Question

Find the determinant of \(\rm\begin{vmatrix} x & \cos\theta &\sin\theta \\ -\cos\theta & -x & 1\\ \sin\theta & 1 & x \end{vmatrix}\) . 
1. 1
2. xsin²θ 
3. x²
4. -x³

Answer 1

Correct Answer - Option 4 : -x³

Concept :

A= \(\rm \begin{vmatrix} a_{11} &a_{12} &a_{13} \\ a_{21}& a_{22} &a_{23} \\ a_{31}& a_{32} & a_{33} \end{vmatrix}\) 

Then determinant of matrix A , | A | = = a₁₁ × {(a₂₂ × a₃₃) – (a₂₃ × a₃₂)} - a₁₂ × {(a₂₁ × a₃₃) – (a₂₃ × a₃₁)} + a₁₃ × {(a₂₁ × a₃₂) – (a₂₂ × a₃₁)} . 

Calculation:

⇒ Δ = \(\rm\begin{vmatrix} x & \cosθ &\sinθ \\ -\cosθ & -x & 1\\ \sinθ & 1 & x \end{vmatrix}\) 

⇒ Δ = \(\rm x(-x^{2}-1)- cosθ (-xcosθ -sinθ) + sinθ (-cosθ+ x sinθ)\) 

⇒ Δ = \(\rm -x^{3}-x + xcos^{2}θ+ sinθ .cosθ -sinθ .cosθ +xsin^{2}θ\) 

⇒ Δ = \(\rm -x^{3}-x+ x (sin^{2}θ +cos^{2}θ )\)               

⇒ Δ = - x³                                        [∵ sin² θ + cos² θ = 1]

The correct option is 4.