Skill Align SACE Stage 2 Specialist Mathematics - Pack 0 - Question Booklet 1
Questions 1-7 | 55 marks | one part of a 100-mark, 130-minute complete examination
- Question booklet
- Question Booklet 1
- Recommended time
- Approximately 65 minutes
- Complete examination
- 130 minutes across both question booklets
- Assessment
- 55 marks
SACE Stage 2 Specialist Mathematics - examination conditions
- Answer all questions and write your answers in this question booklet.
- Examination materials are Question Booklet 1, Question Booklet 2, the current official SACE formula sheet and the candidate registration label. The supervising centre supplies the formula sheet and label separately; neither is bundled in this Skill Align PDF.
- Show appropriate working and steps of logic. State numerical answers correct to three significant figures unless a question instructs otherwise.
- Use black or blue pen. A sharp dark pencil may be used for diagrams and graphs.
- You may bring two unfolded A4 sheets, using all four sides, containing your own handwritten notes.
- You may use either two approved graphics calculators or one approved graphics calculator and one scientific calculator. Computer algebra system (CAS) calculators are not permitted. Scientific-calculator memory must be cleared; graphics-calculator memory does not need to be cleared.
Question Booklet 1
Answer all questions 1-7. Show complete working and steps of logic. Give numerical answers to three significant figures unless otherwise instructed.
Question 1
8 marksQuestion 2
8 marksQuestion 3
8 marksQuestion 4
8 marksQuestion 5
8 marksQuestion 6
7 marksQuestion 7
8 marksWorked Solutions And Marking Guide
Question 1
(a) S_1=2=((1 × 2 × 3) / (3)).
Both sides equal 2.
(b) Assume S_k=frac(k(k+1)(k+2))3.
The hypothesis is stated for an arbitrary positive integer k.
(c) begin(aligned) S_(k+1) &=S_k+(k+1)(k+2) &=((k(k+1)(k+2)) / (3))+(k+1)(k+2) &=(k+1)(k+2)(((k) / (3))+1) &=(((k+1)(k+2)(k+3)) / (3)). end(aligned) Therefore, by mathematical induction, sum_(r=1)^(n)r(r+1)=((n(n+1)(n+2)) / (3)) for every positive integer n.
The displayed equality chain adds the k+1 term, substitutes the induction hypothesis, factors the common product and simplifies to the required k+1 form before the formal conclusion.
Detailed marking criteria
Part a (1 mark)
Award the 1 available mark for the following observable evidence: evaluates both sides at n=1.
Acceptable alternatives: Accept algebraically equivalent exact forms and any logically equivalent sequence of valid steps.; For an approximate answer, accept calculator-supported values that round correctly from unrounded intermediate values.
Do not credit by itself: Do not award evidence that is only asserted when working, proof or interpretation is required.; Do not accept premature rounding that changes the final three-significant-figure answer.
Part b (2 marks)
Award the 2 available marks for the following observable evidence: states the formula at k; identifies the assumption as the induction hypothesis.
Acceptable alternatives: Accept algebraically equivalent exact forms and any logically equivalent sequence of valid steps.; For an approximate answer, accept calculator-supported values that round correctly from unrounded intermediate values.
Do not credit by itself: Do not award evidence that is only asserted when working, proof or interpretation is required.; Do not accept premature rounding that changes the final three-significant-figure answer.
Part c (5 marks)
Award the 5 available marks for the following observable evidence: starts from S_(k+1)=S_k+(k+1)(k+2); substitutes S_k=frac(k(k+1)(k+2))3; factors (k+1)(k+2); simplifies explicitly to frac((k+1)(k+2)(k+3))3; states the conclusion for every positive integer n.
Acceptable alternatives: Accept algebraically equivalent exact forms and any logically equivalent sequence of valid steps.; For an approximate answer, accept calculator-supported values that round correctly from unrounded intermediate values.
Do not credit by itself: Do not award evidence that is only asserted when working, proof or interpretation is required.; Do not accept premature rounding that changes the final three-significant-figure answer.
Question 2
(a) z=4operatorname(cis)(2pi / 3).
The modulus is 4, and the quadrant-II argument is 2pi / 3.
(b) z³=64.
By de Moivre's theorem, z³=4^3operatorname(cis)(2pi)=64.
(c) z⁻¹=((-1-sqrt3,i) / (8)).
Use bar z / |z|²=(-2-2sqrt3 i) / 16.
Detailed marking criteria
Part a (2 marks)
Award the 2 available marks for the following observable evidence: calculates |z|=4; states arg z=2pi / 3.
