| Exam Board | Edexcel |
|---|---|
| Module | AS Paper 1 (AS Paper 1) |
| Year | 2019 |
| Session | June |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Binomial Theorem (positive integer n) |
| Type | Numerical approximation using expansion |
| Difficulty | Moderate -0.8 Part (a) is a straightforward application of the binomial theorem with positive integer n=6, requiring only substitution into the formula and simplification of coefficients. Part (b) tests understanding of how to set up a numerical approximation (recognizing 1.925 = 2 - 0.1 = 2 + 3(-0.1)/4 when x=-0.1) but doesn't require the calculation itself. This is a standard AS-level question testing basic binomial expansion skills with minimal problem-solving demand. |
| Spec | 1.04a Binomial expansion: (a+b)^n for positive integer n |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(2^6\) or \(64\) as the constant term | B1 | Sight of either \(2^6\) or 64 as constant term |
| \(\left(2+\frac{3x}{4}\right)^6 = \ldots + {}^6C_1 2^5\left(\frac{3x}{4}\right)^1 + {}^6C_2 2^4\left(\frac{3x}{4}\right)^2 + \ldots\) | M1 | Correct attempt at second OR third term; correct binomial coefficient with correct power of 2 and correct power of \(\frac{3x}{4}\) |
| \(= \ldots + 6\times2^5\left(\frac{3x}{4}\right)^1 + \frac{6\times5}{2}\times2^4\left(\frac{3x}{4}\right)^2 + \ldots\) | A1 | Correct unsimplified second AND third terms; binomial coefficients processed to numbers |
| \(= 64 + 144x + 135x^2 + \ldots\) | A1 | Ignore any terms after this |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\frac{3x}{4} = -0.075 \Rightarrow x = -0.1\); find value of \(64+144x+135x^2\) with \(x=-0.1\) | B1ft | \(x=-0.1\) or \(-\frac{1}{10}\) with comment about substituting into their \(64+144x+135x^2\) |
## Question 8(a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $2^6$ or $64$ as the constant term | B1 | Sight of either $2^6$ or 64 as constant term |
| $\left(2+\frac{3x}{4}\right)^6 = \ldots + {}^6C_1 2^5\left(\frac{3x}{4}\right)^1 + {}^6C_2 2^4\left(\frac{3x}{4}\right)^2 + \ldots$ | M1 | Correct attempt at second OR third term; correct binomial coefficient with correct power of 2 and correct power of $\frac{3x}{4}$ |
| $= \ldots + 6\times2^5\left(\frac{3x}{4}\right)^1 + \frac{6\times5}{2}\times2^4\left(\frac{3x}{4}\right)^2 + \ldots$ | A1 | Correct unsimplified second AND third terms; binomial coefficients processed to numbers |
| $= 64 + 144x + 135x^2 + \ldots$ | A1 | Ignore any terms after this |
## Question 8(b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\frac{3x}{4} = -0.075 \Rightarrow x = -0.1$; find value of $64+144x+135x^2$ with $x=-0.1$ | B1ft | $x=-0.1$ or $-\frac{1}{10}$ with comment about substituting into their $64+144x+135x^2$ |
---\begin{enumerate}
\item (a) Find the first 3 terms, in ascending powers of $x$, of the binomial expansion of
\end{enumerate}
$$\left( 2 + \frac { 3 x } { 4 } \right) ^ { 6 }$$
giving each term in its simplest form.\\
(b) Explain how you could use your expansion to estimate the value of $1.925 ^ { 6 }$ You do not need to perform the calculation.\\
\hfill \mbox{\textit{Edexcel AS Paper 1 2019 Q8 [5]}}