GATE Electrical Engineering 2020

Question 1
ax^3+bx^2+cx+d is a polynomial on real x over real coefficients a, b, c, d wherein a \neq 0. Which of the following statements is true?
A
d can be chosen to ensure that x = 0 is a root for any given set a, b, c.
B
No choice of coefficients can make all roots identical.
C
a, b, c, d can be chosen to ensure that all roots are complex.
D
c alone cannot ensure that all roots are real.
Engineering Mathematics   Complex Variables
Question 1 Explanation: 
Given Polynomial ax^{3}+bx^{2}+cx+d=0;\; \; \; a\neq 0

Option (A):
If d=0, then the polynomial equation becomes
\begin{aligned} ax^3+bx^2+cx&=0\\ x(ax^2+bx+c)&=0 \\ x=0 \text{ or } ax^2+bx+c&=0 \end{aligned}
d can be choosen to ensure x=0 is a root of given polynomial.
Hence, Option (A) is correct.

Option B:
A third degree polynomial equation with all root equal is given by
(x+\alpha )^3=0
Thus, by selecting suitable values of a, b, c and d we can have all roots identical.
Hence, option (B) is incorrect.

Option (C): Complex roots always occurs in pairs,
So, the given polynomial will have maximum of 2 complex roots and 1 real root.
Hence, option (C) is incorrect.

Option (D): Nature or roots depends on other coefficients also apart from coefficient 'c'.
Hence, option (D) is correct.
Hence, the correct options are (A) and (D).
Question 2
Which of the following is true for all possible non-zero choices of integers m, n; m \neq n, or all possible non-zero choices of real numbers p, q ; p\neq q, as applicable?
A
\frac{1}{\pi}\int_{0}^{\pi}\sin m\theta \sin n\theta \; d\theta =0
B
\frac{1}{2\pi}\int_{-\pi/2}^{\pi/2}\sin p\theta \sin q\theta \; d\theta =0
C
\frac{1}{2\pi}\int_{-\pi}^{\pi}\sin p\theta \cos q\theta \; d\theta =0
D
\lim_{\alpha \to \infty }\frac{1}{2\alpha }\int_{-\alpha }^{\alpha }\sin p\theta \sin q\theta \; d\theta =0
Engineering Mathematics   Complex Variables
Question 2 Explanation: 
\begin{aligned} \because \; p& \neq q\\ &\frac{1}{2\pi}\int_{-\pi}^{\pi} \sin p\theta \cos q\theta d\theta \\ &=\frac{1}{2\pi}\cdot \frac{1}{2}\int_{-\pi}^{\pi} [\sin (p+q)\theta + \sin (p-q)\theta] d\theta \\ &=\frac{1}{4\pi}\left [ \frac{-1}{(p+q)}\cos (p+q)\theta -\frac{1}{(p-q)}\cos (p-q)\theta \right ]_{-\pi}^{\pi}\\ &=\frac{-1}{4\pi} \left \{ \frac{1}{(p+q)}(\cos (p+q) \pi -\cos (p+q)(-\pi)) \right.\\ &+\left. \frac{1}{(p-q)}(\cos (p-q) \pi -\cos (p-q)(-\pi)) \right \}\\ &=0 \end{aligned}
Question 3
Which of the following statements is true about the two sided Laplace transform?
A
It exists for every signal that may or may not have a Fourier transform.
B
It has no poles for any bounded signal that is non-zero only inside a finite time interval.
C
The number of finite poles and finite zeroes must be equal.
D
If a signal can be expressed as a weighted sum of shifted one sided exponentials, then its Laplace Transform will have no poles.
Signals and Systems   Laplace Transform
Question 3 Explanation: 
It has no poles for any bounded signal that is nonzero in a finite time interval. This is true as we know for finite amplitude finite width signal ROC is entire s plane and ROC never includes any pole.
It implies for such signals there is no poles. Hence the correct answer is option (B).
Question 4
Consider a signal x[n]=\left ( \frac{1}{2} \right )^n \; 1[n], where 1[n]=0 if n \lt 0, and 1[n]=1 if n \geq 0. The z-transform of x[n-k], k \gt 0 is \frac{z^{-k}}{1-\frac{1}{2}z^{-1}} with region of convergence being
A
|z| \lt 2
B
|z| \gt 2
C
|z| \lt 1/2
D
|z| \gt 1/2
Signals and Systems   Z-Transform
Question 4 Explanation: 
\begin{aligned}x(n)&=\left (\frac{1}{2} \right )^{n} u(n) , \; \; \; \text{ROC of }x(n):\left | z \right | \gt \frac{1}{2} \\ x(n-k)\rightleftharpoons X(z)&=\frac{z^{-k}}{1-\frac{1}{2}z^{-1}} , \; \; \; \text{ROC of }x(n-k): \left | z \right | \gt \frac{1}{2}\\ \text{For } x(n-k) \; \; \; &\text{ROC will be } \left | z \right |\gt \frac{1}{2}\end{aligned}.
Question 5
The value of the following complex integral, with C representing the unit circle centered at origin in the counterclockwise sense, is:

