question: A thin and uniform rod of mass M and length L is held vertical on a floor with large friction. The rod is released from rest so that it falls by rotating about its contact-point with the floor without slipping. Which of the following statement(s) is/are correct, when the rod makes an angle 60° with vertical? [g is the acceleration due to gravity]

response_choices: (A) The angular speed of the rod will be √(3g/2), (B) The angular acceleration of the rod will be 2g/L, (C) The radial acceleration of the rod's center of mass will be 3g/4, (D) The normal reaction force from the floor on the rod will be Mg/16

answer: B, C

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question_subject: physics


question: A block of mass 2M is attached to a massless spring with spring-constant k. This block is connected to two other blocks of masses M and 2M using two massless pulleys and strings. The accelerations of the blocks are a1, a2 and a3 as shown in the figure. The system is released from rest with the spring in its unstretched state. The maximum extension of the spring is x0. Which of the following option(s) is/are correct? [g is the acceleration due to gravity. Neglect friction]

response_choices:
A) x0 = 4Mg/k
B) When spring achieves an extension of x0/2 for the first time, the speed of the block connected to the spring is 3g sqrt(M/5k)
C) At an extension of x0/4 of the spring, the magnitude of acceleration of the block connected to the spring is 3g/10
D) a2 - a1 = a1 - a3

answer: A, D

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question_subject: physics


question: A small particle of mass m moving inside a heavy, hollow and straight tube along the tube axis undergoes elastic collision at two ends. The tube has no friction and it is closed at one end by a flat surface while the other end is fitted with a heavy movable flat piston as shown in figure. When the distance of the piston from closed end is L = L0, the particle speed is v = v0. The piston is moved inward at a very low speed V such that V << dL/v0 , where dL is the infinitesimal displacement of the piston. Which of the following statement(s) is/are correct?

response_choices:
(A) The rate at which the particle strikes the piston is v/L
(B) After each collision with the piston, the particle speed increases by 2V
(C) If the piston moves inward by dL, the particle speed increases by 2 v dL/L
(D) The particle's kinetic energy increases by a factor of 4 when the piston is moved inward from L0 to 1/2 L0

answer: C

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question_subject: physics


question: An electric dipole with dipole moment p0/(i + j) is held fixed at the origin O in the presence of an uniform electric field of magnitude E0. If the potential is constant on a circle of radius R centered at the origin as shown in figure, then the correct statement(s) is/are: (e0 is permittivity of free space. R >> dipole size)

response_choices: (A) R = (  p0   )^(1/3)         (4πε0E0)          (B) Total electric field at point A is BA =  √2 E0 (i + j)          (C) Total electric field at point B is BB = 0          (D) The magnitude of total electric field on any two points of the circle will be same.

answer: A

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question_subject: physics


question: A mixture of ideal gas containing 5 moles of monatomic gas and 1 mole of rigid diatomic gas is initially at pressure P0, volume V0, and temperature T0. If the gas mixture is adiabatically compressed to a volume V0/4, then the correct statement(s) is/are, (Given  γ^(1/2) = 2.3;  γ^(3/2) = 9.2;  R  is gas constant)

response_choices: (A) The work |W| done during the process is 13RT0          (B) The average kinetic energy of the gas mixture after compression is in between 18RT0 and 19RT0          (C) The final pressure of the gas mixture after compression is in between 9P0 and 10P0          (D) Adiabatic constant of the gas mixture is 1.6

answer: B

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question_subject: physics


question: Three glass cylinders of equal height H = 30 cm and same refractive index n = 1.5 are placed on a horizontal surface as shown in figure. Cylinder I has a flat top, cylinder II has a convex top and cylinder III has a concave top. The radii of curvature of the two curved tops are same ( R = 3m). If H2, H2, and H3 are the apparent depths of a point X on the bottom of the three cylinders, respectively, the correct statement(s) is/are:

answer: D

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question_subject: physics


question: In a Young's double slit experiment, the slit separation d is 0.3 mm and the screen distance D is 1 m. A parallel beam of light of wavelength 600 nm is incident on the slits at angle α as shown in figure. On the screen, the point O is equidistant from the slits and distance P O is 11.0 mm. Which of the following statement(s) is/are correct?

response_choices: (A) For α = 0.3α degree, there will be destructive interference at point O. (B) For α = 0, there will be constructive interference at point P. (C) For α = 0.3α λ degree, there will be destructive interference at point P. (D) Fringe spacing depends on α.

