Question

Let z satisfy $|z|=1$ and $z=1-\overline{z}.$
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 and statement-2 is correct explanation for statement-1.
• B. Statement-1 is true, statement-2 is true and statement-2 is NOT the correct explanation for statement-1.
• C. Statement-1 is false, statement-2 is true.
• D. Statement-1 is true, statement-2 is false.

Answer 1

Answer: (C)

Statement-1 is false, statement-2 is true.

Let $z=x+iy$

$\Rightarrow \overrightarrow{z}=x+iy$

$\Rightarrow x+iy=1-(x-iy)$

$\Rightarrow x+iy+x-iy=1$

$\Rightarrow x=\frac{1}{2}$

$\Rightarrow \sqrt{x^2+y^2}=1=\mid z\mid$

$\Rightarrow x^2+y^2=1$

$\Rightarrow y=\frac{\sqrt{3}}{2}$

$\tan \theta =\frac{\sqrt{3}}{2}\times 2=\sqrt{3}$

$\Rightarrow \theta =\frac{\pi }{3}$

Statement 1: False

Statement 2: True