| Exam Board | AQA |
|---|---|
| Module | C4 (Core Mathematics 4) |
| Year | 2010 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Generalised Binomial Theorem |
| Type | Factoring out constants before expansion |
| Difficulty | Moderate -0.3 This is a straightforward binomial expansion question testing standard C4 technique. Part (a)(i) is direct application of the formula, (a)(ii) requires factoring out 16 first, and part (b) is simple substitution. The multi-step structure adds slight complexity, but each step follows a routine procedure with no problem-solving insight required, making it slightly easier than average. |
| Spec | 1.04c Extend binomial expansion: rational n, |x|<11.04d Binomial expansion validity: convergence conditions |
| Answer | Marks | Guidance |
|---|---|---|
| \((1 + x)^{\frac{1}{3}} = 1 + \frac{1}{3}x + kx^2\) | M1 | |
| \(= 1 + \frac{1}{3}x + \frac{3}{8}x^2\) | A1 | 2 marks |
| Answer | Marks | Guidance |
|---|---|---|
| \((16 + 9x)^{\frac{1}{3}} = 16^{\frac{1}{3}}\left(1 + \frac{9}{16}x\right)^{\frac{1}{3}}\) | B1 | |
| \(= k\left(1 + \frac{3}{16}x + \frac{1}{8}(\frac{9}{16}x)^2\right)\) | M1 | \(x\) replaced by \(\frac{9}{16}x\) or start binomial again; Condone missing brackets |
| \(= 64 + 54x + \frac{243}{32}x^2\) | A1 | 3 marks |
| Answer | Marks | Guidance |
|---|---|---|
| \(x = -\frac{1}{3}\); \(13^{\frac{3}{2}} = 46 + \frac{27}{32}\) | M1 A1 | 2 marks |
**4(a)(i)**
$(1 + x)^{\frac{1}{3}} = 1 + \frac{1}{3}x + kx^2$ | M1
$= 1 + \frac{1}{3}x + \frac{3}{8}x^2$ | A1 | 2 marks
**4(a)(ii)**
$(16 + 9x)^{\frac{1}{3}} = 16^{\frac{1}{3}}\left(1 + \frac{9}{16}x\right)^{\frac{1}{3}}$ | B1
$= k\left(1 + \frac{3}{16}x + \frac{1}{8}(\frac{9}{16}x)^2\right)$ | M1 | $x$ replaced by $\frac{9}{16}x$ or start binomial again; Condone missing brackets
$= 64 + 54x + \frac{243}{32}x^2$ | A1 | 3 marks | Accept 7.59375$x^2$
**4(b)**
$x = -\frac{1}{3}$; $13^{\frac{3}{2}} = 46 + \frac{27}{32}$ | M1 A1 | 2 marks | Use $x = -\frac{1}{3}$; 46 seen with $a = 27$, $b = 32$, or $\left(\frac{k \times 27}{k \times 32}\right)$
---\begin{enumerate}[label=(\alph*)]
\item
\begin{enumerate}[label=(\roman*)]
\item Find the binomial expansion of $(1 + x)^{\frac{3}{2}}$ up to and including the term in $x^2$. [2 marks]
\item Find the binomial expansion of $(16 + 9x)^{\frac{3}{2}}$ up to and including the term in $x^2$. [3 marks]
\end{enumerate}
\item Use your answer to part (a)(ii) to show that $13^{\frac{3}{2}} \approx 46 + \frac{a}{b}$, stating the values of the integers $a$ and $b$. [2 marks]
\end{enumerate}
\hfill \mbox{\textit{AQA C4 2010 Q4 [7]}}