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 2016 June Q4
4 When people visit a certain large shop, on average \(34 \%\) of them do not buy anything, \(53 \%\) spend less than \(
) 50\( and \)13 \%\( spend at least \)\\( 50\).
  1. 15 people visiting the shop are chosen at random. Calculate the probability that at least 14 of them buy something.
  2. \(n\) people visiting the shop are chosen at random. The probability that none of them spends at least \(
    ) 50\( is less than 0.04 . Find the smallest possible value of \)n$.
CAIE S1 2016 June Q5
5 The following are the maximum daily wind speeds in kilometres per hour for the first two weeks in April for two towns, Bronlea and Rogate.
Bronlea21456332733214282413172522
Rogate754152371113261823161034
  1. Draw a back-to-back stem-and-leaf diagram to represent this information.
  2. Write down the median of the maximum wind speeds for Bronlea and find the interquartile range for Rogate.
  3. Use your diagram to make one comparison between the maximum wind speeds in the two towns.
CAIE S1 2016 June Q6
6 The time in minutes taken by Peter to walk to the shop and buy a newspaper is normally distributed with mean 9.5 and standard deviation 1.3.
  1. Find the probability that on a randomly chosen day Peter takes longer than 10.2 minutes.
  2. On \(90 \%\) of days he takes longer than \(t\) minutes. Find the value of \(t\).
  3. Calculate an estimate of the number of days in a year ( 365 days) on which Peter takes less than 8.8 minutes to walk to the shop and buy a newspaper.
CAIE S1 2016 June Q7
7
  1. Find the number of different arrangements which can be made of all 10 letters of the word WALLFLOWER if
    1. there are no restrictions,
    2. there are exactly six letters between the two Ws.
  2. A team of 6 people is to be chosen from 5 swimmers, 7 athletes and 4 cyclists. There must be at least 1 from each activity and there must be more athletes than cyclists. Find the number of different ways in which the team can be chosen.
CAIE S1 2016 June Q1
1 In a group of 30 adults, 25 are right-handed and 8 wear spectacles. The number who are right-handed and do not wear spectacles is 19 .
  1. Copy and complete the following table to show the number of adults in each category.
    Wears spectaclesDoes not wear spectaclesTotal
    Right-handed
    Not right-handed
    Total30
    An adult is chosen at random from the group. Event \(X\) is 'the adult chosen is right-handed'; event \(Y\) is 'the adult chosen wears spectacles'.
  2. Determine whether \(X\) and \(Y\) are independent events, justifying your answer.
CAIE S1 2016 June Q2
2 A group of children played a computer game which measured their time in seconds to perform a certain task. A summary of the times taken by girls and boys in the group is shown below.
MinimumLower quartileMedianUpper quartileMaximum
Girls55.57913
Boys468.51116
  1. On graph paper, draw two box-and-whisker plots in a single diagram to illustrate the times taken by girls and boys to perform this task.
  2. State two comparisons of the times taken by girls and boys.
CAIE S1 2016 June Q3
3 Two ordinary fair dice are thrown. The resulting score is found as follows.
  • If the two dice show different numbers, the score is the smaller of the two numbers.
  • If the two dice show equal numbers, the score is 0 .
    1. Draw up the probability distribution table for the score.
    2. Calculate the expected score.
CAIE S1 2016 June Q4
4 The monthly rental prices, \(
) x$, for 9 apartments in a certain city are listed and are summarised as follows. $$\Sigma ( x - c ) = 1845 \quad \Sigma ( x - c ) ^ { 2 } = 477450$$ The mean monthly rental price is \(
) 2205$.
  1. Find the value of the constant \(c\).
  2. Find the variance of these values of \(x\).
  3. Another apartment is added to the list. The mean monthly rental price is now \(
    ) 2120.50$. Find the rental price of this additional apartment.
CAIE S1 2016 June Q5
5 The heights of school desks have a normal distribution with mean 69 cm and standard deviation \(\sigma \mathrm { cm }\). It is known that 15.5\% of these desks have a height greater than 70 cm .
  1. Find the value of \(\sigma\). When Jodu sits at a desk, his knees are at a height of 58 cm above the floor. A desk is comfortable for Jodu if his knees are at least 9 cm below the top of the desk. Jodu's school has 300 desks.
  2. Calculate an estimate of the number of these desks that are comfortable for Jodu.
CAIE S1 2016 June Q6
6 Find the number of ways all 9 letters of the word EVERGREEN can be arranged if
  1. there are no restrictions,
  2. the first letter is R and the last letter is G ,
  3. the Es are all together. Three letters from the 9 letters of the word EVERGREEN are selected.
