Skill Align QCE Specialist Mathematics Paper 2 - Free Online Pack 0
Full-length Units 3&4 external-assessment-style showcase paper
- Paper
- Paper 2 Technology-active Question and Response Book Showcase
- Reading
- 5 minutes perusal
- Writing
- 90 minutes
- Assessment
- 60 marks
QCAA formula book provided; approved graphics or scientific calculator permitted for Paper 2; CAS calculators are not permitted. No public PDF or formula-book download is supplied with Pack 0.
Section 1
Questions 1-10 are multiple choice. Select the best answer. Technology may be used where appropriate.
Question 1
1 mark- 75.0 and 2.6
- 72.4 and 5.2
- 77.6 and 2.6
- 75.0 and 5.2
Question 2
1 mark- 50mathbf i+15mathbf j
- 30mathbf i-15mathbf j
- 30mathbf i+15mathbf j
- 15mathbf i-30mathbf j
Question 3
1 mark- 3
- 5
- ((11) / (3))
- ((22) / (3))
Question 4
1 mark- 6(1,-1,2)
- (2,-3,2)
- frac13(1,-1,2)
- frac16(1,-1,2)
Question 5
1 mark- (-5,0)
- (5,0)
- (0,-5)
- (0,5)
Question 6
1 mark- doubled
- halved
- quartered
- unchanged
Question 7
1 mark- 0.5
- 15
- 30
- 60
Question 8
1 mark- 1+3i
- 2
- 1+i
- 3+i
Question 9
1 mark- √46
- √14
- 6
- 46
Question 10
1 mark- 0.5000
- 0.8225
- 1.6450
- 9.8700
Section 2
Questions 11-18 are short response. Show working, modelling decisions and interpretation.
Question 11
5 marksQuestion 12
7 marksQuestion 13
6 marksQuestion 14
6 marksQuestion 15
6 marksQuestion 16
7 marksQuestion 17
6 marksQuestion 18
7 marksWorked Solutions And Marking Guide
Section 1 Question 1
Answer: 75.0 and 2.6
Average the endpoints to find the centre, then subtract the centre from either endpoint.
Section 1 Question 2
Answer: 30mathbf i-15mathbf j
Combine signed horizontal and vertical components.
Section 1 Question 3
Answer: ((11) / (3))
Use ((h) / (3))[1+4(4)+5].
Section 1 Question 4
Answer: frac16(1,-1,2)
The dot product is 1 and b dot b is 6.
Section 1 Question 5
Answer: (-5,0)
Use cosine π=-1 and sine π=0.
Section 1 Question 6
Answer: halved
Width is proportional to one over square root n.
Section 1 Question 7
Answer: 30
Set the rate equal to zero.
Section 1 Question 8
Answer: 3+i
Expand (1+i)(2-i)=2-i+2i-i²=3+i.
Section 1 Question 9
Answer: √46
The cross product is (6,-3,1).
Section 1 Question 10
Answer: 0.8225
Compute 1.645 times 6 / 12.
Section 2 Question 11
(a) bar x=((25(102)+75(98)) / (100))=99.
Weight each sample mean by its sample size before dividing by the combined sample size.
(b) ((6) / (√100))=0.6.
Use the combined sample standard deviation for the 100 independent observations.
(c) 99pm1.96(0.6)=(97.824,100.176). No; the interval contains 100.
Use the combined mean and standard error. A value inside the interval cannot be ruled out by this interval.
Detailed marking criteria
Part Part (a) (2 marks)
Award one mark for each required-evidence item, up to the stated maximum. Apply follow-through when an earlier numerical result is used consistently, unless the current mark requires an independent conclusion.
Acceptable alternatives: Equivalent exact forms and correctly rounded decimal forms are acceptable unless the question specifies otherwise.
Do not credit by itself: A correct final value without the required supporting evidence does not earn all available marks.
Part Part (b) (1 mark)
Award one mark for each required-evidence item, up to the stated maximum. Apply follow-through when an earlier numerical result is used consistently, unless the current mark requires an independent conclusion.
Acceptable alternatives: Equivalent exact forms and correctly rounded decimal forms are acceptable unless the question specifies otherwise.
Do not credit by itself: A correct final value without the required supporting evidence does not earn all available marks.
Part Part (c) (2 marks)
Award one mark for each required-evidence item, up to the stated maximum. Apply follow-through when an earlier numerical result is used consistently, unless the current mark requires an independent conclusion.
Acceptable alternatives: Equivalent exact forms and correctly rounded decimal forms are acceptable unless the question specifies otherwise.
