The number of all possible values of `theta`,where `0 lt thetalt 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`
and (xyz)`sin 3 theta = (y + 2z) cos 3theta +y sin 3theta` have a solution `(x_0,y_0,z_0)` with `y_0z_0ne0` is

Options

A. 3
B. 2
C. 1
D. 4

Answer

To solve the problem, we need to find the number of possible values of \(\theta\) in the interval \(0 < \theta < \pi\) for which the given system of equations has a solution \((x_0, y_0, z_0)\) with \(y_0 z_0 \neq 0\). The equations are:

1. \((y + z) \cos 3\theta = (xyz) \sin 3\theta\)
2. \(x \sin 3\theta = \frac{2 \cos 3\theta}{y} + \frac{2 \sin 3\theta}{z}\)
3. \((xyz) \sin 3\theta = (y + 2z) \cos 3\theta + y \sin 3\theta\)

### Step 1: Analyze the first equation
From the first equation, we can rearrange it as follows:

\[ (y + z) \cos 3\theta - (xyz) \sin 3\theta = 0 \]

This implies that either \(\cos 3\theta = 0\) or \((y + z) = (xyz) \tan 3\theta\).

### Step 2: Analyze the second equation
Rearranging the second equation gives:

\[ x \sin 3\theta - \frac{2 \cos 3\theta}{y} - \frac{2 \sin 3\theta}{z} = 0 \]

Multiplying through by \(yz\) to eliminate the denominators:

\[ xyz \sin 3\theta - 2z \cos 3\theta - 2y \sin 3\theta = 0 \]

### Step 3: Analyze the third equation
From the third equation, we rearrange it as:

\[ (xyz) \sin 3\theta - (y + 2z) \cos 3\theta - y \sin 3\theta = 0 \]

This can be simplified to:

\[ xyz \sin 3\theta - y \sin 3\theta - (y + 2z) \cos 3\theta = 0 \]

### Step 4: Equate the equations
Now we have three equations. We can set them equal to each other:

1. From the first and second equations, we can equate the expressions involving \(\tan 3\theta\).
2. From the second and third equations, we can derive another relationship.

### Step 5: Solve for \(\tan 3\theta\)
From the deductions, we find that:

\[ \tan 3\theta = \frac{y + z}{xyz} \]

And from the second equation, we can derive:

\[ \tan 3\theta = \frac{2z \cos 3\theta + 2y \sin 3\theta}{x} \]

### Step 6: Find values of \(\theta\)
Now, we need to find the values of \(\theta\) such that \(\tan 3\theta = 1\):

\[ 3\theta = \frac{\pi}{4} + n\pi \]

This gives:

\[ \theta = \frac{\pi}{12} + \frac{n\pi}{3} \]

### Step 7: Determine valid \(n\) values
We need \(0 < \theta < \pi\):

1. For \(n = 0\):
\[ \theta = \frac{\pi}{12} \]
2. For \(n = 1\):
\[ \theta = \frac{\pi}{12} + \frac{\pi}{3} = \frac{5\pi}{12} \]
3. For \(n = 2\):
\[ \theta = \frac{\pi}{12} + \frac{2\pi}{3} = \frac{9\pi}{12} = \frac{3\pi}{4} \]
4. For \(n = 3\):
\[ \theta = \frac{\pi}{12} + \pi = \frac{13\pi}{12} \quad (\text{not valid as } > \pi) \]

### Conclusion
The valid values of \(\theta\) in the interval \(0 < \theta < \pi\) are:

1. \(\frac{\pi}{12}\)
2. \(\frac{5\pi}{12}\)
3. \(\frac{3\pi}{4}\)

Thus, the number of all possible values of \(\theta\) is **3**.