| Exam Board | Edexcel |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Year | 2008 |
| 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 binomial expansion question requiring direct application of the formula for the first four terms, followed by a routine substitution (x=0.01) to estimate a numerical value. Both parts are standard textbook exercises with no problem-solving or insight required, making it easier than average but not trivial since it involves multiple terms and careful arithmetic. |
| Spec | 1.04a Binomial expansion: (a+b)^n for positive integer n |
| Answer | Marks |
|---|---|
| (a) \[\left(1 + \frac{1}{2}x\right)^{10} = 1 + \binom{10}{1}\left(\frac{1}{2}x\right) + \binom{10}{2}\left(\frac{1}{2}x\right)^2 + \binom{10}{3}\left(\frac{1}{2}x\right)^3\] | M1 A1 |
| \[= 1 + 5x + \frac{45}{4}(or\ 11.25)x^2 + 15x^3\] (coeffs need to be these, i.e., simplified) | A1; A1 (4) |
| Answer | Marks |
|---|---|
| (b) \((1 + \frac{1}{2} \times 0.01)^{10} = 1 + 5(0.01) + \frac{45}{4}or11.25(0.01)^2 + 15(0.01)^3\) | M1 A1 √ |
| \[= 1 + 0.05 + 0.001125 + 0.000015\] | |
| \[= 1.05114\] (cao) | A1 (3) [7] |
**(a)** $$\left(1 + \frac{1}{2}x\right)^{10} = 1 + \binom{10}{1}\left(\frac{1}{2}x\right) + \binom{10}{2}\left(\frac{1}{2}x\right)^2 + \binom{10}{3}\left(\frac{1}{2}x\right)^3$$ | M1 A1 |
$$= 1 + 5x + \frac{45}{4}(or\ 11.25)x^2 + 15x^3$$ (coeffs need to be these, i.e., simplified) | A1; A1 (4) |
[Allow A1A0, if totally correct with unsimplified, single fraction coefficients]
**Notes:**
- (a) For M1 first A1: Consider underlined expression only.
- M1 Requires correct structure for at least two of the three terms:
- (i) Must be attempt at binomial coefficients.
- (ii) Must have increasing powers of $x$.
- (iii) May be listed, need not be added; this applies for all marks.
- First A1: Requires all three correct terms but need not be simplified, allow $1^{10}$ etc., $^{10}C_r$ etc., and condone omission of brackets around powers of $\frac{1}{2}x$
- Second A1: Consider as B1 for $1 + 5x$
**(b)** $(1 + \frac{1}{2} \times 0.01)^{10} = 1 + 5(0.01) + \frac{45}{4}or11.25(0.01)^2 + 15(0.01)^3$ | M1 A1 √ |
$$= 1 + 0.05 + 0.001125 + 0.000015$$ | |
$$= 1.05114$$ (cao) | A1 (3) [7] |
**Notes:**
- (b) For M1: Substituting their (0.01) into their (a) result
- First A1 (f.t.): Substitution of (0.01) into their 4-termed expression in (a)
- Answer with no working scores no marks (calculator gives this answer)
---\begin{enumerate}[label=(\alph*)]
\item Find the first 4 terms of the expansion of $\left(1 + \frac{x}{2}\right)^{10}$ in ascending powers of $x$, giving each term in its simplest form. [4]
\item Use your expansion to estimate the value of $(1.005)^{10}$, giving your answer to 5 decimal places. [3]
\end{enumerate}
\hfill \mbox{\textit{Edexcel C2 2008 Q3 [7]}}