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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]{5e95e0c9-d47d-4f2b-89da-ab949b9661f4-04_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 Q6
8 marks Challenging +1.8
6 \includegraphics[max width=\textwidth, alt={}, center]{5e95e0c9-d47d-4f2b-89da-ab949b9661f4-10_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 2023 November Q2
7 marks Challenging +1.2
2 A ball of mass 2 kg is projected vertically downwards with speed \(5 \mathrm {~ms} ^ { - 1 }\) through a liquid. At time \(t \mathrm {~s}\) after projection, the velocity of the ball is \(v \mathrm {~ms} ^ { - 1 }\) and its displacement from its starting point is \(x \mathrm {~m}\). The forces acting on the ball are its weight and a resistive force of magnitude \(0.2 v ^ { 2 } \mathrm {~N}\).
  1. Find an expression for \(v\) in terms of \(t\).
  2. Deduce what happens to \(v\) for large values of \(t\). \includegraphics[max width=\textwidth, alt={}, center]{e7091f6c-af72-49f3-b825-cdce9fb2c06f-06_803_652_251_703} A uniform square lamina of side \(2 a\) and weight \(W\) is suspended from a light inextensible string attached to the midpoint \(E\) of the side \(A B\). The other end of the string is attached to a fixed point \(P\) on a rough vertical wall. The vertex \(B\) of the lamina is in contact with the wall. The string \(E P\) is perpendicular to the side \(A B\) and makes an angle \(\theta\) with the wall (see diagram). The string and the lamina are in a vertical plane perpendicular to the wall. The coefficient of friction between the wall and the lamina is \(\frac { 1 } { 2 }\). Given that the vertex \(B\) is about to slip up the wall, find the value of \(\tan \theta\). \includegraphics[max width=\textwidth, alt={}, center]{e7091f6c-af72-49f3-b825-cdce9fb2c06f-08_581_576_269_731} A light elastic string has natural length \(8 a\) and modulus of elasticity \(5 m g\). A particle \(P\) of mass \(m\) is attached to the midpoint of the string. The ends of the string are attached to points \(A\) and \(B\) which are a distance \(12 a\) apart on a smooth horizontal table. The particle \(P\) is held on the table so that \(A P = B P = L\) (see diagram). The particle \(P\) is released from rest. When \(P\) is at the midpoint of \(A B\) it has speed \(\sqrt { 80 a g }\).
    1. Find \(L\) in terms of \(a\).
    2. Find the initial acceleration of \(P\) in terms of \(g\).
CAIE S1 2020 June Q1
5 marks Moderate -0.8
1 The score when two fair six-sided dice are thrown is the sum of the two numbers on the upper faces.
  1. Show that the probability that the score is 4 is \(\frac { 1 } { 12 }\).
    The two dice are thrown repeatedly until a score of 4 is obtained. The number of throws taken is denoted by the random variable \(X\).
  2. Find the mean of \(X\).
  3. Find the probability that a score of 4 is first obtained on the 6th throw.
  4. Find \(\mathrm { P } ( X < 8 )\).
CAIE S1 2020 June Q2
6 marks Standard +0.3
2
  1. Find the number of different arrangements that can be made from the 9 letters of the word JEWELLERY in which the three Es are together and the two Ls are together.
  2. Find the number of different arrangements that can be made from the 9 letters of the word JEWELLERY in which the two Ls are not next to each other.
CAIE S1 2020 June Q3
7 marks Moderate -0.8
3 A company produces small boxes of sweets that contain 5 jellies and 3 chocolates. Jemeel chooses 3 sweets at random from a box.
  1. Draw up the probability distribution table for the number of jellies that Jemeel chooses.
    The company also produces large boxes of sweets. For any large box, the probability that it contains more jellies than chocolates is 0.64 . 10 large boxes are chosen at random.
  2. Find the probability that no more than 7 of these boxes contain more jellies than chocolates.
CAIE S1 2020 June Q4
4 marks Standard +0.8
4 In a music competition, there are 8 pianists, 4 guitarists and 6 violinists. 7 of these musicians will be selected to go through to the final. How many different selections of 7 finalists can be made if there must be at least 2 pianists, at least 1 guitarist and more violinists than guitarists?
CAIE S1 2020 June Q5
8 marks Easy -1.2
5 On Mondays, Rani cooks her evening meal. She has a pizza, a burger or a curry with probabilities \(0.35,0.44,0.21\) respectively. When she cooks a pizza, Rani has some fruit with probability 0.3 . When she cooks a burger, she has some fruit with probability 0.8 . When she cooks a curry, she never has any fruit.
  1. Draw a fully labelled tree diagram to represent this information.
  2. Find the probability that Rani has some fruit.
  3. Find the probability that Rani does not have a burger given that she does not have any fruit.
CAIE S1 2020 June Q6
9 marks Standard +0.8
6 The lengths of female snakes of a particular species are normally distributed with mean 54 cm and standard deviation 6.1 cm .
