Questions FP2 (1157 questions)

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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).
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.