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Question

For the expansion (xsinp+x1cosp)10, pR, which among the following satements is (are) CORRECT ?

A
the greatest value of the term independent of x is 10!25(5!)2
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B
the least value of sum of coefficients is zero
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C
the greatest value of sum of coefficients is 32
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D
the least value of the term independent of x occurs when p=(2n+1)π4,nZ
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Solution

The correct option is C the greatest value of sum of coefficients is 32
In the expansion (xsinp+x1cosp)10,
the general term isTr+1= 10Cr(xsinp)10r(x1cosp)r
For this term to be independent of x, 102r=0
r=5.

Independent term in the given expansion is 10C5sin5pcos5p=10C5sin52p32
which is greatest when sin2p=1 i.e., greatest value is 10!25(5!)2
and least when sin2p=1
i.e., p=(4n1)π4,nZ

For sum of coefficients in the expansion of (sinp+cosp)10, by substituting x=1, we get (1+sin2p)5 which is least when sin2p=1
Hence, least sum of coefficients is zero.
Greatest sum of coefficient occurs when sin2p=1.
Hence, greatest sum is 25=32.

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