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

1

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

Electrical Engineering (EE) 2015 [Session 1]

Consider a function \( \vec{f} = \frac{1}{r^2} \hat{r} \), where \( r \) is the distance from the origin and \( \hat{r} \) is the unit vector in the radial direction. The divergence of this function over a sphere of radius R, which includes the origin, is

A

0

B

\(2\pi \)

C

\(4\pi \)

D

\(R\pi \)

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2

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

Electrical Engineering (EE) 2015 [Session 1]

Consider a discrete time signal given by
$x[n] = (-0.25)^n u[n] + (0.5)^n u[-n-1]$
The region of convergence of its Z-transform would be

A

the region inside the circle of radius 0.5 and centered at origin

B

the region outside the circle of radius 0.25 and centered at origin

C

the annular region between the two circles, both centered at origin and having radii 0.25 and 0.5

D

the entire Z plane.

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3

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

Electrical Engineering (EE) 2015 [Session 1]

The transfer function of a second order real system with a perfectly flat magnitude response of unity has a pole at \(2 - j3\). List all the poles and zeroes.

A

Poles at \(2 \pm j3\), no zeroes.

B

Poles at \(\pm 2 - j3\), one zero at origin.

C

Poles at \(2 - j3\), \(-2 + j3\), zeroes at \(-2 - j3\), \(2 + j3\).

D

Poles at \(2 \pm j3\), zeroes at \(-2 \pm j3\).

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4

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

Electrical Engineering (EE) 2015 [Session 1]

The open loop poles of a third order unity feedback system are at \(0, -1, -2\). Let the frequency corresponding to the point where the root locus of the system transits to unstable region be K. Now suppose we introduce a zero in the open loop transfer function at \(-3\), while keeping all the earlier open loop poles intact. Which one of the following is TRUE about the point where the root locus of the modified system transits to unstable region?

A

It corresponds to a frequency greater than K

B

It corresponds to a frequency less than K

C

It corresponds to a frequency K

D

Root locus of modified system never transits to unstable region

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5

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

Electrical Engineering (EE) 2015 [Session 2]

Given \( f(z) = g(z) + h(z) \), where \( f, g, h \) are complex valued functions of a complex variable \( z \). Which one of the following statements is TRUE?

A

If \(f(z) \) is differentiable at \(z_0 \), then \(g(z) \) and \(h(z) \) are also differentiable at \(z_0 \).

B

If \(g(z) \) and \(h(z) \) are differentiable at \(z_0 \), then \(f(z) \) is also differentiable at \(z_0 \).

C

If \(f(z) \) is continuous at \(z_0 \), then it is differentiable at \(z_0 \).

D

If \(f(z) \) is differentiable at \(z_0 \), then so are its real and imaginary parts.

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6

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

Electrical Engineering (EE) 2015 [Session 2]

Given \(f(z) = g(z) + h(z)\), where \(f\), \(g\), \(h\) are complex valued functions of a complex variable \(z\). Which one of the following statements is TRUE?

A

If \(f(z)\) is differentiable at \(z_0\), then \(g(z)\) and \(h(z)\) are also differentiable at \(z_0\).

B

If \(g(z)\) and \(h(z)\) are differentiable at \(z_0\), then \(f(z)\) is also differentiable at \(z_0\).

C

If \(f(z)\) is continuous at \(z_0\), then it is differentiable at \(z_0\).

D

If \(f(z)\) is differentiable at \(z_0\), then so are its real and imaginary parts.

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7

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

Electrical Engineering (EE) 2015 [Session 2]

Consider a signal defined by \(x(t) = \begin{cases} e^{j10t} & \text{for } |t| \leq 1 \\ 0 & \text{for } |t| > 1 \end{cases}\)
Its Fourier Transform is

A

\(\frac{2 \sin (\omega - 10)}{\omega - 10}\)

B

\(2 e^{j10} \frac{\sin (\omega - 10)}{\omega - 10}\)

C

\(\frac{2 \sin \omega}{\omega - 10}\)

D

\(e^{j10\omega} \frac{2 \sin \omega}{\omega}\)

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8

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

Electrical Engineering (EE) 2015 [Session 2]

In the given network \(V_1 = 100\angle 0^{\circ} V\), \(V_2 = 100\angle -120^{\circ} V\), \(V_3 = 100\angle +120^{\circ} V\). The phasor current \(i\) (in Ampere) is

A

\(173.2\angle -60^{\circ}\)

B

\(173.2\angle 120^{\circ}\)

C

\(100.0\angle -60^{\circ}\)

D

\(100.0\angle 120^{\circ}\)

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9

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

Electrical Engineering (EE) 2015 [Session 2]

An open loop transfer function G(s) of a system is
\[G(s) = \frac{K}{s(s+1)(s+2)}\]
For a unity feedback system, the breakaway point of the root loci on the real axis occurs at,

A

\(-0.42\)

B

\(-1.58\)

C

\(-0.42\) and \(-1.58\)

D

none of the above

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10

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

Electrical Engineering (EE) 2016 [Session 2]

Let \(f(x)\) be a real, periodic function satisfying \(f(−x) = −f(x)\). The general form of its Fourier series representation would be

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