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

Let $A=\left[\begin{array}{ccc}1 & 1 & 1 \\ -1 & -1 & -1 \\ 0 & 1 & -1\end{array}\right]$ and $b=\left[\begin{array}{c}1 / 3 \\ -1 / 3 \\ 0\end{array}\right]$, then the system of linear equations $A x=b$ has

A
a unique solution
B
infinitely many solutions
C
a finite number of solutions
D
no solution
2
GATE EE 2025
MCQ (Single Correct Answer)
+1
-0.33

Let $P=\left[\begin{array}{ccc}2 & 1 & 0 \\ -1 & 0 & 0 \\ 0 & 0 & 1\end{array}\right]$ and let $I$ be the identity matrix. Then $P^2$ is equal to

A
$2 P-I$
B
$P$
C
$I$
D
$P+I$
3
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
$\frac{1}{3}$
4
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
$P_Y(y)=\left\{\begin{array}{cc}2 y, & y \in(0,1] \\ 0, & \text { otherwise }\end{array}\right.$
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