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?

d can be chosen to ensure that x = 0 is a root for any given set a, b, c. | |

No choice of coefficients can make all roots identical. | |

a, b, c, d can be chosen to ensure that all roots are complex. | |

c alone cannot ensure that all roots are real. |

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).

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?

\frac{1}{\pi}\int_{0}^{\pi}\sin m\theta \sin n\theta \; d\theta =0 | |

\frac{1}{2\pi}\int_{-\pi/2}^{\pi/2}\sin p\theta \sin q\theta \; d\theta =0 | |

\frac{1}{2\pi}\int_{-\pi}^{\pi}\sin p\theta \cos q\theta \; d\theta =0 | |

\lim_{\alpha \to \infty }\frac{1}{2\alpha }\int_{-\alpha }^{\alpha }\sin p\theta \sin q\theta \; d\theta =0 |

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?

It exists for every signal that may or may not have a Fourier transform. | |

It has no poles for any bounded signal that is non-zero only inside a finite time
interval. | |

The number of finite poles and finite zeroes must be equal. | |

If a signal can be expressed as a weighted sum of shifted one sided exponentials,
then its Laplace Transform will have no poles. |

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).

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

|z| \lt 2 | |

|z| \gt 2 | |

|z| \lt 1/2 | |

|z| \gt 1/2 |

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

\int_{c}\frac{z^2+1}{z^2-2z}dz

8 \pi i | |

-8 \pi i | |

- \pi i | |

\pi i |

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

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?

y_A=kx_A;\; y_R=kx_R | |

y_A=kx_A;\; y_R\neq kx_R | |

y_A \neq kx_A;\; y_R= kx_R | |

y_A \neq kx_A;\; y_R\neq kx_R |

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}

\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?

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?

P is false and Q is true. | |

P is true and Q is false. | |

P is false and Q is false. | |

P is true and Q is true. |

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:

1 | |

3.61 | |

5 | |

6.25 |

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.

=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

solution of economic load dispatch | |

rated power output of the generator | |

rated voltage of the generator | |

base power of the generator |

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

\frac{d^2y(t)}{dt^2}+4y(t)=6r(t)

The poles of this system are at

+2j, -2j | |

+2, -2 | |

+4, -4 | |

+4j, -4j |

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.