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

Let $X$ and $Y$ be continuous random variables with probability density functions $P_X(x)$ and $P_Y(y)$, respectively. Further, let $Y=X^2$ and $P_X(x)=\{\begin{array}{ll}1,& x\in (0,1]\\ 0,& \text{otherwise}\end{array}$
Which one of the following options is correct?

A Boolean function is given as
$f=(\overline{u}+\overline{v}+\overline{w}+\overline{x})\cdot (\overline{u}+\overline{v}+\overline{w}+x)\cdot (\overline{u}+v+\overline{w}+\overline{x})\cdot (\overline{u}+v+\overline{w}+x)$
The simplified form of this function is represented by

In the circuit, $I_{DC}$ is an ideal current source. The transistors $M_1$ and $M_2$ are assumed to be biased in saturation, wherein $V_{in}$ is the input signal and $V_{DC}$ is fixed DC voltage. Both transistors have a small signal resistance of $r_{ds}$ and trans-conductance of $g_m$. The small signal output impedance of this circuit is

In the circuit, shown below, if the values of $R$ and $C$ are very large, the form of the output voltage for a very high frequency square wave input, is best represented by

Let continuous-time signals $x_1(t)$ and $x_2(t)$ be
$x_1(t)=\{\begin{array}{ll}1,& t\in [0,1]\\ 2-t,& t\in [1,2]\\ 0,& \text{otherwise}\end{array}$ and $x_2(t)=\{\begin{array}{ll}t,& t\in [0,1]\\ 2-t,& t\in [1,2]\\ 0,& \text{otherwise}\end{array}$
Consider the convolution $y(t)=x_1(t)\ast x_2(t)$. Then ${\int }_{-\infty }^{\infty }y(t)dt$ is