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.