# If $[x]$ denotes the greatest integer $\leq x$ , then the systemof linear equations $[sin\theta]x+[-cos\theta]y=0$$[cot\theta]x+y=0$ Option 1) Have infinitely many solutions if $\theta\epsilon (\frac{\pi}{2},\frac{2\pi}{3})$ and has a unique solution if $\theta\epsilon (\pi,\frac{7\pi}{6})$. Option 2) has a unique solution if $\theta\epsilon(\frac{\pi}{2},\frac{2\pi}{3})\cup (\pi,\frac{7\pi}{6})$ Option 3) has a unique solution if $\theta\epsilon(\frac{\pi}{2},\frac{2\pi}{3})$ and have infinitely many solutions if $\theta\epsilon (\pi,\frac{7\pi}{6})$ Option 4) infinitely many solutions if $\theta\epsilon(\frac{\pi}{2},\frac{2\pi}{3})\cup$$(\pi,\frac{7\pi}{6})$

P Plabita

linear equations $[sin\theta]x+[-cos\theta]y=0$ &

$[cot\theta]x+y=0$

For infinite many solution,

$\begin{vmatrix} [sin\theta] &[-cos\theta] \\ [cos\theta] & 1 \end{vmatrix}=0$

i.e. $[sin\theta]=-[cos\theta][cot\theta]$...................(1)

* when $\theta\epsilon (\frac{\pi}{2},\frac{2\pi}{3})=>sin\theta\epsilon (0,\frac{1}{2})=>-cos\theta\epsilon (0,\frac{1}{2})$

so,    $cot\theta\epsilon (-\frac{1}{\sqrt3},0)$

* when $\theta\epsilon ({\pi},\frac{7\pi}{6})=>sin\theta\epsilon (-\frac{1}{2},0)=>-cos\theta\epsilon (\frac{\sqrt3}{2},1)$

so,    $cot\theta\epsilon ({\sqrt3},\infty )$

*when $\theta\epsilon (\frac{\pi}{2},\frac{2\pi}{3})=>eqn (1)\: \: satisfies \: \: therefore \: \: infinite\: \: many\: \: \: \: solutions$

*when $\theta\epsilon ({\pi},\frac{7\pi}{6})=>eqn (1)\: \: not\: \: satisfied \: \: therefore \: \: infinite\: \: unique\: \: \: \: solutions$

Option 1)

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

Option 2)

has a unique solution if $\theta\epsilon(\frac{\pi}{2},\frac{2\pi}{3})\cup (\pi,\frac{7\pi}{6})$

Option 3)

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

Option 4)

infinitely many solutions if $\theta\epsilon(\frac{\pi}{2},\frac{2\pi}{3})\cup$$(\pi,\frac{7\pi}{6})$

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