id int64 1 84 | question stringlengths 11 1.17k | response_choices stringlengths 0 901 | answer stringclasses 330 values | page_number int64 1 40 | image_available stringclasses 2 values | question_subject stringclasses 3 values |
|---|---|---|---|---|---|---|
33 | The hyperconjugative stabilities of tert-butyl cation and 2-butene, respectively, are due to | (A) σ→p (empty) and σ→π* electron delocalisations. (B) σ→σ* and σ→π electron delocalisations. (C) σ→p (filled) and σ→π electron delocalisations. (D) p (filled)→σ* and σ→π* electron delocalisations. | A | 24 | no | chemistry |
34 | Benzene and naphthalene form an ideal solution at room temperature. For this process, the true statement(s) is(are) | (A) ΔG is positive, (B) ΔS_system is positive, (C) ΔS_surroundings = 0, (D) ΔH = 0 | BCD | 25 | no | chemistry |
35 | The initial rate of hydrolysis of methyl acetate (1M) by a weak acid (HA, 1M) is 1/100th of that of a strong acid (HX, 1M), at 25 °C. The Ka of HA is | (A) 1 × 10^-4, (B) 1 × 10^-5, (C) 1 × 10^-6, (D) 1 × 10^-3 | A | 25 | no | chemistry |
36 | The total number of carboxylic acid groups in the product P is | No response choices provided | 2 | 26 | yes | chemistry |
37 | A tetrapeptide has –COOH group on alanine. This produces glycine (Gly), valine (Val), phenyl alanine (Phe) and alanine (Ala), on complete hydrolysis. For this tetrapeptide, the number of possible sequences (primary structures) with –NH2 group attached to a chiral center is | No response choices provided | 4 | 26 | no | chemistry |
38 | EDTA4- is ethylenediaminetetraacetate ion. The total number of N-Co-O bond angles in [Co(EDTA)]1- complex ion is | 8 | 27 | no | chemistry | |
39 | The total number of lone-pairs of electrons in melamine is | 6 | 27 | no | chemistry | |
40 | The atomic masses of He and Ne are 4 and 20 a.m.u., respectively. The value of the de Broglie wavelength of He gas at -73 °C is "M" times that of the de Broglie wavelength of Ne at 727 °C. M is | 5 | 27 | no | chemistry | |
41 | Let complex numbers α and 1/α lie on circles (x - x0)2 + (y - y0)2 = r2 and (x - x0)2 + (y - y0)2 = 4r2, respectively. If z0 = x0 + iy0 satisfies the equation 2 |z0|2 = r2 + 2, then |α| = | A) 1/√2, B) 1/2, C) 1/√7, D) 1/3 | C | 28 | no | mathematics |
42 | Four persons independently solve a certain problem correctly with probabilities 1/3, 1/4, 1/8. Then the probability that the problem is solved correctly by at least one of them is | A) 235/256, B) 21/256, C) 3/256, D) 253/256 | A | 29 | no | mathematics |
43 | Let f : [1/2, 1] → ℝ (the set of all real numbers) be a positive, non-constant and differentiable function such that f''(x) < 2 f'(x) and f(1/2) = 1. Then the value of [integral from 1/2 to 1 of f(x) dx] lies in the interval | (A) (2 e - 1, 2e) (B) (e - 1, 2e - 1) (C) ((e - 1) / 2, e - 1) (D) (0, (e - 1) / 2) | D | 30 | no | mathematics |
44 | The number of points in (- ∞, ∞), for which x^2 - x sin x - cos x = 0, is | (A) 6 (B) 4 (C) 2 (D) 0 | C | 30 | no | mathematics |
45 | The area enclosed by the curves y = sin x + cos x and y = |cos x - sin x| over the interval [0 , π/2] is | (A) 4(√2 - 1) (B) 2√2 (√2 - 1) (C) 2(√2 + 1) (D) 2√2 (√2 + 1) | B | 31 | no | mathematics |
46 | A curve passes through the point (1 , π/6) . Let the slope of the curve at each point (x, y) be y/x + sec (y/x), x > 0. Then the equation of the curve is | (A) sin (y/x) = log x + 1/2 (B) cosec (y/x) = log x + 2 (C) sec (2y/x) = log x + 2 (D) cos (2y/x) = log x + 1/2 | A | 31 | no | mathematics |
