A2 October 2021 Paper 2 Q10
10 In this question you must show detailed reasoning.
(a) By using an appropriate Maclaurin series prove that if \(x \gt 0\) then \(\mathrm{e}^x \gt 1 + x\). [2]
(b) Hence, by using a suitable substitution, deduce that \(\mathrm{e}^t \gt \mathrm{e}t\) for \(t \gt 1\). [1]
(c) Using the inequality in part (b), and by making a suitable choice for \(t\), determine which is greater, \(\mathrm{e}^\pi\) or \(\pi^\mathrm{e}\). [3]
| Scheme | Marks | AO |
|---|---|---|
| DR \(\mathrm{e}^x = 1 + x + \dfrac{x^2}{2!} + \dfrac{x^3}{3!} + \ldots = (1 + x) + \left(\dfrac{x^2}{2!} + \dfrac{x^3}{3!} + \ldots\right)\) | M1 | 1.1 |
| \(x \gt 0 \Rightarrow \dfrac{x^2}{2!} + \dfrac{x^3}{3!} + \ldots \gt 0\) \(\Rightarrow \mathrm{e}^x \gt 1 + x\) | A1 | 2.2a |
| [2] |
Notes
M1: Quoting and using the Maclaurin series
A1: AG. Result with sufficient justification
| Scheme | Marks | AO |
|---|---|---|
| DR \(t = x + 1 \Rightarrow \mathrm{e}^{t-1} \gt t \Rightarrow \dfrac{\mathrm{e}^t}{\mathrm{e}} \gt t \Rightarrow \mathrm{e}^t \gt \mathrm{e}t\) | B1 | 3.1a |
| [1] |
Notes
B1: AG
| Scheme | Marks | AO |
|---|---|---|
| DR \(t = \dfrac{\pi}{\mathrm{e}} \gt 1\) since \(2 \lt \mathrm{e} \lt 3\) and \(\pi \gt 3\) | B1 | 3.1a |
| \(\mathrm{e}^{\frac{\pi}{\mathrm{e}}} \gt \mathrm{e} \times \dfrac{\pi}{\mathrm{e}}\ (= \pi)\) | M1 | 3.1a |
| \(\Rightarrow \mathrm{e}^\pi \gt \pi^\mathrm{e}\) (ie \(\mathrm{e}^\pi\) is greater) | A1 | 1.1 |
| [3] |
Notes
B1: Some justification that \(t \gt 1\) is required
M1: Substituting their choice into the inequality
A1: Answer without use of inequality in part (b) scores M0A0
Alternative method
| Scheme | Marks |
|---|---|
| \(t = \ln\pi\) | B1 |
| \(\mathrm{e}^{\ln\pi} \gt \mathrm{e}\ln\pi\) | M1 |
| \(\pi \gt \ln\left(\pi^\mathrm{e}\right)\) \(\mathrm{e}^\pi \gt \pi^\mathrm{e}\) | A1 |
| [3] |
B1: Some justification that \(t \gt 1\) is required