Matrices & Determinants
Determinant Expansion
Grade 12
Question:
<p><strong>Question 83:</strong> Consider the determinant $\Delta = \begin{vmatrix} a_1 + b_1 x^2 & a_1 x^2 + b_1 & c_1 \\ a_2 + b_2 x^2 & a_2 x^2 + b_2 & c_2 \\ a_3 + b_3 x^2 & a_3 x^2 + b_3 & c_3 \end{vmatrix} = 0$, where $a_i, b_i, c_i \in \mathbb{R}$ and $x \in \mathbb{R}$.</p>
<p>(a) Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1</p>
<p>(b) Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1</p>
<p>(c) Statement-1 is true, Statement-2 is false</p>
<p>(d) Statement-1 is false, Statement-2 is true</p>
Step-by-Step Solution
Key Concept: Use column operations to factor out $x^2$ and recognize the structure of the determinant equation.
<p>The determinant can be rewritten by separating columns: $\Delta = \begin{vmatrix} a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3 \end{vmatrix} + x^2 \begin{vmatrix} b_1 & a_1 & c_1 \\ b_2 & a_2 & c_2 \\ b_3 & a_3 & c_3 \end{vmatrix} = 0$</p><p>This is a statement about the structure of the determinant. Statement-1 is verified as correct based on determinant properties. Statement-2 may not directly explain this.</p><p>The answer is (c).</p>
Correct Answer: c