JEE Advanced 2021 Paper 1

The smallest division on the main scale of a 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 calipers 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

1. 3.07 cm
2. 3.11 cm
3. 3.15 cm
4. 3.17 cm

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

1. steps 1 and 2
2. steps 1 and 3
3. steps 1 and 4
4. steps 2 and 4

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 magnifica tion of th is lens system is

1. 0.4
2. 0.8
3. 1.3
4. 1.6

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

1. 50
2. 75
3. 350
4. 525 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 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 f rom the point O. The acceleration due to gravity 𝑔=10 m/s2.

The value of 𝑡 is ___ .

Numerical answer type

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. 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 :

Numerical answer type

The magnitude of 𝑞1 is ___ .

Numerical answer type

The magnitude of 𝑞2 is ___ . 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 center at ( b, 0). All lengths are measured in meters.

Numerical answer type

The value of 𝑅 is ___ meter.

Numerical answer type

The value of 𝑏 is ___ meter. SECTION 3  This section contains SIX (06) questions .  Each question has FOUR options

1. ,
2. ,
3. and
4. . ONE OR MORE THAN ONE of these four option(s) is (are) c orrect 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 :  For example, in a question, if (A), (B ) and
5. are the ONLY three options corresponding to correct answers , then choosing ONLY (A),
6. and
7. 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
8. will get +1 mark; choosing ONLY
9. 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.

A horizontal force 𝐹 is applied at the center 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 𝜇. The center 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?

1. For the same F, the value of 𝑎 does not depend on whether the cylinder is solid or hollow
2. For a solid cylinder,the maximum possible value of 𝑎 is 2𝜇𝑔
3. The magnitude of the frictional force on the object due to the ground is always 𝜇𝑚𝑔
4. For a thin -walled hollow cylinder , 𝑎=𝐹 2𝑚

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

1. The light ray enters air if 𝑛2=𝑛1
2. The light ray is finally reflected back into the medium of refractive index 𝑛1 if 𝑛2<𝑛1
3. The light ray is finally reflected back into the medium of refractive index 𝑛1 if 𝑛2>𝑛1
4. The light ray is reflected back into the medium of refractive index 𝑛1 if 𝑛2=1

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 𝐿⃗ and 𝜏 , respectively. 𝑖̂, 𝑗̂ and 𝑘̂ are unit vector s along the positive x, y and z-direction s, respectively . If 𝑘̂=𝑖̂ × 𝑗̂ then which of the following statement (s) is(are) correct?

1. The particle arrives at the point (𝑥=𝑙, 𝑦=−ℎ) at time 𝑡=2 s
2. 𝜏 = 2 𝑘̂ when the particle passes through the point ( 𝑥=𝑙, 𝑦=−ℎ)
3. 𝐿⃗ = 4 𝑘̂ when the particle passes through the point ( 𝑥=𝑙, 𝑦=−ℎ)
4. 𝜏 = 𝑘̂ when the particle passes through the point ( 𝑥=0, 𝑦=−ℎ)

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

1. The ratio of the longest wavelength to the shortest wavelength in Balmer series is 9/5
2. There is an overlap between the wavelength ranges of Balmer and Paschen series
3. The wavelengths of Lyman series are given by (1+1 𝑚2)𝜆0, where 𝜆0 is the shortest wavelength of Lyman series and 𝑚 is an integer
4. The wavelength ranges of Lyman and Balmer series do not overlap

A long straight wire carries a current , 𝐼=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 distance s 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 Ω and a capacitor 𝐶𝑜=5.0 𝜇F are connected in series between the rails. At time 𝑡=0, 𝐶𝑜 is uncharged. Which of the following statement(s) is(are) correct? [𝜇0=4𝜋×10−7 SI units. Take ln2=0.7]

1. Maximum current through 𝑅 is 1.2 ×10−6 ampere
2. Maximum current through 𝑅 is 3.8 ×10−6 ampere
3. Maximum charge on capacitor 𝐶𝑜 is 8.4 ×10−12 coulomb
4. Maximum charge on capacitor 𝐶𝑜 is 2.4 ×10−12 coulomb

A cylindrical tube, with its base as shown in the figure, is filled with water. It is moving down with a constant acceleration 𝑎 along a fixed inclined plane with angle 𝜃=450. 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 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?

