| Exam Board | Edexcel |
|---|---|
| Module | P4 (Pure Mathematics 4) |
| Year | 2023 |
| Session | October |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Generalised Binomial Theorem |
| Type | Expand and state validity |
| Difficulty | Moderate -0.3 This is a straightforward application of the binomial expansion for negative/fractional powers. Students must rewrite in the form (1+bx)^n, apply the standard formula, and state the validity condition |bx|<1. While it requires careful algebraic manipulation and multiple steps (4 marks typical), it's a routine textbook exercise with no novel problem-solving required, making it slightly easier than average. |
| Spec | 1.04c Extend binomial expansion: rational n, |x|<11.04d Binomial expansion validity: convergence conditions |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \((2-5x)^{-2} = \frac{1}{4}\left(1-\frac{5x}{2}\right)^{-2}\) or e.g. \(\frac{8}{4\left(1-\frac{5x}{2}\right)^2}\) | B1 | For taking out factor of \(2^{-2}\) or \(\frac{1}{4}\) from \((2-5x)^{-2}\) to obtain e.g. \(\frac{1}{4}(1\pm...)^{-2}\). May be implied by constant term of 2 or by e.g. \(2(1\pm...)^{-2}\) |
| \(= 8\times\frac{1}{4}\left(1+(-2)\times\left(-\frac{5x}{2}\right)+\frac{(-2)\times(-3)}{2!}\times\left(-\frac{5x}{2}\right)^2+\frac{(-2)\times(-3)\times(-4)}{3!}\times\left(-\frac{5x}{2}\right)^3\cdots\right)\) | M1A1 | M1: correct structure for either term 3 or term 4, allow slip on sign. Allow \(\frac{(-2)\times(-3)}{2}(\pm ax)^2\) or \(\frac{(-2)\times(-3)\times(-4)}{3!}(\pm ax)^3\) where \(a\neq 1\). Condone missing brackets around \(ax\). A1: any unsimplified or simplified but correct form, ignoring factor preceding bracket |
| \(\frac{8}{(2-5x)^2} = 2+10x+\frac{75}{2}x^2+125x^3...\) | A1 | cao, must be simplified. Allow equivalents for \(\frac{75}{2}\) e.g. 37.5. Do not isw. |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(8\times(2-5x)^{-2} = 8\times\left(2^{-2}+(-2)2^{-3}(-5x)^1+\frac{-2\times-3}{2!}2^{-4}(-5x)^2+\frac{-2\times-3\times-4}{3!}2^{-5}(-5x)^3\right)\) | B1, M1A1 | B1: \((2-5x)^{-2}=2^{-2}+...\) implied by final answer of \(2+...\). M1: correct structure for term 3 or term 4 |
| \(\frac{8}{(2-5x)^2}=2+10x+\frac{75}{2}x^2+125x^3...\) | A1 | cao, must be simplified |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\ | x\ | <\frac{2}{5}\) o.e. |
# Question 1:
## Part (a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $(2-5x)^{-2} = \frac{1}{4}\left(1-\frac{5x}{2}\right)^{-2}$ or e.g. $\frac{8}{4\left(1-\frac{5x}{2}\right)^2}$ | B1 | For taking out factor of $2^{-2}$ or $\frac{1}{4}$ from $(2-5x)^{-2}$ to obtain e.g. $\frac{1}{4}(1\pm...)^{-2}$. May be implied by constant term of 2 or by e.g. $2(1\pm...)^{-2}$ |
| $= 8\times\frac{1}{4}\left(1+(-2)\times\left(-\frac{5x}{2}\right)+\frac{(-2)\times(-3)}{2!}\times\left(-\frac{5x}{2}\right)^2+\frac{(-2)\times(-3)\times(-4)}{3!}\times\left(-\frac{5x}{2}\right)^3\cdots\right)$ | M1A1 | M1: correct structure for either term 3 or term 4, allow slip on sign. Allow $\frac{(-2)\times(-3)}{2}(\pm ax)^2$ or $\frac{(-2)\times(-3)\times(-4)}{3!}(\pm ax)^3$ where $a\neq 1$. Condone missing brackets around $ax$. A1: any unsimplified or simplified but correct form, ignoring factor preceding bracket |
| $\frac{8}{(2-5x)^2} = 2+10x+\frac{75}{2}x^2+125x^3...$ | A1 | cao, must be simplified. Allow equivalents for $\frac{75}{2}$ e.g. 37.5. Do **not** isw. |
**Alternative (direct expansion):**
| Answer/Working | Mark | Guidance |
|---|---|---|
| $8\times(2-5x)^{-2} = 8\times\left(2^{-2}+(-2)2^{-3}(-5x)^1+\frac{-2\times-3}{2!}2^{-4}(-5x)^2+\frac{-2\times-3\times-4}{3!}2^{-5}(-5x)^3\right)$ | B1, M1A1 | B1: $(2-5x)^{-2}=2^{-2}+...$ implied by final answer of $2+...$. M1: correct structure for term 3 or term 4 |
| $\frac{8}{(2-5x)^2}=2+10x+\frac{75}{2}x^2+125x^3...$ | A1 | cao, must be simplified |
## Part (b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\|x\|<\frac{2}{5}$ o.e. | B1 | e.g. $-\frac{2}{5}<x<\frac{2}{5}$, $x>-\frac{2}{5}$ and $x<\frac{2}{5}$, $\left(-\frac{2}{5},\frac{2}{5}\right)$. **Not** $-\frac{2}{5}<\|x\|<\frac{2}{5}$, $x<\frac{2}{5}$, $\left|\frac{5x}{2}\right|<1$, $\left|-\frac{5x}{2}\right|<1$, $-1<\frac{5x}{2}<1$ |
---\begin{enumerate}
\item (a) Find the first four terms, in ascending powers of $x$, of the binomial expansion of
\end{enumerate}
$$\frac { 8 } { ( 2 - 5 x ) ^ { 2 } }$$
writing each term in simplest form.\\
(b) Find the range of values of $x$ for which this expansion is valid.
\hfill \mbox{\textit{Edexcel P4 2023 Q1 [5]}}