AS June 2021 Paper 1 Q7
7 Show that the Maclaurin series for \(\ln(\mathrm{e} + 2\mathrm{e}x)\) is
\[1 + 2x - 2x^2 + ax^3 - \ldots\]where \(a\) is to be determined. [3 marks]
| Scheme | Marks | AO |
|---|---|---|
| Writes the given expression in the form \(m + \ln(1 + \mathrm{f}(x))\) where \(m\) is a non-zero constant (allow unsimplified). Or obtains the first three derivatives of \(\ln(\mathrm{e} + 2\mathrm{e}x)\) in the form \(p(q + rx)^n\) | M1 | 3.1a |
| From an expression of the form \(\ln(1 + \mathrm{f}(x))\), substitutes \(\mathrm{f}(x)\) into the Maclaurin series. Condone sign errors and missing brackets. Or substitutes \(x = 0\) into their \(\mathrm{f}(0)\), \(\mathrm{f}^{\prime}(0)\), \(\mathrm{f}^{\prime\prime}(0)\) and \(\mathrm{f}^{\prime\prime\prime}(0)\) of the form \(p(q + rx)^n\) | M1 | 1.1a |
| Completes a fully correct argument to reach the required result and correctly calculates \(a\). Do not condone missing brackets in their working. | R1 | 2.1 |
| (3 marks) |