question: The potential energy of a particle of mass m at a distance r from a fixed point O is given by V(r) = kr^2/2, where k is a positive constant of appropriate dimensions. This particle is moving in a circular orbit of radius R about the point O. If v is the speed of the particle and L is the magnitude of its angular momentum about O, which of the following statements is (are) true?

answer: C,D

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question_subject: physics


question: Consider a body of mass 1.0 kg at rest at the origin at time t = 0. A force F = (αt i + β j) is applied on the body, where α = 1.0 N/s^-1 and β = 1.0 N. The torque acting on the body about the origin at time t = 1.0 s is τ. Which of the following statements is (are) true?

response_choices:
(A) |τ| = 1/3 N m
(B) The torque τ is in the direction of the unit vector + k
(C) The velocity of the body at t = 1 s is v = 1/2 (i + 2j) m s^-1
(D) The magnitude of displacement of the body at t = 1 s is 1/6 m

answer: B

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question_subject: physics


question: A uniform capillary tube of inner radius r is dipped vertically into a beaker filled with water. The water rises to a height h in the capillary tube above the water surface in the beaker. The surface tension of water is σ. The angle of contact between water and the wall of the capillary tube is θ. Ignore the mass of water in the meniscus. Which of the following statements is (are) true?

response_choices:
(A) For a given material of the capillary tube, h decreases with increase in r
(B) For a given material of the capillary tube, h is independent of σ
(C) If this experiment is performed in a lift going up with a constant acceleration, then h decreases
(D) h is proportional to contact angle θ

answer: A

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question_subject: physics


question: In the figure below, the switches S1 and S2 are closed simultaneously at t = 0 and a current starts to flow in the circuit. Both the batteries have the same magnitude of the electromotive force (emf) and the polarities are as indicated in the figure. Ignore mutual inductance between the inductors. The current I in the middle wire reaches its maximum magnitude Imax at time t = τ. Which of the following statements is (are) true?

response_choices: (A) Imax = V/2R, (B) Imax = V/4R, (C) τ = L/R ln 2, (D) τ = 2L/R ln 2

answer: C

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question_subject: physics


question: Two infinitely long straight wires lie in the xy-plane along the lines x = ±R. The wire located at x = +R carries a constant current I1 and the wire located at x = −R carries a constant current I2. A circular loop of radius R is suspended with its centre at (0, 0, √3R) and in a plane parallel to the xy-plane. This loop carries a constant current I in the clockwise direction as seen from above the loop. The current in the wire is taken to be positive if it is in the +j direction. Which of the following statements regarding the magnetic field B is (are) true?

response_choices: (A) If I1 = I2, then B cannot be equal to zero at the origin (0, 0, 0), (B) If I1 > 0 and I2 < 0, then B can be equal to zero at the origin (0, 0, 0), (C) If I1 < 0 and I2 > 0, then B can be equal to zero at the origin (0, 0, 0), (D) If I1 = I2, then the z-component of the magnetic field at the centre of the loop is (− μ0I/2R)

answer: D

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question_subject: physics


question: One mole of a monatomic ideal gas undergoes a cyclic process as shown in the figure (where V is the volume and T is the temperature). Which of the statements below is (are) true?

response_choices: A) Process I is an isochoric process, B) In process II, gas absorbs heat, C) In process IV, gas releases heat, D) Processes I and III are not isobaric

answer: B

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question_subject: physics


question: Two vectors A and B are defined as A = a i and B = a (cos ωt i + sin ωt j), where a is a constant and ω = π/6 rad s^(-1). If |A + B| = √3|A - B| at time t = τ for the first time, the value of τ, in seconds, is _________.

answer: Numerical response required

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question_subject: mathematics


question: Two men are walking along a horizontal straight line in the same direction. The man in front walks at a speed 1.0 m s^(-1) and the man behind walks at a speed 2.0 m s^(-1). A third man is standing at a height 12 m above the same horizontal line such that all three men are in a vertical plane. The two walking men are blowing identical whistles which emit a sound of frequency 1430 Hz. The speed of sound in air is 330 m s^(-1). At the instant, when the moving men are 10 m apart, the stationary man is equidistant from them. The frequency of beats in Hz, heard by the stationary man at this instant, is _________.

answer: Numerical response required

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question_subject: physics


question: A ring and a disc are initially at rest, side by side, at the top of an inclined plane which makes an angle 60° with the horizontal. They start to roll without slipping at the same instant of time along the shortest path. If the time difference between their reaching the ground is (2 - √3) /√10 s, then the height of the top of the inclined plane, in metres, is _________. Take g = 10 m s^(-2).

