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 2010 June Q6
6
  1. Find the number of different ways that a set of 10 different mugs can be shared between Lucy and Monica if each receives an odd number of mugs.
  2. Another set consists of 6 plastic mugs each of a different design and 3 china mugs each of a different design. Find in how many ways these 9 mugs can be arranged in a row if the china mugs are all separated from each other.
  3. Another set consists of 3 identical red mugs, 4 identical blue mugs and 7 identical yellow mugs. These 14 mugs are placed in a row. Find how many different arrangements of the colours are possible if the red mugs are kept together.
CAIE S1 2010 June Q7
7 In a television quiz show Peter answers questions one after another, stopping as soon as a question is answered wrongly.
  • The probability that Peter gives the correct answer himself to any question is 0.7 .
  • The probability that Peter gives a wrong answer himself to any question is 0.1 .
  • The probability that Peter decides to ask for help for any question is 0.2 .
On the first occasion that Peter decides to ask for help he asks the audience. The probability that the audience gives the correct answer to any question is 0.95 . This information is shown in the tree diagram below.
\includegraphics[max width=\textwidth, alt={}, center]{e7e0fcbe-ab96-4292-b3ad-c57b74f15301-3_394_649_1779_386}
\includegraphics[max width=\textwidth, alt={}, center]{e7e0fcbe-ab96-4292-b3ad-c57b74f15301-3_270_743_2010_1023}
  1. Show that the probability that the first question is answered correctly is 0.89 . On the second occasion that Peter decides to ask for help he phones a friend. The probability that his friend gives the correct answer to any question is 0.65 .
  2. Find the probability that the first two questions are both answered correctly.
  3. Given that the first two questions were both answered correctly, find the probability that Peter asked the audience.
CAIE S1 2010 June Q1
1 The times in minutes for seven students to become proficient at a new computer game were measured. The results are shown below. $$\begin{array} { l l l l l l l } 15 & 10 & 48 & 10 & 19 & 14 & 16 \end{array}$$
  1. Find the mean and standard deviation of these times.
  2. State which of the mean, median or mode you consider would be most appropriate to use as a measure of central tendency to represent the data in this case.
  3. For each of the two measures of average you did not choose in part (ii), give a reason why you consider it inappropriate.
CAIE S1 2010 June Q2
2 The lengths of new pencils are normally distributed with mean 11 cm and standard deviation 0.095 cm .
  1. Find the probability that a pencil chosen at random has a length greater than 10.9 cm .
  2. Find the probability that, in a random sample of 6 pencils, at least two have lengths less than 10.9 cm .
    \includegraphics[max width=\textwidth, alt={}]{7b97cfbe-9960-4f26-8be5-ed393feeb8ae-2_1207_1642_1251_255}
    The birth weights of random samples of 900 babies born in country \(A\) and 900 babies born in country \(B\) are illustrated in the cumulative frequency graphs. Use suitable data from these graphs to compare the central tendency and spread of the birth weights of the two sets of babies.
CAIE S1 2010 June Q4
4 The random variable \(X\) is normally distributed with mean \(\mu\) and standard deviation \(\sigma\).
  1. Given that \(5 \sigma = 3 \mu\), find \(\mathrm { P } ( X < 2 \mu )\).
  2. With a different relationship between \(\mu\) and \(\sigma\), it is given that \(\mathrm { P } \left( X < \frac { 1 } { 3 } \mu \right) = 0.8524\). Express \(\mu\) in terms of \(\sigma\).
CAIE S1 2010 June Q5
5 Two fair twelve-sided dice with sides marked \(1,2,3,4,5,6,7,8,9,10,11,12\) are thrown, and the numbers on the sides which land face down are noted. Events \(Q\) and \(R\) are defined as follows.
\(Q\) : the product of the two numbers is 24 .
\(R\) : both of the numbers are greater than 8 .
  1. Find \(\mathrm { P } ( Q )\).
  2. Find \(\mathrm { P } ( R )\).
  3. Are events \(Q\) and \(R\) exclusive? Justify your answer.
  4. Are events \(Q\) and \(R\) independent? Justify your answer.
CAIE S1 2010 June Q6
6 A small farm has 5 ducks and 2 geese. Four of these birds are to be chosen at random. The random variable \(X\) represents the number of geese chosen.
