JEE Main 2019 — Mathematics Algebra

$[\mathrm{sin}⁡\theta ]x+[-\mathrm{cos}⁡\theta ]y=0$[sin⁡θ]x+[−cos⁡θ]y=0 and $[\mathrm{cos}⁡\theta ]x+y=0$[cos⁡θ]x+y=0 for infinite many solution

$$|\begin{matrix}[\mathrm{sin}⁡\theta ] & [-\mathrm{cos}⁡\theta ] \\ [\mathrm{cot}⁡\theta ] & 1\end{matrix}|=0$$∣[sin⁡θ][cot⁡θ]​[−cos⁡θ]1​∣=0

i.e. $[\mathrm{sin}\theta ]= [-\mathrm{cos}\theta ] [cot\theta ]...(i)$[sinθ]=[−cosθ][cotθ]...(i)

when $\theta \in (\frac{\pi }{2},\frac{2\pi }{3})$θ∈(2π​,32π​) then $sin\theta \in (0,\frac{1}{2})\Rightarrow [\mathrm{sin}\theta ]=0$sinθ∈(0,21​)⇒[sinθ]=0 and $-\mathrm{cos}\theta \in (0,\frac{1}{2})\Rightarrow [-\mathrm{cos}\theta ]=0$−cosθ∈(0,21​)⇒[−cosθ]=0 and

$$\mathrm{cot}\theta \in (-\frac{1}{\sqrt{3}}, 0)\Rightarrow [\mathrm{cot}\theta ]=-1$$cotθ∈(−3​1​,0)⇒[cotθ]=−1

When $\theta \in (\pi ,\frac{7\pi }{6})$θ∈(π,67π​) then $sin\theta \in (-\frac{1}{2}, 0)\Rightarrow [\mathrm{sin}\theta ]=-1$sinθ∈(−21​,0)⇒[sinθ]=−1 and $-\mathrm{cos}\theta \in (\frac{\sqrt{3}}{2}, 1)\Rightarrow [-\mathrm{cos}\theta ]=0$−cosθ∈(23​​,1)⇒[−cosθ]=0 and

$$\mathrm{cot}\theta \in (\sqrt{3}, \infty )\Rightarrow [\mathrm{cot}\theta ]\in {2,3,4,...\infty }$$cotθ∈(3​,∞)⇒[cotθ]∈2,3,4,...∞

When $\theta \in (\frac{\pi }{2},\frac{2\pi }{3})$θ∈(2π​,32π​) then the equation $(i)$(i) satisfied therefore infinitely many solution.

When $\theta \in (\pi ,\frac{7\pi }{6})$θ∈(π,67π​) then the equation $(i)$(i) not satisfied therefore unique solution.