\int_{c}\frac{z^2+1}{z^2-2z}dz
A
8 \pi i
B
-8 \pi i
C
- \pi i
D
\pi i
Engineering Mathematics   Complex Variables
Question 5 Explanation: 
\begin{aligned} I&=\int _C \frac{z^2+1}{z^2-2z}dz\;\;\;|z|=1 \\ \text{Using } & \text{Cauchy's integral theorem}\\ \int _C\frac{F(z)}{z-a}dz&=2 \pi i (Re_{(z=a)})\;\;\;...(i)\\ I&=\int _C \frac{z^2+1}{z(z-2)}dz \end{aligned}
Poles are at z=0 and 2 but only z=0 lies inside the unit circle.
Residue at (z=0)=\lim_{z \to 0}\frac{z^2+1}{z(z-2)}
Re_{(z=0)}=-\frac{1}{2}
Using equation (i)
\int _C \frac{z^2+1}{z^2-2z}dz=2 \pi i \times \left ( \frac{-1}{2} \right )=-\pi i
Question 6
x_R\; and \; x_A are, respectively, the rms and average values of x(t) = x(t - T), and similarly, y_R\; and \; y_A are, respectively, the rms and average values of y(t) = kx(t). k, \;T are independent of t. Which of the following is true?
A
y_A=kx_A;\; y_R=kx_R
B
y_A=kx_A;\; y_R\neq kx_R
C
y_A \neq kx_A;\; y_R= kx_R
D
y_A \neq kx_A;\; y_R\neq kx_R
Electric Circuits   Network Theorems
Question 6 Explanation: 
Given that,
\begin{aligned} x(t)&=x(t-T) \; \text{ i.e. periodic signal}\\ \text{Average of } x(t)&=x_{A}\\ \text{Rms of }x(t)&=x_{R} \\ \text{Average of }y(t)&=y_{A}\\ \text{Rms of } y(t)&=y_{R} \\ y(t)&=k_{x}(t_{0}) \; \; ....(i) \\ &\text{Using equation(i),}\\ \text{Average of }y(t)&=k \times \text{ Average of }x(t) \\ y_{A}&=kx_{A}\\ \text{Power of }y(t)&=|k|^{2}\text{ Power of }x(t) \\ Rms^{2} \text{ of } y(t)&=|k|^{2}\; Rms^{2}\text{ of }x(t) \\ y_{R}^{2}=|k|^{2}\cdot x_{R}^{2} \\ y_{R}=|k|x_{R}\end{aligned}
Question 7
A three-phase cylindrical rotor synchronous generator has a synchronous reactance X_s and a negligible armature resistance. The magnitude of per phase terminal voltage is V_A and the magnitude of per phase induced emf is E_A. Considering the following two statements, P and Q.

P : For any three-phase balanced leading load connected across the terminals of this synchronous generator, V_A is always more than E_A.
Q : For any three-phase balanced lagging load connected across the terminals of this synchronous generator, V_A is always less than E_A.

Which of the following options is correct?
A
P is false and Q is true.
B
P is true and Q is false.
C
P is false and Q is false.
D
P is true and Q is true.
Electrical Machines   Synchronous Machines
Question 7 Explanation: 


For lagging p.f. load :


For all lagging power factor loads: E_A \gt V_A
For unity p.f. load:

Still we can see : E_A \gt V_A
For 'slighlty' load, phasor diagram will be quite similar to that of unity p.f. load, thus E_A will be greater than V_A. Thus P is false.
Question 8
A lossless transmission line with 0.2 pu reactance per phase uniformly distributed along the length of the line, connecting a generator bus to a load bus, is protected up to 80% of its length by a distance relay placed at the generator bus. The generator terminal voltage is 1 pu. There is no generation at the load bus. The threshold pu current for operation of the distance relay for a solid three phase-to-ground fault on the transmission line is closest to:
A
1
B
3.61
C
5
D
6.25
Power Systems   Performance of Transmission Lines, Line Parameters and Corona
Question 8 Explanation: 
I_{f}=\frac{1}{Z_{Th}}=\frac{1}{0.2}
=5 pu for 100% of line
Relay is operated for 80%
Z_{f}=0.8\, Z_{t}\Rightarrow 0.8\times 0.2=0.16\, p.u.
For 80% of line,
I_{f}=\frac{1}{0.16}=6.25\: p.u.
Question 9
Out of the following options, the most relevant information needed to specify the real power (P) at the PV buses in a load flow analysis is
A
solution of economic load dispatch
B
rated power output of the generator
C
rated voltage of the generator
D
base power of the generator
Power Systems   Load Flow Studies
Question 9 Explanation: 
Most relevant information needed to specify P at PV buses is solution of economic load dispatch.
Question 10
Consider a linear time-invariant system whose input r(t) and output y(t) are related by the following differential equation.
\frac{d^2y(t)}{dt^2}+4y(t)=6r(t)
The poles of this system are at
A
+2j, -2j
B
+2, -2
C
+4, -4
D
+4j, -4j
Control Systems   Time Response Analysis
Question 10 Explanation: 
\begin{aligned}\frac{d^2 y(t)}{dt^{2}}+4y(t) &=6 r(t)\\ [s^{2}+4]Y(s)&=6 R(s) \\ \frac{Y(s)}{R(s)}&=\frac{6}{s^{2}+4} \\ \text{Poles: } s^{2}+4&=0 \\ s&=\pm j2 \end{aligned}
There are 10 questions to complete.
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