### Engineering Mathematics - Complex Variables

#### Question - 1

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

• A $\frac { 1 }{ 4! }$
• B $\frac { { 2 }^{ 4 } }{ 4! }$
• C $\frac { { e }^{ 2 } }{ 4! }$
• D $\frac { { e }^{ 4 } }{ 4! }$

#### Question - 2

if $\phi (x,y)and\psi (x,y)$ are functions with continuous second derivative, then
$\phi (x,y)+\psi (x,y)$
can be ecpressed as an analytic funtion of $x+iy=(i=\sqrt { -1) }$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 $\frac { { \partial }^{ 2 }\phi }{ { \partial x }^{ 2 } } +\frac { { \partial }^{ 2 }\phi }{ { \partial y }^{ 2 } } =\frac { { \partial }^{ 2 }\psi }{ { \partial w }^{ 2 } } +\frac { \partial ^{ 2 }\psi }{ { \partial y }^{ 2 } } =1$
• D $\frac { \partial \phi }{ \partial x } +\frac { \partial \phi }{ \partial y } =\frac { \partial \psi }{ \partial x } +\frac { \partial \psi }{ \partial y } =0$

#### Question - 3

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

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

#### Question - 4

The $\lim _{ x\rightarrow 0 }{ \frac { sin\left[ \frac { 2 }{ 3 } x \right] }{ x } }$ is

• A $\frac { 2 }{ 3 }$
• B 1
• C $\frac { 1 }{ 4 }$
• D $\frac { 1 }{ 2 }$

#### Question - 5

The analytic function $f(z)=\frac { z-1 }{ { z }^{ 2 }+1 }$ has singularities at

• A 1 and -1
• B 1 and 1
• C 1 and -i
• D i and -i

#### Question - 6

For an analytic function f(x+iy)=$\mu$ (x,y) + iv(x,y), $\mu$ is given by $\mu$ = 3x2-3y2 expression for v considering K to be at constant is

• A 3y2-3x2+K
• B 6x-6y+K
• C 6x+6y+K
• D 6xy+K

#### Question - 7

The value of the function f(x)=$\lim _{ x\rightarrow 0 }{ \frac { { x }^{ 3 }+{ x }^{ 2 } }{ { 2x }^{ 3 }-{ 7x }^{ 2 } } }$  is

• A 0
• B $-\frac { 1 }{ 7 }$
• C $\frac { 1 }{ 7 }$
• D $\infty$

#### Question - 8

$\lim _{ x\rightarrow \infty }{ \frac { x-sinx }{ x+cosx } }$ equals to

• A 1
• B -1
• C $\infty$
• D -$\infty$

#### Question - 9

The value of the $\oint { \frac { -3z+4 }{ { z }^{ 2 }+4z+5 } } dz$ where C is the circle |z|=1 is given by

• A 0
• B $\frac { 1 }{ 10 }$
• C $\frac { 4 }{ 5 }$
• D 1

#### Question - 10

Which of the following functions would have only odd poweers of x in its Taylor series expansion about the point x=0?

• A sin (x3)
• B sin (x3)
• C cos (x3)
• D cos (x2)