Maths exam papers

Sec 4 Maths exam paper

A free Secondary 4 E-Math paper in the O-Level's two-paper shape: 41 questions for 180 marks, 2 hours 15 minutes a paper, calculator allowed in both. Print it, then check every answer at the end.

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Scholar Tests · Sample paper

Secondary 4 Mathematics

41 questions · 180 marks · 270 minutes · no negative marking

Name ______________________ Date ____________

Paper 1: multiple choice (calculator allowed)

Questions 1 to 8, 3 marks each. Circle the letter of the best answer.

  1. 1.

    In triangle ABC, AB = 8 cm, AC = 7 cm and angle BAC = 53°. Find the length of BC, correct to 3 significant figures.

    1. A14.7 cm
    2. B13.4 cm
    3. C6.75 cm
    4. D45.6 cm
  2. 2.

    ABCD is a cyclic quadrilateral. Angle DAB = 105°. Find angle BCD.

    1. A105°
    2. B255°
    3. C150°
    4. D75°
  3. 3.

    Elena uses the formula v = u + at. Given v = 20, u = 8 and a = 3, what is t?

    1. A4
    2. B12
    3. C7
    4. D28
  4. 4.

    Write 460,000 in standard form.

    1. A4.6 × 10⁻⁵
    2. B46 × 10⁴
    3. C4.6 × 10⁵
    4. D4.6 × 10⁶
  5. 5.

    A sector of a circle has radius 15 cm and angle 210°. Find its arc length, correct to 3 significant figures.

    1. A27.5 cm
    2. B412 cm
    3. C3150 cm
    4. D55.0 cm
  6. 6.

    OA = a and OB = b. M is the midpoint of AB. Express OM in terms of a and b.

    1. A½(a − b)
    2. B½(−a + b)
    3. Ca + b
    4. D½(a + b)
  7. 7.

    Solve the equation x² − 3x + 1 = 0, giving your answers correct to 2 decimal places.

    1. Ax = 3.30 or x = −0.30
    2. Bx = 4.12 or x = 1.88
    3. Cx = −0.38 or x = −2.62
    4. Dx = 2.62 or x = 0.38
  8. 8.

    A ship sails from port P to a buoy B, which is 9 km east and 2 km south of P. Find the bearing of B from P, correct to 1 decimal place where it is not exact.

    1. A282.5°
    2. B257.5°
    3. C102.5°
    4. D347.5°

Paper 1: short answers at 3 marks

Questions 9 to 14, 3 marks each. Write your answer on the line.

  1. 9.

    In a class, ξ is the set of all 27 students, A is the set who play a sport and B the set who play an instrument. n(A) = 20, n(B) = 10 and n(A ∩ B) = 6. Find n(A ∩ B′).

    Answer: ______________

  2. 10.

    y is inversely proportional to x. When x = 4, y = 18. Find y when x = 9.

    Answer: ______________

  3. 11.

    In triangle ABC, AB = 7 cm, AC = 10 cm and BC = 8 cm. Find angle BAC, correct to 1 decimal place. Give your answer in degrees.

    Answer: ______________

  4. 12.

    A cone has a base radius of 5 cm and a height of 12 cm. Find its curved surface area, correct to 3 significant figures. Use the π button on your calculator. Give your answer in cm².

    Answer: ______________

  5. 13.

    Solve 2/(x + 4) = 4/(x − 6).

    Answer: ______________

  6. 14.

    Write 9.3049 correct to 1 decimal place.

    Answer: ______________

Paper 1: short answers at 4 marks

Questions 15 to 26, 4 marks each. Write your answer on the line.

  1. 15.

    AB is a diameter of a circle, and C is a point on the circle. Angle CAB = 73°. Find angle CBA. Give your answer in degrees.

    Answer: ______________

  2. 16.

    In triangle ABC, angle ABC = 90°, angle ACB = 42° and AB = 16 cm. Find x, the length of AC, correct to 3 significant figures. Give your answer in cm.

    Answer: ______________

  3. 17.

    A sofa costs $3300 in cash. On hire purchase, a buyer pays a deposit of 25% of the cash price, then 36 monthly instalments of $81.13. Find the total amount paid on hire purchase. Give your answer in dollars.

    Answer: ______________

  4. 18.

    A sector of a circle has radius 14 cm and angle 70°. Find its arc length, correct to 3 significant figures. Give your answer in cm.

    Answer: ______________

  5. 19.

    Grace has a right-angled ladder. The two shorter sides are 9 cm and 12 cm. How long is the longest side, in centimetres?

