| Exam Board | AQA |
|---|---|
| Module | Further AS Paper 1 (Further AS Paper 1) |
| Year | 2023 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Taylor series |
| Type | Deduce related series from given series |
| Difficulty | Moderate -0.3 This is a straightforward application of standard Maclaurin series techniques. Part (a) requires routine differentiation and substitution into the Maclaurin formula for e^(2x), part (b) is immediate by replacing x with -x, and part (c) uses the definition cosh(2x) = (e^(2x) + e^(-2x))/2 to combine the previous results. While it's Further Maths content, it's a standard textbook exercise with clear scaffolding and no novel insight required, making it slightly easier than average. |
| Spec | 4.08b Standard Maclaurin series: e^x, sin, cos, ln(1+x), (1+x)^n |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(e^{2x} = 1 + 2x + \frac{(2x)^2}{2!} + \frac{(2x)^3}{3!} + \frac{(2x)^4}{4!} + \ldots\) | M1 | Correct unsimplified Maclaurin series for \(e^{2x}\); condone missing brackets |
| \(= 1 + 2x + 2x^2 + \frac{4}{3}x^3 + \frac{2}{3}x^4 + \ldots\) | A1 | Correct simplified series; ignore any higher power terms; ISW |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(e^{-2x} = 1 - 2x + 2x^2 - \frac{4}{3}x^3 + \frac{2}{3}x^4 + \ldots\) | B1F | Rewrites part (a) in the form \(a - bx + cx^2 - dx^3 + ex^4\) where \(a,b,c,d,e\) are non-zero; or obtains correct series for \(e^{-2x}\) evaluating powers of 2 and factorials; ignore higher power terms |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\cosh(2x) = \frac{1}{2}(e^{2x} + e^{-2x})\) | B1 | States the definition of \(\cosh(2x)\); or finds the correct first four derivatives of \(\cosh(2x)\) |
| \(= \frac{1}{2}\left(1+2x+2x^2+\frac{4}{3}x^3+\frac{2}{3}x^4 + 1-2x+2x^2-\frac{4}{3}x^3+\frac{2}{3}x^4\right)+\ldots\) | M1 | Adds their polynomial expansions from parts (a) and (b); condone subtraction; or uses general Maclaurin series |
| \(= 1 + 2x^2 + \frac{2}{3}x^4 + \ldots\) | R1 | Obtains correct simplified series with any equivalent rational coefficients; must come from correct (a) and (b); ignore higher power terms |
## Question 6:
### Part 6(a):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $e^{2x} = 1 + 2x + \frac{(2x)^2}{2!} + \frac{(2x)^3}{3!} + \frac{(2x)^4}{4!} + \ldots$ | M1 | Correct unsimplified Maclaurin series for $e^{2x}$; condone missing brackets |
| $= 1 + 2x + 2x^2 + \frac{4}{3}x^3 + \frac{2}{3}x^4 + \ldots$ | A1 | Correct simplified series; ignore any higher power terms; ISW |
### Part 6(b):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $e^{-2x} = 1 - 2x + 2x^2 - \frac{4}{3}x^3 + \frac{2}{3}x^4 + \ldots$ | B1F | Rewrites part (a) in the form $a - bx + cx^2 - dx^3 + ex^4$ where $a,b,c,d,e$ are non-zero; or obtains correct series for $e^{-2x}$ evaluating powers of 2 and factorials; ignore higher power terms |
### Part 6(c):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\cosh(2x) = \frac{1}{2}(e^{2x} + e^{-2x})$ | B1 | States the definition of $\cosh(2x)$; or finds the correct first four derivatives of $\cosh(2x)$ |
| $= \frac{1}{2}\left(1+2x+2x^2+\frac{4}{3}x^3+\frac{2}{3}x^4 + 1-2x+2x^2-\frac{4}{3}x^3+\frac{2}{3}x^4\right)+\ldots$ | M1 | Adds their polynomial expansions from parts (a) and (b); condone subtraction; or uses general Maclaurin series |
| $= 1 + 2x^2 + \frac{2}{3}x^4 + \ldots$ | R1 | Obtains correct simplified series with any equivalent rational coefficients; must come from correct (a) and (b); ignore higher power terms |
---6
\begin{enumerate}[label=(\alph*)]
\item Find and simplify the first five terms in the Maclaurin series for $\mathrm { e } ^ { 2 x }$\\
6
\item Hence, or otherwise, write down the first five terms in the Maclaurin series for $\mathrm { e } ^ { - 2 x }$\\
6
\item Hence, or otherwise, show that the Maclaurin series for $\cosh ( 2 x )$ is
$$a + b x ^ { 2 } + c x ^ { 4 } + \ldots$$
where $a$, $b$ and $c$ are rational numbers to be determined.
\end{enumerate}
\hfill \mbox{\textit{AQA Further AS Paper 1 2023 Q6 [6]}}