Questions — CAIE (7279 questions)

Browse by module
All questions AEA AS Paper 1 AS Paper 2 AS Pure C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Extra Pure Further Mechanics Further Mechanics A AS Further Mechanics AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics Further Paper 4 Further Pure Core Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Pure with Technology Further Statistics Further Statistics A AS Further Statistics AS Further Statistics B AS Further Statistics Major Further Statistics Minor Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 H240/01 H240/02 H240/03 M1 M2 M3 M4 M5 Mechanics 1 P1 P2 P3 P4 PMT Mocks PURE Paper 1 Paper 2 Paper 3 Pure 1 S1 S2 S3 S4 SPS ASFM SPS ASFM Mechanics SPS ASFM Pure SPS ASFM Statistics SPS FM SPS FM Mechanics SPS FM Pure SPS FM Statistics SPS SM SPS SM Mechanics SPS SM Pure SPS SM Statistics Stats 1 Unit 1 Unit 2 Unit 3 Unit 4
Browse by board
AQA AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Paper 1 Further Paper 2 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics M1 M2 M3 Paper 1 Paper 2 Paper 3 S1 S2 S3 CAIE FP1 FP2 Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 4 M1 M2 P1 P2 P3 S1 S2 Edexcel AEA AS Paper 1 AS Paper 2 C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS M1 M2 M3 M4 M5 P1 P2 P3 P4 PMT Mocks Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 OCR AS Pure C1 C2 C3 C4 D1 D2 FD1 AS FM1 AS FP1 FP1 AS FP2 FP3 FS1 AS Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Mechanics Further Mechanics AS Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Statistics Further Statistics AS H240/01 H240/02 H240/03 M1 M2 M3 M4 Mechanics 1 PURE Pure 1 S1 S2 S3 S4 Stats 1 OCR MEI AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further Extra Pure Further Mechanics A AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Pure Core Further Pure Core AS Further Pure with Technology Further Statistics A AS Further Statistics B AS Further Statistics Major Further Statistics Minor M1 M2 M3 M4 Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 SPS SPS ASFM SPS ASFM Mechanics SPS ASFM Pure SPS ASFM Statistics SPS FM SPS FM Mechanics SPS FM Pure SPS FM Statistics SPS SM SPS SM Mechanics SPS SM Pure SPS SM Statistics WJEC Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 Unit 1 Unit 2 Unit 3 Unit 4
CAIE FP2 2017 November Q11 OR
A large number of people attended a course to improve the speed of their logical thinking. The times taken to complete a particular type of logic puzzle at the beginning of the course and at the end of the course are recorded for each person. The time taken, in minutes, at the beginning of the course is denoted by \(x\) and the time taken, in minutes, at the end of the course is denoted by \(y\). For a random sample of 9 people, the results are summarised as follows. $$\Sigma x = 45.3 \quad \Sigma x ^ { 2 } = 245.59 \quad \Sigma y = 40.5 \quad \Sigma y ^ { 2 } = 195.11 \quad \Sigma x y = 218.72$$ Ken attended the course, but his time to complete the puzzle at the beginning of the course was not recorded. His time to complete the puzzle at the end of the course was 4.2 minutes.
  1. By finding, showing all necessary working, the equation of a suitable regression line, find an estimate for the time that Ken would have taken to complete the puzzle at the beginning of the course.
    The values of \(x - y\) for the sample of 9 people are as follows. $$\begin{array} { l l l l l l l l l } 0.2 & 0.8 & 0.5 & 1.0 & 0.2 & 0.6 & 0.2 & 0.5 & 0.8 \end{array}$$ The organiser of the course believes that, on average, the time taken to complete the puzzle decreases between the beginning and the end of the course by more than 0.3 minutes.
  2. Stating suitable hypotheses and assuming a normal distribution, test the organiser's belief at the \(2 \frac { 1 } { 2 } \%\) significance level.