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question_subject: physics


question: A free hydrogen atom after absorbing a photon of wavelength λin gets excited from the state n = 1 to the state n = 4. Immediately after that the electron jumps to n = m state by emitting a photon of wavelength λe. Let the change in momentum of atom due to the absorption and the emission are Δpₐ and Δpe, respectively. If λₐ/λe = 1/5, which of the option(s) is/are correct? [Use hc = 1242 eV nm ; 1 nm = 10⁻⁹ m, h and c are Planck's constant and speed of light, respectively]

response_choices: (A) m = 2 (B) λe = 418 nm (C) Δpₐ/Δpe = 1/2 (D) The ratio of kinetic energy of the electron in the state n = m to the state n = 1 is 1/4

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question_subject: physics


question: A ball is thrown from ground at an angle θ with horizontal and with an initial speed u0. For the resulting projectile motion, the magnitude of average velocity of the ball up to the point when it hits the ground for the first time is V1. After hitting the ground, the ball rebounds at the same angle θ but with a reduced speed of u0/α. Its motion continues for a long time as shown in figure. If the magnitude of average velocity of the ball for entire duration of motion is 0.8 V1, the value of α is __________.

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question_subject: physics


question: A 10 cm long perfectly conducting wire PQ is moving with a velocity 1 cm/s on a pair of horizontal rails of zero resistance. One side of the rails is connected to an inductor L = 1 mH and a resistance R = 1 Ω as shown in figure. The horizontal rails, L and R lie in the same plane with a uniform magnetic field 𝐵⃗ = 1 T perpendicular to the plane. If the key S is closed at certain instant, the current in the circuit after 1 millisecond is x × 10^-3 A, where the value of x is _______. [Assume the velocity of wire PQ remains constant (1 cm/s) after key S is closed. Given: e^-1 = 0.37, where e is base of the natural logarithm]

response_choices: No options provided

answer: Blank

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question_subject: physics


question: A monochromatic light is incident from air on a refracting surface of a prism of angle 75⁰ and refractive index n₀ = √3. The other refracting surface of the prism is coated by a thin film of material of refractive index n as shown in figure. The light suffers total internal reflection at the coated prism surface for an incidence angle of θ ≤ 60⁰. The value of n^2 is _______.

response_choices: No options provided

answer: Blank

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question_subject: physics


question: A perfectly reflecting mirror of mass M mounted on a spring constitutes a spring-mass system of angular frequency Ω such that 4πMΩ/h = 10^24 m^-2 with h as Planck's constant. N photons of wavelength λ = 8π x 10^-6 m strike the mirror simultaneously at normal incidence such that the mirror gets displaced by 1 μm. If the value of N is x × 10^12, then the value of x is _____. [Consider the spring as massless]

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question_subject: physics


question: Suppose a ²²⁶Ra nucleus at rest and in ground state undergoes α-decay to a ²²²Rn nucleus in its excited state. The kinetic energy of the emitted α particle is found to be 4.44 MeV. ²²²Rn nucleus then goes to its ground state by γ-decay. The energy of the emitted γ photon is ____ keV. [Given: atomic mass of ²²⁶Ra = 226.005 u, atomic mass of ²²²Rn = 222.000 u, atomic mass of α particle = 4.000 u, 1 u = 931 MeV/c², c is speed of the light]

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question_subject: physics


question: An optical bench has 1.5 m long scale having four equal divisions in each cm. While measuring the focal length of a convex lens, the lens is kept at 75 cm mark of the scale and the object pin is kept at 45 cm mark. The image of the object pin on the other side of the lens overlaps with image pin that is kept at 135 cm mark. In this experiment, the percentage error in the measurement of the focal length of the lens is _____.

