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Free Year 10 Maths Test NSW

40 questions covering Stage 5 algebra, trigonometry, bearings, variation, finance and more. Check individual questions beside your working, or complete the test and get a personalised revision plan based on what you attempted.

40 questions100 marks67 answer parts100–120 min suggestedCalculator permittedFree learning plan
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NSW curriculum note: This is an independent practice test, not an official NESA examination. Year 10 schools teach different Core and Path topics in different sequences. Follow your school's own assessment notification.

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Year 10 diagnostic progress
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Section A · Core skills

Questions 1–20 · 24 marks · Algebra, finance, trigonometry and variation.

Question 1 Depreciation 1 mark
A laptop is purchased for $2,400 and depreciates by 18% each year. Find its value after 3 years. Give your answer to the nearest dollar.
Answer & explanation

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Question 2 Linear equations 1 mark
Solve: 5(2x − 3) − 4 = 3(x + 7).
Answer & explanation

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Question 3 Right-angle trigonometry 1 mark
A right-angled triangle has a hypotenuse of 17 cm and one acute angle of 38°. Find the side opposite the 38° angle.
Answer & explanation

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Question 4 Direct variation 1 mark
y varies directly as x. When x = 6, y = 21. Find y when x = 14.
Answer & explanation

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Question 5 Quadratic equations 2 marks
Solve: x² − 11x + 24 = 0.
Answer & explanation

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Question 6 Cosine rule 1 mark
Two sides of a triangle measure 12 cm and 17 cm. The included angle is 64°. Find the third side.
Answer & explanation

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Question 7 Compound interest 1 mark
$8,500 is invested at 4.2% p.a. compound interest. Find the value of the investment after 5 years.
Answer & explanation

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Question 8 Simultaneous equations 2 marks
Solve simultaneously:
2x + 3y = 17
x − y = 1.
Answer & explanation

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Question 9 Right-angle trigonometry 1 mark
A 9 m ladder rests against a vertical wall and makes an angle of 67° with the ground. How high up the wall does the ladder reach?
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Question 10 Inverse variation 1 mark
The time t required to complete a job varies inversely as the number of workers n. Six workers complete a job in 15 hours. How long would 10 workers take?
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Question 11 Inequalities 1 mark
Solve the inequality: 4x − 7 < 2x + 11.
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Question 12 Sine rule 1 mark
In triangle ABC, ∠A = 48°, ∠B = 73°, and BC = 15 cm. Find AC.
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Question 13 Depreciation 1 mark
A car worth $38,000 depreciates at 12% per year. Find its value after 4 years.
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Question 14 Quadratic equations 2 marks
Solve: 2x² + 7x − 15 = 0.
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Question 15 Bearings 2 marks
A walker travels 8 km east and then 11 km north. Find:
(a) the straight-line distance from the starting point
(b) the bearing of the walker from the starting point.
Answer & explanation

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Question 16 Direct variation 1 mark
The cost C of hiring a machine varies directly with the number of hours h. A 6-hour hire costs $330. Find the cost of hiring the machine for 9.5 hours.
Answer & explanation

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Question 17 Linear equations 1 mark
Solve: (x + 3)/4 + (x − 1)/3 = 5.
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Question 18 Cosine rule 1 mark
Triangle PQR has PQ = 13 cm, QR = 18 cm, and PR = 21 cm. Find ∠Q.
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Question 19 Cubic equations 1 mark
Solve: 5x³ = 625.
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Question 20 Area rule 1 mark
A triangular block of land has sides 24 m and 31 m with an included angle of 52°. Calculate its area.
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Section B · Applications

Questions 21–30 · 22 marks · Interpreting results and solving applied problems.

Question 21 Rates of change 2 marks
The distance travelled by a car is represented by the table below. Determine the rate of change and explain what it means in this situation.
Time (h)01234Distance (km)065130195260
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Question 22 Quadratic equations 2 marks
Solve: 3x² − 14x − 5 = 0.
Answer & explanation

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Question 23 Right-angle trigonometry 1 mark
From a point 35 m from the base of a building, the angle of elevation to the top is 41°. Find the height of the building.
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Question 24 Compound interest 2 marks
$12,000 is invested for 6 years at 5.1% p.a. compounded annually. Calculate the interest earned.
Answer & explanation

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Question 25 Simultaneous equations 4 marks
Solve simultaneously:
3x + 2y = 16
5x − y = 19.
Answer & explanation

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Question 26 Bearings and components 4 marks
From point A, a lighthouse is on a bearing of 065°. A boat travels 18 km directly towards the lighthouse. Find:
(a) how far east the boat travels
(b) how far north the boat travels.
Answer & explanation

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Question 27 Inverse variation 3 marks
y varies inversely as x. When x = 8, y = 15. Find:
(a) the variation constant
(b) y when x = 20.
Answer & explanation

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Question 28 Linear equations 1 mark
Simplify and solve: 4(3x − 2) − 2(5x + 3) = 18.
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Question 29 Area rule 1 mark
A triangular roof has sides 6.8 m and 9.2 m with an included angle of 74°. Find its area.
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Question 30 Quadratics and parabolas 2 marks
Solve x² − 4x − 21 = 0, and hence determine the two x-intercepts of y = x² − 4x − 21.
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Section C · Multi-step challenges

Questions 31–40 · 54 marks · Extended problems and combined skills.

