<p><strong>For Problems 10–12</strong><br>\[f(x) = \begin{vmatrix} x+c_1 & x+a & x+a \\ x+b & x+c_2 & x+a \\ x+b & x+b & x+c_3 \end{vmatrix}\] and \(g(x) = (c_1 - x)(c_2 - x)(c_3 - x)\)</p><p>Which of the following is not true?</p>
Step-by-Step Solution
Key Concept: f(x) is a polynomial of degree at most 1, not 3, because subtracting the first row from rows 2 and 3 reveals that the determinant is linear in x. Compare this structure with g(x) which is cubic to identify which statement cannot be true.
<p><strong>Step 1:</strong> Apply row operations. Subtract row 1 from rows 2 and 3:</p><p>R₂ → R₂ - R₁: entries become (b-c₁), (c₂-a), 0</p><p>R₃ → R₃ - R₁: entries become (b-c₁), (b-a), (c₃-a)</p><p><strong>Step 2:</strong> The determinant now has a clear structure. Expanding along column 3 shows f(x) is linear (degree 1), not cubic.</p><p><strong>Step 3:</strong> Since f(x) has degree 1 and g(x) = (c₁-x)(c₂-x)(c₃-x) has degree 3, any statement claiming f(x) = g(x) or that f(x) has three roots cannot be true.</p><p><strong>Step 4:</strong> Check which statement contradicts the linear nature of f(x). The false statement will typically claim f(x) is cubic or equal to g(x).</p><p>∴ Answer: A</p>
Correct Answer: A