This set of Complex Mapping Multiple Choice Questions & Answers (MCQs) focuses on “Transformation w = e^{z}“.

1. What is the region of the w plane into which the rectangular region in the Z plane bounded by the lines x = 0, y = 0, x = 1 and y = 2 is mapped under the transformation w = z + (2-i)?

a) Rectangle

b) Square

c) Triangle

d) Circle

View Answer

Explanation:

Given:w = z + (2-i)

i.e., u + iv = x + iy + (2-i) = (x+2) + i(y-1)

Equating the real and imaginary parts, u = x+2, v = y-1

**Given boundary lines are**

x=0

y=0

x=1

y=2

**Transformed boundary lines are**

u= 0+2 = 2

v = 0-1 = -1

u= 1+2 = 3

v = 2-1 = 1

Hence, the lines x=0 y=0 x=1 and y=2 are mapped into the lines u=2, v=-1, u=3 and v=1 respectively which form a rectangle in the w plane.

2. Which of the following image of the circle |z| = 1 by the transformation w = z + 2 + 4i ?

a) Circle with centre (2, 4) and radius 1

b) Circle with centre (1, 4) and radius 2

c) Circle with centre (1, 4) and radius 3

d) Circle with centre (1, 3) and radius 4

View Answer

Explanation:

Given: w = z + 2 + 4i

i.e., u + iv = x + iy + 2 + 4i

= (x + 2) + i(y+4)

Equating the real and imaginary parts, we get

u = x+2, v = y+4

x = u-2, y = v-4

Given:|z| = 1

i.e., x

^{2}+y

^{2}= 1

(u-2)

^{2}+(v-4)

^{2}= 1

Hence, the circle x

^{2}+y

^{2}= 1 is mapped into the (u-2)

^{2}+(v-4)

^{2}=1 in w plane which is also a circle with centre (2, 4) and radius 1.

3. Which of the following transformations represents both magnification and rotation?

A) w = e^{z}

b) w = c^{z}

c) w = 1^{z}

d) w = z^{2}

View Answer

Explanation:

The transformation w = cz, where c is a complex constant, represents both magnification and rotation.

Let \(z = r\ e^{\text{iθ}}\) \( w = R\ e^{\text{iθ}}\)

and \( c = \alpha e^{\text{iα}}\)

Then Re

^{i∅}\(= \left( \alpha e^{\text{iα}} \right)(re^{\text{iθ}})\)

\( = are^{i(\alpha + \theta)}\)

The transformation equations are

R=ar, ∅=θ+α

Thus, the point (r, θ) in the z-plane is mapped onto the point(ar, θ+α).

This means that the magnitude of the vector representing z is magnified by α=|c| and its direction is rotated through angleα=amp(c).

Hence, the transformation consists of a magnification and a rotation.

4. Which of the following regions of the w-plane into which the triangular region D enclosed by the lines x = 0,y = 0,x+y = 1 is transformation w = 2z?

a) u=0,v=0 and u+v=2

b) u=1, v=0 and u+v=1

c) u=0, v=1 and u+v=2

d)u=1, v=1 and u+v=1

View Answer

Explanation:

Let w = u + iv

z = x+iy

Given: w = 2z

u+iv = 2(x+iy)

u+iv = 2x+i 2y

u=2x →x=u/2, v=2y→y= v/2

In the z plane the line x = 0 is transformed into u = 0 in the w plane.

In the z plane the line y = 0 is transformed into v = 0 in the w plane.

In the z plane the line x+y = 1 is transformed into u+v = 2 in the w plane.

5. Which of the following is the image of the circle |z| = λ under the transformation w=5z?

a) |w| = z

b) |w| = 5z

c) |w| = 3z

d) |w| = 0

View Answer

Explanation:

Given: w = 5z

|w| = 5|z|

i.e., |w| = 5 [|z| = λ] Hence, the image of |z| = λ in z plane is transformed into |w| = 5 λ in the w plane under the transformation w = 5z.

