| Exam Board | Edexcel |
|---|---|
| Module | C12 (Core Mathematics 1 & 2) |
| Year | 2018 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Binomial Theorem (positive integer n) |
| Type | Numerical approximation using expansion |
| Difficulty | Moderate -0.8 This is a straightforward application of the binomial theorem requiring routine expansion and substitution. Part (a) involves direct use of the binomial formula with simple arithmetic, while part (b) requires recognizing that 31/30 = 1 + 1/90 and substituting x = 1/30. The question is easier than average as it follows a standard template with no problem-solving insight needed, though it requires careful arithmetic across multiple steps. |
| Spec | 1.04a Binomial expansion: (a+b)^n for positive integer n |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\left(1+\frac{x}{3}\right)^{18} = 1 + \binom{18}{1}\cdot\left(\frac{x}{3}\right) + \binom{18}{2}\cdot\left(\frac{x}{3}\right)^2 + \binom{18}{3}\cdot\left(\frac{x}{3}\right)^3\ldots\) | M1 | Method mark for binomial attempt to get second, third or fourth term; need to see \(\frac{x}{3}\) with correct power of \(x\) and correct binomial coefficient |
| \(= 1 + 6x + 17x^2 + \frac{272}{9}x^3\ldots\) | B1, A1, A1 [4] | B1: first two terms \(1 + 6x\) (coefficient of \(x\) may be unsimplified, \(1^{18}\) must become 1); A1: \(+17x^2\); A1: \(+\frac{272}{9}x^3\) cao; accept \(30\frac{2}{9}x^3\) or \(30.\dot{2}x^3\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Use \(x = 0.1\) | B1 | Must be seen; states or uses \(x=0.1\) or equivalent such as \(\frac{1}{10}\) or \(\frac{3}{30}\) |
| \(= 1 + 6\times0.1 + 17\times(0.1)^2 + \frac{272}{9}\times(0.1)^3\ldots\) or equivalent | M1 | Fully substituting value into series with at least 4 terms |
| \(= 1.8002\) | A1cao [3] | cao |
# Question 5:
## Part (a)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\left(1+\frac{x}{3}\right)^{18} = 1 + \binom{18}{1}\cdot\left(\frac{x}{3}\right) + \binom{18}{2}\cdot\left(\frac{x}{3}\right)^2 + \binom{18}{3}\cdot\left(\frac{x}{3}\right)^3\ldots$ | M1 | Method mark for binomial attempt to get second, third or fourth term; need to see $\frac{x}{3}$ with correct power of $x$ and correct binomial coefficient |
| $= 1 + 6x + 17x^2 + \frac{272}{9}x^3\ldots$ | B1, A1, A1 [4] | B1: first two terms $1 + 6x$ (coefficient of $x$ may be unsimplified, $1^{18}$ must become 1); A1: $+17x^2$; A1: $+\frac{272}{9}x^3$ cao; accept $30\frac{2}{9}x^3$ or $30.\dot{2}x^3$ |
## Part (b)
| Answer | Marks | Guidance |
|--------|-------|----------|
| Use $x = 0.1$ | B1 | Must be seen; states or uses $x=0.1$ or equivalent such as $\frac{1}{10}$ or $\frac{3}{30}$ |
| $= 1 + 6\times0.1 + 17\times(0.1)^2 + \frac{272}{9}\times(0.1)^3\ldots$ or equivalent | M1 | Fully substituting value into series with at least 4 terms |
| $= 1.8002$ | A1cao [3] | cao |\begin{enumerate}
\item (a) Find the first 4 terms, in ascending powers of $x$, of the binomial expansion of
\end{enumerate}
$$\left( 1 + \frac { x } { 3 } \right) ^ { 18 }$$
giving each term in its simplest form.\\
(b) Use the answer to part (a) to find an estimated value for $\left( \frac { 31 } { 30 } \right) ^ { 18 }$, stating the value of $x$ that you have used and showing your working. Give your estimate to 4 decimal places.
II\\
\hfill \mbox{\textit{Edexcel C12 2018 Q5 [7]}}