Questions — CAIE FP2 (515 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 P1 P2 P3 P4 PMT Mocks PURE Paper 1 Paper 2 Paper 3 Pre-U 9794/1 Pre-U 9794/2 Pre-U 9794/3 Pre-U 9795 Pre-U 9795/1 Pre-U 9795/2 S1 S2 S3 S4 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 PURE 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 PURE S1 S2 S3 S4 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 Pre-U Pre-U 9794/1 Pre-U 9794/2 Pre-U 9794/3 Pre-U 9795 Pre-U 9795/1 Pre-U 9795/2 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
Sort by: Default | Easiest first | Hardest first
CAIE FP2 2017 November Q3
10 marks Challenging +1.8
3 Three uniform small smooth spheres \(A , B\) and \(C\) have equal radii and masses \(m , k m\) and \(m\) respectively, where \(k\) is a constant. The spheres are moving in the same direction along a straight line on a smooth horizontal surface, with \(B\) between \(A\) and \(C\). The speeds of \(A , B\) and \(C\) are \(2 u , u\) and \(\frac { 4 } { 3 } u\) respectively. The coefficient of restitution between any pair of the spheres is \(\frac { 1 } { 2 }\). After sphere \(A\) has collided with sphere \(B\), sphere \(B\) collides with sphere \(C\).
  1. Find an inequality satisfied by \(k\).
  2. Given that \(k = 2\), show that after \(B\) has collided with \(C\) there are no further collisions between any of the three spheres.
CAIE FP2 2017 November Q4
10 marks Challenging +1.8
4 \includegraphics[max width=\textwidth, alt={}, center]{9b520e69-a14e-47e5-97d7-998f5145844b-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]{9b520e69-a14e-47e5-97d7-998f5145844b-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 Q6
6 marks Moderate -0.3
6 A pair of fair dice is thrown repeatedly until a pair of sixes is obtained. The number of throws taken is denoted by the random variable \(X\).
  1. Find the mean value of \(X\).
  2. Find the probability that exactly 12 throws are required to obtain a pair of sixes.
  3. Find the probability that more than 12 throws are required to obtain a pair of sixes.
CAIE FP2 2017 November Q7
7 marks Moderate -0.8
7 The random variable \(X\) has probability density function f given by $$\mathrm { f } ( x ) = \begin{cases} 0.2 \mathrm { e } ^ { - 0.2 x } & x \geqslant 0 \\ 0 & \text { otherwise } \end{cases}$$
  1. Find the distribution function of \(X\).
  2. Find \(\mathrm { P } ( X > 2 )\).
  3. Find the median of \(X\).
CAIE FP2 2017 November Q8
8 marks Standard +0.3
8 Members of a Statistics club are voting to elect a new president of the club. Members must choose to vote either by post or by text or by email. The method of voting chosen by a random sample of 60 male members and 40 female members is given in the following table.
\cline { 3 - 5 } \multicolumn{2}{c|}{}Method of voting
\cline { 3 - 5 } \multicolumn{2}{c|}{}PostTextEmail
\multirow{2}{*}{Gender}Male101238
\cline { 2 - 5 }Female52114
Test, at the \(1 \%\) significance level, whether there is an association between method of voting and gender.
CAIE FP2 2017 November Q9
9 marks Standard +0.8
9 The land areas \(x\) (in suitable units) and populations \(y\) (in millions) for a sample of 8 randomly chosen cities are given in the following table.
Land area \(( x )\)1.04.52.41.63.88.67.56.5
Population \(( y )\)0.88.44.21.62.210.24.25.2
$$\left[ \Sigma x = 35.9 , \Sigma x ^ { 2 } = 216.47 , \Sigma y = 36.8 , \Sigma y ^ { 2 } = 244.96 , \Sigma x y = 212.62 . \right]$$
  1. Find, showing all necessary working, the value of the product moment correlation coefficient for this sample.
  2. Using a \(1 \%\) significance level, test whether there is positive correlation between land area and population of cities.
    The land areas and populations for another randomly chosen sample of cities, this time of size \(n\), give a product moment correlation coefficient of 0.651 . Using a test at the \(1 \%\) significance level, there is evidence of non-zero correlation between the variables.
