Questions — CAIE (7646 questions)

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CAIE M2 Specimen Q5
9 marks Challenging +1.2
5 A particle \(P\) of mass 0.5 kg is projected vertically upwards from a point on a horizontal surface. A resisting force of magnitude \(0.02 v ^ { 2 } \mathrm {~N}\) acts on \(P\), where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the upward velocity of \(P\) when it is a height of \(x \mathrm {~m}\) above the surface. The initial speed of \(P\) is \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  1. Show that, while \(P\) is moving upwards, \(v \frac { \mathrm {~d} v } { \mathrm {~d} x } = - 10 - 0.04 v ^ { 2 }\).
  2. Find the greatest height of \(P\) above the surface.
  3. Find the speed of \(P\) immediately before it strikes the surface after descending.
CAIE M2 Specimen Q6
9 marks Challenging +1.2
6 \includegraphics[max width=\textwidth, alt={}, center]{add3948c-3b45-4e67-ac84-e2ca935afd64-08_442_953_237_596} An object is formed by joining a hemispherical shell of radius 0.2 m and a solid cone with base radius 0.2 m and height \(h \mathrm {~m}\) along their circumferences. The centre of mass, \(G\), of the object is \(d \mathrm {~m}\) from the vertex of the cone on the axis of symmetry of the object. The object rests in equilibrium on a horizontal plane, with the curved surface of the cone in contact with the plane (see diagram). The object is on the point of toppling.
  1. Show that \(d = h + \frac { 0.04 } { h }\).
  2. It is given that the cone is uniform and of weight 4 N , and that the hemispherical shell is uniform and of weight \(W \mathrm {~N}\). Given also that \(h = 0.8\), find \(W\).
CAIE M2 Specimen Q7
11 marks Standard +0.8
7 A particle \(P\) of mass \(M \mathrm {~kg}\) is attached to one end of a light elastic string of natural length 0.8 m and modulus of elasticity 12.5 N . The other end of the string is attached to a fixed point \(A\). The particle is released from rest at \(A\) and falls vertically until it comes to instantaneous rest at the point \(B\). The greatest speed of \(P\) during its descent is \(4.4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when the extension of the string is \(e \mathrm {~m}\).
  1. Show that \(e = 0.64 M\).
  2. Find a second equation in \(e\) and \(M\), and hence find \(M\).
  3. Calculate the distance \(A B\).
CAIE Further Paper 3 2020 November Q1
3 marks Challenging +1.2
1 A particle \(P\) of mass \(m\) is placed on a fixed smooth plane which is inclined at an angle \(\theta\) to the horizontal. A light spring, of natural length \(a\) and modulus of elasticity \(3 m g\), has one end attached to \(P\) and the other end attached to a fixed point \(O\) at the top of the plane. The spring lies along a line of greatest slope of the plane. The system is released from rest with the spring at its natural length. Find, in terms of \(a\) and \(\theta\), an expression for the greatest extension of the spring in the subsequent motion. \includegraphics[max width=\textwidth, alt={}, center]{1c53c407-25ea-43fc-a571-74ba1fffea8f-04_515_707_267_685} A particle \(P\) is attached to one end of a light inextensible string of length \(a\). The other end of the string is attached to a fixed point \(O\). The particle \(P\) is held with the string taut and making an angle \(\theta\) with the downward vertical. The particle \(P\) is then projected with speed \(\frac { 4 } { 5 } \sqrt { 5 a g }\) perpendicular to the string and just completes a vertical circle (see diagram). Find the value of \(\cos \theta\).
CAIE Further Paper 3 2020 November Q3
6 marks Standard +0.8
3 One end of a light elastic string, of natural length \(a\) and modulus of elasticity \(4 m g\), is attached to a fixed point \(O\). The other end of the string is attached to a particle of mass \(m\). The particle moves in a horizontal circle with a constant angular speed \(\sqrt { \frac { \mathrm { g } } { \mathrm { a } } }\) with the string inclined at an angle \(\theta\) to the downward vertical through \(O\). The length of the string during this motion is \(( \mathrm { k } + 1 ) \mathrm { a }\).