Do not credit by itself: A correct final value without the required supporting evidence does not earn all available marks.
Section 2 Question 12
(a) v(1)=√17approx4.123 and v(3)=√43approx6.557.
Substitute the two times into the non-polynomial speed function.
(b) frac13[4+4(4.1231056)+2(4.8989795)+4(6.5574385)+8.9442719]approx21.821 m.
With four subintervals, h=1 and the Simpson weights are 1, 4, 2, 4, 1.
(c) ((|21.821469-21.826807|) / (21.826807)) × 100%approx0.0245%. Exactness is not guaranteed because √(16+t³) is not a polynomial of degree at most 3.
Compare unrounded values and identify the polynomial exactness condition for Simpson's rule.
Detailed marking criteria
Part Part (a) (2 marks)
Award one mark for each required-evidence item, up to the stated maximum. Apply follow-through when an earlier numerical result is used consistently, unless the current mark requires an independent conclusion.
Acceptable alternatives: Equivalent exact forms and correctly rounded decimal forms are acceptable unless the question specifies otherwise.
Do not credit by itself: A correct final value without the required supporting evidence does not earn all available marks.
Part Part (b) (3 marks)
Award one mark for each required-evidence item, up to the stated maximum. Apply follow-through when an earlier numerical result is used consistently, unless the current mark requires an independent conclusion.
Acceptable alternatives: Equivalent exact forms and correctly rounded decimal forms are acceptable unless the question specifies otherwise.
Do not credit by itself: A correct final value without the required supporting evidence does not earn all available marks.
Part Part (c) (2 marks)
Award one mark for each required-evidence item, up to the stated maximum. Apply follow-through when an earlier numerical result is used consistently, unless the current mark requires an independent conclusion.
Acceptable alternatives: Equivalent exact forms and correctly rounded decimal forms are acceptable unless the question specifies otherwise.
Do not credit by itself: A correct final value without the required supporting evidence does not earn all available marks.
Section 2 Question 13
(a) A dilation by sqrt5 and a clockwise rotation through tan⁻¹(2).
The modulus of 1-2i is sqrt5, and its argument is -tan⁻¹(2).
(b) The centre maps to (1-2i)(1+i)+3+i=6, and the radius becomes 2sqrt5, so |w-6|=2sqrt5.
Map the original centre and scale the radius by the modulus of the multiplier.
(c) π(2sqrt5)²=20π.
Use the mapped radius, or multiply the original area by the area scale factor 5.
Detailed marking criteria
Part Part (a) (2 marks)
Award one mark for each required-evidence item, up to the stated maximum. Apply follow-through when an earlier numerical result is used consistently, unless the current mark requires an independent conclusion.
Acceptable alternatives: Equivalent exact forms and correctly rounded decimal forms are acceptable unless the question specifies otherwise.
Do not credit by itself: A correct final value without the required supporting evidence does not earn all available marks.
Part Part (b) (3 marks)
Award one mark for each required-evidence item, up to the stated maximum. Apply follow-through when an earlier numerical result is used consistently, unless the current mark requires an independent conclusion.
Acceptable alternatives: Equivalent exact forms and correctly rounded decimal forms are acceptable unless the question specifies otherwise.
Do not credit by itself: A correct final value without the required supporting evidence does not earn all available marks.
Part Part (c) (1 mark)
Award one mark for each required-evidence item, up to the stated maximum. Apply follow-through when an earlier numerical result is used consistently, unless the current mark requires an independent conclusion.
Acceptable alternatives: Equivalent exact forms and correctly rounded decimal forms are acceptable unless the question specifies otherwise.
Do not credit by itself: A correct final value without the required supporting evidence does not earn all available marks.
Section 2 Question 14
(a) With overrightarrow(AB)=(4,-3,4) and overrightarrow(AC)=(2,0,5), t=((overrightarrow(AC) × overrightarrow(AB)) / (|overrightarrow(AB)|²))=((28) / (41)). Since 0<t<1, the closest point lies on the segment and is H=A+toverrightarrow(AB)=(((153) / (41)),-((2) / (41)),((112) / (41))).
Infer the projection parameter rather than assuming that the closest point is an endpoint.
(b) CH=√(|overrightarrow(AC)|²-(((overrightarrow(AC) × overrightarrow(AB))²) / (|overrightarrow(AB)|²)))=√(29-((784) / (41)))=√(((405) / (41)))approx3.143. The clearance is less than 4, so the requirement is not met.
