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यदि $[ x ]$ महत्तम पूर्णांक $\leq x$ है, तो रैखिक समीकरण निकाय $[\sin \theta] x +[-\cos \theta] y =0$ $[\cot \theta] x + y =0$
के अनन्त हल है यदि $\theta \in\left(\frac{\pi}{2}, \frac{2 \pi}{3}\right)$ तथा मात्र एक हल है यदि $\theta \in\left(\pi, \frac{7 \pi}{6}\right)$
के अनन्त हल है यदि $\theta \in\left(\frac{\pi}{2}, \frac{2 \pi}{3}\right) \cup\left(\pi, \frac{7 \pi}{6}\right)$
का मात्र एक हल है यदि $\theta \in\left(\frac{\pi}{2}, \frac{2 \pi}{3}\right)$ तथा अनन्त हल है यदि $\theta \in\left(\pi, \frac{7 \pi}{6}\right)$
के मात्र एक हल है यदि $\theta \in\left(\frac{\pi}{2}, \frac{2 \pi}{3}\right) \cup\left(\pi, \frac{7 \pi}{6}\right)$
Solution
$\left[ {\sin \theta } \right]x + \left[ { – \cos \theta } \right]y = 0\,\,\,\,\,\,\,…….\left( 1 \right)$
$\left[ {\cot \theta } \right]x + y = 0\,\,\,\,\,\,\,……\left( 2 \right)$
Case $I$
Whene $\theta \in \left( {\frac{\pi }{2},\frac{{2\pi }}{3}} \right)$
$\sin \theta \in \left( {\frac{{\sqrt 3 }}{2},1} \right)$
$\cos \theta \in \left( { – \frac{1}{2},0} \right) – \cos \theta \in \left( {0,\frac{1}{2}} \right)$
$\cot \theta \in \left( { – \frac{1}{{\sqrt 3 }},0} \right)$
$\left[ {\sin \theta } \right] = 0\,\,\,\,\,\left[ { – \cos \theta } \right] = 0\,\,\,\,\,\left[ {\cot \theta } \right] = – 1$
Equation $(1)$ and $(2)$ will
$\left. \begin{array}{l}
0x + 0y = 0\\
– x + y = 0
\end{array} \right]$ ystem will have infinitely many solution
Case $II$
When $\theta \in \left( {\pi ,\frac{{7\pi }}{6}} \right)\,\,\sin \theta \in \left( { – \frac{1}{2},0} \right)$
$\cos \theta \in \left( { – 1,\frac{{\sqrt 3 }}{2}} \right)$
$\cot \theta \in \left( {\sqrt 3 ,\infty } \right)$
$\left[ {\sin \theta } \right] = – 1,\left[ {\cos \theta } \right] = – 1$
$\left[ {\cot \theta } \right] = \left\{ {1,2,3,……} \right\}$
$-x-y=0$
$Ix+y=0$ I={1,2,…..}
It will have unique solution in all cases $x=0,y=0$