GATE Overflow for GATE XE - Recent questions and answers in Functions of Two Variables

GATE XE 2026 | Question: 6

Tue, 24 Feb 2026 15:50:16 +0000

Let $L$ be the lamina of the form $x^{2}+4 y^{2} \leq 64,0 \leq y \leq 4$, with density $\rho(x, y)=\mid x \mid y$. Then, the mass of $L$ (in integer) is $\_\_\_\_$

GATE XE 2025 | Question: 19

Sun, 04 May 2025 19:08:31 +0000

Consider the function

$$ f(x, y)=x^{2} y+2 x y^{2}-2 x^{2} y^{2}.$$

Then which one of the following statements is correct?

$\left(\frac{3}{2}, 0\right)$ is a point of local maxima of $f$.
$\left(0, \frac{3}{4}\right)$ is a point of local minima of $f$.
$\left(\frac{3}{2}, \frac{3}{4}\right)$ is a point of local maxima of $f$.
$\left(\frac{3}{2}, \frac{3}{4}\right)$ is a saddle point of $f$.

GATE XE 2025 | Question: 36

Sun, 04 May 2025 19:07:44 +0000

Consider, $i$ and $j$ are unit vectors along $x$ and $y$ directions of a Cartesian $(x, y)$ coordinate system, respectively and $t$ is time. Temperature ($T$) and fluid velocity $(\vec{V})$ are given for a flow field as:

$$\begin{array}{l} T=x^{2}+y t+35 \\ \vec{V}=(4 x y) i+\left(x t-2 y^{2}\right) j \end{array} $$

The total rate of change of temperature in the flow field (in integer) for time $t=2$ at a point $(2,3)$ is $\_\_\_\_\_\_$

GATE XE 2024 | Question: 10

Sun, 21 Jul 2024 16:42:37 +0000

If
\[
\int_{0}^{\alpha} \int_{\sqrt{x / \alpha}}^{1} e^{y^{3}} d y d x=e-1, \quad \alpha>0,
\]
then the value $\text{(in integer)}$ of $\alpha$ is __________.

GATE XE 2023 | Question: 2

Wed, 14 Feb 2024 18:10:16 +0000

Let $f: \mathbb{R}^{2} \rightarrow \mathbb{R}$ be a function defined by

$f(x, y)=\left\{\begin{array}{cc}\frac{x y}{|x|+y}, & y \neq-|x| \\ 0, & \text { otherwise. }\end{array}\right.$

Then which one of the following statement is TRUE?

$f$ is NOT continuous at $(0,0)$.
$\frac{\partial f}{\partial x}(0,0)=0$, and $\frac{\partial f}{\partial y}(0,0)=1$
$\frac{\partial f}{\partial x}(0,0)=1$, and $\frac{\partial f}{\partial y}(0,0)=0$
$\frac{\partial f}{\partial x}(0,0)=1$, and $\frac{\partial f}{\partial y}(0,0)=1$

GATE XE 2023 | Question: 9

Wed, 14 Feb 2024 18:10:10 +0000

The surface area of the portion of the paraboloid

$z=x^{2}+y^{2}$

that lies between the planes $z=0$ and $z=\frac{1}{4}$ is

$\frac{\pi}{6}(2 \sqrt{2}-1)$
$\frac{\pi}{2}(2 \sqrt{2}-1)$
$\pi(2 \sqrt{2}-1)$
$\frac{\pi}{3}(2 \sqrt{2}-1)$

GATE XE 2022 | Question: 10

Fri, 17 Feb 2023 06:52:31 +0000

Let $f: \mathbb{R}^{2} \rightarrow \mathbb{R}$ be given by $f(x, y)=4 x y-2 x^{2}-y^{4}+1$. The number of critical points where $f$ has local maximum is equal to