Skill Align ACT BSSS Specialist Mathematics T Booklet 2 Practice Assessment Pack 0 - 2026 Edition
Original Skill Align Specialist Mathematics T practice-assessment Pack 0, Booklet 2; not an official ACT BSSS examination and not endorsed by ACT BSSS or the ACT Government.
- Paper
- Booklet 2 Showcase
- Reading
- 10 minutes total planning time across both booklets
- Writing
- 115 minutes total working time across both booklets
- Assessment
- 50 marks
Suggested Skill Align conditions for the two-booklet pair: 10 minutes planning time and 115 minutes working time (125 minutes total). Scientific, graphics and CAS calculators are permitted, and a locally supplied formula sheet may be used. Stored notes or programs, internet access, AI tools, messaging and external communication are not permitted. Sufficient mathematical reasoning must be shown even when technology is used. ACT BSSS assessment is school based. Individual ACT colleges may apply different timing, calculator, formula-sheet and stored-material conditions. No public PDF download is supplied with Pack 0.
Booklet 2
Booklet 2 emphasises applied, technology-supported analysis, modelling, interpretation and communication. Answer all questions. Show complete working, proof, reasoning and interpretation. State exact values unless an approximation is requested, and include units and context where applicable. Suggested Skill Align conditions for the two-booklet pair: 10 minutes planning time and 115 minutes working time (125 minutes total). Scientific, graphics and CAS calculators are permitted, and a locally supplied formula sheet may be used. Stored notes or programs, internet access, AI tools, messaging and external communication are not permitted. Sufficient mathematical reasoning must be shown even when technology is used. ACT BSSS assessment is school based. Individual ACT colleges may apply different timing, calculator, formula-sheet and stored-material conditions.
Question 6
10 marksQuestion 7
10 marksQuestion 8
10 marksQuestion 9
10 marksQuestion 10
10 marksWorked Solutions And Marking Guide
Question 6
(a) T(t)=20+60e^(-kt).
Solve the separable equation and apply the initial condition.
(b) k=ln2 / 10.
At t=10, 30=60e^(-10k), so e^(-10k)=1 / 2.
(c) T(30)=27.5; the temperature approaches 20^circmathrm C.
Since e^(-30k)=(1 / 2)³, add 60 / 8 to 20.
Detailed marking criteria
Part a (3 marks)
Award one mark for each numbered, task-specific statement below, to a maximum of 3. Do not award the final-result mark when the stated mathematical form is incomplete.
Mark-by-mark evidence:
- Separates the temperature difference from time.
- Integrates to an exponential cooling form.
- Uses T(0)=80 to obtain T(t)=20+60e^(-kt).
Acceptable alternatives: For Q6(a), credit an equivalent mathematically correct method only when it establishes all 3 required checkpoints and reaches T(t)=20+60e^(-kt).
Do not credit by itself: Omitting or contradicting this required step: Separates the temperature difference from time. Giving the final result without establishing: Uses T(0)=80 to obtain T(t)=20+60e^(-kt).
Part b (3 marks)
Award one mark for each numbered, task-specific statement below, to a maximum of 3. Apply consequential marking when a correct later method consistently uses an earlier incorrect value, unless the error materially simplifies the task. For the final-result mark, retain the requested exact form.
Mark-by-mark evidence:
- Substitutes T(10)=50.
- Obtains e^(-10k)=1 / 2.
- Solves k=ln2 / 10.
Acceptable alternatives: For Q6(b), credit an equivalent mathematically correct method only when it establishes all 3 required checkpoints and reaches k=ln2 / 10.
Do not credit by itself: Omitting or contradicting this required step: Substitutes T(10)=50. Giving the final result without establishing: Solves k=ln2 / 10.
Part c (4 marks)
Award one mark for each numbered, task-specific statement below, to a maximum of 4. Apply consequential marking when a correct later method consistently uses an earlier incorrect value, unless the error materially simplifies the task. For the final-result mark, include the requested units, context or justification.
Mark-by-mark evidence:
- Uses e^(-30k)=(1 / 2)^3.
- Calculates T(30)=20+60 / 8=27.5^circmathrm C.
- Uses e^(-kt)to0.
- Interprets 20^circmathrm C as the limiting ambient temperature.
