The value of p for which the equation `3 " sin"^(2)x + 12 "cos" x - 3 =p` has at least one solution are

Options

A. `p le 12`
B. `3 le p le9`
C. `-15 le p le 9`
D. none of these

Answer

C

Explanation

We have,
`3 "sin"^(2)x + 12"cos" x - 3 =p`
`rArr 3 "cos"^(2)x - 12 "cos" x + p =0`
`rArr "cos" x = (12+-sqrt(144-12p))/(6)`
This will have a solution, if
`-1 le (12+ sqrt(144-12p))/(6) le 1 " and " 144-12 p ge 0`
`rArr -6 le 12 +- sqrt(144-12p) le 6 " and " p le 12`
`rArr -18 le +- sqrt(144-12p)-le -6 " and " p le 12`
`rArr -18 le - sqrt(144-12 p) le -6 " and " p le 12`
`rArr 6 le sqrt(144-12p) le 18 " and " p le 12`
`rArr 36 le 144-12p le 324 " and " p le 12`
`rArr -108 le -12 p le 180 " and " p le 12`
`rArr -15 le p le 9 " and " p le 12`
`rArr -15 le p le 9`