A-Level AQA Maths Pure General Binomial Expansion: The first three terms, in ascend

The first three terms, in ascending powers of $x$, of the binomial expansion of $(9 + 2x)^2$ are given by $(9 + 2x)^2 ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } a + \frac{x^2}{54} ext{, where } a ext{ is a constant.} 1 (a) State the range of values of $x$ for which this expansion is valid - AQA - A-Level Maths Pure - Question 1 - 2020 - Paper 1

Question 1

$The-first-three-terms,-in-ascending-powers-of-$x$,-of-the-binomial-expansion-of--$(9-+-2x)^2$-are-given-by--$(9-+-2x)^2--ext{-}--ext{-}--ext{-}--ext{-}--ext{-}--ext{-}--ext{-}--ext{-}--ext{-}--ext{-}--ext{-}--ext{-}--ext{-}--ext{-}--ext{-}--ext{-}--ext{-}--ext{-}--ext{-}--ext{-}--ext{-}--ext{-}--ext{-}--ext{-}--ext{-}--ext{-}--ext{-}--ext{-}--ext{-}--ext{-}--ext{-}--ext{-}--ext{-}--ext{-}--ext{-}--ext{-}--ext{-}--ext{-}--ext{-}--ext{-}--ext{-}--ext{-}--ext{-}--ext{-}--ext{-}--ext{-}--ext{-}--ext{-}--ext{-}--ext{-}--ext{-}--ext{-}--ext{-}--ext{-}--ext{-}--ext{-}--ext{-}--ext{-}--ext{-}--ext{-}--ext{-}--ext{-}--ext{-}--ext{-}--ext{-}--ext{-}--ext{-}--ext{-}--ext{-}--ext{-}--ext{-}--ext{-}--ext{-}--ext{-}--ext{-}--ext{-}--ext{-}--ext{-}--ext{-}--ext{-}-a-+-\frac{x^2}{54}--ext{,-where-}-a--ext{-is-a-constant.}--1-(a)-State-the-range-of-values-of-$x$-for-which-this-expansion-is-valid-AQA-A-Level Maths Pure-Question 1-2020-Paper 1.png$

The first three terms, in ascending powers of $x$, of the binomial expansion of $(9 + 2x)^2$ are given by $(9 + 2x)^2 ext{ } ext{ } ext{ } ext{ } ext{ } ext{... show full transcript

Worked Solution & Example Answer:The first three terms, in ascending powers of $x$, of the binomial expansion of $(9 + 2x)^2$ are given by $(9 + 2x)^2 ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } a + \frac{x^2}{54} ext{, where } a ext{ is a constant.} 1 (a) State the range of values of $x$ for which this expansion is valid - AQA - A-Level Maths Pure - Question 1 - 2020 - Paper 1

Step 1

State the range of values of $x$ for which this expansion is valid.

96%

114 rated

Answer

The binomial expansion is valid when the absolute value of the term inside the parentheses is less than 1. Thus, we have:

$|2x| < 9$

This simplifies to:

$|x| < \frac{9}{2}$

Additionally, since we have the term $|2x| < \frac{2}{3}$ from the constant $a$ , we also include the condition:

$|x| < \frac{2}{9}$

Thus, the overall valid range is:

$|x| < \frac{9}{2}$

Step 2

Find the value of $a$.

99%

104 rated

Answer

To find the value of $a$ , we can use the terms from the binomial expansion:

The first three terms are:

The constant term: $9^2 = 81$ ,
The linear term: $2 \cdot 9 \cdot x = 18x$ ,
The quadratic term: $\frac{1}{2} \cdot 2^2 \cdot x^2 = x^2$ .

Combining these, we have:

$81 + 18x + a + \frac{x^2}{54} = 9 + 2x^2$

To find $a$ , we equate the coefficients of $x^2$ and solve:

By matching coefficients, we see that $a = 3$ . Hence,

The value of $a$ is 3.

State the range of values of $x$ for which this expansion is valid.

Find the value of $a$.

Join the A-Level students using SimpleStudy...