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

Moderate

A

have infinitely many solutions if $θ \in (\frac{π}{2}, \frac{2 π}{3})$ and has a unique solutions is $θ \in (π \frac{7 π}{6})$
B

has a unique solution if $θ \in (\frac{π}{2}, \frac{2 π}{3}) \cup (π, \frac{7 π}{6})$
C

has a unique solution if $θ \in (\frac{π}{2}, \frac{2 π}{3})$ and have infinitely many solutions if $θ (π, \frac{7 π}{6})$
D

have infinitely many solutions if $θ \in (\frac{π}{2}, \frac{2 π}{3}) \cup (π, \frac{7 π}{6})$

Solution

Given system of linear equations is
$[\sin θ] x + [- \cos θ] y = 0 ....... (i)$
$And [\cot θ] x + y = 0 - - - - - (i i)$
$Where, [x] denotes the greatest integer \leq x$
$Here, Δ = |\begin{array}{cc} \sin θ] & [- \cos θ] \\ [\cot θ] & 1 \end{array}|$
$\Rightarrow Δ = [\sin θ] - [- \cos θ] [\cot θ]$
$When \sin θ \in (\frac{\sqrt{3}}{2}, 1)$
$\begin{aligned} \Rightarrow [\sin θ] = 0 \\ - \cos θ \in (0, \frac{1}{2}) \\ \Rightarrow [- \cos θ] = 0 \end{aligned}$
$a n d \cot θ \in (- \frac{1}{\sqrt{3}}, 0)$
$\Rightarrow [\cot θ] = - 1$
$So, Δ = [\sin θ] - [- \cos θ] [\cot θ]$
$- (0 \times (- 1)) = 0 [ from Eqs. (iii) ,(iv) and (v)]$
$\underline{\underline{Thus,}} for θ \in (\frac{π}{2}, \frac{2 π}{3}), the given system have infinitely many solutions.$
$When θ \in (π, \frac{7 π}{6}), \sin θ \in (- \frac{1}{2}, 0)$
$\begin{array}{l} \Rightarrow [\sin θ] = - 1 \\ - \cos θ \in (\frac{\sqrt{3}}{2}, 1) \Rightarrow [\cos θ] = 0 \end{array}$
$And \cot θ \in (\sqrt{3}, \infty) \Rightarrow [\cot θ] = n, n \in N .$
$Δ = - 1 - (0 \times n) = - 1$
$Thus, for θ \in (π, \frac{7 π}{6}), the given system has a unique solution.$

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