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Let $z$ satisfy $\left| z \right| = 1$ and $z = 1 - \vec z$.
Statement $1$ : $z$ is a real number
Statement $2$ : Principal argument of $z$ is $\frac{\pi }{3}$
Statement $1$ is true Statement $2$ is true;
Statement $2$ is a correct explanation for Statement $1$.
Statement $1$ is false; Statement $2$ is true
Statement $1$ is true, Statement $2$ is false
Statement $1$ is true; Statement $2$ is true;
Statement $2$ is not a correct explanation for Statement $1$
Solution
Let $z=x+i y$, $\bar{z}=x-i y$
Now, $z=1-\bar{z}$
$\Rightarrow \,\, x+i y=1-(x-i y)$
$\Rightarrow \,\, 2 x=1 \Rightarrow x=\frac{1}{2}$
Now, $|z|=1 \Rightarrow x^{2}+y^{2}=1 \Rightarrow y^{2}=i-x^{2}$
$\Rightarrow \,y=\pm \frac{\sqrt{3}}{2}$
Now, $\tan \theta =\frac{y}{x}$ ( $\theta $ is the argument) $=\frac{\sqrt{3}}{2} \div \frac{1}{2}$
( $+\,ve$ since only principal argument)
$=\sqrt{3}$
$\Rightarrow \theta=\tan ^{-1} \sqrt{3}=\frac{\pi}{3}$
Hence, $z$ is not a real number
So, statement $-1$ is false and $2$ is true.
