(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)

PHYSICS

SECTION-1

• 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)

• 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 If only (all) the correct option(s) is(are) chosen;

Partial Marks : +3 If all the four options are correct but ONLY three options are chosen;

Partial Marks : +2 If three or more options are correct but ONLY two options are chosen, both of which are correct;

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.

• For example, in a question, if (A), (B) and (D) are the ONLY three options corresponding to correct answers, then

choosing ONLY (A), (B) and (D) will get +4 marks;

choosing ONLY (A) and (B) will get +2 marks;

choosing ONLY (A) and (D) will get +2marks;

choosing ONLY (B) and (D) will get +2 marks;

choosing ONLY (A) will get +1 mark;

choosing ONLY (B) will get +1 mark;

choosing ONLY (D) will get +1 mark;

choosing no option(s) (i.e. the question is unanswered) will get 0 marks and

choosing any other option(s) will get −2 marks.

1. One end of a horizontal uniform beam of weight W and length L is hinged on a vertical wall at point O and its other end is supported by a light inextensible rope. The other end of the rope is fixed at point Q, at a height L above the hinge at point O. A block of weight αW is attached at point P of the beam, as shown in the figure (not to scale). The rope can sustain a maximum tension of (2√2) W Which of the following statement(s) is(are) correct?

(A) The vertical component of the reaction force at O does not depend on α

(B) The horizontal component of the reaction force at O is equal to W for α = 0.5

(C) The tension in the rope is 2W for α = 0.5

(D) The rope breaks if α > 1.5

2. A source, approaching with speed u towards the open end of a stationary pipe of length L, is emitting a sound of frequency fs. The farther end of the pipe is closed. The speed of sound in air is v and f0 is the fundamental frequency of the pipe. For which of the following combination(s) of u and fs, will the sound reaching the pipe lead to a resonance?

(A) u = 0.8v and fs = f0

(B) u = 0.8v and fs = 2f0

(C) u = 0.8v and fs = 0.5f0

(D) u = 0.5v and fs = 1.5f0

3. For a prism of prism angle θ = 60º, the refractive indices of the left half and the right half are, respectively, n1 and n2 (n2 ≥ n1) as shown in the figure. The angle of incidence is chosen such that the incident light rays will have minimum deviation if n1 = n2 = n = 1.5. For the case of unequal refractive indices, n1 = n and n2 = n + Δn (where Δn << n), the angle of emergence e = i + Δe. Which of the following statement(s) is(are) correct?

(A) The value of Δe (in radians) is greater than that of Δn

(B) Δe is proportional to Δn

(C) Δe lies between 2.0 and 3.0 milliradians if Δn = 2.8 × 10–3

(D) Δe lies between 1.0 and 1.6 milliradians if Δn = 2.8 × 10–3

4. A physical quantity  where  is electric field,  is magnetic field and μ0 is the permeability of free space. The dimensions of  are the same as the dimensions of which of the following quantity (ies)?

(A)

(B)

(C)

(D)

5. A heavy nucleus N, at rest, undergoes fission N → P + Q, where P and Q are two lighter nuclei. Let δ = MN – MP – MQ, where MP, MQ and MN are the masses of P, Q and N, respectively. EP and EQ are the kinetic energies of P and Q, respectively. The speeds of P and Q are VP and VQ, respectively. If c is the speed of light, which of the following statement(s) is(are) correct?

(A) EP + EQ = c2δ

(B)

(C)

(D) The magnitude of momentum for P as well as Q is

6. Two concentric circular loops, one of radius R and the other of radius 2R lie in the xy-plane with the origin as their common centre, as shown in the figure. The smaller loop carries current I1 in the anti-clockwise direction and the larger loop carries current I2 in the clockwise direction, with I2 > 2I1. denotes the magnetic field at a point (x, y) in the xy-plane. Which of the following statement(s) is(are) correct?

(A) is perpendicular to the xy-plane at any point in the plane

(B)  depends on x and y only through the radial distance

(C) is non-zero at all points for r < R

(D) points normally outward from the xy-plane for all the points between the two loops

SECTION-2

• This section contains THREE (03) question stems.

• There are TWO (02) questions corresponding to each question stem.

• The answer to each question is a NUMERICAL VALUE.

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

• 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 : +2 If ONLY the correct numerical value is entered at the designated place;

Zero Marks : 0 In all other cases.

Question Stem for Question Nos. 7 and 8

A soft plastic bottle, filled with water of density 1 gm/cc, carries an inverted glass test tube with some air (ideal gas) trapped as shown in the figure. The test tube has a mass of 5 gm, and it is made of a thick glass of density 2.5 gm/cc. Initially, the bottle is sealed at atmospheric pressure p0 = 105 Pa so that the volume of the trapped air is V0 = 3.3 cc. When the bottle is squeezed from outside at a constant temperature, the pressure inside rises and the volume of the trapped air reduces. It is found that the test tube begins to sink at pressure p0 + Δp without changing its orientation. At this pressure, the volume of the trapped air is V0 – ΔV.

Let ΔV = X cc and Δp = Y × 103 Pa.

7. The value of X is ______.

8. The value of Y is _____.

Question Stem for Question Nos. 9 and 10

A pendulum consists of a bob of mass m = 0.1 kg and a massless inextensible string of length L = 1.0 m. It is suspended from a fixed point at height H = 0.9 m above a frictionless horizontal floor. Initially, the bob of the pendulum is lying on the floor at rest vertically below the point of suspension. A horizontal impulse P = 0.2 kg-m/s is imparted to the bob at some instant. After the bob slides for some distance, the string becomes taut and the bob lifts off the floor. The magnitude of the angular momentum of the pendulum about the point of suspension just before the bob lifts off is J kg-m2/s. The kinetic energy of the pendulum just after the lift-off is K Joules.

9. The value of J is ________.

10. The value of K is __________.

Question Stem for Question Nos. 11 and 12

In a circuit, a metal filament lamp is connected in series with a capacitor of capacitance C μF across a 200 V, 50 Hz supply. The power consumed by the lamp is 500 W while the voltage drop across it is 100 V. Assume that there is no inductive load in the circuit. Take rms values of the voltages. The magnitude of the phase angle (in degrees) between the current and the supply voltage is ϕ. Assume, π√3 = 5.

11. The value of C is ___.

12. The value of ϕ is ___.

SECTION-3

• This section contains TWO (02) paragraphs. Based on each paragraph, there are TWO (02) 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.

Paragraph Question 13 and 14

A special metal S conducts electricity without any resistance. A closed wire loop, made of S, does not allow any change in flux through itself by inducing a suitable current to generate a compensating flux. The induced current in the loop cannot decay due to its zero resistance. This current gives rise to a magnetic moment which in turn repels the source of magnetic field or flux. Consider such a loop, of radius a, with its centre at the origin. A magnetic dipole of moment m is brought along the axis of this loop from infinity to a point at distance r (>> a) from the centre of the loop with its north pole always facing the loop, as shown in the figure below.

The magnitude of the magnetic field of a dipole m, at a point on its axis at distance r, is   where μ0 is the permeability of free space. The magnitude of the force between two magnetic dipoles with moments, m1 and m2, separated by a distance r on the common axis, with their north poles facing each other, is  where k is a constant of appropriate dimensions. The direction of this force is along the line joining the two dipoles.

13. When the dipole m is placed at a distance r from the centre of the loop (as shown in the figure), the current induced in the loop will be proportional to?

(A) m/r3

(B) m2/r2

(C) m/r2

(D) m2/r

14. The work done in bringing the dipole from infinity to a distance r from the centre of the loop by the given process is proportional to?

(A) m/r5

(B) m2/r5

(C) m2/r6

(D) m2/r7

Paragraph Question 15 and 16

A thermally insulating cylinder has a thermally insulating and frictionless movable partition in the middle, as shown in the figure below. On each side of the partition, there is one mole of an ideal gas, with specific heat at constant volume, CV = 2R. Here, R is the gas constant. Initially, each side has a volume V0 and temperature T0. The left side has an electric heater, which is turned on at very low power to transfer heat Q to the gas on the left side. As a result, the partition moves slowly towards the right, reducing the right side volume to V0/2. Consequently, the gas temperatures on the left and the right sides become TL and TR, respectively. Ignore the changes in the temperatures of the cylinder, heater and partition.

15. The value of TR/T0 is

(A) √2

(B) √3

(C) 2

(D) 3

16. The value of Q/RT0 is

(A) 4(2√2 +1)

(B) 4(2√2 −1)

(C) (5√2 +1)

(D) (5√2 −1)

SECTION-4

• This section contains THREE (03) questions.

• The answer to each question is a NON-NEGATIVE INTEGER.

• 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 : +4 If ONLY the correct integer is entered;

Zero Marks : 0 In all other cases.

17. In order to measure the internal resistance r1 of a cell of emf E, a meter bridge of wire resistance R0 = 50 Ω, a resistance R0/2, another cell of emf E/2 (internal resistance r) and a galvanometer G are used in a circuit, as shown in the figure. If the null point is found at l = 72 cm, then the value of r 1 = ___ Ω.

18. The distance between two stars of masses 3MS and 6MS is 9R. Here R is the mean distance between the centres of the Earth and the Sun, and MS is the mass of the Sun. The two stars orbit around their common centre of mass in circular orbits with period nT, where T is the period of Earth’s revolution around the Sun. The value of n is ___.

19. In a photoemission experiment, the maximum kinetic energies of photoelectrons from metals P, Q and R are EP, EQ and ER, respectively, and they are related by EP = 2EQ = 2ER. In this experiment, the same source of monochromatic light is used for metals P and Q while a different source of monochromatic light is used for metal R. The work functions for metals P, Q and R are 4.0 eV, 4.5 eV and 5.5 eV, respectively. The energy of the incident photon used for metal R, in eV, is _____.

CHEMISTRY

SECTION-1

• 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)

• 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 If only (all) the correct option(s) is(are) chosen;

Partial Marks : +3 If all the four options are correct but ONLY three options are chosen;

Partial Marks : +2 If three or more options are correct but ONLY two options are chosen, both of which are correct;

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.

• For example, in a question, if (A), (B) and (D) are the ONLY three options corresponding to correct answers, then

choosing ONLY (A), (B) and (D) will get +4 marks;

choosing ONLY (A) and (B) will get +2 marks;

choosing ONLY (A) and (D) will get +2marks;

choosing ONLY (B) and (D) will get +2 marks;

choosing ONLY (A) will get +1 mark;

choosing ONLY (B) will get +1 mark;

choosing ONLY (D) will get +1 mark;

choosing no option(s) (i.e. the question is unanswered) will get 0 marks and

choosing any other option(s) will get −2 marks.

1. The reaction sequence(s) that would lead to o-xylene as the major product is(are).

2. Correct option(s) for the following sequence of reactions is(are)

(A) Q = KNO2, W = LiAlH4

(B) R = benzenamine, V = KCN

(C) Q = AgNO2, R = phenylmethanamine

(D) W = LiAlH4, V = AgCN

3. For the following reaction;

the rate of reaction is  Two moles of X are mixed with one mole of Y to make 1.0 L of solution. At 50 s, 0.5 mole of Y is left in the reaction mixture. The correct statement(s) about the reaction is(are).

(Use: ln 2 = 0.693)

(A) The rate constant, k, of the reaction is 13.86 × 10-4 s-1.

