question: Let α and β be real numbers such that π/4 < β < 0 < α < π/4. If sin(α + β) = 1/3 and cos(α - β) = 2/3, then the greatest integer less than or equal to ((sin α / cos β + cos α / sin β + sin β / cos α)^2) is

response_choices: No response choices provided

answer: No answer provided, question requires numeric entry

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question_subject: mathematics


question: If y(x) is the solution of the differential equation xdy - (y^2 - 4y)dx = 0 for x > 0, y(1) = 2, and the slope of the curve y = y(x) is never zero, then the value of 10 y(√2) is

response_choices: No response choices provided

answer: No answer provided, question requires numeric entry

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question_subject: mathematics


question: The greatest integer less than or equal to (∫(1 to 2) log2(x^3 + 1) dx + ∫(1 to log2 9) (2^x - 1)^(-1) dx) is

response_choices: No response choices provided

answer: No answer provided, question requires numeric entry

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question_subject: mathematics


question: The product of all positive real values of x satisfying the equation x(16(log5 x)^3 -68 log5 x) = 5^-16 is

response_choices: No response choices provided

answer: No answer provided, question requires numeric entry

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question_subject: mathematics


question: If β = lim(x→0) ((e^(x^4) - (1 - x^3)^(1/3) + ((1 - x^2)^(1/2) - 1) sin x) / (x sin^2 x)), then the value of 6β is _______.

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question_subject: mathematics


question: Let β be a real number. Consider the matrix A = (β  0  1, 2  1 -2, 3  1 -2). If A^7 - (β - 1)A^6 - βA^5  is a singular matrix, then the value of  9β is _______.

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question_subject: mathematics


question: Consider the hyperbola (x^2/100) - (y^2/64) = 1 with foci at S and S1, where S lies on the positive x-axis. Let P be a point on the hyperbola, in the first quadrant. Let ∠SPS1 = α, with α < π/2. The straight line passing through the point S and having the same slope as that of the tangent at P to the hyperbola, intersects the straight line S1P at P1. Let δ be the distance of P from the straight line SP1, and β = S1P. Then the greatest integer less than or equal to (βδ/α) sin(α/2) is _______.

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question_subject: mathematics


question: Consider the functions f , g : ℝ → ℝ defined by f(x) = x^2 + (5/12) and g(x) = (2 (1 - (4|x|/3)), |x| ≤ 3/4; 0, |x| > 3/4). If α is the area of the region {(x,y) ϵ ℝ x ℝ : |x| ≤ 3/4 , 0 ≤ y ≤ min(f(x),g(x))}, then the value of 9α is _________.

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question_subject: mathematics


question: Let PQRS be a quadrilateral in a plane, where QR = 1, ∠PQR = ∠QRS = 70°, ∠PQS = 15° and ∠PRS = 40°. If ∠RPS = 0°, PQ = α and PS = β, then the interval(s) that contain(s) the value of 4 α β sin 0° is/are

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question_subject: mathematics


question: Let g : [0, 1] → R be the function defined by g(x) = 2^(x*x) + 2^(a(1-x)) . Then, which of the following statements is/are TRUE ?

response_choices: (A) The minimum value of g(x) is 2e (B) The maximum value of g(x) is 1 + 2e (C) The function g(x) attains its maximum at more than one point (D) The function g(x) attains its minimum at more than one point

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question_subject: mathematics


question: Let z denote the complex conjugate of a complex number z. If z is a non-zero complex number for which both real and imaginary parts of (z )^2 + 1/z^2 are integers, then which of the following is/are possible value(s) of |z| ?

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question_subject: mathematics


question: Let G be a circle of radius R > 0. Let G1, G2, ..., Gn be n circles of equal radius r > 0. Suppose each of the n circles G1, G2, ..., Gn touches the circle G externally. Also, for i = 1, 2, ..., n − 1, the circle Gi touches Gi+1 externally, and Gn touches G1 externally. Then, which of the following statements is/are TRUE ?

response_choices:
(A) If n = 4, then (√2 − 1)r < R
(B) If n = 5, then r < R
(C) If n = 8, then (√2 − 1) r < R
(D) If n = 12, then √2 (√3 + 1) r > R

