question: Considering only the principal values of the inverse trigonometric functions, the value of tan(sin^-1(3/5) - 2cos^-1(2/sqrt(5))) is

answer: (B) -7/24

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question_subject: mathematics


question: Let S = {(x, y) in R x R: x ≥ 0, y ≥ 0, y^2 ≤ 4x, y^2 ≤ 12 - 2x and 3y + sqrt(8) x ≤ 5sqrt(8)}. If the area of the region S is α sqrt(2), then α is equal to

answer: (B) 17/3

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question_subject: mathematics


question: Let k ∈ ℝ. If lim (sin(sin kx) + cos x + x)^2 = e^x, then the value of k is

answer: (C) 3

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question_subject: mathematics


question: Let f : ℝ → ℝ be a function defined by f(x) = (x^2 sin(π/x^2), if x ≠ 0, 0, if x = 0). Then which of the following statements is TRUE?

response_choices: (A) f(x) = 0 has infinitely many solutions in the interval (1/10^10, ∞), (B) f(x) = 0 has no solutions in the interval (1/π, ∞), (C) The set of solutions of f(x) = 0 in the interval (0, 1/10^10) is finite, (D) f(x) = 0 has more than 25 solutions in the interval (1/π^2, 1/π)

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question_subject: mathematics


question: Let S be the set of all (α, β) ∈ ℝ × ℝ such that lim[x→∞] (sin(x^2)(log_e x)^α sin(1/x^2))/(x^(β/α)(log_e (1+x))^β) = 0. Then which of the following is (are) correct?

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question_subject: mathematics


question: A straight line drawn from the point P(1, 3, 2), parallel to the line x - 2 = y - 4 = z - 6 / 1 = 2 = 1, intersects the plane L1 : x - y + 3z = 6 at the point Q. Another straight line which passes through Q and is perpendicular to the plane L1 intersects the plane L2 : 2x - y + z = -4 at the point R. Then which of the following statements is (are) TRUE?

response_choices:
(A) The length of the line segment PQ is √6
(B) The coordinates of R are (1, 6, 3)
(C) The centroid of the triangle PQR is (4 14 5 / 3 3 3)
(D) The perimeter of the triangle PQR is √2 + √6 + √11

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question_subject: mathematics


question: Let A, B, C be three points in the xy-plane. Suppose that the lines AC and BC are tangents to the curve y² = 8x at A and B, respectively. If O = (0, 0) and C1 = (-4, 0), then which of the following statements is (are) TRUE?

response_choices:
(A) The length of the line segment OA1 is 4√5
(B) The length of the line segment A1B1 is 16
(C) The orthocenter of the triangle A1B1C1 is (0, 0)
(D) The orthocenter of the triangle A1B1C1 is (1, 0)

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question_subject: mathematics


question: Let f : R → R be a function such that f (x + y) = f (x) + f (y) for all x, y ∈ R, and g : R → (0, ∞) be a function such that g (x + y) = g (x) g (y) for all x, y ∈ R. If f (-3/5) = 12 and g (-1/3) = 2, then the value of f (1/4) + g (-2) - 8 g (0) is _______.

response_choices: No response choices provided

answer: No answer provided, response required

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question_subject: mathematics


question: A bag contains N balls out of which 3 balls are white, 6 balls are green, and the remaining balls are blue. Assume that the balls are identical otherwise. Three balls are drawn randomly one after the other without replacement. For i = 1, 2, 3, let Wi, Gi, and Bi denote the events that the ball drawn in the ith draw is a white ball, green ball, and blue ball, respectively. If the probability P(W1 ∩ G2 ∩ B3) = 2/5N and the conditional probability P(B3 | W1 ∩ G2) = 2/9, then N equals _______.

response_choices: No response choices provided

answer: No answer provided, response required

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question_subject: mathematics


question: Let the function f : R → R be defined by f (x) = (sin x (x^2023 + 2024x + 2025) / e^x) + 2 (x^2023 + 2024x + 2025) / (x^2 - x + 3) e^x. Then the number of solutions of f (x) = 0 in R is ______.

