Matrices & Determinants
Matrices and Determinants
star_batch_jee_advanced_2025
Grade 12
Question:
In a $\triangle ABC$, if $$\begin{vmatrix} 1 & a & b \\ 1 & c & a \\ 1 & b & c \end{vmatrix} = 0$$, then $\sin^2 A + \sin^2 B + \sin^2 C = $_____.
Step-by-Step Solution
Key Concept: A sum of squares equaling zero forces all terms to be zero, yielding the equilateral triangle condition.
From the determinant condition $\begin{vmatrix}1 & a & b\\c & a & 0\\1 & b & c\end{vmatrix} = 0$, expanding gives $a^2 + b^2 + c^2 - ab - bc - ca = 0$, which factors as $(a-b)^2 + (b-c)^2 + (c-a)^2 = 0$. This implies $a = b = c$, so triangle $ABC$ is equilateral with all angles $60°$. Therefore $\sin^2 A + \sin^2 B + \sin^2 C = 3\sin^2 60° = \frac{9}{4}$.
Correct Answer: 2.25