Question

If $f:[0,\frac{\pi }{2})\to R$ is defined as $f\left(\theta \right)=\left|\begin{array}{ccc}1& \tan \theta & 1\\ -\tan \theta & 1& \tan \theta \\ -1& -\tan \theta & 1\end{array}\right|$

Then, the range of $f$ is

• A. $(2,\infty )$
• B. $(-\infty ,-2]$
• C. $[2,\infty )$
• D. $(-\infty ,2]$

Answer 1

The correct option is C $[2,\infty )$

We have,

$f\left(\theta \right)=\left|\begin{array}{ccc}1& \tan \theta & 1\\ -\tan \theta & 1& \tan \theta \\ -1& -\tan \theta & 1\end{array}\right|$

$=1(1+{\tan }^2\theta )-\tan \theta (-\tan \theta +\tan \theta )+1({\tan }^2\theta +1)$

$=2(1+{\tan }^2\theta )-0=2{\sec }^2\theta$

$\therefore$ Range of $f=[2,\infty )$