(B) Half-life of X is 50 s.

(C) At 50 s,

(D) At 100 s,

4. Some standard electrode potentials at 298 K are given below:

Pb2+/Pb –0.13 V

Ni2+/Ni –0.24 V

C2+/Cd –0.40 V

Fe2+/Fe –0.44 V

To a solution containing 0.001 M of X2+ and 0.1 M of Y2+, the metal rods X and Y are inserted (at 298 K) and connected by a conducting wire. This resulted in the dissolution of X.

The correct combination(s) of X and Y, respectively, is(are)

(Given: Gas constant, R = 8.314 J K−1 mol−1, Faraday constant, F = 96500 C mol−1)

(A) Cd and Ni

(B) Cd and Fe

(C) Ni and Pb

(D) Ni and Fe

5. The pair(s) of complexes wherein both exhibit tetrahedral geometry is(are) (Note: py = pyridine, Given: Atomic numbers of Fe, Co, Ni and Cu are 26, 27, 28 and 29, respectively)

(A) [FeCl4] and [Fe(CO)4]2–

(B) [Co(CO)4] and [CoCl4]2–

(C) [Ni(CO)4] and [Ni(CN)4]2–

(D) [Cu(py)4]+ and [Cu(CN)4]3–

6. The correct statement(s) related to oxoacids of phosphorous is(are).

(A) Upon heating, H3PO3 undergoes a disproportionation reaction to produce H3PO4 and PH3.

(B) While H3PO3 can act as a reducing agent, H3PO4 cannot.

(C) H3PO3 is a monobasic acid.

(D) The H atom of the P-H bond in H3PO3 is not ionizable in water.

SECTION-2

• This section contains THREE (03) question stems.

• There are TWO (02) questions corresponding to each question stem.

• The answer to each question is a NUMERICAL VALUE.

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

• 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 : +2 If ONLY the correct numerical value is entered at the designated place;

Zero Marks : 0 In all other cases.

Question Stem for Question Nos. 7 and 8

Question Stem

At 298 K, the limiting molar conductivity of a weak monobasic acid is 4 × 102 S cm2 mol–1. At 298 K, for an aqueous solution of the acid, the degree of dissociation is a and the molar conductivity is y × 102 S cm2 mol–1. At 298 K, upon 20 times dilution with water, the molar conductivity of the solution becomes 3y × 102 S cm2 mol–1.

7. The value of α is _______.

8. The value of y is _______.

Question Stem for Question Nos. 9 and 10

Question Stem

The reaction of x g of Sn with HCl quantitatively produced a salt. The entire amount of the salt reacted with y g of nitrobenzene in the presence of the required amount of HCl to produce 1.29 g of an organic salt (quantitatively).

(Use Molar masses (in g mol–1) of H, C, N, O, Cl and Sn as 1, 12, 14, 16, 35 and 119, respectively).

9. The value of x is ________.

10. The value of y is ________.

Question Stem for Question Nos. 11 and 12

Question Stem

A sample (5.6 g) containing iron is completely dissolved in cold dilute HCl to prepare a 250 mL of solution. Titration of 25.0 mL of this solution requires 12.5 mL of 0.03 M KMnO4 solution to reach the endpoint. Number of moles of Fe2+ present in 250 mL solution is x × 10–2 (consider complete dissolution of FeCl2). The amount of iron present in the sample is y% by weight.

(Assume: KMnO4 reacts only with Fe2+ in the solution Use: Molar mass of iron as 56 g mol–1)

11. The value of x is ________.

12. The value of y is ________.

SECTION-3

• This section contains TWO (02) paragraphs. Based on each paragraph, there are TWO (02) 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.

Paragraph Question 13 and 14

Statement: The amount of energy required to break a bond is the same as the amount of energy released when the same bond is formed. In a gaseous state, the energy required for homolytic cleavage of a bond is called Bond Dissociation Energy (BDE) or Bond Strength. BDE is affected by the s-character of the bond and the stability of the radicals formed. Shorter bonds are typically stronger bonds. BDEs for some bonds are given below:

13. The correct match of the C-H bonds (shown in bold) in Column J with their BDE in Column K is;

(A) P – iii, Q – iv, R – ii, S – i

(B) P – i, Q – ii, R – iii, S – iv

(C) P – iii, Q – ii, R – i, S – iv

(D) P – ii, Q – i, R – iv, S – iii

14. For the following reaction,

the correct statement is

(A) Initiation step is exothermic with DH° = –58 kcal mol–1

(B) Propagation step involving ·CH3 formation is exothermic with DH° = –2 kcal mol–1

(C) Propagation step involving CH3Cl formation is endothermic with DH° = +27 kcal mol–1

(D) The reaction is exothermic with DH° = –25 kcal mol–1

Paragraph Question 15 and 16

The reaction of K3[Fe(CN)6] with freshly prepared FeSO4 solution produces a dark blue precipitate called Turnbull’s blue. The reaction of K4[Fe(CN)6] with the FeSO4 solution in the complete absence of air produces a white precipitate X, which turns blue in the air. Mixing the FeSO4 solution with NaNO3, followed by slow addition of concentrated H2SO4 through the side of the test tube produces a brown ring.

15. Precipitate X is

(A) Fe4[Fe(CN)6]3

(B) Fe4[Fe(CN)6]

(C) K2Fe[Fe(CN)6]

(D) KFe[Fe(CN)6]

16. Among the following, the brown ring is due to the formation of

(A) [Fe(NO)2(SO4)2]2–

(B) [Fe(NO)2(H2O)4]3+

(C) [Fe(NO)4(SO4)2]

(D) [Fe(NO)(H2O)5]2+

SECTION-4

• This section contains THREE (03) questions.

• The answer to each question is a NON-NEGATIVE INTEGER.

• 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 : +4 If ONLY the correct integer is entered;

Zero Marks : 0 In all other cases.

17. One mole of an ideal gas at 900 K, undergoes two reversible processes, I followed by II, as shown below. If the work done by the gas in the two processes are the same, the value of V3/V2 is _________.

(U: internal energy, S: entropy, p: pressure, V: volume, R: gas constant)

(Given: molar heat capacity at constant volume, C of the gas is  )

18. Consider a helium (He) atom that absorbs a photon of wavelength 330 nm. The change in the velocity (in cm s1) of the He atom after the photon absorption is_____.

(Assume: Momentum is conserved when the photon is absorbed.

Use: Planck constant = 6.6 × 1034 J s, Avogadro number = 6 × 1023 mol1, Molar mass of He = 4 g mol1)

19. Ozonolysis of ClO2 produces oxide of chlorine. The average oxidation state of chlorine in this oxide is ____.

MATHEMATICS

SECTION-1

• 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)

• 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 If only (all) the correct option(s) is(are) chosen;

Partial Marks : +3 If all the four options are correct but ONLY three options are chosen;

Partial Marks : +2 If three or more options are correct but ONLY two options are chosen, both of which are correct;

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.

• For example, in a question, if (A), (B) and (D) are the ONLY three options corresponding to correct answers, then

choosing ONLY (A), (B) and (D) will get +4 marks;

choosing ONLY (A) and (B) will get +2 marks;

choosing ONLY (A) and (D) will get +2marks;

choosing ONLY (B) and (D) will get +2 marks;

choosing ONLY (A) will get +1 mark;

choosing ONLY (B) will get +1 mark;

choosing ONLY (D) will get +1 mark;

choosing no option(s) (i.e. the question is unanswered) will get 0 marks and

choosing any other option(s) will get −2 marks.

1. Let;

S1 = {(i, j, k) : i, j, k ∈ {1,2,…,10}}

S2 = {(i, j) : 1 ≤ i < j + 2 ≤ 10,i, j ∈ {1, 2, …, 10}}

S3 = {(i, j, k, l) : 1 ≤ i < j < k < l, i, j, k, l ∈ {1, 2, …, 10}}

S4 = {(i, j, k, l ) : i, j, k and l are distinct elements in {1, 2, …, 10}}.

If the total number of elements in the set Sr is nr, r = 1, 2, 3, 4, then which of the following statements is (are) TRUE?

(A) n1 = 1000

(B) n2 = 44

(C) n3 = 220

(D)

2. Consider a triangle PQR having sides of lengths p, q, and r opposite to the angles P, Q, and R, respectively. Then which of the following statements is (are) TRUE?

(A)

(B)

(C)

(D) If p < q and p < r, then cos Q > p/r and cos R > p/q

3. Let  be a continuous function such that

Then which of the following statements is (are) TRUE?

(A) The equation f(x) − 3 cos 3x = 0 has at least one solution in (0, π/3)

(B) The equation f(x) − 3 sin 3x = −6/π has at least one solution in (0, π/3)

(C)

(D)

4. For any real numbers α and β, let yα, β (x), x ∈ R, be the solution of the differential equation  Let S = {yα,β (x), α, β ∈ R } . Then which of the following functions belong(s) to the set S?

(A)

(B)

(C)

(D)

5. Let O be the origin   and for some λ > 0. If  then which of the following statement is (are) TRUE?

(A) Projection of

(B) Area of the triangle OAB is 9/2

(C) Area of the triangle ABC is 9/2

(D) The acute angle between the diagonals of the parallelogram with adjacent sides

6. Let E denote the parabola y2 = 8x. Let P = (−2, 4), and let Q and Q’ be two distinct points on E such that the lines PQ and PQ’ are tangents to E. Let F be the focus of E. Then which of the following statements is (are) TRUE?

(A) The triangle PFQ is a right-angled triangle

(B) The triangle QPQ’ is a right-angled triangle

(C) The distance between P and F is 5√2

(D) F lies on the line joining Q and Q’

SECTION-2

• This section contains THREE (03) question stems.

• There are TWO (02) questions corresponding to each question stem.

• The answer to each question is a NUMERICAL VALUE.

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

• 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 : +2 If ONLY the correct numerical value is entered at the designated place;

Zero Marks : 0 In all other cases.

Question Stem for Question Nos. 7 and 8

Question Stem

Consider the region R = {(x,y) ∈ R×R : x ≥ 0 and y2 ≤ 4 – x. Let F be the family of all circles that are contained in R and have centres on the x-axis. Let C be the circle that has the largest radius among the circles in F. Let (α, β) be a point where circle C meets the curve y2 = 4 − x.

7. The radius of the circle C is ______.

8. The value of α is ________.

Question Stem for Question Nos. 9 and 10

Question Stem

Let f1 : (0, ∞) → R and f2 : (0, ∞) → R be defined by

and

f2(x) = 98(x – 1)50 – 600(x – 1)49 +2450, x> 0,

where, for any positive integer n and real numbers a1, a2, … an,  denotes the product of a1, a2, … an. Let mi and ni, respectively, denote the number of points of local minima and the number of points of local maxima of function f­i­, i = 1, 2, in the interval (0, ∞).

9. The value 2m1 + 3n1 + m1n1 is ______.

10. The value of 6m2 + 4n2 + 8m2n2 is_____.

Question Stem for Question Nos. 11 and 12

Question Stem

Let  i = 1, 2, and  be functions such that g1(x) = 1, g2(x) = |4x – π| and f(x) = sin2 x, for all

Define

11. The value of  is ______.

12. The value of  is _______.

SECTION-3

• This section contains TWO (02) paragraphs. Based on each paragraph, there are TWO (02) 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.

