Complex Analysis - General Aptitude - General Aptitude (GA) Previous Year Questions

1

2015 · General Aptitude (GA) · General Aptitude · Complex Analysis

Electronics & Communication Engineering (EC) 2015 [Session 1]

Let \( z = x + iy \) be a complex variable. Consider that contour integration is performed along the unit circle in anticlockwise direction. Which one of the following statements is NOT TRUE?

A

The residue of \(\frac{z}{z^2 - 1} \) at \(z = 1 \) is 1/2

B

\(\oint_C z^2 dz = 0 \)

C

\(\frac{1}{2\pi i} \oint_C \frac{1}{z} dz = 1 \)

D

\(\bar{z} \) (complex conjugate of \(z \)) is an analytical function

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2

2015 · General Aptitude (GA) · General Aptitude · Complex Analysis

Electronics & Communication Engineering (EC) 2015 [Session 1]

The polar plot of the transfer function \( G(s) = \frac{10(s+1)}{s+10} \) for \( 0 \le \omega < \infty \) will be in the

A

first quadrant

B

second quadrant

C

third quadrant

D

fourth quadrant

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3

2015 · General Aptitude (GA) · General Aptitude · Complex Analysis

Electronics & Communication Engineering (EC) 2015 [Session 1]

The pole-zero diagram of a causal and stable discrete-time system is shown in the figure. The zero at the origin has multiplicity 4. The impulse response of the system is \(h[n]\). If \(h[0] = 1\), we can conclude

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4

2015 · General Aptitude (GA) · General Aptitude · Complex Analysis

Electronics & Communication Engineering (EC) 2015 [Session 1]

The longitudinal component of the magnetic field inside an air-filled rectangular waveguide made of a perfect electric conductor is given by the following expression \[ H_z(x, y, z, t) = 0.1 \cos(25\pi x) \cos(30.3 \pi y) \cos(12\pi \times 10^9 t - \beta z) \; (A/m) \] The cross-sectional dimensions of the waveguide are given as \(a = 0.08\) m and \(b = 0.033\) m. The mode of propagation inside the waveguide is

A

\(TM_{12}\)

B

\(TM_{21}\)

C

\(TE_{21}\)

D

\(TE_{12}\)

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5

2015 · General Aptitude (GA) · General Aptitude · Complex Analysis

Electronics & Communication Engineering (EC) 2015 [Session 2]

Let \( f(z) = \frac{az+b}{cz+d} \). If \( f(z_1) = f(z_2) \) for all \( z_1 \neq z_2 \), \( a = 2, b = 4 \) and \( c = 5 \), then \( d \) should be equal to ______.

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6

2015 · General Aptitude (GA) · General Aptitude · Complex Analysis

Electronics & Communication Engineering (EC) 2015 [Session 2]

The magnitude and phase of the complex Fourier series coefficients \(a_k\) of a periodic signal \(x(t)\) are shown in the figure. Choose the correct statement from the four choices given. Notation: \(C\) is the set of complex numbers, \(R\) is the set of purely real numbers, and \(P\) is the set of purely imaginary numbers.

A

\(x(t) \in R\)

B

\(x(t) \in P\)

C

\(x(t) \in (C - R)\)

D

The information given is not sufficient to draw any conclusion about \(x(t)\)

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7

2015 · General Aptitude (GA) · General Aptitude · Complex Analysis

Electronics & Communication Engineering (EC) 2015 [Session 2]

Let the signal f(t) = 0 outside the interval [T1, T2], where T1 and T2 are finite. Furthermore, |f(t)| < infinity. The region of convergence (RoC) of the signal's bilateral Laplace transform F(s) is

A

a parallel strip containing the jΩ axis

B

a parallel strip not containing the jΩ axis

C

the entire s-plane

D

a half plane containing the jΩ axis

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8

2015 · General Aptitude (GA) · General Aptitude · Complex Analysis

Electronics & Communication Engineering (EC) 2015 [Session 2]

If \(C\) denotes the counterclockwise unit circle, the value of the contour integral \[ \frac{1}{2\pi j} \oint_C Re\{z\} dz \] is ________.

