| Exam Board | Edexcel |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Marks | 10 |
| 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 requiring students to equate coefficients to find n and a, then substitute values. The algebra is routine (solving n(n-1)/2 = 270 after finding n from the linear term), and part (b) is direct substitution. Slightly above average difficulty due to the two-part structure and need for careful algebraic manipulation, but remains a standard C2 exercise. |
| Spec | 1.04c Extend binomial expansion: rational n, |x|<11.04d Binomial expansion validity: convergence conditions |
| Answer | Marks |
|---|---|
| \((1 + ax)^n = 1 + n(ax) + \frac{n(n-1)}{2}(ax)^2 + \ldots\) | B2 |
| \(\therefore an = -24\) (1) and \(\frac{1}{2}a^2n(n-1) = 270\) (2) | M1 |
| (1) \(\Rightarrow a = \frac{-24}{n}\) sub. (2) | M1 |
| \(\frac{288}{n}(n-1) = 270\) | M1 |
| \(288n - 288 = 270n\) | |
| \(18n = 288\) | |
| \(n = 16, \quad a = -\frac{3}{2}\) | A2 |
| Answer | Marks |
|---|---|
| \(1 - \frac{3}{4}x = 0.9985 \Rightarrow x = 0.001\) | B1 |
| \(\therefore (0.9985)^{16} = 1 - 0.024 + 0.000270 = 0.97627\) (5dp) | M1 A1 |
**Part (a)**
$(1 + ax)^n = 1 + n(ax) + \frac{n(n-1)}{2}(ax)^2 + \ldots$ | B2 |
$\therefore an = -24$ (1) and $\frac{1}{2}a^2n(n-1) = 270$ (2) | M1 |
(1) $\Rightarrow a = \frac{-24}{n}$ sub. (2) | M1 |
$\frac{288}{n}(n-1) = 270$ | M1 |
$288n - 288 = 270n$ |
$18n = 288$ |
$n = 16, \quad a = -\frac{3}{2}$ | A2 |
**Part (b)**
$1 - \frac{3}{4}x = 0.9985 \Rightarrow x = 0.001$ | B1 |
$\therefore (0.9985)^{16} = 1 - 0.024 + 0.000270 = 0.97627$ (5dp) | M1 A1 |
---7. Given that for small values of $x$
$$( 1 + a x ) ^ { n } \approx 1 - 24 x + 270 x ^ { 2 } ,$$
where $n$ is an integer and $n > 1$,
\begin{enumerate}[label=(\alph*)]
\item show that $n = 16$ and find the value of $a$,
\item use your value of $a$ and a suitable value of $x$ to estimate the value of (0.9985) ${ } ^ { 16 }$, giving your answer to 5 decimal places.
\end{enumerate}
\hfill \mbox{\textit{Edexcel C2 Q7 [10]}}