IIT JAM 2025 Mathematics Question Paper with Answer Key PDF Available

The sum of the infinite series \[ \sum_{n=1}^{\infty} \frac{(-1)^{n+1} \pi^{2n+1}}{2^{2n+1} (2n)!} \]
is equal to

• (A) \( -\pi \)
• (B) \( \frac{\pi}{4} \)
• (C) \( \frac{\pi}{2} \)
• (D) \( -\frac{\pi}{4} \)

For which one of the following choices of \( N(x, y) \), is the equation \[ (e^x \sin y - 2y \sin x) \, dx + N(x, y) \, dy = 0 \]
an exact differential equation?

• (A) \( N(x, y) = e^x \sin y + 2 \cos x \)
• (B) \( N(x, y) = e^x \cos y + 2 \cos x \)
• (C) \( N(x, y) = e^x \cos y + 2 \sin x \)
• (D) \( N(x, y) = e^x \sin y + 2 \sin x \)

Let \( f, g : \mathbb{R} \to \mathbb{R} \) be two functions defined by

and

Then, which one of the following is TRUE?

• (A) \( f \) is differentiable at \( x = 0 \), and \( g \) is NOT differentiable at \( x = 0 \)
• (B) \( f \) is NOT differentiable at \( x = 0 \), and \( g \) is differentiable at \( x = 0 \)
• (C) \( f \) is differentiable at \( x = 0 \), and \( g \) is differentiable at \( x = 0 \)
• (D) \( f \) is NOT differentiable at \( x = 0 \), and \( g \) is NOT differentiable at \( x = 0 \)

Let \( f, g : \mathbb{R} \to \mathbb{R} \) be two functions defined by

and

Then, which one of the following is TRUE?

• (A) \( f \) is continuous at \( x = 0 \), and \( g \) is NOT continuous at \( x = 0 \)
• (B) \( f \) is NOT continuous at \( x = 0 \), and \( g \) is continuous at \( x = 0 \)
• (C) \( f \) is continuous at \( x = 0 \), and \( g \) is continuous at \( x = 0 \)
• (D) \( f \) is NOT continuous at \( x = 0 \), and \( g \) is NOT continuous at \( x = 0 \)

Which one of the following is the general solution of the differential equation \[ \frac{d^2 y}{dx^2} - 8 \frac{dy}{dx} + 16y = 2e^{4x} ? \]

• (A) \( \alpha_1 e^{4x} + \alpha_2 x e^{4x} \), where \( \alpha_1, \alpha_2 \in \mathbb{R} \)
• (B) \( \alpha_1 e^{4x} + \alpha_2 x e^{4x} + 2x e^{4x} \), where \( \alpha_1, \alpha_2 \in \mathbb{R} \)
• (C) \( \alpha_1 e^{4x} + \alpha_2 e^{4x} + 2x e^{4x} \), where \( \alpha_1, \alpha_2 \in \mathbb{R} \)
• (D) \( \alpha_1 x e^{-4x} + \alpha_2 x^2 e^{4x} \), where \( \alpha_1, \alpha_2 \in \mathbb{R} \)

Define \( T : \mathbb{R}^3 \to \mathbb{R}^3 \) by \[ T(x, y, z) = (x + z, 2x + 3y + 5z, 2y + 2z), \quad for all (x, y, z) \in \mathbb{R}^3 \]
Then, which one of the following is TRUE?

• (A) \( T \) is one-one and \( T \) is NOT onto
• (B) \( T \) is NOT one-one and \( T \) is onto
• (C) \( T \) is one-one and \( T \) is onto
• (D) \( T \) is NOT one-one and \( T \) is NOT onto

Let 

for some real number \( x \). If 0 is an eigenvalue of \( M \), then is equal to

Let \( T : P_2(\mathbb{R}) \to P_2(\mathbb{R}) \) be the linear transformation defined by \[ T(p(x)) = p(x + 1), \quad for all p(x) \in P_2(\mathbb{R}) \]
If \( M \) is the matrix representation of \( T \) with respect to the ordered basis \( \{1, x, x^2\} \) of \( P_2(\mathbb{R}) \), then which one of the following is TRUE?

• (A) The determinant of \( M \) is 2
• (B) The rank of \( M \) is 2
• (C) 1 is the only eigenvalue of \( M \)
• (D) The nullity of \( M \) is 2

Let \( G \) be a finite abelian group of order 10. Let \( x_0 \) be an element of order 2 in \( G \).
If \( X = \{ x \in G : x^3 = x_0 \} \), then which one of the following is TRUE?

