<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 simultaneously). For a quadratic to have infinitely many solutions, the coefficient of x², coefficient of x, and constant term must all equal zero.
<p><strong>Step 1:</strong> Factor each coefficient:</p><p>• Coefficient of x²: λ² - 4λ + 3 = (λ-1)(λ-3)</p><p>• Coefficient of x: λ² - 5λ + 6 = (λ-2)(λ-3)</p><p>• Constant term: λ² - 9 = (λ-3)(λ+3)</p><p><strong>Step 2:</strong> For more than 2 roots, the equation must be an identity (0·x² + 0·x + 0 = 0). All three coefficients must simultaneously equal zero.</p><p><strong>Step 3:</strong> Find common zeros:</p><p>• (λ-1)(λ-3) = 0 ⟹ λ = 1 or 3</p><p>• (λ-2)(λ-3) = 0 ⟹ λ = 2 or 3</p><p>• (λ-3)(λ+3) = 0 ⟹ λ = 3 or -3</p><p><strong>Step 4:</strong> The only common value is λ = 3.</p><p><strong>Step 5:</strong> Verify: When λ = 3, all three coefficients become zero, giving 0·x² + 0·x + 0 = 0, which is satisfied by infinitely many values (more than 2 roots).</p><p>∴ Answer: <strong>1</strong></p>
Correct Answer: 1