Skill Align QCE Mathematical Methods Paper 1 - Free Online Pack 0
Full-length Units 3&4 external-assessment-style showcase paper
- Paper
- Paper 1 Technology-free Question and Response Book Showcase
- Reading
- 5 minutes perusal
- Writing
- 90 minutes
- Assessment
- 55 marks
QCAA formula book provided; calculators and other technology are not permitted for Paper 1. 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. The QCAA Mathematical Methods formula book is provided. Calculators and other technology are not permitted.
Question 1
1 mark- e^x(x+1)
- xe^x
- e^x(x-1)
- x^2e^x
Question 2
1 mark- xgefrac12
- x>frac12
- x<frac12
- xnefrac12
Question 3
1 mark- 6
- 12
- 9
- 18
Question 4
1 mark- ((π) / (6)),((5π) / (6))
- ((5π) / (6)),((7π) / (6))
- ((4π) / (3)),((5π) / (3))
- ((7π) / (6)),((11π) / (6))
Question 5
1 mark- shifting 3 left, reflecting in the x-axis, stretching vertically by 2, then shifting 1 up
- shifting 3 right, reflecting in the y-axis, stretching horizontally by 2, then shifting 1 up
- shifting 3 left, stretching vertically by 2, then shifting 1 down
- shifting 1 left, reflecting in the x-axis, then shifting 3 up
Question 6
1 mark- 2.1
- 4.2
- 6
- 14
Question 7
1 mark- y-e=x-1
- y=ex-1
- y-1=((x-e) / (e))
- y-1=e(x-e)
Question 8
1 mark- 0.55
- 0.60
- 1.10
- 0.30
Question 9
1 mark- a local maximum
- a local minimum
- a stationary inflection point
- not a stationary point
Question 10
1 mark- frac18
- frac38
- frac14
- frac12
Section 2
Questions 11-19 are short response. Show exact working and relevant mathematical reasoning.
Question 11
5 marksQuestion 12
4 marksQuestion 13
4 marksQuestion 14
4 marksQuestion 15
6 marksQuestion 16
5 marksQuestion 17
6 marksQuestion 18
6 marksQuestion 19
5 marksWorked Solutions And Marking Guide
Section 1 Question 1
Answer: e^x(x+1)
Apply the product rule: f'(x)=e^x+xe^x.
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: x>frac12
The logarithm requires 2x-1>0.
Section 1 Question 3
Answer: 9
The region is a triangle with base 3 and height 6, so its area is frac12(3)(6)=9.
Section 1 Question 4
Answer: ((7π) / (6)),((11π) / (6))
Sine is negative in quadrants III and IV with reference angle π / 6.
Section 1 Question 5
Answer: shifting 3 left, reflecting in the x-axis, stretching vertically by 2, then shifting 1 up
Read horizontal changes inside the function and vertical changes outside it.
Section 1 Question 6
Answer: 4.2
The variance is np(1-p)=20(0.3)(0.7)=4.2.
Section 1 Question 7
Answer: y-1=((x-e) / (e))
The point is (e,1) and the gradient is 1 / e.
Section 1 Question 8
Answer: 0.30
Use P(Acap B)=P(Amid B)P(B)=0.6(0.5)=0.30.
Section 1 Question 9
Answer: a local maximum
Since f''(x)=2x, f''(-2)=-4<0.
Section 1 Question 10
Answer: frac38
There are three arrangements with two heads among eight equally likely outcomes.
Section 2 Question 11
(a) A(t)=π(2+frac t2)^2.
Substitute r(t)=2+t / 2 into A=π r^2.
(b) ((dA) / (dt))=4π m² per minute.
By the chain rule, dA / dt=2π r,dr / dt=2π r(1 / 2)=π r. At t=4, r=4, so dA / dt=4π.
(c) At t=6 minutes, and the area is 25π m^2.
Since dA / dt=π r, the required rate occurs when r=5. Solving 2+t / 2=5 gives t=6, and then A=π(5)²=25π.
Detailed marking criteria
Part (a) (1 mark)
Award one mark for each listed achievement: obtains A(t)=π(2+frac t2)^2. 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 dA / dt=2π r,dr / dt=π r; obtains 4π m² per minute at t=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 (c) (2 marks)
Award one mark for each listed achievement: solves π r=5π to obtain t=6, allowing follow-through from part (b); obtains area 25π m^2. 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) hat p=frac12.
