| Exam Board | Edexcel |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Binomial Theorem (positive integer n) |
| Type | Coefficient relationship between terms |
| Difficulty | Standard +0.8 This is a multi-part binomial expansion problem requiring algebraic manipulation of binomial coefficients to form and solve simultaneous equations. Part (a) requires proving an algebraic relationship, part (b) involves solving a system to find specific values, and part (c) is computational. The conceptual demand of working with parametric coefficients and the multi-step algebraic reasoning elevates this above standard C2 binomial exercises, though it remains within reach for strong students. |
| Spec | 1.04a Binomial expansion: (a+b)^n for positive integer n |
$$f(x) = \left(1 + \frac{x}{k}\right)^n, \quad k, n \in \mathbb{N}, \quad n > 2.$$
Given that the coefficient of $x^3$ is twice the coefficient of $x^2$ in the binomial expansion of f(x),
\begin{enumerate}[label=(\alph*)]
\item prove that $n = 6k + 2$.
Given also that the coefficients of $x^4$ and $x^5$ are equal and non-zero, [3]
\item form another equation in $n$ and $k$ and hence show that $k = 2$ and $n = 14$.
Using these values of $k$ and $n$, [4]
\item expand f(x) in ascending powers of $x$, up to and including the term in $x^5$. Give each coefficient as an exact fraction in its lowest terms [4]
\end{enumerate}
\hfill \mbox{\textit{Edexcel C2 Q8 [11]}}