IIT JAM 2022 Mathematics (MA) Question Paper with Answer Key PDFs (February 13)

Question 1:

Consider the 2\(\times\) 2 matrix \(\in\) \(M_2(\mathbb{R}\)). If the eighth power of M satisfies \(M^8\) , then the value of x is.

• (A) 21
• (B) 22
• (C) 34
• (D) 35

Question 2:

The rank of the 4 \(\times\) 6 matrix A =  with entries in \(\mathbb{R}\), is.

• (A) 1
• (B) 2
• (C) 3
• (D) 4

Question 3:

Let \( V \) be the real vector space consisting of all polynomials in one variable with real coefficients and having degree at most 6, together with the zero polynomial. Then which one of the following is true?

• (A) \( \{ f \in V : f(1/2) \notin \mathbb{Q} \} \) is a subspace of \( V \).
• (B) \( \{ f \in V : f(1/2) = 1 \} \) is a subspace of \( V \).
• (C) \( \{ f \in V : f(1/2) = f(1) \} \) is a subspace of \( V \).
• (D) \( \{ f \in V : f'(1/2) = 13 \} \) is a subspace of \( V \).

Question 4:

Let \( G \) be a group of order 2022. Let \( H \) and \( K \) be subgroups of \( G \) of order 337 and 674, respectively. If \( H \cup K \) is also a subgroup of \( G \), then which one of the following is FALSE?

• (A) \( H \) is a normal subgroup of \( H \cup K \).
• (B) The order of \( H \cup K \) is 1011.
• (C) The order of \( H \cup K \) is 674.
• (D) \( K \) is a normal subgroup of \( H \cup K \).

Question 5:

The radius of convergence of the power series \[ \sum_{n=1}^{\infty} \left( \frac{n^3}{4^n} \right) x^{5n} is. \]

• (A) 4
• (B) \( \frac{\sqrt{4}}{4} \)
• (C) \( \frac{1}{4} \)
• (D) \( \frac{1}{\sqrt{4}} \)

Question 6:

Let \( (x_n) \) and \( (y_n) \) be sequences of real numbers defined by \[ x_1 = 1, \quad y_1 = \frac{1}{2}, \quad x_{n+1} = \frac{x_n + y_n}{2}, \quad and \quad y_{n+1} = \sqrt{x_n y_n} \quad for all \ n \in \mathbb{N}. \]
Then which one of the following is true?

• (A) \( (x_n) \) is convergent, but \( (y_n) \) is not convergent.
• (B) \( (x_n) \) is not convergent, but \( (y_n) \) is convergent.
• (C) Both \( (x_n) \) and \( (y_n) \) are convergent and \( \lim_{n \to \infty} x_n > \lim_{n \to \infty} y_n \).
• (D) Both \( (x_n) \) and \( (y_n) \) are convergent and \( \lim_{n \to \infty} x_n = \lim_{n \to \infty} y_n \).

Question 7:

Suppose \[ a_n = \frac{3n + 3}{5n - 5} \quad and \quad b_n = \frac{1}{(1 + n^2)^{3/4}} \quad for \ n = 2, 3, 4, \dots \]
Then which one of the following is true?

• (A) Both \( \sum_{n=2}^{\infty} a_n \) and \( \sum_{n=2}^{\infty} b_n \) are convergent.
• (B) Both \( \sum_{n=2}^{\infty} a_n \) and \( \sum_{n=2}^{\infty} b_n \) are divergent.
• (C) \( \sum_{n=2}^{\infty} a_n \) is convergent and \( \sum_{n=2}^{\infty} b_n \) is divergent.
• (D) \( \sum_{n=2}^{\infty} a_n \) is divergent and \( \sum_{n=2}^{\infty} b_n \) is convergent.

Question 8:

Consider the series \[ \sum_{n=1}^{\infty} \frac{1}{n^m \left( 1 + \frac{1}{n^p} \right)}, \]
where \( m \) and \( p \) are real numbers.

Under which of the following conditions does the above series converge?

• (A) \( m > 1 \).
• (B) \( 0 < m < 1 \) and \( p > 1 \).
• (C) \( 0 < m \leq 1 \) and \( 0 \leq p \leq 1 \).
• (D) \( m = 1 \) and \( p > 1 \).

Question 9:

Let \( c \) be a positive real number and let \( u: \mathbb{R}^2 \to \mathbb{R} \) be defined by \[ u(x,t) = \frac{1}{2c} \int_{x-ct}^{x+ct} e^{s^2} \, ds \quad for \ (x,t) \in \mathbb{R}^2. \]
Then which one of the following is true?

