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CAIE FP2 2019 November Q9
10 marks Standard +0.8
A random sample of five pairs of values of \(x\) and \(y\) is taken from a bivariate distribution. The values are shown in the following table, where \(p\) and \(q\) are constants.
\(x\)12345
\(y\)4\(p\)\(q\)21
The equation of the regression line of \(y\) on \(x\) is \(y = -0.5x + 3.5\).
  1. Find the values of \(p\) and \(q\). [7]
  2. Find the value of the product moment correlation coefficient. [3]
CAIE FP2 2019 November Q10
10 marks Standard +0.8
The random variable \(X\) has probability density function f given by $$\mathrm{f}(x) = \begin{cases} \frac{1}{30}\left(\frac{8}{x^2} + 3x^2 - 14\right) & 2 \leqslant x \leqslant 4, \\ 0 & \text{otherwise}. \end{cases}$$
  1. Find the distribution function of \(X\). [3]
The random variable \(Y\) is defined by \(Y = X^2\).
  1. Find the probability density function of \(Y\). [4]
  2. Find the value of \(y\) such that \(\mathrm{P}(Y < y) = 0.8\). [3]
CAIE FP2 2019 November Q11
28 marks Challenging +1.2
Answer only one of the following two alternatives. EITHER The points \(A\) and \(B\) are a distance 1.2 m apart on a smooth horizontal surface. A particle \(P\) of mass \(\frac{2}{3}\) kg is attached to one end of a light spring of natural length 0.6 m and modulus of elasticity 10 N. The other end of the spring is attached to the point \(A\). A second light spring, of natural length 0.4 m and modulus of elasticity 20 N, has one end attached to \(P\) and the other end attached to \(B\).
  1. Show that when \(P\) is in equilibrium \(AP = 0.75\) m. [3]
The particle \(P\) is displaced by 0.05 m from the equilibrium position towards \(A\) and then released from rest.
  1. Show that \(P\) performs simple harmonic motion and state the period of the motion. [6]
  2. Find the speed of \(P\) when it passes through the equilibrium position. [2]
  3. Find the speed of \(P\) when its acceleration is equal to half of its maximum value. [3]
OR The number of puncture repairs carried out each week by a small repair shop is recorded over a period of 40 weeks. The results are shown in the following table.
Number of repairs in a week012345\(\geqslant 6\)
Number of weeks61596310
  1. Calculate the mean and variance for the number of repairs in a week and comment on the possible suitability of a Poisson distribution to model the data. [3]
Records over a longer period of time indicate that the mean number of repairs in a week is 1.6. The following table shows some of the expected frequencies, correct to 3 decimal places, for a period of 40 weeks using a Poisson distribution with mean 1.6.
Number of repairs in a week012345\(\geqslant 6\)
Expected frequency8.07612.92110.3375.5132.205\(a\)\(b\)
  1. Show that \(a = 0.706\) and find the value of the constant \(b\). [3]
  2. Carry out a goodness of fit test of a Poisson distribution with mean 1.6, using a 10% significance level. [8]
CAIE M1 2020 June Q1
3 marks Moderate -0.8
Three coplanar forces of magnitudes \(100\text{ N}\), \(50\text{ N}\) and \(50\text{ N}\) act at a point \(A\), as shown in the diagram. The value of \(\cos \alpha\) is \(\frac{4}{5}\). \includegraphics{figure_1} Find the magnitude of the resultant of the three forces and state its direction. [3]
CAIE M1 2020 June Q2
5 marks Moderate -0.8
A car of mass \(1800\text{ kg}\) is towing a trailer of mass \(400\text{ kg}\) along a straight horizontal road. The car and trailer are connected by a light rigid tow-bar. The car is accelerating at \(1.5\text{ m s}^{-2}\). There are constant resistance forces of \(250\text{ N}\) on the car and \(100\text{ N}\) on the trailer.
  1. Find the tension in the tow-bar. [2]
  2. Find the power of the engine of the car at the instant when the speed is \(20\text{ m s}^{-1}\). [3]
CAIE M1 2020 June Q3
7 marks Moderate -0.8
A particle \(P\) is projected vertically upwards with speed \(5\text{ m s}^{-1}\) from a point \(A\) which is \(2.8\text{ m}\) above horizontal ground.
