The number of integral values of `k` for which the equation `7cos x +5 sinx=2k+1` has a solution is

Options

A. 4
B. 8
C. 10
D. 112

Answer

B

Explanation

We know that the equation `a "cos" x + b "sin" x =c` has a solution, if
`|c| le sqrt(a^(2) + b^(2))`
So, the equation `7 "cos" x + 5 "sin" x = 2k + 1` will have a solution, if
`|2k + 1| le sqrt(49 + 25)`
`rArr |2k + 1| le sqrt(74)`
`rArr -sqrt(74) le 2k + 1 le sqrt(74)`
`-8 le 2k + 1 le 8`
`rArr k = -4, -3, -2, -1, 0, 1, 2, 3.`
Thus, there are 8 values of k.