Mathematics - Determinant Question with Solution | TestHub

$A=\left[\begin{array}{ccc}1& \sin \theta & 1\\ -\sin \theta & 1& \sin \theta \\ -1& -\sin \theta & 1\end{array}\right]$
$\Rightarrow \det (A)=\left(11+{\sin }^2\theta \right)-\sin \theta (-\sin \theta +\sin \theta )+1\left({\sin }^2\theta +1\right)$
$=2+2{\sin }^2\theta$
$\because \theta \in \left(\frac{3\pi }{4},\frac{5\pi }{4}\right)$
$\therefore \sin 2\theta \in \left[0,\frac{1}{2}\right]$
$\therefore \det (A)\in [2,3]$ which is a subset of $\left(\frac{3}{2},3\right]$