• (A) \( \frac{\partial^2 u}{\partial t^2} = c^2 \frac{\partial^2 u}{\partial x^2} \) on \( \mathbb{R}^2 \).
• (B) \( \frac{\partial u}{\partial t} = c^2 \frac{\partial^2 u}{\partial x^2} \) on \( \mathbb{R}^2 \).
• (C) \( \frac{\partial u}{\partial t} \frac{\partial u}{\partial x} = 0 \) on \( \mathbb{R}^2 \).
• (D) \( \frac{\partial^2 u}{\partial t \partial x} = 0 \) on \( \mathbb{R}^2 \).

Question 10:

Let \( \theta \in \left( \frac{\pi}{4}, \frac{\pi}{2} \right) \). Consider the functions \[ u: \mathbb{R}^2 - \{(0,0)\} \to \mathbb{R} \quad and \quad v: \mathbb{R}^2 - \{(0,0)\} \to \mathbb{R} \]
given by \[ u(x,y) = x - \frac{x}{x^2 + y^2} \quad and \quad v(x,y) = y + \frac{y}{x^2 + y^2}. \]
The value of the determinant

at the point \( (\cos \theta, \sin \theta) \) is equal to:

• (A) \( 4 \sin \theta \).
• (B) \( 4 \cos \theta \).
• (C) \( 4 \sin^2 \theta \).
• (D) \( 4 \cos^2 \theta \).

Question 11:

Consider the open rectangle \( G = \{ (s,t) \in \mathbb{R}^2 : 0 < s < 1 and 0 < t < 1 \} \) and the map \( T: G \to \mathbb{R}^2 \) given by \[ T(s,t) = \left( \frac{\pi s(1 - t)}{2}, \frac{\pi (1 - s)}{2} \right) \quad for \ (s,t) \in G. \]
Then the area of the image \( T(G) \) of the map \( T \) is equal to:

• (A) \( \frac{\pi}{4} \).
• (B) \( \frac{\pi^2}{4} \).
• (C) \( \frac{\pi^2}{8} \).
• (D) \( 1 \).

Question 12:

Let \( T \) denote the sum of the convergent series \[ 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \frac{1}{5} - \frac{1}{6} + \dots + \frac{(-1)^{n+1}}{n} + \dots \]
and let \( S \) denote the sum of the convergent series \[ 1 - \frac{1}{2} + \frac{1}{4} - \frac{1}{6} + \frac{1}{8} - \frac{1}{10} + \dots = \sum_{n=1}^{\infty} a_n, \]
where \[ a_{3m-2} = \frac{1}{2m-1}, \quad a_{3m-1} = \frac{-1}{4m-2}, \quad a_{3m} = \frac{-1}{4m} \quad for \ m \in \mathbb{N}. \]
Then which one of the following is true?

• (A) \( T = S \) and \( S \neq 0 \).
• (B) \( 2T = S \) and \( S \neq 0 \).
• (C) \( T = 2S \) and \( S \neq 0 \).
• (D) \( T = 0 \).

Question 13:

Let \( u: \mathbb{R} \to \mathbb{R} \) be a twice continuously differentiable function such that \( u(0) > 0 \) and \( u'(0) > 0 \). Suppose \( u \) satisfies \[ u''(x) = \frac{u(x)}{1 + x^2} \quad for all \ x \in \mathbb{R}. \]
Consider the following two statements:

(A) The function \( u u' \) is monotonically increasing on \( [0, \infty) \).
(B) The function \( u \) is monotonically increasing on \( [0, \infty) \).
Then which one of the following is correct?

• (A) Both I and II are false.
• (B) Both I and II are true.
• (C) I is false, but II is true.
• (D) I is true, but II is false.

Question 14:

The value of \[ \lim_{n \to \infty} \sum_{k=2}^{n} \frac{\sqrt{n + 1} - \sqrt{n}}{k (\ln k)^2} \]
is equal to:

• (A) \( \infty \).
• (B) 1.
• (C) \( e \).
• (D) 0.

Question 15:

For \( t \in \mathbb{R} \), let \( \lfloor t \rfloor \) denote the greatest integer less than or equal to \( t \). Define functions \( h: \mathbb{R}^2 \to \mathbb{R} \) and \( g: \mathbb{R} \to \mathbb{R} \) by

Then which one of the following is FALSE?

• (A) \( \lim_{(x,y) \to (\sqrt{2}, \pi)} \cos \left( \frac{x^2 y}{x^2 + 1} \right) = -\frac{1}{2} \).
• (B) \( \lim_{(x,y) \to (\sqrt{2}, 2)} e^{h(x,y)} = 0 \).
• (C) \( \lim_{(x,y) \to (e,e)} \ln(xy - [y]) = e - 2. \)
• (D) \( \lim_{(x,y) \to (0,0)} e^{2y} g(x) = 1. \)

Question 16:

Let \( P \in M_4(\mathbb{R}) \) be such that \( P^4 \) is the zero matrix, but \( P^3 \) is a nonzero matrix. Then which one of the following is FALSE?

