<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) \(a^2 + b^2 + c^2 - ab - bc - ca \le 0\)<br>(II) \(a^2 + b^2 + c^2 + ab + bc + ca \le 0\)<br>(III) \(3(a^2 + b^2 + c^2 + 1) \le 2(a + b + c + ab + bc + ca)\)<br>(IV) \(a^2 + b^2 + c^2 \le 2a + 6b + 4c + 14\)</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>Q. 663.</strong> Which of the following is <strong>correct</strong> combination?</p>
Step-by-Step Solution
Key Concept: Transform algebraic inequalities into conditions on coefficients using sum-of-squares decomposition or completing the square. Then determine the discriminant (for real roots) and analyze the behavior of f(|x|) (for non-derivability points).
<p><strong>Step 1: Analyze Condition (I)</strong></p><p>$a^2 + b^2 + c^2 - ab - bc - ca \le 0$</p><p>Multiply by 2: $2a^2 + 2b^2 + 2c^2 - 2ab - 2bc - 2ca \le 0$</p><p>Rewrite: $(a-b)^2 + (b-c)^2 + (c-a)^2 \le 0$</p><p>Since sum of squares ≥ 0, equality holds: $a = b = c$</p><p>So $f(x) = a(x^2 + x + 1)$. Discriminant: $\Delta = 1 - 4 = -3 < 0$ → <strong>0 real roots</strong></p><p><strong>Step 2: Non-derivability of y = f(|x|)</strong></p><p>For $f(|x|) = a(x^2 + |x| + 1)$:</p><p>• At $x = 0$: left derivative = $a$, right derivative = $a$ (continuous and equal) → <strong>derivable</strong></p><p>• $f(|x|)$ is continuous everywhere, and the critical point $x = 0$ has matching derivatives</p><p>• Since $f(x) = a(x^2 + x + 1)$ has no real roots, $f(|x|)$ never touches x-axis → <strong>0 non-derivability points</strong></p><p><strong>Step 3: Check other conditions similarly</strong></p><p>Condition (II): $a^2 + b^2 + c^2 + ab + bc + ca \le 0$ forces $a = b = c = 0$ (degenerate, infinite roots)</p><p>Condition (III) & (IV): Lead to 1 real root and specific non-derivability counts</p><p><strong>∴ For (I): 0 real roots, 0 non-derivability points = (i, P)</strong></p><p>Answer: C (corresponds to match I→(i,P))</p>
Correct Answer: C