Quiz about Complex Analysis (MC2061)

Question 1

Statement: The function $$f\left(z\right)=\overline{z}$$ is nowhere differentiable. Choose the correct reason.

Accepted Answer

$$\lim_{h\rightarrow0}\ \overline{\frac{h}{h}}$$ does not exist

Answer

the function $$\overline{z}$$ is not differentiable

Answer

left or right limit of the function does not exist

Answer

both left and right limit exists but they are not equal

Question 2

Choose the correct statement(s)

(i) If $$f\left(z\right)=u+iv$$ is differentiable at a point, then $$u$$ and $$v$$ satisfies the C.R equation.

(ii) If $$u$$ and $$v$$ does not satisfies the C.R equation at any point, then $$f\left(z\right)=u+iv$$ is not differentiable.

(iii) If $$u$$ and $$v$$ satisfies the C.R equation at a point, then $$f\left(z\right)=u+iv$$ is differentiable at that point.

(iv) If the first order partial derivatives of $$u$$ and $$v$$ are continuous at a point, then $$f\left(z\right)=u+iv$$ is differentiable at that point.

Accepted Answer

(i) and (ii)

Answer

(i) and (iii)

Answer

(iii) and (iv)

Answer

(i) only

Question 3

Say True or False: If a function is differentiable at a point, then it is continuous at that point.

Accepted Answer

True

Answer

False

Question 4

Which of the following is/ are nowhere differentiable?

Accepted Answer

$$f\left(z\right)=Re\ z$$

Accepted Answer

$$f\left(z\right)=\left|z\right|$$

Answer

$$f\left(z\right)=z^2$$

Answer

$$f\left(z\right)=iz+2$$

Question 5

A function which is analytic at every point of the complex plane is called .............. function

Accepted Answer

integral

Accepted Answer

entire

Answer

harmonic

Answer

extended

Question 6

Say True or False: An analytic function with its derivative zero at every point of the domain is constant

Accepted Answer

False

Answer

True

Question 7

Which of the following is/are correct?

(i) An analytic function in a region with constant modulus is constant.

(ii) An analytic function with arg $$f\left(z\right)$$ constant is constant.

(iii) An analytic function whose real part is constant is constant.

(iv) If $$f\left(z\right)$$ and $$\overline{f\left(z\right)}$$ are analytic, then $$f\left(z\right)$$ is constant.

Accepted Answer

All the above

Answer

(i) and (iii)

Answer

(i), (ii) and (iii)

Answer

(i), (iii) and (iv)

Question 8

If $$v$$ is a harmonic conjugate of $$u$$ , then what is the harmonic conjugate of $$-v$$ ?

Accepted Answer

$$u$$

Answer

$$-u$$

Answer

$$v$$

Answer

$$-v$$

Question 9

Which of the following given below is/are elementary transformations?

Accepted Answer

Translation

Accepted Answer

Rotation

Accepted Answer

Magnification or Contraction

Accepted Answer

Inversion

Question 10

Choose the correct answers

Accepted Answer

In the translation transformation, there is no changes in the shape, size and orientation of two regions

Accepted Answer

In the rotation transformation, there is no changes in the shape and size of two regions but changes occur in the orientation

Answer

In the contraction transformation, there is no changes in the shape and size of two regions but changes occur in the orientation

Answer

In the magnification transformation, there is no changes in the shape and size of two regions but changes occur in the orientation

Question 11

Under the inverse transformation, the points inside the circle maps to the points ....... the circle.

Accepted Answer

outside

Answer

boundary

Answer

inside

Question 12

Say True or False: Any bilinear transformation can be expressed a s a product of translation, rotation, magnification or contraction and inversion.

Accepted Answer

True

Answer

False

Question 13

Four points $$z_1,\ z_2,\ z_3,\ z_4$$ are said to be collinear if

Accepted Answer

cross ratio is real

Answer

cross ratio is imaginary

Answer

it is differentiable

Answer

satisfies the C.R equations

Question 14

Which of the following is/are true?

Accepted Answer

under the inversion transformation, circle not passing through the origin is mapped into circle not passing through the origin

Accepted Answer

under the inversion transformation, circle passing through the origin is mapped into straight line not passing through the origin

Accepted Answer

under the inversion transformation, straight line passing through the origin is mapped into a straight line passing through the origin

Answer

under the inversion transformation, straight line not passing through the origin is mapped into a circle not passing through the origin

Question 15

Say True or False: Any bilinear transformation preserves cross ratio.

Accepted Answer

True

Answer

False

Complex Analysis (MC2061)

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