# Engineering Mathematics

 Question 1
The value of the integral $\iint_{R} x y d x d y$ over the region $R$, given in the figure, is ___
(rounded off to the nearest integer). A 0 B 1 C 2 D 3
GATE EC 2023      Calculus
Question 1 Explanation: \begin{aligned} I & =\iint_{R} x y d x d y \\ & =\int_{y=0}^{1} \int_{x=-y}^{y} x y d x d y+\int_{y=1}^{2} \int_{x=y-2}^{2-y} x y d x d y \\ & =\int_{0}^{1} y\left(\frac{x^{2}}{2}\right)_{-y}^{y} d y+\int_{1}^{2} y\left(\frac{x^{2}}{2}\right)_{y-2}^{2-y} d y \\ & =0+0=0 \end{aligned}
 Question 2
The state equation of a second order system is

$\dot{x}(t)=A x(t), x(0)$ is the initial condition.

Suppose $\lambda_{1}$ and $\lambda_{2}$ are two distinct eigenvalues of $A$ and $v_{1}$ and $v_{2}$ are the corresponding eigenvectors. For constants $\alpha_{1}$ and $\alpha_{2}$, the solution, $x(t)$, of the state equation is
 A $\sum_{i=1}^{2} \alpha_{i} e^{\lambda_{i} t} v_{i}$ B $\sum_{i=1}^{2} \alpha_{i} e^{2 \lambda_{i} t} v_{i}$ C $\sum_{i=1}^{2} \alpha_{i} e^{3 \lambda_{i} t} v_{i}$ D $\sum_{i=1}^{2} \alpha_{i} e^{4 \lambda_{i} t} v_{i}$
GATE EC 2023      Linear Algebra

 Question 3
Let $x$ be an $n \times 1$ real column vector with length $l=\sqrt{x^{T} x}$. The trace of the matrix $P=x x^{T}$ is
 A $l^{2}$ B $\frac{l^{2}}{4}$ C $l$ D $\frac{l^{2}}{2}$
GATE EC 2023      Linear Algebra
Question 3 Explanation:
Given,
$l=\sqrt{x^{T} x}, P=\left(x x^{T}\right)_{n \times n}$

Let
\begin{aligned} (x)_{n \times 1} & =\left[\begin{array}{l} x_{1} \\ x_{2} \\ x_{3} \\ \vdots \\ x_{n} \end{array}\right] \\ l & =\sqrt{x^{T} x}=\sqrt{x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+\ldots x_{n}^{2}} \\ P & =x x^{T} \\ &=\left[\begin{array}{l} x_{1} \\ x_{2} \\ x_{3} \\ \cdot \\ x_{n} \end{array}\right]\left[x_{1} x_{2} x_{3} \ldots x_{n}\right]\\ P&=\left[ \begin{array}{lllll} x_{1}^{2} & & & & \\ & x_{1}^{2} & & & \\ & & - & & \\ & & & - & \\ & & & & x_{n}^{2} \end{array} \right] \end{aligned}
Trace of $P=x_{1}^{2}+x_{2}^{2}+ . . .+ x_{n}^{2}=l^2$
 Question 4
The value of the line integral $\int_{P}^{Q}\left(z^{2} d x+3 y^{2} d y+2 x z d z\right)$ along the straight line joining the points $P(1,1,2)$ and $Q(2,3,1)$ is
 A 20 B 24 C 29 D -5
GATE EC 2023      Calculus
Question 4 Explanation:
$\int_{P}^{Q} z^{2} d x+3 y^{2} d y+2 x z d z$ along the line joining the points $P(1,1,2)$ and $Q(2,3,1)$ is
\begin{aligned} & =\int_{P(1,2)}^{P(2,1)} z^{2} d x+2 x y d z+\int_{y=1}^{3} 3 y^{2} d y \\ & =\left(x z^{2}\right)_{(1,2)}^{(2,1)}+\left(y^{3}\right)_{1}^{3} \\ & =\left(2 \times 1^{2}-1 \times 2^{2}\right)+\left(3^{3}-1^{3}\right) \\ & =-2+26=24 \end{aligned}
 Question 5
A random variable $X$, distributed normally as $N(0,1)$, undergoes the transformation $Y=h(X)$, given in the figure. The form of the probability density function of $Y$ is (In the options given below, $a, b, c$ are non-zero constants and $g(y)$ is piece-wise continuous function) A $a \delta(y-1)+b \delta(y+1)+g(y)$ B $a \delta(y+1)+b \delta(y)+c \delta(y-1)+g(y)$ C $a \delta(y+2)+b \delta(y)+c \delta(y-2)+g(y)$ D $a \delta(y+1)+b \delta(y-2)+g(y)$
GATE EC 2023      Probability and Statistics
Question 5 Explanation:
$X=N(0,1)$ $f_X(x)=\frac{1}{\sqrt{2\pi}}e^{-x^2/2}$ \begin{aligned} Y= & -1 ; x \leq-2 \\ & 0 ;-1 \leq x \leq 1 \\ & 1 ; x \geq 2 \\ & x+1 ;-2 \leq x \leq-1 \\ & x-1 ; 1 \leq x \leq 2 \end{aligned}
$Y$ is taking discrete set of values and a continuous range of values, so it is mixed random variable.
From the given options, density function of '$Y$' will be.
$f_{Y}(y)=a \delta(y+1)+b \delta(y)+c \delta(y-1)+g(y)$

There are 5 questions to complete.