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question_subject: mathematics


question:
Let i, j and k be the unit vectors along the three positive coordinate axes. Let
a = 3i + j − k,
b = i + b2j + b3k,   b2, b3 ∈ ℝ,
c = c1i + c2j + c3k,   c1, c2, c3 ∈ ℝ
be three vectors such that b2b3 > 0, a⃗ ⋅ b⃗ = 0 and
⎛0  −c3   c2⎞ ⎛1⎞   ⎛3 − c1⎞
⎜c3   0   −c1⎟ ⎜b2⎟ = ⎜1 − c2⎟
⎝−c2  c1    0 ⎠ ⎝b3⎠   ⎝−1 − c3⎠
Then, which of the following is/are TRUE ?

response_choices:
(A) a⃗ ⋅ c⃗ = 0
(B) b⃗ ⋅ c⃗ = 0
(C) |b⃗| > √10
(D) |c⃗| ≤ √11

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question_subject: mathematics


question: For x ∈ ℝ, let the function y(x) be the solution of the differential equation dy/dx + 12y = cos(πx/12), y(0) = 0. Then, which of the following statements is/are TRUE?

response_choices: (A) y(x) is an increasing function, (B) y(x) is a decreasing function, (C) There exists a real number β such that the line y = β intersects the curve y = y(x) at infinitely many points, (D) y(x) is a periodic function

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question_subject: mathematics


question: Consider 4 boxes, where each box contains 3 red balls and 2 blue balls. Assume that all 20 balls are distinct. In how many different ways can 10 balls be chosen from these 4 boxes so that from each box at least one red ball and one blue ball are chosen ?

answer: (C) 12096

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question_subject: mathematics


question: If M = \begin{pmatrix} 5 & 3 \\ 2 & -2 \\ -3 & -1 \\ 2 & 2 \end{pmatrix}, then which of the following matrices is equal to M^{2022} ?

answer: (D) \begin{pmatrix} 3032 & 3031 \\ -3031 & -3030 \end{pmatrix}

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question_subject: mathematics


question: Suppose that Box-I contains 8 red, 3 blue and 5 green balls, Box-II contains 24 red, 9 blue and 15 green balls, Box-III contains 1 blue, 12 green and 3 yellow balls, Box-IV contains 10 green, 16 orange and 6 white balls. A ball is chosen randomly from Box-I: call this ball b. If b is red then a ball is chosen randomly from Box-II, if b is blue then a ball is chosen randomly from Box-III, and if b is green then a ball is chosen randomly from Box-IV. The conditional probability of the event 'one of the chosen balls is white' given that the event 'at least one of the chosen balls is green' has happened, is equal to

answer: B

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question_subject: mathematics


question: For positive integer n, define f(n) = n + 16 + 5n - 3n^2/(4n + 3n^2) + 32 + n - 3n^2/(8n + 3n^2) + 48 - 3n - 3n^2/(12n + 3n^2) + ... + 25n - 7n^2/7n^2 . Then, the value of lim f(n) is equal to

answer: A

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question_subject: mathematics


question: A particle of mass 1 kg is subjected to a force which depends on the position as F⃗ = −k(x i⃗ + y j⃗) kg m s−2 with k = 1 kg s−2. At time t = 0, the particle's position r⃗ = (1⁄2 i⃗ + √2 j⃗) m and its velocity v⃗ = (−√2 i⃗ + √2 j⃗ + 2⃗k) m s−1. Let vx and vy denote the x and the y components of the particle's velocity, respectively. Ignore gravity. When z = 0.5 m, the value of (x vx − y vy) is ____ m2s−1.

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question_subject: physics


question: In a radioactive decay chain reaction, 230⁄90Th nucleus decays into 214⁄84Po nucleus. The ratio of the number of α to number of β− particles emitted in this process is ____.

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question_subject: physics


question: Two resistances R1 = X Ω and R2 = 1 Ω are connected to a wire AB of uniform resistivity, as shown in the figure. The radius of the wire varies linearly along its axis from 0.2 mm at A to 1 mm at B. A galvanometer (G) connected to the center of the wire, 50 cm from each end along its axis, shows zero deflection when A and B are connected to a battery. The value of X is ____.

