| Exam Board | SPS |
|---|---|
| Module | SPS FM Pure (SPS FM Pure) |
| Year | 2025 |
| Session | June |
| Marks | 10 |
| Topic | Generalised Binomial Theorem |
| Type | Direct quotient expansion |
| Difficulty | Standard +0.3 This is a straightforward Further Maths binomial expansion question requiring standard techniques: expanding (1+4x)^(1/2) and (1-x)^(-1/2), multiplying series, and understanding convergence conditions (|x|<1). Part (b) tests basic understanding of validity, and part (c) is routine substitution and arithmetic. While it involves multiple steps, each is mechanical with no novel insight required, making it slightly easier than average. |
| Spec | 1.04c Extend binomial expansion: rational n, |x|<11.04d Binomial expansion validity: convergence conditions |
\begin{enumerate}[label=(\alph*)]
\item Use binomial expansions to show that $\sqrt{\frac{1 + 4x}{1 - x}} \approx 1 + \frac{5}{2}x - \frac{5}{8}x^2$
[6]
\end{enumerate}
A student substitutes $x = \frac{1}{2}$ into both sides of the approximation shown in part (a) in an attempt to find an approximation to $\sqrt{6}$
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item Give a reason why the student should not use $x = \frac{1}{2}$
[1]
\item Substitute $x = \frac{1}{11}$ into
$$\sqrt{\frac{1 + 4x}{1 - x}} = 1 + \frac{5}{2}x - \frac{5}{8}x^2$$
to obtain an approximation to $\sqrt{6}$. Give your answer as a fraction in its simplest form.
[3]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS FM Pure 2025 Q2 [10]}}