June 2025 Paper 2 Q6
6 The curve with equation \(y = \dfrac{\mathrm{e}^{\frac{x}{2}}}{x - 3}\) is shown in the diagram.
The region shaded is bounded by the curve, the \(x\)-axis, and the lines \(x = 4\) and \(x = 8\)
The trapezium rule with six ordinates (5 strips) is to be used to find an approximate value for the area of the shaded region.
Some of the values required to obtain this approximation are shown in the table below.
| \(x\) | 4 | 4.8 | 5.6 | 6.4 | 7.2 | 8 |
|---|---|---|---|---|---|---|
| \(y\) | 7.3891 | 6.1240 | 6.3249 | 8.7139 | 10.9196 |
Give your answer to five significant figures.
[3 marks]The student correctly obtains 29.759 as their improved approximation.
The student claims that the exact area must be greater than 29.759
Without further calculation, explain whether or not the student is correct.
[2 marks]| Scheme | Marks | AO |
|---|---|---|
| (i) Obtains AWFW [7.2154, 7.2155] May be seen in the table | B1 | 1.1b |
| (1) | ||
| (ii) States or uses h = 0.8 OE | B1 | 1.1b |
| Substitutes all the given \(y\) values and their answer from 6(a)(i) to achieve \(7.3891 + 10.9196 + 2(6.124 + 6.3249 + 7.2155 + 8.7139)\) FT their 7.2155 PI 75.0653 Accept correct exact values or values to more than 4 decimal places. Condone omission of the bracket after 8.7139 | M1 | 1.1a |
| Obtains AWRT 30.026 | A1 | 1.1b |
| (3) |
Typical solution
(a)(i) 7.2155
(a)(ii)
\[\mathrm{h} = \frac{8 - 4}{5}\]\[\mathrm{A} = \frac{0.8}{2}\big(7.3891 + 10.9196 + 2(6.1240 + 6.3249 + 7.2155 + 8.7139)\big)\]\[= 30.026\]| Scheme | Marks | AO |
|---|---|---|
| States that the curve is convex or concave upwards Accept concave up Or States that \(\dfrac{\mathrm{d}^2y}{\mathrm{d}x^2} \gt 0\) | E1 | 2.4 |
| Explains that the trapezium rule gives an over-estimate and deduces that the student is incorrect | E1 | 2.4 |
| (2) | ||
| (6 marks) |
Typical solution
The curve is convex, so the trapezium rule gives an over-estimate.
Therefore, the student is incorrect.