Quadratic Equations
Quadratic Inequalities
Grade 11
Question:
<p>If <span class="math">0 < f(4a) < f(a^2 - 5)</span>, find the range of <span class="math">a</span>.</p>
<p>(a) <span class="math">a \in (-\infty, -1)</span></p>
<p>(b) <span class="math">a \in (5, \infty)</span></p>
<p>(c) <span class="math">a \in (-1, 5)</span></p>
<p>(d) <span class="math">a \in (-1, 5)</span></p>
Step-by-Step Solution
Key Concept: Convert the function inequality into a quadratic inequality and solve using the sign of the product.
<p><strong>Step 1:</strong> From the given inequality <span class="math">0 < f(4a) < f(a^2 - 5)</span>, we have <span class="math">4a > a^2 - 5</span>.</p><p><strong>Step 2:</strong> Rearranging: <span class="math">a^2 - 4a - 5 < 0</span> which gives <span class="math">(a - 5)(a + 1) < 0</span>.</p><p><strong>Step 3:</strong> This implies <span class="math">a \in (-1, 5)</span>.</p><p>∴ Answer is (d).</p>
Correct Answer: D