Paragraph Question 13 and 14

Let M = {(x, y) ∈ R × R : x2 + y2 ≤ r2}, where r > 0. Consider the geometric progression  Let S0 = 0 and, for n ≥ 1, let Sn denote the sum of the first n terms of this progression. For n ≥ 1 , let Cn denote the circle with center (Sn–1, 0) and radius an, and Dn denote the circle with center (Sn–1, Sn–1) and radius an.

13. Consider M with  Let k be the number of all those circles Cn that are inside M. Let l be the maximum possible number of circles among these k circles such that no two circles intersect. Then,

(A) k + 2l = 22

(B) 2k + l = 26

(C) 2k + 3l = 34

(D) 3k + 2l = 40

14. Consider M with The number of all those circles Dn that are inside M is;

(A) 198

(B) 199

(C) 200

(D) 201

Paragraph Question 15 and 16

Let ψ1 = [0, ∞) → R, ψ2 = [0, ∞) → R, f:[0, ∞) → R and g:[0, ∞) → R be functions such that f(0) = g(0) = 0,

15. Which of the following statements is TRUE?

(A)

(B) For every x > 1, there exists an α ∈ (1, x) such that ψ1(x) = 1 + α x

(C) For every x > 0, there exists a β ∈ (0, x) such that ψ2(x) = 2x (ψ1(β) −1)

(D) f is an increasing function on the interval [0, 3/2]

16. Which of the following statements is TRUE?

(A) ψ1(x) ≤ 1, for all x > 0

(B) ψ2(x) ≤ 0, for all x > 0

(C)

(D)

SECTION-4

• This section contains THREE (03) questions.

• The answer to each question is a NON-NEGATIVE INTEGER.

• 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 : +4 If ONLY the correct integer is entered;

Zero Marks : 0 In all other cases.

17. A number is chosen at random from the set {1, 2, 3……, 2000}. Let p be the probability that the number is a multiple of 3 or a multiple of 7. Then the value of 500p is;

18. Let E be the ellipse  For any three distinct points P, Q and Q’ on E, let M (P, Q) be the mid-point of the line segment joining P and Q, and M(P, Q’) be the mid-point of the line segment joining P and Q’. Then the maximum possible value of the distance between M (P, Q) and M(P, Q’), as P, Q and Q’ vary on E, is

19. For any real number x, let [x] denote the largest integer less than or equal to x. If  then the value of 9I is ______.

PHYSICS

Section 1

• This Section contains Four (04) Questions.

• Each question has FOUR options. 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.

1. The smallest division on the main scale of Vernier calipers is 0.1 cm. Ten divisions of the Vernier scale correspond to nine divisions of the main scale. The figure below on the left shows the reading of this caliper with no gap between its two jaws. The figure on the right shows the reading with a solid sphere held between the jaws. The correct diameter of the sphere is;

(A) 3.07 cm

(B) 3.11 cm

(C) 3.15 cm

(D) 3.17 cm

2. An ideal gas undergoes a four step cycle as shown in the P – V diagram below. During this cycle, heat is absorbed by the gas in;

(A) steps 1 and 2

(B) steps 1 and 3

(C) steps 1 and 4

(D) steps 2 and 4

3. An extended object is placed at point O, 10 cm in front of a convex lens L1 and a concave lens L2 is placed 10 cm behind it, as shown in the figure. The radii of curvature of all the curved surfaces in both the lenses are 20 cm. The refractive index of both the lenses is 1.5. The total magnification of this lens system is;

(A) 0.4

(B) 0.8

(C) 1.3

(D) 1.6

4. A heavy nucleus Q of half-life 20 minutes undergoes alpha-decay with a probability of 60% and beta-decay with a probability of 40%. Initially, the number of Q nuclei is 1000. The number of alpha-decays of Q in the first one hour is;

(A) 50

(B) 75

(C) 350

(D) 525

SECTION-2

• This section contains THREE (03) questions stems.

• There are TWO (02) questions corresponding to each question stem.

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

• If the numerical value has more than two decimal places, truncate/round-off the value of TWO decimal places.

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

Full Marks : +2 If ONLY the correct numerical value is entered at eh designated place;

Zero Marks : 0 In all other cases.

Question Stem for Question Nos. 5 and 6

Question Stem

A projectile is thrown from a point O on the ground at an angle 45° from the vertical and with a speed of 5 √2 m/s. The projectile at the highest point of its trajectory splits into two equal parts. One part falls vertically down to the ground, 0.5 s after the splitting. The other part, t seconds after the splitting, falls to the ground at a distance x meters from the point O. The acceleration due to gravity g = 10 m/s2.

5. The value of t is ______.

6. The value of x is ______.

Question Stem for Question Nos. 7 and 8

Question Stem

In the circuit shown below, the switch S is connected to position P for a long time so that the charge on the capacitor becomes q1 µC. Then S is switched to position Q. After a long time, the charge on the capacitor is q2 µC.

7. The magnitude of q1 _____.

8. The magnitude of q2 ______.

Question Stem for Question Nos. 9 and 10

Question Stem

Two-point charges –Q and +Q/√3 are placed in the xy-plane at the origin (0, 0) and a point (2, 0), respectively, as shown in the figure. This results in an equipotential circle of radius R and potential V = 0 in the xy-plane with its centre at (b, 0). All lengths are measured in meters.

9. The value of R is ____meter.

10. The value of b is _____ meter.

SECTION-3

• This section contains SIX (06) questions.

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

• 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 : +4 If only (all) the correct option(s) is (are) chosen;

Partial Marks : +3 If all four options is correct but ONLY three options are chosen;

Partial Marks : +2 If there 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 questions is unanswered);

Negative Marks: −2 In all other cases.

• For example, in a question, if (A), (B) and (D) are the ONLY three options corresponding to correct answer, then

Choosing ONLY (A), (B) and (D) will get +4 marks;

Choosing ONLY (A) and (B) will get +2 marks;

Choosing ONY (A) and (D) will get +2 marks;

Choosing ONLY (B) and (D) will get +2 marks;

Choosing ONLY (A) will get +1 mark;

Choosing ONLY (B) will get +1 mark;

Choosing ONLY (D) will get +1 mark;

Choosing no option(s) (i.e. the question is unanswered) will get 0 marks and choosing any other option(s) will get −2 marks.

11. A horizontal force F is applied at the centre of mass of a cylindrical object of mass m and radius R, perpendicular to its axis as shown in the figure. The coefficient of friction between the object and the ground is. The centre of mass of the object has an acceleration a. The acceleration due to gravity is g. Given that the object rolls without slipping, which of the following statement(s) is(are) correct?

(A) For the same F, the value of a does not depend on whether the cylinder is solid or hollow

(B) For a solid cylinder, the maximum possible value of a is 2g

(C) The magnitude of the frictional force on the object due to the ground is always mg

(D) For a thin-walled hollow cylinder, a = F/2m

12. A wide slab consisting of two media of refractive indices n1 and n2 is placed in the air as shown in the figure. A ray of light is incident from medium n1 to n2 at an angle, where sin is slightly larger than 1/n1. Take the refractive index of air as 1. Which of the following statement(s) is(are) correct?

(A) The light ray enters air if n2 = n1

(B) The light ray is finally reflected back into the medium of refractive index n1 if n2 < n1

(C) The light ray is finally reflected back into the medium of refractive index n1 if n2 > n1

(D) The light ray is reflected back into the medium of refractive index n1 if n2 = 1

13. A particle of mass M = 0.2 kg is initially at rest in the xy-plane at a point (x = –l, y = –h), where l = 10 m and h = 1 m. The particle is accelerated at time t = 0 with a constant acceleration a = 10 m/s2 along the positive x-direction. Its angular momentum and torque with respect to the origin, in SI units, are represented by  are unit vectors along the positive x, y and z-directions, respectively. If  then which of the following statement(s) is(are) correct?

(A) The particle arrives at the point (x = l, y = –h) at time t = 2s

(B)  when the particle passes through the point (x = l, y = −h)

(C)  when the particle passes through the point ( x = l, y = −h)

(D)  when the particle passes through the point (x = 0, y = −h)

14. Which of the following statement(s) is(are) correct about the spectrum of hydrogen atom?

(A) The ratio of the longest wavelength to the shortest wavelength in the Balmer series is 9/5

(B) There is an overlap between the wavelength ranges of Balmer and Paschen series

(C) The wavelengths of Lyman series are given by  where λ0 is the shortest wavelength of Lyman series and m is an integer

(D) The wavelength ranges of the Lyman and Balmer series do not overlap

15. A long straight wire carries a current, l = 2 ampere. A semi-circular conducting rod is placed beside it on two conducting parallel rails of negligible resistance. Both the rails are parallel to the wire. The wire, the rod and the rails lie in the same horizontal plane, as shown in the figure. Two ends of the semi-circular rod are at distances 1 cm and 4 cm from the wire. At time t = 0, the rod starts moving on the rails with a speed v = 3.0 m/s (see the figure).

A resistor R = 1.4 and a capacitor C0 = 5.0 F are connected in series between the rails. At time t = 0, C0 is uncharged. Which of the following statement(s) is(are) correct? [μ0 = 4 × 10–7 SI units. Take ln 2 = 0.7]

(A) Maximum current through R is 1.2 × 10–6 ampere

(B) Maximum current through R is 3.8 × 10–6 ampere

(C) Maximum charge on capacitor C0 is 8.4 × 10–12 coulomb

(D) Maximum charge on capacitor C0 is 2.4 × 10–12 coulomb

16. A cylindrical tube, with its base as shown in the figure, is filled with water. It is moving down with constant acceleration along a fixed inclined plane with an angle = 45º. P1 and P2 are pressures at points 1 and 2, respectively, located at the base of the tube. Let β = (P1 – P2)/(ρgd), where ρ is the density of water, d is the inner diameter of the tube and g is the acceleration due to gravity. Which of the following statement(s) is(are) correct?

(A) β = 0 when a = g/√2

(B) β ＞0 when a = g/√2

(C)

(D)

SECTION-4

• This section contains THREE (03) questions.

• The answer to each question is a NON-NEGATIVE INTEGER.

• 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        : +4 if ONLY the correct integer is entered;

Zero Marks       : 0 in all other cases.

17. An α -particle (mass 4 amu) and a singly charged sulfur ion (mass 32 amu) are initially at rest. They are accelerated through a potential V and then allowed to pass into a region of a uniform magnetic field which is normal to the velocities of the particles. Within this region, the -particle and the sulfur ion move in circular orbits of radii rα and rs The ratio rs/rα is ________.

18. A thin rod of mass M and length a is free to rotate in a horizontal plane about a fixed vertical axis passing through point O. A thin circular disc of mass M and of radius a/4 is pivoted on this rod with its centre at a distance a/4 from the free end so that it can rotate freely about its vertical axis, as shown in the figure. Assume that both the rod and the disc have uniform density and they remain horizontal during the motion. An outside stationary observer finds the rod rotating with an angular velocity and the disc rotating about its vertical axis with angular velocity 4Ω. The total angular momentum of the system about the point O is

The value of n  is ______.

19. A small object is placed at the centre of a large evacuated hollow spherical container. Assume that the container is maintained at 0 K. At time t = 0, the temperature of the object is 200 K. The temperature of the object becomes 100 K at t = t1 and 50 K at t = t2. Assume the object and the container to be ideal black bodies. The heat capacity of the object does not depend on temperature. The ratio (t1/t2) is ______.

