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CAIE FP2 2015 June Q10 OR
The times taken, in hours, by cyclists from two different clubs, \(A\) and \(B\), to complete a 50 km time trial are being compared. The times taken by a cyclist from club \(A\) and by a cyclist from club \(B\) are denoted by \(t _ { A }\) and \(t _ { B }\) respectively. A random sample of 50 cyclists from \(A\) and a random sample of 60 cyclists from \(B\) give the following summarised data. $$\Sigma t _ { A } = 102.0 \quad \Sigma t _ { A } ^ { 2 } = 215.18 \quad \Sigma t _ { B } = 129.0 \quad \Sigma t _ { B } ^ { 2 } = 282.3$$ Using a 5\% significance level, test whether, on average, cyclists from club \(A\) take less time to complete the time trial than cyclists from club \(B\). A test at the \(\alpha \%\) significance level shows that there is evidence that the population mean time for cyclists from club \(B\) exceeds the population mean time for cyclists from club \(A\) by more than 0.05 hours. Find the set of possible values of \(\alpha\).
CAIE FP2 2016 June Q1
1 A bullet of mass 0.01 kg is fired horizontally into a fixed vertical barrier which exerts a constant resisting force of magnitude 1000 N . The bullet enters the barrier with speed \(320 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and emerges with speed \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). You may assume that the motion takes place in a horizontal straight line. Find
  1. the magnitude of the impulse that acts on the bullet,
  2. the thickness of the barrier,
  3. the time taken for the bullet to pass through the barrier.
CAIE FP2 2016 June Q2
2 A small smooth sphere \(A\) of mass \(m\) is moving with speed \(u\) on a smooth horizontal surface when it collides directly with an identical sphere \(B\) which is initially at rest on the surface. The coefficient of restitution between the spheres is \(e\). Sphere \(B\) subsequently collides with a fixed vertical barrier which is perpendicular to the direction of motion of \(B\). The coefficient of restitution between \(B\) and the barrier is \(\frac { 1 } { 2 }\). Given that \(80 \%\) of the initial kinetic energy is lost as a result of the two collisions, find the value of \(e\).
CAIE FP2 2016 June Q3
3 A particle \(P\) is performing simple harmonic motion with amplitude 0.25 m . During each complete oscillation, \(P\) moves with a speed that is less than or equal to half of its maximum speed for \(\frac { 4 } { 3 }\) seconds. Find the period of the motion and the maximum speed of \(P\).
CAIE FP2 2016 June Q4
4 A particle \(P\) is at rest at the lowest point on the smooth inner surface of a hollow sphere with centre \(O\) and radius \(a\). The particle is projected horizontally with speed \(u\) and begins to move in a vertical circle on the inner surface of the sphere. The particle loses contact with the sphere at the point \(A\), where \(O A\) makes an angle \(\theta\) with the upward vertical through \(O\). Given that the speed of \(P\) at \(A\) is \(\sqrt { } \left( \frac { 3 } { 5 } a g \right)\), find \(u\) in terms of \(a\) and \(g\). Find, in terms of \(a\), the greatest height above the level of \(O\) achieved by \(P\) in its subsequent motion. (You may assume that \(P\) achieves its greatest height before it makes any further contact with the sphere.)
CAIE FP2 2016 June Q5
5
\includegraphics[max width=\textwidth, alt={}, center]{3e224c82-68df-427e-a59b-7dc2bfd716a2-3_727_517_258_813} A thin uniform \(\operatorname { rod } A B\) has mass \(\frac { 3 } { 4 } m\) and length \(3 a\). The end \(A\) of the rod is rigidly attached to a point on the circumference of a uniform disc with centre \(C\), mass \(m\) and radius \(a\). The end \(B\) of the rod is rigidly attached to a point on the circumference of a uniform disc with centre \(D\), mass \(4 m\) and radius \(2 a\). The discs and the rod are in the same plane and \(C A B D\) is a straight line. The mid-point of \(C D\) is \(O\). The object consisting of the two discs and the rod is free to rotate about a fixed smooth horizontal axis \(l\), through \(O\) in the plane of the object and perpendicular to the rod (see diagram). Show that the moment of inertia of the object about \(l\) is \(50 m a ^ { 2 }\). The object hangs in equilibrium with \(D\) vertically below \(C\). It is displaced through a small angle and released from rest, so that it makes small oscillations about the horizontal axis \(l\). Show that it will move in approximate simple harmonic motion and state the period of the motion.