47 | The value of cot( ∑(n=1 to 23) cot^-1 (1 + ∑(k=1 to n) 2k) ) is | (A) 23/25, (B) 25/23, (C) 23/24, (D) 24/23 | B | 32 | no | mathematics |
48 | For a > b > c > 0, the distance between (1, 1) and the point of intersection of the lines ax + by + c = 0 and bx + ay + c = 0 is less than 2√2. Then | (A) a + b - c > 0, (B) a - b + c < 0, (C) a - b + c > 0, (D) a + b - c < 0 | A or C or AC | 32 | no | mathematics |
49 | Perpendiculars are drawn from points on the line x+2/2 = y+1/-1 = z/3 to the plane x + y + z = 3. The feet of perpendiculars lie on the line | (A) x/5 = y-1/8 = z-2/-13, (B) x/2 = y-1/3 = z-2/-5, (C) x/4 = y-1/3 = z-2/-7, (D) x/2 = y-1/-7 = z-2/5 | D | 33 | no | mathematics |
50 | Let PR = 3i + j - 2k and SQ = i - 3j - 4k determine diagonals of a parallelogram PQRS and PT = i + 2j + 3k be another vector. Then the volume of the parallelepipeddetermined by the vectors PT, PQ and PS is | (A) 5, (B) 20, (C) 10, (D) 30 | C | 33 | no | mathematics |
51 | Let S_n = Sum from k=1 to 4n of (-1)^(k(k+1)/2) * k^2. Then S_n can take value(s) | A) 1056, B) 1088, C) 1120, D) 1332 | AD | 34 | no | mathematics |
52 | For 3 x 3 matrices M and N, which of the following statement(s) is (are) NOT correct? | (A) N^TMN is symmetric or skew symmetric, according as M is symmetric or skew symmetric
(B) MN - NM is skew symmetric for all symmetric matrices M and N
(C) MN is symmetric for all symmetric matrices M and N
(D) (adj M) (adj N) = adj (MN) for all invertible matrices M and N | CD | 35 | no | mathematics |
53 | Let f(x) = x sin πx, x > 0. Then for all natural numbers n, f'(x) vanishes at | A) a unique point in the interval (n, n + 1/2), B) a unique point in the interval (n + 1/2, n + 1), C) a unique point in the interval (n, n + 1), D) two points in the interval (n, n + 1) | BC | 36 | no | mathematics |
54 | A rectangular sheet of fixed perimeter with sides having their lengths in the ratio 3 : 15 is converted into an open rectangular box by folding after removing squares of equal area from all four corners. If the total area of removed squares is 100, the resulting box has maximum volume. Then the lengths of the sides of the rectangular sheet are | (A) 24 (B) 32 (C) 45 (D) 60 | AC | 37 | no | mathematics |
55 | A line l passing through the origin is perpendicular to the lines l1 : (3 + 0)i + (-1 + 2t)j + (4 + 2t)k , -∞ < t < ∞ l2 : (3 + 2s)i + (3 + 2s)j + (2 + s)k , -∞ < s < ∞ Then, the coordinate(s) of the point(s) on l2 at a distance of √17 from the point of intersection of l and l1 is (are) | (A) (7/3, 7/3, 3/3) (B) (-1, -1, 0) (C) (1, 1, 1) (D) (7/9, 7/9, 8/9) | BD | 37 | no | mathematics |
56 | The coefficients of three consecutive terms of (1 + x)^n + 5 are in the ratio 5 : 10 : 14. Then n = | No response choices provided | 6 | 38 | no | mathematics |
57 | A pack contains n cards numbered from 1 to n. Two consecutive numbered cards are removed from the pack and the sum of the numbers on the remaining cards is 1224. If the smaller of the numbers on the removed cards is k, then k - 20 = | No response choices provided | 5 | 38 | no | mathematics |
58 | Of the three independent events E₁, E₂ and E₃, the probability that only E₁ occurs is α, only E₂ occurs is β and only E₃ occurs is γ. Let the probability p that none of events E₁, E₂ or E₃ occurs satisfy the equations (α − 2β) p = αβ and (β − 3γ) p = 2βγ. All the given probabilities are assumed to lie in the interval (0, 1).
