Matrices & Determinants
Determinant Applications
Grade 12
Question:
<p>If \(\Delta = \begin{vmatrix} a^2 & b\sin A & C\sin A \\ b\sin A & 1 & \cos A \\ C\sin A & \cos A & 1 \end{vmatrix}\) is independent of which variable? (where \(a, b, c\) are sides of a triangle and \(A, B, C\) are opposite angles)</p>
<p>(a) \(a\)</p>
<p>(b) \(b\)</p>
<p>(c) \(c\)</p>
<p>(d) \(A, B, C\)</p>
Step-by-Step Solution
Key Concept: Apply sine rule and triangle identities to show the determinant is a constant.
<p>Using the sine rule and properties of triangles, the determinant simplifies to a constant independent of all variables.</p>
Correct Answer: D