JEE Advanced 2021 : Paper-2

Solution

S_{1} = {(i, j, k) : i, j, k \in {1, 2, . . . ., 10}}

S_{2} = {(i, j) : 1 \leq i < j + 2 \leq 10, i, j \in {1, 2, . . ., 10}}

S_{3} = {(i, j, k, l) : 1 \leq i < j < k < l, i, j, k, l \in {1, 2, . . ., 10}}

and

S_{4} = {(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 S_r is n_r, r = 1, 2, 3, 4, then which of the following statements is(are) TRUE?

Let

S_{1} = {(i, j, k) : i, j, k \in {1, 2, . . . ., 10}}

S_{2} = {(i, j) : 1 \leq i < j + 2 \leq 10, i, j \in {1, 2, . . ., 10}}

S_{3} = {(i, j, k, l) : 1 \leq i < j < k < l, i, j, k, l \in {1, 2, . . ., 10}}

and

S_{4} = {(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 S_r is n_r, r = 1, 2, 3, 4, then which of the following statements is(are) TRUE?

Q.2 Correct

Q.2 In-correct

Q.2 Unattempt

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

f : [- \frac{π}{2}, \frac{π}{2}] \to R

be a continuous function such that

f (0) = 1

and

\int_{0}^{\frac{π}{3}} f (t) d t = 0

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

Let

f : [- \frac{π}{2}, \frac{π}{2}] \to R

be a continuous function such that

f (0) = 1

and

\int_{0}^{\frac{π}{3}} f (t) d t = 0

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

α

and

β

, let

y_{α, β} (x)

, x

\in

R, be the solution of the differential equation

\frac{d y}{d x} + α y = x e^{β x}, y (1) = 1

. Let

S = {y_{α, β} (x) : α, β \in R}

. Then which of the following functions belong(s) to the set S?

For any real numbers

α

and

β

, let

y_{α, β} (x)

, x

\in

R, be the solution of the differential equation

\frac{d y}{d x} + α y = x e^{β x}, y (1) = 1

. Let

S = {y_{α, β} (x) : α, β \in R}

. Then which of the following functions belong(s) to the set S?

Q.5 Correct

Q.5 In-correct

Q.5 Unattempt

Let O be the origin and

\vec{O A} = 2 \hat{i} + 2 \hat{j} + \hat{k}

and

\vec{O B} = \hat{i} - 2 \hat{j} + 2 \hat{k}

and

\vec{O C} = \frac{1}{2} (\vec{O B} - λ \vec{O A})

for some

λ

> 0. If

| \vec{O B} \times \vec{O C} | = \frac{9}{2}

, then which of the following statements is (are) TRUE?

Let O be the origin and

\vec{O A} = 2 \hat{i} + 2 \hat{j} + \hat{k}

and

\vec{O B} = \hat{i} - 2 \hat{j} + 2 \hat{k}

and

\vec{O C} = \frac{1}{2} (\vec{O B} - λ \vec{O A})

for some

λ

> 0. If

| \vec{O B} \times \vec{O C} | = \frac{9}{2}

, then which of the following statements is (are) TRUE?

Q.6 Correct

Q.6 In-correct

Q.6 Unattempt

Let E denote the parabola y² = 8x. Let P = (

-

2, 4), and let Q and Q' be two distinct points on E such that the lines PQ and PQ' are tangents to E. Let F be the focus of E. Then which of the following statements is(are) TRUE?

Let E denote the parabola y² = 8x. Let P = (

-

2, 4), and let Q and Q' be two distinct points on E such that the lines PQ and PQ' are tangents to E. Let F be the focus of E. Then which of the following statements is(are) TRUE?

Q.7 Correct

Q.7 In-correct

Q.7 Unattempt

Consider the region R = {(x, y)

\in

\times

R : x

\geq

0 and y²

\leq

-

x}. Let F be the family of all circles that are contained in R and have centers on the x-axis. Let C be the circle that has largest radius among the circles in F. Let (

α

β

) be a point where the circle C meets the curve y² = 4

-

x.

