GATE EC 2025 Question Paper (15-Feb-2025) (Shift 2)

Consider a non-negative function $f(x)$ which is continuous and bounded over the interval $[2,8]$. Let $M$ and $m$ denote, respectively, the maximum and the minimum values of $f(x)$ over the interval. Among the combinations of $\alpha$ and $\beta$ given below, choose the one(s) for which the inequality

$\beta \le {\int }_2^{8}f(x)dx\le \alpha$

is guaranteed to hold.

Which of the following statements involving contour integrals (evaluated counter-clockwise) on the unit circle $C$ in the complex plane is/are TRUE?

Consider a system where $x_1(t)$, $x_2(t)$, and $x_3(t)$ are three internal state signals and $u(t)$ is the input signal. The differential equations governing the system are given by

$\frac{d}{dt}$ $\left[\begin{array}{c}{x}_1(t)\\ x_2(t)\\ x_3(t)\end{array}\right]$ $=\left[\begin{array}{ccc}2& 0& 0\\ 0& -2& 0\\ 0& 0& 0\end{array}\right]$ $\left[\begin{array}{c}{x}_1(t)\\ x_2(t)\\ x_3(t)\end{array}\right]$ $+\left[\begin{array}{c}1\\ 1\\ 1\end{array}\right]u(t)$.
Which of the following statements is/are TRUE?

The random variable $X$ takes values in $\{-1,0,1\}$ with probabilities $P(X=-1)=P(X=1)$ and $\alpha$ and $P(X=0)=1-2\alpha$, where $0<\alpha <\frac{1}{2}$. Let $g(\alpha )$ denote the entropy of $X$ (in bits), parameterized by $\alpha$. Which of the following statements is/are TRUE?

Let $f(t)$ be a periodic signal with fundamental period $T_0>0$. Consider the signal $y(t)=f(\alpha t)$, where $\alpha >1$. The Fourier series expansions of $f(t)$ and $y(t)$ are given by

$f(t)={\sum }_{k=-\infty }^{\infty }{C}_k{e}^{j\frac{2\pi }{T_0}kT}$ and $y(t)={\sum }_{k=-\infty }^{\infty }{d}_k{e}^{j\frac{2\pi }{T_0}\alpha kT}$.

Which of the following statements is/are TRUE?