| Exam Board | Edexcel |
|---|---|
| Module | C12 (Core Mathematics 1 & 2) |
| Year | 2016 |
| Session | June |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Binomial Theorem (positive integer n) |
| Type | Find constants from coefficient conditions on terms |
| Difficulty | Moderate -0.8 This is a straightforward application of the binomial theorem requiring students to match coefficients of the first few terms. The calculation involves basic binomial coefficients (8C1 and 8C2) and simple algebra to find p and q. It's easier than average as it's purely procedural with no problem-solving or insight required. |
| Spec | 1.04a Binomial expansion: (a+b)^n for positive integer n |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \((1+px)^8 = 1 + 8(px) + \frac{8\times7}{2!}(px)^2\) | M1 | Uses binomial expansion with correct form for terms 2 and 3; accept correct coefficient with correct power of \(x\); allow missing bracket on \(x^2\) term |
| Compares coefficients in \(x\): \(8p = 12 \Rightarrow p = 1.5\) | M1A1 | M1: sets coefficient in \(x\) equal to 12, must be of form \(kp=12\); A1: \(p=1.5\) or equivalent such as \(\frac{12}{8}\) |
| Compares coefficients in \(x^2\): \(q = 28p^2 \Rightarrow q = 63\) | M1A1 | M1: sets \(q\) equal to coefficient of \(x^2\) (must include \(p\) or \(p^2\)) then substitutes value of \(p\); A1: \(q=63\) |
## Question 1:
| Answer/Working | Mark | Guidance |
|---|---|---|
| $(1+px)^8 = 1 + 8(px) + \frac{8\times7}{2!}(px)^2$ | M1 | Uses binomial expansion with correct form for terms 2 and 3; accept correct coefficient with correct power of $x$; allow missing bracket on $x^2$ term |
| Compares coefficients in $x$: $8p = 12 \Rightarrow p = 1.5$ | M1A1 | M1: sets coefficient in $x$ equal to 12, must be of form $kp=12$; A1: $p=1.5$ or equivalent such as $\frac{12}{8}$ |
| Compares coefficients in $x^2$: $q = 28p^2 \Rightarrow q = 63$ | M1A1 | M1: sets $q$ equal to coefficient of $x^2$ (must include $p$ or $p^2$) then substitutes value of $p$; A1: $q=63$ |
---\begin{enumerate}
\item The first three terms in ascending powers of $x$ in the binomial expansion of $( 1 + p x ) ^ { 8 }$ are given by
\end{enumerate}
$$1 + 12 x + q x ^ { 2 }$$
where $p$ and $q$ are constants.\\
Find the value of $p$ and the value of $q$.\\
\hfill \mbox{\textit{Edexcel C12 2016 Q1 [5]}}