Question 74 - 2019 (12 Apr Shift 2)

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

(1) have infinitely many solutions if $\theta \in \left(\dfrac{\pi}{2}, \dfrac{2\pi}{3}\right)$ and has a unique solution if $\theta \in \left(\pi, \dfrac{7\pi}{6}\right)$.

(2) has a unique solution if $\theta \in \left(\dfrac{\pi}{2}, \dfrac{2\pi}{3}\right) \cup \left(\pi, \dfrac{7\pi}{6}\right)$.

(3) has a unique solution if $\theta \in \left(\dfrac{\pi}{2}, \dfrac{2\pi}{3}\right)$ and have infinitely many solutions if $\theta \in \left(\pi, \dfrac{7\pi}{6}\right)$.

(4) have infinitely many solutions if $\theta \in \left(\dfrac{\pi}{2}, \dfrac{2\pi}{3}\right) \cup \left(\pi, \dfrac{7\pi}{6}\right)$.