# 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 all lagging power factor loads: $E_A \gt V_A$ 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
$\frac{d^2y(t)}{dt^2}+4y(t)=6r(t)$
\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} 