• (A) For every nonzero vector \( v \in \mathbb{R}^4 \), the subset \( \{ v, Pv, P^2 v, P^3 v \} \) of the real vector space \( \mathbb{R}^4 \) is linearly independent.
• (B) The rank of \( P^k \) is \( 4 - k \) for every \( k \in \{ 1, 2, 3, 4 \} \).
• (C) 0 is an eigenvalue of \( P \).
• (D) If \( Q \in M_4(\mathbb{R}) \) is such that \( Q^4 \) is the zero matrix, but \( Q^3 \) is a nonzero matrix, then there exists a nonsingular matrix \( S \in M_4(\mathbb{R}) \) such that \( S^{-1} Q S = P \).

Question 17:

For \( X, Y \in M_2(\mathbb{R}) \), define \( (X, Y) = XY - YX \). Let \( 0 \in M_2(\mathbb{R}) \) denote the zero matrix. Consider the two statements: \[ P : \quad (X, (Y, Z)) + (Y, (Z, X)) + (Z, (X, Y)) = 0 \quad for all \ X, Y, Z \in M_2(\mathbb{R}), \] \[ Q : \quad (X, (Y, Z)) = ((X, Y), Z) \quad for all \ X, Y, Z \in M_2(\mathbb{R}). \]
Then which one of the following is correct?

• (A) Both \( P \) and \( Q \) are true.
• (B) \( P \) is true, but \( Q \) is false.
• (C) \( P \) is false, but \( Q \) is true.
• (D) Both \( P \) and \( Q \) are false.

Question 18:

Consider the system of linear equations \[ x + y + t = 4, \] \[ 2x - 4t = 7, \] \[ x + y + z = 5, \] \[ x - 3y - z - 10t = \lambda, \]
where \( x, y, z, t \) are variables and \( \lambda \) is a constant. Then which one of the following is true?

• (A) If \( \lambda = 1 \), then the system has a unique solution.
• (B) If \( \lambda = 2 \), then the system has infinitely many solutions.
• (C) If \( \lambda = 1 \), then the system has infinitely many solutions.
• (D) If \( \lambda = 2 \), then the system has a unique solution.

Question 19:

Consider the group \( (\mathbb{Q}, +) \) and its subgroup \( (\mathbb{Z}, +) \). For the quotient group \( \mathbb{Q}/\mathbb{Z} \), which one of the following is FALSE?

• (A) \( \mathbb{Q}/\mathbb{Z} \) contains a subgroup isomorphic to \( (\mathbb{Z}, +) \).
• (B) There is exactly one group homomorphism from \( \mathbb{Q}/\mathbb{Z} \) to \( (\mathbb{Q}, +) \).
• (C) For all \( n \in \mathbb{N} \), there exists \( g \in \mathbb{Q}/\mathbb{Z} \) such that the order of \( g \) is \( n \).
• (D) \( \mathbb{Q}/\mathbb{Z} \) is not a cyclic group.

Question 20:

For \( P \in M_5(\mathbb{R}) \) and \( i, j \in \{1, 2, \dots, 5\} \), let \( p_{ij} \) denote the \( (i,j) \)-th entry of \( P \). Let \[ S = \{ P \in M_5(\mathbb{R}) : p_{ij} = p_{rs} \ for \ i, j, r, s \in \{ 1, 2, \dots, 5\} with \ i + r = j + s \}. \]
Then which one of the following is FALSE?

• (A) \( S \) is a subspace of the vector space over \( \mathbb{R} \) of all \( 5 \times 5 \) symmetric matrices.
• (B) The dimension of \( S \) over \( \mathbb{R} \) is 5.
• (C) The dimension of \( S \) over \( \mathbb{R} \) is 11.
• (D) If \( P \in S \) and all the entries of \( P \) are integers, then 5 divides the sum of all the diagonal entries of \( P \).

Question 21:

On the open interval \((-c, c)\), where \(c\) is a positive real number, \(y(x)\) is an infinitely differentiable solution of the differential equation \[ \frac{dy}{dx} = y^2 - 1 + \cos x, \]
with the initial condition \(y(0) = 0\). Then which one of the following is correct?

• (A) \(y(x)\) has a local maximum at the origin
• (B) \(y(x)\) has a local minimum at the origin
• (C) \(y(x)\) is strictly increasing on the open interval \((-\delta, \delta)\) for some positive real number \(\delta\).
• (D) \(y(x)\) is strictly decreasing on the open interval \((-\delta, \delta)\) for some positive real number \(\delta\).

