| Exam Board | AQA |
|---|---|
| Module | C4 (Core Mathematics 4) |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Generalised Binomial Theorem |
| Type | Factor and rescale |
| Difficulty | Standard +0.3 This is a structured multi-part question that guides students through standard binomial expansion techniques with factoring and rescaling. Parts (a) and (b) are routine applications of the binomial theorem, part (c) is standard partial fractions (A-level staple), and part (d) combines previous results. While lengthy, each step follows predictable patterns with no novel insight required, making it slightly easier than average. |
| Spec | 1.02y Partial fractions: decompose rational functions1.04c Extend binomial expansion: rational n, |x|<11.04d Binomial expansion validity: convergence conditions |
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\begin{enumerate}[label=(\alph*)]
\item \begin{enumerate}[label=(\roman*)]
\item Obtain the binomial expansion of $( 1 - x ) ^ { - 1 }$ up to and including the term in $x ^ { 2 }$.
\item Hence, or otherwise, show that
$$\frac { 1 } { 3 - 2 x } \approx \frac { 1 } { 3 } + \frac { 2 } { 9 } x + \frac { 4 } { 27 } x ^ { 2 }$$
for small values of $x$.
\end{enumerate}\item Obtain the binomial expansion of $\frac { 1 } { ( 1 - x ) ^ { 2 } }$ up to and including the term in $x ^ { 2 }$.
\item Given that $\frac { 2 x ^ { 2 } - 3 } { ( 3 - 2 x ) ( 1 - x ) ^ { 2 } }$ can be written in the form $\frac { A } { ( 3 - 2 x ) } + \frac { B } { ( 1 - x ) } + \frac { C } { ( 1 - x ) ^ { 2 } }$, find the values of $A , B$ and $C$.
\item Hence find the binomial expansion of $\frac { 2 x ^ { 2 } - 3 } { ( 3 - 2 x ) ( 1 - x ) ^ { 2 } }$ up to and including the term in $x ^ { 2 }$.
\end{enumerate}
\hfill \mbox{\textit{AQA C4 Q5 [10]}}