# Engineering Mathematics

 Question 1
Let the probability density function of a random variable $x$ be given as
$f(x)=ae^{-2|x|}$
The value of $'a'$ is _________
 A 0.5 B 1 C 1.5 D 2
GATE EE 2022      Probability and Statistics
Question 1 Explanation:
$f(x)=\left\{\begin{matrix} ae^{2x} &x \lt 0 \\ ae^{-2x} &x \gt 0 \end{matrix}\right.$
Therefore,
\begin{aligned} \int_{-\infty }^{\infty }f(x)dx&=1\\ \int_{-\infty }^{0}ae^{2x}dx+\int_{0 }^{\infty }ae^{-2x}dx&=1\\ a\left [ \left [ \frac{e^{2x}}{2} \right ]_{-\infty }^0 +[ \left [ \frac{e^{-2x}}{-2} \right ]^{\infty }_0\right ]&=1\\ a\left [ \frac{1}{2}+\frac{1}{2} \right ]&=1\\ a&=1 \end{aligned}
 Question 2
Let $\vec{E}(x,y,z)=2x^2\hat{i}+5y\hat{j}+3z\hat{k}$. The value of $\int \int \int _V(\vec{\triangledown }\cdot \vec{E})dV$, where $V$ is the volume enclosed by the unit cube defined by $0\leq x\leq 1,0\leq y\leq 1 \text{ and }0\leq z\leq 1$, is
 A 3 B 8 C 10 D 5
GATE EE 2022      Calculus
Question 2 Explanation:
Divergence of $\vec{V}, \triangledown \cdot \vec{V}=4x+5+3=4x+8$
Now,
$\int \int \int( \triangledown \cdot \vec{V})dxdydz=\int_{0}^{1}\int_{0}^{1}\int_{0}^{1}(4x+8)dxdydz=\int_{0}^{1}(4x+8)dx=[2x^2+8x]_0^1=10$
 Question 3
Let $R$ be a region in the first quadrant of the $xy$ plane enclosed by a closed curve $C$ considered in counter-clockwise direction. Which of the following expressions does not represent the area of the region $R$?

 A $\int \int _R dxdy$ B $\oint _c xdy$ C $\oint _c ydx$ D $\frac{1}{2}\oint _c( xdy-ydx)$
GATE EE 2022      Complex Variables
Question 3 Explanation:
Using green theorem?s
$\oint _cF_1dx+F_2dy=\int \int _R\left ( \frac{\partial F_2}{\partial x} -\frac{\partial F_1}{\partial y}\right )dxdy$
Check all the options:
$\oint xdy=\int \int _R\left ( \frac{\partial x}{\partial x} -0\right )dxdy=\int \int _Rdxdy$
$\frac{1}{2}\oint xdy-ydx=\frac{1}{2}\int \int _R(1+1)dxdy=\int \int _Rdxdy$
$\oint ydx=\int \int _R(-1)dxdy=-\int \int _Rdxdy$
Hence, $\oint ydx$ is not represent the area of the region.
 Question 4
Let, $f(x,y,z)=4x^2+7xy+3xz^2$. The direction in which the function $f(x,y,z)$ increases most rapidly at point $P = (1,0,2)$ is
 A $20\hat{i}+7\hat{j}$ B $20\hat{i}+7\hat{j}+12\hat{k}$ C $20\hat{i}+12\hat{j}$ D $20\hat{i}$
GATE EE 2022      Numerical Methods
Question 4 Explanation:
Given: $f(x,y,z)=4x^2+7xy+3xz^2$
The directional derivative at point P is given by
$=\triangledown f|_{point \; P}$
$\therefore \; \triangledown f=(8x+7y+3z^2)\hat{i}+(0+7x+0)\hat{j}+(0+0+6xz)\hat{k}$
at point (1, 0, 2)
$\triangledown f|_{(1,0,2)}=20\hat{i}+7\hat{j}+12\hat{k}$
 Question 5
Consider a matrix $A=\begin{bmatrix} 1 & 0 & 0\\ 0 & 4 & -2\\ 0&1 &1 \end{bmatrix}$
The matrix $A$ satisfies the equation $6A^{-1}=A^2+cA+dI$ where $c$ and $d$ are scalars and $I$ is the identity matrix.
Then $(c+d)$ is equal to
 A 5 B 17 C -6 D 11
GATE EE 2022      Linear Algebra
Question 5 Explanation:
Characteristic equation:
\begin{aligned} |A-\lambda I|&=0 \\ \begin{vmatrix} 1-\lambda & 0 &0 \\ 0 & 4-\lambda & 2\\ 0 & -1 & 1-\lambda \end{vmatrix}&=0 \\ (1-\lambda )[(4-\lambda )(1-\lambda )+2] &=0\\ \lambda ^3-6\lambda ^2+11\lambda -6&=0 \end{aligned}
By cayley hamilton theorem
\begin{aligned} A^3-6A^2+11A-6 &=0 \\ A^2-6A+11I&=6A^{-1} \end{aligned}
On comparison : c= -6 and d = 11
Therefore, c + d = -6 + 11 = 5
 Question 6
Let $f(x)=\int_{0}^{x}e^t(t-1)(t-2)dt$. Then $f(x)$ decreases in the interval
 A $x \in (1,2)$ B $x \in (2,3)$ C $x \in (0,1)$ D $x \in (0.5,1)$
GATE EE 2022      Calculus
Question 6 Explanation:
The function is decreasing, if $f'(x) \lt 0$
$f'(x)=\frac{d}{dx} \int_{0}^{x}e^t(t-1)(t-2)dt$
$\Rightarrow \; e^x(x-1)(x-2) \lt 0$
It is possible in between 1 & 2. Hence $x \in (1,2)$
 Question 7
$e^A$ denotes the exponential of a square matrix A. Suppose $\lambda$ is an eigenvalue and $v$ is the corresponding eigen-vector of matrix A.

