Questions — CAIE (7659 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 Mechanics 1 P1 P2 P3 P4 PMT Mocks PURE Paper 1 Paper 2 Paper 3 Pure 1 S1 S2 S3 S4 Stats 1 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 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 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 2014 November Q2
6 marks Moderate -0.3
2 Find the number of different ways that 6 boys and 4 girls can stand in a line if
  1. all 6 boys stand next to each other,
  2. no girl stands next to another girl.
CAIE S1 2014 November Q3
7 marks Moderate -0.3
3
  1. Four fair six-sided dice, each with faces marked \(1,2,3,4,5,6\), are thrown. Find the probability that the numbers shown on the four dice add up to 5 .
  2. Four fair six-sided dice, each with faces marked \(1,2,3,4,5,6\), are thrown on 7 occasions. Find the probability that the numbers shown on the four dice add up to 5 on exactly 1 or 2 of the 7 occasions. Sharik attempts a multiple choice revision question on-line. There are 3 suggested answers, one of which is correct. When Sharik chooses an answer the computer indicates whether the answer is right or wrong. Sharik first chooses one of the three suggested answers at random. If this answer is wrong he has a second try, choosing an answer at random from the remaining 2 . If this answer is also wrong Sharik then chooses the remaining answer, which must be correct.
CAIE S1 2014 November Q6
9 marks Moderate -0.8
6 On a certain day in spring, the heights of 200 daffodils are measured, correct to the nearest centimetre. The frequency distribution is given below.
Height \(( \mathrm { cm } )\)\(4 - 10\)\(11 - 15\)\(16 - 20\)\(21 - 25\)\(26 - 30\)
Frequency2232784028
  1. Draw a cumulative frequency graph to illustrate the data.
  2. \(28 \%\) of these daffodils are of height \(h \mathrm {~cm}\) or more. Estimate \(h\).
  3. You are given that the estimate of the mean height of these daffodils, calculated from the table, is 18.39 cm . Calculate an estimate of the standard deviation of the heights of these daffodils.
CAIE S1 2014 November Q7
9 marks Moderate -0.3
7 In Marumbo, three quarters of the adults own a cell phone.
  1. A random sample of 8 adults from Marumbo is taken. Find the probability that the number of adults who own a cell phone is between 4 and 6 inclusive.
  2. A random sample of 160 adults from Marumbo is taken. Use an approximation to find the probability that more than 114 of them own a cell phone.
  3. Justify the use of your approximation in part (ii).
CAIE S1 2014 November Q1
3 marks Moderate -0.5
1 Packets of tea are labelled as containing 250 g . The actual weight of tea in a packet has a normal distribution with mean 260 g and standard deviation \(\sigma \mathrm { g }\). Any packet with a weight less than 250 g is classed as 'underweight'. Given that \(1 \%\) of packets of tea are underweight, find the value of \(\sigma\). [3]
CAIE S1 2014 November Q2
5 marks Easy -1.2
2 A traffic camera measured the speeds, \(x\) kilometres per hour, of 8 cars travelling along a certain street, with the following results. $$\begin{array} { l l l l l l l l } 62.7 & 59.6 & 64.2 & 61.5 & 68.3 & 66.9 & 62.0 & 62.3 \end{array}$$
  1. Find \(\Sigma ( x - 62 )\).
  2. Find \(\Sigma ( x - 62 ) ^ { 2 }\).
  3. Find the mean and variance of the speeds of the 8 cars.
CAIE S1 2014 November Q3
5 marks Easy -1.2
3 The number of books read by members of a book club each year has the binomial distribution \(B ( 12,0.7 )\).
  1. State the greatest number of books that could be read by a member of the book club in a particular year and find the probability that a member reads this number of books.
  2. Find the probability that a member reads fewer than 10 books in a particular year.
CAIE S1 2014 November Q4
8 marks Easy -1.3
4 A random sample of 25 people recorded the number of glasses of water they drank in a particular week. The results are shown below.
2319321425
2226364542
4728173815
4618262241
1921282430
  1. Draw a stem-and-leaf diagram to represent the data.
  2. On graph paper draw a box-and-whisker plot to represent the data.
CAIE S1 2014 November Q5
9 marks Standard +0.3
5 Gem stones from a certain mine have weights, \(X\) grams, which are normally distributed with mean 1.9 g and standard deviation 0.55 g . These gem stones are sorted into three categories for sale depending on their weights, as follows. Small: under 1.2 g Medium: between 1.2 g and 2.5 g Large: over 2.5 g
  1. Find the proportion of gem stones in each of these three categories.
  2. Find the value of \(k\) such that \(\mathrm { P } ( k < X < 2.5 ) = 0.8\).
CAIE S1 2014 November Q6
9 marks Standard +0.3
6
  1. Seven fair dice each with faces marked 1,2,3,4,5,6 are thrown and placed in a line. Find the number of possible arrangements where the sum of the numbers at each end of the line add up to 4 .