  3. Find the least possible value of \(n\), justifying your answer.
CAIE FP2 2017 November Q10
13 marks Standard +0.8
10 A factory produces bottles of an energy juice. Two different machines are used to fill empty bottles with the juice. The manager chooses a random sample of 50 bottles filled by machine \(X\) and a random sample of 60 bottles filled by machine \(Y\). The volumes of juice, \(x\) and \(y\) respectively, measured in appropriate units, are summarised by $$\Sigma x = 45.5 , \quad \Sigma ( x - \bar { x } ) ^ { 2 } = 19.56 , \quad \Sigma y = 72.3 , \quad \Sigma ( y - \bar { y } ) ^ { 2 } = 30.25$$ where \(\bar { x }\) and \(\bar { y }\) are the sample means of the volume of juice in the bottles filled by \(X\) and \(Y\) respectively.
  1. Find a 90\% confidence interval for the difference between the mean volume of juice in bottles filled by machine \(X\) and the mean volume of juice in bottles filled by machine \(Y\).
    A test at the \(\alpha \%\) significance level does not provide evidence that there is any difference in the means of the volume of juice in bottles filled by machine \(X\) and the volume of juice in bottles filled by machine \(Y\).
  2. Find the set of possible values of \(\alpha\).
CAIE FP2 2017 November Q11 EITHER
Challenging +1.8
\includegraphics[max width=\textwidth, alt={}]{9b520e69-a14e-47e5-97d7-998f5145844b-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 Q11 OR
Moderate -0.3
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
10 marks Challenging +1.8
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
Challenging +1.8
\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
10 marks Challenging +1.8
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
Challenging +1.8
\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 Q2
9 marks Standard +0.3
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]{f2073c6e-0f76-4246-89a7-2f3a9f7aaff8-04_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 2019 November Q1
5 marks Standard +0.8
1 A particle \(P\) is moving in a circle of radius 2 m . At time \(t\) seconds, its velocity is \(( t - 1 ) ^ { 2 } \mathrm {~m} \mathrm {~s} ^ { - 1 }\). At a particular time \(T\) seconds, where \(T > 0\), the magnitude of the radial component of the acceleration of \(P\) is \(8 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). Find the magnitude of the transverse component of the acceleration of \(P\) at this instant.
[0pt] [5] \includegraphics[max width=\textwidth, alt={}, center]{0f39ff02-a4fc-49ce-b87e-f70bef5a58b6-04_591_805_262_671} A uniform square lamina \(A B C D\) of side \(4 a\) and weight \(W\) rests in a vertical plane with the edge \(A B\) inclined at an angle \(\theta\) to the horizontal, where \(\tan \theta = \frac { 1 } { 3 }\). The vertex \(B\) is in contact with a rough horizontal surface for which the coefficient of friction is \(\mu\). The lamina is supported by a smooth peg at the point \(E\) on \(A B\), where \(B E = 3 a\) (see diagram).
  1. Find expressions in terms of \(W\) for the normal reaction forces at \(E\) and \(B\).
  2. Given that the lamina is about to slip, find the value of \(\mu\).
CAIE FP2 2019 November Q4
9 marks Challenging +1.8
4 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 \(O\) and \(P\) is held with the string taut and horizontal. The particle \(P\) is projected vertically downwards with speed \(\sqrt { } ( 2 a g )\) so that it begins to move along a circular path. The string becomes slack when \(O P\) makes an angle \(\theta\) with the upward vertical through \(O\).
  1. Show that \(\cos \theta = \frac { 2 } { 3 }\).
  2. Find the greatest height, above the horizontal through \(O\), reached by \(P\) in its subsequent motion. \includegraphics[max width=\textwidth, alt={}, center]{0f39ff02-a4fc-49ce-b87e-f70bef5a58b6-10_1049_744_260_696} A thin uniform \(\operatorname { rod } A B\) has mass \(\lambda M\) and length \(2 a\). The end \(A\) of the rod is rigidly attached to the surface of a uniform hollow sphere (spherical shell) with centre \(O\), mass \(3 M\) and radius \(a\). The end \(B\) of the rod is rigidly attached to the surface of a uniform solid sphere with centre \(C\), mass \(5 M\) and radius \(a\). The rod lies along the line joining the centres of the spheres, so that \(C B A O\) is a straight line. The horizontal axis \(L\) is perpendicular to the rod and passes through the point of the rod that is a distance \(\frac { 1 } { 2 } a\) from \(B\) (see diagram). The object consisting of the rod and the two spheres can rotate freely about \(L\).