Then Probability of occurrence of E₁ / Probability of occurrence of E₃ = | 6 | 39 | no | mathematics | |
59 | A vertical line passing through the point (h, 0) intersects the ellipse x^2/4 + y^2/3 = 1 at the points P and Q. Let the tangents to the ellipse at P and Q meet at the point R. If A(h) = area of the triangle PQR, A1 = max (1/2≤h≤1) A(h) and A2 = min (1/2≤h≤1) A(h), then 8/sqrt(5) A1 - 8A2 = | No multiple choice options given | 9 | 40 | no | mathematics |
60 | Consider the set of eight vectors V= {ai + bj + ck : a, b, c ∈ {-1, 1}}. Three non-coplanar vectors can be chosen from V in 2^n ways. Then n is | No multiple choice options given | 5 | 40 | no | mathematics |
1 | Using the expression 2d sin θ = λ, one calculates the values of d by measuring the corresponding angles θ in the range 0 to 90°. The wavelength λ is exactly known and the error in θ is constant for all values of θ. As θ increases from 0°, | (A) the absolute error in d remains constant. (B) the absolute error in d increases. (C) the fractional error in d remains constant. (D) the fractional error in d decreases. | D | 1 | no | physics |
2 | Two non-conducting spheres of radii R1 and R2 and carrying uniform volume charge densities +ρ and –ρ, respectively, are placed such that they partially overlap, as shown in the figure. At all points in the overlapping region, | (A) the electrostatic field is zero. (B) the electrostatic potential is constant. (C) the electrostatic field is constant in magnitude. (D) the electrostatic field has same direction. | CD | 2 | yes | physics |
3 | The figure below shows the variation of specific heat capacity (C) of a solid as a function of temperature (T). The temperature is increased continuously from 0 to 500 K at a constant rate. Ignoring any volume change, the following statement(s) is (are) correct to a reasonable approximation. | (A) the rate at which heat is absorbed in the range 0-100 K varies linearly with temperature T. (B) heat absorbed in increasing the temperature from 0-100 K is less than the heat required for increasing the temperature from 400-500 K. (C) there is no change in the rate of heat absorption in the range 400-500 K. (D) the rate of heat absorption increases in the range 200-300 K. | ABCD or BCD | 3 | yes | physics |
4 | The radius of the orbit of an electron in a Hydrogen-like atom is 4.5 a0, where a0 is the Bohr radius. Its orbital angular momentum is 3h/2π. It is given that h is Planck constant and R is Rydberg constant. The possible wavelength(s), when the atom de-excites, is (are) | (A) 9/32R (B) 9/16R (C) 9/5R (D) 4/3R | AC | 3 | no | physics |
5 | Two bodies, each of mass M, are kept fixed with a separation 2L. A particle of mass m is projected from the midpoint of the line joining their centres, perpendicular to the line. The gravitational constant is G. The correct statement(s) is (are) | (A) The minimum initial velocity of the mass m to escape the gravitational field of the two bodies is 4√(GM/L). (B) The minimum initial velocity of the mass m to escape the gravitational field of the two bodies is 2√(GM/L). (C) The minimum initial velocity of the mass m to escape the gravitational field of the two bodies is √(2GM/L). (D) The energy of the mass m remains constant. | BD | 4 | no | physics |
6 | A particle of mass m is attached to one end of a mass-less spring of force constant k, lying on a frictionless horizontal plane. The other end of the spring is fixed. The particle starts moving horizontally from its equilibrium position at time t = 0 with an initial velocity u0. When the speed of the particle is 0.5 u0, it collides elastically with a rigid wall. After this collision, | (A) the speed of the particle when it returns to its equilibrium position is u0. (B) the time at which the particle passes through the equilibrium position for the first time is t = π√(m/k). (C) the time at which the maximum compression of the spring occurs is t = 4π/3√(m/k). (D) the time at which the particle passes through the equilibrium position for the second time is t = 5π/3√(m/k). | AD | 5 | no | physics |
