UPSC NDA I 2024 Mathematics Paper

Given that in the triangle $A B C, \sin^3 A+ \sin^3 B+ \sin^3 C=3 \sin A \sin B \sin C$, we need to determine the value of the determinant:$\begin{vmatrix} a & b & c \\ b & c & a \\ c & a & b \end{vmatrix}$From the given trigonometric identity, we know that $\sin A, \sin B$, and $\sin C$ satisfy a specific relationship due to the nature of the angles in triangle $A B C$. In this context, the identity $\sin^3 A+ \sin^3 B+ \sin^3 C=3 \sin A \sin B \sin C$ often arises in cases where $A=B=C=\frac{\pi}{3}$, thus making triangle $A B C$ equilateral.In an equilateral triangle, the sides $a, b$, and $c$ are equal. Hence, let's denote the side length of the equilateral triangle as $a$ (i.e., $a=b=c$ ). Substituting these in the determinant, we get:$\begin{vmatrix} a & a & a \\ a & a & a \\ a & a & a \end{vmatrix}$To compute this determinant, notice that each row is identical. A determinant with identical rows is zero. Therefore:$\begin{vmatrix} a & b & c \\ b & c & a \\ c & a & b \end{vmatrix}=0$