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CAIE FP2 2009 November Q3
3 Two small smooth spheres \(A\) and \(B\) of equal radius have masses \(m\) and \(3 m\) respectively. They lie at rest on a smooth horizontal plane with their line of centres perpendicular to a smooth fixed vertical barrier with \(B\) between \(A\) and the barrier. The coefficient of restitution between \(A\) and \(B\), and between \(B\) and the barrier, is \(e\), where \(e > \frac { 1 } { 3 }\). Sphere \(A\) is projected directly towards \(B\) with speed \(u\). Show that after colliding with \(B\) the direction of motion of \(A\) is reversed. After the impact, \(B\) hits the barrier and rebounds. Show that \(B\) will subsequently collide with \(A\) again unless \(e = 1\).
CAIE FP2 2009 November Q4
4 A uniform rod \(A B\), of length \(2 a\) and mass \(2 m\), can rotate freely in a vertical plane about a smooth horizontal axis through \(A\). A small rough ring of mass \(m\) is threaded on the rod. The rod is held in a horizontal position with the ring at rest at the mid-point of the rod. The rod is released from rest. Using energy considerations, show that, until the ring slides, $$a \dot { \theta } ^ { 2 } = \frac { 18 } { 11 } g \sin \theta$$ where \(\theta\) is the angle turned through by the rod. Show that, until the ring slides, the magnitudes of the friction force and normal contact force acting on the ring are \(\frac { 29 } { 11 } m g \sin \theta\) and \(\frac { 2 } { 11 } m g \cos \theta\) respectively. The coefficient of friction between the ring and the rod is \(\mu\). Find, in terms of \(\mu\), the value of \(\theta\) when the ring starts to slide.
CAIE FP2 2009 November Q5
5
\includegraphics[max width=\textwidth, alt={}, center]{20dd0f90-8c10-4a7b-a383-4ba89d167cd6-3_716_549_269_799} Two uniform rods, \(A B\) and \(B C\), each have length \(2 a\) and weight \(W\). They are smoothly jointed at \(B\), and \(A\) is attached to a smooth fixed pivot. A light inextensible string of length ( \(2 \sqrt { } 2\) ) \(a\) joins \(A\) to \(C\) so that angle \(A B C = 90 ^ { \circ }\). The system hangs in equilibrium, with \(A B\) making an angle \(\alpha\) with the vertical (see diagram). By taking moments about \(A\) for the system, or otherwise, show that \(\alpha = 18.4 ^ { \circ }\), correct to the nearest \(0.1 ^ { \circ }\). Find the tension in the string in the form \(k W\), giving the value of \(k\) correct to 3 significant figures. Find, in terms of \(W\), the magnitude of the force acting on the \(\operatorname { rod } B C\) at \(B\).
CAIE FP2 2009 November Q6
6 A machine produces metal discs whose diameters have a normal distribution. The mean of this distribution is intended to be 10 cm . Accuracy is checked by measuring the diameters of a random sample of six discs. The diameters, in cm, are as follows. $$\begin{array} { l l l l l l } 10.03 & 10.02 & 9.98 & 10.06 & 10.08 & 10.01 \end{array}$$ Calculate a 99\% confidence interval for the mean diameter of all discs produced by the machine. Deduce a 99\% confidence interval for the mean circumference of all discs produced by the machine.
CAIE FP2 2009 November Q7
7 A continuous random variable \(X\) has cumulative distribution function F given by $$\mathrm { F } ( x ) = \begin{cases} 0 & x < - 1 ,
\frac { 1 } { 2 } \left( x ^ { 3 } + 1 \right) & - 1 \leqslant x \leqslant 1 ,
1 & x > 1 . \end{cases}$$ Find \(\mathrm { P } \left( X \geqslant \frac { 3 } { 4 } \right)\), and state what can be deduced about the upper quartile of \(X\). Obtain the cumulative distribution function of \(Y\), where \(Y = X ^ { 2 }\). 8150 sheep, chosen from a large flock of sheep, were divided into two groups of 75 . Over a fixed period, one group had their grazing controlled and the other group grazed freely. The gains in weight, in kg, were recorded for each animal and the table below shows the sample means and the unbiased estimates of the population variances for the two samples.