The sample proportion is 32 / 64=1 / 2.
(b) frac1(16).
Substitution gives √((((1 / 2)(1 / 2)) / (64)))=√(1 / 256)=1 / 16.
(c) [frac38,frac58].
The margin is 2(1 / 16)=1 / 8, so 1 / 2pm1 / 8 gives [3 / 8,5 / 8].
(d) The population proportion of households supporting the proposal is estimated to lie between 3 / 8 and 5 / 8.
The interval estimates the unknown population proportion, not the observed sample proportion.
Detailed marking criteria
Part (a) (1 mark)
Award one mark for each listed achievement: obtains hat p=1 / 2. 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 exact standard error 1 / 16. 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 [3 / 8,5 / 8]. 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) (1 mark)
Award one mark for each listed achievement: interprets the interval as an estimate of the population support proportion. 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) x<3.
The logarithm requires 3-x>0, hence x<3.
(b) (2,0).
Set ln(3-x)=0. Then 3-x=e⁰=1, so x=2.
(c) f⁻¹(x)=3-e^x, with domain xinmathbb R.
From y=ln(3-x), exponentiation gives e^y=3-x, so x=3-e^y. Interchanging x and y gives f⁻¹(x)=3-e^x. The range of f is all real numbers, so the inverse has domain mathbb R.
Detailed marking criteria
Part (a) (1 mark)
Award one mark for each listed achievement: states the domain x<3. 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 intercept (2,0). 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 (c) (2 marks)
Award one mark for each listed achievement: rearranges y=ln(3-x) to obtain f⁻¹(x)=3-e^x; states the inverse domain xinmathbb 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.
Section 2 Question 14
(a) P(X=2)=binom52(frac12)⁵=frac5(16).
There are binom52=10 arrangements with two heads, each having probability (1 / 2)^5. Thus P(X=2)=10 / 32=5 / 16.
(b) P(Xge3)=frac12.
For five fair tosses, symmetry gives P(Xge3)=P(Xle2)=1 / 2. Equivalently, sum the probabilities for 3, 4 and 5 heads.
(c) E(X)=frac52.
For Xsim B(5,1 / 2), E(X)=np=5 / 2.
Detailed marking criteria
Part (a) (2 marks)
Award one mark for each listed achievement: forms binom52(1 / 2)⁵; obtains 5 / 16. 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(Xge3)=1 / 2 using symmetry or an exact sum. 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 E(X)=5 / 2. 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) a=frac12 and b=frac52.
Continuity at 1 requires a+b=3. Equal one-sided derivatives require 2a=1, so a=1 / 2. Substitution into the continuity equation gives b=5 / 2.
(b) y-3=x-1, or y=x+2.
The point is (1,3) and differentiability gives gradient 1, so y-3=x-1.
(c) e+((11) / (3)).
Split the integral at 1. Using the values from part (a), int_0¹(frac12x²+frac52),dx=8 / 3. Also int_1²(e^(x-1)+2),dx=e+1. Their sum is e+11 / 3.
Detailed marking criteria
Part (a) (3 marks)
Award one mark for each listed achievement: uses continuity to form a+b=3; uses equal one-sided derivatives to form 2a=1; obtains a=1 / 2 and b=5 / 2. 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: states y=x+2 or an equivalent tangent equation, 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) (2 marks)
Award one mark for each listed achievement: splits the integral at x=1 and uses the appropriate piece on each interval; obtains e+11 / 3, 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.
Section 2 Question 16
(a) Plan A delivers ((80) / (3)) L and Plan B delivers 28 L.
Total volume is the integral of delivery rate. Thus int_0⁴(6t-t²),dt=[3t²-t³ / 3]_0⁴=80 / 3, while int_0⁴(7t / 2),dt=[7t² / 4]_0⁴=28.
(b) Plan A has maximum rate 9 L / min; Plan B has maximum rate 14 L / min.
For Plan A, A'(t)=6-2t=0 at t=3, and comparison with the endpoints gives a maximum of A(3)=9. Plan B is increasing, so its maximum is B(4)=14.
(c) Plan A is suitable: it delivers 80 / 3 L, which exceeds 26 L, and its maximum rate is 9 L / min, which is below 10 L / min. Plan B exceeds the rate limit.
