Matrices & Determinants
Determinant Applications
Grade 12
Question:
<p>If <i>a</i>, <i>b</i> and <i>c</i> are sides of \(\triangle ABC\) such that<br/>\[a^3 + b^3\cos B + c^3\cos A + b^2 + c^2 = 0\]<br/>\[c^3 + a^3\cos B + a^2\cos A + c^2 + a^2 = 0\]<br/>\[b^3 + a^3\cos B + b^3\cos A + a^2 + b^2 = 0\]<br/>where \(\phi, \psi, \omega \in \mathbb{R}\) and \(\angle A, \angle B, \angle C \in [\frac{\pi}{8}, \frac{7\pi}{8}]\), then \(\triangle ABC\) is</p>
<p>(a) an isosceles triangle</p>
<p>(b) an equilateral triangle</p>
<p>(c) can't say</p>
<p>(d) None of these</p>
Step-by-Step Solution
Key Concept: A homogeneous system of equations involving triangle sides and angles that admits a non-trivial solution forces the triangle to be equilateral.
<p>The system of equations represents a homogeneous determinantal condition. For the determinant of coefficients to vanish while satisfying triangle angle constraints and the law of cosines, the triangle must be equilateral ($a = b = c$ and $A = B = C = 60°$).</p>
Correct Answer: B