## Question 9(i):

| Answer | Mark | Guidance |
|--------|------|----------|
| Differentiate to obtain $2e^{2x} - 18$ | B1 | |
| Equate first derivative to zero and use legitimate method to reach equation without e involved | M1 | |
| Confirm $x = \ln 3$ | A1 | AG; necessary detail needed (in particular, for solutions concluding $x = \frac{1}{2}\ln 9 = \ln 3$ or equiv — award A0) |
| **[3]** | | |

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## Question 9(ii):

| Answer | Mark | Guidance |
|--------|------|----------|
| Attempt integration | *M1 | confirmed by at least one correct term |
| Obtain $\frac{1}{2}e^{2x} - 9x^2 + 15x$ | A1 | or equiv |
| Apply limits 0 and $\ln 3$ to obtain exact unsimplified expression | M1 | dep *M |
| Obtain $4 - 9(\ln 3)^2 + 15\ln 3$ | A1 | or exact (maybe unsimplified) equiv perhaps still involving e |
| Attempt area of trapezium or equiv, retaining exactness throughout | M1 | using $\frac{1}{2}\ln 3 \times (y_1 + y_2)$ where $y_1$ is 15 or 16 and $y_2$ is attempt at $y$-coordinate of $Q$; if using alternative approach involving rectangle and triangle, complete attempt needs to be seen for M1; another alternative approach involves equation of $PQ$ ($y = \frac{8-18\ln 3}{\ln 3}\,x + 16$) with integration: M1 for attempting equation and integration, A1 for correct answer |
| Obtain $\frac{1}{2}\ln 3 \times (16 + 24 - 18\ln 3)$ | A1 | or equiv perhaps still including e |
| Subtract areas the right way round, retaining exactness | M1 | dep on award of all three M marks |
| Obtain $5\ln 3 - 4$ | A1 | or similarly simplified exact equiv |
| **[8]** | | |