| Exam Board | Edexcel |
|---|---|
| Module | P4 (Pure Mathematics 4) |
| Year | 2024 |
| Session | January |
| Marks | 4 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Generalised Binomial Theorem |
| Type | Form (1+bx)^n expansion |
| Difficulty | Moderate -0.8 This is a straightforward application of the binomial expansion formula for negative/fractional powers with a simple substitution (b=-4, n=-3). It requires only direct recall of the formula and basic algebraic simplification, making it easier than average but not trivial since it involves negative indices and coefficient calculations. |
| Spec | 1.04c Extend binomial expansion: rational n, |x|<1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \((1-4x)^{-3} = 1\underline{\pm}3\times4x\pm\frac{-3\times-4}{2}(...x)^2\pm\frac{-3\times-4\times-5}{6}(...x)^3+...\) | M1 | Attempts binomial expansion with correct attempts at binomial coefficients for at least two of the \(x, x^2, x^3\) terms; "−4" may be missing or have incorrect sign; allow missing brackets |
| \((1-4x)^{-3} = 1+3\times4x+\frac{-3\times-4}{2}(-4x)^2+\frac{-3\times-4\times-5}{6}(-4x)^3+...\) | A1 | Correct unsimplified expansion (may be in terms of factorials) |
| \(= 1+12x+96x^2+640x^3+...\) | A1 | Any two terms correct and simplified (of the four, including the 1) |
| A1 | Fully correct, all terms simplified; ignore higher order terms; ISW after correct simplified answer |
# Question 1:
| Answer/Working | Mark | Guidance |
|---|---|---|
| $(1-4x)^{-3} = 1\underline{\pm}3\times4x\pm\frac{-3\times-4}{2}(...x)^2\pm\frac{-3\times-4\times-5}{6}(...x)^3+...$ | M1 | Attempts binomial expansion with correct attempts at binomial coefficients for **at least two** of the $x, x^2, x^3$ terms; "−4" may be missing or have incorrect sign; allow missing brackets |
| $(1-4x)^{-3} = 1+3\times4x+\frac{-3\times-4}{2}(-4x)^2+\frac{-3\times-4\times-5}{6}(-4x)^3+...$ | A1 | Correct unsimplified expansion (may be in terms of factorials) |
| $= 1+12x+96x^2+640x^3+...$ | A1 | Any two terms correct and simplified (of the four, including the 1) |
| | A1 | Fully correct, all terms simplified; ignore higher order terms; ISW after correct simplified answer |
---\begin{enumerate}
\item Find, in ascending powers of $x$ up to and including the term in $x ^ { 3 }$, the binomial expansion of
\end{enumerate}
$$( 1 - 4 x ) ^ { - 3 } \quad | x | < \frac { 1 } { 4 }$$
fully simplifying each term.
\hfill \mbox{\textit{Edexcel P4 2024 Q1 [4]}}