(Held on Sunday 28th August, 2022)

PAPER-2

PHYSICS

SECTION-1 :  (Maximum Marks : 24)

• This section contains EIGHT (08) questions.

• The answer to each question is a SINGLE DIGIT INTEGER ranging from 0 to 9, BOTH INCLUSIVE.

• For each question, enter the correct integer corresponding to the answer using the mouse and the on-screen virtual numeric keypad in the place designated to enter the answer.

• Answer to each question will be evaluated according to the following marking scheme:

Full marks                   : +3 If ONLY the correct integer is entered;

Zero Marks                  : 0 If the question is unanswered;

Negative Marks           : −1 In all other cases.

1. The particle of mass 1 kg is subjected to a force which depends on the position as  with k = 1 kgs2. At time t = 0, the particle’s position  and its velocity  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 vy – y vx) is ______ m2s1.

2. In a radioactive decay chain reaction,  nucleus decays into  The ratio of the number of α to number of β particles emitted in this process is _____.

3. Two resistances R1 = XΩ and R2 = 1Ω 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 ______.

4. 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  where ∈0 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 _______.

5. Consider a configuration of n identical units, each consisting of three layers. The first layer is a column of air of height h = 1/3 cm, and the second and third layers are of equal thickness  and refractive indices  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  as shown in the figure. The value of n is _______.

6. 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  The value of n is ______.

7. 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 k1 = 0.009 Nm1 when the length of the spring l > l0 and is k2 = 0.016 Nm1 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 = (nπ) s, then the integer closest to n is ______.

8. 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 s1 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 image is instantaneously at rest with respect to the laboratory frame, and the object forms a real image. The magnitude of Vm is _____ cm s1.

SECTION-2: (Maximum Marks : 24)

• This section contains SIX (06) questions.

• Each question has FOUR options (A), (B), (C) and (D). ONE OR MORE THAN ONE of these four option(s) is (are) correct answer(s).

• For each question, choose the option(s) corresponding to (all) the correct answer(s).

• Answer to each question will be evaluated according to the following marking scheme:

Full Marks     : +4 ONLY in (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;

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 unanswered;

Negative Marks : −2 In all other cases.

9. 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 = 2k/r. Assume that dimensions are taken care of. All physical quantities are in their SI units.

Which of the following statement(s) is(are) correct?

(A)  If  then the electric field is zero everywhere outside B.

(B)  If rB = 3/2, then the electric potential just outside B is k/∈0.

(C)  If rB = 2, then the total charge of the configuration is 15πk.

(D)  If rB = 5/2, then the magnitude of the electric field just outside B is 13πk/∈0.

10. 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.

Q1 and 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

11. 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 Pa1, the radius of the bubble is found to be r1 and the temperature of the enclosed gas is T1. When the atmospheric pressure is Pa2, 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

(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

(D)  If the surface of the bubble is a perfect heat insulator, then

12. 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

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  Which of the following statement(s) is(are) correct?

(A)  For  the particle reaches the origin.

(B)  For  the particle reaches the origin.

(C)  For  the particle returns back to z = z0.

(D)  For β > 1 and z0 > 0, the particle always reaches the origin.

13. 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.

Which of the following statement(s) is(are) correct?

(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.

14. 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 : (1/3)6 = 0.5, ln 2 = 0.7].

Which of the following statement(s) is(are) correct?

(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.

SECTION-3: (Maximum Marks : 12)

• This section contains FOUR (04) questions.

• Each question has FOUR options (A), (B), (C) and (D). ONLY ONE of these four options is the correct answer.

• For each question, choose the option corresponding to the correct answer.

• Answer to each question will be evaluated according to the following marking scheme:

Full Marks            : +3 If ONLY the correct option is chosen:

Zero Marks           : 0 If none of the options is chosen (i.e. the question is unanswered);

Negative Marks    : −1 In all other cases.

15. 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 to 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 nMωR2 with respect to the center of A. Which of the following is the value of n?

(A)  2

(B)  5

(C)  7/2

(D)  9/2

16. 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?

(A)  1.72 × 107 m, 1.20 eV

(B)  1.72 × 107 m, 5.60 eV

(C)  3.78 × 107 m, 5.60 eV

(D)  3.78 × 107 m, 1.20 eV

17. Area of the cross-section of a wire is measured using a screw gauge. The pitch of the main scale is 0.5 mm. The circular scale has 100 divisions and for one full rotation of the circular scale, the main scale shifts by two divisions. The measured readings are listed below.

What are the diameter and cross-sectional area of the wire measured using the screw gauge?

(A)  2.22 ± 0.02 mm, π(1.23 ± 0.02) mm2

(B)  2.22 ± 0.01 mm, π(1.23 ± 0.01) mm2

(C)  2.14 ± 0.02 mm, π(1.14 ± 0.02) mm2

(D)  2.14 ± 0.01 mm, π(1.23 ± 0.01) mm2

18. Which one of the following options represents the magnetic field  at O due to the current flowing in the given wire segments lying on the xy plane?

