| Exam Board | AQA |
|---|---|
| Module | C4 (Core Mathematics 4) |
| Year | 2015 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Generalised Binomial Theorem |
| Type | Factoring out constants first |
| Difficulty | Standard +0.3 This is a straightforward application of the binomial expansion formula for fractional powers. Part (a) is direct substitution, part (b)(i) requires factoring out 8 first (a standard technique), and part (b)(ii) is routine substitution of a value. The question tests procedural fluency rather than problem-solving, making it slightly easier than average for C4 level. |
| Spec | 1.04c Extend binomial expansion: rational n, |x|<11.04d Binomial expansion validity: convergence conditions |
| Answer | Marks | Guidance |
|---|---|---|
| \(= 1 + x - 2x^2 + ...\) | M1 A1 | M1 for correct binomial method, A1 correct expansion |
| 4(b)(i) \((8+3x)^{-\frac{2}{3}} = 8^{-\frac{2}{3}}(1+\frac{3x}{8})^{-\frac{2}{3}} = \frac{1}{4}(1 - \frac{1}{4}x + \frac{5}{96}x^2+...)\) | M1 A1 A1 | |
| 4(b)(ii) Set \(8+3x = \frac{1}{81}\)... \(x = -\frac{647}{243}\) giving estimate | M1 A1 |
## Question 4:
**4(a)** $(1+5x)^{\frac{1}{5}}$
$= 1 + \frac{1}{5}(5x) + \frac{\frac{1}{5}(\frac{1}{5}-1)}{2!}(5x)^2 + ...$
$= 1 + x - 2x^2 + ...$ | M1 A1 | M1 for correct binomial method, A1 correct expansion
**4(b)(i)** $(8+3x)^{-\frac{2}{3}} = 8^{-\frac{2}{3}}(1+\frac{3x}{8})^{-\frac{2}{3}} = \frac{1}{4}(1 - \frac{1}{4}x + \frac{5}{96}x^2+...)$ | M1 A1 A1 |
**4(b)(ii)** Set $8+3x = \frac{1}{81}$... $x = -\frac{647}{243}$ giving estimate | M1 A1 |
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**Would you like me to work through Questions 4 and 5 fully with complete worked solutions?**4
\begin{enumerate}[label=(\alph*)]
\item Find the binomial expansion of $( 1 + 5 x ) ^ { \frac { 1 } { 5 } }$ up to and including the term in $x ^ { 2 }$.
\item \begin{enumerate}[label=(\roman*)]
\item Find the binomial expansion of $( 8 + 3 x ) ^ { - \frac { 2 } { 3 } }$ up to and including the term in $x ^ { 2 }$.
\item Use your expansion from part (b)(i) to find an estimate for $\sqrt [ 3 ] { \frac { 1 } { 81 } }$, giving your answer to four decimal places.\\[0pt]
[2 marks]
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{AQA C4 2015 Q4 [7]}}