Maharashtra BoardEnglish MediumSTD 11 ScienceMathsComplex Numbers2 Marks
MCQ
$\sqrt{-8-6 i}=$
A
$1 \pm 3 i$
✓
$\pm(1-3 i)$
C
$\pm(1+3 i)$
D
$\pm(3- i )$
✓
Answer
Correct option: B.
$\pm(1-3 i)$
(B) Let $\sqrt{-8-6 i }=x+ i y$ $\Rightarrow-8-6 i =(x+ i y)^2$ $\Rightarrow x^2-y^2=-8$ and $2 x y=-6$ By solving, we get $x=1, y=-3$ and $x=-1, y=3$ $\therefore x+ i y= \pm(1-3 i )$ Alternate method: Here, $b <0$ Square root of $z = a + ib$ is $\begin{aligned} \sqrt{ a + ib } & = \pm\left[\sqrt{\frac{| z |+ a }{2}}+ i \sqrt{\frac{| z |- a }{2}}\right], \text { for } b >0 \\ & = \pm\left[\sqrt{\frac{| z |+ a }{2}}- i \sqrt{\frac{| z |- a }{2}}\right], \text { for } b <0\end{aligned}$ $\therefore \sqrt{-8-6 i }= \pm\left[\sqrt{\frac{10-8}{2}}- i \sqrt{\frac{10+8}{2}}\right]= \pm(1-3 i )$
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