Acceptable alternatives: For Q6(c), credit an equivalent mathematically correct method only when it establishes all 4 required checkpoints and reaches T(30)=27.5; the temperature approaches 20^circmathrm C.
Do not credit by itself: Omitting or contradicting this required step: Uses e^(-30k)=(1 / 2)^3. Giving the final result without establishing: Interprets 20^circmathrm C as the limiting ambient temperature.
Question 7
(a) V=2piint_0^2x(4-x²),dx.
A shell has radius x and height 4 minus x squared.
(b) 8π.
2π[2x²-x⁴ / 4]_0²=2π(4)=8π.
(c) At height y, the radius is √(4-y), so piint_0⁴(4-y),dy=8π.
Both methods describe the same solid using perpendicular slices.
Detailed marking criteria
Part a (3 marks)
Award one mark for each numbered, task-specific statement below, to a maximum of 3. Do not award the final-result mark when the stated mathematical form is incomplete.
Mark-by-mark evidence:
- Selects cylindrical shells about the y-axis.
- Uses radius x and height 4-x^2.
- Writes V=2piint_0^2x(4-x²)dx.
Acceptable alternatives: For Q7(a), credit an equivalent mathematically correct method only when it establishes all 3 required checkpoints and reaches V=2piint_0^2x(4-x²),dx.
Do not credit by itself: Omitting or contradicting this required step: Selects cylindrical shells about the y-axis. Giving the final result without establishing: Writes V=2piint_0^2x(4-x²)dx.
Part b (4 marks)
Award one mark for each numbered, task-specific statement below, to a maximum of 4. Apply consequential marking when a correct later method consistently uses an earlier incorrect value, unless the error materially simplifies the task. For the final-result mark, retain the requested exact form.
Mark-by-mark evidence:
- Expands the shell integrand.
- Integrates to 2x²-x⁴ / 4.
- Applies limits 0 and 2.
- Obtains 8π exactly.
Acceptable alternatives: For Q7(b), credit an equivalent mathematically correct method only when it establishes all 4 required checkpoints and reaches 8π.
Do not credit by itself: Omitting or contradicting this required step: Expands the shell integrand. Giving the final result without establishing: Obtains 8π exactly.
Part c (3 marks)
Award one mark for each numbered, task-specific statement below, to a maximum of 3. Apply consequential marking when a correct later method consistently uses an earlier incorrect value, unless the error materially simplifies the task. For the final-result mark, include the requested units, context or justification.
Mark-by-mark evidence:
- Rearranges the curve as x=√(4-y).
- Writes piint_0⁴(4-y)dy.
- Evaluates the disk integral as 8π, confirming the same solid.
Acceptable alternatives: For Q7(c), credit an equivalent mathematically correct method only when it establishes all 3 required checkpoints and reaches At height y, the radius is √(4-y), so piint_0⁴(4-y),dy=8π.
Do not credit by itself: Omitting or contradicting this required step: Rearranges the curve as x=√(4-y). Giving the final result without establishing: Evaluates the disk integral as 8π, confirming the same solid.
Question 8
(a) E(overline X)=mu and operatorname(SE)(overline X)=12 / √36=2 kWh.
Use the sampling-distribution results for a sample mean.
(b) 74.2pm1.96(2)=(70.28,78.12) kWh.
Use the known-standard-deviation normal interval for a population mean.
(c) Across repeated random samples, about 95% of intervals constructed this way would contain mu. Consecutive days may be dependent or seasonally unrepresentative.
Distinguish repeated-sampling coverage from certainty about one realised interval.
Detailed marking criteria
Part a (3 marks)
Award one mark for each numbered, task-specific statement below, to a maximum of 3. For the final-result mark, include the requested units, context or justification.
Mark-by-mark evidence:
- States E(overline X)=mu.
- Uses operatorname(SE)(overline X)=sigma / sqrt n.
- Evaluates 12 / √36=2 kWh.
Acceptable alternatives: For Q8(a), credit an equivalent mathematically correct method only when it establishes all 3 required checkpoints and reaches E(overline X)=mu and operatorname(SE)(overline X)=12 / √36=2 kWh.
Do not credit by itself: Omitting or contradicting this required step: States E(overline X)=mu. Giving the final result without establishing: Evaluates 12 / √36=2 kWh.
