| Exam Board | Edexcel |
|---|---|
| Module | C4 (Core Mathematics 4) |
| Year | 2017 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Generalised Binomial Theorem |
| Type | Finding unknown power and constant |
| Difficulty | Standard +0.3 This is a straightforward application of the binomial expansion formula for negative powers. Part (a) requires simple substitution of x=0, part (b) involves equating the x² coefficient to find k (one equation, one unknown), and part (c) uses the found k value to calculate B. All steps are mechanical with no conceptual challenges beyond knowing the standard binomial expansion formula. |
| Spec | 1.04c Extend binomial expansion: rational n, |x|<1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(A = \frac{1}{8}\) | B1 cao | \(\frac{1}{8}\) or \(2^{-3}\) or \(0.125\), clearly identified as \(A\) or as their answer to part (a) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Uses \(x^2\) term: \(\left(\frac{1}{8}\right)\frac{(-3)(-4)}{2!}\left(\frac{k}{2}\right)^2\) | M1 | Uses the \(x^2\) term; either \(\frac{(-3)(-4)}{2!}\) or \(\left(\frac{k}{2}\right)^2\) or \(\left(\frac{kx}{2}\right)^2\) or \(\frac{(-3)(-4)}{2}\) or \(6\) seen |
| Either (their \(A\))\(\frac{(-3)(-4)}{2!}\left(\frac{k}{2}\right)^2\) or (their \(A\))\(\frac{(-3)(-4)}{2!}\left(\frac{kx}{2}\right)^2\), where (their \(A\)) \(\neq 1\) | M1 o.e. | — |
| \(\frac{3}{16}k^2 = \frac{243}{16} \Rightarrow k^2 = 81\) | — | — |
| \(k = 9\) | A1 cso | \(k = 9\) cao; note \(k = \pm 9\) with no reference to \(k = 9\) only is A0 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\left(\frac{1}{8}\right)(-3)\left(\frac{9}{2}\right) \Rightarrow B = -\frac{27}{16}\) | M1 | Uses \(x\) term: (their \(A\))\((-3)\left(\frac{k}{2}\right)\) or (their \(A\))\((-3)\left(\frac{kx}{2}\right)\); where (their \(A\)) \(\neq 1\), or \((2)^{-4}(-3)(k)\) or \((2)^{-4}(-3)(kx)\) or \(-\frac{3k}{16}\) |
| \(B = -\frac{27}{16}\) | A1 cso | \(-\frac{27}{16}\) or \(-1\frac{11}{16}\) or \(-1.6875\) |
# Question 2:
$$\{(2+kx)^{-3}\} = 2^{-3}\left(1+\frac{kx}{2}\right)^{-3} = \frac{1}{8}\left(1+(-3)\left(\frac{kx}{2}\right)+\frac{(-3)(-3-1)}{2!}\left(\frac{kx}{2}\right)^2+\ldots\right), \quad k>0$$
## Part (a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $A = \frac{1}{8}$ | B1 cao | $\frac{1}{8}$ or $2^{-3}$ or $0.125$, clearly identified as $A$ or as their answer to part (a) |
**[1]**
---
## Part (b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Uses $x^2$ term: $\left(\frac{1}{8}\right)\frac{(-3)(-4)}{2!}\left(\frac{k}{2}\right)^2$ | M1 | Uses the $x^2$ term; either $\frac{(-3)(-4)}{2!}$ or $\left(\frac{k}{2}\right)^2$ or $\left(\frac{kx}{2}\right)^2$ or $\frac{(-3)(-4)}{2}$ or $6$ seen |
| Either (their $A$)$\frac{(-3)(-4)}{2!}\left(\frac{k}{2}\right)^2$ or (their $A$)$\frac{(-3)(-4)}{2!}\left(\frac{kx}{2}\right)^2$, where (their $A$) $\neq 1$ | M1 o.e. | — |
| $\frac{3}{16}k^2 = \frac{243}{16} \Rightarrow k^2 = 81$ | — | — |
| $k = 9$ | A1 cso | $k = 9$ cao; note $k = \pm 9$ with no reference to $k = 9$ only is A0 |
**[3]**
---
## Part (c):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\left(\frac{1}{8}\right)(-3)\left(\frac{9}{2}\right) \Rightarrow B = -\frac{27}{16}$ | M1 | Uses $x$ term: (their $A$)$(-3)\left(\frac{k}{2}\right)$ or (their $A$)$(-3)\left(\frac{kx}{2}\right)$; where (their $A$) $\neq 1$, or $(2)^{-4}(-3)(k)$ or $(2)^{-4}(-3)(kx)$ or $-\frac{3k}{16}$ |
| $B = -\frac{27}{16}$ | A1 cso | $-\frac{27}{16}$ or $-1\frac{11}{16}$ or $-1.6875$ |
**[2] Total: 6**2. $\quad \mathrm { f } ( x ) = ( 2 + k x ) ^ { - 3 } , \quad | k x | < 2$, where $k$ is a positive constant
The binomial expansion of $\mathrm { f } ( x )$, in ascending powers of $x$, up to and including the term in $x ^ { 2 }$ is
$$A + B x + \frac { 243 } { 16 } x ^ { 2 }$$
where $A$ and $B$ are constants.
\begin{enumerate}[label=(\alph*)]
\item Write down the value of $A$.
\item Find the value of $k$.
\item Find the value of $B$.
\end{enumerate}
\hfill \mbox{\textit{Edexcel C4 2017 Q2 [6]}}