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Specialist Mathematics Units 3&4 Free Online - Pack 0

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Paper 1 Technology-free Question and Response Book Showcase

QCE Specialist Mathematics Units 3&4 Free Online Pack 0 — Paper 1 Technology-free Question and Response Book Showcase

Read Paper 1 Technology-free Question and Response Book Showcase online for free, including every question, worked solution, marking note and diagnostic action. No public PDF download or checkout is provided.

QCE Year 12 Final Exam 2026 Edition - Pack 0 v1.0
Paper 1 Technology-free Question and Response Book Showcase is free to read in your browser. There is no public checkout or PDF download.

Exam-pack paper structure

This full-length showcase paper is available to read online.

Paper 1 Technology-free Question and Response Book Showcase

19 questions

60 marks

Estimated duration: Perusal time 5 minutes; working time 90 minutes

Reading: 5 minutes perusal · Writing: 90 minutes

Read Paper 1 Technology-free Question and Response Book Showcase online

Skill Align

Skill Align QCE Specialist Mathematics 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
60 marks

QCAA formula book provided; calculators, technology, notes and other resources 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. Calculators are not permitted.

Question 1

1 mark
For z=-1+sqrt3i, the modulus and principal argument are
  1. 2,((2pi) / (3))
  2. 2,((pi) / (3))
  3. 4,((2pi) / (3))
  4. 2,-((2pi) / (3))

Question 2

1 mark
The determinant of begin(pmatrix)3&2-1&4end(pmatrix) is
  1. 12
  2. 14
  3. 10
  4. -14

Question 3

1 mark
For mathbf a=(1,-2,4) and mathbf b=(3,1,-1), mathbf a × mathbf b=
  1. 1
  2. 3
  3. -3
  4. 9

Question 4

1 mark
An antiderivative of 3x²(1+x³)² is
  1. (1+x³)³+C
  2. x³(1+x³)²+C
  3. 6x(1+x³)+C
  4. (((1+x³)³) / (3))+C

Question 5

1 mark
If ((dy) / (dx))=2xy and y(0)=5, then
  1. y=5e^(x²)
  2. y=5e^(2x)
  3. y=e^(x²)+4
  4. y=5x²+5

Question 6

1 mark
The vector mathbf j × mathbf k is
  1. -mathbf i
  2. mathbf i
  3. mathbf j
  4. -mathbf k

Question 7

1 mark
A confidence interval uses z=2, sigma=4 and n=64. Its margin of error is
  1. 0.5
  2. 2
  3. 1
  4. 8

Question 8

1 mark
The polar point (6,((5pi) / (6))) has Cartesian coordinates
Graph PreviewOPxy
  1. (3,-3sqrt3)
  2. (3sqrt3,3)
  3. (-3,3sqrt3)
  4. (-3sqrt3,3)

Question 9

1 mark
The locus |z-(2-i)|=3 is a circle with
  1. centre (2,-1) and radius 3
  2. centre (-2,1) and radius 3
  3. centre (2,-1) and radius 9
  4. centre (-1,2) and radius 3

Question 10

1 mark
A normal vector to x+3y-2z=4 is
  1. (1,-3,2)
  2. (1,3,-2)
  3. (4,3,-2)
  4. (1,2,3)

Section 2

Questions 11-19 are short response. Show exact working and mathematical reasoning.

Question 11

5 marks
Let z=1+i and w=sqrt3-i.
(a) 3 marks
Express frac zw in modulus-argument form with principal argument.
(b) 2 marks
Hence find (frac zw)⁶ in Cartesian form.

Question 12

6 marks
Two lines are ell_1:mathbf r=(1,0,1)+s(1,1,0) and ell_2:mathbf r=(0,2,0)+t(0,1,1).
(a) 2 marks
Show that the lines are skew.
(b) 2 marks
Find the shortest distance between the lines.
(c) 2 marks
Find a Cartesian equation of the plane containing ell_1 and parallel to ell_2.

Question 13

5 marks
A particle moves on a straight line with acceleration a(t)=6t-4, velocity v(0)=1 and position x(0)=0.
(a) 2 marks
Find v(t) and x(t).
(b) 1 mark
Find the two positive times when the particle is at rest.
(c) 2 marks
Find the distance travelled between these two times.

Question 14

6 marks
A 95% confidence interval for a population mean is centred at 72 and has total width 6. It was formed using z=2 and known population standard deviation sigma=9.
(a) 3 marks
Find the sample size used.
(b) 2 marks
Does the interval provide evidence that the population mean differs from 70? Justify your answer.
(c) 1 mark
State the new total width if the sample size is quadrupled and all else is unchanged.

Question 15

5 marks
Consider int xln x,dx, for x>0.
(a) 3 marks
Use integration by parts to find the indefinite integral.
(b) 2 marks
Evaluate int_1^e xln x,dx.

