Part a (2 marks)

Group the terms to obtain x²(x-3)-9(x-3); factor x-3; factor the resulting difference of squares; States F(x)=(x-3)²(x+3)

Acceptable alternatives: Accept an algebraically equivalent method with the same mathematical evidence.; Apply consequential error follow-through where later work is valid from an earlier stated value.

Do not credit by itself: Unsupported answer, missing required reasoning, or an invalid branch, bound, sign or continuity correction.

Part b (2 marks)

Read the roots from the factorisation; identify the repeated factor (x-3)²; States x=-3 is simple and x=3 is double

Acceptable alternatives: Accept an algebraically equivalent method with the same mathematical evidence.; Apply consequential error follow-through where later work is valid from an earlier stated value.

Do not credit by itself: Unsupported answer, missing required reasoning, or an invalid branch, bound, sign or continuity correction.

Part c (2 marks)

Differentiate the expanded polynomial; factor 3(x²-2x-3); States F'(x)=3(x-3)(x+1)

Acceptable alternatives: Accept an algebraically equivalent method with the same mathematical evidence.; Apply consequential error follow-through where later work is valid from an earlier stated value.

Do not credit by itself: Unsupported answer, missing required reasoning, or an invalid branch, bound, sign or continuity correction.

Part d (3 marks)

Solve F'(x)=0; evaluate F(-1) and F(3); use the derivative sign changes to classify both points; States A local maximum at (-1,32) and a local minimum at (3,0)

Acceptable alternatives: Accept an algebraically equivalent method with the same mathematical evidence.; Apply consequential error follow-through where later work is valid from an earlier stated value.

Do not credit by itself: Unsupported answer, missing required reasoning, or an invalid branch, bound, sign or continuity correction.

Part e (2 marks)

Evaluate F(0)=27; evaluate F'(0)=-9; use point-gradient form; States y=-9x+27

Acceptable alternatives: Accept an algebraically equivalent method with the same mathematical evidence.; Apply consequential error follow-through where later work is valid from an earlier stated value.

Do not credit by itself: Unsupported answer, missing required reasoning, or an invalid branch, bound, sign or continuity correction.

Part f (4 marks)

Use the factorisation to establish F(x)ge0 on [-3,3]; integrate x³-3x²-9x+27; apply the limits -3 and 3; obtain 108; States 108 square units

Acceptable alternatives: Accept an algebraically equivalent method with the same mathematical evidence.; Apply consequential error follow-through where later work is valid from an earlier stated value.

Do not credit by itself: Unsupported answer, missing required reasoning, or an invalid branch, bound, sign or continuity correction.

Part a (2 marks)

Subtract the coordinates of A from B; calculate √(6²+3²)=3sqrt5; States d=(6,3) and |d|=3sqrt5

Acceptable alternatives: Accept an algebraically equivalent method with the same mathematical evidence.; Apply consequential error follow-through where later work is valid from an earlier stated value.

Do not credit by itself: Unsupported answer, missing required reasoning, or an invalid branch, bound, sign or continuity correction.

Part b (2 marks)

Average the corresponding coordinates of A and B; state M=(4,7 / 2); States M=(4,frac72)

Acceptable alternatives: Accept an algebraically equivalent method with the same mathematical evidence.; Apply consequential error follow-through where later work is valid from an earlier stated value.

Do not credit by itself: Unsupported answer, missing required reasoning, or an invalid branch, bound, sign or continuity correction.

Part c (3 marks)

Write P=(1+6t,2+3t); impose 2+3t=1+6t; solve t=1 / 3 and substitute; States t=frac13 and P=(3,3)

Acceptable alternatives: Accept an algebraically equivalent method with the same mathematical evidence.; Apply consequential error follow-through where later work is valid from an earlier stated value.

Do not credit by itself: Unsupported answer, missing required reasoning, or an invalid branch, bound, sign or continuity correction.

Part d (3 marks)

Calculate overrightarrow(AC)=(3,-3); calculate overrightarrow(AC) × d=9 and d × d=45; multiply d by 1 / 5; States operatorname(proj)_(d)overrightarrow(AC)=(frac65,frac35)

Acceptable alternatives: Accept an algebraically equivalent method with the same mathematical evidence.; Apply consequential error follow-through where later work is valid from an earlier stated value.

Do not credit by itself: Unsupported answer, missing required reasoning, or an invalid branch, bound, sign or continuity correction.

Part e (3 marks)

Subtract the projection from overrightarrow(AC); obtain the perpendicular component (9 / 5,-18 / 5); calculate its magnitude 9 / sqrt5; States ((9) / (sqrt5)) units

Acceptable alternatives: Accept an algebraically equivalent method with the same mathematical evidence.; Apply consequential error follow-through where later work is valid from an earlier stated value.

Do not credit by itself: Unsupported answer, missing required reasoning, or an invalid branch, bound, sign or continuity correction.

Part f (1 mark)

Use d × overrightarrow(AC)=9; divide by (3sqrt5)(3sqrt2); States cosangle BAC=frac1(√10)

Acceptable alternatives: Accept an algebraically equivalent method with the same mathematical evidence.; Apply consequential error follow-through where later work is valid from an earlier stated value.

Do not credit by itself: Unsupported answer, missing required reasoning, or an invalid branch, bound, sign or continuity correction.

