Questions — CAIE (7646 questions)

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CAIE S1 2024 November Q3
6 marks Moderate -0.3
3 A fair coin and an ordinary fair six-sided dice are thrown at the same time.The random variable \(X\) is defined as follows.
-If the coin shows a tail,\(X\) is twice the score on the dice.
-If the coin shows a head,\(X\) is the score on the dice if the score is even and \(X\) is 0 otherwise.
  1. Draw up the probability distribution table for \(X\) .
  2. Find \(\operatorname { Var } ( X )\) .
CAIE S1 2024 November Q4
7 marks Standard +0.3
4 The heights, in metres, of white pine trees are normally distributed with mean 19.8 and standard deviation 2.4 . In a certain forest there are 450 white pine trees.
  1. How many of these trees would you expect to have height less than 18.2 metres?
    The heights, in metres, of red pine trees are normally distributed with mean 23.4 and standard deviation \(\sigma\). It is known that \(26 \%\) of red pine trees have height greater than 25.5 metres.
  2. Find the value of \(\sigma\).
CAIE S1 2024 November Q5
6 marks Moderate -0.3
5 In a class of 21 students, there are 10 violinists, 6 guitarists and 5 pianists. A group of 7 is to be chosen from these 21 students. The group will consist of 4 violinists, 2 guitarists and 1 pianist.
  1. In how many ways can the group of 7 be chosen?
    On another occasion a group of 5 will be chosen from the 21 students. The group must contain at least 2 violinists, at least 1 guitarist and at most 1 pianist.
  2. In how many ways can the group of 5 be chosen?
CAIE S1 2024 November Q6
10 marks Easy -1.2
6 Teams of 15 runners took part in a charity run last Saturday. The times taken, in minutes, to complete the course by the runners from the Falcons and the runners from the Kites are shown in the table.
Falcons383942444648505152565859646976
Kites324040454748525458595960616365
  1. Draw a back-to-back stem-and-leaf diagram to represent this information, with the Falcons on the left-hand side.
  2. Find the median and the interquartile range of the times for the Falcons.
    Let \(x\) and \(y\) denote the times, in minutes, of a runner from the Falcons and a runner from the Kites respectively. It is given that $$\sum x = 792 , \quad \sum x ^ { 2 } = 43504 , \quad \sum y = 783 , \quad \sum y ^ { 2 } = 42223 .$$
  3. Find the mean and the standard deviation of the times taken by all 30 runners from the two teams.
CAIE S1 2024 November Q7
11 marks Moderate -0.3
7 In a game,players attempt to score a goal by kicking a ball into a net.The probability that Leno scores a goal is 0.4 on any attempt,independently of all other attempts.The random variable \(X\) denotes the number of attempts that it takes Leno to score a goal.
  1. Find \(\mathrm { P } ( X = 5 )\) .
    ............................................................................................................................................
  2. Find \(\mathrm { P } ( 3 \leqslant X \leqslant 7 )\) .
  3. Find the probability that Leno scores his second goal on or before his 5th attempt. \includegraphics[max width=\textwidth, alt={}, center]{aeb7b26e-6754-4c61-b71e-e8169c617b91-10_2715_33_106_2017} \includegraphics[max width=\textwidth, alt={}, center]{aeb7b26e-6754-4c61-b71e-e8169c617b91-11_2723_33_99_22} Leno has 75 attempts to score a goal.
  4. Use a suitable approximation to find the probability that Leno scores more than 28 goals but fewer than 35 goals.
    If you use the following page to complete the answer to any question, the question number must be clearly shown.
CAIE S1 2024 November Q2
4 marks Moderate -0.8
2 A red fair six-sided dice has faces labelled 1, 1, 1, 2, 2, 2. A blue fair six-sided dice has faces labelled \(1,1,2,2,3,3\). Both dice are thrown. The random variable \(X\) is the product of the scores on the two dice.
