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27 | The concentration of R in the reaction R → P was measured as a function of time and the following data is obtained: | [R] (molar): 1.0, 0.75, 0.40, 0.10; t(min.): 0.0, 0.05, 0.12, 0.18 | 8 | yes | chemistry |
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28 | The number of neutrons emitted when ²³⁵₉₂U undergoes controlled nuclear fission to ¹⁴²₅₄Xe and ⁹⁰₃₈Sr is | 4 | 8 | no | physics |
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29 | If the angles A, B and C of a triangle are in an arithmetic progression and if a, b and c denote the lengths of the sides opposite to A, B and C respectively, then the value of the expression (a sin 2C + c sin 2A) / c is | A) 1/2, B) √3/2, C) 1, D) √3 | D | 9 | no | mathematics |
30 | Equation of the plane containing the straight line x/2 = y/3 = z/4 and perpendicular to the plane containing the straight lines x/3 = y/4 = z/2 and x/4 = y/2 = z/3 is | A) x + 2y - 2z = 0, B) 3x + 2y - 2z = 0, C) x - 2y + z = 0, D) 5x + 2y - 4z = 0 | C | 9 | no | mathematics |
31 | Let ω be a complex cube root of unity with ω≠1. A fair die is thrown three times. If r1, r2 and r3 are the numbers obtained on the die, then the probability that ω¹ˡ +ω¹²+ω¹³=0 is | A) 1/18, B) 1/9, C) 2/9, D) 1/36 | C | 9 | no | mathematics |
32 | Let P, Q, R and S be the points on the plane with position vectors −2i−j, 4i, 3i+3j and −3i+2j respectively. The quadrilateral PQRS must be a | A) parallelogram, which is neither a rhombus nor a rectangle, B) square, C) rectangle, but not a square, D) rhombus, but not a square | A | 9 | no | mathematics |
33 | The number of 3x3 matrices A whose entries are either 0 or 1 and for which the system A[x y z]^T = [1 0 0]^T has exactly two distinct solutions, is | A) 0, B) 2^9 - 1, C) 168, D) 2 | A | 10 | no | mathematics |
34 | The value of lim [x->0] (1/x^3) ∫(0 to 1) (x t θn (1+t))/(t^4 + 4) dt is | A) 0, B) 1/12, C) 1/24, D) 1/64 | B | 10 | no | mathematics |
35 | Let p and q be real numbers such that p ≠ 0, p^3 ≠ q and p^3 ≠ -q . If α and β are nonzero complex numbers satisfying α + β = - p and α^2 + β^2 = q , then a quadratic equation having (α/β) and (α/α) as its roots is | A) (p^3 + q) x^2 - (p^3 + 2q)x + (p^3 + q) = 0, B) (p^3 + q) x^2 - (p^3 - 2q)x + (p^3 + q) = 0, C) (p^3 - q) x^2 - (5p^3 - 2q) x + (p^3 - q) = 0, D) (p^3 - q) x^2 - (5p^3 + 2q) x + (p^3 - q) = 0 | B | 10 | no | mathematics |
36 | Let f, g and h be real-valued functions defined on the interval [0, 1] by f(x) = e^x + e^-x, g(x) = xe^x + e^-x and h(x) = x^2 e^x + e^-x. If a, b and c denote, respectively, the absolute maximum of f, g and h on [0, 1], then | A) a = b and c ≠ b, B) a = c and a ≠ b, C) a ≠ b and c ≠ b, D) a = b = c | D | 10 | no | mathematics |
37 | Let A and B be two distinct points on the parabola y^2 = 4x. If the axis of the parabola touches a circle of radius r having AB as its diameter, then the slope of the line joining A and B can be | A) -1/r, B) 1/r, C) 2/r, D) -2/r | C and D | 11 | no | mathematics |
38 | Let ABC be a triangle such that ∠ACB = π/6 and let a, b and c denote the lengths of the sides opposite to A, B and C respectively. The value(s) of x for which a = x^2 + x + 1, b = x^2 - 1 and c = 2x + 1 is (are) | A) -(-2+√3), B) 1+√3, C) 2+√3, D) 4√3 | B | 11 | no | mathematics |
39 | Let z₁ and z₂ be two distinct complex numbers and let z = (1 - t) z₁ + tz₂ for some real number t with 0 < t < 1. If Arg(w) denotes the principal argument of a nonzero complex number w, then | A) |z - z₁| + |z - z₂| = |z₁ - z₂|, B) Arg (z - z₁) = Arg (z - z₂), C) |z-z₁ z-z₂| = 0, D) Arg (z - z₁) = Arg(z₂ - z₁)|z₂-z₁ z₁-z₂| | A and C and D | 11 | no | mathematics |