CAIE FP2 2016 June Q6
6 The score when two fair dice are thrown is the sum of the two numbers on the upper faces. Two fair dice are thrown repeatedly until a score of 6 is obtained. The number of throws taken is denoted by the random variable \(X\). Find the mean of \(X\). Find the least integer \(N\) such that the probability of obtaining a score of 6 in fewer than \(N\) throws is more than 0.95 .
CAIE FP2 2016 June Q7
7 A random sample of 9 observations of a normal variable \(X\) is taken. The results are summarised as follows. $$\Sigma x = 24.6 \quad \Sigma x ^ { 2 } = 68.5$$ Test, at the \(5 \%\) significance level, whether the population mean is greater than 2.5.
CAIE FP2 2016 June Q8
8 The random variable \(X\) has probability density function f given by $$\mathrm { f } ( x ) = \begin{cases} 2 \mathrm { e } ^ { - 2 x } & x \geqslant 0
0 & \text { otherwise } \end{cases}$$
  1. Find the distribution function of \(X\).
  2. Find the median value of \(X\). The random variable \(Y\) is defined by \(Y = \mathrm { e } ^ { X }\).
  3. Find the probability density function of \(Y\).
CAIE FP2 2016 June Q9
9 Applicants for a national teacher training course are required to pass a mathematics test. Each year, the applicants are tested in groups of 6 and the number of successful applicants in each group is recorded. The overall proportion of successful applicants has remained constant over the years and is equal to \(60 \%\) of the applicants. The results from 150 randomly chosen groups are shown in the following table.
Number of successful applicants0123456
Number of groups13255138302
Test, at the \(5 \%\) significance level, the goodness of fit of the distribution \(\mathbf { B } ( 6,0.6 )\) for the number of successful applicants in a group.
CAIE FP2 2016 June Q10
10 For a random sample of 6 observations of pairs of values \(( x , y )\), where \(0 < x < 21\) and \(0 < y < 14\), the following results are obtained. $$\Sigma x ^ { 2 } = 844.20 \quad \Sigma y ^ { 2 } = 481.50 \quad \Sigma x y = 625.59$$ It is also found that the variance of the \(x\)-values is 36.66 and the variance of the \(y\)-values is 9.69 .
  1. Find the product moment correlation coefficient for the sample.
  2. Find the equations of the regression lines of \(y\) on \(x\) and \(x\) on \(y\).
  3. Use the appropriate regression line to estimate the value of \(x\) when \(y = 6.4\) and comment on the reliability of your estimate.
CAIE FP2 2016 June Q11 EITHER
\includegraphics[max width=\textwidth, alt={}]{3e224c82-68df-427e-a59b-7dc2bfd716a2-5_732_609_431_769}
The end \(A\) of a uniform rod \(A B\), of length \(2 a\) and weight \(W\), is freely hinged to a vertical wall. The end \(B\) of the rod is attached to a light elastic string of natural length \(\frac { 3 } { 2 } a\) and modulus of elasticity \(3 W\). The other end of the string is attached to the point \(C\) on the wall, where \(C\) is vertically above \(A\) and \(A C = 2 a\). A particle of weight \(2 W\) is attached to the rod at the point \(D\), where \(D B = \frac { 1 } { 2 } a\). The angle \(A B C\) is equal to \(\theta\) (see diagram). Show that \(\cos \theta = \frac { 3 } { 4 }\) and find the tension in the string in terms of \(W\). Find the magnitude of the reaction force at the hinge.
CAIE FP2 2016 June Q11 OR
Petra is studying a particular species of bird. She takes a random sample of 12 birds from nature reserve \(A\) and measures the wing span, \(x \mathrm {~cm}\), for each bird. She then calculates a \(95 \%\) confidence interval for the population mean wing span, \(\mu \mathrm { cm }\), for birds of this species, assuming that wing spans are normally distributed. Later, she is not able to find the summary of the results for the sample, but she knows that the \(95 \%\) confidence interval is \(25.17 \leqslant \mu \leqslant 26.83\). Find the values of \(\sum x\) and \(\sum x ^ { 2 }\) for this sample. Petra also measures the wing spans of a random sample of 7 birds from nature reserve \(B\). Their wing spans, \(y \mathrm {~cm}\), are as follows. $$\begin{array} { l l l l l l l } 23.2 & 22.4 & 27.6 & 25.3 & 28.4 & 26.5 & 23.6 \end{array}$$ She believes that the mean wing span of birds found in nature reserve \(A\) is greater than the mean wing span of birds found in nature reserve \(B\). Assuming that this second sample also comes from a normal distribution, with variance the same as the first distribution, test, at the \(10 \%\) significance level, whether there is evidence to support Petra's belief.