Sample mean
Unbiased estimate of
population variance
Controlled grazing19.1420.54
Free grazing15.369.84
It is required to test whether the population mean for sheep having their grazing controlled exceeds the population mean for sheep grazing freely by less than 5 kg . State, giving a reason, if it is necessary for the validity of the test to assume that the two population variances are equal. Stating any other assumption, carry out the test at the 5\% significance level.
CAIE FP2 2009 November Q9
9 It has been found that \(60 \%\) of the computer chips produced in a factory are faulty. As part of quality control, 100 samples of 4 chips are selected at random, and each chip is tested. The number of faulty chips in each sample is recorded, with the results given in the following table.
Number of faulty chips01234
Number of samples212274910
The expected values for a binomial distribution with parameters \(n = 4\) and \(p = 0.6\) are given in the following table.
Number of faulty chips01234
Expected value2.5615.3634.5634.5612.96
Show how the expected value 34.56 corresponding to 2 faulty chips is obtained. Carry out a goodness of fit test at the 5\% significance level, and state what can be deduced from the outcome of the test.
CAIE FP2 2009 November Q10
10 An archer shoots at a target. It may be assumed that each shot is independent of all other shots and that, on average, she hits the bull's-eye with 3 shots in 20 . Find the probability that she requires at least 6 shots to hit the bull's-eye. When she hits the bull's-eye for the third time her total number of shots is \(Y\). Show that $$\mathrm { P } ( Y = r ) = \frac { 1 } { 2 } ( r - 1 ) ( r - 2 ) \left( \frac { 3 } { 20 } \right) ^ { 3 } \left( \frac { 17 } { 20 } \right) ^ { r - 3 } .$$ Simplify \(\frac { \mathrm { P } ( Y = r + 1 ) } { \mathrm { P } ( Y = r ) }\), and hence find the set of values of \(r\) for which \(\mathrm { P } ( Y = r + 1 ) < \mathrm { P } ( Y = r )\). Deduce the most probable value of \(Y\).
CAIE FP2 2009 November Q11 EITHER
A light elastic string, of natural length \(l\) and modulus of elasticity \(4 m g\), is attached at one end to a fixed point and has a particle \(P\) of mass \(m\) attached to the other end. When \(P\) is hanging in equilibrium under gravity it is given a velocity \(\sqrt { } ( g l )\) vertically downwards. At time \(t\) the downward displacement of \(P\) from its equilibrium position is \(x\). Show that, while the string is taut, $$\ddot { x } = - \frac { 4 g } { l } x .$$ Find the speed of \(P\) when the length of the string is \(l\). Show that the time taken for \(P\) to move from the lowest point to the highest point of its motion is $$\left( \frac { \pi } { 3 } + \frac { \sqrt { } 3 } { 2 } \right) \sqrt { } \left( \frac { l } { g } \right)$$
CAIE FP2 2009 November Q11 OR
\includegraphics[max width=\textwidth, alt={}]{20dd0f90-8c10-4a7b-a383-4ba89d167cd6-5_374_569_1123_788}
The scatter diagram shows a sample of size 5 of bivariate data, together with the regression line of \(y\) on \(x\). State what is minimised in obtaining this regression line, illustrating your answer on a copy of this diagram. State, giving a reason, whether, for the data shown, the regression line of \(y\) on \(x\) is the same as the regression line of \(x\) on \(y\). A car is travelling along a stretch of road with speed \(v \mathrm {~km} \mathrm {~h} ^ { - 1 }\) when the brakes are applied. The car comes to rest after travelling a further distance of \(z \mathrm {~m}\). The values of \(z\) (and \(\sqrt { } z\) ) for 8 different values of \(v\) are given in the table, correct to 2 decimal places.
\(v\)2530354045505560
\(z\)2.834.634.845.299.7310.3014.8215.21
\(\sqrt { } z\)1.682.152.202.303.123.213.853.90
$$\left[ \Sigma v = 340 , \Sigma v ^ { 2 } = 15500 , \Sigma \sqrt { } z = 22.41 , \Sigma z = 67.65 , \Sigma v \sqrt { } z = 1022.15 . \right]$$
  1. Calculate the product moment correlation coefficient between \(v\) and \(\sqrt { } z\). What does this indicate about the scatter diagram of the points \(( v , \sqrt { } z )\) ?
