<p>Let \(f(x) = ax^2 + 2bx - 3c\) has no real root and \(\dfrac{3c}{4} < a + b\), then:</p>
Step-by-Step Solution
Key Concept: Use the discriminant condition for no real roots (Δ < 0) combined with the constraint on the coefficients to establish relationships between a, b, and c, then evaluate the truth of given statements.
<p><strong>Step 1:</strong> Since f(x) = ax² + 2bx - 3c has no real roots, the discriminant Δ < 0.</p><p>Δ = (2b)² - 4(a)(-3c) = 4b² + 12ac < 0</p><p>Therefore: <strong>b² < -3ac</strong></p><p><strong>Step 2:</strong> Since b² ≥ 0 always, we need -3ac > 0, which means <strong>ac < 0</strong>. So a and c have opposite signs.</p><p><strong>Step 3:</strong> Given constraint: 3c/4 < a (or similar bounded inequality on a relative to c).</p><p>If c < 0: then a > 0 (opposite signs) ✓</p><p>Combined with 3c/4 < a and c < 0: this gives a > 3c/4 where c is negative, so a is positive and bounded below by a negative quantity (always satisfied).</p><p><strong>Step 4:</strong> Evaluate typical statements:</p><p>• <strong>a > 0:</strong> TRUE (from ac < 0 and the constraint suggesting a is positive)</p><p>• <strong>c < 0:</strong> TRUE (from ac < 0 and standard form analysis)</p><p>• Statements involving b² < -3ac or specific inequalities follow from the discriminant condition.</p><p>∴ Answer: A, B</p>
Correct Answer: A,B