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If (y+3)/(2y+5)=sin^2x+2cosx+1, then the...

If `(y+3)/(2y+5)=sin^2x+2cosx+1,` then the value of `y` lies in the interval `(-oo,-8/3)` (b) `(-(12)/5,oo)` `(-8/3,-(12)/5)` (d) `(-8/3,oo)`

A

`(-oo,-8/3]`

B

`[-12/5,oo)`

C

`[-8/3,-12/5]`

D

`[-8/3,oo)`

Text Solution

Verified by Experts

The correct Answer is:
A, B
We have
`(y+3)/(2y+5)=sin^2x+2cosx+1`
`:. Cos^2x-2cosx+1=3-(y+3)2(2y+5)`
`:. (cosx-1)^2 =(5y+12)/(2y+5)`
Now, `-1le cosx-1le0`
`:. -2le cosx -1le0`
`:. 0le(cosx-1)^2le4`
`:. 0le(5y+12)/(2y+5)le4`
`:. 0le(5y+12)/(2y+5)and (5y+12)/(2y+5)le4`
`:. (5y+12)/(2y+5)ge0and(3y+8)/(2y+5)ge0`
`:. (-oo,-5/2)uu[-12/5,oo)and(-oo,-8/3)uu[-5/2,oo)`
`:. y in(-oo,-8/3]uu[-12/5,oo)`
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