Skill Align ACT BSSS Specialist Methods T Booklet 1 Practice Assessment Pack 0 - 2026 Edition
Original Skill Align Specialist Methods T practice-assessment Pack 0, Booklet 1; not an official ACT BSSS examination and not endorsed by ACT BSSS or the ACT Government.
- Paper
- Booklet 1 Showcase
- Reading
- 10 minutes
- Writing
- 65 minutes
- Assessment
- 50 marks
Suggested Skill Align Booklet 1 conditions: 10 minutes reading time and 65 minutes working time. This is a restricted-technology booklet: a non-CAS scientific calculator may be used, but CAS, graphics-calculator algebra, stored notes or programs, internet access and external communication are prohibited. A locally supplied formula or reference sheet may be used only when authorised by the administering college. ACT BSSS assessment is school based. The administering college may vary timing, technology, stored-note, program and reference-material conditions. No public PDF download is supplied with Pack 0.
Booklet 1
Booklet 1 assesses exact reasoning, proof, mathematical structure, functions, trigonometry, sequences and calculus under restricted technology. Answer all questions. Show complete working, reasoning and interpretation. State exact values unless an approximation is requested, and include units and context where applicable. Suggested Skill Align Booklet 1 conditions: 10 minutes reading time and 65 minutes working time. This is a restricted-technology booklet: a non-CAS scientific calculator may be used, but CAS, graphics-calculator algebra, stored notes or programs, internet access and external communication are prohibited. A locally supplied formula or reference sheet may be used only when authorised by the administering college. ACT BSSS assessment is school based. The administering college may vary timing, technology, stored-note, program and reference-material conditions.
Question 1
10 marksQuestion 2
10 marksQuestion 3
10 marksQuestion 4
10 marksQuestion 5
10 marksWorked Solutions And Marking Guide
Question 1
(a) D(4)=8e^(0.6)approx14.58.
Substitute t=4, retain the exponential value and round only the final demand.
(b) t=ln2 / 0.15approx4.62 months.
Set the model equal to twice its initial value and solve the resulting exponential equation.
(c) The instantaneous relative growth rate is 0.15 per month, or 15% per month; finite demand or capacity makes indefinite exponential growth unrealistic.
Connect the coefficient in the exponent to relative growth and challenge the constant-rate assumption.
Detailed marking criteria
Part a (3 marks)
Award all 3 marks for a complete mathematically correct response to the task. The numbered statements are indicative evidence for one approach, not a compulsory calculation sequence. Award equivalent marks for a valid alternative method that establishes the requested result and justification. For incomplete responses, credit valid progress proportionately using the indicative one-mark evidence or equivalent reasoning. For the final-result mark, carry unrounded values until the final stated accuracy.
Mark-by-mark evidence:
- Substitutes t=4 to obtain D(4)=8e^(0.15(4)).
- Simplifies the exponent to obtain 8e^(0.6).
- Evaluates and rounds the final value to 14.58.
Acceptable alternatives: For Q1(a), award all 3 marks for a mathematically complete alternative solution. Do not require steps belonging only to the displayed method. Retain any method, exact form, accuracy, units or justification explicitly requested in the question.
Do not credit by itself: Using an invalid relationship or leaving the requested result incomplete. Stating a conclusion without the working or justification requested by the task.
Part b (3 marks)
Award all 3 marks for a complete mathematically correct response to the task. The numbered statements are indicative evidence for one approach, not a compulsory calculation sequence. Award equivalent marks for a valid alternative method that establishes the requested result and justification. For incomplete responses, credit valid progress proportionately using the indicative one-mark evidence or equivalent reasoning. 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.
Mark-by-mark evidence:
- Sets 8e^(0.15t)=16 and divides by 8.
- Takes logarithms to obtain 0.15t=ln2.
- States t=ln2 / 0.15approx4.62 months.
Acceptable alternatives: For Q1(b), award all 3 marks for a mathematically complete alternative solution. Do not require steps belonging only to the displayed method. Retain any method, exact form, accuracy, units or justification explicitly requested in the question.
Do not credit by itself: Using an invalid relationship or leaving the requested result incomplete. Stating a conclusion without the working or justification requested by the task.