CHEMISTRY

Section 1

• This Section contains Four (04) Questions.

• Each question has FOUR options. 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.

1. The major product formed in the following reaction is:

2. Among the following, the conformation that corresponds to the most stable conformation of meso-butane-2,3-diol is;

3. For the given close-packed structure of a salt made of cation X and anion Y shown below (ions of only one face are shown for clarity), the packing fraction is approximately

(A) 0.74

(B) 0.63

(C) 0.52

(D) 0.48

4. The calculated spin only magnetic moments of [Cr(NH3)6]3+ and [CuF6]3– in BM, respectively, are (Atomic numbers of Cr and Cu are 24 and 29, respectively).

(A) 3.87 and 2.84

(B) 4.90 and 1.73

(C) 3.87 and 1.73

(D) 4.90 and 2.84

SECTION-2

• This section contains THREE (03) questions stems.

• There are TWO (02) questions corresponding to each question stem.

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

• If the numerical value has more than two decimal places, truncate/round-off the value of TWO decimal places.

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

Full Marks : +2 If ONLY the correct numerical value is entered at eh designated place;

Zero Marks : 0 In all other cases.

Question Stem for Question Nos. 5 and 6

Question Stem

For the following reaction scheme, percentage yields are given along the arrow:

x g and y g are the masses of R and U, respectively. (Use: Molar mass (in g mol–1) of H, C and O as 1, 12 and 16, respectively)

5. The value of x is______.

6. The value of y is ______.

Question Stem for Question Nos. 7 and 8

Question Stem

For the reaction, X(s) ⇌ Y(s) + Z(g), the plot of  is given below (in solid line), where pZ is the pressure (in bar) of the gas Z at temperature T and p = 1 bar.

(Given, where the equilibrium constant,   and the gas constant, R = 8.314 J K1 mol1)

7. The value of standard enthalpy, ∆H (in kJ mol1) for the given reaction is_____.

8. The value of ∆S (in J K−1 mol−1) for the given reaction, at 1000 K is _______.

Question Stem for Question Nos. 9 and 10

Question Stem

The boiling point of water in a 0.1 molal silver nitrate solution (solution A) is x ºC. To this solution A, an equal volume of 0.1 molal aqueous barium chloride solution is added to make a new solution B. The difference in the boiling points of water in the two solutions A and B is y × 102 ºC.

(Assume: Densities of the solutions A and B are the same as that of water and the soluble salts dissociate completely. Use: Molal elevation constant (Ebullioscopic constant), Kb = 0.5 K kg mol1; Boiling point of pure water as 100ºC.)

9. The value of x is _____.

10. The value of |y| is _____.

SECTION-3

• This section contains SIX (06) questions.

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

• 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 : +4 If only (all) the correct option(s) is (are) chosen;

Partial Marks : +3 If all four options is correct but ONLY three options are chosen;

Partial Marks : +2 If there 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 questions is unanswered);

Negative Marks: −2 In all other cases.

• For example, in a question, if (A), (B) and (D) are the ONLY three options corresponding to correct answer, then

Choosing ONLY (A), (B) and (D) will get +4 marks;

Choosing ONLY (A) and (B) will get +2 marks;

Choosing ONY (A) and (D) will get +2 marks;

Choosing ONLY (B) and (D) will get +2 marks;

Choosing ONLY (A) will get +1 mark;

Choosing ONLY (B) will get +1 mark;

Choosing ONLY (D) will get +1 mark;

Choosing no option(s) (i.e. the question is unanswered) will get 0 marks and choosing any other option(s) will get −2 marks.

11. Given:

The compound(s), which on reaction with HNO3 will give the product having a degree of rotation, [α]D = –52.7º is(are);

12. The reaction of Q with PhSNa yields an organic compound (major product) that gives a positive Carius test on treatment with Na2O2 followed by the addition of BaCl2. The correct option(s) for Q is(are).

13. The correct statement(s) related to colloids is(are)

(A) The process of precipitating colloidal sol by an electrolyte is called peptization

(B) Colloidal solution freezes at a higher temperature than the true solution at the same concentration

(C) Surfactants form micelle above critical micelle concentration (CMC). CMC depends on temperature

(D) Micelles are macromolecular colloids

14. An ideal gas undergoes a reversible isothermal expansion from the state I to state II followed by a reversible adiabatic expansion from state II to state III. The correct plot(s) representing the changes from the state I to state III is(are) (p: pressure, V: volume, T: temperature, H: enthalpy, S: entropy)

15. The correct statement(s) related to the metal extraction processes is(are);

(A) A mixture of PbS and PbO undergoes self-reduction to produce Pb and SO2.

(B) In the extraction process of copper from copper pyrites, silica is added to produce copper silicate.

(C) Partial oxidation of sulphide ore of copper by roasting, followed by self-reduction produces blister copper.

(D) In the cyanide process, zinc powder is utilized to precipitate gold from Na[Au(CN)2].

16. A mixture of two salts is used to prepare a solution S, which gives the following results:

The correct option(s) for the salt mixture is(are)

(A) Pb(NO3)2 and Zn(NO3)2

(B) Pb(NO3)2 and Bi(NO3)2

(C) AgNO3 and Bi(NO3)3

(D) Pb(NO3)2 and Hg(NO3)2

SECTION-4

• This section contains THREE (03) questions.

• The answer to each question is a NON-NEGATIVE INTEGER.

• 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        : +4 if ONLY the correct integer is entered;

Zero Marks      : 0 in all other cases.

17. The maximum number of possible isomers (including stereoisomers) which may be formed on mono-bromination of 1-methylcyclohex-1-ene using Br2 and UV light is ______.

18. In the reaction given below, the total number of atoms having sp2 hybridization in the major product P is ______.

19. The total number of possible isomers for [Pt(NH3)4Cl2]Br2 is

MATHEMATICS

Section 1

• This Section contains Four (04) Questions.

• Each question has FOUR options. 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.

1. Consider a triangle Δ whose two sides lie on the x-axis and the line x + y + 1 = 0. If the orthocenter of Δ is (1, 1), then the equation of the circle passing through the vertices of the triangle Δ is;

(A) x2 + y2 − 3x + y = 0

(B) x2 + y2 + x + 3y = 0

(C) x2 + y2 + 2y − 1 = 0

(D) x2 + y2 + x + y = 0

2. The area of the region {(x, y): 0 ≤ x ≤ 9/4, 0 ≤ y ≤ 1, x ≥ 3y, x + y ≥ 2}is

(A) 11/32

(B) 35/96

(C) 37/96

(D) 13/32

3. Consider three sets E1 = {1, 2, 3}, F1 = {1, 3, 4} and G1 = {2, 3, 4, 5}. Two elements are chosen at random, without replacement, from the set E1, and let S1 denote the set of these chosen elements. Let E2 = E1 − S1 and F2 = F1 ⋃ S1. Now two elements are chosen at random, without replacement, from the set F2 and let S2 denote the set of these chosen elements.

Let G2 = G1 ⋃ S2. Finally, two elements are chosen at random, without replacement from the set G2 and let S3 denote the set of these chosen elements. Let E3 = E2 ⋃ S3. Given that E1 = E3, let p be the conditional probability of the event S1 = {1, 2}. Then the value of p is;

(A) 1/5

(B) 3/5

(C) 1/2

(D) 2/5

4. Let θ1, θ2, …., θ10 be positive valued angles (in radian) such that θ1+ θ2+ ….+ θ10 = 2π. Define the complex numbers  for k = 2, 3,  …, 10, where i = √− Consider the statements P and Q given below:

P: |z2 − z1| + |z3 − z2| + …. +|z10 − z9| + |z1 − z10| ≤ 2π

Q: |z22 − z12| + |z32 − z22| + …. +|z102 − z92| + |z12 − z102| ≤ 4π

Then,

(A) P is TRUE and Q is FALSE

(B) Q is TRUE and P is FALSE

(C) Both P and Q are TRUE

(D) Both P and Q are FALSE

SECTION-2

• This section contains THREE (03) questions stems.

• There are TWO (02) questions corresponding to each question stem.

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

• If the numerical value has more than two decimal places, truncate/round-off the value of TWO decimal places.

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

Full Marks : +2 If ONLY the correct numerical value is entered at eh designated place;

Zero Marks : 0 In all other cases.

Question Stem for Question Nos. 5 and 6

Question Stem

Three numbers are chosen at random, one after another with replacement, from the set S = {1, 2, 3, …, 100}. Let p1 be the probability that the maximum of chosen numbers is at least 81 and p2 be the probability that the minimum of chosen numbers is at most 40.

5. The value of  is _____.

6. The value of  is _____.

Question Stem for Question Nos. 7 and 8

Question Stem

Let α, β and γ be real numbers such that the system of linear equations

x + 2y + 3z = α

4x + 5y + 6z = β

7x + 8y + 9z = γ – 1 is consistent.

Let |M| represent the determinant of the matrix.

Let P be the plane containing all those (α, β, γ) for which the above system of linear equations is consistent, and D be the square of the distance of the point (0, 1, 0) from the plane P.

7. The value of |M| is_____.

8. The value of D is ______.

Question Stem for Question Nos. 9 and 10

Question Stem

Consider the lines L1 and L2 defined by

L1 : x√2 + y − 1 = 0 and L2 : x√2 − y + 1 = 0

For a fixed constant λ, let C be the locus of a point P such that the product of the distance of P from L1 and the distance of P from L2 is λ2. The line y = 2x + 1 meets C at two points R and S, where the distance between R and S is √270.

Let the perpendicular bisector of RS meet C at two distinct points R’ and S’. Let D be the square of the distance between R’ and S’.

9. The value of λ2 ______.

10. The value of D is ______.

SECTION-3

• This section contains SIX (06) questions.

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

• 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 : +4 If only (all) the correct option(s) is (are) chosen;

Partial Marks : +3 If all four options is correct but ONLY three options are chosen;

Partial Marks : +2 If there 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 questions is unanswered);

Negative Marks: −2 In all other cases.

• For example, in a question, if (A), (B) and (D) are the ONLY three options corresponding to correct answer, then

Choosing ONLY (A), (B) and (D) will get +4 marks;

Choosing ONLY (A) and (B) will get +2 marks;

Choosing ONY (A) and (D) will get +2 marks;

Choosing ONLY (B) and (D) will get +2 marks;

Choosing ONLY (A) will get +1 mark;

Choosing ONLY (B) will get +1 mark;

Choosing ONLY (D) will get +1 mark;

Choosing no option(s) (i.e. the question is unanswered) will get 0 marks and choosing any other option(s) will get −2 marks.

11. For any 3 × 3 matrix M, let |M| denote the determinant of M. Let

If Q is a nonsingular matrix of order 3 × 3, then which of the following statements is(are) TRUE?

(A) F = PEP and

(B) |EQ + PFQ1| = |EQ| + |PFQ1|

(C) |(EF)3| > |EF|2

(D) Sum of the diagonal entries of P1EP + F is equal to the sum of diagonal entries of E + P1FP

12. Let F: R→ R be defined by

Then which of the following statements is (are) TRUE?

