Equation `cos 2x+7=a(2-sin x)` can have a real solution for

Options

A. all values of a
B. `a in [2,6]`
C. `a in (-oo,2)`
D. `a in (0,oo)`

Answer

B

Explanation

Given ` cos 2x + 7 = a ( 2 - sin x ) `
` rArr 1 - 2 sin ^( 2 ) x + 7 = 2a - a sinx `
` rArr 2 sin ^2 x - a sin x + ( 2a- 8 ) = 0 `
` therefore sin x = (a pm sqrt ( ( - a ) ^ 2 - 8 ( 2a - 3 )))/ ( 2xx 2 ) `
` = ( a pm ( a - 8 )) /( 4 ) `
For (+ ) sign, ` sin x = (a - 4 )/ ( 2) `
For (-) sign,
` sin x = 2 ` which is not possible.
we know ` - 1 le sin x le 1 `
` therefore - 1 lt ( a - 4 ) /(2 ) le 1 rArr 2 le a le 6 `