Question 1 |
The inverse Laplace transform of
H(s)=\frac{s+3}{s^2+2s+1} \; for \; t\geq 0 is
H(s)=\frac{s+3}{s^2+2s+1} \; for \; t\geq 0 is
3te^{-t}+e^{-t} | |
3e^{-t} | |
2te^{-t}+e^{-t} | |
4te^{-t}+e^{-t} |
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
4 and 9 | |
2 and 3 | |
-2 and -3 | |
16 and 81 |
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
\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
heat equation | |
wave equation | |
Poisson's equation | |
Laplace equation |
Question 4 |
Which one of the following functions is analytic in the region |z|\leq 1?
\frac{z^2-1}{z} | |
\frac{z^2-1}{z+2} | |
\frac{z^2-1}{z-0.5} | |
\frac{z^2-1}{z+j0.5} |
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
\frac{kT}{C} | |
\sqrt{\frac{kT}{C}} | |
\frac{C}{kT} | |
\frac{\sqrt{kT}}{C} |
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}}
\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.