UPSC NDA Math Model Paper 10

Let, √(5+12i)=x+iy
On squaring both sides we get,
5+12i=(x+iy)2=(x2−y2) +i(2xy)
On comparing real and imaginary parts on both sides we get
(x2−y2)=5 and 2xy=12⇒xy=6
(x2+y2)=√(x2−y2)+4x2y2 =√(5)2+4×36=√169=13 (∵(x+y)2=(x−y)2+4xy)
Now, (x2−y2)=5 and (x2+y2)=13
⇒2x2=18
⇒x2=9
⇒x=±3
And, 9−y2=5
⇒y2=4
⇒y=±2
Hence, √(5+12i)=±(3+2i)
Hence, option (2) is correct.