Cauchy integral Formula and problems

Q.

The domain of the function $f\left(x\right)={\sin }^{-1}\left[\frac{\left|x\right|+5}{x^2+1}\right]$ is $(-\infty ,-a]\cup [a,\infty )$. Then a is equal to:

1. $\frac{\sqrt{17-1}}{2}$

2. $\frac{\sqrt{17}}{2}$

3. $\frac{1+\sqrt{17}}{2}$

4. $\frac{\sqrt{17}}{2}+1$

Q. The value of ${\int }_c\frac{z^2}{z^4-1}dz$, using Cauchy's integral formula, around the circle $|z+1|=1$ where $z=x+iy$ is

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

Q. The contour integral ${\int }_c{e}^{1/z}dz$ with $C$ as the counter clock wise unit circle in the z-plnae is equal to

1. $0$
2. $2\pi$
3. $2\pi \sqrt{-1}$
4. $\alpha$

Q.

Find the range of function $f:[0,1]\to R,f\left(x\right)=x^3-x^2+4x+2{\sin }^{-1}\left(x\right)$.

1. $[-(\mathrm{\pi }+2),0]$

2. $[0,(4+\mathrm{\pi }\left)\right]$

3. $[2,3]$

4. $[0,(2+\mathrm{\pi }\left)\right]$

Q.

Let $S_1,S_2$ and $S_3$ be three sets defined as $S_1=\left\{z\in C:|z–1|\le \sqrt{2}\right\},S_2=\left\{z\in C:Re\left(\right(1–i\left)z\right)\ge 1\right\},S_3=\left\{z\in C:Im\left(z\right)\le 1\right\}$. Then the set $S_1\cap S_2\cap S_3$

1. Has infinitely many elements

2. Has exactly two elements

3. Has exactly three elements

4. Is a singleton

Q.

The area bounded by the curves $y^2=8x$and $x^2=8y$ is

1. $64$ square units

2. $\frac{64}{3}$ square units

3. $\frac{8}{3}$ square units

4. None of these

Q. The value of the integral ${\int }_{\infty }^{-\infty }\frac{sinx}{x^2+2x+2}dx$ evaluated using contour integration and the residue theorem is

1. $-\pi sin\left(1\right)/e$
2. $-\pi cos\left(1\right)/e$
3. $sin\left(1\right)/e$
4. $cos\left(1\right)/e$

Q.

The domain of the function $f\left(x\right)=\frac{{\sin }^{-1}\left(x-3\right)}{\sqrt{9-x^2}}$ is

1. $\left[2,3\right]$

2. $[2,3)$

3. $\left[1,2\right]$

4. $[1,2)$

Q. The value of the integral ${\oint }_C\frac{-3z+4}{(z^2+4z+5)}dz$ wherer $c$ is the circle $|Z|=1$ is given by

1. $0$
2. $1/10$
3. $4/5$
4. $1$

Q.

Let $S=\left\{t\in R:f\left(x\right)=|x-\pi |(e^{\left|x\right|}-1)\sin x\right\}$ is not differentiable at $t$ . Then the set $S$ is equal to:

1. $\left\{\mathrm{\pi }\right\}$

2. $\left\{0,\pi \right\}$

3. an empty set

4. $\left\{0\right\}$

Q.

Let the function :$R\to R$ be defined by $f\left(x\right)=x^3-x^2+(x-1)\sin x$ and let: $R\to R$ be an arbitrary function. Let$g:R\to R$ be the product function defined by $fg\left(x\right)=f\left(x\right)g\left(x\right)$. Then which of the following statements is/are TRUE?

1. If $g$ is continuous at $x=1$, then $fg$ is differentiable at $x=1$

2. If $g$ is differentiable at $=1$, then $g$ is continuous at $=1$

3. If $g$ is differentiable at $x=1$, then $fg$ is differentiable at $x=1$

4. If $fg$ is differentiable at $=1$, then $g$ is differentiable at $=1$

Q.

If $C$ denotes the counterclockwise unit circle, the value of the contour integral $\frac{1}{2\pi j}{\oint }_{C}Rezdz$ is

1. 0.5

Q.

The value of the integral $\frac{1}{2\pi j}{\oint }_C\frac{z^2+1}{z^2-1}dz$ where $z$ is a complex number and $C$ is a unit circle with center at $1+0j$ in the complex plane is

1. 1

Q. Evaluate $\int \frac{dz}{zsinz},$ which $c$ is $x^2+y^2=1$

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

Q. Evaluate ${\int }_c\frac{1}{(z-1{)}^3.(z-3)}dz$
where $c$ is the rectangular region defined by $z=0,x=4,y=-1$ and $y=1$

1. $1$
2. $0$
3. $\frac{\pi }{2}i$
4. $\pi (3+2i)$

Q.

