| Exam Board | AQA |
|---|---|
| Module | C4 (Core Mathematics 4) |
| Year | 2011 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Topic | Generalised Binomial Theorem |
| Type | Factoring out constants first |
| Difficulty | Standard +0.3 This is a standard C4 binomial expansion question with routine factoring of constants. Part (a)(i) is direct application of the formula, part (a)(ii) requires factoring out 125 = 5³ (a common technique), and part (b) involves substituting x=2 to approximate ∛119. All steps are textbook procedures with no novel problem-solving required, making it slightly easier than average. |
| Spec | 1.04c Extend binomial expansion: rational n, |x|<11.04d Binomial expansion validity: convergence conditions |
3
\begin{enumerate}[label=(\alph*)]
\item \begin{enumerate}[label=(\roman*)]
\item Find the binomial expansion of $( 1 - x ) ^ { \frac { 1 } { 3 } }$ up to and including the term in $x ^ { 2 }$.
\item Hence, or otherwise, show that
$$( 125 - 27 x ) ^ { \frac { 1 } { 3 } } \approx 5 + \frac { m } { 25 } x + \frac { n } { 3125 } x ^ { 2 }$$
for small values of $x$, stating the values of the integers $m$ and $n$.
\end{enumerate}\item Use your result from part (a)(ii) to find an approximate value of $\sqrt [ 3 ] { 119 }$, giving your answer to five decimal places.\\
(2 marks)
\end{enumerate}
\hfill \mbox{\textit{AQA C4 2011 Q3 [7]}}