Question

Assertion :If $z=\sqrt{3+4i}+\sqrt{-3+4i}$, then principal arg of z i.e. arg (z) are $\pm \frac{\pi }{4},\pm \frac{3\pi }{4}$ where $\sqrt{-1}=i$. Reason: If z = A + iB, then $\sqrt{z}=\{\begin{array}{cc}\sqrt{\frac{|z|+Re\left(z\right)}{2}}+i\sqrt{\frac{|z|-Re\left(z\right)}{2}}& ,\text{if}B>0\\ \sqrt{\frac{|z|+Re\left(z\right)}{2}-i\sqrt{\frac{|z|-Re\left(z\right)}{2}}}& ,\text{if}B<0\end{array}$

• 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 (R) is false.
• D. (A) is false (R) is true.

Answer 1

The correct option is B Both (A) & (R) are individually true but (R) is not the correct (proper) explanation of (A).

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