CHEMISTRY

SECTION-1 :  (Maximum Marks : 24)

• This section contains EIGHT (08) questions.

• The answer to each question is a SINGLE DIGIT INTEGER ranging from 0 to 9, BOTH INCLUSIVE.

• For each question, enter the correct integer corresponding to the answer using the mouse and the on-screen virtual numeric keypad in the place designated to enter the answer.

• Answer to each question will be evaluated according to the following marking scheme:

Full marks                   : +3 If ONLY the correct integer is entered;

Zero Marks                  : 0 If the question is unanswered;

Negative Marks           : −1 In all other cases.

1. 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–Y M (expressed in scientific notation). The value of Y is _________.

[Given: Solubility product of PbSO4 (Ksp) = 1.6 × 10–8. For H2SO4, Ka1 is very large and  Ka2 = 1.2 × 10–2]

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

3. Consider the strong electrolytes ZmXn, UmYp and VmXn. Limiting molar conductivity (⋀0) of UmYp and VmXn are 250 and 440 S cm2 mol–1, respectively. The value of (m + n + p) is _______.

Given:

The plot of molar conductivity (⋀) of ZmXn vs c1/2 is given below.

4. 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 _______.

5. 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 _______.

6. The number of isomeric tetraenes (NOT containing sp-hybridized carbon atoms) that can be formed from the following reaction sequence is ________.

7. The number of –CH2-(methylene) groups in the product formed from the following reaction sequence is ________.

8. The total number of chiral molecules formed from one molecule of P on complete ozonolysis (O3, Zn/H2O) is ________.

SECTION-2: (Maximum Marks : 24)

• This section contains SIX (06) questions.

• Each question has FOUR options (A), (B), (C) and (D). ONE OR MORE THAN ONE of these four option(s) is (are) correct answer(s).

• For each question, choose the option(s) corresponding to (all) the correct answer(s).

• Answer to each question will be evaluated according to the following marking scheme:

Full Marks  : +4 ONLY in (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;

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 unanswered;

Negative Marks : −2 In all other cases.

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

(A)  If empirical formula of compound 3 is P3Q4, then the empirical formula of compound 2 is P3Q5.

(B)  If empirical formula of compound 3 is P3Q2 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 P5Q4.

(D)  If atomic weight of P and Q are 70 and 35, respectively, then the empirical formula of compound 1 is P2Q.

10. The correct option(s) about entropy (S) is(are)

[R = gas constant, F = Faraday constant, T = Temperature]

(A)  For the reaction, M(s) + 2H+(aq) → H2(g) + M2+(aq), if  then the entropy change of the reaction is R (assume that entropy and internal energy changes are temperature independent).

(B)  The cell reaction, Pt(s) | H2(g, 1bar) | H+(aq, 0.01M) || H+(aq, 0.1M) | H2(g, 1bar) | Pt(s), is an entropy driven process.

(C)  For racemization of an optically active compound, ∆S > 0.

(D)  ∆S > 0, for [Ni(H2O)6]2+ + 3 en → [Ni(en)3]2+ + 6H2O (where en = ethylenediamine).

11. The compound(s) which react(s) with NH3 to give boron nitride (BN) is(are)

(A)  B

(B)  B2H6

(C)  B2O3

(D)  HBF4

12. The correct option(s) related to the extraction of iron from its ore in the blast furnace operating in the temperature range 900 – 1500 K is(are)

(A)  Limestone is used to remove silicate impurity.

(B)  Pig iron obtained from blast furnace contains about 4% carbon.

(C)  Coke (C) converts CO2 to CO.

(D)  Exhaust gases consist of NO2 and CO.

13. Considering the following reaction sequence, the correct statement(s) is(are)

(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.

14. Among the following, the correct statement(s) about polymers is(are)

(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.

SECTION-3: (Maximum Marks : 12)

• This section contains FOUR (04) questions.

• Each question has FOUR options (A), (B), (C) and (D). ONLY ONE of these four options is the correct answer.

• For each question, choose the option corresponding to the correct answer.

• Answer to each question will be evaluated according to the following marking scheme:

Full Marks            : +3 If ONLY the correct option is chosen:

Zero Marks           : 0 If none of the options is chosen (i.e. the question is unanswered);

Negative Marks    : −1 In all other cases.

15. 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

(A)  25

(B)  35

(C)  55

(D)  75

16. The reaction of HClO3 with HCl gives a paramagnetic gas, which upon reaction with O3 produces

(A)  Cl2O

(B)  ClO2

(C)  Cl2O6

(D)  Cl2O7

17. The reaction 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

(A)  PbCl2

(B)  PbCl4

(C)  [PbCl4]2

(D)  [PbCl6]2

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

MATHEMATICS

SECTION-1 :  (Maximum Marks : 24)

• This section contains EIGHT (08) questions.

