Let n be a positive integer and a complex number with unit modulus is a solution of the equation `Z^n+Z+1=0` , then the value of n can be

Options

A. 62
B. 155
C. 221
D. 196

Answer

To solve the problem, we need to analyze the equation \( Z^n + Z + 1 = 0 \) where \( Z \) is a complex number with unit modulus (i.e., \( |Z| = 1 \)).

### Step-by-Step Solution:

1. **Understanding the Equation**:
We start with the equation:
\[ Z^n + Z + 1 = 0 \]
Rearranging gives us:
\[ Z^n = -Z - 1 \]

2. **Using Unit Modulus**:
Since \( |Z| = 1 \), we know that \( Z^n \) also has unit modulus. Therefore, we can find the modulus of the right-hand side:
\[ |Z^n| = |-Z - 1| = |Z + 1| \]
This implies:
\[ 1 = |Z + 1| \]

3. **Analyzing \( |Z + 1| \)**:
The expression \( |Z + 1| = 1 \) means that the point \( Z \) lies on the circle of radius 1 centered at -1 in the complex plane. This can be interpreted geometrically.

4. **Finding the Condition**:
The equality \( |Z + 1| = 1 \) implies that \( Z \) must be one of the cube roots of unity. The cube roots of unity are:
\[ \omega = e^{2\pi i / 3}, \quad \omega^2 = e^{-2\pi i / 3}, \quad \text{and } 1 \]
We know that:
\[ 1 + \omega + \omega^2 = 0 \]

5. **Relating \( n \) to the Roots**:
For \( Z \) to be a solution of \( Z^n + Z + 1 = 0 \), we can express \( Z^n \) in terms of \( \omega \):
\[ Z^n = \omega^k \quad \text{for } k = 0, 1, 2 \]
This gives us:
\[ \omega^k + \omega + 1 = 0 \]
Therefore, \( Z^n \) must equal \( \omega^2 \) or \( \omega \) or \( 1 \).

6. **Finding Possible Values of \( n \)**:
The condition \( Z^n = \omega^k \) leads us to:
\[ n \equiv k \mod 3 \]
where \( k = 0, 1, 2 \). This means:
- If \( n \equiv 0 \mod 3 \), then \( Z^n = 1 \)
- If \( n \equiv 1 \mod 3 \), then \( Z^n = \omega \)
- If \( n \equiv 2 \mod 3 \), then \( Z^n = \omega^2 \)

7. **Conclusion**:
Therefore, \( n - 2 \) must be a multiple of 3, which means:
\[ n - 2 \equiv 0 \mod 3 \]
Hence, \( n \) can take values such as \( n = 2, 5, 8, 11, \ldots \)

### Checking Options:
Now we check the given options:
- **Option 1**: \( n = 62 \) → \( 62 - 2 = 60 \) (multiple of 3)
- **Option 2**: \( n = 155 \) → \( 155 - 2 = 153 \) (multiple of 3)
- **Option 3**: \( n = 221 \) → \( 221 - 2 = 219 \) (multiple of 3)
- **Option 4**: \( n = 196 \) → \( 196 - 2 = 194 \) (not a multiple of 3)

Thus, the valid values of \( n \) are 62, 155, and 221.