• (A) \( X \) has exactly one element
• (B) \( X \) has exactly two elements
• (C) \( X \) has exactly three elements
• (D) \( X \) is an empty set

The value of \[ \int_0^1 \left( \int_0^{\sqrt{y}} 3e^{x^3} \, dx \right) dy \]
is equal to

• (A) \( e - 1 \)
• (B) \( \frac{e - 1}{2} \)
• (C) \( \sqrt{e} - 1 \)
• (D) \( \frac{\sqrt{e} - 1}{2} \)

Let \( C \) denote the family of curves described by \( yx^2 = \lambda \), for \( \lambda \in (0, \infty) \) and lying in the first quadrant of the \( xy \)-plane. Let \( O \) denote the family of orthogonal trajectories of \( C \).
Which one of the following curves is a member of \( O \), and passes through the point \( (2, 1) \)?

• (A) \( y = \frac{x^2}{4}, \, x > 0, y > 0 \)
• (B) \( x^2 - 2y^2 = 2, \, x > 0, y > 0 \)
• (C) \( x - y = 1, \, x > 0, y > 0 \)
• (D) \( 2x - y^2 = 3, \, x > 0, y > 0 \)

Let \( \varphi : (0, \infty) \to \mathbb{R} \) be the solution of the differential equation \[ x \frac{dy}{dx} = \left( \ln y - \ln x \right) y, \]
satisfying \( \varphi(1) = e^2 \). Then, the value of \( \varphi(2) \) is equal to:

• (A) \( e^2 \)
• (B) \( 2e^3 \)
• (C) \( 3e^2 \)
• (D) \( 6e^3 \)

Let \( X = \{ x \in S_4 : x^3 = id \} \) and \( Y = \{ x \in S_4 : x^2 \neq id \} \).
If \( m \) and \( n \) denote the number of elements in \( X \) and \( Y \), respectively, then which one of the following is TRUE?

• (A) \( m \) is even and \( n \) is even
• (B) \( m \) is odd and \( n \) is even
• (C) \( m \) is even and \( n \) is odd
• (D) \( m \) is odd and \( n \) is odd

Let \( \varphi : \mathbb{R} \to \mathbb{R} \) be the solution of the differential equation \[ x \frac{dy}{dx} = (y - 1)(y - 3), \]
satisfying \( \varphi(0) = 2 \). Then, which one of the following is TRUE?

• (A) \( \lim_{x \to \infty} \varphi(x) = 0 \)
• (B) \( \lim_{x \to \ln \sqrt{2}} \varphi(x) = 1 \)
• (C) \( \lim_{x \to -\infty} \varphi(x) = 3 \)
• (D) \( \lim_{x \to \ln \frac{1}{\sqrt{2}}} \varphi(x) = 6 \)

Let 

Consider the system \( S \) of linear equations given by: \[ 6x_1 + 2x_2 - 6x_3 + 8x_4 = 8 \] \[ 5x_1 + 3x_2 - 9x_3 + 8x_4 = 16 \] \[ 3x_1 + x_2 - 2x_3 + 4x_4 = 32 \]
where \( x_1, x_2, x_3, x_4 \) are unknowns.
Then, which one of the following is TRUE?

• (A) The rank of \( M \) is 3, and the system \( S \) has a solution
• (B) The rank of \( M \) is 3, and the system \( S \) does NOT have a solution
• (C) The rank of \( M \) is 2, and the system \( S \) has a solution
• (D) The rank of \( M \) is 2, and the system \( S \) does NOT have a solution

Let 

for some real number \( x \). Suppose that \( -2 \) and \( 3 \) are eigenvalues of \( M \). If then which one of the following is TRUE?

• (A) \( x = 5 \), and the matrix \( M^2 + M \) is invertible
• (B) \( x \neq 5 \), and the matrix \( M^2 + M \) is invertible
• (C) \( x = 5 \), and the matrix \( M^2 + M \) is NOT invertible
• (D) \( x \neq 5 \), and the matrix \( M^2 + M \) is NOT invertible

Let \( f(x) = 10x^2 + e^x - \sin(2x) - \cos x \), \( x \in \mathbb{R} \). The number of points at which the function \( f \) has a local minimum is:

• (A) 0
• (B) 1
• (C) 2
• (D) greater than or equal to 3

For \( n \in \mathbb{N} \), define \( x_n \) and \( y_n \) by \[ x_n = (-1)^n \cos \frac{1}{n} \quad and \quad y_n = \sum_{k=1}^{n} \frac{1}{n + k}. \]
Then, which one of the following is TRUE?