answer: Numerical response required

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question_subject: physics


question: A spring-block system is resting on a frictionless floor as shown in the figure. The spring constant is 2.0 N m^-1 and the mass of the block is 2.0 kg. Ignore the mass of the spring. Initially the spring is in an unstretched condition. Another block of mass 1.0 kg moving with a speed of 2.0 m s^-1 collides elastically with the first block. The collision is such that the 2.0 kg block does not hit the wall. The distance, in metres, between the two blocks when the spring returns to its unstretched position for the first time after the collision is _________.

response_choices: No multiple choice options given

answer: Numerical answer required

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question_subject: physics


question: Three identical capacitors C1, C2 and C3 have a capacitance of 1.0 μF each and they are uncharged initially. They are connected in a circuit as shown in the figure and C3 is filled completely with a dielectric material of relative permittivity εr. The cell electromotive force (emf) V0 = 8 V. First the switch S1 is closed while the switch S2 is kept open. When the capacitor C3 is fully charged, S1 is opened and S2 is closed simultaneously. When all the capacitors reach equilibrium, the charge on C3 is found to be 5 μC. The value of εr = ___________.

response_choices: No multiple choice options given

answer: Numerical answer required

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question_subject: physics


question: In the xy-plane, the region y > 0 has a uniform magnetic field B1k and the region y < 0 has another uniform magnetic field B2k. A positively charged particle is projected from the origin along the positive y-axis with speed v0 = π m s^-1 at t = 0, as shown in the figure. Neglect gravity in this problem. Let t = T be the time when the particle crosses the x-axis from below for the first time. If B2 = 4B1, the average speed of the particle, in m s^-1, along the x-axis in the time interval T is _________.

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question_subject: physics


question: Sunlight of intensity 1.3 kW m^-2 is incident normally on a thin convex lens of focal length 20 cm. Ignore the energy loss of light due to the lens and assume that the lens aperture size is much smaller than its focal length. The average intensity of light, in kW m^-2, at a distance 22 cm from the lens on the other side is _________.

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question_subject: physics


question: Two conducting cylinders of equal length but different radii are connected in series between two heat baths kept at temperatures T1 = 300 K and T2 = 100 K, as shown in the figure. The radius of the bigger cylinder is twice that of the smaller one and the thermal conductivities of the materials of the smaller and the larger cylinders are K1 and K2 respectively. If the temperature at the junction of the two cylinders in the steady state is 200 K, then K1/K2 = _________.

response_choices: No multiple choice options provided

answer: No answer provided

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question_subject: physics


question: The relation between [E] and [B] is

answer: A

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question_subject: physics


question: The relation between [ϵ0] and [μ0] is

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question_subject: physics


question: Consider the ratio r = (1-a)/(1+a) to be determined by measuring a dimensionless quantity a. If the error in the measurement of a is Δa (Δa/a << 1), then what is the error Δr in determining r?

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question_subject: mathematics


question: In an experiment the initial number of radioactive nuclei is 3000. It is found that 1000 ± 40 nuclei decayed in the first 1.0 s. For |x| ≪ 1, ln(1 + x) = x up to first power in x. The error ∆λ, in the determination of the decay constant λ, in s⁻¹, is

answer: (C) 0.02

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question_subject: physics


question: The compound(s) which generate(s) N2 gas upon thermal decomposition below 300°C is (are)

answer: A

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question_subject: chemistry


question: The correct statement(s) regarding the binary transition metal carbonyl compounds is (are) (Atomic numbers: Fe = 26, Ni = 28)

response_choices: (A) Total number of valence shell electrons at metal centre in Fe(CO)5 or Ni(CO)4 is 16, (B) These are predominantly low spin in nature, (C) Metal–carbon bond strengthens when the oxidation state of the metal is lowered, (D) The carbonyl C–O bond weakens when the oxidation state of the metal is increased

answer: B

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question_subject: chemistry


question: Based on the compounds of group 15 elements, the correct statement(s) is (are)

response_choices: (A) Bi2O5 is more basic than N2O5, (B) NF3 is more covalent than BIF3, (C) PH3 boils at lower temperature than NH3, (D) The N-N single bond is stronger than the P-P single bond

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question_subject: chemistry


question: In the following reaction sequence, the correct structure(s) of X is (are)

