Question

Assertion (A) : If $z = \sqrt{3 + 4i} + \sqrt{-3 + 4i}$ then principal argument of $z$ i. e., $Arg (z)$ are $ \pm \frac{\pi}{4}, \pm \frac{3\pi}{4}$ where $\sqrt{-1} = i$.
Reason (R) : If $z = A + iB$, then
$\sqrt{z} = \begin{cases} \sqrt{\frac{|z| + Re(z)}{2}} + i \sqrt{\frac{|z| - Re(z)}{2}} & \text{if $ B > 0$} \\[2ex] \sqrt{\frac{|z| + Re(z)}{2}} + i \sqrt{\frac{|z| - Re(z)}{2}} & \text{if $ B < 0$} \end{cases}$

• A. Both (A) & (R) are individually true & (R) is correct explanation of (A).
• B. Both (A) & (R) are individually true but (R) is not the correct (proper) explanation of (A).
• C. (A) is true but (R) is false.
• D. (A) is false but (R) is true.

Answer 1

Answer: (B)

Say $z = \sqrt{z_1} + \sqrt{z_2}$, where $z_1= 3 + 4i$ & $z_2 = -3 + 4i$
$\therefore \sqrt{z_{1}} = \sqrt{3 + 4i} = \pm \left\{ \sqrt{\frac{5 + 3}{2}} + i \sqrt{\frac{5-3}{2}} \right\} $
$= \pm \left( 2 + i\right)$
and $\sqrt{z_{2}} = \sqrt{-3+ 4i} = \pm \left\{\sqrt{\frac{5 - 3}{2}} + i\sqrt{\frac{5 + 3}{2}} \right\}$
$ = \pm \left( 1 + 2i\right)$
$\therefore z = \sqrt{z_1} + \sqrt{z_2} = \pm ( 2 + i) \pm ( 1 + 2i)$
$\Rightarrow z = 3 + 3i, 1 - i, -1 + i, -3 - 3i$
$\Rightarrow $ Principal arg of $z$ are $\frac{\pi}{4}, \frac{-\pi}{4}, \frac{3\pi}{4}, \frac{-3\pi}{4}$
Assertion (A) & Reason (R) are correct and Reason (R) is proper explanation of Assertion (A).