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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 |
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Answer» `"^(14)C_(4)` 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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