<p><strong>For Problems 35–37</strong><br>Consider the equation \(x^4 - \lambda x^2 + 9 = 0\).<br><br>If the equation has four real and distinct roots, then \(\lambda\) lies in the interval</p>
Step-by-Step Solution
Key Concept: Substitute y = x² to convert this into a quadratic in y. For four distinct real roots in x, we need two distinct positive roots in y, which requires the quadratic in y to have both roots positive and distinct.
<p><strong>Step 1:</strong> Let y = x². The equation becomes y² - λy + 9 = 0.</p><p><strong>Step 2:</strong> For four real and distinct roots in x, we need two distinct positive roots in y (since each positive y gives ±√y as two real roots in x).</p><p><strong>Step 3:</strong> For two distinct roots: Discriminant > 0<br/>Δ = λ² - 36 > 0<br/>⟹ λ² > 36<br/>⟹ |λ| > 6, so λ < -6 or λ > 6</p><p><strong>Step 4:</strong> For both roots positive, by Vieta's formulas:<br/>• Sum of roots: λ > 0<br/>• Product of roots: 9 > 0 ✓ (always satisfied)</p><p><strong>Step 5:</strong> Combining conditions: λ > 6 AND λ > 0<br/>⟹ λ > 6</p><p><strong>Step 6:</strong> Therefore, λ ∈ (6, ∞)</p><p>∴ Answer: C</p>
Correct Answer: C