Part a (2 marks)

Identify n=200 and p=0.05; calculate np=10; States Xsimoperatorname(Bin)(200,0.05), with mean 10

Acceptable alternatives: Accept an algebraically equivalent method with the same mathematical evidence.; Apply consequential error follow-through where later work is valid from an earlier stated value.

Do not credit by itself: Unsupported answer, missing required reasoning, or an invalid branch, bound, sign or continuity correction.

Part b (2 marks)

Use np(1-p); calculate 200(0.05)(0.95)=9.5; States operatorname(Var)(X)=9.5

Acceptable alternatives: Accept an algebraically equivalent method with the same mathematical evidence.; Apply consequential error follow-through where later work is valid from an earlier stated value.

Do not credit by itself: Unsupported answer, missing required reasoning, or an invalid branch, bound, sign or continuity correction.

Part c (3 marks)

Write P(X=0)=0.95²⁰⁰; write P(X=1)=200(0.05)(0.95)¹⁹⁹=10(0.95)¹⁹⁹; add the two probabilities; States 0.95²⁰⁰+10(0.95)¹⁹⁹

Acceptable alternatives: Accept an algebraically equivalent method with the same mathematical evidence.; Apply consequential error follow-through where later work is valid from an earlier stated value.

Do not credit by itself: Unsupported answer, missing required reasoning, or an invalid branch, bound, sign or continuity correction.

Part d (2 marks)

Calculate both expected category counts; compare each with the applicable minimum of 10; States Both np=10 and n(1-p)=190 are at least 10

Acceptable alternatives: Accept an algebraically equivalent method with the same mathematical evidence.; Apply consequential error follow-through where later work is valid from an earlier stated value.

Do not credit by itself: Unsupported answer, missing required reasoning, or an invalid branch, bound, sign or continuity correction.

Part e (3 marks)

Approximate by Ysim N(10,9.5); use the boundary 14.5; standardise to z=(14.5-10) / √9.5approx1.46; take 1-0.9279=0.0721; States P(Xge15)approx0.0721

Acceptable alternatives: Accept an algebraically equivalent method with the same mathematical evidence.; Apply consequential error follow-through where later work is valid from an earlier stated value.

Do not credit by itself: Unsupported answer, missing required reasoning, or an invalid branch, bound, sign or continuity correction.

Part f (4 marks)

Use the corrected boundary k-0.5; require (k-0.5-10) / √9.5>2.33; obtain k>10.5+2.33sqrt(9.5)approx17.68; select the smallest integer k=18; States k=18

Acceptable alternatives: Accept an algebraically equivalent method with the same mathematical evidence.; Apply consequential error follow-through where later work is valid from an earlier stated value.

Do not credit by itself: Unsupported answer, missing required reasoning, or an invalid branch, bound, sign or continuity correction.

Part a (1 mark)

Set dy / dx=0; since x+1>0, solve 6-y=0; States y=6

Acceptable alternatives: Accept an algebraically equivalent method with the same mathematical evidence.; Apply consequential error follow-through where later work is valid from an earlier stated value.

Do not credit by itself: Unsupported answer, missing required reasoning, or an invalid branch, bound, sign or continuity correction.

Part b (2 marks)

Divide by 6-y; multiply by dx; States ((dy) / (6-y))=(x+1),dx

Acceptable alternatives: Accept an algebraically equivalent method with the same mathematical evidence.; Apply consequential error follow-through where later work is valid from an earlier stated value.

Do not credit by itself: Unsupported answer, missing required reasoning, or an invalid branch, bound, sign or continuity correction.

Part c (3 marks)

Integrate 1 / (6-y) with the negative sign; integrate x+1; include an arbitrary constant; States -ln|6-y|=((x²) / (2))+x+C

Acceptable alternatives: Accept an algebraically equivalent method with the same mathematical evidence.; Apply consequential error follow-through where later work is valid from an earlier stated value.

Do not credit by itself: Unsupported answer, missing required reasoning, or an invalid branch, bound, sign or continuity correction.

Part d (3 marks)

Substitute x=0,y=2 to determine the constant; exponentiate the logarithmic equation; select 6-y>0 from the initial value; States y=6-4e^(-x² / 2-x)

Acceptable alternatives: Accept an algebraically equivalent method with the same mathematical evidence.; Apply consequential error follow-through where later work is valid from an earlier stated value.

Do not credit by itself: Unsupported answer, missing required reasoning, or an invalid branch, bound, sign or continuity correction.

Part e (3 marks)

Use 0<e^(-x² / 2-x)le1 to bound y; infer 6-y>0; combine with x+1>0 in the differential equation; States For xge0, 2le y<6 and dy / dx>0

Acceptable alternatives: Accept an algebraically equivalent method with the same mathematical evidence.; Apply consequential error follow-through where later work is valid from an earlier stated value.

Do not credit by itself: Unsupported answer, missing required reasoning, or an invalid branch, bound, sign or continuity correction.

Part f (3 marks)

Set 6-4e^(-x² / 2-x)=5; obtain x² / 2+x=ln4; solve the quadratic and retain the non-negative root; States x=-1+√(1+2ln4)

Acceptable alternatives: Accept an algebraically equivalent method with the same mathematical evidence.; Apply consequential error follow-through where later work is valid from an earlier stated value.

Do not credit by itself: Unsupported answer, missing required reasoning, or an invalid branch, bound, sign or continuity correction.