• (A) \( \sum_{n=1}^{\infty} x_n \) converges, and \( \sum_{n=1}^{\infty} y_n \) does NOT converge
• (B) \( \sum_{n=1}^{\infty} x_n \) does NOT converge, and \( \sum_{n=1}^{\infty} y_n \) converges
• (C) \( \sum_{n=1}^{\infty} x_n \) converges, and \( \sum_{n=1}^{\infty} y_n \) converges
• (D) \( \sum_{n=1}^{\infty} x_n \) does NOT converge, and \( \sum_{n=1}^{\infty} y_n \) does NOT converge

Let \( x_1 = \frac{5}{2} \) and for \( n \in \mathbb{N} \), define \[ x_{n+1} = \frac{1}{5} \left( x_n^2 + 6 \right). \]
Then, which one of the following is TRUE?

• (A) \( (x_n) \) is an increasing sequence, and \( (x_n) \) is NOT a bounded sequence
• (B) \( (x_n) \) is NOT an increasing sequence, and \( (x_n) \) is NOT a bounded sequence
• (C) \( (x_n) \) is NOT a decreasing sequence, and \( (x_n) \) is a bounded sequence
• (D) \( (x_n) \) is a decreasing sequence, and \( (x_n) \) is a bounded sequence

Let \( x_1 = 2 \) and \( x_{n+1} = 2 + \frac{1}{2x_n} \) for all \( n \in \mathbb{N} \).
Then, which one of the following is TRUE?

• (A) \( x_{n+1} \geq \frac{4}{x_n} \) for all \( n \in \mathbb{N} \), and \( (x_n) \) is a Cauchy sequence
• (B) \( x_{n+1} = \frac{4}{x_n} \) for some \( n \in \mathbb{N} \), and \( (x_n) \) is a Cauchy sequence
• (C) \( x_{n+1} = \frac{4}{x_n} \) for all \( n \in \mathbb{N} \), and \( (x_n) \) is NOT a Cauchy sequence
• (D) \( x_{n+1} \leq \frac{4}{x_n} \) for some \( n \in \mathbb{N} \), and \( (x_n) \) is NOT a Cauchy sequence

For \( n \in \mathbb{N} \), define \( x_n \) and \( y_n \) by \[ x_n = (-1)^n \frac{3n}{n^3} \quad and \quad y_n = \left(4n^3 + (-1)^n 3n^3 \right)^{1/n}. \]
Then, which one of the following is TRUE?

• (A) \( (x_n) \) has a convergent subsequence, and NO subsequence of \( (y_n) \) is convergent.
• (B) NO subsequence of \( (x_n) \) is convergent, and \( (y_n) \) has a convergent subsequence.
• (C) \( (x_n) \) has a convergent subsequence, and \( (y_n) \) has a convergent subsequence.
• (D) NO subsequence of \( (x_n) \) is convergent, and NO subsequence of \( (y_n) \) is convergent.

Let \( M = (m_{ij}) \) be a \( 3 \times 3 \) real, invertible matrix and \( \sigma \in S_3 \) be the permutation defined by \( \sigma(1) = 2, \sigma(2) = 3 \) and \( \sigma(3) = 1 \). The matrix \( M_\sigma \) is defined by \( n_{ij} = m_{i\sigma(j)} \) for all \( i,j \in \{1, 2, 3\} \).
Then, which one of the following is TRUE?

• (A) \( \det(M) = \det(M_\sigma) \), and the nullity of the matrix \( M - M_\sigma \) is 0
• (B) \( \det(M) = -\det(M_\sigma) \), and the nullity of the matrix \( M - M_\sigma \) is 1
• (C) \( \det(M) = \det(M_\sigma) \), and the nullity of the matrix \( M - M_\sigma \) is 1
• (D) \( \det(M) = -\det(M_\sigma) \), and the nullity of the matrix \( M - M_\sigma \) is 0

Let \( \mathbb{R}/\mathbb{Z} \) denote the quotient group, where \( \mathbb{Z} \) is considered as a subgroup of the additive group of real numbers \( \mathbb{R} \).

