Questions — CAIE (7659 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 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 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 M2 2019 March Q7
11 marks Challenging +1.2
7 A particle \(P\) is projected horizontally from a point \(O\) on a rough horizontal surface. The coefficient of friction between the particle and the surface is 0.2 . A horizontal force of magnitude \(0.06 t \mathrm {~N}\) directed away from \(O\) acts on \(P\), where \(t \mathrm {~s}\) is the time after projection. \(P\) comes to rest when \(t = 4\).
  1. The particle begins to move again when \(t = 8\). Show that the mass of \(P\) is 0.24 kg .
  2. Show that, for \(0 \leqslant t \leqslant 4 , \frac { \mathrm {~d} v } { \mathrm {~d} t } = 0.25 t - 2\), and find the speed of projection of \(P\).
  3. Find the distance from \(O\) at which \(P\) comes to rest.
    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 M2 2002 November Q1
3 marks Moderate -0.5
1 \includegraphics[max width=\textwidth, alt={}, center]{fcf239a6-6558-43ec-b404-70aa349af6a9-2_373_552_260_799} A uniform isosceles triangular lamina \(A B C\) is right-angled at \(B\). The length of \(A C\) is 24 cm . The lamina rotates in a horizontal plane, about a vertical axis through the mid-point of \(A C\), with angular speed \(5 \mathrm { rad } \mathrm { s } ^ { - 1 }\) (see diagram). Find the speed with which the centre of mass of the lamina is moving.
[0pt] [3]
CAIE M2 2002 November Q2
5 marks Standard +0.3
2 \includegraphics[max width=\textwidth, alt={}, center]{fcf239a6-6558-43ec-b404-70aa349af6a9-2_319_874_968_639} A uniform rod \(A B\), of length 2 m and mass 10 kg , is freely hinged to a fixed point at the end \(B\). A light elastic string, of modulus of elasticity 200 N , has one end attached to the end \(A\) of the rod and the other end attached to a fixed point \(O\), which is in the same vertical plane as the rod. The rod is horizontal and in equilibrium, with \(O A = 3 \mathrm {~m}\) and angle \(O A B = 150 ^ { \circ }\) (see diagram). Find
  1. the tension in the string,
  2. the natural length of the string.
CAIE M2 2002 November Q3
6 marks Moderate -0.3
3 \includegraphics[max width=\textwidth, alt={}, center]{fcf239a6-6558-43ec-b404-70aa349af6a9-2_502_789_1742_680} A stone is projected horizontally, with speed \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), from the top of a vertical cliff of height 45 m above sea level (see diagram). At time \(t \mathrm {~s}\) after projection the horizontal and vertically upward displacements of the stone from the top of the cliff are \(x \mathrm {~m}\) and \(y \mathrm {~m}\) respectively.
  1. Write down expressions for \(x\) and \(y\) in terms of \(t\), and hence obtain the equation of the stone's trajectory.
  2. Find the angle the trajectory makes with the horizontal at the point where the stone reaches sea level.
CAIE M2 2002 November Q4
6 marks Standard +0.3
4 \includegraphics[max width=\textwidth, alt={}, center]{fcf239a6-6558-43ec-b404-70aa349af6a9-3_604_490_258_831} A small ball \(B\) of mass 0.5 kg is attached to points \(P\) and \(Q\) on a fixed vertical axis by two light inextensible strings of equal length. Both of the strings are taut and each is inclined at \(60 ^ { \circ }\) to the vertical, as shown in the diagram. The ball moves with constant speed \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in a horizontal circle of radius 0.8 m . Find the tension in the string \(P B\).
CAIE M2 2002 November Q5
9 marks Standard +0.8
5 A light elastic string has natural length 2 m and modulus of elasticity 1.5 N . One end of the string is attached to a fixed point \(O\) of a smooth plane which is inclined at \(30 ^ { \circ }\) to the horizontal. The other end of the string is attached to a particle \(P\) of mass \(0.075 \mathrm {~kg} . P\) is released from rest at \(O\). Find
  1. the distance of \(P\) from \(O\) when \(P\) is at its lowest point,
  2. the acceleration with which \(P\) starts to move up the plane immediately after it has reached its lowest point.