CAIE FP2 2019 November Q1
5 marks Standard +0.8
1 A particle \(P\) is moving in a circle of radius 2 m . At time \(t\) seconds, its velocity is \(( t - 1 ) ^ { 2 } \mathrm {~m} \mathrm {~s} ^ { - 1 }\). At a particular time \(T\) seconds, where \(T > 0\), the magnitude of the radial component of the acceleration of \(P\) is \(8 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). Find the magnitude of the transverse component of the acceleration of \(P\) at this instant.
[0pt] [5] \includegraphics[max width=\textwidth, alt={}, center]{4240c99e-10ba-443e-8021-1872e6e64ccf-04_591_805_262_671} A uniform square lamina \(A B C D\) of side \(4 a\) and weight \(W\) rests in a vertical plane with the edge \(A B\) inclined at an angle \(\theta\) to the horizontal, where \(\tan \theta = \frac { 1 } { 3 }\). The vertex \(B\) is in contact with a rough horizontal surface for which the coefficient of friction is \(\mu\). The lamina is supported by a smooth peg at the point \(E\) on \(A B\), where \(B E = 3 a\) (see diagram).
  1. Find expressions in terms of \(W\) for the normal reaction forces at \(E\) and \(B\).
  2. Given that the lamina is about to slip, find the value of \(\mu\).
CAIE FP2 2019 November Q4
9 marks Challenging +1.8
4 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 \(O\) and \(P\) is held with the string taut and horizontal. The particle \(P\) is projected vertically downwards with speed \(\sqrt { } ( 2 a g )\) so that it begins to move along a circular path. The string becomes slack when \(O P\) makes an angle \(\theta\) with the upward vertical through \(O\).
  1. Show that \(\cos \theta = \frac { 2 } { 3 }\).
  2. Find the greatest height, above the horizontal through \(O\), reached by \(P\) in its subsequent motion. \includegraphics[max width=\textwidth, alt={}, center]{4240c99e-10ba-443e-8021-1872e6e64ccf-10_1051_744_258_696} A thin uniform \(\operatorname { rod } A B\) has mass \(\lambda M\) and length \(2 a\). The end \(A\) of the rod is rigidly attached to the surface of a uniform hollow sphere (spherical shell) with centre \(O\), mass \(3 M\) and radius \(a\). The end \(B\) of the rod is rigidly attached to the surface of a uniform solid sphere with centre \(C\), mass \(5 M\) and radius \(a\). The rod lies along the line joining the centres of the spheres, so that \(C B A O\) is a straight line. The horizontal axis \(L\) is perpendicular to the rod and passes through the point of the rod that is a distance \(\frac { 1 } { 2 } a\) from \(B\) (see diagram). The object consisting of the rod and the two spheres can rotate freely about \(L\).
CAIE FP2 2017 Specimen Q2
10 marks Standard +0.8
2 A small uniform sphere \(A\), of mass \(2 m\), is moving with speed \(u\) on a smooth horizontal surface when it collides directly with a small uniform sphere \(B\), of mass \(m\), which is at rest. The spheres have equal radii and the coefficient of restitution between them is \(e\).
  1. Find expressions for the speeds of \(A\) and \(B\) immediately after the collision.
    Subsequently \(B\) collides with a vertical wall which is perpendicular to the direction of motion of \(B\). The coefficient of restitution between \(B\) and the wall is 0.4 . After \(B\) has collided with the wall, the speeds of \(A\) and \(B\) are equal.
  2. Find \(e\).
  3. Initially \(B\) is at a distance \(d\) from the wall. Find the distance of \(B\) from the wall when it next collides with \(A\). \(3 A\) and \(B\) are two fixed points on a smooth horizontal surface, with \(A B = 3 a \mathrm {~m}\). One end of a light elastic string, of natural length \(a\) m and modulus of elasticity \(m g \mathrm {~N}\), is attached to the point \(A\). The other end of this string is attached to a particle \(P\) of mass \(m \mathrm {~kg}\). One end of a second light elastic string, of natural length \(k a \mathrm {~m}\) and modulus of elasticity \(2 m g \mathrm {~N}\), is attached to \(B\). The other end of this string is attached to \(P\). It is given that the system is in equilibrium when \(P\) is at \(M\), the mid-point of \(A B\).
