Question

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

• A. $\frac{1}{2}$
• B. $\frac{\sqrt{3}}{2}$
• C. $\sqrt{2}$
• D. $\frac{3\sqrt{2}}{4}$

Answer 1

The correct option is A $\frac{1}{2}$

$\Delta =\left|\begin{array}{ccc}1& 1& 1\\ 1& 1+\sin \theta & 1\\ 1& 1& 1+\cos \theta \end{array}\right|$

Applying $R_2\to R_2-R_1,R_3\to R_3-R_1$

$\Delta =\left|\begin{array}{ccc}1& 1& 1\\ 0& \sin \theta & 0\\ 0& 0& \cos \theta \end{array}\right|=1\left(\sin \theta \cos \theta \right)=\frac{\sin 2\theta }{2}$

Therefore, max of $\Delta$ is $\frac{1}{2}$

Hence, option 'A' is correct.