| Exam Board | Edexcel |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Year | 2015 |
| Session | June |
| Marks | 4 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Binomial Theorem (positive integer n) |
| Type | Expansion up to x^2 term |
| Difficulty | Moderate -0.8 This is a straightforward application of the binomial theorem requiring only substitution into the formula and simplification. It's a standard C2 question with no problem-solving element—students simply need to recall the binomial expansion formula and perform routine algebraic manipulation to find three terms. |
| Spec | 1.04a Binomial expansion: (a+b)^n for positive integer n |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Mark | Guidance |
| \(2^{10}+\binom{10}{1}2^9\left(-\frac{1}{4}x\right)+\binom{10}{2}2^8\left(-\frac{1}{4}x\right)^2+\ldots\) | M1 | For either the \(x\) term or the \(x^2\) term including a correct binomial coefficient with a correct power of \(x\) |
| First term of 1024 | B1 | Award for 1024 as distinct constant term (not \(1024x^0\)) and not \(1+1024\) |
| Either \(-1280x\) or \(720x^2\) (Allow \(\pm 1280x\)) | A1 | For one correct simplified term in \(x\) |
| Both \(-1280x\) and \(720x^2\) (Do not allow \(\pm 1280x\)) | A1 | Both correct simplified terms |
| \(=1024-1280x+720x^2\) | [4] |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Mark | Guidance |
| \(\left(2-\frac{x}{4}\right)^{10}=2^k\left(1-10\times\frac{x}{8}+\frac{10\times9}{2}\left(-\frac{x}{8}\right)^2\right)\) | M1 | Correct structure for at least one underlined term |
| \(1024(1\pm\ldots)\) | B1 | Needs \(1024(1\ldots)\) to become 1024 |
| \(=1024-1280x+720x^2\) | A1 A1 | As before [4] |
# Question 1:
## Expansion of $\left(2-\frac{x}{4}\right)^{10}$
| Working/Answer | Mark | Guidance |
|---|---|---|
| $2^{10}+\binom{10}{1}2^9\left(-\frac{1}{4}x\right)+\binom{10}{2}2^8\left(-\frac{1}{4}x\right)^2+\ldots$ | M1 | For **either** the $x$ term **or** the $x^2$ term including a correct binomial coefficient with a correct power of $x$ |
| First term of 1024 | B1 | Award for 1024 as distinct constant term (not $1024x^0$) and not $1+1024$ |
| Either $-1280x$ **or** $720x^2$ (Allow $\pm 1280x$) | A1 | For one correct simplified term in $x$ |
| Both $-1280x$ and $720x^2$ (Do not allow $\pm 1280x$) | A1 | Both correct simplified terms |
| $=1024-1280x+720x^2$ | | **[4]** |
**Way 2:**
| Working/Answer | Mark | Guidance |
|---|---|---|
| $\left(2-\frac{x}{4}\right)^{10}=2^k\left(1-10\times\frac{x}{8}+\frac{10\times9}{2}\left(-\frac{x}{8}\right)^2\right)$ | M1 | Correct structure for at least one underlined term |
| $1024(1\pm\ldots)$ | B1 | Needs $1024(1\ldots)$ to become 1024 |
| $=1024-1280x+720x^2$ | A1 A1 | As before **[4]** |
---\begin{enumerate}
\item Find the first 3 terms, in ascending powers of $x$, of the binomial expansion of
\end{enumerate}
$$\left( 2 - \frac { x } { 4 } \right) ^ { 10 }$$
giving each term in its simplest form.\\
\hfill \mbox{\textit{Edexcel C2 2015 Q1 [4]}}