| Exam Board | Edexcel |
|---|---|
| Module | Paper 2 (Paper 2) |
| Session | Specimen |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Generalised Binomial Theorem |
| Type | Expand and state validity |
| Difficulty | Moderate -0.3 This is a straightforward application of the binomial expansion with a fractional power, requiring rewriting as 2(1-x/4)^(1/2), expanding to find k, and checking validity |x/4|<1. The validity check is routine. Slightly easier than average due to being a standard textbook exercise with clear steps. |
| Spec | 1.04c Extend binomial expansion: rational n, |x|<11.04d Binomial expansion validity: convergence conditions |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\sqrt{(4-x)} = 2\left(1 - \frac{1}{4}x\right)^{\frac{1}{2}}\) | M1 | Take out factor of 4; write as \(2(1\pm\ldots)^{\frac{1}{2}}\) |
| \(\left(1-\frac{1}{4}x\right)^{\frac{1}{2}} = 1 + \frac{1}{2}\left(-\frac{1}{4}x\right) + \frac{\left(\frac{1}{2}\right)\left(-\frac{1}{2}\right)}{2!}\left(-\frac{1}{4}x\right)^2 + \ldots\) | M1 | Attempt binomial expansion with \(n = \frac{1}{2}\) |
| \(\sqrt{(4-x)} = 2\left(1 - \frac{1}{8}x - \frac{1}{128}x^2 + \ldots\right)\) | A1 | Correct expression inside bracket (may be unsimplified) |
| \(\sqrt{(4-x)} = 2 - \frac{1}{4}x - \frac{1}{64}x^2 + \ldots\) and \(k = -\frac{1}{64}\) | A1 | Fully simplified with correct \(k\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| The expansion is valid for \( | x | < 4\), so \(x=1\) can be used |
## Question 7(a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\sqrt{(4-x)} = 2\left(1 - \frac{1}{4}x\right)^{\frac{1}{2}}$ | M1 | Take out factor of 4; write as $2(1\pm\ldots)^{\frac{1}{2}}$ |
| $\left(1-\frac{1}{4}x\right)^{\frac{1}{2}} = 1 + \frac{1}{2}\left(-\frac{1}{4}x\right) + \frac{\left(\frac{1}{2}\right)\left(-\frac{1}{2}\right)}{2!}\left(-\frac{1}{4}x\right)^2 + \ldots$ | M1 | Attempt binomial expansion with $n = \frac{1}{2}$ |
| $\sqrt{(4-x)} = 2\left(1 - \frac{1}{8}x - \frac{1}{128}x^2 + \ldots\right)$ | A1 | Correct expression inside bracket (may be unsimplified) |
| $\sqrt{(4-x)} = 2 - \frac{1}{4}x - \frac{1}{64}x^2 + \ldots$ and $k = -\frac{1}{64}$ | A1 | Fully simplified with correct $k$ |
## Question 7(b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| The expansion is valid for $|x| < 4$, so $x=1$ can be used | B1 | Must state valid range $|x|<4$ and confirm $x=1$ satisfies it |\begin{enumerate}
\item (a) Use the binomial expansion, in ascending powers of $x$, to show that
\end{enumerate}
$$\sqrt { ( 4 - x ) } = 2 - \frac { 1 } { 4 } x + k x ^ { 2 } + \ldots$$
where $k$ is a rational constant to be found.
A student attempts to substitute $x = 1$ into both sides of this equation to find an approximate value for $\sqrt { 3 }$.\\
(b) State, giving a reason, if the expansion is valid for this value of $x$.
\hfill \mbox{\textit{Edexcel Paper 2 Q7 [5]}}