Skill Align QCE Mathematical Methods 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
- 55 marks
QCAA formula book provided; QCAA-approved technology is permitted for Paper 2. No public PDF or formula-book download is supplied with Pack 0.
Section 1
Questions 1-10 are multiple choice. Select the best answer for each question. Technology may be used where appropriate.
Question 1
1 mark- t=((ln5) / (0.4))
- t=0.4ln5
- t=ln4.6
- t=((0.4) / (ln5))
Question 2
1 mark- 0.0228
- 0.0668
- 0.1587
- 0.9332
Question 3
1 mark- ((1) / (x²+1))
- 2xln(x²+1)
- ((2x) / (x²+1))
- ((x²+1) / (2x))
Question 4
1 mark- 4
- 12
- 16
- 8
Question 5
1 mark- (0.8)⁸+8(0.2)(0.8)⁷
- 8(0.2)(0.8)⁷
- 1-(0.8)⁸
- (0.2)⁸+(0.8)⁸
Question 6
1 mark- 0.035
- 0.069
- 0.055
- 0.096
Question 7
1 mark- 1
- e
- e-1
- ln e
Question 8
1 mark- 0.67
- 4
- 6
- 1.5
Question 9
1 mark- ((ln2) / (ln1.06))
- ((ln1.06) / (ln2))
- 2ln1.06
- 120ln2
Question 10
1 mark- 3
- 5
- 4
- 6
Section 2
Questions 11-19 are short response. Show working, modelling decisions and contextual interpretation.
Question 11
6 marksQuestion 12
7 marksQuestion 13
4 marksQuestion 14
6 marksQuestion 15
3 marksQuestion 16
4 marksQuestion 17
4 marksQuestion 18
5 marksQuestion 19
6 marksWorked Solutions And Marking Guide
Section 1 Question 1
Answer: t=((ln5) / (0.4))
Take natural logarithms and divide by 0.4.
Mark allocation
- Equivalent mathematically valid methods should be accepted. Reasonable rounding tolerances apply. Follow-through marks may be awarded where an earlier error is used consistently and the subsequent reasoning remains mathematically valid. Contextual conclusions must be consistent with the student's calculated result.
Section 1 Question 2
Answer: 0.0668
The z-score is -1.5, so the lower-tail probability is about 0.0668.
Section 1 Question 3
Answer: ((2x) / (x²+1))
Use the chain rule for ln u.
Section 1 Question 4
Answer: 8
The period is 2π / (π / 4)=8.
Section 1 Question 5
Answer: (0.8)⁸+8(0.2)(0.8)⁷
Add P(X=0) and P(X=1).
Section 1 Question 6
Answer: 0.069
Compute 1.96sqrt(0.55(0.45) / 200).
Section 1 Question 7
Answer: e-1
An antiderivative is e^x.
Section 1 Question 8
Answer: 1.5
z=(74-68) / 4=1.5.
Section 1 Question 9
Answer: ((ln2) / (ln1.06))
Set the growth factor equal to 2 and take logarithms.
Section 1 Question 10
Answer: 5
The mean of a uniform distribution is the midpoint.
Section 2 Question 11
(a) f(x)=-3cos x+sin(2x)+4.
Integrating gives f(x)=-3cos x+sin(2x)+C. Since f(0)=-3+C=1, C=4.
(b) y-4=x-fracpi2.
At x=fracpi2, f(x)=4 and f'(x)=3sin(fracpi2)+2cospi=1. Hence the tangent is y-4=1(x-fracpi2).
Detailed marking criteria
Part (a) (4 marks)
Award one mark for each listed achievement: integrates 3sin x to -3cos x; integrates 2cos(2x) to sin(2x); uses f(0)=1 to obtain C=4; states f(x)=-3cos x+sin(2x)+4. 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. Apply follow-through when an earlier numerical result is used consistently, unless the current mark requires an independent conclusion.
Do not credit by itself: A correct final value without the required supporting evidence does not earn all available marks.
Part (b) (2 marks)
Award one mark for each listed achievement: obtains the point (fracpi2,4) and gradient 1, allowing follow-through from part (a); states y-4=x-fracpi2 or an equivalent equation. 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. Apply follow-through when an earlier numerical result is used consistently, unless the current mark requires an independent conclusion.
