Skill Align SACE Stage 2 Specialist Mathematics - Pack 0 - Question Booklet 2
Questions 8-10 | 45 marks | one part of a 100-mark, 130-minute complete examination
- Question booklet
- Question Booklet 2
- Recommended time
- Approximately 65 minutes
- Complete examination
- 130 minutes across both question booklets
- Assessment
- 45 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 2
Answer all questions 8-10. Show complete working and steps of logic. Give numerical answers to three significant figures unless otherwise instructed.
Question 8
15 marksQuestion 9
14 marksQuestion 10
16 marksWorked Solutions And Marking Guide
Question 8
(a) t=2, at (5,0,1).
Substitution gives 5t-6=4, so t=2.
(b) sin⁻¹(5 / 6)=56.4^circ.
With direction (2,1,-1) and normal (1,2,-1), sintheta=|5| / (sqrt6sqrt6).
(c) 4 / sqrt6=2sqrt6 / 3.
Use |-4| / √(1²+2²+(-1)²).
(d) 2x+y-z=9.
Use the path direction as the new plane's normal and pass through (5,0,1).
(e) Speed =sqrt6 units / s; the particle reaches the plane after 2 s.
The constant velocity is (2,1,-1), whose magnitude is sqrt6.
Detailed marking criteria
Part a (4 marks)
Award the 4 available marks for the following observable evidence: writes the three parametric coordinates; substitutes them into the plane; solves 5t-6=4; states the intersection point.
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 direction and normal vectors; uses the line-plane angle formula; reports 56.4^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 c (2 marks)
Award the 2 available marks for the following observable evidence: uses the point-to-plane distance formula; simplifies the exact distance.
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 (3 marks)
Award the 3 available marks for the following observable evidence: uses normal (2,1,-1); uses the intersection point; expands to 2x+y-z=9.
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 e (3 marks)
Award the 3 available marks for the following observable evidence: identifies the velocity vector; calculates its magnitude; interprets t=2 in context.
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 9
(a) z_k=2operatorname(cis)(π / 6+kpi / 2), k=0,1,2,3.
Take fourth roots of the modulus and use (2π / 3+2kpi) / 4.
(b) The roots are vertices of a square centred at the origin on |z|=2.
Plot at arguments π / 6,2π / 3,7π / 6,5π / 3.
(c) A dilation by sqrt2 and rotation through -π / 4 about the origin.
1-i=sqrt2operatorname(cis)(-π / 4).
(d) -16operatorname(cis)(2π / 3)=16operatorname(cis)(5π / 3).
For z⁴-A=0, the product of the roots is -A.
Detailed marking criteria
Part a (5 marks)
Award the 5 available marks for the following observable evidence: takes the fourth root of 16; forms the complete argument rule; uses increment π / 2; uses k=0,1,2,3; lists four 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 b (3 marks)
Award the 3 available marks for the following observable evidence: plots all four arguments; keeps radius 2; identifies the square geometry.
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 (3 marks)
Award the 3 available marks for the following observable evidence: expresses 1-i in polar form; identifies the dilation; identifies the clockwise rotation.
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 (3 marks)
Award the 3 available marks for the following observable evidence: uses the monic-polynomial root product; substitutes A=16operatorname(cis)(2π / 3); gives an equivalent standard-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.
Question 10
(a) T=18+60e^(-kt).
Separate variables to obtain ln|T-18|=-kt+C, then use T(0)=78.
(b) k=((ln2) / (10)).
30=60e^(-10k), so e^(-10k)=1 / 2.
(c) 28.6^circC.
Use T(25)=18+60e^(-25ln2 / 10)=28.6066ldots.
(d) t=((10ln10) / (ln2))=33.2 minutes.
Set 6=60e^(-kt), then solve e^(-kt)=0.1.
(e) The limit is 18^circC; the room temperature and heat-transfer conditions are assumed constant.
As ttoinfty, the exponential term tends to zero.
Detailed marking criteria
Part a (4 marks)
Award the 4 available marks for the following observable evidence: separates the variables; integrates both sides; exponentiates the relation; uses the initial condition.
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: substitutes the second observation; isolates the exponential factor; solves exactly for k.
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: substitutes t=25; uses the exact parameter value; rounds to 28.6^circC.
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 (3 marks)
Award the 3 available marks for the following observable evidence: forms the threshold equation; solves it exactly for t; reports 33.2 minutes.
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 e (3 marks)
Award the 3 available marks for the following observable evidence: uses the exponential limit; states 18^circC; identifies a relevant constant-environment assumption.
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 |
|---|---|---|---|---|
| Vectors in Three Dimensions | Q8 | 15 | ___ | Rework Q8: practise line plane intersection angle distance perpendicular plane. Check every labelled part against its observable evidence and retain exact or unrounded values until the final answer. |
| Complex Numbers | Q9 | 14 | ___ | Rework Q9: practise complex roots argand transformation product. Check every labelled part against its observable evidence and retain exact or unrounded values until the final answer. |
| Rates of Change and Differential Equations | Q10 | 16 | ___ | Rework Q10: practise differential equation parameter prediction threshold assumption. Check every labelled part against its observable evidence and retain exact or unrounded values until the final answer. |