The value of {5+12i}, Where i={-1}

Correct Answer is : ± (3 + 2i)

Correct Answer is : ± (3 + 2i)

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UPSC NDA Math Model Paper 10

Question : 14 of 120

Marks: +1, -0

The value of

\sqrt{5+12i}

i=\sqrt{-1}

Solution:

Let,

\sqrt{5+12i}=x+iy

On squaring both sides we get,

5+12i=(x+iy)^{2}=(x^{2}-y^{2})

+i(2xy)

On comparing real and imaginary parts on both sides we get

(x^{2}-y^{2})=5

and

2xy=12 \Rightarrow xy=6

(x^{2}+y^{2})=\sqrt{(x^{2}-y^{2})+4x^{2}y^{2}}

=\sqrt{(5)^{2}+4 \times 36}=\sqrt{169}=13

(\because (x+y)^{2}=(x-y)^{2}+4xy)

Now,

(x^{2}-y^{2})=5

and

(x^{2}+y^{2})=13

\Rightarrow 2x^{2}=18

\Rightarrow x^{2}=9

\Rightarrow x=\pm 3

And,

9-y^{2}=5

\Rightarrow y^{2}=4

\Rightarrow y=\pm 2

Hence,

\sqrt{5+12i}=\pm(3+2i)

Hence, option (2) is correct.

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