| Exam Board | Edexcel |
|---|---|
| Module | C4 (Core Mathematics 4) |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Generalised Binomial Theorem |
| Type | Direct single expansion substitution |
| Difficulty | Moderate -0.3 This is a straightforward application of the binomial expansion with standard follow-through parts. Part (a) is simple substitution/algebra, part (b) requires routine application of (1-x)^(-1/2) expansion, and parts (c)-(d) are mechanical calculations. While it spans multiple parts, each step is procedural with no novel insight required, making it slightly easier than average. |
| Spec | 1.04c Extend binomial expansion: rational n, |x|<1 |
| Answer | Marks | Guidance |
|---|---|---|
| (a) \(f\left(\frac{1}{10}\right) = \frac{3}{\sqrt{1-\frac{1}{10}}} = \frac{3}{\sqrt{\frac{9}{10}}} = \frac{3}{\frac{3}{\sqrt{10}}} = \sqrt{10}\) | M1 A1 | |
| (b) \(3(1-x)^{-\frac{1}{2}} = 3\left[1 + \left(-\frac{1}{2}\right)(-x) + \frac{(-\frac{1}{2})(-\frac{3}{2})}{2}(-x)^2 + \frac{(-\frac{1}{2})(-\frac{3}{2})(-\frac{5}{2})}{3 \cdot 2}(-x)^3 + \ldots\right]\) | M1 | |
| \(= 3 + \frac{3}{2}x + \frac{9}{8}x^2 + \frac{15}{16}x^3 + \ldots\) | A2 | |
| (c) \(\sqrt{10} = f\left(\frac{1}{10}\right) = 3 + \frac{3}{30} + \frac{9}{800} + \frac{15}{16000} = 3.1621875\) (8sf) | B1 | |
| (d) \(\frac{\sqrt{10} - 3.1621875}{\sqrt{10}} \times 100\% = 0.003\%\) (1sf) | M1 A1 | (8 marks) |
**(a)** $f\left(\frac{1}{10}\right) = \frac{3}{\sqrt{1-\frac{1}{10}}} = \frac{3}{\sqrt{\frac{9}{10}}} = \frac{3}{\frac{3}{\sqrt{10}}} = \sqrt{10}$ | M1 A1 |
**(b)** $3(1-x)^{-\frac{1}{2}} = 3\left[1 + \left(-\frac{1}{2}\right)(-x) + \frac{(-\frac{1}{2})(-\frac{3}{2})}{2}(-x)^2 + \frac{(-\frac{1}{2})(-\frac{3}{2})(-\frac{5}{2})}{3 \cdot 2}(-x)^3 + \ldots\right]$ | M1 |
$= 3 + \frac{3}{2}x + \frac{9}{8}x^2 + \frac{15}{16}x^3 + \ldots$ | A2 |
**(c)** $\sqrt{10} = f\left(\frac{1}{10}\right) = 3 + \frac{3}{30} + \frac{9}{800} + \frac{15}{16000} = 3.1621875$ (8sf) | B1 |
**(d)** $\frac{\sqrt{10} - 3.1621875}{\sqrt{10}} \times 100\% = 0.003\%$ (1sf) | M1 A1 | (8 marks)
---2.
$$f ( x ) = \frac { 3 } { \sqrt { 1 - x } } , | x | < 1$$
\begin{enumerate}[label=(\alph*)]
\item Show that $\mathrm { f } \left( \frac { 1 } { 10 } \right) = \sqrt { 10 }$.
\item Expand $\mathrm { f } ( x )$ in ascending powers of $x$ up to and including the term in $x ^ { 3 }$, simplifying each coefficient.
\item Use your expansion to find an approximate value for $\sqrt { 10 }$, giving your answer to 8 significant figures.
\item Find, to 1 significant figure, the percentage error in your answer to part (c).
\end{enumerate}
\hfill \mbox{\textit{Edexcel C4 Q2 [8]}}