The number of all possible values of θ where 0<θ<π for which the system of equations (y+z)cos3θ=(xyz)sin3θ,xsin3θ=2cos3θy+2sin3θz, (xyz) sin3θ=(y+2z)cos3θ+ysin3θ has a solution (x0,y0,z0) with y0z0≠0 is

Solving given equations we getsin3θ=cos3θ ⇒tan3θ=1=tanπ4 ⇒3θ=nπ+π4,n∈z ⇒θ=nπ3+π12,n∈z No. of solutions in (0,π) are 3For n=1,θ=π12n=2,θ=5π12 n=3,θ=3π4

The number of all possible values of θ where 0<θ<π for which the system of equations (y+z)cos3θ=(xyz)sin3θ,xsin3θ=2cos3θy+2sin3θz, (xyz) sin3θ=(y+2z)cos3θ+ysin3θ has a solution (x0,y0,z0) with y0z0≠0 is

Solving given equations we getsin3θ=cos3θ ⇒tan3θ=1=tanπ4 ⇒3θ=nπ+π4,n∈z ⇒θ=nπ3+π12,n∈z No. of solutions in (0,π) are 3For n=1,θ=π12n=2,θ=5π12 n=3,θ=3π4

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The number of all possible values of $θ$ where $0 < θ < π$ for which the system of equations $(y + z) \cos 3 θ = (x y z) \sin 3 θ,$ $x \sin 3 θ = \frac{2 \cos 3 θ}{y} + \frac{2 \sin 3 θ}{z},$ $(x y z) \sin 3 θ = (y + 2 z) \cos 3 θ + y \sin 3 θ$ has a solution $(x_{0}, y_{0}, z_{0})$ with $y_{0} z_{0} \neq 0$ is

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answer is D.

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Detailed Solution

Solving given equations we get
$\sin 3 θ = \cos 3 θ \Rightarrow \tan 3 θ = 1 = \tan \frac{π}{4} \Rightarrow 3 θ = n π + \frac{π}{4}, n \in z \Rightarrow θ = \frac{n π}{3} + \frac{π}{12}, n \in z$
No. of solutions in $(0, π)$ are 3
For $n = 1, θ = \frac{π}{12}$
$n = 2, θ = \frac{5 π}{12} n = 3, θ = \frac{3 π}{4}$