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CAIE FP2 2018 June Q4
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 angle between the rod and the horizontal is \(\theta\), where \(\tan \theta = \frac { 4 } { 3 }\). One end of a light inextensible rope is attached to a point \(C\) on the rod. The other end is attached to a point where the vertical wall and the horizontal ground meet. The rope is taut and perpendicular to the rod. The rope and rod are in a vertical plane perpendicular to the wall.
  1. Show that \(A C = \frac { 18 } { 25 } a\).
    The magnitude of the frictional force at \(A\) is equal to one quarter of the magnitude of the normal reaction force at \(A\).
  2. Show that the tension in the rope is \(\frac { 1 } { 4 } W\).
  3. Find expressions, in terms of \(W\), for the magnitudes of the normal reaction forces at \(A\) and \(B\).
CAIE FP2 2018 June Q5
5
\includegraphics[max width=\textwidth, alt={}, center]{1b542910-a57e-4f58-a19f-92e67ee9bdf7-08_323_515_260_813} 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 with the string taut and horizontal. It is projected downwards with speed \(\sqrt { } ( 12 a g )\). At the lowest point of its motion, \(P\) collides directly with a particle \(Q\) of mass \(k m\) which is at rest (see diagram). In the collision, \(P\) and \(Q\) coalesce. The tension in the string immediately after the collision is half of its value immediately before the collision. Find the possible values of \(k\).
CAIE FP2 2018 June Q6
6 A random sample of 15 observations of pairs of values of two variables gives a product moment correlation coefficient of 0.430 .
  1. Test at the \(10 \%\) significance level whether there is evidence of non-zero correlation between the variables.
    A second random sample of \(N\) observations gives a product moment correlation coefficient of 0.615 . Using a 5\% significance level, there is evidence of positive correlation between the variables.
  2. Find the least possible value of \(N\), justifying your answer.
CAIE FP2 2018 June Q7
7 The probability that a driver passes an advanced driving test has a fixed value \(p\) for each attempt. A driver keeps taking the test until he passes. The random variable \(X\) denotes the number of attempts required for the driver to pass. The variance of \(X\) is 3.75 .
  1. Show that \(15 p ^ { 2 } + 4 p - 4 = 0\) and hence find the value of \(p\).
  2. Find \(\mathrm { P } ( X = 5 )\).
  3. Find \(\mathrm { P } ( 3 \leqslant X \leqslant 7 )\).
CAIE FP2 2018 June Q8
8 For a random sample of 6 observations of pairs of values \(( x , y )\), the equation of the regression line of \(y\) on \(x\) is \(y = b x + 1.306\), where \(b\) is a constant. The corresponding equation of the regression line of \(x\) on \(y\) is \(x = 0.6331 y + d\), where \(d\) is a constant. The values of \(x\) from the sample are $$\begin{array} { l l l l l l } 2.3 & 2.8 & 3.7 & p & 6.1 & 6.4 \end{array}$$ and the sum of the values of \(y\) is 46.5 . The product moment correlation coefficient is 0.9797 .
  1. Find the value of \(b\) correct to 3 decimal places.
  2. Find the value of \(p\).
  3. Use the equation of the regression line of \(x\) on \(y\) to estimate the value of \(x\) when \(y = 8.5\).
CAIE FP2 2018 June Q9
9 The continuous random variable \(X\) has probability density function given by $$f ( x ) = \begin{cases} \frac { 1 } { 20 } \left( 3 - \frac { 1 } { \sqrt { } x } \right) & 1 \leqslant x \leqslant 9
0 & \text { otherwise } \end{cases}$$ The random variable \(Y\) is defined by \(Y = \sqrt { } X\).
  1. Show that the probability density function of \(Y\) is given by $$\operatorname { g } ( y ) = \begin{cases} \frac { 1 } { 10 } ( 3 y - 1 ) & 1 \leqslant y \leqslant 3
    0 & \text { otherwise } \end{cases}$$
  2. Find the mean value of \(Y\).
CAIE FP2 2018 June Q10
10 During the summer months, all members of a large swimming club take part in intensive training. The times taken to swim 50 metres at the beginning of the summer and at the end of the summer are recorded for each member of the club. The time taken, in seconds, at the beginning of the summer is denoted by \(x\) and the time taken at the end of the summer is denoted by \(y\). For a random sample of 9 members the results are shown in the following table.
