Complex Variables and Numbers

Question 1

Multiple choice general knowledge math & puzzles

The conjugate of a complex number is 1/i-1. Then the complex number is

-1/i-1
1/i+1
1/i-1
-1/i+1

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Answer

$\dfrac{1}{2}$<$\left | z \right|$< 3

Answer

$\frac{5}{6} < |z| < \frac{6}{5}$

Answer

set of concentric circles

Answer

- ^{$\frac{1}{32}$}

Answer

2$\pi$jc₁

Answer

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

Answer

e^–y sin (x)

Answer

^{_{- 4}$\pi$_{(3 + j2)}}

Answer

is ^{$\frac{1}{3}$}> |z| > ^{$\frac{1}{9}$}

Answer

$\frac{1}{2}$, 1 and –3/2

Question 2

Multiple choice

If x[n]=(1/3)^{| n |} - (1/2)ⁿ u[n], then the region of convergence (ROC) of its Z-transform in the z- Plane will be Z - plane will be

$\dfrac{1}{3}$<$\left | z \right|$< 3
$\dfrac{1}{3}$<$\left | z \right|$< $\dfrac{1}{2}$
$\dfrac{1}{3}$<$\left | z \right|$

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Question 3

Multiple choice

The z-transform X [z] of a sequence x[n] is given by x[z] =$\dfrac{0.5}{1-2z^{-1}}$. It is given that the region of convergence of X[z] includes the unit circle. The value of x[0] is

- 0.5
0
0.25
0.5

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Question 4

Multiple choice

The region of convergence of Z-transform of the sequence $\bigg(\frac{5}{6}\bigg)^n u(n) - \bigg(\frac{6}{5}\bigg)^n u(n - 1)$ must be

$|z| < \frac{5}{6}$
$|z| > \frac{6}{5}$
$\frac{6}{5} < |z| < \infty$

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Question 5

Multiple choice

For the function of a complex variable W = ln Z (where, W = u + jv and Z = x + jy), the u = constant lines get mapped in Z-plane as

set of radial straight lines
set of confocal hyperbolas
set of confocal ellipses

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Question 6

Multiple choice

The residue of the function f (z) =^{$\frac{1}{(z + 2)^2(z - 2)^2}$}at z = 2 is

- ^{$\frac{1}{16}$}
$\frac{1}{16}$
$\frac{1}{32}$

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Question 7

Multiple choice

If f (z) = c₀ + c₁z ^-1, then $\oint \frac{1 + f(z)}{z}$dz is given by

2$\pi$c₁
2$\pi$ (1 + c₀)
2$\pi$ (1 - c₀)

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Question 8

Multiple choice

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

The residue of ^{$\frac{z}{z^2 - 1}$} at z = 1 is ^{$\frac{1}{2}$}.
$\oint_c z^2 dz = 0$
$\frac{1}{2 \pi i}\oint_c \frac{1}{z} dz = 1$

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Question 9

Multiple choice

The real part of an analytic function f (z) where z = x + jy is given by e^–y cos (x). The imaginary part of f (z) is

e^y cos (x)
– e^y sin(x)
– e^–y sin (x)

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Question 10

Multiple choice

C is a closed path in the z-plane by |z| = 3. The value of the integral ^{$\oint_c \bigg(\frac{z^2 - z + 4j}{z + 2j}\bigg)dz$} is

^{_{- 4}$\pi$_{(1 + j2)}}
^{₄$\pi$_{(3 - j2)}}
_⁴^$\pi$_{^{(1 - j2)}}

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Question 11

Multiple choice

Let x [n] = ^{$\bigg(- \frac{1}{9} \bigg)^n$} u (n) – ^{$\bigg(- \frac{1}{3} \bigg)^n$} u (– n – 1). The Region of Convergence (ROC) of the z-transform of x [n]

is |z| > ^{$\frac{1}{9}$}
is |z| < ^{$\frac{1}{3}$}
Does not exist

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Question 12

Multiple choice

If C is a circle of radius r with centre z₀ in the complex z-plane and if n is a non-zero integer, then ^{$\oint_c \frac{dz}{(z - z_0)^{n+1}}$}equals

2^_$\pi$nj
0
$\frac{nj}{2 \pi}$
2^_$\pi$n

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Question 13

Multiple choice

The residues of a complex function X(z) = $\frac{1 - 12z}{z(z - 1)(z - 2)}$at its poles are

$\frac{1}{2}$, $\frac{1}{2}$ and 1
$\frac{1}{2}$, $\frac{1}{2}$and – 1
$\frac{1}{2}$, – 1 and $\frac{2}{3}$

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Question 14

Multiple choice

If two complex numbers Z₁ = 5 + (5^$\sqrt{3}$) i and ^{$z_2=\frac{2}{\sqrt{3}}+2i$}, the argument of ^{$\frac{z_1}{z_2}$}in degrees is

0
30
60
90

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Question 15

Multiple choice

The analytic function f(z) = ^{$\frac{z - 1}{z^2 + 1}$}has singularities at

1 and - 1
1 and i
1 and - i
i and - i

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Complex Variables and Numbers

Complex Variables and Numbers Questions