Jee advance (03 october)2021 answer key

JEE Advance 2021 answer key

 Chemistry Paper 1

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

Ans: B

Q2.Among the following, the conformation that corresponds to the most stable

conformation of meso-butane-2,3-diol is

Ans:

Q.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 (packing fraction =packing efficiency/100)

(A) 0.74       (B) 0.63      (C) 0.52          (D) 0.48

Ans B 0.63

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

Ans.A

Q5.For the following reaction scheme, percentage yields are given along the arrow:

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)

Q.5 The value of x is ___.

Q.6 The value of y is ___.

For the reaction, 𝐗(𝑠) ⇌ 𝐘(𝑠) + 𝐙(𝑔), the plot of ln 𝑝𝐙𝑝oversus104𝑇is given below (insolid line), where 𝑝𝐙

is the pressure (in bar) of the gas Z at temperature T and 𝑝o = 1 bar.(Given, d (ln 𝐾)

d (1𝑇)= −∆𝐻o𝑅

, where the equilibrium constant, 𝐾 =𝑝𝑧𝑝oand

the gas constant, R = 8.314 J K–1 mol–1

)

Q.7 The value of standard enthalpy, ∆𝐻o

(in kJ mol–1) for the given reaction is ___.

Ans is : -2×8.314×10 kj/mol

Q.8 The value of ∆𝑆o(in J K–1 mol–1

) for the given reaction, at 1000 K is ___.

 Ans:Delta s = 17×8.314

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 salts dissociate completely.

Use: Molal elevation constant (Ebullioscopic Constant), 𝐾𝑏= 0.5 K kg mol1

;

Boiling point of pure water as 100 C.)

Q.9 The value of x is ___.

Ans: 100.1°c

Q.10 The value of |y| is ___.

Ans: 2.5

Q.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 higher temperature than the true solution at the

same concentration.

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

(D) Micelles are macromolecular colloids.

Q.14 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)

Ans is option A, Band D

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

Ans is A, C, D

Q.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 ___.

Q.19 The total number of possible isomers for [Pt(NH3)4Cl2]Br2 is ___.  Ans is 6

Jee advance 2021 Math paper 1

Q.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}}

and

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) 𝑛4/12= 420

 Ans is abd option

Q2

Q3.

  Ans :( acd )

Q18. E be the ellipse 𝑥^2/16+𝑦^2/9= 1. 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 ___ .

Ans:' (c)  4

Jee advance 2021  solved Mathe paper 2

Q.1 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

(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

Ans: c 

Q2.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

Q.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

Q.4  Let 𝜃1, 𝜃2

, … , 𝜃10 be positive valued angles (in radian) such that

𝜃1 + 𝜃2 + ⋯ + 𝜃10 = 2𝜋. Define the complex numbers 𝑧1 = 𝑒

𝑖𝜃1, 𝑧𝑘 = 𝑧𝑘−1𝑒

𝑖𝜃𝑘

for 𝑘 = 2, 3, … , 10, where 𝑖 = √−1 . 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

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.

Q.5 The value of 625/4𝑝1 is ___ .

Q.6 The value of 125/4𝑝2 is ____ .

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 = [𝛼     2       𝛾   ]

         𝛽      1       0

         -1      0      1

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.

Q.7 The value of |M| is ___ .

Q.8 The value of 𝐷 is ___ .

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 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^2is ___ .

Q.10 The value of 𝐷 is ___ .

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

𝐸 = [1    2      3

         2   3      4

        8   13    18],

 𝑃 = [

          1   0    0

          0   0     1

          0   1     0]

and 𝐹 = [

1     3      2

8   18      13

2     4       3]

If 𝑄 is a nonsingular matrix of order 3 × 3, then which of the following statements

is (are) TRUE ?

(A) 𝐹 = 𝑃𝐸𝑃 and

 𝑃^2 = [

1     0      0

0     1      0

0     0      1

]

(B) |𝐸𝑄 + 𝑃𝐹𝑄−1| = |𝐸𝑄| + |𝑃𝐹𝑄−1|

(C) |(𝐸𝐹)3| > |𝐸𝐹|2

(D) Sum of the diagonal entries of 𝑃

−1𝐸𝑃 + 𝐹 is equal to the sum of diagonal

entries of 𝐸 + 𝑃−1𝐹𝑃

Q.12 Let 𝑓: ℝ → ℝ be defined by

𝑓(𝑥)= 𝑥^2 − 3𝑥 − 6/ 𝑥^2 + 2𝑥 + 4

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]

Ans: A, D

Q.13 Let 𝐸, 𝐹 and G be three events having probabilities

𝑃(𝐸) =18 , 𝑃(𝐹) =16  and 𝑃(𝐺) =14

, and let 𝑃(𝐸 ∩ 𝐹 ∩ 𝐺) =110

.

For any event 𝐻, if 𝐻

𝑐 denotes its complement, then which of the following

statements is (are) TRUE ?

(A) 𝑃(𝐸 ∩ 𝐹 ∩ 𝐺𝑐) ≤140

(B) 𝑃(𝐸𝑐 ∩ 𝐹 ∩ 𝐺) ≤115

(C) 𝑃(𝐸 ∪ 𝐹 ∪ 𝐺) ≤1324

(D) 𝑃(𝐸𝑐 ∩ 𝐹𝑐 ∩ 𝐺𝑐) ≤512

Q.14 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 statements is (are)

TRUE ?