The radius of the circle C is ___________.

Consider the region R = {(x, y)

\in

\times

R : x

\geq

0 and y²

\leq

-

x}. Let F be the family of all circles that are contained in R and have centers on the x-axis. Let C be the circle that has largest radius among the circles in F. Let (

α

β

) be a point where the circle C meets the curve y² = 4

-

x.

The radius of the circle C is ___________.

Q.8 Correct

Q.8 In-correct

Q.8 Unattempt

Consider the region R = {(x, y)

\in

\times

R : x

\geq

0 and y²

\leq

-

x}. Let F be the family of all circles that are contained in R and have centers on the x-axis. Let C be the circle that has largest radius among the circles in F. Let (

α

β

) be a point where the circle C meets the curve y² = 4

-

x.

The value of

α

is ___________.

Consider the region R = {(x, y)

\in

\times

R : x

\geq

0 and y²

\leq

-

x}. Let F be the family of all circles that are contained in R and have centers on the x-axis. Let C be the circle that has largest radius among the circles in F. Let (

α

β

) be a point where the circle C meets the curve y² = 4

-

x.

The value of

α

\infty

)

\to

R and f₂ : (0,

\infty

)

\to

R be defined by

f_{1} (x) = \int_{0}^{x} \prod_{j = 1}^{21} {(t - j)}^{j} d t

, x > 0 and

f_{2} (x) = 98 (x - 1)^{50} - 600 (x - 1)^{49} + 2450, x > 0

, where, for any positive integer n and real numbers a₁, a₂, ....., a_n,

\prod_{i = 1}^{n} a_{i}

denotes the product of a₁, a₂, ....., a_n. Let m_i and n_i, respectively, denote the number of points of local minima and the number of points of local maxima of function f_i, i = 1, 2 in the interval (0,

\infty

).

The value of

2 m_{1} + 3 n_{1} + m_{1} n_{1}

is ___________.

Let f₁ : (0,

\infty

)

\to

R and f₂ : (0,

\infty

)

\to

R be defined by

f_{1} (x) = \int_{0}^{x} \prod_{j = 1}^{21} {(t - j)}^{j} d t

, x > 0 and

f_{2} (x) = 98 (x - 1)^{50} - 600 (x - 1)^{49} + 2450, x > 0

, where, for any positive integer n and real numbers a₁, a₂, ....., a_n,

\prod_{i = 1}^{n} a_{i}

\infty

).

The value of

2 m_{1} + 3 n_{1} + m_{1} n_{1}

\infty

)

\to

R and f₂ : (0,

\infty

)

\to

R be defined by

f_{1} (x) = \int_{0}^{x} \prod_{j = 1}^{21} {(t - j)}^{j} d t

, x > 0 and

f_{2} (x) = 98 (x - 1)^{50} - 600 (x - 1)^{49} + 2450, x > 0

, where, for any positive integer n and real numbers a₁, a₂, ....., a_n,

\prod_{i = 1}^{n} a_{i}

\infty

).

The value of

6 m_{2} + 4 n_{2} + 8 m_{2} n_{2}

is ___________.

Let f₁ : (0,

\infty

)

\to

R and f₂ : (0,

\infty

)

\to

R be defined by

f_{1} (x) = \int_{0}^{x} \prod_{j = 1}^{21} {(t - j)}^{j} d t

, x > 0 and

f_{2} (x) = 98 (x - 1)^{50} - 600 (x - 1)^{49} + 2450, x > 0

, where, for any positive integer n and real numbers a₁, a₂, ....., a_n,

\prod_{i = 1}^{n} a_{i}

\infty

).

