Questions — CAIE S1 (785 questions)

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AQA AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Paper 1 Further Paper 2 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics M1 M2 M3 Paper 1 Paper 2 Paper 3 S1 S2 S3 CAIE FP1 FP2 Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 4 M1 M2 P1 P2 P3 S1 S2 Edexcel AEA AS Paper 1 AS Paper 2 C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS M1 M2 M3 M4 M5 P1 P2 P3 P4 PMT Mocks Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 OCR AS Pure C1 C2 C3 C4 D1 D2 FD1 AS FM1 AS FP1 FP1 AS FP2 FP3 FS1 AS Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Mechanics Further Mechanics AS Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Statistics Further Statistics AS H240/01 H240/02 H240/03 M1 M2 M3 M4 Mechanics 1 PURE Pure 1 S1 S2 S3 S4 Stats 1 OCR MEI AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further Extra Pure Further Mechanics A AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Pure Core Further Pure Core AS Further Pure with Technology Further Statistics A AS Further Statistics B AS Further Statistics Major Further Statistics Minor M1 M2 M3 M4 Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 SPS SPS ASFM SPS ASFM Mechanics SPS ASFM Pure SPS ASFM Statistics SPS FM SPS FM Mechanics SPS FM Pure SPS FM Statistics SPS SM SPS SM Mechanics SPS SM Pure SPS SM Statistics WJEC Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 Unit 1 Unit 2 Unit 3 Unit 4
CAIE S1 2017 November Q3
3 A box contains 6 identical-sized discs, of which 4 are blue and 2 are red. Discs are taken at random from the box in turn and not replaced. Let \(X\) be the number of discs taken, up to and including the first blue one.
  1. Show that \(\mathrm { P } ( X = 3 ) = \frac { 1 } { 15 }\).
  2. Draw up the probability distribution table for \(X\).
CAIE S1 2017 November Q4
4 A fair tetrahedral die has faces numbered \(1,2,3,4\). A coin is biased so that the probability of showing a head when thrown is \(\frac { 1 } { 3 }\). The die is thrown once and the number \(n\) that it lands on is noted. The biased coin is then thrown \(n\) times. So, for example, if the die lands on 3 , the coin is thrown 3 times.
  1. Find the probability that the die lands on 4 and the number of times the coin shows heads is 2 .
  2. Find the probability that the die lands on 3 and the number of times the coin shows heads is 3 .
  3. Find the probability that the number the die lands on is the same as the number of times the coin shows heads.
CAIE S1 2017 November Q5
5 Blank CDs are packed in boxes of 30 . The probability that a blank CD is faulty is 0.04 . A box is rejected if more than 2 of the blank CDs are faulty.
  1. Find the probability that a box is rejected.
  2. 280 boxes are chosen randomly. Use an approximation to find the probability that at least 30 of these boxes are rejected.
CAIE S1 2017 November Q6
6
  1. Find the number of different 3-digit numbers greater than 300 that can be made from the digits \(1,2,3,4,6,8\) if
    1. no digit can be repeated,
    2. a digit can be repeated and the number made is even.
  2. A team of 5 is chosen from 6 boys and 4 girls. Find the number of ways the team can be chosen if
    1. there are no restrictions,
    2. the team contains more boys than girls.
CAIE S1 2017 November Q7
7 In Jimpuri the weights, in kilograms, of boys aged 16 years have a normal distribution with mean 61.4 and standard deviation 12.3.
  1. Find the probability that a randomly chosen boy aged 16 years in Jimpuri weighs more than 65 kilograms.
  2. For boys aged 16 years in Jimpuri, \(25 \%\) have a weight between 65 kilograms and \(k\) kilograms, where \(k\) is greater than 65 . Find \(k\).
    In Brigville the weights, in kilograms, of boys aged 16 years have a normal distribution. \(99 \%\) of the boys weigh less than 97.2 kilograms and \(33 \%\) of the boys weigh less than 55.2 kilograms.
  3. Find the mean and standard deviation of the weights of boys aged 16 years in Brigville.
CAIE S1 2017 November Q1
1 A statistics student asks people to complete a survey. The probability that a randomly chosen person agrees to complete the survey is 0.2 . Find the probability that at least one of the first three people asked agrees to complete the survey.
CAIE S1 2017 November Q2
2 Tien measured the arm lengths, \(x \mathrm {~cm}\), of 20 people in his class. He found that \(\Sigma x = 1218\) and the standard deviation of \(x\) was 4.2. Calculate \(\Sigma ( x - 45 )\) and \(\Sigma ( x - 45 ) ^ { 2 }\).
CAIE S1 2017 November Q3
3 At the end of a revision course in mathematics, students have to pass a test to gain a certificate. The probability of any student passing the test at the first attempt is 0.85 . Those students who fail are allowed to retake the test once, and the probability of any student passing the retake test is 0.65 .