Part c (4 marks)
Award all 4 marks for a complete mathematically correct response to the task. The numbered statements are indicative evidence for one approach, not a compulsory calculation sequence. Award equivalent marks for a valid alternative method that establishes the requested result and justification. For incomplete responses, credit valid progress proportionately using the indicative one-mark evidence or equivalent reasoning. 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:
- Identifies 0.15 as the continuous relative growth-rate parameter.
- States the associated rate as 15% per month.
- Identifies a finite-demand, resource or capacity constraint.
- Explains why that constraint invalidates indefinite constant relative growth.
Acceptable alternatives: For Q1(c), award all 4 marks for a mathematically complete alternative solution. Do not require steps belonging only to the displayed method. Retain any method, exact form, accuracy, units or justification explicitly requested in the question.
Do not credit by itself: Using an invalid relationship or leaving the requested result incomplete. Stating a conclusion without the working or justification requested by the task.
Question 2
(a) f'(x)=xe^(-x)(2-x).
Apply the product and chain rules, then factorise.
(b) The stationary points are (0,0) and (2,4e⁻²).
Solve the factorised derivative equation and evaluate the original function.
(c) f increases on (0,2), decreases on (2,infty), and tends to 0; hence the global maximum is 4e⁻² at x=2.
Use derivative signs, the endpoint and limiting behaviour.
Detailed marking criteria
Part a (3 marks)
Award all 3 marks for a complete mathematically correct response to the task. The numbered statements are indicative evidence for one approach, not a compulsory calculation sequence. Award equivalent marks for a valid alternative method that establishes the requested result and justification. For incomplete responses, credit valid progress proportionately using the indicative one-mark evidence or equivalent reasoning. Do not award the final-result mark when the stated mathematical form is incomplete.
Mark-by-mark evidence:
- Applies the product rule to x^2e^(-x).
- Uses d(e^(-x)) / dx=-e^(-x) to obtain 2xe^(-x)-x^2e^(-x).
- Factorises to f'(x)=xe^(-x)(2-x).
Acceptable alternatives: For Q2(a), award all 3 marks for a mathematically complete alternative solution. Do not require steps belonging only to the displayed method. Retain any method, exact form, accuracy, units or justification explicitly requested in the question.
Do not credit by itself: Using an invalid relationship or leaving the requested result incomplete. Stating a conclusion without the working or justification requested by the task.
Part b (3 marks)
Award all 3 marks for a complete mathematically correct response to the task. The numbered statements are indicative evidence for one approach, not a compulsory calculation sequence. Award equivalent marks for a valid alternative method that establishes the requested result and justification. For incomplete responses, credit valid progress proportionately using the indicative one-mark evidence or equivalent reasoning. 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:
- Sets xe^(-x)(2-x)=0.
- Uses e^(-x)>0 to obtain x=0 or x=2.
- Evaluates f to state (0,0) and (2,4e⁻²).
Acceptable alternatives: For Q2(b), award all 3 marks for a mathematically complete alternative solution. Do not require steps belonging only to the displayed method. Retain any method, exact form, accuracy, units or justification explicitly requested in the question.
Do not credit by itself: Using an invalid relationship or leaving the requested result incomplete. Stating a conclusion without the working or justification requested by the task.
Part c (4 marks)
Award all 4 marks for a complete mathematically correct response to the task. The numbered statements are indicative evidence for one approach, not a compulsory calculation sequence. Award equivalent marks for a valid alternative method that establishes the requested result and justification. For incomplete responses, credit valid progress proportionately using the indicative one-mark evidence or equivalent reasoning. 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:
- Shows f'(x)>0 for 0<x<2.
- Shows f'(x)<0 for x>2.
- Checks f(0)=0 and lim_(xtoinfty)x^2e^(-x)=0.
- Concludes that the global maximum is f(2)=4e⁻².
Acceptable alternatives: For Q2(c), award all 4 marks for a mathematically complete alternative solution. Do not require steps belonging only to the displayed method. Retain any method, exact form, accuracy, units or justification explicitly requested in the question.
Do not credit by itself: Using an invalid relationship or leaving the requested result incomplete. Stating a conclusion without the working or justification requested by the task.
Question 3
(a) S'(r)=4π r-1000π / r^2.
Write the reciprocal term as a negative power before differentiating.
(b) r=√3(250) cm.
Solve the positive stationary-point equation.
(c) h=2sqrt(3)(250)approx12.60 cm; Stoinfty as rto0^+ and as rtoinfty, so the stationary point is the global minimum.
