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CAIE Further Paper 3 2024 November Q4
3 marks Standard +0.8
When the object is suspended from \(A\), the angle between \(AB\) and the vertical is \(\theta\), where \(\tan \theta = \frac{1}{2}\).
  1. Given that \(h = \frac{8}{3}a\), find the possible values of \(k\). [3]
CAIE Further Paper 3 2024 November Q5
4 marks Challenging +1.2
A particle \(P\) of mass \(2 \text{ kg}\) moving on a horizontal straight line has displacement \(x \text{ m}\) from a fixed point \(O\) on the line and velocity \(v \text{ m s}^{-1}\) at time \(t \text{ s}\). The only horizontal force acting on \(P\) is a variable force \(F \text{ N}\) which can be expressed as a function of \(t\). It is given that $$\frac{v}{x} = \frac{3-t}{1+t}$$ and when \(t = 0\), \(x = 5\).
  1. Find an expression for \(x\) in terms of \(t\). [4]
CAIE Further Paper 3 2024 November Q6
3 marks Standard +0.8
\includegraphics{figure_6} A particle \(P\) of mass \(0.05 \text{ kg}\) is attached to one end of a light inextensible string of length \(1 \text{ m}\). The other end of the string is attached to a fixed point \(O\). A particle \(Q\) of mass \(0.04 \text{ kg}\) is attached to one end of a second light inextensible string. The other end of this string is attached to \(P\). The particle \(P\) moves in a horizontal circle of radius \(0.8 \text{ m}\) with angular speed \(\omega \text{ rad s}^{-1}\). The particle \(Q\) moves in a horizontal circle of radius \(1.4 \text{ m}\) also with angular speed \(\omega \text{ rad s}^{-1}\). The centres of the circles are vertically below \(O\), and \(O\), \(P\) and \(Q\) are always in the same vertical plane. The strings \(OP\) and \(PQ\) remain at constant angles \(\alpha\) and \(\beta\) respectively to the vertical (see diagram).
  1. Find the tension in the string \(OP\). [3]
CAIE Further Paper 3 2024 November Q7
4 marks Challenging +1.8
A particle \(P\) is projected with speed \(u\) at an angle \(\tan^{-1}\left(\frac{4}{3}\right)\) above the horizontal from a point \(O\) on a horizontal plane and moves freely under gravity. When \(P\) is moving horizontally, it strikes a smooth inclined plane at the point \(A\). This plane is inclined to the horizontal at an angle \(\alpha\), and the line of greatest slope through \(A\) lies in the vertical plane through \(O\) and \(A\). As a result of the impact, \(P\) moves vertically upwards. The coefficient of restitution between \(P\) and the inclined plane is \(e\).
  1. Show that \(e \tan^2 \alpha = 1\). [4]
CAIE Further Paper 3 2020 Specimen Q1
4 marks Standard +0.8
A child's toy consists of an iron disc of radius \(r\) and a vertical bead with \(3r\) at rail that is rigidly fixed to the disc so that the toy rocks as it rolls. The circumference of the disc is such that the disc and bead have the same material. Show that the centre of mass of the toy is at a distance \(\frac{27r}{10}\) from the centre of the disc. [4]
CAIE Further Paper 3 2020 Specimen Q2
8 marks Standard +0.3
A light elastic string of natural length \(a\) and modulus of elasticity \(2mg\). One end of the string is attached to a fixed point \(A\). The other end of the string is attached to a particle of mass \(2m\).
  1. Find, in terms of \(a\), the extension of the string when the particle hangs freely in equilibrium below \(A\). [2]
  2. The particle is released from rest at \(A\). Find, in terms of \(a\), the distance of the particle below \(A\) when it first comes to instantaneous rest. [6]
CAIE Further Paper 3 2020 Specimen Q3
10 marks Challenging +1.2
A particle \(P\) of mass \(mk\) falls from rest due to gravity. There is a resistance force of magnitude \(mkv^2\) N, where \(v\) ms\(^{-1}\) is the speed of \(P\) after it has fallen a distance \(x\) m and \(k\) is a positive constant.
  1. By using \(v \frac{dv}{dx} = \frac{dv}{dt}\) and appropriate differential equation, show that $$v^2 = \frac{g}{k}(1 - e^{-2kx}).$$ [7] It is given that \(k = 0.01\). The speed of \(P\) when \(x = 0.2\) comes to approximately \(v\) ms\(^{-1}\).