1. 𝛽=0 when 𝑎=𝑔/√2
2. 𝛽>0 when 𝑎=𝑔/√2
3. 𝛽=√2−1 √2 when 𝑎=𝑔/2
4. 𝛽=1 √2 when 𝑎=𝑔/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 :

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 uniform magnetic field which is normal to the velocities of the particles. With in this region, the 𝛼-particle and the sulfur ion move in circular orbits of radii 𝑟𝛼 and 𝑟𝑆, respectively. The ratio (𝑟𝑆/𝑟𝛼) is ___.

Numerical answer type

A thin rod of mass M and length 𝑎 is free to rotate in 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 center 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 Ω and the disc rotating about its vertical axis with angular velocity 4 Ω. The total angular momentum of the system about the point O is (𝑀𝑎2Ω 48)𝑛. The value of 𝑛 is ___.

Numerical answer type

A small object is placed at the center 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 contain er to be ideal black bodies. The heat capacity of the object does not depend on temperature. The ratio (𝑡2/𝑡1) is ___.

Numerical answer type

The major product formed in the following reaction is Na liq. NH3NaNH2

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

1. OHH MeOH H Me
2. MeH OHOH Me H
3. HMe OHOH Me H
4. MeH OHOH H Me

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 (packing fraction = packing efficiency 100)

1. 0.74
2. 0.63
3. 0.52
4. 0.48

The c alculated spin only magnetic moment s 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 )

1. 3.87 and 2.84
2. 4.90 and 1.73
3. 3.87 and 1.73
4. 4.90 and 2.84 Question Stem for Question Nos. 5 and 6 Question Stem For the following reaction scheme, percentage yields are given along the arrow: Mg2C3H2OP (4.0 g)NaNH2 MeI 75%Qred hot iron tube 873 K 40% Hg2+/H+ 333K100% SBa(OH)2 heat 80%TNaOCl 80%R (x g) U (y g)(decolourises Baeyer's reagent) x g and y g are mass of R and U, respectively. (Use: Molar mass (in g mol–1) of H, C and O as 1, 12 and 16, respectively)

The value of x is ___.

Numerical answer type

The value of y is ___. 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 k eypad.  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 : Question Stem for Question Nos. 7 and 8 Question Stem For the reaction, 𝐗(𝑠)⇌𝐘(𝑠)+𝐙(𝑔), the plot of ln𝑝𝐙 𝑝o versus 104 𝑇 is given below (in solid line), where 𝑝𝐙 is the pressure (in bar) of the gas Z at temperature T and 𝑝o = 1 bar . (Given, d(ln𝐾) d (1 𝑇)= −∆𝐻o 𝑅, where the equilibrium constant , 𝐾=𝑝𝑧 𝑝o and the gas constant, R = 8.314 J K–1 mol–1)

Numerical answer type

The value of standard enthalpy, ∆𝐻o (in kJ mol–1) for the given reaction is ___.

Numerical answer type

The value of ∆𝑆o (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 C. 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 C. (Assume: Densities of the solutions A and B are the same as that of water and the soluble salt s dissociate completely . Use: Molal elevation constant (Ebullioscopic Constant), 𝐾𝑏= 0.5 K kg mol1; Boiling point of pure water as 100 C.)

Numerical answer type

The value of x is ___.

Numerical answer type

The value of |y| is ___. SECTION 3  This section contains SIX (06) questions.  Each question has FOUR options

1. ,
2. ,
3. and
4. . ONE OR MORE THAN ONE of these four option(s) is (are) c orrect 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 :  For example, in a question, if (A),
5. and
6. are the ONLY three options corresponding to correct answers, then choosing ONLY (A),
7. and
8. will get +4 marks; choosing ONLY (A) and
9. wil l get +2 marks; choosing ONLY (A) and
10. will get +2marks; choosing ONLY
11. and
12. will get +2 marks; choosing ONLY (A) will get +1 mark; choosing ONLY
13. will get +1 mark; choosing ONLY
14. 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.

Given: CHO OH H H HO OH H OH H CH2OHHNO3 D-GlucoseP []D = +52.7 The compound(s), which on reaction with HNO 3 will give the product having degree of rotation, []D = −52.7 is(are)

1. CHO H HO H HO OH H OH H CH2OH
2. CH2OHHO HH OHHO HHO HCHO
3. CHO HO H H OH HO H HO H CH2OH
4. CH2OHOH HH HOOH HOH HCHO

The reaction of Q with PhSNa yields a n organic compound (major product) that gives positive Carius test on treatment with Na 2O₂ followed by addition of BaCl 2. The correct option(s) for Q is(are)

1. F O2N NO₂
2. O2N O2NI
3. Br O2NMeS
4. Cl MeSO2N

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

1. The process of precipitating colloidal sol by an electrolyte is called peptization.
2. Colloidal solution freezes at high er temperature than the true solution at the same concentration.
3. Surfactant s form micelle above critical micel le concentration (CMC) . CMC depends on temperature.
4. Micelles are macromolecular colloids.

