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

Consider those continuous functions \( f : \mathbb{R} \to \mathbb{R} \) that have the property that for every \(x \in \mathbb{R},\) \[ f(x) \in \mathbb{Q} \text{ if and only if } f(x + 1) \notin \mathbb{Q}. \] The number of such functions is _________.

Let \(y : \left(\frac{9}{10}, 3\right) \to \mathbb{R}\) be a differentiable function satisfying \[ (x - 2y)\frac{dy}{dx} + (2x + y) = 0, \quad x \in \left(\frac{9}{10}, 3\right), \] and \(y(1) = 1.\) Then \(y(2)\) equals _________.

The least possible value of \(k\), accurate up to two decimal places, for which the following problem has a solution is: \[ y''(t) + 2y'(t) + ky(t) = 0, \quad t \in \mathbb{R}, \] with \(y(0) = 0,\ y(1) = 0,\ y(1/2) = 1.\)

Let \(m>1\) and \(n>1\) be integers. Let \(A\) be an \(m \times n\) matrix such that for some \(m \times 1\) matrix \(b_1,\) the equation \(A x = b_1\) has infinitely many solutions. Let \(b_2\) denote an \(m \times 1\) matrix different from \(b_1.\) Then \(A x = b_2\) has

Consider the set \[ A = \{ a \in \mathbb{R} : x^2 = a(a+1)(a+2) \text{ has a real root} \}. \] The number of connected components of \(A\) is _________.

Let \[ B = \{(x,y,z) \in \mathbb{R}^3 : x^2 + y^2 + z^2 \le 1 \} \] and define \[ u(x,y,z) = \sin\!\left(\pi(1 - x^2 - y^2 - z^2)^2\right) \] for \((x,y,z) \in B.\) Then the value of \[ \iiint_B \left( \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} + \frac{\partial^2 u}{\partial z^2} \right) \, dx\,dy\,dz \] is _________.

Consider the subset \( S = \{ (x, y) : x^2 + y^2>0 \} \) of \(\mathbb{R}^2.\) Let \[ P(x,y) = \frac{y}{x^2 + y^2}, \quad Q(x,y) = \frac{-x}{x^2 + y^2}. \] For \((x, y) \in S.\) If \(C\) denotes the unit circle traversed in the counter-clockwise direction, then the value of \[ \frac{1}{\pi} \int_C (P\,dx + Q\,dy) \] is _________.

The number of cycles of length 4 in \(S_6\) is _________.

The value of \[ \lim_{n \to \infty} \left(3^n + 5^n + 7^n \right)^{\tfrac{1}{n}} \] is _________.

Let \(V\) be a finite-dimensional vector space and \(T: V \to V\) be a linear transformation. Let \(\mathcal{R}(T)\) denote the range of \(T\) and \(\mathcal{N}(T)\) denote the null space of \(T\). If \(\operatorname{rank}(T) = \operatorname{rank}(T^2)\), then which of the following is/are necessarily true?

Let \(G\) be a finite group of order \(28.\) Assume that \(G\) contains a subgroup of order \(7.\) Which of the following statements is/are true?

Which of the following subsets of \(\mathbb{R}\) are connected?

Let \( D = \mathbb{R}^2 \setminus \{(0,0)\} \). Consider the two functions \( u,v : D \to \mathbb{R} \) defined by \[ u(x,y) = x^2 - y^2 \quad \text{and} \quad v(x,y) = xy. \] Consider the gradients \(\nabla u\) and \(\nabla v\) of the functions \(u\) and \(v\), respectively. Then

Let \( f : (a,b) \to \mathbb{R} \) be a differentiable function on \((a,b)\). Which of the following statements is/are true?

Consider the two functions \( f(x,y) = x + y \) and \( g(x,y) = xy - 16 \) defined on \( \mathbb{R}^2 \). Then

Let \(G\) be a finite abelian group of odd order. Consider the following two statements: I. The map \( f : G \to G \) defined by \( f(g) = g^2 \) is a group isomorphism.
II. The product \( \displaystyle \prod_{g \in G} g = e. \)
Which one of the following statements is true?

Let \( g \) be an element of \( S_7 \) such that \(g\) commutes with the element \((2,6,4,3)\). The number of such \(g\) is

Let \( f : \mathbb{R} \to \mathbb{R} \) be a function with the property that for every \( y \in \mathbb{R} \), the value of the expression \[ \sup_{x \in \mathbb{R}} [xy - f(x)] \] is finite. Define \( g(y) = \sup_{x \in \mathbb{R}} [xy - f(x)] \) for \(y \in \mathbb{R}\). Then

Let \( n \ge 2 \) be an integer. Let \( A : \mathbb{C}^n \to \mathbb{C}^n \) be the linear transformation defined by \[ A(z_1, z_2, \ldots, z_n) = (z_n, z_1, z_2, \ldots, z_{n-1}). \] Which one of the following statements is true for every \( n \ge 2 \)?

Consider the two series \[ \text{I. } \sum_{n=1}^{\infty} \frac{1}{n^{1 + (1/n)}} \quad \text{and} \quad \text{II. } \sum_{n=1}^{\infty} \frac{1}{n(\ln n)^{1/n}}. \] Which one of the following holds?

Consider the equation \[ x^{2021} + x^{2020} + \cdots + x + 1 = 0. \] Then

Which one of the following statements is true?

Let \( f : [0,1] \to [0,1] \) be a non-constant continuous function such that \( f \circ f = f. \) Define \[ E_f = \{x \in [0,1] : f(x) = x\}. \] Then

Let \( f : [0,1] \to [0, \infty) \) be a continuous function such that \[ (f(t))^2<1 + 2 \int_0^t f(s)\,ds, \quad \text{for all } t \in [0,1]. \] Then

Let \( y \) be a twice differentiable function on \( \mathbb{R} \) satisfying \[ y''(x) = 2 + e^{-|x|}, \quad x \in \mathbb{R}, \quad y(0) = -1, \quad y'(0) = 0. \] Then