The variance of numbers `x_(1),x_(2),x_(3),…..x_(n)` is V. Consider the following statements:
If every `x_(1)` is increased by 2, the variance of the new set of the new set of numbers is V.
2 If the numbers `x_(i)` is squared, the variance of the new set is `V^(2)`.
Which of the following statements is/are correct?

Options

A. 1 only
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 the variance of a set of numbers \( x_1, x_2, x_3, \ldots, x_n \) with variance \( V \).

### Step 1: Analyze the first statement
The first statement claims: "If every \( x_i \) is increased by 2, the variance of the new set of numbers is \( V \)."

**Solution:**
- Variance is a measure of how spread out the numbers are in a dataset. It is calculated as the average of the squared differences from the mean.
- When we increase every number in the dataset by a constant (in this case, 2), the spread of the numbers does not change; only the position of the numbers shifts.
- Therefore, the variance of the new set of numbers remains the same as the original variance \( V \).

**Conclusion for Statement 1:** The first statement is **true**.

### Step 2: Analyze the second statement
The second statement claims: "If the numbers \( x_i \) are squared, the variance of the new set is \( V^2 \)."

**Solution:**
- Squaring the numbers changes both their values and their spread. The variance is not simply the square of the original variance.
- The variance of a new set of numbers formed by squaring each element is not equal to the square of the original variance. Instead, it involves a more complex relationship that depends on the mean and the original variance.
- Therefore, we cannot conclude that the variance of the new set is \( V^2 \).

**Conclusion for Statement 2:** The second statement is **false**.

### Final Conclusion:
- The first statement is true, and the second statement is false. Thus, the correct option is that only the first statement is correct.