Question

If $\Delta =\left|\begin{array}{ccc}1& 2cos\theta & 1\\ sin\theta & 1& 3cos\theta \\ 1& sin\theta & 1\end{array}\right|$, then maximum value of $\Delta$ is

• A. 1
• B. 9
• C. 16
• D. none of these

Answer 1

The correct option is D none of these

$\Delta =\left|\begin{array}{ccc}1& 2cos\theta & 1\\ sin\theta & 1& 3cos\theta \\ 1& sin\theta & 1\end{array}\right|$ applying $R_3\to R_3-R_1$
$=\left|\begin{array}{ccc}1& 2cos\theta & 1\\ sin\theta & 1& 3cos\theta \\ 1& sin\theta -3cos\theta & 1\end{array}\right|=-\left(\sin \theta -3\cos \theta \right)\left(3\cos \theta -\sin \theta \right)={\left(3\cos \theta -\sin \theta \right)}^2$
Now, $-\sqrt{9+1}\le 3\cos \theta -\sin \theta \le \sqrt{9+1}\Rightarrow {\left(3\cos \theta -\sin \theta \right)}^2\le 10$