| Exam Board | Edexcel |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Year | 2013 |
| Session | January |
| Marks | 4 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Binomial Theorem (positive integer n) |
| Type | Expansion up to x^2 term |
| Difficulty | Easy -1.2 This is a straightforward application of the binomial theorem requiring only substitution into the formula and basic arithmetic. It's a standard C2 question with no problem-solving element—students simply need to recall the binomial expansion formula and compute three terms mechanically. |
| Spec | 1.04a Binomial expansion: (a+b)^n for positive integer n |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(2^6 = 64\) | B1 | Award when first seen (not \(64x^0\)) |
| \(+6\times(2)^5(-5x)+\frac{6\times5}{2}(2)^4(-5x)^2\) | M1 | Attempt binomial expansion with correct structure for at least one of these terms. Term of the form: \(\binom{6}{p}\times(2)^{6-p}(-5x)^p\) with \(p=1\) or \(p=2\) consistently. Condone sign errors. Condone missing brackets if later work implies correct structure. Allow alternative forms for binomial coefficients e.g. \(^6C_1\) or \(\binom{6}{1}\) or even \(\left(\frac{6}{1}\right)\) |
| \(-960x\) | A1 (first) | Not \(+-960x\) |
| \((+)6000x^2\) | A1 (second) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(64(1\pm\ldots)\) | B1 | 64 and \((1\pm\ldots)\) — award when first seen |
| \(\left(1-\frac{5x}{2}\right)^6 = 1 \underline{-6\times\frac{5x}{2}}+\frac{6\times5}{2}\left(\underline{-\frac{5x}{2}}\right)^2\) | M1 | Correct structure for at least one underlined terms. Term of the form \(\binom{6}{p}\times(kx)^p\) with \(p=1\) or \(p=2\) consistently and \(k\neq\pm5\). Condone sign errors. Condoned missing brackets if later work implies correct structure, but must be an expansion of \((1-kx)^6\) where \(k\neq\pm5\) |
| \(-960x\) | A1 | Not \(+-960x\) |
| \((+)6000x^2\) | A1 |
## Question 1: $(2-5x)^6$
**Way 1:**
| Answer/Working | Mark | Guidance |
|---|---|---|
| $2^6 = 64$ | B1 | Award when first seen (not $64x^0$) |
| $+6\times(2)^5(-5x)+\frac{6\times5}{2}(2)^4(-5x)^2$ | M1 | Attempt binomial expansion with correct structure for at least one of these terms. Term of the form: $\binom{6}{p}\times(2)^{6-p}(-5x)^p$ with $p=1$ or $p=2$ consistently. Condone sign errors. Condone missing brackets if later work implies correct structure. Allow alternative forms for binomial coefficients e.g. $^6C_1$ or $\binom{6}{1}$ or even $\left(\frac{6}{1}\right)$ |
| $-960x$ | A1 (first) | **Not** $+-960x$ |
| $(+)6000x^2$ | A1 (second) | |
**Way 2:**
| Answer/Working | Mark | Guidance |
|---|---|---|
| $64(1\pm\ldots)$ | B1 | 64 **and** $(1\pm\ldots)$ — award when first seen |
| $\left(1-\frac{5x}{2}\right)^6 = 1 \underline{-6\times\frac{5x}{2}}+\frac{6\times5}{2}\left(\underline{-\frac{5x}{2}}\right)^2$ | M1 | Correct structure for at least one underlined terms. Term of the form $\binom{6}{p}\times(kx)^p$ with $p=1$ or $p=2$ consistently and $k\neq\pm5$. Condone sign errors. Condoned missing brackets if later work implies correct structure, but must be an expansion of $(1-kx)^6$ where $k\neq\pm5$ |
| $-960x$ | A1 | **Not** $+-960x$ |
| $(+)6000x^2$ | A1 | |
**(4 marks total)**\begin{enumerate}
\item Find the first 3 terms, in ascending powers of $x$, in the binomial expansion of
\end{enumerate}
$$( 2 - 5 x ) ^ { 6 }$$
Give each term in its simplest form.\\
\hfill \mbox{\textit{Edexcel C2 2013 Q1 [4]}}