Questions — CAIE (7646 questions)

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CAIE Further Paper 3 2024 November Q3
7 marks Challenging +1.8
\includegraphics{figure_3} The diagram shows two identical smooth uniform spheres \(A\) and \(B\) of equal radii and each of mass \(m\). The two spheres are moving on a smooth horizontal surface when they collide with speeds \(2u\) and \(3u\) respectively. Immediately before the collision, \(A\)'s direction of motion makes an angle \(\theta\) with the line of centres and \(B\)'s direction of motion is perpendicular to that of \(A\). After the collision, \(B\) moves perpendicular to the line of centres. The coefficient of restitution between the spheres is \(\frac{1}{3}\).
  1. Find the value of \(\tan \theta\). [3]
  2. Find the total loss of kinetic energy as a result of the collision. [2]
  3. Find, in degrees, the angle through which the direction of motion of \(A\) is deflected as a result of the collision. [2]
CAIE Further Paper 3 2024 November Q4
7 marks Challenging +1.8
\includegraphics{figure_4} The end \(A\) of a uniform rod \(AB\) of length \(6a\) and weight \(W\) is in contact with a rough vertical wall. One end of a light inextensible string of length \(3a\) is attached to the midpoint \(C\) of the rod. The other end of the string is attached to a point \(D\) on the wall, vertically above \(A\). The rod is in equilibrium when the angle between the rod and the wall is \(\theta\), where \(\tan \theta = \frac{3}{4}\). A particle of weight \(W\) is attached to the point \(E\) on the rod, where the distance \(AE\) is equal to \(ka\) (\(3 < k < 6\)) (see diagram). The rod and the string are in a vertical plane perpendicular to the wall. The coefficient of friction between the rod and the wall is \(\frac{1}{3}\). The rod is about to slip down the wall.
  1. Find the value of \(k\). [5]
  2. Find, in terms of \(W\), the magnitude of the frictional force between the rod and the wall. [2]
CAIE Further Paper 3 2024 November Q5
8 marks Challenging +1.2
A particle \(P\) is projected from a point \(O\) on horizontal ground with speed \(u\) at an angle \(\theta\) above the horizontal, where \(\tan \theta = \frac{1}{3}\). The particle \(P\) moves freely under gravity and passes through the point with coordinates \((3a, \frac{4}{5}a)\) relative to horizontal and vertical axes through \(O\) in the plane of the motion.
  1. Use the equation of the trajectory to show that \(u^2 = 25ag\). [2]
  2. Express \(V^2\) in the form \(kag\), where \(k\) is a rational number. [6]
At the instant when \(P\) is moving horizontally, a particle \(Q\) is projected from \(O\) with speed \(V\) at an angle \(\alpha\) above the horizontal. The particles \(P\) and \(Q\) reach the ground at the same point and at the same time.
CAIE Further Paper 3 2024 November Q6
10 marks Challenging +1.8
\includegraphics{figure_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 \(P\) is held with the string taut and the string makes an angle \(\theta\) with the downward vertical through \(O\). The particle is projected at right angles to the string with speed \(\frac{1}{3}\sqrt{10ag}\) and begins to move downwards along a circular path. When the string is vertical, it strikes a small smooth peg at the point \(A\) which is vertically below \(O\). The circular path and the point \(A\) are in the same vertical plane. After the string strikes the peg, the particle \(P\) begins to move in a vertical circle with centre \(A\). When the string makes an angle \(\theta\) with the upward vertical through \(A\) the string becomes slack (see diagram). The distance of \(A\) below \(O\) is \(\frac{5}{6}a\).
  1. Find the value of \(\cos \theta\). [6]
  2. Find the ratio of the tensions in the string immediately before and immediately after it strikes the peg. [4]
CAIE Further Paper 3 2024 November Q7
11 marks Challenging +1.2
A particle \(P\) of mass \(m\) kg is held at rest at a point \(O\) and released so that it moves vertically under gravity against a resistive force of magnitude \(0.1mv^2\) N, where \(v\) m s\(^{-1}\) is the velocity of \(P\) at time \(t\) s.
  1. Find an expression for \(v\) in terms of \(t\). [6]
  2. Find an expression for \(v^2\) in terms of \(x\). [5]
The displacement of \(P\) from \(O\) at time \(t\) s is \(x\) m.
CAIE Further Paper 3 2024 November Q1
5 marks Challenging +1.2
A particle \(P\) is projected with speed \(u \text{ m s}^{-1}\) at an angle \(\tan^{-1} 2\) above the horizontal from a point \(O\) on a horizontal plane and moves freely under gravity. When \(P\) has travelled a distance \(56 \text{ m}\) horizontally from \(O\), it is at a vertical height \(H \text{ m}\) above the plane. When \(P\) has travelled a distance \(84 \text{ m}\) horizontally from \(O\), it is at a vertical height \(\frac{1}{2}H \text{ m}\) above the plane. Find, in either order, the value of \(u\) and the value of \(H\). [5]
CAIE Further Paper 3 2024 November Q2
5 marks Challenging +1.2
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 \(P\) is held at the point \(A\) with the string taut. It is given that \(OA\) makes an angle \(\theta\) with the downward vertical through \(O\), where \(\tan \theta = \frac{3}{4}\). The particle \(P\) is projected perpendicular to \(OA\) in an upwards direction with speed \(\sqrt{5ag}\), and it starts to move along a circular path in a vertical plane. When \(P\) is at the point \(B\), where angle \(AOB\) is a right angle, the tension in the string is \(T\). Find \(T\) in terms of \(m\) and \(g\). [5]
CAIE Further Paper 3 2024 November Q3
6 marks Challenging +1.2
A particle \(P\) of mass \(m \text{ kg}\) is attached to one end of a light elastic string of natural length \(2 \text{ m}\) and modulus of elasticity \(2mg \text{ N}\). The other end of the string is attached to a fixed point \(O\). The particle \(P\) hangs in equilibrium vertically below \(O\). The particle \(P\) is pulled down vertically a distance \(d \text{ m}\) below its equilibrium position and released from rest.
  1. Given that the particle just reaches \(O\) in the subsequent motion, find the value of \(d\). [6]
CAIE Further Paper 3 2024 November Q3
2 marks Moderate -0.5
  1. Hence find the speed of \(P\) when it is \(2 \text{ m}\) below \(O\). [2]
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]