I : KCl

II : CH3(CH2)11OSO3-Na+ III : CH3OH

(D) I : CH3OH

II : KCl

III : CH3(CH2)11OSO3-Na+

answer: D

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question_subject: chemistry


question: In the following reaction sequence in aqueous solution, the species X, Y and Z, respectively, are

answer: A

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question_subject: chemistry


question: The geometries of the ammonia complexes of Ni2+, Pt2+ and Zn2+, respectively, are

response_choices:
(A) octahedral, square planar and tetrahedral
(B) square planar, octahedral and tetrahedral
(C) tetrahedral, square planar and octahedral
(D) octahedral, tetrahedral and square planar

answer: (A)

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question_subject: chemistry


question: The correct order of acidity for the following compounds is

response_choices:
(A) I > II > III > IV
(C) III > IV > II > I
(B) III > I > II > IV
(D) I > III > IV > II

answer: (A)

image_available: yes

question_subject: chemistry


question: The major product of the following reaction sequence is

answer: (A)

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question_subject: chemistry


question: According to Molecular Orbital Theory,

response_choices:
(A) C2+ is expected to be diamagnetic
(B) O2+ is expected to have a longer bond length than O2
(C) N2+ and N2- have the same bond order
(D) He2+ has the same energy as two isolated He atoms

answer: (A) and (C)

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question_subject: chemistry


question: Mixture(s) showing positive deviation from Raoult's law at 35 °C is(are)

response_choices:
(A) carbon tetrachloride + methanol
(B) carbon disulphide + acetone
(C) benzene + toluene
(D) phenol + aniline

answer: (A) and (B)

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question_subject: chemistry


question: The CORRECT statement(s) for cubic close packed (ccp) three dimensional structure is(are)

response_choices:
(A) The number of the nearest neighbours of an atom present in the topmost layer is 12
(B) The efficiency of atom packing is 74%
(C) The number of octahedral and tetrahedral voids per atom are 1 and 2, respectively
(D) The unit cell edge length is 2√2 times the radius of the atom

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

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question_subject: chemistry


question: Extraction of copper from copper pyrite (CuFeS₂) involves

response_choices:
(A) crushing followed by concentration of the ore by froth-flotation
(B) removal of iron as slag
(C) self-reduction step to produce 'blister copper' following evolution of SO₂
(D) refining of 'blister copper' by carbon reduction

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

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question_subject: chemistry


question: The nitrogen containing compound produced in the reaction of HNO₃ with P₄O₁₀

response_choices:
(A) can also be prepared by reaction of P₄ and HNO₃
(B) is diamagnetic
(C) contains one N-N bond
(D) reacts with Na metal producing a brown gas

answer: (B) and (D)

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question_subject: chemistry


question:
For 'invert sugar', the correct statement(s) is(are)
(Given: specific rotations of (+)-sucrose, (+)-maltose, L-(-)-glucose and L-(+)-fructose in aqueous solution are +66°, +140°, -52° and +92°, respectively)

response_choices:
(A) 'invert sugar' is prepared by acid catalyzed hydrolysis of maltose
(B) 'invert sugar' is an equimolar mixture of D-(+)-glucose and D-(-)-fructose
(C) specific rotation of 'invert sugar' is -20°
(D) on reaction with Br₂ water, 'invert sugar' forms saccharic acid as one of the products

answer: (B) and (C)

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question_subject: chemistry


question: Reagent(s) which can be used to bring about the following transformation is(are)

response_choices: (A) LiAlH4 in (C2H5)2O, (B) BH4 in THF, (C) NaBH4 in C2H5OH, (D) Raney Ni/H2 in THF

answer: (C) and (D)

image_available: yes

question_subject: chemistry


question: Among the following, reaction(s) which gives(give) tert-butyl benzene as the major product is(are)

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

image_available: yes

question_subject: chemistry


question: The equilibrium constant Kp for this reaction at 298 K, in terms of βequilibrium, is

response_choices: (A) 8β²equilibrium/(2 - βequilibrium) (B) 8β²equilibrium/(4 - β²equilibrium) (C) 4β²equilibrium/(2 - βequilibrium) (D) 4β²equilibrium/(4 - β²equilibrium)

answer: B

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question_subject: chemistry


question: The INCORRECT statement among the following, for this reaction, is

response_choices: (A) Decrease in the total pressure will result in formation of more moles of gaseous X (B) At the start of the reaction, dissociation of gaseous X₂ takes place spontaneously (C) βequilibrium = 0.7 (D) Kc < 1

answer: C

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question_subject: chemistry


question: The compound R is

response_choices: (A) NH2, (B) Br, (C) NHBr, (D) NBr

answer: (A)

image_available: yes

question_subject: chemistry


question: The compound T is

response_choices: (A) glycine, (B) alanine, (C) valine, (D) serine

answer: (B)

