1
GATE EE 2025
MCQ (Single Correct Answer)
+1
-0.33

Consider discrete random variable $X$ and $Y$ with probabilities as follows:

$$ \begin{aligned} & P(X=0 \text { and } Y=0)=\frac{1}{4} \\ & P(X=1 \text { and } Y=1)=\frac{1}{8} \\ & P(X=0 \text { and } Y=1)=\frac{1}{2} \\ & P(X=1 \text { and } Y=1)=\frac{1}{8} \end{aligned} $$

Given $X=1$, the expected value of $Y$ is

A
$\frac{1}{4}$
B
$\frac{1}{2}$
C
$\frac{1}{8}$
D
$\frac{1}{3}$
2
GATE EE 2025
MCQ (Single Correct Answer)
+2
-0.67

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 $P_X(x)=\left\{\begin{array}{cc}1, & x \in(0,1] \\ 0, & \text { otherwise }\end{array}\right.$.

Which one of the following options is correct?

A
$P_Y(y)=\left\{\begin{array}{cc}\frac{1}{2 \sqrt{y}}, & y \in(0,1] \\ 0, & \text { otherwise }\end{array}\right.$
B
$P_Y(y)=\left\{\begin{array}{lc}1, & y \in(0,1] \\ 0, & \text { otherwise }\end{array}\right.$
C
$P_Y(y)=\left\{\begin{array}{cc}1.5 \sqrt{y}, & y \in(0,1] \\ 0, & \text { otherwise }\end{array}\right.$
D
$P_Y(y)=\left\{\begin{array}{cc}2 y, & y \in(0,1] \\ 0, & \text { otherwise }\end{array}\right.$
3
GATE EE 2025
MCQ (More than One Correct Answer)
+2
-0

Let $a_R$ be the unit radial vector in the spherical co-ordinate system. For which of the following value(s) of $n$, the divergence of the radial vector field $f(R)=a_R \frac{1}{R^n}$ is independent of $R$ ?

A
-2
B
-1
C
1
D
2
4
GATE EE 2025
Numerical
+2
-0
Let $(x, y) \in \Re^2$. The rate of change of the real valued function, $V(x, y)=x^2+x+y^2+1$ at the origin in the direction of the point $(1,2)$ is _________ (round off to the nearest integer)
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