Pearson Edexcel GCSE (9–1) Mathematics
1MA1
Original practice paper written in the style of Edexcel papers — not a Pearson publication. This pack is untiered and can include both Foundation and Higher content.
Name: __________________________________
Class: __________________________________
Answer every question. Show all stages of your working. The total number of marks is 105.
This topic pack mixes calculator and non-calculator practice. Each question states whether a calculator may be used.
1
120 people were asked whether they own a bicycle. 75 of them were adults and the rest were children. Of the adults, 45 own a bicycle. Of the children, 27 own a bicycle.
Either calculator policy
(a)How many of the 120 people were children?
(1)
(b)How many of the 120 people do not own a bicycle?
(2)
(c)One of the 120 people is chosen at random. Write down the probability that this person owns a bicycle.
(1)
(Total for Question 1 is 4 marks)
2
A bag contains 5 red counters and 3 blue counters. A counter is taken at random, its colour recorded, and then it is put back in the bag. A second counter is then taken at random.
Either calculator policy
(a)Write down the probability that the first counter taken is red.
(1)
(b)Work out the probability that both counters taken are red.
(2)
(Total for Question 2 is 3 marks)
3
A spinner can land on red, blue, green or white. The table shows the probabilities of some of the colours. P(red) = 0.4, P(blue) = 0.25, P(green) = 0.15.
Either calculator policy
(a)Work out the probability that the spinner lands on white.
(2)
(b)The spinner is spun 200 times. Work out an estimate for the number of times it lands on red.
(2)
(Total for Question 3 is 4 marks)
4
Two fair ordinary six-sided dice are rolled. The scores on the two dice are added together to give a total.
Non-calculator
(a)Work out the probability that the total is 7.
(2)
(b)Work out the probability that the total is greater than 9.
(2)
(Total for Question 4 is 4 marks)
5
A biased coin is flipped 200 times. It lands on heads 130 times.
Either calculator policy
(a)Work out the relative frequency (experimental probability) of the coin landing on heads. Give your answer as a decimal.
(2)
(b)The coin is to be flipped 500 times. Work out an estimate for the number of times it will land on heads.
(1)
(c)Explain why this estimate is likely to be reliable.
(1)
(Total for Question 5 is 4 marks)
6
A café offers a set lunch. A customer chooses one sandwich (cheese, ham or tuna) and one drink (tea or coffee).
Either calculator policy
(a)List all the possible combinations of one sandwich and one drink.
(2)
(b)One of these combinations is chosen at random. All combinations are equally likely. Work out the probability that it includes tea.
(1)
(Total for Question 6 is 3 marks)
7
90 students travel to school by bus or on foot. 54 of the students are in Year 10 and the rest are in Year 11. Of the Year 10 students, 21 walk to school. Of the Year 11 students, 18 walk to school.
Non-calculator
(a)Work out the number of Year 11 students.
(1)
(b)Work out the total number of students who walk to school.
(2)
(Total for Question 7 is 3 marks)
8
160 households are surveyed. 92 households own a car. Of the households that own a car, 48 also own a bicycle. Of the households that do not own a car, 34 own a bicycle.
Either calculator policy
(a)Work out the number of households that do not own a bicycle.
(2)
(b)One of the households is chosen at random. Work out the probability that it owns a bicycle. Give your answer as a fraction in its simplest form.
(2)
(Total for Question 8 is 4 marks)
9
A fair spinner has 8 equal sections. Three sections are blue. The spinner is spun 240 times.
Non-calculator
Work out how many times the spinner is expected to land on blue.
(2)
(Total for Question 9 is 2 marks)
10
A spinner has three equal sections labelled A, B and C. It is claimed that the spinner is fair. In 300 spins it lands on A 126 times, on B 93 times and on C 81 times.
Either calculator policy
(a)If the spinner is fair, work out the expected number of times it lands on each letter.
(1)
(b)Do the results suggest that the spinner may not be fair? Give a reason for your answer.
(2)
(Total for Question 10 is 3 marks)
11
A tennis player gets 52 of her first serves in during 80 attempts.
Calculator
(a)Work out the relative frequency of the player getting her first serve in.
(2)
(b)Use this relative frequency to estimate how many first serves she will get in during 200 attempts.
(1)
(Total for Question 11 is 3 marks)
12
A biased spinner can land on red or blue. Leila spins it 30 times and gets red 18 times. Marcus spins the same spinner 400 times and gets red 264 times.
Calculator
(a)Work out Leila's estimate for the probability of red.
