Free NSW Maths Practice Test

Free Year 9 Maths Test NSW – Full-Year Practice Exam & Answers

Take this free Year 9 maths practice test covering algebra, inequalities, the number plane, gradient, Pythagoras, trigonometry, indices, financial mathematics, probability, statistics and measurement.

This is a broad full-year practice paper for students who have completed most of Year 9 mathematics. Use pen and paper for your working, then enter your final answers below.

Questions 29

Total marks 80

Suggested time 90 minutes

Calculator Permitted

The Year 9 practice exam starts immediately below. Attempt the questions on paper where necessary and enter only your final answers on the webpage. Answers and worked solutions are shown after you submit.

Section A – Multiple Choice

Questions 1–15 · 15 marks · Choose the best answer.

1. Solve 5x − 7 = 18.

1 mark

A. x = 3 B. x = 5 C. x = 7 D. x = 11

Answer & solution

5x − 7 = 18 → 5x = 25 → x = 5.

2. If v = u + at, which expression gives a?

1 mark

A. a = (v − u)/t B. a = (u − v)/t C. a = v − u/t D. a = vt − u

Answer & solution

v − u = at, therefore a = (v − u)/t.

3. Solve the inequality −3x + 4 > 16.

1 mark

A. x > 4 B. x < 4 C. x > −4 D. x < −4

Answer & worked solution

−3x + 4 > 16
−3x > 12
Dividing both sides by −3 reverses the inequality sign:
x < −4.

4. A right-angled triangle has shorter sides of 6 cm and 8 cm. What is the length of its hypotenuse?

1 mark

A. 10 cm B. 12 cm C. 14 cm D. 15 cm

Answer & solution

c² = 6² + 8² = 36 + 64 = 100. Therefore c = 10 cm.

5. In a right-angled triangle, relative to angle θ, the opposite side is 7 and the adjacent side is 9. Which ratio should be used to find θ?

1 mark

A. sin θ = 7/9 B. cos θ = 7/9 C. tan θ = 7/9 D. tan θ = 9/7

Answer & explanation

tan θ = opposite ÷ adjacent, therefore tan θ = 7/9.

6. A rectangular prism is 5 cm × 4 cm × 3 cm. What is its total surface area?

1 mark

A. 47 cm² B. 60 cm² C. 74 cm² D. 94 cm²

Answer & solution

TSA = 2(lw + lh + wh) = 2(20 + 15 + 12) = 2(47) = 94 cm².

7. Simplify x³ × x⁵.

1 mark

A. x⁸ B. x¹⁵ C. 2x⁸ D. x²

Answer & explanation

When multiplying powers with the same base, add the indices: x³ × x⁵ = x⁸.

8. Write 0.00072 in scientific notation.

1 mark

A. 7.2 × 10⁻⁴ B. 7.2 × 10⁻³ C. 72 × 10⁻⁴ D. 0.72 × 10⁻³

Answer & explanation

Move the decimal four places to the right: 7.2 × 10⁻⁴.

9. Find the simple interest on $800 at 5% p.a. for 3 years.

1 mark

A. $40 B. $80 C. $120 D. $920

Answer & solution

I = Prt = 800 × 0.05 × 3 = $120.

10. An investment grows from $1000 to $1102.50 in 2 years at 5% p.a. compounded annually. How much interest was earned?

1 mark

A. $50.00 B. $100.00 C. $102.50 D. $1102.50

Answer & explanation

Interest = final amount − principal = 1102.50 − 1000 = $102.50.

11. What is the gradient of the line through (1, 3) and (5, 11)?

1 mark

A. 1 B. 2 C. 3 D. 4

Answer & solution

m = (11 − 3)/(5 − 1) = 8/4 = 2.

12. The straight line below passes through A(0, 1) and B(2, 5). What is the gradient of the line?

1 mark

A. 1 B. 2 C. 3 D. 4

Answer & worked solution

Gradient = rise ÷ run.
Using A(0,1) and B(2,5):
m = (5 − 1)/(2 − 0)
= 4/2
= 2.

13. A bag contains 4 red, 3 blue and 3 green counters. Find P(blue).

1 mark

A. 3/7 B. 3/10 C. 1/3 D. 7/10

Answer & explanation

There are 10 counters in total and 3 are blue. Therefore P(blue) = 3/10.

14. Find the median of 4, 7, 8, 10, 15.

1 mark

A. 7 B. 8 C. 8.8 D. 10

Answer

The middle value is 8.

15. A square-based pyramid has a base side length of 6 cm and a slant height of 5 cm. What is its total surface area?

1 mark

A. 66 cm² B. 81 cm² C. 96 cm² D. 120 cm²

Answer & worked solution

Base area:
6 × 6 = 36 cm²

Area of one triangular face:
½ × 6 × 5 = 15 cm²

There are four triangular faces:
4 × 15 = 60 cm²

Total surface area:
36 + 60 = 96 cm².

