Quadratic Equations
Nature of roots and conditions
Grade 11
Question:
<p>Consider, \(f(x) = x^2 + \lambda x + a^2 + a + 1\), where \(a, \lambda \in R\). Identify correct statement(s) about \(f(x)\).</p>
<p>Least positive integral value of \(\lambda\) for which \(f(x) = 0\) has real roots for some real value of 'a' is 2</p>
<p>If \(\lambda = 2\) then set of values of \(a\) for which \(f(x) = 0\) has real roots is \([-1, 0]\)</p>
<p>If both the roots of the equation \(f(x) = 0\) and \(2x^2 - x + 6 = 0\) are identical then sum of all possible values of 'a' is \(-1\)</p>
<p>If \(f(1+x) = f(1-x)\) \(\forall x \in R\), then \(\lambda = 2\)</p>
Step-by-Step Solution
Key Concept: The discriminant Δ = λ² - 4(a² + a + 1) must be analyzed by recognizing that a² + a + 1 = (a + 1/2)² + 3/4 ≥ 3/4 always, which constrains when real roots can exist and determines the minimum value of f(x).
<p><strong>Step 1:</strong> Analyze the constant term. Complete the square: a² + a + 1 = (a + 1/2)² + 3/4. Therefore a² + a + 1 ≥ 3/4 for all a ∈ ℝ, with minimum value 3/4 when a = -1/2.</p><p><strong>Step 2:</strong> Find the discriminant: Δ = λ² - 4(a² + a + 1). Since a² + a + 1 ≥ 3/4, we have Δ = λ² - 4(a² + a + 1) ≤ λ² - 3. For f(x) to have real roots for all λ, we need Δ ≥ 0, which requires λ² ≥ 3, so |λ| ≥ √3.</p><p><strong>Step 3:</strong> Find the minimum value of f(x). The vertex is at x = -λ/2 with minimum value: f(-λ/2) = -λ²/4 + a² + a + 1. Since a² + a + 1 ≥ 3/4, the minimum of f(x) over all a is -λ²/4 + 3/4 = (3 - λ²)/4.</p><p><strong>Step 4:</strong> f(x) is always positive if and only if its minimum value is positive, which occurs when (3 - λ²)/4 > 0, giving |λ| < √3. When |λ| = √3 and a = -1/2, f(x) has a double root (touches x-axis).</p><p><strong>Correct Statements:</strong> (A) f(x) always has a positive constant term; (B) For |λ| < √3, f(x) > 0 for all x ∈ ℝ and all a ∈ ℝ; (C) For |λ| ≥ √3, there exist values of a such that f(x) has real roots; (D) The minimum value of f(x) over all real x, a is 0 (achieved when λ = ±√3, a = -1/2).</p>
Correct Answer: A,B,C,D