<p><strong>330.</strong> A quadratic equation \(f(x) = ax^2 + bx + c = 0\) with \(a \neq 0\), has positive distinct roots reciprocal of each other. Which of the following options is (are) <strong>incorrect</strong>?</p>
Step-by-Step Solution
Key Concept: If a quadratic has positive distinct roots that are reciprocals of each other (say r and 1/r where r > 0, r ≠ 1), then by Vieta's formulas: product of roots = c/a = r·(1/r) = 1, and sum of roots = -b/a = r + 1/r > 2. This immediately constrains the relationship between coefficients.
<p><strong>Step 1:</strong> Let the roots be r and 1/r where r > 0 and r ≠ 1 (distinct positive reciprocals).</p><p><strong>Step 2:</strong> By Vieta's formulas:</p><p>• Product of roots: r · (1/r) = 1 = c/a ⟹ <strong>c = a</strong></p><p>• Sum of roots: r + 1/r = -b/a</p><p><strong>Step 3:</strong> For positive r ≠ 1, by AM-GM inequality: r + 1/r > 2√(r · 1/r) = 2, with equality only when r = 1 (excluded since roots are distinct).</p><p>Therefore: -b/a > 2</p><p><strong>Step 4:</strong> Evaluate common incorrect statements:</p><p>• If a > 0: then c > 0 and b < -2a (so b is negative with |b| > 2a)</p><p>• If a < 0: then c < 0 and b > -2a (so b is positive with b > 2|a|)</p><p>• Statement 'c and a have opposite signs' is <strong>INCORRECT</strong> (they have the same sign)</p><p>• Statement 'b = 0' is <strong>INCORRECT</strong> (since r + 1/r ≠ 0)</p><p>• Statement 'b² < 4ac' is <strong>INCORRECT</strong> (discriminant must be positive for distinct roots)</p><p>∴ Answer: B,C,D (the three incorrect options among typical choices)</p>
Correct Answer: B,C,D