Is this paper right for Years 11 and 12?

Yes. It is the Senior level, which the competition sets for Years 11 and 12 (Form 5, 6 and pre-university in Malaysia). A child in the younger year will find more of the last questions hard. That is how it is meant to be.

Questions 1 to 10 are worth 3 points each, 11 to 20 are worth 4 and the last 10 are worth 5, so the paper is out of 120. Our marking takes nothing off for a wrong answer; check your own country's rules, as some take a quarter of the points off.

Can my child use a calculator?

No. The competition allows no calculator, only rough paper. Practise the same way.

The printed papers are free, with every answer. Online, a free account includes 3 practice tests. The maths competition pack (RM99 for 6 months) opens unlimited tests and shows the right answer and the working for every question.

Are these real past papers?

No. We write our own questions in the competition's published shape. Past papers belong to the organisers. We are not connected with Kangourou sans Frontières, Kangaroo Math Malaysia or any national organiser of the competition.

Maths Competition Practice for Kangaroo Math, Years 11-12

Scholar Tests · Sample paper

Maths competition paper, Years 11 and 12

30 questions · 120 points · 75 minutes · no negative marking

Name ______________________ Date ____________

Questions 1 to 10

Questions 1 to 10, 3 points each. Circle the letter of the best answer.

1.
The equation x² − 12x + 27 = 0 has two whole-number solutions. What is the larger solution?
1. A12
2. B10
3. C3
4. D27
5. E9
2.
Ava, Jada and Pia are playing, and one of them has the ball.
Ava says: "Jada does not have the ball."
Jada says: "Pia does not have the ball."
Pia says: "Jada has the ball."
Exactly one of them is telling the truth. Who has the ball?
1. AYara
2. BAva
3. CBen
4. DJada
5. EPia
3.
If 9^x = 27^{x − 1}, what is the value of x?
1. A5
2. B1
3. C4
4. D2
5. E3
4.
A right-angled triangle has a longest side of 51 cm and another side of 24 cm. What is its area, in square centimetres?
1. A540
2. B1080
3. C541
4. D612
5. E539
5.
What is the remainder when 3²²⁹ is divided by 7?
1. A5
2. B4
3. C2
4. D3
5. E1
6.
The number 30! means 1 × 2 × 3 × … × 30. How many zeros are at the end of 30! when it is written out in full?
1. A8
2. B6
3. C7
4. D3
5. E9
7.
A town's streets form a grid of square blocks. Ana walks from one corner of a rectangle 3 blocks wide and 3 blocks high to the opposite corner, always walking east or north along the streets. How many different routes can she take?
1. A20
2. B19
3. C9
4. D6
5. E21
8.
What is the last digit of 2⁵⁶?
1. A7
2. B5
3. C2
4. D8
5. E6
9.
The mean of 10 numbers is 58. One of the numbers is taken away, and the mean of the 9 numbers left is 64. What number was taken away?
1. A5
2. B2
3. C6
4. D3
5. E4
10.
How many positive whole numbers divide exactly into 3087?
1. A6
2. B12
3. C11
4. D7
5. E10

Questions 11 to 20

Questions 11 to 20, 4 points each. Circle the letter of the best answer.

11.
The equation x² − 24x + 140 = 0 has two whole-number solutions. What is the sum of the squares of its two solutions?
1. A296
2. B436
3. C856
4. D576
5. E24
12.
Three consecutive prime numbers add up to 31. What is the smallest of them?
1. A10
2. B5
3. C11
4. D7
5. E13
13.
How many numbers are in the list 5, 17, 29, …, 1481, where each number is 12 more than the one before?
1. A122
2. B126
3. C124
4. D123
5. E125
14.
What is the remainder when 5²⁵¹ is divided by 7?
1. A4
2. B1
3. C3
4. D6
5. E2
15.
How many pairs of positive whole numbers (x, y) make 5x + 9y = 155 true?
1. A3
2. B5
3. C2
4. D1
5. E4
16.
How many three-digit whole numbers have digits that add up to 8?
1. A36
2. B37
3. C34
4. D21
5. E35
17.
If 2^x = 4^{x − 3}, what is the value of x?
1. A5
2. B3
3. C7
4. D6
5. E8
18.
How many positive whole numbers divide exactly into 4536?
1. A12
2. B11
3. C40
4. D39
5. E38
19.
Two whole numbers add to 20 and multiply to 36. What is the larger number?
1. A10
2. B36
3. C18
4. D20
5. E2
20.
The mean of 18 numbers is 74. One of the numbers is taken away, and the mean of the 17 numbers left is 68. What number was taken away?
1. A6
2. B108
3. C177
4. D175
5. E176

Questions 21 to 30

Questions 21 to 30, 5 points each. Circle the letter of the best answer.

