About this O-Level Math practice test

O-Level Maths splits into non-calculator Paper 1 and calculator Paper 2, and doing well means being fast and accurate on both. This set targets the high-yield topics — from algebra to probability trees — with every intermediate step shown, exactly the kind of working Cambridge examiners award marks for.

O-Level Mathematics Practice Test sample questions

These starter questions help you launch a math mock test quickly. Swap them with your own worksheet, notebook, or textbook questions any time.

1. 1. Evaluate (3.2 × 10^5) × (5 × 10^-2), giving your answer in standard form.

2. 2. A trader buys a box of 240 mangoes for $60. She finds that 1/8 of the mangoes are rotten and throws them away. She sells the remaining mangoes at $0.50 each. Calculate her percentage profit.

3. 3. Express the recurring decimal 0.457457457... as a fraction in its simplest form.

4. 4. The sizes of the three angles in a triangle are in the ratio 2:3:4. Find the size of the largest angle.

5. 5. It takes 8 workers 15 days to build a wall, all working at the same constant rate. How many days would it take 12 workers to build the same wall?

6. 6. A cyclist travels 18 km at an average speed of 12 km/h, then a further 10 km at an average speed of 8 km/h. Calculate her average speed for the whole journey, correct to 1 decimal place.

7. 7. Solve the equation (x+3)/4 - (x-2)/3 = 1.

8. 8. Factorise completely: 2x² - 7x - 15.

9. 9. The nth term of a sequence is given by T(n) = n² - 2n + 3. Find the value of the 6th term, and determine whether 83 is a term in this sequence.

10. 10. A sequence of patterns is made from square tiles: Pattern 1 uses 4 tiles, Pattern 2 uses 7 tiles, Pattern 3 uses 10 tiles, and so on, increasing by the same amount each time. Find an expression for the number of tiles in Pattern n, and use it to find the number of tiles in Pattern 15.

11. 11. Solve the simultaneous equations: 5x - 2y = 16, 3x + 4y = 20.

12. 12. Simplify (2x³y²)³ ÷ (4x²y⁵), giving your answer in the form ax^b y^c.

13. 13. Simplify fully: (x² - 9) / (x² + x - 6).

14. 14. The interior angle of a regular polygon is 156°. Calculate the number of sides of the polygon.

15. 15. ABCD is a cyclic quadrilateral. Angle ABC = 108° and angle BCD = 95°. Find angle ADC and angle DAB, giving reasons.

16. 16. Triangle PQR and triangle STU have PQ = ST = 6 cm, QR = TU = 8 cm, and angle PQR = angle STU = 70°. State, giving a reason, whether the two triangles are congruent.

17. 17. Two similar triangles have corresponding sides in the ratio 4:5. The area of the smaller triangle is 32 cm². Find the area of the larger triangle.

18. 18. In triangle ABC, angle B = 90°, AB = 9 cm, and angle C = 32°. Calculate the length of BC, correct to 1 decimal place.

19. 19. A ship sails from a port P on a bearing of 040° for 15 km to point Q. It then changes course and sails on a bearing of 130° for 20 km to point R. Show that angle PQR = 90°, and calculate the distance PR.

20. 20. A goat is tied by a rope of length 7 m to a fixed post at point P in the middle of a large flat field. Describe fully the locus of points the goat can reach, and calculate the area it can graze, correct to the nearest square metre.

21. 21. A sector of a circle has radius 10 cm and angle 72°. Calculate (a) the arc length and (b) the area of the sector, giving your answers in terms of π.

22. 22. A solid metal sphere of radius 6 cm is melted down and recast into a solid cylinder of radius 4 cm. Calculate the height of the cylinder. [Volume of sphere = (4/3)πr³]

23. 23. A solid cone has base radius 5 cm and slant height 13 cm. Calculate the total surface area of the cone, in terms of π. [Curved surface area of cone = πrl]

24. 24. A window is in the shape of a rectangle 80 cm wide and 120 cm tall, topped with a semicircle of diameter 80 cm. Calculate the total area of the window, correct to the nearest cm².

25. 25. The table shows the number of goals scored by a football team in 20 matches: Goals 0,1,2,3,4 with frequencies 4,6,5,3,2 respectively. Calculate the mean number of goals scored per match.

26. 26. The table shows the times, t minutes, taken by 60 students to complete a puzzle: 0<t≤10 (freq 8), 10<t≤20 (freq 14), 20<t≤30 (freq 20), 30<t≤40 (freq 12), 40<t≤50 (freq 6). Use interpolation to estimate the median time.

27. 27. The table shows the masses, m kg, of 50 parcels: 0<m≤5 (freq 10), 5<m≤15 (freq 20), 15<m≤25 (freq 15), 25<m≤50 (freq 5). A histogram is drawn to represent this data. Calculate the frequency density of the bar representing the class 25<m≤50.

