GATE EC 2026 Question Paper (15-Feb-2026) (Shift 1)

Consider the two series, $S_A$ and $S_B$, where

$$S_A=\sum _{n=1}^{\infty }\frac{n^2}{2^n}$$

$$S_B=1+\frac{1}{2}+\frac{1}{8}+\frac{1}{16}+\frac{1}{64}+\frac{1}{128}+\frac{1}{512}+\cdots$$
Which of the following statements is correct for the two given series?

The continuous time signal $x(t)$ is real, periodic with period $T$ and satisfies the Dirichlet conditions.
The Fourier series representation of $x(t)={\sum }_{-\infty }^{\infty }{a}_n{e}^{j\left(\frac{2\pi nt}{T}\right)}$ and $x(t)$ satisfies the following: $x\left(t-\frac{T}{2}\right)=-x(t)$.
For any integer $m$, which of the following options is correct?

Let $X$, $N$, $Y$ and $Z$ be random variables. The variables $X$ and $N$ are independent of each other. $X$ is uniformly distributed between -1 and 1; $N$ follows Normal distribution with zero mean and unity variance.
$Y$ and $Z$ are defined as, $Y=X+N$ and $Z=X^2+N$.
Which of the following pairs represents the values of correlation between $X$ and $Y$ and that between $X$ and $Z$?

In the given circuit, $L=1\mu \text{H}$ and $C=1\mu \text{F}$. The phasor diagram for $I_C$ and $I_L$ is also shown. Assume that the phase $({\theta }_1+{\theta }_2)$ is $90^{\circ }$ at a frequency of 159.15 kHz.
Among the following options, what is the nearest integer value of $R_C\times R_L$?

For the control system shown in the Figure, the transfer function of a plant,

$$G(s)=\frac{1}{(s+1)(s+2)}$$
is connected in cascade with a compensator

$$C(s)=K(s+\alpha ),$$
where $K$ and $\alpha$ are positive real valued constants.
Which of the following pairs $(K,\alpha )$ represent the correct values for the closed loop system to have poles at $(-3\pm j\sqrt{5})$?