response_choices: No response choices provided

answer: No answer provided, response required

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question_subject: mathematics


question: Let p = 2i + j + 3k and q = i - j + k. If for some real numbers α, β, and γ, we have 15i + 10 j + 6k = α (2p + q) + β (p - 2q) + γ (p × q), then the value of γ is _______.

response_choices: No response choices provided

answer: No answer provided, response required

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question_subject: mathematics


question: A normal with slope 1/sqrt(6) is drawn from the point (0, -α) to the parabola x^2 = -4αy, where α > 0. Let L be the line passing through (0, -α) and parallel to the directrix of the parabola. Suppose that L intersects the parabola at two points A and B. Let r denote the length of the latus rectum and s denote the square of the length of the line segment AB. If r : s = 1 : 16, then the value of 24α is _____.

response_choices: No multiple choice options given

answer: No selected answer given

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question_subject: mathematics


question:
Let the function f : [1, ∞) → ℝ be defined by

f(t) = {    (-1)^(t+1) 2,                       if t = 2n - 1, n ∈ ℕ,
          (2n + 1 - t)            {t - (2n - 1)}
          ---------------- + -----------------------, if 2n - 1 < t < 2n + 1, n ∈ ℕ.
                2                     2

Define g(x) = ∫ f(t) dt, x ∈ (1, ∞). Let α denote the number of solutions of the equation

g(x) = 0 in the interval (1, 8] and β = lim g(x). Then the value of α + β is equal to _____.

response_choices: No multiple choice options given

answer: No selected answer given

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question_subject: mathematics


question: If n(X) = n C6, then the value of m is __________.

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question_subject: mathematics


question: If the value of n(Y) + n(Z) is k^2, then |k | is _____________.

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question_subject: mathematics


question: The value of ∫[0 to π/2] f(x)g(x)dx - ∫[0 to π/2] g(x)dx is ___________.

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question_subject: mathematics


question: The value of (16/π^3) ∫[0 to π/2] f(x)g(x)dx is ___________.

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question_subject: mathematics


question: A region in the form of an equilateral triangle (in x-y plane) of height L has a uniform magnetic field B pointing in the +z-direction. A conducting loop PQR, in the form of an equilateral triangle of the same height L, is placed in the x-y plane with its vertex P at x = 0 in the orientation shown in the figure. At t = 0, the loop starts entering the region of the magnetic field with a uniform velocity v along the +x-direction. The plane of the loop and its orientation remain unchanged throughout its motion. Which of the following graph best depicts the variation of the induced emf (E) in the loop as a function of the distance (x) starting from x = 0?

answer: (B)

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question_subject: physics


question: A particle of mass m is under the influence of the gravitational field of a body of mass M (>> m). The particle is moving in a circular orbit of radius r0 with time period T0 around the mass M. Then, the particle is subjected to an additional central force, corresponding to the potential energy Vc(r) = mα/r3, where α is a positive constant of suitable dimensions and r is the distance from the center of the orbit. If the particle moves in the same circular orbit of radius r0 in the combined gravitational potential due to M and Vc(r), but with a new time period T1, then (T1^2 - T0^2)/T1^2 is given by

answer: C

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question_subject: physics


question: A metal target with atomic number Z = 46 is bombarded with a high energy electron beam. The emission of X-rays from the target is analyzed. The ratio r of the wavelengths of the Kα-line and the cut-off is found to be r = 2. If the same electron beam bombards another metal target with Z = 41, the value of r will be

answer: B

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question_subject: physics


question: A thin stiff insulated metal wire is bent into a circular loop with its two ends extending tangentially from the same point of the loop. The wire loop has mass m and radius r and it is in a uniform vertical magnetic field B0, as shown in the figure. Initially, it hangs vertically downwards, because of acceleration due to gravity g, on two conducting supports at P and Q. When a current I is passed through the loop, the loop turns about the line PQ by an angle θ given by