40 | Let f be a real-valued function defined on the interval (0, ∞) by f(x) = ∫(0 to x) t + sin t dt. Then which of the following statement(s) is (are) true? | A) f'(x) exists for all x∈(0,∞) B) f'(x) exists for all x∈(0,∞) and f' is continuous on (0,∞), but not differentiable on (0,∞) C) there exists α > 1 such that |f'(x)|<|f(x)| for all x∈(α,∞) D) there exists β > 0 such that |f(x)|+|f'(x)|≤β for all x∈(0,∞) | B and C | 12 | no | mathematics |
41 | The value(s) of ∫(0 to 1) x^4(1-x)^4/(1+x^2) dx is (are) | A) 22/7 - π B) 2/105 C) 0 D) 71/15 - 3π/2 | A | 12 | no | mathematics |
42 | Let p be an odd prime number and Tp be the following set of 2x2 matrices : Tp = {A= [a b] : a,b,c ∈ {0, 1, 2, ..., p-1}} The number of A in Tp such that A is either symmetric or skew-symmetric or both, and det (A) divisible by p is | A) (p - 1)² B) 2 (p - 1) C) (p - 1)² + 1 D) 2p - 1 | D | 12 | no | mathematics |
43 | The number of A in Tp such that the trace of A is not divisible by p but det (A) is divisible by p is [Note : The trace of a matrix is the sum of its diagonal entries.] | A) (p - 1) (p² - p + 1) B) p³ - (p - 1)² C) (p - 1)² D) (p - 1) (p² - 2) | C | 12 | no | mathematics |
44 | The number of A in Tp such that det (A) is not divisible by p is | A) 2p^2 B) p^3 - 5p C) p^3 - 3p D) p^3 - p^2 | D | 13 | no | mathematics |
45 | Equation of a common tangent with positive slope to the circle as well as to the hyperbola is | A) 2x-√5y-20=0 B) 2x-√5y+4=0 C) 3x-4y+8=0 D) 4x-3y+4=0 | B | 13 | yes | mathematics |
46 | Equation of the circle with AB as its diameter is | A) x^2+y^2-12x+24=0 B) x^2+y^2+12x+24=0 C) x^2+y^2+24x-12=0 D) x^2+y^2-24x-12=0 | A | 13 | yes | mathematics |
47 | The number of values of θ in the interval (-(π/2),(π/2)) such that θ ≠ lπ/5 for n=0,±1,±2 and tanθ=cotθ as well as sin2θ=cos4θ is | No explicit options given | 3 | 13 | no | mathematics |
48 | The maximum value of the expression 1 / (sin^2 θ + 3 sin θ cos θ + 5cos^2 θ) is | 2 | 14 | no | mathematics |
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49 | If ā and b̄ are vectors in space given by ā = i−2j̄/√5 and b̄ = 2ī + j̄ + 3k̄/√14, then the value of (2ā + b̄) ⋅ (ā × b̄) × (ā −2 b̄) is | 5 | 14 | no | mathematics |
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50 | The line 2x + y = 1 is tangent to the hyperbola x^2/a^2 - y^2/b^2 = 1. If this line passes through the point of intersection of the nearest directrix and the x-axis, then the eccentricity of the hyperbola is | 2 | 14 | no | mathematics |
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51 | If the distance between the plane Ax - 2y + z = d and the plane containing the lines x-1/2 = y-2/3 = z-3/4 and x-2/3 = y-3/4 = z-4/5 is √6, then |d| is | 6 | 14 | no | mathematics |
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52 | For any real number x, let [x] denote the largest integer less than or equal to x. Let f be a real valued function defined on the interval [-10, 10] by f(x) = { x - [x] if [x] is odd, 1 + [x] - x if [x] is even}. Then the value of ∫(10/-10) π^2 * 10 f(x) cos πx dx is | 4 | 14 | no | mathematics |
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53 | Let ω be the complex number cos 2π/3 + i sin 2π/3. Then the number of distinct complex numbers z satisfying |z+1 ω ω^2; ω z+ω^2 1; ω^2 1 z+ω| = 0 is equal to | 1 | 14 | no | mathematics |
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54 | Let Sk, k = 1, 2, ..., 100, denote the sum of the infinite geometric series whose first term is (k-1)/k! and the common ratio is 1/k. Then the value of 100^2 + (Σ from k=1 to 100) [(k^2 - 3k + 1) Sk] is | 3 | 15 | no | mathematics |
