First slide

Introduction to probability

Question

If a is an integer lying in [- 5, 30], then the probability that the graph of y = x² + 2 (a + 4)x - 5a + 64 is strictly above the x-axis is

Moderate

A

$\frac{1}{6}$
B

$\frac{7}{36}$
C

$\frac{2}{9}$
D

$\frac{3}{5}$

Solution

$\frac{3}{5}$ $x^{2} + 2 (a + 4) x - 5 a + 64 \geq 0$
If D < 0, then
$(a + 4)^{2} - (- 5 a + 64) < 0$
$\begin{array}{l} or a^{2} + 13 a - 48 < 0 \\ or (a + 16) (a - 3) < 0 \\ \Rightarrow - 16 < a < 3 \Leftrightarrow - 5 \leq a \leq 2 \end{array}$
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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