# Question 1:

| Answer/Working | Marks | Guidance |
|---|---|---|
| Let roots be $\alpha, \alpha, \beta$ | | N.B. Not $\alpha, \beta, \gamma$ |
| (1) $2\alpha + \beta = 0$ | M1 | Writes down sum of roots |
| (2) $2\alpha\beta + \alpha^2 = p$ | A1 | Sum of products in pairs |
| (3) $\alpha^2\beta = -q$ | A1 | Product of roots |
| From (1) $\beta = -2\alpha$ | | |
| (2) $\Rightarrow -4\alpha^2 + \alpha^2 = p \Rightarrow p = -3\alpha^2$ | M1 | Eliminates $\beta$ (or $\alpha$) |
| (3) $\Rightarrow -2\alpha^3 = -q \Rightarrow q = 2\alpha^3$ | | |
| $\alpha^6 = \left(-\frac{p}{3}\right)^3 = \left(\frac{q}{2}\right)^2 \Rightarrow 4p^3 + 27q^2 = 0$ (AG) | A1 | Equates power of $\alpha$ (or $\beta$); Total: 5 marks |

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