| Exam Board | Edexcel |
|---|---|
| Module | AS Paper 1 (AS Paper 1) |
| Year | 2024 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Binomial Theorem (positive integer n) |
| Type | Numerical approximation using expansion |
| Difficulty | Standard +0.3 This is a straightforward binomial expansion question requiring students to find coefficients by comparing terms, then substitute a value for numerical approximation. The algebra is routine (finding a from the x coefficient, then k from x²), and part (c) requires recognizing 17/16 = 1 + 1/16 to substitute x = 1/8. All steps are standard textbook procedures with no novel insight required, making it slightly easier than average. |
| Spec | 1.04a Binomial expansion: (a+b)^n for positive integer n1.04c Extend binomial expansion: rational n, |x|<11.04d Binomial expansion validity: convergence conditions |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Mark | Guidance |
| \(^{12}C_1 \times a = -\frac{15}{2}\) | M1 | Correct equation to find \(a\), e.g. \(12a = -\frac{15}{2}\) or \(^{12}C_1 \times a = -7.5\) or \(\frac{12!}{11!}a = -\frac{15}{2}\). Condone if \(x\) present both sides. |
| \(12a = -\frac{15}{2} \Rightarrow a = -\frac{5}{8}\) * | A1* | Must rearrange with no errors and sufficient steps; binomial coefficient evaluated first. Minimum acceptable: \(12a = -\frac{15}{2} \Rightarrow a = -\frac{5}{8}\). Note: \(^{12}C_1 \times a = -7.5 \Rightarrow a = -\frac{5}{8}\) is M1A0* |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Mark | Guidance |
| \(^{12}C_2 \times a^2 = k \Rightarrow k = \ldots\) | M1 | Correct expression/equation to find \(k\). May be implied by correct answer \(\frac{825}{32}x^2\). Allow use of \(\frac{5}{8}\) for \(a\). If expression seen in (a) it must be used in (b). |
| \(k = \frac{825}{32}\) | A1 | o.e. e.g. 25.78125. isw if they round after correct answer seen. Do not accept \(\frac{825}{32}x^2\) but allow coefficient circled/underlined. |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Mark | Guidance |
| \(1 - \frac{5}{8}x = \frac{17}{16} \Rightarrow x = -0.1\) then substitutes into \(1 - \frac{15}{2}x + kx^2\) | M1 | Correct strategy to find \(x\), e.g. \(1-\frac{5}{8}x = \frac{17}{16} \Rightarrow x=\ldots(=-0.1)\) AND substitutes into expansion using their \(k\). Accept sign slip on substitution if embedded values seen. Alternatively attempts \(\left(1+\frac{1}{16}\right)^{12} = 1+12\times\frac{1}{16}+\frac{12\times11}{2}\times\left(\frac{1}{16}\right)^2\) |
| \(= \text{awrt } 2.0078\) | A1 | Full answer 2.0078125 or allow \(\frac{257}{128}\). Value with no working can score both marks. Note: including \(x^3\) term gives awrt 2.0615 (scores both). Note: \(\left(\frac{17}{16}\right)^{12} = 2.06988999\ldots\) likely scores M0A0 unless method shown. |
## Question 6:
### Part (a):
| Working | Mark | Guidance |
|---------|------|----------|
| $^{12}C_1 \times a = -\frac{15}{2}$ | M1 | Correct equation to find $a$, e.g. $12a = -\frac{15}{2}$ or $^{12}C_1 \times a = -7.5$ or $\frac{12!}{11!}a = -\frac{15}{2}$. Condone if $x$ present both sides. |
| $12a = -\frac{15}{2} \Rightarrow a = -\frac{5}{8}$ * | A1* | Must rearrange with no errors and sufficient steps; binomial coefficient evaluated first. Minimum acceptable: $12a = -\frac{15}{2} \Rightarrow a = -\frac{5}{8}$. Note: $^{12}C_1 \times a = -7.5 \Rightarrow a = -\frac{5}{8}$ is M1A0* |
### Part (b):
| Working | Mark | Guidance |
|---------|------|----------|
| $^{12}C_2 \times a^2 = k \Rightarrow k = \ldots$ | M1 | Correct expression/equation to find $k$. May be implied by correct answer $\frac{825}{32}x^2$. Allow use of $\frac{5}{8}$ for $a$. If expression seen in (a) it must be used in (b). |
| $k = \frac{825}{32}$ | A1 | o.e. e.g. 25.78125. isw if they round after correct answer seen. Do not accept $\frac{825}{32}x^2$ but allow coefficient circled/underlined. |
### Part (c):
| Working | Mark | Guidance |
|---------|------|----------|
| $1 - \frac{5}{8}x = \frac{17}{16} \Rightarrow x = -0.1$ then substitutes into $1 - \frac{15}{2}x + kx^2$ | M1 | Correct strategy to find $x$, e.g. $1-\frac{5}{8}x = \frac{17}{16} \Rightarrow x=\ldots(=-0.1)$ AND substitutes into expansion using their $k$. Accept sign slip on substitution if embedded values seen. Alternatively attempts $\left(1+\frac{1}{16}\right)^{12} = 1+12\times\frac{1}{16}+\frac{12\times11}{2}\times\left(\frac{1}{16}\right)^2$ |
| $= \text{awrt } 2.0078$ | A1 | Full answer 2.0078125 or allow $\frac{257}{128}$. Value with no working can score both marks. Note: including $x^3$ term gives awrt 2.0615 (scores both). Note: $\left(\frac{17}{16}\right)^{12} = 2.06988999\ldots$ likely scores M0A0 unless method shown. |
---\begin{enumerate}
\item The binomial expansion of
\end{enumerate}
$$( 1 + a x ) ^ { 12 }$$
up to and including the term in $x ^ { 2 }$ is
$$1 - \frac { 15 } { 2 } x + k x ^ { 2 }$$
where $a$ and $k$ are constants.\\
(a) Show that $a = - \frac { 5 } { 8 }$\\
(b) Hence find the value of $k$
Using the expansion and making your method clear,\\
(c) find an estimate for the value of $\left( \frac { 17 } { 16 } \right) ^ { 12 }$, giving your answer to 4 decimal places.
\hfill \mbox{\textit{Edexcel AS Paper 1 2024 Q6 [6]}}