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CAIE FP2 2017 June Q5
5
\includegraphics[max width=\textwidth, alt={}, center]{c437c752-5518-4185-b02f-74206dc4b13c-10_445_735_264_696} A particle 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 point \(A\) is such that \(O A = a\) and \(O A\) makes an angle \(\alpha\) with the upward vertical through \(O\). The particle is held at \(A\) and then projected downwards with speed \(\sqrt { } ( a g )\) so that it begins to move in a vertical circle with centre \(O\). There is a small smooth peg at the point \(B\) which is at the same horizontal level as \(O\) and at a distance \(\frac { 1 } { 3 } a\) from \(O\) on the opposite side of \(O\) to \(A\) (see diagram).
  1. Show that, when the string first makes contact with the peg, the speed of the particle is \(\sqrt { } ( \operatorname { ag } ( 1 + 2 \cos \alpha ) )\).
    The particle now begins to move in a vertical circle with centre \(B\). When the particle is at the point \(C\) where angle \(C B O = 150 ^ { \circ }\), the tension in the string is the same as it was when the particle was at the point \(A\).
  2. Find the value of \(\cos \alpha\).
CAIE FP2 2017 June Q2
2 A particle \(P\) moves on a straight line in simple harmonic motion. The centre of the motion is \(O\), and the amplitude of the motion is 2.5 m . The points \(L\) and \(M\) are on the line, on opposite sides of \(O\), with \(O L = 1.5 \mathrm {~m}\). The magnitudes of the accelerations of \(P\) at \(L\) and at \(M\) are in the ratio 3:4.
  1. Find the distance \(O M\).
    ................................................................................................................................. .
    The time taken by \(P\) to travel directly from \(L\) to \(M\) is 2 s .
  2. Find the period of the motion.
  3. Find the speed of \(P\) when it passes through \(L\).
CAIE FP2 2017 June Q3
3 Two uniform small smooth spheres \(A\) and \(B\) have equal radii and each has mass \(m\). Sphere \(A\) is moving with speed \(u\) on a smooth horizontal surface when it collides directly with sphere \(B\) which is at rest. The coefficient of restitution between the spheres is \(\frac { 2 } { 3 }\). Sphere \(B\) is initially at a distance \(d\) from a fixed smooth vertical wall which is perpendicular to the direction of motion of \(A\). The coefficient of restitution between \(B\) and the wall is \(\frac { 1 } { 3 }\).
  1. Show that the speed of \(B\) after its collision with the wall is \(\frac { 5 } { 18 } u\).
  2. Find the distance of \(B\) from the wall when it collides with \(A\) for the second time.
    \includegraphics[max width=\textwidth, alt={}, center]{b10d2991-abff-4d2b-b470-1df844d1c7ee-08_743_673_258_737} A uniform rod \(A B\) of length \(3 a\) and weight \(W\) is freely hinged to a fixed point at the end \(A\). The end \(B\) is below the level of \(A\) and is attached to one end of a light elastic string of natural length 4a. The other end of the string is attached to a point \(O\) on a vertical wall. The horizontal distance between \(A\) and the wall is \(5 a\). The string and the rod make angles \(\theta\) and \(2 \theta\) respectively with the horizontal (see diagram). The system is in equilibrium with the rod and the string in the same vertical plane. It is given that \(\sin \theta = \frac { 3 } { 5 }\) and you may use the fact that \(\cos 2 \theta = \frac { 7 } { 25 }\).
CAIE FP2 2017 June Q6
6 The independent variables \(X\) and \(Y\) have distributions with the same variance \(\sigma ^ { 2 }\). Random samples of \(N\) observations of \(X\) and \(2 N\) observations of \(Y\) are taken, and the results are summarised by $$\Sigma x = 4 , \quad \Sigma x ^ { 2 } = 10 , \quad \Sigma y = 8 , \quad \Sigma y ^ { 2 } = 102 .$$ These data give a pooled estimate of 10 for \(\sigma ^ { 2 }\). Find \(N\).
