NIMCET 2008 — Mathematics PYQ

1. Identify the Structure of the Expression

The given expression is a product of $n$ binomials of the form $(x+a_i)$.

Let's look at the constants: $1,4,9,16,\dots ,400$.

These are perfect squares: $1^2,2^2,3^2,4^2,\dots ,20^2$.

So, there are 20 terms in the product:

2. Understand Coefficient Rules

For a polynomial $P(x)=(x+a_1)(x+a_2)\dots (x+a_n)$:

• The coefficient of $x^n$ is $1$.

• The coefficient of $x^{n-1}$ is the sum of the constants $(a_1+a_2+\cdots +a_n)$.

In our case, $n=20$. We are asked to find the coefficient of $x^{19}$ (which is $x^{n-1}$).

3. Calculate the Sum of Constants

The coefficient of $x^{19}$ is the sum of the squares from $1$ to $20$:

4. Apply the Sum of Squares Formula

The formula for the sum of squares of the first $n$ natural numbers is:

Substituting $n=20$:

Now, simplify the fraction:

Conclusion

The coefficient of $x^{19}$ is the sum of the square constants in each bracket, which totals $2870$.

Correct Option: (a)