GATE Overflow for GATE XE - Questions without answers in Vector Calculus

GATE XE 2025 | Question: 15

Sun, 04 May 2025 19:08:37 +0000

Let $C$ be the positively oriented boundary of the domain bounded by the curves $y=2 x^{2}$ and $y^{2}=4 x$. Then the value of the line integral

$$\oint_{C}\left(2 y^{2}+2 x y+4 y\right) d x+\left(x^{2}+4 x y+8 x\right) d y $$ is equal to

$\frac{8}{3}$
$\frac{2}{3}$
$\frac{4}{3}$
$\frac{1}{3}$

GATE XE 2024 | Question: 11

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

Consider the vector field $\vec{F}=\left(2 x+y^{2}\right) \hat{\imath}+(2 x y+3 y) \hat{\jmath}$ and $\alpha_{m}=\int_{C_{m}} \vec{F} \cdot d \vec{r}$, $m=1,2$, where $C_{1}$ is an arc of the unit circle connecting the points $(1,0)$ and $(0,1)$ and $C_{2}$ is the straight line connecting the points $(1,0)$ and $(0,1)$. Then, the value $\text{(in integer)}$ of $2\left(\alpha_{1}^{2}+3 \alpha_{2}^{2}\right)$ is ___________.

GATE XE 2023 | Question: 7

Wed, 14 Feb 2024 18:10:12 +0000

The value of $m$ for which the vector field

$\vec{F}(x, y)=\left(4 x^{m} y^{2}-2 x y^{m}\right) \hat{\imath}+\left(2 x^{4} y-3 x^{2} y^{2}\right) \hat{\jmath}$

is a conservative vector field, is __________$\text{(in integer)}$.