GATE EE 2025 Question Paper (02-Feb-2025) (Shift-2)

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 $R$?

Consider two coupled circuits having self inductances $L_1$ and $L_2$ that carry non-zero currents $I_1$ and $I_2$ respectively. The mutual inductance between the circuit is $M$ with unity coupling coefficient. The stored magnetic energy of the coupled circuit is minimum at which of the following value(s) of $I_1/I_2$?

Let $(x,y)\in {\mathbb{R}}^2$. 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)

Consider ordinary differential equations given by ${\dot{x}}_1(t)=2x_2(t)$, ${\dot{x}}_2(t)=r(t)$ with initial conditions $x_1(0)=1$ and $x_2(0)=0$. If
$r(t)=\{\begin{array}{ll}1,& t\ge 0\\ 0,& t<0\end{array}$, then $t=1$, $x_1(t)=$ __________. (Round off to the nearest integer).

Let $C$ be a clockwise oriented closed curve in the complex plane defined by $|z|=1$. Further, let $f(z)=jz$ be a complex function, where $j=\sqrt{-1}$. Then, ${\oint }_{C}f(z)dz=$ _________ .