JEE Mathematics 2023 Question with Solution

Given: The numbers are $100$ consecutive positive integers starting from $a_1$, so they are $a_1, a_1+1, a_1+2, \dots, a_1+99$.

Find: The set of all values of $a_1$ for which the mean deviation about the mean is $25$.

The solution states that $a_1$ can be any natural number.

First compute the mean:

Now the mean deviation about the mean is

For these equally spaced consecutive integers, the deviations from the mean are symmetric, so the expression becomes

The sum of the first $50$ odd numbers is

Hence

This value is independent of $a_1$. Therefore, the condition is satisfied for every natural number $a_1$. The correct option is C.

Given: The numbers are consecutive integers shifted by $a_1$.

Find: Whether the mean deviation depends on $a_1$.

Adding the same constant to every observation shifts the mean by the same amount, but leaves all deviations from the mean unchanged. Therefore, the mean deviation about the mean for

is the same as that for

For $100$ consecutive integers, the data are symmetric about the mean, and the mean deviation evaluates to $25$. Hence it is true for every natural number $a_1$, so $S=\mathbb{N}$. The correct option is C.