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) h=((500) / (r²)), so C(r)=0.04π r²+0.02(2π rh)=0.04π r²+((20π) / (r)).
From π r^2h=500π, h=500 / r^2. The base costs 0.04π r² dollars and the side costs 0.02(2π rh)=20π / r dollars.
(b) r=√3(250)approx6.30 cm and happrox12.60 cm; this gives a minimum.
C'(r)=0.08π r-20π / r^2. Solving C'(r)=0 gives r³=250, so rapprox6.30. Then h=500 / r^2approx12.60. Since C''(r)=0.08π+40π / r³>0 for r>0, the stationary value is a minimum.
(c) Use r=6 cm: C(6)approx$15.00, compared with C(7)approx$15.13.
Substitution gives C(6)=0.04π(6)²+20π / 6approx14.996 dollars and C(7)approx15.134 dollars, so radius 6 cm is cheaper.
Detailed marking criteria
Part (a) (2 marks)
Award one mark for each listed achievement: uses the volume constraint to obtain h=500 / r²; forms C(r)=0.04π r²+20π / r with r>0. 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. Apply follow-through when an earlier numerical result is used consistently, unless the current mark requires an independent conclusion.
Do not credit by itself: A correct final value without the required supporting evidence does not earn all available marks.
Part (b) (3 marks)
Award one mark for each listed achievement: differentiates and solves C'(r)=0 to obtain r=√3(250); obtains rapprox6.30 cm and happrox12.60 cm; justifies a minimum using C''(r)>0, a valid sign test or end behaviour. 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. Apply follow-through when an earlier numerical result is used consistently, unless the current mark requires an independent conclusion.
Do not credit by itself: A correct final value without the required supporting evidence does not earn all available marks.
Part (c) (2 marks)
Award one mark for each listed achievement: obtains costs of approximately $15.00 and $15.13, allowing follow-through from part (a); recommends the 6 cm radius because it has the lower cost. 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. Apply follow-through when an earlier numerical result is used consistently, unless the current mark requires an independent conclusion.
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) hat p=0.40.
The pilot estimate is 32 / 80=0.40.
(b) n=577.
Require 1.96sqrt(0.4(0.6) / n)le0.04. Squaring and rearranging gives nge576.24, so the minimum whole-number sample size is 577.
(c) The sample should be randomly selected and observations should be independent.
A random, representative selection with independent observations is required; a larger biased sample would not repair selection bias.
Detailed marking criteria
Part (a) (1 mark)
Award one mark for each listed achievement: obtains hat p=0.40. 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. Apply follow-through when an earlier numerical result is used consistently, unless the current mark requires an independent conclusion.
Do not credit by itself: A correct final value without the required supporting evidence does not earn all available marks.
Part (b) (2 marks)
Award one mark for each listed achievement: forms 1.96sqrt(0.4(0.6) / n)le0.04; obtains 577 after rounding the lower bound up. 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. Apply follow-through when an earlier numerical result is used consistently, unless the current mark requires an independent conclusion.
Do not credit by itself: A correct final value without the required supporting evidence does not earn all available marks.
Part (c) (1 mark)
Award one mark for each listed achievement: states a valid random-selection, representativeness or independence condition. 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. Apply follow-through when an earlier numerical result is used consistently, unless the current mark requires an independent conclusion.
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) x=e^(0.8)-1approx1.226.
Exponentiating ln(x+1)=0.8 gives x+1=e^(0.8), hence xapprox1.226.
(b) y-ln2=frac12(x-1).
Since f(1)=ln2 and f'(x)=1 / (x+1), the gradient at 1 is 1 / 2.
(c) 0.057 square units.
The logarithmic curve is concave down, so its tangent lies above it. Numerically evaluate int_0¹[ln2+(x-1) / 2-ln(x+1)],dxapprox0.0568528, giving 0.057 square units to three decimal places.
Detailed marking criteria
Part (a) (1 mark)
Award one mark for each listed achievement: obtains x=e^(0.8)-1approx1.226. 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. Apply follow-through when an earlier numerical result is used consistently, unless the current mark requires an independent conclusion.
Do not credit by itself: A correct final value without the required supporting evidence does not earn all available marks.
