# Complex Variables

 Question 1
The value of $k$ that makes the complex-valued function

$f(z)=e^{-kx}(\cos 2y -i \sin 2y)$

analytic, where $z=x+iy$, is _________. (Answer in integer)
 A 1 B 2 C 3 D 4
GATE ME 2023   Engineering Mathematics
Question 1 Explanation:
\begin{aligned} & f(z)=e^{-k x} \cos 2 y-i e^{-k x} \sin 2 y \\ & \text { Suppose }=e^{-k x} \cos 2 y=4(x, y) \\ & =-i e^{-k x} \sin 2 y=v(x, y) \end{aligned}
If function is analytical then it satisfy the equation
\begin{aligned} & \frac{\partial u}{\partial x}=-k e^{-k x} \cos 2 y &...(i)\\ & \frac{\partial u}{\partial y}=-2 e^{-k x} \sin 2 y &...(ii)\\ & \frac{\partial v}{\partial x}=-k e^{-k x} \sin 2 y &...(iii)\\ & \frac{\partial v}{\partial y}=-2 e^{-k x} \cos 2 y &...(iv) \end{aligned}
Cauchy-Riemann equation $\frac{\partial u}{\partial x}=\frac{\partial v}{\partial y}$
by putting equation (i), (iv) and solve then $k=2$
 Question 2
Let $w^{4}=16 j$. Which of the following cannot be a value of $w$ ?
 A $2 e^{\frac{j 2 \pi}{8}}$ B $2 e^{\frac{j \pi}{8}}$ C $2 e^{\frac{j 5 \pi}{8}}$ D $2 e^{\frac{j 9 \pi}{8}}$
GATE EC 2023   Engineering Mathematics
Question 2 Explanation:
$w=(2) j^{1 / 4}$
$w=2(0+j)^{1 / 4}$
$w=2\left[e^{j(2 n+1) \pi / 2}\right]^{1 / 4}$ $=2\left[e^{j(2 n+1) \pi / 8}\right]$

For $n=0$, $w=e^{j \pi / 8}$
For $n=2$, $w=2 e^{5 \pi j / 8}$
For $n=4$, $w=2 e^{9 \pi j / 8}$

 Question 3
Given $z=x+iy,i=\sqrt{-1}$ is a circle of radius 2 with the centre at the origin. If the contour C is traversed anticlockwise, then the value of the integral $\frac{1}{2}\int_{c}^{}\frac{1}{(z-i)(z+4i)}dz$ is _______(round off to one decimal place).
 A 0.2 B 0.4 C 0.6 D 0.8
GATE ME 2022 SET-2   Engineering Mathematics
Question 3 Explanation:
Given contour is a circle at centre (0,0) and radius 2 given function is $\frac{1}{(z-i)(z+4i)}$ here $z = i$ is a singular point lies now by caucus integral formula \begin{aligned} &\int \frac{1}{(z-i)(z+4i)}dz=\int \frac{\left ( \frac{1}{z+4} \right )}{z-i}dz=2 \pi i \times f(i)\\ &f(z)=\frac{1}{z+4i}\\ &f(i)=\frac{1}{5i}\\ &2 \pi i \times f(i)=2 \pi i \times \frac{1}{5i}=\frac{2\pi}{5}\\ &\text{Now }\frac{1}{2 \pi}\int_{c}\frac{1}{(z-1)(z+4i)}dz=\frac{1}{2 \pi}+\frac{2 \pi}{5}=\frac{1}{5} \end{aligned}
 Question 4
Let $R$ be a region in the first quadrant of the $xy$ plane enclosed by a closed curve $C$ considered in counter-clockwise direction. Which of the following expressions does not represent the area of the region $R$? A $\int \int _R dxdy$ B $\oint _c xdy$ C $\oint _c ydx$ D $\frac{1}{2}\oint _c( xdy-ydx)$
GATE EE 2022   Engineering Mathematics
Question 4 Explanation:
Using green theorem?s
$\oint _cF_1dx+F_2dy=\int \int _R\left ( \frac{\partial F_2}{\partial x} -\frac{\partial F_1}{\partial y}\right )dxdy$
Check all the options:
$\oint xdy=\int \int _R\left ( \frac{\partial x}{\partial x} -0\right )dxdy=\int \int _Rdxdy$
$\frac{1}{2}\oint xdy-ydx=\frac{1}{2}\int \int _R(1+1)dxdy=\int \int _Rdxdy$
$\oint ydx=\int \int _R(-1)dxdy=-\int \int _Rdxdy$
Hence, $\oint ydx$ is not represent the area of the region.
 Question 5
Consider the following series:
$\sum_{n=1}^{\infty }\frac{n^d}{c^n}$
For which of the following combinations of $c,d$ values does this series converge?
 A $c=1,d=-1$ B $c=2,d=1$ C $c=0.5,d=-10$ D $c=1,d=-2$
GATE EC 2022   Engineering Mathematics
Question 5 Explanation:
\begin{aligned} \Sigma u_n&=\Sigma \frac{n}{2^n}\\ &\text{Ratio test; }\\ \lim_{n \to \infty }\frac{u_{n+1}}{u_n}&=\lim_{n \to \infty }\frac{n+1}{2^{n+1}} \times \frac{2^n}{n}=\frac{1}{2}\\ \frac{1}{2}& \lt 2 \end{aligned}
$\therefore$ By ratio test, $\Sigma u_n$ is convergent.
(A) c=1, d=-1
$\Sigma u_n=\Sigma \frac{1}{n}$ is divergent by P-test
(B) c=0.5, d=-10
\begin{aligned} \Sigma u_n&=\Sigma \frac{n^{-10}}{(0.5)^n}\\ &\text{Ratio test; }\\ \lim_{n \to \infty }\frac{u_{n+1}}{u_n}&=\lim_{n \to \infty }\frac{(n+1)^{-10}}{(0.5)^{n+1}} \times \frac{(0.5)^n}{n^{10}}\\ &=\frac{1}{0.5}=2\\ 2& \gt 1 \end{aligned}
$\therefore$ By ratio test, $\Sigma u_n$ is divergent.
(D) c=1,d=-2 $\Sigma u_n=\Sigma \frac{n^{-2}}{(1)^n}=\Sigma \frac{1}{n^2}$
$\Sigma u_n$ is convergent by P-test.

There are 5 questions to complete.

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