  1. Draw up the probability distribution table for \(X\).
  2. Find \(\mathrm { E } ( X )\).
CAIE S1 2024 November Q3
7 marks Moderate -0.8
3 In Molimba, the heights, in cm , of adult males are normally distributed with mean 176 cm and standard deviation 4.8 cm .
  1. Find the probability that a randomly chosen adult male in Molimba has a height greater than 170 cm .
    60\% of adult males in Molimba have a height between 170 cm and \(k \mathrm {~cm}\), where \(k\) is greater than 170 .
  2. Find the value of \(k\), giving your answer correct to 1 decimal place.
CAIE S1 2024 November Q4
11 marks Moderate -0.8
4 On a certain day, the heights of 150 sunflower plants grown by children at a local school are measured, correct to the nearest cm . These heights are summarised in the following table.
Height
\(( \mathrm { cm } )\)
\(10 - 19\)\(20 - 29\)\(30 - 39\)\(40 - 44\)\(45 - 49\)\(50 - 54\)\(55 - 59\)
Frequency1018324228146
  1. Draw a cumulative frequency graph to illustrate the data. \includegraphics[max width=\textwidth, alt={}, center]{915661eb-2544-4293-af72-608fedb43d70-06_1600_1301_760_383}
  2. Use your graph to estimate the 30th percentile of the heights of the sunflower plants. \includegraphics[max width=\textwidth, alt={}, center]{915661eb-2544-4293-af72-608fedb43d70-07_2723_35_101_20}
  3. Calculate estimates for the mean and the standard deviation of the heights of the 150 sunflower plants.
CAIE S1 2024 November Q5
10 marks Standard +0.8
5 A factory produces chocolates. 30\% of the chocolates are wrapped in gold foil, 25\% are wrapped in red foil and the remainder are unwrapped. Indigo chooses 8 chocolates at random from the production line.
  1. Find the probability that she obtains no more than 2 chocolates that are wrapped in red foil.
    Jake chooses chocolates one at a time at random from the production line.
  2. Find the probability that the first time he obtains a chocolate that is wrapped in red foil is before the 7th choice. \includegraphics[max width=\textwidth, alt={}, center]{915661eb-2544-4293-af72-608fedb43d70-08_2720_35_106_2015} \includegraphics[max width=\textwidth, alt={}, center]{915661eb-2544-4293-af72-608fedb43d70-09_2717_29_105_22} Keifa chooses chocolates one at a time at random from the production line.
  3. Find the probability that the second chocolate chosen is the first one wrapped in gold foil given that the fifth chocolate chosen is the first unwrapped chocolate.
CAIE S1 2024 November Q6
11 marks Challenging +1.2
6
  1. Find the number of different arrangements of the 9 letters in the word HAPPINESS.
  2. Find the number of different arrangements of the 9 letters in the word HAPPINESS in which the first and last letters are not the same as each other. \includegraphics[max width=\textwidth, alt={}, center]{915661eb-2544-4293-af72-608fedb43d70-10_2715_35_110_2012}
  3. Find the number of different arrangements of the 9 letters in the word HAPPINESS in which the two Ps are together and there are exactly two letters between the two Ss.
    The 9 letters in the word HAPPINESS are divided at random into a group of 5 and a group of 4 .
  4. Find the probability that both Ps are in one group and both Ss are in the other group.
    If you use the following page to complete the answer to any question, the question number must be clearly shown.
CAIE S1 2003 June Q1
5 marks Easy -1.8
1
  1. \begin{figure}[h]
    \captionsetup{labelformat=empty} \caption{Sales of Superclene Toothpaste} \includegraphics[alt={},max width=\textwidth]{df20f053-8d67-428d-bb19-9447049deed5-2_725_1073_347_497}
    \end{figure} The diagram represents the sales of Superclene toothpaste over the last few years. Give a reason why it is misleading.
