Question

Let \( z \) satisfy \( |z| = 1, \ z = 1 - \overline{z} \text{ and } \operatorname{Im}(z)>0 \)

Then consider:

Statement-I: \( z \) is a real number

Statement-II: Principal argument of \( z \) is \( \dfrac{\pi}{3} \)

Then:

• A. Statement-I is true, Statement-II is true and Statement-II is a correct explanation of Statement-I
• B. Statement-I is true, Statement-II is true, but Statement-II is not a correct explanation of Statement-I
• C. Statement-I is false, Statement-II is true
• D. Statement-I is true, Statement-II is false

Hint:

For complex number problems, always use the identities: \( z + \overline{z} = 2\operatorname{Re}(z) \) and \( z\overline{z} = |z|^2 \). These simplify many expressions.

Answer 1

The Correct Option is C

Given: \( |z| = 1 \), so \( z \) lies on the unit circle.
Also, \( z = 1 - \overline{z} \Rightarrow z + \overline{z} = 1 \Rightarrow 2\operatorname{Re}(z) = 1 \Rightarrow \operatorname{Re}(z) = \frac{1}{2} \)
Since \( |z| = 1 \), and \( \operatorname{Re}(z) = \frac{1}{2} \), we find: \[ |z|^2 = \left(\frac{1}{2}\right)^2 + (\operatorname{Im}(z))^2 = 1 \Rightarrow \operatorname{Im}(z)^2 = \frac{3}{4} \Rightarrow \operatorname{Im}(z) = \frac{\sqrt{3}}{2} \] So: \[ z = \frac{1}{2} + i\frac{\sqrt{3}}{2} \Rightarrow \text{arg}(z) = \tan^{-1}\left(\frac{\sqrt{3}/2}{1/2}\right) = \tan^{-1}(\sqrt{3}) = \frac{\pi}{3} \]
Therefore:
Statement-I is false (since \( \operatorname{Im}(z)>0 \Rightarrow z \) is not real)
Statement-II is true \[ \boxed{\text{Correct Option: (3)}} \] % Tip