In the expression , the coefficient of is:

Explanation: 1. Identify the Structure of the Expression The given expression is a product of binomials of the form . Let's look at the constants: . These are perfect squares: . So, there are 20 terms in the product: 2. Understand Coefficient Rules For a polynomial : The coefficient of is . The coefficient of is the sum of the constants . In our case, . We are asked to find the coefficient of (which is ). 3. Calculate the Sum of Constants The coefficient of is the sum of the squares from to : 4. Apply the Sum of Squares Formula The formula for the sum of squares of the first natural numbers is: Substituting : Now, simplify the fraction: Conclusion The coefficient of is the sum of the square constants in each bracket, which totals . Correct Option: (a)

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NIMCET 2008 — Mathematics PYQ

NIMCET | Mathematics | 2008

In the expression $(x + 1)(x + 4)(x + 9)(x + 16) \dots (x + 400)$ , the coefficient of $x^{19}$ is:

Choose the correct answer:

Explanation

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:

P(x) = (x + 1^2)(x + 2^2)(x + 3^2) \dots (x + 20^2)

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 $(a_1 + a_2 + \dots + a_n)$ .

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 $20$ :

\text{Coefficient} = 1^2 + 2^2 + 3^2 + \dots + 20^2

\text{Coefficient} = \sum_{r=1}^{20} r^2

4. Apply the Sum of Squares Formula

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

\sum_{r=1}^{n} r^2 = \frac{n(n + 1)(2n + 1)}{6}

Substituting $n = 20$ :

\text{Coefficient} = \frac{20(20 + 1)(2 \times 20 + 1)}{6}

\text{Coefficient} = \frac{20 \times 21 \times 41}{6}

Now, simplify the fraction:

\text{Coefficient} = \frac{20}{2} \times \frac{21}{3} \times 41

\text{Coefficient} = 10 \times 7 \times 41

\text{Coefficient} = 70 \times 41 = 2870

Conclusion

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

Correct Option: (a)