If [x] denotes the greatest integer leq x,the | vedclass

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(A) The given system of equations is:$[\sin 	heta ]x + [-\cos 	heta ]y = 0 \dots (1)$$[\cot 	heta ]x + y = 0 \dots (2)$Case $I$: When $	heta \in \

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have infinitely many solutions if $	heta \in \left( {\frac{\pi }{2},\frac{{2\pi }}{3}} ight)$ and has a unique solution if $	heta \in \left( {\pi ,\frac{{7\pi }}{6}} ight)$

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(A) The given system of equations is:$[\sin 	heta ]x + [-\cos 	heta ]y = 0 \dots (1)$$[\cot 	heta ]x + y = 0 \dots (2)$Case $I$: When $	heta \in \left( \frac{\pi }{2}, \frac{2\pi }{3} ight)$$\sin 	heta \in \left( \frac{\sqrt{3}}{2}, 1 ight) \implies [\sin 	heta ] = 0$$-\cos 	heta \in \left( 0, \frac{1}{2} ight) \implies [-\cos 	heta ] = 0$$\cot 	heta \in \left( -\frac{1}{\sqrt{3}}, 0 ight) \implies [\cot 	heta ] = -1$Substituting these into the equations:$0x + 0y = 0$$-x + y = 0$The first equation is satisfied for all $(x, y)$,and the second implies $y = x$. Thus,the system has infinitely many solutions.Case $II$: When $	heta \in \left( \pi, \frac{7\pi}{6} ight)$$\sin 	heta \in \left( -\frac{1}{2}, 0 ight) \implies [\sin 	heta ] = -1$$-\cos 	heta \in \left( -\frac{\sqrt{3}}{2}, 1 ight) \implies [-\cos 	heta ] = 0$$\cot 	heta \in (\sqrt{3}, \infty) \implies [\cot 	heta ] = k$,where $k \in \{1, 2, 3, \dots\}$Substituting these into the equations:$-x + 0y = 0 \implies x = 0$$kx + y = 0 \implies k(0) + y = 0 \implies y = 0$Thus,the system has a unique solution $(0, 0)$.

If $[x]$ denotes the greatest integer $\leq x$,then the system of linear equations
$[\sin \theta ] x + [-\cos \theta ] y = 0$
$[\cot \theta ] x + y = 0$

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If $[x]$ denotes the greatest integer $\leq x$,then the system of linear equations$[\sin \theta ] x + [-\cos \theta ] y = 0$$[\cot \theta ] x + y = 0$