[Note: sgn z and [z] denotes signum function and greatest integer less than or equal to z respectively.]

Step-by-step solution

(P) Clearly, number of solutions is 2 . (Q) aligned & f(x)= ⁻¹ ( x^2 1+x^2 )= ⁻¹ (1- 1 1+x^2 ) & t=1- 1 1+x^2 t [0,1) & ⁻¹ t (0, 2 ]=(a, b] [a+b]=1 aligned (R) aligned & 12 ( ^2 ( ⁻¹ )+ ^2 ( ⁻¹ ) )=12 ( ^2+ ^2 ) = & 12 (( + )^2-2 ) = & 12 ( 9 4 -2 1 )=3 aligned aligned & (S) ( ⁻¹ x )= ( 2 - ⁻¹ x )= ( ⁻¹ x )=t & f(x)= ( ⁻¹ t+ ⁻¹ t )= 2 =1 & _ x=1 ^4 f ( 3 x 16 )=1+1+1+1=4 aligned