Let \( m \) denote the number of injective (one-one) group homomorphisms from \( \mathbb{Z}_3 \) to \( \mathbb{R}/\mathbb{Z} \) and \( n \) denote the number of group homomorphisms from \( \mathbb{R}/\mathbb{Z} \) to \( \mathbb{Z}_3 \).

Then, which one of the following is TRUE?

• (A) \( m = 2 \) and \( n = 1 \)
• (B) \( m = 3 \) and \( n = 3 \)
• (C) \( m = 2 \) and \( n = 3 \)
• (D) \( m = 1 \) and \( n = 1 \)

Let \( f_1, f_2, f_3 \) be nonzero linear transformations from \( \mathbb{R}^4 \) to \( \mathbb{R} \) and \[ \ker(f_1) \subset \ker(f_2) \cap \ker(f_3). \]
Let \( T : \mathbb{R}^4 \to \mathbb{R}^3 \) be the linear transformation defined by \[ T(v) = (f_1(v), f_2(v), f_3(v)) \quad for all v \in \mathbb{R}^4. \]
Then, the nullity of \( T \) is equal to:

• (A) 1
• (B) 2
• (C) 3
• (D) 4

Let \( x_1 = 1 \). For \( n \in \mathbb{N} \), define \[ x_{n+1} = \left( \frac{1}{2} + \frac{\sin^2 n}{n} \right) x_n. \]
Then, which one of the following is TRUE?

• (A) \( \sum_{n=1}^{\infty} x_n \) converges
• (B) \( \sum_{n=1}^{\infty} x_n \) does NOT converge
• (C) \( \sum_{n=1}^{\infty} x_n^2 \) does NOT converge
• (D) \( \sum_{n=1}^{\infty} x_n x_{n+1} \) does NOT converge

Let \( x_1 > 0 \). For \( n \in \mathbb{N} \), define \[ x_{n+1} = x_n + 4. \]
If \[ \lim_{n \to \infty} \left( \frac{1}{x_1 x_2 x_3} + \frac{1}{x_2 x_3 x_4} + \cdots + \frac{1}{x_{n+1} x_{n+2} x_{n+3}} \right) = \frac{1}{24}, \]
then the value of \( x_1 \) is equal to:

• (A) 1
• (B) 2
• (C) 3
• (D) 8

Let \( f : \mathbb{R}^2 \to \mathbb{R} \) be defined by \[ f(x, y) = e^{y}(x^2 + y^2) \quad for all (x, y) \in \mathbb{R}^2. \]
Then, which one of the following is TRUE?

• (A) The number of points at which \( f \) has a local minimum is 2
• (B) The number of points at which \( f \) has a local maximum is 2
• (C) The number of points at which \( f \) has a local minimum is 1
• (D) The number of points at which \( f \) has a local maximum is 1

Let \( \Omega \) be the bounded region in \( \mathbb{R}^3 \) lying in the first octant \( (x \geq 0, y \geq 0, z \geq 0) \), and bounded by the surfaces \( z = x^2 + y^2 \), \( z = 4 \), \( x = 0 \) and \( y = 0 \).
Then, the volume of \( \Omega \) is equal to:

• (A) \( \pi \)
• (B) \( 2\pi \)
• (C) \( 3\pi \)
• (D) \( 4\pi \)

Let \( x_1 = 1 \). For \( n \in \mathbb{N} \), define \[ x_{n+1} = \left( \frac{1}{2} + \frac{\sin^2 n}{n} \right) x_n. \]
Then, which one of the following is TRUE?

• (A) \( \sum_{n=1}^{\infty} x_n \) converges
• (B) \( \sum_{n=1}^{\infty} x_n \) does NOT converge
• (C) \( \sum_{n=1}^{\infty} x_n^2 \) does NOT converge
• (D) \( \sum_{n=1}^{\infty} x_n x_{n+1} \) does NOT converge

The number of elements in the set \[ \{ x \in \mathbb{R} : 8x^2 + x^4 + x^8 = \cos x \} \]
is equal to:

• (A) 0
• (B) 1
• (C) 2
• (D) greater than or equal to 3

Let \( f : \mathbb{R}^2 \to \mathbb{R} \) be defined by

Then, which of the following is/are TRUE?

