The maximum value of $\left| {\begin{array}{cccccccccccccccccccc}1&1&1\\1&{1 + \sin \theta }&1\\1&1&{1 + \cos \theta }\end{array}} \right|$ is $\frac{1}{2}$.

Correct Answer True

Since, $\left| {\begin{array}{cccccccccccccccccccc}1&1&1\\0&{\sin \theta }&0\\0&0&{\cos \theta }\end{array}} \right|$

and $\left. {{R_3} \to {R_3} - {R_1}} \right]$

On expanding along third row, we get the value of the determinant

$= \cos \theta \cdot \sin \theta = \frac{1}{2}\sin 2\theta = \frac{1}{2}$

[when $\theta$ is ${45^\circ }$ which gives maximum value]