CAIE FP2 2017 Specimen Q4
13 marks Challenging +1.2
4 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 \(O\). When \(P\) is hanging at rest vertically below \(O\), it is projected horizontally. In the subsequent motion \(P\) completes a vertical circle. The speed of \(P\) when it is at its highest point is \(u\).
  1. Show that the least possible value of \(u\) is \(\sqrt { } ( a g )\).
    It is now given that \(u = \sqrt { } ( a g )\). When \(P\) passes through the lowest point of its path, it collides with, and coalesces with, a stationary particle of mass \(\frac { 1 } { 4 } m\).
  2. Find the speed of the combined particle immediately after the collision.
    In the subsequent motion, when \(O P\) makes an angle \(\theta\) with the upward vertical the tension in the string is \(T\).
  3. Find an expression for \(T\) in terms of \(m , g\) and \(\theta\).
  4. Find the value of \(\cos \theta\) when the string becomes slack.
CAIE FP2 2017 Specimen Q5
5 marks Standard +0.3
5 A random sample of 10 observations of a normal variable \(X\) gave the following summarised data, where \(\bar { x }\) is the sample mean. $$\Sigma x = 222.8 \quad \Sigma ( x - \bar { x } ) ^ { 2 } = 4.12$$ Find a \(95 \%\) confidence interval for the population mean.
CAIE FP2 2017 Specimen Q6
8 marks Standard +0.3
6 A biased coin is tossed repeatedly until a head is obtained. The random variable \(X\) denotes the number of tosses required for a head to be obtained. The mean of \(X\) is equal to twice the variance of \(X\).
  1. Show that the probability that a head is obtained when the coin is tossed once is \(\frac { 2 } { 3 }\). \includegraphics[max width=\textwidth, alt={}, center]{3b311657-f609-4e8d-81e6-b0cbc7a8cbae-11_69_1571_450_328}
  2. Find \(\mathrm { P } ( X = 4 )\).
  3. Find \(\mathrm { P } ( X > 4 )\).
  4. Find the least integer \(N\) such that \(\mathrm { P } ( X \leqslant N ) > 0.999\).
CAIE FP2 2017 Specimen Q7
9 marks Standard +0.8
7 The continuous random variable \(X\) has probability density function given by $$f ( x ) = \begin{cases} \frac { 1 } { 21 } x ^ { 2 } & 1 \leqslant x \leqslant 4 \\ 0 & \text { otherwise } \end{cases}$$ The random variable \(Y\) is defined by \(Y = X ^ { 2 }\).
  1. Show that \(Y\) has probability density function given by $$g ( y ) = \begin{cases} \frac { 1 } { 42 } y ^ { \frac { 1 } { 2 } } & 1 \leqslant y \leqslant 16 \\ 0 & \text { otherwise } \end{cases}$$
  2. Find the median value of \(Y\).
  3. Find the expected value of \(Y\).
CAIE FP2 2017 Specimen Q8
10 marks Challenging +1.2
8 The number of goals scored by a certain football team was recorded for each of 100 matches, and the results are summarised in the following table.
Number of goals0123456 or more
Frequency121631251330
Fit a Poisson distribution to the data, and test its goodness of fit at the 5\% significance level.
CAIE FP2 2017 Specimen Q9
11 marks Standard +0.8
9 A random sample of 8 students is chosen from those sitting examinations in both Mathematics and French. Their marks in Mathematics, \(x\), and in French, \(y\), are summarised as follows. $$\Sigma x = 472 \quad \Sigma x ^ { 2 } = 29950 \quad \Sigma y = 400 \quad \Sigma y ^ { 2 } = 21226 \quad \Sigma x y = 24879$$ Another student scored 72 marks in the Mathematics examination but was unable to sit the French examination.
  1. Estimate the mark that this student would have obtained in the French examination.
  2. Test, at the \(5 \%\) significance level, whether there is non-zero correlation between marks in Mathematics and marks in French.