response_choices: (A) tan θ = πr1B0/(mg) (B) tan θ = 2πr1B0/(mg) (C) tan θ = πr1B0/(2mg) (D) tan θ = mg/(πr1B0)

answer: B

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question_subject: physics


question: A small electric dipole p0, having a moment of inertia I about its center, is kept at a distance r from the center of a spherical shell of radius R. The surface charge density σ is uniformly distributed on the spherical shell. The dipole is initially oriented at a small angle θ as shown in the figure. While staying at a distance r, the dipole is free to rotate about its center. If released from rest, then which of the following statement(s) is(are) correct?

response_choices: [ε0 is the permittivity of free space.] (A) The dipole will undergo small oscillations at any finite value of r. (B) The dipole will undergo small oscillations at any finite value of r > R. (C) The dipole will undergo small oscillations with an angular frequency of √(2σp0/εoI) at r = 2R. (D) The dipole will undergo small oscillations with an angular frequency of √(σp0/100εoI) at r = 10R.

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question_subject: physics


question: A table tennis ball has radius (3/2) × 10^−2 m and mass (22/7) × 10^−3 kg. It is slowly pushed down into a swimming pool to a depth of d = 0.7 m below the water surface and then released from rest. It emerges from the water surface at speed v, without getting wet, and rises up to a height H. Which of the following option(s) is(are) correct? [Given: π = 22/7, g = 10 m s^-2, density of water = 1 × 10³ kg m^-3, viscosity of water = 1 × 10^−3 Pa.s.]

response_choices: (A) The work done in pushing the ball to the depth d is 0.077 J. (B) If we neglect the viscous force in water, then the speed v = 7 m/s. (C) If we neglect the viscous force in water, then the height H = 1.4 m. (D) The ratio of the magnitudes of the net force excluding the viscous force to the maximum viscous force in water is 500/9.

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question_subject: physics


question: A positive, singly ionized atom of mass number AM is accelerated from rest by the voltage 192 V. Thereafter, it enters a rectangular region of width w with magnetic field B0 = 0.1T Tesla, as shown in the figure. The ion finally hits a detector at the distance x below its starting trajectory.

response_choices:
(A) The value of x for H+ ion is 4 cm.
(B) The value of x for an ion with AM = 144 is 48 cm.
(C) For detecting ions with 1 ≤ AM ≤ 196, the minimum height (x1 – x0) of the detector is 55 cm.
(D) The minimum width w of the region of the magnetic field for detecting ions with AM = 196 is 56 cm.

answer: C

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question_subject: physics


question: The dimensions of a cone are measured using a scale with a least count of 2 mm. The diameter of the base and the height are both measured to be 20.0 cm. The maximum percentage error in the determination of the volume is _______.

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question_subject: mathematics


question: A ball is thrown from the location (x0, y0) = (0,0) of a horizontal playground with an initial speed v0 at an angle θ0 from the +x-direction. The ball is to be hit by a stone, which is thrown at the same time from the location (x1, y1) = (I, 0). The stone is thrown at an angle (180 − θ1) from the +x-direction with a suitable initial speed. For a fixed v0, when (θ0, θ1) = (45°, 45°), the stone hits the ball after time T1, and when (θ0, θ1) = (60°, 30°), it hits the ball after time T2. In such a case, (T1/T2)² is _______.

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question_subject: physics


question: A charge is kept at the central point P of a cylindrical region. The two edges subtend a half-angle θ at P, as shown in the figure. When θ = 30°, then the electric flux through the curved surface of the cylinder is Φ. If θ = 60°, then the electric flux through the curved surface becomes Φ/√n, where the value of n is_______.

