Questions S1 (1967 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 2020 June Q3
3 A company produces small boxes of sweets that contain 5 jellies and 3 chocolates. Jemeel chooses 3 sweets at random from a box.
  1. Draw up the probability distribution table for the number of jellies that Jemeel chooses.
    The company also produces large boxes of sweets. For any large box, the probability that it contains more jellies than chocolates is 0.64 . 10 large boxes are chosen at random.
  2. Find the probability that no more than 7 of these boxes contain more jellies than chocolates.
CAIE S1 2020 June Q4
4 In a music competition, there are 8 pianists, 4 guitarists and 6 violinists. 7 of these musicians will be selected to go through to the final. How many different selections of 7 finalists can be made if there must be at least 2 pianists, at least 1 guitarist and more violinists than guitarists?
CAIE S1 2020 June Q5
5 On Mondays, Rani cooks her evening meal. She has a pizza, a burger or a curry with probabilities \(0.35,0.44,0.21\) respectively. When she cooks a pizza, Rani has some fruit with probability 0.3 . When she cooks a burger, she has some fruit with probability 0.8 . When she cooks a curry, she never has any fruit.
  1. Draw a fully labelled tree diagram to represent this information.
  2. Find the probability that Rani has some fruit.
  3. Find the probability that Rani does not have a burger given that she does not have any fruit.
CAIE S1 2020 June Q6
6 The lengths of female snakes of a particular species are normally distributed with mean 54 cm and standard deviation 6.1 cm .
  1. Find the probability that a randomly chosen female snake of this species has length between 50 cm and 60 cm .
    The lengths of male snakes of this species also have a normal distribution. A scientist measures the lengths of a random sample of 200 male snakes of this species. He finds that 32 have lengths less than 45 cm and 17 have lengths more than 56 cm .
  2. Find estimates for the mean and standard deviation of the lengths of male snakes of this species.
CAIE S1 2020 June Q7
7 The numbers of chocolate bars sold per day in a cinema over a period of 100 days are summarised in the following table.
Number of chocolate bars sold\(1 - 10\)\(11 - 15\)\(16 - 30\)\(31 - 50\)\(51 - 60\)
Number of days182430208
  1. Draw a histogram to represent this information.
    \includegraphics[max width=\textwidth, alt={}, center]{3ada18de-c4f7-4049-9032-46b796be83c3-12_1203_1399_833_415}
  2. What is the greatest possible value of the interquartile range for the data?
  3. Calculate estimates of the mean and standard deviation of the number of chocolate bars sold.
    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 2004 June Q1
1 Two cricket teams kept records of the number of runs scored by their teams in 8 matches. The scores are shown in the following table.
Team \(A\)150220773029811816057
Team \(B\)1661421709311113014886
  1. Find the mean and standard deviation of the scores for team \(A\). The mean and standard deviation for team \(B\) are 130.75 and 29.63 respectively.
  2. State with a reason which team has the more consistent scores.
CAIE S1 2004 June Q2
2 In a recent survey, 640 people were asked about the length of time each week that they spent watching television. The median time was found to be 20 hours, and the lower and upper quartiles were 15 hours and 35 hours respectively. The least amount of time that anyone spent was 3 hours, and the greatest amount was 60 hours.
  1. On graph paper, show these results using a fully labelled cumulative frequency graph.
  2. Use your graph to estimate how many people watched more than 50 hours of television each week.
CAIE S1 2004 June Q3
3 Two fair dice are thrown. Let the random variable \(X\) be the smaller of the two scores if the scores are different, or the score on one of the dice if the scores are the same.
  1. Copy and complete the following table to show the probability distribution of \(X\).
    \(x\)123456
    \(\mathrm { P } ( X = x )\)
  2. Find \(\mathrm { E } ( X )\).
CAIE S1 2004 June Q4
4 Melons are sold in three sizes: small, medium and large. The weights follow a normal distribution with mean 450 grams and standard deviation 120 grams. Melons weighing less than 350 grams are classified as small.
  1. Find the proportion of melons which are classified as small.
  2. The rest of the melons are divided in equal proportions between medium and large. Find the weight above which melons are classified as large.
CAIE S1 2004 June Q5
5
  1. The menu for a meal in a restaurant is as follows. \begin{displayquote} Starter Course
    Melon
    or
    Soup
    or
    Smoked Salmon \end{displayquote} \begin{displayquote} Main Course
    Chicken
    or
    Steak
    or
    Lamb Cutlets
    or
    Vegetable Curry
    or
    Fish \end{displayquote} \begin{displayquote} Dessert Course
    Cheesecake
    or
    Ice Cream
    or
    Apple Pie
    All the main courses are served with salad and either
    new potatoes or french fries.
    1. How many different three-course meals are there?
    2. How many different choices are there if customers may choose only two of the three courses?
  2. In how many ways can a group of 14 people eating at the restaurant be divided between three tables seating 5, 5 and 4? \end{displayquote}
CAIE S1 2004 June Q6
6 When Don plays tennis, \(65 \%\) of his first serves go into the correct area of the court. If the first serve goes into the correct area, his chance of winning the point is \(90 \%\). If his first serve does not go into the correct area, Don is allowed a second serve, and of these, \(80 \%\) go into the correct area. If the second serve goes into the correct area, his chance of winning the point is \(60 \%\). If neither serve goes into the correct area, Don loses the point.