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9

2015 · General Aptitude (GA) · General Aptitude · Complex Analysis

Electronics & Communication Engineering (EC) 2015 [Session 2]

The electric field of a plane wave propagating in a lossless non-magnetic medium is given by the following expression
\[\mathbf{E}(z, t) = \mathbf{a}_x 5 \cos(2\pi \times 10^9 t + \beta z) + \mathbf{a}_y 3 \cos\left(2\pi \times 10^9 t + \beta z - \frac{\pi}{2}\right)\]
The type of the polarization is

A

Right Hand Circular.

B

Left Hand Elliptical.

C

Right Hand Elliptical.

D

Linear.

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10

2015 · General Aptitude (GA) · General Aptitude · Complex Analysis

Electronics & Communication Engineering (EC) 2015 [Session 3]

The transfer function of a first-order controller is given as
\( G_c(s) = \frac{K(s+a)}{s+b} \)
where \( K \), \( a \) and \( b \) are positive real numbers. The condition for this controller to act as a phase lead compensator is

A

\(a < b \)

B

\(a > b \)

C

\(K < ab \)

D

\(K > ab \)

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11

2015 · General Aptitude (GA) · General Aptitude · Complex Analysis

Electronics & Communication Engineering (EC) 2015 [Session 3]

Suppose x[n] is an absolutely summable discrete-time signal. Its z-transform is a rational function with two poles and two zeroes. The poles are at z = ±2j. Which one of the following statements is TRUE for the signal x[n]?

A

It is a finite duration signal.

B

It is a causal signal.

C

It is a non-causal signal.

D

It is a periodic signal.

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12

2015 · General Aptitude (GA) · General Aptitude · Complex Analysis

Electronics & Communication Engineering (EC) 2015 [Session 3]

Let \(\tilde{x}[n] = 1 + \cos\left(\frac{\pi n}{8}\right)\) be a periodic signal with period 16. Its DFS coefficients are defined by \(a_k = \frac{1}{16}\sum_{n=0}^{15}\tilde{x}[n]\exp\left(-j\frac{\pi}{8} kn\right)\) for all \(k\). The value of the coefficient \(a_{31}\) is ______.

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13

2015 · General Aptitude (GA) · General Aptitude · Complex Analysis

Electronics & Communication Engineering (EC) 2015 [Session 3]

The complex envelope of the bandpass signal \(x(t) = -\sqrt{2} \left( \frac{\sin(\pi t / 5)}{\pi t / 5} \right) \sin(\pi t - \frac{\pi}{4})\), centered about \(f = \frac{1}{2}\) Hz, is

A

\(\left( \frac{\sin(\pi t / 5)}{\pi t / 5} \right) e^{j \frac{\pi}{4}}\)

B

\(\left( \frac{\sin(\pi t / 5)}{\pi t / 5} \right) e^{-j \frac{\pi}{4}}\)

C

\(\sqrt{2} \left( \frac{\sin(\pi t / 5)}{\pi t / 5} \right) e^{j \frac{\pi}{4}}\)

D

\(\sqrt{2} \left( \frac{\sin(\pi t / 5)}{\pi t / 5} \right) e^{-j \frac{\pi}{4}}\)

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14

2016 · General Aptitude (GA) · General Aptitude · Complex Analysis

Electronics & Communication Engineering (EC) 2016 [Session 1]

Which one of the following is an eigen function of the class of all continuous-time, linear, time-invariant 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

\(\sin(\omega_0 t) \)

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15

2016 · General Aptitude (GA) · General Aptitude · Complex Analysis

Electronics & Communication Engineering (EC) 2016 [Session 1]

Consider the sequence \( x[n] = a^n u[n] + b^n u[n] \), where \( u[n] \) denotes the unit-step sequence and \( 0 < |a| < |b| < 1 \). The region of convergence (ROC) of the z-transform of \( x[n] \) is

A

\(|z| > |a| \)

B

\(|z| > |b| \)

C

\(|z| < |a| \)

D

\(|a| < |z| < |b| \)

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