| Exam Board | AQA |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Year | 2013 |
| Session | January |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Binomial Theorem (positive integer n) |
| Type | Product with reciprocal term binomial |
| Difficulty | Moderate -0.3 This is a straightforward binomial expansion question with standard techniques. Part (a) is trivial algebra, part (b) is routine application of binomial coefficients with n=8, and part (c) requires multiplying two expansions and collecting the x term—methodical but not challenging. Slightly easier than average due to small powers and clear structure. |
| Spec | 1.04a Binomial expansion: (a+b)^n for positive integer n |
| Answer | Marks | Guidance |
|---|---|---|
| \(\left(1 + \frac{4}{x}\right)^2 = 1 + \frac{8}{x} + \frac{16}{x^2}\) (or \(1 + 8x^{-1} + 16x^{-2}\)) | B1 (1 mark) | Unsimplified equivalent answers e.g. \(1 + \frac{4}{x} + \frac{4}{x} + \left(\frac{4}{x}\right)^2\) must be correctly simplified in part (c) |
| Answer | Marks | Guidance |
|---|---|---|
| \(\left(1 + \frac{x}{4}\right)^8 = \{1+\} \binom{8}{1}\left(\frac{x}{4}\right) + \binom{8}{2}\left(\frac{x}{4}\right)^2 + \binom{8}{3}\left(\frac{x}{4}\right)^3 + \ldots\) | M1 | Any valid method. PI by a correct value for either \(a\) or \(b\) or \(c\) |
| \(= \{1+\} 2x + \frac{7}{4}x^2 + \frac{7}{8}x^3 + \ldots\) | A1A1A1 (4 marks) | A1 for each of \(a, b, c\); SC \(a=8, b=28, c=56\) or \(a=32, b=448, c=3584\) either explicitly or within expansion (M1A0) |
| Answer | Marks | Guidance |
|---|---|---|
| \(\left(1 + \frac{8}{x} + \frac{16}{x^2}\right)\left(1 + 2x + \frac{7}{4}x^2 + \frac{7}{8}x^3\right)\) | M1 | Product of c's two expansions either stated explicitly or used |
| \(x\) terms from expansion are \(ax\) and '\(8\)'\(bx\) and '\(16\)'\(cx\) | m1 | Any two of the three; ft from products of non-zero terms using c's two expansions. May just use the coefficients. |
| \(ax +\) '\(8\)'\(bx +\) '\(16\)'\(cx\) | A1F | Ft on c's non-zero values for \(a\), \(b\) and \(c\) and also ft on c's non-zero coefficients of \(1/x\) and \(1/x^2\) in part (a); Accept \(x\)'s missing i.e. sum of coefficients. PI by correct final answer. |
| Coefficient of \(x\) is \(2 + 14 + 14 = 30\) | A1 (4 marks) | OE Condone answer left as \(30x\). Ignore terms in other powers of \(x\) in the expansion. |
# Question 8:
## Part 8(a):
$\left(1 + \frac{4}{x}\right)^2 = 1 + \frac{8}{x} + \frac{16}{x^2}$ (or $1 + 8x^{-1} + 16x^{-2}$) | B1 (1 mark) | Unsimplified equivalent answers e.g. $1 + \frac{4}{x} + \frac{4}{x} + \left(\frac{4}{x}\right)^2$ must be correctly simplified in part (c)
## Part 8(b):
$\left(1 + \frac{x}{4}\right)^8 = \{1+\} \binom{8}{1}\left(\frac{x}{4}\right) + \binom{8}{2}\left(\frac{x}{4}\right)^2 + \binom{8}{3}\left(\frac{x}{4}\right)^3 + \ldots$ | M1 | Any valid method. PI by a correct value for either $a$ or $b$ or $c$
$= \{1+\} 2x + \frac{7}{4}x^2 + \frac{7}{8}x^3 + \ldots$ | A1A1A1 (4 marks) | A1 for each of $a, b, c$; SC $a=8, b=28, c=56$ or $a=32, b=448, c=3584$ either explicitly or within expansion (M1A0)
$\{a = 2,\ b = 1.75 \text{ OE},\ c = 0.875 \text{ OE}\}$
## Part 8(c):
$\left(1 + \frac{8}{x} + \frac{16}{x^2}\right)\left(1 + 2x + \frac{7}{4}x^2 + \frac{7}{8}x^3\right)$ | M1 | Product of c's two expansions either stated explicitly or used
$x$ terms from expansion are $ax$ and '$8$'$bx$ and '$16$'$cx$ | m1 | Any two of the three; ft from products of non-zero terms using c's two expansions. May just use the coefficients.
$ax +$ '$8$'$bx +$ '$16$'$cx$ | A1F | Ft on c's non-zero values for $a$, $b$ and $c$ and also ft on c's non-zero coefficients of $1/x$ and $1/x^2$ in part (a); Accept $x$'s missing i.e. sum of coefficients. PI by correct final answer.
Coefficient of $x$ is $2 + 14 + 14 = 30$ | A1 (4 marks) | OE Condone answer left as $30x$. Ignore terms in other powers of $x$ in the expansion.
---8
\begin{enumerate}[label=(\alph*)]
\item Expand $\left( 1 + \frac { 4 } { x } \right) ^ { 2 }$.
\item The first four terms of the binomial expansion of $\left( 1 + \frac { x } { 4 } \right) ^ { 8 }$ in ascending powers of $x$ are $1 + a x + b x ^ { 2 } + c x ^ { 3 }$. Find the values of the constants $a , b$ and $c$.
\item Hence find the coefficient of $x$ in the expansion of $\left( 1 + \frac { 4 } { x } \right) ^ { 2 } \left( 1 + \frac { x } { 4 } \right) ^ { 8 }$.
\end{enumerate}
\hfill \mbox{\textit{AQA C2 2013 Q8 [9]}}