(A) f is decreasing in the interval (−2, −1)

(B) f is increasing in the interval (1, 2)

(C) f is onto

(D) Range of f is [−3/2, 2]

13. Let E, F and G be three events having probabilities P(E) = 1/8, P(F) = ⅙ and P(G) = ¼, and P(E⋂F⋂G) = 1/10. For any event H, if Hc denotes its complement, then which of the following statements is(are) TRUE?

(A) P(E ⋂ F ⋂ Gc) ≤ 1/40

(B) P(Ec ⋂ F ⋂ G) ≤ 1/15

(C) P(E ⋃ F ⋃ G) ≤ 13/24

(D) P(Ec ⋂ Fc ⋂ Gc) ≤ 5/12

14. For any 3 × 3 matrix M, let |M| denote the determinant of M. Let I be the 3 × 3 identify matrix. Let E and F be two 3 × 3 matrices such that (I − EF) is invertible. If G = (I − EF)–1, then which of the following statements is(are) TRUE?

(A) |FE| = |I − FE| |FGE|

(B) (I − FE) (I + FGE) = I

(C) EFG = GEF

(D) (I − FE) (I − FGE) = I

15. For any positive integer n, let Sn : (0, ∞) R be defined by

where for any x ∈ R, cot1 (x) ∈ (0, π) and  Then which of the following statements is (are) TRUE?

(A)

(B)

(C) Then equation  has a root in (0, ∞)

(D)  for all n ≥ 1 and x > 0

16. For any complex number w = c + id, let arg(w) ∈ (-π, π], where i = √−1 . Let α and β be real numbers such that for all complex numbers z = x + iy satisfying  the ordered pair (x, y) lies on the circle x2 + y2 + 5x − 3y + 4 = 0. Then which of the following statements is (are) TRUE?

(A) α = −1

(B) αβ = 4

(C) αβ = −4

(D) β = 4

SECTION-4

• This section contains THREE (03) questions.

• The answer to each question is a NON-NEGATIVE INTEGER.

• 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        : +4 if ONLY the correct integer is entered;

Zero Marks      : 0 in all other cases.

17. For x ∈ R, the number of real roots of the equation 3x2 – 4|x2 – 1| + x – 1 = 0 is

18. In a triangle ABC, let AB = √23, and BC = 3 and CA = 4. Then the value of  is _____.

19. Let  be vectors in three-dimensional space, where  are unit vectors which are not perpendicular to each other and

If the volume of the parallelepiped, whose adjacent sides are represented by the vectors  is √2, then the value of  is ____.

## JEE Advanced Exam 2020 Paper-2 Code-E Question Paper With Answer Key

PHYSICS

Section 1

• This Section contains Four (06) Questions.

• The answer to each question is a SINGLE DIGIT INTEGER ranging from 0 to 9

• 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 in unanswered;

Negative Marks : −1 In all other cases.

1. A large square container with thin transparent vertical walls and filled with water (refractive index 4/3) is kept on a horizontal table. A student holds a thin straight wire vertically inside the water 12 cm from one of its corners, as shown schematically in the figure. Looking at the wire from this corner, another student sees two images of the wire, located symmetrically on each side of the line of sight as shown. The separation (in cm) between these images is ____________.

2. A train with cross-sectional area St is moving with speed νt inside a long tunnel of cross-sectional area S0 (S0 = 4St). Assume that almost all the air (density ρ) in front of the train flows back between its sides and the walls of the tunnel. Also, the airflow with respect to the train is steady and laminar. Take the ambient pressure and that inside the train to be p0. If the pressure in the region between the sides of the train and the tunnel walls is p, then  The value of N is ________.

3. Two large circular discs separated by a distance of 0.01 m are connected to a battery via a switch as shown in the figure. Charged oil drops of density 900 kg m−3 are released through a tiny hole at the centre of the top disc. Once some oil drops achieve terminal velocity, the switch is closed to apply a voltage of 200 V across the discs. As a result, an oil drop of radius 8 × 10−7 m stops moving vertically and floats between the discs. The number of electrons present in this oil drop is ________. (neglect the buoyancy force, take acceleration due to gravity = 10 ms−2 and charge on an electron (e) = 1.6×10–19 C)

4. A hot air balloon is carrying some passengers, and a few sandbags of mass 1 kg each so that its total mass is 480 kg. Its effective volume giving the balloon its buoyancy is V. The balloon is floating at an equilibrium height of 100 m. When N number of sandbags are thrown out, the balloon rises to a new equilibrium height close to 150 m with its volume V remaining unchanged. If the variation of the density of air with height h from the ground is  where ρ0 = 1.25 kg m3 and h0 = 6000 m, the value of N is __________.

5. A point charge q of mass m is suspended vertically by a string of length l. A point dipole of dipole moment  is now brought towards q from infinity so that the charge moves away. The final equilibrium position of the system including the direction of the dipole, the angles and distances is shown in the figure below. If the work done in bringing the dipole to this position is N × (mgh), where g is the acceleration due to gravity, then the value of N is _________ . (Note that for three coplanar forces keeping a point mass in equilibrium, F/sin θ is the same for all forces, where F is any one of the forces and θ is the angle between the other two forces)

6. A thermally isolated cylindrical closed vessel of height 8 m is kept vertically. It is divided into two equal parts by a diathermic (perfect thermal conductor) frictionless partition of mass 8.3 kg. Thus the partition is held initially at a distance of 4 m from the top, as shown in the schematic figure below. Each of the two parts of the vessel contains 0.1 mole of an ideal gas at temperature 300 K. The partition is now released and moves without any gas leaking from one part of the vessel to the other. When equilibrium is reached, the distance of the partition from the top (in m) will be _______ (take the acceleration due to gravity = 10 ms−2 and the universal gas constant = 8.3 J mol−1K−1).

SECTION 2

• This section contains SIX (06) questions.

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

• 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       : +4 If only (all) the correct option(s) is (are) chosen;

Partial Marks : +3 If all four options is correct but ONLY three options are chosen;

Partial Marks : +2 If there 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 questions is unanswered);

Negative Marks: −2 In all other cases.

7. A beaker of radius r is filled with water (refractive index 4/3) up to a height H as shown in the figure on the left. The beaker is kept on a horizontal table rotating with angular speed & omega;. This makes the water surface curved so that the difference in the height of water level at the centre and at the circumference of the beaker is h(h << H, h << r), as shown in the figure on the right. Take this surface to be approximately spherical with a radius of curvature R. Which of the following is/are correct? (g is the acceleration due to gravity)

(A)

(B)

(C) Apparent depth of the bottom of the beaker is close to

(D) Apparent depth of the bottom of the beaker is close to

8. A student skates up a ramp that makes an angle 30° with the horizontal. He/she starts (as shown in the figure) at the bottom of the ramp with speed 𝑣0 and wants to turn around over a semi-circular path xyz of radius R during which he/she reaches a maximum height h (at point y) from the ground as shown in the figure. Assume that the energy loss is negligible and the force required for this turn at the highest point is provided by his/her weight only. Then (g is the acceleration due to gravity)

(A)

(B)

(C) the centripetal force required at points x and z is zero

(D) the centripetal force required is maximum at points x and z

9. A rod of mass m and length L, pivoted at one of its ends, is hanging vertically. A bullet of the same mass moving at speed v strikes the rod horizontally at a distance x from its pivoted end and gets embedded in it. The combined system now rotates with angular speed ω about the pivot. The maximum angular speed ωM is achieved for x = xM. Then

(A)

(B)

(C)

(D)

10. In an X-ray tube, electrons emitted from a filament (cathode) carrying current I hit a target (anode) at a distance d from the cathode. The target is kept at a potential V higher than the cathode resulting in the emission of continuous and characteristic X-rays. If the filament current𝐼 is decreased to 1/2, the d potential difference Vis increased to 2v, and the separation distance d is reduced to d/2, then

(A) the cut-off wavelength will reduce to half, and the wavelengths of the characteristic X-rays will remain the same

(B) the cut-off wavelength, as well as the wavelengths of the characteristic X-rays, will remain the same

(C) the cut-off wavelength will reduce to half, and the intensities of all the X-rays will decrease

(D) the cut-off wavelength will become two times larger, and the intensity of all the X-rays will decrease

11. Two identical non-conducting solid spheres of same mass and charge are suspended in the air from a common point by two non-conducting, massless strings of the same length. At equilibrium, the angle between the strings is α. The spheres are now immersed in a dielectric liquid of density 800 kg m−3 and dielectric constant 21. If the angle between the strings remains the same after the immersion, then

(A) electric force between the spheres remains unchanged

(B) electric force between the spheres reduces

(C) mass density of the spheres is 840 kg m−3

(D) the tension in the strings holding the spheres remains unchanged

12. Starting at time t = 0 from the origin with speed 1 ms−1, a particle follows a two-dimensional trajectory in the x-y plane so that its coordinates are related by the equation y = x2/2. The x and y components of its acceleration are denoted by ax and ay, respectively. Then

(A) ax = 1 ms−2 implies that when the particle is at the origin, ay = 1 ms−2

(B) ax = 0 implies ay = 1 ms−2 at all times

(C) at t = 0, the particle’s velocity points in the x-direction

(D) ax = 0 implies that at t = 1 s, the angle between the particle’s velocity and the x-axis is 45°

Answer: (A, B, C, D or B, C, D)

SECTION-3

• This section contains SIX (06) 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 : +4 If ONLY the correct numerical value is entered;

Zero Marks : 0 In all other cases.

13. A spherical bubble inside water has radius R. Take the pressure inside the bubble and the water pressure to be p0. The bubble now gets compressed radially in an adiabatic manner so that its radius becomes (R − a). For a << R the magnitude of the work done in the process is given by (4 πp0Ra2)X, where X is a constant and γ = Cp/CV = 41⁄30. The value of X is________.

14. In the balanced condition, the values of the resistances of the four arms of a Wheatstone bridge are shown in the figure below. The resistance R3 has temperature coefficient 0.0004℃−1. If the temperature of R3 is increased by 100℃, the voltage developed between S and T will be __________ volt.

15. Two capacitors with capacitance values C1 = 2000 ± 10 pF and C2 = 3000 ± 15 pF are connected in series. The voltage applied across this combination is V = 5.00 ± 0.02 V. The percentage error in the calculation of the energy stored in this combination of capacitors is _______.

16. A cubical solid aluminium  block has an edge length of 1m on the surface of the earth. It is kept on the floor of a 5 km deep ocean. Taking the average density of water and the acceleration due to gravity to be 103 kg m−3 and 10 ms−2, respectively, the change in the edge length of the block in mm is _____.

17. The inductors of two LR circuits are placed next to each other, as shown in the figure. The values of the self-inductance of the inductors, resistances, mutual-inductance and applied voltages are specified in the given circuit. After both the switches are closed simultaneously, then total work done by the batteries against the induced EMF in the inductors by the time the currents reach their steady-state values is ________ mJ.

18. A container with 1 kg of water in it is kept in sunlight, which causes the water to get warmer than the surroundings. The average energy per unit time per unit area received due to the sunlight is 700 Wm−2 and it is absorbed by the water over an effective area of 0.05 m2. Assuming that the heat loss from the water to the surroundings is governed by Newton’s law of cooling, the difference (in ℃) in the temperature of water and the surroundings after a long time will be _____________. (Ignore the effect of the container, and take constant for Newton’s law of cooling = 0.001 s−1, the Heat capacity of water = 4200 J kg−1 K−1)

CHEMISTRY

Section 1

• This Section contains Four (06) Questions.