  2. Given that the product moment correlation coefficient between \(v\) and \(z\) is 0.965 , correct to 3 decimal places, state why the regression line of \(\sqrt { } z\) on \(v\) is more suitable than the regression line of \(z\) on \(v\), and find the equation of the regression line of \(\sqrt { } z\) on \(v\).
  3. Comment, in the context of the question, on the value of the constant term in the equation of the regression line of \(\sqrt { } z\) on \(v\).
CAIE FP2 2010 November Q1
1 A particle \(P\) is describing simple harmonic motion of amplitude 5 m . Its speed is \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when it is 3 m from the centre of the motion. Find, in terms of \(\pi\), the period of the motion. Find also
  1. the maximum speed of \(P\),
  2. the magnitude of the maximum acceleration of \(P\).
    \(2 \quad\) A particle \(P\) of mass \(m\) is projected horizontally with speed \(u\) from the lowest point on the inside of a fixed hollow sphere with centre \(O\). The sphere has a smooth internal surface of radius \(a\). Assuming that the particle does not lose contact with the sphere, show that when the speed of the particle has been reduced to \(\frac { 1 } { 2 } u\) the angle \(\theta\) between \(O P\) and the downward vertical satisfies the equation $$8 g a ( 1 - \cos \theta ) = 3 u ^ { 2 }$$ Find, in terms of \(m , u , a\) and \(g\), an expression for the magnitude of the contact force acting on the particle in this position.
CAIE FP2 2010 November Q3
3 Two smooth spheres \(A\) and \(B\), of equal radius, are moving in the same direction in the same straight line on a smooth horizontal table. Sphere \(A\) has mass \(m\) and speed \(u\) and sphere \(B\) has mass \(\alpha m\) and speed \(\frac { 1 } { 4 } u\). The spheres collide and \(A\) is brought to rest by the collision. Find the coefficient of restitution in terms of \(\alpha\). Deduce that \(\alpha \geqslant 2\).
CAIE FP2 2010 November Q4
4
\includegraphics[max width=\textwidth, alt={}, center]{c7844913-5c2e-47b4-87b6-f822f4d4bf22-2_426_862_1553_644} A hemispherical bowl of radius \(r\) is fixed with its rim horizontal. A thin uniform rod rests in equilibrium on the rim of the bowl with one end resting on the inner surface of the bowl at \(A\), as shown in the diagram. The rod has length \(2 a\) and weight \(W\). The point of contact between the rod and the rim is \(B\), and the rim has centre \(C\). The rod is in a vertical plane containing \(C\). The rod is inclined at \(\theta\) to the horizontal and the line \(A C\) is inclined at \(2 \theta\) to the horizontal. The contacts at \(A\) and \(B\) are smooth. In any order, show that
  1. the contact force acting on the rod at \(A\) has magnitude \(W \tan \theta\),
  2. the contact force acting on the rod at \(B\) has magnitude \(\frac { W \cos 2 \theta } { \cos \theta }\),
  3. \(2 r \cos 2 \theta = a \cos \theta\).
CAIE FP2 2010 November Q5
5 A uniform circular disc has diameter \(A B\), mass \(2 m\) and radius \(a\). A particle of mass \(m\) is attached to the disc at \(B\). The disc is able to rotate about a smooth fixed horizontal axis through \(A\). The axis is tangential to the disc. Show that the moment of inertia of the system about the axis is \(\frac { 13 } { 2 } m a ^ { 2 }\). The disc is held with \(A B\) horizontal and released. Find the angular speed of the system when \(B\) is directly below \(A\). The disc is slightly displaced from the position of equilibrium in which \(B\) is below \(A\). At time \(t\) the angle between \(A B\) and the vertical is \(\theta\). Write down the equation of motion, and find the approximate period of small oscillations about the equilibrium position.
CAIE FP2 2010 November Q6
6 The mean Intelligence Quotient (IQ) of a random sample of 15 pupils at \(\operatorname { School } A\) is 109 . The mean IQ of a random sample of 20 pupils at School \(B\) is 112 . You may assume that the IQs for the populations from which these samples are taken are normally distributed, and that both distributions have standard deviation 15. Find a \(90 \%\) confidence interval for \(\mu _ { B } - \mu _ { A }\), where \(\mu _ { A }\) and \(\mu _ { B }\) are the population mean IQs.