Acceptable alternatives: Accept algebraically equivalent exact forms and any logically equivalent sequence of valid steps.; For an approximate answer, accept calculator-supported values that round correctly from unrounded intermediate values.
Do not credit by itself: Do not award evidence that is only asserted when working, proof or interpretation is required.; Do not accept premature rounding that changes the final three-significant-figure answer.
Part b (3 marks)
Award the 3 available marks for the following observable evidence: cubes the modulus; triples the argument; converts the result to 64.
Acceptable alternatives: Accept algebraically equivalent exact forms and any logically equivalent sequence of valid steps.; For an approximate answer, accept calculator-supported values that round correctly from unrounded intermediate values.
Do not credit by itself: Do not award evidence that is only asserted when working, proof or interpretation is required.; Do not accept premature rounding that changes the final three-significant-figure answer.
Part c (3 marks)
Award the 3 available marks for the following observable evidence: uses the conjugate-over-modulus-squared rule; uses |z|²=16; simplifies both Cartesian components.
Acceptable alternatives: Accept algebraically equivalent exact forms and any logically equivalent sequence of valid steps.; For an approximate answer, accept calculator-supported values that round correctly from unrounded intermediate values.
Do not credit by itself: Do not award evidence that is only asserted when working, proof or interpretation is required.; Do not accept premature rounding that changes the final three-significant-figure answer.
Question 3
(a) x=-2 and y=2.
Write f(x)=2-frac5(x+2).
(b) (1 / 2,0) and (0,-1 / 2).
Set the numerator to zero, then substitute x=0.
(c) f'(x)=frac5((x+2)²)>0; there are no stationary points.
Differentiate 2-5 / (x+2).
(d) mathbb Rsetminus(2).
The equation f(x)=2 has no solution, while every other real value is attained.
Detailed marking criteria
Part a (2 marks)
Award the 2 available marks for the following observable evidence: identifies x=-2; identifies y=2.
Acceptable alternatives: Accept algebraically equivalent exact forms and any logically equivalent sequence of valid steps.; For an approximate answer, accept calculator-supported values that round correctly from unrounded intermediate values.
Do not credit by itself: Do not award evidence that is only asserted when working, proof or interpretation is required.; Do not accept premature rounding that changes the final three-significant-figure answer.
Part b (2 marks)
Award the 2 available marks for the following observable evidence: finds the x-intercept; finds the y-intercept.
Acceptable alternatives: Accept algebraically equivalent exact forms and any logically equivalent sequence of valid steps.; For an approximate answer, accept calculator-supported values that round correctly from unrounded intermediate values.
Do not credit by itself: Do not award evidence that is only asserted when working, proof or interpretation is required.; Do not accept premature rounding that changes the final three-significant-figure answer.
Part c (2 marks)
Award the 2 available marks for the following observable evidence: obtains 5 / (x+2)²; uses its positivity to exclude stationary points.
Acceptable alternatives: Accept algebraically equivalent exact forms and any logically equivalent sequence of valid steps.; For an approximate answer, accept calculator-supported values that round correctly from unrounded intermediate values.
Do not credit by itself: Do not award evidence that is only asserted when working, proof or interpretation is required.; Do not accept premature rounding that changes the final three-significant-figure answer.
Part d (2 marks)
Award the 2 available marks for the following observable evidence: excludes y=2; states all other real values are in the range.
Acceptable alternatives: Accept algebraically equivalent exact forms and any logically equivalent sequence of valid steps.; For an approximate answer, accept calculator-supported values that round correctly from unrounded intermediate values.
Do not credit by itself: Do not award evidence that is only asserted when working, proof or interpretation is required.; Do not accept premature rounding that changes the final three-significant-figure answer.
Question 4
(a) -8=8operatorname(cis)pi.
Use modulus 8 and argument pi.
(b) 2operatorname(cis)(pi / 3), 2operatorname(cis)pi, 2operatorname(cis)(5pi / 3).
Use arguments (pi+2kpi) / 3, k=0,1,2.
(c) 1+sqrt3i, -2, 1-sqrt3i.
Convert each polar root using exact trigonometric values.
(d) Three equally spaced points on the circle |w|=2 at the stated arguments.
Plot (1,sqrt3), (-2,0) and (1,-sqrt3).
Detailed marking criteria
Part a (1 mark)
Award the 1 available mark for the following observable evidence: states a valid modulus-argument form.
