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

 

  • 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})

 

Answers (1)

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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