CAIE M2 2002 November Q6
10 marks Standard +0.3
6 \includegraphics[max width=\textwidth, alt={}, center]{fcf239a6-6558-43ec-b404-70aa349af6a9-3_177_880_1658_635} A particle \(P\) of mass \(\frac { 1 } { 10 } \mathrm {~kg}\) travels in a straight line on a smooth horizontal surface. It passes through the fixed point \(O\) with velocity \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at time \(t = 0\). After \(t\) seconds its displacement from \(O\) is \(x \mathrm {~m}\) and its velocity is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 } . P\) is subject to a single force of magnitude \(\frac { v } { 200 } \mathrm {~N}\) which acts in a direction opposite to the motion (see diagram).
  1. Find an expression for \(v\) in terms of \(x\).
  2. Find an expression for \(x\) in terms of \(t\).
  3. Show that the value of \(x\) is less than 100 for all values of \(t\).
  4. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{fcf239a6-6558-43ec-b404-70aa349af6a9-4_477_684_264_774} \captionsetup{labelformat=empty} \caption{Fig. 1}
    \end{figure} Fig. 1 shows the cross section through the centre of mass \(C\) of a uniform L-shaped prism. \(C\) is \(x \mathrm {~cm}\) from \(O Y\) and \(y \mathrm {~cm}\) from \(O X\). Find the values of \(x\) and \(y\).
  5. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{fcf239a6-6558-43ec-b404-70aa349af6a9-4_257_428_1064_902} \captionsetup{labelformat=empty} \caption{Fig. 2}
    \end{figure} The prism is placed on a rough plane with \(O X\) in contact with the plane. The plane is tilted from the horizontal so that \(O X\) lies along a line of greatest slope, as shown in Fig. 2. When the angle of inclination of the plane is sufficiently great the prism starts to slide (without toppling). Show that the coefficient of friction between the prism and the plane is less than \(\frac { 7 } { 5 }\).
  6. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{fcf239a6-6558-43ec-b404-70aa349af6a9-4_303_414_1710_909} \captionsetup{labelformat=empty} \caption{Fig. 3}
    \end{figure} The prism is now placed on a rough plane with \(O Y\) in contact with the plane. The plane is tilted from the horizontal so that \(O Y\) lies along a line of greatest slope, as shown in Fig. 3. When the angle of inclination of the plane is sufficiently great the prism starts to topple (without sliding). Find the least possible value of the coefficient of friction between the prism and the plane. [3]
CAIE M2 2003 November Q1
3 marks Moderate -0.8
1 A railway engine of mass 50000 kg travels at a constant speed of \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) on a horizontal circular track of radius 1250 m . Find the magnitude of the horizontal force on the engine.
CAIE M2 2003 November Q2
6 marks Standard +0.3
2 \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{be83d46f-bf5b-4382-b424-bb5067626adc-2_376_569_559_466} \captionsetup{labelformat=empty} \caption{Fig. 1}
\end{figure} \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{be83d46f-bf5b-4382-b424-bb5067626adc-2_485_456_450_1226} \captionsetup{labelformat=empty} \caption{Fig. 2}
\end{figure} A uniform solid cone has height 20 cm and base radius 10 cm . It is placed with its axis vertical on a rough horizontal plane (see Fig. 1). The plane is slowly tilted and the cone remains in equilibrium until the angle of inclination of the plane reaches \(\theta ^ { \circ }\), when the cone begins to topple without sliding (see Fig. 2).
  1. Find the value of \(\theta\).
  2. What can you say about the value of the coefficient of friction between the cone and the plane?
CAIE M2 2003 November Q3
8 marks Standard +0.3
3 \includegraphics[max width=\textwidth, alt={}, center]{be83d46f-bf5b-4382-b424-bb5067626adc-2_433_446_1635_854} One end of a light elastic spring, of natural length 0.4 m and modulus of elasticity 88 N , is attached to a fixed point \(O\). A particle \(P\) of mass 0.2 kg is attached to the other end of the spring and is held, with the spring compressed, at a point 0.3 m vertically above \(O\), as shown in the diagram. \(P\) is now released from rest and moves vertically upwards.
