Questions — CAIE (7646 questions)

Browse by module
All questions AEA AS Paper 1 AS Paper 2 AS Pure 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 Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Extra Pure Further Mechanics Further Mechanics A AS Further Mechanics AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics Further Paper 4 Further Pure Core Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Pure with Technology Further Statistics Further Statistics A AS Further Statistics AS Further Statistics B AS Further Statistics Major Further Statistics Minor Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 H240/01 H240/02 H240/03 M1 M2 M3 M4 M5 P1 P2 P3 P4 PMT Mocks PURE Paper 1 Paper 2 Paper 3 Pre-U 9794/1 Pre-U 9794/2 Pre-U 9794/3 Pre-U 9795 Pre-U 9795/1 Pre-U 9795/2 S1 S2 S3 S4 Unit 1 Unit 2 Unit 3 Unit 4
Browse by board
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 PURE 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 PURE S1 S2 S3 S4 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 Pre-U Pre-U 9794/1 Pre-U 9794/2 Pre-U 9794/3 Pre-U 9795 Pre-U 9795/1 Pre-U 9795/2 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
Sort by: Default | Easiest first | Hardest first
CAIE S1 2006 June Q2
6 marks Moderate -0.8
2 The probability that Henk goes swimming on any day is 0.2 . On a day when he goes swimming, the probability that Henk has burgers for supper is 0.75 . On a day when he does not go swimming the probability that he has burgers for supper is \(x\). This information is shown on the following tree diagram. \includegraphics[max width=\textwidth, alt={}, center]{14e8a601-2180-4491-9336-cafd262f2596-2_693_1038_845_555} The probability that Henk has burgers for supper on any day is 0.5 .
  1. Find \(x\).
  2. Given that Henk has burgers for supper, find the probability that he went swimming that day.
CAIE S1 2006 June Q3
8 marks Standard +0.3
3 The lengths of fish of a certain type have a normal distribution with mean 38 cm . It is found that \(5 \%\) of the fish are longer than 50 cm .
  1. Find the standard deviation.
  2. When fish are chosen for sale, those shorter than 30 cm are rejected. Find the proportion of fish rejected.
  3. 9 fish are chosen at random. Find the probability that at least one of them is longer than 50 cm .
CAIE S1 2006 June Q4
8 marks Moderate -0.8
4 \includegraphics[max width=\textwidth, alt={}, center]{14e8a601-2180-4491-9336-cafd262f2596-3_277_682_274_733} The diagram shows the seating plan for passengers in a minibus, which has 17 seats arranged in 4 rows. The back row has 5 seats and the other 3 rows have 2 seats on each side. 11 passengers get on the minibus.
  1. How many possible seating arrangements are there for the 11 passengers?
  2. How many possible seating arrangements are there if 5 particular people sit in the back row? Of the 11 passengers, 5 are unmarried and the other 6 consist of 3 married couples.
  3. In how many ways can 5 of the 11 passengers on the bus be chosen if there must be 2 married couples and 1 other person, who may or may not be married?
CAIE S1 2006 June Q5
7 marks Moderate -0.8
5 Each father in a random sample of fathers was asked how old he was when his first child was born. The following histogram represents the information. \includegraphics[max width=\textwidth, alt={}, center]{14e8a601-2180-4491-9336-cafd262f2596-3_789_1627_1468_260}
  1. What is the modal age group?
  2. How many fathers were between 25 and 30 years old when their first child was born?
  3. How many fathers were in the sample?
  4. Find the probability that a father, chosen at random from the group, was between 25 and 30 years old when his first child was born, given that he was older than 25 years. 632 teams enter for a knockout competition, in which each match results in one team winning and the other team losing. After each match the winning team goes on to the next round, and the losing team takes no further part in the competition. Thus 16 teams play in the second round, 8 teams play in the third round, and so on, until 2 teams play in the final round.
