mention of $-5$ as a square root of 25 or $(-5)^2 = 25$ | M1 | condone $-5^2 = 25$

$-5 -5 \neq 0$ o.e. or $x + 5 = 0$ | M1 | or, dep on first M1 being obtained, allow M1 for showing that 5 is the only soln of $x - 5 = 0$

| | | allow M2 for $x^2 - 25 = 0$ $(x + 5)(x - 5) = [0]$ so $x - 5 = 0$ or $x + 5 = 0$

## Question 10(i)

$(2x - 3)(x + 1)$ | M2 | M1 for factors with one sign error or giving two terms correct allow M1 for $2(x - 1.5)(x + 1)$ with no better factors seen

$x = 3/2$ and $-1$ obtained | B1 | or ft their factors

## Question 10(ii)

graph of quadratic the correct way up and crossing both axes | B1 |

crossing $x$-axis only at $3/2$ and $-1$ or ft from their roots in (i), or their factors if roots not given | B1 | for $x = 3/2$ condone 1 and 2 marked on axis and crossing roughly halfway between; intns must be shown labelled or worked out nearby

crossing $y$-axis at $-3$ | B1 |

## Question 10(iii)

use of $b^2 - 4ac$ with numbers subst (condone one error in substitution) (may be in quadratic formula) | M1 | may be in formula or $c - 2.5)^2 = 6.25 - 10$ or $(x - 2.5)^2 + 3.75 = 0$ oe (condone one error)

$25 - 40 < 0$ or $-15$ obtained | A1 | or $\sqrt{-15}$ seen in formula or $(x - 2.5)^2 = -3.75$ oe or $x = 2.5 \pm \sqrt{-3.75}$ oe

## Question 10(iv)

$2x^2 - x - 3 = x^2 - 5x + 10$ o.e. | M1 | attempt at eliminating y by subst or subtraction

$x^2 + 4x - 13 [= 0]$ | M1 | or $(x + 2)^2 = 17$; for rearranging to form $ax^2 + bx + c = [0]$ or to completing square form condone one error for each of 2nd and 3rd M1s

use of quad. formula on resulting eqn (do not allow for original quadratics used) | M1 | or $x + 2 = \pm\sqrt{17}$ o.e. 2nd and 3rd M1s may be earned for good attempt at completing square as far as roots obtained

$-2 \pm \sqrt{17}$ cao | A1 |

## Question 11(i)

$\text{grad } AB = \frac{1 - 3}{5 - (-1)} = [-1/3]$ | M1 |

$y - 3 = \text{their grad } (x - (-1))$ or $y - 1 = \text{their grad } (x - 5)$ | M1 | or use of $y = \text{their gradient } x + c$ with coords of A or B or M2 for $\frac{y - 3}{1 - 3} = \frac{x - (-1)}{5 - (-1)}$ o.e.

$y = -1/3x + 8/3$ or $3y = -x + 8$ o.e. isw | A1 | o.e. eg $x + 3y - 8 = 0$ or $6y = 16 - 2x$ allow B3 for correct eqn www

## Question 11(ii)

when $y = 0$, $x = 8$; when $x = 0$, $y = 8/3$ or ft their (i) | M1 | allow $8/3$ used without explanation if already seen in eqn in (i)

$[\text{Area} =] 1/2 \times 8/3 \times 8$ o.e. cao isw | M1 | NB answer $32/3$ given; allow $4 \times 8/3$ if first M1 earned; or M1 for $\int_0^8 [\frac{1}{3}(8 - x)]\text{d}x = [\frac{1}{3}(8x - \frac{1}{2}x^2)]_0^8$ and M1 dep for $\frac{1}{3}(64 - 32[- 0])$