CAIE FP2 2017 June Q7
7 A random sample of twelve pairs of values of \(x\) and \(y\) is taken from a bivariate distribution. The equations of the regression lines of \(y\) on \(x\) and of \(x\) on \(y\) are respectively $$y = 0.46 x + 1.62 \quad \text { and } \quad x = 0.93 y + 8.24$$
  1. Find the value of the product moment correlation coefficient for this sample.
  2. Using a \(5 \%\) significance level, test whether there is non-zero correlation between the variables.
CAIE FP2 2017 June Q8
8 The number, \(x\), of beech trees was counted in each of 50 randomly chosen regions of equal size in beech forests in country \(A\). The number, \(y\), of beech trees was counted in each of 40 randomly chosen regions of the same equal size in beech forests in country \(B\). The results are summarised as follows. $$\Sigma x = 1416 \quad \Sigma x ^ { 2 } = 41100 \quad \Sigma y = 888 \quad \Sigma y ^ { 2 } = 20140$$ Find a 95\% confidence interval for the difference between the mean number of beech trees in regions of this size in country \(A\) and in country \(B\).
CAIE FP2 2017 June Q9
9 The continuous random variable \(X\) has probability density function f given by $$\mathrm { f } ( x ) = \begin{cases} 0 & x < 0 ,
a \mathrm { e } ^ { - x \ln 2 } & x \geqslant 0 , \end{cases}$$ where \(a\) is a positive constant.
  1. Find the value of \(a\).
  2. State the value of \(\mathrm { E } ( X )\).
  3. Find the interquartile range of \(X\).
    The variable \(Y\) is related to \(X\) by \(Y = 2 ^ { X }\).
  4. Find the probability density function of \(Y\).
CAIE FP2 2017 June Q10
10 Roberto owns a small hotel and offers accommodation to guests. Over a period of 100 nights, the numbers of rooms, \(x\), that are occupied each night at Roberto's hotel and the corresponding frequencies are shown in the following table.
Number of rooms
occupied \(( x )\)
0123456\(\geqslant 7\)
Number of nights491826201670
  1. Show that the mean number of rooms that are occupied each night is 3.25 .
    The following table shows most of the corresponding expected frequencies, correct to 2 decimal places, using a Poisson distribution with mean 3.25.
    Number of rooms
    occupied \(( x )\)
    		0123456\(\geqslant 7\)
    Observed frequency491826201670
    Expected frequency3.8812.6020.4822.1818.0211.72
  2. Show how the expected value of 22.18 , for \(x = 3\), is obtained and find the expected values for \(x = 6\) and for \(x \geqslant 7\).
  3. Use a goodness-of-fit test at the \(5 \%\) significance level to determine whether the Poisson distribution is a suitable model for the number of rooms occupied each night at Roberto's hotel.
CAIE FP2 2017 June Q11 EITHER
\includegraphics[max width=\textwidth, alt={}]{b10d2991-abff-4d2b-b470-1df844d1c7ee-20_312_787_440_678}
The diagram shows a uniform thin rod \(A B\) of length \(3 a\) and mass \(8 m\). The end \(A\) is rigidly attached to the surface of a sphere with centre \(O\) and radius \(a\). The rod is perpendicular to the surface of the sphere. The sphere consists of two parts: an inner uniform solid sphere of mass \(\frac { 3 } { 2 } m\) and radius \(a\) surrounded by a thin uniform spherical shell of mass \(m\) and also of radius \(a\). The horizontal axis \(l\) is perpendicular to the rod and passes through the point \(C\) on the rod where \(A C = a\).
  1. Show that the moment of inertia of the object, consisting of rod, shell and inner sphere, about the axis \(l\) is \(\frac { 289 } { 15 } m a ^ { 2 }\).
    The object is free to rotate about the axis \(l\). The object is held so that \(C A\) makes an angle \(\alpha\) with the downward vertical and is released from rest.
  2. Given that \(\cos \alpha = \frac { 1 } { 6 }\), find the greatest speed achieved by the centre of the sphere in the subsequent motion.
CAIE FP2 2017 June Q11 OR
The times taken to run 200 metres at the beginning of the year and at the end of the year are recorded for each member of a large athletics club. The time taken, in seconds, at the beginning of the year is denoted by \(x\) and the time taken, in seconds, at the end of the year is denoted by \(y\). For a random sample of 8 members, the results are shown in the following table.
