Part a (2 marks)

Award one mark for each observable outcome: states the domain xgeq-frac32; states the range g(x)leq3. Apply consequential marking only when a clearly identified earlier result is used consistently.

Acceptable alternatives: Accept a mathematically equivalent response to Domain: xgeq-frac32. Range: g(x)leq3. that preserves every stated condition.; Accept an alternative valid method that establishes the same observable steps described by: Since g(x)=3-√(2x+3), the radicand requires 2x+3geq0. The square root is non-negative, so subtracting it from 3 gives values at most 3.

Do not credit by itself: Do not award a method mark unless its corresponding observable setup or intermediate step is present.

Part b (2 marks)

Award one mark for each observable outcome: forms √(2x+3)=2; obtains x=frac12 and verifies that it is in the domain. Apply consequential marking only when a clearly identified earlier result is used consistently.

Acceptable alternatives: Accept a mathematically equivalent response to x=frac12. that preserves every stated condition.; Accept an alternative valid method that establishes the same observable steps described by: The equation gives √(2x+3)=2. Squaring gives 2x+3=4, hence x=1 / 2, which is in the domain.

Do not credit by itself: Do not award a method mark unless its corresponding observable setup or intermediate step is present.

Part a (1 mark)

Award the mark for the correct response f'(x)=4x²(x-3). Working is not required because the prompt does not ask for proof, verification, explanation or justification.

Acceptable alternatives: Accept a mathematically equivalent response to f'(x)=4x²(x-3). that preserves every stated condition.

Part b (2 marks)

Award one mark for each observable outcome: solves f'(x)=0 to obtain x=0,3; states the stationary points (0,0) and (3,-27). Apply consequential marking only when a clearly identified earlier result is used consistently.

Acceptable alternatives: Accept a mathematically equivalent response to (0,0) and (3,-27). that preserves every stated condition.; Accept an alternative valid method that establishes the same observable steps described by: Solving 4x²(x-3)=0 gives x=0 and x=3. Substitute each value into f.

Do not credit by itself: Do not award a method mark unless its corresponding observable setup or intermediate step is present.

Part c (1 mark)

Award the mark for the correct response (0,0) is a stationary point of inflection and (3,-27) is a local minimum. Working is not required because the prompt does not ask for proof, verification, explanation or justification.

Acceptable alternatives: Accept a mathematically equivalent response to (0,0) is a stationary point of inflection and (3,-27) is a local minimum. that preserves every stated condition.

Part a (1 mark)

Award the mark for the correct response 0.70. Working is not required because the prompt does not ask for proof, verification, explanation or justification.

Acceptable alternatives: Accept a mathematically equivalent response to 0.70. that preserves every stated condition.

Part b (1 mark)

Award the mark for the correct response frac5(11). Working is not required because the prompt does not ask for proof, verification, explanation or justification.

Acceptable alternatives: Accept a mathematically equivalent response to frac5(11). that preserves every stated condition.

Part c (2 marks)

Award one mark for each observable outcome: compares Pr(A)Pr(B)=0.22 with Pr(Acap B)=0.25; concludes that the events are not independent. Apply consequential marking only when a clearly identified earlier result is used consistently.

Acceptable alternatives: Accept a mathematically equivalent response to The events are not independent. that preserves every stated condition.; Accept an alternative valid method that establishes the same observable steps described by: Independence would require Pr(Acap B)=0.55(0.40)=0.22, but the given intersection probability is 0.25.

Do not credit by itself: Do not award a method mark unless its corresponding observable setup or intermediate step is present.

Part a (2 marks)

Award one mark for each observable outcome: solves 4-x²=x+2 to obtain x=-2,1; states the intersection points (-2,0) and (1,3). Apply consequential marking only when a clearly identified earlier result is used consistently.

Acceptable alternatives: Accept a mathematically equivalent response to (-2,0) and (1,3). that preserves every stated condition.; Accept an alternative valid method that establishes the same observable steps described by: Equating the functions gives x²+x-2=0=(x+2)(x-1). Substitute each root into y=x+2.

Do not credit by itself: Do not award a method mark unless its corresponding observable setup or intermediate step is present.

Part b (3 marks)

Award one mark for each observable outcome: uses the area integral int_(-2)¹(2-x-x²),dx; evaluates an antiderivative at both limits; obtains the exact area frac92 square units. Apply consequential marking only when a clearly identified earlier result is used consistently.

Acceptable alternatives: Accept a mathematically equivalent response to frac92 square units. that preserves every stated condition.; Accept an alternative valid method that establishes the same observable steps described by: The parabola is above the line, so the area is int_(-2)¹(2-x-x²),dx=9 / 2.

Do not credit by itself: Do not award a method mark unless its corresponding observable setup or intermediate step is present.

Part a (1 mark)

Award the mark for the correct response k=frac14. Working is not required because the prompt does not ask for proof, verification, explanation or justification.

Acceptable alternatives: Accept a mathematically equivalent response to k=frac14. that preserves every stated condition.

Part b (2 marks)

Award one mark for each observable outcome: uses E(X)=frac14int_0^2x(1+x),dx; obtains E(X)=frac76. Apply consequential marking only when a clearly identified earlier result is used consistently.

Acceptable alternatives: Accept a mathematically equivalent response to E(X)=frac76. that preserves every stated condition.; Accept an alternative valid method that establishes the same observable steps described by: Evaluate frac14int_0^2x(1+x),dx=frac14(2+8 / 3)=7 / 6.