  4. Find the number of selections which contain no Es and exactly 1 R .
  5. Find the number of selections which contain no Es.
CAIE S1 2016 June Q7
7 Passengers are travelling to Picton by minibus. The probability that each passenger carries a backpack is 0.65 , independently of other passengers. Each minibus has seats for 12 passengers.
  1. Find the probability that, in a full minibus travelling to Picton, between 8 passengers and 10 passengers inclusive carry a backpack.
  2. Passengers get on to an empty minibus. Find the probability that the fourth passenger who gets on to the minibus will be the first to be carrying a backpack.
  3. Find the probability that, of a random sample of 250 full minibuses travelling to Picton, more than 54 will contain exactly 7 passengers carrying backpacks.
CAIE S1 2017 June Q1
1 Kadijat noted the weights, \(x\) grams, of 30 chocolate buns. Her results are summarised by $$\Sigma ( x - k ) = 315 , \quad \Sigma ( x - k ) ^ { 2 } = 4022$$ where \(k\) is a constant. The mean weight of the buns is 50.5 grams.
  1. Find the value of \(k\).
  2. Find the standard deviation of \(x\).
CAIE S1 2017 June Q2
2 Ashfaq throws two fair dice and notes the numbers obtained. \(R\) is the event 'The product of the two numbers is 12 '. \(T\) is the event 'One of the numbers is odd and one of the numbers is even'. By finding appropriate probabilities, determine whether events \(R\) and \(T\) are independent.
CAIE S1 2017 June Q3
3 Redbury United soccer team play a match every week. Each match can be won, drawn or lost. At the beginning of the soccer season the probability that Redbury United win their first match is \(\frac { 3 } { 5 }\), with equal probabilities of losing or drawing. If they win the first match, the probability that they win the second match is \(\frac { 7 } { 10 }\) and the probability that they lose the second match is \(\frac { 1 } { 10 }\). If they draw the first match they are equally likely to win, draw or lose the second match. If they lose the first match, the probability that they win the second match is \(\frac { 3 } { 10 }\) and the probability that they draw the second match is \(\frac { 1 } { 20 }\).
  1. Draw a fully labelled tree diagram to represent the first two matches played by Redbury United in the soccer season.
  2. Given that Redbury United win the second match, find the probability that they lose the first match.
CAIE S1 2017 June Q4
4 The times taken, \(t\) seconds, by 1140 people to solve a puzzle are summarised in the table.
Time \(( t\) seconds \()\)\(0 \leqslant t < 20\)\(20 \leqslant t < 40\)\(40 \leqslant t < 60\)\(60 \leqslant t < 100\)\(100 \leqslant t < 140\)
Number of people320280220220100
  1. On the grid, draw a histogram to illustrate this information.
    \includegraphics[max width=\textwidth, alt={}, center]{7652f36c-59b5-4fcd-b17b-d796dc82aec0-05_812_1406_804_411}
  2. Calculate an estimate of the mean of \(t\).
CAIE S1 2017 June Q5
5 Eggs are sold in boxes of 20. Cracked eggs occur independently and the mean number of cracked eggs in a box is 1.4 .
  1. Calculate the probability that a randomly chosen box contains exactly 2 cracked eggs.
  2. Calculate the probability that a randomly chosen box contains at least 1 cracked egg.
  3. A shop sells \(n\) of these boxes of eggs. Find the smallest value of \(n\) such that the probability of there being at least 1 cracked egg in each box sold is less than 0.01 .
CAIE S1 2017 June Q6
6
  1. The random variable \(X\) has a normal distribution with mean \(\mu\) and standard deviation \(\sigma\). You are given that \(\sigma = 0.25 \mu\) and \(\mathrm { P } ( X < 6.8 ) = 0.75\).
    1. Find the value of \(\mu\).
    2. Find \(\mathrm { P } ( X < 4.7 )\).
  2. The lengths of metal rods have a normal distribution with mean 16 cm and standard deviation 0.2 cm . Rods which are shorter than 15.75 cm or longer than 16.25 cm are not usable. Find the expected number of usable rods in a batch of 1000 rods.
CAIE S1 2017 June Q7
7
  1. Eight children of different ages stand in a random order in a line. Find the number of different ways this can be done if none of the three youngest children stand next to each other.
  2. David chooses 5 chocolates from 6 different dark chocolates, 4 different white chocolates and 1 milk chocolate. He must choose at least one of each type. Find the number of different selections he can make.