Member\(A\)\(B\)\(C\)\(D\)\(E\)\(F\)\(G\)\(H\)\(I\)
\(x\)38.540.232.335.136.241.432.038.238.2
\(y\)37.438.131.634.734.238.631.836.336.8
The swimming coach believes that, on average, the time taken by a swimmer to swim 50 metres will decrease by more than one second as a result of the intensive training.
  1. Stating suitable hypotheses and assuming a normal distribution, test the coach's belief at the \(10 \%\) significance level.
  2. Find a 95\% confidence interval for the population mean time taken to swim 50 metres after the intensive training, assuming a normal distribution.
CAIE FP2 2018 June Q11 EITHER
\includegraphics[max width=\textwidth, alt={}]{1b542910-a57e-4f58-a19f-92e67ee9bdf7-18_588_609_438_769}
An object is formed from a square frame \(A B C D\) with a square lamina attached in one corner of the frame. The frame consists of four identical thin rods, each of mass \(M\) and length \(2 a\). The lamina has mass \(k M\) and edges of length \(a\). It has one vertex at \(C\) and adjacent sides in contact with \(C B\) and \(C D\) (see diagram).
  1. Show that the moment of inertia of the object about an axis \(l\) through \(A\) perpendicular to the plane of the object is \(\frac { 2 } { 3 } M a ^ { 2 } ( 7 k + 20 )\).
    The object is released from rest with the edge \(A B\) horizontal and \(D\) vertically above \(A\). The object rotates freely about the fixed axis \(l\). The angular speed of the object is \(\frac { 1 } { 2 } \sqrt { } \left( \frac { 5 g } { a } \right)\) when \(D\) is first vertically below \(A\).
  2. Find the value of \(k\).
CAIE FP2 2018 June Q11 OR
A scientist carries out an experiment to investigate the quantity \(X\), which takes the values \(0,1,2,3,4\), 5 or 6 . He believes that the values taken by \(X\) follow a binomial distribution. He conducts 250 trials. His results are summarised in the following table.
\(x\)0123456
Observed frequency228372531730
  1. Show that unbiased estimates of the mean and variance for these results are 1.876 and 1.266 respectively, correct to 3 decimal places. By evaluating the mean and variance of the distribution B(6, 0.313), explain why \(X\) could have this distribution.
    The expected frequencies corresponding to the distribution \(\mathrm { B } ( 6,0.313 )\) are shown in the following table.
    \(x\)0123456
    Observed frequency228372531730
    Expected frequency26.371.981.849.717.03.10.2
  2. Show how the expected frequency for \(x = 4\) is calculated.
  3. Test at the \(5 \%\) significance level whether the scientist's belief is correct.
    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 2019 June Q1
1 A particle \(P\) moves along an arc of a circle with centre \(O\) and radius 2 m . At time \(t\) seconds, the angle POA is \(\theta\), where \(\theta = 1 - \cos 2 t\), and \(A\) is a fixed point on the arc of the circle.
  1. Show that the magnitude of the radial component of the acceleration of \(P\) when \(t = \frac { 1 } { 6 } \pi\) is \(6 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
    \includegraphics[max width=\textwidth, alt={}, center]{2aaf3493-6509-4668-91a2-9f4708bbbb58-03_65_1573_488_324}
  2. Find the magnitude of the transverse component of the acceleration of \(P\) when \(t = \frac { 1 } { 6 } \pi\).
CAIE FP2 2019 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 \(O\) such that \(O A = 3.5 \mathrm {~m}\) and \(O B = 1 \mathrm {~m}\). The speed of \(P\) when it is at \(B\) is twice its speed when it is at \(A\). The maximum acceleration of \(P\) is \(1 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