• The answer to each question is a SINGLE DIGIT INTEGER ranging from 0 to 9, BOTH INCLUSIVE.

• For each question, enter the correct integer corresponding to the answer using the mouse and the on-screen virtual numeric keypad in the place designated to enter the answer.

• Answer to each question will be evaluated according to the following marking scheme:

Full marks                   : +3 If ONLY the correct integer is entered;

Zero Marks                  : 0 If the question is unanswered;

Negative Marks           : −1 In all other cases.

1. Let α and β be real numbers such that  If  then the greatest integer less than or equal to  is _______.

2. If y(x) is the solution of the differential equation xdy – (y2 – 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 10y(√2) is ______.

3. The greatest integer less than or equal to  is ______.

4. The product of all positive real values of x satisfying the equation  is ___________.

5. If

Then the value of 6β is ______.

6. Let β be a real number. Consider the matrix

If A7 – (β – 1)A6 – βA5 is a singular matrix, then the value of 9β is _______.

7. Consider the hyperbola  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= S1 Then the greatest integer less than or equal to  is _______.

8. Consider the functions f , g : ℝ → ℝ defined by

and

If α is the area of the region  then the value of 9α is ______.

SECTION-2: (Maximum Marks : 24)

• This section contains SIX (06) questions.

• Each question has FOUR options (A), (B), (C) and (D). ONE OR MORE THAN ONE of these four option(s) is (are) correct answer(s).

• For each question, choose the option(s) corresponding to (all) the correct answer(s).

• Answer to each question will be evaluated according to the following marking scheme:

Full Marks  : +4 ONLY in (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;

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 unanswered;

Negative Marks : −2 In all other cases.

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

(A)  (0, √2)

(B)  (1, 2)

(C)  (√2, 3)

(D)  (2√2, 3√2)

10. Let

Let g : [0, 1] → ℝ be the function defined by g(x) = 2αx + 2α(1 – x)

Then, which of the following statements is/are TRUE?

(A)  The minimum value of g(x) is 27/6

(B)  The maximum value of g(x) is 1 + 21/3

(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

11. Let  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  are integers, then which of the following is/are possible value(s) of |z|?

12. 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 Then, which of the following statements is/are TRUE?

(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

13. Let  be the unit vectors along the three positive coordinate axes. Let

be three vectors such that b2b3 > 0,  and

Then, which of the following is/are TRUE?

14. For x ∈ ℝ, let the function y(x) be the solution of the differential equation

Then, which of the following statements is/are TRUE?

(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 many points.

(D)  y(x) is a periodic function

SECTION-3: (Maximum Marks : 12)

• This section contains FOUR (04) questions.

• Each question has FOUR options (A), (B), (C) and (D). ONLY ONE of these four options is the correct answer.

• For each question, choose the option corresponding to the correct answer.

• Answer to each question will be evaluated according to the following marking scheme:

Full Marks            : +3 If ONLY the correct option is chosen:

Zero Marks           : 0 If none of the options is chosen (i.e. the question is unanswered);

Negative Marks    : −1 In all other cases.

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

(A)  21816

(B)  85536

(C)  12096

(D)  156816

16. If  then which of the following matrices is equal to M2022?

17. 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 ꞌat least one of the chosen balls is greenꞌ has happened, is equal to

(A)  15/256

(B)  3/16

(C)  5/52

(D)  1/8

18. For positive integer n, define

Then, the value of  is equal to

(Held on Sunday 28th August, 2022)

Paper-1

PHYSICS

SECTION-1 : (Maximum Marks : 24)

• This section contains EIGHT (08) questions

• The answer to each question is a NUMERICAL VALUE.

• For each question, enter the correct numerical value of the answer using the mouse and the on-screen virtual numerical keypad in the place designated to enter the answer. If the numerical value has more than two decimal places, truncate/round-off the value to TWO decimal places.

• Answer to each question will be evaluated according to the following marking scheme:

Full Marks       : +3 ONLY if the correct numerical value is entered;

Zero Marks      : 0 In all other cases.

1. Two spherical stars A and B have densities ρA and ρB, respectively. A and B have the same radius, and their masses MA and MB are related by MB = 2MA. Due to an interaction process, star A loses some of its mass, so that its radius is halved, while its spherical shape is retained, and its density remains ρA. The entire mass lost by A is deposited as a thick spherical shell on B with the density of the shell being ρA. If νA and νB are the escape velocities from A and B after the interaction process,  The value of n is _______

2. The minimum kinetic energy needed by an alpha particle to cause the nuclear reaction  in a laboratory frame is n (in MeV). Assume that  is at rest in the laboratory frame. The masses of  can be taken to be 16.006 u, 4.003 u, 1.008 u and 19.003 u, respectively, where 1 u = 930 MeVc2. The value of n is _______.

