| Exam Board | SPS |
|---|---|
| Module | SPS FM (SPS FM) |
| Year | 2023 |
| Session | February |
| Marks | 9 |
| Topic | Generalised Binomial Theorem |
| Type | Product with quadratic or higher term |
| Difficulty | Standard +0.3 This is a standard binomial expansion question with routine algebraic manipulation. Part (a) requires direct application of the binomial series formula for (1+2x)^(1/2), part (b) involves multiplying by the geometric series expansion of 1/(1+9x²), and part (c) asks for the intersection of convergence radii. While it requires careful algebra and understanding of validity conditions, it follows a well-practiced template with no novel problem-solving 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 Expand $\sqrt{1 + 2x}$ in ascending powers of x, up to and including the term in $x^3$. [4]
\item Hence expand $\frac{\sqrt{1 + 2x}}{1 + 9x^2}$ in ascending powers of x, up to and including the term in $x^3$. [3]
\item Determine the range of values of x for which the expansion in part (b) is valid. [2]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS FM 2023 Q5 [9]}}