1
GATE EE 2025
MCQ (More than One Correct Answer)
+2
-0

Let $a_R$ be the unit radial vector in the spherical co-ordinate system. For which of the following value(s) of $n$, the divergence of the radial vector field $f(R)=a_R \frac{1}{R^n}$ is independent of $R$ ?

A
-2
B
-1
C
1
D
2
2
GATE EE 2025
Numerical
+2
-0
Let $(x, y) \in \Re^2$. The rate of change of the real valued function, $V(x, y)=x^2+x+y^2+1$ at the origin in the direction of the point $(1,2)$ is _________ (round off to the nearest integer)
Your input ____
3
GATE EE 2025
Numerical
+2
-0
Consider ordinary differential equations given by $\dot{x}_1(t)=2 x_2(t), \dot{x}_2(t)=r(t)$ with initial conditions $x_1(0)=1$ and $x_2(0)=0$. If $r(t)=\left\{\begin{array}{ll}1, & t \geq 0 \\ 0, & t<0\end{array}\right.$, then $t=1, x_1(t)=$ _____________ (Round off to the nearest integer).
Your input ____
4
GATE EE 2025
Numerical
+2
-0
Let $C$ be a clockwise oriented closed curve in the complex plane defined by $|\lambda|=1$. Further, let $f(x)=j z$ be a complex function, where $j=\sqrt{-1}$. Then, $\oint_C f(z) d z=$ ___________ .
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