  1. Find the speed of \(P\) when it is at \(O\).
    \includegraphics[max width=\textwidth, alt={}, center]{2aaf3493-6509-4668-91a2-9f4708bbbb58-04_64_1566_492_328}
  2. Find the time taken by \(P\) to travel directly from \(A\) to \(B\).
CAIE FP2 2019 June Q3
3 Three uniform small spheres \(A , B\) and \(C\) have equal radii and masses \(2 m , 4 m\) and \(m\) respectively. The spheres are moving in a straight line on a smooth horizontal surface, with \(B\) between \(A\) and \(C\). The coefficient of restitution between each pair of spheres is \(e\). Spheres \(A\) and \(B\) are moving towards each other with speeds \(2 u\) and \(u\) respectively. The first collision is between \(A\) and \(B\).
  1. Find the velocities of \(A\) and \(B\) after this collision.
    Sphere \(C\) is moving towards \(B\) with speed \(\frac { 4 } { 3 } u\) and now collides with it. As a result of this collision, \(B\) is brought to rest.
  2. Find the value of \(e\).
  3. Find the total kinetic energy lost by the three spheres as a result of the two collisions.
CAIE FP2 2019 June Q4
4
\includegraphics[max width=\textwidth, alt={}, center]{2aaf3493-6509-4668-91a2-9f4708bbbb58-08_677_812_258_664} A uniform rod \(A B\) of length \(4 a\) and weight \(W\) rests with the end \(A\) in contact with a rough vertical wall. A light inextensible string of length \(\frac { 5 } { 2 } a\) has one end attached to the point \(C\) on the rod, where \(A C = \frac { 5 } { 2 } a\). The other end of the string is attached to a point \(D\) on the wall, vertically above \(A\). The vertical plane containing the \(\operatorname { rod } A B\) is perpendicular to the wall. The angle between the rod and the wall is \(\theta\), where \(\tan \theta = 2\) (see diagram). The end \(A\) of the rod is on the point of slipping down the wall and the coefficient of friction between the rod and the wall is \(\mu\). Find, in either order, the tension in the string and the value of \(\mu\).
CAIE FP2 2019 June Q5
5
\includegraphics[max width=\textwidth, alt={}, center]{2aaf3493-6509-4668-91a2-9f4708bbbb58-10_809_778_258_680} A thin uniform \(\operatorname { rod } A B\) has mass \(k M\) and length \(2 a\). The end \(A\) of the rod is rigidly attached to the surface of a uniform hollow sphere with centre \(O\), mass \(k M\) and radius \(2 a\). The end \(B\) of the rod is rigidly attached to the circumference of a uniform ring with centre \(C\), mass \(M\) and radius \(a\). The points \(C , B , A , O\) lie in a straight line. The horizontal axis \(L\) passes through the mid-point of the rod and is perpendicular to the rod and in the plane of the ring (see diagram). The object consisting of the rod, the ring and the hollow sphere can rotate freely about \(L\).
  1. Show that the moment of inertia of the object about \(L\) is \(\frac { 3 } { 2 } ( 8 k + 3 ) M a ^ { 2 }\).
    The object performs small oscillations about \(L\), with the ring above the sphere as shown in the diagram.
  2. Find the set of possible values of \(k\) and the period of these oscillations in terms of \(k\).
CAIE FP2 2019 June Q6
6 A fair six-sided die is thrown until a 3 or a 4 is obtained. The number of throws taken is denoted by the random variable \(X\).
  1. State the mean value of \(X\).
  2. Find the probability that obtaining a 3 or a 4 takes exactly 6 throws.
  3. Find the probability that obtaining a 3 or a 4 takes more than 4 throws.
  4. Find the greatest integer \(n\) such that the probability of obtaining a 3 or a 4 in fewer than \(n\) throws is less than 0.95.
