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If log(cosx) sinx>=2 and x in [0,3pi] th...

If `log_(cosx) sinx>=2` and `x in [0,3pi]` then `sinx` lies in the interval

A

`[(sqrt(5)-1)/(2), 1]`

B

`(0, (sqrt(5)-1)/(2)]`

C

`[0, 1//2]`

D

none of these

Text Solution

Verified by Experts

The correct Answer is:
B
We observe that `"log"_("cos"x)sin x` is defined for ` x in (0, pi//2) cup (2pi, 5pi//2)`.
Now,
`"log_("cos"x) "sin"x ge 2`
`rArr "sin" x le ("cos" x)^(2) " "[because 0 lt "cos" x lt 1]`
`rArr "sin"^(2) x + "sin"x -1 lt 0`
`rArr ("sin" x + (1)/(2))^(2) - (5)/(4) le 0`
`rArr ("sin" x + (1)/(2) - (sqrt(5))/(2))("sin" x + (1)/(2) + (sqrt(5))/(2)) le 0`
`rArr "sin" x + (1)/(2) -(sqrt(5))/(2) le 0 " " [because "sin" x + (1)/(2) + (sqrt(5))/(2) gt 0]`
`rArr "sin" x le (sqrt(5)-1)/(2) rArr "sin" x in (0, (sqrt(5)-1)/(2)]`
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