Question

Find the maximum value of \[\begin{vmatrix}1 & 1 & 1 \\ 1 & 1 + \sin \theta & 1 \\ 1 & 1 & 1 + \cos \theta\end{vmatrix}\]

Answer 1

Let Δ = \[\begin{vmatrix}1 & 1 & 1 \\ 1 & 1 + \sin \theta & 1 \\ 1 & 1 & 1 + \cos \theta\end{vmatrix}\]

Applying

\[R_2 \to R_2 - R_1\] and 

\[R_3 \to R_3 - R_1\], we get

\[∆ = \begin{vmatrix}1 & 1 & 1 \\ 0 & \sin\theta & 0 \\ 0 & 0 & \cos\theta\end{vmatrix}\] 

\[= \sin\theta\cos\theta\]

\[ = \frac{\sin2\theta}{2}\]

We know that −1 ≤ sin2θ ≤ 1.
∴ Maximum value of ∆ = \[\frac{1}{2} \times 1 = \frac{1}{2}\]