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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.
Solution
Expanding along R1, we get
\[ \Rightarrow - x^3 - x + x \sin^2 \theta + \sin\theta\cos\theta - \sin\theta\cos\theta + x \cos^2 \theta = 8\]
\[ \Rightarrow - x^3 - x + x\left( \sin^2 \theta + \cos^2 \theta \right) = 8\]
\[ \Rightarrow - x^3 - x + x = 8\]
\[ \Rightarrow x^3 + 8 = 0\]
\[ \Rightarrow \left( x + 2 \right)\left( x^2 - 2x + 4 \right) = 0\]
\[ \Rightarrow x + 2 = 0 \left[ \because x^2 - 2x + 4 > 0 \forall x \right]\]
\[ \Rightarrow x = - 2\]
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