(1)
(b)Work out Marcus's estimate for the probability of red.
(1)
(c)Use all the results to work out a combined estimate for the probability of red. Give your answer to 3 decimal places.
(2)
(d)Explain why Marcus's estimate is likely to be more reliable than Leila's estimate.
(1)
(Total for Question 12 is 5 marks)
13
A spinner can land on A, B, C or D. P(A) = 0.18, P(B) = 0.27 and P(C) = 0.31.
Non-calculator
(a)Work out P(D).
(2)
(b)Work out the probability that the spinner does not land on A.
(1)
(Total for Question 13 is 3 marks)
14
A bag contains only red, yellow and green counters. P(red) = 2x, P(yellow) = x + 0.1 and P(green) = 0.3.
Non-calculator
(a)Form an equation in x and solve the equation.
(3)
(b)Write down P(red).
(1)
(Total for Question 14 is 4 marks)
15
The same coin is tossed in two experiments. In Experiment A the coin is tossed 20 times. In Experiment B the coin is tossed 500 times.
Either calculator policy
Which experiment is more likely to give a relative frequency close to the true probability of heads? Give a reason for your answer.
(2)
(Total for Question 15 is 2 marks)
16
A spinner lands on red 14 times in its first 40 spins. It is then spun 360 more times. Across all 400 spins it lands on red 134 times.
Calculator
(a)Work out the relative frequency of red after the first 40 spins.
(1)
(b)Work out the relative frequency of red after all 400 spins.
(2)
(c)Use the result from all 400 spins to estimate the number of reds in a further 1200 spins.
(2)
(Total for Question 16 is 5 marks)
17
Nadia chooses one hat and one scarf. The hats are black and grey. The scarves are red, blue and green.
Non-calculator
(a)List all the possible combinations of one hat and one scarf.
(2)
(b)How many combinations are possible?
(1)
(Total for Question 17 is 3 marks)
18
A code is made by choosing one digit from 1, 2, 3 and 4, followed by one letter from A, B and C.
Non-calculator
(a)Work out the total number of different codes.
(2)
(b)List all the codes that begin with an even digit.
(2)
(Total for Question 18 is 4 marks)
19
A three-digit number is made using each of the digits 2, 4 and 7 exactly once.
Either calculator policy
(a)List all the possible three-digit numbers.
(2)
(b)One of the possible numbers is chosen at random. Work out the probability that it is greater than 400.
(2)
(Total for Question 19 is 4 marks)
20
A fair coin is tossed and a fair spinner numbered 1, 2 and 3 is spun.
Non-calculator
(a)List the six possible outcomes. Use H and T for the coin.
(1)
(b)Work out the probability of getting heads and a number greater than 1.
(2)
(Total for Question 20 is 3 marks)
21
Two fair four-sided spinners are each numbered 1, 2, 3 and 4. Both spinners are spun.
Either calculator policy
(a)Work out the probability that the two scores differ by 2.
(2)
(b)Work out the probability that the total score is greater than 5.
(2)
(Total for Question 21 is 4 marks)
22
Spinner A is fair and numbered 1, 2, 3 and 4. Spinner B is fair and numbered 1, 2, 3, 4 and 5. Each spinner is spun once.
Either calculator policy
(a)Work out the probability that the total score is a prime number.
(2)
(b)Work out the probability that the product of the two scores is a multiple of 6.
(2)
(c)Which event is more likely: a prime total or a product that is a multiple of 6?
(1)
(Total for Question 22 is 5 marks)
23
A box contains 4 cards with vowels on them and 6 cards with consonants on them. A card is chosen at random, replaced, and then a second card is chosen.
Non-calculator
Work out the probability that exactly one of the two cards has a vowel on it.
(3)
(Total for Question 23 is 3 marks)
24
A bag contains 5 red counters, 3 green counters and 2 blue counters. Two counters are taken at random without replacement.
Calculator
(a)Work out the probability that the first counter is red and the second counter is green.
(2)
(b)Work out the probability that the two counters are the same colour.
(3)
(Total for Question 24 is 5 marks)
25
A bag contains n red counters and 4 blue counters. Two counters are taken at random without replacement. The probability that both counters are blue is 1/6.
Calculator
Work out the total number of counters in the bag. Show your algebraic working.
(5)
(Total for Question 25 is 5 marks)
26
120 students are surveyed. 68 students learn a musical instrument, 45 students play a school sport, and 30 students do both.
Either calculator policy
(a)A student who learns a musical instrument is chosen at random. Work out the probability that this student also plays a school sport.