Question 16

6 marks
A ray starts at P=(1,-1,2) with direction mathbf d=(2,1,-1) and reflects from the plane x+2y+2z=9. On reflection, the component parallel to the plane is unchanged and the component normal to the plane reverses.
(a) 2 marks
Find the point where the ray meets the plane.
(b) 3 marks
Find a direction vector for the reflected ray.
(c) 1 mark
Verify that reflection preserves the magnitude of the direction vector before rescaling.

Question 17

5 marks
A linear transformation has matrix A=begin(pmatrix)1&11&-1end(pmatrix).
(a) 1 mark
Find the image of (2,-1).
(b) 2 marks
Find A⁻¹.
(c) 2 marks
Find the point whose image is (4,2).

Question 18

6 marks
A curve satisfies ((dy) / (dx))=((x+1) / (y+2)) and passes through (0,0).
(a) 4 marks
Find y in terms of x, selecting the branch that satisfies the initial condition.
(b) 2 marks
Find the equation of the tangent to the curve at x=1.

Question 19

6 marks
The polar curve r=1+2costheta, for 0lethetale2pi, has an inner loop. The tracing interval for the inner loop is not given.
(a) 2 marks
Determine the interval over which the inner loop is traced.
(b) 4 marks
Find the exact area enclosed by the inner loop.

Queensland Certificate of Education (QCE) subjects and external assessments are administered by the Queensland Curriculum and Assessment Authority (QCAA). Skill Align is an independent publisher and is not affiliated with, authorised by, sponsored by, approved by, or endorsed by QCAA or the Queensland Government.

Copyright (c) 2026 Skill Align. Free for personal, non-commercial online viewing at https://skillalign.au. You may share the Skill Align page link. Except as permitted by law or with Skill Align's prior written permission, the pack itself must not be resold, copied, redistributed, republished, automatically extracted, or uploaded to a question bank.

Worked Solutions And Marking Guide

Section 1 Question 1

Answer: 2,((2pi) / (3))

The point lies in quadrant II and has modulus 2.

Section 1 Question 2

Answer: 14

The determinant is 3(4)-2(-1)=14.

Section 1 Question 3

Answer: -3

Compute mathbf a × mathbf b=1(3)+(-2)(1)+4(-1)=-3.

Section 1 Question 4

Answer: (((1+x³)³) / (3))+C

Let u=1+x³, so du=3x²,dx, then integrate u^2.

Section 1 Question 5

Answer: y=5e^(x²)

Separation gives ln y=x squared plus a constant.

Section 1 Question 6

Answer: mathbf i

Use the cyclic right-hand order i, j, k.

Section 1 Question 7

Answer: 1

The margin is 2(4 / √64)=1.

Section 1 Question 8

Answer: (-3sqrt3,3)

Use x=rcostheta and y=rsintheta.

Section 1 Question 9

Answer: centre (2,-1) and radius 3

The modulus gives the distance from z to the fixed point 2-i.

Section 1 Question 10

Answer: (1,3,-2)

The coordinate coefficients form a normal vector.

Section 2 Question 11

(a) z=sqrt2operatorname(cis)(fracpi4) and w=2operatorname(cis)(-fracpi6), so frac zw=frac1(sqrt2)operatorname(cis)(((5pi) / (12))).

Divide the moduli and subtract the arguments.

(b) (frac1(sqrt2))^6operatorname(cis)(((5pi) / (2)))=frac18operatorname(cis)(fracpi2)=frac i8.

Apply de Moivre's theorem and reduce the argument modulo two pi.

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.

Section 2 Question 12

(a) The directions (1,1,0) and (0,1,1) are not parallel. Equating coordinates gives s=-1, then t=-3 from y but t=1 from z, so the lines do not intersect. Hence they are skew.

Establish both non-parallel directions and the absence of an intersection.

(b) With mathbf d_1 × mathbf d_2=(1,-1,1) and mathbf q-mathbf p=(-1,2,-1), the distance is ((|(-1,2,-1) × (1,-1,1)|) / (sqrt3))=frac4(sqrt3).

Project the vector between points on the lines onto their common normal.

(c) A normal is (1,-1,1). Through (1,0,1), the plane is (x-1)-y+(z-1)=0, or x-y+z=2.

Use both direction vectors in the plane to construct its normal.

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) (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) v(t)=3t²-4t+1 and x(t)=t³-2t²+t.

Integrate twice and apply the two initial conditions.

(b) 3t²-4t+1=(3t-1)(t-1)=0, so t=frac13 and t=1.

Set velocity equal to zero.

(c) The velocity is negative between the roots. Since x(frac13)=frac4(27) and x(1)=0, the distance is frac4(27).

Use the sign of velocity so that displacement is converted to distance.

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 14

(a) The margin is 3, so 3=((2(9)) / (sqrt n)). Hence sqrt n=6 and n=36.

Use half the total width as the margin of error.

(b) The interval is (69,75). No; 70 lies inside the interval, so this interval does not rule out a population mean of 70.

Construct the endpoints before interpreting the claim.

(c) The width is halved to 3.

Interval width is inversely proportional to the square root of sample size.

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 15

(a) Take u=ln x and dv=x,dx. Then du=frac1x,dx, v=((x²) / (2)), and int xln x,dx=((x²) / (2))ln x-frac12int x,dx=((x²) / (2))ln x-((x²) / (4))+C.