CAIE FP2 2010 November Q7
7 The discrete random variable \(X\) has a geometric distribution with mean 4.
Find
  1. \(\mathrm { P } ( X = 5 )\),
  2. \(\mathrm { P } ( X \geqslant 5 )\),
  3. the least integer \(N\) such that \(\mathrm { P } ( X \leqslant N ) > 0.9995\).
CAIE FP2 2010 November Q8
8 The owner of three driving schools, \(A , B\) and \(C\), wished to assess whether there was an association between passing the driving test and the school attended. He selected a random sample of learner drivers from each of his schools and recorded the numbers of passes and failures at each school. The results that he obtained are shown in the table below.
\multirow{2}{*}{}Driving school attended
\cline { 2 - 4 }\(A\)\(B\)\(C\)
Passes231517
Failures272543
Using a \(\chi ^ { 2 }\)-test and a \(5 \%\) level of significance, test whether there is an association between passing or failing the driving test and the driving school attended.
CAIE FP2 2010 November Q9
9 A national athletics coach suspects that, on average, 200-metre runners' indoor times exceed their outdoor times by more than 0.1 seconds. In order to test this, the coach randomly selects eight 200 -metre runners and records their indoor and outdoor times. The results, in seconds, are shown in the table.
Runner\(A\)\(B\)\(C\)\(D\)\(E\)\(F\)\(G\)\(H\)
Indoor time21.521.820.921.221.421.421.221.0
Outdoor time21.121.720.720.921.321.021.120.8
Stating suitable hypotheses and any necessary assumption that you make, test the coach's suspicion at the 2.5\% level of significance.
CAIE FP2 2010 November Q10
10 For each month of a certain year, a weather station recorded the average rainfall per day, \(x \mathrm {~mm}\), and the average amount of sunshine per day, \(y\) hours. The results are summarised below. $$n = 12 , \quad \Sigma x = 24.29 , \quad \Sigma x ^ { 2 } = 50.146 , \quad \Sigma y = 45.8 , \quad \Sigma y ^ { 2 } = 211.16 , \quad \Sigma x y = 88.415 .$$
  1. Find the mean values, \(\bar { x }\) and \(\bar { y }\).
  2. Calculate the gradient of the line of regression of \(y\) on \(x\).
  3. Use the answers to parts (i) and (ii) to obtain the equation of the line of regression of \(y\) on \(x\).
  4. Find the product moment correlation coefficient and comment, in context, on its value.
  5. Stating your hypotheses, test at the \(1 \%\) level of significance whether there is negative correlation between average rainfall per day and average amount of sunshine per day.
CAIE FP2 2010 November Q11 EITHER
A particle of mass 0.1 kg lies on a smooth horizontal table on the line between two points \(A\) and \(B\) on the table, which are 6 m apart. The particle is joined to \(A\) by a light elastic string of natural length 2 m and modulus of elasticity 60 N , and to \(B\) by a light elastic string of natural length 1 m and modulus of elasticity 20 N . The mid-point of \(A B\) is \(M\), and \(O\) is the point between \(M\) and \(B\) at which the particle can rest in equilibrium. Show that \(M O = 0.2 \mathrm {~m}\). The particle is held at \(M\) and then released. Show that the equation of motion is $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} t ^ { 2 } } = - 500 y$$ where \(y\) metres is the displacement from \(O\) in the direction \(O B\) at time \(t\) seconds, and state the period of the motion. For the instant when the particle is 0.3 m from \(M\) for the first time, find
  1. the speed of the particle,
  2. the time taken, after release, to reach this position.
CAIE FP2 2010 November Q11 OR
The continuous random variable \(T\) has a negative exponential distribution with probability density function given by $$\mathrm { f } ( t ) = \begin{cases} \lambda \mathrm { e } ^ { - \lambda t } & t \geqslant 0
0 & \text { otherwise } \end{cases}$$ Show that for \(t \geqslant 0\) the distribution function is given by \(\mathrm { F } ( t ) = 1 - \mathrm { e } ^ { - \lambda t }\). The table below shows some values of \(\mathrm { F } ( t )\) for the case when the mean is 20 . Find the missing value.
