GATE Questions & Answers of Complex variables Mechanical Engineering

$ F\left(z\right) $ is a function of the complex variable $ z=x+iy $ given by

                              $ F\left(z\right)=i\;z\;+\;k\;Re\;\left(z\right)+i\;Im\left(z\right) $.

For what value of k will $ F\left(z\right) $ satisfy the Cauchy-Riemann equations?

•  (A) 0
•  (B) 1
•  (C) -1
•  (D) $ y $

Let $z$ be a complex variable. For a counter-clockwise integration around a unit circle $C$ , centred at origin,

                                                              $ \oint\limits_c\;\frac1{5z-4}\;dz=A\pi i\;, $

the value of $A$ is

•  (A) 2/5
•  (B) 1/2
•  (C) 2
•  (D) 4/5

The argument of the complex number $\frac{1+i}{1-i}$ , where $i=\sqrt{-1},$,is

•  (A) $-\mathrm{\pi }$
•  (B) $-\frac{\mathrm{\pi }}{2}$
•  (C) $\frac{\mathrm{\pi }}{2}$
•  (D) $\mathrm{\pi }$

An analytic function of a complex variable $z=x+iy$ is expressed as $f\left(z\right)=u\left(x,y\right)+iv\left(x,y\right),$, where $i=\sqrt{-1}$ . If $u\left(x,y\right)=2xy$, then $v\left(x,y\right)$ must be

•  (A) x² + y² + constant
•  (B) x² - y² + constant
•  (C) - x² + y² + constant
•  (D) - x² - y² + constant

An analytic function of a complex variable z = x + i y is expressed as f (z) = u(x, y) + i v(x, y) ,where i = $\sqrt{-1}$ . If u(x, y) = x² − y ² , then expression for v(x, y) in terms of x, y and a general constant c would be

•  (A)  xy + c
•  (B) $\frac{x^2+y^2}{2}+c$
•  (C) 2xy + c
•  (D) $\frac{{\left(x-y\right)}^2}{2}+c$

If z is a complex variable, the value of $\int\limits_5^{3i}\frac{dz}z$ is

•  (A) − 0.511−1.57i
•  (B) − 0.511+1.57i
•  (C) 0.511− 1.57i
•  (D) 0.511+1.57i

The product of two complex numbers 1 +  i  and 2 - 5i is 

•  (A) 7 - 3i
•  (B) 3 - 4i
•  (C) - 3 - 4i
•  (D) 7 + 3i

The modulus of the complex number $\left(\frac{3+4i}{1-2i}\right)$ is

•  (A) 5
•  (B) $\sqrt{5}$
•  (C) $\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\sqrt{5}$}\right.$
•  (D) $\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$5$}\right.$

An analytic function of a complex variable  z  =  x  + iy  is expressed as f(z) = u(x,y) + i v(x,y) where i = $\sqrt{-1}$. If u = xy, the expression for v should be

•  (A) $\frac{{\left(x+y\right)}^2}{2}+k$
•  (B) $\frac{x^2-y^2}{2}+k$
•  (C) $\frac{y^2-x^2}{2}+k$
•  (D) $\frac{{\left(x-y\right)}^2}{2}+k$

In the Taylor series expansion of e^x about x = 2, the coefficient of (x- 2)⁴ is

•  (A) $\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4!$}\right.$
•  (B) $\raisebox{1ex}{$2^4$}\!\left/ \!\raisebox{-1ex}{$4!$}\right.$
•  (C) $\raisebox{1ex}{$e^2$}\!\left/ \!\raisebox{-1ex}{$4!$}\right.$
•  (D) $\raisebox{1ex}{$e^4$}\!\left/ \!\raisebox{-1ex}{$4!$}\right.$

The integral $\oint f\left(z\right)dz$ evaluated around the unit circle on the complex plane for $f\left(z\right)=\frac{\cos z}{z}$ is

•  (A) $2\mathrm{\pi i}$
•  (B) $4\mathrm{\pi i}$
•  (C) $-2\mathrm{\pi i}$
•  (D) 0

If $\phi \left(x,y\right)$ and $\psi \left(x,y\right)$ are functions with continuous second derivatives, then $\phi \left(x,y\right)+i\psi \left(x,y\right)$ can be expressed as an analytic function of $x+iy\left(i=\sqrt{-1}\right)$, when

•  (A) $\frac{\partial \phi }{\partial x}=-\frac{\partial \psi }{\partial x};\frac{\partial \phi }{\partial y}=\frac{\partial \psi }{\partial y}$
•  (B) $\frac{\partial \phi }{\partial y}=-\frac{\partial \psi }{\partial x};\frac{\partial \phi }{\partial x}=\frac{\partial \psi }{\partial y}$
•  (C) $\frac{{\partial }^2\phi }{\partial x^2}+\frac{{\partial }^2\phi }{\partial y^2}=\frac{{\partial }^2\psi }{\partial x^2}+\frac{{\partial }^2\psi }{\partial y^2}=1$
•  (D) $\frac{\partial \phi }{\partial x}+\frac{\partial \phi }{\partial y}=\frac{\partial \psi }{\partial x}+\frac{\partial \psi }{\partial y}=0$