  1. Find the initial acceleration of \(P\).
  2. Find the initial elastic potential energy of the spring.
  3. Find the speed of \(P\) when the distance \(O P\) is 0.4 m . \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{be83d46f-bf5b-4382-b424-bb5067626adc-3_362_657_269_744} \captionsetup{labelformat=empty} \caption{Fig. 1}
    \end{figure} Fig. 1 shows a uniform lamina \(A B C D\) with dimensions \(A B = 15.5 \mathrm {~cm} , B C = 8 \mathrm {~cm}\) and \(C D = 9.5 \mathrm {~cm}\). Angles \(A B C\) and \(B C D\) are right angles.
CAIE M2 2003 November Q5
11 marks Standard +0.8
5 A stone is projected from a point on horizontal ground with a speed of \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(\alpha ^ { \circ }\) above the horizontal. The stone is moving horizontally when it hits a vertical wall at a point 7.2 m above the ground.
  1. Find the value of \(\alpha\). After rebounding at right angles from the wall the speed of the stone is halved. Find
  2. the distance from the wall of the point at which the stone hits the ground,
  3. the angle which the direction of motion of the stone makes with the horizontal, immediately before the stone hits the ground.
CAIE M2 2003 November Q6
12 marks Standard +0.3
6 A cyclist and his machine have a total mass of 80 kg . The cyclist starts from rest and rides from the bottom to the top of a straight slope inclined at an angle \(\theta\) to the horizontal, where \(\sin \theta = 0.1\). The cyclist exerts a constant force of magnitude 120 N . There is a resisting force of magnitude \(8 v \mathrm {~N}\) acting on the cyclist, where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the cyclist's speed at time \(t \mathrm {~s}\) after the start.
  1. Show that \(\left( \frac { 1 } { 5 - v } \right) \frac { \mathrm { d } v } { \mathrm {~d} t } = \frac { 1 } { 10 }\).
  2. Solve this differential equation and hence show that \(v = 5 \left( 1 - \mathrm { e } ^ { - \frac { 1 } { 10 } t } \right)\).
  3. Given that the cyclist takes 20 s to reach the top of the slope, find the length of the slope.
CAIE M2 2004 November Q1
6 marks Standard +0.3
1 A light elastic string has natural length 1.5 m and modulus of elasticity 60 N . The string is stretched between two fixed points \(A\) and \(B\), which are at the same horizontal level and 2 m apart.
  1. Find the tension in the string. A particle of weight \(W \mathrm {~N}\) is now attached to the mid-point of the string and the particle is in equilibrium at a point 0.75 m vertically below the mid-point of \(A B\).
  2. Find the value of \(W\).
CAIE M2 2004 November Q2
6 marks Standard +0.3
2 \includegraphics[max width=\textwidth, alt={}, center]{81411376-b926-4857-bc9b-ac85d7957f3d-2_333_737_762_705} A uniform rod \(A B\) of length 1.2 m and weight 30 N is in equilibrium with the end \(A\) in contact with a vertical wall. \(A B\) is held at right angles to the wall by a light inextensible string. The string has one end attached to the rod at \(B\) and the other end attached to a point \(C\) of the wall. The point \(C\) is 0.5 m vertically above \(A\) (see diagram). Find
  1. the tension in the string,
  2. the horizontal and vertical components of the force exerted on the rod by the wall at \(A\).
CAIE M2 2004 November Q3
6 marks Standard +0.8
3 A car of mass 1000 kg is moving on a straight horizontal road. The driving force of the car is \(\frac { 28000 } { v } \mathrm {~N}\) and the resistance to motion is \(4 \nu \mathrm {~N}\), where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the speed of the car \(t\) seconds after it passes a fixed point on the road.
  1. Show that \(\frac { \mathrm { d } v } { \mathrm {~d} t } = \frac { 7000 - v ^ { 2 } } { 250 v }\). The car passes points \(A\) and \(B\) with speeds \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(40 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) respectively.
