Intermediate maths challenge practice

Intermediate maths challenge practice paper for Year 11

A free paper in the intermediate challenge's shape: 25 questions in 60 minutes, five choices each, for 135 marks. Print it, then check every answer and its working at the end.

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Scholar Tests · Sample paper

Intermediate maths challenge-style paper, Year 11

25 questions · 135 marks · 60 minutes · no negative marking

Name ______________________ Date ____________

Questions 1 to 15

Questions 1 to 15, 5 marks each. Circle the letter of the best answer.

  1. 1.

    Ava rolls a die 10 times and gets: 1, 4, 6, 2, 4, 6, 2, 5, 1, 5. From these results, what is the experimental probability of rolling a 1?

    1. A3/10
    2. B1/10
    3. C1/4
    4. D1/5
    5. E1/6
  2. 2.

    How many numbers are in the list 8, 19, 30, …, 206, where each number is 11 more than the one before?

    1. A21
    2. B18
    3. C17
    4. D19
    5. E20
  3. 3.

    The mean of 7 numbers is 54. One of the numbers is taken away, and the mean of the 6 numbers left is 50. What number was taken away?

    1. A28
    2. B79
    3. C4
    4. D78
    5. E77
  4. 4.

    A right-angled triangle has a longest side of 15 cm and another side of 9 cm. What is its area, in square centimetres?

    1. A108
    2. B55
    3. C54
    4. D90
    5. E53
  5. 5.

    What is the highest common factor of 4, 6?

    1. A12
    2. B4
    3. C2
    4. D3
    5. E10
  6. 6.

    A box holds 19 cards numbered 1 to 19. Cards 1 to 4 are yellow, 5 to 11 are purple and 12 to 19 are orange. One card is taken out without looking. What is the chance that it is purple and its number is a factor of 36?

    1. A2/7
    2. B2/19
    3. C8/19
    4. D15/19
    5. E7/19
  7. 7.

    How many two-digit whole numbers have digits that add up to 10?

    1. A7
    2. B11
    3. C8
    4. D10
    5. E9
  8. 8.

    What is the last digit of 783?

    1. A4
    2. B3
    3. C1
    4. D7
    5. E9
  9. 9.

    The number 398! means 1 × 2 × 3 × … × 398. How many zeros are at the end of 398! when it is written out in full?

    1. A39
    2. B79
    3. C96
    4. D97
    5. E98
  10. 10.

    How many factors does 100 have?

    1. A9
    2. B5
    3. C2
    4. D7
    5. E10
  11. 11.

    At a holiday camp, 14 children have a cat, 11 play football and 5 do both. Everyone does at least one of the two. How many children are there?

    1. A25 children
    2. B30 children
    3. C15 children
    4. D20 children
    5. E14 children
  12. 12.

    Sara, Kai, Wes, Pia and Hana are playing, and one of them has the ball.
    Sara says: "I do not have the ball."
    Kai says: "I have the ball."
    Wes says: "Kai does not have the ball."
    Pia says: "Hana has the ball."
    Hana says: "I have the ball."
    Exactly one of them is telling the truth. Who has the ball?

    1. AHana
    2. BPia
    3. CKai
    4. DSara
    5. EWes
  13. 13.

    How many positive whole numbers divide exactly into 27440?

    1. A11
    2. B40
    3. C38
    4. D12
    5. E39
  14. 14.

    In the drawing below, B, C and D lie on a straight line, and the sides with a tick are equal. What is the size of the angle marked x?

    1. A64°
    2. B58°
    3. C61°
    4. D122°
    5. E29°
  15. 15.

    A town's streets form a grid of square blocks. Ana walks from one corner of a rectangle 6 blocks wide and 4 blocks high to the opposite corner, always walking east or north along the streets. How many different routes can she take?

    1. A211
    2. B209
    3. C10
    4. D24
    5. E210

Questions 16 to 25

Questions 16 to 25, 6 marks each. Circle the letter of the best answer.

  1. 16.

    The equation x2 − 12x + 32 = 0 has two whole-number solutions. What is the sum of the squares of its two solutions?

    1. A112
    2. B144
    3. C208
    4. D80
    5. E12
  2. 17.

    The number 287! means 1 × 2 × 3 × … × 287. How many zeros are at the end of 287! when it is written out in full?

    1. A28
    2. B57
    3. C71
    4. D70
    5. E69
  3. 18.

    How many positive whole numbers divide exactly into 2646?

    1. A6
    2. B24
    3. C22
    4. D25
    5. E9
  4. 19.

    What is the remainder when 2177 is divided by 9?

    1. A4
    2. B9
    3. C8
    4. D7
    5. E6
  5. 20.

    What is the last digit of 12383?

    1. A8
    2. B7
    3. C4
    4. D2
    5. E6
  6. 21.

    The mean of 9 numbers is 21. One of the numbers is taken away, and the mean of the 8 numbers left is 23.5. What number was taken away?

    1. A0
    2. B3
    3. C2
    4. D4
    5. E1
  7. 22.

    How many pairs of positive whole numbers (x, y) make 4x + 8y = 92 true?

    1. A10
    2. B12
    3. C9
    4. D13
    5. E11
  8. 23.

    Lily rolls a die 14 times and gets: 3, 1, 5, 3, 6, 4, 6, 2, 5, 5, 6, 5, 5, 2. From these results, what is the experimental probability of rolling a prime number?

