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 2013 November Q2
2 A factory produces flower pots. The base diameters have a normal distribution with mean 14 cm and standard deviation 0.52 cm . Find the probability that the base diameters of exactly 8 out of 10 randomly chosen flower pots are between 13.6 cm and 14.8 cm .
CAIE S1 2013 November Q3
3 In a large consignment of mangoes, 15\% of mangoes are classified as small, 70\% as medium and \(15 \%\) as large.
  1. Yue-chen picks 14 mangoes at random. Find the probability that fewer than 12 of them are medium or large.
  2. Yue-chen picks \(n\) mangoes at random. The probability that none of these \(n\) mangoes is small is at least 0.1 . Find the largest possible value of \(n\).
CAIE S1 2013 November Q4
4 Barry weighs 20 oranges and 25 lemons. For the oranges, the mean weight is 220 g and the standard deviation is 32 g . For the lemons, the mean weight is 118 g and the standard deviation is 12 g .
  1. Find the mean weight of the 45 fruits.
  2. The individual weights of the oranges in grams are denoted by \(x _ { o }\), and the individual weights of the lemons in grams are denoted by \(x _ { l }\). By first finding \(\Sigma x _ { o } ^ { 2 }\) and \(\Sigma x _ { l } ^ { 2 }\), find the variance of the weights of the 45 fruits.
CAIE S1 2013 November Q5
5
  1. The random variable \(X\) is normally distributed with mean 82 and standard deviation 7.4. Find the value of \(q\) such that \(\mathrm { P } ( 82 - q < X < 82 + q ) = 0.44\).
  2. The random variable \(Y\) is normally distributed with mean \(\mu\) and standard deviation \(\sigma\). It is given that \(5 \mu = 2 \sigma ^ { 2 }\) and that \(\mathrm { P } \left( Y < \frac { 1 } { 2 } \mu \right) = 0.281\). Find the values of \(\mu\) and \(\sigma\).
CAIE S1 2013 November Q6
6
  1. Find the number of different ways that the 9 letters of the word AGGREGATE can be arranged in a line if the first letter is \(R\).
  2. Find the number of different ways that the 9 letters of the word AGGREGATE can be arranged in a line if the 3 letters G are together, both letters A are together and both letters E are together.
  3. The letters G, R and T are consonants and the letters A and E are vowels. Find the number of different ways that the 9 letters of the word AGGREGATE can be arranged in a line if consonants and vowels occur alternately.
  4. Find the number of different selections of 4 letters of the word AGGREGATE which contain exactly 2 Gs or exactly 3 Gs.
CAIE S1 2013 November Q7
7 Dayo chooses two digits at random, without replacement, from the 9-digit number 113333555.
  1. Find the probability that the two digits chosen are equal.
  2. Find the probability that one digit is a 5 and one digit is not a 5 .
  3. Find the probability that the first digit Dayo chose was a 5, given that the second digit he chose is not a 5 .
  4. The random variable \(X\) is the number of 5s that Dayo chooses. Draw up a table to show the probability distribution of \(X\).
CAIE S1 2014 November Q1
1 Find the mean and variance of the following data. $$\begin{array} { l l l l l l l l l l } 5 & - 2 & 12 & 7 & - 3 & 2 & - 6 & 4 & 0 & 8 \end{array}$$
CAIE S1 2014 November Q2
2 The number of phone calls, \(X\), received per day by Sarah has the following probability distribution.
\(x\)01234\(\geqslant 5\)
\(\mathrm { P } ( X = x )\)0.240.35\(2 k\)\(k\)0.050
  1. Find the value of \(k\).
  2. Find the mode of \(X\).
  3. Find the probability that the number of phone calls received by Sarah on any particular day is more than the mean number of phone calls received per day.
CAIE S1 2014 November Q3
3 Jodie tosses a biased coin and throws two fair tetrahedral dice. The probability that the coin shows a head is \(\frac { 1 } { 3 }\). Each of the dice has four faces, numbered \(1,2,3\) and 4 . Jodie's score is calculated from the numbers on the faces that the dice land on, as follows:
  • if the coin shows a head, the two numbers from the dice are added together;
  • if the coin shows a tail, the two numbers from the dice are multiplied together.
Find the probability that the coin shows a head given that Jodie's score is 8 .
CAIE S1 2014 November Q4
4 The following back-to-back stem-and-leaf diagram shows the times to load an application on 61 smartphones of type \(A\) and 43 smartphones of type \(B\).
(7)
Type \(A\)Type \(B\)
976643321358
55442223044566667889
998887664322040112368899
655432110525669
973061389
874410757
766653321081244
86555906
Key: 3 | 2 | 1 means 0.23 seconds for type \(A\) and 0.21 seconds for type \(B\).
  1. Find the median and quartiles for smartphones of type \(A\). You are given that the median, lower quartile and upper quartile for smartphones of type \(B\) are 0.46 seconds, 0.36 seconds and 0.63 seconds respectively.
  2. Represent the data by drawing a pair of box-and-whisker plots in a single diagram on graph paper.
  3. Compare the loading times for these two types of smartphone.
CAIE S1 2014 November Q5
5 Screws are sold in packets of 15. Faulty screws occur randomly. A large number of packets are tested for faulty screws and the mean number of faulty screws per packet is found to be 1.2 .
  1. Show that the variance of the number of faulty screws in a packet is 1.104 .
  2. Find the probability that a packet contains at most 2 faulty screws. Damien buys 8 packets of screws at random.
  3. Find the probability that there are exactly 7 packets in which there is at least 1 faulty screw.
CAIE S1 2014 November Q6
6 A farmer finds that the weights of sheep on his farm have a normal distribution with mean 66.4 kg and standard deviation 5.6 kg .
  1. 250 sheep are chosen at random. Estimate the number of sheep which have a weight of between 70 kg and 72.5 kg .
  2. The proportion of sheep weighing less than 59.2 kg is equal to the proportion weighing more than \(y \mathrm {~kg}\). Find the value of \(y\). Another farmer finds that the weights of sheep on his farm have a normal distribution with mean \(\mu \mathrm { kg }\) and standard deviation 4.92 kg . 25\% of these sheep weigh more than 67.5 kg .
  3. Find the value of \(\mu\).
CAIE S1 2014 November Q7
7 A committee of 6 people is to be chosen from 5 men and 8 women. In how many ways can this be done
  1. if there are more women than men on the committee,
  2. if the committee consists of 3 men and 3 women but two particular men refuse to be on the committee together? One particular committee consists of 5 women and 1 man.
  3. In how many different ways can the committee members be arranged in a line if the man is not at either end?
CAIE S1 2014 November Q1
1 The 50 members of a club include both the club president and the club treasurer. All 50 members want to go on a coach tour, but the coach only has room for 45 people. In how many ways can 45 members be chosen if both the club president and the club treasurer must be included?
CAIE S1 2014 November Q2
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
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
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
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
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
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
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
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
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
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
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.