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 2018 November Q1
1 A group consists of 5 men and 2 women. Find the number of different ways that the group can stand in a line if the women are not next to each other.
CAIE S1 2018 November Q2
2 A fair 6 -sided die has the numbers \(- 1 , - 1,0,0,1,2\) on its faces. A fair 3 -sided spinner has edges numbered \(- 1,0,1\). The die is thrown and the spinner is spun. The number on the uppermost face of the die and the number on the edge on which the spinner comes to rest are noted. The sum of these two numbers is denoted by \(X\).
  1. Draw up a table showing the probability distribution of \(X\).
  2. Find \(\operatorname { Var } ( X )\).
CAIE S1 2018 November Q3
3 A box contains 3 red balls and 5 blue balls. One ball is taken at random from the box and not replaced. A yellow ball is then put into the box. A second ball is now taken at random from the box.
  1. Complete the tree diagram to show all the outcomes and the probability for each branch. First ball
    Second ball
    \includegraphics[max width=\textwidth, alt={}, center]{7dc85f33-2647-4f73-8093-524b70f99767-04_655_392_688_474}
    \includegraphics[max width=\textwidth, alt={}, center]{7dc85f33-2647-4f73-8093-524b70f99767-04_785_387_703_1110}
  2. Find the probability that the two balls taken are the same colour.
  3. Find the probability that the first ball taken is red, given that the second ball taken is blue.
CAIE S1 2018 November Q4
4 Out of a class of 8 boys and 4 girls, a group of 7 people is chosen at random.
  1. Find the probability that the group of 7 includes one particular boy.
  2. Find the probability that the group of 7 includes at least 2 girls.
CAIE S1 2018 November Q5
4 marks
5 The weights of apples sold by a store can be modelled by a normal distribution with mean 120 grams and standard deviation 24 grams. Apples weighing less than 90 grams are graded as 'small'; apples weighing more than 140 grams are graded as 'large'; the remainder are graded as 'medium'.
  1. Show that the probability that an apple chosen at random is graded as medium is 0.692 , correct to 3 significant figures.
  2. Four apples are chosen at random. Find the probability that at least two are graded as medium. [4]
CAIE S1 2018 November Q6
6 The lifetimes, in hours, of a particular type of light bulb are normally distributed with mean 2000 hours and standard deviation \(\sigma\) hours. The probability that a randomly chosen light bulb of this type has a lifetime of more than 1800 hours is 0.96 .
  1. Find the value of \(\sigma\).
    New technology has resulted in a new type of light bulb. It is found that on average one in five of these new light bulbs has a lifetime of more than 2500 hours.
  2. For a random selection of 300 of these new light bulbs, use a suitable approximate distribution to find the probability that fewer than 70 have a lifetime of more than 2500 hours.
  3. Justify the use of your approximate distribution in part (ii).
CAIE S1 2018 November Q7
7 The heights, in cm, of the 11 members of the Anvils athletics team and the 11 members of the Brecons swimming team are shown below.
Anvils173158180196175165170169181184172
Brecons166170171172172178181182183183192
  1. Draw a back-to-back stem-and-leaf diagram to represent this information, with Anvils on the left-hand side of the diagram and Brecons on the right-hand side.
  2. Find the median and the interquartile range for the heights of the Anvils.
    The heights of the 11 members of the Anvils are denoted by \(x \mathrm {~cm}\). It is given that \(\Sigma x = 1923\) and \(\Sigma x ^ { 2 } = 337221\). The Anvils are joined by 3 new members whose heights are \(166 \mathrm {~cm} , 172 \mathrm {~cm}\) and 182 cm .
  3. Find the standard deviation of the heights of all 14 members of the Anvils.
    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 2019 November Q1
1 When Shona goes to college she either catches the bus with probability 0.8 or she cycles with probability 0.2 . If she catches the bus, the probability that she is late is 0.4 . If she cycles, the probability that she is late is \(x\). The probability that Shona is not late for college on a randomly chosen day is 0.63 . Find the value of \(x\).
CAIE S1 2019 November Q2
2 Annan has designed a new logo for a sportswear company. A survey of a large number of customers found that \(42 \%\) of customers rated the logo as good.