Question 22:

Let \(H : \mathbb{R} \to \mathbb{R}\) be the function given by \(H(x) = \frac{1}{2} \left(e^x + e^{-x}\right)\) for \(x \in \mathbb{R}\).

Let \(f : \mathbb{R} \to \mathbb{R}\) be defined by \[ f(x) = \int_0^{\pi} H(x \sin \theta) d\theta \quad for \quad x \in \mathbb{R}. \]
Then which one of the following is true?

• (A) \(x f''(x) + f'(x) + x f(x) = 0\) for all \(x \in \mathbb{R}\).
• (B) \(x f''(x) - f'(x) + x f(x) = 0\) for all \(x \in \mathbb{R}\).
• (C) \(x f''(x) + f'(x) - x f(x) = 0\) for all \(x \in \mathbb{R}\).
• (D) \(x f''(x) - f'(x) - x f(x) = 0\) for all \(x \in \mathbb{R}\).

Question 23:

Consider the differential equation \[ y'' + a y' + y = \sin x \quad for \quad x \in \mathbb{R}. \]
Then which one of the following is true?

• (A) If \(a = 0\), then all the solutions of \((**)\) are unbounded over \(\mathbb{R}\).
• (B) If \(a = 1\), then all the solutions of \((**)\) are unbounded over \((0, \infty)\).
• (C) If \(a = 1\), then all the solutions of \((**)\) tend to zero as \(x \to \infty\).
• (D) If \(a = 2\), then all the solutions of \((**)\) are bounded over \((-\infty, 0)\).

Question 24:

For \(g \in \mathbb{Z}\), let \(\bar{g} \in \mathbb{Z}_{37}\) denote the residue class of \(g\) modulo 37. Consider the group \(U_{37} = \left\{g \in \mathbb{Z}_{37} : 1 \leq g \leq 37 and \gcd(g, 37) = 1\right\}\) with respect to multiplication modulo 37. Then which one of the following is FALSE?

• (A) The set \(\{\bar{g} \in U_{37} : \bar{g} = \bar{g}^{-1}\}\) contains exactly 2 elements.
• (B) The order of the element \(10\) in \(U_{37}\) is 36.
• (C) There is exactly one group homomorphism from \(U_{37}\) to \((\mathbb{Z}, +)\).
• (D) There is exactly one group homomorphism from \(U_{37}\) to \((\mathbb{Q}, +)\).

Question 25:

For some real number \(c\) with \(0 < c < 1\), let \(\varphi : (1 - c, 1 + c) \to (0, \infty)\) be a differentiable function such that \(\varphi(1) = 1\) and \(y = \varphi(x)\) is a solution of the differential equation \[ (x^2 + y^2) dx - 4xy \, dy = 0. \]
Then which one of the following is true?

• (A) \(\left(3(\varphi(x))^2 + x^2\right)^2 = 4x\).
• (B) \(\left(3(\varphi(x))^2 - x^2\right)^2 = 4x\).
• (C) \(\left(3(\varphi(x))^2 + x^2\right)^2 = 4\varphi(x)\).
• (D) \(\left(3(\varphi(x))^2 - x^2\right)^2 = 4\varphi(x)\).

Question 26:

For a \(4 \times 4\) matrix \(M \in M_4(\mathbb{C})\), let \(\overline{M}\) denote the matrix obtained from \(M\) by replacing each entry of \(M\) by its complex conjugate. Consider the real vector space \[ H = \{ M \in M_4(\mathbb{C}) : M^T = \overline{M} \}, \]
where \(M^T\) denotes the transpose of \(M\). The dimension of \(H\) as a vector space over \(\mathbb{R}\) is equal to

• (A) 6.
• (B) 16.
• (C) 15.
• (D) 12.

Question 27:

Let \(a, b\) be positive real numbers such that \(a < b\). Given that \[ \lim_{N \to \infty} \int_0^N e^{-t^2} dt = \frac{\sqrt{\pi}}{2}, \]
the value of \[ \lim_{N \to \infty} \int_0^N \frac{1}{t^2} \left( e^{-at^2} - e^{-bt^2} \right) dt \]
is equal to

• (A) \(\sqrt{\pi} \left( \sqrt{a} - \sqrt{b} \right)\).
• (B) \(\sqrt{\pi} \left( \sqrt{a} + \sqrt{b} \right)\).
• (C) \(-\sqrt{\pi} \left( \sqrt{a} + \sqrt{b} \right)\).
• (D) \(\sqrt{\pi} \left( \sqrt{b} - \sqrt{a} \right)\).

