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If p^(4)+q^(3)=2(p gt 0, q gt 0), then t...

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 (a) `"^(14)C_(4)` (b) `"^(14)C_(6)` (c) `"^(14)C_(7)` (d) `"^(14)C_(12)`

A

`"^(14)C_(4)`

B

`"^(14)C_(6)`

C

`"^(14)C_(7)`

D

`"^(14)C_(12)`

Text Solution

Verified by Experts

The correct Answer is:
B
`(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)`.
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