| Exam Board | Edexcel |
|---|---|
| Module | C34 (Core Mathematics 3 & 4) |
| Year | 2016 |
| Session | January |
| Marks | 4 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Generalised Binomial Theorem |
| Type | Factoring out constants first |
| Difficulty | Moderate -0.3 This is a straightforward application of the binomial expansion for negative/fractional powers. Students must factor out the constant (3) first, then apply the standard formula with n=-4. While it requires careful algebraic manipulation and simplification of fractions, it's a routine textbook exercise with no problem-solving insight needed—slightly easier than average due to its mechanical nature. |
| Spec | 1.04c Extend binomial expansion: rational n, |x|<1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \((3-2x)^{-4} = 3^{-4}\left(1-\frac{2}{3}x\right)^{-4}\), showing \(3^{-4}\) or \(\frac{1}{81}\) | B1 | For taking out factor of \(3^{-4}\); evidence is seeing either \(3^{-4}\) or \(\frac{1}{81}\) before bracket |
| \(= \frac{1}{81}\times\left(1+(-4)\left(-\frac{2}{3}x\right)+\frac{(-4)(-5)}{2}\left(-\frac{2}{3}x\right)^2+\ldots\right)\) | M1A1 | M1: correct binomial form with \(n=-4\) and term \((kx)\); sufficient to see 2nd and 3rd terms with correct coefficients; condone sign slips. A1: any unsimplified correct expansion, ignore factor before bracket |
| \(= \frac{1}{81}+\frac{8}{243}x+\frac{40}{729}x^2+\ldots\) | A1 | cao; ignore any further terms |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(3^{-4}+(-4)(3)^{-5}(-2x)+\frac{(-4)(-5)}{2}(3)^{-6}(-2x)^2+\ldots\) | B1 M1 A1 | B1: first term \(3^{-4}\) or \(\frac{1}{81}\); M1: 2nd and 3rd terms (unsimplified); A1: unsimplified correct expansion |
| \(=\frac{1}{81}+\frac{8}{243}x+\frac{40}{729}x^2+\ldots\) | A1 | Must be simplified cao |
# Question 1:
| Answer/Working | Mark | Guidance |
|---|---|---|
| $(3-2x)^{-4} = 3^{-4}\left(1-\frac{2}{3}x\right)^{-4}$, showing $3^{-4}$ or $\frac{1}{81}$ | B1 | For taking out factor of $3^{-4}$; evidence is seeing either $3^{-4}$ or $\frac{1}{81}$ before bracket |
| $= \frac{1}{81}\times\left(1+(-4)\left(-\frac{2}{3}x\right)+\frac{(-4)(-5)}{2}\left(-\frac{2}{3}x\right)^2+\ldots\right)$ | M1A1 | M1: correct binomial form with $n=-4$ and term $(kx)$; sufficient to see 2nd and 3rd terms with correct coefficients; condone sign slips. A1: any unsimplified correct expansion, ignore factor before bracket |
| $= \frac{1}{81}+\frac{8}{243}x+\frac{40}{729}x^2+\ldots$ | A1 | cao; ignore any further terms |
**Alternative:**
| Answer/Working | Mark | Guidance |
|---|---|---|
| $3^{-4}+(-4)(3)^{-5}(-2x)+\frac{(-4)(-5)}{2}(3)^{-6}(-2x)^2+\ldots$ | B1 M1 A1 | B1: first term $3^{-4}$ or $\frac{1}{81}$; M1: 2nd and 3rd terms (unsimplified); A1: unsimplified correct expansion |
| $=\frac{1}{81}+\frac{8}{243}x+\frac{40}{729}x^2+\ldots$ | A1 | Must be simplified cao |
---1.
$$f ( x ) = ( 3 - 2 x ) ^ { - 4 } , \quad | x | < \frac { 3 } { 2 }$$
Find the binomial expansion of $\mathrm { f } ( x )$, in ascending powers of $x$, up to and including the term in $x ^ { 2 }$, giving each coefficient as a simplified fraction.\\
\begin{center}
\end{center}
\hfill \mbox{\textit{Edexcel C34 2016 Q1 [4]}}