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Maths Olympiad Sample Paper 2 (Practice)

Maths Olympiad topic 33: Maths Olympiad mock test 2 · Set 653 · 40 questions
TalentJR

How to solve it (Name the topic, then solve): Name the topic first: angles, counting, worst case, probability, digits, mensuration, data, coordinates, AP, algebra, primes or Venn diagrams. Use that lesson’s method; draw a quick sketch, list or Venn diagram on rough paper. Rule out options of the wrong kind — a probability above 1, or a triangle angle of 180° or more. Check by working backwards or with a second method when you can.

NameDateTime takenScore ___ / 40
  1. 1.What is the sum of the first 9 terms of the AP 7, 10, 13, …? (a) 342 (b) 171 (c) 174 (d) 31
  2. 2.In an isosceles triangle, the angle between the two equal sides is 130°. What is each of the other two angles? (a) 50° (b) 130° (c) 25° (d) 35°
  3. 3.A book has 51 pages, numbered 1, 2, 3 and so on. How many digits are printed to number all the pages? (a) 94 (b) 93 (c) 84 (d) 102
  4. 4.If a + b = 16 and ab = 48, what is a² + b²? (a) 256 (b) 160 (c) 208 (d) 352
  5. 5.What are the roots of x² − 15x + 54 = 0? (a) 1 and 14 (b) 2 and 27 (c) 6 and 9 (d) 1 and 54
  6. 6.What is the mean of 13, 8, 16, 3, 11 and 15? (a) 10 (b) 11 (c) 13 (d) 12
  7. 7.How many prime numbers are there between 80 and 100? (a) 2 (b) 3 (c) 4 (d) 5
  8. 8.How many factors does 85 have? (a) 4 (b) 3 (c) 2 (d) 5
  9. 9.A fair dice is rolled once. What is the probability of getting an even number? (a) 1/6 (b) 1/2 (c) 2/3 (d) 1/3
  10. 10.A bag has 4 yellow balls and 3 blue balls. Without looking, how many balls must you take out to be sure of getting at least one ball of each colour? (a) 5 (b) 7 (c) 4 (d) 2
  11. 11.What is the midpoint of the line joining (10, 10) and (22, 22)? (a) (6, 6) (b) (32, 32) (c) (16, 16) (d) (17, 16)
  12. 12.What is the prime factorisation of 56? (a) 2³ × 7 (b) 2⁴ × 7 (c) 2 × 7³ (d) 2² × 7
  13. 13.A fair dice is rolled once. What is the probability of getting a multiple of 3? (a) 1/2 (b) 1/3 (c) 2/3 (d) 1/6
  14. 14.What is the distance between the points (9, 4) and (18, 16)? (a) 16 (b) 21 (c) 15 (d) 225
  15. 15.What is the 35th term of the AP 10, 13, 16, 19, …? (a) 112 (b) 105 (c) 102 (d) 115
  16. 16.A team of 2 players is to be chosen from 10 players. How many different teams are possible? (a) 45 (b) 90 (c) 100 (d) 20
  17. 17.What is the mean of 11, 20, 3, 6 and 20? (a) 12 (b) 13 (c) 15 (d) 10
  18. 18.Using (a + b)² = a² + 2ab + b², what is 34²? (a) 1166 (b) 676 (c) 1156 (d) 916
  19. 19.How far is the point (4, 3) from the origin (0, 0)? (a) 6 (b) 25 (c) 5 (d) 7
  20. 20.A bag has 4 blue balls and 3 black balls. Without looking, how many balls must you take out to be sure of getting at least one ball of each colour? (a) 5 (b) 7 (c) 2 (d) 4
  21. 21.Three coins are tossed together. What is the probability of getting at least one head? (a) 7/8 (b) 1/4 (c) 1/8 (d) 3/8
  22. 22.In a class of 47 children, 26 like maths, 22 like science and 9 like both. How many like only maths? (a) 13 (b) 17 (c) 25 (d) 26
  23. 23.How many two-digit numbers have digits that add up to 13? (a) 7 (b) 5 (c) 6 (d) 14
  24. 24.In a class of 29 children, 23 like tea, 10 like coffee and 3 like neither. How many like both? (a) 3 (b) 7 (c) 26 (d) 4
  25. 25.Which is the correct factorisation of x² + 13x + 42? (a) (x + 6)(x + 7) (b) (x + 1)(x + 42) (c) (x − 6)(x − 7) (d) (x + 1)(x + 12)
  26. 26.What is the sum of the first 12 terms of the AP 8, 13, 18, …? (a) 852 (b) 456 (c) 426 (d) 63
  27. 27.A drawer has lots of blue, red, white, green and yellow socks, all mixed up. In the dark, how many socks must you take out to be sure of getting a matching pair? (a) 5 (b) 10 (c) 2 (d) 6
  28. 28.A team of 2 players is to be chosen from 11 players. How many different teams are possible? (a) 121 (b) 110 (c) 55 (d) 22
  29. 29.A bag has 12 white balls and 7 red balls. Without looking, how many balls must you take out to be sure of getting 2 red balls? (a) 2 (b) 14 (c) 19 (d) 13
  30. 30.In a class of 27 children, 19 like tea, 9 like coffee and 3 like both. How many like neither? (a) 1 (b) 2 (c) 3 (d) 5
  31. 31.A bag has 4 blue, 3 white and 2 red balls. One ball is taken out without looking. What is the probability that it is white? (a) 1/3 (b) 4/9 (c) 1/2 (d) 2/3
  32. 32.A canteen sells 5 kinds of sandwich and 6 kinds of juice. In how many ways can you choose one sandwich and one juice? (a) 35 (b) 11 (c) 30 (d) 60
  33. 33.How far is the point (24, 7) from the origin (0, 0)? (a) 25 (b) 31 (c) 625 (d) 26
  34. 34.In which quadrant does the point (−5, −3) lie? (a) First quadrant (b) Second quadrant (c) Third quadrant (d) Fourth quadrant
  35. 35.What is the smallest number of children in a group that makes sure at least three of them were born in the same day of the week? (a) 15 (b) 14 (c) 21 (d) 22
  36. 36.What is the sum of the roots of x² − 11x + 24 = 0? (a) 11 (b) 12 (c) 10 (d) 24
  37. 37.How many two-digit numbers have digits that add up to 15? (a) 5 (b) 3 (c) 16 (d) 4
  38. 38.In a class of 35 children, 20 like dance, 14 like music and 7 like both. How many like at least one of the two? (a) 20 (b) 34 (c) 27 (d) 35
  39. 39.Two angles of a triangle are 65° and 90°. What is the third angle? (a) 25° (b) 155° (c) 90° (d) 115°
  40. 40.In an isosceles triangle, the two equal angles are 75° each. What is the third angle? (a) 15° (b) 30° (c) 75° (d) 105°

Answer key

  1. 171
  2. 25°
  3. 93
  4. 160
  5. 6 and 9
  6. 11
  7. 3
  8. 4
  9. 1/2
  10. 5
  11. (16, 16)
  12. 2³ × 7
  13. 1/3
  14. 15
  15. 112
  16. 45
  17. 12
  18. 1156
  19. 5
  20. 5
  21. 7/8
  22. 17
  23. 6
  24. 7
  25. (x + 6)(x + 7)
  26. 426
  27. 6
  28. 55
  29. 14
  30. 2
  31. 1/3
  32. 30
  33. 25
  34. Third quadrant
  35. 15
  36. 11
  37. 4
  38. 27
  39. 25°
  40. 30°

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