# If [x] denotes the greatest integer

Question:

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 $\theta \in\left(\frac{\pi}{2}, \frac{2 \pi}{3}\right) \cup\left(\pi, \frac{7 \pi}{6}\right)$

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

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

4. 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 Option: , 2

Solution:

$[\sin \theta] x+[-\cos \theta] y=0$ and $[\cos \theta] x+y=0$ for infinite many solution

$\left|\left[\begin{array}{c}\sin \theta\end{array}\right]\left[\begin{array}{c}-\cos \theta\end{array}\right]\right|=0$

ie $[\sin \theta]=-[\cos \theta][\cot \theta]$   ...............(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 $\theta \in\left(\pi, \frac{7 \pi}{6}\right) \Rightarrow \sin \theta \in\left(-\frac{1}{2}, 0\right)$

$-\cos \theta \in\left(\frac{\sqrt{3}}{2}, 1\right)$

$\cot \theta \in(\sqrt{3}, \infty)$

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.