| Exam Board | SPS |
|---|---|
| Module | SPS SM Pure (SPS SM Pure) |
| Year | 2024 |
| Session | June |
| Marks | 6 |
| Topic | Binomial Theorem (positive integer n) |
| Type | Ratio of coefficients condition |
| Difficulty | Standard +0.3 This is a straightforward binomial expansion with a parameter, requiring students to find coefficients and apply an arithmetic progression condition. Part (a) is routine application of the binomial theorem, and part (b) involves setting up a simple algebraic equation (2a₂ = a₁ + a₃) that leads to a quadratic. The algebra is uncomplicated and the question follows a standard template for this topic, making it slightly easier than average. |
| Spec | 1.04a Binomial expansion: (a+b)^n for positive integer n1.04h Arithmetic sequences: nth term and sum formulae |
\begin{enumerate}
\item In this question you must show detailed reasoning.
\end{enumerate}
Solutions relying entirely on calculator technology are not acceptable.
$$f ( x ) = \left( 2 + \frac { k x } { 8 } \right) ^ { 7 } \quad \text { where } k \text { is a non-zero constant }$$
(a) Find the first 4 terms, in ascending powers of $x$, of the binomial expansion of $\mathrm { f } ( x )$. Give each term in simplest form.
Given that, in the binomial expansion of $\mathrm { f } ( x )$, the coefficients of $x , x ^ { 2 }$ and $x ^ { 3 }$ are the first 3 terms of an arithmetic progression,\\
(b) find, using algebra, the possible values of $k$.\\
(Solutions relying entirely on calculator technology are not acceptable.)\\
\hfill \mbox{\textit{SPS SPS SM Pure 2024 Q14 [6]}}