| Exam Board | SPS |
|---|---|
| Module | SPS FM Pure (SPS FM Pure) |
| Year | 2023 |
| Session | June |
| Marks | 8 |
| Topic | Generalised Binomial Theorem |
| Type | Direct quotient expansion |
| Difficulty | Standard +0.3 This is a straightforward binomial expansion question with standard steps: (i) routine application of the binomial series formula, (ii) recall of validity condition |2x| < 1, (iii) combining two expansions using multiplication, and (iv) substitution to approximate a square root. All techniques are standard A-level Further Maths fare 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=(\roman*)]
\item Use the binomial expansion to show that $(1 - 2x)^{-\frac{1}{4}} \approx 1 + x + \frac{5}{8}x^2$ for sufficiently small values of $x$. [2]
\item For what values of $x$ is the expansion valid? [1]
\item Find the expansion of $\sqrt{\frac{1+2x}{1-2x}}$ in ascending powers of $x$ as far as the term in $x^2$. [3]
\item Use $x = \frac{1}{20}$ in your answer to part (iii) to find an approximate value for $\sqrt{11}$. [2]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS FM Pure 2023 Q9 [8]}}