• (A) The iterated limits \( \lim_{x \to 0} \left( \lim_{y \to 0} f(x, y) \right) \) and \( \lim_{y \to 0} \left( \lim_{x \to 0} f(x, y) \right) \) exist.
• (B) Exactly one of the partial derivatives \( \frac{\partial f}{\partial x} \) and \( \frac{\partial f}{\partial y} \) exists at \( (0, 0) \).
• (C) Both the partial derivatives \( \frac{\partial f}{\partial x} \) and \( \frac{\partial f}{\partial y} \) exist at \( (0, 0) \).
• (D) \( f \) is NOT differentiable at \( (0, 0) \).

If \( M, N, \mu, w : \mathbb{R}^2 \to \mathbb{R} \) are differentiable functions with continuous partial derivatives, satisfying \[ \mu(x, y) M(x, y) \, dx + \mu(x, y) N(x, y) \, dy = dw, \]
then which one of the following is TRUE?

• (A) \( \mu w \) is an integrating factor for \( M(x, y) \, dx + N(x, y) \, dy = 0 \)
• (B) \( \mu w^2 \) is an integrating factor for \( M(x, y) \, dx + N(x, y) \, dy = 0 \)
• (C) \( w(x, y) = w(0,0) + \int_0^x \mu M(s) \, ds + \int_0^y \mu N(t) \, dt \), for all \( (x, y) \in \mathbb{R}^2 \)
• (D) \( w(x, y) = w(0,0) + \int_0^x \mu M(s, y) \, ds + \int_0^y \mu N(x, t) \, dt \), for all \( (x, y) \in \mathbb{R}^2 \)

Let \( \varphi : (-1, \infty) \to (0, \infty) \) be the solution of the differential equation \[ \frac{dy}{dx} = 2 y e^x = 2 e^x \sqrt{y}, \]
satisfying \( \varphi(0) = 1 \). Then, which of the following is/are TRUE?

• (A) \( \varphi \) is an unbounded function.
• (B) \( \lim_{x \to \ln 2} \varphi(x) = (2e - 1)^2 \).
• (C) \( \lim_{x \to \ln 2} \varphi(x) = \sqrt{2e - 1} \).
• (D) \( \varphi \) is a strictly increasing function on the interval \( (0, \infty) \).

Let \( f : \mathbb{R}^2 \to \mathbb{R} \) be defined by

Then, which of the following is/are TRUE?

• (A) \( \lim_{(x, y) \to (0, 0)} f(x, y) \) exists and \( \lim_{(x, y) \to (0, 0)} f(x, y) = 1 \).
• (B) \( \lim_{(x, y) \to (0, 0)} f(x, y) \) exists and \( \lim_{(x, y) \to (0, 0)} f(x, y) = 0 \).
• (C) \( f \) is differentiable at \( (0, 0) \).
• (D) \( f \) is NOT differentiable at \( (0, 0) \).

Let \( u_1 = (1, 0, 0, -1) \), \( u_2 = (2, 0, 0, -1) \), \( u_3 = (0, 0, 1, -1) \), \( u_4 = (0, 0, 0, 1) \) be elements in the real vector space \( \mathbb{R}^4 \).
Then, which of the following is/are TRUE?

• (A) \( \{u_1, u_2, u_3, u_4\} \) is a linearly independent set in \( \mathbb{R}^4 \).
• (B) \( \{u_1 - u_2, u_3 - u_4, u_4 - u_1 \} \) is NOT a linearly independent set in \( \mathbb{R}^4 \).
• (C) \( \{u_1, -u_2, u_3, -u_4 \} \) is NOT a linearly independent set in \( \mathbb{R}^4 \).
• (D) \( \{u_1 + u_2, u_2 + u_3, u_3 + u_4, u_4 + u_1 \} \) is a linearly independent set in \( \mathbb{R}^4 \).

For \( n \in \mathbb{N} \), let \[ x_n = \sum_{k=1}^{n} \frac{k}{n^2 + k}. \]
Then, which of the following is/are TRUE?

• (A) The sequence \( (x_n) \) converges.
• (B) The series \( \sum_{n=1}^{\infty} x_n \) converges.
• (C) The series \( \sum_{n=1}^{\infty} x_n \) does NOT converge.
• (D) The series \( \sum_{n=1}^{\infty} x_n^n \) converges.