(A) |𝐹𝐸| = |𝐼 − 𝐹𝐸||𝐹𝐺𝐸|

 (B) (𝐼 − 𝐹𝐸)(𝐼 + 𝐹𝐺𝐸) = 𝐼

(C) 𝐸𝐹𝐺 = 𝐺𝐸𝐹

 (D) (𝐼 − 𝐹𝐸)(𝐼 − 𝐹𝐺𝐸) = 𝐼

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

𝑆𝑛

(𝑥) = ∑ cot−1 (1 + 𝑘(𝑘 + 1)𝑥2𝑥)𝑛𝑘=1,

where for any 𝑥 ∈ ℝ, cot−1

(𝑥) ∈ (0, 𝜋) and tan−1

(𝑥) ∈ (−𝜋2,π2). Then which of

the following statements is (are) TRUE ?

(A) 𝑆10(𝑥) =𝜋2− tan−1(1+11𝑥2 10𝑥

), for all 𝑥 > 0

(B) lim

𝑛→∞

cot(𝑆𝑛(𝑥)) = 𝑥, for all 𝑥 > 0

(C) The equation 𝑆3

(𝑥) =

𝜋

4

has a root in (0, ∞)

(D) tan(𝑆𝑛

(𝑥)) ≤

1

2

, for all 𝑛 ≥ 1 and 𝑥 > 0

Q.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

arg (

𝑧+𝛼

𝑧+𝛽

) =

𝜋

4

, 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

Ans is B, D

Q.17 For 𝑥 ∈ ℝ, the number of real roots of the equation

3𝑥^2 − 4|𝑥^2 − 1| + 𝑥 − 1 = 0 is ___ .

Ans:      (-1/6 ,  -13/12) number of sollution is 4

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

cot 𝐴 + cot 𝐶

cot 𝐵

is ___ .

Q.19 Let 𝑢⃗ , 𝑣⃗ and 𝑤⃗ be vectors in three-dimensional space, where 𝑢⃗ and 𝑣⃗ are unit

vectors which are not perpendicular to each other and

𝑢⃗ ⋅ 𝑤⃗ = 1, 𝑣⃗ ⋅ 𝑤⃗ = 1, 𝑤⃗ ⋅ 𝑤⃗ = 4

If the volume of the parallelopiped, whose adjacent sides are represented by the

vectors 𝑢⃗ , 𝑣⃗ and 𝑤⃗ , is √2 , then the value of |3 𝑢⃗ +5 𝑣⃗ | is ___ .

Jee advance 2021 physices paper 2

Q1.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 the 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 reaction force at O does not depend on 𝛼

(B) The horizontal component of 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

Ans is A, B, D

Q.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

Ans is A, D

Q.3 For a prism of prism angle 𝜃 = 60°, 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 𝑖 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 + ∆𝑛 (where ∆𝑛 ≪ 𝑛), the angle of emergence 𝑒 = 𝑖 + ∆𝑒. Which of

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

(A) The value of ∆𝑒 (in radians) is greater than that of ∆𝑛

(B) ∆𝑒 is proportional to Δ𝑛

(C) Δ𝑒 lies between 2.0 and 3.0 milliradians, if ∆𝑛 = 2.8 × 10−3

(D) Δ𝑒 lies between 1.0 and 1.6 milliradians, if ∆𝑛 = 2.8 × 10−3

Ans is C, D

Q.4 A physical quantity 𝑆⃗ is defined as 𝑆⃗ = (𝐸⃗⃗ × 𝐵⃗⃗)/𝜇0

, where 𝐸⃗⃗ is electric field, 𝐵⃗⃗ is

magnetic field and 𝜇0

is the permeability of free space. The dimensions of 𝑆⃗ are the

same as the dimensions of which of the following quantity(ies) ?

(A) 𝐸𝑛𝑒𝑟𝑔𝑦 /𝐶ℎ𝑎𝑟𝑔𝑒 × 𝐶𝑢𝑟𝑟𝑒𝑛𝑡

(B) 𝐹𝑜𝑟𝑐𝑒 /𝐿𝑒𝑛𝑔𝑡ℎ × 𝑇𝑖𝑚𝑒

(C) 𝐸𝑛𝑒𝑟𝑔𝑦/𝑉𝑜𝑙𝑢𝑚𝑒

(D) 𝑃𝑜𝑤𝑒𝑟/𝐴𝑟𝑒𝑎

Ans is B, D

Q.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 = 𝑐^2𝛿

(B) 𝐸𝑃 = (Mp/𝑀𝑃+𝑀𝑄) 𝑐^2𝛿

(C) 𝑣𝑃/𝑣𝑄=𝑀𝑄/𝑀𝑃

(D) The magnitude of momentum for 𝑃 as well as 𝑄 is 𝑐√2𝜇𝛿, where 𝜇 =𝑀𝑃𝑀𝑄 /(𝑀𝑃+𝑀𝑄)

Ans is A, C, D

Q.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 center, 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

. 𝐵⃗⃗(𝑥, 𝑦) denotes the magnetic

field at a point (𝑥, 𝑦) in the xy-plane. Which of the following statement(s) is(are)

correct?

(A) 𝐵⃗⃗(𝑥, 𝑦) is perpendicular to the xy-plane at any point in the plane

(B) |𝐵⃗⃗(𝑥, 𝑦)| depends on x and y only through the radial distance 𝑟 = √𝑥^2 + 𝑦^2

(C) |𝐵⃗⃗ (𝑥, 𝑦)| is non-zero at all points for 𝑟 < 𝑅

(D) 𝐵⃗⃗ (𝑥, 𝑦) points normally outward from the xy-plane for all the points between

the two loops

Ans is A,B

Question stem

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 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.

Q.7 The value of 𝑋 is ___ . ans is 0.3

Q.8 The value of 𝑌 is ___.ans is 9.09

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.

Q.9 The value of 𝐽 is ___ .

Q.10 The value of 𝐾 is ___.