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question_subject: chemistry


question: The reaction(s) leading to the formation of 1,3,5-trimethylbenzene is (are)

answer: (C)

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question_subject: chemistry


question: A reversible cyclic process for an ideal gas is shown below. Here, P, V, and T are pressure, volume and temperature, respectively. The thermodynamic parameters q, w, H and U are heat, work, enthalpy and internal energy, respectively.

response_choices:
(A) qAC = ΔUBC and wAB = P2(V2 – V1)
(B) wBC = P2(V2 – V1) and qBC = ΔHAC
(C) ΔHCA < ΔUCA and qAC = ΔUBC
(D) qBC = ΔHAC and ΔHCA > ΔUCA

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question_subject: physics


question: Among the species given below, the total number of diamagnetic species is ____.

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question_subject: chemistry


question: The ammonia prepared by treating ammonium sulphate with calcium hydroxide is completely used by NiCl2.6H2O to form a stable coordination compound. Assume that both the reactions are 100% complete. If 1584 g of ammonium sulphate and 952 g of NiCl2.6H2O are used in the preparation, the combined weight (in grams) of gypsum and the nickel-ammonia coordination compound thus produced is ____.

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question_subject: chemistry


question: Consider an ionic solid MX with NaCl structure. Construct a new structure (Z) whose unit cell is constructed from the unit cell of MX following the sequential instructions given below. Neglect the charge balance. (i) Remove all the anions (X) except the central one (ii) Replace all the face centered cations (M) by anions (X) (iii) Remove all the corner cations (M) (iv) Replace the central anion (X) with cation (M) The value of (number of anions/number of cations) in Z is _____.

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question_subject: chemistry


question: For the electrochemical cell, Mg(s) | Mg2+ (aq, 1 M) || Cu2+ (aq, 1 M) | Cu(s) the standard emf of the cell is 2.70 V at 300 K. When the concentration of Mg2+ is changed to x M, the cell potential changes to 2.67 V at 300 K. The value of x is _____. (given, F = 11500 K V^-1, where F is the Faraday constant and R is the gas constant, ln(10) = 2.30)

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question_subject: chemistry


question: A closed tank has two compartments A and B, both filled with oxygen (assumed to be ideal gas). The partition separating the two compartments is fixed and is a perfect heat insulator (Figure 1). If the old partition is replaced by a new partition which can slide and conduct heat but does NOT allow the gas to leak across (Figure 2), the volume (in m3) of the compartment A after the system attains equilibrium is _____.

response_choices: Multiple choice options not provided

answer: No answer provided

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question_subject: physics


question: Liquids A and B form ideal solution over the entire range of composition. At temperature T, equimolar binary solution of liquids A and B has vapour pressure 45 Torr. At the same temperature, a new solution of A and B having mole fractions xA and xB, respectively, has vapour pressure of 22.5 Torr. The value of xA/xB in the new solution is _____. (given that the vapour pressure of pure liquid A is 20 Torr at temperature T)

response_choices: Multiple choice options not provided

answer: No answer provided

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question_subject: chemistry


question: The solubility of a salt of weak acid (AB) at pH 3 is Y×10^-3 mol L^-1. The value of Y is _____.

response_choices: No response choices provided

answer: (Given that the value of solubility product of AB (Ksp) = 2×10^-10 and the value of ionization constant of HB (Ka) = 1×10^-8)

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question_subject: chemistry


question: The plot given below shows P – T curves (where P is the pressure and T is the temperature) for two solvents X and Y and isomolal solutions of NaCl in these solvents. NaCl completely dissociates in both the solvents. On addition of equal number of moles of a non-volatile solute S in equal amount (in kg) of these solvents, the elevation of boiling point of solvent X is three times that of solvent Y. Solute S is known to undergo dimerization in these solvents. If the degree of dimerization is 0.7 in solvent Y, the degree of dimerization in solvent X is _____.

response_choices: No response choices provided

answer: The image shows curves numbered 1, 2, 3, and 4 representing: 1. solvent X, 2. solution of NaCl in solvent X, 3. solvent Y, 4. solution of NaCl in solvent Y

image_available: yes

question_subject: chemistry


question: The compound Y is

response_choices: A) COBr, B) OH with Br and HO attached to a ring, C) Ring with triple bond, D) Br attached to a ring with COBr attached

answer: B

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question_subject: chemistry


question: The compound Z is

response_choices: (A) [molecular structure 1] (B) [molecular structure 2] (C) [molecular structure 3] (D) [molecular structure 4]

answer: B

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question_subject: chemistry


question: The compound R is

answer: (B)

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question_subject: chemistry


question: The compound S is

answer: D

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question_subject: chemistry


question: This section contains SIX (06) questions.

