Question

If maximum and minimum values of $D=\left|\begin{array}{ccc}1& -\cos \theta & 1\\ \cos \theta & 1& -\cos \theta \\ 1& \cos \theta & 1\end{array}\right|$ are $p$ and $q$ respectively, then the value of $2p+3q$ is _________.

• A. $16$
• B. $6$
• C. $14$
• D. $8$

Answer 1

The correct option is C $14$

$D=\left|\begin{array}{ccc}1& -\cos \theta & 1\\ \cos \theta & 1& -\cos \theta \\ 1& \cos \theta & 1\end{array}\right|$

$\Rightarrow D=1(1+{\cos }^2\theta )+\cos \theta (\cos \theta +\cos \theta )-1({\cos }^2\theta -1)$
$=1+{\cos }^2\theta +2{\cos }^2\theta -{\cos }^2\theta +1$
$=2(1+{\cos }^2\theta )$

Now, $-1\le \cos \theta \le 1\Rightarrow 0\le {\cos }^2\theta \le 1$
$\therefore p=4,q=2$
$\therefore 2p+3q=14$.