Rajasthan 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 $\sqrt{7-24 i}=x+i y$ $\begin{array}{l}\Rightarrow 7-24 i=(x+i y)^2 \\ \Rightarrow 7-24 i=\left(x^2-y^2\right)+2 i x y \\ \Rightarrow x^2-y^2=7 \ldots \text { (i) } \\ \text { and } 2 x y=-24 \ldots \text { (ii) }\end{array}$ $\begin{array}{l}\text { Now, }\left(x^2+y^2\right)^2=\left(x^2-y^2\right)^2+4 x^2 y^2 \\ \Rightarrow\left(x^2+y^2\right)^2=49+576=625\left[\because x^2+y^2>0\right] \\ \Rightarrow x^2+y^2=25 \ldots \text { (iii) }\end{array}$ add (i) and (iii), we get $\begin{array}{l}2 x^2=32 \\ \Rightarrow x^2=16 \\ \Rightarrow x= \pm 4\end{array}$ 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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