# 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
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 3 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 4
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 4 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 5
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 5 Explanation:
logz is not analytic at all points.
 Question 6
An analytic function f(z) of complex variable z=x+iy may be written as $f(z)=u(x,y)+iv(x,y)$. Then u(x,y) and v(x,y) must satisfy
 A $\frac{\partial u}{\partial x}=\frac{\partial v}{\partial y}$ and $\frac{\partial u}{\partial y}=\frac{\partial v}{\partial x}$ B $\frac{\partial u}{\partial x}=\frac{\partial v}{\partial y}$ and $\frac{\partial u}{\partial y}=-\frac{\partial v}{\partial x}$ C $\frac{\partial u}{\partial x}=-\frac{\partial v}{\partial y}$ and $\frac{\partial u}{\partial y}=\frac{\partial v}{\partial x}$ D $\frac{\partial u}{\partial x}=-\frac{\partial v}{\partial y}$ and $\frac{\partial u}{\partial y}=-\frac{\partial v}{\partial x}$
GATE ME 2019 SET-2   Engineering Mathematics
Question 6 Explanation:
Given that the complex function f(z)=u(x,y)+ i v(x,y) is an analytic function.
$\Rightarrow$ the Cauchy-Riemann equation will satisfy for u(x,y) & v(x,y)
$\therefore \frac{\partial \mathrm{u}}{\partial \mathrm{x}}=\frac{\partial \mathrm{v}}{\partial \mathrm{y}} \text{ and } \frac{\partial \mathrm{u}}{\partial \mathrm{x}}=-\frac{\partial \mathrm{v}}{\partial \mathrm{y}}$
 Question 7
Let z be a complex variable. For a counter-clockwise integration around a unit circle C ,centred at origin,
$\mathrm{\oint_{c} \frac{1}{5z-4}dz}=A\pi i$
the value of A is
 A $\frac{2}{5}$ B $\frac{1}{2}$ C 2 D $\frac{4}{5}$
GATE ME 2018 SET-2   Engineering Mathematics
Question 7 Explanation:
\begin{aligned} 5 z-4&=0 \\ z&=\frac{4}{5} \text{ lies inside circle,}\\ |z|&=1\\ \int \frac{1}{(5 z-4)} d z &=A \pi i \\ \frac{1}{5} \int \frac{1}{\left(z-\frac{4}{5}\right)} d z &=A \pi i \\ \int \frac{\left(\frac{1}{5}\right)}{\left(z-\frac{4}{5}\right)} d z &=2 \pi i \cdot f\left(\frac{4}{5}\right) \\ & =2 \pi i \times\left(\frac{1}{5}\right)\\ &=\frac{2}{5} \pi i \\ A &=\frac{2}{5}=0.4 \end{aligned}
 Question 8
F(z) is a function of the complex variable z = x + iy given by
$F(z)=iz+kRe(z)+ i \; lm(z)$
For what value of k will F(z) satisfy the Cauchy-Riemann equations?
 A 0 B 1 C -1 D y
GATE ME 2018 SET-1   Engineering Mathematics
Question 8 Explanation:
\begin{aligned} F(z) &=i z+k \operatorname{Re}(z)+i \operatorname{Im}(z) \\ u+i v &=i(x+i y)+k x+i y \\ u+i v &=k x-y+i(x+y) \\ u &=k x-y, v=x+y \\ u_{x} &=k, u_{y}=-1 \\ V &=x+y \\ v_{x} &=1 \\ v_{y} &=1 \\ u_{x} &=v_{y} \\ k &=1 \end{aligned}
 Question 9
If $f(z)=(x^{2}+ay^{2})+ibxy$ is a complex analytic function of $z=x+iy$ ,where $i=\sqrt{-1}$, then
 A a=-1,b=-1 B a=-1,b=2 C a=1,b=2 D a=2,b=2
GATE ME 2017 SET-2   Engineering Mathematics
Question 9 Explanation:
Given that the analytic function
\begin{aligned} f(2) &=\left(x^{2}+a y^{2}\right)+i b x y \\ u+i v &=\left(x^{2}+a y^{2}\right)+i(b x y) \\ u &=x^{2}+a y^{2} ; \quad v=b x y \\ u_{x} &=2 x ; \quad u_{y}=2 a y \\ v_{x} &=b y ; \quad v_{y}=b x \\ u_{x} &=v_{y} ; \quad u_{y}=-v_{x} \\ 2 x &=b x ; \quad 2 a y=-b y \\ b &=2 ; \quad 2 a=-b \quad \text { since } b=2 \\ 2 a &=-2 \\ a &=-1 \end{aligned}
 Question 10
The area (in percentage) under standard normal distribution curve of random variable Z within limits from -3 to +3 is __________
 A 55.2 B 88.6 C 99.8 D 44.6
GATE ME 2016 SET-3   Engineering Mathematics
Question 10 Explanation:

There are 10 questions to complete.