If `a` is an integer lying in `[-5,30]`, then the probability that the probability the graph of `y=x^2+2(a+4)x-5a+64` is strictly above the x-axis is a. `1//6` b. `7//36` c. `2//9` d. `3//5`

Options

A. `1//6`
B. `7//36`
C. `2//9`
D. `3//5`

Answer

C

Explanation

`x^(2) + 2(a + 4) x - 5a + 64 ge 0`
If `D lt 0`, then
`(a + 4)^(2) - (-5a + 64) lt 0`
or `a^(2) + 13 a - 48 lt 0`
or `(a + 16) (a - 3) lt 0`
implies `-16 lt a lt 3 iff -5 le a le 2`
Then, the favorable cases is equal to the number of integers in the interval [-5, 2], i.e., 8.
Total number of cases is equal to the number of integers in the interval [-5, 30], i.e., 8.
Hence, the required probability is `8//36` = `2//9`.