| Exam Board | Edexcel |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Topic | Binomial Theorem (positive integer n) |
| Type | Find constants from coefficient conditions on terms |
| Difficulty | Standard +0.8 This is a multi-part binomial theorem question requiring algebraic manipulation to form and solve simultaneous equations from coefficient conditions, followed by expansion. Part (a) requires setting up coefficient ratios and algebraic proof; part (b) involves solving simultaneous equations with binomial coefficients; part (c) is computational but requires careful fraction arithmetic. The problem-solving aspect and algebraic complexity elevate this above routine C2 exercises. |
| 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 $n = 6k + 2$. [3]
\end{enumerate}
Given also that the coefficients of $x^4$ and $x^5$ are equal and non-zero,
\begin{enumerate}[label=(\alph*)]
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\item form another equation in $n$ and $k$ and hence show that $k = 2$ and $n = 14$. [4]
\end{enumerate}
Using these values of $k$ and $n$,
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{2}
\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 Q19 [11]}}