(2)
(b)Work out the number of students who do neither activity.
(2)
(Total for Question 26 is 4 marks)
27
In a group of 80 students, 46 study French, 38 study Spanish and 22 study both languages.
Calculator
(a)A student who studies Spanish is chosen at random. Work out the probability that this student also studies French.
(2)
(b)A student who studies at least one of the two languages is chosen at random. Work out the probability that this student studies exactly one of the languages.
(3)
(Total for Question 27 is 5 marks)
28
A factory makes 150 items on the morning shift and 250 items on the evening shift. Nine morning-shift items and 10 evening-shift items are defective. One of the 400 items is chosen at random.
Calculator
(a)Work out the probability that the chosen item is defective.
(2)
(b)Given that the chosen item is defective, work out the probability that it was made on the evening shift.
(2)
(Total for Question 28 is 4 marks)
Pearson Edexcel GCSE (9–1) Mathematics
1
(a)
Answer: 45
B1 for 45 (from 120 − 75).
(b)
Answer: 48
M1 for (75 − 45) + (45 − 27), or 120 − (45 + 27). A1 for 48.
(c)
Answer: 72/120 (= 3/5)
B1 for 72/120 or any correct equivalent (e.g. 3/5, 0.6).
4 marks
2
(a)
Answer: 5/8
B1 for 5/8.
(b)
Answer: 25/64
M1 for 5/8 × 5/8 (independent events, with replacement). A1 for 25/64.
3 marks
3
(a)
Answer: 0.2
M1 for 1 − (0.4 + 0.25 + 0.15) (probabilities sum to 1). A1 for 0.2.
(b)
Answer: 80
M1 for 0.4 × 200. A1 for 80.
4 marks
4
(a)
Answer: 1/6
M1 for identifying that there are 36 equally likely outcomes and 6 give a total of 7 (e.g. a listed possibility space). A1 for 6/36 or 1/6.
(b)
Answer: 1/6
M1 for identifying the 6 outcomes with total 10, 11 or 12 out of 36. A1 for 6/36 or 1/6.
4 marks
5
(a)
Answer: 0.65
M1 for 130 ÷ 200. A1 for 0.65.
(b)
Answer: 325
B1 for 325 (0.65 × 500, ft their probability).
(c)
Answer: The relative frequency was found from a large number of trials (200 flips), so it should be close to the true probability.
B1 for a valid reason referring to the large number of trials making the relative frequency a good estimate of the true probability.
4 marks
6
(a)
Answer: cheese & tea, cheese & coffee, ham & tea, ham & coffee, tuna & tea, tuna & coffee (6 combinations)
B2 for all 6 correct combinations with none repeated or missing. B1 for a systematic attempt giving at least 4 correct combinations.
(b)
Answer: 1/2
B1 for 3/6 or 1/2 (ft their list).
3 marks
7
(a)
Answer: 36
B1 for 36.
(b)
Answer: 39
M1 for 21 + 18. A1 for 39.
3 marks
8
(a)
Answer: 78
M1 for (92 - 48) + (68 - 34), or an equivalent complete frequency-tree calculation. A1 for 78.
(b)
Answer: 41/80
M1 for (48 + 34)/160. A1 for 41/80.
4 marks
9
Answer: 90
M1 for 3/8 × 240. A1 for 90.
2 marks
10
(a)
Answer: 100 times on each letter
B1 for 100 on each letter.
(b)
Answer: Yes. The observed frequencies are not close to the expected 100 each, particularly the 126 results for A. The results are evidence, but do not prove, that the spinner is biased.
B1 for a conclusion supported by a comparison with 100, e.g. A occurred 126 times rather than about 100. B1 for recognising that experimental variation is possible, so the results suggest bias but do not prove it.
3 marks
11
(a)
Answer: 0.65
M1 for 52 ÷ 80. A1 for 0.65.
(b)
Answer: 130
B1 for 130 (0.65 × 200), ft their relative frequency.
3 marks
12
(a)
Answer: 0.6
B1 for 18/30 or 0.6.
(b)
Answer: 0.66
B1 for 264/400 or 0.66.
(c)
Answer: 0.656
M1 for (18 + 264)/(30 + 400). A1 for 0.656 to 3 decimal places.
(d)
Answer: Marcus used many more spins, so random variation has less effect and his relative frequency is likely to be closer to the true probability.
B1 for explaining that the larger number of trials makes the estimate more reliable / likely to be closer to the true probability.