  1. Find the greatest height above the ground reached by \(P\). [3]
  2. Find the length of time for which \(P\) is at a height of more than \(3.6\text{ m}\) above the ground. [4]
CAIE M1 2020 June Q4
7 marks Standard +0.3
The diagram shows a ring of mass \(0.1\text{ kg}\) threaded on a fixed horizontal rod. The rod is rough and the coefficient of friction between the ring and the rod is \(0.8\). A force of magnitude \(T\text{ N}\) acts on the ring in a direction at \(30°\) to the rod, downwards in the vertical plane containing the rod. Initially the ring is at rest. \includegraphics{figure_4}
  1. Find the greatest value of \(T\) for which the ring remains at rest. [4]
  2. Find the acceleration of the ring when \(T = 3\). [3]
CAIE M1 2020 June Q5
7 marks Moderate -0.3
A child of mass \(35\text{ kg}\) is swinging on a rope. The child is modelled as a particle \(P\) and the rope is modelled as a light inextensible string of length \(4\text{ m}\). Initially \(P\) is held at an angle of \(45°\) to the vertical (see diagram). \includegraphics{figure_5}
  1. Given that there is no resistance force, find the speed of \(P\) when it has travelled half way along the circular arc from its initial position to its lowest point. [4]
  2. It is given instead that there is a resistance force. The work done against the resistance force as \(P\) travels from its initial position to its lowest point is \(X\text{ J}\). The speed of \(P\) at its lowest point is \(4\text{ m s}^{-1}\). Find \(X\). [3]
CAIE M1 2020 June Q6
11 marks Standard +0.3
A particle moves in a straight line \(AB\). The velocity \(v\text{ m s}^{-1}\) of the particle \(t\text{ s}\) after leaving \(A\) is given by \(v = t(5 - 2t)\) where \(k\) is a constant. The displacement of the particle from \(A\), in the direction towards \(B\), is \(2.5\text{ m}\) when \(t = 3\) and is \(2.4\text{ m}\) when \(t = 6\).
  1. Find the value of \(k\). Hence find an expression, in terms of \(t\), for the displacement of the particle from \(A\). [7]
  2. Find the displacement of the particle from \(A\) when its velocity is a minimum. [4]
CAIE M1 2020 June Q7
10 marks Standard +0.3
A particle \(P\) of mass \(0.3\text{ kg}\), lying on a smooth plane inclined at \(30°\) to the horizontal, is released from rest. \(P\) slides down the plane for a distance of \(2.5\text{ m}\) and then reaches a horizontal plane. There is no change in speed when \(P\) reaches the horizontal plane. A particle \(Q\) of mass \(0.2\text{ kg}\) lies at rest on the horizontal plane \(1.5\text{ m}\) from the end of the inclined plane (see diagram). \(P\) collides directly with \(Q\). \includegraphics{figure_7}
  1. It is given that the horizontal plane is smooth and that, after the collision, \(P\) continues moving in the same direction, with speed \(2\text{ m s}^{-1}\). Find the speed of \(Q\) after the collision. [5]
  2. It is given instead that the horizontal plane is rough and that when \(P\) and \(Q\) collide, they coalesce and move with speed \(1.2\text{ m s}^{-1}\). Find the coefficient of friction between \(P\) and the horizontal plane. [5]
CAIE M1 2020 June Q1
6 marks Moderate -0.8
A tram starts from rest and moves with uniform acceleration for 20 s. The tram then travels at a constant speed, \(V \text{ ms}^{-1}\), for 170 s before being brought to rest with a uniform deceleration of magnitude twice that of the acceleration. The total distance travelled by the tram is 2.775 km.