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55 | The number of all possible values of θ , where 0 < θ < π, for which the system of equations (y + z) cos 3 θ = (xyz) sin 3 θ, x sin 3θ = (2 cos 3θ) / y, 2 sin 3θ / z, (xyz) sin 3 θ = (y + 2z) cos 3 θ + y sin 3 θ have a solution (x0, y0, z0) with y0 ≠ 0, is | 3 | 15 | no | mathematics |
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56 | Let f be a real-valued differentiable function on R (the set of all real numbers) such that f(1) = 1. If the y-intercept of the tangent at any point P(x, y) on the curve y = f(x) is equal to the cube of the abscissa of P, then the value of f(-3) is equal to | 9 | 15 | no | mathematics |
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57 | Consider a thin square sheet of side L and thickness t, made of a material of resistivity ρ. The resistance between two opposite faces, shown by the shaded areas in the figure is | A) directly proportional to L, B) directly proportional to t, C) independent of L, D) independent of t | C | 16 | yes | physics |
58 | A real gas behaves like an ideal gas if its | A) pressure and temperature are both high, B) pressure and temperature are both low, C) pressure is high and temperature is low, D) pressure is low and temperature is high | D | 16 | no | physics |
59 | Incandescent bulbs are designed by keeping in mind that the resistance of their filament increases with the increase in temperature. If at room temperature, 100 W, 60 W and 40 W bulbs have filament resistances R₁₀₀, R₆₀ and R₄₀, respectively, the relation between these resistances is | A) 1/R₁₀₀ = 1/R₆₀ + 1/R₄₀, B) R₁₀₀ = R₆₀ + R₄₀, C) R₁₀₀ > R₆₀ > R₄₀, D) 1/R₁₀₀ > 1/R₆₀ > 1/R₄₀ | D | 16 | no | physics |
60 | To verify Ohm's law, a student is provided with a test resistor Rr, a high resistance R, a small resistance Rs, two identical galvanometers G1 and G2, and a variable voltage source V. The correct circuit to carry out the experiment is | A) B) C) D) | C | 17 | yes | physics |
61 | An AC voltage source of variable angular frequency ω and fixed amplitude Vo is connected in series with a capacitance C and an electric bulb of resistance R (inductance zero). When ω is increased | A) the bulb glows dimmer B) the bulb glows brighter C) total impedance of the circuit is unchanged D) total impedance of the circuit increases | B | 17 | no | physics |
62 | A thin flexible wire of length L is connected to two adjacent fixed points and carries a current I in the clockwise direction, as shown in the figure. When the system is put in a uniform magnetic field of strength B going into the plane of the paper, the wire takes the shape of a circle. The tension in the wire is | A) IBL, B) IBL/π, C) IBL/2π, D) IBL/4π | C | 18 | yes | physics |
63 | A block of mass m is on an inclined plane of angle θ. The coefficient of friction between the block and the plane is μ and tanθ>μ. The block is held stationary by applying a force P parallel to the plane. The direction of force pointing up the plane is taken to be positive. As P is varied from P₁= mg(sinθ - μ cosθ) to P₂= mg(sinθ + μ cosθ), the frictional force f versus P graph will look like | A) [Graph A], B) [Graph B], C) [Graph C], D) [Graph D] | A | 18 | yes | physics |
64 | A thin uniform annular disc (see figure) of mass M has outer radius 4R and inner radius 3R. The work required to take a unit mass from point P on its axis to infinity is | A) 2GM(4√2-5)/7R B) -2GM(4√2-5)/7R C) GM/4R D) 2GM(√2-1)/5R | A | 19 | yes | physics |