  1. Draw a fully labelled tree diagram to show all the outcomes.
  2. Given that a student gains the certificate, find the probability that this student fails the test on the first attempt.
CAIE S1 2017 November Q4
4 A fair die with faces numbered \(1,2,2,2,3,6\) is thrown. The score, \(X\), is found by squaring the number on the face the die shows and then subtracting 4.
  1. Draw up a table to show the probability distribution of \(X\).
  2. Find \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).
CAIE S1 2017 November Q5
5 The number of Olympic medals won in the 2012 Olympic Games by the top 27 countries is shown below.
1048882654438353428
281818171714131312
1210101096522
  1. Draw a stem-and-leaf diagram to illustrate the data.
  2. Find the median and quartiles and draw a box-and-whisker plot on the grid.
    \includegraphics[max width=\textwidth, alt={}, center]{4c2afa86-960c-473e-970c-ed16c8434fec-07_1006_1406_1007_411}
CAIE S1 2017 November Q6
5 marks
6 A car park has spaces for 18 cars, arranged in a line. On one day there are 5 cars, of different makes, parked in randomly chosen positions and 13 empty spaces.
  1. Find the number of possible arrangements of the 5 cars in the car park.
  2. Find the probability that the 5 cars are not all next to each other.
    On another day, 12 cars of different makes are parked in the car park. 5 of these cars are red, 4 are white and 3 are black. Elizabeth selects 3 of these cars.
    [0pt]
  3. Find the number of selections Elizabeth can make that include cars of at least 2 different colours. [5]
CAIE S1 2017 November Q7
7 Josie aims to catch a bus which departs at a fixed time every day. Josie arrives at the bus stop \(T\) minutes before the bus departs, where \(T \sim \mathrm {~N} \left( 5.3,2.1 ^ { 2 } \right)\).
  1. Find the probability that Josie has to wait longer than 6 minutes at the bus stop.
    On \(5 \%\) of days Josie has to wait longer than \(x\) minutes at the bus stop.
  2. Find the value of \(x\).
  3. Find the probability that Josie waits longer than \(x\) minutes on fewer than 3 days in 10 days.
  4. Find the probability that Josie misses the bus.
CAIE S1 2018 November Q2
2 A random variable \(X\) has the probability distribution shown in the following table, where \(p\) is a constant.
\(x\)- 10124
\(\mathrm { P } ( X = x )\)\(p\)\(p\)\(2 p\)\(2 p\)0.1
  1. Find the value of \(p\).
  2. Given that \(\mathrm { E } ( X ) = 1.15\), find \(\operatorname { Var } ( X )\).
CAIE S1 2018 November Q3
3 marks
3 In an orchestra, there are 11 violinists, 5 cellists and 4 double bass players. A small group of 6 musicians is to be selected from these 20.
  1. How many different selections of 6 musicians can be made if there must be at least 4 violinists, at least 1 cellist and no more than 1 double bass player?
    The small group that is selected contains 4 violinists, 1 cellist and 1 double bass player. They sit in a line to perform a concert.
    [0pt]
  2. How many different arrangements are there of these 6 musicians if the violinists must sit together? [3]
CAIE S1 2018 November Q4
4
  1. It is given that \(X \sim \mathrm {~N} ( 31.4,3.6 )\). Find the probability that a randomly chosen value of \(X\) is less than 29.4.
  2. The lengths of fish of a particular species are modelled by a normal distribution. A scientist measures the lengths of 400 randomly chosen fish of this species. He finds that 42 fish are less than 12 cm long and 58 are more than 19 cm long. Find estimates for the mean and standard deviation of the lengths of fish of this species.
CAIE S1 2018 November Q5
5 At the Nonland Business College, all students sit an accountancy examination at the end of their first year of study. On average, \(80 \%\) of the students pass this examination.
  1. A random sample of 9 students who will take this examination is chosen. Find the probability that at most 6 of these students will pass the examination.
  2. A random sample of 200 students who will take this examination is chosen. Use a suitable approximate distribution to find the probability that more than 166 of them will pass the examination.
  3. Justify the use of your approximate distribution in part (ii).
CAIE S1 2018 November Q6
6 The daily rainfall, \(x \mathrm {~mm}\), in a certain village is recorded on 250 consecutive days. The results are summarised in the following cumulative frequency table.
