Question

Find the maximum value of $\left|\begin{array}{ccc}1& 1& 1\\ 1& 1+\sin \mathrm{\theta }& 1\\ 1& 1& 1+\cos \mathrm{\theta }\end{array}\right|$

Answer 1

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

Applying $R_2\to R_2-R_1$ and $R_3\to R_3-R_1$, we get

$∆=\left|\begin{array}{ccc}1& 1& 1\\ 0& \sin \theta & 0\\ 0& 0& \cos \theta \end{array}\right|$

$$\begin{gathered} =\sin \theta \cos \theta \\ =\frac{\sin 2\theta }{2} \end{gathered}$$
We know that −1 ≤ sin2θ ≤ 1.
∴ Maximum value of ∆ = $\frac{1}{2}\times 1=\frac{1}{2}$