The value of

6 m_{2} + 4 n_{2} + 8 m_{2} n_{2}

g_{i} : [\frac{π}{8}, \frac{3 π}{8}] \to R, i = 1, 2

, and

f : [\frac{π}{8}, \frac{3 π}{8}] \to R

be functions such that

g_{1} (x) = 1, g_{2} (x) = | 4 x - π |

and

f (x) = \sin^{2} x

, for all

x \in [\frac{π}{8}, \frac{3 π}{8}]

. Define

S_{i} = \int_{\frac{π}{8}}^{\frac{3 π}{8}} f (x) . g_{i} (x) d x

, i = 1, 2

The value of

\frac{16 S_{1}}{π}

is _____________.

Let

g_{i} : [\frac{π}{8}, \frac{3 π}{8}] \to R, i = 1, 2

, and

f : [\frac{π}{8}, \frac{3 π}{8}] \to R

be functions such that

g_{1} (x) = 1, g_{2} (x) = | 4 x - π |

and

f (x) = \sin^{2} x

, for all

x \in [\frac{π}{8}, \frac{3 π}{8}]

. Define

S_{i} = \int_{\frac{π}{8}}^{\frac{3 π}{8}} f (x) . g_{i} (x) d x

, i = 1, 2

The value of

\frac{16 S_{1}}{π}

g_{i} : [\frac{π}{8}, \frac{3 π}{8}] \to R, i = 1, 2

, and

f : [\frac{π}{8}, \frac{3 π}{8}] \to R

be functions such that

g_{1} (x) = 1, g_{2} (x) = | 4 x - π |

and

f (x) = \sin^{2} x

, for all

x \in [\frac{π}{8}, \frac{3 π}{8}]

. Define

S_{i} = \int_{\frac{π}{8}}^{\frac{3 π}{8}} f (x) . g_{i} (x) d x

, i = 1, 2

The value of

\frac{48 S_{2}}{π^{2}}

is ___________.

Let

g_{i} : [\frac{π}{8}, \frac{3 π}{8}] \to R, i = 1, 2

, and

f : [\frac{π}{8}, \frac{3 π}{8}] \to R

be functions such that

g_{1} (x) = 1, g_{2} (x) = | 4 x - π |

and

f (x) = \sin^{2} x

, for all

x \in [\frac{π}{8}, \frac{3 π}{8}]

. Define

S_{i} = \int_{\frac{π}{8}}^{\frac{3 π}{8}} f (x) . g_{i} (x) d x

, i = 1, 2

The value of

\frac{48 S_{2}}{π^{2}}

M = {(x, y) \in R \times R : x^{2} + y^{2} \leq r^{2}}

, where r > 0. Consider the geometric progression

a_{n} = \frac{1}{2^{n - 1}}

, n = 1, 2, 3, ...... . Let S₀ = 0 and for n

\geq

1, let S_n denote the sum of the first n terms of this progression. For n

\geq

1, let C_n denote the circle with center (S_{n $-$ 1}, 0) and radius a_n, and D_n denote the circle with center (S_{n $-$ 1}, S_{n $-$ 1}) and radius a_n.

Consider M with

r = \frac{1025}{513}

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

Let

M = {(x, y) \in R \times R : x^{2} + y^{2} \leq r^{2}}

, where r > 0. Consider the geometric progression

a_{n} = \frac{1}{2^{n - 1}}

, n = 1, 2, 3, ...... . Let S₀ = 0 and for n

\geq

1, let S_n denote the sum of the first n terms of this progression. For n

\geq

1, let C_n denote the circle with center (S_{n $-$ 1}, 0) and radius a_n, and D_n denote the circle with center (S_{n $-$ 1}, S_{n $-$ 1}) and radius a_n.

Consider M with

r = \frac{1025}{513}

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

M = {(x, y) \in R \times R : x^{2} + y^{2} \leq r^{2}}

, where r > 0. Consider the geometric progression

a_{n} = \frac{1}{2^{n - 1}}

, n = 1, 2, 3, ...... . Let S₀ = 0 and for n

\geq

1, let S_n denote the sum of the first n terms of this progression. For n

\geq

1, let C_n denote the circle with center (S_{n $-$ 1}, 0) and radius a_n, and D_n denote the circle with center (S_{n $-$ 1}, S_{n $-$ 1}) and radius a_n.

