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 2023 November Q3
3 A farmer sells eggs. The weights, in grams, of the eggs can be modelled by a normal distribution with mean 80.5 and standard deviation 6.6. Eggs are classified as small, medium or large according to their weight. A small egg weighs less than 76 grams and \(40 \%\) of the eggs are classified as medium.
  1. Find the percentage of eggs that are classified as small.
  2. Find the least possible weight of an egg classified as large.
    150 of the eggs for sale last week were weighed.
  3. Use an approximation to find the probability that more than 68 of these eggs were classified as medium.
CAIE S1 2023 November Q4
4 The times, to the nearest minute, of 150 athletes taking part in a charity run are recorded. The results are summarised in the table.
Time in minutes\(101 - 120\)\(121 - 130\)\(131 - 135\)\(136 - 145\)\(146 - 160\)
Frequency1848343218
  1. Draw a histogram to represent this information.
    \includegraphics[max width=\textwidth, alt={}, center]{e8c2b51e-d788-4917-829e-1b056a24f520-08_1493_1397_936_415}
  2. Calculate estimates for the mean and standard deviation of the times taken by the athletes.
CAIE S1 2023 November Q5
5 A red spinner has four sides labelled \(1,2,3,4\). When the spinner is spun, the score is the number on the side on which it lands. The random variable \(X\) denotes this score. The probability distribution table for \(X\) is given below.
\(x\)1234
\(\mathrm { P } ( X = x )\)0.28\(p\)\(2 p\)\(3 p\)
  1. Show that \(p = 0.12\).
    A fair blue spinner and a fair green spinner each have four sides labelled 1, 2, 3, 4. All three spinners (red, blue and green) are spun at the same time.
  2. Find the probability that the sum of the three scores is 4 or less.
  3. Find the probability that the product of the three scores is 4 or less given that \(X\) is odd.
CAIE S1 2023 November Q6
6
\includegraphics[max width=\textwidth, alt={}, center]{e8c2b51e-d788-4917-829e-1b056a24f520-12_291_809_255_667} In a restaurant, the tables are rectangular. Each table seats four people: two along each of the longer sides of the table (see diagram). Eight friends have booked two tables, \(X\) and \(Y\). Rajid, Sue and Tan are three of these friends.
  1. The eight friends will be divided into two groups of 4, one group for table \(X\) and one group for table \(Y\). Find the number of ways in which this can be done if Rajid and Sue must sit at the same table as each other and Tan must sit at the other table.
    When the friends arrive at the restaurant, Rajid and Sue now decide to sit at table \(X\) on the same side as each other. Tan decides that he does not mind at which table he sits.
  2. Find the number of different seating arrangements for the 8 friends.
    As they leave the restaurant, the 8 friends stand in a line for a photograph.
  3. Find the number of different arrangements if Rajid and Sue stand next to each other, but neither is at an end of the line.
    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 2023 November Q1
1 A competitor in a throwing event has three attempts to throw a ball as far as possible. The random variable \(X\) denotes the number of throws that exceed 30 metres. The probability distribution table for \(X\) is shown below.
\(x\)0123
\(\mathrm { P } ( X = x )\)0.4\(p\)\(r\)0.15
  1. Given that \(\mathrm { E } ( X ) = 1.1\), find the value of \(p\) and the value of \(r\).
  2. Find the numerical value of \(\operatorname { Var } ( X )\).
CAIE S1 2023 November Q2
2 George has a fair 5 -sided spinner with sides labelled 1,2,3,4,5. He spins the spinner and notes the number on the side on which the spinner lands.
  1. Find the probability that it takes fewer than 7 spins for George to obtain a 5 .
    George spins the spinner 10 times.
  2. Find the probability that he obtains a 5 more than 4 times but fewer than 8 times.
CAIE S1 2023 November Q3
3 A factory produces a certain type of electrical component. It is known that \(15 \%\) of the components produced are faulty. A random sample of 200 components is chosen. Use an approximation to find the probability that more than 40 of these components are faulty.
