First slide

Theorems of probability

Question

$If a is an integer lying in [- 5, 30], then the probability that the graph of y = x^{2} + 2 (a + 4) x - 5 a + 64 is strictly$ $above the x -axis is$

Moderate

A

$1 / 6$
B

$7 / 36$
C

$2 / 9$
D

$3 / 5$

Solution

$x^{2} + 2 (a + 4) x - 5 a + 64 \geq 0$
$If D \leq 0, then$
$\begin{array}{l} (a + 4)^{2} - (- 5 a + 64) < 0 \\ OR a^{2} + 13 a - 48 < 0 \\ OR (a + 16) (a - 3) < 0 \\ \Rightarrow - 16 < a < 3 \Leftrightarrow - 5 \leq a \leq 2 \end{array}$
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., 36.
Hence, the required probability is 8/36 = 2/9.

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