  1. Find the value of \(k\).
  2. Find the value of \(\cos \theta\). \includegraphics[max width=\textwidth, alt={}, center]{1c53c407-25ea-43fc-a571-74ba1fffea8f-06_584_695_264_667} The diagram shows the cross-section \(A B C D\) of a uniform solid object which is formed by removing a cone with cross-section \(D C E\) from the top of a larger cone with cross-section \(A B E\). The perpendicular distance between \(A B\) and \(D C\) is \(h\), the diameter \(A B\) is \(6 r\) and the diameter \(D C\) is \(2 r\).
    1. Find an expression, in terms of \(h\), for the distance of the centre of mass of the solid object from \(A B\).
      The object is freely suspended from the point \(B\) and hangs in equilibrium. The angle between \(A B\) and the downward vertical through \(B\) is \(\theta\).
    2. Given that \(h = \frac { 13 } { 4 } r\), find the value of \(\tan \theta\).
CAIE S1 2020 June Q1
3 marks Easy -1.8
1 For \(n\) values of the variable \(x\), it is given that $$\Sigma ( x - 50 ) = 144 \quad \text { and } \quad \Sigma x = 944 .$$ Find the value of \(n\).
CAIE S1 2020 June Q2
5 marks Moderate -0.8
2 A total of 500 students were asked which one of four colleges they attended and whether they preferred soccer or hockey. The numbers of students in each category are shown in the following table.
\cline { 2 - 4 } \multicolumn{1}{c|}{}SoccerHockeyTotal
Amos543286
Benn8472156
Canton225678
Devar12060180
Total280220500
  1. Find the probability that a randomly chosen student is at Canton college and prefers hockey.
  2. Find the probability that a randomly chosen student is at Devar college given that he prefers soccer.
  3. One of the students is chosen at random. Determine whether the events 'the student prefers hockey' and 'the student is at Amos college or Benn college' are independent, justifying your answer.
CAIE S1 2020 June Q3
8 marks Easy -1.2
3 Two machines, \(A\) and \(B\), produce metal rods of a certain type. The lengths, in metres, of 19 rods produced by machine \(A\) and 19 rods produced by machine \(B\) are shown in the following back-to-back stem-and-leaf diagram. \begin{table}[h]
\(A\)\(B\)
21124
76302224556
8743112302689
55532243346
4310256
\captionsetup{labelformat=empty} \caption{Key: 7 | 22 | 4 means 0.227 m for machine \(A\) and 0.224 m for machine \(B\).}
\end{table}
  1. Find the median and the interquartile range for machine \(A\).
    It is given that for machine \(B\) the median is 0.232 m , the lower quartile is 0.224 m and the upper quartile is 0.243 m .
  2. Draw box-and-whisker plots for \(A\) and \(B\). \includegraphics[max width=\textwidth, alt={}, center]{a3b3ebd1-db9e-4552-9abe-bfdeba786d02-05_812_1205_616_511}
  3. Hence make two comparisons between the lengths of the rods produced by machine \(A\) and those produced by machine \(B\).
CAIE S1 2020 June Q4
8 marks Moderate -0.8
4 Trees in the Redian forest are classified as tall, medium or short, according to their height. The heights can be modelled by a normal distribution with mean 40 m and standard deviation 12 m . Trees with a height of less than 25 m are classified as short.
  1. Find the probability that a randomly chosen tree is classified as short.
    Of the trees that are classified as tall or medium, one third are tall and two thirds are medium.
  2. Show that the probability that a randomly chosen tree is classified as tall is 0.298 , correct to 3 decimal places.
  3. Find the height above which trees are classified as tall.
CAIE S1 2020 June Q5
8 marks Moderate -0.8
5 A fair three-sided spinner has sides numbered 1, 2, 3. A fair five-sided spinner has sides numbered \(1,1,2,2,3\). Both spinners are spun once. For each spinner, the number on the side on which it lands is noted. The random variable \(X\) is the larger of the two numbers if they are different, and their common value if they are the same.