• The answer to each question is a SINGLE DIGIT INTEGER ranging from 0 to 9

• 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 in unanswered;

Negative Marks : −1 In all other cases.

1. The 1st, 2nd, and the 3rd ionization enthalpies, I1, I2, and I3, of four atoms with atomic numbers n, n + 1, n + 2, and n + 3, where n < 10, are tabulated below. What is the value of n?

2. Consider the following compounds in the liquid form:

O2, HF, H2O, NH3, H2O2, CCl4, CHCl3, C6H6, C6H5Cl.

When a charged comb is brought near their flowing stream, how many of them show deflection as per the following figure?

3. In the chemical reaction between stoichiometric quantities of KMnO4 and KI in weakly basic solution, what is the number of moles of I2 released for 4 moles of KMnO4 consumed?

4. An acidified solution of potassium chromate was layered with an equal volume of amyl alcohol. When it was shaken after the addition of 1 mL of 3% H2O2, a blue alcohol layer was obtained. The blue color is due to the formation of a chromium (VI) compound ‘X’. What is the number of oxygen atoms bonded to chromium through only single bonds in a molecule of X?

5. The structure of a peptide is given below.

If the absolute values of the net charge of the peptide at pH = 2, pH = 6, and pH = 11 are |z1|, |z2|, and |z3|, respectively, then what is |z1| + |z2| + |z3|?

6. An organic compound (C8H10O2) rotates plane-polarized light. It produces pink color with neutral FeCl3 What is the total number of all the possible isomers for this compound?

SECTION 2

• This section contains SIX (06) questions.

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

• 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       : +4 If only (all) the correct option(s) is (are) chosen;

Partial Marks : +3 If all four options is correct but ONLY three options are chosen;

Partial Marks : +2 If there 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 questions is unanswered);

Negative Marks: −2 In all other cases.

7. In an experiment, m grams of a compound X (gas/liquid/solid) taken in a container is loaded in a balance as shown in figure I below. In the presence of a magnetic field, the pan with X is either deflected upwards (figure II), or deflected downwards (figure III), depending on the compound X. Identify the correct statement(s).

(A) If X is H2O(l), deflection of the panis upwards.

(B) If X is K4[Fe(CN)6](s), deflection of the panis upwards.

(C) If X is O2 (g), deflection of the panis downwards.

(D) If X is C6H6(l), deflection of the panis downwards.

8. Which of the following plots is (are) correct for the given reaction?

([P]0 is the initial concentration of P)

9. Which among the following statement(s) is(are) true for the extraction of aluminium from bauxite?

(A) Hydrated Al2O3 precipitates, when CO2 is bubbled through a solution of sodium aluminate.

(B) Addition of Na3AlF6 lowers the melting point of alumina.

(C) CO2 is evolved at the anode during electrolysis.

(D) The cathode is a steel vessel with a lining of carbon.

10. Choose the correct statement(s) among the following.

(A) SnCl2.2H2O is a reducing agent.

(B) SnO2 reacts with KOH to form K2[Sn(OH)6].

(C) A solution of PbCl2 in HCl contains Pb2+ and Cl ions.

(D) The reaction of Pb3O4 with hot dilute nitric acid to give PbO2 is a redox reaction.

11. Consider the following four compounds I, II, III, and IV.

Choose the correct statement(s).

(A) The order of basicity is II >I >III >IV.

(B) The magnitude of pKb difference between I and II is more than that between III and IV.

(C) Resonance effect is more in III than in IV.

(D) Steric effect makes compound IV more basic than III.

12. Consider the following transformations of a compound P.

SECTION-3

• This section contains SIX (06) 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 : +4 If ONLY the correct numerical value is entered;

Zero Marks : 0 In all other cases.

13. A solution of 0.1 M weak base (B) is titrated with 0.1 M of a strong acid (HA). The variation of pH of the solution with the volume of HA added is shown in the figure below. What is the p𝐾b of the base? The neutralization reaction is given by B + HA &rarr; BH+ + A.

14. Liquids A and B form ideal solution for all compositions of A and B at 25℃. Two such solutions with 0.25 and 0.50 mole fractions of A have the total vapor pressures of 0.3 and 0.4 bar, respectively. What is the vapor pressure of pure liquid B in bar?

15. The figure below is the plot of potential energy versus internuclear distance (d) of H2 molecule in the electronic ground state. What is the value of the net potential energy E0 (as indicated in the figure) in kJ mol−1, for d = d0 at which the electron-electron repulsion and the nucleus-nucleus repulsion energies are absent? As reference, the potential energy of H atom is taken as zero when its electron and the nucleus are infinitely far apart.

Use Avogadro constant as 6.023 × 1023 mol−1.

16. Consider the reaction sequence from P to Q shown below. The overall yield of the major product Q from P is 75%. What is the amount in grams of Q obtained from 9.3 mL of P? (Use density of P = 1.00 g mL−1; Molar mass of C = 12.0, H =1.0, O =16.0 and N = 14.0 g mol−1)

17. Tin is obtained from cassiterite by reduction with coke. Use the data given below to determine the minimum temperature (in K) at which the reduction of cassiterite by coke would take place.

At 298 K: ∆fH0 (SnO2(s)) = −581.0 kJ mol−1, ∆fH0(CO2(g)) = −394.0 kJ mol−1,

S0(SnO2(s)) = 56.0J K−1 mol−1, S0(Sn(s)) = 52.0 J K−1mol−1,

S0(C(𝑠)) = 6.0J K−1mol−1, S0(CO2(g)) = 210.0 J K−1mol−1.

Assume that the enthalpies and the entropies are temperature independent.

18. An acidified solution of 0.05 MZn2+ is saturated with 0.1 M H2 What is the minimum molar concentration (M) of H+ required to prevent the precipitation of ZnS?

Use Ksp(ZnS) = 1.25 x 1022 and overall dissociation constant of H2S,

KNET = K1K2 = 1 × 1021.

MATHEMATICS

Section 1

• This Section contains Four (06) Questions.

• The answer to each question is a SINGLE DIGIT INTEGER ranging from 0 to 9

• 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 in unanswered;

Negative Marks : −1 In all other cases.

1. For a complex number z, let Re(z) denote the real part of z. Let S be the set of all complex numbers z satisfying z4 − |z|4 = 4 iz2, where i = √(−1) . Then the minimum possible value of |z1 − z2|2, where z1, z2 ∈ S with Re(𝑧1) > 0 and Re(z2) < 0, is _____

2. The probability that a missile hits a target successfully is 0.75. In order to destroy the target completely, at least three successful hits are required. Then the minimum number of missiles that have to be fired so that the probability of completely destroying the target is NOT less than 0.95, is_____.

3. Let O be the centre of the circle x2 + y2 = r2, where r > √5/2 . Suppose PQ is a chord of this circle and the equation of the line passing through P and Q is 2x + 4𝑦 = 5. If the centre of the circumcircle of the triangle OPQ lies on the line 𝑥 + 2𝑦 = 4, then the value of r is_____

4. The trace of a square matrix is defined to be the sum of its diagonal entries. If A is a 2 × 2 matrix such that the trace of A is 3 and the trace of A3 is −18, then the value of the determinant of A is_____

5. Let the functions f: (−1,1) → R and g: (−1, 1) → (−1, 1) be defined by f(x) = |2x − 1| + |2x + 1|and g (x) = x − [x], where [x] denotes the greatest integer less than or equal to x. Let fog: (−1,1) → R be the composite function defined by (fog)(x) = f(g(x)). Suppose C is the number of points in the interval (−1, 1) at which fog is NOT continuous, and suppose d is the number of points in the interval (−1,1) at which fog is NOT differentiable. Then the value of c + d is _____

6. The value of the limit  is ______

SECTION 2

• This section contains SIX (06) questions.

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

• 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       : +4 If only (all) the correct option(s) is (are) chosen;

Partial Marks : +3 If all four options is correct but ONLY three options are chosen;

Partial Marks : +2 If there 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 questions is unanswered);

Negative Marks: −2 In all other cases.

7. Let b be a nonzero real number. Suppose f = ℝ → ℝ is a differentiable function such that f(0) = 1. If the derivative f’ of f satisfies the equation

for all x ∈ ℝ, then which of the following statements is/are TRUE?

(A) If b > 0, then f is an increasing function

(B) If b < 0, then f is a decreasing function

(C) f(x) f(−x) = 1 for all x ∈ ℝ

(D) f(x) − f(− x) = 0 for all x ∈ ℝ

8. Let a and b be positive real numbers such that a > 1 and b < a. Let P be a point in the first quadrant that lies on the hyperbola  Suppose the tangent to the hyperbola at P passes through the point (1, 0), and suppose the normal to the hyperbola at P cuts off equal intercepts on the coordinate axes. Let Δ denote the area of the triangle formed by the tangent at P, the normal at P and the x-axis. If e denotes the eccentricity of the hyperbola, then which of the following statements is/are TRUE?

(A) 1 < e < √2

(B) √2 < e < 2

(C) ∆ = a4

(D) ∆ = b4

9. Let f: R → R and g: R → R be functions satisfying f(x + y) = f(x) + f(y) + f(x)f(y) and f(x) = xg(x)

For all x, y ∈ R. If  then which of the following statements is/are TRUE?

(A) f is differentiable at every x ∈ R

(B) If g(0) = 1,then g is differentiable at every x ∈ R

(C) The derivative f′(1) is equal to 1

(D) The derivative f′(0) is equal to 1

10. Let α, β, γ, δ be real numbers such that α2 + β2 + γ2 ≠ 0 and α + γ = 1. Suppose the point (3, 2, −1) is the mirror image of the point (1, 0, −1) with respect to the plane αx + βy + γz = δ. Then which of the following statements is/are TRUE?

(A) α + β = 2

(B) δ − γ = 3

(C) δ + β = 4

(D) α + β + γ = δ

11. Let a and b be positive real numbers. Suppose  are adjacent sides of a parallelogram PQRS. Let  be the projection vectors of   If  and if the area of parallelogram PQRS is 8, then which of the following statements is/are TRUE?

(A) a + b = 4

(B) a – b = 2

(C) The length of the diagonal PR of the parallelogram PQRS is 4

(D)  is an angle bisector of the vectors

12. For nonnegative integers s and r, let

For positive integers m and n, let

where for any nonnegative integer p,

Then which of the following statements is/are TRUE?

(A) (m, n) = (n, m) for all positive integers m, n

(B) (m, n + 1) = (m + 1, n) for all positive integers m, n

(C) (2m, 2n) = 2 (m, n) for all positive integers m, n

(D) (2m, 2n) = ((m, n))2 for all positive integers m, n

SECTION-3

• This section contains SIX (06) 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 : +4 If ONLY the correct numerical value is entered;

Zero Marks : 0 In all other cases.

13. An engineer is required to visit a factory for exactly four days during the first 15 days of every month and it is mandatory that no two visits take place on consecutive days. Then the number of all possible ways in which such visits to the factory can be made by the engineer during 1-15 June 2021 is_____.

14. In a hotel, four rooms are available. Six persons are to be accommodated in these four rooms in such a way that each of these rooms contains at least one person and at most two persons. Then the number of all possible ways in which this can be done is_____.

