NIMCET 2008 — Mathematics PYQ

1. Understand the Given Functional Property

We are given the equation:

This tells us that if two inputs sum to $1$ (i.e., $x$ and $1-x$), the sum of their functional values is always $2$.

2. Analyze the Series

Let the required sum be $S$:

The series contains $2000$ terms in total.

3. Pair the Symmetric Terms

We can pair the first term with the last term, the second term with the second-to-last term, and so on.

• Pair 1: $f\left(\frac{1}{2001}\right)+f\left(\frac{2000}{2001}\right)$

  Since $\frac{1}{2001}+\frac{2000}{2001}=1$, using the given property $f(x)+f(1-x)=2$:

  $$f\left(\frac{1}{2001}\right)+f\left(\frac{2000}{2001}\right)=2$$

• Pair 2: $f\left(\frac{2}{2001}\right)+f\left(\frac{1999}{2001}\right)$

  Since $\frac{2}{2001}+\frac{1999}{2001}=1$:

  $$f\left(\frac{2}{2001}\right)+f\left(\frac{1999}{2001}\right)=2$$

4. Calculate Total Number of Pairs

Since there are $2000$ terms in the series and we are grouping them in pairs of two, the total number of pairs is:

Each pair contributes a value of $2$ to the total sum.

5. Final Summation

Conclusion

By pairing terms that sum to unity, we find that the total sum of the $2000$ terms is $2000$.

Correct Option: (a)