  1. Show that \(\mathrm { P } ( X = 3 ) = \frac { 7 } { 15 }\). \includegraphics[max width=\textwidth, alt={}, center]{a3b3ebd1-db9e-4552-9abe-bfdeba786d02-08_69_1569_541_328}
  2. Draw up the probability distribution table for \(X\).
  3. Find \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).
CAIE S1 2020 June Q6
9 marks Moderate -0.3
6
  1. Find the number of different ways in which the 10 letters of the word SUMMERTIME can be arranged so that there is an E at the beginning and an E at the end.
  2. Find the number of different ways in which the 10 letters of the word SUMMERTIME can be arranged so that the Es are not together.
  3. Four letters are selected from the 10 letters of the word SUMMERTIME. Find the number of different selections if the four letters include at least one M and exactly one E .
CAIE S1 2020 June Q7
9 marks Moderate -0.3
7 On any given day, the probability that Moena messages her friend Pasha is 0.72 .
  1. Find the probability that for a random sample of 12 days Moena messages Pasha on no more than 9 days.
  2. Moena messages Pasha on 1 January. Find the probability that the next day on which she messages Pasha is 5 January.
  3. Use an approximation to find the probability that in any period of 100 days Moena messages Pasha on fewer than 64 days.
    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 2020 June Q1
6 marks Easy -1.2
1 Juan goes to college each day by any one of car or bus or walking. The probability that he goes by car is 0.2 , the probability that he goes by bus is 0.45 and the probability that he walks is 0.35 . When Juan goes by car, the probability that he arrives early is 0.6 . When he goes by bus, the probability that he arrives early is 0.1 . When he walks he always arrives early.
  1. Draw a fully labelled tree diagram to represent this information.
  2. Find the probability that Juan goes to college by car given that he arrives early.
CAIE S1 2020 June Q2
4 marks Moderate -0.8
2 In a certain large college, \(22 \%\) of students own a car.
  1. 3 students from the college are chosen at random. Find the probability that all 3 students own a car.
  2. 16 students from the college are chosen at random. Find the probability that the number of these students who own a car is at least 2 and at most 4 .
CAIE S1 2020 June Q3
5 marks Moderate -0.8
3 In a certain town, the time, \(X\) hours, for which people watch television in a week has a normal distribution with mean 15.8 hours and standard deviation 4.2 hours.
  1. Find the probability that a randomly chosen person from this town watches television for less than 21 hours in a week.
  2. Find the value of \(k\) such that \(\mathrm { P } ( X < k ) = 0.75\).
CAIE S1 2020 June Q4
6 marks Moderate -0.8
4 A fair four-sided spinner has edges numbered 1, 2, 2, 3. A fair three-sided spinner has edges numbered \(- 2 , - 1,1\). Each spinner is spun and the number on the edge on which it comes to rest is noted. The random variable \(X\) is the sum of the two numbers that have been noted.
  1. Draw up the probability distribution table for \(X\).
  2. Find \(\operatorname { Var } ( X )\).
CAIE S1 2020 June Q5
9 marks Standard +0.3
5 A pair of fair coins is thrown repeatedly until a pair of tails is obtained. The random variable \(X\) denotes the number of throws required to obtain a pair of tails.
  1. Find the expected value of \(X\).
  2. Find the probability that exactly 3 throws are required to obtain a pair of tails.
  3. Find the probability that fewer than 6 throws are required to obtain a pair of tails.
    On a different occasion, a pair of fair coins is thrown 80 times.
  4. Use an approximation to find the probability that a pair of tails is obtained more than 25 times.
CAIE S1 2020 June Q6
10 marks Easy -1.3
6 The annual salaries, in thousands of dollars, for 11 employees at each of two companies \(A\) and \(B\) are shown below.
