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

Consider a system represented by the block diagram shown below. Which of the following signal flow graphs represent(s) this system? Choose the correct option(s).

All the diodes in the circuit given below are ideal.
Which of the following plots is/are correct when $V_i$ (in Volts) is swept from $-M$ to $M$?

Two fair dice (with faces labeled 1, 2, 3, 4, 5, and 6) are rolled. Let the random variable $X$ denote the sum of the outcomes obtained.
The expectation of $X$ is __________ (rounded off to two decimal places).

Consider the vectors $a=\left[\begin{array}{c}1\\ 1\end{array}\right]$, $b=\left[\begin{array}{c}0\\ 3\sqrt{2}\end{array}\right]$
For real-valued scalar variable $x$, the value of ${\min }_x||ax-b|{|}_2$ is __________ (rounded off to two decimal places).
$||\cdot |{|}_2$ denotes the Euclidean norm, i.e., for $y=\left[\begin{array}{c}{y}_1\\ y_2\end{array}\right]$, $||y|{|}_2=\sqrt{y_1^2+y_2^2}$.

$X$ and $Y$ are Bernoulli random variables taking values in $\{0,1\}$. The joint probability mass function of the random variables is given by:
$P(X=0,Y=0)=0.06$
$P(X=0,Y=1)=0.14$
$P(X=1,Y=0)=0.24$
$P(X=1,Y=1)=0.56$
The mutual information $I(X;Y)$ is __________ (rounded off to two decimal places).