Let C be the set of complex number and `z_(1),z_(2)` are in C.
1. agr `(z_(1))=agr(z_(2))impliesz_(1)=z_(2)`
2. `|z_(1)|=|z_(2)|impliesz_(1)=z_(2)`
Which of the statements givben above is/are correct?

Options

A. 1only
B. 2 only
C. Both 1 and 2
D. neither 1 nor 2

Explanation

To solve the problem, we need to analyze the two statements regarding complex numbers \( z_1 \) and \( z_2 \).

### Step 1: Analyze the first statement
The first statement is:
1. If \( \text{arg}(z_1) = \text{arg}(z_2) \), then \( z_1 = z_2 \).

Let \( z_1 = a_1 + i b_1 \) and \( z_2 = a_2 + i b_2 \). The argument of a complex number is defined as:
\[ \text{arg}(z) = \tan^{-1}\left(\frac{b}{a}\right) \]
Thus, if \( \text{arg}(z_1) = \text{arg}(z_2) \), we have:
\[ \tan^{-1}\left(\frac{b_1}{a_1}\right) = \tan^{-1}\left(\frac{b_2}{a_2}\right) \]
This implies:
\[ \frac{b_1}{a_1} = \frac{b_2}{a_2} \]
However, this does not necessarily mean that \( a_1 = a_2 \) and \( b_1 = b_2 \). For example, if \( z_1 = 1 + 3i \) and \( z_2 = 2 + 6i \), both have the same argument, but they are not equal. Therefore, the first statement is **false**.

### Step 2: Analyze the second statement
The second statement is:
2. If \( |z_1| = |z_2| \), then \( z_1 = z_2 \).

The modulus of a complex number is given by:
\[ |z| = \sqrt{a^2 + b^2} \]
So, if \( |z_1| = |z_2| \), we have:
\[ \sqrt{a_1^2 + b_1^2} = \sqrt{a_2^2 + b_2^2} \]
Squaring both sides gives:
\[ a_1^2 + b_1^2 = a_2^2 + b_2^2 \]
This means that the sum of the squares of the real and imaginary parts of \( z_1 \) and \( z_2 \) are equal. However, this does not imply that \( a_1 = a_2 \) and \( b_1 = b_2 \). For example, \( z_1 = 1 + i \) and \( z_2 = -1 - i \) both have a modulus of \( \sqrt{2} \) but are not equal. Therefore, the second statement is also **false**.

### Conclusion
Both statements are false. Thus, the answer is that neither statement is correct.