If $∆=\left|\begin{array}{ccc}1& \cos \mathrm{\theta }& 1\\ -\cos \mathrm{\theta }& 1& \cos \mathrm{\theta }\\ -1& -\cos \mathrm{\theta }& 1\end{array}\right|, \mathrm{then} ∆ \mathrm{lies} \mathrm{in} \mathrm{the} \mathrm{interval}$

$\begin{array}{rcl}∆& =& \left|\begin{array}{ccc}1& \cos \mathrm{\theta }& 1\\ -\cos \mathrm{\theta }& 1& \cos \mathrm{\theta }\\ -1& -\cos \mathrm{\theta }& 1\end{array}\right|\\ {\mathrm{R}}_1& \to & {\mathrm{R}}_1+{\mathrm{R}}_3\\ & =& \left|\begin{array}{ccc}0& 0& 2\\ -\mathrm{cos\theta }& 1& \mathrm{cos\theta }\\ -1& -\mathrm{cos\theta }& 1\end{array}\right|\\ & =& 2\left({\cos }^2\mathrm{\theta }+1\right)\\ & =& 2{\cos }^2\mathrm{\theta }+2\\ & & \mathrm{wkt}, \\ 0& \le & {\cos }^2\mathrm{\theta }\le 1\\ 0& \le & 2{\cos }^2\mathrm{\theta }\le 2\\ 2& \le & 2{\cos }^2\mathrm{\theta }+2\le 4\\ & & \therefore ∆ \mathrm{lies} \mathrm{in} [2, 4]\end{array}$