If $C=2 \cos θ$, then the value of the determinant $Δ=\begin{vmatrix}C &1& 0\\1 &C& 1\\6&1& C\end{vmatrix}$, is

none of these

We have,

$Δ=\begin{vmatrix}C &1& 0\\1 &C& 1\\6&1& C\end{vmatrix}$

$⇒Δ=\begin{vmatrix}0 &1& 0\\1-C^2 &C& 1\\6-C&1& C\end{vmatrix}$  [Applying $C_1→C_1-CC_2$]

$⇒Δ=-\begin{vmatrix}1-C^2 &1\\6-C&C\end{vmatrix}$  [Expanding along $R_1$]

$⇒Δ=-(C-C^3-6+C)$

$⇒Δ=C^3 -2C +6$

$⇒Δ= 8 \cos^3 θ-4 \cos θ + 6$