| Exam Board | Edexcel |
|---|---|
| Module | C34 (Core Mathematics 3 & 4) |
| Year | 2018 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Generalised Binomial Theorem |
| Type | Finding unknown constant from coefficient |
| Difficulty | Standard +0.3 This is a standard binomial expansion question with a coefficient comparison. Part (a) is routine application of the generalized binomial theorem. Part (b) requires multiplying the expansion by (3+4x) and equating coefficients, which is a common textbook exercise. Part (c) is straightforward substitution once p is found. The question involves multiple steps but uses well-practiced techniques with no novel insight required, making it slightly easier than average. |
| Spec | 1.04c Extend binomial expansion: rational n, |x|<11.04d Binomial expansion validity: convergence conditions |
4. (a) Find the binomial expansion of
$$( 1 + p x ) ^ { - 4 } , \quad | p x | < 1$$
in ascending powers of $x$, up to and including the term in $x ^ { 3 }$, giving each coefficient as simply as possible in terms of the constant $p$.
$$f ( x ) = \frac { 3 + 4 x } { ( 1 + p x ) ^ { 4 } } \quad | p x | < 1$$
where $p$ is a positive constant.
In the series expansion of $\mathrm { f } ( x )$, the coefficient of $x ^ { 2 }$ is twice the coefficient of $x$.\\
(b) Find the value of $p$.\\
(c) Hence find the coefficient of $x ^ { 3 }$ in the series expansion of $\mathrm { f } ( x )$, giving your answer as a simplified fraction.\\
\begin{center}
\end{center}
\hfill \mbox{\textit{Edexcel C34 2018 Q4 [10]}}