    Answer: ______________

  6. 20.

    Write 6.058 correct to 1 decimal place.

    Answer: ______________

  7. 21.

    The matrix N = (4 14 27 ; 12 19 18) shows the numbers of small pizzas, large pizzas, drinks (in that order) sold on Friday (row 1) and sold on Saturday (row 2). The matrix P = (13 ; 21 ; 4) gives each price in dollars. Find the total takings on Saturday, the second entry of NP. Give your answer in dollars.

    Answer: ______________

  8. 22.

    In a class, ξ is the set of all 43 students, A is the set who play a sport and B the set who play an instrument. n(A) = 17, n(B) = 19 and n(A ∩ B) = 8. Find n(A ∩ B′).

    Answer: ______________

  9. 23.

    The numbers of minutes 7 students took to walk to school were 8, 2, 25, 16, 23, 13, 19. Find the standard deviation, correct to 3 significant figures.

    Answer: ______________

  10. 24.

    A pack holds 3 starred cards and 5 plain cards. Two cards are taken out one after the other, and the first is not put back. What is the chance that both are starred?

    Answer: ______________

  11. 25.

    The bearing of B from A is 287°. Find the bearing of A from B. Give your answer in degrees.

    Answer: ______________

  12. 26.

    Two similar solid bottles are made of the same glass. Their heights are 25 cm and 35 cm. The smaller bottle has a mass of 2500 g. Find the mass of the larger bottle. Give your answer in g.

    Answer: ______________

Paper 2: longer problems (calculator allowed)

Questions 1 to 15, 6 marks each. Write your answer on the line.

  1. 1.

    In triangle ABC, angle ABC = 90°, angle ACB = 63° and BC = 5 cm. Find x, the length of AB, correct to 3 significant figures. Give your answer in cm.

    Answer: ______________

  2. 2.

    Chloe sets off along a straight road walking at 6 km/h. 60 minutes later James sets off from the same place along the same road, cycling at 15 km/h. How many minutes after James sets off does James catch up with Chloe?

    Answer: ______________

  3. 3.

    The bearing of B from A is 217°. Find the bearing of A from B. Give your answer in degrees.

    Answer: ______________

  4. 4.

    A sports centre had 240 students. Then the number of boys rose by 30%, the number of girls fell by 10%, and the total number of students rose by 25%. How many boys were there at first?

    Answer: ______________

  5. 5.

    In triangle ABC, angle ABC = 63°, angle ACB = 38° and AC = 5 cm. Find the length of BC, correct to 3 significant figures. Give your answer in cm.

    Answer: ______________

  6. 6.

    A sector of a circle has radius 5 cm and angle 190°. Find its area, correct to 3 significant figures. Give your answer in cm².

    Answer: ______________

  7. 7.

    A rectangular deck measures 8 m by 6 m. Timber costs $50 per square metre. What is the total cost of covering the deck? Give your answer in dollars.

    Answer: ______________

  8. 8.

    4 folders and 4 pens cost $100. 1 folder and 4 pens cost $70. What does one pen cost? Give your answer in dollars.

    Answer: ______________

  9. 9.

    Two similar solid bottles are made of the same glass. Their heights are 5 cm and 35 cm. The smaller bottle has a mass of 20 g. Find the mass of the larger bottle. Give your answer in g.

    Answer: ______________

  10. 10.

    A hose fills a paddling pool that holds 180 L. In the first minute it pours in 24 L, and each minute after that it pours in 1 L less than the minute before. The paddling pool becomes full exactly at the end of a minute. How many minutes does it take to fill? Give your answer in minutes.

    Answer: ______________

  11. 11.

    A solid is made of a cone on top of a hemisphere. Both have radius 9 cm, and the cone's height is 12 cm. Find the volume of the solid, correct to 3 significant figures. Use the π button on your calculator. Give your answer in cm³.

    Answer: ______________

  12. 12.

    $13,500 is put in a bank account that pays compound interest at 4.5% per year, compounded yearly. Find the interest earned after 2 years, correct to the nearest cent. Give your answer in dollars.

    Answer: ______________

  13. 13.

    A small gear wheel of radius 3 cm rolls without slipping all the way round the outside of a large fixed gear wheel of radius 9 cm, once. How many times does the small gear wheel turn round?

    Answer: ______________

  14. 14.

    The numbers of minutes 6 students took to walk to school were 23, 10, 18, 8, 7, 2. Find the standard deviation, correct to 3 significant figures.

    Answer: ______________

  15. 15.