Question 28:

For \(-1 \leq x \leq 1\), if \(f(x)\) is the sum of the convergent power series \[ x + \frac{x^2}{2^2} + \frac{x^3}{3^2} + \dots + \frac{x^n}{n^2} + \dots \]
then \(f\left(\frac{1}{2}\right)\) is equal to

• (A) \(\int_0^{\frac{1}{2}} \frac{\ln(1 - t)}{t} dt\).
• (B) \(-\int_0^{\frac{1}{2}} \frac{\ln(1 - t)}{t} dt\).
• (C) \(\int_0^{\frac{1}{2}} \ln(1 + t) dt\).
• (D) \(\int_0^{\frac{1}{2}} t \ln(1 - t) dt\).

Question 29:

For \(n \in \mathbb{N}\) and \(x \in [1, \infty)\), let \[ f_n(x) = \int_0^\pi \left( x^2 + (\cos \theta) \sqrt{x^2 - 1} \right)^n \, d\theta. \]
Then which one of the following is true?

• (A) \(f_n(x)\) is not a polynomial in \(x\) if \(n\) is odd and \(n \geq 3\).
• (B) \(f_n(x)\) is not a polynomial in \(x\) if \(n\) is even and \(n \geq 4\).
• (C) \(f_n(x)\) is a polynomial in \(x\) for all \(n \in \mathbb{N}\).
• (D) \(f_n(x)\) is not a polynomial in \(x\) for any \(n \geq 3\).

Question 30:

Let \(P\) be a \(3 \times 3\) real matrix having eigenvalues \(\lambda_1 = 0, \lambda_2 = 1\), and \(\lambda_3 = -1\). Further,

are eigenvectors of the matrix \(P\) corresponding to the eigenvalues \(\lambda_1, \lambda_2\) and \(\lambda_3\), respectively. Then the entry in the first row and the third column of \(P\) is

• (A) 0.
• (B) 1.
• (C) -1.
• (D) 2.

Question 31:

Let \((-c, c)\) be the largest open interval in \(\mathbb{R}\) (where \(c\) is either a positive real number or \(c = \infty\)) on which the solution \(y(x)\) of the differential equation \[ \frac{dy}{dx} = x^2 + y^2 + 1 \]
with initial condition \(y(0) = 0\) exists and is unique. Then which of the following is/are true?

• (A) \(y(x)\) is an odd function on \((-c, c)\).
• (B) \(y(x)\) is an even function on \((-c, c)\).
• (C) \((y(x))^2\) has a local minimum at 0.
• (D) \((y(x))^2\) has a local maximum at 0.

Question 32:

Let \(S\) be the set of all continuous functions \(f: [-1, 1] \to \mathbb{R}\) satisfying the following three conditions:

[(i)] \(f\) is infinitely differentiable on the open interval \((-1, 1)\),
[(ii)] the Taylor series
\[ f(0) + f'(0)x + \frac{f''(0)}{2!}x^2 + \dots \]
of \(f\) at 0 converges to \(f(x)\) for each \(x \in (-1, 1)\),
[(iii)] \(f^{(n)}(1) = 0\) for all \(n \in \mathbb{N}\).

Then which of the following is true?

• (A) \(f(0) = 0\) for every \(f \in S\).
• (B) \(f'\left(\frac{1}{2}\right) = 0\) for every \(f \in S\).
• (C) There exists \(f \in S\) such that \(f'\left(\frac{1}{2}\right) \neq 0\).
• (D) There exists \(f \in S\) such that \(f(x) \neq 0\) for some \(x \in [-1, 1]\).

Question 33:

Define

and define
Then which of the following is/are true?

• (A) \(f\) is Riemann integrable on \([0, 1]\).
• (B) \(g\) is Riemann integrable on \([0, 1]\).
• (C) The composite function \(f \circ g\) is Riemann integrable on \([0, 1]\).
• (D) The composite function \(g \circ f\) is Riemann integrable on \([0, 1]\).

Question 34:

Let \(S\) be the set of all functions \(f: \mathbb{R} \to \mathbb{R}\) satisfying \[ |f(x) - f(y)|^2 \leq |x - y|^3 \quad for all x, y \in \mathbb{R}. \]
Then which of the following is/are true?

• (A) Every function in \(S\) is differentiable.
• (B) There exists a function \(f \in S\) such that \(f\) is differentiable, but \(f\) is not twice differentiable.
• (C) There exists a function \(f \in S\) such that \(f\) is twice differentiable, but is not thrice differentiable.
• (D) Every function in \(S\) is infinitely differentiable.

