UPSC NDA Math Model Paper 1

Given, 3‌sin‌θ−5‌cos‌θ=3
Squaring on both side, we get
(3‌sin‌θ−5‌cos‌θ)2=9
⇒9sin2θ−30‌sin‌θ‌cos‌θ +25cos2θ=9 ......(1)
To find: 5‌sin‌θ+3‌cos‌θ
Consider, 5‌sin‌θ+3‌cos‌θ=x
Squaring on both side, we get
(5‌sin‌θ+3‌cos‌θ)2=x2
⇒25sin2θ+30‌sin‌θ‌cos‌θ +9cos2θ=x2 .......(2)
Adding (1) and(2), we get
⇒34sin2θ+34cos2θ=x2+9
⇒34(sin2θ+cos2θ)=x2+9
⇒34=x2+9
⇒x2=25
⇒x=±5
5‌sin‌θ+3‌cos‌θ=±5
Hence, If 3‌sin‌θ−5‌cos‌θ=3, then the value of 5‌sin‌θ+3‌cos‌θ is ±5