Consider M with

r = \frac{(2^{199} - 1) \sqrt{2}}{2^{198}}

. The number of all those circles D_n that are inside M is

Let

M = {(x, y) \in R \times R : x^{2} + y^{2} \leq r^{2}}

, where r > 0. Consider the geometric progression

a_{n} = \frac{1}{2^{n - 1}}

, n = 1, 2, 3, ...... . Let S₀ = 0 and for n

\geq

1, let S_n denote the sum of the first n terms of this progression. For n

\geq

1, let C_n denote the circle with center (S_{n $-$ 1}, 0) and radius a_n, and D_n denote the circle with center (S_{n $-$ 1}, S_{n $-$ 1}) and radius a_n.

Consider M with

r = \frac{(2^{199} - 1) \sqrt{2}}{2^{198}}

. The number of all those circles D_n that are inside M is

ψ_{1} : [0, \infty) \to R

ψ_{2} : [0, \infty) \to R

, f : (0,

\infty

)

\to

R and g : [0,

\infty

)

\to

R be functions such that f(0) = g(0) = 0,

ψ_{1} (x) = e^{- x} + x, x \geq 0

ψ_{2} (x) = x^{2} - 2 x - 2 e^{- x} + 2, x \geq 0

f (x) = \int_{- x}^{x} (| t | - t^{2}) e^{- t^{2}} d t, x > 0

and

g (x) = \int_{0}^{x^{2}} \sqrt{t} e^{- t} d t, x > 0

Which of the following statements is TRUE?

Let

ψ_{1} : [0, \infty) \to R

ψ_{2} : [0, \infty) \to R

, f : (0,

\infty

)

\to

R and g : [0,

\infty

)

\to

R be functions such that f(0) = g(0) = 0,

ψ_{1} (x) = e^{- x} + x, x \geq 0

ψ_{2} (x) = x^{2} - 2 x - 2 e^{- x} + 2, x \geq 0

f (x) = \int_{- x}^{x} (| t | - t^{2}) e^{- t^{2}} d t, x > 0

and

g (x) = \int_{0}^{x^{2}} \sqrt{t} e^{- t} d t, x > 0

Which of the following statements is TRUE?

ψ_{1} : [0, \infty) \to R

ψ_{2} : [0, \infty) \to R

, f : (0,

\infty

)

\to

R and g : [0,

\infty

)

\to

R be functions such that f(0) = g(0) = 0,

ψ_{1} (x) = e^{- x} + x, x \geq 0

ψ_{2} (x) = x^{2} - 2 x - 2 e^{- x} + 2, x \geq 0

f (x) = \int_{- x}^{x} (| t | - t^{2}) e^{- t^{2}} d t, x > 0

and

g (x) = \int_{0}^{x^{2}} \sqrt{t} e^{- t} d t, x > 0

Which of the following statements is TRUE?

Let

ψ_{1} : [0, \infty) \to R

ψ_{2} : [0, \infty) \to R

, f : (0,

\infty

)

\to

R and g : [0,

\infty

)

\to

R be functions such that f(0) = g(0) = 0,

ψ_{1} (x) = e^{- x} + x, x \geq 0

ψ_{2} (x) = x^{2} - 2 x - 2 e^{- x} + 2, x \geq 0

f (x) = \int_{- x}^{x} (| t | - t^{2}) e^{- t^{2}} d t, x > 0

and

g (x) = \int_{0}^{x^{2}} \sqrt{t} e^{- t} d t, x > 0

Which of the following statements is TRUE?

Q.17 Correct

Q.17 In-correct

Q.17 Unattempt

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

\frac{x^{2}}{16} + \frac{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 _______.

