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) x-y=(x-y) × 1.
Both sides reduce to the same first-degree difference.
(b) Assume x^k-y^k=(x-y)(x^(k-1)+x^(k-2)y+ × s+xy^(k-2)+y^(k-1)) for some integer kgeq1.
State the complete factor identity at an arbitrary admissible integer k.
(c) begin(aligned) x^(k+1)-y^(k+1) &=x(x^k-y^k)+y^k(x-y) &=(x-y)[x(x^(k-1)+x^(k-2)y+ × s+y^(k-1))+y^k] &=(x-y)(x^k+x^(k-1)y+ × s+xy^(k-1)+y^k). end(aligned) This is the required identity for n=k+1. Therefore the identity holds for every positive integer n by mathematical induction.
Decompose the next difference so that the induction-hypothesis factor appears, then append the final term of the required polynomial factor.
Detailed marking criteria
Part a (1 mark)
Award the 1 available mark for the following observable evidence: verifies the base case explicitly.
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 required final accuracy.
Part b (2 marks)
Award the 2 available marks for the following observable evidence: states the identity at k; identifies it 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 required final accuracy.
Part c (5 marks)
Award the 5 available marks for the following observable evidence: starts from a valid decomposition of x^(k+1)-y^(k+1); substitutes the induction hypothesis; factors out x-y; obtains the complete polynomial factor for k+1; states the formal induction conclusion.
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 required final accuracy.
Question 2
(a) z=4operatorname(cis)(2π / 3).
The modulus is 4, and the quadrant-II argument is 2π / 3.
(b) z³=64.
By de Moivre's theorem, z³=4^3operatorname(cis)(2π)=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=2π / 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 required final accuracy.
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.
Do not credit by itself: Do not award evidence that is only asserted when working, proof or interpretation is required. Do not accept a decimal approximation when the part requires an exact final 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.
Do not credit by itself: Do not award evidence that is only asserted when working, proof or interpretation is required. Do not accept a decimal approximation when the part requires an exact final 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 required final accuracy.
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 required final accuracy.
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 required final accuracy.
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 required final accuracy.
Question 4
(a) -8=8operatorname(cis)π.
Use modulus 8 and argument π.
(b) 2operatorname(cis)(π / 3), 2operatorname(cis)π, 2operatorname(cis)(5π / 3).
Use arguments (π+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 required final accuracy.
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 required final accuracy.
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.
Do not credit by itself: Do not award evidence that is only asserted when working, proof or interpretation is required. Do not accept a decimal approximation when the part requires an exact final 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 required final accuracy.
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 required final accuracy.
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 required final accuracy.
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 required final accuracy.
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 required final accuracy.
Question 6
(a) xtan x+ln|cos x|+C.
Take u=x and dv=sec^2x,dx. Then v=tan x, and inttan x,dx=-ln|cos x|.
(b) I=fracpi4-frac12ln2.
At x=π / 4, the antiderivative is π / 4+ln(sqrt2 / 2)=π / 4-frac12ln2; at x=0 it is zero.
Detailed marking criteria
Part a (4 marks)
Award the 4 available marks for the following observable evidence: chooses u=x; finds v=tan x; applies integration by parts; simplifies the antiderivative.
Acceptable alternatives: Accept algebraically equivalent exact forms and any logically equivalent sequence of valid steps.
Do not credit by itself: Do not award evidence that is only asserted when working, proof or interpretation is required. Do not accept a decimal approximation when the part requires an exact final answer.
Part b (3 marks)
Award the 3 available marks for the following observable evidence: substitutes the upper bound exactly; uses ln(sqrt2 / 2)=-frac12ln2; subtracts the lower-bound value.
Acceptable alternatives: Accept algebraically equivalent exact forms and any logically equivalent sequence of valid steps.
Do not credit by itself: Do not award evidence that is only asserted when working, proof or interpretation is required. Do not accept a decimal approximation when the part requires an exact final 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 required final accuracy.
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.
Do not credit by itself: Do not award evidence that is only asserted when working, proof or interpretation is required. Do not accept a decimal approximation when the part requires an exact final 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 required final accuracy.
Diagnostic Checklist
| Topic | Questions | Marks | Marks Lost | Action |
|---|---|---|---|---|
| Mathematical Induction | Q1 | 8 | ___ | Rework Q1: practise polynomial factor identity induction. 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 trigonometric exact. 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. |