| Exam Board | AQA |
|---|---|
| Module | C4 (Core Mathematics 4) |
| Year | 2008 |
| Session | January |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Generalised Binomial Theorem |
| Type | Substitute expression for variable |
| Difficulty | Moderate -0.3 This is a standard C4 binomial expansion question with routine substitution steps. Part (a) is direct application of the formula with n=1/2, part (b) requires simple substitution of 3x/2 for x, and part (c) involves algebraic manipulation to match the given form. While it requires careful algebra across multiple parts, it follows a predictable pattern with no novel problem-solving or insight required, making it slightly easier than average. |
| Spec | 1.04c Extend binomial expansion: rational n, |x|<11.04d Binomial expansion validity: convergence conditions |
| Answer | Marks |
|---|---|
| \((1+x)^{\frac{1}{2}} = 1 + \frac{1}{2}x + kx^2\) | M1 |
| \(= 1 + \frac{1}{2}x - \frac{1}{8}x^2\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| \(\left(1 + \frac{3}{2}x\right)^{\frac{1}{4}} = 1 + \frac{1}{2}\left(\frac{3}{2}x\right) - \frac{1}{8}\left(\frac{3}{2}x\right)^2\) | M1 | \(x\) replaced by \(\frac{3}{2}x\) – condone missing brackets, but not incorrectly placed brackets eg \(\left(\frac{3}{2}\right)x^2\); alternatively, start again and find correct expression |
| \(= 1 + \frac{3}{4}x - \frac{9}{32}x^2\) | A1 | correct evaluation |
| Answer | Marks | Guidance |
|---|---|---|
| \(\sqrt{\frac{2+3x}{8}} = \sqrt{\frac{2+3x}{4 \times 2}} = k\left(1 + \frac{3}{2}x\right)^{\frac{1}{4}}\) | M1 | manipulation to \(k \times\) (answer to (b)) and evaluated \(\Rightarrow a + bx + cx^2\) |
| \(= \frac{1}{2} + \frac{3}{8}x - \frac{9}{64}x^2\) | A1 | \(a, b, c\) fractions or decimals only; Or use \((a+x)^n\) formula (condone one error for M1) |
**3(a)**
| $(1+x)^{\frac{1}{2}} = 1 + \frac{1}{2}x + kx^2$ | M1 | |
| $= 1 + \frac{1}{2}x - \frac{1}{8}x^2$ | A1 | |
**3(b)**
| $\left(1 + \frac{3}{2}x\right)^{\frac{1}{4}} = 1 + \frac{1}{2}\left(\frac{3}{2}x\right) - \frac{1}{8}\left(\frac{3}{2}x\right)^2$ | M1 | $x$ replaced by $\frac{3}{2}x$ – condone missing brackets, but not incorrectly placed brackets eg $\left(\frac{3}{2}\right)x^2$; alternatively, start again and find correct expression |
| $= 1 + \frac{3}{4}x - \frac{9}{32}x^2$ | A1 | correct evaluation |
**3(c)**
| $\sqrt{\frac{2+3x}{8}} = \sqrt{\frac{2+3x}{4 \times 2}} = k\left(1 + \frac{3}{2}x\right)^{\frac{1}{4}}$ | M1 | manipulation to $k \times$ (answer to (b)) and evaluated $\Rightarrow a + bx + cx^2$ |
| $= \frac{1}{2} + \frac{3}{8}x - \frac{9}{64}x^2$ | A1 | $a, b, c$ fractions or decimals only; Or use $(a+x)^n$ formula (condone one error for M1) |
**Total: 6 marks**
---3
\begin{enumerate}[label=(\alph*)]
\item Obtain the binomial expansion of $( 1 + x ) ^ { \frac { 1 } { 2 } }$ up to and including the term in $x ^ { 2 }$.
\item Hence obtain the binomial expansion of $\sqrt { 1 + \frac { 3 } { 2 } x }$ up to and including the term in $x ^ { 2 }$.
\item Hence show that $\sqrt { \frac { 2 + 3 x } { 8 } } \approx a + b x + c x ^ { 2 }$ for small values of $x$, where $a , b$ and $c$ are constants to be found.
\end{enumerate}
\hfill \mbox{\textit{AQA C4 2008 Q3 [6]}}