Use the volume constraint and endpoint behaviour on the positive domain.
Detailed marking criteria
Part a (3 marks)
Award all 3 marks for a complete mathematically correct response to the task. The numbered statements are indicative evidence for one approach, not a compulsory calculation sequence. Award equivalent marks for a valid alternative method that establishes the requested result and justification. For incomplete responses, credit valid progress proportionately using the indicative one-mark evidence or equivalent reasoning. Do not award the final-result mark when the stated mathematical form is incomplete.
Mark-by-mark evidence:
- Rewrites 1000π / r as 1000π r⁻¹.
- Differentiates 2π r² to 4π r.
- Obtains S'(r)=4π r-1000π r⁻².
Acceptable alternatives: For Q3(a), award all 3 marks for a mathematically complete alternative solution. Do not require steps belonging only to the displayed method. Retain any method, exact form, accuracy, units or justification explicitly requested in the question.
Do not credit by itself: Using an invalid relationship or leaving the requested result incomplete. Stating a conclusion without the working or justification requested by the task.
Part b (3 marks)
Award all 3 marks for a complete mathematically correct response to the task. The numbered statements are indicative evidence for one approach, not a compulsory calculation sequence. Award equivalent marks for a valid alternative method that establishes the requested result and justification. For incomplete responses, credit valid progress proportionately using the indicative one-mark evidence or equivalent reasoning. 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; include the requested units, context or justification.
Mark-by-mark evidence:
- Sets 4π r-1000π / r²=0.
- Multiplies by r² / (4π) to obtain r³=250.
- Uses r>0 to state r=√3(250) cm.
Acceptable alternatives: For Q3(b), award all 3 marks for a mathematically complete alternative solution. Do not require steps belonging only to the displayed method. Retain any method, exact form, accuracy, units or justification explicitly requested in the question.
Do not credit by itself: Using an invalid relationship or leaving the requested result incomplete. Stating a conclusion without the working or justification requested by the task.
Part c (4 marks)
Award all 4 marks for a complete mathematically correct response to the task. The numbered statements are indicative evidence for one approach, not a compulsory calculation sequence. Award equivalent marks for a valid alternative method that establishes the requested result and justification. For incomplete responses, credit valid progress proportionately using the indicative one-mark evidence or equivalent reasoning. 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 π r^2h=500π to obtain h=500 / r^2.
- Uses r³=250 to simplify h=2r=2sqrt(3)(250)approx12.60 cm.
- Shows S(r)toinfty as rto0^+.
- Shows S(r)toinfty as rtoinfty and concludes the sole stationary point is the global minimum.
Acceptable alternatives: For Q3(c), award all 4 marks for a mathematically complete alternative solution. Do not require steps belonging only to the displayed method. Retain any method, exact form, accuracy, units or justification explicitly requested in the question. AM-GM gives S(r)=2π(r²+250 / r+250 / r)geq6π250^(2 / 3), with equality precisely when r³=250. Together with h=500 / r²=2sqrt(3)(250)approx12.60 cm, this earns all 4 marks. Do not require either endpoint limit when a valid global bound, strict-convexity proof or derivative-sign argument establishes the minimum.
Do not credit by itself: Using an invalid relationship or leaving the requested result incomplete. Stating a conclusion without the working or justification requested by the task.
Question 4
(a) F'(x)=3-x², so the stationary value is x=sqrt3.
Apply the fundamental theorem and solve on the stated interval.
(b) F(x)=3x-x³ / 3-8 / 3 and F(sqrt3)=2sqrt3-8 / 3.
Evaluate the prescribed polynomial antiderivative at variable and fixed limits.
(c) F increases on [1,sqrt3], decreases on [sqrt3,3], and the endpoint values are F(1)=0 and F(3)=-8 / 3; hence F(sqrt3) is the global maximum.
Use the derivative sign and both endpoints rather than an unprescribed integration technique.
Detailed marking criteria
Part a (3 marks)
Award all 3 marks for a complete mathematically correct response to the task. The numbered statements are indicative evidence for one approach, not a compulsory calculation sequence. Award equivalent marks for a valid alternative method that establishes the requested result and justification. For incomplete responses, credit valid progress proportionately using the indicative one-mark evidence or equivalent reasoning. Do not award the final-result mark when the stated mathematical form is incomplete.