15. Two fair dice, each with faces numbered 1, 2, 3, 4, 5 and 6, are rolled together and the sum of the numbers on the faces is observed. This process is repeated till the sum is either a prime number or a perfect square. Suppose the sum turns out to be a perfect square before it turns out to be a prime number. If p is the probability that this perfect square is an odd number, then the value of 14 p is _____

16. Let the function f: [0, 1] → R be defined by

Then the value of  is______

17. Let f:R → R be a differentiable function such that its derivative f’ is continuous and f(π) = − If F: [0, π ] → R is defined by  and if  then the value of f(0) is _______

18. Let the function f: (0, π) → R be defined by f(θ) = (sin θ + cos θ)2 + (sin θ − cos θ)4 . Suppose, the function f has a local minimum at θ precisely when θ ∈ {λ1 π, … , λr π}, where 0 < λ1 < ⋯ < λr < 1.Then the value of λ1 + ⋯ + λr is _____

## JEE Advanced Exam 2020 Paper-1 Code-E Question Paper With Answer Key

PHYSICS

Section 1 (Maximum Marks:18)

• This Section contains Four (06) Questions.

• Each question has FOUR options. 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.

1. A football of radius R is kept on a hole of radius r (r < R) made on a plank kept horizontally. One end of the plank is now lifted so that it gets tilted making an angle θ from the horizontal as shown in the figure below. The maximum value of θ so that the football does not start rolling down the plank satisfies (the figure is schematic and not drawn to scale)

(A) sin θ = r/R

(B) tan θ = r/R

(C) sin θ = r/2R

(D) cos θ = r/2R

2. A light disc made of aluminium (a nonmagnetic material) is kept horizontally and is free to rotate about its axis as shown in the figure. A strong magnet is held vertically at a point above the disc away from its axis. On revolving the magnet about the axis of the disc, the disc will (the figure is schematic and not drawn to scale);

(A) rotate in the direction opposite to the direction of magnet’s motion

(B) rotate in the same direction as the direction of magnet’s motion

(C) not rotate and its temperature will remain unchanged

(D) not rotate but its temperature will slowly rise

3. A small roller of diameter 20 cm has an axle of diameter 10 cm (see figure below on the left). It is on a horizontal floor and a meter scale is positioned horizontally on its axle with one edge of the scale on top of the axle (see figure on the right). The scale is now pushed slowly on the axle so that it moves without slipping on the axle, and the roller starts rolling without slipping. After the roller has moved 50 cm, the position of the scale will look like (figures are schematic and not drawn to scale);

4. A circular coil of radius R and N turns has negligible resistance. As shown in the schematic figure, its two ends are connected to two wires and it is hanging by those wires with its plane being vertical. The wires are connected to a capacitor with charge Q through a switch. The coil is in a horizontal uniform magnetic field Bo parallel to the plane of the coil. When the switch is closed, the capacitor gets discharged through the coil in a very short time. By the time the capacitor is discharged fully, magnitude of the angular momentum gained by the coil will be (assume that the discharge time is so short that the coil has hardly rotated during this time)

(A)

(B) πNQB0R2

(C) 2πNQB0R2

(D) 4πNQB0R2

5. A parallel beam of light strikes a piece of transparent glass having cross section as shown in the figure below. Correct shape of the emergent wave front will be (figures are schematic and not drawn to scale)

6. An open-ended U-tube of uniform cross-sectional area contains water (density 103kg m−3). Initially, the water level stands at 0.29 m from the bottom in each arm. Kerosene oil (a water-immiscible liquid) of density 800 kg m−3 is added to the left arm until its length is 0.1 m, as shown in the schematic figure below. The ratio (h1/ h2) of the heights of the liquid in the two arms is

(A) 15/14

(B) 35/33

(C) 7/6

(D) 5/4

SECTION 2 (Maximum Marks: 24)

• This section contains SIX (06) questions.

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

• 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       : +4 If only (all) the correct option(s) is (are) chosen;

Partial Marks : +3 If all four options is correct but ONLY three options are chosen;

Partial Marks : +2 If there 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 questions is unanswered);

Negative Marks: −2 In all other cases.

7. A particle of mass m moves in circular orbits with potential energy V(r) = Fr, where F is a positive constant and r is its distance from the origin. Its energies are calculated using the Bohr model. If the radius of the particle’s orbit is denoted by R and its speed and energy are denoted by v and E, respectively, then for the nth orbit (here h is the Planck’s constant)

(A) R ∝ 11/3 and v ∝ n2/3

(B) R ∝ n2/3 and v ∝ n1/3

(C)

(D)

8. The filament of a light bulb has a surface area 64 mm2. The filament can be considered as a black body at temperature 2500 K emitting radiation like a point source when viewed from far. At night the light bulb is observed from a distance of 100 m. Assume the pupil of the eyes of the observer to be circular with radius 3 mm. Then

(Take Stefan-Boltzmann constant = 5.67 × 10−8 Wm−2K−4,

Wien’s displacement constant = 2.90 × 10−3m-K,

Planck’s constant = 6.63 × 10−34Js,

speed of light in vacuum = 3.00 ×108 ms−1)

(A) power radiated by the filament is in the range 642 W to 645 W

(B) radiated power entering into one eye of the observer is in the range 3.15 × 10−8 W to 3.25 × 10−8 W

(C) the wavelength corresponding to the maximum intensity of light is 1160 nm

(D) taking the average wavelength of emitted radiation to be 1740 nm, the total number of photons entering per second into one eye of the observer is in the range 2.75 × 1011 to 2.85 × 1011

9. Sometimes it is convenient to construct a system of units so that all quantities can be expressed in terms of only one physical quantity. In one such system, dimensions of different quantities are given in terms of a quantity X as follows: [position] = [Xα]; [speed] = [Xβ]; [acceleration] =[Xp]; [linear momentum] = [Xq]; [force] = [Xr]. Then

(A) α + 𝑝 = 2β

(B) p + q − r = β

(C) − q + r = α

(D) p + q + r = β

10. A uniform electric field,  is applied in a region. A charged particle of mass m carrying positive charge q is projected in this region with an initial speed of 2√10 × 106 ms1. his particle is aimed to hit a target T, which is 5 m away from its entry point into the field as shown schematically in the figure. Take q/m = 1010CKg1. Then

(A) the particle will hit T if projected at an angle 45° from the horizontal

(B) the particle will hit T if projected either at an angle 30° or 60° from the horizontal

(C) time taken by the particle to hit T could be

(D) time taken by the particle to hit T is

11. Shown in the figure is a semicircular metallic strip that has thickness t and resistivity ρ. Its inner radius is R1 and outer radius is R2. If a voltage V0 is applied between its two ends, a current I flows in it. In addition, it is observed that a transverse voltage ∆V develops between its inner and outer surfaces due to purely kinetic effects of moving electrons (ignore any role of the magnetic field due to the current). Then (figure is schematic and not drawn to scale)

(A)

(B) the outer surface is at a higher voltage than the inner surface

(C) the outer surface is at a lower voltage than the inner surface

(D) ∆VαI2

12. As shown schematically in the figure, two vessels contain water solutions (at temperature T) of potassium permanganate (KMnO4) of different concentrations 𝑛1 and n2(n1 > n2) molecules per unit volume with ∆𝑛 = (n1 – n2) ≪ n1. When they are connected by a tube of small length l and cross-sectional area S, KMnO4 starts to diffuse from the left to the right vessel through the tube. Consider the collection of molecules to behave as dilute ideal gases and the difference in their partial pressure in the two vessels causing the diffusion. The speed v of the molecules is limited by the viscous force −βv on each molecule, where β is a constant. Neglecting all terms of the order (∆n)2, which of the following is/are correct? (kB is the Boltzmann constant)

(A) the force causing the molecules to move across the tube is ∆nkBTS

(B) force balance implies n1βvl = ∆nkBT

(C) total number of molecules going across the tube per sec is

(D) rate of molecules getting transferred through the tube does not change with time

SECTION-3 (Maximum Marks: 24)

• This section contains SIX (06) 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 : +4 If ONLY the correct numerical value is entered;

Zero Marks : 0 In all other cases.

13. Put a uniform meter scale horizontally on your extended index fingers with the left one at 0.00 cm and the right one at 90.00 cm. When you attempt to move both the fingers slowly towards the centre, initially only the left finger slips with respect to the scale and the right finger does not. After some distance, the left finger stops and the right one start slipping. Then the right finger stops at a distance xR from the centre (50.00 cm) of the scale and the left one starts slipping again. This happens because of the difference in the frictional forces on the two fingers. If the coefficients of static and dynamic friction between the fingers and the scale are 0.40 and 0.32, respectively, the value of xR (in cm) is ______.

14. When water is filled carefully in a glass, one can fill it to a height h above the rim of the glass due to the surface tension of water. To calculate h just before water starts flowing, model the shape of the water above the rim as a disc of thickness h having semicircular edges, as shown schematically in the figure. When the pressure of water at the bottom of this disc exceeds what can be withstood due to the surface tension, the water surface breaks near the rim and water starts flowing from there. If the density of water, its surface tension and the acceleration due to gravity are 103kg m−3, 0.07 Nm−1 and 10 ms−2, respectively, the value of h (in mm) is _________.

15. One end of a spring of negligible unstretched length and spring constant k is fixed at the origin (0, 0). A point particle of mass m carrying a positive charge q is attached at its other end. The entire system is kept on a smooth horizontal surface. When a point dipole 𝑝 pointing towards the charge q is fixed at the origin, the spring gets stretched to a length l and attains a new equilibrium position (see figure below). If the point mass is now displaced slightly by ∆l ≪ 𝑙from its equilibrium position and released, it is found to oscillate at frequency  The value of δ is ______.

16. Consider one mole of helium gas enclosed in a container at initial pressure P1 and volume V1. It expands isothermally to volume 4V1. After this, the gas expands adiabatically and its volume becomes 32V1. The work done by the gas during isothermal and adiabatic expansion processes are Wiso and Wadia, respectively. If the ratio  then f is ________.

17. A stationary tuning fork is in resonance with an air column in a pipe. If the tuning fork is moved with a speed of 2 ms−1 in front of the open end of the pipe and parallel to it, the length of the pipe should be changed for the resonance to occur with the moving tuning fork. If the speed of sound in air is 320 ms−1, the smallest value of the percentage change required in the length of the pipe is ____________.

18. A circular disc of radius R carries surface charge density  where 𝜎0 is a constant and r is the distance from the center of the disc. Electric flux through a large spherical surface that encloses the charged disc completely is ϕ0. Electric flux through another spherical surface of radius R/4 and concentric with the disc is ϕ. Then the ratio  is _________.

CHEMISTRY

Section 1 (Maximum Marks:18)

• This Section contains Four (06) Questions.

• Each question has FOUR options. 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.

1. If the distribution of molecular speeds of gas is as per the figure shown below, then the ratio of the most probable, the average, and the root mean square speeds, respectively, is

(A) 1 : 1 : 1

(B) 1 : 1 : 1.224

(C) 1 : 1.128 : 1.224

(D) 1 : 1.128 : 1

2. Which of the following liberates O2 upon hydrolysis?

(A) Pb3O2

(B) KO2

(C) Na2O2

(D) Li2O2

3. A colourless aqueous solution contains nitrates of two metals, X and Y. When it was added to an aqueous solution of NaCl, a white precipitate was formed. This precipitate was found to be partially soluble in hot water to give a residue P and a solution Q. The residue P was soluble in aq. NH3 and also in excess sodium thiosulfate. The hot solution Q gave a yellow precipitate with KI. The metals X and Y, respectively, are

(A) Ag and Pb

(B) Ag and Cd

(C) Cd and Pb

(D) Cd and Zn

4. Newman projections P, Q, R and S are shown below:

Which one of the following options represents identical molecules?

