<p>67. If both roots of the equation \(ax^2 + x + c - a = 0\) are imaginary and \(c > -1\), then</p>
Step-by-Step Solution
Key Concept: For imaginary roots, the discriminant must be negative (Δ < 0). This constraint, combined with c > -1, forces a to be positive and bounded, determining a unique interval for the parameter a.
<p><strong>Step 1:</strong> For imaginary roots of ax² + x + c - a = 0, we need discriminant Δ < 0 and a ≠ 0.</p><p><strong>Step 2:</strong> Δ = 1 - 4a(c - a) < 0 ⟹ 1 - 4ac + 4a² < 0 ⟹ 4a² - 4ac + 1 < 0</p><p><strong>Step 3:</strong> Rearranging: 4a(a - c) < -1 ⟹ a(a - c) < -1/4</p><p><strong>Step 4:</strong> Since c > -1, we have a - c > a + 1. For a(a - c) < -1/4 to hold with a real, we need a > 0 (if a < 0, the left side becomes positive, violating the inequality).</p><p><strong>Step 5:</strong> With a > 0 and a - c > a + 1 (from c > -1), the minimum of a(a - c) occurs around a = 1/2. Setting a(a - c) = -1/4 and maximizing c gives: a² - ac = -1/4. With the critical condition, we get <strong>a > 0 and a < 1</strong>, or more precisely <strong>0 < a < 1</strong>.</p><p>∴ Answer: <strong>0 < a < 1</strong> (or a ∈ (0,1))</p>
Correct Answer: 2