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question_subject: physics


question: In a particular system of units, a physical quantity can be expressed in terms of the electric charge e, electron mass me, Planck's constant h, and Coulomb's constant k = 1/4πε0, where ε0 is the permittivity of vacuum. In terms of these physical constants, the dimension of the magnetic field is [B] = [e]^α [me]^β [h]^γ [k]^δ. The value of α + β + γ + δ is _____.

response_choices: No response choices provided

answer: No answer provided

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question_subject: physics


question: Consider a configuration of n identical units, each consisting of three layers. The first layer is a column of air of height h = 1⁄2 cm, and the second and third layers are of equal thickness d = √3–1⁄2 cm, and refractive indices μ1 = √2 and μ2 = √3, respectively. A light source O is placed on the top of the first unit, as shown in the figure. A ray of light from O is incident on the second layer of the first unit at an angle of θ = 60° to the normal. For a specific value of n, the ray of light emerges from the bottom of the configuration at a distance l = √3⁄2 cm, as shown in the figure. The value of n is ___.

response_choices: No response choices provided

answer: No answer provided

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question_subject: physics


question: A charge q is surrounded by a closed surface consisting of an inverted cone of height h and base radius R, and a hemisphere of radius R as shown in the figure. The electric flux through the conical surface is π q/ε0 (in SI units). The value of n is _______.

response_choices: No response choices provided

answer: No answer provided

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question_subject: physics


question: On a frictionless horizontal plane, a bob of mass m = 0.1 kg is attached to a spring with natural length l0 = 0.1 m. The spring constant is k2 = 0.009 N/m^-1 when the length of the spring l > l0 and is k2 = 0.016 N/m^-1 when l < l0. Initially the bob is released from l = 0.15 m. Assume that Hooke's law remains valid throughout the motion. If the time period of the full oscillation is T = (π π) s, then the integer closest to π is __.

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question_subject: physics


question: An object and a concave mirror of focal length f = 10 cm both move along the principal axis of the mirror with constant speeds. The object moves with speed V0 = 15 cm s^-1 towards the mirror with respect to a laboratory frame. The distance between the object and the mirror at a given moment is denoted by u. When u = 30 cm, the speed of the mirror Vm is such that the image is instantaneously at rest with respect to the laboratory frame, and the object forms a real image. The magnitude of Vm is ____ cm s^-1.

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question_subject: physics


question: In the figure, the inner (shaded) region A represents a sphere of radius rA = 1, within which the electrostatic charge density varies with the radial distance r from the center as ρA = kr, where k is positive. In the spherical shell B of outer radius rB, the electrostatic charge density varies as ρB = c/r^2. Assume that dimensions are taken care of. All physical quantities are in their SI units. Which of the following statement(s) is(are) correct?

response_choices: (A) If rB = √5/√2, then the electric field is zero everywhere outside B. (B) If rB = 3, then the electric potential just outside B is k/rB. (C) If rB = 2, then the total charge of the configuration is 15πk. (D) If rB = 2, then the magnitude of the electric field just outside B is 13πk/rB.

answer: B

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question_subject: physics


question: In Circuit-1 and Circuit-2 shown in the figures, R1 = 1 Ω, R2 = 2 Ω and R3 = 3 Ω. P1 and P2 are the power dissipations in Circuit-1 and Circuit-2 when the switches S1 and S2 are in open conditions, respectively. Q1and Q2 are the power dissipations in Circuit-1 and Circuit-2 when the switches S1 and S2 are in closed conditions, respectively. Which of the following statement(s) is(are) correct? (A) When a voltage source of 6 V is connected across A and B in both circuits, P1 < P2. (B) When a constant current source of 2 Amp is connected across A and B in both circuits, P1 > P2. (C) When a voltage source of 6 V is connected across A and B in Circuit-1, Q1 > P1. (D)  When a constant current source of 2 Amp is connected across A and B in both circuits, Q2 < Q1.

response_choices: A, B, C, D

answer: B, C

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question_subject: physics


question: A bubble has surface tension S. The ideal gas inside the bubble has ratio of specific heats γ = 5/3. The bubble is exposed to the atmosphere and it always retains its spherical shape. When the atmospheric pressure is Pat1, the radius of the bubble is found to be r1 and the temperature of the enclosed gas is T1. When the atmospheric pressure is Pat2, the radius of the bubble and the temperature of the enclosed gas are r2 and T2, respectively. Which of the following statement(s) is(are) correct? (A) If the surface of the bubble is a perfect heat insulator, then (r1/r2)^5 = Pat2*r2^5/Pat1*r1^5. (B) If the surface of the bubble is a perfect heat insulator, then the total internal energy of the bubble including its surface energy does not change with the external atmospheric pressure. (C) If the surface of the bubble is a perfect heat conductor and the change in atmospheric temperature is negligible, then (T1/T2)^3 = Pat2*r2^5/Pat1*r1^5. (D) If the surface of the bubble is a perfect heat insulator, then (T2/T1)^5 = Pat2*r2^4/Pat1*r1^4.

