Question

If \[\begin{vmatrix}x & \sin \theta & \cos \theta \\ - \sin \theta & - x & 1 \\ \cos \theta & 1 & x\end{vmatrix} = 8\] , write the value of x.

Answer 1

\[\begin{vmatrix}x & \sin \theta & \cos \theta \\ - \sin \theta & - x & 1 \\ \cos \theta & 1 & x\end{vmatrix} = 8\]
Expanding along R₁, we get

\[x\left( - x^2 - 1 \right) - \sin\theta\left( - x\sin\theta - \cos\theta \right) + \cos\theta\left( - \sin\theta + x\cos\theta \right) = 8\]
\[ \Rightarrow - x^3 - x + x \sin^2 \theta + \sin\theta\cos\theta - \sin\theta\cos\theta + x \cos^2 \theta = 8\]
\[ \Rightarrow - x^3 - x + x\left( \sin^2 \theta + \cos^2 \theta \right) = 8\]
\[ \Rightarrow - x^3 - x + x = 8\]
\[ \Rightarrow x^3 + 8 = 0\]
\[ \Rightarrow \left( x + 2 \right)\left( x^2 - 2x + 4 \right) = 0\]
\[ \Rightarrow x + 2 = 0 \left[ \because x^2 - 2x + 4 > 0 \forall x \right]\]
\[ \Rightarrow x = - 2\]