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Question

If [x] denotes the greatest integer x, then the system of linear equations [sinθ]x+[cosθ]y=0, [cotθ]x+y=0

A
has a unique solution if θ(π2,2π3)(π,7π6)
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B
have infinitely many solution if θ(π2,2π3)(π,7π6)
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C
has a unique solution if θ(π2,2π3) and have infinitely many solutions if θ(π,7π6)
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D
have infinitely many solutions if θ(π2,2π3) and has a unique solution if θ(π,7π6)
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Solution

The correct option is D have infinitely many solutions if θ(π2,2π3) and has a unique solution if θ(π,7π6)
[sinθ]x+[cosθ]y=0 ...(1)
[cotθ]x+y=0 ...(2)
Case I
θ(π2,2π3)sinθ(32,1)[sinθ]=0cosθ(12,0)[cosθ]=0cotθ(13,0)[cotθ]=1
Equation (1) and (2) will be
0x+0y=0x+y=0
So, the system will have infinitely many solution

Case II
θ(π,7π6)sinθ(12,0)[sinθ]=1cosθ(1,32)[cosθ]=0cotθ(3,)[cotθ]={1,2,3,..}
Equation (1) and (2) will be
[sinθ]x+[cosθ]y=0x=0[cotθ]x+y=0[I]x+y=0 I={1,2,3,..}x+y=0, 2x+y=0,
Each line will intersect x=0 at only one point i.e. y=0
So, the system will have unique solution.

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