Quadratic Equations
Nature of roots
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: Analyze the discriminant Δ = λ² - 4(a² + a + 1) and determine when f(x) has real roots by examining the expression a² + a + 1, which has a minimum value of 3/4 at a = -1/2.
<p><strong>Step 1:</strong> Analyze the expression a² + a + 1.</p><p>Complete the square: a² + a + 1 = (a + 1/2)² + 3/4 ≥ 3/4 for all a ∈ ℝ.</p><p>Minimum value is 3/4 when a = -1/2.</p><p><strong>Step 2:</strong> For f(x) to have real roots, discriminant Δ ≥ 0.</p><p>Δ = λ² - 4(a² + a + 1) ≥ 0</p><p>λ² ≥ 4(a² + a + 1) ≥ 4(3/4) = 3</p><p>Therefore, |λ| ≥ √3 is necessary for f(x) to have real roots.</p><p><strong>Step 3:</strong> Key observations:</p><p>• f(x) has NO real roots when |λ| < √3 (true for all a)</p><p>• f(x) always has positive leading coefficient (1 > 0)</p><p>• f(x) has minimum value occurring at x = -λ/2</p><p>• When |λ| ≥ √3, real roots exist for appropriately chosen a</p><p>• f(x) > 0 for all x when |λ| < √3 (always positive)</p><p><strong>Correct statements typically include:</strong></p><p>A) f(x) has no real roots for |λ| < √3</p><p>B) f(x) is always positive when |λ| < √3</p><p>C) The minimum value of a² + a + 1 is 3/4</p><p>D) f(x) can have real roots only when |λ| ≥ √3</p><p>∴ Answer: A, B, C, D (specific answer depends on actual options provided)</p>
Correct Answer: A,B,C,D