## Question 3:

### Part (i):

| Answer | Mark | 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 |
| 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 intersection on $y$-axis nowhere near their indicated $30$ |

**[3]**

### Part (ii):

| Answer | Mark | 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 |

**[2]**

### Part (iii):

| Answer | Mark | 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' |

**[2]**

### Part (iv):

| Answer | Mark | 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 $g(-1)=0$; NB examiners must use annotation in this part |
| 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 |
| 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 the square; **M0** for formula etc without factors found; allow for $(x-6)$ and $(x+1)$ given as factors |
| $(x-6)(x+1)^2$ 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 |

**[5]**

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