  2. Find the number of ways in which 9 different computer games can be shared out between Wainah, Jingyi and Hebe so that each person receives an odd number of computer games.
CAIE S1 2014 November Q7
11 marks Standard +0.3
7 A box contains 2 green apples and 2 red apples. Apples are taken from the box, one at a time, without replacement. When both red apples have been taken, the process stops. The random variable \(X\) is the number of apples which have been taken when the process stops.
  1. Show that \(\mathrm { P } ( X = 3 ) = \frac { 1 } { 3 }\).
  2. Draw up the probability distribution table for \(X\). Another box contains 2 yellow peppers and 5 orange peppers. Three peppers are taken at random from the box without replacement.
  3. Given that at least 2 of the peppers taken from the box are orange, find the probability that all 3 peppers are orange.
CAIE S1 2015 November Q1
4 marks Moderate -0.8
1 In a certain town, 76\% of cars are fitted with satellite navigation equipment. A random sample of 11 cars from this town is chosen. Find the probability that fewer than 10 of these cars are fitted with this equipment.
CAIE S1 2015 November Q2
4 marks Moderate -0.3
2 The random variable \(X\) has the distribution \(\mathrm { N } \left( \mu , \sigma ^ { 2 } \right)\). It is given that \(\mathrm { P } ( X < 54.1 ) = 0.5\) and \(\mathrm { P } ( X > 50.9 ) = 0.8665\). Find the values of \(\mu\) and \(\sigma\).
CAIE S1 2015 November Q3
6 marks Easy -1.8
3 Robert has a part-time job delivering newspapers. On a number of days he noted the time, correct to the nearest minute, that it took him to do his job. Robert used his results to draw up the following table; two of the values in the table are denoted by \(a\) and \(b\).
Time \(( t\) minutes \()\)\(60 - 62\)\(63 - 64\)\(65 - 67\)\(68 - 71\)
Frequency (number of days)396\(b\)
Frequency density1\(a\)21.5
  1. Find the values of \(a\) and \(b\).
  2. On graph paper, draw a histogram to represent Robert's times.
CAIE S1 2015 November Q4
7 marks Moderate -0.8
4
  1. Amy measured her pulse rate while resting, \(x\) beats per minute, at the same time each day on 30 days. The results are summarised below. $$\Sigma ( x - 80 ) = - 147 \quad \Sigma ( x - 80 ) ^ { 2 } = 952$$ Find the mean and standard deviation of Amy's pulse rate.
  2. Amy's friend Marok measured her pulse rate every day after running for half an hour. Marok's pulse rate, in beats per minute, was found to have a mean of 148.6 and a standard deviation of 18.5. Assuming that pulse rates have a normal distribution, find what proportion of Marok's pulse rates, after running for half an hour, were above 160 beats per minute.
CAIE S1 2015 November Q5
9 marks Moderate -0.8
5
  1. Find the number of ways in which all nine letters of the word TENNESSEE can be arranged
    1. if all the letters E are together,
    2. if the T is at one end and there is an S at the other end.
  2. Four letters are selected from the nine letters of the word VENEZUELA. Find the number of possible selections which contain exactly one E .
CAIE S1 2015 November Q6
9 marks Moderate -0.8
6 Nadia is very forgetful. Every time she logs in to her online bank she only has a \(40 \%\) chance of remembering her password correctly. She is allowed 3 unsuccessful attempts on any one day and then the bank will not let her try again until the next day.
  1. Draw a fully labelled tree diagram to illustrate this situation.
  2. Let \(X\) be the number of unsuccessful attempts Nadia makes on any day that she tries to log in to her bank. Copy and complete the following table to show the probability distribution of \(X\).
    \(x\)0123
    \(\mathrm { P } ( X = x )\)0.24
  3. Calculate the expected number of unsuccessful attempts made by Nadia on any day that she tries to \(\log\) in.
CAIE S1 2015 November Q7
11 marks Standard +0.3
7 The faces of a biased die are numbered \(1,2,3,4,5\) and 6 . The probabilities of throwing odd numbers are all the same. The probabilities of throwing even numbers are all the same. The probability of throwing an odd number is twice the probability of throwing an even number.
