If the coefficient of `x^(7)` in `[ax^(2) + (1/(bx))]^(11)` equals the coefficient of `x^(-7)` in `[ax - (1/(bx^(2)))]^(11)`, then `a` and `b` satisfy the relation

Options

A. `a+b=1`
B. `a-b=1`
C. `ab=1`
D. `a/b=1`

Answer

C

Explanation

For `(ax^(2)+(1/(bx)))^(11)`.
`T_(r+1)=.^(13)C_(r)(ax^(2))^(11-r)(1/(bx))^(r) = .^(11)C_(r ) a^(11-r) (1)/(b^(r)) x^(22-3r)`
For `x^(7)`,
`22-3r = 7`
or `3r = 15`
or `r = 5`
`rArr T_(6) = .^(11)C_(5)a^(6)(1)/(b^(5)) x^(7)`
`rArr` Coefficient of `x^(7)` is `.^(11)C_(5) (a^(6))/(b^(5))`
Similarly, coefficient of `x^(-7)`in `(ax- (1/(bx^(2))))^(11)` is `.^(11)C_(6)(a^(5))/(b^(6))`.
Given that `.^(11)C_(5) (a^(6))/(b^(5)) = .^(11)C_(6) (a^(5))/(b_(6))`
`rArr a= 1/b`
or `ab = 1`