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 March Q1
1 For 10 values of \(x\) the mean is 86.2 and \(\Sigma ( x - a ) = 362\). Find the value of
  1. \(\Sigma x\),
  2. the constant \(a\).
CAIE S1 2016 March Q2
2 A flower shop has 5 yellow roses, 3 red roses and 2 white roses. Martin chooses 3 roses at random. Draw up the probability distribution table for the number of white roses Martin chooses.
CAIE S1 2016 March Q3
3 A fair eight-sided die has faces marked \(1,2,3,4,5,6,7,8\). The score when the die is thrown is the number on the face the die lands on. The die is thrown twice.
  • Event \(R\) is 'one of the scores is exactly 3 greater than the other score'.
  • Event \(S\) is 'the product of the scores is more than 19'.
    1. Find the probability of \(R\).
    2. Find the probability of \(S\).
    3. Determine whether events \(R\) and \(S\) are independent. Justify your answer.
CAIE S1 2016 March Q4
4 A survey was made of the journey times of 63 people who cycle to work in a certain town. The results are summarised in the following cumulative frequency table.
Journey time (minutes)\(\leqslant 10\)\(\leqslant 25\)\(\leqslant 45\)\(\leqslant 60\)\(\leqslant 80\)
Cumulative frequency018505963
  1. State how many journey times were between 25 and 45 minutes.
  2. Draw a histogram on graph paper to represent the data.
  3. Calculate an estimate of the mean journey time.
CAIE S1 2016 March Q5
5 In a certain town, 35\% of the people take a holiday abroad and 65\% take a holiday in their own country. Of those going abroad \(80 \%\) go to the seaside, \(15 \%\) go camping and \(5 \%\) take a city break. Of those taking a holiday in their own country, \(20 \%\) go to the seaside and the rest are divided equally between camping and a city break.
  1. A person is chosen at random. Given that the person chosen goes camping, find the probability that the person goes abroad.
  2. A group of \(n\) people is chosen randomly. The probability of all the people in the group taking a holiday in their own country is less than 0.002 . Find the smallest possible value of \(n\).
CAIE S1 2016 March Q6
6 Hannah chooses 5 singers from 15 applicants to appear in a concert. She lists the 5 singers in the order in which they will perform.
  1. How many different lists can Hannah make? Of the 15 applicants, 10 are female and 5 are male.
  2. Find the number of lists in which the first performer is male, the second is female, the third is male, the fourth is female and the fifth is male. Hannah's friend Ami would like the group of 5 performers to include more males than females. The order in which they perform is no longer relevant.
  3. Find the number of different selections of 5 performers with more males than females.
  4. Two of the applicants are Mr and Mrs Blake. Find the number of different selections that include Mr and Mrs Blake and also fulfil Ami's requirement.
CAIE S1 2016 March Q7
7 The times taken by a garage to fit a tow bar onto a car have a normal distribution with mean \(m\) hours and standard deviation 0.35 hours. It is found that \(95 \%\) of times taken are longer than 0.9 hours.
  1. Find the value of \(m\).
  2. On one day 4 cars have a tow bar fitted. Find the probability that none of them takes more than 2 hours to fit. The times in hours taken by another garage to fit a tow bar onto a car have the distribution \(\mathrm { N } \left( \mu , \sigma ^ { 2 } \right)\) where \(\mu = 3 \sigma\).
  3. Find the probability that it takes more than \(0.6 \mu\) hours to fit a tow bar onto a randomly chosen car at this garage.
CAIE S1 2017 March Q1
4 marks
1 Twelve values of \(x\) are shown below.
1761.61758.51762.31761.41759.41759.1
1762.51761.91762.41761.91762.81761.0
Find the mean and standard deviation of \(( x - 1760 )\). Hence find the mean and standard deviation of \(x\). [4]
CAIE S1 2017 March Q2
2 A bag contains 10 pink balloons, 9 yellow balloons, 12 green balloons and 9 white balloons. 7 balloons are selected at random without replacement. Find the probability that exactly 3 of them are green.