Member\(A\)\(B\)\(C\)\(D\)\(E\)\(F\)\(G\)\(H\)
\(x\)24.223.822.825.124.524.023.822.8
\(y\)23.923.622.824.524.223.523.622.7
$$\left[ \Sigma x = 191 , \quad \Sigma x ^ { 2 } = 4564.46 , \quad \Sigma y = 188.8 , \quad \Sigma y ^ { 2 } = 4458.4 , \quad \Sigma x y = 4510.99 . \right]$$
  1. Find, showing all necessary working, the equation of the regression line of \(y\) on \(x\).
    The athletics coach believes that, on average, the time taken by an athlete to run 200 metres decreases between the beginning and the end of the year by more than 0.2 seconds.
  2. Stating suitable hypotheses and assuming a normal distribution, test the coach's belief at the \(10 \%\) significance level.
CAIE FP2 2018 June Q1
1 A bullet of mass \(m \mathrm {~kg}\) is fired horizontally into a fixed vertical block of material. It enters the block horizontally with speed \(250 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and emerges horizontally with speed \(70 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) after 0.04 s . The block offers a constant horizontal resisting force of magnitude 450 N . Find the value of \(m\).
CAIE FP2 2018 June Q2
2 A particle \(P\) moves on a straight line in simple harmonic motion. The centre of the motion is \(O\). The points \(A\) and \(B\) are on the line, on opposite sides of \(O\), with \(O A = 1.6 \mathrm {~m}\) and \(O B = 1.2 \mathrm {~m}\). The ratio of the speed of \(P\) at \(A\) to its speed at \(B\) is \(3 : 4\).
  1. Find the amplitude of the motion.
    The maximum speed of \(P\) during its motion is \(\frac { 1 } { 3 } \pi \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  2. Find the period of the motion.
  3. Find the time taken for \(P\) to travel directly from \(A\) to \(B\).
CAIE FP2 2018 June Q3
3 Two identical uniform small spheres \(A\) and \(B\), each of mass \(m\), are moving towards each other in a straight line on a smooth horizontal surface. Their speeds are \(u\) and \(k u\) respectively, and they collide directly. The coefficient of restitution between the spheres is \(e\). Sphere \(B\) is brought to rest by the collision.
  1. Show that \(e = \frac { k - 1 } { k + 1 }\).
  2. Given that \(60 \%\) of the total initial kinetic energy is lost in the collision, find the values of \(k\) and \(e\).
CAIE FP2 2018 June Q4
3 marks
4 A uniform \(\operatorname { rod } A B\) has length \(2 a\) and weight \(W\). The end \(A\) rests on rough horizontal ground and the end \(B\) rests against a smooth vertical wall. The rod is in a vertical plane that is perpendicular to the wall. The angle between the rod and the horizontal is \(\theta\). A particle of weight \(5 W\) hangs from the rod at the point \(C\), with \(A C = x a\), where \(0 < x < 1\).
  1. By taking moments about \(A\), show that the magnitude of the normal reaction at \(B\) is \(\frac { W ( 5 x + 1 ) } { 2 \tan \theta }\).
    [0pt] [3]
    The particle of weight \(5 W\) is now moved a distance \(a\) up the rod, so that \(A C = ( x + 1 ) a\). This results in the magnitude of the normal reaction at \(B\) being double its previous value. The system remains in equilibrium with the rod at angle \(\theta\) with the horizontal.
  2. Show that \(x = \frac { 4 } { 5 }\).
    The coefficient of friction between the rod and the ground is \(\frac { 2 } { 3 }\).
  3. Given that the rod is about to slip when the particle of weight \(5 W\) is in its second position, find the value of \(\tan \theta\).
    \begin{figure}[h]
    \captionsetup{labelformat=empty} \caption{Axis \(l\)} \includegraphics[alt={},max width=\textwidth]{0eb3892f-628f-449a-b022-b38170754d89-08_462_693_301_731}
    \end{figure} Three thin uniform rings \(A , B\) and \(C\) are joined together, so that each ring is in contact with each of the other two rings. Ring \(A\) has radius \(2 a\) and mass \(3 M\); rings \(B\) and \(C\) each have radius \(3 a\) and mass \(2 M\). The rings lie in the same plane and the centres of the rings are at the vertices of an isosceles triangle. The object consisting of the three rings is free to rotate about the horizontal axis \(l\) which is tangential to ring \(A\), in the plane of the object and perpendicular to the line of symmetry of the object (see diagram).