CAIE FP2 2017 November Q4
4
\includegraphics[max width=\textwidth, alt={}, center]{1651d08b-b20f-4f2e-9f47-0a1a5d0a839a-06_465_663_262_742} A small ring \(P\) of weight \(W\) is free to slide on a rough horizontal wire, one end of which is attached to a vertical wall at \(Q\). The end \(A\) of a thin uniform \(\operatorname { rod } A B\) of length \(2 a\) and weight \(\frac { 5 } { 2 } W\) is freely hinged to the wall at the point \(A\) which is a distance \(a\) vertically below \(Q\). A light elastic string of natural length \(2 a\) has one end attached to the ring \(P\) and the other end attached to the rod at \(B\). The string is at right angles to the rod and \(A , B , P\) and \(Q\) lie in a vertical plane. The system is in limiting equilibrium with \(A B\) making an angle \(\theta\) with the horizontal, where \(\sin \theta = \frac { 3 } { 5 }\) (see diagram).
  1. Find the tension in the string in terms of \(W\).
  2. Find the coefficient of friction between the ring and the wire.
  3. Find the magnitude of the resultant force on the rod at the hinge in terms of \(W\).
  4. Find the modulus of elasticity of the string in terms of \(W\).
    \includegraphics[max width=\textwidth, alt={}, center]{1651d08b-b20f-4f2e-9f47-0a1a5d0a839a-08_862_698_260_721} A uniform picture frame of mass \(m\) is made by removing a rectangular lamina \(E F G H\) in which \(E F = 4 a\) and \(F G = 2 a\) from a larger rectangular lamina \(A B C D\) in which \(A B = 6 a\) and \(B C = 4 a\). The side \(E F\) is parallel to the side \(A B\). The point of intersection of the diagonals \(A C\) and \(B D\) coincides with the point of intersection of the diagonals \(E G\) and \(F H\). One end of a light inextensible string of length \(10 a\) is attached to \(A\) and the other end is attached to \(B\). The frame is suspended from the mid-point \(O\) of the string. A small object of mass \(\frac { 11 } { 12 } m\) is fixed to the mid-point of \(A B\) (see diagram).
CAIE FP2 2017 November Q11 EITHER
\includegraphics[max width=\textwidth, alt={}]{1651d08b-b20f-4f2e-9f47-0a1a5d0a839a-18_552_588_438_776}
A particle \(P\) of mass \(m\) is free to move on the smooth inner surface of a fixed hollow sphere of radius \(a\). The centre of the sphere is \(O\). The points \(A\) and \(A ^ { \prime }\) are on the inner surface of the sphere, on opposite sides of the vertical through \(O\); the radius \(O A\) makes an angle \(\alpha\) with the downward vertical and the radius \(O A ^ { \prime }\) makes an angle \(\beta\) with the upward vertical. The point \(B\) is on the inner surface of the sphere, vertically below \(O\). The point \(B ^ { \prime }\) is on the inner surface of the sphere and such that \(O B ^ { \prime }\) makes an angle \(2 \beta\) with the upward vertical through \(O\) (see diagram). It is given that \(\cos \alpha = \frac { 1 } { 16 }\).
  1. \(P\) is projected from \(A\) with speed \(u\) along the surface of the sphere downwards towards \(B\). Subsequently it loses contact with the sphere at \(A ^ { \prime }\). Show that \(u ^ { 2 } = \frac { 1 } { 8 } a g ( 1 + 24 \cos \beta )\).
  2. \(P\) is now projected from \(B\) with speed \(u\) along the surface of the sphere towards \(B ^ { \prime }\). Subsequently it loses contact with the sphere at \(B ^ { \prime }\). Find \(\cos \beta\).
  3. In part (i), the reaction of the sphere on \(P\) when it is initially projected at \(A\) is \(R\). Find \(R\) in terms of \(m\) and \(g\).