Use both requirements. Plan A meets the total-volume and maximum-rate constraints; Plan B meets the volume requirement but fails the maximum-rate requirement.
Detailed marking criteria
Part (a) (2 marks)
Award one mark for each listed achievement: recognises total volume as the integral of each rate over the interval; obtains totals 80 / 3 L and 28 L. 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 Plan A maximum rate 9 L / min using a stationary-point and endpoint check; obtains the Plan B maximum rate 14 L / min. 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: recommends Plan A and justifies the decision against both stated constraints. 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) The period is π and the range is [-3,1].
The coefficient 2 inside the sine gives period 2π / 2=π. Since -1lesin(2x)le1, -3le g(x)le1.
(b) x=fracpi(12),((5π) / (12)),((13π) / (12)),((17π) / (12)).
The equation becomes sin(2x)=1 / 2. For 0le2xle4π, 2x=π / 6,5π / 6,13π / 6,17π / 6. Dividing by 2 gives the four solutions.
(c) x=fracpi4.
The maximum occurs when sin(2x)=1. The first solution is 2x=π / 2, hence x=π / 4.
Detailed marking criteria
Part (a) (2 marks)
Award one mark for each listed achievement: states period π; states range [-3,1]. 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: reduces the equation to sin(2x)=1 / 2; identifies all four valid arguments over two cycles; obtains x=π / 12,5π / 12,13π / 12,17π / 12. 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 the first maximum at x=π / 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.
Section 2 Question 18
(a) k=frac18.
A density integrates to 1, so int_0^4kx,dx=8k=1, giving k=1 / 8.
(b) The median is 2sqrt2 hours.
For median m, P(Xle m)=1 / 2. Thus int_0^m x / 8,dx=m² / 16=1 / 2, so m²=8 and m=2sqrt2 within the support.
(c) Plan B meets both requirements. Plan A pays on 7 / 16 of deliveries and has expected cost $21 / 4, while Plan B pays on 3 / 4 of deliveries and has expected cost $3.
Using F(x)=x² / 16, Plan A pays with probability 1-F(3)=7 / 16, so its expected cost is 12(7 / 16)=21 / 4 dollars; it fails both requirements. Plan B pays with probability 1-F(2)=3 / 4, so its expected cost is 4(3 / 4)=3 dollars; it meets both requirements.
Detailed marking criteria
Part (a) (1 mark)
Award one mark for each listed achievement: normalises the density to obtain k=1 / 8. 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 m² / 16=1 / 2; obtains the valid median 2sqrt2 hours. 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: obtains Plan A probability 7 / 16 and expected cost $21 / 4; obtains Plan B probability 3 / 4 and expected cost $3; recommends Plan B because it satisfies both requirements and rejects Plan 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.
Section 2 Question 19
(a) The point of tangency is (1,e) and k=e.
Let the tangency occur at x=a. Equal gradients give k=e^a, while equal y-values give e^a=ka. Substitution gives e^a=e^a a, so a=1. Hence the point is (1,e) and k=e.
(b) frac e2-1 square units.
On [0,1], the convex exponential curve lies above its tangent line. The area is int_0¹(e^x-ex),dx=[e^x-ex² / 2]_0¹=e / 2-1.
Detailed marking criteria
Part (a) (3 marks)
Award one mark for each listed achievement: uses equal gradients to obtain k=e^a; uses equal y-values to obtain e^a=ka and hence a=1; states the point (1,e) and k=e. 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 int_0¹(e^x-ex),dx, allowing follow-through from part (a); obtains e / 2-1 square units. 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 |
|---|---|---|---|---|
| Functions, Graphs and Algebra | Q2, Q5, Q13, Q19 | 11 | ___ | Review domains, inverse functions, transformations and connections between logarithmic curves and tangents. |
| Calculus and Applications | Q1, Q3, Q7, Q9, Q11, Q15-Q16 | 20 | ___ | Review differentiation, accumulation from rates, related rates, piecewise smoothness and exact integration. |
| Trigonometry, Probability and Inference | Q4, Q6, Q8, Q10, Q12, Q14, Q17-Q18 | 24 | ___ | Review exact trigonometry, binomial and continuous distributions, independence and confidence intervals. |