<p>If \(a\) is an integer lying in \([-5, 30]\), then the probability that the graph of \(y = x^2 + 2(a+4)x - 5a + 64\) is strictly above the \(x\)-axis is</p>
Step-by-Step Solution
Key Concept: A parabola with positive leading coefficient lies strictly above the x-axis if and only if its discriminant is negative (no real roots). You must count integers in [-5, 30] where Δ < 0.
<p><strong>Step 1:</strong> For y = x² + 2(a+4)x - 5a + 64 to be strictly above x-axis, we need Δ < 0 (no real roots, positive leading coefficient).</p><p><strong>Step 2:</strong> Calculate discriminant:<br/>Δ = [2(a+4)]² - 4(1)(-5a + 64)<br/>Δ = 4(a+4)² + 20a - 256<br/>Δ = 4(a² + 8a + 16) + 20a - 256<br/>Δ = 4a² + 32a + 64 + 20a - 256<br/>Δ = 4a² + 52a - 192</p><p><strong>Step 3:</strong> Require Δ < 0:<br/>4a² + 52a - 192 < 0<br/>a² + 13a - 48 < 0<br/>(a + 16)(a - 3) < 0<br/>-16 < a < 3</p><p><strong>Step 4:</strong> Find integers in [-5, 30] that satisfy -16 < a < 3:<br/>a ∈ {-5, -4, -3, -2, -1, 0, 1, 2}<br/>Count: 8 integers</p><p><strong>Step 5:</strong> Total integers in [-5, 30]: 30 - (-5) + 1 = 36</p><p><strong>Step 6:</strong> Probability = 8/36 = 2/9</p><p>∴ Answer: C</p>
Correct Answer: C