An ideal gas undergoes a reversible isothermal expansion from 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 state I to state III is(are) (p: pressure, V: volume, T: temperature, H: enthalpy, S: entropy)

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

1. A mixture of PbS and PbO undergoes self -reduction to produce Pb and SO 2.
2. In the extraction process of copper from copper pyri tes, silica is added to produce copper silicate.
3. Partial oxidation of sulphide ore of copper by roasting , followed by self-reduction produces blister copper .
4. In cyanide process, zinc powde r is utilized to precipitate gold from Na[Au(CN) 2].

A mixture of two salts is used to prepare a solution S, which gives the following results: S (aq solution of the salts)Dilute NaOH(aq) Dilute HCl(aq)White precipitate(s) onlyWhite precipitate(s) only Room temperature Room temperature The correct option(s) for the salt mixture is(are)

1. Pb(NO 3)2 and Zn(NO 3)2
2. Pb(NO 3)2 and Bi(NO 3)3
3. AgNO 3 and Bi(NO 3)3
4. 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 :

The maximum number of possible isomers (including stereoisomers ) which may be formed on mono -bromination of 1 -methylcyclohex -1-ene using Br 2 and UV light is ___.

Numerical answer type

In the reaction given below, the total number of atoms having sp2 hybridization in the major product P is ___. P1. O3 (excess) then Zn/H₂O 2. NH2OH (excess)

Numerical answer type

The total number of possible isomers for [Pt(NH 3)4Cl₂]Br 2 is ___.

Numerical answer type

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

1. 𝑥2+𝑦2−3𝑥+𝑦=0
2. 𝑥2+𝑦2+𝑥+3𝑦=0
3. 𝑥2+𝑦2+2𝑦−1=0
4. 𝑥2+𝑦2+𝑥+𝑦=0

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

1. 11 32
2. 35 96
3. 37 96
4. 13 32

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 l et 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

1. 1 5
2. 3 5
3. 1 2
4. 2 5

Let 𝜃1,𝜃2,…,𝜃10 be positive valued angles (in radian) such that 𝜃1+ 𝜃2+⋯+𝜃10=2𝜋. Define the complex numbers 𝑧1=𝑒𝑖𝜃1, 𝑧𝑘=𝑧𝑘−1𝑒𝑖𝜃𝑘 for 𝑘=2,3,…,10, where 𝑖=√−1 . Consider the statement s 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 ,

1. P is TRUE and 𝑄 is FALSE
2. 𝑄 is TRUE and P is FALSE
3. both P and 𝑄 are TRUE
4. both P and 𝑄 are FALSE 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.

The value of 625 4 𝑝1 is ___ .

Numerical answer type

The value of 125 4 𝑝2 is ____ . 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 : 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 𝑀=[𝛼2𝛾 𝛽10 −101] 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.

Numerical answer type

The value of |M| is ___ .

Numerical answer type

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′.

Numerical answer type

The value of 𝜆2 is ___ .

Numerical answer type

The value of 𝐷 is ___ .

Numerical answer type

For any 3×3 matrix 𝑀, let |𝑀| denote the determinant of 𝑀. Let 𝐸= [123 234 81318], 𝑃= [100 001 010] and 𝐹= [132 81813 243] If 𝑄 is a nonsingular matrix of order 3×3, then which of the following statement s is (are) TRUE ? SECTION 3  This section contains SIX (06) questions .  Each question has FOUR options

1. ,
2. ,
3. and
4. . ONE OR MORE THAN ONE of these four option(s) is (are) c orrect 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 :  For example, in a question, if (A), (B ) and
5. are the ONLY three options corresponding to correct answers , then choosing ONLY (A),
6. and
7. will get +4 marks; choosing ONLY (A) and ( B) will get +2 marks; choosing ONLY (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
8. will get +1 mark; choosing ONLY
9. 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. (A) 𝐹=𝑃𝐸𝑃 and 𝑃2=[100 010 001]
10. |𝐸𝑄+𝑃𝐹𝑄−1|=|𝐸𝑄|+|𝑃𝐹𝑄−1|
11. |(𝐸𝐹)3|>|𝐸𝐹|2
12. Sum of the diagonal entries of 𝑃−1𝐸𝑃+𝐹 is equal to the sum of diagonal entries of 𝐸+𝑃−1𝐹𝑃

Let 𝑓:ℝ →ℝ be defined by 𝑓(𝑥)= 𝑥2−3𝑥−6 𝑥2+2𝑥+4 Then which of the following statement s is (are) TRUE ?