Question 35:

A real-valued function \(y(x)\) defined on \(\mathbb{R}\) is said to be periodic if there exists a real number \(T > 0\) such that \(y(x + T) = y(x)\) for all \(x \in \mathbb{R}\).

Consider the differential equation \[ \frac{d^2y}{dx^2} + 4y = \sin(ax), \quad x \in \mathbb{R}, \]
where \(a \in \mathbb{R}\) is a constant. Then which of the following is/are true?

• (A) All solutions of \((*)\) are periodic for every choice of \(a\).
• (B) All solutions of \((*)\) are periodic for every choice of \(a \in \mathbb{R} \setminus \{-2, 2\}\).
• (C) All solutions of \((*)\) are periodic for every choice of \(a \in \mathbb{R} \setminus \{-2, 2\}\).
• (D) If \(a \in \mathbb{R} \setminus \mathbb{Q}\), then there is a unique periodic solution of \((*)\).

Question 36:

Let \(M\) be a positive real number and let \(u, v: \mathbb{R}^2 \to \mathbb{R}\) be continuous functions satisfying \[ \sqrt{(u(x, y))^2 + (v(x, y))^2} \geq M \sqrt{x^2 + y^2} \quad for all (x, y) \in \mathbb{R}^2. \]
Let \[ F: \mathbb{R}^2 \to \mathbb{R}^2 \quad be given by \quad F(x, y) = (u(x, y), v(x, y)) \quad for \quad (x, y) \in \mathbb{R}^2. \]
Then which of the following is/are true?

• (A) \(F\) is injective.
• (B) If \(K\) is open in \(\mathbb{R}^2\), then \(F(K)\) is open in \(\mathbb{R}^2\).
• (C) If \(K\) is closed in \(\mathbb{R}^2\), then \(F(K)\) is closed in \(\mathbb{R}^2\).
• (D) If \(E\) is closed and bounded in \(\mathbb{R}^2\), then \(F^{-1}(E)\) is closed and bounded in \(\mathbb{R}^2\).

Question 37:

Let \(G\) be a finite group of order at least two and let \(e\) denote the identity element of \(G\). Let \(\sigma : G \to G\) be a bijective group homomorphism that satisfies the following two conditions:

[(i)] If \(\sigma(g) = g\) for some \(g \in G\), then \(g = e\),
[(ii)] \((\sigma \circ \sigma)(g) = g\) for all \(g \in G\).

Then which of the following is/are correct?

• (A) For each \(g \in G\), there exists \(h \in G\) such that \(h^{-1} \sigma(h) = g\).
• (B) There exists \(x \in G\) such that \(x \sigma(x) \neq e\).
• (C) The map \(\sigma\) satisfies \(\sigma(x) = x^{-1}\) for every \(x \in G\).
• (D) The order of the group \(G\) is an odd number.

Question 38:

Let \((x_n)\) be a sequence of real numbers. Consider the set \[ P = \{n \in \mathbb{N} : x_n > x_m for all m \in \mathbb{N} with m > n \}. \]
Then which of the following is true?

• (A) If \(P\) is finite, then \((x_n)\) has a monotonically increasing subsequence.
• (B) If \(P\) is finite, then no subsequence of \((x_n)\) is monotonically increasing.
• (C) If \(P\) is infinite, then \((x_n)\) has a monotonically decreasing subsequence.
• (D) If \(P\) is infinite, then no subsequence of \((x_n)\) is monotonically decreasing.

Question 39:

Let \(V\) be the real vector space consisting of all polynomials in one variable with real coefficients and having degree at most 5, together with the zero polynomial.

Let \(T: V \to \mathbb{R}\) be the linear map defined by \(T(1) = 1\) and \[ T(x(x - 1)(x - 2) \dots (x - k + 1)) = 1 \quad for \quad 1 \leq k \leq 5. \]
Then which of the following is/are true?

• (A) \(T(x^4) = 15\).
• (B) \(T(x^3) = 5\).
• (C) \(T(x^2) = 14\).
• (D) \(T(x^3) = 3\).

Question 40:

Let \(P\) be a fixed \(3 \times 3\) matrix with entries in \(\mathbb{R}\). Which of the following maps from \(M_3(\mathbb{R})\) to \(M_3(\mathbb{R})\) is/are linear?