Mark-by-mark evidence:
- Uses the fundamental theorem to state F'(x)=3-x^2.
- Sets 3-x²=0.
- Uses 1leq xleq3 to obtain x=sqrt3.
Acceptable alternatives: For Q4(a), award all 3 marks for a mathematically complete alternative solution. Do not require steps belonging only to the displayed method. Retain any method, exact form, accuracy, units or justification explicitly requested in the question.
Do not credit by itself: Using an invalid relationship or leaving the requested result incomplete. Stating a conclusion without the working or justification requested by the task.
Part b (3 marks)
Award all 3 marks for a complete mathematically correct response to the task. The numbered statements are indicative evidence for one approach, not a compulsory calculation sequence. Award equivalent marks for a valid alternative method that establishes the requested result and justification. For incomplete responses, credit valid progress proportionately using the indicative one-mark evidence or equivalent reasoning. 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:
- Uses the antiderivative 3t-t³ / 3.
- Evaluates the lower limit to obtain F(x)=3x-x³ / 3-8 / 3.
- Substitutes x=sqrt3 to obtain 2sqrt3-8 / 3.
Acceptable alternatives: For Q4(b), award all 3 marks for a mathematically complete alternative solution. Do not require steps belonging only to the displayed method. Retain any method, exact form, accuracy, units or justification explicitly requested in the question.
Do not credit by itself: Using an invalid relationship or leaving the requested result incomplete. Stating a conclusion without the working or justification requested by the task.
Part c (4 marks)
Award all 4 marks for a complete mathematically correct response to the task. The numbered statements are indicative evidence for one approach, not a compulsory calculation sequence. Award equivalent marks for a valid alternative method that establishes the requested result and justification. For incomplete responses, credit valid progress proportionately using the indicative one-mark evidence or equivalent reasoning. 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:
- Shows F'(x)>0 for 1leq x<sqrt3.
- Shows F'(x)<0 for sqrt3<xleq3.
- Evaluates F(1)=0 and F(3)=-8 / 3.
- Compares the stationary and endpoint values to prove the global maximum occurs at x=sqrt3.
Acceptable alternatives: For Q4(c), award all 4 marks for a mathematically complete alternative solution. Do not require steps belonging only to the displayed method. Retain any method, exact form, accuracy, units or justification explicitly requested in the question.
Do not credit by itself: Using an invalid relationship or leaving the requested result incomplete. Stating a conclusion without the working or justification requested by the task.
Question 5
(a) Since f'(x)=1+1 / x>0 for x>0, f is strictly increasing and one-to-one.
Use derivative sign to establish strict monotonicity.
(b) The unique solution is x=1.
Verify the solution and use strict monotonicity for uniqueness.
(c) g(1)=1 and g'(1)=1 / 2.
Use the inverse relation and differentiate f(g(y))=y.
Detailed marking criteria
Part a (3 marks)
Award all 3 marks for a complete mathematically correct response to the task. The numbered statements are indicative evidence for one approach, not a compulsory calculation sequence. Award equivalent marks for a valid alternative method that establishes the requested result and justification. For incomplete responses, credit valid progress proportionately using the indicative one-mark evidence or equivalent reasoning. For the final-result mark, include the requested units, context or justification.
Mark-by-mark evidence:
- Differentiates to obtain f'(x)=1+1 / x.
- Shows 1+1 / x>0 for every x>0.
- Concludes that f is strictly increasing and therefore one-to-one.
Acceptable alternatives: For Q5(a), award all 3 marks for a mathematically complete alternative solution. Do not require steps belonging only to the displayed method. Retain any method, exact form, accuracy, units or justification explicitly requested in the question.
Do not credit by itself: Using an invalid relationship or leaving the requested result incomplete. Stating a conclusion without the working or justification requested by the task.
Part b (3 marks)
Award all 3 marks for a complete mathematically correct response to the task. The numbered statements are indicative evidence for one approach, not a compulsory calculation sequence. Award equivalent marks for a valid alternative method that establishes the requested result and justification. For incomplete responses, credit valid progress proportionately using the indicative one-mark evidence or equivalent reasoning. 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:
- Substitutes x=1 to verify 1+ln1=1.
- Uses the strict increase proved in part (a).
- Concludes that x=1 is the unique solution.
