| Exam Board | AQA |
|---|---|
| Module | C4 (Core Mathematics 4) |
| Year | 2011 |
| Session | January |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Generalised Binomial Theorem and Partial Fractions |
| Type | Partial fractions with validity range |
| Difficulty | Standard +0.3 This is a standard C4 partial fractions question with routine binomial expansion. Part (a) is mechanical algebra, part (b) applies standard binomial theorem to two simple terms, and part (c) requires finding the intersection of two convergence conditions |x|<1 and |5x/3|<1. Slightly above average due to the multi-step nature and the validity range requiring careful consideration of both fractions, but all techniques are standard textbook exercises with no novel insight required. |
| Spec | 1.02y Partial fractions: decompose rational functions1.04c Extend binomial expansion: rational n, |x|<1 |
| Answer | Marks |
|---|---|
| \(3+9x \equiv A(3+5x) + B(1+x)\) | M1 |
| \(x = -1\): \(-6 = -2A \Rightarrow A = 3\) | A1 |
| \(x = -\frac{3}{5}\): \(\frac{-12}{5} = \frac{2B}{5} \Rightarrow B = -6\) | A1 |
| Answer | Marks |
|---|---|
| \(\frac{3}{1+x} = 3(1+x)^{-1} = 3\left(1 - x + x^2 - \cdots\right)\) | M1 A1 |
| \(\frac{-6}{3+5x} = \frac{-6}{3}\left(1+\frac{5x}{3}\right)^{-1} = -2\left(1 - \frac{5x}{3} + \frac{25x^2}{9} - \cdots\right)\) | M1 A1 |
| Combining: \(1 + \frac{1}{3}x - \frac{23}{9}x^2\) | M1 A1 A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Valid for \( | x | < 1\) and \(\left |
| \(-\frac{3}{5} < x < \frac{3}{5}\) | A1 |
# Question 3:
## Part (a)
| $3+9x \equiv A(3+5x) + B(1+x)$ | M1 | |
|---|---|---|
| $x = -1$: $-6 = -2A \Rightarrow A = 3$ | A1 | |
| $x = -\frac{3}{5}$: $\frac{-12}{5} = \frac{2B}{5} \Rightarrow B = -6$ | A1 | |
## Part (b)
| $\frac{3}{1+x} = 3(1+x)^{-1} = 3\left(1 - x + x^2 - \cdots\right)$ | M1 A1 | |
|---|---|---|
| $\frac{-6}{3+5x} = \frac{-6}{3}\left(1+\frac{5x}{3}\right)^{-1} = -2\left(1 - \frac{5x}{3} + \frac{25x^2}{9} - \cdots\right)$ | M1 A1 | |
| Combining: $1 + \frac{1}{3}x - \frac{23}{9}x^2$ | M1 A1 A1 | |
## Part (c)
| Valid for $|x| < 1$ and $\left|\frac{5x}{3}\right| < 1 \Rightarrow |x| < \frac{3}{5}$ | M1 | |
|---|---|---|
| $-\frac{3}{5} < x < \frac{3}{5}$ | A1 | |
---3
\begin{enumerate}[label=(\alph*)]
\item Express $\frac { 3 + 9 x } { ( 1 + x ) ( 3 + 5 x ) }$ in the form $\frac { A } { 1 + x } + \frac { B } { 3 + 5 x }$, where $A$ and $B$ are integers.
\item Hence, or otherwise, find the binomial expansion of $\frac { 3 + 9 x } { ( 1 + x ) ( 3 + 5 x ) }$ up to and including the term in $x ^ { 2 }$.
\item Find the range of values of $x$ for which the binomial expansion of $\frac { 3 + 9 x } { ( 1 + x ) ( 3 + 5 x ) }$ is valid.\\
(2 marks)
\end{enumerate}
\hfill \mbox{\textit{AQA C4 2011 Q3 [12]}}