CAIE FP2 2018 June Q6
6 The continuous random variable \(X\) has distribution function given by $$\mathrm { F } ( x ) = \begin{cases} 1 - \mathrm { e } ^ { - 0.4 x } & x \geqslant 0
0 & \text { otherwise } \end{cases}$$
  1. Find \(\mathrm { P } ( X > 2 )\).
  2. Find the interquartile range of \(X\).
CAIE FP2 2018 June Q7
7 A large number of athletes are taking part in a competition. The masses, in kg , of a random sample of 7 athletes are as follows. $$\begin{array} { l l l l l l l } 98.1 & 105.0 & 92.2 & 89.8 & 99.9 & 95.4 & 101.2 \end{array}$$ Assuming that masses are normally distributed, test, at the \(10 \%\) significance level, whether the mean mass of athletes in this competition is equal to 94 kg .
CAIE FP2 2018 June Q8
8 A manufacturer produces three types of car: hatchbacks, saloons and estates. Each type of car is available in one of three colours: silver, blue and red. The manufacturer wants to know whether the popularity of the colour of the car is related to the type of car. A random sample of 300 cars chosen by customers gives the information summarised in the following table.
\cline { 3 - 5 } \multicolumn{2}{c|}{}Colour of car
\cline { 3 - 5 } \multicolumn{2}{c|}{}SilverBlueRed
\multirow{3}{*}{Type of car}Hatchback533641
\cline { 2 - 5 }Saloon294031
\cline { 2 - 5 }Estate282418
Test at the \(10 \%\) significance level whether the colour of car chosen by customers is independent of the type of car.
CAIE FP2 2018 June Q9
9 At a ski resort, the probability of snow on any particular day is constant and equal to \(p\). The skiing season begins on 1 November. The random variable \(X\) denotes the day of the skiing season on which the first snowfall occurs. (For example, if the first snowfall is on 5 November, then \(X = 5\).) The variance of \(X\) is \(\frac { 4 } { 9 }\).
  1. Show that \(4 p ^ { 2 } + 9 p - 9 = 0\) and hence find the value of \(p\).
  2. Find the probability that the first snowfall will be on 3 November.
  3. Find the probability that the first snowfall will not be before 4 November.
  4. Find the least integer \(N\) so that the probability of the first snowfall being on or before the \(N\) th day of November is more than 0.999 .
CAIE FP2 2018 June Q10
10 The times taken to run 400 metres by students at two large colleges \(P\) and \(Q\) are being compared. There is no evidence that the population variances are equal. The time taken by a student at college \(P\) and the time taken by a student at college \(Q\) are denoted by \(x\) seconds and \(y\) seconds respectively. A random sample of 50 students from college \(P\) and a random sample of 60 students from college \(Q\) give the following summarised data. $$\Sigma x = 2620 \quad \Sigma x ^ { 2 } = 138200 \quad \Sigma y = 3060 \quad \Sigma y ^ { 2 } = 157000$$
  1. Using a 10\% significance level, test whether, on average, students from college \(P\) take longer to run 400 metres than students from college \(Q\).
  2. Find a \(90 \%\) confidence interval for the difference in the mean times taken to run 400 metres by students from colleges \(P\) and \(Q\).
CAIE FP2 2018 June Q11 EITHER
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 is held so that the string is taut, with \(O P\) horizontal. The particle is projected downwards with speed \(\sqrt { } \left( \frac { 2 } { 5 } a g \right)\) and begins to move in a vertical circle. The string breaks when its tension is equal to \(\frac { 11 } { 5 } m g\).
  1. Show that the string breaks when \(O P\) makes an angle \(\theta\) with the downward vertical through \(O\), where \(\cos \theta = \frac { 3 } { 5 }\). Find the speed of \(P\) at this instant.
  2. For the subsequent motion after the string breaks, find the distance \(O P\) when the particle \(P\) is vertically below \(O\).
CAIE FP2 2018 June Q11 OR
The regression line of \(y\) on \(x\), obtained from a random sample of 6 pairs of values of \(x\) and \(y\), has equation $$y = 0.25 x + k$$ where \(k\) is a constant. The values from the sample are shown in the following table.
\(x\)45781014
\(y\)58\(p\)7\(p\)9
  1. Find the value of \(p\) and the value of \(k\).
  2. Find the product moment correlation coefficient for the data.
  3. Test, at the \(5 \%\) significance level, whether there is evidence of positive correlation between the variables.
    If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.