CAIE S1 2006 June Q7
9 marks Standard +0.3
7 A survey of adults in a certain large town found that \(76 \%\) of people wore a watch on their left wrist, \(15 \%\) wore a watch on their right wrist and \(9 \%\) did not wear a watch.
  1. A random sample of 14 adults was taken. Find the probability that more than 2 adults did not wear a watch.
  2. A random sample of 200 adults was taken. Using a suitable approximation, find the probability that more than 155 wore a watch on their left wrist.
CAIE S1 2007 June Q1
4 marks Easy -1.2
1 The length of time, \(t\) minutes, taken to do the crossword in a certain newspaper was observed on 12 occasions. The results are summarised below. $$\Sigma ( t - 35 ) = - 15 \quad \Sigma ( t - 35 ) ^ { 2 } = 82.23$$ Calculate the mean and standard deviation of these times taken to do the crossword.
CAIE S1 2007 June Q2
6 marks Moderate -0.8
2 Jamie is equally likely to attend or not to attend a training session before a football match. If he attends, he is certain to be chosen for the team which plays in the match. If he does not attend, there is a probability of 0.6 that he is chosen for the team.
  1. Find the probability that Jamie is chosen for the team.
  2. Find the conditional probability that Jamie attended the training session, given that he was chosen for the team.
CAIE S1 2007 June Q3
7 marks Standard +0.3
3
  1. The random variable \(X\) is normally distributed. The mean is twice the standard deviation. It is given that \(\mathrm { P } ( X > 5.2 ) = 0.9\). Find the standard deviation.
  2. A normal distribution has mean \(\mu\) and standard deviation \(\sigma\). If 800 observations are taken from this distribution, how many would you expect to be between \(\mu - \sigma\) and \(\mu + \sigma\) ?
CAIE S1 2007 June Q4
7 marks Easy -1.3
4 The lengths of time in minutes to swim a certain distance by the members of a class of twelve 9 -year-olds and by the members of a class of eight 16 -year-olds are shown below.
9-year-olds:13.016.116.014.415.915.114.213.716.716.415.013.2
16-year-olds:14.813.011.411.716.513.712.812.9
  1. Draw a back-to-back stem-and-leaf diagram to represent the information above.
  2. A new pupil joined the 16 -year-old class and swam the distance. The mean time for the class of nine pupils was now 13.6 minutes. Find the new pupil's time to swim the distance.
CAIE S1 2007 June Q5
7 marks Moderate -0.3
5
  1. Find the number of ways in which all twelve letters of the word REFRIGERATOR can be arranged
    1. if there are no restrictions,
    2. if the Rs must all be together.
    3. How many different selections of four letters from the twelve letters of the word REFRIGERATOR contain no Rs and two Es?
CAIE S1 2007 June Q6
9 marks Standard +0.3
6 The probability that New Year's Day is on a Saturday in a randomly chosen year is \(\frac { 1 } { 7 }\).
  1. 15 years are chosen randomly. Find the probability that at least 3 of these years have New Year's Day on a Saturday.
  2. 56 years are chosen randomly. Use a suitable approximation to find the probability that more than 7 of these years have New Year's Day on a Saturday.
CAIE S1 2007 June Q7
10 marks Standard +0.3
7 A vegetable basket contains 12 peppers, of which 3 are red, 4 are green and 5 are yellow. Three peppers are taken, at random and without replacement, from the basket.
  1. Find the probability that the three peppers are all different colours.
  2. Show that the probability that exactly 2 of the peppers taken are green is \(\frac { 12 } { 55 }\).
  3. The number of green peppers taken is denoted by the discrete random variable \(X\). Draw up a probability distribution table for \(X\).
CAIE S1 2008 June Q1
4 marks Easy -1.2
1 The stem-and-leaf diagram below represents data collected for the number of hits on an internet site on each day in March 2007. There is one missing value, denoted by \(x\).