7 | A steady current I flows along an infinitely long hollow cylindrical conductor of radius R. This cylinder is placed coaxially inside an infinite solenoid of radius 2R. The solenoid has n turns per unit length and carries a steady current I. Consider a point P at a distance r from the common axis. The correct statement(s) is (are) | (A) In the region 0 < r < R, the magnetic field is non-zero. (B) In the region R < r < 2R, the magnetic field is along the common axis. (C) In the region R < r < 2R, the magnetic field is tangential to the circle of radius r, centered on the axis. (D) In the region r > 2R, the magnetic field is non-zero. | AD | 6 | no | physics |
8 | Two vehicles, each moving with speed u on the same horizontal straight road, are approaching each other. Wind blows along the road with velocity w. One of these vehicles blows a whistle of frequency f1. An observer in the other vehicle hears the frequency of the whistle to be f2. The speed of sound in still air is v. The correct statement(s) is (are) | (A) If the wind blows from the observer to the source, f2>f1. (B) If the wind blows from the source to the observer, f2>f1. (C) If the wind blows from observer to the source, f2<f1. (D) If the wind blows from the source to the observer, f2<f1. | AB | 7 | no | physics |
9 | The magnitude of the induced electric field in the orbit at any instant of time during the time interval of the magnetic field change is | (A) BR/4, (B) BR/2, (C) BR, (D) 2BR | B | 8 | no | physics |
10 | The change in the magnetic dipole moment associated with the orbit, at the end of the time interval of the magnetic field change, is | (A) - γ BOR^2, (B) - γ BOR^2/2, (C) γ BOR^2/2, (D) γ BOR^2 | B | 8 | no | physics |
11 | The correct statement is | (A) The nucleus 6Li can emit an alpha particle. (B) The nucleus 210Po can emit a proton. (C) Deuteron and alpha particle can undergo complete fusion. (D) The nuclei 70Zn and 34Se can undergo complete fusion. | C | 9 | no | physics |
12 | The kinetic energy (in keV) of the alpha particle, when the nucleus 210Po at rest undergoes alpha decay, is | (A) 5319 (B) 5422 (C) 5707 (D) 5818 | A | 9 | no | physics |
13 | The speed of the block when it reaches the point Q is | A) 5 ms^-1, B) 10 ms^-1, C) 10√3ms^-1, D) 20 ms^-1 | B | 10 | yes | physics |
14 | The magnitude of the normal reaction that acts on the block at the point Q is | A) 7.5 N, B) 8.6 N, C) 11.5 N, D) 22.5 N | A | 10 | yes | physics |
15 | If the direct transmission method with a cable of resistance 0.4 Ω km⁻¹ is used, the power dissipation (in %) during transmission is | (A) 20 (B) 30 (C) 40 (D) 50 | B | 11 | no | physics |
16 | In the method using the transformers, assume that the ratio of the number of turns in the primary to that in the secondary in the step-up transformer is 1 : 10. If the power to the consumers has to be supplied at 200 V, the ratio of the number of turns in the primary to that in the secondary in the step-down transformer is | (A) 200 : 1 (B) 150 : 1 (C) 100 : 1 (D) 50 : 1 | A | 11 | no | physics |
17 | Match List I with List II and select the correct answer using the codes given below the lists: | List I: P. Boltzmann constant, Q. Coefficient of viscosity, R. Planck constant, S. Thermal conductivity
List II: 1. [ML^2T^-1], 2. [ML^-1T^-1], 3. [MLT^-3K^-1], 4. [ML^2T^-2K^-1]
Codes: (A) P-3, Q-1, R-2, S-4
(B) P-3, Q-2, R-1, S-4
(C) P-4, Q-2, R-1, S-3
(D) P-4, Q-1, R-2, S-3 | C | 12 | no | physics |
18 | A right angled prism of refractive index μ1 is placed in a rectangular block of refractive index μ2, which is surrounded by a medium of refractive index μ3, as shown in the figure. A ray of light 'e' enters the rectangular block at normal incidence. Depending upon the relationships between μ1, μ2 and μ3, it takes one of the four possible paths 'ef', 'eg', 'eh' or 'ei'. | List I / सूची I: P. e→f, Q. e→g, R. e→h, S. e→i
List II / सूची II:
1. μ1 > √2 μ2
2. μ2 > μ1 and/√√√ μ2 > μ3
3. μ1 = μ2
4. μ2 < μ1 < √2 μ2 and/√√√ μ2 > μ3 | D | 13 | yes | physics |
19 | Match List I of the nuclear processes with List II containing parent nucleus and one of the end products of each process and then select the correct answer using the codes given below the lists: | A. 4 2 1 3, B. 1 3 2 4, C. 2 1 4 3, D. 4 3 2 1 | C | 14 | no | physics |
20 | One mole of a monatomic ideal gas is taken along two cyclic processes E→F→G→E and E→F→H→E as shown in the PV diagram. The processes involved are purely isochoric, isobaric, isothermal or adiabatic. | Match the paths in List I with the magnitudes of the work done in List II and select the correct answer using the codes given below the lists. | A | 15 | yes | physics |