    1. Find \(V\) correct to 2 decimal places. [1]
    2. Hence find how far \(P\) has fallen when its speed is \(\frac{1}{2}V\) ms\(^{-1}\). [2]
CAIE Further Paper 3 2020 Specimen Q4
9 marks Challenging +1.2
\includegraphics{figure_4} Two uniform smooth spheres \(A\) and \(B\) of equal radii have masses \(m\) and \(2m\) respectively. Sphere \(B\) is at rest on a smooth horizontal surface. Sphere \(A\) is moving on the surface with speed \(u\) at an angle of \(30°\) to the line of centres of \(A\) and \(B\) when it collides with \(B\) (see diagram). The coefficient of restitution between the spheres is \(e\).
  1. Show that the speed of \(B\) after the collision is \(\frac{\sqrt{3}}{6}u(1 + e)\) and find the speed of \(A\) after the collision. [6]
  2. Given that \(e = \frac{1}{2}\), find the loss of kinetic energy as a result of the collision. [3]
CAIE Further Paper 3 2020 Specimen Q5
10 marks Standard +0.8
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 \(O\).
  1. \includegraphics{figure_5a} The particle \(P\) moves in a horizontal circle with a constant angular speed \(\omega\) with the string inclined at \(60°\) to the downward vertical through \(O\) (see diagram). Show that \(\omega^2 = \frac{2g}{a}\). [4]
  2. The particle now hangs at rest and is then projected horizontally so that it begins to move in a vertical circle with centre \(O\). When the string makes an angle \(\theta\) with the downward vertical through \(O\), the angular speed of \(P\) is \(\sqrt{\frac{2g}{a}}\). The string first goes slack when \(OP\) makes an angle \(\theta\) with the upward vertical through \(O\). Find the value of \(\cos \theta\). [6]
CAIE Further Paper 3 2020 Specimen Q6
9 marks Standard +0.3
A particle \(P\) is projected with speed \(u\) at an angle \(\alpha\) above the horizontal from a point \(O\) on a horizontal plane and moves freely under gravity. The horizontal and vertical displacements of \(P\) from \(O\) at a time \(t\) are denoted by \(x\) and \(y\) respectively.
  1. Derive the equation of the trajectory of \(P\) in the form $$y = x \tan \alpha - \frac{gx^2}{2u^2} \sec^2 \alpha.$$ [3]
  2. The greatest height of \(P\) above the plane is denoted by \(H\). When \(P\) is at a height of \(\frac{3}{4}H\), it is travelling at a horizontal distance \(d\). Given that \(\tan \alpha = 3\) and in terms of \(H\), the two possible values of \(d\). [6]
CAIE S1 2023 March Q1
8 marks Moderate -0.8
Each year the total number of hours, \(x\), of sunshine in Kintoo is recorded during the month of June. The results for the last 60 years are summarised in the table.
\(x\)\(30 \leqslant x < 60\)\(60 \leqslant x < 90\)\(90 \leqslant x < 110\)\(110 \leqslant x < 140\)\(140 \leqslant x < 180\)\(180 \leqslant x \leqslant 240\)
Number of years48142572
  1. Draw a cumulative frequency graph to illustrate the data. [3]
  2. Use your graph to estimate the 70th percentile of the data. [2]
  3. Calculate an estimate for the mean number of hours of sunshine in Kintoo during June over the last 60 years. [3]
CAIE S1 2023 March Q2
7 marks Moderate -0.3
Alisha has four coins. One of these coins is biased so that the probability of obtaining a head is 0.6. The other three coins are fair. Alisha throws the four coins at the same time. The random variable \(X\) denotes the number of heads obtained.
  1. Show that the probability of obtaining exactly one head is 0.225. [3]
  2. Complete the following probability distribution table for \(X\). [2]
    \(x\)01234
    P(\(X = x\))0.050.2250.075
  3. Given that E(\(X\)) = 2.1, find the value of Var(\(X\)). [2]
CAIE S1 2023 March Q3
6 marks Moderate -0.8
80\% of the residents of Kinwawa are in favour of a leisure centre being built in the town. 20 residents of Kinwawa are chosen at random and asked, in turn, whether they are in favour of the leisure centre.
