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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 2002 June Q1
4 marks Easy -1.2
1 Events \(A\) and \(B\) are such that \(\mathrm { P } ( A ) = 0.3 , \mathrm { P } ( B ) = 0.8\) and \(\mathrm { P } ( A\) and \(B\) )=0.4. State, giving a reason in each case, whether events \(A\) and \(B\) are
  1. independent,
  2. mutually exclusive.
CAIE S1 2002 June Q2
6 marks Easy -1.8
2 The manager of a company noted the times spent in 80 meetings. The results were as follows.
Time ( \(t\) minutes)\(0 < t \leqslant 15\)\(15 < t \leqslant 30\)\(30 < t \leqslant 60\)\(60 < t \leqslant 90\)\(90 < t \leqslant 120\)
Number of meetings4724387
Draw a cumulative frequency graph and use this to estimate the median time and the interquartile range.
CAIE S1 2002 June Q3
7 marks Moderate -0.3
3 A fair cubical die with faces numbered \(1,1,1,2,3,4\) is thrown and the score noted. The area \(A\) of a square of side equal to the score is calculated, so, for example, when the score on the die is 3 , the value of \(A\) is 9 .
  1. Draw up a table to show the probability distribution of \(A\).
  2. Find \(\mathrm { E } ( A )\) and \(\operatorname { Var } ( A )\).
  3. In a spot check of the speeds \(x \mathrm {~km} \mathrm {~h} ^ { - 1 }\) of 30 cars on a motorway, the data were summarised by \(\Sigma ( x - 110 ) = - 47.2\) and \(\Sigma ( x - 110 ) ^ { 2 } = 5460\). Calculate the mean and standard deviation of these speeds.
  4. On another day the mean speed of cars on the motorway was found to be \(107.6 \mathrm {~km} \mathrm {~h} ^ { - 1 }\) and the standard deviation was \(13.8 \mathrm {~km} \mathrm {~h} ^ { - 1 }\). Assuming these speeds follow a normal distribution and that the speed limit is \(110 \mathrm {~km} \mathrm {~h} ^ { - 1 }\), find what proportion of cars exceed the speed limit.
CAIE S1 2002 June Q5
8 marks Moderate -0.3
5 The digits of the number 1223678 can be rearranged to give many different 7 -digit numbers. Find how many different 7-digit numbers can be made if
  1. there are no restrictions on the order of the digits,
  2. the digits 1,3,7 (in any order) are next to each other,
  3. these 7 -digit numbers are even.
  4. In a normal distribution with mean \(\mu\) and standard deviation \(\sigma , \mathrm { P } ( X > 3.6 ) = 0.5\) and \(\mathrm { P } ( X > 2.8 ) = 0.6554\). Write down the value of \(\mu\), and calculate the value of \(\sigma\).
  5. If four observations are taken at random from this distribution, find the probability that at least two observations are greater than 2.8.
CAIE S1 2002 June Q7
10 marks Standard +0.3
7
  1. A garden shop sells polyanthus plants in boxes, each box containing the same number of plants. The number of plants per box which produce yellow flowers has a binomial distribution with mean 11 and variance 4.95.
    (a) Find the number of plants per box.
    (b) Find the probability that a box contains exactly 12 plants which produce yellow flowers.
  2. Another garden shop sells polyanthus plants in boxes of 100 . The shop's advertisement states that the probability of any polyanthus plant producing a pink flower is 0.3 . Use a suitable approximation to find the probability that a box contains fewer than 35 plants which produce pink flowers.
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. $$52,73,34,85,62,79,89,50,45,83,84,91,85,84,87,44,86,41,35,73,86 \text {. }$$ (a) Construct a stem-and-leaf diagram to illustrate the data.
    (b) 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 Q1
5 marks Standard +0.8
1 A particle is projected with speed \(u\) at an angle \(\alpha\) above the horizontal from a point \(O\) on a horizontal plane. The particle moves freely under gravity.
  1. Write down the horizontal and vertical components of the velocity of the particle at time \(T\) after projection.
    At time \(T\) after projection, the direction of motion of the particle is perpendicular to the direction of projection.