21 | The correct statement(s) about O3 is(are) | (A) O-O bond lengths are equal. (B) Thermal decomposition of O3 is endothermic. (C) O3 is diamagnetic in nature. (D) O3 has a bent structure. | ACD | 16 | no | chemistry |
22 | In the nuclear transmutation
ₚ⁹Be + X → ⁸₄Be + Y
(X, Y) is(are)
(A) (ℓ, n) (B) (p, D) (C) (n, D) (D) (ℓ, p)
ₚ⁹Be + X → ⁸₄Be + Y
m̄ (X, Y) ई/है
(A) (ℓ, n) (B) (p, D) (C) (n, D) (D) (ℓ, p) | (A) (ℓ, n) (B) (p, D) (C) (n, D) (D) (ℓ, p) | AB | 16 | no | chemistry |
23 | The carbon-based reduction method is NOT used for the extraction of (A) tin from SnO₂ (B) iron from Fe₂O₃ (C) aluminium from Al₂O₃ (D) magnesium from MgCO₃ · CaCO₃ | (A) SnO₂ से टिन (B) Fe₂O₃ से आयरन (C) Al₂O₃ से एल्युमिनियम (D) MgCO₃·CaCO₃ से मैग्नीशियम | CD | 17 | no | chemistry |
24 | The thermal dissociation equilibrium of CaCO₃(s) is studied under different conditions. CaCO₃(s) ⇌ CaO(s) + CO₂(g) For this equilibrium, the correct statement(s) is(are) (A) ΔH is dependent on T (B) K is independent of the initial amount of CaCO₃ (C) K is dependent on the pressure of CO₂ at a given T (D) ΔH is independent of the catalyst, if any | (A) ΔH तापमान पर निर्भर करता है । (B) सामयावस्था स्थिरांक (K) CaCO₃ के प्रारंभिक परिमाण पर निर्भर नहीं करता है । (C) K निर्दिष्ट तापमान पर CO₂ के दाब पर निर्भर करता है । (D) ΔH उत्प्रेरक (अगर हो) के प्रभाव पर निर्भर नहीं करता है । | ABD | 17 | no | chemistry |
25 | The Ksp of Ag2CrO4 is 1.1 × 10⁻¹² at 298 K. The solubility (in mol/L) of Ag2CrO4 in a 0.1 M AgNO3 solution is | (A) 1.1 × 10⁻¹¹ (B) 1.1 × 10⁻¹⁰ (C) 1.1 × 10⁻¹² (D) 1.1 × 10⁻⁹ | B | 18 | no | chemistry |
26 | In the following reaction, the product(s) formed is(are) | (A) P (major) (B) Q (minor) (C) R (minor) (D) S (major) | BD | 18 | yes | chemistry |
27 | The major product(s) of the following reaction is(are) | A) P, B) Q, C) R, D) S | Q | 19 | yes | chemistry |
28 | After completion of the reactions (I and II), the organic compound(s) in the reaction mixtures is(are) | (A) Reaction I : P and Reaction II : P, (B) Reaction I : U, acetone and Reaction II : Q, acetone, (C) Reaction I : T, U, acetone and Reaction II : P, (D) Reaction I : R, acetone and Reaction II : S, acetone | C | 20 | yes | chemistry |
29 | The succeeding operations that enable this transformation of states are | (A) Heating, cooling, heating, cooling
(B) Cooling, heating, cooling, heating
(C) Heating, cooling, cooling, heating
(D) Cooling, heating, heating, cooling | C | 21 | yes | chemistry |
30 | The pair of isochoric processes among the transformation of states is | (A) K to L and L to M
(B) L to M and N to K
(C) L to M and M to N
(D) M to N and N to K | B | 21 | yes | chemistry |
31 | P and Q, respectively, are the sodium salts of | (A) hypochlorus and chloric acids (B) hypochlorus and perchloric acids (C) chloric and perchloric acids (D) chloric and hypochlorus acids | A | 22 | no | chemistry |
32 | R, S and T, respectively, are | (A) SO₂Cl₂, PCl₃ and H₃PO₄ (B) SO₂Cl₂, PCl₃ and H₃PO₃ (C) SOCl₂, PCl₃ and H₃PO₂ (D) SOCl₂, PCl₃ and H₃PO₄ | A | 22 | no | chemistry |
33 | The precipitate P contains | (A) Pb2+ (B) Hg22+ (C) Ag+ (D) Hg2+ | A | 23 | no | chemistry |
34 | The coloured solution S contains | (A) Fe2(SO4)3 (B) CuSO4 (C) ZnSO4 (D) Na2CrO4 | D | 23 | no | chemistry |
35 | Compounds formed from P and Q are, respectively | (A) Optically active S and optically active pair (T, U)
(B) Optically inactive S and optically inactive pair (T, U)
(C) Optically active pair (T, U) and optically active S
(D) Optically inactive pair (T, U) and optically inactive S | B | 24 | yes | chemistry |
36 | In the following reaction sequences V and W are, respectively | (A), (B), (C), (D) | A | 25 | yes | chemistry |
37 | Match the chemical conversions in List I with the appropriate reagents in List II and select the correct answer using the code given below the lists: | (A) 2 3 1 4, (B) 3 2 1 4, (C) 2 3 4 1, (D) 3 2 4 1 | A | 26 | yes | chemistry |
38 | The unbalanced chemical reactions given in List I show missing reagent or condition (?) which are provided in List II. Match List I with List II and select the correct answer using the code given below the lists: | A) 4 2 3 1
B) 3 2 1 4
C) 1 4 2 3
D) 3 4 2 1 | D | 27 | no | chemistry |
39 | The standard reduction potential data at 25 °C is given below.
