List of top Questions asked in IIT JAM MA- 2019

The set \[ \left\{ \frac{1}{m} + \frac{1}{n} : m, n \in \mathbb{N} \right\} \cup \{0\}, \text{ as a subset of } \mathbb{R}, \text{ is} \]

Let \( g: \mathbb{R} \to \mathbb{R} \) be a twice differentiable function. Define \( f: \mathbb{R}^3 \to \mathbb{R} \) by \[ f(x, y, z) = g(x^2 + y^2 - 2z^2). \] Then \[ \frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2} + \frac{\partial^2 f}{\partial z^2} \] is equal to

Let \( \{a_n\}_{n=0}^{\infty} \) and \( \{b_n\}_{n=0}^{\infty} \) be sequences of positive real numbers such that \( n a_n<b_n<n^2 a_n \), for all \( n \geq 2 \). If the radius of convergence of the power series \[ \sum_{n=0}^{\infty} a_n x^n \] is 4, then the power series \[ \sum_{n=0}^{\infty} b_n x^n \] is

Let \( S \) be the set of all limit points of the set \[ \left\{ \frac{n}{\sqrt{2}} + \frac{\sqrt{2}}{n} : n \in \mathbb{N} \right\}. \] Let \( \mathbb{Q}_+ \) be the set of all positive rational numbers. Then

If \( x^h y^k \) is an integrating factor of the differential equation \[ y(1 + xy) \, dx + x(1 - xy) \, dy = 0, \] then the ordered pair \( (h, k) \) is equal to

The equation of the tangent plane to the surface \[ x^2 z + \sqrt{8 - x^2 - y^4} = 6 \text{ at the point } (2, 0, 1) \] is

Let \( a_1 = b_1 = 0 \), and for each \( n \geq 2 \), let \( a_n \) and \( b_n \) be real numbers given by \[ a_n = \sum_{m=2}^{n} (-1)^{m} m (\log(m))^m \] \[ b_n = \sum_{m=2}^{n} \frac{1}{(\log(m))^m}. \] Then which one of the following is TRUE about the sequences \( \{a_n\} \) and \( \{b_n\} \)?

Let \( M_{n \times p}(\mathbb{R}) \) be the subspace of \( M_{n \times p}(\mathbb{R}) \) defined by \[ V = \{ X \in M_{n \times p}(\mathbb{R}) : TX = 0 \}. \] Then the dimension of \( V \) is