00156
1135668
2112344489
31222\(x\)89
425679
Key: 1 | 5 represents 15 hits
  1. Find the median and lower quartile for the number of hits each day.
  2. The interquartile range is 19 . Find the value of \(x\).
CAIE S1 2008 June Q2
5 marks Moderate -0.8
2 In country \(A 30 \%\) of people who drink tea have sugar in it. In country \(B 65 \%\) of people who drink tea have sugar in it. There are 3 million people in country \(A\) who drink tea and 12 million people in country \(B\) who drink tea. A person is chosen at random from these 15 million people.
  1. Find the probability that the person chosen is from country \(A\).
  2. Find the probability that the person chosen does not have sugar in their tea.
  3. Given that the person chosen does not have sugar in their tea, find the probability that the person is from country \(B\).
CAIE S1 2008 June Q3
6 marks Moderate -0.8
3 Issam has 11 different CDs, of which 6 are pop music, 3 are jazz and 2 are classical.
  1. How many different arrangements of all 11 CDs on a shelf are there if the jazz CDs are all next to each other?
  2. Issam makes a selection of 2 pop music CDs, 2 jazz CDs and 1 classical CD. How many different possible selections can be made?
CAIE S1 2008 June Q4
7 marks Moderate -0.3
4 In a certain country the time taken for a common infection to clear up is normally distributed with mean \(\mu\) days and standard deviation 2.6 days. \(25 \%\) of these infections clear up in less than 7 days.
  1. Find the value of \(\mu\). In another country the standard deviation of the time taken for the infection to clear up is the same as in part (i), but the mean is 6.5 days. The time taken is normally distributed.
  2. Find the probability that, in a randomly chosen case from this country, the infection takes longer than 6.2 days to clear up.
CAIE S1 2008 June Q5
8 marks Moderate -0.8
5 As part of a data collection exercise, members of a certain school year group were asked how long they spent on their Mathematics homework during one particular week. The times are given to the nearest 0.1 hour. The results are displayed in the following table.
Time spent \(( t\) hours \()\)\(0.1 \leqslant t \leqslant 0.5\)\(0.6 \leqslant t \leqslant 1.0\)\(1.1 \leqslant t \leqslant 2.0\)\(2.1 \leqslant t \leqslant 3.0\)\(3.1 \leqslant t \leqslant 4.5\)
Frequency1115183021
  1. Draw, on graph paper, a histogram to illustrate this information.
  2. Calculate an estimate of the mean time spent on their Mathematics homework by members of this year group.
CAIE S1 2008 June Q6
9 marks Moderate -0.8
6 Every day Eduardo tries to phone his friend. Every time he phones there is a \(50 \%\) chance that his friend will answer. If his friend answers, Eduardo does not phone again on that day. If his friend does not answer, Eduardo tries again in a few minutes' time. If his friend has not answered after 4 attempts, Eduardo does not try again on that day.
  1. Draw a tree diagram to illustrate this situation.
  2. Let \(X\) be the number of unanswered phone calls made by Eduardo on a day. Copy and complete the table showing the probability distribution of \(X\).
    \(x\)01234
    \(\mathrm { P } ( X = x )\)\(\frac { 1 } { 4 }\)
  3. Calculate the expected number of unanswered phone calls on a day.
CAIE S1 2008 June Q7
11 marks Standard +0.3
7 A die is biased so that the probability of throwing a 5 is 0.75 and the probabilities of throwing a 1,2 , 3 , 4 or 6 are all equal.
  1. The die is thrown three times. Find the probability that the result is a 1 followed by a 5 followed by any even number.
  2. Find the probability that, out of 10 throws of this die, at least 8 throws result in a 5 .
  3. The die is thrown 90 times. Using an appropriate approximation, find the probability that a 5 is thrown more than 60 times.
CAIE S1 2009 June Q1
5 marks Standard +0.3
1 The volume of milk in millilitres in cartons is normally distributed with mean \(\mu\) and standard deviation 8. Measurements were taken of the volume in 900 of these cartons and it was found that 225 of them contained more than 1002 millilitres.
  1. Calculate the value of \(\mu\).
  2. Three of these 900 cartons are chosen at random. Calculate the probability that exactly 2 of them contain more than 1002 millilitres.
