Question

If $\Delta =\left|\begin{array}{ccc}\cos \theta /2& 1& 1\\ 1& \cos \theta /2& -\cos \theta /2\\ -\cos \theta /2& 1& -1\end{array}\right|$, If the minimun of $\Delta$ is $m_1$ and maximum of $\Delta$ is $m_2$, then $[m_1,m_2]$ are related

• A. [-4, -2]
• B. [2, 4]
• C. [-4, 0]
• D. [0, 2]

Answer 1

The correct option is B [2, 4]

$\Delta =\left|\begin{array}{ccc}\cos \theta /2& 1& 1\\ 1& \cos \theta /2& -\cos \theta /2\\ -\cos \theta /2& 1& -1\end{array}\right|$
$=-1(-1-{\cos }^2\theta /2)+1(1+{\cos }^2\theta /2)$
$=2+2{\cos }^2\theta /2$
$=3+\cos \theta$
Now, $\because -1\le \cos \theta \le 1$
$\Rightarrow 2\le 3+\cos \theta \le 4$
Hence, $m_1=2,m_2=4$