CAIE FP2 2017 June Q1
3 marks
1 A bullet of mass 0.08 kg is fired horizontally into a fixed vertical barrier. It enters the barrier horizontally with speed \(300 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and emerges horizontally after 0.02 s . There is a constant horizontal resisting force of magnitude 1000 N . Find the speed with which the bullet emerges from the barrier.
[0pt] [3]
\includegraphics[max width=\textwidth, alt={}, center]{1dba0ab0-f3a4-4e7e-a67a-00fd37223cc7-04_748_561_260_794} A uniform smooth disc with centre \(O\) and radius \(a\) is fixed at the point \(D\) on a horizontal surface. A uniform rod of length \(3 a\) and weight \(W\) rests on the disc with its end \(A\) in contact with a rough vertical wall. The rod and the disc lie in a vertical plane that is perpendicular to the wall. The wall meets the horizontal surface at the point \(E\) such that \(A E = a\) and \(E D = \frac { 5 } { 4 } a\). A particle of weight \(k W\) is hung from the rod at \(B\) (see diagram). The coefficient of friction between the rod and the wall is \(\frac { 1 } { 8 }\) and the system is in limiting equilibrium. Find the value of \(k\).
CAIE FP2 2017 June Q3
3 Two uniform small smooth spheres \(A\) and \(B\) have equal radii and masses \(3 m\) 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. Find, in terms of \(u\) and \(e\), expressions for the velocities of \(A\) and \(B\) after the collision.
    Sphere \(B\) continues to move until it strikes 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 \(\frac { 3 } { 4 }\). When the spheres subsequently collide, \(A\) is brought to rest.
  2. Find the value of \(e\).
    \includegraphics[max width=\textwidth, alt={}, center]{1dba0ab0-f3a4-4e7e-a67a-00fd37223cc7-08_608_652_258_744} Three identical uniform discs, \(A , B\) and \(C\), each have mass \(m\) and radius \(a\). They are joined together by uniform rods, each of which has mass \(\frac { 1 } { 3 } m\) and length \(2 a\). The discs lie in the same plane and their centres form the vertices of an equilateral triangle of side \(4 a\). Each rod has one end rigidly attached to the circumference of a disc and the other end rigidly attached to the circumference of an adjacent disc, so that the rod lies along the line joining the centres of the two discs (see diagram).
CAIE FP2 2017 June Q5
5
\includegraphics[max width=\textwidth, alt={}, center]{1dba0ab0-f3a4-4e7e-a67a-00fd37223cc7-10_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 Q6
6 A fair die is thrown repeatedly until a 6 is obtained.
  1. Find the probability that obtaining a 6 takes no more than four throws.
  2. Find the least integer \(N\) such that the probability of obtaining a 6 before the \(N\) th throw is more than 0.95.
CAIE FP2 2017 June Q7
7 A farmer grows a particular type of fruit tree. On average, the mass of fruit produced per tree has been 6.2 kg . He has developed a new kind of soil and claims that the mean mass of fruit produced per tree when growing in this new soil has increased. A random sample of 10 trees grown in the new soil is chosen. The masses, \(x \mathrm {~kg}\), of fruit produced are summarised as follows. $$\Sigma x = 72.0 \quad \Sigma x ^ { 2 } = 542.0$$ Test at the \(5 \%\) significance level whether the farmer's claim is justified, assuming a normal distribution.
CAIE FP2 2017 June Q8
8 The continuous random variable \(X\) has probability density function f given by $$f ( x ) = \begin{cases} \frac { 1 } { 4 } ( x - 1 ) & 2 \leqslant x \leqslant 4
0 & \text { otherwise } \end{cases}$$
  1. Find the distribution function of \(X\).
    The random variable \(Y\) is defined by \(Y = ( X - 1 ) ^ { 3 }\).
  2. Find the probability density function of \(Y\).
  3. Find the median value of \(Y\).
CAIE FP2 2017 June Q9
10 marks
9 Two fish farmers \(X\) and \(Y\) produce a particular type of fish. Farmer \(X\) chooses a random sample of 8 of his fish and records the masses, \(x \mathrm {~kg}\), as follows. $$\begin{array} { l l l l l l l l } 1.2 & 1.4 & 0.8 & 2.1 & 1.8 & 2.6 & 1.5 & 2.0 \end{array}$$ Farmer \(Y\) chooses a random sample of 10 of his fish and summarises the masses, \(y \mathrm {~kg}\), as follows. $$\Sigma y = 20.2 \quad \Sigma y ^ { 2 } = 44.6$$ You should assume that both distributions are normal with equal variances. Test at the \(10 \%\) significance level whether the mean mass of fish produced by farmer \(X\) differs from the mean mass of fish produced by farmer \(Y\).