CAIE S1 2023 November Q4
4 The heights, in cm, of the 11 players in each of two teams, the Aces and the Jets, are shown in the following table.
Aces180174169182181166173182168171164
Jets175174188168166174181181170188190
  1. Draw a back-to-back stem-and-leaf diagram to represent this information with the Aces on the left-hand side of the diagram.
  2. Find the median and the interquartile range of the heights of the players in the Aces.
  3. Give one comment comparing the spread of the heights of the Aces with the spread of the heights of the Jets.
CAIE S1 2023 November Q5
5
  1. The heights of the members of a club are normally distributed with mean 166 cm and standard deviation 10 cm .
    1. Find the probability that a randomly chosen member of the club has height less than 170 cm .
    2. Given that \(40 \%\) of the members have heights greater than \(h \mathrm {~cm}\), find the value of \(h\) correct to 2 decimal places.
  2. The random variable \(X\) is normally distributed with mean \(\mu\) and standard deviation \(\sigma\). Given that \(\sigma = \frac { 2 } { 3 } \mu\), find the probability that a randomly chosen value of \(X\) is positive.
CAIE S1 2023 November Q6
6 Freddie has two bags of marbles.
Bag \(X\) contains 7 red marbles and 3 blue marbles.
Bag \(Y\) contains 4 red marbles and 1 blue marble.
Freddie chooses one of the bags at random. A marble is removed at random from that bag and not replaced. A new red marble is now added to each bag. A second marble is then removed at random from the same bag that the first marble had been removed from.
  1. Draw a tree diagram to represent this information, showing the probability on each of the branches.
  2. Find the probability that both of the marbles removed from the bag are the same colour.
  3. Find the probability that bag \(Y\) is chosen given that the marbles removed are not both the same colour.
CAIE S1 2023 November Q7
7
  1. Find the number of different arrangements of the 9 letters in the word ANDROMEDA in which no consonant is next to another consonant. (The letters D, M, N and R are consonants and the letters A, E and O are not consonants.)
  2. Find the number of different arrangements of the 9 letters in the word ANDROMEDA in which there is an A at each end and the Ds are not together.
    Four letters are selected at random from the 9 letters in the word ANDROMEDA.
  3. Find the probability that this selection contains at least one D and exactly one A .
    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 2023 November Q1
1 Becky sometimes works in an office and sometimes works at home. The random variable \(X\) denotes the number of days that she works at home in any given week. It is given that $$\mathrm { P } ( X = x ) = k x ( x + 1 )$$ where \(k\) is a constant and \(x = 1,2,3\) or 4 only.
  1. Draw up the probability distribution table for \(X\), giving the probabilities as numerical fractions.
  2. Find \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).
CAIE S1 2023 November Q2
2 The weights of large bags of pasta produced by a company are normally distributed with mean 1.5 kg and standard deviation 0.05 kg .
  1. Find the probability that a randomly chosen large bag of pasta weighs between 1.42 kg and 1.52 kg .
    The weights of small bags of pasta produced by the company are normally distributed with mean 0.75 kg and standard deviation \(\sigma \mathrm { kg }\). It is found that \(68 \%\) of these small bags have weight less than 0.9 kg .
  2. Find the value of \(\sigma\).
CAIE S1 2023 November Q3
4 marks
3 Tim has two bags of marbles, \(A\) and \(B\).
Bag \(A\) contains 8 white, 4 red and 3 yellow marbles.
Bag \(B\) contains 6 white, 7 red and 2 yellow marbles.
Tim also has an ordinary fair 6 -sided dice. He rolls the dice. If he obtains a 1 or 2 , he chooses two marbles at random from bag \(A\), without replacement. If he obtains a \(3,4,5\) or 6 , he chooses two marbles at random from bag \(B\), without replacement.
  1. Find the probability that both marbles are white.
  2. Find the probability that the two marbles come from bag \(B\) given that one is white and one is red. [4]
CAIE S1 2023 November Q4
6 marks
4 The weights, \(x \mathrm {~kg}\), of 120 students in a sports college are recorded. The results are summarised in the following table.