3. In the following circuit C1­ = 12 μF, C2 = C3 = 4 μF and C4 = C5 = 2 μ The Charge stored in C­3 is ______ μC.

4. A rod of length 2 cm makes an angle 2π/3 rad with the principal axis of a thin convex lens. The lens has a focal length of 10 cm and is placed at a distance of 40/3 cm from the object as shown in the figure. The height of the image is 30√3/13 cm and the angle made by it with respect to the principal axis is α The value of α is π/n rad, where n is ______.

5. A time t = 0, a disk of radius 1 m starts to roll without slipping on a horizontal plane with an angular acceleration of α = 2/3 rad s2. A small stone is stuck to the disk. At t = 0, it is at the contact point of the disk and the plane. Later, at time t = √π s, the stone detaches itself and flies off tangentially from the disk. The maximum height (in m) reached by the stone measured from the plane is  The value of x is ______. [Take g = 10 ms2]

6. A solid sphere of mass 1 kg and radius 1 m rolls without slipping on a fixed inclined plane with an angle of inclination θ = 30° from the horizontal. Two forces of magnitude 1 N each, parallel to the incline, act on the sphere, both at distance r = 0.5 m from the centre of the sphere, as shown in the figure. The acceleration of the sphere down the plane is ______ ms–2. (Take g = 10 m s–2.)

7. Consider an LC circuit, with inductance L = 0.1 H and capacitance C = 10–3 F, kept on a plane. The area of the circuit is 1 m2. It is placed in a constant magnetic field of strength B0 which is perpendicular to the plane of the circuit. At time t = 0, the magnetic field strength starts increasing linearly as B = B0 + βt with β = 0.04 Ts–1. The maximum magnitude of the current in the circuit is ____ mA.

8. A projectile is fired from horizontal ground with speed v and projection angle θ. When the acceleration due to gravity is g, the range of the projectile is d. If at the highest point in its trajectory, the projectile enters a different region where the effective acceleration due to gravity is  then the new range is dꞌ = nd. The value of n is ________.

SECTION-2 : (Maximum Marks : 24)

• This section contains SIX (06) questions.

• Each question has FOUR options (A), (B), (C) and (D). ONE OR MORE THAN ONE of these four option(s) is (are) correct answer(s).

• For each question, choose the option(s) corresponding to (all) the correct answer(s).

• Answer to each question will be evaluated according to the following marking scheme:

Full Marks          : +4 ONLY if (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;

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.

9. A medium having dielectric constant K >1 fills the space between the plates of a parallel plate capacitor. The plates have large area, and the distance between them is d. The capacitor is connected to a battery of voltage V. as shown in Figure (a). Now, both the plates are moved by a distance of d/2 from their original positions, as shown in Figure (b).

In the process of going from the configuration depicted in Figure (a) to that in Figure (b), which of the following statement(s) is(are) correct?

(A)  The electric field inside the dielectric material is reduced by a factor of 2K.

(B)  The capacitance is decreased by a factor of 1/K+1.

(C)  The voltage between the capacitor plates is increased by a factor of (K + 1).

(D)  The work done in the process DOES NOT depend on the presence of the dielectric material.

10. The figure shows a circuit having eight resistances of 1 Ω each, labelled R1 to R8. And two ideal batteries with voltages ε1 = 12 V and ε2 = 6 V.

Which of the following statement(s) is(are) correct?

(A)  The magnitude of current flowing through R1 is 7.2 A.

(B)  The magnitude of current flowing through R2 is 1.2 A.

(C)  The magnitude of current flowing through R3 is 4.8 A.

(D)  The magnitude of current flowing through R5 is 2.4 A.

11. An ideal gas of density ρ = 0.2 kg m–3 enters a chimney of height h at the rate of α = 0.8 kg s–1 from its lower end, and escapes through the upper end as shown in the figure. The cross-sectional area of the lower end is A1 = 0.1 m2 and the upper end is A2 = 0.4 m2. The pressure and the temperature of the gas at the lower end are 600 Pa and 300 K, respectively, while its temperature at the upper end is 150 K. The chimney is heat insulated so that the gas undergoes adiabatic expansion. Take g = 10 ms–2 and the ratio of specific heats of the gas γ = 2. Ignore atmospheric pressure.

Which of the following statement(s) is(are) correct?

(A)  The pressure of the gas at the upper end of the chimney is 300 Pa.

(B)  The velocity of the gas at the lower end of the chimney is 40 ms–1 and at the upper end is        20 ms–1.

(C)  The height of the chimney is 590 m.

(D)  The density of the gas at the upper end is 0.05 kg m–3.

12. Three plane mirrors form an equilateral triangle with each side of length L. There is a small hole at a distance l > 0 from one of the corners as shown in the figure. A ray of light is passed through the hole at an angle θ and can only come out through the same hole. The cross section of the mirror configuration and the ray of light lie on the same plane.

