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Let f (x) = log e (sinx ), ( 0 lt x lt p...

Let `f (x) = log _e (sinx ), ( 0 lt x lt pi ) and g(x) = sin ^(-1) (e ^(-x)), (x ge 0)`. If `alpha` is a positive real number such that ` a = ( fog)' ( alpha ) and b = (fog ) ( alpha )`, then

A

`a alpha^(2) + b alpha -a = -2 alpha^(2)`

B

`a alpha^(2) + b alpha + a = 0`

C

`a alpha^(2) - b alpha -a = 1`

D

`a alpha^(2) - b alpha - a = 0`

Text Solution

Verified by Experts

The correct Answer is:
C
`f(x) = log_(e) (sin x)`
`g(x) = sin^(-1) (e^(-x))`
`f(g(x)) = log (sin sin^(-1) e^(-x)) = -x`
`(fog)'= -1 = a`
`(fog) (alpha) = -alpha = b`
`a = -1 and b = -alpha` satisfies option (3).
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