Part b (4 marks)
Award one mark for each numbered, task-specific statement below, to a maximum of 4. Apply consequential marking when a correct later method consistently uses an earlier incorrect value, unless the error materially simplifies the task. For the final-result mark, carry unrounded values until the final stated accuracy; include the requested units, context or justification.
Mark-by-mark evidence:
- Selects overline xpm1.96sigma / sqrt n.
- Substitutes 74.2pm1.96(2).
- Calculates the margin 3.92 kWh.
- States (70.28,78.12) kWh to two decimal places.
Acceptable alternatives: For Q8(b), credit an equivalent mathematically correct method only when it establishes all 4 required checkpoints and reaches 74.2pm1.96(2)=(70.28,78.12) kWh.
Do not credit by itself: Omitting or contradicting this required step: Selects overline xpm1.96sigma / sqrt n. Giving the final result without establishing: States (70.28,78.12) kWh to two decimal places.
Part c (3 marks)
Award one mark for each numbered, task-specific statement below, to a maximum of 3. Apply consequential marking when a correct later method consistently uses an earlier incorrect value, unless the error materially simplifies the task. For the final-result mark, include the requested units, context or justification.
Mark-by-mark evidence:
- Refers to repeated random samples and intervals constructed by the same method.
- States that about 95% of those intervals contain the population mean.
- Identifies dependence or seasonal bias from consecutive-day sampling.
Acceptable alternatives: For Q8(c), credit an equivalent mathematically correct method only when it establishes all 3 required checkpoints and reaches Across repeated random samples, about 95% of intervals constructed this way would contain mu. Consecutive days may be dependent or seasonally unrepresentative.
Do not credit by itself: Omitting or contradicting this required step: Refers to repeated random samples and intervals constructed by the same method. Giving the final result without establishing: Identifies dependence or seasonal bias from consecutive-day sampling.
Question 9
(a) w=5+i.
(1+i)(2-i)+2=3+i+2.
(b) A dilation by sqrt2 and an anticlockwise rotation by π / 4.
1+i=sqrt2operatorname(cis)(π / 4).
(c) A circle centred at 2 on the real axis with radius sqrt2.
Multiplication scales the radius and the final addition translates the centre.
Detailed marking criteria
Part a (3 marks)
Award one mark for each numbered, task-specific statement below, to a maximum of 3. Do not award the final-result mark when the stated mathematical form is incomplete.
Mark-by-mark evidence:
- Expands (1+i)(2-i).
- Combines real and imaginary parts as 3+i.
- Adds the translation 2 to obtain w=5+i.
Acceptable alternatives: For Q9(a), credit an equivalent mathematically correct method only when it establishes all 3 required checkpoints and reaches w=5+i.
Do not credit by itself: Omitting or contradicting this required step: Expands (1+i)(2-i). Giving the final result without establishing: Adds the translation 2 to obtain w=5+i.
Part b (3 marks)
Award one mark for each numbered, task-specific statement below, to a maximum of 3. Apply consequential marking when a correct later method consistently uses an earlier incorrect value, unless the error materially simplifies the task. Do not award the final-result mark when the stated mathematical form is incomplete.
Mark-by-mark evidence:
- Writes 1+i=sqrt2operatorname(cis)(π / 4).
- Identifies scale factor sqrt2.
- Identifies an anticlockwise rotation by π / 4.
Acceptable alternatives: For Q9(b), credit an equivalent mathematically correct method only when it establishes all 3 required checkpoints and reaches A dilation by sqrt2 and an anticlockwise rotation by π / 4.
Do not credit by itself: Omitting or contradicting this required step: Writes 1+i=sqrt2operatorname(cis)(π / 4). Giving the final result without establishing: Identifies an anticlockwise rotation by π / 4.
Part c (4 marks)
Award one mark for each numbered, task-specific statement below, to a maximum of 4. Apply consequential marking when a correct later method consistently uses an earlier incorrect value, unless the error materially simplifies the task. Do not award the final-result mark when the stated mathematical form is incomplete.
Mark-by-mark evidence:
- Maps the original centre 0 to 2.
- Scales the unit radius to sqrt2.
- Notes rotation preserves circular shape.
- States a circle centred at 2 with radius sqrt2.
Acceptable alternatives: For Q9(c), credit an equivalent mathematically correct method only when it establishes all 4 required checkpoints and reaches A circle centred at 2 on the real axis with radius sqrt2.