Consider the following two statements:
Statement 1: $e^\lambda$ is an eigenvalue of $e^A$.
Statement 2: $v$is an eigen-vector of $e^A$.

Which one of the following options is correct?
 A Statement 1 is true and statement 2 is false. B Statement 1 is false and statement 2 is true C Both the statements are correct. D Both the statements are false.
GATE EE 2022      Linear Algebra
Question 7 Explanation:
Eigen value will change but eigen vector not change.
 Question 8
Consider a 3 x 3 matrix A whose (i,j)-th element, $a_{i,j}=(i-j)^3$. Then the matrix A will be
 A symmetric. B skew-symmetric. C unitary D null.
GATE EE 2022      Linear Algebra
Question 8 Explanation:
$for \; i=j\Rightarrow a_{ij}=(i-i)^3=0\forall i$
$for \; i\neq j\Rightarrow a_{ij}=(i-j)^3=(-(j-i))^3=-(j-i)^3=-a_{ji}$
$\therefore \; A_{3 \times 3 }$ is skew symmetric matrix.
 Question 9
Let A be a $10\times10$ matrix such that $A^{5}$ is a null matrix, and let I be the $10\times10$ identity matrix. The determinant of $\text{A+I}$ is ___________________.
 A 1 B 2 C 4 D 8
GATE EE 2021      Linear Algebra
Question 9 Explanation:
\begin{aligned} \text{Given}:\qquad A^{5}&=0\\ A x &=\lambda x \\ \Rightarrow \qquad \qquad A^{5} x &=\lambda^{5} x \quad(\because x \neq 0) \\ \Rightarrow \qquad \qquad \lambda^{5} &=0 \\ \Rightarrow \qquad \qquad \lambda &=0 \end{aligned}
Eigen values of $A+I$ given $\lambda+1$
$\because$ Eigen values of $I_{A}=1$
Hence $|A+I|=$ Product of eigen values $=1 \times 1 \times 1 \times \ldots 10$times
$=1$
 Question 10
Let $\left ( -1 -j \right ), \left ( 3 -j \right ), \left ( 3 + j \right )$ and $\left ( -1+ j \right )$ be the vertices of a rectangle C in the complex plane. Assuming that C is traversed in counter-clockwise direction, the value of the contour integral $\oint _{C}\dfrac{dz}{z^{2}\left ( z-4 \right )}$ is
 A $j\pi /2$ B 0 C $-j\pi /8$ D $j\pi /16$
GATE EE 2021      Complex Variables
Question 10 Explanation:
$\oint_{C} \frac{d z}{z^{2}(z-4)}$

Singularities are given by $z^{2}(z-4)=0$
$\Rightarrow \qquad\qquad z=0,4$
$z=0$ is pole of order $m=2$ lies inside contour 'c'
$z=4$ is pole of order $m=1$ lies outside 'c'
\begin{aligned} \text{Res}_{0} &=\frac{1}{(2-1) !} \text{lt}_{\rightarrow 0} \frac{d^{2-1}}{d z^{2-1}}\left[(z-0)^{2} \frac{1}{z^{2}(z-4)}\right] \\ &=\frac{-1}{(0.4)^{2}}=\frac{-1}{16}\\ \text{By CRT}\\ \oint_{C} f(z) d z &=2 \pi j \text{Res}_{0}=2 \pi j\left[\frac{-1}{16}\right] \\ &=\frac{-j \pi }{8} \end{aligned}

There are 10 questions to complete.