<p>One root lies inside the unit circle and one outside if</p>
<p>(1) \(-1 < \lambda < 1\)</p>
<p>(2) \(\lambda > 1\)</p>
<p>(3) \(\lambda < 1\)</p>
<p>(4) none of these</p>
Step-by-Step Solution
Key Concept: For a quadratic with real coefficients, roots lie on opposite sides of the unit circle if and only if the product of roots has modulus not equal to 1, specifically when |product of roots| ≠ 1. Use the relationship between coefficients and roots: if roots are α and β, then |αβ| = |c/a|, and one root inside/outside the circle requires |αβ| ≠ 1.
<p><strong>Step 1:</strong> For a quadratic equation az² + bz + c = 0 with roots α and β, by Vieta's formulas: αβ = c/a.</p><p><strong>Step 2:</strong> For one root inside unit circle (|α| < 1) and one outside (|β| > 1), we need |α||β| ≠ 1.</p><p><strong>Step 3:</strong> This means |αβ| ≠ 1, which translates to |c/a| ≠ 1, or equivalently |c| ≠ |a|.</p><p><strong>Step 4:</strong> For the specific condition with real coefficients, one root inside and one outside the unit circle occurs when the product of moduli satisfies: |c/a| < 1 and |c/a| > 0, which requires |c| < |a| (assuming the quadratic has the standard form).</p><p>∴ Answer: A</p>
Correct Answer: A