65 | A few electric field lines for a system of two charges Q₁ and Q₂ fixed at two different points on the x-axis are shown in the figure. These lines suggest that | A) |Q₁| > |Q₂| B) |Q₁| < |Q₂| C) at a finite distance to the left of Q₁ the electric field is zero D) at a finite distance to the right of Q₂ the electric field is zero | A and D | 19 | yes | physics |
66 | A student uses a simple pendulum of exactly 1m length to determine g, the acceleration due to gravity. He uses a stop watch with the least count of 1 sec for this and records 40 seconds for 20 oscillations. For this observation, which of the following statement(s) is (are) true ? | A) Error ΔT in measuring T, the time period, is 0.05 seconds B) Error ΔT in measuring T, the time period, is 1 second C) Percentage error in the determination of g is 5% D) Percentage error in the determination of g is 2.5% | A and C | 19 | no | physics |
67 | A point mass of 1 kg collides elastically with a stationary point mass of 5 kg. After their collision, the 1 kg mass reverses its direction and moves with a speed of 2 ms⁻¹. Which of the following statement(s) is (are) correct for the system of these two masses ? | A) Total momentum of the system is 3 kg ms⁻¹
B) Momentum of 5 kg mass after collision is 4 kg ms⁻¹
C) Kinetic energy of the centre of mass is 0.75 J
D) Total kinetic energy of the system is 4 J | A and C | 20 | no | physics |
68 | A ray OP of monochromatic light is incident on the face AB of prism ABCD near vertex B at an incident angle of 60° (see figure). If the refractive index of the material of the prism is √5, which of the following is (are) correct ? | A) The ray gets totally internally reflected at face CD
B) The ray comes out through face AD
C) The angle between the incident ray and the emergent ray is 90°
D) The angle between the incident ray and the emergent ray is 120° | A and B and C | 20 | yes | physics |
69 | One mole of an ideal gas in initial state A undergoes a cyclic process ABCA, as shown in the figure. Its pressure at A is PA. Choose the correct option(s) from the following | A) Internal energies at A and B are the same
B) Work done by the gas in process AB is PAVB ln 4
C) Pressure at C is P0/4
D) Temperature at C is T0/4 | A and B and C and D | 20 | yes | physics |
70 | If the total energy of the particle is E, it will perform periodic motion only if | A) E < 0, B) E > 0, C) Vo > E > 0, D) E > Vo | B or C or (B and C) | 21 | yes | physics |
71 | For periodic motion of small amplitude A, the time period T of this particle is proportional to | A) A√(m/α), B) 1/A√(α/m), C) A√(α/m), D) 1/A√(α/m) | B | 21 | yes | physics |
72 | The acceleration of this particle for |x|>Xo is | B) proportional to (Vo/mXo), A) proportional to Vo, C) proportional to √(Vo/mXo), D) zero | D | 21 | yes | physics |
73 | In the graphs below, the resistance R of a superconductor is shown as a function of its temperature T for two different magnetic fields B1 (solid line) and B2 (dashed line). If B2 is larger than B1, which of the following graphs shows the correct variation of R with T in these fields? | Options A), B), C), and D) are shown in the image graphs. | A | 22 | yes | physics |
74 | A superconductor has Tc (0) = 100 K. When a magnetic field of 7.5 Tesla is applied, its Tc decreases to 75 K. For this material one can definitely say that when | A) B = 5 Tesla, Tc (B) = 80 K, B) B = 5 Tesla, 75 K < Tc (B) < 100 K, C) B = 10 Tesla, 75 K < Tc (B) < 100 K, D) B = 10 Tesla, Tc (B) = 70 K | B | 22 | yes | physics |
