Two forces P and Q of magnitude 2F and 3F. . .

Question : Two forces P and Q of magnitude 2F and 3F respectively are at angle with each other. If the force Q is doubled, then their resultant also gets doubled. The angle is _____

(a) 30°

(b) 60°

(c) 90°

(d) 120°

Doubt by Mouli 
[JEE Mains 2019]

Solution :
 
Case I

P=2F
Q=3F
R=R
We know, 
R²=P²+Q²+2PQcosθ
R²=(2F)²+(3F)²+2(2F)(3F)cosθ
R²=4F²+9F²+12F²cosθ — (1) 

Case II 

P=2F
Q=2(3F) = 6F
R=2R
We know, 
R²=P²+Q²+2PQcosθ
(2R)²=(2F)²+(6F)²+2(2F)(6F)cosθ
4R²=4F²+36F²+24F²cosθ — (2) 
4[4F²+9F²+12F²cosθ]=4F²+36F²+24F²cosθ 
Substituting the value of R² from equation (1) 

16F²+36F²+48F²cosθ=4F²+36F²+24F²cosθ
52F²+48F²cosθ=40F²+24F²cosθ
48F²cosθ-24F²cosθ=40F²-52F²
24F²cosθ=-12F²
cosθ=-12F²/24F²
cosθ=-1/2
cosθ=cos120°
θ=120°

Hence, (d) 120°, would be the correct option.