GATE EE 2015 Set 2 | Complex Variable Question 8 | Engineering Mathematics

GATE EE 2015 Set 2

MCQ (Single Correct Answer)

-0.3

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

If $$f(z)$$ is differentiable at $${z_0},$$ then $$g(z)$$ & $$h(z)$$ are also differentiable at $${z_0}.$$

If $$g(z)$$ & $$h(z)$$ are differentiable at $${z_0},$$ then $$f(z)$$ is also differentiable at $${z_0}.$$

If $$f(z)$$ is continuous at $${z_0},$$ then it is differentiable at $${z_0}.$$

If $$f(z)$$ is differentiable at $${z_0},$$ then so are its real and imaginary parts.

GATE EE 2014 Set 3

MCQ (Single Correct Answer)

-0.3

Integration of the complex function $$f\left( z \right) = {{{z^2}} \over {{z^2} - 1}},$$ in the counterclockwise direction, around $$\left| {z - 1} \right| = 1,$$ is

$$ - \pi i$$

$$0$$

$$\pi i$$

$$2\pi i$$

GATE EE 2014 Set 2

MCQ (Single Correct Answer)

-0.3

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

purely imaginary

real and non negative

on the unit circle

equal in real and imaginary parts.

GATE EE 2013

MCQ (Single Correct Answer)

-0.3

Square roots of $$-i,$$ where $$i = \sqrt { - 1} $$ are

$$i, -i$$

$$\eqalign{ & \cos \left( { - {\pi \over 4}} \right) + \sin \left( { - {\pi \over 4}} \right), \cr & \cos \left( {{{3\pi } \over 4}} \right) + i\,\sin \left( {{{3\pi } \over 4}} \right) \cr} $$

$$\eqalign{ & \cos \left( {{\pi \over 4}} \right) + i\sin \left( {{{3\pi } \over 4}} \right), \cr & \cos \left( {{{3\pi } \over 4}} \right) + i\sin \left( {{\pi \over 4}} \right) \cr} $$

$$\eqalign{ & \cos \left( {{{3\pi } \over 4}} \right) + i\sin \left( { - {{3\pi } \over 4}} \right), \cr & \cos \left( { - {{3\pi } \over 4}} \right) + i\sin \left( {{{3\pi } \over 4}} \right) \cr} $$

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