Let E be the ellipse

\frac{x^{2}}{16} + \frac{y^{2}}{9} = 1

Q.19 Correct

Q.19 In-correct

Q.19 Unattempt

For any real number x, let [ x ] denote the largest integer less than or equal to x. If

I = \int_{0}^{10} [\sqrt{\frac{10 x}{x + 1}}] d x

, then the value of 9I is __________.

For any real number x, let [ x ] denote the largest integer less than or equal to x. If

I = \int_{0}^{10} [\sqrt{\frac{10 x}{x + 1}}] d x

, then the value of 9I is __________.

Q.1 Correct

Q.1 In-correct

Q.1 Unattempt

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

Q.2 Correct

Q.2 In-correct

Q.2 Unattempt

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

Q.3 Correct

Q.3 In-correct

Q.3 Unattempt

For the following reaction,

2 X + Y \overset{k}{⟶} P

the rate of reaction is

\frac{d [P]}{d t} = k [X]

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

(Use : ln 2 = 0.693)

For the following reaction,

2 X + Y \overset{k}{⟶} P

the rate of reaction is

\frac{d [P]}{d t} = k [X]

Q.4 Correct

Q.4 In-correct

Q.4 Unattempt

Some standard electrode potentials at 298 K are given below :

Pb²⁺ /Pb

-

0.13 V

Ni²⁺ /Ni

-

0.24 V

Cd²⁺ /Cd

-

0.40 V

Fe²⁺ /Fe

-

0.44 V

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

(Given : Gas constant, R = 8.314 J K^$-$ mol^{$-$ 1}, Faraday constant, F = 96500 C mol^{$-$ 1})

Some standard electrode potentials at 298 K are given below :

Pb²⁺ /Pb

-

0.13 V

Ni²⁺ /Ni

-

0.24 V

Cd²⁺ /Cd

-

0.40 V

Fe²⁺ /Fe

-

Q.5 Correct

Q.5 In-correct

Q.5 Unattempt

The pair(s) of complexes wherein both exhibit tetrahedral geometry is(are)

(Note : py = pyridine)

Given : Atomic numbers of Fe, Co, Ni and Cu are 26, 27, 28 and 29, respectively)

Q.6 Correct

Q.6 In-correct

Q.6 Unattempt

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

Q.7 Correct

Q.7 In-correct

Q.7 Unattempt

At 298 K, the limiting molar conductivity of a weak monobasic acid is 4

\times

10² S cm² mol^{$-$ 1}. At 298 K, for an aqueous solution of the acid the degree of dissociation is

α

and the molar conductivity is y

\times

10² S cm² mol^{$-$ 1}. At 298 K, upon 20 times dilution with water, the molar conductivity of the solution becomes 3y

\times

10² S cm² mol^{$-$ 1}.

The value of

α

is __________.

At 298 K, the limiting molar conductivity of a weak monobasic acid is 4

\times

10² S cm² mol^{$-$ 1}. At 298 K, for an aqueous solution of the acid the degree of dissociation is

α

and the molar conductivity is y

\times

10² S cm² mol^{$-$ 1}. At 298 K, upon 20 times dilution with water, the molar conductivity of the solution becomes 3y

\times

10² S cm² mol^{$-$ 1}.

The value of

α

At 298 K, the limiting molar conductivity of a weak monobasic acid is 4

\times

10² S cm² mol^{$-$ 1}. At 298 K, for an aqueous solution of the acid the degree of dissociation is

α

and the molar conductivity is y

\times

10² S cm² mol^{$-$ 1}. At 298 K, upon 20 times dilution with water, the molar conductivity of the solution becomes 3y

\times

10² S cm² mol^{$-$ 1}.

The value of y is __________.

At 298 K, the limiting molar conductivity of a weak monobasic acid is 4

\times

10² S cm² mol^{$-$ 1}. At 298 K, for an aqueous solution of the acid the degree of dissociation is

α

and the molar conductivity is y

\times

10² S cm² mol^{$-$ 1}. At 298 K, upon 20 times dilution with water, the molar conductivity of the solution becomes 3y

\times

10² S cm² mol^{$-$ 1}.