28. 28. A box contains 6 red pens, 4 blue pens, and 5 black pens. A pen is chosen at random from the box. Find the probability that the pen chosen is red or black.

29. 29. A drawer contains 5 white socks and 3 black socks. Two socks are taken at random from the drawer, one after another, without replacement. Calculate the probability that both socks are the same colour.

30. 30. The probability that Aisha wins her first tennis match is 3/5. If she wins the first match, the probability she wins the second match is 2/3. If she loses the first match, the probability she wins the second match is 1/4. Calculate the probability that Aisha wins exactly one of the two matches.

Answer key & quick explanations

Short answers for the sample questions above. Use this to self-check before generating a fresh AI-built mock test.

1. 1. Standard form multiplication

   1.6 × 10^4

   Multiply the coefficients (3.2×5=16) and add the indices (5+(-2)=3) to get 16×10³. Since 16 is not between 1 and 10, rewrite it as 1.6×10¹, giving 1.6×10^4 overall.

2. 2. Percentage profit on mangoes

   75%

   Removing 1/8 of 240 mangoes (30) leaves 210 to sell. Revenue = 210 × $0.50 = $105, so profit = $105 − $60 = $45. Percentage profit = (45/60) × 100 = 75%.

3. 3. Recurring decimal to fraction

   457/999

   Let x = 0.457457457... Since the repeating block has 3 digits, multiply by 1000: 1000x = 457.457457... Subtracting gives 999x = 457, so x = 457/999, already in lowest terms since 457 is prime.

4. 4. Angle ratio in a triangle

   80°

   The ratio parts 2+3+4 = 9 represent the angle sum of 180°, so each part = 20°. The largest angle has 4 parts = 4 × 20° = 80°.

5. 5. Inverse proportion (workers and days)

   10 days

   The total work is constant: 8 workers × 15 days = 120 worker-days. With 12 workers, days = 120 ÷ 12 = 10 (inverse proportion: more workers means fewer days).

6. 6. Average speed over two stages

   10.2 km/h

   Time for stage 1 = 18/12 = 1.5 h; time for stage 2 = 10/8 = 1.25 h; total time = 2.75 h. Average speed = total distance ÷ total time = 28/2.75 = 10.2 km/h (1 dp).

7. 7. Linear equation with fractions

   x = 5

   Multiply every term by 12 (LCM of 4 and 3): 3(x+3) − 4(x−2) = 12. Expanding gives 3x+9−4x+8=12, so −x+17=12, giving x=5. Check: (8/4)−(3/3)=2−1=1 ✓.

8. 8. Factorising a quadratic

   (2x + 3)(x − 5)

   Find two numbers multiplying to 2×(−15)=−30 and summing to −7: these are −10 and 3. Split the middle term: 2x²−10x+3x−15, then group: 2x(x−5)+3(x−5) = (2x+3)(x−5).

9. 9. Quadratic sequence formula

   T(6) = 27; yes, 83 is the 10th term

   Substituting n=6 gives 36−12+3=27. To test 83, solve n²−2n−80=0, which factorises as (n−10)(n+8)=0, giving n=10 (rejecting the negative root), so 83 is the 10th term.

10. 10. Linear sequence from a tile pattern

    T(n) = 3n + 1; Pattern 15 has 46 tiles

    The pattern increases by 3 tiles each time, so T(n) = 3n + c. Using Pattern 1 (T=4): 3(1)+c=4, so c=1, giving T(n)=3n+1. Substituting n=15: 3(15)+1=46.

11. 11. Simultaneous equations by elimination

    x = 4, y = 2

    Multiply the first equation by 2: 10x − 4y = 32. Adding this to 3x + 4y = 20 eliminates y: 13x = 52, so x = 4. Substituting into 5x − 2y = 16 gives 20 − 2y = 16, so y = 2.

12. 12. Simplifying indices

    2x⁷y

    Cube the bracket first: (2x³y²)³ = 2³x⁹y⁶ = 8x⁹y⁶. Dividing by 4x²y⁵: coefficients 8÷4=2, and subtract indices, x^(9−2)=x⁷ and y^(6−5)=y¹, giving 2x⁷y.

13. 13. Simplifying an algebraic fraction

    (x − 3)/(x − 2)

    Factorise numerator and denominator: x²−9=(x−3)(x+3) (difference of two squares), and x²+x−6=(x+3)(x−2). The common factor (x+3) cancels, leaving (x−3)/(x−2).

14. 14. Interior/exterior angles of a polygon

    15 sides

    Each exterior angle = 180° − 156° = 24°. Since exterior angles of a regular polygon sum to 360°, the number of sides = 360 ÷ 24 = 15.

15. 15. Cyclic quadrilateral angles

    Angle ADC = 72°, Angle DAB = 85°

    Opposite angles in a cyclic quadrilateral sum to 180°. So angle ADC = 180° − 108° = 72° (opposite ABC), and angle DAB = 180° − 95° = 85° (opposite BCD).

