## Question 10(i):

| Answer | Marks | Guidance |
|--------|-------|----------|
| | B1 | Use given substitution correctly in LHS of equation |
| | M1 | Rearrange and square to eliminate $\sqrt{u}$ or multiply by $u^{\frac{3}{2}} + 4u^{\frac{1}{2}} - 3$ |
| $u^3 + 8u^2 + 16u - 9 = 0$ | A1 | Obtain correct answer, must be an equation $= 0$ |
| **[3]** | | |

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## Question 10(ii):

**Either:**

| Answer | Marks | Guidance |
|--------|-------|----------|
| $\alpha\beta\gamma = -3$ | B1 | State or use correct result |
| $\sum\alpha^2 = -8\quad \sum\alpha^2\beta^2 = 16$ | B1 B1 | Use correct result, using correct (i) or using an identity involving $\sum\alpha=0,\ \sum\alpha\beta=4$ |
| $\left(\sum\alpha^2\right)^2 = \sum\alpha^4 + 2\sum\alpha^2\beta^2$ | M1* | Obtain an identity connecting $\sum\alpha^4$ and $\left(\sum\alpha^2\right)^2$ |
| | A1 | Obtain a correct answer |
| $29$ | DM1 | Use their values in their expression |
| | A1 | Obtain correct answer, c.w.o. |
| **[7]** | | |

**Or:**

| Answer | Marks | Guidance |
|--------|-------|----------|
| $\alpha\beta\gamma = -3$ | B1 | State or use correct result |
| $\sum\alpha=0,\ \sum\alpha\beta=4,\ \sum\alpha^2=-8,\ \sum\alpha^2\beta^2=16$ | B1 B1 | Use any 2 correct B1, other 2 correct B1 |
| $\sum\alpha^4 + 4\sum\alpha^2\cdot\sum\alpha\beta + 6\sum\alpha^2\beta^2 + 8\alpha\beta\gamma\sum\alpha$ | M1 | Expand $(\alpha+\beta+\gamma)^4$ and get expression involving symmetric functions only |
| | A1 | Obtain correct expression |
| | M1 | Use their values in their expression |
| | A1 | Obtain correct answer, c.w.o. |

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