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CAIE FP2 2018 November Q11
28 marks Moderate -0.5
Answer only one of the following two alternatives. **EITHER** One end of a light elastic spring, of natural length 0.8 m and modulus of elasticity 40 N, is attached to a fixed point \(O\). The spring hangs vertically, at rest, with particles of masses 2 kg and \(M\) kg attached to its free end. The \(M\) kg particle becomes detached from the spring, and as a result the 2 kg particle begins to move upwards. \begin{enumerate}[label=(\roman*)] \item Show that the 2 kg particle performs simple harmonic motion about its equilibrium position with period \(\frac{2\pi}{5}\) s. State the distance below \(O\) of the centre of the oscillations. [7] \item The speed of the 2 kg particle is 0.4 m s\(^{-1}\) when its displacement from the centre of oscillation is 0.06 m. Find the amplitude of the motion. [3] \item Deduce the value of \(M\). [4] \end{enumerate] **OR** In a particular country, large numbers of ducks live on lakes \(A\) and \(B\). The mass, in kg, of a duck on lake \(A\) is denoted by \(x\) and the mass, in kg, of a duck on lake \(B\) is denoted by \(y\). A random sample of 8 ducks is taken from lake \(A\) and a random sample of 10 ducks is taken from lake \(B\). Their masses are summarised as follows. \(\Sigma x = 10.56\) \(\quad\) \(\Sigma x^2 = 14.1775\) \(\quad\) \(\Sigma y = 12.39\) \(\quad\) \(\Sigma y^2 = 15.894\) A scientist claims that ducks on lake \(A\) are heavier on average than ducks on lake \(B\). \begin{enumerate}[label=(\roman*)] \item Test, at the 10% significance level, whether the scientist's claim is justified. You should assume that both distributions are normal and that their variances are equal. [9] \item A second random sample of 8 ducks is taken from lake \(A\) and their masses are summarised as \(\Sigma x = 10.24\) \(\quad\) and \(\quad\) \(\Sigma(x - \bar{x})^2 = 0.294\), where \(\bar{x}\) is the sample mean. The scientist now claims that the population mean mass of ducks on lake \(A\) is greater than \(p\) kg. A test of this claim is carried out at the 10% significance level, using only this second sample from lake \(A\). This test supports the scientist's claim. Find the greatest possible value of \(p\). [5] \end{enumerate]
CAIE FP2 2019 November Q1
5 marks Standard +0.3
A particle \(P\) is moving in a circle of radius 2 m. At time \(t\) seconds, its velocity is \((t - 1)^2\) m s\(^{-1}\). At a particular time \(T\) seconds, where \(T > 0\), the magnitude of the radial component of the acceleration of \(P\) is 8 m s\(^{-2}\). Find the magnitude of the transverse component of the acceleration of \(P\) at this instant. [5]
CAIE FP2 2019 November Q2
8 marks Challenging +1.2
\includegraphics{figure_2} A uniform square lamina \(ABCD\) of side \(4a\) and weight \(W\) rests in a vertical plane with the edge \(AB\) inclined at angle \(\theta\) to the horizontal, where \(\tan \theta = \frac{1}{4}\). The vertex \(B\) is in contact with a rough horizontal surface for which the coefficient of friction is \(\mu\). The lamina is supported by a smooth peg at the point \(E\) on \(AB\), where \(BE = 3a\) (see diagram).
  1. Find expressions in terms of \(W\) for the normal reaction forces at \(E\) and \(B\). [5]
  2. Given that the lamina is about to slip, find the value of \(\mu\). [3]
CAIE FP2 2019 November Q3
9 marks Standard +0.8
Three uniform small spheres \(A\), \(B\) and \(C\) have equal radii and masses \(5m\), \(5m\) and \(3m\) respectively. The spheres are at rest on a smooth horizontal surface, in a straight line, with \(B\) between \(A\) and \(C\). The coefficient of restitution between each pair of spheres is \(e\). Sphere \(A\) is projected directly towards \(B\) with speed \(u\).