  1. Draw a tree diagram to represent this information.
  2. Using your tree diagram, find the probability that Don loses the point.
  3. Find the conditional probability that Don's first serve went into the correct area, given that he loses the point.
CAIE S1 2004 June Q7
7 A shop sells old video tapes, of which 1 in 5 on average are known to be damaged.
  1. A random sample of 15 tapes is taken. Find the probability that at most 2 are damaged.
  2. Find the smallest value of \(n\) if there is a probability of at least 0.85 that a random sample of \(n\) tapes contains at least one damaged tape.
  3. A random sample of 1600 tapes is taken. Use a suitable approximation to find the probability that there are at least 290 damaged tapes.
CAIE S1 2005 June Q1
1 It is known that, on average, 2 people in 5 in a certain country are overweight. A random sample of 400 people is chosen. Using a suitable approximation, find the probability that fewer than 165 people in the sample are overweight.
CAIE S1 2005 June Q2
2 The following table shows the results of a survey to find the average daily time, in minutes, that a group of schoolchildren spent in internet chat rooms.
Time per day
\(( t\) minutes \()\)
Frequency
\(0 \leqslant t < 10\)2
\(10 \leqslant t < 20\)\(f\)
\(20 \leqslant t < 40\)11
\(40 \leqslant t < 80\)4
The mean time was calculated to be 27.5 minutes.
  1. Form an equation involving \(f\) and hence show that the total number of children in the survey was 26 .
  2. Find the standard deviation of these times.
CAIE S1 2005 June Q3
3 A fair dice has four faces. One face is coloured pink, one is coloured orange, one is coloured green and one is coloured black. Five such dice are thrown and the number that fall on a green face are counted. The random variable \(X\) is the number of dice that fall on a green face.
  1. Show that the probability of 4 dice landing on a green face is 0.0146 , correct to 4 decimal places.
  2. Draw up a table for the probability distribution of \(X\), giving your answers correct to 4 decimal places.
CAIE S1 2005 June Q4
4 The following back-to-back stem-and-leaf diagram shows the cholesterol count for a group of 45 people who exercise daily and for another group of 63 who do not exercise. The figures in brackets show the number of people corresponding to each set of leaves.
People who exercisePeople who do not exercise
(9)98764322131577(4)
(12)9888766533224234458(6)
(9)87776533151222344567889(13)
(7)6666432612333455577899(14)
(3)8417245566788(9)
(4)95528133467999(9)
(1)4914558(5)
(0)10336(3)
Key: 2 | 8 | 1 represents a cholesterol count of 8.2 in the group who exercise and 8.1 in the group who do not exercise.
  1. Give one useful feature of a stem-and-leaf diagram.
  2. Find the median and the quartiles of the cholesterol count for the group who do not exercise. You are given that the lower quartile, median and upper quartile of the cholesterol count for the group who exercise are 4.25, 5.3 and 6.6 respectively.
  3. On a single diagram on graph paper, draw two box-and-whisker plots to illustrate the data.
CAIE S1 2005 June Q5
5 Data about employment for males and females in a small rural area are shown in the table.
\cline { 2 - 3 } \multicolumn{1}{c|}{}UnemployedEmployed
Male206412
Female358305
A person from this area is chosen at random. Let \(M\) be the event that the person is male and let \(E\) be the event that the person is employed.
  1. Find \(\mathrm { P } ( M )\).
  2. Find \(\mathrm { P } ( M\) and \(E )\).
  3. Are \(M\) and \(E\) independent events? Justify your answer.
  4. Given that the person chosen is unemployed, find the probability that the person is female.
CAIE S1 2005 June Q6
6 Tyre pressures on a certain type of car independently follow a normal distribution with mean 1.9 bars and standard deviation 0.15 bars.
  1. Find the probability that all four tyres on a car of this type have pressures between 1.82 bars and 1.92 bars.
  2. Safety regulations state that the pressures must be between \(1.9 - b\) bars and \(1.9 + b\) bars. It is known that \(80 \%\) of tyres are within these safety limits. Find the safety limits.
CAIE S1 2005 June Q7
7
  1. A football team consists of 3 players who play in a defence position, 3 players who play in a midfield position and 5 players who play in a forward position. Three players are chosen to collect a gold medal for the team. Find in how many ways this can be done
    1. if the captain, who is a midfield player, must be included, together with one defence and one forward player,
    2. if exactly one forward player must be included, together with any two others.
  2. Find how many different arrangements there are of the nine letters in the words GOLD MEDAL
    1. if there are no restrictions on the order of the letters,
    2. if the two letters D come first and the two letters L come last.
CAIE S1 2006 June Q1
1 The salaries, in thousands of dollars, of 11 people, chosen at random in a certain office, were found to be: $$40 , \quad 42 , \quad 45 , \quad 41 , \quad 352 , \quad 40 , \quad 50 , \quad 48 , \quad 51 , \quad 49 , \quad 47 .$$ Choose and calculate an appropriate measure of central tendency (mean, mode or median) to summarise these salaries. Explain briefly why the other measures are not suitable.
CAIE S1 2006 June Q2
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
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
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
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
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.