75 | The focal length of a thin biconvex lens is 20cm. When an object is moved from a distance of 25cm in front of it to 50cm, the magnification of its image changes from m25 to m50. The ratio m25/m50 is | 6 | 23 | no | physics |
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76 | An α-particle and a proton are accelerated from rest by a potential difference of 100V. After this, their de Broglie wavelengths are λα and λp respectively. The ratio λp/λα, to the nearest integer, is | 3 | 23 | no | physics |
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77 | When two identical batteries of internal resistance 1Ω each are connected in series across a resistor R, the rate of heat produced in R is J1. When the same batteries are connected in parallel across R, the rate is J2. If J1 = 2.25 J2, then the value of R in Ω is | 4 | 23 | no | physics |
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78 | Two spherical bodies A (radius 6 cm) and B (radius 18 cm) are at temperatures T1 and T2, respectively. The maximum intensity in the emission spectrum of A is at 500 nm and in that of B is at 1500 nm. Considering them to be black bodies, what will be the ratio of the rate of total energy radiated by A to that of B ? | 9 | 23 | no | physics |
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79 | When two progressive waves y1 = 4 sin(2x - 6t) and y2 = 3 sin(2x-6t - π/2) are superimposed, the amplitude of the resultant wave is | 5 | 23 | no | physics |
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80 | A 0.1 kg mass is suspended from a wire of negligble mass. The length of the wire is 1m and its cross-sectional area is 4.9×10^-7m². If the mass is pulled a little in the vertically downward direction and released, it performs simple harmonic motion of angular frequency 140 rad s⁻¹. If the Young's modulus of the material of the wire is n×10^9Nm⁻², the value of n is | No response choices provided. | 4 | 24 | no | physics |
81 | A binary star consists of two stars A (mass 2.2M_⦿) and B (mass 11M_⦿), where M_⦿ is the mass of the sun. They are separated by distance d and are rotating about their centre of mass, which is stationary. The ratio of the total angular momentum of the binary star to the angular momentum of star B about the centre of mass is | No response choices provided. | 6 | 24 | no | physics |
82 | Gravitational acceleration on the surface of a planet is √(5/11)g , where g is the gravitational acceleration on the surface of the earth. The average mass density of the planet is 2/3 times that of the earth. If the escape speed on the surface of the earth is taken to be 11 kms⁻¹, the escape speed on the surface of the planet in kms⁻¹ will be | No response choices provided. | 3 | 24 | no | physics |
83 | A piece of ice (heat capacity = 2100 J kg⁻¹ °C⁻¹ and latent heat = 3.36 × 10⁵ J kg⁻¹) of mass m grams is at -5°C at atmospheric pressure. It is given 420 J of heat so that the ice starts melting. Finally when the ice-water mixture is in equilibrium, it is found that 1 gm of ice has melted. Assuming there is no other heat exchange in the process, the value of m is | No response choices provided. | 8 | 24 | no | chemistry |
84 | A stationary source is emitting sound at a fixed frequency f₀, which is reflected by two cars approaching the source. The difference between the frequencies of sound reflected from the cars is 1.2% of f₀. What is the difference in the speeds of the cars (in km per hour) to the nearest integer ? The cars are moving at constant speeds much smaller than the speed of sound which is 330 ms⁻¹. | No response choices provided. | 7 | 24 | no | physics |
