<p>The given equation is \(-3(x - [x])^2 + 2(x - [x]) + a^2 = 0\). The values of \(a\) for which the equation has non-integral solutions satisfy:</p>
Step-by-Step Solution
Key Concept: Substitute y = x - [x] (the fractional part, where 0 ≤ y < 1) to transform into a quadratic in y. For non-integral solutions, y must be in (0,1), excluding 0.
<p><strong>Step 1:</strong> Let y = x - [x], where 0 ≤ y < 1 (fractional part of x). The equation becomes: -3y² + 2y + a² = 0, or 3y² - 2y - a² = 0</p><p><strong>Step 2:</strong> For non-integral x, we need y ≠ 0. Using quadratic formula: y = (2 ± √(4 + 12a²))/6 = (1 ± √(1 + 3a²))/3</p><p><strong>Step 3:</strong> For valid solutions, we need 0 < y < 1. The positive root y₊ = (1 + √(1 + 3a²))/3 must satisfy y₊ < 1:</p><p>1 + √(1 + 3a²) < 3</p><p>√(1 + 3a²) < 2</p><p>1 + 3a² < 4</p><p>a² < 1, so |a| < 1</p><p><strong>Step 4:</strong> Also need y₊ > 0, which is always true. The negative root y₋ = (1 - √(1 + 3a²))/3 is always negative, so it's invalid.</p><p><strong>Step 5:</strong> For exactly one non-integral solution per value of a, we need 0 < a² < 1</p><p>∴ Answer: A (typically stated as 0 < |a| < 1 or a² ∈ (0,1))</p>
Correct Answer: A