List of top Mathematics Questions asked in Indian Institute Of Technology Joint Admission Test for MSc

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 \[ T(v) = (f_1(v), f_2(v), f_3(v)) \quad \text{for all } v \in \mathbb{R}^4. \] Then, the nullity of \( T \) is equal to:

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:

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?

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 \( (2, 1) \)?

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:

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?

Let \( T : P_2(\mathbb{R}) \to P_2(\mathbb{R}) \) be the linear transformation defined by \[ T(p(x)) = p(x + 1), \quad \text{for all } p(x) \in P_2(\mathbb{R}) \] 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?

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

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

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?

Define the sequence \[ s_n = \begin{cases} \dfrac{1}{2^n}\displaystyle\sum_{j=0}^{n-2} 2^{2j}, & \text{if } n \text{ is even and } n \gt 0, \\[8pt] \dfrac{1}{2^n}\displaystyle\sum_{j=0}^{n-1} 2^{2j}, & \text{if } n \text{ is odd and } n \gt 0. \end{cases} \] Define \[ \sigma_m = \frac{1}{m}\sum_{n=1}^{m} s_n. \] The number of limit points of the sequence \(\{\sigma_m\}\) is _________.

The determinant of the matrix \[ \begin{pmatrix} 2021 & 2020 & 2020 & 2020 \\ 2021 & 2021 & 2020 & 2020 \\ 2021 & 2021 & 2021 & 2020 \\ 2021 & 2021 & 2021 & 2021 \end{pmatrix} \] is _________.

Let \[ A = \begin{pmatrix} 2 & -1 & 3 \\ 2 & -1 & 3 \\ 3 & 2 & -1 \end{pmatrix}. \] Then the largest eigenvalue of \(A\) is _________.

Let \[ A = \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & -1 \end{pmatrix}. \] Consider the linear map \(T_A : M_4(\mathbb{R}) \to M_4(\mathbb{R})\) defined by \[ T_A(X) = AX - XA. \] Then the dimension of the range of \(T_A\) is _________.

Consider the four functions from \(\mathbb{R}\) to \(\mathbb{R}\): \[ f_1(x) = x^4 + 3x^3 + 7x + 1, \quad f_2(x) = x^3 + 3x^2 + 4x, \quad f_3(x) = \arctan(x), \] and \[ f_4(x) = \begin{cases} x, & \text{if } x \notin \mathbb{Z}, \\ 0, & \text{if } x \in \mathbb{Z}. \end{cases} \] Which of the following subsets of \(\mathbb{R}\) are open?

Let \( A \) be an \( n \times n \) invertible matrix and \( C \) be an \( n \times n \) nilpotent matrix. If \[ X = \begin{pmatrix} X_{11} & X_{12} \\ X_{21} & X_{22} \end{pmatrix} \] is a \( 2n \times 2n \) matrix (each \( X_{ij} \) is \( n \times n \)) that commutes with the \( 2n \times 2n \) matrix \[ B = \begin{pmatrix} A & 0 \\ 0 & C \end{pmatrix}, \] then

Consider the function \[ f(x) = \begin{cases} 1, & \text{if } x \in (\mathbb{R} \setminus \mathbb{Q}) \cup \{0\}, \\ 1 - \frac{1}{p}, & \text{if } x = \frac{n}{p},\ n \in \mathbb{Z}\setminus\{0\},\ p \in \mathbb{N},\ \gcd(n,p)=1 . \end{cases} \] Then

Let \(S\) be the surface defined by \[ \{(x, y, z) \in \mathbb{R}^3 : z = 1 - x^2 - y^2,\, z \ge 0\}. \] Let \[ \vec{F} = -y\hat{i} + (x - 1)\hat{j} + z^2\hat{k}, \] and let \(\hat{n}\) be the continuous unit normal field to the surface \(S\) with positive \(z\)-component. Then the value of \[ \frac{1}{\pi} \iint_S (\nabla \times \vec{F}) \cdot \hat{n}\, dS \] is _________.

The value of \[ \lim_{n \to \infty} \int_0^1 e^{x^2} \sin(nx)\,dx \] is _________.

The largest positive number \(a\) such that \[ \int_0^5 f(x)\,dx + \int_0^3 f^{-1}(x)\,dx \ge a \] for every strictly increasing surjective continuous function \(f : [0, \infty) \to [0, \infty)\) is _________.

The number of elements of order two in the group \(S_4\) is equal to _________.

Let \[ \vec{F} = (y + 1)e^y \cos(x)\,\hat{i} + (y + 2)e^y \sin(x)\,\hat{j} \] be a vector field in \(\mathbb{R}^2,\) and \(C\) be a continuously differentiable path with starting point \((0,1)\) and end point \(\left(\frac{\pi}{2}, 0\right).\) Then \[ \int_C \vec{F} \cdot d\vec{r} \] equals _________.

The value of \[ \frac{\pi}{2} \lim_{n \to \infty} \cos\!\left(\frac{\pi}{4}\right) \cos\!\left(\frac{\pi}{8}\right) \cos\!\left(\frac{\pi}{16}\right) \cdots \cos\!\left(\frac{\pi}{2^{n+1}}\right) \] is _________.

Let \(V\) be the real vector space of all continuous functions \( f : [0,2] \to \mathbb{R} \) such that the restriction of \(f\) to the interval \([0,1]\) is a polynomial of degree \(\le 2,\) the restriction of \(f\) to \([1,2]\) is a polynomial of degree \(\le 3,\) and \(f(0) = 0.\) Then the dimension of \(V\) is _________.

The number of group homomorphisms from the group \(\mathbb{Z}_4\) to the group \(S_3\) is _________.