Use the perpendicular component of AC and interpret the result against the stated constraint.
Detailed marking criteria
Part Part (a) (3 marks)
Award one mark for each required-evidence item, up to the stated maximum. Apply follow-through when an earlier numerical result is used consistently, unless the current mark requires an independent conclusion.
Acceptable alternatives: Equivalent exact forms and correctly rounded decimal forms are acceptable unless the question specifies otherwise.
Do not credit by itself: A correct final value without the required supporting evidence does not earn all available marks.
Part Part (b) (3 marks)
Award one mark for each required-evidence item, up to the stated maximum. Apply follow-through when an earlier numerical result is used consistently, unless the current mark requires an independent conclusion.
Acceptable alternatives: Equivalent exact forms and correctly rounded decimal forms are acceptable unless the question specifies otherwise.
Do not credit by itself: A correct final value without the required supporting evidence does not earn all available marks.
Section 2 Question 15
(a) ((dT) / (T-20))=-k,dt, so ln(T-20)=-kt+C and T=20+Ce^(-kt). From T(0)=80, C=60. Then 50=20+60e^(-10k), so e^(-10k)=frac12 and k=((ln2) / (10)). Thus T=20+60e^(-(ln2)t / 10).
Separate and integrate before applying both temperature observations.
(b) 25=20+60e^(-kt) gives e^(-kt)=frac1(12), so t=((ln12) / (k))=((10ln12) / (ln2))approx35.85 minutes.
Solve the calibrated model for time.
Detailed marking criteria
Part Part (a) (4 marks)
Award one mark for each required-evidence item, up to the stated maximum. Apply follow-through when an earlier numerical result is used consistently, unless the current mark requires an independent conclusion.
Acceptable alternatives: Equivalent exact forms and correctly rounded decimal forms are acceptable unless the question specifies otherwise.
Do not credit by itself: A correct final value without the required supporting evidence does not earn all available marks.
Part Part (b) (2 marks)
Award one mark for each required-evidence item, up to the stated maximum. Apply follow-through when an earlier numerical result is used consistently, unless the current mark requires an independent conclusion.
Acceptable alternatives: Equivalent exact forms and correctly rounded decimal forms are acceptable unless the question specifies otherwise.
Do not credit by itself: A correct final value without the required supporting evidence does not earn all available marks.
Section 2 Question 16
(a) The initial conditions give x(t)=3cos(2t). Since the angular frequency is 2, the period is T=2π / 2=π seconds.
Use the simple-harmonic-motion solution with both initial conditions, then identify the angular frequency.
(b) Solve 3cos(2t)=-1. At the first trigger, 2t=cos⁻¹(-1 / 3), so t=frac12cos⁻¹(-1 / 3)approx0.955 seconds.
Choose the first positive solution of the cosine equation.
(c) Since v(t)=-6sin(2t) and sin(2t)=√(1-(-1 / 3)²)=2sqrt2 / 3 at the first trigger, v=-4sqrt2approx-5.657 m s⁻¹. The carriage is moving in the negative x-direction.
Differentiate the position function and use the first-trigger quadrant to select the sign of the sine value.
Detailed marking criteria
Part Part (a) (2 marks)
Award one mark for each required-evidence item, up to the stated maximum. Apply follow-through when an earlier numerical result is used consistently, unless the current mark requires an independent conclusion.
Acceptable alternatives: Equivalent exact forms and correctly rounded decimal forms are acceptable unless the question specifies otherwise.
Do not credit by itself: A correct final value without the required supporting evidence does not earn all available marks.
Part Part (b) (2 marks)
Award one mark for each required-evidence item, up to the stated maximum. Apply follow-through when an earlier numerical result is used consistently, unless the current mark requires an independent conclusion.
Acceptable alternatives: Equivalent exact forms and correctly rounded decimal forms are acceptable unless the question specifies otherwise.
Do not credit by itself: A correct final value without the required supporting evidence does not earn all available marks.
Part Part (c) (3 marks)
Award one mark for each required-evidence item, up to the stated maximum. Apply follow-through when an earlier numerical result is used consistently, unless the current mark requires an independent conclusion.
Acceptable alternatives: Equivalent exact forms and correctly rounded decimal forms are acceptable unless the question specifies otherwise.
Do not credit by itself: A correct final value without the required supporting evidence does not earn all available marks.
Section 2 Question 17
(a) 2(1.96)((12) / (sqrt n))le3, so nge(((1.96(12)) / (1.5)))²=245.8624. Therefore the minimum total sample size is 246.
