| Exam Board | AQA |
|---|---|
| Module | C4 (Core Mathematics 4) |
| Year | 2005 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Generalised Binomial Theorem |
| Type | Substitute expression for variable |
| Difficulty | Moderate -0.3 This is a straightforward application of the binomial theorem for negative/fractional powers. Part (a) is direct recall, part (b) requires the standard substitution technique (replacing x with 2x), and part (c) is routine numerical verification. All steps are textbook-standard with no problem-solving 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 | Guidance |
|---|---|---|
| (a) \((1 + x)^{-\frac{1}{2}} = 1 + \left(-\frac{1}{2}\right)x + \frac{\left(-\frac{1}{2}\right)\left(-\frac{3}{2}\right)}{2}x^2\) | M1A1 | \(1 - \frac{1}{2}x + \frac{3}{8}x^2\) but simplification not required |
| (b) \(\frac{1}{\sqrt{1+2x}} = (1 + 2x)^{-\frac{1}{2}}\) | B1 | |
| \(= 1 - \frac{1}{2}(2x) + \frac{3}{8}(2x)^2\) | M1 | Condone missing brackets, if recovered |
| \(= 1 - x + \frac{3}{2}x^2\) | A1 | CAO |
| (c) \(1 - (-0.1) + \frac{3}{2}(-0.1)^2 (= 1.115)\) | M1 | Attempt to substitute in |
| \((1 - 0.2)^{-\frac{1}{2}} = \frac{\sqrt{5}}{2}\) | M1 | Link between \(\frac{1}{\sqrt{1+2x}}\) and \(\frac{\sqrt{5}}{2}\) |
| \(2 \times 1.115 = 2.23 \approx \sqrt{5}\) | A1 | AG; convincingly obtained |
**(a)** $(1 + x)^{-\frac{1}{2}} = 1 + \left(-\frac{1}{2}\right)x + \frac{\left(-\frac{1}{2}\right)\left(-\frac{3}{2}\right)}{2}x^2$ | M1A1 | $1 - \frac{1}{2}x + \frac{3}{8}x^2$ but simplification not required
**(b)** $\frac{1}{\sqrt{1+2x}} = (1 + 2x)^{-\frac{1}{2}}$ | B1 |
$= 1 - \frac{1}{2}(2x) + \frac{3}{8}(2x)^2$ | M1 | Condone missing brackets, if recovered
$= 1 - x + \frac{3}{2}x^2$ | A1 | CAO
**(c)** $1 - (-0.1) + \frac{3}{2}(-0.1)^2 (= 1.115)$ | M1 | Attempt to substitute in
$(1 - 0.2)^{-\frac{1}{2}} = \frac{\sqrt{5}}{2}$ | M1 | Link between $\frac{1}{\sqrt{1+2x}}$ and $\frac{\sqrt{5}}{2}$
$2 \times 1.115 = 2.23 \approx \sqrt{5}$ | A1 | AG; convincingly obtained
**Total: 8 marks**
---4
\begin{enumerate}[label=(\alph*)]
\item Find the binomial expansion of $( 1 + x ) ^ { - \frac { 1 } { 2 } }$ up to the term in $x ^ { 2 }$.
\item Hence, or otherwise, obtain the binomial expansion of $\frac { 1 } { \sqrt { 1 + 2 x } }$ up to the term in $x ^ { 2 }$, in simplified form.
\item Use your answer to part (b) with $x = - 0.1$ to show that $\sqrt { 5 } \approx 2.23$.
\end{enumerate}
\hfill \mbox{\textit{AQA C4 2005 Q4 [8]}}