Complex Function ( Limit, Analytic Function)

Q.

If$y=a{e}^{mx}+b{e}^{-mx}$, then $\frac{d^{2}y}{d{x}^2}-m^{2}y$ equals to ?

1. $1$

2. $0$

3. $-1$

4. None of these

Q.

If $z\ne 1$ and $\frac{z^2}{z-1}$ is real, then the point represented by the complex number $z$ lies

1. either on the real axis or on a circle passing through the origin

2. on a circle with centre at the origin

3. either on the real axis or on a circle not passing through the origin

4. on the imaginary axis

Q. The bilinear transformation $w=\frac{z-1}{z+1}$

1. maps the inside of the unit circle in the z-plane to the left half of the w-plane
2. maps the outside the unit circle in the z-plane to the left half of the w-plane
3. maps the inside of the unit circle in the z-plane to right half of the w-plane
4. maps the outside the unit circule in the z-plane to the right half of the w-plane.

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

1. $x^2+y^2+$ constnat
2. $x^2-y^2+$ constant
3. $-x^2+y^2+$ constant
4. $-x^2-y^2+$ constant

Q. What will be the value of $li{m}_{z\to 0}\frac{\overline{z}}{z}$?

1. 0
2. 1
3. $\frac{1}{2}$
4. does not exist

Q. $F\left(z\right)$ is a function of the complex variable $z=x+iy$ given by
$F\left(z\right)=iz+kRe\left(z\right)+iIm\left(z\right)$
For what value of $k$ will $F\left(z\right)$ satisfy the Cauchy-Riemann equations?

1. $0$
2. $1$
3. $-1$
4. $y$

Q. If $x=\sqrt{-1}$, then the value of $x^x$ is

1. $e^{-\pi /2}$
2. $e^{-\pi /2}$
3. $x$
4. $1$

Q. Potential funtion $\varphi$ is givne as $\varphi =x^2-y^2$. What will be the stream funciton $\left(\Psi \right)$ with the condition $\Psi =0$ at $x=y=0$?

1. $2xy$
2. $x^2+y^2$
3. $x^2-y^2$
4. $2x^2{y}^2$

Q. For an analytic function, $f(x+iy)=u(x,y)+iv(x,y),u$ is given by $u=3x^2-3y^2$. Th expression for $v$, considering $K$ to be a constant is

1. $3y^2-3x^2+K$
2. $6x-6y+K$
3. $6y-6x+K$
4. $6xy+K$

Q. The equation $sin\left(z\right)=10$ has

1. no real (or) complex solution
2. exactly two distinct complex solution
3. a unique solution
4. an infinite number of complex solutions

Q. Given $f\left(z\right)=g\left(z\right)+h\left(z\right)$ where $f,g,h$ are complex valued function of a complex variable $z$. Which one of the following statements is TRUE?

1. If $f\left(z\right)$ is differentiable at $z_o$, then $g\left(z\right)$ and $h\left(z\right)$ are also differentiable at $z_0$
2. If $g\left(z\right)$ and $h\left(z\right)$ are differentiable at $z_0$, then $f\left(z\right)$ is also differentiable at $z_o$
3. If $f\left(z\right)$ is continuous at $z_0$ then it is differentiable at $z_0$
4. If $f\left(z\right)$ is diffeentiable at $z_0$, then so are its real and imaginary parts.

Q. Which one of the following is not true for complex number $z_1$ and $z_2$?

1. $\frac{z_1}{z_2}=\frac{z_1{\overline{z}}_1}{|z_2{|}^2}$
2. $|z_1+z_2|\le |z_1|+|z_2|$
3. $|z_1-z_2|\le |z_1|-|z_2|$
4. $|z_1+z_2{|}^2+|z_1-z_2{|}^2=2|z_1{|}^2+2|z_2{|}^2$

Q. If $f\left(z\right)=(x^2+a{y}^2)+ibxy$ is a complex analytic function of $z=x+iy$, where $i=\sqrt{-1}$, then

1. $a=-1,b=-1$
2. $a=-1,b=2$
3. $a=1,b=2$
4. $a=2,b=2$

Q. Let $z^3=\overline{z}$, wherer $z$ is a complex number not equal to zero. Then $z$ is a solution of

1. $z^2=1$
2. $z^3=1$
3. $z^4=1$
4. $z^9=1$

Q. Let $j=\sqrt{-1}$. Then one value of $j^j$ is

1. $\sqrt{3}$
2. $-1$
3. $\frac{\sqrt{1}}{2}$
4. $e^{\frac{-\pi }{2}}$

Q.