CAIE FP2 2017 November Q4
4
\includegraphics[max width=\textwidth, alt={}, center]{2ab1a594-6c37-4c78-b53c-33c13bf6eb21-06_465_663_262_742} A small ring \(P\) of weight \(W\) is free to slide on a rough horizontal wire, one end of which is attached to a vertical wall at \(Q\). The end \(A\) of a thin uniform \(\operatorname { rod } A B\) of length \(2 a\) and weight \(\frac { 5 } { 2 } W\) is freely hinged to the wall at the point \(A\) which is a distance \(a\) vertically below \(Q\). A light elastic string of natural length \(2 a\) has one end attached to the ring \(P\) and the other end attached to the rod at \(B\). The string is at right angles to the rod and \(A , B , P\) and \(Q\) lie in a vertical plane. The system is in limiting equilibrium with \(A B\) making an angle \(\theta\) with the horizontal, where \(\sin \theta = \frac { 3 } { 5 }\) (see diagram).
  1. Find the tension in the string in terms of \(W\).
  2. Find the coefficient of friction between the ring and the wire.
  3. Find the magnitude of the resultant force on the rod at the hinge in terms of \(W\).
  4. Find the modulus of elasticity of the string in terms of \(W\).
    \includegraphics[max width=\textwidth, alt={}, center]{2ab1a594-6c37-4c78-b53c-33c13bf6eb21-08_862_698_260_721} A uniform picture frame of mass \(m\) is made by removing a rectangular lamina \(E F G H\) in which \(E F = 4 a\) and \(F G = 2 a\) from a larger rectangular lamina \(A B C D\) in which \(A B = 6 a\) and \(B C = 4 a\). The side \(E F\) is parallel to the side \(A B\). The point of intersection of the diagonals \(A C\) and \(B D\) coincides with the point of intersection of the diagonals \(E G\) and \(F H\). One end of a light inextensible string of length \(10 a\) is attached to \(A\) and the other end is attached to \(B\). The frame is suspended from the mid-point \(O\) of the string. A small object of mass \(\frac { 11 } { 12 } m\) is fixed to the mid-point of \(A B\) (see diagram).
CAIE FP2 2017 November Q11 EITHER
\includegraphics[max width=\textwidth, alt={}]{2ab1a594-6c37-4c78-b53c-33c13bf6eb21-18_552_588_438_776}
A particle \(P\) of mass \(m\) is free to move on the smooth inner surface of a fixed hollow sphere of radius \(a\). The centre of the sphere is \(O\). The points \(A\) and \(A ^ { \prime }\) are on the inner surface of the sphere, on opposite sides of the vertical through \(O\); the radius \(O A\) makes an angle \(\alpha\) with the downward vertical and the radius \(O A ^ { \prime }\) makes an angle \(\beta\) with the upward vertical. The point \(B\) is on the inner surface of the sphere, vertically below \(O\). The point \(B ^ { \prime }\) is on the inner surface of the sphere and such that \(O B ^ { \prime }\) makes an angle \(2 \beta\) with the upward vertical through \(O\) (see diagram). It is given that \(\cos \alpha = \frac { 1 } { 16 }\).
  1. \(P\) is projected from \(A\) with speed \(u\) along the surface of the sphere downwards towards \(B\). Subsequently it loses contact with the sphere at \(A ^ { \prime }\). Show that \(u ^ { 2 } = \frac { 1 } { 8 } a g ( 1 + 24 \cos \beta )\).
  2. \(P\) is now projected from \(B\) with speed \(u\) along the surface of the sphere towards \(B ^ { \prime }\). Subsequently it loses contact with the sphere at \(B ^ { \prime }\). Find \(\cos \beta\).
  3. In part (i), the reaction of the sphere on \(P\) when it is initially projected at \(A\) is \(R\). Find \(R\) in terms of \(m\) and \(g\).
CAIE FP2 2018 November Q1
1 A particle \(P\) oscillates in simple harmonic motion between the points \(A\) and \(B\), where \(A B = 6 \mathrm {~m}\). The period of the motion is \(\frac { 1 } { 2 } \pi \mathrm {~s}\). Find the speed of \(P\) when it is 2 m from \(B\).