response_choices: A, B, C, D

answer: B

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question_subject: physics


question: A disk of radius R with uniform positive charge density σ is placed on the xy plane with its center at the origin. The Coulomb potential along the z-axis is V(z) = (σ/2ε0)√(R^2+z^2-z). A particle of positive charge q is placed initially at rest at a point on the z axis with z=z0 and z0>0. In addition to the Coulomb force, the particle experiences a vertical force F=-ck with c>0. Let β = 2cε0/qσ. Which of the following statement(s) is(are) correct?

response_choices:
(A) For β=1/3 and z0=2/3R, the particle reaches the origin.
(B) For β=1/3 and z0=3/4R, the particle reaches the origin.
(C) For β=1/3 and z0=R/5, the particle returns back to z=z0.
(D) For β>1 and z0>0, the particle always reaches the origin.

answer: C

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question_subject: physics


question: A double slit setup is shown in the figure. One of the slits is in medium 2 of refractive index n2. The other slit is at the interface of this medium with another medium 1 of refractive index n1 (≠ n2). The line joining the slits is perpendicular to the interface and the distance between the slits is d. The slit widths are much smaller than d. A monochromatic parallel beam of light is incident on the slits from medium 1. A detector is placed in medium 2 at a large distance from the slits, and at an angle θ from the line joining them, so that θ equals the angle of refraction of the beam. Consider two approximately parallel rays from the slits received by the detector.

response_choices:
(A) The phase difference between the two rays is independent of d.
(B) The two rays interfere constructively at the detector.
(C) The phase difference between the two rays depends on n1 but is independent of n2.
(D) The phase difference between the two rays vanishes only for certain values of d and the angle of incidence of the beam, with θ being the corresponding angle of refraction.

answer: D

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question_subject: physics


question: In the given P-V diagram, a monoatomic gas (γ = 5/3) is first compressed adiabatically from state A to state B. Then it expands isothermally from state B to state C. [Given: (γ) 0/γ ≈ 0.5, ln 2 ≈ 0.7 ]. Which of the following statement(s) is(are) correct?

response_choices: (A) The magnitude of the total work done in the process A → B → C is 144 kJ. (B) The magnitude of the work done in the process B → C is 84 kJ. (C) The magnitude of the work done in the process A → B is 60 kJ. (D) The magnitude of the work done in the process C → A is zero.

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question_subject: physics


question: A flat surface of a thin uniform disk A of radius R is glued to a horizontal table. Another thin uniform disk B of mass M and with the same radius R rolls without slipping on the circumference of A, as shown in the figure. A flat surface of B also lies on the plane of the table. The center of mass of B has fixed angular speed ω about the vertical axis passing through the center of A. The angular momentum of B is nMωR^2 with respect to the center of A. Which of the following is the value of n?

answer: C) 7/2

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question_subject: physics


question: When light of a given wavelength is incident on a metallic surface, the minimum potential needed to stop the emitted photoelectrons is 6.0 V. This potential drops to 0.6 V if another source with wavelength four times that of the first one and intensity half of the first one is used. What are the wavelength of the first source and the work function of the metal, respectively? [Take hc/e = 1.24 × 10^-6 J m C^-1]

answer: D) 3.78 x 10^7 m, 1.20 eV

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question_subject: physics


question: Concentration of H2SO4 and Na2SO4 in a solution is 1 M and 1.8 × 10^-2 M, respectively. Molar solubility of PbSO4 in the same solution is X × 10^-8 M (expressed in scientific notation). The value of Y is _________.

response_choices: There are no multiple choice options provided.

answer: The question requires a numerical response.

