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  • Let z satisfy  and <img alt="z=1\; -\; \overline{z}" src="http://entrancecorner.oncodecogs.com/gi

Let z satisfy |z|=1 and z=1\; -\; \overline{z}.

Statement I : z is a real number.

Statement II: Principal argument of z is \frac{\pi}{3}.

Option: 1

Statement I is true;Statement II is true;Statement II is a correct explanation for statement I.


Option: 2

Statement I is false; Statement II is true .


Option: 3

Statement I is true;Statement II is false.


Option: 4

Statement I is true;Statement II is true;Statement II is not a correct explanation for satement I.


Answers (1)

best_answer

\\ \text { Let } z=x+i y,\; \bar{z}=x-i y \\ \text { Now, } z=1-\bar{z} \\ \Rightarrow x+i y=1-(x-i y) \\ \Rightarrow 2 x=1 \Rightarrow x=\frac{1}{2} \\\text { Now, }|z|=1 \Rightarrow x^{2}+y^{2}=1 \Rightarrow y^{2}=1-x^{2} \\ \Rightarrow y=\pm \frac{\sqrt{3}}{2}

\\\text { Now, } \tan \theta=\frac{y}{x}\;\;\;\;(\theta \text { is the argument })\\\tan\theta=\frac{\sqrt3/2}{1/2}=\sqrt3

\\\theta=\tan ^{-1} \sqrt{3}=\frac{\pi}{3} \\

Hence, z is not a real number
So, statement-1 is false and 2 is true.

Posted by

vinayak

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