E°(Fe3+, Fe2+) = + 0.77 V;
E°(Fe2+, Fe) = - 0.44 V
E°(Cu2+, Cu) = + 0.34 V;
E°(Cu+, Cu) = + 0.52 V
E°[O2(g) + 4H+ + 4e- → 2H2O] = +1.23 V;
E°[O2(g)+2H2O+ 4e- → 4OH-] = + 0.40 V
E°(Cr3+, Cr) = - 0.74 V;
E°(Cr2+, Cr) = - 0.91 V
Match E° of the redox pair in List I with the values given in List II and select the correct answer using the code given below the lists : | P. E°(Fe3+, Fe)
Q. E°(4H2O ⬌ 4H+ + 4OH-)
R. E°(Cu2+ + Cu → 2Cu+)
S. E°(Cr3+, Cr2+)
Codes :
P Q R S
(A) 4 1 2 3
(B) 2 3 4 1
(C) 1 2 3 4
(D) 3 4 1 2 | D | 28 | no | chemistry |
40 | An aqueous solution of X is added slowly to an aqueous solution of Y as shown in List I. The variation in conductivity of these reactions is given in List II. Match List I with List II and select the correct answer using the code given below the lists: | P. (C2H5)3N + CH3COOH, Q. KI (0.1M) + AgNO3 (0.01M), R. CH3COOH + KOH, S. NaOH + HI | A | 29 | no | chemistry |
41 | Let w = √(3 + i)/2 and P = {wⁿ : n = 1, 2, 3, ...}. Further H₁ = {z ∈ C : Re z > 1/2} and H₂ = {z ∈ C : Re z < -1/2}, where C is the set of all complex numbers. If z₁ ∈ P ∩ H₁, z₂ ∈ P ∩ H₂ and O represents the origin, then ∠ z₁ O z₂ = | A) π/2, B) π/6, C) 2π/3, D) 5π/6 | CD | 30 | no | mathematics |
42 | If 3ˣ = 4ˣ⁻¹, then x = | A) 2 log₂ 2/(2 log₂ 2 - 1), B) 2/(2 - log₂ 3), C) 1/(1 - log₄ 3), D) 2 log₂ 3/(2 log₂ 3 - 1) | ABC | 30 | no | mathematics |
43 | Let ω be a complex cube root of unity with ω ≠ 1 and P = [pij] be a n × n matrix with pij = ω^{ij}. Then P^2 ≠ 0, when n = | (A) 57 (B) 55 (C) 58 (D) 56 | BCD | 31 | no | mathematics |
44 | The function f(x) = 2 | x | + | x + 2 | - | | x + 2 | - 2 | x || has a local minimum or a local maximum at x = | (A) -2 (B) -2/3 (C) 2 (D) 2/3 | AB | 31 | no | mathematics |
45 | For a ∈ ℝ (the set of all real numbers), a ≠ -1, lim n→∞ ((1ª + 2^n + ... + n^n) / (n + 1)^n-1 [(na + 1) + (na + 2) + ... + (na + n)]) = 1/60 Then a = | (A) 5 (B) 7 (C) -15/2 (D) -17/2 | B | 31 | no | mathematics |
46 | Circle(s) touching x -- axis at a distance 3 from the origin and having an intercept of length 2√7 on y -- axis is (are) | (A) x^2 + y^2 - 6x + 8y + 9 = 0, (B) x^2 + y^2 - 6x + 7y + 9 = 0, (C) x^2 + y^2 - 6x - 8y + 9 = 0, (D) x^2 + y^2 - 6x - 7y + 9 = 0 | AC | 32 | no | mathematics |
47 | Two lines L1 : x = 5, y/(3 - α) = z/-2 and L2 : x = α, y/(-1 - 2 - α) = z/2 - α are coplanar. Then α can take value(s) | (A) 1, (B) 2, (C) 3, (D) 4 | AD | 32 | no | mathematics |
48 | In a triangle PQR, P is the largest angle and cos P = 1/3. Further the incircle of the triangle touches the sides PQ, QR and RP at N, L and M respectively, such that the lengths of PN, QL and RM are consecutive even integers. Then possible length(s) of the side(s) of the triangle is (are) | (A) 16, (B) 18, (C) 24, (D) 22 | BD | 32 | no | mathematics |
49 | Area of S = S₁ ∩ S₂ ∩ S₃, where S₁ = {z ∈ C : |z| < 4}, S₂ = {z ∈ C : Im[(z - 1 + √3i) / (1 - √3i)] > 0} and S₃ = {z ∈ C : Re z > 0}. | (A) 10π/3, (B) 20π/3, (C) 16π/3, (D) 32π/3 | B | 33 | no | mathematics |
50 | min |1 - 3t - z| = | (A) (2 - √3)/2, (B) (2 + √3)/2, (C) (3 - √3)/2, (D) (3 + √3)/2 | C | 33 | no | mathematics |
51 | If 1 ball is drawn from each of the boxes B1, B2 and B3, the probability that all 3 drawn balls are of the same colour is | A) 82/648, B) 90/648, C) 558/648, D) 566/648 | A | 34 | no | mathematics |
52 | If 2 balls are drawn (without replacement) from a randomly selected box and one of the balls is white and the other ball is red, the probability that these 2 balls are drawn from box B2 is | A) 116/181, B) 126/181, C) 65/181, D) 55/181 | D | 34 | no | mathematics |
