<p>Number of real values of \(\lambda\) such that \((\lambda^2 - 4\lambda + 3)x^2 + (\lambda^2 - 5\lambda + 6)x + (\lambda^2 - 9) = 0\) has more than 2 roots is:</p>
Step-by-Step Solution
Key Concept: A polynomial equation can have more than 2 roots only if it becomes an identity (all coefficients are zero), since a quadratic has at most 2 roots. Set all three coefficients equal to zero simultaneously and find common values of λ.
<p><strong>Step 1:</strong> For the equation to have more than 2 roots, all coefficients must be zero (identity condition).</p><p><strong>Step 2:</strong> Set coefficient of x²: λ² - 4λ + 3 = 0 → (λ-1)(λ-3) = 0 → λ = 1 or λ = 3</p><p><strong>Step 3:</strong> Set coefficient of x: λ² - 5λ + 6 = 0 → (λ-2)(λ-3) = 0 → λ = 2 or λ = 3</p><p><strong>Step 4:</strong> Set constant term: λ² - 9 = 0 → (λ-3)(λ+3) = 0 → λ = 3 or λ = -3</p><p><strong>Step 5:</strong> Find common value in all three sets: {1,3} ∩ {2,3} ∩ {3,-3} = {3}</p><p><strong>Step 6:</strong> Verify at λ = 3: (0)x² + (0)x + (0) = 0 ✓ (identity, infinite roots)</p><p>∴ <strong>Answer: 1</strong></p>
Correct Answer: 1