Acceptable alternatives: For Q5(b), award all 3 marks for a mathematically complete alternative solution. Do not require steps belonging only to the displayed method. Retain any method, exact form, accuracy, units or justification explicitly requested in the question.
Do not credit by itself: Using an invalid relationship or leaving the requested result incomplete. Stating a conclusion without the working or justification requested by the task.
Part c (4 marks)
Award all 4 marks for a complete mathematically correct response to the task. The numbered statements are indicative evidence for one approach, not a compulsory calculation sequence. Award equivalent marks for a valid alternative method that establishes the requested result and justification. For incomplete responses, credit valid progress proportionately using the indicative one-mark evidence or equivalent reasoning. 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 f(1)=1 to state g(1)=1.
- Differentiates f(g(y))=y to obtain f'(g(y))g'(y)=1.
- Substitutes y=1 and g(1)=1.
- Uses f'(1)=2 to obtain g'(1)=1 / 2.
Acceptable alternatives: For Q5(c), award all 4 marks for a mathematically complete alternative solution. Do not require steps belonging only to the displayed method. Retain any method, exact form, accuracy, units or justification explicitly requested in the question.
Do not credit by itself: Using an invalid relationship or leaving the requested result incomplete. Stating a conclusion without the working or justification requested by the task.
Diagnostic Checklist
| Topic | Questions | Marks | Marks Lost | Action |
|---|---|---|---|---|
| Unit 2 — Exponential functions — Evaluate an exponential model at a stated time | Q1(a) | 3 | ___ | Q1(a): solve the task independently using a valid method. Indicative evidence for one approach: Substitutes t=4 to obtain D(4)=8e^(0.15(4)); Simplifies the exponent to obtain 8e^(0.6); Evaluates and rounds the final value to 14.58. Check that your method establishes the requested result and justification; these steps are not compulsory for an alternative method. |
| Unit 3 — The logarithmic function — Solve an exponential doubling-time equation | Q1(b) | 3 | ___ | Q1(b): solve the task independently using a valid method. Indicative evidence for one approach: Sets 8e^(0.15t)=16 and divides by 8; Takes logarithms to obtain 0.15t=ln2; States t=ln2 / 0.15approx4.62 months. Check that your method establishes the requested result and justification; these steps are not compulsory for an alternative method. |
| Unit 2 — Exponential functions — Interpret a continuous growth parameter and critique extrapolation | Q1(c) | 4 | ___ | Q1(c): solve the task independently using a valid method. Indicative evidence for one approach: Identifies 0.15 as the continuous relative growth-rate parameter; States the associated rate as 15% per month; Identifies a finite-demand, resource or capacity constraint; Explains why that constraint invalidates indefinite constant relative growth. Check that your method establishes the requested result and justification; these steps are not compulsory for an alternative method. |
| Unit 3 — Further differentiation and applications — Differentiate and factorise an exponential product | Q2(a) | 3 | ___ | Q2(a): solve the task independently using a valid method. Indicative evidence for one approach: Applies the product rule to x^2e^(-x); Uses d(e^(-x)) / dx=-e^(-x) to obtain 2xe^(-x)-x^2e^(-x); Factorises to f'(x)=xe^(-x)(2-x). Check that your method establishes the requested result and justification; these steps are not compulsory for an alternative method. |
| Unit 3 — Further differentiation and applications — Solve a factorised stationary-point equation | Q2(b) | 3 | ___ | Q2(b): solve the task independently using a valid method. Indicative evidence for one approach: Sets xe^(-x)(2-x)=0; Uses e^(-x)>0 to obtain x=0 or x=2; Evaluates f to state (0,0) and (2,4e⁻²). Check that your method establishes the requested result and justification; these steps are not compulsory for an alternative method. |
| Unit 3 — Further differentiation and applications — Justify a global maximum on an unbounded domain | Q2(c) | 4 | ___ | Q2(c): solve the task independently using a valid method. Indicative evidence for one approach: Shows f'(x)>0 for 0<x<2; Shows f'(x)<0 for x>2; Checks f(0)=0 and lim_(xtoinfty)x^2e^(-x)=0; Concludes that the global maximum is f(2)=4e⁻². Check that your method establishes the requested result and justification; these steps are not compulsory for an alternative method. |
