NIMCET 2011 — Mathematics PYQ

The sum of the squares of the first $n$ natural numbers is given by:

2. Formula for the mean:

The mean is the total sum divided by the number of terms ($n$):

3. Set up the equation:

We are given that the mean is 11.

4. Solve for $n$:

Multiply both sides by 6:

Rearrange into a quadratic equation:

5. Factorize the equation:

We need to split the middle term ($3n$) such that the factors multiply to $2\times -65=-130$. The numbers are $13$ and $-10$.

This gives two possible values:

• $2n+13=0\Rightarrow n=-\frac{13}{2}$

• $n-5=0\Rightarrow n=5$

Conclusion:

Since $n$ represents the number of natural numbers, it must be a positive integer. Therefore, $n=5$.

The correct option is (c) 5.