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question_subject: physics


question: If the tension in each string is T0, the correct match for the highest fundamental frequency in f0 units will be,

response_choices:
(A) I → P, II → Q, III → T, IV → S
(B) I → P, II → R, III → S, IV → Q
(C) I → Q, II → S, III → R, IV → P
(D) I → Q, II → P, III → R, IV → T

answer: (B) I → P, II → R, III → S, IV → Q

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question_subject: physics


question: The length of the strings 1, 2, 3 and 4 are kept fixed at L0, 3L0/2 , 5L0/4 and 7L0/4 , respectively. Strings 1, 2, 3, and 4 are vibrated at their 1st, 3rd , 5th, and 14th harmonics, respectively such that all the strings have same frequency. The correct match for the tension in the four strings in the units of T0 will be,

response_choices:
(A) I → P, II → R, III → T, IV → U
(B) I → P, II → Q, III → T, IV → U
(C) I → P, II → Q, III → R, IV → T
(D) I → T, II → Q, III → R, IV → U

answer: (B) I → P, II → Q, III → T, IV → U

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question_subject: physics


question: Answer Q.17 and Q.18 by appropriately matching the lists based on the information given in the paragraph

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question_subject: physics


question: In a thermodynamic process on an ideal monatomic gas, the infinitesimal heat absorbed by the gas is given by TdX, where T is the temperature of the system and dX is the infinitesimal change in a thermodynamic quantity X of the system. For a mole of monatomic ideal gas X = (3/2)R ln (Vd/Va) + R ln (Vd/Va). Here, R is gas constant, V is volume of gas, Td and Va are constants.

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question_subject: physics


question: The List-I below gives some quantities involved in a process and List-II gives some possible values of these quantities.

response_choices: List-I: (I) Work done by the system in process 1->2->3, (II) Change in internal energy in process 1->2->3, (III) Heat absorbed by the system in process 1->2->3, (IV) Heat absorbed by the system in process 1->2; List-II: (P) 1/2 RTo ln 2, (Q) 1/3 RTo, (R) RTo, (S) 4/3 RTo, (T) 1/3 RTo(3 + ln 2), (U) 5/3 RTo

answer: (I) P, (II) Q, (III) R, (IV) S

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question_subject: physics


question: If the process carried out on one mole of monatomic ideal gas is as shown in figure in the PV-diagram with P0V0 = 4/3 RT0, the correct match is,

response_choices: (A) I → Q, II → R, III → P, IV → U; (B) I → Q, II → R, III → S, IV → U; (C) I → S, II → R, III → Q, IV → T; (D) I → Q, II → S, III → R, IV → U

answer: (D) I → Q, II → S, III → R, IV → U

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question_subject: physics


question: If the process on one mole of monatomic ideal gas is as shown in the TV-diagram with P0V0 = 1/3 RT0, the correct match is,

response_choices: (A) I → P, II → R, III → T, IV → S; (B) I → P, II → T, III → Q, IV → T; (C) I → P, II → R, III → T, IV → P; (D) I → S, II → T, III → Q, IV → U

answer: (C) I → P, II → R, III → T, IV → P

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question_subject: physics


question: The cyanide process of gold extraction involves leaching out gold from its ore with CN⁻ in the presence of O in water to form R. Subsequently, R is treated with T to obtain Au and Z. Choose the correct option(s)

response_choices: (A) O is O₂, (B) T is Zn, (C) Z is [Zn(CN)₄]²⁻, (D) R is [Au(CN)₂]⁻

answer: C

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question_subject: chemistry


question: With reference to aqua regia, choose the correct option(s)

response_choices: (A) Aqua regia is prepared by mixing conc. HCl and conc. HNO₃ in 3:1 (v/v) ratio, (B) Reaction of gold with aqua regia produces an anion having Au in +3 oxidation state, (C) Reaction of gold with aqua regia produces NO₂ in the absence of air, (D) The yellow colour of aqua regia is due to the presence of NOCl and Cl₂

answer: A,B,D

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question_subject: chemistry


question:
Consider the following reactions (unbalanced)

