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. $\operatorname{Re}(z)=0$ p. $\operatorname{Re}\left(z^{2}\right)=0$
B. $\arg (z)=\frac{\pi}{4}$ q. $\operatorname{Im}\left(z^{2}\right)=0$
r. $\operatorname{Re}\left(z^{2}\right)=\operatorname{Im}\left(z^{2}\right)$
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Answer:

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

Solution:

  1. Let $z=a+i b$.

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

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

Therefore, $A \rightarrow q$

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

Let

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

Then,

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

Therefore, $\operatorname{Re}\left(z^{2}\right)=0 \Rightarrow B \rightarrow p$.

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

$[\because a, b \leftarrow(0,1)]$



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