  1. Sketch a velocity-time graph for the motion, stating the total time for which the tram is moving. [2]
  2. Find \(V\). [2]
  3. Find the magnitude of the acceleration. [2]
CAIE M1 2020 June Q2
6 marks Standard +0.3
\includegraphics{figure_2} Coplanar forces of magnitudes 20 N, \(P\) N, \(3P\) N and \(4P\) N act at a point in the directions shown in the diagram. The system is in equilibrium. Find \(P\) and \(\theta\). [6]
CAIE M1 2020 June Q3
8 marks Standard +0.8
\includegraphics{figure_3} A particle of mass 2.5 kg is held in equilibrium on a rough plane inclined at 20° to the horizontal by a force of magnitude \(T\) N making an angle of 60° with a line of greatest slope of the plane (see diagram). The coefficient of friction between the particle and the plane is 0.3. Find the greatest and least possible values of \(T\). [8]
CAIE M1 2020 June Q4
10 marks Standard +0.3
Small smooth spheres \(A\) and \(B\), of equal radii and of masses 4 kg and 2 kg respectively, lie on a smooth horizontal plane. Initially \(B\) is at rest and \(A\) is moving towards \(B\) with speed \(10 \text{ ms}^{-1}\). After the spheres collide \(A\) continues to move in the same direction but with half the speed of \(B\).
  1. Find the speed of \(B\) after the collision. [2]
A third small smooth sphere \(C\), of mass 1 kg and with the same radius as \(A\) and \(B\), is at rest on the plane. \(B\) now collides directly with \(C\). After this collision \(B\) continues to move in the same direction but with one third the speed of \(C\).
  1. Show that there is another collision between \(A\) and \(B\). [3]
  2. \(A\) and \(B\) coalesce during this collision. Find the total loss of kinetic energy in the system due to the three collisions. [5]
CAIE M1 2020 June Q5
10 marks Standard +0.3
A car of mass 1250 kg is moving on a straight road.
  1. On a horizontal section of the road, the car has a constant speed of \(32 \text{ ms}^{-1}\) and there is a constant force of 750 N resisting the motion.
    1. Calculate, in kW, the power developed by the engine of the car. [2]
    2. Given that this power is suddenly decreased by 8 kW, find the instantaneous deceleration of the car. [3]
  2. On a section of the road inclined at \(\sin^{-1} 0.096\) to the horizontal, the resistance to the motion of the car is \((1000 + 8v)\) N when the speed of the car is \(v \text{ ms}^{-1}\). The car travels up this section of the road at constant speed with the engine working at 60 kW. Find this constant speed. [5]
CAIE M1 2020 June Q6
10 marks Moderate -0.8
A particle \(P\) moves in a straight line. The velocity \(v \text{ ms}^{-1}\) at time \(t\) s is given by $$v = 2t + 1 \quad \text{for } 0 \leqslant t \leqslant 5,$$ $$v = 36 - t^2 \quad \text{for } 5 \leqslant t \leqslant 7,$$ $$v = 2t - 27 \quad \text{for } 7 \leqslant t \leqslant 13.5.$$
  1. Sketch the velocity-time graph for \(0 \leqslant t \leqslant 13.5\). [3]
  2. Find the acceleration at the instant when \(t = 6\). [2]
  3. Find the total distance travelled by \(P\) in the interval \(0 \leqslant t \leqslant 13.5\). [5]
CAIE M1 2021 June Q1
3 marks Moderate -0.5
A winch operates by means of a force applied by a rope. The winch is used to pull a load of mass 50 kg up a line of greatest slope of a plane inclined at 60° to the horizontal. The winch pulls the load a distance of 5 m up the plane at constant speed. There is a constant resistance to motion of 100 N. Find the work done by the winch. [3]
CAIE M1 2021 June Q2
6 marks Standard +0.3
\includegraphics{figure_2} Two particles \(A\) and \(B\) have masses \(m\) kg and 0.1 kg respectively, where \(m > 0.1\). The particles are attached to the ends of a light inextensible string. The string passes over a fixed smooth pulley and the particles hang vertically below it. Both particles are at a height of 0.9 m above horizontal ground (see diagram). The system is released from rest, and while both particles are in motion the tension in the string is 1.5 N. Particle \(B\) does not reach the pulley.
  1. Find \(m\). [4]
  2. Find the speed at which \(A\) reaches the ground. [2]
CAIE M1 2021 June Q3
6 marks Standard +0.3
Three particles \(P\), \(Q\) and \(R\), of masses 0.1 kg, 0.2 kg and 0.5 kg respectively, are at rest in a straight line on a smooth horizontal plane. Particle \(P\) is projected towards \(Q\) at a speed of \(5 \text{ m s}^{-1}\). After \(P\) and \(Q\) collide, \(P\) rebounds with speed \(1 \text{ m s}^{-1}\).