6. Which of the following is the image of the circle |z| = 3 under the transformation w = 2z?

a) |w|= 2

b) |w|= 6

c) |w| = 3

d) |w| = 1

View Answer

Explanation:

Given: w=2z , |z|=3

|w| = (2)|z|

=(2)(3) , since |z| = 3

=6

Hence, the image of z= 3 in the |z| plane is transformed into |w| = 6 in the w plane under the transformation w = 2z.

7. Which of the following is an image of the region y>1 under the transformation w = (1-i)z?

a) u+v>1

b) u+v>0

c) u+v>2

d) u+v>3

View Answer

Explanation:

w = (1-i)z

u + iv = (1 – i)(x + iy)

=x + iy – ix + y

=(x+y)+ i(y-x)

i.e., u = x+y, v = y-x

u + v = 2y u – v = 2x

y = (u + v)/2 x = (u – v)/2

Hence, image region y > 1 is (u+v)/2 > 1 i.e., u + v > 2 in the w plane.

8. Which of the following image represents inversion under the transformation w = 1/z ?

a) Parabola

b) Rectangle

c) Circle

d) Hyperbola

View Answer

Explanation:

The transformation w = 1/z represents inversion with respect to the unit circle |z| = 1, followed by reflection in the real axis.

w = 1/z

(or) z = 1/w

\(x + iy = \frac{1}{u + iv}\)

\(x + iy = \frac{u – iv}{u^{2} + v^{2}}\)

\(x = \frac{u}{u^{2} + v^{2}} \ldots.(1)\)

\(y = \frac{- v}{u^{2} + v^{2}}\ldots.(2)\)

We know that, the general equation of circle in z plane is

x

^{2}+ y

^{2}+ 2gx + 2fy + c = 0 …(3)

Substitute (1) and (2) in (3) we get

\(\frac{u^{2}}{{(u^{2} + v^{2})}^{2}} + \frac{v^{2}}{{(u^{2} + v^{2})}^{2}} + 2g\ \left( \frac{u}{u^{2} + v^{2}} \right) + 2f\left( \frac{- v}{u^{2} + v^{2}} \right) + c = 0\)

i.e., c(u

^{2}+v

^{2})+ 2gu – 2fv + 1 = 0…….(4)

which is the equation of circle in w plane.

9. Which of the following is the image of the circle |z| = 4 under the transformation w = 5z?

a) |w| = 20

b) |w| = 12

c) |w| = 9

d) |w| = 14

View Answer

Explanation:

Given: w = 5z ,|z| = 4

|w| = (5)|z|

=(5)(4) , since |z| = 4

=20

Hence, the image of |z| = 4 in the z plane is transformed into the |w| = 20 in the w plane under the transformation w = 5z.

10. Which of the following is a correct statement about conformal mapping?

a) A mapping w=f(z) is said to be conformal at z = z_{0}, if f'(z_{0}) ≠0

b) A mapping w=f(z) is said to be conformal at z = z_{0}, if f'(z_{0}) = 0

c) A mapping w=f(z) is said to be conformal at z = z_{0}, if f'(z_{0}) ≠1

d) A mapping w=f(z) is said to be conformal at z = z_{0}, if f'(z_{0}) =1

View Answer

Explanation:

In general linear transformation w = az+b where a and b are complex constants, represents magnification, rotation and translation.

A transformation under which angles between every pair of curves through a point are preserved in magnitude, but altered in sense is said to be an isogonal at that point.

A mapping w = f(z) is said to be conformal at z = z

_{0}, if f'(z

_{0}) ≠0.

11. Which of the following is known as the point, at which the mapping w = f(z) is not conformal i.e., f(z) = 0?

a) Critical point

b) Fixed point

c) Conformal point

d) Invariant point

View Answer

Explanation:

The point, at which the mapping w = f(z) is not conformal, i.e., f(z) = 0 is called a critical point of the mapping. If the transformation w = f(z) is conformal at a point, the inverse transformation z = f

^{-1}(w) is also conformal at the corresponding point. The critical points of z = f

^{-1}(w) are given by dz/dw=0.