Acceptable alternatives: Accept algebraically equivalent exact forms and any logically equivalent sequence of valid steps.; For an approximate answer, accept calculator-supported values that round correctly from unrounded intermediate values.
Do not credit by itself: Do not award evidence that is only asserted when working, proof or interpretation is required.; Do not accept premature rounding that changes the final three-significant-figure answer.
Part b (3 marks)
Award the 3 available marks for the following observable evidence: takes the cube root of the modulus; uses the complete root-angle formula; lists three distinct roots.
Acceptable alternatives: Accept algebraically equivalent exact forms and any logically equivalent sequence of valid steps.; For an approximate answer, accept calculator-supported values that round correctly from unrounded intermediate values.
Do not credit by itself: Do not award evidence that is only asserted when working, proof or interpretation is required.; Do not accept premature rounding that changes the final three-significant-figure answer.
Part c (2 marks)
Award the 2 available marks for the following observable evidence: converts the upper and lower roots; includes the real root -2.
Acceptable alternatives: Accept algebraically equivalent exact forms and any logically equivalent sequence of valid steps.; For an approximate answer, accept calculator-supported values that round correctly from unrounded intermediate values.
Do not credit by itself: Do not award evidence that is only asserted when working, proof or interpretation is required.; Do not accept premature rounding that changes the final three-significant-figure answer.
Part d (2 marks)
Award the 2 available marks for the following observable evidence: plots all three roots at the correct coordinates; shows their equal radius and 120^circ spacing.
Acceptable alternatives: Accept algebraically equivalent exact forms and any logically equivalent sequence of valid steps.; For an approximate answer, accept calculator-supported values that round correctly from unrounded intermediate values.
Do not credit by itself: Do not award evidence that is only asserted when working, proof or interpretation is required.; Do not accept premature rounding that changes the final three-significant-figure answer.
Question 5
(a) -4.
1(2)+2(-1)+(-2)(2)=-4.
(b) |mathbf a|=|mathbf b|=3.
Each squared magnitude is 9.
(c) 116^circ.
costheta=-4 / 9, so theta=116.388ldots^circ.
(d) operatorname(proj)_(mathbf b)mathbf a=(-8 / 9,4 / 9,-8 / 9).
Multiply mathbf b by (mathbf a × mathbf b) / |mathbf b|²=-4 / 9.
Detailed marking criteria
Part a (2 marks)
Award the 2 available marks for the following observable evidence: forms the component products; sums them to -4.
Acceptable alternatives: Accept algebraically equivalent exact forms and any logically equivalent sequence of valid steps.; For an approximate answer, accept calculator-supported values that round correctly from unrounded intermediate values.
Do not credit by itself: Do not award evidence that is only asserted when working, proof or interpretation is required.; Do not accept premature rounding that changes the final three-significant-figure answer.
Part b (2 marks)
Award the 2 available marks for the following observable evidence: calculates |mathbf a|=3; calculates |mathbf b|=3.
Acceptable alternatives: Accept algebraically equivalent exact forms and any logically equivalent sequence of valid steps.; For an approximate answer, accept calculator-supported values that round correctly from unrounded intermediate values.
Do not credit by itself: Do not award evidence that is only asserted when working, proof or interpretation is required.; Do not accept premature rounding that changes the final three-significant-figure answer.
Part c (2 marks)
Award the 2 available marks for the following observable evidence: uses costheta=(mathbf a × mathbf b) / (|mathbf a||mathbf b|); reports 116^circ.
Acceptable alternatives: Accept algebraically equivalent exact forms and any logically equivalent sequence of valid steps.; For an approximate answer, accept calculator-supported values that round correctly from unrounded intermediate values.
Do not credit by itself: Do not award evidence that is only asserted when working, proof or interpretation is required.; Do not accept premature rounding that changes the final three-significant-figure answer.
Part d (2 marks)
Award the 2 available marks for the following observable evidence: finds the scalar projection coefficient -4 / 9; multiplies mathbf b componentwise.
Acceptable alternatives: Accept algebraically equivalent exact forms and any logically equivalent sequence of valid steps.; For an approximate answer, accept calculator-supported values that round correctly from unrounded intermediate values.
Do not credit by itself: Do not award evidence that is only asserted when working, proof or interpretation is required.; Do not accept premature rounding that changes the final three-significant-figure answer.
Question 6
(a) ((x²) / (2))ln x-((x²) / (4))+C.
Take u=ln x and dv=x,dx, then integrate the remaining x / 2 term.
(b) ((e²+1) / (4)).
