Q102 / 155 · WBJEE 2025 · Mathematics

Let $f(\theta)=\left|\begin{array}{ccc}1 & \cos \theta & -1 \\ -\sin \theta & 1 & -\cos \theta \\ -1 & \sin \theta & 1\end{array}\right|$. Suppose $A$ and $B$ are respectively maximum and minimum values of $f(\theta)$. Then $(A, B)$ is equal to

  • A
    $(2,1)$
  • Correct answer
    B
    $(2,0)$
  • C
    $(\sqrt{2}, 1)$
  • D
    $\left(2, \frac{1}{\sqrt{2}}\right)$

Solution

$\begin{aligned} & f(\theta)=\left|\begin{array}{ccc}1 & \cos \theta & -1 \\ -\sin \theta & 1 & -\cos \theta \\ -1 & \sin \theta & 1\end{array}\right| \text { (after expansion) } \\ & f(\theta)=1+\sin 2 \theta \\ & A=2, B=0 \quad(A, B)=(2,0)\end{aligned}$

Hence, the correct answer is option: 'B'.

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