5 marks
13
(a)
Answer: 0.24
M1 for 1 - (0.18 + 0.27 + 0.31). A1 for 0.24.
(b)
Answer: 0.82
B1 for 0.82.
3 marks
14
(a)
Answer: 2x + (x + 0.1) + 0.3 = 1, so x = 0.2
M1 for 2x + x + 0.1 + 0.3 = 1. M1 for reducing to 3x = 0.6. A1 for x = 0.2.
(b)
Answer: 0.4
B1 for 0.4, ft 2 × their positive value of x.
4 marks
15
Answer: Experiment B, because it uses many more trials, so the effect of random variation is smaller.
B1 for Experiment B. B1 for a reason referring to the larger number of trials reducing random variation / giving a more reliable estimate.
2 marks
16
(a)
Answer: 0.35
B1 for 14/40 or 0.35.
(b)
Answer: 0.335
M1 for 134 ÷ 400. A1 for 0.335.
(c)
Answer: 402
M1 for 0.335 × 1200. A1 for 402, ft their answer to part (b).
5 marks
17
(a)
Answer: black-red, black-blue, black-green, grey-red, grey-blue, grey-green
B2 for all 6 combinations with none missing or repeated. B1 for a systematic list containing 4 or 5 correct combinations.
(b)
Answer: 6
B1 for 6, ft a complete list.
3 marks
18
(a)
Answer: 12
M1 for 4 × 3. A1 for 12.
(b)
Answer: 2A, 2B, 2C, 4A, 4B, 4C
B2 for all 6 correct codes with none missing or repeated. B1 for 4 or 5 correct codes.
4 marks
19
(a)
Answer: 247, 274, 427, 472, 724, 742
B2 for all 6 numbers with none missing or repeated. B1 for a systematic list containing 4 or 5 correct numbers.
(b)
Answer: 2/3
M1 for identifying 4 favourable numbers out of 6. A1 for 4/6 or 2/3.
4 marks
20
(a)
Answer: H1, H2, H3, T1, T2, T3
B1 for all 6 outcomes with none missing or repeated.
(b)
Answer: 1/3
M1 for identifying H2 and H3 as 2 outcomes out of 6. A1 for 2/6 or 1/3.
3 marks
21
(a)
Answer: 1/4
M1 for identifying 4 favourable ordered pairs, (1,3), (2,4), (3,1), (4,2), out of 16. A1 for 4/16 or 1/4.
(b)
Answer: 3/8
M1 for identifying 6 favourable ordered pairs out of 16. A1 for 6/16 or 3/8.
4 marks
22
(a)
Answer: 1/2
M1 for a complete 4 by 5 possibility space or equivalent systematic method. A1 for 10/20 or 1/2.
(b)
Answer: 1/5
M1 for identifying the 4 favourable ordered pairs (2,3), (3,2), (3,4), (4,3) out of 20. A1 for 4/20 or 1/5.
(c)
Answer: A prime total
B1 for a prime total, with probabilities compared consistently.
5 marks
23
Answer: 12/25
M1 for 4/10 × 6/10 and 6/10 × 4/10. M1 for adding the two mutually exclusive orders. A1 for 48/100 or 12/25.
3 marks
24
(a)
Answer: 1/6
M1 for 5/10 × 3/9. A1 for 15/90 or 1/6.
(b)
Answer: 14/45
M1 for 5/10 × 4/9, 3/10 × 2/9 and 2/10 × 1/9. M1 for adding the three probabilities. A1 for 28/90 or 14/45.
5 marks
25
Answer: 9
M1 for 4/(n + 4) × 3/(n + 3) = 1/6. M1 for (n + 4)(n + 3) = 72. M1 for n² + 7n - 60 = 0. A1 for n = 5, rejecting n = -12. A1 for the total 9.
5 marks
26
(a)
Answer: 15/34
M1 for 30/68. A1 for 15/34.
(b)
Answer: 37
M1 for 120 - (68 + 45 - 30). A1 for 37.
4 marks
27
(a)
Answer: 11/19
M1 for 22/38. A1 for 11/19.
(b)
Answer: 20/31
M1 for exactly one = (46 - 22) + (38 - 22) = 40. M1 for at least one = 46 + 38 - 22 = 62. A1 for 40/62 or 20/31.
5 marks
28
(a)
Answer: 19/400
M1 for (9 + 10)/400. A1 for 19/400.
(b)
Answer: 10/19
M1 for using the 19 defective items as the conditional sample space. A1 for 10/19.
4 marks