16. 16. Congruence condition (SAS)

    Yes, congruent by SAS

    Two sides (PQ=ST=6cm and QR=TU=8cm) and the included angle between them (70°) are equal in both triangles, satisfying the Side-Angle-Side condition, so the triangles are congruent.

17. 17. Area ratio of similar triangles

    50 cm²

    For similar shapes the area ratio equals the square of the length ratio: (4:5)² = 16:25. So the larger area = 32 × (25/16) = 50 cm².

18. 18. Right-angled trigonometry (tan)

    BC ≈ 14.4 cm

    With the right angle at B, AB is opposite angle C and BC is adjacent to it. Using tan(C) = opposite/adjacent: tan(32°) = 9/BC, so BC = 9/tan(32°) = 14.4 cm (1 dp).

19. 19. Bearings and Pythagoras

    Angle PQR = 90°; PR = 25 km

    The back bearing of P from Q is 040°+180°=220°. Angle PQR = 220°−130°=90°, confirming the right angle at Q. By Pythagoras, PR² = 15²+20² = 625, so PR = 25 km.

20. 20. Locus of a grazing goat

    A circle of radius 7 m centred at P; area ≈ 154 m²

    Since the rope has fixed length 7 m anchored at P, the goat can reach every point exactly 7 m or less from P, forming a circle of radius 7 m. Grazing area = πr² = π×49 = 153.9... ≈ 154 m².

21. 21. Arc length and sector area

    Arc length = 4π cm; Area = 20π cm²

    The sector is 72/360 = 1/5 of the full circle. Arc length = (1/5)×2π×10 = 4π cm. Sector area = (1/5)×π×10² = 20π cm².

22. 22. Volume: sphere recast as cylinder

    18 cm

    Volume of sphere = (4/3)π(6)³ = 288π cm³. Setting this equal to the cylinder's volume π(4)²h = 16πh gives 16πh = 288π, so h = 18 cm.

23. 23. Total surface area of a cone

    90π cm²

    Curved surface area = πrl = π×5×13 = 65π cm². Adding the base area πr² = π×25 = 25π cm² gives total surface area = 65π + 25π = 90π cm².

24. 24. Composite area: rectangle plus semicircle

    ≈ 12113 cm²

    Rectangle area = 80×120 = 9600 cm². The semicircle has radius 40 cm, so its area = ½π(40)² = 800π ≈ 2513.3 cm². Total area = 9600 + 2513.3 ≈ 12113 cm² (nearest cm²).

25. 25. Mean from a frequency table

    1.65 goals

    Multiply each goal value by its frequency and sum: (0×4)+(1×6)+(2×5)+(3×3)+(4×2) = 33. Divide by the total number of matches, 20: mean = 33/20 = 1.65.

26. 26. Median by interpolation (cumulative frequency)

    24 minutes

    With n=60, the median lies at position 30. Cumulative frequency reaches 22 by t=20 and 42 by t=30, so the median lies in the 20<t≤30 class. Interpolating: median ≈ 20 + (30−22)/20 × 10 = 24 minutes.

27. 27. Frequency density for a histogram

    0.2

    Frequency density = frequency ÷ class width. For 25<m≤50, the class width is 25 and the frequency is 5, so frequency density = 5/25 = 0.2.

28. 28. Basic probability (or event)

    11/15

    Total pens = 6+4+5 = 15. Pens that are red or black = 6+5 = 11, and these outcomes are mutually exclusive, so P(red or black) = 11/15.

29. 29. Combined probability without replacement

    13/28

    P(both white) = (5/8)×(4/7) = 20/56, and P(both black) = (3/8)×(2/7) = 6/56. These are mutually exclusive events, so total = 20/56 + 6/56 = 26/56 = 13/28.

30. 30. Tree diagram: exactly one win

    3/10

    P(win then lose) = (3/5)×(1/3) = 1/5, since P(lose 2nd | won 1st) = 1−2/3 = 1/3. P(lose then win) = (2/5)×(1/4) = 1/10. Adding these mutually exclusive paths gives 1/5 + 1/10 = 3/10.

Curriculum Mapping & Learning Guide

Use this breakdown to identify which skills each question tests and guide post-test review.

O-Level Math units covered

1. Chapter 1: Number: integers, fractions, decimals, percentages, standard form
2. Chapter 2: Number: ratio, proportion, speed/distance/time
3. Chapter 3: Algebra: expressions, equations, factorisation, sequences
4. Chapter 4: Algebra: simultaneous equations, indices, algebraic fractions
5. Chapter 5: Geometry: angle properties, circle theorems, congruence, similarity
6. Chapter 6: Geometry: trigonometry (SOH-CAH-TOA), bearings, loci and constructions
7. Chapter 7: Mensuration: area, arc length, sector area, volume, surface area
8. Chapter 8: Statistics: averages, cumulative frequency, histograms, scatter graphs
9. Chapter 9: Probability: basic, combined, tree diagrams

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