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Maths Olympiad Sample Paper 2 (Practice)

Maths Olympiad topic 33: Maths Olympiad mock test 2 · Set 22 · 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.Which is the correct factorisation of x² + 12x + 35? (a) (x + 1)(x + 35) (b) (x + 1)(x + 11) (c) (x − 5)(x − 7) (d) (x + 5)(x + 7)
  2. 2.In a league, 11 teams each play every other team once. How many matches are played? (a) 110 (b) 121 (c) 55 (d) 66
  3. 3.What is the prime factorisation of 30? (a) 2² × 3 × 5 (b) 2 × 3 × 5 (c) 2³ × 3 × 5 (d) 2 × 3 × 5²
  4. 4.Two dice are rolled together. What is the probability that the two numbers add up to 4? (a) 1/9 (b) 1/11 (c) 1/12 (d) 3/11
  5. 5.Two coins are tossed together. What is the probability of getting at least one head? (a) 1/2 (b) 1/3 (c) 3/4 (d) 1/4
  6. 6.Two dice are rolled together. What is the probability that the two numbers add up to 7? (a) 1/11 (b) 6/11 (c) 7/36 (d) 1/6
  7. 7.14 friends meet and each one shakes hands once with every other friend. How many handshakes are there? (a) 91 (b) 196 (c) 105 (d) 182
  8. 8.How many prime numbers are there between 80 and 100? (a) 2 (b) 4 (c) 5 (d) 3
  9. 9.Two angles of a triangle are 75° and 85°. What is the third angle? (a) 160° (b) 20° (c) 105° (d) 95°
  10. 10.What are the roots of x² − 17x + 72 = 0? (a) 1 and 16 (b) 8 and 9 (c) 1 and 72 (d) 2 and 36
  11. 11.How many two-digit numbers have digits that add up to 6? (a) 5 (b) 6 (c) 10 (d) 7
  12. 12.In a league, 14 teams each play every other team once. How many matches are played? (a) 196 (b) 91 (c) 105 (d) 182
  13. 13.Two angles of a triangle are 55° and 30°. What is the third angle? (a) 85° (b) 95° (c) 125° (d) 150°
  14. 14.What is the sum of all the whole numbers from 1 to 56? (a) 1540 (b) 1568 (c) 1596 (d) 3192
  15. 15.Which term of the AP 10, 14, 18, 22, … is 166? (a) 41st (b) 42nd (c) 39th (d) 40th
  16. 16.A canteen sells 8 kinds of sandwich and 6 kinds of juice. In how many ways can you choose one sandwich and one juice? (a) 56 (b) 14 (c) 48 (d) 96
  17. 17.A fair dice is rolled once. What is the probability of getting a multiple of 3? (a) 1/3 (b) 2/3 (c) 1/6 (d) 1/2
  18. 18.Two angles of a triangle are 20° and 135°. What is the third angle? (a) 155° (b) 45° (c) 160° (d) 25°
  19. 19.What is the midpoint of the line joining (6, 10) and (14, 0)? (a) (5, 10) (b) (4, 5) (c) (10, 5) (d) (20, 10)
  20. 20.In how many different ways can the letters of the word MATHS be arranged? (a) 25 (b) 120 (c) 20 (d) 240
  21. 21.If a + b = 12 and ab = 35, what is a² + b²? (a) 214 (b) 144 (c) 109 (d) 74
  22. 22.Using (a − b)² = a² − 2ab + b², what is 59²? (a) 3721 (b) 3491 (c) 3481 (d) 3599
  23. 23.In an isosceles triangle, the two equal angles are 45° each. What is the third angle? (a) 45° (b) 91° (c) 90° (d) 135°
  24. 24.In how many different ways can the letters of the word PEN be arranged? (a) 12 (b) 6 (c) 3 (d) 9
  25. 25.Two coins are tossed together. What is the probability of getting a head on every coin? (a) 1/2 (b) 1/3 (c) 3/4 (d) 1/4
  26. 26.Using (a + b)² = a² + 2ab + b², what is 51²? (a) 2401 (b) 2501 (c) 2601 (d) 2611
  27. 27.In a league, 7 teams each play every other team once. How many matches are played? (a) 49 (b) 42 (c) 21 (d) 28
  28. 28.A bag has 4 red balls and 10 black balls. Without looking, how many balls must you take out to be sure of getting 2 black balls? (a) 2 (b) 14 (c) 5 (d) 6
  29. 29.How many times is the digit 8 written when you write all the numbers from 1 to 51? (a) 3 (b) 4 (c) 6 (d) 5
  30. 30.What is the sum of the first 6 terms of the AP 5, 11, 17, …? (a) 138 (b) 35 (c) 240 (d) 120
  31. 31.How many prime numbers are there between 70 and 100? (a) 7 (b) 5 (c) 8 (d) 6
  32. 32.Two angles of a triangle are 50° and 95°. What is the exterior angle at the third corner? (a) 130° (b) 145° (c) 85° (d) 35°
  33. 33.Two angles of a triangle are 20° and 55°. What is the third angle? (a) 160° (b) 105° (c) 125° (d) 75°
  34. 34.A book has 116 pages, numbered 1, 2, 3 and so on. How many digits are printed to number all the pages? (a) 241 (b) 240 (c) 232 (d) 231
  35. 35.A team of 2 players is to be chosen from 6 players. How many different teams are possible? (a) 36 (b) 30 (c) 12 (d) 15
  36. 36.Which is the correct factorisation of x² + 11x + 30? (a) (x − 5)(x − 6) (b) (x + 1)(x + 30) (c) (x + 5)(x + 6) (d) (x + 1)(x + 10)
  37. 37.In a class of 39 children, 16 like maths, 15 like science and 2 like both. How many like neither? (a) 2 (b) 8 (c) 12 (d) 10
  38. 38.A book has 39 pages, numbered 1, 2, 3 and so on. How many digits are printed to number all the pages? (a) 70 (b) 60 (c) 69 (d) 78
  39. 39.Two angles of a triangle are 70° and 80°. What is the third angle? (a) 110° (b) 30° (c) 150° (d) 100°
  40. 40.What is the sum of the first 10 terms of the AP 9, 10, 11, …? (a) 135 (b) 270 (c) 140 (d) 18

Answer key

  1. (x + 5)(x + 7)
  2. 55
  3. 2 × 3 × 5
  4. 1/12
  5. 3/4
  6. 1/6
  7. 91
  8. 3
  9. 20°
  10. 8 and 9
  11. 6
  12. 91
  13. 95°
  14. 1596
  15. 40th
  16. 48
  17. 1/3
  18. 25°
  19. (10, 5)
  20. 120
  21. 74
  22. 3481
  23. 90°
  24. 6
  25. 1/4
  26. 2601
  27. 21
  28. 6
  29. 5
  30. 120
  31. 6
  32. 145°
  33. 105°
  34. 240
  35. 15
  36. (x + 5)(x + 6)
  37. 10
  38. 69
  39. 30°
  40. 135

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