| Exam Board | Edexcel |
|---|---|
| Module | C12 (Core Mathematics 1 & 2) |
| Year | 2015 |
| Session | January |
| 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 2.025 = 2 + 0.1/4 and substituting x = 0.1. The calculations are mechanical with no problem-solving insight needed, making it easier than average but not trivial due to the arithmetic involved. |
| Spec | 1.04a Binomial expansion: (a+b)^n for positive integer n |
| Answer | Marks | Guidance |
|---|---|---|
| \(\left(2+\frac{x}{4}\right)^{10} = 2^{10}+\binom{10}{1}2^9\cdot\left(\frac{x}{4}\right)+\binom{10}{2}2^8\left(\frac{x}{4}\right)^2+\binom{10}{3}2^7\left(\frac{x}{4}\right)^3\ldots\) | M1 | Attempt at Binomial with correct coefficient multiplied by correct power of \(x\) |
| \(= 1024 + 1280x + 720x^2 + 240x^3\ldots\) | B1, A1, A1 [4] | B1: simplified to 1024. A1: for two correct from \(1280x\), \(720x^2\), \(240x^3\). A1: all three correct |
| Answer | Marks | Guidance |
|---|---|---|
| State or use \(x=0.1\) | B1 | |
| Estimate \(= 1024 + 1280\times0.1 + 720\times(0.1)^2 + 240\times(0.1)^3\ldots\) | M1 | Uses \(\frac{x}{4}=0.025\) substituted into series |
| \(= 1159.44\) or \(1159.440\) or \(1159\) or \(1159.4\) | A1 [3] |
# Question 4:
## Part (a)
$\left(2+\frac{x}{4}\right)^{10} = 2^{10}+\binom{10}{1}2^9\cdot\left(\frac{x}{4}\right)+\binom{10}{2}2^8\left(\frac{x}{4}\right)^2+\binom{10}{3}2^7\left(\frac{x}{4}\right)^3\ldots$ | M1 | Attempt at Binomial with correct coefficient multiplied by correct power of $x$
$= 1024 + 1280x + 720x^2 + 240x^3\ldots$ | B1, A1, A1 [4] | B1: simplified to 1024. A1: for two correct from $1280x$, $720x^2$, $240x^3$. A1: all three correct
## Part (b)
State or use $x=0.1$ | B1 |
Estimate $= 1024 + 1280\times0.1 + 720\times(0.1)^2 + 240\times(0.1)^3\ldots$ | M1 | Uses $\frac{x}{4}=0.025$ substituted into series
$= 1159.44$ or $1159.440$ or $1159$ or $1159.4$ | A1 [3] |\begin{enumerate}
\item (a) Find the first 4 terms in ascending powers of $x$ of the binomial expansion of
\end{enumerate}
$$\left( 2 + \frac { x } { 4 } \right) ^ { 10 }$$
giving each term in its simplest form.\\
(b) Use your expansion to find an estimated value for $2.025 ^ { 10 }$, stating the value of $x$ which you have used and showing your working.\\
\hfill \mbox{\textit{Edexcel C12 2015 Q4 [7]}}