  1. Find the speed of \(Q\) immediately after the collision with \(P\). [3]
\(Q\) now collides with \(R\). Immediately after the collision with \(Q\), \(R\) begins to move with speed \(V \text{ m s}^{-1}\).
  1. Given that there is no subsequent collision between \(P\) and \(Q\), find the greatest possible value of \(V\). [3]
CAIE M1 2021 June Q4
7 marks Moderate -0.3
Two cyclists, Isabella and Maria, are having a race. They both travel along a straight road with constant acceleration, starting from rest at point \(A\). Isabella accelerates for 5 s at a constant rate \(a \text{ m s}^{-2}\). She then travels at the constant speed she has reached for 10 s, before decelerating to rest at a constant rate over a period of 5 s. Maria accelerates at a constant rate, reaching a speed of \(5 \text{ m s}^{-1}\) in a distance of 27.5 m. She then maintains this speed for a period of 10 s, before decelerating to rest at a constant rate over a period of 5 s.
  1. Given that \(a = 1.1\), find which cyclist travels further. [5]
  2. Find the value of \(a\) for which the two cyclists travel the same distance. [2]
CAIE M1 2021 June Q5
8 marks Standard +0.3
A particle moving in a straight line starts from rest at a point \(A\) and comes instantaneously to rest at a point \(B\). The acceleration of the particle at time \(t\) s after leaving \(A\) is \(a \text{ m s}^{-2}\), where $$a = 6t^{\frac{1}{2}} - 2t.$$
  1. Find the value of \(t\) at point \(B\). [3]
  2. Find the distance travelled from \(A\) to the point at which the acceleration of the particle is again zero. [5]
CAIE M1 2021 June Q6
9 marks Standard +0.3
\includegraphics{figure_6} Three coplanar forces of magnitudes 10 N, 25 N and 20 N act at a point \(O\) in the directions shown in the diagram.
  1. Given that the component of the resultant force in the \(x\)-direction is zero, find \(\alpha\), and hence find the magnitude of the resultant force. [4]
  2. Given instead that \(\alpha = 45\), find the magnitude and direction of the resultant of the three forces. [5]
CAIE M1 2021 June Q7
11 marks Standard +0.3
\includegraphics{figure_7} A slide in a playground descends at a constant angle of 30° for 2.5 m. It then has a horizontal section in the same vertical plane as the sloping section. A child of mass 35 kg, modelled as a particle \(P\), starts from rest at the top of the slide and slides straight down the sloping section. She then continues along the horizontal section until she comes to rest (see diagram). There is no instantaneous change in speed when the child goes from the sloping section to the horizontal section. The child experiences a resistance force on the horizontal section of the slide, and the work done against the resistance force on the horizontal section of the slide is 250 J per metre.
  1. It is given that the sloping section of the slide is smooth.
    1. Find the speed of the child when she reaches the bottom of the sloping section. [3]
    2. Find the distance that the child travels along the horizontal section of the slide before she comes to rest. [2]
  2. It is given instead that the sloping section of the slide is rough and that the child comes to rest on the slide 1.05 m after she reaches the horizontal section. Find the coefficient of friction between the child and the sloping section of the slide. [6]
CAIE M1 2022 June Q1
6 marks Moderate -0.8
A car starts from rest and moves in a straight line with constant acceleration for a distance of 200 m, reaching a speed of 25 m s\(^{-1}\). The car then travels at this speed for 400 m, before decelerating uniformly to rest over a period of 5 s.
  1. Find the time for which the car is accelerating. [2]
  2. Sketch the velocity–time graph for the motion of the car, showing the key points. [2]
  3. Find the average speed of the car during its motion. [2]
CAIE M1 2022 June Q2
5 marks Moderate -0.8
Two particles \(P\) and \(Q\), of masses 0.5 kg and 0.3 kg respectively, are connected by a light inextensible string. The string is taut and \(P\) is vertically above \(Q\). A force of magnitude 10 N is applied to \(P\) vertically upwards. Find the acceleration of the particles and the tension in the string connecting them. [5]