  1. Find the probability that a randomly chosen female snake of this species has length between 50 cm and 60 cm .
    The lengths of male snakes of this species also have a normal distribution. A scientist measures the lengths of a random sample of 200 male snakes of this species. He finds that 32 have lengths less than 45 cm and 17 have lengths more than 56 cm .
  2. Find estimates for the mean and standard deviation of the lengths of male snakes of this species.
CAIE S1 2020 June Q7
11 marks Moderate -0.8
7 The numbers of chocolate bars sold per day in a cinema over a period of 100 days are summarised in the following table.
Number of chocolate bars sold\(1 - 10\)\(11 - 15\)\(16 - 30\)\(31 - 50\)\(51 - 60\)
Number of days182430208
  1. Draw a histogram to represent this information. \includegraphics[max width=\textwidth, alt={}, center]{3ada18de-c4f7-4049-9032-46b796be83c3-12_1203_1399_833_415}
  2. What is the greatest possible value of the interquartile range for the data?
  3. Calculate estimates of the mean and standard deviation of the number of chocolate bars sold.
    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 2004 June Q1
4 marks Easy -1.2
1 Two cricket teams kept records of the number of runs scored by their teams in 8 matches. The scores are shown in the following table.
Team \(A\)150220773029811816057
Team \(B\)1661421709311113014886
  1. Find the mean and standard deviation of the scores for team \(A\). The mean and standard deviation for team \(B\) are 130.75 and 29.63 respectively.
  2. State with a reason which team has the more consistent scores.
CAIE S1 2004 June Q2
5 marks Easy -1.8
2 In a recent survey, 640 people were asked about the length of time each week that they spent watching television. The median time was found to be 20 hours, and the lower and upper quartiles were 15 hours and 35 hours respectively. The least amount of time that anyone spent was 3 hours, and the greatest amount was 60 hours.
  1. On graph paper, show these results using a fully labelled cumulative frequency graph.
  2. Use your graph to estimate how many people watched more than 50 hours of television each week.
CAIE S1 2004 June Q3
5 marks Moderate -0.8
3 Two fair dice are thrown. Let the random variable \(X\) be the smaller of the two scores if the scores are different, or the score on one of the dice if the scores are the same.
  1. Copy and complete the following table to show the probability distribution of \(X\).
    \(x\)123456
    \(\mathrm { P } ( X = x )\)
  2. Find \(\mathrm { E } ( X )\).
CAIE S1 2004 June Q4
8 marks Moderate -0.3
4 Melons are sold in three sizes: small, medium and large. The weights follow a normal distribution with mean 450 grams and standard deviation 120 grams. Melons weighing less than 350 grams are classified as small.
  1. Find the proportion of melons which are classified as small.
  2. The rest of the melons are divided in equal proportions between medium and large. Find the weight above which melons are classified as large.
CAIE S1 2004 June Q5
8 marks Moderate -0.8
5
  1. The menu for a meal in a restaurant is as follows. \begin{displayquote} Starter Course
    Melon
    or
    Soup
    or
    Smoked Salmon \end{displayquote} \begin{displayquote} Main Course
    Chicken
    or
    Steak
    or
    Lamb Cutlets
    or
    Vegetable Curry
    or
    Fish \end{displayquote} \begin{displayquote} Dessert Course
    Cheesecake
    or
    Ice Cream
    or
    Apple Pie
    All the main courses are served with salad and either
    new potatoes or french fries.
    1. How many different three-course meals are there?
    2. How many different choices are there if customers may choose only two of the three courses?
  2. In how many ways can a group of 14 people eating at the restaurant be divided between three tables seating 5, 5 and 4? \end{displayquote}
CAIE S1 2004 June Q6
9 marks Moderate -0.3
6 When Don plays tennis, \(65 \%\) of his first serves go into the correct area of the court. If the first serve goes into the correct area, his chance of winning the point is \(90 \%\). If his first serve does not go into the correct area, Don is allowed a second serve, and of these, \(80 \%\) go into the correct area. If the second serve goes into the correct area, his chance of winning the point is \(60 \%\). If neither serve goes into the correct area, Don loses the point.
  1. Draw a tree diagram to represent this information.
  2. Using your tree diagram, find the probability that Don loses the point.
  3. Find the conditional probability that Don's first serve went into the correct area, given that he loses the point.
CAIE S1 2004 June Q7
11 marks Standard +0.3
7 A shop sells old video tapes, of which 1 in 5 on average are known to be damaged.
  1. A random sample of 15 tapes is taken. Find the probability that at most 2 are damaged.
  2. Find the smallest value of \(n\) if there is a probability of at least 0.85 that a random sample of \(n\) tapes contains at least one damaged tape.
