The determinant \(\begin{vmatrix} x & \sin\theta & \cos\theta \\ -\sin\theta & -x & 1 \\ \cos\theta & 1 & x \end{vmatrix}\) is:
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → independent of \(\theta\) only
Expanding along the first row:
\(\Delta = x\left(-x^2 - 1\right) - \sin\theta\left(-x\sin\theta - \cos\theta\right) + \cos\theta\left(-\sin\theta + x\cos\theta\right)\)
\(= -x^3 - x + x\sin^2\theta + \sin\theta\cos\theta - \sin\theta\cos\theta + x\cos^2\theta\)
\(= -x^3 - x + x\left(\sin^2\theta + \cos^2\theta\right)\)
\(= -x^3 - x + x\)
\(= -x^3\)
The value \(-x^3\) contains no \(\theta\) but changes with x, so the determinant is independent of \(\theta\) only.
NCERT (2026-27 edition):
Determinants
