List of top Questions asked in IIT JAM MA

The volume of the solid \[ \left\{ (x, y, z) \in \mathbb{R}^3 \mid 1 \leq x \leq 2, 0 \leq y \leq 2/x, 0 \leq z \leq x \right\} \] is expressible as

Let \( f: \mathbb{R}^2 \to \mathbb{R} \) be a function. Then which of the following statements is/are TRUE?

Let \( y(x) \) be the solution of the differential equation \[ (xy + y + e^{-x}) \, dx + (x + e^{-x}) \, dy = 0 \] satisfying \( y(0) = 1 \). Then \( y(-1) \) is equal to

A particular integral of the differential equation \[ \frac{d^2y}{dx^2} - 2 \frac{dy}{dx} = e^{2x} \sin x \] is

Which of the following is TRUE?

Evaluate the limit \[ \lim_{n \to \infty} \frac{1}{\sqrt{n}} \left( \frac{1}{\sqrt{3} + \sqrt{6}} + \frac{1}{\sqrt{6} + \sqrt{9}} + \cdots + \frac{1}{\sqrt{3n} + \sqrt{3n + 3}} \right). \]

The sum of the series \[ \sum_{n=1}^{\infty} \tan^{-1} \left( \frac{2}{n^2} \right) \] is

Let \( S \) be an infinite subset of \( \mathbb{R} \) such that \( S \setminus \{a\} \) is compact for some \( a \in S \). Then which one of the following is TRUE?

Let \( P_3 \) denote the real vector space of all polynomials with real coefficients of degree at most 3. Consider the map \[ T: P_3 \to P_3 \quad \text{given by} \quad T(p(x)) = p'(x) + p(x). \] Then

The interval of convergence of the power series \[ \sum_{n=1}^{\infty} \frac{1}{(-3)^n + 2} \frac{(4x - 12)^n}{n^2 + 1} \] is

Let \( f: \mathbb{R} \to [0, \infty) \) be a continuous function. Then which one of the following is NOT TRUE? \[ \text{There exists } x \in \mathbb{R} \text{ such that } f(x) = \frac{f(0) + f(1)}{2}. \]

Let \( f: \mathbb{R} \to \mathbb{R} \) be a differentiable function such that \( f(2) = 2 \) and \[ |f(x) - f(y)| \leq 5| |x - y|^{3/2}. \] For all \( x \in \mathbb{R}, y \in \mathbb{R} \), let \( g(x) = x^3 f(x) \). Then} \[ g'(2) = \]

The flux of \[ \mathbf{F} = y \hat{i} - x \hat{j} + 2z \hat{k} \] along the outward normal, across the surface of the solid \[ \left\{ (x, y, z) \in \mathbb{R}^3 \mid 0 \leq x \leq 1, 0 \leq y \leq 1, 0 \leq z \leq \sqrt{2 - x^2 - y^2} \right\} \] is equal to \[ \iint_S \mathbf{F} \cdot \hat{n} \, dS = \]

The line integral of the vector field \[ \mathbf{F} = zx \hat{i} + xy \hat{j} + yz \hat{k} \] along the boundary of the triangle with vertices \( (1,0,0), (0,1,0) \) and \( (0,0,1) \), oriented anti-clockwise, when viewed from the point \( (2,2,2) \), is \[ \int_C \mathbf{F} \cdot d\mathbf{r} = \]

Let \[ \mathbf{F} = (3 + 2xy) \hat{i} + (x^2 - 3y^2) \hat{j} \] and let \( \mathbf{L} \) be the curve} \[ \mathbf{r}(t) = e^t \sin t \, \hat{i} + e^t \cos t \, \hat{j}, \quad 0 \leq t \leq \pi. \] Then \[ \int_{\mathbf{L}} \mathbf{F} \cdot d\mathbf{r} = \]

Let \[ M = \begin{bmatrix} 1 & 1 \\ 2 & 4 \end{bmatrix} \quad \text{and} \quad x = \begin{bmatrix} 3 \\ 4 \end{bmatrix}. \] Then \[ \lim_{n \to \infty} M^n x \]

Let \( \mathcal{M} \) be the set of all invertible \( 5 \times 5 \) matrices with entries 0 and 1. For each \( M \in \mathcal{M} \), let \( n_1(M) \) and \( n_0(M) \) denote the number of 1's and 0's in \( M \), respectively. Then \[ \min_{M \in \mathcal{M}} |n_1(M) - n_0(M)| = \]

If
\[ f(x) = \begin{cases} 1 + x & \text{if } x < 0, \\ (1 - x)(px + q) & \text{if } x \geq 0, \end{cases} \] satisfies the assumptions of Rolle’s theorem in the interval \( [-1, 1] \), then the ordered pair \( (p, q) \) is

Let \( f(x) = \frac{x + |x|(1 + x)}{x} \sin \left( \frac{1}{x} \right) \), for \( x \neq 0 \). Write \( L = \lim_{x \to 0^-} f(x) \) and \( R = \lim_{x \to 0^+} f(x) \). Then which one of the following is TRUE?

Let \( f_1(x), f_2(x), g_1(x), g_2(x) \) be differentiable functions on \( \mathbb{R} \). Let
\[ F(x) = \begin{vmatrix} f_1(x) & f_2(x) \\ g_1(x) & g_2(x) \end{vmatrix}. \] Then \( F'(x) \) is equal to

Let \( f: \mathbb{R} \to \mathbb{R} \) be a twice differentiable function. If \( g(u, v) = f(u^2 - v^2) \), then
\[ \frac{\partial^2 g}{\partial u^2} + \frac{\partial^2 g}{\partial v^2} = \]

Let \( \varphi: \mathbb{R} \to \mathbb{R} \) be a differentiable function such that \( \varphi'(1) = 0 \). Let \( \alpha \) and \( \beta \) denote the minimum and maximum values of \( \varphi(x) \) on the interval [2, 3], respectively. Then which one of the following is TRUE?

Consider the function \( f(x, y) = 5 - 4 \sin x + y^2 \) for \( 0<x<2\pi \) and \( y \in \mathbb{R} \). The set of critical points of \( f(x, y) \) consists of

Let \[ f(x, y) = \frac{xz y}{x^2 + y^2 + z^2}, \quad (x, y) \neq (0, 0). \] Then \[ \frac{\partial f}{\partial x} \text{ and } f \text{ are bounded and unbounded.} \]

Let \( 0<a_1<b_1 \), For \( n \geq 1 \), define \[ a_{n+1} = \sqrt{a_n b_n} \quad \text{and} \quad b_{n+1} = \frac{a_n + b_n}{2}. \] Then which one of the following is NOT TRUE?