The number of all possible values of `theta,` where, for which the system of equations `(y+z)cos3theta=(x y z)sin3theta` `xsin3theta=(2cos3theta)/y+(2sin3theta)/z` `(x y z)sin3theta=(y+2z)cos3theta+ysin3theta` has a solution `(x_0,y_0,z_0)` with `y_0z_0!=0` is

Answer

3

Explanation

We have
`(y+z) cos 3 theta=(xyz) sin 3 theta` ...(1)
`xyz sin 3 theta=(2 cos 3 theta)z+(2 sin 3 theta)y` ...(ii)
`(xyz) sin 3 theta=(y+2z) cos 3 theta+y sin 3 theta` ...(iii)
`:. (y+z) cos 3 theta=(2 cos 3 theta)z+(2 sin 3 theta)y`
`=(y+2z) cos 3 theta+y sin 3 theta`
`rArr y(cos 3 theta-2 sin 3 theta)=z cos 3 theta`
and `y(sin 3 theta- cos 3 theta)=0`
Since, `y, z ne 0, sin 3 theta-cos 3 theta=0` or `tan 3 theta=0`
`rArr 3 theta=npi +pi/4, n in Z`
`rArr theta=((4n+1)pi)/12, n in Z`
`rArr theta=pi/12, (5pi)/12, (9pi)/12`