The derivative of ${\tan }^{-1}\left(\frac{\sqrt{x+1}-\sqrt{x-1}}{\sqrt{x+1}+\sqrt{x-1}}\right)$ is

1. $\sqrt{\left(1-{\mathrm{x}}^2\right)}$

2. $\frac{1}{\sqrt{\left(1-{\mathrm{x}}^2\right)}}$

3. $\frac{1}{2\sqrt{\left(1-{\mathrm{x}}^2\right)}}$

4. $\mathrm{x}$

Q.

In the following integral, th contour $C$ encloses the points $2\pi j$ and $-2\pi j$.
$-\frac{1}{2\pi }{\oint }_C\frac{sinz}{(z-2\pi j{)}^3}dz$
The value of the integral is

1. 133.87

Q.

The angle between the curves $y=2x$ and $x^3+y^3=3xy$ at the point other than the origin is

Q. If $z$ is a complex variable, the value of ${\int }_5^{3i}\frac{dz}{z}$ is

1. $-0.511-1.57i$
2. $-0.511+1.57i$
3. $0.511-1.57i$
4. $0.511+1.57i$

Q. The value of ${\oint }_r\frac{3z-5}{(z-1)(z-2)}dz$ along a closed path $\Gamma$ is equal ot $4\pi i$, where $z=x+iy$ and $i=\sqrt{-1}$.
The correct $\Gamma$ is

1. 
2. 
3. 
4. 

Q. An integra $I$ over a counterclockwise circle $C$ is given by $I={\oint }_C\frac{z^2-1}{z^2+1}{e}^{z}dz$
If $C$ is defined as $|z|=3$, then the value of $I$ is

1. $-\pi isin\left(1\right)$
2. $-2\pi isin\left(1\right)$
3. $-3\pi isin\left(1\right)$
4. $-4\pi isin\left(1\right)$

Q.

Let $f:R\to R$ be defined as:

$f\left(x\right)=\left\{\begin{array}{l}2\sin \left(\frac{-\pi x}{2}\right)ifx<-1\\ \left|a{x}^2+x+b\right|if-1\le x\le 1\\ \sin \left(\pi x\right)ifx>1\end{array}\right.$

If $f\left(x\right)$ is continuous on $R$, then $a+b$equals to:

1. $3$

2. $-1$

3. $-3$

4. $1$

Q. The value of the integral ${\oint }_C\frac{z+1}{z^2-4}dz$ in counter clockwise direction around a circle $C$ of radius $1$ with center at the point $z=-2$

1. $\frac{\pi i}{2}$
2. $2\pi i$
3. $-\frac{\pi i}{2}$
4. $-2\pi i$

Q. The value of ${\oint }_C\frac{sinz}{z}dz,$ where the contour of the integration is a simple closed curve around the origin is

1. $0$
2. $2\pi j$
3. $\alpha$
4. $\frac{1}{2\pi j}$

Q.

The contour $C$ given below is on the complex plane $z=x+iy$, where $j=\sqrt{-1}$.



The value of the integral $\frac{1}{\pi j}{\oint }_C\frac{dz}{z^2-1}$ is

1. 2

Q. Given $f\left(z\right)=\frac{1}{z+1}-\frac{2}{z+3}$. If $C$ is a counterclockwise path in the $z-$ plane such that $|z+1|=1$, the value of $\frac{1}{2\pi j}{\oint }_{C}f\left(z\right)dz$ is

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

Q.

The coefficient of correlation between two variable $x$ and $y$ is given by

1. $r=[{{\sigma }_x}^2+{{\sigma }_y}^2+{{\sigma }_{x-y}}^2]/[2{\sigma }_x{\sigma }_Y]$

2. $r=[{{\sigma }_x}^2+{{\sigma }_y}^2-{{\sigma }_{x-y}}^2]/[2{\sigma }_x{\sigma }_Y]$

3. $r=[{{\sigma }_x}^2+{{\sigma }_y}^2+{{\sigma }_{x-y}}^2]/[{\sigma }_x{\sigma }_Y]$

4. $r=[{{\sigma }_x}^2+{{\sigma }_y}^2-{{\sigma }_{x-y}}^2]/[{\sigma }_x{\sigma }_Y]$

Q. The vlaue of ${\oint }_C\frac{1}{z^2}dz$ where the contour is the unit circle traversed clockwise is

1. $-2\pi i$
2. $0$
3. $-2\pi i$
4. $4\pi i$

Q. Consider likely applicability of Cauchy's integral theorem to evaluate the following integral counter clockwise arround the unit circle $C,I=\oint seczdz,z$ beign a complex variable. The value of I will be

1. $I=0$, singularities set $=\varphi$
2. $I=0$, singularties set $=\{\pm \frac{2n+1}{2}\pi ;n=0,1,2\}$
3. $I=\frac{\pi }{2}$, singulariteis set $=\pm n\pi ;n=0,1,\mathrm{2...}$
4. None of the abvoe

Q. $C$ is a closed path in the $z-$plane given by $|z|=3$. The value of the integral ${\oint }_C\left(\frac{z^2-z+4j}{z+2j}\right)dz$ is

1. $-4\pi (1+j2)$
2. $4\pi (3+j2)$
3. $-4\pi (3+j2)$
4. $4\pi (1-j2)$