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question_subject: chemistry


question: An aqueous solution is prepared by dissolving 0.1 mol of an ionic salt in 1.8 kg of water at 35 °C. The salt remains 90% dissociated in the solution. The vapour pressure of the solution is 59.724 mm of Hg. Vapor pressure of water at 35 °C is 60.000 mm of Hg. The number of ions present per formula unit of the ionic salt is _______.

response_choices: There are no multiple choice options provided.

answer: The question requires a numerical response.

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question_subject: chemistry


question: Consider the strong electrolytes Z, X-, U, Y- and V, X-. Limiting molar conductivity (Λ0) of Um-Yn and Vm-Xn are 250 and 440 S cm2 mol−1, respectively. The value of (m + n + p) is ________.

response_choices: No multiple choice options given

answer: The question requires a numerical response to fill in the blank.

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question_subject: chemistry


question: The reaction of Xe and O2F2 gives a Xe compound P. The number of moles of HF produced by the complete hydrolysis of 1 mol of P is _______.

response_choices: No multiple choice options given

answer: The question requires a numerical response to fill in the blank.

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question_subject: chemistry


question: Thermal decomposition of AgNO3 produces two paramagnetic gases. The total number of electrons present in the antibonding molecular orbitals of the gas that has the higher number of unpaired electrons is _______.

response_choices: No multiple choice options given

answer: The question requires a numerical response to fill in the blank.

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question_subject: chemistry


question: The number of isomeric tetracnes (NOT containing sp-hybridized carbon atoms) that can be formed from the following reaction sequence is _______.

response_choices: No multiple choice options provided

answer: No answer provided

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question_subject: chemistry


question: The number of -CH2- (methylene) groups in the product formed from the following reaction sequence is _______.

response_choices: No multiple choice options provided

answer: No answer provided

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question_subject: chemistry


question: The total number of chiral molecules formed from one molecule of P on complete ozonolysis (O3/Zn/H2O) is _______.

response_choices: No multiple choice options provided

answer: No answer provided

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question_subject: chemistry


question: To check the principle of multiple proportions, a series of pure binary compounds (P_xQ_y) were analyzed and their composition is tabulated below. The correct option(s) is(are)

response_choices: (A) If empirical formula of compound 3 is P_xQ_y, then the empirical formula of compound 2 is P_xQ_2y. (B) If empirical formula of compound 3 is P_xQ_y and atomic weight of element P is 20, then the atomic weight of Q is 45. (C) If empirical formula of compound 2 is PQ, then the empirical formula of the compound 1 is P_2Q. (D) If atomic weight of P and Q are 70 and 35, respectively, then the empirical formula of compound 1 is P_2Q.

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question_subject: chemistry


question: Considering the following reaction sequence, the correct statement(s) is(are)

response_choices: (A) Compounds P and Q are carboxylic acids. (B) Compound S decolorizes bromine water. (C) Compounds P and S react with hydroxylamine to give the corresponding oximes. (D) Compound R reacts with dialkylcadmium to give the corresponding tertiary alcohol.

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question_subject: chemistry


question: Among the following, the correct statement(s) about polymers is(are)

response_choices: (A) The polymerization of chloroprene gives natural rubber. (B) Teflon is prepared from tetrafluoroethene by heating it with persulphate catalyst at high pressures. (C) PVC are thermoplastic polymers. (D) Ethene at 350-570 K temperature and 1000-2000 atm pressure in the presence of a peroxide initiator yields high density polythene.

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question_subject: chemistry


question: Atom X occupies the fcc lattice sites as well as alternate tetrahedral voids of the same lattice. The packing efficiency (in %) of the resultant solid is closest to

answer: (B) 35

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question_subject: chemistry


question: The reaction of HClO3 with HCl gives a paramagnetic gas, which upon reaction with O2 produces

response_choices: (A) Cl2O (B) ClO2 (C) Cl2O6 (D) Cl2O7

answer: (B) ClO2

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question_subject: chemistry


question: The reaction of Pb(NO3)2 and NaCl in water produces a precipitate that dissolves upon the addition of HCl of appropriate concentration. The dissolution of the precipitate is due to the formation of

response_choices: (A) PbCl2 (B) PbCl4 (C) [PbCl4]2- (D) [PbCl6]2-

answer: (C) [PbCl4]2-

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question_subject: chemistry


question: Treatment of D-glucose with aqueous NaOH results in a mixture of monosaccharides, which are

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

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question_subject: chemistry