Zn + hot conc. H₂SO₄ → G + R + X
Zn + conc. NaOH → T + Q
G + H₂S + NH₄OH → Z (a precipitate) + X + Y

response_choices:
(A) Z is dirty white in colour

(B) The oxidation state of Zn in T is +1

(C) R is a V-shaped molecule

(D) Bond order of Q is 1 in its ground state

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question_subject: chemistry


question: The ground state energy of hydrogen atom is  -13.6 eV.  Consider an electronic state Ψ'of He⁺ whose energy, azimuthal quantum number and magnetic quantum number are  -3.4 eV, 2 and 0, respectively.  Which of the following statement(s) is(are) true for the state Ψ'?

response_choices:
(A) It is a 4d state

(B) It has 2 angular nodes

(C) It has 3 radial nodes

(D) The nuclear charge experienced by the electron in this state is less than 2e, where e is the magnitude of the electronic charge

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question_subject: chemistry


question: Which of the following reactions produce(s) propane as a major product?

response_choices:
(A)

H₃C⎻⎻⎻COONa  NaOH, CaO, Δ

(B)

H₃C⎻⎻⎻COONa + H₂O electrolysis

(C)

H₃C⎻⎻⎻Cl     Zn, dil. HCl

(D)

Br
|
H₃C⎻⎻⎻Br  Zn

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question_subject: chemistry


question: Choose the correct option(s) that give(s) an aromatic compound as the major product

response_choices: (A) i) alc. KOH, ii) NaNH2, iii) red hot iron tube, 873 K, (B) NaOMe, (C) Br, NaOEt, (D) + Cl2 (excess), UV, 500 K

answer: D

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question_subject: chemistry


question: Choose the correct option(s) for the following reaction sequence

response_choices:
A)

MeO Q

MeO S

B)

MeO R

MeO S

C)

MeO Q

MeO R

D)

MeO R

MeO S

answer: B

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question_subject: chemistry


question: Choose the correct option(s) from the following

response_choices:
A) Natural rubber is polyisoprene containing trans alkene units

B) Nylon-6 has amide linkages

C) Teflon is prepared by heating tetrafluoroethene in presence of a persulphate catalyst at high pressure

D) Cellulose has only α-D-glucose units that are joined by glycosidic linkages

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question_subject: chemistry


question:
The amount of water produced (in g) in the oxidation of 1 mole of rhombic sulphur by conc. HNO3 to a compound with the highest oxidation state of sulphur is _____
(Given data: Molar mass of water = 18 g mol^-1)

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question_subject: chemistry


question:
Total number of cis N-Mn-Cl bond angles (that is, Mn-N and Mn-Cl bonds in cis positions) present in a molecule of cis-[Mn(en)2Cl2] complex is _____
(en = NH2CH2CH2NH2)

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question_subject: chemistry


question: The decomposition reaction 2N2O5(g) → 2N2O4(g) + O2(g) is started in a closed cylinder under isothermal isochoric condition at an initial pressure of 1 atm. After Y × 10^8 s, the pressure inside the cylinder is found to be 1.45 atm. If the rate constant of the reaction is 5 × 10^-4 s^-1, assuming ideal gas behavior, the value of Y is _____

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question_subject: chemistry


question:
The mole fraction of urea in an aqueous urea solution containing 900 g of water is 0.05. If the density of the solution is 1.2 g cm^-3, the molarity of urea solution is _____
(Given data: Molar masses of urea and water are 60 g mol^-1 and 18 g mol^-1, respectively)

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question_subject: chemistry


question: Total number of hydroxyl groups present in a molecule of the major product P is ____

response_choices: No multiple choice options provided

answer: No answer marked

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question_subject: chemistry


question: Total number of isomers, considering both structural and stereoisomers, of cyclic ethers with the molecular formula C4H8O is ____

response_choices: No multiple choice options provided

answer: No answer marked

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question_subject: chemistry


question: Which of the following options has the correct combination considering List-I and List-II?

