**JEE-Advanced-2021-Question-Paper-1**

**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 L_{1} and a concave lens L_{2} 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/s^{2}.

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 q_{1} µC. Then S is switched to position Q. After a long time, the charge on the capacitor is q_{2} µC.

7. The magnitude of q_{1 _____.}

8. The magnitude of q_{2 ______.}

**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 n_{1} and n_{2} is placed in the air as shown in the figure. A ray of light is incident from medium n1 to n_{2} at an angle, where sin is slightly larger than 1/n_{1}. Take the refractive index of air as 1. Which of the following statement(s) is(are) correct?

(A) The light ray enters air if n_{2} = n_{1}

(B) The light ray is finally reflected back into the medium of refractive index n1 if n_{2} < n_{1}

(C) The light ray is finally reflected back into the medium of refractive index n_{1} if n_{2} > n_{1}

(D) The light ray is reflected back into the medium of refractive index n_{1} if n_{2} = 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/s^{2} 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 λ_{0} 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 C_{0} = 5.0 F are connected in series between the rails. At time t = 0, C_{0} is uncharged. Which of the following statement(s) is(are) correct? [μ_{0} = 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 C_{0} is 8.4 × 10^{–12} coulomb

(D) Maximum charge on capacitor C_{0} 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. P_{1} and P_{2} are pressures at points 1 and 2, respectively, located at the base of the tube. Let β = (P_{1} – P_{2})/(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 r_{α} and r_{s} The ratio r_{s}/r_{α} 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 = t_{1} and 50 K at t = t_{2}. Assume the object and the container to be ideal black bodies. The heat capacity of the object does not depend on temperature. The ratio (t_{1}/t_{2}) 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(NH_{3})_{6}]^{3+} and [CuF_{6}]^{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 p_{Z} 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 K^{−}^{1} mol^{−}^{1})

7. The value of standard enthalpy, ∆H^{∅} (in kJ mol^{−}^{1}) 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 × 10^{−}^{2} 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), K_{b} = 0.5 K kg mol^{−}^{1}; 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 HNO_{3} 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 Na_{2}O_{2} followed by the addition of BaCl_{2}. 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 SO_{2}.

(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(NO_{3})_{2} and Zn(NO_{3})_{2}

(B) Pb(NO_{3})_{2} and Bi(NO_{3})_{2}

(C) AgNO_{3} and Bi(NO_{3})_{3}

(D) Pb(NO_{3})_{2} and Hg(NO_{3})_{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 sp^{2} hybridization in the major product P is ______.

19. The total number of possible isomers for [Pt(NH_{3})_{4}Cl_{2}]Br_{2} 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) x^{2} + y^{2} − 3x + y = 0

(B) x^{2} + y^{2} + x + 3y = 0

(C) x^{2} + y^{2} + 2y − 1 = 0

(D) x^{2} + y^{2} + 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 E_{1} = {1, 2, 3}, F_{1} = {1, 3, 4} and G_{1} = {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 E_{2} = E_{1} − S_{1} and F_{2} = F_{1} ⋃ S_{1}. Now two elements are chosen at random, without replacement, from the set F_{2} and let S_{2} denote the set of these chosen elements.

Let G_{2} = G_{1} ⋃ S_{2}. Finally, two elements are chosen at random, without replacement from the set G_{2} and let S_{3} denote the set of these chosen elements. Let E_{3} = E_{2} ⋃ S_{3}. Given that E_{1} = E_{3}, let p be the conditional probability of the event S_{1} = {1, 2}. Then the value of p is;

(A) 1/5

(B) 3/5

(C) 1/2

(D) 2/5

4. Let θ_{1}, θ_{2}, …., θ_{10} be positive valued angles (in radian) such that θ_{1}+ θ_{2}+ ….+ θ_{10} = 2π. Define the complex numbers for k = 2, 3, …, 10, where i = √− Consider the statements P and Q given below:

P: |z_{2} − z_{1}| + |z_{3} − z_{2}| + …. +|z_{10} − z_{9}| + |z_{1} − z_{10}| ≤ 2π

Q: |z_{2}^{2} − z_{1}^{2}| + |z_{3}^{2} − z_{2}^{2}| + …. +|z_{10}^{2} − z_{9}^{2}| + |z_{1}^{2} − z_{10}^{2}| ≤ 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 p_{1} be the probability that the maximum of chosen numbers is at least 81 and p_{2} 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 L_{1} and L_{2} defined by

L_{1} : x√2 + y − 1 = 0 and L_{2} : 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 L_{1} and the distance of P from L_{2} is λ_{2}. 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 λ^{2 }______.

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 + PFQ^{−}^{1}| = |EQ| + |PFQ^{−}^{1}|

(C) |(EF)^{3}| > |EF|^{2}

(D) Sum of the diagonal entries of P^{−}^{1}EP + F is equal to the sum of diagonal entries of E + P^{−}^{1}FP

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 H^{c} denotes its complement, then which of the following statements is(are) TRUE?

(A) P(E ⋂ F ⋂ G^{c}) ≤ 1/40

(B) P(E^{c} ⋂ F ⋂ G) ≤ 1/15

(C) P(E ⋃ F ⋃ G) ≤ 13/24

(D) P(E^{c} ⋂ F^{c }⋂ G^{c}) ≤ 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 S_{n} : (0, ∞) → R be defined by

where for any x ∈ R, cot^{−}^{1} (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 x^{2} + y^{2} + 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 3x^{2} – 4|x^{2} – 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 ____.

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