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question_subject: mathematics


question: Each question has FOUR options for correct answer(s). ONE OR MORE THAN ONE of these four option(s) is (are) correct option(s).

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question_subject: mathematics


question: For each question, choose the correct option(s) to answer the question.

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question_subject: mathematics


question: Answer to each question will be evaluated according to the following marking scheme: Full Marks : +4 If only (all) the correct option(s) is (are) chosen. Partial Marks : +3 If all the four options are correct but ONLY three options are chosen. Partial Marks : +2 If three or more options are correct but ONLY two options are chosen, both of which are correct options. Partial Marks : +1 If two or more options are correct but ONLY one option is chosen and it is a correct option. Zero Marks : 0 If none of the options is chosen (i.e. the question is unanswered). Negative Marks : -2 In all other cases.

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question_subject: mathematics


question: For Example: If first, third and fourth are the ONLY three correct options for a question with second option being an incorrect option; selecting only all the three correct options will result in +4 marks. Selecting only two of the three correct options (e.g. the first and fourth options), without selecting any incorrect option (second option in this case), will result in +2 marks. Selecting only one of the three correct options (either first or third or fourth option) , without selecting any incorrect option (second option in this case), will result in +1 marks. Selecting any incorrect option(s) (second option in this case), with or without selection of any correct option(s) will result in -2 marks.

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question_subject: mathematics


question: For a non-zero complex number z, let arg(z) denote the principal argument with - π < arg(z) ≤ π. Then, which of the following statement(s) is (are) FALSE?

response_choices:
(A) arg(−1 − i) = π/4, where i = √−1

(B) The function f : ℝ → (−π, π], defined by f(t) = arg(−1 + it) for all t ∈ ℝ, is continuous at all points of ℝ, where i = √−1

(C) For any two non-zero complex numbers z1 and z2, arg(z1/z2) − arg(z1) + arg(z2) is an integer multiple of 2π

(D) For any three given distinct complex numbers z1, z2 and z3, the locus of the point z satisfying the condition arg((z−z1)(z2−z3)/((z−z3)(z2−z1))) = π, lies on a straight line

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question_subject: mathematics


question: In a triangle PQR, let ∠PQR = 30° and the sides PQ and QR have lengths 10√3 and 10, respectively. Then, which of the following statement(s) is (are) TRUE?

response_choices:
(A) ∠QPR = 45°

(B) The area of the triangle PQR is 25√3 and ∠QRP = 120°

(C) The radius of the incircle of the triangle PQR is 10√3 − 15

(D) The area of the circumcircle of the triangle PQR is 100 π

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question_subject: mathematics


question: Let P1: 2x + y - z = 3 and P2: x + 2y + z = 2 be two planes. Then, which of the following statement(s) is (are) TRUE?

response_choices:
(A) The line of intersection of P1 and P2 has direction ratios 1, 2, -1
(B) The line
3x - 4 = 1 - 3y = z
9       9       3
is perpendicular to the line of intersection of P1 and P2
(C) The acute angle between P1 and P2 is 60°
(D) If P3 is the plane passing through the point (4, 2, -2) and perpendicular to the line of intersection of P1 and P2, then the distance of the point (2, 1, 1) from the plane P3 is 2/√3

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question_subject: mathematics


question: For every twice differentiable function f: R → [-2, 2] with (f'(0))^2 + (f''(0))^2 = 85, which of the following statement(s) is (are) TRUE?

response_choices:
(A) There exist r, s ∈ R, where r < s, such that f is one-one on the open interval (r, s)
(B) There exists x0 ∈ (-4, 0) such that |f''(x0)| ≤ 1
(C) lim f(x) = 1
x→∞
(D) There exists α ∈ (-4, 4) such that f(α) + f''(α) = 0 and f'(α) ≠ 0

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question_subject: mathematics


question:
Let f: R → R and g: R → R be two non-constant differentiable functions. If
f'(x) = (e^(f(x)-g(x))) g'(x) for all x ∈ R,
and f(1) = g(2) = 1, then which of the following statement(s) is (are) TRUE?