    1. A9/5
    2. B9/14
    3. C5/14
    4. D5/7
    5. E1/2
  9. 24.

    A jar holds 17 counters numbered 1 to 17. Counters 1 to 7 are pink, 8 to 10 are brown and 11 to 17 are silver. One counter is taken out without looking. What is the chance that it is brown and its number is a factor of 30?

    1. A7/17
    2. B1/17
    3. C3/17
    4. D10/17
    5. E1/3
  10. 25.

    If 8x = 32x − 6, what is the value of x?

    1. A30
    2. B15
    3. C12
    4. D6
    5. E14

Answers and working

Questions 1 to 15

  1. 1.D (1/5). 2 of the 10 rolls were a 1, and 2/10 in simplest form is 1/5. So the answer is 1/5.
  2. 2.D (19). From 8 to 206 is 198, which is 18 steps of 11. The list has one more number than steps: 19.
  3. 3.D (78). The 7 numbers add up to 7 × 54 = 378. The 6 left add up to 6 × 50 = 300. The number taken away is the difference, 78.
  4. 4.C (54). By Pythagoras, the third side is √(152 − 92) = 12 cm. The area is half of 9 × 12, which is 54.
  5. 5.C (2). 4 = 2 × 2, 6 = 2 × 3. The primes they all share multiply to 2.
  6. 6.B (2/19). Of the purple ones, 5 to 11, the ones whose number is a factor of 36: 6 and 9. That is 2 of the 19: 2/19.
  7. 7.E (9). Fix the first digit, then count the ways the other digit can make up the rest of 10. Adding the cases gives 9.
  8. 8.B (3). The last digit of the powers of 7 repeat in a cycle. Find the cycle, then find where the power 83 falls in it: the answer is 3.
  9. 9.D (97). Each zero at the end comes from a 10 = 2 × 5, and there are far more 2s than 5s, so count the 5s. There are 79 multiples of 5, 15 multiples of 25, 3 multiples of 125, so 398! ends in 97 zeros.
  10. 10.A (9). The factors of 100 are 1, 2, 4, 5, 10, 20, 25, 50, 100: 9 of them.
  11. 11.D (20 children). 14 + 11 counts the 5 who do both twice, so take them off once: 14 + 11 − 5. So the answer is 20 children.
  12. 12.D (Sara). Try each child as the one with the ball and count the true statements. Only when Sara has it is exactly one statement true.
  13. 13.B (40). 27440 = 24 × 51 × 73. A divisor takes each prime to any power from 0 up, so there are 5 × 2 × 4 = 40 divisors.
  14. 14.E (29°). Angle ACB = 58°, so angle ACD = 122°, and the second triangle's equal angles are 29° each. The angle marked x is 29°.
  15. 15.E (210). Every route is 10 blocks long, 6 of them east. Choosing which 6 are east gives 210 routes.

Questions 16 to 25

  1. 16.D (80). The solutions add to 12 and multiply to 32. The sum of the squares is 122 − 2 × 32 = 80.
  2. 17.D (70). Each zero at the end comes from a 10 = 2 × 5, and there are far more 2s than 5s, so count the 5s. There are 57 multiples of 5, 11 multiples of 25, 2 multiples of 125, so 287! ends in 70 zeros.
  3. 18.B (24). 2646 = 21 × 33 × 72. A divisor takes each prime to any power from 0 up, so there are 2 × 4 × 3 = 24 divisors.
  4. 19.C (8). The remainders of 21, 22, 23, … when divided by 9 repeat in a cycle. Find where 177 falls in that cycle: the remainder is 8.
  5. 20.A (8). The last digit of the powers of 12 repeat in a cycle. Find the cycle, then find where the power 383 falls in it: the answer is 8.
  6. 21.E (1). The 9 numbers add up to 9 × 21 = 189. The 8 left add up to 8 × 23.5 = 188. The number taken away is the difference, 1.
  7. 22.E (11). Try each positive x with 4x below 92, and keep it when 92 − 4x is a positive multiple of 8. Once one pair is found, the next comes by adding 8 to x and taking 4 from y. There are 11 pairs.
  8. 23.B (9/14). 9 of the 14 rolls were a prime number, and 9/14 in simplest form is 9/14. So the answer is 9/14.
  9. 24.B (1/17). Of the brown ones, 8 to 10, the ones whose number is a factor of 30: 10. That is 1 of the 17: 1/17.
  10. 25.B (15). 8 = 23 and 32 = 25, so 3x = 5(x − 6). That gives 30 = 2x, so x = 15.

Written by Scholar Tests in the intermediate maths challenge's published shape. Not a past paper; we are not connected with the UK Mathematics Trust.

Sit a new paper online

Every online paper is new, timed like the real thing, and marked the moment it ends, 5 marks for each of the first 15 and 6 for the rest.

  • Year 11

    25 questions · 60 minutes

    Number theory, algebra, counting and geometry, the hardest last. No calculator.

Questions parents ask

Is this paper right for Year 11?
Yes. This is the top year for the intermediate challenge, and the last ten lean on the hardest algebra and counting.
How is it marked?
Questions 1 to 15 are worth 5 marks each and 16 to 25 are worth 6, so the paper is out of 135. Our marking takes nothing off for a wrong answer; check the rules printed on the real paper, as some have taken marks off late in the paper.
Can my child use a calculator?
No. The challenge allows no calculator or measuring instruments, only rough paper. Practise the same way.
Are these real past papers?
No. We write our own questions in the challenge's published shape. Past papers belong to the UK Mathematics Trust, and we are not connected with it.

Papers for other years

Year 9 paperYear 10 paper

The other maths challenges

Primary Maths ChallengeJunior Maths Challenge