Section B – Short Answer

Questions 16–25 · 45 marks · Use paper for your working and enter your final answers below.

16. Solve each equation.

4 marks

a) 3x + 11 = 35

x =

b) 5(2x − 3) = 4x + 21

x =

Answers & worked solutions

a) 3x + 11 = 35 → 3x = 24 → x = 8.

b) 10x − 15 = 4x + 21 → 6x = 36 → x = 6.

17. The formula for the perimeter of a rectangle is P = 2l + 2w.

4 marks

a) Rearrange the formula to make w the subject.

Formula for w

b) Find w when P = 54 cm and l = 17 cm.

w (cm)

Answers & worked solutions

a) P − 2l = 2w, therefore w = (P − 2l)/2.

b) w = (54 − 34)/2 = 10 cm.

18. A right-angled triangle has a horizontal side of 12 m and a vertical side of 5 m.

5 marks

a) Find the length of the hypotenuse.

Hypotenuse (m)

b) Find the angle between the 12 m side and the hypotenuse.

Angle θ (degrees)

Answers & worked solutions

a) c² = 12² + 5² = 169, therefore c = 13 m.

b) tan θ = 5/12. θ = tan⁻¹(5/12) ≈ 22.62°.

19. Triangle A has side lengths 6 cm, 8 cm and 10 cm. Triangle B has side lengths 15 cm, 20 cm and 25 cm.

4 marks

a) Explain why the triangles are similar.

Write your explanation on paper.

b) State the scale factor from Triangle A to Triangle B.

Scale factor

c) If an angle in Triangle A is 53°, what is the corresponding angle in Triangle B?

Angle (degrees)

Answers & model explanation

a) Corresponding sides are proportional: 15/6 = 20/8 = 25/10 = 2.5. Therefore the triangles are similar by SSS similarity.

b) Scale factor = 2.5.

c) Corresponding angles in similar triangles are equal, therefore 53°.

20. A cylindrical water tank has radius 1.2 m and height 2.5 m. Use π = 3.1416.

5 marks

a) Calculate its volume in cubic metres.

Volume (m³)

b) Convert your answer to litres. (1 m³ = 1000 L)

Volume (L)

Answers & worked solutions

a) V = πr²h = 3.1416 × 1.2² × 2.5 = 11.30976 m³ ≈ 11.31 m³.

b) 11.30976 × 1000 = 11,309.76 L.

21. Simplify or evaluate.

5 marks

a) a⁷ ÷ a³

b) (m²)⁴

c) 2⁻³

d) (4.8 × 10⁵) ÷ (1.2 × 10²)

Answers & worked solutions

a) a⁴

b) m⁸

c) 2⁻³ = 1/2³ = 1/8

d) (4.8/1.2) × 10³ = 4 × 10³ = 4000.

22. Sofia invests $2400 at 4.5% p.a.

5 marks

a) Find the simple interest earned after 3 years.

Interest ($)

b) Find the total amount after 3 years if the interest is compounded annually instead.

Total amount ($)

Answers & worked solutions

a) I = 2400 × 0.045 × 3 = $324.

b) A = 2400(1.045)³ ≈ $2738.80.

23. A straight line passes through A(−2, 1) and B(4, 13).

5 marks

a) Calculate the gradient.

Gradient

b) Find the equation of the line in the form y = mx + b.

Equation

c) Find y when x = 10.

y =

Answers & worked solutions

a) m = (13 − 1)/(4 − (−2)) = 12/6 = 2.

b) y = 2x + b. Using (−2,1), 1 = −4 + b, therefore b = 5. y = 2x + 5.

c) y = 2(10) + 5 = 25.

24. A spinner has 8 equal sectors: 3 red, 2 blue, 2 green and 1 yellow.

4 marks

a) P(red)

b) P(not blue)

c) Expected number of green results in 120 spins

Answers & worked solutions

a) 3/8

b) 6/8 = 3/4

c) 120 × 2/8 = 30 times.

25. The test scores of eight students are: 12, 15, 18, 18, 19, 21, 23, 34.

4 marks

a) Mean

b) Median

c) Which measure, mean or median, better represents a typical score? Give a reason.

Write your explanation on paper.

Answers & model explanation

a) Mean = 160/8 = 20.

b) Median = (18 + 19)/2 = 18.5.

c) The median is more representative because 34 is a relatively high outlier that pulls the mean upwards.

Section C – Problem Solving & Reasoning

Questions 26–29 · 20 marks · Connect more than one mathematical idea and explain your reasoning.