Part (b) (2 marks)
Award one mark for each listed achievement: obtains the point (1,ln2) and gradient 1 / 2; states y-ln2=frac12(x-1) or an equivalent equation. 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. Apply follow-through when an earlier numerical result is used consistently, unless the current mark requires an independent conclusion.
Do not credit by itself: A correct final value without the required supporting evidence does not earn all available marks.
Part (c) (3 marks)
Award one mark for each listed achievement: identifies the tangent as the upper function on the interval; forms int_0¹[ln2+(x-1) / 2-ln(x+1)],dx, allowing follow-through from part (b); uses permitted technology to obtain 0.057 square units to three decimal places. 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. Apply follow-through when an earlier numerical result is used consistently, unless the current mark requires an independent conclusion.
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) P(X<100)approx0.0912.
Standardising gives z=(100-120) / 15=-1.333ldots, so the lower-tail probability is approximately 0.0912.
(b) Approximately 204 batteries.
By symmetry, P(100<X<140)approx0.8176. The expected count is 250(0.8176)=204.4, so approximately 204 batteries.
Detailed marking criteria
Part (a) (1 mark)
Award one mark for each listed achievement: obtains a lower-tail probability of approximately 0.0912. 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. Apply follow-through when an earlier numerical result is used consistently, unless the current mark requires an independent conclusion.
Do not credit by itself: A correct final value without the required supporting evidence does not earn all available marks.
Part (b) (2 marks)
Award one mark for each listed achievement: obtains the between-limits probability of approximately 0.8176; uses the probability to estimate approximately 204 batteries. 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. Apply follow-through when an earlier numerical result is used consistently, unless the current mark requires an independent conclusion.
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) ((dA) / (dt))=0.6π r.
By the chain rule, dA / dt=(dA / dr)(dr / dt)=2π r(0.3)=0.6π r.
(b) 2.4π m² per year.
Substitute r=4 into dA / dt=0.6π r.
(c) After 10 years.
Solve 1+0.3t=4, giving t=10.
Detailed marking criteria
Part (a) (2 marks)
Award one mark for each listed achievement: uses dA / dt=(dA / dr)(dr / dt); obtains dA / dt=0.6π r. 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. Apply follow-through when an earlier numerical result is used consistently, unless the current mark requires an independent conclusion.
Do not credit by itself: A correct final value without the required supporting evidence does not earn all available marks.
Part (b) (1 mark)
Award one mark for each listed achievement: obtains 2.4π m² per year, allowing follow-through from part (a). 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. Apply follow-through when an earlier numerical result is used consistently, unless the current mark requires an independent conclusion.
Do not credit by itself: A correct final value without the required supporting evidence does not earn all available marks.
Part (c) (1 mark)
Award one mark for each listed achievement: obtains 10 years. 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. Apply follow-through when an earlier numerical result is used consistently, unless the current mark requires an independent conclusion.
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) P(replace)approx0.2121.
For Xsim B(20,0.08), P(Xge3)=1-[P(X=0)+P(X=1)+P(X=2)]approx0.2121.
(b) Approximately 0.9078.
If qapprox0.2120538 is the unrounded result from part (a), then P(at least one)=1-(1-q)¹⁰approx0.9078.
Detailed marking criteria
Part (a) (2 marks)
Award one mark for each listed achievement: models the failure count as B(20,0.08); obtains 1-[P(0)+P(1)+P(2)]approx0.2121. 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. Apply follow-through when an earlier numerical result is used consistently, unless the current mark requires an independent conclusion.
Do not credit by itself: A correct final value without the required supporting evidence does not earn all available marks.
Part (b) (2 marks)
Award one mark for each listed achievement: uses the complement 1-(1-q)¹⁰, allowing follow-through from part (a); obtains approximately 0.9078 using an unrounded batch probability. 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. Apply follow-through when an earlier numerical result is used consistently, unless the current mark requires an independent conclusion.
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) Approximately (0.3716,0.4684).
For Survey A, hat p=168 / 400=0.42. The margin is 1.96sqrt(0.42(0.58) / 400)approx0.0484, giving approximately (0.3716,0.4684).
(b) Approximately (0.4455,0.4945).
For Survey B, hat p=752 / 1600=0.47 and the calculated margin is approximately 0.0245, giving (0.4455,0.4945).