CAIE FP2 2018 November Q2
2 Two uniform small smooth spheres \(A\) and \(B\) have equal radii and masses \(5 m\) and \(2 m\) respectively. Sphere \(A\) is moving with speed \(u\) on a smooth horizontal surface when it collides directly with sphere \(B\) which is moving towards it with speed \(2 u\). The coefficient of restitution between the spheres is \(e\).
  1. Show that the speed of \(B\) after the collision is \(\frac { 1 } { 7 } u ( 1 + 15 e )\) and find an expression for the speed of \(A\).
    In the collision, the speed of \(A\) is halved and its direction of motion is reversed.
  2. Find the value of \(e\).
  3. For this collision, find the ratio of the loss of kinetic energy of \(A\) to the loss of kinetic energy of \(B\).
    \includegraphics[max width=\textwidth, alt={}, center]{89a83576-4568-46c8-8872-f59f2397627d-04_630_332_264_900} A uniform disc, of radius \(a\) and mass \(2 M\), is attached to a thin uniform rod \(A B\) of length \(6 a\) and mass \(M\). The rod lies along a diameter of the disc, so that the centre of the disc is a distance \(x\) from \(A\) (see diagram).
  4. Find the moment of inertia of the object, consisting of disc and rod, about a fixed horizontal axis \(l\) through \(A\) and perpendicular to the plane of the disc.
    The object is free to rotate about the axis \(l\). The object is held with \(A B\) horizontal and is released from rest. When \(A B\) makes an angle \(\theta\) with the vertical, where \(\cos \theta = \frac { 3 } { 5 }\), the angular speed of the object is \(\sqrt { } \left( \frac { 2 g } { 5 a } \right)\).
  5. Find the possible values of \(x\).
CAIE FP2 2018 November Q4
4 A uniform rod \(A B\) of length \(4 a\) and weight \(W\) is smoothly hinged to a vertical wall at the end \(A\). The rod is held at an angle \(\theta\) above the horizontal by a light elastic string. One end of the string is attached to the point \(C\) on the rod, where \(A C = 3 a\). The other end of the string is attached to a point \(D\) on the wall, with \(D\) vertically above \(A\) and such that angle \(A C D = 2 \theta\). A particle of weight \(\frac { 1 } { 2 } W\) is attached to the rod at \(B\). It is given that \(\tan \theta = \frac { 8 } { 15 }\).
  1. Show that the tension in the string is \(\frac { 17 } { 12 } W\).
  2. Find the magnitude and direction of the reaction at the hinge.
  3. Given that the natural length of the string is \(2 a\), find its modulus of elasticity.
CAIE FP2 2018 November Q5
5 The fixed points \(A\) and \(B\) are on a smooth horizontal surface with \(A B = 2.6 \mathrm {~m}\). One end of a light elastic spring, of natural length 1.25 m and modulus of elasticity \(\lambda \mathrm { N }\), is attached to \(A\). The other end is attached to a particle \(P\) of mass 0.4 kg . One end of a second light elastic spring, of natural length 1.0 m and modulus of elasticity \(0.6 \lambda \mathrm {~N}\), is attached to \(B\); its other end is attached to \(P\). The system is in equilibrium with \(P\) on the surface at the point \(E\).
  1. Show that \(A E = 1.4 \mathrm {~m}\).
    The particle \(P\) is now displaced slightly from \(E\), along the line \(A B\).
  2. Show that, in the subsequent motion, \(P\) performs simple harmonic motion.
  3. Given that the period of the motion is \(\frac { 1 } { 7 } \pi \mathrm {~s}\), find the value of \(\lambda\).
CAIE FP2 2018 November Q6
6 The continuous random variable \(X\) has probability density function f given by $$f ( x ) = \begin{cases} \frac { 1 } { 80 } \left( 3 \sqrt { } x - \frac { 8 } { \sqrt { } x } \right) & 4 \leqslant x \leqslant 16
0 & \text { otherwise } \end{cases}$$
  1. Find the distribution function of \(X\).
    The random variable \(Y\) is defined by \(Y = \sqrt { } X\).