CAIE S1 2017 March Q3
3 It is found that \(10 \%\) of the population enjoy watching Historical Drama on television. Use an appropriate approximation to find the probability that, out of 160 people chosen randomly, more than 17 people enjoy watching Historical Drama on television.
CAIE S1 2017 March Q4
4 The weights in kilograms of packets of cereal were noted correct to 4 significant figures. The following stem-and-leaf diagram shows the data.
7473\(( 1 )\)
748125779\(( 6 )\)
749022235556789\(( 12 )\)
750112223445677889\(( 15 )\)
7510023344455779\(( 13 )\)
75200011223444\(( 11 )\)
7532\(( 1 )\)
Key: 748 | 5 represents 0.7485 kg .
  1. On the grid, draw a box-and-whisker plot to represent the data.
    \includegraphics[max width=\textwidth, alt={}, center]{556a1cc2-47ef-4ef7-a8f6-42850c303531-05_814_1604_1336_299}
  2. Name a distribution that might be a suitable model for the weights of this type of cereal packet. Justify your answer.
CAIE S1 2017 March Q5
5
  1. A plate of cakes holds 12 different cakes. Find the number of ways these cakes can be shared between Alex and James if each receives an odd number of cakes.
  2. Another plate holds 7 cup cakes, each with a different colour icing, and 4 brownies, each of a different size. Find the number of different ways these 11 cakes can be arranged in a row if no brownie is next to another brownie.
  3. A plate of biscuits holds 4 identical chocolate biscuits, 6 identical shortbread biscuits and 2 identical gingerbread biscuits. These biscuits are all placed in a row. Find how many different arrangements are possible if the chocolate biscuits are all kept together.
CAIE S1 2017 March Q6
6 Pack \(A\) consists of ten cards numbered \(0,0,1,1,1,1,1,3,3,3\). Pack \(B\) consists of six cards numbered \(0,0,2,2,2,2\). One card is chosen at random from each pack. The random variable \(X\) is defined as the sum of the two numbers on the cards.
  1. Show that \(\mathrm { P } ( X = 2 ) = \frac { 2 } { 15 }\).
    \includegraphics[max width=\textwidth, alt={}, center]{556a1cc2-47ef-4ef7-a8f6-42850c303531-08_59_1569_497_328}
  2. Draw up the probability distribution table for \(X\).
  3. Given that \(X = 3\), find the probability that the card chosen from pack \(A\) is a 1 .
CAIE S1 2017 March Q7
7
  1. The lengths, in centimetres, of middle fingers of women in Raneland have a normal distribution with mean \(\mu\) and standard deviation \(\sigma\). It is found that \(25 \%\) of these women have fingers longer than 8.8 cm and \(17.5 \%\) have fingers shorter than 7.7 cm .
    1. Find the values of \(\mu\) and \(\sigma\).
      The lengths, in centimetres, of middle fingers of women in Snoland have a normal distribution with mean 7.9 and standard deviation 0.44. A random sample of 5 women from Snoland is chosen.
    2. Find the probability that exactly 3 of these women have middle fingers shorter than 8.2 cm .
  2. The random variable \(X\) has a normal distribution with mean equal to the standard deviation. Find the probability that a particular value of \(X\) is less than 1.5 times the mean.
CAIE S1 2019 March Q1
1 On each day that Tamar goes to work, he wears either a blue suit with probability 0.6 or a grey suit with probability 0.4 . If he wears a blue suit then the probability that he wears red socks is 0.2 . If he wears a grey suit then the probability that he wears red socks is 0.32 .
  1. Find the probability that Tamar wears red socks on any particular day that he is at work.
  2. Given that Tamar is not wearing red socks at work, find the probability that he is wearing a grey suit.
CAIE S1 2019 March Q2
2 For 40 values of the variable \(x\), it is given that \(\Sigma ( x - c ) ^ { 2 } = 3099.2\), where \(c\) is a constant. The standard deviation of these values of \(x\) is 3.2 .