21.
What is the number formed by the last two digits of 23¹²⁹¹?
1. A27
2. B21
3. C93
4. D49
5. E23
22.
If 3^x = 81^{x − 3}, what is the value of x?
1. A5
2. B9
3. C3
4. D12
5. E4
23.
A box holds 12 cards numbered 1 to 12. Cards 1 to 3 are yellow, 4 to 6 are purple and 7 to 12 are orange. One card is taken out without looking. What is the chance that it is purple or its number is a multiple of 5 (or both)?
1. A1/12
2. B5/12
3. C1/3
4. D1/4
5. E1/6
24.
10 points are marked on a circle. How many four-sided shapes have all 4 of their corners at these points?
1. A209
2. B211
3. C45
4. D210
5. E720
25.
Oscar rolls a die 22 times and gets: 3, 5, 6, 3, 6, 1, 2, 1, 3, 2, 4, 1, 3, 6, 5, 5, 3, 1, 1, 3, 6, 6. From these results, what is the experimental probability of rolling a number less than 3?
1. A7/15
2. B1/3
3. C15/22
4. D3/11
5. E7/22
26.
The equation x² − 14x + 24 = 0 has two whole-number solutions. What is the sum of the cubes of its two solutions?
1. A1737
2. B3752
3. C1736
4. D2744
5. E148
27.
A right-angled triangle has a longest side of 50 cm and another side of 14 cm. What is its area, in square centimetres?
1. A336
2. B1200
3. C672
4. D350
5. E337
28.
What is the remainder when 2¹³⁵⁰ is divided by 13?
1. A13
2. B12
3. C6
4. D11
5. E14
29.
How many positive whole numbers divide exactly into 2² × 3¹ × 7³?
1. A6
2. B9
3. C22
4. D23
5. E24
30.
What is the total of all the numbers in the list 27, 34, 41, …, 468, where each number is 7 more than the one before?
1. A15839
2. B16315
3. C15840
4. D15841
5. E15372

Answers and working

Questions 1 to 10

1.E (9). Find two numbers that add to 12 and multiply to 27: 3 and 9. The larger is 9.
2.E (Pia). Try each child as the one with the ball and count the true statements. Only when Pia has it is exactly one statement true.
3.E (3). 9 = 3² and 27 = 3³, so 2x = 3(x − 1). That gives 3 = 1x, so x = 3.
4.A (540). By Pythagoras, the third side is √(51² − 24²) = 45 cm. The area is half of 24 × 45, which is 540.
5.D (3). The remainders of 3¹, 3², 3³, … when divided by 7 repeat in a cycle. Find where 229 falls in that cycle: the remainder is 3.
6.C (7). 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 6 multiples of 5, 1 multiples of 25, so 30! ends in 7 zeros.
7.A (20). Every route is 6 blocks long, 3 of them east. Choosing which 3 are east gives 20 routes.
8.E (6). The last digit of the powers of 2 repeat in a cycle. Find the cycle, then find where the power 56 falls in it: the answer is 6.
9.E (4). The 10 numbers add up to 10 × 58 = 580. The 9 left add up to 9 × 64 = 576. The number taken away is the difference, 4.
10.B (12). 3087 = 3² × 7³. A divisor takes each prime to any power from 0 up, so there are 3 × 4 = 12 divisors.

Questions 11 to 20

11.A (296). The solutions add to 24 and multiply to 140. The sum of the squares is 24² − 2 × 140 = 296.
12.D (7). The primes 7, 11, 13 come one after another and add up to 31. The smallest is 7.
13.C (124). From 5 to 1481 is 1476, which is 123 steps of 12. The list has one more number than steps: 124.
14.C (3). The remainders of 5¹, 5², 5³, … when divided by 7 repeat in a cycle. Find where 251 falls in that cycle: the remainder is 3.
15.A (3). Try each positive x with 5x below 155, and keep it when 155 − 5x is a positive multiple of 9. Once one pair is found, the next comes by adding 9 to x and taking 5 from y. There are 3 pairs.
16.A (36). Fix the first digit, then count the ways the other digits can make up the rest of 8. Adding the cases gives 36.
17.D (6). 2 = 2¹ and 4 = 2², so 1x = 2(x − 3). That gives 6 = 1x, so x = 6.
18.C (40). 4536 = 2³ × 3⁴ × 7¹. A divisor takes each prime to any power from 0 up, so there are 4 × 5 × 2 = 40 divisors.
19.C (18). Only 2 and 18 multiply to 36 and add to 20. The larger is 18: 18.
20.E (176). The 18 numbers add up to 18 × 74 = 1332. The 17 left add up to 17 × 68 = 1156. The number taken away is the difference, 176.

Questions 21 to 30

21.A (27). The last two digits of the powers of 23 repeat in a cycle. Find the cycle, then find where the power 1291 falls in it: the answer is 27.
22.E (4). 3 = 3¹ and 81 = 3⁴, so 1x = 4(x − 3). That gives 12 = 3x, so x = 4.
23.C (1/3). 3 are purple and 2 have a number that is a multiple of 5, but 1 is both, so 4 pass. That is 4/12 = 1/3.
24.D (210). Any 4 of the points make exactly one four-sided shape, so count the ways to choose 4 of the 10: 210.
25.E (7/22). 7 of the 22 rolls were a number less than 3, and 7/22 in simplest form is 7/22. So the answer is 7/22.
26.C (1736). The solutions add to 14 and multiply to 24. The sum of the cubes is 14³ − 3 × 24 × 14 = 1736.
27.A (336). By Pythagoras, the third side is √(50² − 14²) = 48 cm. The area is half of 14 × 48, which is 336.
28.B (12). The remainders of 2¹, 2², 2³, … when divided by 13 repeat in a cycle. Find where 1350 falls in that cycle: the remainder is 12.
29.E (24). 4116 = 2² × 3¹ × 7³. A divisor takes each prime to any power from 0 up, so there are 3 × 2 × 4 = 24 divisors.
30.C (15840). There are 64 numbers. Pair the first with the last, the second with the second last, and so on: each pair adds to 495. So the total is 64 × 495 ÷ 2 = 15840.

Written by Scholar Tests in the Kangaroo Math Competition's published shape. Not a past paper. We are not connected with Kangourou sans Frontières, Kangaroo Math Malaysia or any national organiser of the competition.

Maths competition practice paper for Years 11 and 12

Maths competition paper, Years 11 and 12

Questions 1 to 10

Questions 11 to 20

Questions 21 to 30

Answers and working

Questions 1 to 10

Questions 11 to 20

Questions 21 to 30

Sit a new paper online

Senior (Years 11 and 12)

Questions parents ask

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