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: 
For reciprocal network AD-BC= 1
|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.