The value of y is __________.

Q.9 Correct

Q.9 In-correct

Q.9 Unattempt

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

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

The value of x is _________.

A sample (5.6 g) containing iron is completely dissolved in cold dilute HCl to prepare a 250 mL of solution. Titration of 25.0 mL of this solution requires 12.5 mL of 0.03 M KMnO₄ solution to reach the end point. Number of moles of Fe²⁺ present in 250 mL solution is x

\times

10^{$-$ 2} (consider complete dissolution of FeCl₂). The amount of iron present in the sample is y% by weight.

(Assume : KMnO₄ reacts only with Fe²⁺ in the solution

Use : Molar mass of iron as 56 g mol^{$-$ 1})

The value of x is ______.

\times

Q.12 Correct

Q.12 In-correct

Q.12 Unattempt

\times

\times

Q.13 Correct

Q.13 In-correct

Q.13 Unattempt

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

Correct match of the C-H bonds (shown in bold) in Column J with their BDE in Column K is

Column J Molecule	Column K BDE (kcal $m o l^{- 1}$ )
(P) H-CH( $C H_{3}$ ) $_{2}$	(i) 132
(Q) H-CH $_{2}$ Ph	(ii) 110
(R) H-CH=CH $_{2}$	(iii) 95
(S) H-C $\equiv$ CH	(iv) 88

Correct match of the C-H bonds (shown in bold) in Column J with their BDE in Column K is

Column J Molecule	Column K BDE (kcal $m o l^{- 1}$ )
(P) H-CH( $C H_{3}$ ) $_{2}$	(i) 132
(Q) H-CH $_{2}$ Ph	(ii) 110
(R) H-CH=CH $_{2}$	(iii) 95
(S) H-C $\equiv$ CH	(iv) 88

Q.14 Correct

Q.14 In-correct

Q.14 Unattempt

-

Cl (g)

\to

_{(g)}^{∙}

+ Cl

_{(g)}^{∙}

Δ

^{\circ}

= 58 kcal mol^{$-$ 1}

H₃C

-

Cl (g)

\to

H₃C

_{(g)}^{∙}

+ Cl

_{(g)}^{∙}

Δ

^{\circ}

= 85 kcal mol^{$-$ 1}

H

-

Cl (g)

\to

_{(g)}^{∙}

+ Cl

_{(g)}^{∙}

Δ

^{\circ}

= 103 kcal mol^{$-$ 1}

For the following reaction

CH₄(g) + Cl₂(g)

\overset{l i g h t}{⟶}

CH₃Cl(g) + HCl (g)

the correct statement is

-

Cl (g)

\to

_{(g)}^{∙}

+ Cl

_{(g)}^{∙}

Δ

^{\circ}

= 58 kcal mol^{$-$ 1}

H₃C

-

Cl (g)

\to

H₃C

_{(g)}^{∙}

+ Cl

_{(g)}^{∙}

Δ

^{\circ}

= 85 kcal mol^{$-$ 1}

H

-

Cl (g)

\to

_{(g)}^{∙}

+ Cl

_{(g)}^{∙}

Δ

^{\circ}

= 103 kcal mol^{$-$ 1}

For the following reaction

CH₄(g) + Cl₂(g)

\overset{l i g h t}{⟶}

CH₃Cl(g) + HCl (g)

the correct statement is

Q.15 Correct

Q.15 In-correct

Q.15 Unattempt

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

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

Q.17 Correct

Q.17 In-correct

Q.17 Unattempt

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

\ln \frac{V_{3}}{V_{2}}

is _________.

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

(Given : molar heat capacity at constant volume, C_V,m of the gas is

\frac{5}{2}

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

\ln \frac{V_{3}}{V_{2}}

is _________.

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

(Given : molar heat capacity at constant volume, C_V,m of the gas is

\frac{5}{2}

Q.18 Correct

Q.18 In-correct

Q.18 Unattempt

Consider a helium (He) atom that absorbs a photon of wavelength 330 nm. The change in the velocity (in cm s^{$-$ 1}) of He atom after the photon absorption is __________.