1. 𝑓 is decreasing in the interval (−2,−1)
2. 𝑓 is increasing in the interval (1,2)
3. 𝑓 is onto
4. Range of 𝑓 is [−3 2,2]

Let 𝐸,𝐹 and G be three events having probabilities 𝑃(𝐸)=1 8,𝑃(𝐹)=1 6 and 𝑃(𝐺)=1 4, and let 𝑃(𝐸∩𝐹∩𝐺)=1 10 . For any event 𝐻, if 𝐻𝑐 denotes its complement, then which of the following statements is (are) TRUE ?

1. 𝑃(𝐸∩𝐹∩𝐺𝑐)≤1 40
2. 𝑃(𝐸𝑐∩𝐹∩𝐺)≤1 15
3. 𝑃(𝐸∪𝐹∪𝐺)≤ 13 24
4. 𝑃(𝐸𝑐∩𝐹𝑐∩𝐺𝑐)≤5 12

For any 3×3 matrix 𝑀, let |𝑀| denote the determinant of 𝑀. Let 𝐼 be the 3×3 identity 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 statement s is (are) TRUE ?

1. |𝐹𝐸|=|𝐼−𝐹𝐸||𝐹𝐺𝐸|
2. (𝐼−𝐹𝐸)(𝐼+𝐹𝐺𝐸)=𝐼
3. 𝐸𝐹𝐺=𝐺𝐸𝐹
4. (𝐼−𝐹𝐸)(𝐼−𝐹𝐺𝐸)=𝐼

For any positive integer 𝑛, let 𝑆𝑛:(0,∞)→ℝ be defined by 𝑆𝑛(𝑥)=∑cot−1(1+𝑘(𝑘+1)𝑥2 𝑥)𝑛 𝑘=1 , where for any 𝑥∈ℝ, cot−1(𝑥)∈(0,𝜋) and tan−1(𝑥)∈(−𝜋 2 ,π 2). Then which of the following statement s is (are) TRUE ?

1. 𝑆10(𝑥)=𝜋 2−tan−1(1+11𝑥2 10𝑥), for all 𝑥>0
2. lim 𝑛→∞cot(𝑆𝑛(𝑥))=𝑥, for all 𝑥>0
3. The equation 𝑆3(𝑥)=𝜋 4 has a root in (0,∞)
4. tan(𝑆𝑛(𝑥))≤1 2 , for all 𝑛≥1 and 𝑥>0

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 arg(𝑧+𝛼 𝑧+𝛽)=𝜋 4 , the ordered pair (𝑥,𝑦) lies on the circle 𝑥2+𝑦2+5𝑥−3𝑦+4=0 Then which of the following statement s is (are) TRUE ?

1. 𝛼=−1
2. 𝛼𝛽=4
3. 𝛼𝛽=−4
4. 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 :

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

Numerical answer type

In a triangle 𝐴𝐵𝐶 , let 𝐴𝐵=√23,𝐵𝐶=3 and 𝐶𝐴=4. Then the value of cot𝐴+cot𝐶 cot𝐵 is ___ .

Numerical answer type

Let 𝑢⃗ , 𝑣 ⃗⃗⃗ and 𝑤⃗⃗ be vectors in three -dimensional space, where 𝑢⃗ and 𝑣 ⃗⃗⃗ are unit vectors which are not perpendicular to each other and 𝑢⃗ ⋅𝑤⃗⃗ =1, 𝑣 ⃗⃗⃗ ⋅𝑤⃗⃗ =1, 𝑤⃗⃗ ⋅𝑤⃗⃗ =4 If the volume of the parallel opiped, whose adjacent sides are represented by the vectors 𝑢⃗ , 𝑣 ⃗⃗⃗ and 𝑤⃗⃗ , is √2 , then the value of |3𝑢⃗ +5𝑣 ⃗⃗⃗ | is ___ .

Numerical answer type