$6n + 9$ isw or $3(2n + 3)$ | B1 |
$6n$ is even [but 9 is odd], even + odd = odd or $2n + 3$ is odd since even + odd = odd and odd × odd = odd | B1 dep | this mark is dependent on the previous B1; accept equiv. general statements using either $6n + 9$ or $3(2n + 3)$
'$n$ is a multiple of 3' or '$n$ is divisible by 3'' without additional incorrect statement(s) | B2 | B2 for 'it is divisible by 9, so $n$ is divisible by 3'; M1 for '$6n$ is divisible by 9' or '$2n + 3$ is divisible by 3' or '$n$ is a multiple of 3'' oe with additional incorrect statement(s) | B2 for just 'it is divisible by 3'' but M1 for 'it is divisible by 9, so it is divisible by 3'; eg M1 for '$n$ is divisible by 9, so $n$ is divisible by 3''; N.B. 0 for '$n$ is a factor of 3'' (but M1 may be earned earlier)
| [4] |

# Question 10(i):

$AB^2 = (1-(-1))^2 + (5-1)^2$ | M1 | oe, or square root of this; condone poor notation re roots; condone $(1 + 1)^2$ instead of $(1-(-1))^2$; allow M1 for vector $\vec{AB} = \begin{pmatrix}-2\\-4\end{pmatrix}$, condoning poor notation, or triangle with hyp AB and lengths 2 and 4 correctly marked
$BC^2 = (3-(-1))^2 + (-1-1)^2$ | M1 | oe, or square root of this; condone poor notation re roots; condone $(3 + 1)^2$ instead of $(3-(-1))^2$ oe; allow M1 for vector $\vec{BC} = \begin{pmatrix}4\\-2\end{pmatrix}$, condoning poor notation, or triangle with hyp BC and lengths 4 and 2 correctly marked
shown equal eg $AB^2 = 2^2 + 4^2 = [20]$ and $BC^2 = 4^2 + 2^2 = [20]$ with correct notation for final comparison | A1 | or statement that AB and BC are each the hypotenuse of a right-angled triangle with sides 2 and 4 so are equal; SC2 for just $AB^2 = 2^2 + 4^2$ and $BC^2 = 4^2 + 2^2$ (or roots of these) with no clearer earlier working; condone poor notation | eg A0 for $AB = 20$ etc
| [3] |

# Question 10(ii):

$[\text{grad. of } AC =] \frac{5-(-1)}{1-3}$ or $\frac{6}{-2}$ oe | M1 | award at first step shown even if errors after
$[\text{grad. of } BD =] \frac{5-1}{11-(-1)}$ or $\frac{4}{12}$ oe | M1 | if one or both of grad $AC = -3$ and grad $BD = 1/3$ seen without better working for both gradients, award one M1 only. For M1M1 it must be clear that they are obtained independently
showing or stating product of gradients $= -1$ or that one gradient is the negative reciprocal of the other oe | B1 | eg accept $m_1 \times m_2 = -1$ or 'one gradient is negative reciprocal of the other'; B0 for 'opposite' used instead of 'negative' or 'reciprocal' | may be earned independently of correct gradients, but for all 3 marks to be earned the work must be fully correct
| [3] |

# Question 10(iii):

midpoint E of AC $= (2, 2)$ www | B1 | condone missing brackets for both B1s
eqn BD is $y = \frac{1}{4}x + \frac{4}{3}$ oe | M1 | accept any correct form isw or correct ft their gradients or their midpt F of BD; this mark will often be gained on the first line of their working for BD | may be earned using $(2, 2)$ but then must independently show that B or D or $(5, 3)$ is on this line to be eligible for A1
eqn AC is $y = -3x + 8$ oe | M1 | accept any correct form isw or correct ft their gradients or their midpt E of AC; this mark will often be gained on the first line of their working for AC | if equation(s) of lines are seen in part ii, allow the M1s if seen/used in this part
using both lines and obtaining intersection E is $(2, 2)$ (NB must be independently obtained from midpt of AC) | A1 | [see appendix for alternative ways of gaining these last two marks in different methods]
midpoint F of BD $= (5, 3)$ | B1 | this mark is often earned earlier | see the appendix for some common alternative methods for this question; for all methods show annotations M1 B1 etc then omission mark or A0 if that mark has not been earned
| [5] |

# Question 11(i):

$(2x + 1)(x + 2)(x - 5)$ | M1 | or $(x + 1/2)(x + 2)(x - 5)$; need not be written as product
correct expansion of two linear factors of their product of three linear factors | M1 | dep on first M1; ft one error in previous expansion; condone one error in this expansion
expansion of their linear and quadratic factors | M1 | or for direct expansion of all three factors, allow M2 for $2x^3 - 10x^2 + 4x^2 + x^2 - 20x - 5x + 2x - 10$ [or half all these], or M1 if one or two errors,
$[y =] 2x^3 - 5x^2 - 23x - 10$ or $a = -5, b = -23$ and $c = -10$ | A1 | condone poor notation when 'doubling' to reach expression with $2x^3\ldots$; for an attempt at setting up three simultaneous equations in $a, b,$ and $c$: M1 for at least two of the three equations then M2 for correctly eliminating any two variables or M1 for correctly eliminating one variable to get two equations in two unknowns; and then A1 for values.
| [4] |

