| Exam Board | Edexcel |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2024 |
| Session | January |
| Marks | 3 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Binomial Theorem (positive integer n) |
| Type | Standard binomial expansion coefficient |
| Difficulty | Moderate -0.8 This is a straightforward application of the binomial theorem requiring identification of the correct term (r=7), substitution into nCr·a^(n-r)·b^r, and arithmetic simplification. It's more routine than average since it involves direct formula application with no problem-solving or conceptual challenges, though the fractional coefficient requires careful calculation. |
| Spec | 1.04a Binomial expansion: (a+b)^n for positive integer n |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Mark | Guidance |
| Term in \(x^7\) is \({}^{12}C_7\left(\dfrac{3}{8}\right)^5(4x)^7\) or coefficient of \(x^7\) is \({}^{12}C_7\left(\dfrac{3}{8}\right)^5 \cdot 4^7\) | M1A1 | M1: Combines a correct binomial coefficient with \(\left(\dfrac{3}{8}\right)^5\) and either \((4x)^7\), \(4^7\) or \(x^7\). Binomial coefficient of the form \(\binom{12}{7}\), \({}^{12}C_7\) or 792. Condone \(\binom{12}{5}\) or \({}^{12}C_5\). Condone invisible brackets e.g. \({}^{12}C_7 \times \frac{3}{8}^5 \times 4x^7\). A1: Correct unsimplified term or coefficient e.g. \({}^{12}C_7\left(\dfrac{3}{8}\right)^5 \times 4^7\). Invisible brackets may be implied by further work. |
| Coefficient is \(96228\) | A1 | For \(96228\); condone \(96\,228\,x^7\) isw once correct answer seen. Term or coefficient must be identified if more than one term present. A correct answer alone with no incorrect working scores full marks. Note: \({}^{12}C_7\left(\frac{3}{8}\right)^5 \times 4x^7 = 23.493... \approx 23.5\) scores M1A0A0 (multiplying by 4 instead of \(4^7\)) |
## Question 2:
$\left(\dfrac{3}{8} + 4x\right)^{12}$
| Working/Answer | Mark | Guidance |
|---|---|---|
| Term in $x^7$ is ${}^{12}C_7\left(\dfrac{3}{8}\right)^5(4x)^7$ or coefficient of $x^7$ is ${}^{12}C_7\left(\dfrac{3}{8}\right)^5 \cdot 4^7$ | M1A1 | M1: Combines a correct binomial coefficient with $\left(\dfrac{3}{8}\right)^5$ and either $(4x)^7$, $4^7$ or $x^7$. Binomial coefficient of the form $\binom{12}{7}$, ${}^{12}C_7$ or 792. Condone $\binom{12}{5}$ or ${}^{12}C_5$. Condone invisible brackets e.g. ${}^{12}C_7 \times \frac{3}{8}^5 \times 4x^7$. A1: Correct unsimplified term or coefficient e.g. ${}^{12}C_7\left(\dfrac{3}{8}\right)^5 \times 4^7$. Invisible brackets may be implied by further work. |
| Coefficient is $96228$ | A1 | For $96228$; condone $96\,228\,x^7$ isw once correct answer seen. Term or coefficient must be identified if more than one term present. A correct answer alone with no incorrect working scores full marks. Note: ${}^{12}C_7\left(\frac{3}{8}\right)^5 \times 4x^7 = 23.493... \approx 23.5$ scores M1A0A0 (multiplying by 4 instead of $4^7$) |
---\begin{enumerate}
\item Find the coefficient of the term in $x ^ { 7 }$ of the binomial expansion of
\end{enumerate}
$$\left( \frac { 3 } { 8 } + 4 x \right) ^ { 12 }$$
giving your answer in simplest form.
\hfill \mbox{\textit{Edexcel P2 2024 Q2 [3]}}