| Exam Board | AQA |
|---|---|
| Module | C4 (Core Mathematics 4) |
| Year | 2006 |
| Session | January |
| Marks | 15 |
| 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 generalized binomial theorem, part (c) is standard partial fractions (A-level staple), and part (d) combines previous results. While it requires multiple steps, each component is a textbook exercise with clear scaffolding, making it slightly easier than average overall. |
| Spec | 1.02y Partial fractions: decompose rational functions1.04c Extend binomial expansion: rational n, |x|<11.04d Binomial expansion validity: convergence conditions |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \((1-x)^{-1} = 1 + (-1)(-x) + \frac{(-1)(-2)}{2}(-x)^2\) | M1 | First two terms \(+ kx^2\) |
| \(= 1 + x + x^2\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\frac{1}{(3-2x)} = \frac{1}{3}\left(1 - \frac{2}{3}x\right)^{-1}\) | B1 | Or directly substitute into formula |
| \(\approx \frac{1}{3}\left(1 + \frac{2}{3}x + \left(\frac{2}{3}x\right)^2\right)\) | M1 | M1 power of 3; M1 other coefficients (allow one error) |
| \(\approx \frac{1}{3} + \frac{2}{9}x + \frac{4}{27}x^2\) | A1 | AG convincingly obtained |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \((1-x)^{-2} = 1 + (-2)(-x) + \frac{(-2)(-3)(-x)^2}{2}\) | M1 | First two terms \(+ kx^2\) |
| \(= 1 + 2x + 3x^2\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(2x^2 - 3 = A(1-x)^2 + B(3-2x)(1-x) + C(3-2x)\) | M1 | Or by equating coefficients |
| \(x=1\): \(-1 = C \times 1\); \(x=\frac{3}{2}\): \(\frac{3}{2} = A \times \frac{1}{4}\) | M1 | M1 same; A1 collect terms; M1 equate coefficients |
| \(C = -1\), \(A = 6\) | A1 | Follow on \(A\) and \(C\); A1 2 correct; A1 3 correct |
| \(x=0\): \((-3 = 6 + 3B - 3)\) or other value \(\Rightarrow\) equation in \(A,B,C\) | m1 | |
| \(B = -2\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\frac{6}{3-2x} - \frac{2}{1-x} - \frac{1}{(1-x)^2}\) | ||
| \(\approx \frac{6}{3}\left(1 + \frac{2}{3}x + \frac{4}{9}x^2\right) - 2(1 + x + x^2) - (1 + 2x + 3x^2)\) | M1A1F | Follow on \(A\ B\ C\) and expansions |
| \(\approx -1 - \frac{8}{3}x - \frac{37}{9}x^2\) | A1 | CAO |
## Question 5:
### Part (a)(i)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $(1-x)^{-1} = 1 + (-1)(-x) + \frac{(-1)(-2)}{2}(-x)^2$ | M1 | First two terms $+ kx^2$ |
| $= 1 + x + x^2$ | A1 | |
### Part (a)(ii)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\frac{1}{(3-2x)} = \frac{1}{3}\left(1 - \frac{2}{3}x\right)^{-1}$ | B1 | Or directly substitute into formula |
| $\approx \frac{1}{3}\left(1 + \frac{2}{3}x + \left(\frac{2}{3}x\right)^2\right)$ | M1 | M1 power of 3; M1 other coefficients (allow one error) |
| $\approx \frac{1}{3} + \frac{2}{9}x + \frac{4}{27}x^2$ | A1 | AG convincingly obtained |
### Part (b)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $(1-x)^{-2} = 1 + (-2)(-x) + \frac{(-2)(-3)(-x)^2}{2}$ | M1 | First two terms $+ kx^2$ |
| $= 1 + 2x + 3x^2$ | A1 | |
### Part (c)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $2x^2 - 3 = A(1-x)^2 + B(3-2x)(1-x) + C(3-2x)$ | M1 | Or by equating coefficients |
| $x=1$: $-1 = C \times 1$; $x=\frac{3}{2}$: $\frac{3}{2} = A \times \frac{1}{4}$ | M1 | M1 same; A1 collect terms; M1 equate coefficients |
| $C = -1$, $A = 6$ | A1 | Follow on $A$ and $C$; A1 2 correct; A1 3 correct |
| $x=0$: $(-3 = 6 + 3B - 3)$ or other value $\Rightarrow$ equation in $A,B,C$ | m1 | |
| $B = -2$ | A1 | |
### Part (d)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\frac{6}{3-2x} - \frac{2}{1-x} - \frac{1}{(1-x)^2}$ | | |
| $\approx \frac{6}{3}\left(1 + \frac{2}{3}x + \frac{4}{9}x^2\right) - 2(1 + x + x^2) - (1 + 2x + 3x^2)$ | M1A1F | Follow on $A\ B\ C$ and expansions |
| $\approx -1 - \frac{8}{3}x - \frac{37}{9}x^2$ | A1 | CAO |
---5
\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 $x ^ { 2 }$.\\
(2 marks)
\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 2006 Q5 [15]}}