  1. Find the probability that more than 17 of these residents are in favour of the leisure centre. [3]
  2. Find the probability that the 5th person asked is the first person who is not in favour of the leisure centre. [1]
  3. Find the probability that the 7th person asked is the second person who is not in favour of the leisure centre. [2]
CAIE S1 2023 March Q4
3 marks Standard +0.3
The probability that it will rain on any given day is \(x\). If it is raining, the probability that Aran wears a hat is 0.8 and if it is not raining, the probability that he wears a hat is 0.3. Whether it is raining or not, if Aran wears a hat, the probability that he wears a scarf is 0.4. If he does not wear a hat, the probability that he wears a scarf is 0.1. The probability that on a randomly chosen day it is not raining and Aran is not wearing a hat or a scarf is 0.36. Find the value of \(x\). [3]
CAIE S1 2023 March Q5
3 marks Standard +0.8
Marco has four boxes labelled \(K\), \(L\), \(M\) and \(N\). He places them in a straight line in the order \(K\), \(L\), \(M\), \(N\) with \(K\) on the left. Marco also has four coloured marbles: one is red, one is green, one is white and one is yellow. He places a single marble in each box, at random. Events \(A\) and \(B\) are defined as follows. \(A\): The white marble is in either box \(L\) or box \(M\). \(B\): The red marble is to the left of both the green marble and the yellow marble. Determine whether or not events \(A\) and \(B\) are independent. [3]
CAIE S1 2023 March Q6
11 marks Standard +0.3
In a cycling event the times taken to complete a course are modelled by a normal distribution with mean 62.3 minutes and standard deviation 8.4 minutes.
  1. Find the probability that a randomly chosen cyclist has a time less than 74 minutes. [2]
  2. Find the probability that 4 randomly chosen cyclists all have times between 50 and 74 minutes. [4]
In a different cycling event, the times can also be modelled by a normal distribution. 23\% of the cyclists have times less than 36 minutes and 10\% of the cyclists have times greater than 54 minutes.
  1. Find estimates for the mean and standard deviation of this distribution. [5]
CAIE S1 2023 March Q7
12 marks Standard +0.3
  1. Find the number of different arrangements of the 9 letters in the word DELIVERED in which the three Es are together and the two Ds are not next to each other. [4]
  2. Find the probability that a randomly chosen arrangement of the 9 letters in the word DELIVERED has exactly 4 letters between the two Ds. [5]
Five letters are selected from the 9 letters in the word DELIVERED.
  1. Find the number of different selections if the 5 letters include at least one D and at least one E. [3]
CAIE S1 2002 June Q1
4 marks Easy -1.2
Events \(A\) and \(B\) are such that \(\text{P}(A) = 0.3\), \(\text{P}(B) = 0.8\) and \(\text{P}(A \text{ and } B) = 0.4\). State, giving a reason in each case, whether events \(A\) and \(B\) are
  1. independent, [2]
  2. mutually exclusive. [2]
CAIE S1 2002 June Q2
6 marks Easy -1.2
The manager of a company noted the times spent in 80 meetings. The results were as follows.
Time (\(t\) minutes)\(0 < t \leq 15\)\(15 < t \leq 30\)\(30 < t \leq 60\)\(60 < t \leq 90\)\(90 < t \leq 120\)
Number of meetings4724387
Draw a cumulative frequency graph and use this to estimate the median time and the interquartile range. [6]
CAIE S1 2002 June Q3
7 marks Moderate -0.8
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 \(A\). [3]
  2. Find \(\text{E}(A)\) and \(\text{Var}(A)\). [4]
CAIE S1 2002 June Q4
7 marks Moderate -0.8
  1. In a spot check of the speeds \(x \text{ km h}^{-1}\) 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]
  2. On another day the mean speed of cars on the motorway was found to be \(107.6 \text{ km h}^{-1}\) and the standard deviation was \(13.8 \text{ km h}^{-1}\). Assuming these speeds follow a normal distribution and that the speed limit is \(110 \text{ km h}^{-1}\), find what proportion of cars exceed the speed limit. [3]
CAIE S1 2002 June Q5
8 marks Moderate -0.3
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]
  2. the digits 1, 3, 7 (in any order) are next to each other, [3]
  3. these 7-digit numbers are even. [3]
CAIE S1 2002 June Q6
8 marks Standard +0.3
  1. In a normal distribution with mean \(\mu\) and standard deviation \(\sigma\), \(\text{P}(X > 3.6) = 0.5\) and \(\text{P}(X > 2.8) = 0.6554\). Write down the value of \(\mu\), and calculate the value of \(\sigma\). [4]
  2. If four observations are taken at random from this distribution, find the probability that at least two observations are greater than 2.8. [4]
CAIE S1 2002 June Q7
10 marks Moderate -0.3
  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.
    1. Find the number of plants per box. [4]
    2. Find the probability that a box contains exactly 12 plants which produce yellow flowers. [2]
  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. [4]
CAIE S1 2010 June Q1
5 marks Moderate -0.8
The times in minutes for seven students to become proficient at a new computer game were measured. The results are shown below. $$15 \quad 10 \quad 48 \quad 10 \quad 19 \quad 14 \quad 16$$
  1. Find the mean and standard deviation of these times. [2]
  2. State which of the mean, median or mode you consider would be most appropriate to use as a measure of central tendency to represent the data in this case. [1]
  3. For each of the two measures of average you did not choose in part (ii), give a reason why you consider it inappropriate. [2]