  1. Find the probability of throwing a 3 .
  2. The die is thrown three times. Find the probability of throwing two 5 s and one 4 .
  3. The die is thrown 100 times. Use an approximation to find the probability that an even number is thrown at most 37 times.
CAIE S1 2015 November Q1
3 marks Easy -1.2
1 For \(n\) values of the variable \(x\), it is given that \(\Sigma ( x - 100 ) = 216\) and \(\Sigma x = 2416\). Find the value of \(n\).
CAIE S1 2015 November Q2
3 marks Moderate -0.8
2 A committee of 6 people is to be chosen at random from 7 men and 9 women. Find the probability that there are no men on the committee.
CAIE S1 2015 November Q3
6 marks Easy -1.2
3 One plastic robot is given away free inside each packet of a certain brand of biscuits. There are four colours of plastic robot (red, yellow, blue and green) and each colour is equally likely to occur. Nick buys some packets of these biscuits. Find the probability that
  1. he gets a green robot on opening his first packet,
  2. he gets his first green robot on opening his fifth packet. Nick's friend Amos is also collecting robots.
  3. Find the probability that the first four packets Amos opens all contain different coloured robots.
CAIE S1 2015 November Q4
7 marks Standard +0.3
4 A group of 8 friends travels to the airport in two taxis, \(P\) and \(Q\). Each taxi can take 4 passengers.
  1. The 8 friends divide themselves into two groups of 4, one group for taxi \(P\) and one group for taxi \(Q\), with Jon and Sarah travelling in the same taxi. Find the number of different ways in which this can be done. \includegraphics[max width=\textwidth, alt={}, center]{e2f57f0f-d9dd-4506-afdd-77d61bd47e4b-2_284_467_1491_495} \includegraphics[max width=\textwidth, alt={}, center]{e2f57f0f-d9dd-4506-afdd-77d61bd47e4b-2_286_471_1489_1183} Each taxi can take 1 passenger in the front and 3 passengers in the back (see diagram). Mark sits in the front of taxi \(P\) and Jon and Sarah sit in the back of taxi \(P\) next to each other.
  2. Find the number of different seating arrangements that are now possible for the 8 friends.
CAIE S1 2015 November Q5
9 marks Easy -1.3
5 The weights, in kilograms, of the 15 rugby players in each of two teams, \(A\) and \(B\), are shown below.
Team \(A\)9798104841001091159912282116968410791
Team \(B\)75799410196771111088384861158211395
  1. Represent the data by drawing a back-to-back stem-and-leaf diagram with team \(A\) on the lefthand side of the diagram and team \(B\) on the right-hand side.
  2. Find the interquartile range of the weights of the players in team \(A\).
  3. A new player joins team \(B\) as a substitute. The mean weight of the 16 players in team \(B\) is now 93.9 kg . Find the weight of the new player.
CAIE S1 2015 November Q6
9 marks Moderate -0.8
6 A fair spinner \(A\) has edges numbered \(1,2,3,3\). A fair spinner \(B\) has edges numbered \(- 3 , - 2 , - 1,1\). Each spinner is spun. The number on the edge that the spinner comes to rest on is noted. Let \(X\) be the sum of the numbers for the two spinners.
  1. Copy and complete the table showing the possible values of \(X\).
    Spinner \(A\)
    \cline { 2 - 6 }1233
    Spinner \(B\)- 2
    - 21
    - 1
    1
  2. Draw up a table showing the probability distribution of \(X\).
  3. Find \(\operatorname { Var } ( X )\).
  4. Find the probability that \(X\) is even, given that \(X\) is positive.
CAIE S1 2015 November Q7
13 marks Standard +0.3
7
  1. A petrol station finds that its daily sales, in litres, are normally distributed with mean 4520 and standard deviation 560.
    1. Find on how many days of the year ( 365 days) the daily sales can be expected to exceed 3900 litres. The daily sales at another petrol station are \(X\) litres, where \(X\) is normally distributed with mean \(m\) and standard deviation 560. It is given that \(\mathrm { P } ( X > 8000 ) = 0.122\).
    2. Find the value of \(m\).
    3. Find the probability that daily sales at this petrol station exceed 8000 litres on fewer than 2 of 6 randomly chosen days.
  2. The random variable \(Y\) is normally distributed with mean \(\mu\) and standard deviation \(\sigma\). Given that \(\sigma = \frac { 2 } { 3 } \mu\), find the probability that a random value of \(Y\) is less than \(2 \mu\).