SATHEE: Complex Numbers 2 Question 41

Complex Numbers 2 Question 41

42.

Find all non-zero complex numbers $z$ satisfying $\bar{z} = i z^{2}$ .

(1996, 2M)

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Answer:

Correct Answer: 42. $(z = i, \pm \frac{\sqrt{3}}{2} - \frac{i}{2})$

Solution:

$\begin{aligned} z & = x + i y . \\ \bar{z} & = i z^{2} \\ (\overset{―}{x + i y}) & = i (x + i y)^{2} \\ x - i y & = i (x^{2} - y^{2} + 2 i x y) \\ x - i y & = - 2 x y + i (x^{2} - y^{2}) \end{aligned}$

NOTE: It is a compound equation, therefore we can generate from it more than one primary equations.

On equating real and imaginary parts, we get

$x = - 2 x y and - y = x^{2} - y^{2}$

$\Rightarrow x + 2 x y = 0 and x^{2} - y^{2} + y = 0$

$\Rightarrow x (1 + 2 y) = 0$

$\Rightarrow x = 0 or y = - 1 / 2$

When $x = 0, x^{2} - y^{2} + y = 0 \Rightarrow 0 - y^{2} + y = 0$

$\Rightarrow y (1 - y) = 0$

$\Rightarrow y = 0$ or $y = 1$

When, $y = - 1 / 2, x^{2} - y^{2} + y = 0$

$\Rightarrow x^{2} - \frac{1}{4} - \frac{1}{2} = 0$

$\Rightarrow x^{2} = \frac{3}{4}$

$\Rightarrow x = \pm \frac{\sqrt{3}}{2}$

Therefore, $z = 0 + i 0, 0 + i; \pm \frac{\sqrt{3}}{2} - \frac{i}{2}$

$\Rightarrow z = i, \pm \frac{\sqrt{3}}{2} - \frac{i}{2}$

Complex Numbers 2 Question 41

42.

Answer:

Engineering Preparation

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Complex Numbers 2 Question 41

42.

Answer:

Engineering Preparation

Medical Preparation

NCERT Books & Solutions

Important Resources

Other Resources

Syllabus

Test Platform

Sample Mock Test

Mentorship

NCERT Exercise Video Solution

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