GATE Electrical Engineering 2019

 Question 1
The inverse Laplace transform of
$H(s)=\frac{s+3}{s^2+2s+1} \; for \; t\geq 0$ is
 A $3te^{-t}+e^{-t}$ B $3e^{-t}$ C $2te^{-t}+e^{-t}$ D $4te^{-t}+e^{-t}$
Signals and Systems   Laplace Transform
Question 1 Explanation:
\begin{aligned} L^{-1}\left ( \frac{s+3}{s^2+2s+1} \right )&=L^{-1}\left ( \frac{s+1+2}{(s+1)^2} \right )\\ &=L^{-1}\left ( \frac{1}{s+1}+\frac{2}{(s+1)^2} \right )\\ &=e^{-t}+2te^{-t} \end{aligned}
 Question 2
M is a 2x2 matrix with eigenvalues 4 and 9. The eigenvalues of $M^2$ are
 A 4 and 9 B 2 and 3 C -2 and -3 D 16 and 81
Engineering Mathematics   Linear Algebra
Question 2 Explanation:
M is 2 x 2 matrix with eigen values 4 and 9. The eigen values of $M^2$ are 16 and 81.

 Question 3
The partial differential equation
$\frac{\partial^2 u}{\partial t^2}-c^2\left ( \frac{\partial^2 u}{\partial x^2}+\frac{\partial^2 u}{\partial y^2} \right )=0$; where $c\neq 0$
is known as
 A heat equation B wave equation C Poisson's equation D Laplace equation
Engineering Mathematics   Differential Equations
 Question 4
Which one of the following functions is analytic in the region $|z|\leq 1$?
 A $\frac{z^2-1}{z}$ B $\frac{z^2-1}{z+2}$ C $\frac{z^2-1}{z-0.5}$ D $\frac{z^2-1}{z+j0.5}$
Engineering Mathematics   Complex Variables
Question 4 Explanation:

By Cauchy integral theorem,
$\int \frac{z^2-1}{z+2}dz=0$
Therefore, $\frac{z^2-1}{z+2}$ is analytic in the region $|z|\leq 1$.
 Question 5
The mean-square of a zero-mean random process is $\frac{kT}{C}$, where k is Boltzmann's constant, T is the absolute temperature, and C is a capacitance. The standard deviation of the random process is
 A $\frac{kT}{C}$ B $\sqrt{\frac{kT}{C}}$ C $\frac{C}{kT}$ D $\frac{\sqrt{kT}}{C}$
Engineering Mathematics   Probability and Statistics
Question 5 Explanation:
Given that,
\begin{aligned} E(x^2)&=\frac{kT}{C}\\ E(x)&=0\\ &=E(x^2)-(E(x))^2\\ Var(x)&=\frac{kT}{C}-0=\frac{kT}{C} \end{aligned}
Standard deviation $=\sqrt{\frac{kT}{C}}$

There are 5 questions to complete.