  2. Find the probability density function of \(Y\).
CAIE FP2 2018 November Q7
7 The random variable \(T\) is the lifetime, in hours, of a particular type of battery. It is given that \(T\) has a negative exponential distribution with mean 500 hours.
  1. Write down the probability density function of \(T\).
  2. Find the probability that a randomly chosen battery of this type has a lifetime of more than 750 hours.
  3. Find the median value of \(T\).
CAIE FP2 2018 November Q8
8 The weekly salaries of employees at two large electronics companies, \(A\) and \(B\), are being compared. The weekly salaries of an employee from company \(A\) and an employee from company \(B\) are denoted by \(
) x\( and \)\\( y\) respectively. A random sample of 50 employees from company \(A\) and a random sample of 40 employees from company \(B\) give the following summarised data. $$\Sigma x = 5120 \quad \Sigma x ^ { 2 } = 531000 \quad \Sigma y = 3760 \quad \Sigma y ^ { 2 } = 375135$$
  1. The population mean salaries of employees from companies \(A\) and \(B\) are denoted by \(
    ) \mu _ { A }\( and \)\\( \mu _ { B }\) respectively. Using a \(5 \%\) significance level, test the null hypothesis \(\mu _ { A } = \mu _ { B }\) against the alternative hypothesis \(\mu _ { A } \neq \mu _ { B }\).
  2. State, with a reason, whether any assumptions about the distributions of employees' salaries are needed for the test in part (i).
CAIE FP2 2018 November Q9
9 There are a large number of students at a particular college. The heights, in metres, of a random sample of 8 students are as follows. $$\begin{array} { l l l l l l l l } 1.75 & 1.72 & 1.62 & 1.70 & 1.82 & 1.75 & 1.68 & 1.84 \end{array}$$ You may assume that heights of students are normally distributed.
  1. Test, at the \(5 \%\) significance level, whether the population mean height of students at this college is greater than 1.70 metres.
  2. Find a \(95 \%\) confidence interval for the population mean height of students at this college.
CAIE FP2 2018 November Q10
10 For a random sample of 10 observations of pairs of values \(( x , y )\), the equation of the regression line of \(y\) on \(x\) is \(y = 1.1664 + 0.4604 x\). It is given that $$\Sigma x ^ { 2 } = 1419.98 \quad \text { and } \quad \Sigma y ^ { 2 } = 439.68 .$$ The mean value of \(y\) is 6.24 .
  1. Find the equation of the regression line of \(x\) on \(y\).
  2. Find the product moment correlation coefficient.
  3. Test at the \(5 \%\) significance level whether there is evidence of positive correlation between the two variables.
CAIE FP2 2018 November Q11 EITHER
6 marks
A particle \(P\) of mass \(m\) is free to move on the smooth inner surface of a fixed hollow sphere of radius \(a\). The centre of the sphere is \(O\) and the point \(C\) is on the inner surface of the sphere, vertically below \(O\). The points \(A\) and \(B\) on the inner surface of the sphere are the ends of a diameter of the sphere. The diameter \(A O B\) makes an acute angle \(\alpha\) with the vertical, where \(\cos \alpha = \frac { 4 } { 5 }\), with \(A\) below the horizontal level of \(B\). The particle is projected from \(A\) with speed \(u\), and moves along the inner surface of the sphere towards \(C\). The normal reaction forces on the particle at \(A\) and \(C\) are in the ratio \(8 : 9\).
