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
$\sqrt{z}=\{\begin{array}{ll}\sqrt{\frac{|z|+Re(z)}{2}}+i\sqrt{\frac{|z|-Re(z)}{2}}& \text{if B>0}\\ \sqrt{\frac{|z|+Re(z)}{2}}+i\sqrt{\frac{|z|-Re(z)}{2}}& \text{if B<0}\end{array}$

• 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).