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If x(1), x(2) and x(3) are the positive ...

If `x_(1)`, `x_(2)` and `x_(3)` are the positive roots of the equation `x^(3)-6x^(2)+3px-2p=0`, `p inR`, then the value of `sin^(-1)((1)/(x_(1))+(1)/(x_(2)))+cos^(-1)((1)/(x_(2))+(1)/(x_(3)))-tan^(-1)((1)/(x_(3))+(1)/(x_(1)))` is equal to

A

`(pi)/(4)`

B

`(pi)/(2)`

C

`(3pi)/(4)`

D

`pi`

Text Solution

Verified by Experts

The correct Answer is:
A
`(a)` `x^(3)-6x^(2)+3px-2p=0`
`A.M.=(x_(1)+x_(2)+x_(3))/(3)=(6)/(3)=2`
`H.M.=(3)/((1)/(x_(1))+(1)/(x_(2))+(1)/(x_(3)))=(3x_(1)x_(2)x_(3))/(sumx_(1)x_(2))=2`
`:, A.M.=H.M.impliesx_(1)=x_(2)=x_(3)=2`
`sin^(-1)((1)/(x_(1))+(1)/(x_(2)))+cos^(-1)((1)/(x_(2))+(1)/(x_(3)))-tan^(-1)((1)/(x_(3))+(1)/(x_(1)))`
`=(pi)/(2)+0-(pi)/(4)=(pi)/(4)`
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