| Exam Board | SPS |
|---|---|
| Module | SPS SM Pure (SPS SM Pure) |
| Year | 2023 |
| Session | September |
| Marks | 6 |
| Topic | Binomial Theorem (positive integer n) |
| Type | Product with reciprocal term binomial |
| Difficulty | Moderate -0.8 Part (a) is a straightforward binomial expansion requiring direct application of the formula with fractional coefficients. Part (b) requires multiplying the expansion by a simple quadratic and collecting terms, which is a standard technique. Both parts are routine exercises with no conceptual challenges beyond basic binomial theorem application. |
| Spec | 1.04a Binomial expansion: (a+b)^n for positive integer n1.04c Extend binomial expansion: rational n, |x|<1 |
In all questions you must show all stages of your working, justifying solutions and not relying solely on calculator technology.
\begin{enumerate}[label=(\alph*)]
\item Find the first four terms, in ascending powers of $x$, of the binomial expansion of $\left(1+\frac{x}{2}\right)^7$, giving each coefficient in exact simplified form. [3]
\item Hence determine the coefficient of $x$ in the expansion of
$$\left(1+\frac{2}{x}\right)^2\left(1+\frac{x}{2}\right)^7.$$ [3]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS SM Pure 2023 Q1 [6]}}