Statement-1, If `z_(1),z_(2),z_(3),……………….,z_(n)` are uni-modular complex numbers, then
`|z_(1)+z+(2)+…………+z_(n)|=|1/z_(1)+1/z_(2)+…………..+1/z_(n)|`
Statement-2: For any complex number z, `zbarz=|z|^(2)`

Options

A. Statement-1 is True, Statement-2 is True: Statement-2 is a correct exp,anation for statement-1.
B. Statement-1 is true, statement -2 is true, Statement-2 is not a correct explanation for statement-1.
C. Statement-1 is True, statement-2 is false,
D. statement-1 is False, Statement-2 is true.

Answer

To solve the problem, we need to prove that if \( z_1, z_2, \ldots, z_n \) are uni-modular complex numbers, then

\[ |z_1 + z_2 + \ldots + z_n| = \left| \frac{1}{z_1} + \frac{1}{z_2} + \ldots + \frac{1}{z_n} \right| \]

### Step-by-step Solution:

1. **Understanding Uni-modular Complex Numbers**:
- A complex number \( z \) is called uni-modular if its modulus is 1, i.e., \( |z| = 1 \).
- Therefore, for each \( z_i \), we have \( |z_i| = 1 \).

2. **Expressing the Conjugate**:
- For any complex number \( z \), the conjugate \( \overline{z} \) is given by \( \overline{z} = \frac{1}{z} \) when \( |z| = 1 \).
- Thus, we can write:
\[ \overline{z_1} = \frac{1}{z_1}, \quad \overline{z_2} = \frac{1}{z_2}, \quad \ldots, \quad \overline{z_n} = \frac{1}{z_n} \]

3. **Taking the Modulus of the Sum**:
- We need to compute \( |z_1 + z_2 + \ldots + z_n| \).
- The modulus of a sum of complex numbers can be expressed as:
\[ |z_1 + z_2 + \ldots + z_n| = |(z_1 + z_2 + \ldots + z_n)| \]

4. **Taking the Conjugate of the Sum**:
- The conjugate of the sum is:
\[ \overline{z_1 + z_2 + \ldots + z_n} = \overline{z_1} + \overline{z_2} + \ldots + \overline{z_n} \]
- Substituting the expressions for the conjugates:
\[ \overline{z_1 + z_2 + \ldots + z_n} = \frac{1}{z_1} + \frac{1}{z_2} + \ldots + \frac{1}{z_n} \]

5. **Equating the Moduli**:
- Since \( |z| = |\overline{z}| \) for any complex number \( z \), we have:
\[ |z_1 + z_2 + \ldots + z_n| = \left| \frac{1}{z_1} + \frac{1}{z_2} + \ldots + \frac{1}{z_n} \right| \]

6. **Conclusion**:
- Therefore, we have shown that:
\[ |z_1 + z_2 + \ldots + z_n| = \left| \frac{1}{z_1} + \frac{1}{z_2} + \ldots + \frac{1}{z_n} \right| \]
- This proves Statement 1 is true.