answer: C

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question_subject: physics


question: Which of the following options has the correct combination considering List-I and List-II?

answer: B

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question_subject: physics


question: Answer Q.17 and Q.18 by appropriately matching the lists based on the information given in the paragraph

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question_subject: chemistry


question: List-I includes starting materials and reagents of selected chemical reactions. List-II gives structures of compounds that may be formed as intermediate products and/or final products from the reactions of List-I.

response_choices: (I) i) DIBAL-H, ii) dil. HCl, iii) NaBH4, iv) conc. H2SO4; (II) i) O3, ii) Zn, H2O, iii) NaBH4, iv) conc. H2SO4; (III) i) KCN, ii) H3O+, Δ, iii) LiAlH4, iv) conc. H2SO4; (IV) i) LiAlH4, ii) conc. H2SO4; (P) CHO-CO2H; (Q) OH-OH; (R) O; (S) OH-CO2H; (T) CO2H-CO2H; (U) O-O

answer: (I) P, (II) Q, (III) R, (IV) S, (V) T

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question_subject: chemistry


question: Which of the following options has correct combination considering List-I and List-II?

answer: (B) (II), (P), (S), (T)

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question_subject: mathematics


question: Which of the following options has correct combination considering List-I and List-II?

answer: (B) (IV), (Q), (U)

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question_subject: mathematics


question: Let P1 = I = [1 0 0; 0 1 0; 0 0 1], P2 = [1 0 0; 0 0 1; 0 1 0], P3 = [0 1 0; 1 0 0; 0 0 1], P4 = [0 1 0; 0 0 1; 1 0 0], P5 = [0 0 1; 1 0 0; 0 1 0], P6 = [0 0 1; 0 1 0; 1 0 0] and X = ∑(6 k=1) Pk [2 1 3; 1 0 2; 2 2 1]PkT where PkT denotes the transpose of the matrix Pk. Then which of the following options is/are correct? (A) If X [1 1 1]T = α [1 1 1]T, then α = 30 (B) X is a symmetric matrix (C) The sum of diagonal entries of X is 18 (D) X − 30I is an invertible matrix

response_choices: (A) If X [1 1 1]T = α [1 1 1]T, then α = 30, (B) X is a symmetric matrix, (C) The sum of diagonal entries of X is 18, (D) X − 30I is an invertible matrix

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question_subject: mathematics


question: Let x ∈ ℝ and let P = [1 1 1, 0 2 2, 0 0 3], Q = [2 x x, 0 4 0] and R = PQP^(-1). Then which of the following options is/are correct?

response_choices: (A) There exists a real number x such that PQ = QP, (B) det R = det [0 4 0] + 8, for all x ∈ ℝ, (C) For x = 0, if R [a] = 6 [a], then a + b = 5, (D) For x = 1, there exists a unit vector αi + βj + γk for which R [β] = [0]

answer: (B)

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question_subject: mathematics


question: For non-negative integers n, let f(n) = (Σ from k=0 to n) sin((k+1)π/(n+2)π) sin((k+2)/(n+2)π) / (Σ from k=0 to n) sin^2((k+1)π/(n+2)π). Assuming cos^(-1)x takes values in [0,π], which of the following options is/are correct?

answer: (B),(D)

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question_subject: mathematics


question: Let f: R → R be a function. We say that f has PROPERTY 1 if lim [f(h) - f(0)]/sqrt(|h|) exists and is finite, and PROPERTY 2 if lim [f(h) - f(0)]/h^2 exists and is finite. Then which of the following options is/are correct?

response_choices: (A) f(x) = |x| has PROPERTY 1, (B) f(x) = x^(2/3) has PROPERTY 1, (C) f(x) = x|x| has PROPERTY 2, (D) f(x) = sin x has PROPERTY 2

