| Exam Board | AQA |
|---|---|
| Module | C4 (Core Mathematics 4) |
| Year | 2014 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Generalised Binomial Theorem |
| Type | Product of separate expansions |
| Difficulty | Moderate -0.3 This is a standard three-part binomial expansion question requiring routine application of the generalized binomial theorem formula. Parts (a) and (b) are direct substitutions into the formula, while part (c) requires multiplying two series and collecting terms—a common textbook exercise with no novel insight needed. Slightly easier than average due to its mechanical nature. |
| Spec | 1.04c Extend binomial expansion: rational n, |x|<11.04d Binomial expansion validity: convergence conditions |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \((1-4x)^{1/4} = 1 + \frac{1}{4}(-4x) + \frac{(\frac{1}{4})(-\frac{3}{4})}{2!}(-4x)^2 + \cdots\) | M1 | Correct method |
| \(= 1 - x - \frac{3}{2}x^2 + \cdots\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \((2+3x)^{-3} = 2^{-3}\left(1+\frac{3x}{2}\right)^{-3}\) | M1 | Factor out \(2^{-3}\) |
| \(= \frac{1}{8}\left[1+(-3)\left(\frac{3x}{2}\right) + \frac{(-3)(-4)}{2!}\left(\frac{3x}{2}\right)^2+\cdots\right]\) | M1 | Correct binomial expansion |
| \(= \frac{1}{8}\left[1 - \frac{9x}{2} + \frac{27x^2}{2}+\cdots\right] = \frac{1}{8} - \frac{9x}{16} + \frac{27x^2}{16}+\cdots\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Multiply expansions from (a) and (b) | M1 | |
| \(= \frac{1}{8} - \frac{9x}{16} + \frac{27x^2}{16} - \frac{x}{8} + \frac{9x^2}{16} - \frac{3x^2}{16}+\cdots\) | Collecting terms | |
| \(= \frac{1}{8} - \frac{11x}{16} + \frac{33x^2}{16}+\cdots\) | A1 |
# Question 3:
## Part (a)
| Answer/Working | Mark | Guidance |
|---|---|---|
| $(1-4x)^{1/4} = 1 + \frac{1}{4}(-4x) + \frac{(\frac{1}{4})(-\frac{3}{4})}{2!}(-4x)^2 + \cdots$ | M1 | Correct method |
| $= 1 - x - \frac{3}{2}x^2 + \cdots$ | A1 | |
## Part (b)
| Answer/Working | Mark | Guidance |
|---|---|---|
| $(2+3x)^{-3} = 2^{-3}\left(1+\frac{3x}{2}\right)^{-3}$ | M1 | Factor out $2^{-3}$ |
| $= \frac{1}{8}\left[1+(-3)\left(\frac{3x}{2}\right) + \frac{(-3)(-4)}{2!}\left(\frac{3x}{2}\right)^2+\cdots\right]$ | M1 | Correct binomial expansion |
| $= \frac{1}{8}\left[1 - \frac{9x}{2} + \frac{27x^2}{2}+\cdots\right] = \frac{1}{8} - \frac{9x}{16} + \frac{27x^2}{16}+\cdots$ | A1 | |
## Part (c)
| Answer/Working | Mark | Guidance |
|---|---|---|
| Multiply expansions from (a) and (b) | M1 | |
| $= \frac{1}{8} - \frac{9x}{16} + \frac{27x^2}{16} - \frac{x}{8} + \frac{9x^2}{16} - \frac{3x^2}{16}+\cdots$ | | Collecting terms |
| $= \frac{1}{8} - \frac{11x}{16} + \frac{33x^2}{16}+\cdots$ | A1 | |3
\begin{enumerate}[label=(\alph*)]
\item Find the binomial expansion of $( 1 - 4 x ) ^ { \frac { 1 } { 4 } }$ up to and including the term in $x ^ { 2 }$.\\[0pt]
[2 marks]
\item Find the binomial expansion of $( 2 + 3 x ) ^ { - 3 }$ up to and including the term in $x ^ { 2 }$.
\item Hence find the binomial expansion of $\frac { ( 1 - 4 x ) ^ { \frac { 1 } { 4 } } } { ( 2 + 3 x ) ^ { 3 } }$ up to and including the term in $x ^ { 2 }$.\\[0pt]
[2 marks]
\end{enumerate}
\hfill \mbox{\textit{AQA C4 2014 Q3 [7]}}