The maximum value of \( \Delta=\ \begin{vmatrix}1 & \sin\theta & 1\\-\sin\theta & 1 & \sin\theta\\-1 & -\sin\theta & 1\end{vmatrix} \) is ( \( \theta \) is real number)
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → 4
Expanding along the first row:
\( \displaystyle \Delta=1(1+\sin^2\theta)-\sin\theta(-\sin\theta+\sin\theta)+1(\sin^2\theta+1) \)
\( \displaystyle =2+2\sin^2\theta \)
Since \( 0\le\sin^2\theta\le 1 \), the maximum value is 2 + 2 = 4.