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question_subject: mathematics


question: Let f(x) = sin πx/x^2, x > 0. Let x1 < x2 < x3 < ... < xn < ... be all the points of local maximum of f and y1 < y2 < y3 < ... < yn < ... be all the points of local minimum of f. Then which of the following options is/are correct?

response_choices: (A) x1 < y1, (B) xn+1 - xn > 2 for every n, (C) xn ∈ (2n,2n + 1) for every n, (D) |xn - yn| > 1 for every n

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question_subject: mathematics


question: For a ∈ R, |a| > 1, let lim (1 + sqrt(2) + ... + sqrt(n))/(n^(7/3) ((an+1)^2 + (an+2)^2 + ... + (an+n)^2)) = 54. Then the possible value(s) of a is/are

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question_subject: mathematics


question: Let f: ℝ → ℝ be given by f(x) = (x - 1)(x - 2)(x - 5). Define F(x) = ∫[x to ∞] f(t) dt, x > 0. Then which of the following options is/are correct?

response_choices:
(A) F has a local minimum at x = 1
(B) F has a local maximum at x = 2
(C) F has two local maxima and one local minimum in (0, ∞)
(D) F(x) ≠ 0 for all x ∈ (0, 5)

answer: C

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question_subject: mathematics


question: Three lines l1: r⃗ = λi⃗, λ ∈ ℝ, l2: r⃗ = k⃗ + μj⃗, μ ∈ ℝ and l3: r⃗ = i⃗ + j⃗ + νk⃗, ν ∈ ℝ are given. For which point(s) Q on l2 can we find a point P on l1 and a point R on l3 so that P, Q and R are collinear?

response_choices:
(A) k⃗ - 1/2j⃗
(B) k⃗
(C) k⃗ + 1/2j⃗
(D) k⃗ + j⃗

answer: C

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question_subject: mathematics


question: Suppose det [ (Sum from k=0 to n) k, (Sum from k=0 to n) nCk k^2 ] = 0 [ (Sum from k=0 to n) nCk k, (Sum from k=0 to n) nCk 3k ] holds for some positive integer n. Then (Sum from k=0 to n) nCk / (k + 1) equals _

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question_subject: mathematics


question: Five persons A, B, C, D and E are seated in a circular arrangement. If each of them is given a hat of one of the three colours red, blue and green, then the number of ways of distributing the hats such that the persons seated in adjacent seats get different coloured hats is _

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question_subject: mathematics


question: Let |X| denote the number of elements in a set X. Let S = {1, 2, 3, 4, 5, 6} be a sample space, where each element is equally likely to occur. If A and B are independent events associated with S, then the number of ordered pairs (A, B) such that 1 ≤ |B| < |A|, equals _

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question_subject: mathematics


question: The value of sec^-1 (1/4 (sum from k=0 to 10) sec (2π/12 + kπ/2) + sec (7π/12 + (k + 1)π/2)) in the interval [-π 3π/4, π/4] equals ___

response_choices: No response choices provided

answer: No answer provided

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question_subject: mathematics


question: The value of the integral from 0 to π/2 of (3 √(cos θ)) / ((√(cos θ) + √(sin θ))^2) dθ equals ___

response_choices: No response choices provided

answer: No answer provided

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question_subject: mathematics


question: Let â = 2î + j - k̂ and b̂ = î + 2j + k be two vectors. Consider a vector ĉ = aâ + βb̂, α, β ∈ ℝ. If the projection of ĉ on the vector (â + b̂) is 3√2, then the minimum value of (ĉ - (â × b̂)) ⋅ ĉ equals___

response_choices: No response choices provided

answer: No answer provided

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question_subject: mathematics


question: Which of the following is the only CORRECT combination?

answer: B

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question_subject: mathematics


question: Which of the following is the only CORRECT combination?

answer: B

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question_subject: mathematics


question: Which of the following is the only CORRECT combination?

answer: (C) (II), (Q)

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question_subject: mathematics


question: Which of the following is the only INCORRECT combination?

answer: (D) (IV), (S)

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question_subject: mathematics