At e the antiderivative is e² / 4; at 1 it is -1 / 4.
Detailed marking criteria
Part a (4 marks)
Award the 4 available marks for the following observable evidence: selects u=ln x; selects v=x² / 2; applies the integration-by-parts formula; simplifies the antiderivative.
Acceptable alternatives: Accept algebraically equivalent exact forms and any logically equivalent sequence of valid steps.; For an approximate answer, accept calculator-supported values that round correctly from unrounded intermediate values.
Do not credit by itself: Do not award evidence that is only asserted when working, proof or interpretation is required.; Do not accept premature rounding that changes the final three-significant-figure answer.
Part b (3 marks)
Award the 3 available marks for the following observable evidence: substitutes the upper bound exactly; substitutes the lower bound exactly; subtracts to obtain (e²+1) / 4.
Acceptable alternatives: Accept algebraically equivalent exact forms and any logically equivalent sequence of valid steps.; For an approximate answer, accept calculator-supported values that round correctly from unrounded intermediate values.
Do not credit by itself: Do not award evidence that is only asserted when working, proof or interpretation is required.; Do not accept premature rounding that changes the final three-significant-figure answer.
Question 7
(a) (0,0) and (2,4).
Solve x²=2x.
(b) 4 / 3 square units.
int_0²(2x-x²),dx=[x²-x³ / 3]_0²=4 / 3.
(c) The maximum separation is 1, at x=1.
For d(x)=2x-x², d'(x)=2-2x=0 at x=1, and d''=-2<0.
Detailed marking criteria
Part a (2 marks)
Award the 2 available marks for the following observable evidence: finds x=0,2; states both coordinate pairs.
Acceptable alternatives: Accept algebraically equivalent exact forms and any logically equivalent sequence of valid steps.; For an approximate answer, accept calculator-supported values that round correctly from unrounded intermediate values.
Do not credit by itself: Do not award evidence that is only asserted when working, proof or interpretation is required.; Do not accept premature rounding that changes the final three-significant-figure answer.
Part b (3 marks)
Award the 3 available marks for the following observable evidence: identifies the upper-minus-lower integrand; uses the correct bounds; evaluates the exact area.
Acceptable alternatives: Accept algebraically equivalent exact forms and any logically equivalent sequence of valid steps.; For an approximate answer, accept calculator-supported values that round correctly from unrounded intermediate values.
Do not credit by itself: Do not award evidence that is only asserted when working, proof or interpretation is required.; Do not accept premature rounding that changes the final three-significant-figure answer.
Part c (3 marks)
Award the 3 available marks for the following observable evidence: defines the vertical separation; finds its stationary point x=1; classifies and evaluates the maximum.
Acceptable alternatives: Accept algebraically equivalent exact forms and any logically equivalent sequence of valid steps.; For an approximate answer, accept calculator-supported values that round correctly from unrounded intermediate values.
Do not credit by itself: Do not award evidence that is only asserted when working, proof or interpretation is required.; Do not accept premature rounding that changes the final three-significant-figure answer.
Diagnostic Checklist
| Topic | Questions | Marks | Marks Lost | Action |
|---|---|---|---|---|
| Mathematical Induction | Q1 | 8 | ___ | Rework Q1: practise induction base hypothesis algebraic step. Check every labelled part against its observable evidence and retain exact or unrounded values until the final answer. |
| Complex Numbers | Q2 | 8 | ___ | Rework Q2: practise modulus argument de moivre reciprocal. Check every labelled part against its observable evidence and retain exact or unrounded values until the final answer. |
| Functions, Sketching and Graphs | Q3 | 8 | ___ | Rework Q3: practise rational asymptotes intercepts derivative range. Check every labelled part against its observable evidence and retain exact or unrounded values until the final answer. |
| Complex Numbers | Q4 | 8 | ___ | Rework Q4: practise complex roots exact cartesian argand plot. Check every labelled part against its observable evidence and retain exact or unrounded values until the final answer. |
| Vectors in Three Dimensions | Q5 | 8 | ___ | Rework Q5: practise dot product angle projection. Check every labelled part against its observable evidence and retain exact or unrounded values until the final answer. |
| Integration Techniques and Applications | Q6 | 7 | ___ | Rework Q6: practise integration by parts exact definite integral. Check every labelled part against its observable evidence and retain exact or unrounded values until the final answer. |
| Integration Techniques and Applications | Q7 | 8 | ___ | Rework Q7: practise intersection area maximum separation. Check every labelled part against its observable evidence and retain exact or unrounded values until the final answer. |