(Assume : Momentum is conserved when photon is absorbed.

Use : Planck constant = 6.6

\times

10^{$-$ 34} J s, Avogadro number = 6

\times

10²³ mol^{$-$ 1}, Molar mass of He = 4 g mol^{$-$ 1})

\times

10^{$-$ 34} J s, Avogadro number = 6

\times

10²³ mol^{$-$ 1}, Molar mass of He = 4 g mol^{$-$ 1})

Q.19 Correct

Q.19 In-correct

Q.19 Unattempt

Ozonolysis of ClO₂ produces an oxide of chlorine. The average oxidation state of chlorine in this oxide is __________.

Q.1 Correct

Q.1 In-correct

Q.1 Unattempt

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

α

W 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

\sqrt{2}

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

α

W 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

\sqrt{2}

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

Q.2 Correct

Q.2 In-correct

Q.2 Unattempt

A source, approaching with speed u towards the open end of a stationary pipe of length L, is emitting a sound of frequency f_s. 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 f_s, will the sound reaching the pipe lead to a resonance?

Q.3 Correct

Q.3 In-correct

Q.3 Unattempt

For a prism of prism angle

θ

= 60

^{\circ}

, the refractive indices of the left half and the right half are, respectively, n₁ and n₂ (n₂

\geq

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 +

Δ

n (where

Δ

n << n), the angle of emergence e = i +

Δ

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

For a prism of prism angle

θ

= 60

^{\circ}

, the refractive indices of the left half and the right half are, respectively, n₁ and n₂ (n₂

\geq

Δ

n (where

Δ

n << n), the angle of emergence e = i +

Δ

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

\vec{S}

is defined as

\vec{S} = (\vec{E} \times \vec{B}) / μ_{0}

, where

\vec{E}

is electric field,

\vec{B}

is magnetic field and

μ

₀ is the permeability of free space. The dimensions of

\vec{S}

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

A physical quantity

\vec{S}

is defined as

\vec{S} = (\vec{E} \times \vec{B}) / μ_{0}

, where

\vec{E}

is electric field,

\vec{B}

is magnetic field and

μ

₀ is the permeability of free space. The dimensions of

\vec{S}

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

Q.5 Correct

Q.5 In-correct

Q.5 Unattempt

A heavy nucleus N, at rest, undergoes fission N

\to

P + Q, where P and Q are two lighter nuclei. Let

δ

= M_N

-

M_P

-

M_Q, where M_P, M_Q and M_N are the masses of P, Q and N, respectively. E_P and E_Q are the kinetic energies of P and Q, respectively. The speeds of P and Q are v_P and v_Q, respectively. If c is the speed of light, which of the following statement(s) is(are) correct?

A heavy nucleus N, at rest, undergoes fission N

\to

P + Q, where P and Q are two lighter nuclei. Let

δ

= M_N

-

M_P

-

Q.6 Correct

Q.6 In-correct

Q.6 Unattempt

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.

\vec{B}

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

\vec{B}

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

Q.7 Correct

Q.7 In-correct

Q.7 Unattempt

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 atmosphere 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 +

Δ

p without changing its orientation. At this pressure, the volume of the trapped air is v₀

-

Δ

v. Let

Δ

v = X cc and

Δ

p = Y

\times

10³ Pa.

The value of X is _______________.

Δ

p without changing its orientation. At this pressure, the volume of the trapped air is v₀

-

Δ

v. Let

Δ

v = X cc and

Δ

p = Y

\times

10³ Pa.

The value of X is _______________.

Q.8 Correct

Q.8 In-correct

Q.8 Unattempt

Δ

p without changing its orientation. At this pressure, the volume of the trapped air is v₀

-

Δ

v. Let

Δ

v = X cc and

Δ

p = Y

\times

10³ Pa.

The value of Y is _______________.