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question_subject: chemistry


question: Let P = [1 0 0; 4 1 0; 16 4 1] and I be the identity matrix of order 3. If Q = [qij] is a matrix such that P^50 - Q = I, then q31 + q32 / q21 equals

answer: B

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question_subject: mathematics


question: Let bi > 1 for i = 1, 2, ..., 101. Suppose log2 b1, log2 b2, ..., log2 b101 are in Arithmetic Progression (A.P.) with the common difference log2 2. Suppose a1, a2, ..., a101 are in A.P. such that a1 = b1 and a51 = b51. If t = b1 + b2 + ... + b51 and s = a1 + a2 + ... + a51, then

answer: B

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question_subject: mathematics


question: The value of Σ (1 / sin(π/4 + (k - 1)π/6) * sin(π/4 + kπ/6)) from k=1 to 3 is equal to

answer: C

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question_subject: mathematics


question: The value of ∫(x^2 cos x / 1 + e^(x/2)) dx is equal to

answer: A

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question_subject: mathematics


question: Area of the region {(x, y) ∈ R^2 : y ≥ √(x + 3), 5y ≤ x + 9 ≤ 15} is equal to

answer: C

image_available: yes

question_subject: mathematics


question: Let P be the image of the point (3, 1, 7) with respect to the plane x - y + z = 3. Then the equation of the plane passing through P and containing the straight line x/2 = y/2 = z/1 is

answer: C

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question_subject: mathematics


question: Let f(x) = lim [n^(x+n)/(x+n/2)*...(x+n)/n!] / [n!*(x^2+n^2)*(x^2+n^2/4)*...(x^2+n^4/n^2)], for all x > 0. Then

answer: (B) and (C)

image_available: yes

question_subject: mathematics


question: Let a, b ∈ ℝ and f : ℝ→ℝ be defined by f(x) = a cos(|x^3 - x|) + b|x|sin(|x^3 + x|). Then f is

response_choices: (A) differentiable at x = 0 if a = 0 and b = 1 (B) differentiable at x = 1 if a = 1 and b = 0 (C) NOT differentiable at x = 0 if a = 1 and b = 0 (D) NOT differentiable at x = 1 if a = 1 and b = 1

answer: (A) and (B)

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question_subject: mathematics


question: Let f : R →(0, ∞) and g : R → R be twice differentiable functions such that f'' and g'' are continuous functions on R. Suppose f''(2)=g(2)=0, f''(2)≠0 and g'(2)≠0. If lim(x→2) f(x)/g(x) = -1, then

response_choices:
(A) f has a local minimum at x = 2
(B) f has a local maximum at x = 2
(C) f''(2) > f(2)
(D) f(x) - f''(x) = 0 for at least one x ∈ R

answer: (A) and (D)

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question_subject: mathematics


question: Let f : [1/2, 2] → R and g : [-1/2, 2] → R be functions defined by f(x) = [x^2 - 3] and g(x) = [x] f(x) + [4x - 7] f(x), where [y] denotes the greatest integer less than or equal to y for y ∈ R. Then

response_choices:
(A) f is discontinuous exactly at three points in [-1/2, 2]
(B) f is discontinuous exactly at four points in [-1/2, 2]
(C) g is NOT differentiable exactly at four points in (-1/2, 2)
(D) g is NOT differentiable exactly at five points in (-1/2, 2)

answer: (B) and (C)

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question_subject: mathematics


question: Let a, b ∈ ℝ and a^2 + b^2 ≠ 0. Suppose S = {z ∈ ℂ : z = (1 +/- i sqrt(-1)t) / (a +/- ibt), t ∈ ℝ, t ≠ 0}. If z = x + iy and z ∈ S, then (x, y) lies on

response_choices:
(A) the circle with radius 1/2a and centre (1/2a, 0) for a > 0, b ≠ 0
(B) the circle with radius -1/2a and centre (-1/2a, 0) for a < 0, b ≠ 0
(C) the x-axis for a ≠ 0, b = 0
(D) the y-axis for a = 0, b ≠ 0

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

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question_subject: mathematics


question: Let P be the point on the parabola y^2 = 4x which is at the shortest distance from the center S of the circle x^2 + y^2 - 4x - 16y + 64 = 0. Let Q be the point on the circle dividing the line segment SP internally. Then

response_choices:
(A) SP = 2√5
(B) SQ : QP = (√5 + 1) : 2
(C) the x-intercept of the normal to the parabola at P is 6
(D) the slope of the tangent to the circle at Q is 1/2

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

image_available: no

question_subject: mathematics


question: Let α, λ, μ ∈ ℝ. Consider the system of linear equations αx + 2y = λ 3x - 2y = μ Which of the following statement(s) is(are) correct?

response_choices: (A) If α = -3, then the system has infinitely many solutions for all values of λ and μ (B) If α ≠ -3, then the system has a unique solution for all values of λ and μ (C) If λ + μ = 0, then the system has infinitely many solutions for α = -3 (D) If λ + μ ≠ 0, then the system has no solution for α = -3

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

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question_subject: mathematics


question: Let ū = u₁i + u₂j + u₃k be a unit vector in ℝ³ and ŵ = (1/√6)(i + j + 2k). Given that there exists a vector v̄ in ℝ³ such that |ū × v̄| = 1 and ŵ · (ū × v̄) = 1. Which of the following statement(s) is(are) correct?

response_choices: (A) There is exactly one choice for such v̄ (B) There are infinitely many choices for such v̄ (C) If ū lies in the xy-plane then |u₁| = |u₂| (D) If ū lies in the xz-plane then 2|u₁| = |u₃|

answer: (B) and (C)

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question_subject: mathematics


question: P(X > Y) is

answer: B

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question_subject: mathematics


question: P(X = Y) is

answer: C

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question_subject: mathematics


question: The orthocentre of the triangle FiMN is

answer: (A)

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question_subject: mathematics


question: If the tangents to the ellipse at M and N meet at R and the normal to the parabola at M meets the x-axis at Q, then the ratio of area of the triangle MQR to area of the quadrilateral MFiNF2 is

answer: (C)

image_available: yes

question_subject: mathematics