# 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)
$\int _C \frac{z^2+1}{z^2-2z}dz=2 \pi i \times \left ( \frac{-1}{2} \right )=-\pi i$
 Question 6
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 6 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 7
$ax^3+bx^2+cx+d$ 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 7 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).
 Question 8
The closed loop line integral
$\oint _{|z|=5}\frac{z^3+z^2+8}{z+2}dz$
evaluated counter-clockwise, is
 A $+8j\pi$ B $-8j\pi$ C $-4j\pi$ D $+4j\pi$
GATE EE 2019   Engineering Mathematics
Question 8 Explanation:
\begin{aligned} \oint _{|z|=5}\frac{z^3+z^2+8}{2+2}dz&=2 \pi j \text{ (sum of residues)} \\ &=2 \pi j \times \left [ \lim_{z \to -2} \frac{(z+2)(z^3+z^2+8)}{(z+2)}\right ] \\ &= 2 \pi j \times \left [ \frac{-8+4+8}{1} \right ]=8 \pi j \end{aligned}
 Question 9
Which one of the following functions is analytic in the region $|z|\leq 1$?
 A $\frac{z^2-1}{z}$ B $\frac{z^2-1}{z+2}$ C $\frac{z^2-1}{z-0.5}$ D $\frac{z^2-1}{z+j0.5}$
GATE EE 2019   Engineering Mathematics
Question 9 Explanation:

By Cauchy integral theorem,
$\int \frac{z^2-1}{z+2}dz=0$
Therefore, $\frac{z^2-1}{z+2}$ is analytic in the region $|z|\leq 1$.
 Question 10
If C is a circle |z|=4 and $f(z)=\frac{z^{2}}{(z^{2}-3z+2)^{2}}$, then $\oint_{C}f(z)dz$ is
 A 1 B 0 C -1 D -2
GATE EE 2018   Engineering Mathematics
Question 10 Explanation:

\begin{aligned} &\int \frac{z^2}{(z^2-3z+2)^2}dz\\ &\int \frac{z^2}{(z-1)^2(z-2)^2}dz\\ \end{aligned}
\begin{aligned} \text{Res. }f(z)_{z=1}&=\lim_{z \to 1}\frac{1}{1!}\frac{d}{dz} \left ((z-1)^2\frac{z^2}{(z-1)^2(z-2)^2} \right )\\ &=\lim_{z \to 1}\left (\frac{2z(z-2)^2-2z^2(z-2)}{((z-2)^4} \right )\\ &=\lim_{z \to 1}\left (\frac{2z(z-2)-2z^2}{((z-2)^3} \right )=\frac{-4}{-1}=4\\ \text{Res. }f(z)_{z=2}&=\lim_{z \to 2}\frac{1}{1!}\frac{d}{dz} \left ((z-2)^2\frac{z^2}{(z-1)^2(z-2)^2} \right )\\ &=\lim_{z \to 2}\left (\frac{(z-1)^2\cdot 2z-z^22(z-1)}{((z-1)^4} \right )\\ &=\lim_{z \to 2}\left (\frac{2z(z-1)-2z^2}{((z-1)^3} \right )=\frac{4-8}{1}=-4\\ \end{aligned}
By residue theorem, $I=2 \pi i (4-4)=0$
There are 10 questions to complete.