    One painter would paint the fence on its own in 60 hours. A second painter would paint it on its own in 15 hours. If both work at the same time, how long do they take together?

    Answer: ______________

Answers and working

Paper 1: multiple choice (calculator allowed)

  1. 1.C (6.75 cm). Cosine rule: BC² = 8² + 7² − 2 × 8 × 7 × cos 53° = 45.60. Take the square root. The answer is 6.75 cm.
  2. 2.D (75°). Opposite angles of a cyclic quadrilateral add to 180°. 180° − 105°. The answer is 75°.
  3. 3.A (4). v − u = at, so 20 − 8 = 12. Then t = 12 ÷ 3 = 4.
  4. 4.C (4.6 × 10⁵). Move the decimal point so one non-zero digit sits before it: 4.6. The point moved 5 places left, so the power is positive: 10⁵. The answer is 4.6 × 10⁵.
  5. 5.D (55.0 cm). Arc length = 210/360 × 2π × 15. The answer is 55.0 cm.
  6. 6.D (½(a + b)). OM = OA + ½AB = a + ½(b − a) = ½(a + b). The answer is ½(a + b).
  7. 7.D (x = 2.62 or x = 0.38). Here a = 1, b = −3 and c = 1. x = (−b ± √(b² − 4ac)) / 2a = (3 ± √5) / 2. The answer is x = 2.62 or x = 0.38.
  8. 8.C (102.5°). The angle from the north–south line is tan⁻¹(9/2) = 77.5°. B is south-east of P, so measure clockwise from north to reach it. The answer is 102.5°.

Paper 1: short answers at 3 marks

  1. 9.14. Fill the Venn diagram from the middle: A ∩ B = 6, A only = 20 − 6 = 14, B only = 10 − 6 = 4, outside both = 27 − 24 = 3. Read off n(A ∩ B′). The answer is 14.
  2. 10.8. Write y = k/x, find k from x = 4, y = 18, then put in x = 9: y = 8.
  3. 11.52.6. Cosine rule: cos A = (7² + 10² − 8²) / (2 × 7 × 10) = 0.6071. A = cos⁻¹(0.6071). The answer is 52.6°.
  4. 12.204. Find the slant height first: l = √(5² + 12²) = 13 cm. Curved surface area = πrl = π × 5 × 13. This is 204 cm².
  5. 13.−14. Cross-multiply: 2(x − 6) = 4(x + 4). Expand: 2x − 12 = 4x + 16. Collect: −2x = 28, so x = −14. The answer is −14.
  6. 14.9.3. Keep 1 digit after the decimal point and look at the next one to round. The answer is 9.3.

Paper 1: short answers at 4 marks

  1. 15.17. The angle in a semicircle is 90°, so angle ACB = 90°. Angle CBA = 180° − 90° − 73°. The answer is 17°.
  2. 16.23.9. sin 42° = AB/AC, so AC = 16 ÷ sin 42° = 23.9 cm.
  3. 17.3745.68. Deposit = 25% of $3300.00 = $825.00. Instalments = 36 × $81.13 = $2920.68. Total paid = $3745.68. The answer is $3745.68.
  4. 18.17.1. Arc length = 70/360 × 2π × 14. The answer is 17.1 cm.
  5. 19.15. 9² + 12² = 81 + 144 = 225, and 15² = 225, so the longest side is 15 cm.
  6. 20.6.1. Keep 1 digit after the decimal point and look at the next one to round. The answer is 6.1.
  7. 21.627. Multiply row by column: 12 × 13 + 19 × 21 + 18 × 4. = 627. The answer is $627.
  8. 22.9. Fill the Venn diagram from the middle: A ∩ B = 8, A only = 17 − 8 = 9, B only = 19 − 8 = 11, outside both = 43 − 28 = 15. Read off n(A ∩ B′). The answer is 9.
  9. 23.7.59. Mean = 106/7 = 15.14. Standard deviation = √(Σ(x − mean)²/n) = √(402.9/7). The answer is 7.59.
  10. 24.3/28. The first draw is 3 out of 8. One starred is now gone, so the second is 2 out of 7. Both must happen, so multiply: 3/8 × 2/7 = 3/28.
  11. 25.107. Turn round: 287° − 180° = 107°. The answer is 107°.
  12. 26.6860. The length ratio is 5 : 7, so the volume ratio is 125 : 343. Mass goes with volume: 2500 × 343/125. The answer is 6860 g.