Do not credit by itself: Omitting or contradicting this required step: Maps the original centre 0 to 2. Giving the final result without establishing: States a circle centred at 2 with radius sqrt2.
Question 10
(a) (1,2,1) × (1,-1,1)=0, so the line direction is perpendicular to the plane normal.
Use the scalar product of the direction and normal vectors.
(b) At (2,-1,0), x-y+z=3ne4.
Test a point on the line in the plane equation.
(c) 1 / sqrt3.
Use |2-(-1)+0-4| / √(1²+(-1)²+1²).
Detailed marking criteria
Part a (3 marks)
Award one mark for each numbered, task-specific statement below, to a maximum of 3. Do not award the final-result mark when the stated mathematical form is incomplete.
Mark-by-mark evidence:
- Identifies line direction (1,2,1).
- Identifies plane normal (1,-1,1).
- Calculates their scalar product as 0 and concludes parallelism.
Acceptable alternatives: For Q10(a), credit an equivalent mathematically correct method only when it establishes all 3 required checkpoints and reaches (1,2,1) × (1,-1,1)=0, so the line direction is perpendicular to the plane normal.
Do not credit by itself: Omitting or contradicting this required step: Identifies line direction (1,2,1). Giving the final result without establishing: Calculates their scalar product as 0 and concludes parallelism.
Part b (3 marks)
Award one mark for each numbered, task-specific statement below, to a maximum of 3. Apply consequential marking when a correct later method consistently uses an earlier incorrect value, unless the error materially simplifies the task. Do not award the final-result mark when the stated mathematical form is incomplete.
Mark-by-mark evidence:
- Uses the line point (2,-1,0).
- Evaluates x-y+z=3.
- Compares 3ne4 and concludes the line is not contained in the plane.
Acceptable alternatives: For Q10(b), credit an equivalent mathematically correct method only when it establishes all 3 required checkpoints and reaches At (2,-1,0), x-y+z=3ne4.
Do not credit by itself: Omitting or contradicting this required step: Uses the line point (2,-1,0). Giving the final result without establishing: Compares 3ne4 and concludes the line is not contained in the plane.
Part c (4 marks)
Award one mark for each numbered, task-specific statement below, to a maximum of 4. Apply consequential marking when a correct later method consistently uses an earlier incorrect value, unless the error materially simplifies the task. Do not award the final-result mark when the stated mathematical form is incomplete.
Mark-by-mark evidence:
- Uses the point-plane distance formula.
- Substitutes (2,-1,0) and plane coefficients.
- Obtains numerator 1 and denominator sqrt3.
- States 1 / sqrt3.
Acceptable alternatives: For Q10(c), credit an equivalent mathematically correct method only when it establishes all 4 required checkpoints and reaches 1 / sqrt3.
Do not credit by itself: Omitting or contradicting this required step: Uses the point-plane distance formula. Giving the final result without establishing: States 1 / sqrt3.
Diagnostic Checklist
| Topic | Questions | Marks | Marks Lost | Action |
|---|---|---|---|---|
| Unit 4: Specialist Mathematics — Rates of change and differential equations — rates-change-differential-equations: Show that \(T(t)=20+60e^{-kt}\) | Q6(a) | 3 | ___ | Q6(a): reproduce this exact process without the solution: Separates the temperature difference from time; then Integrates to an exponential cooling form; then Uses T(0)=80 to obtain T(t)=20+60e^(-kt). |
| Unit 4: Specialist Mathematics — Rates of change and differential equations — rates-change-differential-equations: Find k exactly | Q6(b) | 3 | ___ | Q6(b): reproduce this exact process without the solution: Substitutes T(10)=50; then Obtains e^(-10k)=1 / 2; then Solves k=ln2 / 10. |
| Unit 4: Specialist Mathematics — Rates of change and differential equations — rates-change-differential-equations: Find \(T(30)\) and interpret the limiting temperature | Q6(c) | 4 | ___ | Q6(c): reproduce this exact process without the solution: Uses e^(-30k)=(1 / 2)³; then Calculates T(30)=20+60 / 8=27.5^circmathrm C; then Uses e^(-kt)to0; then Interprets 20^circmathrm C as the limiting ambient temperature. |