  1. Find the value of \(\Sigma ( x - c )\).
  2. Given that \(c = 50\), find the mean of these values of \(x\).
CAIE S1 2019 March Q3
3 The times taken, in minutes, for trains to travel between Alphaton and Beeton are normally distributed with mean 140 and standard deviation 12.
  1. Find the probability that a randomly chosen train will take less than 132 minutes to travel between Alphaton and Beeton.
  2. The probability that a randomly chosen train takes more than \(k\) minutes to travel between Alphaton and Beeton is 0.675 . Find the value of \(k\).
CAIE S1 2019 March Q4
4 The random variable \(X\) takes the values \(- 1,1,2,3\) only. The probability that \(X\) takes the value \(x\) is \(k x ^ { 2 }\), where \(k\) is a constant.
  1. Draw up the probability distribution table for \(X\), in terms of \(k\), and find the value of \(k\).
  2. Find \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).
CAIE S1 2019 March Q5
5 The weights, in kg, of the 11 members of the Dolphins swimming team and the 11 members of the Sharks swimming team are shown below.
Dolphins6275698263806565738272
Sharks6884597071647780667472
  1. Draw a back-to-back stem-and-leaf diagram to represent this information, with Dolphins on the left-hand side of the diagram and Sharks on the right-hand side.
  2. Find the median and interquartile range for the Dolphins.
CAIE S1 2019 March Q6
6 The results of a survey by a large supermarket show that \(35 \%\) of its customers shop online.
  1. Six customers are chosen at random. Find the probability that more than three of them shop online.
  2. For a random sample of \(n\) customers, the probability that at least one of them shops online is greater than 0.95 . Find the least possible value of \(n\).
  3. For a random sample of 100 customers, use a suitable approximating distribution to find the probability that more than 39 shop online.
CAIE S1 2019 March Q7
7 Find the number of different arrangements that can be made of all 9 letters in the word CAMERAMAN in each of the following cases.
  1. There are no restrictions.
  2. The As occupy the 1st, 5th and 9th positions.
  3. There is exactly one letter between the Ms.
    Three letters are selected from the 9 letters of the word CAMERAMAN.
  4. Find the number of different selections if the three letters include exactly one M and exactly one A.
  5. Find the number of different selections if the three letters include at least one M.
    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 2002 November Q1
1 The discrete random variable \(X\) has the following probability distribution.
\(x\)1357
\(\mathrm { P } ( X = x )\)0.3\(a\)\(b\)0.25
  1. Write down an equation satisfied by \(a\) and \(b\).
  2. Given that \(\mathrm { E } ( X ) = 4\), find \(a\) and \(b\).
CAIE S1 2002 November Q2
2 Ivan throws three fair dice.
  1. List all the possible scores on the three dice which give a total score of 5 , and hence show that the probability of Ivan obtaining a total score of 5 is \(\frac { 1 } { 36 }\).
  2. Find the probability of Ivan obtaining a total score of 7.
CAIE S1 2002 November Q3
3 The distance in metres that a ball can be thrown by pupils at a particular school follows a normal distribution with mean 35.0 m and standard deviation 11.6 m .
  1. Find the probability that a randomly chosen pupil can throw a ball between 30 and 40 m .
  2. The school gives a certificate to the \(10 \%\) of pupils who throw further than a certain distance. Find the least distance that must be thrown to qualify for a certificate.
CAIE S1 2002 November Q4
4 In a certain hotel, the lock on the door to each room can be opened by inserting a key card. The key card can be inserted only one way round. The card has a pattern of holes punched in it. The card has 4 columns, and each column can have either 1 hole, 2 holes, 3 holes or 4 holes punched in it. Each column has 8 different positions for the holes. The diagram illustrates one particular key card with 3 holes punched in the first column, 3 in the second, 1 in the third and 2 in the fourth.
\includegraphics[max width=\textwidth, alt={}, center]{2bcbd4d3-0d41-48fa-8f70-192b158c0bbe-2_410_214_1811_968}
  1. Show that the number of different ways in which a column could have exactly 2 holes is 28 .
  2. Find how many different patterns of holes can be punched in a column.
  3. How many different possible key cards are there?