Paper 2: longer problems (calculator allowed)

  1. 1.9.81. tan 63° = AB/BC, so AB = 5 × tan 63° = 9.81 cm.
  2. 2.40. In 60 minutes Chloe covers 6 km. James gains 15 − 6 = 9 km every hour, so closing that gap takes 6 ÷ 9 of an hour = 40 minutes.
  3. 3.037. Turn round: 217° − 180° = 37°. The answer is 037°.
  4. 4.210. Boys : girls = 7 : 1, so each of the 8 parts is 30. So the answer is 210.
  5. 5.5.51. Angle BAC = 180° − 63° − 38° = 79°. Sine rule: BC / sin 79° = 5 / sin 63°, so BC = 5 sin 79° / sin 63°. The answer is 5.51 cm.
  6. 6.41.5. Area = 190/360 × π × 5². The answer is 41.5 cm².
  7. 7.2400. Area = 8 × 6 = 48 m². Cost = 48 × $50 = $2400.
  8. 8.15. Both orders hold the same 4 pens, so the difference between them is folders alone: $100 − $70 = $30 buys 4 − 1 = 3 folders. One folder is therefore $30 ÷ 3 = $10. Putting that back into the second order: 1 folder cost $10, leaving $70 − $10 = $60 for 4 pens, so one pen is $15.
  9. 9.6860. The length ratio is 1 : 7, so the volume ratio is 1 : 343. Mass goes with volume: 20 × 343/1. The answer is 6860 g.
  10. 10.9. The hose pours 24 L, then 23 L, then 22 L, and the minutes add up to 180 L at the end of minute 9. So the answer is 9 minutes.
  11. 11.2540. Cone: 1/3 × π × 9² × 12. Hemisphere: 2/3 × π × 9³. Add the two volumes. This is 2540 cm³.
  12. 12.1242.34. Amount = P(1 + r/100)ⁿ. = 13500 × (1 + 4.5/100)^2 = $14,742.34. Interest = $14,742.34 − $13,500.00. The answer is $1242.34.
  13. 13.4. The centre travels a circle of radius 9 + 3, so the wheel turns (9 + 3) ÷ 3 = 4 times.
  14. 14.7.06. Mean = 68/6 = 11.33. Standard deviation = √(Σ(x − mean)²/n) = √(299.3/6). The answer is 7.06.
  15. 15.12. Count what each does in one hour rather than in whole jobs: 1/60 and 1/15. Together that is 75/900 of the job each hour, so the whole job takes 900 ÷ 75 = 12 hours.

Written by Scholar Tests to MOE's Secondary 4 Mathematics syllabus. Not a school paper. Scholar Tests is not connected with MOE, SEAB or any school.

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  • Secondary 4 Mathematics Paper 1

    26 questions · 135 minutes

    Secondary 4 E-Math in the O-Level Paper 1 shape. Calculator allowed.

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    15 questions · 135 minutes

    Secondary 4 E-Math longer problems. Calculator allowed.

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Questions parents ask

What does the Sec 4 paper cover?
The Secondary 4 Mathematics topics in MOE's syllabus: the whole SEC G3 Mathematics syllabus: Secondary 3's topics, plus vectors, matrices and standard deviation.
How is it marked?
As the O-Level papers are, each paper is worth 90: in Paper 1 the first 14 questions are worth 3 marks and the other 12 are worth 4; in Paper 2 each of the 15 longer problems is worth 6, for 180 in all. The real Paper 2 splits each question into parts and gives marks for method; we mark one final answer per problem, correct to 3 significant figures (an angle to 1 decimal place) where it is not exact. The parts a computer cannot mark, a graph to draw, a construction, a chart that misleads and an extended real-world task, are a separate written paper of 30 marks in the practice room, which you mark yourself against a model answer.
Is there practice for graphs, constructions and the extended task?
Yes, in the practice room: Paper 2's written questions, one graph to plot and read, one ruler-and-compass construction, one chart or statistic that misleads, and one extended real-world task, 30 marks in 45 minutes. They are done on paper, then marked against a model answer, one tick a mark.
Can my child use a calculator?
Yes, in both papers, as in the O-Level E-Math exam. Set it to degrees for angles, and use its π button.
Is it free?
A free account includes 3 practice tests. The Singapore pack (S$99 for 6 months) opens unlimited tests and shows the right answer and the working for every question.
Is this a real O-Level paper?
No. It is written by us in the shape of SEAB's SEC G3 Mathematics papers. Past O-Level papers are copyright and are not used. Scholar Tests is not connected with MOE, SEAB or any school.

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