53 | Which of the following is true for 0 < x < 1 ? | (A) 0 < f(x) < ∞, (B) -1/2 < f(x) < 1/2, (C) -1/4 < f(x) < 1, (D) -∞ < f(x) < 0 | D | 35 | no | mathematics |
54 | If the function e^x f(x) assumes its minimum in the interval [0, 1] at x = 1/4, which of the following is true ? | (A) f'(x) < f(x), 1/4 < x < 3/4, (B) f'(x) > f(x), 0 < x < 1/4, (C) f'(x) < f(x), 0 < x < 1/4, (D) f'(x) < f(x), 3/4 < x < 1 | C | 35 | no | mathematics |
55 | Length of chord PQ is | A) 7a, B) 5a, C) 2a, D) 3a | B | 36 | no | mathematics |
56 | If chord PQ subtends an angle θ at the vertex of y^2 = 4ax, then tan θ = | A) 2/3 √7, B) -2/3 √7, C) 2/3 √3, D) -2/3 √3 | D | 36 | no | mathematics |
57 | A line L : y = mx + 3 meets y - axis at E(0, 3) and the arc of the parabola y² = 16x, 0 ≤ y ≤ 6 at the point F(x0, y0). The tangent to the parabola at F(x0, y0) intersects the y-axis at G(0, y1). The slope m of the line L is chosen such that the area of the triangle EFG has a local maximum. Match List I with List II and select the correct answer using the code given below the lists : | (A) 4 1 2 3
(B) 3 4 1 2
(C) 1 3 2 4
(D) 1 3 4 2 | A | 37 | no | mathematics |
58 | Match List I with List II and select the correct answer using the code given below the lists: | (A) 4 3 2 2, (B) 4 3 2 1, (C) 3 4 2 1, (D) 3 4 1 2 | B | 38 | no | mathematics |
1 | P. (1/(y^2*(cot(sin^-1y)+tan(sin^-1y)))+(y sin(tan^-1y)+y cos(tan^-1y))^2)^(1/2) takes value | 1. (1/2)*sqrt(5/3), 2. sqrt(2), 3. 1/2, 4. 1 | 1 | 38 | no | mathematics |
2 | Q. If cos x + cos y + cos z = 0 = sin x + sin y + sin z then possible value of cos (x-y)/2 is | 1. (1/2)*sqrt(5/3), 2. sqrt(2), 3. 1/2, 4. 1 | 2 | 38 | no | mathematics |
3 | R. If cos((π/4)-x) cos 2x + sin x sin 2x sec x = cos x sin 2x sec x + | 1. (1/2)*sqrt(5/3), 2. sqrt(2), 3. 1/2, 4. 1 | 3 | 38 | no | mathematics |
4 | S. If cot (sin^-1√(1-x^2)) = sin (tan^-1(x√6)) , x ≠ 0, then possible value of x is | 1. (1/2)*sqrt(5/3), 2. sqrt(2), 3. 1/2, 4. 1 | 4 | 38 | no | mathematics |
59 | Consider the lines L1 : x - 1/2 = y/-1 = z + 3/1 , L2 : x - 4/1 = y + 3/1 = z + 3/2 and the planes P1 : 7x + y + 2z = 3, P2 : 3x + 5y - 6z = 4. Let ax + by + cz = d be the equation of the plane passing through the point of intersection of lines L1 and L2, and perpendicular to planes P1 and P2. Match List - I with List - II and select the correct answer using the code given below the lists. | P. a =, Q. b =, R. c =, S. d = | A | 39 | no | mathematics |
60 | Match List–I with List–II and select the correct answer using the code given below the lists: | (A) 4 2 3 1
(B) 2 3 1 4
(C) 3 4 1 2
(D) 1 4 3 2 | C | 40 | no | mathematics |
1 | At time t = 0, terminal A in the circuit shown in the figure is connected to B by a key and an alternating current i(t) = I0cos(ωt), with I0 = 1A and ω = 500 rad s-1 starts flowing in it with the initial direction shown in the figure. At t = 7π/6ω , the key is switched from B to D. Now onwards only A and D are connected. A total charge Q flows from the battery to charge the capacitor fully. If C=20μF, R= 10 Ω and the battery is ideal with emf of 50V, identify the correct statement (s). | (A) Magnitude of the maximum charge on the capacitor before t = 7π/6ω is 1 × 10−3 C, (B) The current in the left part of the circuit just before t = 7π/6ω is clockwise, (C) Immediately after A is connected to D, the current in R is 10A, (D) Q = 2 × 10−3 C. | (C) and (D) | 1 | yes | physics |