26. Mobile phone plans

6 marks

Plan A costs $18 per month plus $0.08 per GB of data used. Plan B costs $24 per month and includes unlimited data.

a) Write an equation for the monthly cost C of Plan A in terms of data d (GB).

Equation

b) Find the cost of Plan A for 50 GB.

Cost ($)

c) Find the data usage at which both plans cost the same.

Data usage (GB)

d) Which plan is cheaper for 90 GB?

Plan A cost at 90 GB ($)

Cheaper plan

Answers & worked solutions

a) C = 18 + 0.08d

b) C = 18 + 0.08(50) = $22.

c) 18 + 0.08d = 24 → d = 75 GB.

d) Plan A = $25.20 at 90 GB. Plan B costs $24, so Plan B is cheaper.

27. Water tank application

6 marks

A cylindrical tank has radius 0.8 m and height 1.5 m. It is initially 25% full. Water flows in at 18 litres per minute. Use π = 3.1416.

a) Full capacity (L)

b) Litres required to fill

c) Time required (minutes)

Answers & worked solutions

a) V = π(0.8)²(1.5) = 3.015936 m³ = 3015.94 L.

b) 75% remains to be filled: 0.75 × 3015.94 = 2261.95 L.

c) 2261.95 ÷ 18 ≈ 125.66 minutes, so approximately 126 minutes.

28. Comparing savings options

4 marks

Ethan has $3000 to invest for 2 years. Bank X offers 6% p.a. simple interest. Bank Y offers 5.8% p.a. compound interest, compounded annually.

Final amount – Bank X ($)

Final amount – Bank Y ($)

Better return

Answers & worked solutions

Bank X: 3000 + 3000(0.06)(2) = $3360.

Bank Y: 3000(1.058)² ≈ $3358.09.

Bank X gives the better return by approximately $1.91.

29. Using data to make a decision

4 marks

Two Year 9 classes completed the same quiz. Class A had a mean of 72 and median of 73. Class B had a mean of 72 and median of 66.

a) What does the lower median for Class B suggest about its score distribution compared with Class A?

Write your response on paper.

b) A teacher says, “Both classes performed equally because their means are equal.” Explain why this conclusion may be misleading.

Write your response on paper.

Model answers

a) Class B's lower median suggests that the typical student score was lower. Some relatively high scores may have increased the mean, so its results may be more uneven or skewed.

b) Equal means do not guarantee similar distributions. The different medians indicate a difference in typical student performance, and the spread of scores may also differ.

Your objective answers will be checked automatically. Written reasoning questions should be compared with the model answers after submission.

Your Year 9 Maths Test Result

0 / 73

0% on the automatically checked questions

The complete assessment is worth 80 marks. Seven marks come from written explanations in Questions 19(a), 25(c) and 29. Compare your written responses with the model answers shown beneath those questions.

What does this Year 9 Maths practice test cover?

This practice exam samples a broad range of Year 9 and Stage 5 mathematical skills, including equations, inequalities, the Cartesian or number plane, straight-line graphs, gradient, Pythagoras' theorem, right-angled trigonometry, similarity, measurement, surface area, volume, indices, scientific notation, financial mathematics, probability and statistics.

Schools may teach these topics in different sequences, which is why this resource is designed as a full-year Year 9 maths test rather than a universal Term 1, Term 2, Term 3 or Term 4 paper.

Revise your Year 9 Maths topics

If a section of the test was difficult, use the related Aussie Math Tutor NSW guide below before attempting the questions again.

Year 9 Maths NSW Syllabus Guide Number Plane, Gradient & Straight Lines Guide Indices Laws, Examples & Practice Pythagoras Theorem Guide & Practice Test Trigonometry & SOH CAH TOA Guide Year 9 2D Spatial Relations Review Year 8 Maths Foundations Free Maths Guides & Theory

Common Year 9 Maths difficulties I see while tutoring

Test scores are useful, but they do not always explain why a student made a mistake. These are some patterns that repeatedly appear when working with Year 9 students.

Number Plane, Gradient and Equation of a Line

The number plane and straight-line graphs can be one of the most intimidating Year 9 topics. Students often know isolated formulas but do not understand how coordinates, gradient and the equation of a line are connected.

Tutor tip: Identify the gradient and y-intercept separately before trying to write the equation y = mx + b.

Recognising When to Use Pythagoras

Students sometimes remember the formula a² + b² = c² but fail to recognise that Pythagoras' theorem is applicable when a right-angled triangle appears in a problem.

Tutor tip: When you see a right-angled triangle and need a missing side, ask yourself whether Pythagoras can be used before searching for another formula.