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question_subject: physics


question: Two equilateral-triangular prisms P1 and P2 are kept with their sides parallel to each other, in vacuum, as shown in the figure. A light ray enters prism P1 at an angle of incidence θ such that the outgoing ray undergoes minimum deviation in prism P2. If the respective refractive indices of P1 and P2 are √2 and √3, then θ = sin^-1[β/√2 sin(π/β)], where the value of β is _______.

response_choices: No response choices provided for this question.

answer: No answer provided for this question.

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question_subject: physics


question: An infinitely long thin wire, having a uniform charge density per unit length of 5 nC/m, is passing through a spherical shell of radius 1 m, as shown in the figure. A 10 nC charge is distributed uniformly over the spherical shell. If the configuration of the charges remains static, the magnitude of the potential difference between points P and R, in Volt, is ______.

response_choices: No response choices provided for this question.

answer: No answer provided for this question.

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question_subject: physics


question: A spherical soap bubble inside an air chamber at pressure P0 = 10^5 Pa has a certain radius so that the excess pressure inside the bubble is ΔP = 144 Pa. Now, the chamber pressure is reduced to 8P0/27 so that the bubble radius and its excess pressure change. In this process, all the temperatures remain unchanged. Assume air to be an ideal gas and the excess pressure ΔP in both the cases to be much smaller than the chamber pressure. The new excess pressure ΔP' in Pa is _______.

response_choices: No response choices provided for this question.

answer: No answer provided for this question.

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question_subject: physics


question: The 8th bright fringe above the point O oscillates with time between two extreme positions. The separation between these two extreme positions, in micrometer (μm), is _______.

response_choices: Numerical Response Question

answer: 0.8

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question_subject: physics


question: The maximum speed in μm/s at which the 8th bright fringe will move is __________.

response_choices: No response choices provided

answer: No answer provided

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question_subject: physics


question: If the collision occurs at time t0 = 0, the value of vcm/(aω) will be __________.

response_choices: No response choices provided

answer: No answer provided

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question_subject: physics


question: If the collision occurs at time t0 = π/(2ω), then the value of 4b2/a2 will be _________.

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question_subject: physics


question: According to Bohr's model, the highest kinetic energy is associated with the electron in the

response_choices:
(A) first orbit of H atom
(B) first orbit of He+
(C) second orbit of He+
(D) second orbit of Li2+

answer: A

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question_subject: chemistry


question: In a metal deficient oxide sample, MxY2O4 (M and Y are metals), M is present in both +2 and +3 oxidation states and Y is in +3 oxidation state. If the fraction of M2+ ions present in M is 1/3, the value of X is _____.

answer: B

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question_subject: chemistry


question: In the following reaction sequence, the major product Q is

response_choices:
(A)

Cl
Cl
Cl
Cl

(B)

Cl
Cl

Cl

(C)

Cl

Cl

(D)

Cl
Cl

Cl
Cl
Cl

answer: B

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question_subject: chemistry


question: The species formed on fluorination of phosphorus pentachloride in a polar organic solvent are

response_choices: (A) [PF4]+[PF6]- and [PCl4]+[PF6]- (B) [PCl4]+[PCl3F2]- and [PCl2]+[PF6]- (C) PF5 and PCl5 (D) PF5 and PCl3

answer: B

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question_subject: chemistry


question: An aqueous solution of hydrazine (N₂H₄) is electrochemically oxidized by O₂, thereby releasing chemical energy in the form of electrical energy. One of the products generated from the electrochemical reaction is N₂(g). Choose the correct statement(s) about the above process

response_choices: (A) OH⁻ ions react with N₂H₄ at the anode to form N₂(g) and water, releasing 4 electrons to the anode. (B) At the cathode, N₂H₄ breaks to N₂(g) and nascent hydrogen released at the electrode reacts with oxygen to form water. (C) At the cathode, molecular oxygen gets converted to OH⁻. (D) Oxides of nitrogen are major by-products of the electrochemical process.