  1. A random sample of 10 customers is chosen. Find the probability that fewer than 8 of them rate the logo as good.
  2. On another occasion, a random sample of \(n\) customers of the company is chosen. Find the smallest value of \(n\) for which the probability that at least one person rates the logo as good is greater than 0.995 .
CAIE S1 2019 November Q3
3 The mean and standard deviation of 20 values of \(x\) are 60 and 4 respectively.
  1. Find the values of \(\Sigma x\) and \(\Sigma x ^ { 2 }\).
    Another 10 values of \(x\) are such that their sum is 550 and the sum of their squares is 40500 .
  2. Find the mean and standard deviation of all these 30 values of \(x\).
CAIE S1 2019 November Q4
4 In a probability distribution the random variable \(X\) takes the values \(- 1,0,1,2,4\). The probability distribution table for \(X\) is as follows.
\(x\)- 10124
\(\mathrm { P } ( X = x )\)\(\frac { 1 } { 4 }\)\(p\)\(p\)\(\frac { 3 } { 8 }\)\(4 p\)
  1. Find the value of \(p\).
  2. Find \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).
  3. Given that \(X\) is greater than zero, find the probability that \(X\) is equal to 2 .
CAIE S1 2019 November Q5
5 Ransha measured the lengths, in centimetres, of 160 palm leaves. His results are illustrated in the cumulative frequency graph below.
\includegraphics[max width=\textwidth, alt={}, center]{7ea494c0-5e1a-4da9-a189-30128654fa1d-08_1090_1424_404_356}
  1. Estimate how many leaves have a length between 14 and 24 centimetres.
  2. \(10 \%\) of the leaves have a length of \(L\) centimetres or more. Estimate the value of \(L\).
  3. Estimate the median and the interquartile range of the lengths.
    Sharim measured the lengths, in centimetres, of 160 palm leaves of a different type. He drew a box-and-whisker plot for the data, as shown on the grid below.
    \includegraphics[max width=\textwidth, alt={}, center]{7ea494c0-5e1a-4da9-a189-30128654fa1d-09_540_1287_1181_424}
  4. Compare the central tendency and the spread of the two sets of data.
CAIE S1 2019 November Q6
6
  1. Find the number of different ways in which all 12 letters of the word STEEPLECHASE can be arranged so that all four Es are together.
  2. Find the number of different ways in which all 12 letters of the word STEEPLECHASE can be arranged so that the Ss are not next to each other.
    Four letters are selected from the 12 letters of the word STEEPLECHASE.
  3. Find the number of different selections if the four letters include exactly one \(S\).
CAIE S1 2019 November Q7
7 The shortest time recorded by an athlete in a 400 m race is called their personal best (PB). The PBs of the athletes in a large athletics club are normally distributed with mean 49.2 seconds and standard deviation 2.8 seconds.
  1. Find the probability that a randomly chosen athlete from this club has a PB between 46 and 53 seconds.
  2. It is found that \(92 \%\) of athletes from this club have PBs of more than \(t\) seconds. Find the value of \(t\).
    Three athletes from the club are chosen at random.
  3. Find the probability that exactly 2 have PBs of less than 46 seconds.
    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 2019 November Q1
1 Twelve tourists were asked to estimate the height, in metres, of a new building. Their estimates were as follows. $$\begin{array} { l l l l l l l l l l l l } 50 & 45 & 62 & 30 & 40 & 55 & 110 & 38 & 52 & 60 & 55 & 40 \end{array}$$
  1. Find the median and the interquartile range for the data.
  2. Give a disadvantage of using the mean as a measure of the central tendency in this case.
CAIE S1 2019 November Q2
2 Benju cycles to work each morning and he has two possible routes. He chooses the hilly route with probability 0.4 and the busy route with probability 0.6 . If he chooses the hilly route, the probability that he will be late for work is \(x\) and if he chooses the busy route the probability that he will be late for work is \(2 x\). The probability that Benju is late for work on any day is 0.36 .
  1. Show that \(x = 0.225\).
  2. Given that Benju is not late for work, find the probability that he chooses the hilly route.
CAIE S1 2019 November Q3
3 The speeds, in \(\mathrm { km } \mathrm { h } ^ { - 1 }\), of 90 cars as they passed a certain marker on a road were recorded, correct to the nearest \(\mathrm { km } \mathrm { h } ^ { - 1 }\). The results are summarised in the following table.
