# GATE Questions & Answers of Complex variables Mechanical Engineering

#### Complex variables 12 Question(s)

$F\left(z\right)$ 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?

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

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

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

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

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

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

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

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

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

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