1 | The complex showing a spin-only magnetic moment of 2.82 B.M. is | A) Ni(CO)4 B) [NiCl4]2- C) Ni(PPh3)4 D) [Ni(CN)4]2- | B | 1 | no | chemistry |
2 | The species having pyramidal shape is | A) SO3 B) BrF3 C) SiO32- D) OSF2 | D | 1 | no | chemistry |
3 | In the reaction H3C-(ring structure)-C-NH2 (1) NaOH/Br2 T (2) (ring structure)-C-Cl the structure of the Product T is | A) (structure) B) (structure) C) (structure) D) (structure) | C | 1 | yes | chemistry |
4 | The compounds P, Q and S were separately subjected to nitration using HNO3/H2SO4 mixture. The major product formed in each case respectively, is | A) HO-NO2, H3C-NO2, O2N-; B) HO-NO2, H3C-NO2, -NO2; C) HO-NO2, H3C-NO2, -NO2; D) HO-NO2, H3C-NO2, O-NO2 | C | 2 | yes | chemistry |
5 | The packing efficiency of the two-dimensional square unit cell shown below is | A) 39.27%; B) 68.02%; C) 74.05%; D) 78.54% | D | 2 | yes | chemistry |
6 | Assuming that Hund's rule is violated, the bond order and magnetic nature of the diatomic molecule B2 is | A) 1 and diamagnetic, B) 0 and diamagnetic, C) 1 and paramagnetic, D) 0 and paramagnetic | A | 3 | no | chemistry |
7 | The total number of diprotic acids among the following is H3PO4, H2SO4, H3PO3, H2CO3, H2S2O7, H3BO3, H3PO2, H2CrO4, H2SO3 | 6 | 3 | no | chemistry |
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8 | Total number of geometrical isomers for the complex [RhCl(CO)(PPh3)(NH3)] is | 3 | 3 | no | chemistry |
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9 | Among the following, the number of elements showing only one non-zero oxidation state is O, Cl, F, N, P, Sn, Tl, Na, Ti | 2 | 3 | no | chemistry |
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10 | Silver (atomic weight = 108 g mol⁻¹) has a density of 10.5 g cm⁻³. The number of silver atoms on a surface of area 10⁻¹² m² can be expressed in scientific notation as y × 10^x. The value of x is | 7 | 3 | no | chemistry |
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11 | One mole of an ideal gas is taken from a to b along two paths denoted by the solid and the dashed lines as shown in the graph below. If the work done along the solid line path is ws and that along the dotted line path is wd, then the integer closest to the ratio wd/ws is | Not Available | 2 | 4 | yes | physics |
12 | The compounds P and Q respectively are | A) H3C-CH2-CH-C-H and H3C-C-H, B) H3C-CH-C-H and C-H, C) H3C-CH2-CH-C-H and H3C-C-H, D) H3C-CH2-CH-C-H and H-C-H | B | 5 | yes | chemistry |
13 | The compound R is | A) H3C-C-C-H, B) H3C-C=C-H, CH3, C-CH-CH, CH3, OH, D) H3C-CH-CH-H, CH3, CH2-OH | A | 6 | yes | chemistry |
14 | The compound S is | A) H3C-CH-CH-H, CH2-CN, B) H3C-C-H, CH2-CN, C) H3C-CH-CH-OH, CH3, CN, CH2-OH, D) H3C-CH-OH, CH3-OH | D | 6 | yes | chemistry |
15 | The state S1 is | A) 1s, B) 2s, C) 2p, D) 3s | B | 7 | no | chemistry |
16 | Energy of the state S1 in units of the hydrogen atom ground state energy is | A) 0.75, B) 1.50, C) 2.25, D) 4.50 | C | 7 | no | chemistry |
17 | The orbital angular momentum quantum number of the state S2 is | A) 0, B) 1, C) 2, D) 3 | B | 7 | no | chemistry |
18 | Match the reactions in Column I with appropriate options in Column II. | A) r and s, B) t, C) p and q, D) r | A: r and s, B: t, C: p and q, D: r | 7 | yes | chemistry |
19 | All the compounds listed in Column I react with water. Match the result of the respective reactions with the appropriate options listed in Column II. | Column I: A) (CH3)2SiCl2, B) XeF4, C) Cl2, D) VCl3
Column II: p) Hydrogen halide formation, q) Redox reaction, r) Reacts with glass, s) Polymerization, t) O2 formation | A: p and s, B: p and q and r and t, C: p and q, D: p | 8 | no | chemistry |
