A) \[\left( 1,\frac{5}{2} \right]\]
B) \[\left[ \frac{5}{2},4 \right)\]
C) \[\left( 0,\frac{3}{2} \right]\]
D) \[\left[ 2,3 \right)\]
Correct Answer: D
Solution :
| \[\left| A \right|=\left| \begin{matrix} 1 & \sin \theta & 1 \\ -\sin \theta & 1 & \sin \theta \\ -1 & -\sin \theta & 1 \\ \end{matrix} \right|\] |
| \[=2(1+si{{n}^{2}}\theta )\] |
| \[\theta \in \left( \frac{3\pi }{4},\frac{5\pi }{4} \right)\]\[\Rightarrow \]\[\frac{1}{\sqrt{2}}<\sin \theta <\frac{1}{2}\]\[\Rightarrow \]\[0\le {{\sin }^{2}}\theta <\frac{1}{2}\] |
| \[1+0\le 1+{{\sin }^{2}}\theta <\frac{1}{2}+1\] |
| \[1\le {{\sin }^{2}}\theta <\frac{3}{2}\] |
| \[2\le 2(1+si{{n}^{2}}\theta )<3.\] |
| \[\therefore \] \[\left| A \right|\in [2,3).\] |
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