Convert total width to a margin of error and round the required sample size up.
(b) Adding 246-64=182 observations costs 182(8)=$1456. Calibration gives width 2(1.96)(6 / 8)=2.94 and costs $900. Calibrate, because it meets the target at the lower stated cost.
Verify that each option meets the target before comparing its cost.
(c) The observations must be independent and representative of the same stable process, and the planning standard deviations must be reasonable.
State one relevant modelling or sampling assumption.
Detailed marking criteria
Part Part (a) (3 marks)
Award one mark for each required-evidence item, up to the stated maximum. Apply follow-through when an earlier numerical result is used consistently, unless the current mark requires an independent conclusion.
Acceptable alternatives: Equivalent exact forms and correctly rounded decimal forms are acceptable unless the question specifies otherwise.
Do not credit by itself: A correct final value without the required supporting evidence does not earn all available marks.
Part Part (b) (2 marks)
Award one mark for each required-evidence item, up to the stated maximum. Apply follow-through when an earlier numerical result is used consistently, unless the current mark requires an independent conclusion.
Acceptable alternatives: Equivalent exact forms and correctly rounded decimal forms are acceptable unless the question specifies otherwise.
Do not credit by itself: A correct final value without the required supporting evidence does not earn all available marks.
Part Part (c) (1 mark)
Award one mark for each required-evidence item, up to the stated maximum. Apply follow-through when an earlier numerical result is used consistently, unless the current mark requires an independent conclusion.
Acceptable alternatives: Equivalent exact forms and correctly rounded decimal forms are acceptable unless the question specifies otherwise.
Do not credit by itself: A correct final value without the required supporting evidence does not earn all available marks.
Section 2 Question 18
(a) The resultant force is (2,4,2), so mathbf a=(1,2,1). Hence mathbf v(t)=(1+t,-1+2t,2+t) and mathbf r(t)=(t+frac12t²,1-t+t²,2t+frac12t²).
Use Newton's second law, then integrate while applying both initial vectors.
(b) Substitution gives 1+2t+2t²=5, so (t-1)(t+2)=0. For tge0, the first crossing is at t=1, at (frac32,1,frac52).
Reject the negative-time root after solving the plane-intersection condition.
(c) At t=1, mathbf v=(2,1,3), so the speed is √14 m s⁻¹. With plane normal mathbf n=(1,1,1), sinalpha=((|mathbf v × mathbf n|) / (|mathbf v||mathbf n|))=frac6(√42), giving alphaapprox67.8^circ.
Use the complementary relationship between the angle to a plane and the angle to its normal.
Detailed marking criteria
Part Part (a) (3 marks)
Award one mark for each required-evidence item, up to the stated maximum. Apply follow-through when an earlier numerical result is used consistently, unless the current mark requires an independent conclusion.
Acceptable alternatives: Equivalent exact forms and correctly rounded decimal forms are acceptable unless the question specifies otherwise.
Do not credit by itself: A correct final value without the required supporting evidence does not earn all available marks.
Part Part (b) (2 marks)
Award one mark for each required-evidence item, up to the stated maximum. Apply follow-through when an earlier numerical result is used consistently, unless the current mark requires an independent conclusion.
Acceptable alternatives: Equivalent exact forms and correctly rounded decimal forms are acceptable unless the question specifies otherwise.
Do not credit by itself: A correct final value without the required supporting evidence does not earn all available marks.
Part Part (c) (2 marks)
Award one mark for each required-evidence item, up to the stated maximum. Apply follow-through when an earlier numerical result is used consistently, unless the current mark requires an independent conclusion.
Acceptable alternatives: Equivalent exact forms and correctly rounded decimal forms are acceptable unless the question specifies otherwise.
Do not credit by itself: A correct final value without the required supporting evidence does not earn all available marks.
Diagnostic Checklist
| Topic | Questions | Marks | Marks Lost | Action |
|---|---|---|---|---|
| Statistical Inference and Normal Models | Q1, Q6, Q10-Q11, Q17 | 14 | ___ | Review combined samples, interval width and sample-size decisions. |
| Forces, Vectors and Geometry | Q2, Q4, Q9, Q14, Q18 | 16 | ___ | Review projection, constrained distance, resultant force and three-dimensional motion. |
| Numerical Methods, Complex Numbers, Differential Equations and Motion | Q3, Q5, Q7-Q8, Q12-Q13, Q15-Q16 | 30 | ___ | Review Simpson's rule, complex mappings, non-logistic differential equations and simple harmonic motion. |