(A) P and Q

(B) Q and S

(C) Q and R

(D) R and S

5. Which one of the following structures has the IUPAC name 3-ethynyl-2-hydroxy-4-methylhex-3-en-5-ynoic acid?

6. The Fischer projection of D-erythrose is shown below.

D-Erythrose and its isomers are listed as P, Q, R, and S in Column-I. Choose the correct relationship of P, Q, R, and S with D-erythrose from Column II.

(A) P→2, Q→3, R→2, S→2

(B) P→3, Q→1, R→1, S→2

(C) P→2, Q→1, R→1, S→3

(D) P→2, Q→3, R→3, S→1

SECTION 2 (Maximum Marks: 24)

• This section contains SIX (06) questions.

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

• 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       : +4 If only (all) the correct option(s) is (are) chosen;

Partial Marks : +3 If all four options is correct but ONLY three options are chosen;

Partial Marks : +2 If there 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 questions is unanswered);

Negative Marks: −2 In all other cases.

7. In thermodynamics, the P − V work done is given by w = −∫dVPext

For a system undergoing a particular process, the work done is,

This equation is applicable to a

(A) System that satisfies the van der Waals equation of state.

(B) Process that is reversible and isothermal.

(C) Process that is reversible and adiabatic.

(D) Process that is irreversible and at constant pressure.

8. With respect to the compounds I-V, choose the correct statement(s).

(A) The acidity of compound I is due to delocalization in the conjugate base.

(B) The conjugate base of compound IV is aromatic.

(C) Compound II becomes more acidic, when it has a -NO2 substituent.

(D) The acidity of compounds follows the order I >IV>V>II>III.

9. In the reaction scheme shown below, Q, R, and S are the major products.

The correct structure of

10. Choose the correct statement(s) among the following:

(A) [FeCl4] has tetrahedral geometry.

(B) [Co(en)(NH3)2Cl2]+has 2 geometrical isomers.

(C) [FeCl4] has higher spin-only magnetic moment than [Co(en)(NH3)2Cl2]+.

(D) The cobalt ion in [Co(en)(NH3)2Cl2]+ has sp3d2 hybridization.

11. With respect to hypochlorite, chlorate and perchlorate ions, choose the correct statement(s).

(A) The hypochlorite ion is the strongest conjugate base.

(B) The molecular shape of only chlorate ion is influenced by the lone pair of electrons of Cl.

(C) The hypochlorite and chlorate ions disproportionate to give rise to an identical set of ions.

(D) The hypochlorite ion oxidizes the sulfite ion.

12. The cubic unit cell structure of a compound containing cation M and anion X is shown below. When compared to the anion, the cation has a smaller ionic radius. Choose the correct statement(s).

(A) The empirical formula of the compound is MX.

(B) The cation M and anion X have different coordination geometries.

(C) The ratio of M-X bond length to the cubic unit cell edge length is 0.866.

(D) The ratio of the ionic radii of cation M to anion X is 0.414.

SECTION-3 (Maximum Marks: 24)

• This section contains SIX (06) 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 : +4 If ONLY the correct numerical value is entered;

Zero Marks : 0 In all other cases.

13. 5.00 mL of 0.10 M oxalic acid solution taken in a conical flask is titrated against NaOH from a burette using phenolphthalein indicator. The volume of NaOH required for the appearance of permanent faint pink color is tabulated below for five experiments. What is the concentration, in molarity, of the NaOH solution?

 Exp. No. Vol. of NaOH (mL) 1 12.5 2 10.5 3 9.0 4 9.0 5 9.0

14. Consider the reaction A ⇌ B at 1000 K. At time 𝑡’, the temperature of the system was increased to 2000 K and the system was allowed to reach equilibrium. Throughout this experiment, the partial pressure of A was maintained at 1 bar. Given below is the plot of the partial pressure of B with time.

What is the ratio of the standard Gibbs energy of the reaction at 1000 K to that at 2000 K?

15. Consider a 70% efficient hydrogen-oxygen fuel cell working under standard conditions at 1 bar and 298 K. Its cell reaction is

The work derived from the cell on the consumption of 1.0 × 10−3 mol of H2(g) is used to compress 1.00 mol of a monoatomic ideal gas in a thermally insulated container. What is the change in the temperature (in K) of the ideal gas?

The standard reduction potentials for the two half-cells are given below.

O2(g) + 4H+(aq) + 4e → 2H2O (l), E° = 1.23 V,

2H+(aq) + 2e → H2(g), E° = 0.00 V

Use F = 96500 C mol1, R = 8.314 J mol1 K1.

16. Aluminum reacts with sulfuric acid to form aluminum sulfate and hydrogen. What is the volume of hydrogen gas in liters (L) produced at 300 K and 1.0 atm pressure, when 5.4 g of aluminum and 50.0 mL of 5.0 M sulfuric acid are combined for the reaction?

(Use molar mass of aluminum as 27.0 g mol−1, R = 0.082 atm L mol−1 K−1)

17. is known to undergo radioactive decay to form  by emitting alpha and beta particles. A rock initially contained 68 × 10−6 g of  If the number of alpha particles that it would emit during its radioactive decay of  in three half-lives is Z × 1018, then what is the value of Z?

18. In the following reaction, compound Q is obtained from compound P via an ionic intermediate.

What is the degree of unsaturation of Q?

MATHEMATICS

Section 1 (Maximum Marks:18)

• This Section contains Four (06) Questions.

• Each question has FOUR options. 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.

1. Suppose a, b denote the distinct real roots of the quadratic polynomial x2 + 20x − 2020 and suppose c, d denote the distinct complex roots of the quadratic polynomial x2 – 20x + 2020. Then the value of ac(a – c) + ad(a – d) + bc(b – c) + bd(b – d) is

(A) 0

(B) 8000

(C) 8080

(D) 16000

2. If the function f: R ⟶ R is defined by (x) = |x| (x – sinx), then which of the following statements is TRUE?

(A) f is one-one, but NOT onto

(B) f is onto, but NOT one-one

(C) f is BOTH one-one and onto

(D) f is NEITHER one-one NOR onto

3. Let the functions: R ⟶ R and g : R ⟶ R be defined by

Then, the area of the region in the first quadrant bounded by the curves y = (x), y = g(x) and x = 0 is

(A)

(B)

(C)

(D)

4. Let a, b and λ be positive real numbers. Suppose P is an end point of the latus rectum of the parabola y2 = 4λx, and suppose the ellipse  passes through the point𝑃. If the tangents to the parabola and the ellipse at the point P are perpendicular to each other, then the eccentricity of the ellipse is

(A) 1/√2

(B) 1/2

(C) 1/3

(D) 2/5

5. Let C1 and C2 be two biased coins such that the probabilities of getting head in a single toss are 2/3 and 1/3, respectively. Suppose α is the number of heads that appear when C1 is tossed twice, independently, and suppose α is the number of heads that appear when C2 is tossed twice, independently. Then the probability that the roots of the quadratic polynomial x2 − αx + β are real and equal, is

(A) 40/81

(B) 20/81

(C) 1/2

(D) 1/4

6. Consider all rectangles lying in the region and having one side on the x-axis. The area of the rectangle which has the maximum perimeter among all such rectangles, is

(A) 3π/2

(B) π

(C) π/2√3

(D) π√3/2

SECTION 2 (Maximum Marks: 24)

• This section contains SIX (06) questions.

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

• 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       : +4 If only (all) the correct option(s) is (are) chosen;

Partial Marks : +3 If all four options is correct but ONLY three options are chosen;

Partial Marks : +2 If there 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 questions is unanswered);

Negative Marks: −2 In all other cases.

7. Let the function f : R → R be defined by (x) = x3 − x2 + (x − 1) sin 𝑥 and let g : R → R be an arbitrary function. Let fg : R → R be the product function defined by (fg)(𝑥) = f(x)g(x). Then which of the following statements is/are TRUE?

(A) If g is continuous at x = 1, then fg is differentiable at x = 1

(B) If fg is differentiable at x = 1, then g is continuous at x = 1

(C) If g is differentiable at x = 1, then fg is differentiable at x = 1

(D) If fg is differentiable at x = 1, then g is differentiable at x = 1

8. Let M be a 3 × 3 invertible matrix with real entries and let I denote the 3 × 3 identity matrix. If M−1 = adj (adj M), then which of the following statements is/are ALWAYS TRUE?

(A) M = I

(B) det M = 1

(C) M2 = I

9. Let S be the set of all complex numbers z satisfying |z2 + z + 1| = 1. Then which of the following statements is/are TRUE?

(A)

(B) |z| ≤ 2 for all z ∈ S

(C)

(D) The set S has exactly four elements

10. Let x, y and z be positive real numbers. Suppose x, y and z are the lengths of the sides of a triangle opposite to its angles X, Y and Z, respectively. If  then which of the following statements is/are TRUE?

(A) 2Y = X + Z

(B) Y = X + Z

(C)

(D) x2 + z2 – y2 = xz

11. Let L1 and L2 be the following straight line .

Suppose the straight line

Lies in the plane containing L1 and L2, and passes through the point of intersection of L1 and L2. If the line L bisects the acute angle between the lines L1 and L2, then which of the following statements is/are TRUE?

(A) α − γ

(B) l + m = 2

(C) α – γ  = 1

(D) l + m = 0

12. Which of the following inequalities is/are TRUE?

(A)

(B)

(C)

(D)

SECTION-3 (Maximum Marks: 24)

• This section contains SIX (06) 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 : +4 If ONLY the correct numerical value is entered;

Zero Marks : 0 In all other cases.

13. Let m be the minimum possible value of  where y1, y2, y3 are real numbers for which y1 + y2 + y3 = 9. Let M be the maximum possible value of (log3x1 + log3x2 + log3x3), where x1, x2, x3 are positive real numbers for which x1 + x2 + x3 = 9. Then value of log2(m3) + log3(M2) is _______

14. Let a1, a2, a3,… be a sequence of positive integers in arithmetic progression with common difference 2. Also, let b1, b2, b3,… be a sequence of positive integers in geometric progression with common ratio 2. If a1 = b1 = c, then the number of all possible values of c, for which the equality 2(a1 + a2 + ⋯ + an) = b1 + b2 + ⋯ + bn holds for some positive integer n, is _____

15. Let f : [0, 2] → R be the function defined by

If α, β ∈ [0, 2] are such that {x ∈ [0, 2] : f(x) ≥ 0} = [α, β], then the value of β – α is ________

16. In a triangle PQR, let  If and

Then the value of  is _________

17. For a polynomial g(x) with real coefficients, let 𝑚𝑔 denote the number of distinct real roots of g(x). Suppose S is the set of polynomials with real coefficients defined by S = {(x2 – 1)2(a0 + a1x + a2x2 + a3x3) : a0, a1, a2, a3 ∈ R}

For a polynomial f, let f’ and f” denote its first and second order derivatives, respectively.

Then the minimum possible value of (mf + mf), where f ∈ S, is _____