| Unit 3 — Further differentiation and applications — Differentiate a reciprocal surface-area model | Q3(a) | 3 | ___ | Q3(a): solve the task independently using a valid method. Indicative evidence for one approach: Rewrites 1000π / r as 1000π r⁻¹; Differentiates 2π r² to 4π r; Obtains S'(r)=4π r-1000π r⁻². Check that your method establishes the requested result and justification; these steps are not compulsory for an alternative method. |
| Unit 3 — Further differentiation and applications — Solve a constrained stationary-point equation exactly | Q3(b) | 3 | ___ | Q3(b): solve the task independently using a valid method. Indicative evidence for one approach: Sets 4π r-1000π / r²=0; Multiplies by r² / (4π) to obtain r³=250; Uses r>0 to state r=√3(250) cm. Check that your method establishes the requested result and justification; these steps are not compulsory for an alternative method. |
| Unit 3 — Further differentiation and applications — Recover a constrained dimension and prove global minimality | Q3(c) | 4 | ___ | Q3(c): solve the task independently using a valid method. Indicative evidence for one approach: Uses π r^2h=500π to obtain h=500 / r²; Uses r³=250 to simplify h=2r=2sqrt(3)(250)approx12.60 cm; Shows S(r)toinfty as rto0^+; Shows S(r)toinfty as rtoinfty and concludes the sole stationary point is the global minimum. Check that your method establishes the requested result and justification; these steps are not compulsory for an alternative method. A global inequality with an attained equality case can establish the minimum without endpoint limits. |
| Unit 3 — Integrals — Apply the fundamental theorem to an accumulation function | Q4(a) | 3 | ___ | Q4(a): solve the task independently using a valid method. Indicative evidence for one approach: Uses the fundamental theorem to state F'(x)=3-x²; Sets 3-x²=0; Uses 1leq xleq3 to obtain x=sqrt3. Check that your method establishes the requested result and justification; these steps are not compulsory for an alternative method. |
| Unit 3 — Integrals — Evaluate a variable-limit polynomial integral exactly | Q4(b) | 3 | ___ | Q4(b): solve the task independently using a valid method. Indicative evidence for one approach: Uses the antiderivative 3t-t³ / 3; Evaluates the lower limit to obtain F(x)=3x-x³ / 3-8 / 3; Substitutes x=sqrt3 to obtain 2sqrt3-8 / 3. Check that your method establishes the requested result and justification; these steps are not compulsory for an alternative method. |
| Unit 3 — Integrals — Establish a global extremum of an accumulation function | Q4(c) | 4 | ___ | Q4(c): solve the task independently using a valid method. Indicative evidence for one approach: Shows F'(x)>0 for 1leq x<sqrt3; Shows F'(x)<0 for sqrt3<xleq3; Evaluates F(1)=0 and F(3)=-8 / 3; Compares the stationary and endpoint values to prove the global maximum occurs at x=sqrt3. Check that your method establishes the requested result and justification; these steps are not compulsory for an alternative method. |
| Unit 1 — Functions and graphs — Prove that a function is one-to-one | Q5(a) | 3 | ___ | Q5(a): solve the task independently using a valid method. Indicative evidence for one approach: Differentiates to obtain f'(x)=1+1 / x; Shows 1+1 / x>0 for every x>0; Concludes that f is strictly increasing and therefore one-to-one. Check that your method establishes the requested result and justification; these steps are not compulsory for an alternative method. |
| Unit 3 — The logarithmic function — Solve and justify a logarithmic equation structurally | Q5(b) | 3 | ___ | Q5(b): solve the task independently using a valid method. Indicative evidence for one approach: Substitutes x=1 to verify 1+ln1=1; Uses the strict increase proved in part (a); Concludes that x=1 is the unique solution. Check that your method establishes the requested result and justification; these steps are not compulsory for an alternative method. |
| Unit 1 — Functions and graphs — Differentiate an inverse relation at a known point | Q5(c) | 4 | ___ | Q5(c): solve the task independently using a valid method. Indicative evidence for one approach: Uses f(1)=1 to state g(1)=1; Differentiates f(g(y))=y to obtain f'(g(y))g'(y)=1; Substitutes y=1 and g(1)=1; Uses f'(1)=2 to obtain g'(1)=1 / 2. Check that your method establishes the requested result and justification; these steps are not compulsory for an alternative method. |