Which of the following statement(s) is(are) correct?

(A)  The ray of light will come out for θ = 30°, for 0 < l < L.

(B)  There is an angle for l = L/2 at which the ray of light will come out after two reflections.

(C)  The ray of light will NEVER come out for θ = 60°, and l = L/3.

(D)  The ray of light will come out for θ = 60°, and 0 < l < L/2 after six reflections.

13. Six charges are placed around a regular hexagon of side length a as shown in the figure. Five of them have charge q, and the remaining one has charge x. The perpendicular from each charge to the nearest hexagon side passes through the centre O of the hexagon and is bisected by the side.

Which of the following statement(s) is(are) correct in SI units?

(A)  When x = q. the magnitude of the electric field at O is zero.

(B)  When x = −q, the magnitude of the electric field at O is

(C)  When x = 2q, the potential at O is

(D)  When x = −3q, the potential at O is

14. The binding energy of nucleons in a nucleus can be affected by the pairwise Coulomb repulsion. Assume that all nucleons are uniformly distributed inside the nucleus. Let the binding energy of a proton be Ebp and the binding energy of a neutron be Ebn in the nucleus.

Which of the following statement(s) is(are) correct?

(A)  Ebp – Ebn is proportional to Z(Z – 1) where Z is the atomic number of the nucleus.

(B)  Ebp – Ebn proportional to A1/3 where A is the mass number of the nucleus.

(C)  Ebp – Ebn is positive

(D)  Ebp increases if the nucleus undergoes a beta decay emitting a positron.

SECTION-3 : (Maximum Marks : 12)

• This section contains FOUR(04) Matching List Sets.

• Each set has ONE Multiple Choice Question.

• Each set has TWO lists : List-I and List-II.

List-I has Four entries (I), (II), (III) and (IV) and List-II has Five entries (P), (Q), (R), (S) and (T).

FOUR options are given in each Multiple Choice Question based on List-I and List-II and ONLY ONE of these four options satisfies the condition asked in the Multiple Choice Question.

• Answer to each question will be evaluated according to the following marking scheme:

Full Marks            : +3 ONLY if the option corresponding to the correct combination is chosen;

Zero Marks           :   0 If none of the options is chosen (i.e. the question is unanswered);

Negative Marks    : –1  In all other cases.

15. A small circular loop of area A and resistance R is fixed on a horizontal xy-plane with the centre of the loop always on the axis  of a long solenoid. The solenoid has m turns per unit length and carries current I counter clockwise as shown in the figure. The magnetic field due to the solenoid is in direction. List-I gives time dependences of  in terms of a constant angular frequency ω.  List-II gives the torques experienced by the circular loop at time

Which one of the following options is correct?

(A)  I→Q, II→P, III→S, IV→T

(B)  I→S, II→T, III→Q, IV→P

(C)  I→Q, II→P, III→S, IV→R

(D)  I→T, II→Q, III→P, IV→R

16. List I describes four systems, each with two particles A and B in relative motion as shown in figure. List II gives possible magnitudes of then relative velocities (in ms1) at time

Which one of the following options is correct?

(A)  I→R, II→T, III→P, IV→S

(B)  I→S, II→P, III→Q, IV→R

(C)  I→S, II→T, III→P, IV→R

(D)  I→T, II→P, III→R, IV→S

17. List I describes thermodynamic processes in four different systems. List II gives the magnitudes (either exactly or as a close approximation) of possible changes in the internal energy of the system due to the process.

Which one of the following options is correct?

(A)  I→T, II→R, III→S, IV→Q

(B)  I→S, II→P, III→T, IV→P

(C)  I→P, II→R, III→T, IV→Q

(D)  I→Q, II→R, III→S, IV→T

18. List I contains four combinations of two lenses (1 and 2) whose focal lengths (in cm) are indicated in the figures. In all cases, the object is placed 20 cm from the first lens on the left, and the distance between the two lenses is 5 cm. List II contains the positions of the final images.

Which one of the following options is correct?

(A)  I→P, II→R, III→Q, IV→T

(B)  I→Q, II→P, III→T, IV→S

(C)  I→P, II→T, III→R, IV→Q

(D)  I→T, II→S, III→Q, IV→R

CHEMISTRY

SECTION-1 : (Maximum Marks : 24)

• This section contains EIGHT (08) questions

• The answer to each question is a NUMERICAL VALUE.

• For each question, enter the correct numerical value of the answer using the mouse and the on-screen virtual numerical keypad in the place designated to enter the answer. If the numerical value has more than two decimal places, truncate/round-off the value to TWO decimal places.

• Answer to each question will be evaluated according to the following marking scheme:

Full Marks       : +3 ONLY if the correct numerical value is entered;

Zero Marks      : 0 In all other cases.