  1. Show that the speed of \(A\) after its collision with \(B\) is \(\frac{1}{2}u(1 - e)\) and find the speed of \(B\). [3]
Sphere \(B\) now collides with sphere \(C\). Subsequently there are no further collisions between any of the spheres.
  1. Find the set of possible values of \(e\). [6]
CAIE FP2 2019 November Q4
9 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 \(O\) and \(P\) is held with the string taut and horizontal. The particle \(P\) is projected vertically downwards with speed \(\sqrt{(2ag)}\) so that it begins to move along a circular path. The string becomes slack when \(OP\) makes an angle \(\theta\) with the upward vertical through \(O\).
  1. Show that \(\cos \theta = \frac{2}{3}\). [5]
  2. Find the greatest height, above the horizontal through \(O\), reached by \(P\) in its subsequent motion. [4]
CAIE FP2 2019 November Q5
12 marks Challenging +1.8
\includegraphics{figure_5} A thin uniform rod \(AB\) has mass \(\lambda M\) and length \(2a\). The end \(A\) of the rod is rigidly attached to the surface of a uniform hollow sphere (spherical shell) with centre \(O\), mass \(3M\) and radius \(a\). The end \(B\) of the rod is rigidly attached to the surface of a uniform solid sphere with centre \(C\), mass \(5M\) and radius \(a\). The rod lies along the line joining the centres of the spheres, so that \(CBAO\) is a straight line. The horizontal axis \(L\) is perpendicular to the rod and passes through the point of the rod that is a distance \(\frac{1}{2}a\) from \(B\) (see diagram). The object consisting of the rod and the two spheres can rotate freely about \(L\).
  1. Show that the moment of inertia of the object about \(L\) is \(\left(\frac{408 + 7\lambda}{12}\right)Ma^2\). [6]
The period of small oscillations of the object about \(L\) is \(5\pi\sqrt{\left(\frac{2a}{g}\right)}\).
  1. Find the value of \(\lambda\). [6]
CAIE FP2 2019 November Q6
7 marks Challenging +1.2
A random sample of 9 members is taken from the large number of members of a sports club, and their heights are measured. The heights of all the members of the club are assumed to be normally distributed. A 95% confidence interval for the population mean height, \(\mu\) metres, is calculated from the data as \(1.65 \leqslant \mu \leqslant 1.85\).
  1. Find an unbiased estimate for the population variance. [3]
  2. Denoting the height of a member of the club by \(x\) metres, find \(\Sigma x^2\) for this sample of 9 members. [4]
CAIE FP2 2019 November Q7
7 marks Standard +0.3
The time, \(T\) days, before an electrical component develops a fault has distribution function F given by $$\mathrm{F}(t) = \begin{cases} 1 - e^{-at} & t \geqslant 0, \\ 0 & \text{otherwise}, \end{cases}$$ where \(a\) is a positive constant. The mean value of \(T\) is 200.
  1. Write down the value of \(a\). [1]
  2. Find the probability that an electrical component of this type develops a fault in less than 150 days. [2]
A piece of equipment contains \(n\) of these components, which develop faults independently of each other. The probability that, after 150 days, at least one of the \(n\) components has not developed a fault is greater than 0.99.
  1. Find the smallest possible value of \(n\). [4]
CAIE FP2 2019 November Q8
9 marks Challenging +1.2
A random sample of 8 elephants from region \(A\) is taken and their weights, \(x\) tonnes, are recorded. (1 tonne = 1000 kg.) The results are summarised as follows. $$\Sigma x = 32.4 \quad \Sigma x^2 = 131.82$$ A random sample of 10 elephants from region \(B\) is taken. Their weights give a sample mean of 3.78 tonnes and an unbiased variance estimate of 0.1555 tonnes\(^2\). The distributions of the weights of elephants in regions \(A\) and \(B\) are both assumed to be normal with the same population variance. Test at the 10% significance level whether the mean weight of elephants in region \(A\) is the same as the mean weight of elephants in region \(B\). [9]
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]