If `p^(4)+q^(3)=2(p gt 0, q gt 0)`, then the maximum value of term independent of `x` in the expansion of `(px^((1)/(12))+qx^(-(1)/(9)))^(14)` is

Options

A. `"^(14)C_(4)`
B. `"^(14)C_(6)`
C. `"^(14)C_(7)`
D. `"^(14)C_(12)`

Answer

B

Explanation

`(b)` `(px^((1)/(12))+qx^(-(1)/(9)))^(14)`
General term `T_(r+1)=14C_(r )(px^((1)/(12)))^(14-r)(qx^((-1)/(9)))^(r )`
`=^(14)C_(r )p^(14-r)q^(r )x^((14-r)/(12)-(r )/(9))`
Term is independent of `r`, then `(14-r)/(12)-(r )/(9)=0`
`:.r=6`
`:.` Term independent of `x` is `"^(14)C_(5)p^(8)q^(6)=^(14)C_(6)(p^(4)q^(3))^(2)`
Now `p^(4)`, `q^(3)` are positive
Using `AM ge GM`
`(p^(4)+q^(3))/(2) ge (p^(4)q^(3))^(1//2)implies(p^(4)q^(3))^(2) le 1`
`implies` Maximum value of term independent of `x` is `"^(14)C_(6)`.