June 2023 Paper 2 Q3
3 In this question you must show detailed reasoning.
Find the exact area of the region enclosed by the curve \(y = \dfrac{1}{x + 2}\), the two axes and the line \(x = 2.5\). [3]
| Scheme | Marks | AO |
|---|---|---|
| DR \(\displaystyle\int \frac{1}{x + 2}\,\mathrm{d}x = \ln(x + 2)\) | M1 | 1.1 |
| \(\Big[\ln(x + 2)\Big]_0^{2.5} = \ln 4.5 - \ln 2\) or \(\ln\dfrac{4.5}{2}\) | M1 | 1.1 |
| \(= \ln\dfrac{9}{4}\) or \(\ln 2.25\) or \(2\ln\left(\frac{3}{2}\right)\) etc | A1 | 1.1 |
| [3] |
Notes
M1: Integrate and obtain expression involving ln. This may be combined with the next step. Ignore limits. Brackets \((x + 2)\) soi
If a substitution is used then only award this mark when \(\ln(x + 2)\) reached or an equivalent integral with appropriate limits (typically \(\ln x\) with limits 2 and 4.5).
M1: Substitute and use limits 0 & 2.5 (or appropriately changed limits in the case of a substitution) in their ln integral. Must see this step.
A1: www, any equivalent exact form, but must be a single term expression. Do not accept \(\ln\frac{4.5}{2}\) for this mark.
Correct answer with no working SC B1 [1/3]
Ignore use of modulus signs throughout.