(c) Use Survey A. Survey B's voluntary-response design can create selection bias, so its narrower calculated interval does not establish greater trustworthiness for the population.
Survey A uses random selection, which supports inference to the population. Survey B's larger sample reduces the formula's nominal random sampling error, but it does not remove self-selection bias; therefore the usual confidence-interval interpretation is not justified for Survey B.
Detailed marking criteria
Part (a) (2 marks)
Award one mark for each listed achievement: obtains hat p_A=0.42 and a margin of approximately 0.0484; states the interval (0.3716,0.4684), allowing equivalent rounding. 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. Apply follow-through when an earlier numerical result is used consistently, unless the current mark requires an independent conclusion.
Do not credit by itself: A correct final value without the required supporting evidence does not earn all available marks.
Part (b) (1 mark)
Award one mark for each listed achievement: obtains the nominal interval (0.4455,0.4945), allowing equivalent rounding. 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. Apply follow-through when an earlier numerical result is used consistently, unless the current mark requires an independent conclusion.
Do not credit by itself: A correct final value without the required supporting evidence does not earn all available marks.
Part (c) (2 marks)
Award one mark for each listed achievement: explains that a larger voluntary sample does not remove self-selection bias; recommends Survey A because random selection supports population inference. 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. Apply follow-through when an earlier numerical result is used consistently, unless the current mark requires an independent conclusion.
Do not credit by itself: A correct final value without the required supporting evidence does not earn all available marks.
Section 2 Question 19
(a) P(+)=0.0858.
The false-positive probability is 1-0.94=0.06. Hence P(+)=0.03(0.92)+0.97(0.06)=0.0276+0.0582=0.0858.
(b) Approximately 0.3217.
By conditional probability, P(Cmid+)=0.03(0.92) / 0.0858approx0.3217.
(c) P(+,+)=0.028884.
Conditional independence gives P(+,+)=0.03(0.92)²+0.97(0.06)²=0.028884.
(d) Yes. P(Cmid+,+)=((0.03(0.92)²) / (0.028884))approx0.8791>0.80.
The probability of both positives coming from a person with the condition is 0.03(0.92)²=0.025392. Dividing by P(+,+)=0.028884 gives approximately 0.8791, which exceeds the referral threshold.
Detailed marking criteria
Part (a) (2 marks)
Award one mark for each listed achievement: identifies the false-positive probability as 0.06; obtains P(+)=0.03(0.92)+0.97(0.06)=0.0858. 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. Apply follow-through when an earlier numerical result is used consistently, unless the current mark requires an independent conclusion.
Do not credit by itself: A correct final value without the required supporting evidence does not earn all available marks.
Part (b) (1 mark)
Award one mark for each listed achievement: obtains P(Cmid+)approx0.3217, allowing follow-through from part (a). 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. Apply follow-through when an earlier numerical result is used consistently, unless the current mark requires an independent conclusion.
Do not credit by itself: A correct final value without the required supporting evidence does not earn all available marks.
Part (c) (1 mark)
Award one mark for each listed achievement: obtains P(+,+)=0.03(0.92)²+0.97(0.06)²=0.028884. 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. Apply follow-through when an earlier numerical result is used consistently, unless the current mark requires an independent conclusion.
Do not credit by itself: A correct final value without the required supporting evidence does not earn all available marks.
Part (d) (2 marks)
Award one mark for each listed achievement: obtains P(Cmid+,+)approx0.8791, allowing follow-through from part (c); concludes that the probability exceeds 0.80 and the referral rule is met. 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. Apply follow-through when an earlier numerical result is used consistently, unless the current mark requires an independent conclusion.
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 |
|---|---|---|---|---|
| Calculus, Functions and Modelling | Q1, Q3-Q4, Q7, Q9, Q11-Q12, Q14, Q16 | 28 | ___ | Review derivatives, antiderivatives, logarithmic functions, optimisation, related rates and modelling. |
| Probability Distributions | Q2, Q5, Q8, Q10, Q15, Q17, Q19 | 17 | ___ | Review binomial, normal, uniform and conditional probability models, including repeated-test reasoning. |
| Statistical Inference | Q6, Q13, Q18 | 10 | ___ | Review confidence intervals for proportions, sample-size planning and the effect of sampling design. |