AS June 2023 Paper 1 Q6
6
(a) Find and simplify the first five terms in the Maclaurin series for \(\mathrm{e}^{2x}\) [2 marks]
(b) Hence, or otherwise, write down the first five terms in the Maclaurin series for \(\mathrm{e}^{-2x}\) [1 mark]
(c) Hence, or otherwise, show that the Maclaurin series for \(\cosh(2x)\) is\[a + bx^2 + cx^4 + \ldots\]
where \(a\), \(b\) and \(c\) are rational numbers to be determined. [3 marks]
| Scheme | Marks | AO |
|---|---|---|
| Obtains a correct unsimplified Maclaurin series for \(\mathrm{e}^{2x}\) eg substitutes \(2x\) into the Maclaurin series for \(\mathrm{e}^x\) (condone missing brackets) | M1 | 1.1a |
| Obtains correct series, evaluating powers of 2 and factorials. Ignore any higher power terms. ISW | A1 | 1.1b |
| (2) |
Typical solution
\[\begin{aligned}\mathrm{e}^{2x} &= 1 + 2x + \frac{(2x)^2}{2!} + \frac{(2x)^3}{3!} + \frac{(2x)^4}{4!} + \ldots \\ &= 1 + 2x + \frac{4x^2}{2} + \frac{8x^3}{6} + \frac{16x^4}{24} + \ldots \\ &= 1 + 2x + 2x^2 + \frac{4}{3}x^3 + \frac{2}{3}x^4 + \ldots\end{aligned}\]| Scheme | Marks | AO |
|---|---|---|
| Rewrites their part (a) of the form \(a + bx + cx^2 + dx^3 + ex^4\) as \(a - bx + cx^2 - dx^3 + ex^4\) where \(a, b, c, d, e\) are non-zero. Ignore any higher power terms. or Obtains a correct series for \(\mathrm{e}^{-2x}\) evaluating powers of 2 and factorials. | B1F | 1.1b |
| (1) |
Typical solution
\[\mathrm{e}^{-2x} = 1 - 2x + 2x^2 - \frac{4}{3}x^3 + \frac{2}{3}x^4 + \ldots\]| Scheme | Marks | AO |
|---|---|---|
| States the definition of \(\cosh(2x)\) Or finds the correct first four derivatives of \(\cosh(2x)\) | B1 | 1.1b |
| Adds their polynomial expansions from parts (a) and (b). Condone subtraction. Or uses the general Maclaurin series to find the first five terms. | M1 | 3.1a |
| Obtains the correct simplified series with any equivalent rational coefficients. Must come from correct (a) and (b) Ignore any higher power terms. | R1 | 2.1 |
| (3) | ||
| (6 marks) |