| Exam Board | Edexcel |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Year | 2005 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Binomial Theorem (positive integer n) |
| Type | Two equations from coefficients |
| Difficulty | Standard +0.3 This is a straightforward binomial expansion question requiring students to write down the first three terms using the standard formula, then solve two simultaneous equations from the given coefficient conditions. While it involves algebraic manipulation, the steps are routine and the problem-solving required is minimal—slightly easier than average for A-level. |
| Spec | 1.04a Binomial expansion: (a+b)^n for positive integer n |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(1 + 12px + \frac{12 \times 11}{2}(px)^2\) | B1, B1 | Terms can be listed rather than added; first B1: simplified form must be seen, but may be in (b). Total: 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(12p(x) = -q(x)\) and \(66p^2(x^2) = 11q(x^2)\) (equate terms or coefficients) | M1 | First M: may still have \(\binom{12}{2}\) or \({}^{12}C_2\) |
| \(\Rightarrow 66p^2 = -132p\) (equation in \(p\) or \(q\) only) | M1 | Second M: not with \(\binom{12}{2}\) or \({}^{12}C_2\); dependent on having \(p\)'s in each term |
| \(p = -2\), \(q = 24\) | A1, A1 | Zero solutions must be rejected for final A mark. Total: 4 |
## Question 4:
### Part (a):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $1 + 12px + \frac{12 \times 11}{2}(px)^2$ | B1, B1 | Terms can be listed rather than added; first B1: simplified form must be seen, but may be in (b). **Total: 2** |
### Part (b):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $12p(x) = -q(x)$ and $66p^2(x^2) = 11q(x^2)$ (equate terms or coefficients) | M1 | First M: may still have $\binom{12}{2}$ or ${}^{12}C_2$ |
| $\Rightarrow 66p^2 = -132p$ (equation in $p$ or $q$ only) | M1 | Second M: not with $\binom{12}{2}$ or ${}^{12}C_2$; dependent on having $p$'s in each term |
| $p = -2$, $q = 24$ | A1, A1 | Zero solutions must be rejected for final A mark. **Total: 4** |
---\begin{enumerate}[label=(\alph*)]
\item Write down the first three terms, in ascending powers of $x$, of the binomial expansion of $( 1 + p x ) ^ { 12 }$, where $p$ is a non-zero constant.
Given that, in the expansion of $( 1 + p x ) ^ { 12 }$, the coefficient of $x$ is $( - q )$ and the coefficient of $x ^ { 2 }$ is $11 q$,
\item find the value of $p$ and the value of $q$.
\end{enumerate}
\hfill \mbox{\textit{Edexcel C2 2005 Q4 [6]}}