CBSE BoardEnglish MediumSTD 11 ScienceMathsModel Paper 63 Marks
Question
Find the square roots $: 7 - 24i.$
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Answer
Let $\sqrt{7-24 i}=x+ iy$.
Then $\sqrt{7-24 i}=x+i y$
$\Rightarrow 7-24 i=(x+i y)^2$
$\Rightarrow 7-24 i=\left(x^2-y^2\right)+2 i x y$
$\Rightarrow x^2-y^2=7 \ldots \text { (i) }$
and $2 x y=-24 \ldots \text { (ii) }$
Now, $\left(x^2+y^2\right)^2=\left(x^2-y^2\right)^2+4 x^2 y^2$
$\Rightarrow\left(x^2+y^2\right)^2=49+576=625\left[\because x^2+y^2>0\right]$
$\Rightarrow x^2+y^2=25 \ldots \text { (iii) }$
add $(i)$ and $(iii),$
we get $2 x^2=32$
$\Rightarrow x^2=16$
$\Rightarrow x= \pm 4$
put value of $x$ in $(I)$,
we get $y^2=9$
$\Rightarrow y= \pm 3$
From $(ii)$ we observe that $2xy$ is negative.
So, $x$ and $y$ are of opposite signs.
Hence, $\sqrt{7-24 i}= \pm(4-3 i)$
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