Speed \(\left( \mathrm { km } \mathrm { h } ^ { - 1 } \right)\)\(10 - 29\)\(30 - 39\)\(40 - 49\)\(50 - 59\)\(60 - 89\)
Frequency1024301412
  1. On the grid, draw a histogram to illustrate the data in the table.
    \includegraphics[max width=\textwidth, alt={}, center]{5307cf3d-3d3a-441a-83d7-4adad917e168-04_1594_1198_657_516}
  2. Calculate an estimate for the mean speed of these 90 cars as they pass the marker.
CAIE S1 2019 November Q4
5 marks
4 In Quarendon, \(66 \%\) of households are satisfied with the speed of their wifi connection.
  1. Find the probability that, out of 10 households chosen at random in Quarendon, at least 8 are satisfied with the speed of their wifi connection.
  2. A random sample of 150 households in Quarendon is chosen. Use a suitable approximation to find the probability that more than 84 are satisfied with the speed of their wifi connection. [5]
CAIE S1 2019 November Q5
5 A fair red spinner has four sides, numbered 1, 2, 3, 3. A fair blue spinner has three sides, numbered \(- 1,0,2\). When a spinner is spun, the score is the number on the side on which it lands. The spinners are spun at the same time. The random variable \(X\) denotes the score on the red spinner minus the score on the blue spinner.
  1. Draw up the probability distribution table for \(X\).
  2. Find \(\operatorname { Var } ( X )\).
CAIE S1 2019 November Q6
6 The heights, in metres, of fir trees in a large forest have a normal distribution with mean 40 and standard deviation 8 .
  1. Find the probability that a fir tree chosen at random in this forest has a height less than 45 metres.
  2. Find the probability that a fir tree chosen at random in this forest has a height within 5 metres of the mean.
    In another forest, the heights of another type of fir tree are modelled by a normal distribution. A scientist measures the heights of 500 randomly chosen trees of this type. He finds that 48 trees are less than 10 m high and 76 trees are more than 24 m high.
  3. Find the mean and standard deviation of the heights of trees of this type.
CAIE S1 2019 November Q7
7
  1. Find the number of different ways in which the 9 letters of the word TOADSTOOL can be arranged so that all three Os are together and both Ts are together.
  2. Find the number of different ways in which the 9 letters of the word TOADSTOOL can be arranged so that the Ts are not together.
  3. Find the probability that a randomly chosen arrangement of the 9 letters of the word TOADSTOOL has a T at the beginning and a T at the end.
  4. Five letters are selected from the 9 letters of the word TOADSTOOL. Find the number of different selections if the five letters include at least 2 Os and at least 1 T .
    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 2019 November Q1
1 There are 300 students at a music college. All students play exactly one of the guitar, the piano or the flute. The numbers of male and female students that play each of the instruments are given in the following table.
GuitarPianoFlute
Female students623543
Male students784042
  1. Find the probability that a randomly chosen student at the college is a male who does not play the piano.
  2. Determine whether the events 'a randomly chosen student is male' and 'a randomly chosen student does not play the piano' are independent, justifying your answer.
CAIE S1 2019 November Q2
2
  1. How many different arrangements are there of the 9 letters in the word CORRIDORS?
  2. How many different arrangements are there of the 9 letters in the word CORRIDORS in which the first letter is D and the last letter is R or O ?
CAIE S1 2019 November Q3
3 A sports team of 7 people is to be chosen from 6 attackers, 5 defenders and 4 midfielders. The team must include at least 3 attackers, at least 2 defenders and at least 1 midfielder.
  1. In how many different ways can the team of 7 people be chosen?
    The team of 7 that is chosen travels to a match in two cars. A group of 4 travel in one car and a group of 3 travel in the other car.
  2. In how many different ways can the team of 7 be divided into a group of 4 and a group of 3 ?
CAIE S1 2019 November Q4
4 The heights of students at the Mainland college are normally distributed with mean 148 cm and standard deviation 8 cm .
  1. The probability that a Mainland student chosen at random has a height less than \(h \mathrm {~cm}\) is 0.67 . Find the value of \(h\).
    120 Mainland students are chosen at random.
  2. Find the number of these students that would be expected to have a height within half a standard deviation of the mean.