## Question 4:

| Answer | Marks | Guidance |
|--------|-------|----------|
| Use correct quotient rule | M1 | Allow use of correct product rule on $x \times (1 + \ln x)^{-1}$ |
| Obtain correct derivative in any form: $\frac{dy}{dx} = \frac{(1+\ln x) - x \times \frac{1}{x}}{(1+\ln x)^2} = \left(\frac{1}{1+\ln x} - \frac{1}{(1+\ln x)^2}\right)$ | A1 | |
| Equate derivative to $\frac{1}{4}$ and obtain a quadratic in $\ln x$ or $(1+\ln x)$ | M1 | Horizontal form. Accept $\ln x = \frac{1}{4}(1+\ln x)^2$ |
| Reduce to $(\ln x)^2 - 2\ln x + 1 = 0$ | A1 | Or 3-term equivalent. Condone $\ln x^2$ if later used correctly |
| Solve a 3-term quadratic in $\ln x$ for $x$ | M1 | Must see working if solving incorrect quadratic |
| Obtain answer $x = e$ | A1 | Accept $e^1$ |
| Obtain answer $y = \frac{1}{2}e$ | A1 | Exact only with no decimals seen before the exact value. Accept $\frac{e^1}{2}$ but not $\frac{e}{1+\ln e}$ |
| **Total** | **7** | |

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