Rainfall, \(x \mathrm {~mm}\)\(x \leqslant 20\)\(x \leqslant 30\)\(x \leqslant 40\)\(x \leqslant 50\)\(x \leqslant 70\)\(x \leqslant 100\)
Cumulative frequency5294142172222250
  1. On the grid, draw a cumulative frequency graph to illustrate the data.
  2. On 100 of the days, the rainfall was \(k \mathrm {~mm}\) or more. Use your graph to estimate the value of \(k\).
  3. Calculate estimates of the mean and standard deviation of the daily rainfall in this village.
CAIE S1 2018 November Q7
7 In a group of students, the numbers of boys and girls studying Art, Music and Drama are given in the following table. Each of these 160 students is studying exactly one of these subjects.
ArtMusicDrama
Boys244032
Girls151237
  1. Find the probability that a randomly chosen student is studying Music.
  2. Determine whether the events 'a randomly chosen student is a boy' and 'a randomly chosen student is studying Music' are independent, justifying your answer.
  3. Find the probability that a randomly chosen student is not studying Drama, given that the student is a girl.
  4. Three students are chosen at random. Find the probability that exactly 1 is studying Music and exactly 2 are boys.
    If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.
CAIE S1 2018 November Q1
1
  1. How many different arrangements are there of the 11 letters in the word MISSISSIPPI?
  2. Two letters are chosen at random from the 11 letters in the word MISSISSIPPI. Find the probability that these two letters are the same.
CAIE S1 2018 November Q2
2 The following back-to-back stem-and-leaf diagram shows the reaction times in seconds in an experiment involving two groups of people, \(A\) and \(B\).
\(A\)\(B\)
(4)420020567(3)
(5)9850021122377(6)
(8)98753222221356689(7)
(6)8765212345788999(8)
(3)863242456788(7)
(1)0250278(4)
Key: 5 | 22 | 6 means a reaction time of 0.225 seconds for \(A\) and 0.226 seconds for \(B\)
  1. Find the median and the interquartile range for group \(A\).
    The median value for group \(B\) is 0.235 seconds, the lower quartile is 0.217 seconds and the upper quartile is 0.245 seconds.
  2. Draw box-and-whisker plots for groups \(A\) and \(B\) on the grid.
    \includegraphics[max width=\textwidth, alt={}, center]{62812433-baee-490a-bad4-b6b0f917c234-03_805_1495_1729_365}
CAIE S1 2018 November Q3
3 Jake attempts the crossword puzzle in his daily newspaper every day. The probability that he will complete the puzzle on any given day is 0.75 , independently of all other days.
  1. Find the probability that he will complete the puzzle at least three times over a period of five days.
    Kenny also attempts the puzzle every day. The probability that he will complete the puzzle on a Monday is 0.8 . The probability that he will complete it on a Tuesday is 0.9 if he completed it on the previous day and 0.6 if he did not complete it on the previous day.
  2. Find the probability that Kenny will complete the puzzle on at least one of the two days Monday and Tuesday in a randomly chosen week.
CAIE S1 2018 November Q4
4
  1. Find the number of different ways that 5 boys and 6 girls can stand in a row if all the boys stand together and all the girls stand together.
  2. Find the number of different ways that 5 boys and 6 girls can stand in a row if no boy stands next to another boy.
CAIE S1 2018 November Q5
5 The Quivers Archery club has 12 Junior members and 20 Senior members. For the Junior members, the mean age is 15.5 years and the standard deviation of the ages is 1.2 years. The ages of the Senior members are summarised by \(\Sigma y = 910\) and \(\Sigma y ^ { 2 } = 42850\), where \(y\) is the age of a Senior member in years.
  1. Find the mean age of all 32 members of the club.
  2. Find the standard deviation of the ages of all 32 members of the club.
CAIE S1 2018 November Q6
6 A fair red spinner has 4 sides, numbered 1,2,3,4. A fair blue spinner has 3 sides, numbered 1,2,3. When a spinner is spun, the score is the number on the side on which it lands. The spinners are spun at the same time. The random variable \(X\) denotes the score on the red spinner minus the score on the blue spinner.
  1. Draw up the probability distribution table for \(X\).
  2. Find \(\operatorname { Var } ( X )\).
  3. Find the probability that \(X\) is equal to 1 , given that \(X\) is non-zero.
CAIE S1 2018 November Q7
7
  1. The time, \(X\) hours, for which students use a games machine in any given day has a normal distribution with mean 3.24 hours and standard deviation 0.96 hours.
    1. On how many days of the year ( 365 days) would you expect a randomly chosen student to use a games machine for less than 4 hours?
    2. Find the value of \(k\) such that \(\mathrm { P } ( X > k ) = 0.2\).
    3. Find the probability that the number of hours for which a randomly chosen student uses a games machine in a day is within 1.5 standard deviations of the mean.
  2. The variable \(Y\) is normally distributed with mean \(\mu\) and standard deviation \(\sigma\), where \(4 \sigma = 3 \mu\) and \(\mu \neq 0\). Find the probability that a randomly chosen value of \(Y\) is positive.
    If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.