CAIE S1 2009 June Q2
6 marks Moderate -0.3
2 Gohan throws a fair tetrahedral die with faces numbered \(1,2,3,4\). If she throws an even number then her score is the number thrown. If she throws an odd number then she throws again and her score is the sum of both numbers thrown. Let the random variable \(X\) denote Gohan's score.
  1. Show that \(\mathrm { P } ( X = 2 ) = \frac { 5 } { 16 }\).
  2. The table below shows the probability distribution of \(X\).
    \(x\)234567
    \(\mathrm { P } ( X = x )\)\(\frac { 5 } { 16 }\)\(\frac { 1 } { 16 }\)\(\frac { 3 } { 8 }\)\(\frac { 1 } { 8 }\)\(\frac { 1 } { 16 }\)\(\frac { 1 } { 16 }\)
    Calculate \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).
CAIE S1 2009 June Q3
8 marks Standard +0.3
3 On a certain road \(20 \%\) of the vehicles are trucks, \(16 \%\) are buses and the remainder are cars.
  1. A random sample of 11 vehicles is taken. Find the probability that fewer than 3 are buses.
  2. A random sample of 125 vehicles is now taken. Using a suitable approximation, find the probability that more than 73 are cars.
CAIE S1 2009 June Q4
8 marks Standard +0.8
4 A choir consists of 13 sopranos, 12 altos, 6 tenors and 7 basses. A group consisting of 10 sopranos, 9 altos, 4 tenors and 4 basses is to be chosen from the choir.
  1. In how many different ways can the group be chosen?
  2. In how many ways can the 10 chosen sopranos be arranged in a line if the 6 tallest stand next to each other?
  3. The 4 tenors and 4 basses in the group stand in a single line with all the tenors next to each other and all the basses next to each other. How many possible arrangements are there if three of the tenors refuse to stand next to any of the basses?
CAIE S1 2009 June Q5
9 marks Moderate -0.8
5 At a zoo, rides are offered on elephants, camels and jungle tractors. Ravi has money for only one ride. To decide which ride to choose, he tosses a fair coin twice. If he gets 2 heads he will go on the elephant ride, if he gets 2 tails he will go on the camel ride and if he gets 1 of each he will go on the jungle tractor ride.
  1. Find the probabilities that he goes on each of the three rides. The probabilities that Ravi is frightened on each of the rides are as follows: $$\text { elephant ride } \frac { 6 } { 10 } , \quad \text { camel ride } \frac { 7 } { 10 } , \quad \text { jungle tractor ride } \frac { 8 } { 10 } .$$
  2. Draw a fully labelled tree diagram showing the rides that Ravi could take and whether or not he is frightened. Ravi goes on a ride.
  3. Find the probability that he is frightened.
  4. Given that Ravi is not frightened, find the probability that he went on the camel ride.
CAIE S1 2009 June Q6
14 marks Moderate -0.8
6 During January the numbers of people entering a store during the first hour after opening were as follows.
Time after opening,
\(x\) minutes
Frequency
Cumulative
frequency
\(0 < x \leqslant 10\)210210
\(10 < x \leqslant 20\)134344
\(20 < x \leqslant 30\)78422
\(30 < x \leqslant 40\)72\(a\)
\(40 < x \leqslant 60\)\(b\)540
  1. Find the values of \(a\) and \(b\).
  2. Draw a cumulative frequency graph to represent this information. Take a scale of 2 cm for 10 minutes on the horizontal axis and 2 cm for 50 people on the vertical axis.
  3. Use your graph to estimate the median time after opening that people entered the store.
  4. Calculate estimates of the mean, \(m\) minutes, and standard deviation, \(s\) minutes, of the time after opening that people entered the store.
  5. Use your graph to estimate the number of people entering the store between ( \(m - \frac { 1 } { 2 } s\) ) and \(\left( m + \frac { 1 } { 2 } s \right)\) minutes after opening.