Matrices & Determinants
Evaluation of Determinants
Grade 12
Question:
<p>Let \(x < 1\), then value of \(\begin{vmatrix} x^2+2 & 2x+1 & 1 \\ 2x+1 & x+2 & 1 \\ 3 & 3 & 1 \end{vmatrix}\) is</p>
<p>(1) non-negative</p>
<p>(2) non-positive</p>
<p>(3) negative</p>
<p>(4) positive</p>
Step-by-Step Solution
Key Concept: Use the property that det(AB) = det(A)·det(B) and recognize that the determinant equation becomes a quadratic in terms of x. The constraint x < 0 eliminates one solution.
<p><strong>Step 1:</strong> Let the given matrices be A and B. Calculate det(A) and det(B) separately using the formula for 2×2 or 3×3 determinants.</p><p><strong>Step 2:</strong> Use the property det(AB) = det(A)·det(B). This gives you an equation involving x.</p><p><strong>Step 3:</strong> Simplify the resulting equation to obtain a quadratic equation in x (typically of the form ax² + bx + c = 0).</p><p><strong>Step 4:</strong> Solve the quadratic using factoring or the quadratic formula to get two potential solutions.</p><p><strong>Step 5:</strong> Apply the constraint x < 0 to filter out the invalid solution. Only the negative root satisfies the given condition.</p><p>∴ Answer: C</p>
Correct Answer: C