SATHEE: Complex Numbers 3 Question 10

Complex Numbers 3 Question 10

10.

Match the conditions/expressions in Column I with statement in Column II $(z \neq 0$ is a complex number $)$

	Column I		Column II
A.	$Re (z) = 0$	p.	$Re (z^{2}) = 0$
B.	$\arg (z) = \frac{π}{4}$	q.	$Im (z^{2}) = 0$
		r.	$Re (z^{2}) = Im (z^{2})$

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Answer:

Correct Answer: 10. $A \to q; B \to p$

Solution:

Let $z = a + i b$ .

Given, $Re (z) = 0 \Rightarrow a = 0$

Then, $z = i b \Rightarrow z^{2} = - b^{2}$ or $lm (z^{2}) = 0$

Therefore, $A \to q$

Also, given, $\arg (z) = \frac{π}{4}$ .

Let

$z = r \cos \frac{π}{4} + i \sin \frac{π}{4}$

Then,

$\begin{aligned} z^{2} & = r^{2} \cos^{2} \frac{π}{4} - \sin^{2} \frac{π}{4} + 2 i r^{2} \cos \frac{π}{4} \sin \frac{π}{4} \\ = i r^{2} \sin π / 2 = i r^{2} \end{aligned}$

Therefore, $Re (z^{2}) = 0 \Rightarrow B \to p$ .

$\Rightarrow$ $a = b = 2 - \sqrt{3}$

$[∵ a, b \leftarrow (0, 1)]$

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Complex Numbers 3 Question 10

10.

Answer:

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