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question_subject: chemistry


question: The option(s) with correct sequence of reagents for the conversion of P to Q is(are)

response_choices: (A) i) Lindlar's catalyst, H₂; ii) SnCl₂/HCl; iii) NaBH₄; iv) H₂O⁻, (B) i) Lindlar's catalyst, H₂; ii) H₂O⁻; iii) SnCl₂/HCl; iv) NaBH₄, (C) i) NaBH₄; ii) SnCl₂/HCl; iii) H₂O⁻; iv) Lindlar's catalyst, H₂, (D) i) Lindlar's catalyst, H₂; ii) NaBH₄; iii) SnCl₂/HCl; iv) H₂O⁻

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question_subject: chemistry


question: The compound(s) having peroxide linkage is(are)

response_choices: (A) H₂S₂O₇, (B) H₂S₂O₈, (C) H₂S₂O₅, (D) H₂SO₅

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question_subject: chemistry


question: To form a complete monolayer of acetic acid on 1g of charcoal, 100 mL of 0.5 M acetic acid was used. Some of the acetic acid remained unadsrobed. To neutralize the unadsrobed acetic acid, 40 mL of 1 M NaOH solution was required. If each molecule of acetic acid occupies P × 10^-23 m^2 surface area on charcoal, the value of P is _____.

response_choices: No response choices provided for this type of question.

answer: Numerical response expected.

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question_subject: chemistry


question: Vessel-1 contains w2 g of a non-volatile solute X dissolved in w1 g of water. Vessel-2 contains w2 g of another non-volatile solute Y dissolved in w1 g of water. Both the vessels are at the same temperature and pressure. The molar mass of X is 80% of that of Y. The van't Hoff factor for X is 1.2 times of that of Y for their respective concentrations. The elevation of boiling point for solution in Vessel-1 is _____ % of the solution in Vessel-2.

response_choices: No response choices provided for this type of question.

answer: Numerical response expected.

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question_subject: chemistry


question: For a double strand DNA, one strand is given below: 5' A G T C A C G T A A G T C 3' The amount of energy required to split the double strand DNA into two single strands is _____ kcal mol^-1. [Given: Average energy per H-bond for A-T base pair = 1.0 kcal mol^-1, G-C base pair = 1.5 kcal mol^-1, and A-U base pair = 1.25 kcal mol^-1. Ignore electrostatic repulsion between the phosphate groups.]

response_choices: No response choices provided for this type of question.

answer: Numerical response expected.

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question_subject: chemistry


question: A sample initially contains only U-238 isotope of uranium. With time, some of the U-238 radioactively decays into Pb-206 while the rest of it remains undisintegrated. When the age of the sample is P x10^8 years, the ratio of mass of Pb-206 to that of U-238 in the sample is found to be 7. The value of P is _____. [Given: Half-life of U-238 is 4.5 x10^9 years; log₂2 = 0.693]

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question_subject: physics


question: Among [Co(CN)₆]⁴⁻, [Co(CO)₃(NO)], XeF₄, [PtCl₄]²⁻, [PdCl₄]²⁻, [IrCl₆]³⁻, [Cu(CN)₄]³⁻ and P₄ the total number of species with tetrahedral geometry is _____.

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question_subject: chemistry


question: An organic compound P having molecular formula C₈H₈O₃ gives ferric chloride test and does not have intramolecular hydrogen bond. The compound P reacts with 3 equivalents of NH₂OH to produce oxime Q. Treatment of P with excess methyl iodide in the presence of KOH produces compound R as the major product. Reaction of R with excess iso-butylmagnesium bromide followed by treatment with H₃O⁺ gives compound S as the major product. The total number of methyl (–CH₃) group(s) in compound S is _____.

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question_subject: chemistry


question: Sum of number of oxygen atoms in S and T is _____.

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question_subject: chemistry


question: The molecular weight of U is _____.

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question_subject: chemistry


question: The number of moles of potassium iodide required to produce two moles of P is _____.

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question_subject: chemistry


question: The number of zinc ions present in the molecular formula of Q is _____.

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question_subject: chemistry