If [x] denotes the greatest integer ≤ x, then the system of linear equations [sinθ]x + [-cosθ]y = 0, [cotθ]x + y = 0. A. Have infinitely many solutions if θ∈(π/2, 2π/3) and has a unique solution if θ∈(π, 7π/6) B. Have infinitely many solutions if θ∈(π/2, 2π/3)∪(π, 7π/6) C. Has a unique solution if θ∈(π/2, 2π/3) and have infinitely many solutions if θ∈(π, 7π/6) D. Has a unique solution if θ∈(π/2, 2π/3)∪(π, 7π/6)

Correct option is B. Have infinitely many solutions if θ∈(π/2, 2π/3)∪(π, 7π/6)
[sinθ]x + [-cosθ]y = 0 and [cotθ]x + y = 0
For infinite many solutions |
[sinθ][-cosθ]
[cotθ] 1
| = 0 i.e [sinθ] = -[cosθ][cotθ].
When θ∈(π/2, 2π/3) ⇒ sinθ∈(1/2, 1), -cosθ∈(0, 1/2), cotθ∈(-1/√3, 0)
When θ∈(π, 7π/6) ⇒ sinθ∈(-1/2, 0), -cosθ∈(√3/2, 1), cotθ∈(√3, ∞)
When θ∈(π/2, 2π/3) then equation (i) satisfied therefore infinite many solutions.
When θ∈(π, 7π/6) then equation (i) not satisfied therefore infinite unique solution.