The number of all possible values of `theta`, where `0 lt theta lt pi`, for which the system of equations
`(y+z) cos 3 theta = (xyz) sin 3 theta`
`x sin 3 theta=(2 cos 3 theta)/y+(2 sin 3 theta)/z`
`(xyz) sin 3 theta = (y+2z) cos 3 theta +y sin 3 theta`
has a solution `(x_(0), y_(0), z_(0))` with `y_(0)z_(0) ne 0` is _______.

Options

A. 0
B. 2
C. 3
D. 4

Answer

C

Explanation

We have,
`(y +z) "cos" 3theta = xyz "sin" 3theta " "..(i)`
`x "sin" 3theta = (2"cos" 3theta)/(y) + (2"sin" 3theta)/(z) " "… (ii)`
`xyz "sin" 3theta = (y +2z)"cos" 3theta + y"sin"3theta " "....(iii)`
From (i) and (ii), we get
`y ("cos" 3theta-2"sin" 3theta)-z"cos" 3theta = 0 " "...(iv)`
From (ii) and (iii), we get
`y("cos" 3theta- "sin" 3theta) = 0" "...(v)`
Since the given system of equations have a solutoin `(x_(0), y_(0), z_(0))` such that `y_(0)z_(0) ne 0`
`therefore "cos" 3theta -"sin" 3theta = 0 " "["From"(v)]`
`rArr "tan" 3theta = 1`
`rArr 3theta = (pi)/(4), (5pi)/(4), (9pi)/(4) " " [because 0 lt theta lt pi]`
`rArr theta = (pi)/(12), (5pi)/(12), (3pi)/(4)`
Hence, theta are three values of `theta`.