Question 31 Bearings and cosine rule 2 marks
A boat leaves harbour A on a bearing of 035° and travels 18 km to B. It then travels 25 km on a bearing of 120° to C. Calculate the straight-line distance AC.
Answer & explanation

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Question 32 Ambiguous trigonometry 4 marks
A triangular field has sides 38 m and 47 m. Its area is 720 m². Find the possible included angle(s) between the two sides.
Answer & explanation

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Question 33 Depreciation model 6 marks
A car is purchased for $52,000. It depreciates by 16% per year.
(a) Write an expression for its value after n years.
(b) Find its value after 5 years.
(c) Determine the first whole year in which its value is less than $20,000.
Answer & explanation

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Question 34 Inverse proportion 8 marks
The variables x and y are inversely proportional. When x = 12, y = 18.
(a) Find the equation connecting x and y.
(b) Find y when x = 7.5.
(c) Find x when y = 32.
(d) Explain what happens to y when x is doubled.
Answer & explanation

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Question 35 Simultaneous non-linear equations 8 marks
Solve simultaneously:
y = x² − 5x + 4
y = 2x − 2.
Give the coordinates of all points of intersection.
Answer & explanation

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Question 36 Two-position trigonometry 2 marks
A vertical communications tower stands on level ground. From point A, the angle of elevation to the top is 32°. After walking 45 m directly towards the tower to point B, the angle of elevation is 51°. Find the height of the tower.
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Question 37 Sine rule and ambiguity 4 marks
In triangle ABC, AB = 18 cm, AC = 25 cm, and ∠B = 42°. Find all possible values of ∠C. You must consider whether more than one triangle is possible.
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Question 38 Quadratic inequalities 6 marks
Solve: 2x² − 9x − 5 = 0. Then use your solutions to determine the values of x satisfying 2x² − 9x − 5 ≤ 0.
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Question 39 3D Pythagoras and trigonometry 6 marks
A rectangular box measures 12 cm × 9 cm × 16 cm. A straight line joins one bottom corner of the box to the diagonally opposite top corner.
(a) Find the diagonal of the base.
(b) Find the length of the three-dimensional diagonal.
(c) Find the angle the three-dimensional diagonal makes with the base of the box.
Answer & explanation

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Question 40 Bearings, sine and cosine rules 8 marks
A rescue vessel travels 32 km from A to B on a bearing of 048°. It then travels 27 km from B to C on a bearing of 135°.
(a) Find ∠ABC.
(b) Calculate the direct distance AC.
(c) Find ∠BAC.
(d) Hence determine the bearing of C from A.
Answer & explanation

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Year 10 Maths Practice for NSW Students

This 40-question paper samples NSW Stage 5 concepts including algebraic equations, quadratics, simultaneous equations, right-angle and non-right-angle trigonometry, bearings, inverse and direct variation, rates, compound interest, depreciation, quadratic graphs and three-dimensional applications. Questions 31–40 emphasise multi-step problem solving.

About the RoSA: NSW's Record of School Achievement is not one standard statewide Year 10 maths exam. Refer to your school's assessment advice and the official NESA syllabus to confirm the topics that apply to you.

Free resources to revisit tricky topics

Algebra explainedQuadratic equationsTrigonometry guideParabolas & graphsRatios and ratesFinancial mathematicsMore maths practice testsYear 10 syllabus guide

Year 10 maths test: common questions

Is this an official RoSA mathematics paper?

No. This is an independent practice paper for students in NSW. Schools set their own Year 10 assessment tasks.

Will I lose marks for topics I have not learnt?

Not in the attempted-answers accuracy or learning plan. Leave unanswered questions blank or select “Not yet taught”. The test still shows a possible score out of 100 for the full paper.

Can I see explanations while practising?

Yes. Choose “Check & explain” beside an attempted question for immediate marking and the worked answer alongside it, or directly below on smaller screens.

Can I return to this test later?

Yes, provided this browser allows local storage and you use the same browser/device. Progress does not sync between devices.

Curriculum context: NESA Mathematics K–10 syllabus overview · NSW RoSA information.