Weight \(( x \mathrm {~kg} )\)\(x \leqslant 40\)\(x \leqslant 60\)\(x \leqslant 65\)\(x \leqslant 70\)\(x \leqslant 85\)\(x \leqslant 100\)
Cumulative frequency0143860106120
  1. Draw a cumulative frequency graph to represent this information.
    \includegraphics[max width=\textwidth, alt={}, center]{82c36c11-878c-47d1-a07f-fbf8b2a22d97-06_1390_1389_660_418}
  2. It is found that \(35 \%\) of the students weigh more than \(W \mathrm {~kg}\). Use your graph to estimate the value of \(W\).
  3. Calculate estimates for the mean and standard deviation of the weights of the 120 students. [6]
CAIE S1 2023 November Q5
5 marks
5 The probability that a driver passes an advanced driving test is 0.3 on any given attempt.
  1. Dipak keeps taking the test until he passes. The random variable \(X\) denotes the number of attempts required for Dipak to pass the test.
    1. Find \(\mathrm { P } ( 2 \leqslant X \leqslant 6 )\).
    2. Find \(\mathrm { E } ( X )\).
      Five friends will each take their advanced driving test tomorrow.
  2. Find the probability that at least three of them will pass tomorrow.
    75 people will take their advanced driving test next week.
    [0pt]
  3. Use an approximation to find the probability that more than 20 of them will pass next week. [5]
CAIE S1 2023 November Q6
6 Jai and his wife Kaz are having a party. Jai has invited five friends and each friend will bring his wife.
  1. At the beginning of the party, the 12 people will stand in a line for a photograph.
    1. How many different arrangements are there of the 12 people if Jai stands next to Kaz and each friend stands next to his own wife?
    2. How many different arrangements are there of the 12 people if Jai and Kaz occupy the two middle positions in the line, with Jai's five friends on one side and the five wives of the friends on the other side?
  2. For a competition during the party, the 12 people are divided at random into a group of 5, a group of 4 and a group of 3 . Find the probability that Jai and Kaz are in the same group as each other.
    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 2024 November Q1
1 Nicola throws an ordinary fair six-sided dice. The random variable \(X\) is the number of throws that she takes to obtain a 6.
  1. Find \(\mathrm { P } ( X < 8 )\).
  2. Find the probability that Nicola obtains a 6 for the second time on her 8th throw.
    \includegraphics[max width=\textwidth, alt={}, center]{ad3a6a8a-23fe-415a-b2f4-7c49136ccc6c-02_2717_35_109_2012}
CAIE S1 2024 November Q2
2 The random variable \(X\) takes the values \(- 2 , - 1,0,2,3\). It is given that \(\mathrm { P } ( X = x ) = k \left( x ^ { 2 } + 2 \right)\), where \(k\) is a positive constant.
  1. Draw up the probability distribution table for \(X\), giving the probabilities as numerical fractions.
  2. Find the value of \(\operatorname { Var } ( X )\).
CAIE S1 2024 November Q3
3 The time taken, in minutes, to walk to school was recorded for 200 pupils at a certain school. These times are summarised in the following table.
Time taken
\(( t\) minutes \()\)
\(t \leqslant 15\)\(t \leqslant 25\)\(t \leqslant 30\)\(t \leqslant 40\)\(t \leqslant 50\)\(t \leqslant 70\)
Cumulative
frequency
184688140176200
  1. Draw a cumulative frequency graph to illustrate the data.
    \includegraphics[max width=\textwidth, alt={}, center]{ad3a6a8a-23fe-415a-b2f4-7c49136ccc6c-04_1217_1509_705_278}
  2. Use your graph to estimate the median and the interquartile range of the data.
    \includegraphics[max width=\textwidth, alt={}, center]{ad3a6a8a-23fe-415a-b2f4-7c49136ccc6c-05_2723_35_101_20}
  3. Calculate an estimate for the mean value of the times taken by the 200 pupils to walk to school.
CAIE S1 2024 November Q4
4 Rahul has two bags, \(X\) and \(Y\). Bag \(X\) contains 4 red marbles and 2 blue marbles. Bag \(Y\) contains 3 red marbles and 4 blue marbles. Rahul also has a coin which is biased so that the probability of obtaining a head when it is thrown is \(\frac { 1 } { 4 }\). Rahul throws the coin.
