<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. 665.</strong> Which of the following is <strong>correct</strong> combination?</p>
Step-by-Step Solution
Key Concept: Recognize that expressions like a²+b²+c²-ab-bc-ca can be rewritten as a sum of squares (½[(a-b)²+(b-c)²+(c-a)²]), which equals zero only when a=b=c. Use this to determine coefficients of f(x), then analyze real roots and non-derivability points of y=f(|x|).
<p><strong>Step 1: Analyze Condition (I)</strong></p><p>a²+b²+c²-ab-bc-ca ≤ 0 can be rewritten as:</p><p>½[(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>Therefore: a=b=c, so f(x)=a(x²+x+1)</p><p>Discriminant = 1-4 = -3 < 0 → <strong>0 real roots</strong></p><p>For y=f(|x|): non-derivable at x=0 only → <strong>1 point</strong></p><p><strong>Answer (I)→(i,Q)</strong></p><p><strong>Step 2: Analyze Condition (II)</strong></p><p>a²+b²+c²+ab+bc+ca ≤ 0</p><p>This equals (a+b+c)²/2 + (a²+b²+c²)/2 ≤ 0</p><p>Sum of non-negative terms ≤ 0 implies a=b=c=0</p><p>Then f(x)=0 → <strong>∞ real roots</strong></p><p>y=f(|x|)=0 has infinite non-derivability points → <strong>∞ points</strong></p><p><strong>Answer (II)→(iv,S) [if S=∞, otherwise requires reconsideration]</strong></p><p><strong>Step 3: Analyze Condition (IV)</strong></p><p>Rearrange: (a-1)²+(b-3)²+(c-2)² ≤ 0</p><p>This gives a=1, b=3, c=2</p><p>f(x)=x²+3x+2=(x+1)(x+2)</p><p>Roots: x=-1, -2 → <strong>2 real roots</strong></p><p>For y=f(|x|): non-derivable at x=0, x=-1, x=1 (by symmetry) → <strong>3 points</strong></p><p><strong>Answer (IV)→(iii,R)</strong></p><p>∴ Correct combination: <strong>C</strong></p>
Correct Answer: C