| Exam Board | Edexcel |
|---|---|
| Module | C12 (Core Mathematics 1 & 2) |
| Year | 2017 |
| Session | June |
| Marks | 8 |
| 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 find the first three terms using the standard formula, then solve two simple linear equations using coefficient matching. The algebra is routine and the problem-solving is minimal—slightly easier than average since it's mostly mechanical application of the binomial theorem with no conceptual challenges. |
| Spec | 1.04a Binomial expansion: (a+b)^n for positive integer n |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(2^{10}\) OR 1024 seen as the constant term | B1 | |
| \(\left(2-\frac{x}{8}\right)^{10} = 2^{10} + {}^{10}C_1 2^9\left(-\frac{x}{8}\right)^1 + {}^{10}C_2 2^8\left(-\frac{x}{8}\right)^2 + \ldots\) | M1A1 | Correct attempt at binomial expansion with \(a=2\), \(b=\pm\frac{x}{8}\), \(n=10\). Condone missing brackets. Accept any unsimplified term in \(x\) as evidence. Accept power series form \(\left(1\pm kx\right)^{10} = 1+10(\pm kx)+\frac{10\times9}{2}(\pm kx)^2\) condoning missing brackets |
| \(= 1024 - 640x + 180x^2\) | A1 | Accept \(1024 + -640x + 180x^2\). Can be listed with commas or on separate lines. Accept in reverse order |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(1024a = 256 \Rightarrow a = \frac{1}{4}\) | M1A1 | Sets their '\(1024\)'\(\times a = 256\). Accept equivalents such as 0.25. Accept by substituting \(x=0\) into both sides as long as not found from incorrect method |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(1024b - 640a = 352 \Rightarrow b = \frac{1}{2}\) | M1A1 | Sets their '\(1024\)'\(\times b \pm\) their'\(640\)'\(a = 352\). Accept 0.5 |
# Question 10:
## Part (a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $2^{10}$ OR 1024 seen as the constant term | B1 | |
| $\left(2-\frac{x}{8}\right)^{10} = 2^{10} + {}^{10}C_1 2^9\left(-\frac{x}{8}\right)^1 + {}^{10}C_2 2^8\left(-\frac{x}{8}\right)^2 + \ldots$ | M1A1 | Correct attempt at binomial expansion with $a=2$, $b=\pm\frac{x}{8}$, $n=10$. Condone missing brackets. Accept any unsimplified term in $x$ as evidence. Accept power series form $\left(1\pm kx\right)^{10} = 1+10(\pm kx)+\frac{10\times9}{2}(\pm kx)^2$ condoning missing brackets |
| $= 1024 - 640x + 180x^2$ | A1 | Accept $1024 + -640x + 180x^2$. Can be listed with commas or on separate lines. Accept in reverse order |
## Part (b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $1024a = 256 \Rightarrow a = \frac{1}{4}$ | M1A1 | Sets their '$1024$'$\times a = 256$. Accept equivalents such as 0.25. Accept by substituting $x=0$ into both sides as long as not found from incorrect method |
## Part (c):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $1024b - 640a = 352 \Rightarrow b = \frac{1}{2}$ | M1A1 | Sets their '$1024$'$\times b \pm$ their'$640$'$a = 352$. Accept 0.5 |
---\begin{enumerate}
\item (a) Find the first 3 terms, in ascending powers of $x$, of the binomial expansion of
\end{enumerate}
$$\left( 2 - \frac { x } { 8 } \right) ^ { 10 }$$
giving each term in its simplest form.
$$\mathrm { f } ( x ) = \left( 2 - \frac { x } { 8 } \right) ^ { 10 } ( a + b x ) , \text { where } a \text { and } b \text { are constants }$$
Given that the first two terms, in ascending powers of $x$ in the series expansion of $\mathrm { f } ( x )$, are 256 and $352 x$,\\
(b) find the value of $a$,\\
(c) find the value of $b$.\\
\hfill \mbox{\textit{Edexcel C12 2017 Q10 [8]}}