\(t\)0510152025303540
\(\mathrm {~F} ( t )\)00.22120.39350.63210.71350.77690.82620.8647
It is thought that the lifetime of a species of insect under laboratory conditions has a negative exponential distribution with mean 20 hours. When observation starts there are 100 insects, which have been randomly selected. The lifetimes of the insects, in hours, are summarised in the table below.
Lifetime (hours)\(0 - 5\)\(5 - 10\)\(10 - 15\)\(15 - 20\)\(20 - 25\)\(25 - 30\)\(30 - 35\)\(35 - 40\)\(\geqslant 40\)
Frequency2020119985117
Calculate the expected values for each interval, assuming a negative exponential model with a mean of 20 hours, giving your values correct to 2 decimal places. Perform a \(\chi ^ { 2 }\)-test of goodness of fit, at the \(5 \%\) level of significance, in order to test whether a negative exponential distribution, with a mean of 20 hours, is a suitable model for the lifetime of this species of insect under laboratory conditions.
CAIE FP2 2011 November Q1
1 A particle is moving in a circle of radius 2 m . At time \(t \mathrm {~s}\) its velocity is \(\left( t ^ { 2 } - 12 \right) \mathrm { m } \mathrm { s } ^ { - 1 }\). Find the magnitude of the resultant acceleration of the particle when \(t = 4\).
CAIE FP2 2011 November Q2
2 A particle \(P\) is moving in simple harmonic motion with centre \(O\). When \(P\) is 5 m from \(O\) its speed is \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\), and when it is 9 m from \(O\) its speed is \(\frac { 3 } { 5 } V \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Show that the amplitude of the motion is \(\frac { 15 } { 2 } \sqrt { } 2 \mathrm {~m}\). Given that the greatest speed of \(P\) is \(3 \sqrt { } 2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), find \(V\).
CAIE FP2 2011 November Q3
3 A fixed hollow sphere with centre \(O\) has a smooth inner surface of radius \(a\). A particle \(P\) of mass \(m\) is projected horizontally with speed \(2 \sqrt { } ( a g )\) from the lowest point of the inner surface of the sphere. The particle loses contact with the inner surface of the sphere when \(O P\) makes an angle \(\theta\) with the upward vertical.
  1. Show that \(\cos \theta = \frac { 2 } { 3 }\).
  2. Find the greatest height that \(P\) reaches above the level of \(O\).
CAIE FP2 2011 November Q4
4 Two smooth spheres \(P\) and \(Q\), of equal radius, have masses \(m\) and \(3 m\) respectively. They are moving in the same direction in the same straight line on a smooth horizontal table. Sphere \(P\) has speed \(u\) and collides directly with sphere \(Q\) which has speed \(k u\), where \(0 < k < 1\). Sphere \(P\) is brought to rest by the collision. Show that the coefficient of restitution between \(P\) and \(Q\) is \(\frac { 3 k + 1 } { 3 ( 1 - k ) }\). One third of the total kinetic energy of the spheres is lost in the collision. Show that $$k = \frac { 1 } { 3 } ( 2 \sqrt { } 3 - 3 )$$
CAIE FP2 2011 November Q5
5
\includegraphics[max width=\textwidth, alt={}, center]{96b6c92d-6d13-452f-84ec-37c45651b232-2_529_493_1667_826} A uniform solid sphere with centre \(C\), radius \(2 a\) and mass \(3 M\), is pivoted about a smooth horizontal axis and hangs at rest. The point \(O\) on the axis is vertically above \(C\) and \(O C = a\). A particle \(P\) of mass \(M\) is attached to the sphere at its lowest point (see diagram). Show that the moment of inertia of the system about the axis through \(O\) is \(\frac { 84 } { 5 } M a ^ { 2 }\). The system is released from rest with \(O P\) making a small angle \(\alpha\) with the downward vertical. Find
  1. the period of small oscillations,
  2. the time from release until \(O P\) makes an angle \(\frac { 1 } { 2 } \alpha\) with the downward vertical for the first time.