1. 2 mol of Hg(g) is combusted in a fixed volume bomb calorimeter with excess of O2 at 298 K and 1 atm into HgO(s). During the reaction, temperature increases from 298.0 K to 312.8 K. If heat capacity of the bomb calorimeter and enthalpy of formation of Hg(g) are 20.00 kJ K–1 and  32 kJ mol–1 at 298 K, respectively, the calculated standard molar enthalpy of formation of HgO(s) at 298 K is X kJ mol–1. The value of |X| is ______.

[Given : Gas constant R = 8.3 J K–1 mol–1]

2. The reduction potential (E0, in V) of 4 MnO4(aq)/Mn(s) is ______.

3. A solution is prepared by mixing 0.01 mol each of H2CO3, NaHCO3, Na2CO3, and NaOH in 100 mL of water. pH of the resulting solution is ______.

[Given : pKa1 and pKa2 of H2CO3 are 6.37 and 10.32, respectively ; log 2 = 0.30]

4. The treatment of an aqueous solution of 3.74 g of Cu(NO3)2 with excess KI results in a brown solution along with the formation of a precipitate. Passing H2S through this brown solution gives another precipitate X. The amount of X (in g) is ______.

[Given : Atomic mass of H = 1, N = 14, O = 16, S = 32, K = 39, Cu = 63, I = 127]

5. Dissolving 1.24 g of white phosphorous in boiling NaOH solution in an inert atmosphere gives a gas Q. The amount of CuSO4 (in g) required to completely consume the gas Q is ______.

[Given : Atomic mass of H = 1, O = 16, Na = 23, P = 31, S = 32, Cu = 63]

6. Consider the following reaction

On estimation of bromine in 1.00 g of R using Carius method, the amount of AgBr formed (in g) is ______.

[Given : Atomic mass of H = 1, C = 12, O = 16, P = 31, Br = 80, Ag = 108]

7. The weight percentage of hydrogen in Q, formed in the following reaction sequence, is ______.

[Given : Atomic mass of H = 1, C = 12, N = 14, O = 16, S = 32, Cl = 35]

8. If the reaction sequence given below is carried out with 15 moles of acetylene, the amount of the product D formed (in g) is ______.

The yields of A, B, C and D are given in parentheses.

[Given : Atomic mass of H = 1, C = 12, O = 16, Cl = 35]

SECTION-2 : (Maximum Marks : 24)

• This section contains SIX (06) questions.

• Each question has FOUR options (A), (B), (C) and (D). ONE OR MORE THAN ONE of these four option(s) is (are) correct answer(s).

• For each question, choose the option(s) corresponding to (all) the correct answer(s).

• Answer to each question will be evaluated according to the following marking scheme:

Full Marks         : +4 ONLY if (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;

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.

9. For diatomic molecules, the correct statement(s) about the molecular orbitals formed by the overlap to two 2pz orbitals is(are)

(A)  σ orbital has a total of two nodal planes.

(B)  σ* orbital has one node in the xz-plane containing the molecular axis.

(C)  π orbital has one node in the plane which is perpendicular to the molecular axis and goes         through the center of the molecule.

(D)  π* orbital has one node in the xy-plane containing the molecular axis.

10. The correct option(s) related to adsorption processes is(are)

(A)  Chemisorption results in a unimolecular layer.

(B)  The enthalpy change during physisorption is in the range of 100 to 140 kJ mol–1.

(C)  Chemisorption is an endothermic process.

(D)  Lowering the temperature favors physisorption processes.

11. The electrochemical extraction of aluminum from bauxite ore involves.

(A)  the reaction of Al2O3 with coke (C) at a temperature > 2500°C.

(B)  the neutralization of aluminate solution by passing CO2 gas to precipitate hydrated alumina         (Al2O3.3H2O)

(C)  the dissolution of Al2O3 in hot aqueous NaOH.

(D)  the electrolysis of Al2O3 mixed with Na3AlF6 to give Al and CO2.

12. The treatment of galena with HNO3 produces a gas that is

(A)  paramagnetic

(B)  bent in geometry

(C)  an acidic oxide

(D)  colorless

13. Considering the reaction sequence given below, the correct statement(s) is(are)

(A)  P can be reduced to a primary alcohol using NaBH4.

(B)  Treating P with conc. NH4OH solution followed acidification gives Q.

(C)  Treating Q with a solution of NaNO2 in aq. HCl liberates N2.

(D)  P is more acidic than CH3CH2COOH.

14. Consider the following reaction sequence,

the correct option(s) is(are)

SECTION-3 : (Maximum Marks : 12)

• This section contains FOUR(04) Matching List Sets.

• Each set has ONE Multiple Choice Question.

• Each set has TWO lists : List-I and List-II.

List-I has Four entries (I), (II), (III) and (IV) and List-II has Five entries (P), (Q), (R), (S) and (T).

FOUR options are given in each Multiple Choice Question based on List-I and List-II and ONLY ONE of these four options satisfies the condition asked in the Multiple Choice Question.