CAIE FP2 2019 June Q7
7 The continuous random variable \(X\) has probability density function f given by $$f ( x ) = \begin{cases} \frac { 3 } { 4 x ^ { 2 } } + \frac { 1 } { 4 } & 1 \leqslant x \leqslant 3
0 & \text { otherwise } \end{cases}$$
  1. Find the distribution function of \(X\).
  2. Find the exact value of the interquartile range of \(X\).
CAIE FP2 2019 June Q8
8 marks
8 A large number of runners are attending a summer training camp. A random sample of 6 runners is chosen and their times to run 1500 m at the beginning of the camp and at the end of the camp are recorded. Their times, in minutes, are shown in the following table.
Runner\(A\)\(B\)\(C\)\(D\)\(E\)\(F\)
Time at beginning of camp3.823.623.553.713.753.92
Time at end of camp3.723.553.523.683.543.73
The organiser of the training camp claims that a runner's time will improve by more than 0.05 minutes between the beginning and end of the camp. Assuming that differences in time over the two runs are normally distributed, test at the \(10 \%\) significance level whether the organiser's claim is justified. [8]
CAIE FP2 2019 June Q9
9 A random sample of 50 observations of the continuous random variable \(X\) was taken and the values are summarised in the following table.
Interval\(0 \leqslant x < 0.8\)\(0.8 \leqslant x < 1.6\)\(1.6 \leqslant x < 2.4\)\(2.4 \leqslant x < 3.2\)\(3.2 \leqslant x < 4\)
Observed frequency1816862
It is required to test the goodness of fit of the distribution with probability density function \(f\) given by $$f ( x ) = \begin{cases} \frac { 3 } { 16 } ( 4 - x ) ^ { \frac { 1 } { 2 } } & 0 \leqslant x < 4
0 & \text { otherwise. } \end{cases}$$ The relevant expected frequencies, correct to 2 decimal places, are given in the following table.
Interval\(0 \leqslant x < 0.8\)\(0.8 \leqslant x < 1.6\)\(1.6 \leqslant x < 2.4\)\(2.4 \leqslant x < 3.2\)\(3.2 \leqslant x < 4\)
Expected frequency14.2212.5410.598.184.47
  1. Show how the expected frequency for \(1.6 \leqslant x < 2.4\) is obtained.
  2. Carry out a goodness of fit test at the \(5 \%\) significance level.
CAIE FP2 2019 June Q10
10 The values from a random sample of five pairs \(( x , y )\) taken from a bivariate distribution are shown below.
\(x\)34468
\(y\)57\(q\)67
The equation of the regression line of \(x\) on \(y\) is given by \(x = \frac { 5 } { 4 } y + c\).
  1. Given that \(q\) is an integer, find its value.
  2. Find the value of \(c\).
  3. Find the value of the product moment correlation coefficient.
CAIE FP2 2019 June Q11 EITHER
A particle \(P\), of mass \(m\), is able to move in a vertical circle on the smooth inner surface of a sphere with centre \(O\) and radius \(a\). Points \(A\) and \(B\) are on the inner surface of the sphere and \(A O B\) is a horizontal diameter. Initially, \(P\) is projected vertically downwards with speed \(\sqrt { } \left( \frac { 21 } { 2 } a g \right)\) from \(A\) and begins to move in a vertical circle. At the lowest point of its path, vertically below \(O\), the particle \(P\) collides with a stationary particle \(Q\), of mass \(4 m\), and rebounds. The speed acquired by \(Q\), as a result of the collision, is just sufficient for it to reach the point \(B\).
  1. Find the speed of \(P\) and the speed of \(Q\) immediately after their collision.
    In its subsequent motion, \(P\) loses contact with the inner surface of the sphere at the point \(D\), where the angle between \(O D\) and the upward vertical through \(O\) is \(\theta\).
  2. Find \(\cos \theta\).
CAIE FP2 2019 June Q11 OR
A farmer grows two different types of cherries, Type \(A\) and Type \(B\). He assumes that the masses of each type are normally distributed. He chooses a random sample of 8 cherries of Type \(A\). He finds that the sample mean mass is 15.1 g and that a \(95 \%\) confidence interval for the population mean mass, \(\mu \mathrm { g }\), is \(13.5 \leqslant \mu \leqslant 16.7\).