  2. The following data represent the daily ticket sales at a small theatre during three weeks. $$52,73,34,85,62,79,89,50,45,83,84,91,85,84,87,44,86,41,35,73,86 \text {. }$$
    1. Construct a stem-and-leaf diagram to illustrate the data.
    2. Use your diagram to find the median of the data.
CAIE S1 2003 June Q2
6 marks Moderate -0.8
2 A box contains 10 pens of which 3 are new. A random sample of two pens is taken.
  1. Show that the probability of getting exactly one new pen in the sample is \(\frac { 7 } { 15 }\).
  2. Construct a probability distribution table for the number of new pens in the sample.
  3. Calculate the expected number of new pens in the sample.
CAIE S1 2003 June Q3
6 marks Moderate -0.3
3
  1. The height of sunflowers follows a normal distribution with mean 112 cm and standard deviation 17.2 cm . Find the probability that the height of a randomly chosen sunflower is greater than 120 cm .
  2. When a new fertiliser is used, the height of sunflowers follows a normal distribution with mean 115 cm . Given that \(80 \%\) of the heights are now greater than 103 cm , find the standard deviation.
CAIE S1 2003 June Q4
7 marks Moderate -0.3
4 Kamal has 30 hens. The probability that any hen lays an egg on any day is 0.7 . Hens do not lay more than one egg per day, and the days on which a hen lays an egg are independent.
  1. Calculate the probability that, on any particular day, Kamal's hens lay exactly 24 eggs.
  2. Use a suitable approximation to calculate the probability that Kamal's hens lay fewer than 20 eggs on any particular day.
CAIE S1 2003 June Q5
8 marks Moderate -0.8
5 A committee of 5 people is to be chosen from 6 men and 4 women. In how many ways can this be done
  1. if there must be 3 men and 2 women on the committee,
  2. if there must be more men than women on the committee,
  3. if there must be 3 men and 2 women, and one particular woman refuses to be on the committee with one particular man?
CAIE S1 2003 June Q6
9 marks Moderate -0.8
6 The people living in 3 houses are classified as children ( \(C\) ), parents ( \(P\) ) or grandparents ( \(G\) ). The numbers living in each house are shown in the table below.
House number 1House number 2House number 3
\(4 C , 1 P , 2 G\)\(2 C , 2 P , 3 G\)\(1 C , 1 G\)
  1. All the people in all 3 houses meet for a party. One person at the party is chosen at random. Calculate the probability of choosing a grandparent.
  2. A house is chosen at random. Then a person in that house is chosen at random. Using a tree diagram, or otherwise, calculate the probability that the person chosen is a grandparent.
  3. Given that the person chosen by the method in part (ii) is a grandparent, calculate the probability that there is also a parent living in the house.
CAIE S1 2003 June Q7
9 marks Easy -1.2
7 A random sample of 97 people who own mobile phones was used to collect data on the amount of time they spent per day on their phones. The results are displayed in the table below.
Time spent per
day \(( t\) minutes \()\)
\(0 \leqslant t < 5\)\(5 \leqslant t < 10\)\(10 \leqslant t < 20\)\(20 \leqslant t < 30\)\(30 \leqslant t < 40\)\(40 \leqslant t < 70\)
Number
of people
11203218106
  1. Calculate estimates of the mean and standard deviation of the time spent per day on these mobile phones.
  2. On graph paper, draw a fully labelled histogram to represent the data.
CAIE Further Paper 3 2021 November Q3
6 marks Challenging +1.2
3 A light elastic string has natural length \(a\) and modulus of elasticity 12 mg . One end of the string is attached to a fixed point \(O\). The other end of the string is attached to a particle of mass \(m\). The particle hangs in equilibrium vertically below \(O\). The particle is pulled vertically down and released from rest with the extension of the string equal to \(e\), where \(\mathrm { e } > \frac { 1 } { 3 } \mathrm { a }\). In the subsequent motion the particle has speed \(\sqrt { 2 \mathrm { ga } }\) when it has ascended a distance \(\frac { 1 } { 3 } a\). Find \(e\) in terms of \(a\). \includegraphics[max width=\textwidth, alt={}, center]{b10c65ef-dafd-4746-be5b-789130b7d030-06_488_496_269_781} A uniform lamina \(A E C F\) is formed by removing two identical triangles \(B C E\) and \(C D F\) from a square lamina \(A B C D\). The square has side \(3 a\) and \(E B = D F = h\) (see diagram).
