A) 0
B) {2, 4}
C) [2, 4]
D) None of the above
Correct Answer: C
Solution :
\[\left| A \right|=\left| \begin{matrix} 1 & \sin \theta & 1 \\ -\sin \theta & 1 & \sin \theta \\ -1 & -\sin \theta & 1 \\ \end{matrix} \right|\] \[\Rightarrow \] \[\left| A \right|=2(1+{{\sin }^{2}}\theta )\] Now,\[0\le {{\sin }^{2}}\theta \le 1\]for all\[\theta \in [0,\,\,2\pi ]\] \[\Rightarrow \]\[2\le 2+2{{\sin }^{2}}\theta \le 4\]for all\[\theta \in [0,\,\,2\pi ]\]. \[\therefore \]The range of\[|A|\]is [2, 4].
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