GATE EE 2026 Question Paper (14-Feb-2026) (Shift 1)

Consider the circuit shown. Assume that the diode ($D$) is ideal.



Given $v_s=100\sin (2\pi 50t)\text{ V}$, $V_{dc}=50\text{ V}$, and $R=10\Omega$, the average value of the current through the diode is ___________ A

(Round off to two decimal places)

The magnitude of the contour integral
$${\oint }_C\left(\frac{(z+1{)}^2}{(z-i)(z-2)}\right)dz$$
over the contour $C:|z-2-i|=3/2$ is ______

(Round off to two decimal places)

Note: $z$ is a complex variable and $i=\sqrt{-1}$

A uniform spherical volume charge distribution of radius 2 m, centered at the origin, has a strength of $\frac{3}{\pi }\times 10^{-6}{\text{ C/m}}^3$. A point charge of strength $\pi \times 8.854\times 10^{-12}\text{ C}$ is moved from (– 3, 0, – 4) to (0, 0, 4) in Cartesian coordinate system. The relative permittivity of the medium is 1 and the coordinate values are in meters.

The work done during the process is ______ μ J

(Round off to two decimal places)

Let X and Y be two real-valued random variables with

$\text{E}(X)=1$, $\text{E}(Y)=2$, $\text{E}(X^2)=4$, $\text{E}(Y^2)=9$, and $\text{E}(XY)=0.9$,

where $\text{E}$ denotes the expectation operator.

The value of $\alpha$ that minimizes $\text{E}((X-\alpha Y{)}^2)$ is __________

(Round off to one decimal place)

The integral
$$\frac{1}{\pi }{\int }_0^{\infty }\frac{x^{2026}}{(1+x^{2026})(1+x^2)}dx$$
evaluates to _________

(Round off to two decimal places)