## Question 10(i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Graph of cubic correct way up | B1 | B0 if stops at $x$-axis; must not have any ruled sections; no curving back; condone slight 'flicking out' at ends but not approaching a turning point; allow max on $y$-axis or in 1st or 2nd quadrants; condone some 'doubling' or 'feathering' |
| Crossing $x$-axis at $-3$, $2$ and $5$ | B1 | On graph or nearby; may be in coordinate form; mark intent for intersections with both axes; allow if no graph, but marked on $x$-axis; condone intercepts for $x$ and/or $y$ given as reversed coordinates |
| Crossing $y$-axis at $30$ | B1 | or $x=0$, $y=30$ seen if consistent with graph drawn; allow if no graph, but eg B0 for graph with intn on $y$-axis nowhere near their indicated 30 |

## Question 10(ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Correct expansion of two of the linear factors | M1 | May be 3 or 4 terms; condone lack of brackets if correct expansions as if they were there |
| Correct expansion and completion to given answer $x^3 - 4x^2 - 11x + 30$ | A1 | Must be working for this step before given answer; or for direct expansion of all three factors, allow M1 for $x^3 + 3x^2 - 2x^2 - 5x^2 - 6x - 15x + 10x + 30$, condoning an error in one term, and A1 if no error for completion by stating given answer |

## Question 10(iii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Translation | B1 | 0 for shift or move etc without stating translation; 0 if eg stretch also mentioned |
| $\begin{pmatrix} 0 \\ -36 \end{pmatrix}$ | B1 | or 36 down, or $-36$ in $y$ direction oe; if conflict, eg between '$-36$ in $y$ direction' and wrong vector, award B0; 0 for '$-36$ down' |

## Question 10(iv):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $-1 - 4 + 11 - 6 = 0$ | B1 | or B1 for correct division by $(x+1)$ or for the quadratic factor found by inspection, and the conclusion that no remainder means that $g(-1) = 0$; NB examiners must use annotation in this part; a tick where each mark is earned is sufficient |
| Attempt at division by $(x+1)$ as far as $x^3 + x^2$ in working | M1 | or inspection with at least two terms of three-term quadratic factor correct; or finding $f(6) = 0$; M0 for trials of factors to give cubic unless correct answer found with clear correct working, in which case award the M1A1M1A1 |
| Correctly obtaining $x^2 - 5x - 6$ | A1 | or $(x-6)$ found as factor |
| Factorising the correct quadratic factor $x^2 - 5x - 6$, that has been correctly obtained | M1 | For factors giving two terms of quadratic correct or for factors ft one error in quadratic formula or completing square; M0 for formula etc without factors found; for those who have used the factor theorem to find $(x-6)$, M1 for working with cubic to find that $(x+1)$ is repeated; allow for $(x-6)$ and $(x+1)$ given as factors eg after quadratic formula etc |
| $(x-6)(x+1)^2$ oe isw | A1 | Condone inclusion of $'=0'$; isw roots found, even if stated as factors; just the answer $(x-6)(x+1)^2$ oe gets last 4 marks |