Let `omega ne 1` be a complex cube root of unity. If `(3-3omega+2omega^(2))^(4n+3) + (2+3omega-3omega^(2))^(4n+3)+(-3+2omega+3omega^(2))^(4n+3)=0`, then the set of possible value(s) of n is are

Options

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

Answer

To solve the equation

\[ (3 - 3\omega + 2\omega^2)^{4n+3} + (2 + 3\omega - 3\omega^2)^{4n+3} + (-3 + 2\omega + 3\omega^2)^{4n+3} = 0 \]

where \(\omega\) is a complex cube root of unity (with \(\omega \neq 1\)), we will follow these steps:

### Step 1: Simplify the Terms
Recall that the properties of cube roots of unity are:
1. \(\omega^3 = 1\)
2. \(1 + \omega + \omega^2 = 0\)

Using these properties, we can simplify the terms in the equation.

### Step 2: Analyze Each Term
Let's denote:
- \(A = 3 - 3\omega + 2\omega^2\)
- \(B = 2 + 3\omega - 3\omega^2\)
- \(C = -3 + 2\omega + 3\omega^2\)

We need to analyze \(A\), \(B\), and \(C\).

#### For \(A\):
\[ A = 3 - 3\omega + 2\omega^2 \]

#### For \(B\):
\[ B = 2 + 3\omega - 3\omega^2 \]

#### For \(C\):
\[ C = -3 + 2\omega + 3\omega^2 \]

### Step 3: Substitute and Simplify
We can express \(B\) and \(C\) in terms of \(A\) using the properties of \(\omega\).

After substituting and simplifying, we find that:
\[ B = -\omega A \]
\[ C = -\omega^2 A \]

### Step 4: Substitute Back into the Equation
Now substituting these into the original equation gives:
\[ A^{4n+3} + (-\omega A)^{4n+3} + (-\omega^2 A)^{4n+3} = 0 \]

This simplifies to:
\[ A^{4n+3} + (-1)^{4n+3} \omega^{4n+3} A^{4n+3} + (-1)^{4n+3} \omega^{2(4n+3)} A^{4n+3} = 0 \]

### Step 5: Factor Out \(A^{4n+3}\)
Factoring out \(A^{4n+3}\) gives:
\[ A^{4n+3} \left(1 + (-1)^{4n+3} \omega^{4n+3} + (-1)^{4n+3} \omega^{2(4n+3)}\right) = 0 \]

### Step 6: Solve the Inner Equation
The inner equation must equal zero:
\[ 1 + (-1)^{4n+3} \omega^{4n+3} + (-1)^{4n+3} \omega^{2(4n+3)} = 0 \]

### Step 7: Analyze the Conditions
Since \((-1)^{4n+3} = -1\) for all integers \(n\), we can rewrite:
\[ 1 - \omega^{4n+3} - \omega^{2(4n+3)} = 0 \]

Using the property \(1 + \omega + \omega^2 = 0\), we can analyze the values of \(n\).

### Step 8: Check Values of \(n\)
We need to check for which values of \(n\) the equation holds:
1. For \(n = 1\): \(1 + \omega + \omega^2 = 0\) (True)
2. For \(n = 2\): \(1 + \omega^8 + \omega^{4} = 0\) (True)
3. For \(n = 3\): \(1 + \omega^{12} + \omega^{6} = 3 \neq 0\) (False)
4. For \(n = 4\): \(1 + \omega^{16} + \omega^{8} = 0\) (True)

### Conclusion
The possible values of \(n\) that satisfy the equation are:
\[ n = 1, 2, 4 \]