State the parts and complete the remaining elementary integral.

(b) [((x²) / (2))ln x-((x²) / (4))]_1^e=((e²) / (4))-(-frac14)=((e²+1) / (4)).

Evaluate the antiderivative at both bounds.

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.

Section 2 Question 16

(a) On mathbf r=P+tmathbf d, the plane expression is 3+2t. Thus 3+2t=9, so t=3 and the point is H=(7,2,-1).

Form and solve the line-plane intersection without assuming a supplied parameter value.

(b) A plane normal is mathbf n=(1,2,2). The normal component is operatorname(proj)_(mathbf n)mathbf d=((mathbf d × mathbf n) / (mathbf n × mathbf n))mathbf n=frac29(1,2,2). Hence mathbf d_(text(ref))=mathbf d-2operatorname(proj)_(mathbf n)mathbf d=frac19(14,1,-17), so (14,1,-17) is a direction vector.

Decompose the incident direction into normal and parallel components, then reverse only the normal component.

(c) |mathbf d|²=6, while |frac19(14,1,-17)|²=((196+1+289) / (81))=6, so the magnitudes are equal.

Compare squared magnitudes to avoid unnecessary radicals.

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 17

(a) Abinom2(-1)=binom13, so the image is (1,3).

Multiply the matrix by the column vector.

(b) det A=-2, so A⁻¹=begin(pmatrix)frac12&frac12frac12&-frac12end(pmatrix).

Use the two-by-two inverse formula and the non-zero determinant.

(c) A⁻¹binom42=binom31, so the preimage is (3,1).

Apply the inverse transformation.

Detailed marking criteria

Part Part (a) (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 (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.

Section 2 Question 18

(a) (y+2),dy=(x+1),dx, so frac12(y+2)²=frac12(x+1)²+C. Using (0,0) gives C=frac32, hence (y+2)²=(x+1)²+3. Since y+2=2>0 at x=0, y=-2+√((x+1)²+3).

Separate, integrate, apply the initial condition and justify the square-root branch.

(b) At x=1, y=-2+sqrt7 and ((dy) / (dx))=frac2(sqrt7). Thus y+2-sqrt7=frac2(sqrt7)(x-1).

Use the differential equation for the gradient after finding the point.

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 19

(a) The curve is at the pole when 1+2costheta=0, so costheta=-frac12. The consecutive pole crossings are theta=((2pi) / (3)) and theta=((4pi) / (3)); the inner loop is traced between them.

Identify the consecutive zeros of r that bound the inner loop.

(b) A=frac12int_(2pi / 3)^(4pi / 3)(1+2costheta)²,dtheta=frac12[3theta+4sintheta+sin2theta]_(2pi / 3)^(4pi / 3)=pi-((3sqrt3) / (2)).

Use the interval found in part (a), expand the squared radius and evaluate exactly.

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) (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.

Diagnostic Checklist

TopicQuestionsMarksMarks LostAction
Complex Numbers and Polar Form Q1, Q8-Q9, Q11, Q19 14 ___ Review modulus-argument form, powers and polar-loop geometry.
Vectors, Matrices and Geometry Q2-Q3, Q6, Q10, Q12, Q16-Q17 21 ___ Review determinants, skew lines, matrix transformations and vector reflection.
Calculus, Differential Equations and Inference Q4-Q5, Q7, Q13-Q15, Q18 25 ___ Review motion, confidence intervals, integration by parts and separable equations.

What is included

Paper 1 Technology-free Question and Response Book Showcase questions (60 marks)

Paper 2 Technology-active Question and Response Book Showcase questions (60 marks)

Worked solutions and marking guidance shown online

Diagnostic checklist shown online

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Queensland Certificate of Education (QCE) subjects and external assessments are administered by the Queensland Curriculum and Assessment Authority (QCAA). Skill Align is an independent publisher and is not affiliated with, authorised by, sponsored by, approved by, or endorsed by QCAA or the Queensland Government.

Each exam pack is listed with a pack label so parents do not buy the same pack twice. Future packs will use the next label for that state or curriculum.

Related Online Practice and curriculum

Pack 0 is a free online resource. These links open the related subscription practice, curriculum coverage, and free public sample questions.

QCE Year 12 Specialist Mathematics practice questions Browse QCE Specialist Mathematics curriculum coverage Try QCE Specialist Mathematics public sample questions

Questions about this exam pack

What is included in Specialist Mathematics Units 3&4 Free Online - Pack 0?

Pack 0 includes 2 full-length showcase papers, worked solutions, marking guidance and diagnostic checklists, all shown online.

Is Pack 0 really free?

Yes. Pack 0 can be read online without checkout or a monthly subscription.

Can I download Pack 0 as a PDF?

No. Pack 0 is intentionally online-only and no downloadable PDF is provided.

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No. The questions are original Skill Align material. Skill Align is independent and is not affiliated with, authorised by, sponsored by, approved by, or endorsed by any state assessment authority.