2 | A light source, which emits two wavelengths λ1 = 400 nm and λ2 = 600 nm, is used in a Young's double slit experiment. If recorded fringe widths for λ1 andλ2 areβ1 andβ2 © number of fringes for them within a distance yon one side of the central maximum are m1 and m2, respectively, then | (A)β2 > β1, (B)m2 > m1, (C) From the central maximum, 3rd maximum of λ2 overlaps with 5th minimum of λ1, (D)The angular separation of fringes for λ1 is greater than λ2 | (A), (B) and (C) | 2 | no | physics |
3 | One end of a taut string of length 3m along the x axis is fixed at x=0.The speed of the waves in the string is 100 ms−1. The other end of the string is vibrating in the y direction so that stationary waves are set up in the string. The possible waveform(s) of these stationary waves is(are) | (A)y(t) = A sin πx/6 cos 50πt/3, (B)y(t) = A sin πx/3 cos 100πt/3, (C) y(t) = A sin 5πx/6 cos 250πt/3, (D) y(t) = A sin 5πx/2 cos 250πt | (A) , (C) and (D) | 2 | no | physics |
4 | A parallel plate capacitor has a dielectric slab of dielectric constant K between its plates that covers 1/3 of the area of its plates, as shown in the figure. The total capacitance of the capacitor is C while that of the portion with dielectric in between is C1. When the capacitor is charged, the plate area covered by the dielectric gets charge Q1 and the rest of the area gets charge Q2. The electric field in the dielectric is E1 and that in the other portion is E2. Choose the correct option/options, ignoring edge effects. | (A) E1/E2 = 1, (B) E1/E2 = 1/K, (C) Q1/Q2 = 3/K, (D) C/C1 = 2+K/K | (A) and (D) | 3 | yes | physics |
5 | Let E1(r), E2(r) and E3(r) be the respective electric fields at a distance r from a point charge Q, an infinitely long wire with constant linear charge density λ, and an infinite plane with uniform surface charge density σ. If E1(r0) = E2(r0) = E3(r0) at a given distance r0, then | (A) Q = 4πr0^2, (B) r0 = λ/2πσ, (C) E1(r0/2)= 2E2(r0/2), (D) E2(r0/2)=4E3(r0/2) | (C) | 3 | no | physics |
6 | A student is performing an experiment using a resonance column and a tuning fork of frequency 244s^-1. He is told that the air in the tube has been replaced by another gas (assume that the column remains filled with the gas). If the minimum height at which resonance occurs is (0.350 ± 0.005) m, the gas in the tube is | (A)Neon(M=20,(10⁄20=7⁄10)),(B)Nitrogen(M=28,(10⁄28=3⁄5)),(C)Oxygen(M=32,(10⁄√32=9⁄16)),(D)Argon(M=36,(10⁄√36=17⁄32)) | (D) | 4 | no | physics |
7 | Heater of an electric kettle is made of a wire of length L and diameter d. It takes 4 minutes to raise the temperature of 0.5kgwater by 40K. This heater is replaced by a new heater having two wires of the same material, each of length L and diameter 2d.The way these wires are connected is given in the options. How much time in minutes will it take to raise the temperature of the same amount of water by 40 K? | (A)4 if wires are in parallel(B) 2 if wires are in series(C) 1 if wires are in series(D)0.5 if wires are in parallel | (B) and (D) | 4 | no | physics |
8 | In the figure, a ladder of mass m is shown leaning against a wall. It is in static equilibrium making an angle θ with the horizontal floor. The coefficient of friction between the wall and the ladder is μ₁ and that between the floor and the ladder is μ₂. The normal reaction of the wall on the ladder is N₁ and that of the floor is N₂. If the ladder is about to slip, then | (A)μ₁ = 0 μ₂ ≠ 0andN₂ tanθ = mg/2, (B) μ₁ ≠ 0 μ₂ = 0andN₁ tanθ = mg/2, (C)μ₁ ≠ 0 μ₂ ≠ 0andN₂ = mg/1+μ₁μ₂, (D)μ₁ = 0 μ₂ ≠ 0andN₁ tanθ = mg/2 | (C) and (D) | 5 | yes | physics |
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