  1. Find the distance of the centre of mass of the lamina \(A E C F\) from \(A D\) and from \(A B\), giving your answers in terms of \(a\) and \(h\).
    The lamina \(A E C F\) is placed vertically on its edge \(A E\) on a horizontal plane.
  2. Find, in terms of \(a\), the set of values of \(h\) for which the lamina remains in equilibrium.
CAIE Further Paper 3 2021 November Q6
8 marks Challenging +1.8
6 A particle \(P\), of mass \(m\), 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 \(P\) moves in complete vertical circles about \(O\) with the string taut. The points \(A\) and \(B\) are on the path of \(P\) with \(A B\) a diameter of the circle. \(O A\) makes an angle \(\theta\) with the downward vertical through \(O\) and \(O B\) makes an angle \(\theta\) with the upward vertical through \(O\). The speed of \(P\) when it is at \(A\) is \(\sqrt { 5 a g }\). The ratio of the tension in the string when \(P\) is at \(A\) to the tension in the string when \(P\) is at \(B\) is \(9 : 5\).
  1. Find the value of \(\cos \theta\).
  2. Find, in terms of \(a\) and \(g\), the greatest speed of \(P\) during its motion. \includegraphics[max width=\textwidth, alt={}, center]{b10c65ef-dafd-4746-be5b-789130b7d030-12_613_718_251_676} The smooth vertical walls \(A B\) and \(C B\) are at right angles to each other. A particle \(P\) is moving with speed \(u\) on a smooth horizontal floor and strikes the wall \(C B\) at an angle \(\alpha\). It rebounds at an angle \(\beta\) to the wall \(C B\). The particle then strikes the wall \(A B\) and rebounds at an angle \(\gamma\) to that wall (see diagram). The coefficient of restitution between each wall and \(P\) is \(e\).
    1. Show that \(\tan \beta = e \tan \alpha\).
    2. Express \(\gamma\) in terms of \(\alpha\) and explain what this result means about the final direction of motion of \(P\).
      As a result of the two impacts the particle loses \(\frac { 8 } { 9 }\) of its initial kinetic energy.
    3. Given that \(\alpha + \beta = 90 ^ { \circ }\), find the value of \(e\) and the value of \(\tan \alpha\).
      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 Further Paper 3 2022 November Q1
3 marks Moderate -0.5
1 A particle of mass 2 kg is attached to one end of a light inextensible string of length 0.6 m . The other end of the string is attached to a fixed point on a smooth horizontal surface. The particle is moving in a circular path on the surface. The tension in the string is 20 N . Find how many revolutions the particle makes per minute.
CAIE Further Paper 3 2022 November Q2
6 marks Standard +0.3
2 A light elastic string has natural length \(a\) and modulus of elasticity 4 mg . One end of the string is fixed to a point \(O\) on a smooth horizontal surface. A particle \(P\) of mass \(m\) is attached to the other end of the string. The particle \(P\) is projected along the surface in the direction \(O P\). When the length of the string is \(\frac { 5 } { 4 } a\), the speed of \(P\) is \(v\). When the length of the string is \(\frac { 3 } { 2 } a\), the speed of \(P\) is \(\frac { 1 } { 2 } v\).
  1. Find an expression for \(v\) in terms of \(a\) and \(g\).
  2. Find, in terms of \(g\), the acceleration of \(P\) when the stretched length of the string is \(\frac { 3 } { 2 } a\). \includegraphics[max width=\textwidth, alt={}, center]{7febbd80-4cbb-4b2e-b022-d6a20e7e13aa-04_552_1059_264_502} A smooth cylinder is fixed to a rough horizontal surface with its axis of symmetry horizontal. A uniform rod \(A B\), of length \(4 a\) and weight \(W\), rests against the surface of the cylinder. The end \(A\) of the rod is in contact with the horizontal surface. The vertical plane containing the rod \(A B\) is perpendicular to the axis of the cylinder. The point of contact between the rod and the cylinder is \(C\), where \(A C = 3 a\). The angle between the rod and the horizontal surface is \(\theta\) where \(\tan \theta = \frac { 3 } { 4 }\) (see diagram). The coefficient of friction between the rod and the horizontal surface is \(\frac { 6 } { 7 }\). A particle of weight \(k W\) is attached to the rod at \(B\). The rod is about to slip. The normal reaction between the rod and the cylinder is \(N\).
