List of top Questions asked in IIT JAM MA

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 _________.

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 \( D = \mathbb{R}^2 \setminus \{(0,0)\} \). 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

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

Let \( f : (a,b) \to \mathbb{R} \) be a differentiable function on \((a,b)\). Which of the following statements is/are 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 \( g \) be an element of \( S_7 \) such that \(g\) commutes with the element \((2,6,4,3)\). The number of such \(g\) is

Let \(G\) 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 \( n \ge 2 \) 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?

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

Consider the equation \[ x^{2021} + x^{2020} + \cdots + x + 1 = 0. \] 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 \( D \subseteq \mathbb{R}^2 \) be defined by \[ D = \mathbb{R}^2 \setminus \{(x,0) : x \in \mathbb{R}\}. \] Consider the function \( f : D \to \mathbb{R} \) defined by \[ f(x,y) = x \sin\!\left(\frac{1}{y}\right). \] Then

Which one of the following statements is true?

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

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

Consider the following statements. I. The group \((\mathbb{Q}, +)\) has no proper subgroup of finite index.
II. The group \((\mathbb{C} \setminus \{0\}, \cdot)\) has no proper subgroup of finite index.
Which one of the following statements is true?

Let \( f : \mathbb{N} \to \mathbb{N} \) be a bijective map such that \[ \sum_{n=1}^{\infty} \frac{f(n)}{n^2}<+\infty. \] The number of such bijective maps is

Define \[ S = \lim_{n \to \infty} \left(1 - \frac{1}{2^2}\right)\left(1 - \frac{1}{3^2}\right)\cdots\left(1 - \frac{1}{n^2}\right). \] Then

Let \( f : \mathbb{R} \to \mathbb{R} \) be an infinitely differentiable function such that for all \(a, b \in \mathbb{R}\) with \(a<b\), \[ \frac{f(b) - f(a)}{b - a} = f''\left(\frac{a+b}{2}\right). \] Then

Consider the family of curves \(x^2 - y^2 = ky\) with parameter \(k \in \mathbb{R}\). The equation of the orthogonal trajectory to this family passing through \((1,1)\) is given by

Which one of the following statements is true?

For every \( n \in \mathbb{N} \), let \( f_n : \mathbb{R} \to \mathbb{R} \) be a function. From the given choices, pick the statement that is the negation of \[ \text{“For every } x \in \mathbb{R} \text{ and for every real number } \varepsilon>0, \text{ there exists an integer } N>0 \text{ such that } \sum_{i=1}^p |f_{N+i}(x)|<\varepsilon \text{ for every integer } p>0.” \]