## Question 11(iii)

grad perp $= -1/\text{grad } AB$ stated, or used after their grad AB stated in this part | M1 | or showing $3 \times (-1/3) = -1$ if (i) is wrong, allow the first M1 here ft, provided the answer is correct ft

midpoint [of AB] $= (2, 2)$ | M1 | must state 'midpoint' or show working

$y - 2 = \text{their grad perp } (x - 2)$ or ft their midpoint | M1 | for M3 this must be correct, starting from grad AB $= -1/3$, and also needs correct completion to given ans $y = 3x - 4$

alt method working back from ans: grad perp $= -1/\text{grad } AB$ and showing/stating same as given line | M1 | mark one method or the other, to benefit of candidate, not a mixture eg stating $-1/3 \times 3 = -1$

finding intn of their $y = -1/3x - 8/3$and $y = 3x - 4$ is $(2, 2)$ | M1 | or showing that $(2, 2)$ is on $y = 3x - 4$, having found $(2, 2)$ first

showing midpt of AB is $(2, 2)$ | M1 | [for both methods: for M3 must be fully correct]

## Question 11(iv)

subst $x = 3$ into $y = 3x - 4$ and obtaining centre $= (3, 5)$ | M1 | or using $(-1 - 3)^2 + (3 - b)^2 = (5 - 3)^2 + (1 - b)^2$ and finding $(3, 5)$

$r^2 = (5 - 3)^2 + (1 - 5)^2$ o.e. | M1 | or $(-1 - 3)^2 + (3 - 5)^2$ oe or ft their centre using A or B

$r = \sqrt{20}$ o.e. cao | A1 |

eqn is $(x - 3)^2 + (y - 5)^2 = 20$ or ft their $r$ and $y$-coord of centre | B1 | condone $(x - 3)^2 + (y - \text{their } 5)^2 = r^2$ o.e. or $(x - 3)^2 + (y - \text{their } 5)^2 = r^2$ o.e. (may be seen earlier)

## Question 12(i)

trials of at calculating $f(x)$ for at least one factor of 30 | M1 | M0 for division or inspection used

details of calculation for $f(2)$ or $f(-3)$ or $f(-5)$ | A1 |

attempt at division by $(x - 2)$ as far as $x^3 - 2x^2$ in working | M1 | or equiv for $(x + 3)$ or $(x + 5)$; or inspection with at least two terms of quadratic factor correct

correctly obtaining $x^2 + 8x + 15$ | A1 | or B2 for another factor found by factor theorem

factorising a correct quadratic factor | M1 | for factors giving two terms of quadratic correct; M0 for formula without factors found

$(x - 2)(x + 3)(x + 5)$ | A1 | condone omission of first factor found; ignore '= 0' seen allow last four marks for $(x - 2)(x + 3)(x + 5)$ obtained; for all 6 marks must see factor theorem use first

## Question 12(ii)

sketch of cubic right way up, with two turning points | B1 | 0 if stops at $x$-axis

values of intns on $x$ axis shown, correct $(-5, -3,$ and $2)$ or ft from their factors/roots in (i) | B1 | on graph or nearby in this part mark intent for intersections with both axes

$y$-axis intersection at $-30$ | B1 | or $x = 0, y = -30$ seen in this part if consistent with graph drawn

## Question 12(iii)

$(x - 1)$ substituted for $x$ in either form of eqn for $y = f(x)$ | M1 | correct or ft their (i) or (ii) for factorised form; condone one error; allow for new roots stated as $-4, -2$ and 3 or ft

$(x - 1)^3$ expanded correctly (need not be simplified) or two of their factors multiplied correctly | M1 dep | or M1 for correct or correct ft multiplying out of all 3 brackets at once, condoning one error $[x^3 - 3x^2 + 4x^2 + 2x^2 + 8x - 6x - 12x - 24]$

correct completion to given answer [condone omission of 'y ='] | M1 | unless all 3 brackets already expanded, must show at least one further interim step allow SC1 for $(x + 1)$ subst and correct exp of $(x + 1)^3$ or two of their factors ft or, for those using given answer: M1 for roots stated or used as $-4, -2$ and 3 or ft A1 for showing all 3 roots satisfy given eqn B1 for comment re coefft of $x^3$ or product of roots to show that eqn of translated graph is not a multiple of RHS of given eqn