  • If he obtains a head, he chooses at random a marble from bag \(X\). He notes the colour and replaces the marble in bag \(X\). He then chooses at random a second marble from bag \(X\).
  • If he obtains a tail, he chooses at random a marble from bag \(Y\). He notes the colour and discards the marble. He then chooses at random a second marble from bag \(Y\).
    1. Find the probability that the two marbles that Rahul chooses are the same colour.
      \includegraphics[max width=\textwidth, alt={}, center]{ad3a6a8a-23fe-415a-b2f4-7c49136ccc6c-06_2717_33_109_2014}
      \includegraphics[max width=\textwidth, alt={}, center]{ad3a6a8a-23fe-415a-b2f4-7c49136ccc6c-07_2725_35_99_20}
    2. Find the probability that the two marbles that Rahul chooses are both from bag \(Y\) given that both marbles are blue.
CAIE S1 2024 November Q5
5 The weights of the green apples sold by a shop are normally distributed with mean 90 grams and standard deviation 8 grams.
  1. Find the probability that a randomly chosen green apple weighs between 83 grams and 95 grams.
    \includegraphics[max width=\textwidth, alt={}, center]{ad3a6a8a-23fe-415a-b2f4-7c49136ccc6c-09_2717_29_105_22}
  2. The shop also sells red apples. \(60 \%\) of the red apples sold by the shop weigh more than 80 grams. 160 red apples are chosen at random from the shop. Use a suitable approximation to find the probability that fewer than 105 of the chosen red apples weigh more than 80 grams.
CAIE S1 2024 November Q6
6 The heights of the female students at Breven college are normally distributed:
  • \(90 \%\) of the female students have heights less than 182.7 cm .
  • \(40 \%\) of the female students have heights less than 162.5 cm .
    1. Find the mean and the standard deviation of the heights of the female students at Breven college.
      \includegraphics[max width=\textwidth, alt={}, center]{ad3a6a8a-23fe-415a-b2f4-7c49136ccc6c-10_2715_41_110_2008}
      \includegraphics[max width=\textwidth, alt={}, center]{ad3a6a8a-23fe-415a-b2f4-7c49136ccc6c-11_2723_35_101_20}
Ten female students are chosen at random from those at Breven college.
  • Find the probability that fewer than 8 of these 10 students have heights more than 162.5 cm .
    •
    CAIE S1 2024 November Q7
    4 marks
    7
    1. How many different arrangements are there of the 9 letters in the word INTELLECT in which the two Ts are together?
    2. How many different arrangements are there of the 9 letters in the word INTELLECT in which there is a T at each end and the two Es are not next to each other?
      Four letters are selected at random from the 9 letters in the word INTELLECT.
      [0pt]
    3. Find the percentage of the possible selections which contain at least one E and exactly one T. [4]
      If you use the following page to complete the answer to any question, the question number must be clearly shown.
      \includegraphics[max width=\textwidth, alt={}, center]{ad3a6a8a-23fe-415a-b2f4-7c49136ccc6c-14_2715_31_106_2016}
    CAIE S1 2024 November Q1
    1 At a college, the students choose exactly one of tennis, hockey or netball to play. The table shows the numbers of students in Year 1 and Year 2 at the college playing each of these sports.
    TennisHockeyNetball
    Year 1162212
    Year 2241828
    One student is chosen at random from the 120 students. Events \(X\) and \(N\) are defined as follows:
    \(X\) : the student is in Year 1
    \(N\) : the student plays netball.
    1. Find \(\mathrm { P } ( X \mid N )\).
    2. Find \(\mathrm { P } ( N \mid X )\).
    3. Determine whether or not \(X\) and \(N\) are independent events.
      One of the students who plays netball takes 8 shots at goal. On each shot, the probability that she will succeed is 0.15 , independently of all other shots.
    4. Find the probability that she succeeds on fewer than 3 of these shots.