Question

The probability of selecting integers a ∈ [–5, 30] such that x² + 2(a + 4)x – 5a + 64 > 0, for all x ∈ R, is ______.

Options

• `1/4`

• `7/36`

• `2/9`

• `1/6`

Answer 1

The probability of selecting integers a ∈ [–5, 30] such that x² + 2(a + 4)x – 5a + 64 > 0, for all x ∈ R, is `underlinebb(2/9)`.

Explanation:

Given that x² + 2(a + 4)x – (5a – 64) > 0

Comparing given quadratic equation with general form, Ax² + Bx + c

Here, A = 1, B = 2(a + 4), C = –(5a – 64)

So, D < 0

⇒ B² – 4AC < 0

⇒ 4(a + 4)² + 4(5a – 64) < 0

⇒ (a + 4)² + (5a – 64) < 0

⇒ a² + 13a – 48 < 0

∴ a = `(13 +- sqrt(169 + 192))/2` = –16, 3

So, a ∈ (–16, 3)

Since a is integer ⇒ a = –5, –4, –3, –2, –1, 0, 1, 2 as a ∈ [–5, 30]

∴ Required probability = `8/36 = 2/9`