  3. A password for Chelsea's computer consists of 4 characters in a particular order. The characters are chosen from the following.
    • The 26 capital letters A to Z
    • The 9 digits 1 to 9
    • The 5 symbols \# \~{} * ? !
    The password must include at least one capital letter, at least one digit and at least one symbol. No character can be repeated. Find the number of different passwords that Chelsea can make.
CAIE S1 2017 June Q1
1 Rani and Diksha go shopping for clothes.
  1. Rani buys 4 identical vests, 3 identical sweaters and 1 coat. Each vest costs \(
    ) 5.50\( and the coat costs \)\\( 90\). The mean cost of Rani's 8 items is \(
    ) 29\(. Find the cost of a sweater.
  2. Diksha buys 1 hat and 4 identical shirts. The mean cost of Diksha's 5 items is \)\\( 26\) and the standard deviation is \(
    ) 0\(. Explain how you can tell that Diksha spends \)\\( 104\) on shirts.
CAIE S1 2017 June Q2
2 Anabel measured the lengths, in centimetres, of 200 caterpillars. Her results are illustrated in the cumulative frequency graph below.
\includegraphics[max width=\textwidth, alt={}, center]{184a04ac-4396-4a0f-8fa8-ab11a4b6df39-03_1173_1195_356_466}
  1. Estimate the median and the interquartile range of the lengths.
  2. Estimate how many caterpillars had a length of between 2 and 3.5 cm .
  3. 6\% of caterpillars were of length \(l\) centimetres or more. Estimate \(l\).
CAIE S1 2017 June Q3
3 In a probability distribution the random variable \(X\) takes the value \(x\) with probability \(k x ^ { 2 }\), where \(k\) is a constant and \(x\) takes values \(- 2 , - 1,2,4\) only.
  1. Show that \(\mathrm { P } ( X = - 2 )\) has the same value as \(\mathrm { P } ( X = 2 )\).
  2. Draw up the probability distribution table for \(X\), in terms of \(k\), and find the value of \(k\).
  3. Find \(\mathrm { E } ( X )\).
CAIE S1 2017 June Q4
4 Two identical biased triangular spinners with sides marked 1,2 and 3 are spun. For each spinner, the probabilities of landing on the sides marked 1,2 and 3 are \(p , q\) and \(r\) respectively. The score is the sum of the numbers on the sides on which the spinners land. You are given that \(\mathrm { P } (\) score is \(6 ) = \frac { 1 } { 36 }\) and \(\mathrm { P } (\) score is \(5 ) = \frac { 1 } { 9 }\). Find the values of \(p , q\) and \(r\).
CAIE S1 2017 June Q5
5 The lengths of videos of a certain popular song have a normal distribution with mean 3.9 minutes. \(18 \%\) of these videos last for longer than 4.2 minutes.
  1. Find the standard deviation of the lengths of these videos.
  2. Find the probability that the length of a randomly chosen video differs from the mean by less than half a minute.
    The lengths of videos of another popular song have a normal distribution with the same mean of 3.9 minutes but the standard deviation is twice the standard deviation in part (i). The probability that the length of a randomly chosen video of this song differs from the mean by less than half a minute is denoted by \(p\).
  3. Without any further calculation, determine whether \(p\) is more than, equal to, or less than your answer to part (ii). You must explain your reasoning.
CAIE S1 2017 June Q6
6 A library contains 4 identical copies of book \(A , 2\) identical copies of book \(B\) and 5 identical copies of book \(C\). These 11 books are arranged on a shelf in the library.
  1. Calculate the number of different arrangements if the end books are either both book \(A\) or both book \(B\).
  2. Calculate the number of different arrangements if all the books \(A\) are next to each other and none of the books \(B\) are next to each other.
CAIE S1 2017 June Q7
7 During the school holidays, each day Khalid either rides on his bicycle with probability 0.6 , or on his skateboard with probability 0.4 . Khalid does not ride on both on the same day. If he rides on his bicycle then the probability that he hurts himself is 0.05 . If he rides on his skateboard the probability that he hurts himself is 0.75 .
  1. Find the probability that Khalid hurts himself on any particular day.
  2. Given that Khalid hurts himself on a particular day, find the probability that he is riding on his skateboard.
  3. There are 45 days of school holidays. Show that the variance of the number of days Khalid rides on his skateboard is the same as the variance of the number of days that Khalid rides on his bicycle.
  4. Find the probability that Khalid rides on his skateboard on at least 2 of 10 randomly chosen days in the school holidays.