Inequalities

Inequalities can initially look completely different from algebra, but most of the solving process is very similar. The major additional rule students need to remember is that the inequality sign reverses when multiplying or dividing both sides by a negative number.

Example: −2x > 8 becomes x < −4 after dividing by −2.

Indices

With index laws , the difficulty is often remembering which rule applies. Students may confuse adding indices during multiplication with multiplying indices in a power of a power.

Tutor tip: Before calculating, identify the specific index law being tested.

Surface Area and Volume

Three-dimensional measurement can become confusing, particularly with prisms, pyramids and cones. Students need to distinguish between dimensions such as vertical height, radius and slant height, and decide which surfaces or faces are actually included.

Tutor tip: Sketch and label the solid before substituting values into a surface-area or volume formula.

Trigonometry

Basic SOH CAH TOA and trigonometry often becomes manageable once students correctly label the opposite, adjacent and hypotenuse sides. Greater confusion can arise when trigonometry is combined with bearings or more involved diagrams.

Tutor tip: Label the sides relative to the angle before deciding whether to use sine, cosine or tangent.

NSW Year 9 Maths curriculum references

This practice test is an independent learning resource. For official curriculum information and current NSW Mathematics requirements, refer directly to the NSW curriculum sources below.

NSW Curriculum – Mathematics K–10 Course Overview NSW Department of Education – Stage 5 Scope & Sequence

About this Year 9 Maths practice resource Questions were developed with AI-assisted drafting and reviewed for educational use by Adi Jadhav, Maths Tutor at Aussie Math Tutor NSW. The resource is designed for practice and diagnosis and is not an official NESA, NSW Department of Education or school examination.

Did this test identify gaps in your child's Year 9 Maths?

A low score does not always mean the current Year 9 topic is the only problem. Earlier gaps in algebra, fractions, multiplication, number skills or graphing can make later mathematics much harder.

Aussie Math Tutor NSW provides personalised maths support for students who need concepts broken down step by step.

Learn About Private Maths Tutoring

Year 9 Maths Test – Frequently Asked Questions

Is this a real Year 9 maths practice paper?

Yes. The complete 29-question practice paper is displayed directly on this webpage. Students can attempt the questions using pen and paper and enter their final answers online.

Is this Year 9 maths test free?

Yes. The test, answer checking and worked solutions are free to use and do not require an account or login.

When should a student attempt this Year 9 maths test?

It is designed as a broad full-year practice assessment, so it is most useful after a student has covered a substantial portion of Year 9 mathematics or when preparing for a mid-to-late-year or end-of-year assessment.

Is this an official NSW Year 9 examination?

No. It is an independent Aussie Math Tutor NSW learning resource. NSW schools can organise Stage 5 content in different sequences, so students should also follow their school's own assessment and revision requirements.

Can students use a calculator?

A scientific calculator may be used for this practice paper. Students should still perform their working on paper before entering the final answer.

Are answers and worked solutions included?

Yes. After pressing “Check My Answers”, the automatically checked questions are marked and answer explanations or model responses become available beneath each question.

What should a student do after completing the test?

Focus on the topics that caused the most difficulty rather than only looking at the overall score. The revision links above lead to detailed Aussie Math Tutor NSW resources for number plane, indices, Pythagoras, trigonometry and other Year 9 topics.

Free exam papers

Free Year 9 Maths Test NSW – Full-Year Practice Exam & Answers

Section A – Multiple Choice

Section B – Short Answer

Section C – Problem Solving & Reasoning

Your Year 9 Maths Test Result

What does this Year 9 Maths practice test cover?

Revise your Year 9 Maths topics

Common Year 9 Maths difficulties I see while tutoring

Number Plane, Gradient and Equation of a Line

Recognising When to Use Pythagoras

Inequalities

Indices

Surface Area and Volume

Trigonometry

NSW Year 9 Maths curriculum references

Did this test identify gaps in your child's Year 9 Maths?

Year 9 Maths Test – Frequently Asked Questions

Choose a suitable time for your free assessment.

Book Your Free Assessment

Free Year 9 Maths Test NSW – Full-Year Practice Exam & Answers

Section A – Multiple Choice

Section B – Short Answer

Section C – Problem Solving & Reasoning

Your Year 9 Maths Test Result

What does this Year 9 Maths practice test cover?

Revise your Year 9 Maths topics

Common Year 9 Maths difficulties I see while tutoring

Number Plane, Gradient and Equation of a Line

Recognising When to Use Pythagoras

Inequalities

Indices

Surface Area and Volume

Trigonometry

NSW Year 9 Maths curriculum references

Did this test identify gaps in your child's Year 9 Maths?

Year 9 Maths Test – Frequently Asked Questions

Book Your Free Assessment

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