• (A) \(T_1 : M_3(\mathbb{R}) \to M_3(\mathbb{R})\) given by \(T_1(M) = MP - PM\) for \(M \in M_3(\mathbb{R})\).
• (B) \(T_2 : M_3(\mathbb{R}) \to M_3(\mathbb{R})\) given by \(T_2(M) = M^2P - P^2M\) for \(M \in M_3(\mathbb{R})\).
• (C) \(T_3 : M_3(\mathbb{R}) \to M_3(\mathbb{R})\) given by \(T_3(M) = MP^2 + P^2M\) for \(M \in M_3(\mathbb{R})\).
• (D) \(T_4 : M_3(\mathbb{R}) \to M_3(\mathbb{R})\) given by \(T_4(M) = MP^2 - P M^2\) for \(M \in M_3(\mathbb{R})\).

Question 41:

The value of the limit \[ \lim_{n \to \infty} \left( \frac{1^1 + 2^2 + \cdots + n^n}{n^5} + \frac{1}{\sqrt{n}} \left( \frac{1}{\sqrt{n+1}} + \frac{1}{\sqrt{n+2}} + \cdots + \frac{1}{\sqrt{4n}} \right) \right) \]
is equal to ........... (Rounded off to two decimal places)

Question 42:

Consider the function \( u: \mathbb{R}^3 \to \mathbb{R} \) given by \[ u(x_1, x_2, x_3) = x_1^2 x_2^3 - x_1^3 x_4 - 26x_1^2 x_2^2 x_3^3 \]
Let \( c \in \mathbb{R} \) and \( k \in \mathbb{N} \) be such that \[ x_1 \frac{\partial u}{\partial x_2} + 2x_2 \frac{\partial u}{\partial x_3} evaluated at the point (t, t^2, t^3), equals ct^k for every t \in \mathbb{R}. \]
Then the value of \( k \) is equal to .........

Question 43:

Let \( y(x) \) be the solution of the differential equation \[ \frac{dy}{dx} + 3x^2 y = x^2, \quad for x \in \mathbb{R}, \]
satisfying the initial condition \( y(0) = 4 \).
Then \( \lim_{x \to \infty} y(x)\) is equal to ....... (Rounded off to two decimal places)

Question 44:

The sum of the series \[ \sum_{n=1}^{\infty} \frac{1}{(4n - 3)(4n + 1)} \]
is equal to ......... (Rounded off to two decimal places)

Question 45:

The number of distinct subgroups of \( \mathbb{Z}_{999} \) is .........

Question 46:

The number of elements of order 12 in the symmetric group \( S_7 \) is ........

Question 47:

Let \( y(x) \) be the solution of the differential equation \[ xy^2 y' + y^3 = \frac{\sin x}{x} \quad for \quad x > 0, \]
satisfying \( y\left( \frac{\pi}{2} \right) = 0 \).
Then the value of \( y\left( \frac{5\pi}{2} \right) \) is equal to ........ (Rounded off to two decimal places)

Question 48:

Consider the region \[ G = \left\{ (x, y, z) \in \mathbb{R}^3 : 0 < z < x^2 - y^2, x^2 + y^2 < 1 \right\}. \]
Then the volume of \( G \) is equal to ........ (Rounded off to two decimal places)

Question 49:

Given that \( y(x) \) is a solution of the differential equation \[ x^2 y'' + x y' - 4y = x^2 \quad on the interval \quad (0, \infty) \]
such that \[ \lim_{x \to 0^+} y(x) exists and y(1) = 1. \]
The value of \( y'(1) \) is equal to ......... (Rounded off to two decimal places)

Question 50:

Consider the family \( \mathcal{F}_1 \) of curves lying in the region \[ \left\{ (x, y) \in \mathbb{R}^2 : y > 0 and 0 < x < \pi \right\} \]
and given by \[ y = \frac{c(1 - \cos x)}{\sin x}, \quad where c is a positive real number. \]
Let \( \mathcal{F}_2 \) be the family of orthogonal trajectories to \( \mathcal{F}_1 \). Consider the curve \( C \) belonging to the family \( \mathcal{F}_2 \) passing through the point \( \left( \frac{\pi}{3}, 1 \right) \). If \( a \) is a real number such that \( \left( \frac{\pi}{4}, a \right) lies on C, \textbf{ then the value of} a^4 \textbf{is equal to ..........} \) (Rounded off to two decimal places)

Question 51:

For \( t \in \mathbb{R} \), let \( \lfloor t \rfloor \) denote the greatest integer less than or equal to \( t \).
Let \[ D = \{ (x, y) \in \mathbb{R}^2 : x^2 + y^2 < 4 \}. \]
Let \( f: D \to \mathbb{R} \) and \( g: D \to \mathbb{R} \) be defined by \[ f(x, y) = \left[ x^2 + y^2 \right]^2, \quad g(x, y) = \frac{x y}{x^2 + y^2} \]
for \( (x, y) \neq (0, 0) \). Let \( E \) be the set of points of \( D \) at which both \( f \) and \( g \) are discontinuous. The number of elements in the set \( E \) is ........