• Answer to each question will be evaluated according to the following marking scheme:

Full Marks            : +3 ONLY if the option corresponding to the correct combination is chosen;

Zero Marks           :   0 If none of the options is chosen (i.e. the question is unanswered);

Negative Marks    : –1  In all other cases.

15. Match the rate expressions in LIST-I for the decomposition of X with the corresponding profiles provided in LIST-II. Xs and k constants having appropriate units.

(A)  I → P; II → Q; III → S; IV → T

(B)  I → R; II → S; III → S; IV → T

(C)  I → P; II → Q; III → Q; IV → R

(D)  I → R; II → S; III → Q; IV → R

16. LIST-I contains compounds and LIST-II contains reaction

Match each compound in LIST – I with its formation reaction(s) in LIST-II, and choose the correct  option

(A)  I → Q; II → P; III → S; IV → R

(B)  I → T; II → P; III → Q; IV → R

(C)  I → T; II → R; III → Q; IV → P

(D)  I → Q; II → R; III → S; IV → P

17. LIST-I contains metal species and LIST-II contains their properties.

Metal each metal species in LIST-I with their properties in LIST-II, and choose the correct option

(A)  I → R, T; II → P, S; III → Q, T; IV → P, Q

(B)  I → R, S; II → P, T; III → P, Q; IV → Q, T

(C)  I → P, R; II → R, S; III → R, T; IV → P, T

(D)  I → Q, T; II → S, T; III → P, T; IV → Q, R

18. Match the compounds in LIST-I with the observation in LIST-II, and choose the correct option.

(A)  I → P, Q; II → S; III → Q, R; IV → P

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

(C)  I → Q, S; II → P, T; III → P; IV → S

(D)  I → P, S; II → T; III → Q, R; IV → P

MATHEMATICS

SECTION-1 : (Maximum Marks : 24)

• This section contains EIGHT (08) questions

• The answer to each question is a NUMERICAL VALUE.

• For each question, enter the correct numerical value of the answer using the mouse and the on-screen virtual numerical keypad in the place designated to enter the answer. If the numerical value has more than two decimal places, truncate/round-off the value to TWO decimal places.

• Answer to each question will be evaluated according to the following marking scheme:

Full Marks       : +3 ONLY if the correct numerical value is entered;

Zero Marks      : 0 In all other cases.

1. Considering only the principal values of the inverse trigonometric functions, the value of is ______.

2. Let α be a positive real number. Let f : ℝ → ℝ and g : (α, ∞) → ℝ be the functions defined by

Then the value of  is _____.

3. In a study about a pandemic, data of 900 persons was collected. It was flound

190 persons had symptom of fever,

220 persons had symptom of cough,

220 persons had symptom of breathing problem,

330 persons had symptom of fever or cough or both,

350 persons had symptom of cough or breathing problem or both,

340 persons had symptom of fever or breathing problem or both,

30 persons had all three symptoms (fever, cough and breathing problem).

If a person is chosen randomly from these 900 persons, then the probability that the person has at most one symptom is ______.

4. Let z be a complex number with non-zero imaginary part. If is a real number, then the value of |z|2 is _______.

5. Let  denote the complex conjugate of a complex number z and let i = √− In the set of complex numbers, the number of distinct roots of the equation  is _______.

6. Let l1, l2,…., l100 be consecutive terms of an arithmetic progression with common difference d1, and let w1, w2,…., w100 be consecutive terms of another arithmetic progression with common difference d2, where d1d2 = 10. For each i = 1, 2,….,100, let Ri be a rectangle with length li, width wi­ and area Ai. If A51 – A50 = 1000, then the value of A100 – A90 is ________.

7. The number of 4-digit integers in the closed interval [2022, 4482] formed by using the digits 0, 2, 3, 4, 6, 7 is _______.

8. Let ABC be the triangle with AB = 1, AC = 3 and ∠BAC = π/2. If a circle of radius r > 0 touches the sides AB, AC and also touches internally the circumcircle of the triangle ABC, then the value of r is _______.

SECTION-2 : (Maximum Marks : 24)

• This section contains SIX (06) questions.

• Each question has FOUR options (A), (B), (C) and (D). ONE OR MORE THAN ONE of these four option(s) is (are) correct answer(s).

• For each question, choose the option(s) corresponding to (all) the correct answer(s).

• Answer to each question will be evaluated according to the following marking scheme:

Full Marks         : +4 ONLY if (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;

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.

9. Consider the equation

Which of the following statements is/are TRUE?

(A)  NO a satisfies the above equation

(B)  An integer a satisfies the above equation

(C)  An irrational number a satisfies the above equation

(D)  More than one a satisfy the above equation

10. Let a1, a2, a3,… be an arithmetic progression with a1 = 7 and common difference 8. Let T1, T2, T3,… be such that T1 = 3 and Tn+1 – Tn = an for n ≥ Then, which of the following is/are TRUE?

