Consider the following statements in respect of an arbitrary complex number Z :
1. The difference of Z and its conjugate is an imaginary number.
2. The sum of Z and its conjugate is a real number.
Which of the above statements is/are correct?

Options

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

Answer

To solve the problem, we need to analyze the two statements regarding an arbitrary complex number \( Z \).

Let’s denote the complex number \( Z \) as:
\[ Z = a + ib \]
where \( a \) and \( b \) are real numbers, and \( i \) is the imaginary unit.

### Step 1: Analyze the first statement
The first statement is: "The difference of \( Z \) and its conjugate is an imaginary number."

1. **Find the conjugate of \( Z \)**:
The conjugate of \( Z \) is given by:
\[ \overline{Z} = a - ib \]

2. **Calculate the difference \( Z - \overline{Z} \)**:
\[ Z - \overline{Z} = (a + ib) - (a - ib) = a + ib - a + ib = 2ib \]

3. **Determine the nature of the result**:
The result \( 2ib \) is purely imaginary since it has no real part (it is of the form \( bi \) where \( b \) is a real number).

Thus, the first statement is **correct**.

### Step 2: Analyze the second statement
The second statement is: "The sum of \( Z \) and its conjugate is a real number."

1. **Calculate the sum \( Z + \overline{Z} \)**:
\[ Z + \overline{Z} = (a + ib) + (a - ib) = a + ib + a - ib = 2a \]

2. **Determine the nature of the result**:
The result \( 2a \) is purely real since it has no imaginary part (it is of the form \( 2a \) where \( a \) is a real number).

Thus, the second statement is also **correct**.

### Conclusion
Both statements are correct. Therefore, the answer is:
**Both statements 1 and 2 are correct.**