Statement – 1: The variance of first n even natural number is n² - 1 / 4

Statement – 2: The sum of first n natural number is n(n + 1) / 2 and the sum of squares of first n natural number is n(n + 1)(2n + 1) / 6.

• (a) Statement – 1 is true, statement – 2 is true; statement – 2 is a correct explanation 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.

To evaluate the statements provided, let's analyze each one carefully.

Understanding Statement 1

Statement 1 claims that the variance of the first n even natural numbers is given by the formula n² - 1 / 4. This statement is indeed true. The variance measures how spread out the numbers are from their mean, and for the first n even numbers, this formula accurately represents that spread.

Examining Statement 2

Statement 2 states that the sum of the first n natural numbers is n(n + 1) / 2 and the sum of the squares of the first n natural numbers is n(n + 1)(2n + 1) / 6. Both parts of this statement are also true and are well-known formulas in mathematics.

Linking the Statements

Now, we need to determine if Statement 2 serves as a correct explanation for Statement 1. While both statements are true, the sum and sum of squares do not directly explain the variance of the first n even numbers. Therefore, Statement 2 does not provide a correct explanation for Statement 1.

Final Evaluation

Based on the analysis:

• Statement 1 is true.
• Statement 2 is true.
• Statement 2 does not explain Statement 1.

The correct answer is: (b) Statement – 1 is true, statement – 2 is true; statement – 2 is not a correct explanation for statement – 1.