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question_subject: mathematics


question: Let f: [0, ∞) → ℝ be a continuous function such that f(x) = 1 - 2x + ∫(0 to x) e^(x-t) f(t) dt for all x ∈ [0, ∞). Then, which of the following statement(s) is (are) TRUE?

response_choices: (A) The curve y = f(x) passes through the point (1, 2) (B) The curve y = f(x) passes through the point (2, -1) (C) The area of the region {(x, y) ∈ [0, 1] × ℝ : f(x) ≤ y ≤ √(1 - x^2) } is π-2/4 (D) The area of the region {(x, y) ∈ [0,1] × ℝ : f(x) ≤ y ≤ √(1 - x^2) } is π-1/4

answer: No selected answer provided

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question_subject: mathematics


question: The value of ((log2 9)^2)^(log2 (log2 9) × (√7)^(log4 7)) is _______.

answer: None provided (numerical response required)

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question_subject: mathematics


question: The number of 5 digit numbers which are divisible by 4, with digits from the set {1, 2, 3, 4, 5} and the repetition of digits is allowed, is _____ .

answer: None provided (numerical response required)

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question_subject: mathematics


question: Let X be the set consisting of the first 2018 terms of the arithmetic progression 1, 6, 11, ... , and Y be the set consisting of the first 2018 terms of the arithmetic progression 9, 16, 23, ... . Then, the number of elements in the set X ∪ Y is _____.

answer: None provided (numerical response required)

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question_subject: mathematics


question: The number of real solutions of the equation sin^-1(∑(∞)(l=1) x^(l+1) - x ∑(∞)(i=1) (x^i)/(i!)) = (π/2 - cos^-1(∑(∞)(i=1) ((-x)^i)/2 - ∑(∞)(i=1) (-x)^i)) lying in the interval ((-1/2, 2/1)) is _____ . (Here, the inverse trigonometric functions sin^-1 x and cos^-1 x assume values in [(-π/2, π/2)] and [0, π], respectively.)

answer: None provided (numerical response required)

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question_subject: mathematics


question: For each positive integer n, let yn = 1/n ((n + 1)(n + 2)...(n + n))^(1/n). For x ∈ ℝ, let [x] be the greatest integer less than or equal to x. If lim(n→∞) yn = L, then the value of [L] is _____ .

answer: None provided (numerical response required)

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question_subject: mathematics


question: Let â and b̂ be two unit vectors such that â · b̂ = 0. For some x, y ∈ R, let c ̂ = x â + y b̂ + (â × b̂). If |ĉ| = 2 and the vector ĉ is inclined at the same angle α to both â and b̂, then the value of 8 cos^2 α is _____.

response_choices: No options given

answer: No answer given

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question_subject: mathematics


question: Let a, b, c be three non-zero real numbers such that the equation √3 a cos x + 2 b sin x = c, x ∈ [- π/2, π/2], has two distinct real roots α and β with α + β = π/3. Then, the value of b/a is _____.

response_choices: No options given

answer: No answer given

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question_subject: mathematics


question: A farmer F₁ has a land in the shape of a triangle with vertices at P(0, 0), Q(1, 1) and R(2, 0). From this land, a neighbouring farmer F₂ takes away the region which lies between the side PQ and a curve of the form y = x^n (n > 1). If the area of the region taken away by the farmer F₂ is exactly 30% of the area of ΔPQR, then the value of n is _____.

response_choices: No options given

answer: No answer given

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question_subject: mathematics


question: Let E1E2 and F1F2 be the chords of S passing through the point P0 (1, 1) and parallel to the x-axis and the y-axis, respectively. Let G1G2 be the chord of S passing through P0 and having slope −1. Let the tangents to S at E1 and E2 meet at E3, the tangents to S at F1 and F2 meet at F3, and the tangents to S at G1 and G2 meet at G3. Then, the points E3, F3, and G3 lie on the curve

answer: (B) (x − 4)2 + (y − 4)2 = 16

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question_subject: mathematics


question: Let P be a point on the circle S with both coordinates being positive. Let the tangent to S at P intersect the coordinate axes at the points M and N. Then, the mid-point of the line segment MN must lie on the curve

answer: (B) x2/3 + y2/3 = 24/3

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question_subject: mathematics


question: The probability that, on the examination day, the student S1 gets the previously allotted seat R1, and NONE of the remaining students gets the seat previously allotted to him/her is

answer: C

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question_subject: mathematics


question: For i = 1, 2, 3, 4, let Ti denote the event that the students Si and Si+1 do NOT sit adjacent to each other on the day of the examination. Then, the probability of the event T1 ∩ T2 ∩ T3 ∩ T4 is

answer: B

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question_subject: mathematics