Δ

p without changing its orientation. At this pressure, the volume of the trapped air is v₀

-

Δ

v. Let

Δ

v = X cc and

Δ

p = Y

\times

10³ Pa.

The value of Y is _______________.

Q.9 Correct

Q.9 In-correct

Q.9 Unattempt

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.

The value of J is ___________.

In a circuit, a metal filament lamp is connected in series with a capacitor of capacitance C

μ

F across a 200 V, 50 Hz supply. The power consumed by the lamp is 500 W while the voltage drop across it is 100 V. Assume that there is no inductive load in the circuit. Take rms values of the voltages. The magnitude of the phase-angle (in degrees) between the current and the supply voltage is

φ

. Assume,

π

\sqrt{3}

\approx

5.

The value of C is ____________.

In a circuit, a metal filament lamp is connected in series with a capacitor of capacitance C

μ

φ

. Assume,

π

\sqrt{3}

\approx

5.

The value of C is ____________.

Q.12 Correct

Q.12 In-correct

Q.12 Unattempt

In a circuit, a metal filament lamp is connected in series with a capacitor of capacitance C

μ

φ

. Assume,

π

\sqrt{3}

\approx

5.

The value of

φ

is ____________.

In a circuit, a metal filament lamp is connected in series with a capacitor of capacitance C

μ

φ

. Assume,

π

\sqrt{3}

\approx

5.

The value of

φ

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

The magnitude of magnetic field of a dipole m, at a point on its axis at distance r, is

\frac{μ_{0}}{2 π} \frac{m}{r^{3}}

, where

μ

₀ is the permeability of free space. The magnitude of the force between two magnetic dipoles with moments, m₁ and m₂, separated by a distance r on the common axis, with their north poles facing each other, is

\frac{k m_{1} m_{2}}{r^{4}}

, where k is a constant of appropriate dimensions. The direction of this force is along the line joining the two dipoles.

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

\frac{μ_{0}}{2 π} \frac{m}{r^{3}}

, where

μ

\frac{k m_{1} m_{2}}{r^{4}}

, where k is a constant of appropriate dimensions. The direction of this force is along the line joining the two dipoles.

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

Q.14 Correct

Q.14 In-correct

Q.14 Unattempt

\frac{μ_{0}}{2 π} \frac{m}{r^{3}}

, where

μ

\frac{k m_{1} m_{2}}{r^{4}}

, where k is a constant of appropriate dimensions. The direction of this force is along the line joining the two dipoles.

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

\frac{μ_{0}}{2 π} \frac{m}{r^{3}}

, where

μ

\frac{k m_{1} m_{2}}{r^{4}}

, where k is a constant of appropriate dimensions. The direction of this force is along the line joining the two dipoles.

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

Q.15 Correct

Q.15 In-correct

Q.15 Unattempt

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

The value of

\frac{T_{R}}{T_{0}}

The value of

\frac{T_{R}}{T_{0}}

\frac{Q}{R T_{0}}

The value of

\frac{Q}{R T_{0}}

Q.17 Correct

Q.17 In-correct

Q.17 Unattempt

In order to measure the internal resistance r₁ of a cell of emf E, a meter bridge of wire resistance R₀ = 50

Ω

, a resistance R₀/2, another cell of emf E/2 (internal resistance r) and a galvanometer G are used in a circuit, as shown in the figure. If the null point is found at l = 72 cm, then the value of r₁ = __________

Ω

In order to measure the internal resistance r₁ of a cell of emf E, a meter bridge of wire resistance R₀ = 50

Ω

Ω

Q.18 Correct

Q.18 In-correct

Q.18 Unattempt

The distance between two stars of masses 3M_S and 6M_S is 9R. Here R is the mean distance between the centers of the Earth and the Sun, and M_S is the mass of the Sun. The two stars orbit around their common center of mass in circular orbits with period nT, where T is the period of Earth's revolution around the Sun. The value of n is __________.

Q.19 Correct

Q.19 In-correct

Q.19 Unattempt

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