Find the coefficient of ` x^(7)` in the expansion of `(ax^(2) + (1)/(bx))^(11)` . (ii) the coefficient of ` x^(-7)` in the expansion of `(ax + (1)/(bx^2))^(11)` . Also , find the relation between a and b , so that these coefficients are equal .

Explanation

Here , ` T_(R +1) = ""^(11)C_(r) (ax^(2))^(11-r) ((1)/(bx))^(n)`
`= ""^(11)C_(r) . (a^(11-r))/(b^(r)) . x^(22-3r) ...(i)`
Now , in order to find out the coefficient of ` x^(7), 22 - 3r ` must be 7 .
i.e. 22 - 3r = 7
` therefore r = 5 `
Hence , putting r = 5 in Eq. (i) , we get
Required coefficient `= ""^(11)C_(r) .(a^(6))/(b^(5))`
(ii) Here , ` T_(R+1)= ""^(11)C_(R) (ax)^(11-R) ((1)/(bx^(2)))^(R)`
` = ""^(11)C_(n) (a)^(11 -R) (-(1)/(b))^(R) . x^(11-3R)`
` = (-1)^(R). ""^(11)C_(R) . (a^(11-R))/(b^(R)) x^(11-3R) `...(ii)