Conisder the complex valued funciton $f\left(z\right)=2z^3+b|z{|}^3$ where $z$ is a complex variable. The value of $b$ for which the function $f\left(z\right)$ is analytic is

1. 0

Q. Consider the function $f\left(z\right)=z+z^{\ast }$ where $z$ is a complex variable and $z^{\ast }$ denotes its complex conjugate.Which oneof the following is TRUE?

1. $f\left(z\right)$ is both continuous and analytic
2. $f\left(z\right)$ is a continuous but not analytic
3. $f\left(z\right)$ is not continuous but is analytic
4. $f\left(z\right)$ is neither continuous nor analytic

Q. If a complex number $\omega$ satisfies the equation ${\omega }^3=1$ then value of $1+\omega +\frac{1}{\omega }$ is

1. $0$
2. $1$
3. $2$
4. $4$

Q. Let $S$ be the set of points in the complex plane corresponding to the unit circle. That is $S=[z:|z|=1]$. Consider the function $f\left(z\right)=z{z}^{\ast }$ wherer $z^{\ast }$ denotes wth complex conjugate of $z$. The $f\left(z\right)$ maps S to which one of the following in the complex plane

1. unit circle
2. horizontal axis line segment from orgin to ($1,0$)
3. the point $(1,0)$
4. the entire horizontal axis

Q. One of the roots of equation $x^3=j$, where $j$ is the positve square roots of $-1$ is

1. $j$
2. $\frac{\sqrt{3}}{2}+\frac{j}{2}$
3. $\frac{\sqrt{3}}{2}-\frac{j}{2}$
4. $-\frac{\sqrt{3}}{2}-\frac{j}{2}$

Q. A complex variable $z=x+y\left(0.1\right)$ has its real part $x$ varying in the range $-\infty$ to $\infty$. Which one of the following is the locus (show in thick lines) of $\frac{1}{z}$ in the complex palne?

1. 
2. 
3. 
4. 

Q. For a complex number $z$, ${lim}_{z\to i}\frac{z^2+1}{z^3+2z-i(z^2+2)}$ is

1. $-2i$
2. $-i$
3. $i$
4. $2i$

Q. The real part of an analytic funciton $f\left(z\right)$ where $z=x+jy$ is givne by $e^{-y}cos\left(x\right)$. THe imaginary part of $f\left(x\right)$ is

1. $e^{y}cos\left(x\right)$
2. $e^{-y}sin\left(x\right)$
3. $-e^{y}sin\left(x\right)$
4. $-e^{-y}sin\left(x\right)$

Q. If $f(x+iy)=x^3-3x{y}^2+i\varphi (x,y)$ where $i=\sqrt{-1}$ and $f(x+iy)$ is an analytic fucntion then $\varphi (x,y)$ is

1. $y^3-3x^{2}y$
2. $3x^{2}y-y^3$
3. $x^4-4x^{2}y$
4. $xy-y^2$

Q. A point $z$ has been potted in the complex plane, as shown in figure below



$\frac{1}{Z}$ lies in the curve

1. 
2. 
3. 
4. 

Q. An analytic function $f\left(z\right)$ of complex variable $z=x+iy$ may be written as $f\left(z\right)=u(x,y)+iv(x,y)$. Then $u(x,y)$ and $v(x,y)$ must satisfy.

1. $\frac{\partial u}{\partial x}=-\frac{\partial v}{\partial y}$ and $\frac{\partial u}{\partial y}=\frac{\partial v}{\partial x}$
2. $\frac{\partial u}{\partial x}=\frac{\partial v}{\partial y}$ and $\frac{\partial u}{\partial y}=-\frac{\partial v}{\partial x}$
3. $\frac{\partial u}{\partial x}=-\frac{\partial v}{\partial y}$ and $\frac{\partial u}{\partial y}=-\frac{\partial v}{\partial x}$
4. $\frac{\partial u}{\partial x}=\frac{\partial v}{\partial y}$ and $\frac{\partial u}{\partial y}=\frac{\partial v}{\partial x}$

Q. $e^z$ is a periodic with a period of

1. $2\pi$
2. $2\pi i$
3. $\pi$
4. $i\pi$

Q. A function $f$ of the complex variable $z=x+iy$, is given as $f(x,y)=u(x,y)+iv(x,y),whereu(x,y)=2kxy$ and $v(x,y)=x^2-y^2$. The value of $k$, for which the function is analytic, is

1. -1

Q. If $Z=x+jy$ where $x,y$ are real when the value of $|e^{iz}|$ is

1. $1$
2. $e^{\sqrt{x^2+y^2}}$
3. $e^y$
4. $e^{-y}$

Q. All the values of the multi-valued complex funciton $1^i$ where $i=\sqrt{-1}$, are

1. purely umaginary
2. real and non-negative
3. on the unit circle
4. equal in real and imaginary parts