If the sum of the coefficients of x^6 and x^3 in the expansion of (2x^2 - a x )^9 is zero, where a > 0 , then the coefficient of x^9 in this expansion is

Step-by-step solution

The general term in the expansion of (2x^2 - a x )^9 is given by T_ r+1 = ⁹C_ r (2x^2)^ 9-r (- a x )^r = ⁹C_ r 2^ 9-r (-a)^r x^ 18-3r For the coefficient of x^6 , we set 18 - 3r = 6 r = 4 . Coefficient of x^6 = ⁹C₄ 2^5 (-a)^4 = 126 32 a^4 For the coefficient of x^3 , we set 18 - 3r = 3 r = 5 . Coefficient of x^3 = ⁹C₅ 2^4 (-a)^5 = -126 16 a^5 Given that the sum of these coefficients is zero: 126 32 a^4 - 126 16 a^5 = 0 Since a > 0 , dividing by 126 16 a^4 gives: 2 - a = 0 a = 2 Now, for the coefficient of x^9 , we