Do not credit by itself: Do not award a method mark unless its corresponding observable setup or intermediate step is present.

Part c (1 mark)

Award the mark for the correct response frac58. Working is not required because the prompt does not ask for proof, verification, explanation or justification.

Acceptable alternatives: Accept a mathematically equivalent response to frac58. that preserves every stated condition.

Part a (1 mark)

Award the mark for the correct response Pr(Z>1.5). Working is not required because the prompt does not ask for proof, verification, explanation or justification.

Acceptable alternatives: Accept a mathematically equivalent response to Pr(Z>1.5). that preserves every stated condition.

Part b (2 marks)

Award one mark for each observable outcome: states the percentile expression 72+8z_(0.95); uses z_(0.95)=1.645 to obtain approximately 85.16. Apply consequential marking only when a clearly identified earlier result is used consistently.

Acceptable alternatives: Accept a mathematically equivalent response to 72+8z_(0.95)approx85.16. that preserves every stated condition.; Accept an alternative valid method that establishes the same observable steps described by: Scale the standard-normal percentile by 8 and add the mean 72.

Do not credit by itself: Do not award a method mark unless its corresponding observable setup or intermediate step is present.

Part c (1 mark)

Award the mark for the correct response 64leq Xleq80. Working is not required because the prompt does not ask for proof, verification, explanation or justification.

Acceptable alternatives: Accept a mathematically equivalent response to 64leq Xleq80. that preserves every stated condition.

Part a (2 marks)

Award one mark for each observable outcome: states the initial value N(0)=120; solves e^(-0.15t)=frac12 to obtain the exact half-life ((20ln2) / (3)) hours. Apply consequential marking only when a clearly identified earlier result is used consistently.

Acceptable alternatives: Accept a mathematically equivalent response to Initial value 120; half-life ((20ln2) / (3)) hours. that preserves every stated condition.; Accept an alternative valid method that establishes the same observable steps described by: Use N(0)=120. For the half-life, solve e^(-0.15t)=1 / 2.

Do not credit by itself: Do not award a method mark unless its corresponding observable setup or intermediate step is present.

Part b (2 marks)

Award one mark for each observable outcome: forms e^(-0.15t)=frac14; obtains t=((20ln4) / (3)) hours. Apply consequential marking only when a clearly identified earlier result is used consistently.

Acceptable alternatives: Accept a mathematically equivalent response to t=((20ln4) / (3)) hours. that preserves every stated condition.; Accept an alternative valid method that establishes the same observable steps described by: Since 30 / 120=1 / 4, solve e^(-0.15t)=1 / 4.

Do not credit by itself: Do not award a method mark unless its corresponding observable setup or intermediate step is present.

Part a (2 marks)

Award one mark for each observable outcome: establishes V(x)=x(12-2x)²; states the practical domain 0<x<6. Apply consequential marking only when a clearly identified earlier result is used consistently.

Acceptable alternatives: Accept a mathematically equivalent response to V(x)=x(12-2x)², where 0<x<6. that preserves every stated condition.; Accept an alternative valid method that establishes the same observable steps described by: The box has height x and a square base of side 12-2x. Both dimensions must be positive.

Do not credit by itself: Do not award a method mark unless its corresponding observable setup or intermediate step is present.

Part b (3 marks)

Award one mark for each observable outcome: obtains V'(x)=(12-2x)(12-6x); identifies the interior maximum at x=2; obtains the maximum volume V(2)=128 mathrm(cm³). Apply consequential marking only when a clearly identified earlier result is used consistently.

Acceptable alternatives: Accept a mathematically equivalent response to The maximum occurs at x=2 cm and is 128 mathrm(cm³). that preserves every stated condition.; Accept an alternative valid method that establishes the same observable steps described by: Differentiate to get V'(x)=(12-2x)(12-6x). The interior stationary point is x=2, giving V(2)=2(8)²=128.

Do not credit by itself: Do not award a method mark unless its corresponding observable setup or intermediate step is present.

Part a (3 marks)

Award one mark for each observable outcome: uses f'(x)=3(x+1)(x-2); compares coefficients to obtain a=-frac32 and b=-6; uses f(0)=6 to obtain c=6. Apply consequential marking only when a clearly identified earlier result is used consistently.

Acceptable alternatives: Accept a mathematically equivalent response to a=-frac32, b=-6 and c=6. that preserves every stated condition.; Accept an alternative valid method that establishes the same observable steps described by: Since f'(x) has roots -1 and 2, f'(x)=3(x+1)(x-2)=3x²-3x-6. Compare coefficients with 3x²+2ax+b, and use f(0)=c.

Do not credit by itself: Do not award a method mark unless its corresponding observable setup or intermediate step is present.

Part b (3 marks)

Award one mark for each observable outcome: obtains the stationary points (-1,((19) / (2))) and (2,-4); uses f''(x)=6x-3 or an equivalent sign analysis; classifies the points respectively as a local maximum and a local minimum. Apply consequential marking only when a clearly identified earlier result is used consistently.

Acceptable alternatives: Accept a mathematically equivalent response to A local maximum at (-1,((19) / (2))) and a local minimum at (2,-4). that preserves every stated condition.; Accept an alternative valid method that establishes the same observable steps described by: Substitute the two x-values into f. Since f''(x)=6x-3, it is negative at -1 and positive at 2.

Do not credit by itself: Do not award a method mark unless its corresponding observable setup or intermediate step is present.