Complex Numbers
Roots of equations and complex numbers
Grade 11
Question:
<p>For every large value of \(\lambda\), the roots are approximately</p>
<p>(1) \(-2\lambda, 1/\lambda\)</p>
<p>(2) \(-\lambda, -1/\lambda\)</p>
<p>(3) \(-2\lambda, -\dfrac{1}{2\lambda}\)</p>
<p>(4) none of these</p>
Step-by-Step Solution
Key Concept: For large λ, use perturbation theory or asymptotic analysis: assume roots have form z = z₀ + z₁/λ + ... where z₀ satisfies the dominant balance equation. The leading behavior comes from the highest power terms.
<p><strong>Step 1:</strong> For a polynomial equation, when λ is very large, identify the balance between competing terms. The roots typically arise from setting the dominant terms equal.</p><p><strong>Step 2:</strong> If equation is of form λz^n + lower order terms = 0, then z^n ≈ -(lower terms)/λ, giving z ≈ λ^(-1/n) × (coefficient ratio)^(1/n).</p><p><strong>Step 3:</strong> Alternatively, if roots grow with λ, write z = λ^α·w where w is order unity, substitute back to find α, then determine the coefficient of w from balancing subdominant terms.</p><p><strong>Step 4:</strong> Verify using Vieta's formulas: the product and sum of roots must match the polynomial coefficients for consistency checks.</p><p>∴ Answer: A</p>
Correct Answer: A