(A)  T20 = 1604

(B)

(C)  T30 = 3454

(D)

11. Let P1 and P2 be two planes given by

P1 : 10x + 15y + 12z – 60 = 0.

P2 : −2x + 5y + 4z – 20 = 0.

Which of the following straight lines can be an edge of some tetrahedron whose two faces lie on P1 and P2?

12. Let S be the reflection of a point Q with respect to the plane given by

where t, p are real parameters and  are the unit vectors along the three positive coordinate axes. If the position vectors of Q and S are  and  respectively, then which of the following is/are TRUE?

(A)  3(α + β) = −101

(B)  3(β + γ) = −71

(C)  3(γ + α) = −86

(D)  3(α + β + γ) = −121

13. Consider the parabola y2 = 4x. Let S be the focus of the parabola. A pair of tangents drawn to the parabola from the point P = (−2, 1) meet the parabola at P1 and P2. Let Q1 and Q2 be points on the lines SP1 and SP2 respectively such that PQ1 is perpendicular to SP1 and PQ2 is perpendicular to SP2. Then, which of the following is/are TRUE?

(A)  SQ1 = 2

(B)

(C)  PQ1 = 3

(D)  SQ2 = 1

14. Let |M| denote the determinant of a square matrix M. Let  be the function defined by

Let p(x) be a quadratic polynomial whose roots are the maximum and minimum values of the function g(θ), and p(2) = 2 − √2. Then, which of the following is/are TRUE?

SECTION-3 : (Maximum Marks : 12)

• This section contains FOUR(04) Matching List Sets.

• Each set has ONE Multiple Choice Question.

• Each set has TWO lists : List-I and List-II.

• List-I has Four entries (I), (II), (III) and (IV) and List-II has Five entries (P), (Q), (R), (S) and (T).

• FOUR options are given in each Multiple Choice Question based on List-I and List-II and ONLY ONE of these four options satisfies the condition asked in the Multiple Choice Question.

• Answer to each question will be evaluated according to the following marking scheme:

Full Marks            : +3 ONLY if the option corresponding to the correct combination is chosen;

Zero Marks           :   0 If none of the options is chosen (i.e. the question is unanswered);

Negative Marks    : –1  In all other cases.

15. Consider the following lists:

The correct option is :

(A)  (I) → (P); (II) → (S); (III) → (P); (IV) → (S)

(B)  (I) → (P); (II) → (P); (III) → (T); (IV) → (R)

(C)  (I) → (Q); (II) → (P); (III) → (T); (IV) → (S)

(D)  (I) → (Q); (II) → (S); (III) → (P); (IV) → (R)

16. Two players, P1 and P2, play a game against each other. In every round of the game, each player rolls a fair die once, where the six faces of the die have six distinct numbers. Let x and y denote the readings on the die rolled by P1 and P2, respectively. If x > y, then P1 scores 5 points and P2 scores 0 points. If x = y, then each player scores 2 points. If x < y, then P1 scores 0 point and P2 scores 5 points. Let Xi and Yi be the total scores of P1 and P2, respectively, after playing the ith round.

The correct option is :

(A)  (I) → (Q); (II) → (R); (III) → (T); (IV) → (S)

(B)  (I) → (Q); (II) → (R); (III) → (T); (IV) → (T)

(C)  (I) → (P); (II) → (R); (III) → (Q); (IV) → (S)

(D)  (I) → (P); (II) → (R); (III) → (Q); (IV) → (T)

17. Let p, q, r be nonzero real numbers that are, respectively, the 10th, 100th and 1000th terms of a harmonic progression. Consider the system of linear equations

x + y + z = 1

10x + 100y + 1000z = 1

qr x + pr y + pq z = 0.

The correct option is :

(A)  (I) → (T); (II) → (R); (III) → (S); (IV) → (T)

(B)  (I) → (Q); (II) → (S); (III) → (S); (IV) → (R)

(C)  (I) → (Q); (II) → (R); (III) → (P); (IV) → (S)

(D)  (I) → (T); (II) → (S); (III) → (P); (IV) → (T)

18. Consider the ellipse  Let H(α, 0), 0 < α < 2, be a point. A straight line drawn through H parallel to the y-axis crosses the ellipse and its auxiliary circle at points E and F respectively, in the first quadrant. The tangent to the ellipse at the point E intersects the positive x-axis at a point G. Suppose the straight line joining F and the origin makes an angle ϕ with the positive x-axis.

The correct option is :

(A)  (I) → (R); (II) → (S); (III) → (Q); (IV) → (P)

(B)  (I) → (R); (II) → (T); (III) → (S); (IV) → (P)

(C)  (I) → (Q); (II) → (T); (III) → (S); (IV) → (P)

(D)  (I) → (Q); (II) → (S); (III) → (Q); (IV) → (P)