12. Which of the following points is mapped onto themselves are ‘kept fixed’ under the mapping?

a) Invariant point

b) Critical point

c) Conformal point

d) Transformation point

View Answer

Explanation: Fixed or Invariant point of a mapping w = f(z) are points that are mapped onto themselves, are ‘kept fixed’ under the mapping. Thus they are obtained from w = f(z) = z. The identity mapping w = z has every point as a fixed point. The mapping w = z has infinitely many fixed

13. Which of the following are the critical points of the transformation w^{2} = (z-α)(z-β)?

a) \(\frac{\alpha + \beta}{2},\) α and β

b) 0

c) 1

d) ∞, -∞

View Answer

Explanation:

Given: w

^{2}= (z-α)(z-β) ….(1)

Critical points occur at dw/dz = 0 and dz/dw = 0

Differentiating (1) with respect to z, we get

\(2w\ \frac{\text{dw}}{\text{dz}} = (z – \alpha) + (z – \beta)\)

\( = 2z – (\alpha + \beta)\ldots(2)\)

\(\frac{\text{dw}}{\text{dz}} = \frac{2z – (\alpha + \beta)}{2w}\ldots(3)\)

Case(i) \(\frac{\text{dw}}{\text{dz}} = 0 \rightarrow \ \frac{2z – (\alpha + \beta)}{2w} = 0\)

\(2z – (\alpha + \beta) = 0\)

\(2z = \ \alpha + \beta\)

\(z = \ \frac{\alpha + \beta}{2}\ \)

Case (ii) \( \frac{\text{dz}}{\text{dw}} = 0\ \ \ \rightarrow \ \frac{2w}{2z – (\alpha + \beta)} = 0\ \rightarrow \ \frac{w}{z – \frac{\alpha + \beta}{2}} = 0\)

w = 0 → (z-α)(z-β) = 0

z = α, β

14. Which of the following is the critical point of the

transformation\(w = z^{2} + \frac{1}{z^{2}}\)?

a) ±1, ±i, 0

b) ±∞

c) ± i

d) 0

View Answer

Explanation:

Given: \(w = \ z^{2} + \frac{1}{z^{2}}\ldots(1)\)

Critical points occur at dw/dz = 0 and dz/dw = 0

Differentiating 1 w.r.t z, we get,

\(\frac{\text{dw}}{\text{dz}} = 2z – \frac{2}{z^{2}} – \frac{2z^{4} – 2}{z^{3}}\)

Case(i) \( \frac{\text{dw}}{\text{dz}} = 0 \rightarrow \frac{2z^{4} – 2}{z^{3}} = 0 \rightarrow 2z^{4} – 2 = 0\)

z

^{4}– 1 = 0

z = ±1 ,±i

Case(ii) \( \frac{\text{dz}}{\text{dw}} = 0 \rightarrow \ \frac{z^{3}}{2z^{4} – 2} = 0 \rightarrow \ z^{3} = 0 \rightarrow z = 0\)

The critical points are ±1, ±i, 0

15. Which of the following is the critical point of the transformation w = z + 1/z?

A) 0, ±1

b) ±1

c) ±∞

d) ±i

View Answer

Explanation:

Given: w = z + 1/z …(1)

Critical points occur at dw/dz = 0 and dz/dw = 0

Differentiating (1) w.r.t z, we get

\(\frac{\text{dw}}{\text{dz}} = 1 – \frac{1}{z^{2}} = \ \frac{z^{2} – 1}{z^{2}}\)

Case(i) \(\frac{\text{dw}}{\text{dz}} = 0 \rightarrow \frac{z^{2} – 1}{z^{2}\ } = 0 \rightarrow \ z^{2} – 1 = 0 \rightarrow z = \pm 1 \)

Case(ii) \( \frac{\text{dz}}{\text{dw}} = 0 \rightarrow \frac{z^{2}}{z^{2} – 1} = 0 \rightarrow \ z^{2} = 0 \rightarrow z = 0\)

The critical points are 0, ±1.

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