# Question 11(ii):

graph of cubic correct way up | B1 | must not be ruled; no curving back; condone slight 'flicking out' at ends; allow min on y axis or in 3rd or 4th quadrants; condone some 'doubling' or 'feathering' (deleted work still may show in scans)
crossing x axis at −2, −1/2 and 5 | B1 | B0 if stops at x-axis on graph or nearby in this part | allow if no graph, but marked on x-axis
crossing y axis at −10 or ft their cubic in (i) | B1 | mark intent for intersections with both axes; or $x = 0, y = -10$ or ft in this part if consistent with graph drawn;
| [3] |

# Question 11(iii):

$(0, -18)$; accept $-18$ or ft their constant − 8 | [1] | or ft their intn on y-axis − 8
| [1] |

# Question 11(iv):

roots at 2.5, 1, 8 | M1 | or attempt to substitute $(x - 3)$ in $(2x + 1)(x + 2)(x - 5)$ or in $(x + 1/2)(x + 2)(x - 5)$ or in their unfactorised form of f(x)– attempt need not be simplified
$(2x - 5)(x - 1)(x - 8)$ | A1 | accept $2(x - 2.5)$ oe instead of $(2x - 5)$ | M0 for use of $(x + 3)$ or roots $-3.5, -5, 2$ but then allow SC1 for $(2x + 7)(x + 5)(x - 2)$
$(0, -40)$; accept $-40$ | B2 | M1 for $-5 \times -1 \times -8$ or ft for their f(−3) attempted or g(0) attempted or for their answer ft from their factorised form | eg M1 for $(0, -70)$ or $-70$ after $(2x + 7)(x + 5)(x - 2)$ after M0, allow SC1 for f(3) = −70
| [4] |

# Question 12(i):

$(-1, 6) (0.1) (1,-2) (2,-3) (3,-2) (4, 1) (5,6)$ seen plotted | B2 | or for a curve within 2 mm of these points; B1 for 3 correct plots or for at least 3 of the pairs of values seen eg in table | use overlay; scroll down to spare copy of graph to see if used [or click 'fit height']; also allow B1 for $(2 ± \sqrt{3}, 0)$ and $(2, -3)$ seen or plotted and curve not through other correct points
smooth curve through all 7 points | B1 dep | dep on correct points; tolerance 2 mm; condone some feathering/doubling (deleted work still may show in scans); curve should not be flat-bottomed or go to a point at min. or curve back in at top;
$(0.3$ to $0.5, -0.3$ to $-0.5)$ and $(2.5$ to $2.7, -2.5$ to $-2.7)$ and $(4, 1)$ | B2 | may be given in form $x = \ldots, y = \ldots$; B1 for two intersections correct or for all the x values given correctly
| [5] |

# Question 12(ii):

$\frac{1}{x-3} = x^2 - 4x + 1$ | M1 |
$1 = (x - 3)(x^2 - 4x + 1)$ | M1 | condone omission of brackets only if used correctly afterwards, with at most one error; allow for terms expanded correctly with at most one error
at least one further correct interim step with '=1' or '=0' ,as appropriate, leading to given answer, which must be stated correctly | A1 | there may also be a previous step of expansion of terms without an equation, eg in grid; if M0, allow SC1 for correct division of given cubic by quadratic to gain $(x - 3)$ with remainder −1, or vice-versa | NB mark method not answer - given answer is $x^3 - 7x + 13x - 4 = 0$
quadratic factor is $x^2 - 3x + 1$ | B2 | found by division or inspection; allow M1 for division by $x - 4$ as far as $x^3 - 4x^2$ in the working, or for inspection with two terms correct
substitution into quadratic formula or for completing the square used as far as $\left(x - \frac{3}{2}\right)^2 = \frac{3}{4}$ | M1 | condone one error | no ft from a wrong 'factor';
$\frac{3 ± \sqrt{5}}{2}$ oe | A2 | A1 if one error in final numerical expression, but only if roots are real | isw factors
| [5] |

# Question 12(iii):

quadratic factor is $x^2 - 3x + 1$ | B2 | found by division or inspection; allow M1 for division by $x - 4$ as far as $x^3 - 4x^2$ in the working, or for inspection with two terms correct
substitution into quadratic formula or for completing the square used as far as $\left(x - \frac{3}{2}\right)^2 = \frac{3}{4}$ | M1 | condone one error | no ft from a wrong 'factor';
$\frac{3 ± \sqrt{5}}{2}$ oe | A2 | A1 if one error in final numerical expression, but only if roots are real | isw factors
| [5] |