  3. A random sample of 1600 tapes is taken. Use a suitable approximation to find the probability that there are at least 290 damaged tapes.
CAIE S1 2005 June Q1
5 marks Standard +0.3
1 It is known that, on average, 2 people in 5 in a certain country are overweight. A random sample of 400 people is chosen. Using a suitable approximation, find the probability that fewer than 165 people in the sample are overweight.
CAIE S1 2005 June Q2
6 marks Moderate -0.8
2 The following table shows the results of a survey to find the average daily time, in minutes, that a group of schoolchildren spent in internet chat rooms.
Time per day
\(( t\) minutes \()\)
Frequency
\(0 \leqslant t < 10\)2
\(10 \leqslant t < 20\)\(f\)
\(20 \leqslant t < 40\)11
\(40 \leqslant t < 80\)4
The mean time was calculated to be 27.5 minutes.
  1. Form an equation involving \(f\) and hence show that the total number of children in the survey was 26 .
  2. Find the standard deviation of these times.
CAIE S1 2005 June Q3
7 marks Moderate -0.8
3 A fair dice has four faces. One face is coloured pink, one is coloured orange, one is coloured green and one is coloured black. Five such dice are thrown and the number that fall on a green face are counted. The random variable \(X\) is the number of dice that fall on a green face.
  1. Show that the probability of 4 dice landing on a green face is 0.0146 , correct to 4 decimal places.
  2. Draw up a table for the probability distribution of \(X\), giving your answers correct to 4 decimal places.
CAIE S1 2005 June Q4
8 marks Easy -1.2
4 The following back-to-back stem-and-leaf diagram shows the cholesterol count for a group of 45 people who exercise daily and for another group of 63 who do not exercise. The figures in brackets show the number of people corresponding to each set of leaves.
People who exercisePeople who do not exercise
(9)98764322131577(4)
(12)9888766533224234458(6)
(9)87776533151222344567889(13)
(7)6666432612333455577899(14)
(3)8417245566788(9)
(4)95528133467999(9)
(1)4914558(5)
(0)10336(3)
Key: 2 | 8 | 1 represents a cholesterol count of 8.2 in the group who exercise and 8.1 in the group who do not exercise.
  1. Give one useful feature of a stem-and-leaf diagram.
  2. Find the median and the quartiles of the cholesterol count for the group who do not exercise. You are given that the lower quartile, median and upper quartile of the cholesterol count for the group who exercise are 4.25, 5.3 and 6.6 respectively.
  3. On a single diagram on graph paper, draw two box-and-whisker plots to illustrate the data.
CAIE S1 2005 June Q5
8 marks Easy -1.3
5 Data about employment for males and females in a small rural area are shown in the table.
\cline { 2 - 3 } \multicolumn{1}{c|}{}UnemployedEmployed
Male206412
Female358305
A person from this area is chosen at random. Let \(M\) be the event that the person is male and let \(E\) be the event that the person is employed.
  1. Find \(\mathrm { P } ( M )\).
  2. Find \(\mathrm { P } ( M\) and \(E )\).
  3. Are \(M\) and \(E\) independent events? Justify your answer.
  4. Given that the person chosen is unemployed, find the probability that the person is female.
CAIE S1 2005 June Q6
8 marks Moderate -0.3
6 Tyre pressures on a certain type of car independently follow a normal distribution with mean 1.9 bars and standard deviation 0.15 bars.
  1. Find the probability that all four tyres on a car of this type have pressures between 1.82 bars and 1.92 bars.
  2. Safety regulations state that the pressures must be between \(1.9 - b\) bars and \(1.9 + b\) bars. It is known that \(80 \%\) of tyres are within these safety limits. Find the safety limits.
CAIE S1 2005 June Q7
8 marks Moderate -0.8
7
  1. A football team consists of 3 players who play in a defence position, 3 players who play in a midfield position and 5 players who play in a forward position. Three players are chosen to collect a gold medal for the team. Find in how many ways this can be done
    1. if the captain, who is a midfield player, must be included, together with one defence and one forward player,
    2. if exactly one forward player must be included, together with any two others.
  2. Find how many different arrangements there are of the nine letters in the words GOLD MEDAL
    1. if there are no restrictions on the order of the letters,
    2. if the two letters D come first and the two letters L come last.
CAIE S1 2006 June Q1
3 marks Easy -1.2
1 The salaries, in thousands of dollars, of 11 people, chosen at random in a certain office, were found to be: $$40 , \quad 42 , \quad 45 , \quad 41 , \quad 352 , \quad 40 , \quad 50 , \quad 48 , \quad 51 , \quad 49 , \quad 47 .$$ Choose and calculate an appropriate measure of central tendency (mean, mode or median) to summarise these salaries. Explain briefly why the other measures are not suitable.