CAIE Further Paper 3 2022 November Q4
8 marks Challenging +1.2
4 A particle of mass 0.5 kg moves along a horizontal straight line. Its velocity is \(v \mathrm {~ms} ^ { - 1 }\) at time \(t \mathrm {~s}\). The forces acting on the particle are a driving force of magnitude 50 N and a resistance of magnitude \(2 v ^ { 2 } \mathrm {~N}\). The initial velocity of the particle is \(3 \mathrm {~ms} ^ { - 1 }\).
  1. Find an expression for \(v\) in terms of \(t\).
  2. Deduce the limiting value of \(v\).
CAIE Further Paper 3 2022 November Q5
8 marks Challenging +1.2
5 A particle \(P\) of mass \(m\) 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 string is held taut with \(O P\) horizontal. The particle \(P\) is projected vertically downwards with speed \(\sqrt { \frac { 1 } { 3 } \mathrm { ag } }\) and starts to move in a vertical circle. \(P\) passes through the lowest point of the circle and reaches the point \(Q\) where \(O Q\) makes an angle \(\theta\) with the downward vertical. At \(Q\) the speed of \(P\) is \(\sqrt { \mathrm { kag } }\) and the tension in the string is \(\frac { 11 } { 6 } \mathrm { mg }\).
  1. Find the value of \(k\) and the value of \(\cos \theta\).
    At \(Q\) the particle \(P\) becomes detached from the string.
  2. In the subsequent motion, find the greatest height reached by \(P\) above the level of the lowest point of the circle.
CAIE Further Paper 3 2022 November Q6
8 marks Challenging +1.8
6 \includegraphics[max width=\textwidth, alt={}, center]{7febbd80-4cbb-4b2e-b022-d6a20e7e13aa-10_426_1191_267_438} Two uniform smooth spheres \(A\) and \(B\) of equal radii have masses \(m\) and \(k m\) respectively. The two spheres are moving on a horizontal surface with speeds \(u\) and \(\frac { 5 } { 8 } u\) respectively. Immediately before the spheres collide, \(A\) is travelling along the line of centres, and \(B\) 's direction of motion makes an angle \(\alpha\) with the line of centres (see diagram). The coefficient of restitution between the spheres is \(\frac { 2 } { 3 }\) and \(\tan \alpha = \frac { 3 } { 4 }\). After the collision, the direction of motion of \(B\) is perpendicular to the line of centres.
  1. Find the value of \(k\).
  2. Find the loss in the total kinetic energy as a result of the collision.
CAIE Further Paper 3 2022 November Q7
10 marks Challenging +1.2
7 A particle \(P\) is projected with speed \(\mathrm { Vms } ^ { - 1 }\) at an angle \(75 ^ { \circ }\) above the horizontal from a point \(O\) on a horizontal plane. It then moves freely under gravity.
  1. Show that the total time of flight, in seconds, is \(\frac { 2 \mathrm {~V} } { \mathrm {~g} } \sin 75 ^ { \circ }\).
    A smooth vertical barrier is now inserted with its lower end on the plane at a distance 15 m from \(O\). The particle is projected as before but now strikes the barrier, rebounds and returns to \(O\). The coefficient of restitution between the barrier and the particle is \(\frac { 3 } { 5 }\).
  2. Explain why the total time of flight is unchanged.
  3. Find an expression for \(V\) in terms of \(g\).
    If you use the following page to complete the answer to any question, the question number must be clearly shown.