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