# GATE EC 2016 SET-1

 Question 1
Let $M^{4}=I$, (where $I$ denotes the identity matrix) and $M \neq I$, $M^{2} \neq I$ and $M^{3} \neq I$. Then, for any natural number $k,M^{-1}$ equals
 A $M^{4k+1}$ B $M^{4k+2}$ C $M^{4k+3}$ D $M^{4k}$
Engineering Mathematics   Linear Algebra
Question 1 Explanation:
Given that $M^{4}=I$ or $M^{4 k}=I$ or $M^{4(k+1)}=I$
\begin{aligned} \therefore \quad M^{-1} \times I & =M^{4(k+1)} \times M^{-1} \\ \therefore \quad M^{-1} & =M^{4 k+3}\end{aligned}
 Question 2
The second moment of a Poisson-distributed random variable is 2. The mean of the random variable is _______
 A 0.5 B 1 C 2 D 3
Engineering Mathematics   Probability and Statistics
Question 2 Explanation:
In Poisson distribution,
Mean = First moment $=\lambda$
secondmoment $=\lambda^{2}+\lambda$
Given, that second moment is 2
$\begin{array}{r} \lambda^{2}+\lambda=2 \\ \lambda^{2}+\lambda-2=0 \\ (\lambda+2)(\lambda-1)=0 \\ \lambda=1 \end{array}$
 Question 3
Given the following statements about a function $f:\mathbb{R}\rightarrow \mathbb{R}$, select the right option:
P: If f(x) is continuous at x=$x_{0}$, then it is also differentiable at x =$x_{0}$
Q: If f(x) is continuous at x = $x_{0}$, then it may not be differentiable at x= $x_{0}$.
R: If f(x) is differentiable at x= $x_{0}$, then it is also continuous at x= $x_{0}$.
 A P is true, Q is false, R is false B P is false, Q is true, R is true C P is false, Q is true, R is false D P is true, Q is false, R is true
Engineering Mathematics   Calculus
Question 3 Explanation:
P: If f(x) is continuous at $x=x_{0},$ then it is also differentiable at $x=x_{0}$
Q: If f(x) is continuous at $x=x_{0},$ then it may or may not be derivable at $x=x_{0}$
R: If f(x) is differentiable at $x=x_{0}$, then it is also continuous at $x=x_{0}$
P is false
Q is true
R is true
Option (B) is correct
 Question 4
Which one of the following is a property of the solutions to the Laplace equation: $\bigtriangledown ^{2} f= 0$?
 A The solutions have neither maxima nor minima anywhere except at the boundaries. B The solutions are not separable in the coordinates. C The solutions are not continuous D The solutions are not dependent on the boundary conditions
Signals and Systems   Laplace Transform
 Question 5
Consider the plot of $f(x)$ versus $x$ as shown below.

Suppose F(x)=$\int_{-5}^{x}f(y)dy$ . Which one of the following is a graph of F(x) ?
 A A B B C C D D
Engineering Mathematics   Calculus
Question 5 Explanation:
$F^{\prime}(x)=f(x)$ which is density function
$F^{\prime}(x)=f(x) \lt 0$ when $x \lt 0$
$\therefore \quad F(x)$ is decreasing for $x \lt 0$
$F^{\prime}(x)=f(x) \gt 0$
when $x\gt 0$
$\therefore \quad F(x)$ is increasing for $x\gt 0$.
 Question 6
Which one of the following is an eigen function of the class of all continuous-time, linear, timeinvariant systems (u(t) denotes the unit-step function)?
 A $e^{j\omega_{0}t }u(t)$ B $cos(\omega _{0}t)$ C $e^{j\omega_{0}t }$ D $sin(\omega _{0}t)$
Signals and Systems   LTI Systems Continuous and Discrete
Question 6 Explanation:
If the input to the system is eigen signal output is also the same eigen signal.
 Question 7
A continuous-time function x(t) is periodic with period T. The function is sampled uniformly with a sampling period $T_{s}$. In which one of the following cases is the sampled signal periodic?
 A $T=\sqrt{2}T_{s}$ B $T=1.2T_{s}$ C Always D Never
Signals and Systems   Basics of Signals and Systems
Question 7 Explanation:
A signal is said to be periodic if $\frac{T}{T_{s}}$ is a rational number.
Here, $T=1.2 T_{s}$
$\Rightarrow \frac{T}{T_{s}}=\frac{6}{5} \quad$ Which is a rational number
 Question 8
Consider the sequence $x[n]=a^{n}u[n]+b^{n}u[n]$ , where u[n] denotes the unit-step sequence and 0 $\lt$ |a| $\lt$ |b| $\lt$ 1. The region of convergence (ROC) of the z-transform of x[n] is
 A |z|$\gt$|a| B |z|$\gt$|b| C |z|$\lt$|a| D |a|$\lt$|z|$\lt$|b|
Signals and Systems   Z-Transform
Question 8 Explanation:
\begin{array}{l} \text { Given, } x[n]=a^{n} u[n]+b^{n} u[n] \\ \text { Also given, } \quad \begin{aligned} 0 &<|a|<|b|<1 \\ & A O C=(|z|>|a|) \text { and }(|z|>|b|) \\ & R O C=|z|>|b| \end{aligned} \end{array}
 Question 9
Consider a two-port network with the transmission matrix: $T=\begin{pmatrix} A & B\\ C& D \end{pmatrix}$. If the network is reciprocal, then
 A $T^{-1}=T$ B $T^{2}=T$ C Determinant (T) = 0 D Determinant (T) = 1
Network Theory   Two Port Networks
Question 9 Explanation:
|T|=1
 Question 10
A continuous-time sinusoid of frequency 33 Hz is multiplied with a periodic Dirac impulse train of frequency 46 Hz. The resulting signal is passed through an ideal analog low-pass filter with a cutoff frequency of 23 Hz. The fundamental frequency (in Hz) of the output is _________
 A 5 B 12 C 18 D 24
Signals and Systems   Sampling
Question 10 Explanation:
If x(t) is a message signal and y(t) is a sampled signal, then y(t) is related to x(t) as
$y(t)=x(t) \sum_{n=-\infty}^{\infty} \delta\left(t-n T_{s}\right)$
$r(f)=f_{s} \sum_{n=--\infty}^{\infty} X\left(f-n f_{s}\right)$
Spectrum of X(f) and Y(f) are as shown

Cut off frequency of LPF = 23 Hz
Hence, frequency at the output is 13 Hz
There are 10 questions to complete.