Question

If $f:[0,\pi /2)\to R$ 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 $[2,\infty )$

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

Using determinant expansion, we get
$f\left(\theta \right)=1(1+{\tan }^2\theta )-\tan \theta (-\tan \theta +\tan \theta )+1({\tan }^2\theta +1)\Rightarrow f\left(\theta \right)=2({\tan }^2\theta +1)\Rightarrow f\left(\theta \right)=2{\sec }^2\theta$
$\sec \theta \in (-\infty ,-1]\cup [1,\infty )\Rightarrow {\sec }^2\theta \in [1,\infty )$
Therefore, $R_f=[2,\infty )$.