Let \( f : \mathbb{R} \to \mathbb{R} \) be a twice differentiable function such that \[ f(0) = 0, \, f'(0) = 2, \, f(1) = -3. \]
Then, which of the following is/are TRUE?

• (A) \( |f'(x)| \leq 2 \) for all \( x \in [0, 1] \).
• (B) \( |f'(x_1)| > 2 \) for some \( x_1 \in [0, 1] \).
• (C) \( |f''(x)| < 10 \) for all \( x \in [0, 1] \).
• (D) \( |f''(x_2)| \geq 10 \) for some \( x_2 \in [0, 1] \).

Let \( f : \mathbb{R} \to \mathbb{R} \) be a twice differentiable function such that \[ f(0) = 4, \, f(1) = -2, \, f(2) = 8, \, f(3) = 2. \]
Then, which of the following is/are TRUE?

• (A) \( |f'(x)| < 5 \) for all \( x \in [0, 1] \).
• (B) \( |f'(x_1)| \geq 5 \) for some \( x_1 \in [0, 1] \).
• (C) \( f'(x_2) = 0 \) for some \( x_2 \in [0, 3] \).
• (D) \( f''(x_3) = 0 \) for some \( x_3 \in [0, 3] \).

For \( n \in \mathbb{N} \), consider the set \( U(n) = \{ x \in \mathbb{Z}_n : \gcd(x, n) = 1 \} \) as a group under multiplication modulo \( n \).
Then, which of the following is/are TRUE?

• (A) \( U(8) \) is a cyclic group.
• (B) \( U(5) \) is a cyclic group.
• (C) \( U(12) \) is a cyclic group.
• (D) \( U(9) \) is a cyclic group.

Consider the following subspaces of the real vector space \( \mathbb{R}^3 \):

Then, which of the following is/are TRUE?

• (A) \( V_1 \cup V_2 \) is a subspace of \( \mathbb{R}^3 \).
• (B) \( V_1 \cup V_3 \) is a subspace of \( \mathbb{R}^3 \).
• (C) \( V_1 \cup V_4 \) is a subspace of \( \mathbb{R}^3 \).
• (D) \( V_1 \cup V_5 \) is a subspace of \( \mathbb{R}^3 \).

The radius of convergence of the power series \[ \sum_{n=1}^{\infty} \frac{(x + \frac{1}{4})^n}{(-2)^n n^2} \]
about \( x = -\frac{1}{4} \) is equal to ___________ (rounded off to two decimal places).

The value of \[ \lim_{n \to \infty} 8n \left( \left( e^{\frac{1}{2n}} - 1 \right) \left( \sin \frac{1}{2n} + \cos \frac{1}{2n} \right) \right) \]
is equal to ______________ (rounded off to two decimal places).

Let \( \alpha \) be the real number such that \[ \lim_{x \to 0} \frac{(1 - \cos x)(22x^2 + x - 4)}{x^3} = \alpha \ln 2. \]
Then, the value of \( \alpha \) is equal to ___________ (rounded off to two decimal places).

Let \( \varphi : \mathbb{R} \to \mathbb{R} \) be the solution of the differential equation \[ 4 \frac{d^2 y}{dx^2} + 16 \frac{dy}{dx} + 25y = 0 \]
satisfying \( \varphi(0) = 1 \) and \( \varphi'(0) = -\frac{1}{2} \).
Then, the value of \( \lim_{x \to \infty} e^{2x} \varphi(x) \) is equal to ___________ (rounded off to two decimal places).

Let \( S \) be the surface area of the portion of the plane \( z = x + y + 3 \), which lies inside the cylinder \( x^2 + y^2 = 1 \).
Then, the value of \( \left( \frac{S}{\pi} \right)^2 \) is equal to ____________ (rounded off to two decimal places).

Consider the following subspaces of \( \mathbb{R}^4 \): \[ V_1 = \left\{ (x, y, z, w) \in \mathbb{R}^4 : x + y + 2w = 0 \right\}, \quad V_2 = \left\{ (x, y, z, w) \in \mathbb{R}^4 : 2y + z + w = 0 \right\}, \quad V_3 = \left\{ (x, y, z, w) \in \mathbb{R}^4 : x + 3y + z + 3w = 0 \right\}. \]
Then, the dimension of the subspace \( V_1 \cap V_2 \cap V_3 \) is equal to ______________ (rounded off to two decimal places).