  1. Show that \(u ^ { 2 } = 4 a g\).
  2. Determine whether \(P\) reaches \(B\) without losing contact with the inner surface of the sphere. [6]
CAIE FP2 2018 November Q11 OR
A machine is used to produce metal rods. When the machine is working efficiently, the lengths, \(x \mathrm {~cm}\), of the rods have a normal distribution with mean 150 cm and standard deviation 1.2 cm . The machine is checked regularly by taking random samples of 200 rods. The latest results are shown in the following table.
Interval\(146 \leqslant x < 147\)\(147 \leqslant x < 148\)\(148 \leqslant x < 149\)\(149 \leqslant x < 150\)
Observed frequency122352
\(150 \leqslant x < 151\)\(151 \leqslant x < 152\)\(152 \leqslant x < 153\)\(153 \leqslant x < 154\)
6936152
As a first check, the sample is used to calculate an estimate for the mean.
  1. Show that an estimate for the mean from this sample is close to 150 cm .
    As a second check, the results are tested for goodness of fit of the normal distribution with mean 150 cm and standard deviation 1.2 cm . The relevant expected frequencies, found using the normal distribution function given in the List of Formulae (MF10), are shown in the following table.
    Interval\(x < 147\)\(147 \leqslant x < 148\)\(148 \leqslant x < 149\)\(149 \leqslant x < 150\)
    Observed frequency122352
    Expected frequency1.248.3230.9459.50
    \(150 \leqslant x < 151\)\(151 \leqslant x < 152\)\(152 \leqslant x < 153\)\(153 \leqslant x\)
    6936152
    59.5030.948.321.24
  2. Show how the expected frequency for \(151 \leqslant x < 152\) is obtained.
  3. Test, at the \(5 \%\) significance level, the goodness of fit of the normal distribution to the results.
    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 November Q1
1 The point \(O\) is on the fixed horizontal line \(l\). Points \(A\) and \(B\) on \(l\) are such that \(O A = 0.1 \mathrm {~m}\) and \(O B = 0.5 \mathrm {~m}\), with \(A\) between \(O\) and \(B\). A particle \(P\) oscillates on \(l\) in simple harmonic motion with centre \(O\). The kinetic energy of \(P\) when it is at \(A\) is twice its kinetic energy when it is at \(B\). Find the amplitude of the motion.
CAIE FP2 2018 November Q2
2 Two uniform small smooth spheres \(A\) and \(B\) have equal radii and masses \(2 m\) 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 \(\frac { 2 } { 3 }\).
  1. Find, in terms of \(u\), the speeds of \(A\) and \(B\) after this collision.
    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 \(\frac { 1 } { 2 }\).
  2. Find, in terms of \(d\) and \(u\), the time that elapses between the first and second collisions between \(A\) and \(B\).
    \includegraphics[max width=\textwidth, alt={}, center]{a092cd45-dc19-476d-adf7-0198fbb2116e-04_440_518_262_810} 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, where \(\tan \alpha = \frac { 12 } { 5 }\). The particle is projected downwards from \(A\) with speed \(u\) perpendicular to the string and moves in a vertical plane (see diagram). The string becomes slack after the string has rotated through \(270 ^ { \circ }\) from its initial position, with the particle now at the point \(B\).
CAIE FP2 2018 November Q5
5 An object is formed from a uniform circular disc, of radius \(2 a\) and mass \(3 M\), and a uniform \(\operatorname { rod } A B\), of length \(3 a\) and mass \(k M\), where \(k\) is a constant. The centre of the disc is \(O\). The end \(B\) of the rod is rigidly joined to a point on the circumference of the disc so that \(O B A\) is a straight line. The fixed horizontal axis \(l\) is in the plane of the object, passes through \(A\) and is perpendicular to \(A B\).
  1. Show that the moment of inertia of the object about the axis \(l\) is \(3 M a ^ { 2 } ( 26 + k )\).
    The object is free to rotate about \(l\).
  2. Show that small oscillations of the object about \(l\) are approximately simple harmonic. Given that the period of these oscillations is \(4 \pi \sqrt { } \left( \frac { a } { g } \right)\), find the value of \(k\).
