Question

Let $z$ satisfy $\left|z\right|=1andz=1-\overline{z}.$
Statement 1 : $z$ is a real number.
Statement 2 : Principal argument of $z$ is $\frac{\pi }{3}$

• A. Statement 1 is true Statement 2 is true; Statement 2 is a correct explanation for Statement 1.
• B. Statement 1 is false; Statement 2 is true
• C. Statement 1 is true, Statement 2 is false.
• D. Statement 1 is true; Statement 2 is true; Statement 2 is not a correct explanation for Statement 1.

Answer 1

Answer: (B)

Let $z=x+iy,\overline{z}=x-iy$
Now, $z=1-\overline{z}$
$\Rightarrow x+iy=1-(x-iy)$
$\Rightarrow 2x=1$
$\Rightarrow x=\frac{1}{2}$
Now, $\left|z\right|=1$
$\Rightarrow x^2+y^2=1$
$\Rightarrow y^2=1-x^2$
$\Rightarrow y=\pm \frac{\sqrt{3}}{2}$
Now, tan $\theta =\frac{y}{x}$ $(\theta$ is the argument
$=\frac{\sqrt{3}}{2}÷\frac{1}{2}$
(+ve since only principal argument)
$=\sqrt{3}$
$\Rightarrow \theta =ta{n}^{-1}\sqrt{3}=\frac{\pi }{3}$
Hence, $z$ is not a real number
So, statement-1 is false and 2 is true.