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In a triangle $ABC$, if the determinant $\begin{vmatrix}\sin A & \sin^2 A\\ \sin B & \sin^2 B\\ \sin C & \sin^2 C\end{vmatrix}=0$, then the triangle must be

1. equilateral or isosceles
2. equilateral or right-angled
3. right-angled or isosceles
4. none of these

Correct answer: equilateral or isosceles

Solution

Using $A+B+C=\pi$, the rows are built from $\sin\theta$ and $\sin^2\theta$. The determinant becomes zero when two angles are equal (isosceles) or when all three are equal (equilateral), which is included in isosceles as a special case; among the given choices, the intended answer is equilateral or isosceles.

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