## Question 4:

### Part (i)(A):

| Answer | Mark | Guidance |
|--------|------|----------|
| Sketch of cubic correct way up and with two turning points, crossing $x$-axis in 3 distinct points | B1 | **0** if stops at $x$-axis; condone not crossing $y$-axis; no section to be ruled; no curving back; condone slight 'flicking out' at ends; condone some doubling |
| Crossing $x$-axis at $1$, $2.5$ and $4$ | B1 | Intersections labelled on graph or shown nearby; mark intent for intersections with both axes; allow $2.5$ indicated by graph crossing halfway between their marked $2$ and $3$ on scale |
| Crossing $y$-axis at $-20$ | B1 | Or $x=0$, $y=-20$ seen in this part if consistent with graph drawn; allow if no graph, but eg **B0** for graph with intersection on $+$ve $y$-axis or nowhere near their indicated $-20$ |

### Part (i)(B):

| Answer | Mark | Guidance |
|--------|------|----------|
| Correct expansion of two brackets | M1 | Or **M2** for all 3 brackets multiplied at once, showing all 8 terms (**M1** if error in one term): $2x^3-8x^2-2x^2-5x^2+8x+ x+20x-20$; eg **M1** for $(2x-5)(x^2-5x+4)$; condone missing brackets if intent clear/used correctly |
| Correct interim step(s) multiplying out linear and quadratic factors before given answer | M1 | |
| **Or** showing that $1$, $2.5$ and $4$ all satisfy $f(x)=0$ for cubic in $2x^3\ldots$ form | M1 | Or **M1** for dividing $2x^3\ldots$ form by one of the linear factors and **M1** for factorising the resultant quadratic factor |
| Comparing coefficients of eg $x^3$ in the two forms | M1 | |

### Part (ii)(A):

| Answer | Mark | Guidance |
|--------|------|----------|
| $250-375+165-40$ isw | B1 | Or showing that $x-5$ is a factor by eg division and then stating that $x=5$ is root or that $g(5)=0$; '$2\times125+15\times25+33\times5-40$' is not sufficient; or $[g(5)=]\ f(5)-20=5\times4\times1-20\ [=0]$ |

### Part (ii)(B):

| Answer | Mark | Guidance |
|--------|------|----------|
| $(x-5)$ seen or used as linear factor | M1 | May be in attempt at division; allow if seen in (ii)(A) |
| Division by $(x-5)$ as far as $2x^3-10x^2$ seen in working | M1 | Or inspection/equating coefficients with two terms correct eg $(2x^2\ldots+8)$; for division: condone signs of $2x^3-10x^2$ changed for subtraction, or subtraction sign in front of first term |
| $2x^2-5x+8$ obtained isw | A1 | Eg may be seen in grid; condone $g(x)$ not expressed as product |

### Part (ii)(C):

| Answer | Mark | Guidance |
|--------|------|----------|
| $b^2-4ac$ used on their quadratic factor | M1 | May be in formula |
| $(-5)^2-4\times2\times8$ oe and negative [or $-39$] so no [real] root [may say only one [real] root, thinking of $x=5$] | A1 | [Or allow **2** marks for complete correct attempt at completing the square and conclusion, or using calculus to show min value is above $x$-axis and comment re curve all above $x$-axis]; no ft for A mark from wrong quadratic factor; condone error in working out $-39$ if correct unsimplified expression seen and neg result obtained; $-5^2-4\times2\times8$ evaluated correctly with comment is eligible for **A1**, otherwise bod for the **M1** only |

### Part (iii):

| Answer | Mark | Guidance |
|--------|------|----------|
| Translation | B1 | NB 'Moves' not sufficient for this first mark |
| $\begin{pmatrix} 0 \\ -20 \end{pmatrix}$ | B1 | Or 20 down; **B0** for second mark if choice of one wrong, one right description |