Question 52:

If \( G \) is the region in \( \mathbb{R}^2 \) given by \[ G = \left\{ (x, y) \in \mathbb{R}^2 : x^2 + y^2 < 1, \quad \frac{x}{\sqrt{3}} < y < \sqrt{3} x, \quad x > 0, \quad y > 0 \right\}, \]
then the value of \[ \frac{200}{\pi} \int \int_G x^2 \, dx \, dy \]
is equal to ............ (Rounded off to two decimal places)

Question 53:

Let
and let \( A^T \) denote the transpose of \( A \). Let be column vectors with entries in \(\mathbb{R}\) such that \[ u_1^2 + u_2^2 = 1 and v_1^2 + v_2^2 + v_3^2 = 1. \]
Suppose \[ A u = \sqrt{2} v \quad and \quad A^T v = \sqrt{2} u. \]
Then \( |u_1 + 2\sqrt{2} v_1| \) is equal to ............ (Rounded off to two decimal places)

Question 54:

Let \( f : [0, \pi] \to \mathbb{R} \) be the function defined by

Then the value of \[ \int_0^\pi f(x) \, dx \]
is equal to ........ (Rounded off to two decimal places)

Question 55:

Let \( r \) be the radius of convergence of the power series \[ \frac{1}{3} \cdot \frac{x}{5} + \frac{x^2}{32} + \frac{x^3}{52} + \frac{x^4}{33} + \frac{x^5}{53} + \frac{x^6}{34} + \frac{x^7}{54} + \dots \]
Then the value of \( r^2 \) is equal to ............ (Rounded off to two decimal places)

Question 56:

Define \( f: \mathbb{R}^2 \to \mathbb{R} \) by \[ f(x, y) = x^2 + 2y^2 - x \quad for \quad (x, y) \in \mathbb{R}^2. \]
Let \[ D = \{(x, y) \in \mathbb{R}^2 : x^2 + y^2 \leq 1 \} \quad and \quad E = \left\{ (x, y) \in \mathbb{R}^2 : \frac{x^2}{4} + \frac{y^2}{9} \leq 1 \right\}. \]
Consider the sets \[ D_{max} = \{(a, b) \in D : f has absolute maximum on D at (a, b)\}, \] \[ D_{min} = \{(a, b) \in D : f has absolute minimum on D at (a, b)\}, \] \[ E_{max} = \{(c, d) \in E : f has absolute maximum on E at (c, d)\}, \] \[ E_{min} = \{(c, d) \in E : f has absolute minimum on E at (c, d)\}. \]
Then the total number of elements in the set \[ D_{max} \cup D_{min} \cup E_{max} \cup E_{min} \]
is equal to .............

Question 57:

Consider the \( 4 \times 4 \) matrix

Then the value of the determinant of \( M \) is equal to .........

Question 58:

Let \( \sigma \) be the permutation in the symmetric group \( S_5 \) given by \[ \sigma(1) = 2, \quad \sigma(2) = 3, \quad \sigma(3) = 1, \quad \sigma(4) = 5, \quad \sigma(5) = 4. \]
Define \[ N(\sigma) = \{ \tau \in S_5 : \sigma \circ \tau = \tau \circ \sigma \}. \]
Then the number of elements in \( N(\sigma) \) is equal to .........

Question 59:

Let \( f : (-1, 1) \to \mathbb{R} \) and \( g : (-1, 1) \to \mathbb{R} \) be thrice continuously differentiable functions such that \( f(x) \neq g(x) \) for every nonzero \( x \in (-1, 1) \). Suppose \[ f(0) = \ln 2, \quad f'(0) = \pi, \quad f''(0) = \pi^2, \quad f^{(3)}(0) = \pi^9, \]
and \[ g(0) = \ln 2, \quad g'(0) = \pi, \quad g''(0) = \pi^2, \quad g^{(3)}(0) = \pi^3. \]
Then the value of the limit \[ \lim_{x \to 0} \frac{e^{f(x)} - e^{g(x)}}{f(x) - g(x)} \]
is equal to ........ (Rounded off to two decimal places)

Question 60:

If \( f: [0, \infty) \to \mathbb{R} \) and \( g: [0, \infty) \to [0, \infty) \) are continuous functions such that \[ \int_0^{x^3 + x^2} f(t) \, dt = x^2 \quad and \quad \int_0^{g(x)} t^2 \, dt = 9(x + 1)^3 \quad for all \quad x \in [0, \infty), \]
then the value of \[ f(2) + g(2) + 16 f(12) is equal to \_\_\_\_\_\_\_ . (Rounded off to two decimal places) \]