| Unit 4: Specialist Mathematics — Integration and applications of integration — integration-applications: Write a cylindrical-shell integral for the volume | Q7(a) | 3 | ___ | Q7(a): reproduce this exact process without the solution: Selects cylindrical shells about the y-axis; then Uses radius x and height 4-x²; then Writes V=2piint_0^2x(4-x²)dx. |
| Unit 4: Specialist Mathematics — Integration and applications of integration — integration-applications: Evaluate the exact volume | Q7(b) | 4 | ___ | Q7(b): reproduce this exact process without the solution: Expands the shell integrand; then Integrates to 2x²-x⁴ / 4; then Applies limits 0 and 2; then Obtains 8π exactly. |
| Unit 4: Specialist Mathematics — Integration and applications of integration — integration-applications: Explain why a disk integral with respect to y gives the same result | Q7(c) | 3 | ___ | Q7(c): reproduce this exact process without the solution: Rearranges the curve as x=√(4-y); then Writes piint_0⁴(4-y)dy; then Evaluates the disk integral as 8π, confirming the same solid. |
| Unit 4: Specialist Mathematics — Statistical inference — Determine the sampling distribution parameters of a sample mean | Q8(a) | 3 | ___ | Q8(a): reproduce this exact process without the solution: States E(overline X)=mu; then Uses operatorname(SE)(overline X)=sigma / sqrt n; then Evaluates 12 / √36=2 kWh. |
| Unit 4: Specialist Mathematics — Statistical inference — Construct a normal confidence interval for a population mean | Q8(b) | 4 | ___ | Q8(b): reproduce this exact process without the solution: Selects overline xpm1.96sigma / sqrt n; then Substitutes 74.2pm1.96(2); then Calculates the margin 3.92 kWh; then States (70.28,78.12) kWh to two decimal places. |
| Unit 4: Specialist Mathematics — Statistical inference — Interpret confidence level and assess sampling dependence | Q8(c) | 3 | ___ | Q8(c): reproduce this exact process without the solution: Refers to repeated random samples and intervals constructed by the same method; then States that about 95% of those intervals contain the population mean; then Identifies dependence or seasonal bias from consecutive-day sampling. |
| Unit 3: Specialist Mathematics — Complex numbers — complex-numbers: Find w when \(z=2-i\) | Q9(a) | 3 | ___ | Q9(a): reproduce this exact process without the solution: Expands (1+i)(2-i); then Combines real and imaginary parts as 3+i; then Adds the translation 2 to obtain w=5+i. |
| Unit 3: Specialist Mathematics — Complex numbers — complex-numbers: Describe the dilation and rotation caused by multiplication by 1+i | Q9(b) | 3 | ___ | Q9(b): reproduce this exact process without the solution: Writes 1+i=sqrt2operatorname(cis)(π / 4); then Identifies scale factor sqrt2; then Identifies an anticlockwise rotation by π / 4. |
| Unit 3: Specialist Mathematics — Complex numbers — complex-numbers: State the image of the unit circle centred at the origin | Q9(c) | 4 | ___ | Q9(c): reproduce this exact process without the solution: Maps the original centre 0 to 2; then Scales the unit radius to sqrt2; then Notes rotation preserves circular shape; then States a circle centred at 2 with radius sqrt2. |
| Unit 3: Specialist Mathematics — Vectors in three dimensions — vectors-three-dimensions: Show that the line is parallel to the plane | Q10(a) | 3 | ___ | Q10(a): reproduce this exact process without the solution: Identifies line direction (1,2,1); then Identifies plane normal (1,-1,1); then Calculates their scalar product as 0 and concludes parallelism. |
| Unit 3: Specialist Mathematics — Vectors in three dimensions — vectors-three-dimensions: Show that the line does not lie in the plane | Q10(b) | 3 | ___ | Q10(b): reproduce this exact process without the solution: Uses the line point (2,-1,0); then Evaluates x-y+z=3; then Compares 3ne4 and concludes the line is not contained in the plane. |
| Unit 3: Specialist Mathematics — Vectors in three dimensions — vectors-three-dimensions: Find the perpendicular distance between the line and the plane | Q10(c) | 4 | ___ | Q10(c): reproduce this exact process without the solution: Uses the point-plane distance formula; then Substitutes (2,-1,0) and plane coefficients; then Obtains numerator 1 and denominator sqrt3; then States 1 / sqrt3. |