CAIE FP2 2018 November Q6
6 The heights, in metres, of a random sample of 8 trees of a particular type are as follows.
8.4
9.2
10.8
11.3
11.5
12.8
12.1
14.2 Assuming that heights of trees of this type are normally distributed, calculate a \(95 \%\) confidence interval for the mean height of trees of this type.
CAIE FP2 2018 November Q7
7 The continuous random variable \(X\) has distribution function given by $$\mathrm { F } ( x ) = \begin{cases} 0 & x < 0 ,
\frac { 1 } { 90 } \left( x ^ { 2 } + x ^ { 4 } \right) & 0 \leqslant x \leqslant 3 ,
1 & x > 3 . \end{cases}$$ The random variable \(Y\) is defined by \(Y = X ^ { 2 }\).
  1. Find the probability density function of \(Y\).
  2. Find the mean value of \(Y\).
CAIE FP2 2018 November Q8
8 Lan starts a new job on Monday. He will catch the bus to work every day from Monday to Friday inclusive. The probability that he will get a seat on the bus has the constant value \(p\). The random variable \(X\) denotes the number of days that Lan will catch the bus until he is able to get a seat. The probability that Lan will not get a seat on the Monday, Tuesday, Wednesday or Thursday of his first week is 0.4096 .
  1. Show that \(p = 0.2\).
  2. Find the probability that Lan first gets a seat on Monday of the second week in his new job.
  3. Find the least integer \(N\) such that \(\mathrm { P } ( X \leqslant N ) > 0.9\), and identify the day and the week that corresponds to this value of \(N\).
CAIE FP2 2018 November Q9
9 For a random sample of 5 observations of pairs of values \(( x , y )\), the equation of the regression line of \(y\) on \(x\) is \(y = 4.2 + c x\) and the equation of the regression line of \(x\) on \(y\) is \(x = 10.8 + d y\), where \(c\) and \(d\) are constants. The product moment correlation coefficient is - 0.7214 and the mean value of \(x\) is 7.018.
  1. Test at the \(5 \%\) significance level whether there is evidence of non-zero correlation between the variables.
  2. Find the values of \(c\) and \(d\).
  3. Use an appropriate regression line to estimate the value of \(x\) when \(y = 3.5\), and comment on the reliability of your estimate.
CAIE FP2 2018 November Q10
10 The number of accidents, \(x\), that occur each day on a motorway are recorded over a period of 40 days. The results are shown in the following table.
Number of accidents0123456\(\geqslant 7\)
Observed frequency358105720
  1. Show that the mean number of accidents each day is 2.95 and calculate the variance for this sample. Explain why these values suggest that a Poisson distribution might fit the data.
    A Poisson distribution with mean 2.95, as found from the data, is used to calculate the expected frequencies, correct to 2 decimal places. The results are shown in the following table.
    Number of accidents0123456\(\geqslant 7\)
    Observed frequency358105720
    Expected frequency2.096.189.118.966.613.901.921.23
  2. Show how the expected frequency of 6.61 for \(x = 4\) is obtained.
  3. Test at the \(5 \%\) significance level the goodness of fit of this Poisson distribution to the data.
CAIE FP2 2018 November Q11 EITHER
One end of a light elastic spring, of natural length 0.8 m and modulus of elasticity 40 N , is attached to a fixed point \(O\). The spring hangs vertically, at rest, with particles of masses 2 kg and \(M \mathrm {~kg}\) attached to its free end. The \(M \mathrm {~kg}\) particle becomes detached from the spring, and as a result the 2 kg particle begins to move upwards.
  1. Show that the 2 kg particle performs simple harmonic motion about its equilibrium position with period \(\frac { 2 } { 5 } \pi \mathrm {~s}\). State the distance below \(O\) of the centre of the oscillations.
    The speed of the 2 kg particle is \(0.4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when its displacement from the centre of oscillation is 0.06 m .
  2. Find the amplitude of the motion.
  3. Deduce the value of \(M\).