Question

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

• A. have infinitely many solutions if $\theta \in (\frac{\pi}{2} , \frac{2\pi}{3}) \cup (\pi , \frac{7 \pi}{6})$
• B. have infinitely many solutions if $\theta \in (\frac{\pi}{2} , \frac{2\pi}{3}$and has a unique solution if $\theta \in (\pi , \frac{7\pi}{6})$
• C. has a unique solution if $\theta \in (\frac{\pi}{2} , \frac{2\pi}{3})$and have infinitely many solutions if $\theta \in (\pi, \frac{7 \pi}{6})$
• D. has a unique solution if $\theta \in (\frac{\pi}{2} , \frac{2\pi}{3}) \cup (\pi, \frac{7\pi}{6})$

Answer 1

The Correct Option is B

[sin$\theta$]x + [-cos$\theta$]y = 0 and [cos$\theta$] x + y = 0
for infinite many solution
$\left|\left[\begin{array}{cc}\sin \theta \\ \cos \theta\end{array}\right] \quad \begin{array}{c}{[-\cos \theta]} \\ 1\end{array}\right|=0$
ie [sin $\theta$] = - [cos $\theta$] [cot $\theta$] (1)
when $\theta \in (\frac{\pi}{2} , \frac{2\pi}{3}) \Rightarrow \ \ sin \theta \in \bigg(0, \frac{1}{2}\bigg)$
$ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ -cos\theta \in \bigg(0, \frac{1}{2}\bigg)$
$ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ cot\theta \in \bigg(-\frac{1}{\sqrt{3}} , 0\bigg)$
when $\theta \in \bigg(\pi , \frac{7 \pi}{6}\bigg) \ \Rightarrow \ \ sin \theta \in \bigg(- \frac{1}{2} , 0\bigg)$
$ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ -cos\theta \in \bigg(\frac{\sqrt{3}}{2} , 1\bigg)$
$ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ cot \theta \in (\sqrt{3} , \infty)$
when $\theta \in (\frac{\pi}{2} , \frac{2\pi}{3})$ then equation (i) satisfied
there fore infinite many solution.
when $\theta \in \bigg(\pi , \frac{7\pi}{6}\bigg)$ then equation (i) not
satisfied there fore infinite unique solution.