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If cos^(-1) alpha + cos^(-1) beta + cos^...

If `cos^(-1) alpha + cos^(-1) beta + cos^(-1) gamma = 3pi`, then `alpha (beta + gamma) + beta(gamma + alpha) + gamma(alpha + beta)` equal to

A

`0`

B

`1`

C

`6`

D

`12`

Text Solution

Verified by Experts

The correct Answer is:
C
We have, `cos^(-1) alpha + cos^(-1) beta + cos^(-1) gamma = 3 pi`
We know that `0 le cos^(-1) x le pi`
`rArr cos^(-1) alpha + cos^(-1) beta + cos^(-1) gamma = 3 pi`
If and only if, `cos^(-1) alpha + cos^(-1) beta + cos^(-1) gamma = pi`
`rArr cos pi = alpha = beta = gamma`
`rArr -1 = alpha = beta = gamma`
`rArr alpha = beta = gamma = - 1`
`:. alpha(beta + gamma) + beta(gamma + alpha) + gamma (alpha + beta)`
`= - 1 (-1-1)-1(-1-1)-1(-1-1)`
`= 2+2+2 = 6`
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