**JEE (Advanced Examination – 2022)**

**(Held on Sunday 28 ^{th} 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 ρ_{A} and ρ_{B}, respectively. A and B have the same radius, and their masses M_{A} and M_{B} are related by M_{B} = 2M_{A}. 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 ρ_{A}. The entire mass lost by A is deposited as a thick spherical shell on B with the density of the shell being ρ_{A}. If ν_{A} and ν_{B} 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 MeVc^{−}^{2}. The value of n is _______.

3. In the following circuit C_{1} = 12 μF, C_{2} = C_{3} = 4 μF and C_{4} = C_{5} = 2 μ The Charge stored in C_{3} 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 s^{−}^{2}. 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 ms^{−}^{2}]

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 m^{2}. It is placed in a constant magnetic field of strength B_{0} which is perpendicular to the plane of the circuit. At time t = 0, the magnetic field strength starts increasing linearly as B = B_{0} + 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 R_{1} to R_{8}. And two ideal batteries with voltages ε_{1} = 12 V and ε_{2} = 6 V.

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

(A) The magnitude of current flowing through R_{1} is 7.2 A.

(B) The magnitude of current flowing through R_{2} is 1.2 A.

(C) The magnitude of current flowing through R_{3} is 4.8 A.

(D) The magnitude of current flowing through R_{5} 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 A_{1} = 0.1 m^{2} and the upper end is A_{2} = 0.4 m^{2}. 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 E_{b}^{p} and the binding energy of a neutron be E_{b}^{n} in the nucleus.

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

(A) E_{b}^{p} – E_{b}^{n} is proportional to Z(Z – 1) where Z is the atomic number of the nucleus.

(B) E_{b}^{p} – E_{b}^{n} proportional to A^{−}^{1/3} where A is the mass number of the nucleus.

(C) E_{b}^{p} – E_{b}^{n} is positive

(D) E_{b}^{p} 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 ms^{−}^{1}) 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 O_{2} 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 (E^{0}, in V) of 4 MnO_{4}^{−}(aq)/Mn(s) is ______.

3. A solution is prepared by mixing 0.01 mol each of H_{2}CO_{3}, NaHCO_{3}, Na_{2}CO_{3}, and NaOH in 100 mL of water. pH of the resulting solution is ______.

[Given : pKa_{1} and pKa_{2} of H_{2}CO_{3} are 6.37 and 10.32, respectively ; log 2 = 0.30]

4. The treatment of an aqueous solution of 3.74 g of Cu(NO_{3})_{2} with excess KI results in a brown solution along with the formation of a precipitate. Passing H_{2}S 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 2_{pz} 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 Al_{2}O_{3} with coke (C) at a temperature > 2500°C.

(B) the neutralization of aluminate solution by passing CO_{2} gas to precipitate hydrated alumina (Al_{2}O_{3}.3H_{2}O)

(C) the dissolution of Al_{2}O_{3} in hot aqueous NaOH.

(D) the electrolysis of Al_{2}O_{3} mixed with Na_{3}AlF_{6} to give Al and CO_{2}.

12. The treatment of galena with HNO_{3} 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 NaBH_{4}.

(B) Treating P with conc. NH_{4}OH solution followed acidification gives Q.

(C) Treating Q with a solution of NaNO_{2} in aq. HCl liberates N_{2}.

(D) P is more acidic than CH_{3}CH_{2}COOH.

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. X_{s} 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 l_{1}, l_{2},…., l_{100} be consecutive terms of an arithmetic progression with common difference d_{1}, and let w_{1}, w_{2},…., w_{100} be consecutive terms of another arithmetic progression with common difference d_{2}, where d_{1}d_{2} = 10. For each i = 1, 2,….,100, let R_{i} be a rectangle with length l_{i}, width w_{i} and area A_{i}. If A_{51} – A_{50} = 1000, then the value of A_{100} – A_{90} 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 a_{1}, a_{2}, a_{3},… be an arithmetic progression with a_{1} = 7 and common difference 8. Let T_{1}, T_{2}, T_{3},… be such that T_{1} = 3 and T_{n+1} – T_{n} = a_{n} for n ≥ Then, which of the following is/are TRUE?

(A) T_{20} = 1604

(B)

(C) T_{30} = 3454

(D)

11. Let P_{1} and P_{2} be two planes given by

P_{1} : 10x + 15y + 12z – 60 = 0.

P_{2} : −2x + 5y + 4z – 20 = 0.

Which of the following straight lines can be an edge of some tetrahedron whose two faces lie on P_{1} and P_{2}?

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 y^{2} = 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 P_{1} and P_{2}. Let Q_{1} and Q_{2} be points on the lines SP_{1} and SP_{2} respectively such that PQ_{1} is perpendicular to SP_{1} and PQ_{2} is perpendicular to SP_{2}. Then, which of the following is/are TRUE?

(A) SQ_{1} = 2

(B)

(C) PQ_{1} = 3

(D) SQ_{2} = 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, P_{1} and P_{2}, 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 P_{1} and P_{2}, respectively. If x > y, then P_{1} scores 5 points and P_{2} scores 0 points. If x = y, then each player scores 2 points. If x < y, then P_{1} scores 0 point and P_{2} scores 5 points. Let X_{i} and Y_{i} be the total scores of P_{1} and P_{2}, respectively, after playing the i^{th} 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 10^{th}, 100^{th} and 1000^{th} 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)

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