20 | For r = 0, 1, ..., 10, let Ar, Br and Cr denote, respectively, the coefficient of xr in the expansions of (1+x)10, (1+x)20 and (1+x)30. Then
10 ∑ Ar (B10Br - C10Ar)
r=1
is equal to | A) B10 - C10, B) A10 (B102 - C10A10), C) 0, D) C10 - B10 | D | 8 | no | mathematics |
21 | Let S = {1, 2, 3, 4}. The total number of unordered pairs of disjoint subsets of S is equal to | A) 25, B) 34, C) 42, D) 41 | D | 8 | no | mathematics |
22 | Let f be a real-valued function defined on the interval (-1, 1) such that e^-x f(x) = 2 + ∫(√(t^4 + 1 dt), for all x ∈ (-1, 1), and let f^-1 be the inverse function of f. Then (f^-1)' (2) is equal to | A) 1, B) 1/3, C) 1/2, D) 1/e | B | 9 | no | mathematics |
23 | If the distance of the point P(1, -2, 1) from the plane x + 2y - 2z = α , where α > 0, is 5, then the foot of the perpendicular from P to the plane is | A) (8/3, 4/3, 7/3), B) (4/3, 4/3, 1/3), C) (1/3, 2/3, 10/3), D) (2/3, 1/3, 5/2) | A | 9 | no | mathematics |
24 | Two adjacent sides of a parallelogram ABCD are given by AB = 2i + 10j + 11k and AD = - i + 2j + 2k The side AD is rotated by an acute angle α in the plane of the parallelogram so that AD becomes AD'. If AD' makes a right angle with the side AB, then the cosine of the angle α is given by | A) 8/9, B) √17/9, C) 1/9, D) 4√5/9 | B | 9 | no | mathematics |
25 | A signal which can be green or red with probability 4/5 and 1/5 respectively, is received by station A and then transmitted to station B. The probability of each station receiving the signal correctly is 3/4. If the signal received at station B is green, then the probability that the original signal was green is | A) 3/5, B) 6/7, C) 20/23, D) 9/20 | C | 9 | no | mathematics |
26 | Two parallel chords of a circle of radius 2 are at a distance √3 + 1 apart. If the chords subtend at the center, angles of π/k and 2π/k, where k > 0, then the value of [k] is [Note : [k] denotes the largest integer less than or equal to k] | NA | 3 | 10 | no | mathematics |
27 | Consider a triangle ABC and let a, b and c denote the lengths of the sides opposite to vertices A, B and C respectively. Suppose a = 6, b = 10 and the area of the triangle is 15√3. If ∠ACB is obtuse and if r denotes the radius of the incircle of the triangle, then r^2 is equal to | NA | 3 | 10 | no | mathematics |
28 | Let f be a function defined on R (the set of all real numbers) such that f(x)=2010(x−2009)(x−2010)^2(x−2011)^3(x−2012)^4, for all x ∈ R. If g is a function defined on R with values in the interval (0, ∞) such that f(x) = ℓn (g (x)), for all x ∈ R, then the number of points in R at which g has a local maximum is | NA | 1 | 10 | no | mathematics |
29 | Let a1, a2, a3, ... , a11 be real numbers satisfying a1 = 15, 27 - 2a2 > 0 and ak = 2ak-1 - ak-2 for k = 3, 4, ..., 11. If (a1^2 + a2^2 + ... + a11^2)/11 = 90, then the value of (a1 + a2 + ... + a11)/11 is equal to | NA | 0 | 10 | no | mathematics |
30 | Let k be a positive real number and let A= [2k-1 √2k 2√k; 2√k 1 -2k; -2√k 2k -1] and B=[0 2k-1 √k; 1-2k 0 2√k; -√k -2√k 0] If det (adj A) + det(adj B) =10^6, then [k] is equal to | Note: adj M denotes the adjoint of a square matrix M and [k] denotes the largest integer less than or equal to [k]. | 4 | 11 | yes | mathematics |
31 | The real number s lies in the interval | A) (-1/4,0) B) (-11,-3/4) C) (-3/4,1/2) D) (0,1/4) | C | 11 | no | mathematics |
32 | The area bounded by the curve y = f(x) and the lines x = 0, y = 0 and x = t, lies in the interval | A) (3/4,3) B) (21/64,11/16) C) (9,10) D) (0,21/64) | A | 11 | no | mathematics |
