<p>\((\lambda^2 + \lambda - 2)x^2 + (\lambda + 2)x - 1 < 0\) for all \(x \in R\). Find the number of integral values of \(\lambda\).</p>
Step-by-Step Solution
Key Concept: For this to be a quadratic equation, the coefficient of x² must be non-zero. Factor λ² + λ - 2 and identify which values of λ make it zero, then exclude those values.
<p><strong>Step 1:</strong> For the given expression to be a quadratic equation, we need the coefficient of x² to be non-zero.</p><p><strong>Step 2:</strong> Factor the coefficient: λ² + λ - 2 = (λ + 2)(λ - 1)</p><p><strong>Step 3:</strong> Set (λ + 2)(λ - 1) = 0 to find when the equation degenerates.</p><p><strong>Step 4:</strong> This gives λ = -2 or λ = 1. When λ takes these values, the equation is no longer quadratic.</p><p><strong>Step 5:</strong> Therefore, for a valid quadratic equation: λ ≠ -2 and λ ≠ 1, or equivalently λ ∈ ℝ \ {-2, 1}</p><p>∴ Answer: B</p>
Correct Answer: B