  1. Draw up the probability distribution of \(X\).
  2. Show that \(\mathrm { E } ( X ) = \frac { 8 } { 7 }\) and calculate \(\operatorname { Var } ( X )\).
  3. When the farmer's dog is let loose, it chases either the ducks with probability \(\frac { 3 } { 5 }\) or the geese with probability \(\frac { 2 } { 5 }\). If the dog chases the ducks there is a probability of \(\frac { 1 } { 10 }\) that they will attack the dog. If the dog chases the geese there is a probability of \(\frac { 3 } { 4 }\) that they will attack the dog. Given that the dog is not attacked, find the probability that it was chasing the geese.
CAIE S1 2010 June Q7
7 Nine cards, each of a different colour, are to be arranged in a line.
  1. How many different arrangements of the 9 cards are possible? The 9 cards include a pink card and a green card.
  2. How many different arrangements do not have the pink card next to the green card? Consider all possible choices of 3 cards from the 9 cards with the 3 cards being arranged in a line.
  3. How many different arrangements in total of 3 cards are possible?
  4. How many of the arrangements of 3 cards in part (iii) contain the pink card?
  5. How many of the arrangements of 3 cards in part (iii) do not have the pink card next to the green card?
CAIE S1 2010 June Q1
1 A bottle of sweets contains 13 red sweets, 13 blue sweets, 13 green sweets and 13 yellow sweets. 7 sweets are selected at random. Find the probability that exactly 3 of them are red.
CAIE S1 2010 June Q2
2 The heights, \(x \mathrm {~cm}\), of a group of 82 children are summarised as follows. $$\Sigma ( x - 130 ) = - 287 , \quad \text { standard deviation of } x = 6.9 .$$
  1. Find the mean height.
  2. Find \(\Sigma ( x - 130 ) ^ { 2 }\).
CAIE S1 2010 June Q3
3 Christa takes her dog for a walk every day. The probability that they go to the park on any day is 0.6 . If they go to the park there is a probability of 0.35 that the dog will bark. If they do not go to the park there is a probability of 0.75 that the dog will bark.
  1. Find the probability that they go to the park on more than 5 of the next 7 days.
  2. Find the probability that the dog barks on any particular day.
  3. Find the variance of the number of times they go to the park in 30 days.
CAIE S1 2010 June Q4
4 Three identical cans of cola, 2 identical cans of green tea and 2 identical cans of orange juice are arranged in a row. Calculate the number of arrangements if
  1. the first and last cans in the row are the same type of drink,
  2. the 3 cans of cola are all next to each other and the 2 cans of green tea are not next to each other.
CAIE S1 2010 June Q5
5 Set \(A\) consists of the ten digits \(0,0,0,0,0,0,2,2,2,4\).
Set \(B\) consists of the seven digits \(0,0,0,0,2,2,2\).
One digit is chosen at random from each set. The random variable \(X\) is defined as the sum of these two digits.
  1. Show that \(\mathrm { P } ( X = 2 ) = \frac { 3 } { 7 }\).
  2. Tabulate the probability distribution of \(X\).
  3. Find \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).
  4. Given that \(X = 2\), find the probability that the digit chosen from set \(A\) was 2 .
CAIE S1 2010 June Q6
6 The lengths of some insects of the same type from two countries, \(X\) and \(Y\), were measured. The stem-and-leaf diagram shows the results.
Country \(X\)Country \(Y\)
(10)976664443280
(18)888776655544333220811122333556789(13)
(16)999887765532210082001233394566788(15)
(16)87655533222111008301224444556677789(17)
(11)8765544331184001244556677789(15)
85\(12 r 335566788\)(12)
8601223555899(11)
Key: 5 | 81 | 3 means an insect from country \(X\) has length 0.815 cm and an insect from country \(Y\) has length 0.813 cm .
  1. Find the median and interquartile range of the lengths of the insects from country \(X\).
  2. The interquartile range of the lengths of the insects from country \(Y\) is 0.028 cm . Find the values of \(q\) and \(r\).
  3. Represent the data by means of a pair of box-and-whisker plots in a single diagram on graph paper.
  4. Compare the lengths of the insects from the two countries.
CAIE S1 2010 June Q7
7 The heights that children of a particular age can jump have a normal distribution. On average, 8 children out of 10 can jump a height of more than 127 cm , and 1 child out of 3 can jump a height of more than 135 cm .
  1. Find the mean and standard deviation of the heights the children can jump.
  2. Find the probability that a randomly chosen child will not be able to jump a height of 145 cm .
  3. Find the probability that, of 8 randomly chosen children, at least 2 will be able to jump a height of more than 135 cm .
CAIE S1 2011 June Q1
1 Biscuits are sold in packets of 18. There is a constant probability that any biscuit is broken, independently of other biscuits. The mean number of broken biscuits in a packet has been found to be 2.7 . Find the probability that a packet contains between 2 and 4 (inclusive) broken biscuits.