  1. Find an unbiased estimate for the population variance of the masses of cherries of Type \(A\).
    The farmer now chooses a random sample of 6 cherries of Type \(B\) and records their masses as follows.
    12.2
    13.3
    13.9
    14.0
    15.4
    16.4
  2. Test at the \(5 \%\) significance level whether the mean mass of cherries of Type \(B\) is less than the mean mass of cherries of Type \(A\). You should assume that the population variances for the two types of cherry are equal.
    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 2019 June Q1
1 A particle \(P\) moves along an arc of a circle with centre \(O\) and radius 2 m . At time \(t\) seconds, the angle POA is \(\theta\), where \(\theta = 1 - \cos 2 t\), and \(A\) is a fixed point on the arc of the circle.
  1. Show that the magnitude of the radial component of the acceleration of \(P\) when \(t = \frac { 1 } { 6 } \pi\) is \(6 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
    \includegraphics[max width=\textwidth, alt={}, center]{bbb7dbf7-1322-42d4-a0af-3850a4ea95ac-03_65_1573_488_324}
  2. Find the magnitude of the transverse component of the acceleration of \(P\) when \(t = \frac { 1 } { 6 } \pi\).
CAIE FP2 2019 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 \(O\) such that \(O A = 3.5 \mathrm {~m}\) and \(O B = 1 \mathrm {~m}\). The speed of \(P\) when it is at \(B\) is twice its speed when it is at \(A\). The maximum acceleration of \(P\) is \(1 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
  1. Find the speed of \(P\) when it is at \(O\).
    \includegraphics[max width=\textwidth, alt={}, center]{bbb7dbf7-1322-42d4-a0af-3850a4ea95ac-04_64_1566_492_328}
  2. Find the time taken by \(P\) to travel directly from \(A\) to \(B\).
CAIE FP2 2019 June Q4
4
\includegraphics[max width=\textwidth, alt={}, center]{bbb7dbf7-1322-42d4-a0af-3850a4ea95ac-08_677_812_258_664} A uniform rod \(A B\) of length \(4 a\) and weight \(W\) rests with the end \(A\) in contact with a rough vertical wall. A light inextensible string of length \(\frac { 5 } { 2 } a\) has one end attached to the point \(C\) on the rod, where \(A C = \frac { 5 } { 2 } a\). The other end of the string is attached to a point \(D\) on the wall, vertically above \(A\). The vertical plane containing the \(\operatorname { rod } A B\) is perpendicular to the wall. The angle between the rod and the wall is \(\theta\), where \(\tan \theta = 2\) (see diagram). The end \(A\) of the rod is on the point of slipping down the wall and the coefficient of friction between the rod and the wall is \(\mu\). Find, in either order, the tension in the string and the value of \(\mu\).
CAIE FP2 2019 June Q5
5
\includegraphics[max width=\textwidth, alt={}, center]{bbb7dbf7-1322-42d4-a0af-3850a4ea95ac-10_809_778_258_680} A thin uniform \(\operatorname { rod } A B\) has mass \(k M\) and length \(2 a\). The end \(A\) of the rod is rigidly attached to the surface of a uniform hollow sphere with centre \(O\), mass \(k M\) and radius \(2 a\). The end \(B\) of the rod is rigidly attached to the circumference of a uniform ring with centre \(C\), mass \(M\) and radius \(a\). The points \(C , B , A , O\) lie in a straight line. The horizontal axis \(L\) passes through the mid-point of the rod and is perpendicular to the rod and in the plane of the ring (see diagram). The object consisting of the rod, the ring and the hollow sphere can rotate freely about \(L\).
  1. Show that the moment of inertia of the object about \(L\) is \(\frac { 3 } { 2 } ( 8 k + 3 ) M a ^ { 2 }\).
    The object performs small oscillations about \(L\), with the ring above the sphere as shown in the diagram.
  2. Find the set of possible values of \(k\) and the period of these oscillations in terms of \(k\).