Verified Solution • 2025–26 Session moderate

QJEEM19Ma80b9: JEE Main 2019 Mathematics - Class 11 JEEM19Ma80b9 Solution: If [x] denotes the greatest integer x, then the system of…

If [x] denotes the greatest integer ≤\le x, then the system of linear equations [sin θ\theta ]x + [–cos θ\theta ]y = 0, [cot θ\theta ]x + y = 0

Multiple Choice Options

  • A has a unique solution if θ∈(π2,2π3)\theta \in \left( {{\pi \over 2},{{2\pi } \over 3}} \right) and have infinitely many solutions if θ∈(π,7π6)\theta \in \left( {\pi ,{{7\pi } \over 6}} \right)
  • B have infinitely many solutions if θ∈(π2,2π3)\theta \in \left( {{\pi \over 2},{{2\pi } \over 3}} \right) and has a unique solution if θ∈(π,7π6)\theta \in \left( {\pi ,{{7\pi } \over 6}} \right)
  • C have infinitely many solutions if θ∈(π2,2π3)∪(π,7π6)\theta \in \left( {{\pi \over 2},{{2\pi } \over 3}} \right) \cup \left( {\pi ,{{7\pi } \over 6}} \right)
  • D has a unique solution if θ∈(π2,2π3)∪(π,7π6)\theta \in \left( {{\pi \over 2},{{2\pi } \over 3}} \right) \cup \left( {\pi ,{{7\pi } \over 6}} \right)

Step-by-Step Verified Solution

if θ\theta ∈\in (π2,2π3)\left( {{\pi \over 2},{{2\pi } \over 3}} \right) ⇒cos⁡θ∈(−12,0)⇒−cos⁡θ∈(0,12)\Rightarrow \cos \theta \in \left( { - {1 \over 2},0} \right) \Rightarrow - \cos \theta \in \left( {0,{1 \over 2}} \right)

sin⁡θ∈(32,1)\sin \theta \in \left( {{{\sqrt 3 } \over 2},1} \right) and cot⁡θ∈(−13,0)\cot \theta \in \left( { - {1 \over {\sqrt 3 }},0} \right)

If θ∈(π,7π6)⇒cos⁡θ∈(−1,−32)⇒−cos⁡θ∈(32,1)\theta \in \left( {\pi ,{{7\pi } \over 6}} \right) \Rightarrow \cos \theta \in \left( { - 1,{{ - \sqrt 3 } \over 2}} \right) \Rightarrow - \cos \theta \in \left( {{{\sqrt 3 } \over 2},1} \right)

sin⁡θ∈(−12,0)\sin \theta \in \left( { - {1 \over 2},0} \right) and cot⁡θ∈(3,∞)\cot \theta \in \left( {\sqrt 3 ,\infty } \right)

Then in θ\theta ∈\in (π2,2π3)\left( {{\pi \over 2},{{2\pi } \over 3}} \right) ⇒\Rightarrow [sin θ\theta ] = 0; [-cos θ\theta ] = 0; [cot θ\theta ] = -1;

Hence 0 = 0 and -x + y = 0 [have infinitely solutions]

and in θ∈(π,7π6)\theta \in \left( {\pi ,{{7\pi } \over 6}} \right) ⇒\Rightarrow [sin θ\theta ] = -1; [-cos θ\theta ] = 0; [cot θ\theta ] = 1, 2. 3 ...........;

Then -x - 0.y = 0 ⇒\Rightarrow x = 0

[cot⁡θ]x+y=0[\cot \theta ]x + y = 0

⇒\Rightarrow 1.x + y = 0 or 2x + y = 0 or ....... each of the line will cut x = 0 at exactly one point.

Hence unique solutions.

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