| Exam Board | Edexcel |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Year | 2014 |
| Session | January |
| 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 three terms. The calculation involves basic binomial coefficient formulas (12C1 and 12C2) 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 |
| \(1+\binom{12}{1}\times px+\binom{12}{2}\times(px)^2\) or \(1+12px+\frac{12\cdot11}{2}(px)^2\) | M1 | Correct structure for at least 1 of the underlined terms, including coefficients. Could be implied by e.g. \(12p=18\) |
| \(=(1+12px+66p^2x^2)\) | ||
| \(12p=18 \Rightarrow p=\) | M1 | Compare coefficients of \(x\) and solve for \(p\) |
| \(p=\frac{18}{12}\left(=\frac{3}{2}\right)\) | A1 | Correct value for \(p\) |
| \(q=66\times\left(\text{their } \frac{3}{2}\right)^2\) | M1 | Substitutes their value of \(p\) into their coefficient of \(x^2\) to find \(q\) |
| \(q=148.5\) or equivalent | A1 | cao |
## Question 1: $(1+px)^{12}$
| Answer/Working | Mark | Guidance |
|---|---|---|
| $1+\binom{12}{1}\times px+\binom{12}{2}\times(px)^2$ or $1+12px+\frac{12\cdot11}{2}(px)^2$ | M1 | Correct structure for at least 1 of the underlined terms, including coefficients. Could be implied by e.g. $12p=18$ |
| $=(1+12px+66p^2x^2)$ | | |
| $12p=18 \Rightarrow p=$ | M1 | Compare coefficients of $x$ and solve for $p$ |
| $p=\frac{18}{12}\left(=\frac{3}{2}\right)$ | A1 | Correct value for $p$ |
| $q=66\times\left(\text{their } \frac{3}{2}\right)^2$ | M1 | Substitutes their value of $p$ into their coefficient of $x^2$ to find $q$ |
| $q=148.5$ or equivalent | A1 | cao |
**Note:** Failing to square $p$ in the $x^2$ term could score M1M1A1M1A0 (4/5) (Gives $q=99$)
**Total: 5 marks**\begin{enumerate}
\item The first three terms in ascending powers of $x$ in the binomial expansion of $( 1 + p x ) ^ { 12 }$ are given by
\end{enumerate}
$$1 + 18 x + q x ^ { 2 }$$
where $p$ and $q$ are constants.\\
Find the value of $p$ and the value of $q$.\\
\hfill \mbox{\textit{Edexcel C2 2014 Q1 [5]}}