<p><strong>Question nos. 663 to 665</strong><br>Match the condition of column-I with corresponding number of real roots of \(f(x) = 0\) in column-II and number of points of non-derivability of \(y = f(|x|)\) in column-III, where \(f(x) = ax^2 + bx + c\).</p><p><strong>Column-I</strong><br>(I) \(a^2 + b^2 + c^2 - ab - bc - ca \leq 0\)<br>(II) \(a^2 + b^2 + c^2 + ab + bc + ca \leq 0\)<br>(III) \(3(a^2 + b^2 + c^2 + 1) \leq 2(a + b + c + ab + bc + ca)\)<br>(IV) \(a^2 + b^2 + c^2 \leq 2a + 6b + 4c + 14\)</p><p><strong>Column-II</strong><br>(i) 0 (ii) 1 (iii) 2 (iv) \(\infty\)</p><p><strong>Column-III</strong><br>(P) 0 (Q) 1 (R) 3 (S) 5</p><p><strong>Q663.</strong> Which of the following is <strong>correct</strong> combination?</p>
Step-by-Step Solution
Key Concept: Recognize that conditions in Column-I involve quadratic forms that can be rewritten using algebraic identities. The condition a² + b² + c² - ab - bc - ca ≤ 0 factors as ½[(a-b)² + (b-c)² + (c-a)²] ≤ 0, which forces a = b = c, making f(x) a constant function.
<p><strong>Step 1: Analyze condition (I): a² + b² + c² - ab - bc - ca ≤ 0</strong></p><p>Multiply by 2: 2a² + 2b² + 2c² - 2ab - 2bc - 2ca ≤ 0</p><p>Rewrite: (a-b)² + (b-c)² + (c-a)² ≤ 0</p><p>Since sum of squares ≥ 0, equality holds only when a = b = c.</p><p><strong>Step 2: When a = b = c</strong></p><p>f(x) = a(x² + x + 1). The discriminant Δ = 1 - 4 = -3 < 0</p><p>So f(x) = 0 has <strong>0 real roots</strong> → Column-II: (i)</p><p><strong>Step 3: Find non-derivability points of y = f(|x|)</strong></p><p>f(|x|) = a(|x|² + |x| + 1) = a(x² + |x| + 1)</p><p>The absolute value |x| has derivative discontinuity at x = 0</p><p>Also, if f(x) = 0 has no real roots, f(|x|) = 0 has no real solutions</p><p>Points of non-derivability occur at:</p><ul><li>x = 0 (from |x|)</li><li>Points where f(x) = 0 (if they exist)</li></ul><p>Since f(x) = 0 has no real roots, only x = 0 is non-derivable → Column-III: (Q) = 1</p><p><strong>Answer: (I) → (i), (Q)</strong> which is <strong>C</strong></p>
Correct Answer: C