[0pt] [10]
CAIE FP2 2017 June Q10
10 A random sample of 5 pairs of values \(( x , y )\) is given in the following table.
\(x\)12458
\(y\)75864
  1. Find, showing all necessary working, the equation of the regression line of \(y\) on \(x\).
  2. Find, showing all necessary working, the value of the product moment correlation coefficient for this sample.
  3. Test, at the \(10 \%\) significance level, whether there is evidence of non-zero correlation between the variables.
CAIE FP2 2017 June Q11 EITHER
A particle \(P\) of mass \(3 m\) is attached to one end of a light elastic spring of natural length \(a\) and modulus of elasticity \(k m g\). The other end of the spring is attached to a fixed point \(O\) on a smooth plane that is inclined to the horizontal at an angle \(\alpha\), where \(\sin \alpha = \frac { 2 } { 3 }\). The system rests in equilibrium with \(P\) on the plane at the point \(E\). The length of the spring in this position is \(\frac { 5 } { 4 } a\).
  1. Find the value of \(k\).
    The particle \(P\) is now replaced by a particle \(Q\) of mass \(2 m\) and \(Q\) is released from rest at the point \(E\).
  2. Show that, in the resulting motion, \(Q\) performs simple harmonic motion. State the centre and the period of the motion.
  3. Find the least tension in the spring and the maximum acceleration of \(Q\) during the motion.
CAIE FP2 2017 June Q11 OR
A shop is supplied with large quantities of plant pots in packs of six. These pots can be damaged easily if they are not packed carefully. The manager of the shop is a statistician and he believes that the number of damaged pots in a pack of six has a binomial distribution. He chooses a random sample of 250 packs and records the numbers of damaged pots per pack. His results are shown in the following table.
Number of damaged
pots per pack \(( x )\)
0123456
Frequency486978322210
  1. Show that the mean number of damaged pots per pack in this sample is 1.656 .
    The following table shows some of the expected frequencies, correct to 2 decimal places, using an appropriate binomial distribution.
    Number of damaged
    pots per pack \(( x )\)
    		0123456
    Expected frequency36.0182.36\(a\)39.89\(b\)1.740.11
  2. Find the values of \(a\) and \(b\), correct to 2 decimal places
  3. Use a goodness-of-fit test at the \(1 \%\) significance level to determine whether the manager's belief is justified.
CAIE FP2 2017 June Q1
3 marks
1 A bullet of mass 0.08 kg is fired horizontally into a fixed vertical barrier. It enters the barrier horizontally with speed \(300 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and emerges horizontally after 0.02 s . There is a constant horizontal resisting force of magnitude 1000 N . Find the speed with which the bullet emerges from the barrier.
[0pt] [3]
\includegraphics[max width=\textwidth, alt={}, center]{c437c752-5518-4185-b02f-74206dc4b13c-04_748_561_260_794} A uniform smooth disc with centre \(O\) and radius \(a\) is fixed at the point \(D\) on a horizontal surface. A uniform rod of length \(3 a\) and weight \(W\) rests on the disc with its end \(A\) in contact with a rough vertical wall. The rod and the disc lie in a vertical plane that is perpendicular to the wall. The wall meets the horizontal surface at the point \(E\) such that \(A E = a\) and \(E D = \frac { 5 } { 4 } a\). A particle of weight \(k W\) is hung from the rod at \(B\) (see diagram). The coefficient of friction between the rod and the wall is \(\frac { 1 } { 8 }\) and the system is in limiting equilibrium. Find the value of \(k\).
CAIE FP2 2017 June Q3
3 Two uniform small smooth spheres \(A\) and \(B\) have equal radii and masses \(3 m\) 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. Find, in terms of \(u\) and \(e\), expressions for the velocities of \(A\) and \(B\) after the collision.
    Sphere \(B\) continues to move until it strikes 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 \(\frac { 3 } { 4 }\). When the spheres subsequently collide, \(A\) is brought to rest.
  2. Find the value of \(e\).
    \includegraphics[max width=\textwidth, alt={}, center]{c437c752-5518-4185-b02f-74206dc4b13c-08_608_652_258_744} Three identical uniform discs, \(A , B\) and \(C\), each have mass \(m\) and radius \(a\). They are joined together by uniform rods, each of which has mass \(\frac { 1 } { 3 } m\) and length \(2 a\). The discs lie in the same plane and their centres form the vertices of an equilateral triangle of side \(4 a\). Each rod has one end rigidly attached to the circumference of a disc and the other end rigidly attached to the circumference of an adjacent disc, so that the rod lies along the line joining the centres of the two discs (see diagram).