If the coefficient of x¹⁰ in the expansion of (x^3 + k x )^6 is equal to the coefficient of x⁻¹⁰ in the expansion of (4x - k x^3 )^6 , and k > 0 , then the value of k is

Step-by-step solution

For the expansion of (x^3 + k x )^6 , the general term is given by: T_ r+1 = ⁶C_ r (x^3)^ 6-r ( k x )^r = ⁶C_ r k^r x^ 18-4r To find the coefficient of x¹⁰ , we set the exponent of x to 10 : 18 - 4r = 10 4r = 8 r = 2 So, the coefficient of x¹⁰ is ⁶C₂ k^2 = 15k^2 . For the expansion of (4x - k x^3 )^6 , the general term is: T_ r+1 = ⁶C_ r (4x)^ 6-r (- k x^3 )^r = ⁶C_ r 4^ 6-r (-k)^r x^ 6-4r To find the coefficient of x⁻¹⁰ , we set the exponent of x to -10 : 6 - 4r = -10 4r = 16 r = 4 So, the coefficient of x⁻¹⁰ is ⁶