Consider the real vector space \( \mathbb{R}^3 \). Let \( T : \mathbb{R}^3 \to \mathbb{R} \) be a linear transformation such that \[ T(1, 1, 1) = 0, \quad T(1, -1, 1) = 0, \quad T(0, 0, 1) = 16. \]
Then, the value of \( T \left( \frac{1}{2}, \frac{2}{3}, \frac{3}{4} \right) \) is equal to ______________ (rounded off to two decimal places).

Let \( T \) denote the triangle in the \( xy \)-plane bounded by the \( x \)-axis and the lines \( y = x \) and \( x = 1 \). The value of the double integral (over \( T \)) \[ \iint_T (5 - y) \, dx \, dy \]
is equal to ____________ (rounded off to two decimal places).

Let \( T, S : P_4(\mathbb{R}) \to P_4(\mathbb{R}) \) be the linear transformations defined by \[ T(p(x)) = xp'(x), \quad S(p(x)) = (x + 1)p'(x) \]
for all \( p(x) \in P_4(\mathbb{R}) \).
Then, the nullity of the composition \( S \circ T \) is ______________

Let \( f : \mathbb{R}^2 \to \mathbb{R} \) be defined by

Then, the value of \( \frac{\partial f}{\partial y}(0, 0) \) and \( \frac{\partial f}{\partial x}(0, 0) \) is equal to ___________ (rounded off to two decimal places).

Let \( f : \mathbb{R} \to \mathbb{R} \) be a continuous function satisfying \[ \int_0^{\frac{\pi}{4}} \left( \sin(x) f(x) + \cos(x) \int_0^x f(t) \, dt \right) \, dx = \sqrt{2}. \]
Then, the value of \[ \int_0^{\frac{\pi}{4}} f(x) \, dx \]
is equal to ________ (rounded off to two decimal places).

Let \( \sigma \in S_4 \) be the permutation defined by \( \sigma(1) = 2 \), \( \sigma(2) = 3 \), \( \sigma(3) = 1 \), and \( \sigma(4) = 4 \).
The number of elements in the set \[ \{ \tau \in S_4 : \tau \circ \sigma^{-1} = \sigma \} \]
is equal to ___________

Let \( f(x) = 2x - \sin(x) \), for all \( x \in \mathbb{R} \). Let \( k \in \mathbb{N} \) be such that \[ \lim_{x \to 0} \left( \frac{1}{x} \sum_{i=1}^{k} i^2 f \left( \frac{x}{i} \right) \right) = 45. \]
Then, the value of \( k \) is equal to ___________

The value of the infinite series \[ \sum_{n=1}^{\infty} n \left( \frac{3}{4} \right)^{2n-1} \]
is equal to ___________ (rounded off to two decimal places).

Let \( \varphi : (0, \infty) \to \mathbb{R} \) be the solution of the differential equation \[ x^2 \frac{d^2 y}{dx^2} - x \frac{dy}{dx} + y = 6x \ln x, \]
satisfying \( \varphi(1) = -3 \) and \( \varphi(e) = 0 \).
Then, the value of \( \varphi'(1) \) is equal to ___________ (rounded off to two decimal places).

Let \( \varphi : \mathbb{R} \to \mathbb{R} \) be the solution of the differential equation \[ \frac{dy}{dx} + 2xy = 2 + 4x^2, \]
satisfying \( \varphi(0) = 0 \).
Then, the value of \( \varphi(2) \) is equal to ___________ (rounded off to two decimal places).

Let \( \Omega \) be the solid bounded by the planes \( z = 0 \), \( y = 0 \), \( x = \frac{1}{2} \), \( 2y = x \) and \( 2x + y + z = 4 \).
If \( V \) is the volume of \( \Omega \), then the value of \( 64V \) is equal to ___________ (rounded off to two decimal places).

Let the subspace \( H \) of \( P_3(\mathbb{R}) \) be defined as \[ H = \{ p(x) \in P_3(\mathbb{R}) : xp'(x) = 3p(x) \}. \]
Then, the dimension of \( H \) is equal to ___________

Let \( G \) be an abelian group of order 35. Let \( m \) denote the number of elements of order 5 in \( G \), and let \( n \) denote the number of elements of order 7 in \( G \).
Then, the value of \( m + n \) is equal to ___________

The number of surjective (onto) group homomorphisms from \( S_4 \) to \( \mathbb{Z}_6 \) is equal to ______________