Company \(A\)3032354141424749525364
Company \(B\)2647305241383542493142
  1. Represent the data by drawing a back-to-back stem-and-leaf diagram with company \(A\) on the left-hand side of the diagram.
  2. Find the median and the interquartile range of the salaries of the employees in company \(A\). [3]
    A new employee joins company \(B\). The mean salary of the 12 employees is now \(\\) 38500$.
  3. Find the salary of the new employee.
CAIE S1 2020 June Q7
10 marks Moderate -0.3
7
  1. Find the number of different possible arrangements of the 9 letters in the word CELESTIAL.
  2. Find the number of different arrangements of the 9 letters in the word CELESTIAL in which the first letter is C, the fifth letter is T and the last letter is E.
  3. Find the probability that a randomly chosen arrangement of the 9 letters in the word CELESTIAL does not have the two Es together.
    5 letters are selected at random from the 9 letters in the word CELESTIAL.
  4. Find the number of different selections if the 5 letters include at least one E and at most one L .
    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 2021 June Q1
4 marks Moderate -0.3
1 A bag contains 12 marbles, each of a different size. 8 of the marbles are red and 4 of the marbles are blue. How many different selections of 5 marbles contain at least 4 marbles of the same colour?
CAIE S1 2021 June Q2
5 marks Moderate -0.8
2 A company produces a particular type of metal rod. The lengths of these rods are normally distributed with mean 25.2 cm and standard deviation 0.4 cm . A random sample of 500 of these rods is chosen. How many rods in this sample would you expect to have a length that is within 0.5 cm of the mean length?
CAIE S1 2021 June Q3
8 marks Moderate -0.3
3
  1. How many different arrangements are there of the 8 letters in the word RELEASED?
  2. How many different arrangements are there of the 8 letters in the word RELEASED in which the letters LED appear together in that order?
  3. An arrangement of the 8 letters in the word RELEASED is chosen at random. Find the probability that the letters A and D are not together.
CAIE S1 2021 June Q4
7 marks Moderate -0.8
4 To gain a place at a science college, students first have to pass a written test and then a practical test.
Each student is allowed a maximum of two attempts at the written test. A student is only allowed a second attempt if they fail the first attempt. No student is allowed more than one attempt at the practical test. If a student fails both attempts at the written test, then they cannot attempt the practical test. The probability that a student will pass the written test at the first attempt is 0.8 . If a student fails the first attempt at the written test, the probability that they will pass at the second attempt is 0.6 . The probability that a student will pass the practical test is always 0.3 .
  1. Draw a tree diagram to represent this information, showing the probabilities on the branches.
  2. Find the probability that a randomly chosen student will succeed in gaining a place at the college.
    [0pt] [2]
  3. Find the probability that a randomly chosen student passes the written test at the first attempt given that the student succeeds in gaining a place at the college.
CAIE S1 2021 June Q5
8 marks Moderate -0.3
5 The times taken by 200 players to solve a computer puzzle are summarised in the following table.
Time \(( t\) seconds \()\)\(0 \leqslant t < 10\)\(10 \leqslant t < 20\)\(20 \leqslant t < 40\)\(40 \leqslant t < 60\)\(60 \leqslant t < 100\)
Number of players1654783220
  1. Draw a histogram to represent this information. \includegraphics[max width=\textwidth, alt={}, center]{1a27e2ca-9be5-48a0-a1aa-01844573f4d4-08_1397_1198_808_516}
  2. Calculate an estimate of the mean time taken by these 200 players.
  3. Find the greatest possible value of the interquartile range of these times.
CAIE S1 2021 June Q6
9 marks Moderate -0.8
6 In Questa, 60\% of the adults travel to work by car.
  1. A random sample of 12 adults from Questa is taken. Find the probability that the number who travel to work by car is less than 10 .
  2. A random sample of 150 adults from Questa is taken. Use an approximation to find the probability that the number who travel to work by car is less than 81 .
  3. Justify the use of your approximation in part (b).