If _ x 1 [4] x⁻³ +a [4] x^5 x-1 =-2 , then the coefficient of x in the expansion of ( [4] x⁻³ +a [4] x^5 )^4 is

Step-by-step solution

aligned & _ x 1 [4] x⁻³ +a [4] x^5 x-1 =-2 & _ x 1 [4] x⁻³ (1+a x^2 ) x-1 =-2 aligned This is possible if and if a=1 aligned & ( [4] x⁻³ +a [4] x^5 )^4= ( [4] x⁻³ + [4] x^5 )^4 & = ^4 C₀ ( [4] x⁻³ )^0 ( [4] x^5 )^4+ ^4 C₁ ( [4] x⁻³ )^1 ( [4] x^5 )^3 & + ^4 C₂ ( [4] x⁻³ )^2 ( [4] x^5 )^2+ ^4 C₃ ( [4] x⁻³ )^3 ( [4] x^5 ) & + ^4 C ( [4] x⁻³ )^4 ( [4] x^5 )^0 & The coefficient of x= ^4 C₂= 4 ! 2 ! 2 ! = 4 3 2 1 =6 aligned