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)
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