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For the expansion (x sin p + x^(-1)p)^(10), (p in R), |
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Answer» the greatest value of the term independent of x is `10!//2^(5)(5!)^(2)` The general term in the expansion is `T_(R+1) = .^(10)C_(r)(xsin p)^(10-r)(x^(-1) COSP)^(r)` For the term independent of `x`, we have `10-2r = 0` or `r = 5`, Hence, the independent term is `.^(10)C_(5)sin^(5)p cos^(5)p -.^(10)C_(5)(sin^(5)2p)/(32)` which is the greatest when sin `2p= 1`. The least value of `.^(10)C_(5)(sin^(5)2p)/(32)` is `-(10!)/(2^(5)(5!)^(2))` when `sin 2p = - 1` or `p = (4n-1)(pi)/(4), n in Z`. Sum of coefficient is `(sin p + cosp)^(10)`, when `x =1` or `(1+sin2p)^(5)`, which is least when `sin 2p = - 1`. Hence, least sum of COEFFICIENTS is zero. Greatest sum of coefficient occurs when `sin 2 p = 1`, Hence, greatest sum is `2^(5) = 32`. |
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