CAIE FP2 2018 June Q4
3 marks
4 A uniform \(\operatorname { rod } A B\) has length \(2 a\) and weight \(W\). The end \(A\) rests on rough horizontal ground and the end \(B\) rests against a smooth vertical wall. The rod is in a vertical plane that is perpendicular to the wall. The angle between the rod and the horizontal is \(\theta\). A particle of weight \(5 W\) hangs from the rod at the point \(C\), with \(A C = x a\), where \(0 < x < 1\).
  1. By taking moments about \(A\), show that the magnitude of the normal reaction at \(B\) is \(\frac { W ( 5 x + 1 ) } { 2 \tan \theta }\).
    [0pt] [3]
    The particle of weight \(5 W\) is now moved a distance \(a\) up the rod, so that \(A C = ( x + 1 ) a\). This results in the magnitude of the normal reaction at \(B\) being double its previous value. The system remains in equilibrium with the rod at angle \(\theta\) with the horizontal.
  2. Show that \(x = \frac { 4 } { 5 }\).
    The coefficient of friction between the rod and the ground is \(\frac { 2 } { 3 }\).
  3. Given that the rod is about to slip when the particle of weight \(5 W\) is in its second position, find the value of \(\tan \theta\).
    \begin{figure}[h]
    \captionsetup{labelformat=empty} \caption{Axis \(l\)} \includegraphics[alt={},max width=\textwidth]{c6c8e0fd-6af2-40c9-9513-6581e26e2aec-08_462_693_301_731}
    \end{figure} Three thin uniform rings \(A , B\) and \(C\) are joined together, so that each ring is in contact with each of the other two rings. Ring \(A\) has radius \(2 a\) and mass \(3 M\); rings \(B\) and \(C\) each have radius \(3 a\) and mass \(2 M\). The rings lie in the same plane and the centres of the rings are at the vertices of an isosceles triangle. The object consisting of the three rings is free to rotate about the horizontal axis \(l\) which is tangential to ring \(A\), in the plane of the object and perpendicular to the line of symmetry of the object (see diagram).
CAIE FP2 2018 June Q1
1 A particle \(P\) is moving in a fixed circle of radius 0.8 m . At time \(t \mathrm {~s}\) its velocity is \(\left( t ^ { 2 } - t + 2 \right) \mathrm { m } \mathrm { s } ^ { - 1 }\). Find the magnitudes of the radial and the transverse components of the acceleration of \(P\) when \(t = 2\). Radial component
Transverse component \(\_\_\_\_\)
CAIE FP2 2018 June Q2
2 Two uniform small spheres \(A\) and \(B\) have equal radii and masses \(4 m\) and \(m\) respectively. Sphere \(A\) is moving with speed \(u\) on a smooth horizontal surface when it collides directly with sphere \(B\) which is at rest. The coefficient of restitution between the spheres is \(e\).
  1. Show that after the collision \(A\) moves with speed \(\frac { 1 } { 5 } u ( 4 - e )\) and find the speed of \(B\).
    Sphere \(B\) continues to move until it collides with a fixed smooth vertical barrier which is perpendicular to the direction of motion of \(B\). The coefficient of restitution between \(B\) and the barrier is \(\frac { 3 } { 4 } e\). After this collision, the speeds of \(A\) and \(B\) are equal.
  2. Find the value of \(e\).
    The spheres \(A\) and \(B\) now collide directly again.
  3. Determine whether sphere \(B\) collides with the barrier for a second time.
CAIE FP2 2018 June Q3
3 A particle \(P\) moves on the positive \(x\)-axis in simple harmonic motion. The centre of the motion is a distance \(d \mathrm {~m}\) from the origin \(O\), where \(0 < d < 6.5\). The points \(A\) and \(B\) are on the positive \(x\)-axis, with \(O A = 6.5 \mathrm {~m}\) and \(O B = 7.5 \mathrm {~m}\). The magnitude of the acceleration of \(P\) when it is at \(B\) is twice the magnitude of the acceleration of \(P\) when it is at \(A\).
  1. Find \(d\).
    The period of the motion is \(\pi \mathrm { s }\) and the maximum acceleration of \(P\) during the motion is \(10 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
  2. Find the speed of \(P\) when it is 7 m from \(O\).
  3. Find the time taken by \(P\) to travel directly from \(A\) to \(B\).