Complex Variables

Question 1
Value of (1+i)^8, where i=\sqrt{-1}, is equal to
A
4
B
16
C
4i
D
16i
GATE ME 2021 SET-2   Engineering Mathematics
Question 1 Explanation: 
\begin{aligned} (1+i)^{8} & \\ \mathrm{z} &=1+i \mathrm{r}=|z|=\sqrt{2} \\ \theta &=\frac{\pi}{4} \\ (1+i)^{8} &=\left(\sqrt{2} e^{i \frac{\pi}{4}}\right)^{8} \\ &=16 \times e^{i \times 2 \pi} \\ &=16(\cos 2 \pi+i \sin 2 \pi) \\ &=16 \times 1=16 \end{aligned}
Question 2
Let C represent the unit circle centered at origin in the complex plane, and complex variable, z=x+iy. The value of the contour integral \oint _C\frac{\cosh 3z}{2z}dz (where integration is taken counter clockwise) is
A
0
B
2
C
\pi i
D
2 \pi i
GATE ME 2021 SET-1   Engineering Mathematics
Question 2 Explanation: 
Pole of f(c) is z=0 simple pole.
Reduce at z=0
R_{b} f(z)=\lim _{z \rightarrow 0}(z-0) f(z)=\lim _{z \rightarrow 0} \frac{\cosh (3 z)}{2}=\lim _{z \rightarrow 0} \frac{e^{3 z}+e^{-3 z}}{2 \times 2}=\frac{1}{2}
By Cauchy Riemann theorem,
I=2 \pi\left(\frac{1}{2}\right)=\pi i
Question 3
Let p\left ( z\right )=z^{3}+\left ( 1+j \right )z^{2}+\left ( 2+j \right )z+3, where z is a complex number.
Which one of the following is true?
A
\text{conjugate}\:\left \{ p\left ( z \right ) \right \}=p\left ( \text{conjugate} \left \{ z \right \} \right ) for all z
B
The sum of the roots of p\left ( z \right )=0 is a real number
C
The complex roots of the equation p\left ( z \right )=0 come in conjugate pairs
D
All the roots cannot be real
GATE EE 2021   Engineering Mathematics
Question 3 Explanation: 
Since sum of the roots is a complex number
\Rightarrow absent one root is complex
So all the roots cannot be real.
Question 4
The function f(z) of complex variable z = x + iy, where i = \sqrt{-1}, is given as f(z) = (x^3 - 3xy^2) + iv(x, y). For this function to be analytic, v(x, y) should be
A
(3xy^2-y^3)+constant
B
(3 x^2 y^2-y^3)+constant
C
(x^3-3x^2 y^3)+constant
D
(3x^2y-y^3)+constant
GATE ME 2020 SET-2   Engineering Mathematics
Question 4 Explanation: 
\begin{aligned} f(z) &=u+i v \\ u &=x^{3}-3 x y^{2}, \quad v=v(x, y)\\ &\text{For }f(z) \text{ to be Analytical,}\\ u_{x} &=3 x^{2}-3 y^{2}=v_{y} \\ u_{y} &=-6 x y=-v_{x} \\ v_{x} &=6 x y \text { by integrating w.r.t } x \Rightarrow v=3 x^{2} y+C_{1} \\ v_{y} &=3 x^{2}-3 y^{2} \text { by integrating w.r.t } y \Rightarrow v=3 x^{2} y-y^{3}+C_{2} \\ v &=\left(3 x^{2} y-y^{3}\right)+\text { constant }\left(C_{1}=-y^{3}\right) \end{aligned}
Question 5
An analytic function of a complex variable z=x+iy(i=\sqrt{-1}) is defined as

f(z)=x^2-y^2+i\psi (x,y)

where \psi (x,y) is a real function. The value of the imaginary part of f(z) at z=(1+i) is ___________ (round off to 2 decimal places)
A
1
B
2
C
3
D
4
GATE ME 2020 SET-1   Engineering Mathematics
Question 5 Explanation: 
\begin{aligned} f(z) &=\phi+i \psi \text { is analytic } \\ &\text{C-R equations} \\ \phi _x&=\psi _x \\ \phi_{y}&=-\psi _y \\ \phi &=x^{2}-y^{2} \\ \phi_{x} &=2 x=\psi_{y} \\ \phi_{y} &=-2 y=-\psi_{x} \\ \psi_{x} &=2 y \quad \Rightarrow \psi=2 x y+C_{1} \\ \psi_{y} &=2 x \quad \Rightarrow \psi=2 x y+C_{2}\\ \text{Comparing }\psi&=2 x y+C \text{ valid for all C put C=0 }\\ \psi(1+i) &\Rightarrow(x=1 \quad y=1)\\ \therefore \psi&=2 \end{aligned}
Question 6
The real numbers, x and y with y = 3x^2 + 3x + 1, the maximum and minimum value of y for x \in [-2, 0] are respectively ________
A
7 and 1/4
B
7 and 1
C
-2 and -1/2
D
1 and 1/4
GATE EE 2020   Engineering Mathematics
Question 6 Explanation: 
\begin{aligned} y&=3x^{2}+3x+1 \; \; in \; [-2,0] \\ \frac{\partial y}{\partial x}&=6x+3,\; \; \frac{\partial^2 y}{\partial x^2}=6 \\ \frac{\mathrm{d} y}{\mathrm{d} x}&=0\\ 6x+3&=0 \\ x&=\frac{-1}{2} \\ \frac{d^{2} y}{dx^{2}}&=6 \gt 0\text{ minimum} \end{aligned}

Maximum value of y in [-2, 0] is maximum {f(-2), f(0)}
max{7, 1} = 7

Minimum value of y in [-2, 0]
min\begin{Bmatrix} f(-2) & f(0) &f(-\frac{1}{2}) \\ \downarrow, &\downarrow, &\downarrow \\ 7& 1 & \frac{1}{4} \end{Bmatrix}+=\frac{1}{4}
Maximum value 7, minimum value \frac{1}{4}
Question 7
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
GATE EE 2020   Engineering Mathematics
Question 7 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 8
Which of the following function f(z), of the complex variable z, is NOT analytic at all the points of the complex plane?
A
f(z)=z^2
B
f(z)=e^z
C
f(z)=\sin z
D
f(z)=\log z
GATE ME 2020 SET-1   Engineering Mathematics
Question 8 Explanation: 
logz is not analytic at all points.
Question 9
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
GATE EE 2020   Engineering Mathematics
Question 9 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 10
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.
GATE EE 2020   Engineering Mathematics
Question 10 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).
There are 10 questions to complete.

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