CAIE S1 2011 June Q2
2 When Ted is looking for his pen, the probability that it is in his pencil case is 0.7 . If his pen is in his pencil case he always finds it. If his pen is somewhere else, the probability that he finds it is 0.2 . Given that Ted finds his pen when he is looking for it, find the probability that it was in his pencil case.
CAIE S1 2011 June Q3
3 The possible values of the random variable \(X\) are the 8 integers in the set \(\{ - 2 , - 1,0,1,2,3,4,5 \}\). The probability of \(X\) being 0 is \(\frac { 1 } { 10 }\). The probabilities for all the other values of \(X\) are equal. Calculate
  1. \(\mathrm { P } ( X < 2 )\),
  2. the variance of \(X\),
  3. the value of \(a\) for which \(\mathrm { P } ( - a \leqslant X \leqslant 2 a ) = \frac { 17 } { 35 }\).
CAIE S1 2011 June Q4
4 A cricket team of 11 players is to be chosen from 21 players consisting of 10 batsmen, 9 bowlers and 2 wicketkeepers. The team must include at least 5 batsmen, at least 4 bowlers and at least 1 wicketkeeper.
  1. Find the number of different ways in which the team can be chosen. Each player in the team is given a present. The presents consist of 5 identical pens, 4 identical diaries and 2 identical notebooks.
  2. Find the number of different arrangements of the presents if they are all displayed in a row.
  3. 10 of these 11 presents are chosen and arranged in a row. Find the number of different arrangements that are possible.
CAIE S1 2011 June Q5
5
  1. The random variable \(X\) is normally distributed with mean \(\mu\) and standard deviation \(\sigma\). It is given that \(3 \mu = 7 \sigma ^ { 2 }\) and that \(\mathrm { P } ( X > 2 \mu ) = 0.1016\). Find \(\mu\) and \(\sigma\).
  2. It is given that \(Y \sim \mathrm {~N} ( 33,21 )\). Find the value of \(a\) given that \(\mathrm { P } ( 33 - a < Y < 33 + a ) = 0.5\).
CAIE S1 2011 June Q6
6 There are 5000 schools in a certain country. The cumulative frequency table shows the number of pupils in a school and the corresponding number of schools.
Number of pupils in a school\(\leqslant 100\)\(\leqslant 150\)\(\leqslant 200\)\(\leqslant 250\)\(\leqslant 350\)\(\leqslant 450\)\(\leqslant 600\)
Cumulative frequency20080016002100410047005000
  1. Draw a cumulative frequency graph with a scale of 2 cm to 100 pupils on the horizontal axis and a scale of 2 cm to 1000 schools on the vertical axis. Use your graph to estimate the median number of pupils in a school.
  2. \(80 \%\) of the schools have more than \(n\) pupils. Estimate the value of \(n\) correct to the nearest ten.
  3. Find how many schools have between 201 and 250 (inclusive) pupils.
  4. Calculate an estimate of the mean number of pupils per school.
CAIE S1 2011 June Q7
7
    1. Find the probability of getting at least one 3 when 9 fair dice are thrown.
    2. When \(n\) fair dice are thrown, the probability of getting at least one 3 is greater than 0.9. Find the smallest possible value of \(n\).
  2. A bag contains 5 green balls and 3 yellow balls. Ronnie and Julie play a game in which they take turns to draw a ball from the bag at random without replacement. The winner of the game is the first person to draw a yellow ball. Julie draws the first ball. Find the probability that Ronnie wins the game.
CAIE S1 2011 June Q1
1 A biased die was thrown 20 times and the number of 5 s was noted. This experiment was repeated many times and the average number of 5 s was found to be 4.8 . Find the probability that in the next 20 throws the number of 5 s will be less than three.
CAIE S1 2011 June Q2
2 In Scotland, in November, on average \(80 \%\) of days are cloudy. Assume that the weather on any one day is independent of the weather on other days.
  1. Use a normal approximation to find the probability of there being fewer than 25 cloudy days in Scotland in November (30 days).
  2. Give a reason why the use of a normal approximation is justified.
CAIE S1 2011 June Q3
3 A sample of 36 data values, \(x\), gave \(\Sigma ( x - 45 ) = - 148\) and \(\Sigma ( x - 45 ) ^ { 2 } = 3089\).
  1. Find the mean and standard deviation of the 36 values.
  2. One extra data value of 29 was added to the sample. Find the standard deviation of all 37 values.