A) have infinitely many solutions if \[\theta \in \left( \frac{\pi }{2},\frac{2\pi }{3} \right)\cup \left( \pi ,\frac{7\pi }{6} \right)\]
B) have infinitely many solutions if\[\theta \in \left( \frac{\pi }{2},\frac{2\pi }{3} \right)\]and has a unique solution if\[\theta \in \left( \pi ,\frac{7\pi }{6} \right)\]
C) has a unique solution if \[\theta \in \left( \frac{\pi }{2},\frac{2\pi }{3} \right)\]and have infinitely many solutions if \[\theta \in \left( \pi ,\frac{7\pi }{6} \right)\]
D) has a unique solution if\[\theta \in \left( \frac{\pi }{2},\frac{2\pi }{3} \right)\cup \left( \pi ,\frac{7\pi }{6} \right)\]
Correct Answer: B
Solution :
\[[\sin \theta ]x+[-cos\theta ]y=0\]and \[[cos\theta ]x+y=0\]for infinite many solution \[\left| \begin{matrix} \left[ \sin \theta \right] & \left[ -\cos \theta \right] \\ \left[ \cos \theta \right] & 1 \\ \end{matrix} \right|=0\] ie\[\left[ \sin \theta \right]=-\left[ \cos \theta \right]\left[ \cot \theta \right]\] (1) when\[\theta \in \left( \frac{\pi }{2},\frac{2\pi }{3} \right)\Rightarrow \sin \theta \in \left( 0,\frac{1}{2} \right)\] \[-\cos \theta \in \left( 0,\frac{1}{2} \right)\] \[\cot \theta \in \left( -\frac{1}{\sqrt{3}},0 \right)\] when\[\theta \in \left( \pi ,\frac{7\pi }{6} \right)\Rightarrow \sin \theta \in \left( -\frac{1}{2},0 \right)\] \[-\cos \theta \in \left( \frac{\sqrt{3}}{2},1 \right)\] \[\cot \theta \in \left( \sqrt{3},\infty \right)\] when\[\theta \in \left( \frac{\pi }{2},\frac{2\pi }{3} \right)\]then equation (i) satisfied there fore infinite many solution. when\[\theta \in \left( \pi ,\frac{7\pi }{6} \right)\]then equation (i) not satisfied there fore infinite unique solution.
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