# Complex Variables

 Question 1
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 1 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 2
Let $\left ( -1 -j \right ), \left ( 3 -j \right ), \left ( 3 + j \right )$ and $\left ( -1+ j \right )$ be the vertices of a rectangle C in the complex plane. Assuming that C is traversed in counter-clockwise direction, the value of the contour integral $\oint _{C}\dfrac{dz}{z^{2}\left ( z-4 \right )}$ is
 A $j\pi /2$ B 0 C $-j\pi /8$ D $j\pi /16$
GATE EE 2021   Engineering Mathematics
Question 2 Explanation:
$\oint_{C} \frac{d z}{z^{2}(z-4)}$ Singularities are given by $z^{2}(z-4)=0$
$\Rightarrow \qquad\qquad z=0,4$
$z=0$ is pole of order $m=2$ lies inside contour 'c'
$z=4$ is pole of order $m=1$ lies outside 'c'
\begin{aligned} \text{Res}_{0} &=\frac{1}{(2-1) !} \text{lt}_{\rightarrow 0} \frac{d^{2-1}}{d z^{2-1}}\left[(z-0)^{2} \frac{1}{z^{2}(z-4)}\right] \\ &=\frac{-1}{(0.4)^{2}}=\frac{-1}{16}\\ \text{By CRT}\\ \oint_{C} f(z) d z &=2 \pi j \text{Res}_{0}=2 \pi j\left[\frac{-1}{16}\right] \\ &=\frac{-j \pi }{8} \end{aligned}

 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 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 4 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 5
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 5 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)
$\int _C \frac{z^2+1}{z^2-2z}dz=2 \pi i \times \left ( \frac{-1}{2} \right )=-\pi i$

There are 5 questions to complete.