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

detailed solution

Correct option is A

Given system of linear equations is sinθx+−cosθy=0              .......(i) And  [cot⁡θ]x+y=0-----ii Where, [x] denotes the greatest integer ≤xHere ,Δ=sin⁡θ][−cos⁡θ][cot⁡θ]1⇒  Δ =sin θ−−cosθcotθ When sin⁡θ∈32,1⇒[sin⁡θ]=0−cos⁡θ∈0,12⇒[−cos⁡θ]=0and   cotθ∈−13,0⇒  cotθ=−1 So,  Δ=[sin⁡θ]−[−cos⁡θ][cot⁡θ]−(0×(−1))=0 [ from Eqs. (iii) ,(iv) and (v) ]  Thus, __ for θ∈π2,2π3, the given system have infinitely many solutions.  When θ∈π,7π6,sin⁡θ∈−12,0⇒[sin⁡θ]=−1−cos⁡θ∈32,1⇒[cos⁡θ]=0 And  cot⁡θ∈(3,∞)⇒[cot⁡θ]=n,n∈N. Δ=−1−0×n=−1 Thus, for θ∈π,7π6, the given system has a unique solution.