  2. Express \(T\) in terms of \(u , g\) and \(\alpha\).
  3. Deduce that \(\mathrm { T } > \frac { \mathrm { u } } { \mathrm { g } }\).
CAIE Further Paper 3 2021 November Q2
6 marks Standard +0.8
2 A light spring \(A B\) has natural length \(a\) and modulus of elasticity 5 mg . The end \(A\) of the spring is attached to a fixed point on a smooth horizontal surface. A particle \(P\) of mass \(m\) is attached to the end \(B\) of the spring. The spring and particle \(P\) are at rest on the surface. Another particle \(Q\) of mass \(k m\) is moving with speed \(\sqrt { 4 \mathrm { ga } }\) along the horizontal surface towards \(P\) in the direction \(B A\). The particles \(P\) and \(Q\) collide directly and coalesce. In the subsequent motion the greatest amount by which the spring is compressed is \(\frac { 1 } { 5 } a\). Find the value of \(k\). \includegraphics[max width=\textwidth, alt={}, center]{e34abb4b-1c6c-4f39-836d-467ed18345eb-04_307_1088_274_470} Particles \(A\) and \(B\), of masses \(m\) and \(3 m\) respectively, are connected by a light inextensible string of length \(a\) that passes through a fixed smooth ring \(R\). Particle \(B\) hangs in equilibrium vertically below the ring. Particle \(A\) moves in horizontal circles with speed \(v\). Particles \(A\) and \(B\) are at the same horizontal level. The angle between \(A R\) and \(B R\) is \(\theta\) (see diagram).
  1. Show that \(\cos \theta = \frac { 1 } { 3 }\).
  2. Find an expression for \(v\) in terms of \(a\) and \(g\). \includegraphics[max width=\textwidth, alt={}, center]{e34abb4b-1c6c-4f39-836d-467ed18345eb-06_597_803_258_625} An object is formed by removing a solid cylinder, of height \(k a\) and radius \(\frac { 1 } { 2 } a\), from a uniform solid hemisphere of radius \(a\). The axes of symmetry of the hemisphere and the cylinder coincide and one circular face of the cylinder coincides with the plane face of the hemisphere. \(A B\) is a diameter of the circular face of the hemisphere (see diagram).
CAIE Further Paper 3 2021 November Q6
9 marks Challenging +1.8
6 A particle \(P\) of mass 2 kg moves along a horizontal straight line. The point \(O\) is a fixed point on this line. At time \(t\) s the velocity of \(P\) is \(v \mathrm {~ms} ^ { - 1 }\) and the displacement of \(P\) from \(O\) is \(x \mathrm {~m}\).
A force of magnitude \(\left( 8 x - \frac { 128 } { x ^ { 3 } } \right) \mathrm { N }\) acts on \(P\) in the direction \(O P\). When \(\mathrm { t } = 0 , \mathrm { x } = 8\) and \(\mathrm { v } = - 15\).
  1. Show that \(\mathrm { v } = - \frac { 2 } { \mathrm { x } } \left( \mathrm { x } ^ { 2 } - 4 \right)\).
  2. Find an expression for \(x\) in terms of \(t\).
CAIE Further Paper 3 2021 November Q7
8 marks Challenging +1.8
7 One end of a light inextensible string of length \(a\) is attached to a fixed point \(O\). The other end of the string is attached to a particle \(P\) of mass \(m\). The particle \(P\) is held vertically below \(O\) with the string taut and then projected horizontally. When the string makes an angle of \(60 ^ { \circ }\) with the upward vertical, \(P\) becomes detached from the string. In its subsequent motion, \(P\) passes through the point \(A\) which is a distance \(a\) vertically above \(O\).
  1. The speed of \(P\) when it becomes detached from the string is \(V\). Use the equation of the trajectory of a projectile to find \(V\) in terms of \(a\) and \(g\).
  2. Find, in terms of \(m\) and \(g\), the tension in the string immediately after \(P\) is initially projected horizontally.
    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 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 \(e\).
  3. Show that \(\tan \beta = e \tan \alpha\).
  4. 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.
  5. 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.