List of top Questions asked in IIT JAM MA

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?

The area of the surface \[ z = \frac{xy}{3} \] intercepted by the cylinder \[ x^2 + y^2 \leq 16 \] lies in the interval \[ \left( 20\pi, 22\pi \right) \]

For \( a>0, b>0 \), let \[ \mathbf{F} = \frac{xj - yk}{b x^2 + a y^2}. \] Let \[ C = \{(x, y) \in \mathbb{R}^2 | x^2 + y^2 = a^2 + b^2\}. \] Then the line integral \[ \oint_C \mathbf{F} \cdot d\mathbf{r} = \]

If \( \lim_{t \to \infty} \int_0^t e^{-x^2} dx = \frac{\sqrt{\pi}}{2} \), then \[ \lim_{t \to \infty} \int_0^t x^2 e^{-x^2} dx = \]

The flux of the vector field \[ \mathbf{F} = \left( \frac{2\pi x + 2x^2 y^2}{\pi} \right) \hat{i} + \left( \frac{2\pi x y - 4y}{\pi} \right) \hat{j} \] along the outward normal, across the ellipse \( x^2 + 16y^2 = 4 \) is equal to

The number of generators of the additive group \( \mathbb{Z}_{36} \) is equal to

Find the limit: \[ \lim_{n \to \infty} \sum_{k=1}^{n} \sin \left( \frac{\pi}{2} + \frac{5\pi}{2} \cdot \frac{k}{n} \right) = \]

Evaluate the integral: \[ \int_0^1 \int_x^1 \sin(y^2) \, dy \, dx \]