The number of integral values of a for which the equation `cos2x+a sin x=2a-7` possessess a solution.

Options

A. `a lt 2`
B. ` a ge 8`
C. ` a gt 8`
D. `"a is any integar" lt -2`

Answer

B

Explanation

Given equation `cos 2x + a sin x = 2a -7` can be written as
`cos^(2)x - sin^(2)x + a sin x = 2a - 7`
`rArr 1- sin^(2) x - sin^(2)x+ a sin x = 2a - 7 `
`rArr 2sin ^(2)x - a sin x + (2a - 7 - 1) = 0`
`rArr 2sin^(2)- a sin x + 2a-8 = 0 `
This is quadratic equation in sin x and its discriminant `ge 0.`
Here, `a=2, b=-a, c=2a - 8 `
`rArr a^(2) - 4.2. (2a-8) ge 0`
` rArr a^(2) - 16a + 64 ge 0`
`rArr (a-8)^(2) ge 0 rArr a ge 8 `