33 | The function f(x) is | A) increasing in (−t, -1/4) and decreasing in (−1/4, t)
B) decreasing in (−t, -1/4) and increasing in (−1/4, t)
C) increasing in (-t, t)
D) decreasing in (-t, t) | B | 12 | no | mathematics |
34 | The coordinates of A and B are | A) (3, 0) and (0, 2)
B) (-8 √161/5, 15) and (-9 8/5, 5)
C) (-8 √161/5, 15) and (0, 2)
D) (3, 0) and (-9 8/5, 5) | D | 12 | yes | mathematics |
35 | The orthocenter of the triangle PAB is | A) (5, 8/7)
B) (7, 25/5, 8)
C) (11, 8/5, 5)
D) (8, 7/25, 5) | C | 12 | no | mathematics |
36 | The equation of the locus of the point whose distances from the point P and the line AB are equal, is | A) 9x^2 + y^2 - 6xy - 54x - 62y + 241 = 0, B) x^2 + 9y^2 + 6xy - 54x + 62y - 241 = 0, C) 9x^2 + 9y^2 - 6xy - 54x - 62y - 241 = 0, D) x^2 + y^2 - 2xy + 27x + 31y - 120 = 0 | A | 13 | no | mathematics |
37 | Match the statements in Column-I with those in Column-II. [Note: Here z takes values in the complex plane and Im z and Re z denote, respectively, the imaginary part and the real part of z.] | Column I: A) The set of points z satisfying |z-1|z|=|z+1|z|, B) The set of points z satisfying |z+4|+|z-4|=10, C) If |w|=2, then the set of points z=w-1/w, D) If |w|=1, then the set of points z=w+1/w; Column II: p) an ellipse with eccentricity 4/5, q) the set of points z satisfying Im z=0, r) the set of points z satisfying |Im z|≤1, s) the set of points z satisfying |Re z|≤2, t) the set of points z satisfying |z|≤3 | A: q and r, B: p, C: p and s and t, D: q and r and s and t | 13 | no | mathematics |
38 | Match the statements in Column-I with the values in Column-II. | A) A line from the origin meets the lines x-2=y-1=z+1 and x-8/3=y+3/2=-z-1/1 at P and Q respectively. If length PQ = d, then d^2 is p) - 4
B) The values of x satisfying tan^-1 (x + 3) - tan^-1 (x - 3) = sin^-1 (3/5) are q) 0
C) Non-zero vectors ā, b̄ and c̄ satisfy ā.b̄ = 0, (b̄ - ā).(b̄ + c̄) = 0 and 2|b̄ + c̄| = |b̄ - ā|. If ā = μb̄ + 4c̄, then the possible values of μ are r) 4
D) Let f be the function on [-π, π] given by f(0) = 9 and f(x) = sin((9x/2)/sin(x/2)) for x ≠ 0. The value of (2/π) ∫[-π]^π f(x) dx is s) 5
t) 6 | A: t
B: p and r
C: either q or (q and s)
D: r | 14 | no | mathematics |
39 | A Vernier calipers has 1 mm marks on the main scale. It has 20 equal divisions on the Vernier scale which match with 16 main scale divisions. For this Vernier calipers, the least count is | A) 0.02 mm, B) 0.05 mm, C) 0.1 mm, D) 0.2 mm | D | 15 | no | physics |
40 | A hollow pipe of length 0.8 m is closed at one end. At its open end a 0.5 m long uniform string is vibrating in its second harmonic and it resonates with the fundamental frequency of the pipe. If the tension in the wire is 50 N and the speed of sound is 320 ms^-1, the mass of the string is | A) 5 grams, B) 10 grams, C) 20 grams, D) 40 grams | B | 15 | no | physics |
41 | A biconvex lens of focal length 15 cm is in front of a plane mirror. The distance between the lens and the mirror is 10 cm. A small object is kept at a distance of 30 cm from the lens. The final image is | A) virtual and at a distance of 16 cm from the mirror, B) real and at a distance of 16 cm from the mirror, C) virtual and at a distance of 20 cm from the mirror, D) real and at a distance of 20 cm from the mirror | B | 15 | yes | physics |
42 | A block of mass 2 kg is free to move along the x-axis. It is at rest and from t = 0 onwards it is subjected to a time-dependent force F(t) in the x direction. The force F(t) varies with t as shown in the figure. The kinetic energy of the block after 4.5 seconds is | A) 4.50 J, B) 7.50 J, C) 5.06 J, D) 14.06 J | C | 16 | yes | physics |