Question

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

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

Answer 1

The correct option is D have infinitely many solutions if $\theta \in (\frac{\pi }{2},\frac{2\pi }{3})$ and has a unique solution if $\theta \in (\pi ,\frac{7\pi }{6})$

$\left[\sin \theta \right]x+[–\cos \theta ]y=0...\left(1\right)$
$\left[\cot \theta \right]x+y=0...\left(2\right)$
Case $\text{I}$
$\theta \in (\frac{\pi }{2},\frac{2\pi }{3})\sin \theta \in (\frac{\sqrt{3}}{2},1)\Rightarrow \left[\sin \theta \right]=0\cos \theta \in (-\frac{1}{2},0)\Rightarrow [-\cos \theta ]=0\cot \theta \in (-\frac{1}{\sqrt{3}},0)\Rightarrow \left[\cot \theta \right]=-1$
Equation $\left(1\right)$ and $\left(2\right)$ will be
$0x+0y=0-x+y=0$
So, the system will have infinitely many solution

Case $\text{II}$
$\theta \in (\pi ,\frac{7\pi }{6})\sin \theta \in (-\frac{1}{2},0)\Rightarrow \left[\sin \theta \right]=-1\cos \theta \in (-1,-\frac{\sqrt{3}}{2})\Rightarrow [-\cos \theta ]=0\cot \theta \in (\sqrt{3},\infty )\Rightarrow \left[\cot \theta \right]=\{1,2,3,..\}$
Equation $\left(1\right)$ and $\left(2\right)$ will be
$\left[\sin \theta \right]x+[-\cos \theta ]y=0\Rightarrow x=0\left[\cot \theta \right]x+y=0\Rightarrow \left[I\right]x+y=0I=\{1,2,3,..\}\Rightarrow x+y=0,2x+y=0,\cdots \infty$
Each line will intersect $x=0$ at only one point i.e. $y=0$
So, the system will have unique solution.