  2. Find the time taken for the car to travel from \(A\) to \(B\).
CAIE M2 2004 November Q4
7 marks Moderate -0.3
4 A particle is projected from a point \(O\) on horizontal ground with speed \(50 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle \(\theta\) to the horizontal. Given that the speed of the particle when it is at its highest point is \(40 \mathrm {~m} \mathrm {~s} ^ { - 1 }\),
  1. show that \(\cos \theta = 0.8\),
  2. find, in either order,
    (a) the greatest height reached by the particle,
    (b) the distance from \(O\) at which the particle hits the ground.
CAIE M2 2004 November Q5
7 marks Standard +0.8
5 One end of a light elastic string of natural length 0.4 m and modulus of elasticity 16 N is attached to a fixed point \(O\) of a horizontal table. A particle \(P\) of mass 0.8 kg is attached to the other end of the string. The particle \(P\) is released from rest on the table, at a point which is 0.5 m from \(O\). The coefficient of friction between the particle and the table is 0.2 . By considering work and energy, find the speed of \(P\) at the instant the string becomes slack.
CAIE M2 2004 November Q6
8 marks Moderate -0.3
6 A horizontal turntable rotates with constant angular speed \(\omega\) rad s \(^ { - 1 }\) about its centre \(O\). A particle \(P\) of mass 0.08 kg is placed on the turntable. The particle moves with the turntable and no sliding takes place.
  1. It is given that \(\omega = 3\) and that the particle is about to slide on the turntable when \(O P = 0.5 \mathrm {~m}\). Find the coefficient of friction between the particle and the turntable.
  2. Given instead that the particle is about to slide when its speed is \(1.2 \mathrm {~ms} ^ { - 1 }\), find \(\omega\).
CAIE M2 2004 November Q7
10 marks Standard +0.3
7 \includegraphics[max width=\textwidth, alt={}, center]{81411376-b926-4857-bc9b-ac85d7957f3d-3_327_1006_1037_573} A light container has a vertical cross-section in the form of a trapezium. The container rests on a horizontal surface. Grain is poured into the container to a depth of \(y \mathrm {~m}\). As shown in the diagram, the cross-section \(A B C D\) of the grain is such that \(A B = 0.4 \mathrm {~m}\) and \(D C = ( 0.4 + 2 y ) \mathrm { m }\).
  1. When \(y = 0.3\), find the vertical height of the centre of mass of the grain above the base of the container.
  2. Find the value of \(y\) for which the container is about to topple.
CAIE M2 2005 November Q1
3 marks Standard +0.8
1 \includegraphics[max width=\textwidth, alt={}, center]{a20a6641-d771-4c89-b40f-168a0c61f99d-2_552_604_264_772} A uniform solid cone has vertical height 28 cm and base radius 6 cm . The cone is held with a point of the circumference of its base in contact with a horizontal table, and with the base making an angle of \(\theta ^ { \circ }\) with the horizontal (see diagram). When the cone is released, it moves towards the equilibrium position in which its base is in contact with the table. Show that \(\theta < 40.6\), correct to 1 decimal place.
CAIE M2 2005 November Q2
5 marks Moderate -0.5
2 \includegraphics[max width=\textwidth, alt={}, center]{a20a6641-d771-4c89-b40f-168a0c61f99d-2_456_871_1228_635} An aircraft flies horizontally at a constant speed of \(220 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Initially it is flying due east. On reaching a point \(A\) it flies in a circular arc from \(A\) to \(B\), taking 50 s . At \(B\) the aircraft is flying due south (see diagram).
  1. Show that the radius of the arc is approximately 7000 m .
  2. Find the magnitude of the acceleration of the aircraft while it is flying between \(A\) and \(B\).
CAIE M2 2005 November Q3
6 marks Standard +0.8
3 \includegraphics[max width=\textwidth, alt={}, center]{a20a6641-d771-4c89-b40f-168a0c61f99d-3_293_1045_267_550} A uniform lamina \(A B C D\) is in the form of a trapezium in which \(A B\) and \(D C\) are parallel and have lengths 2 m and 3 m respectively. \(B D\) is perpendicular to the parallel sides and has length 1 m (see diagram).
  1. Find the distance of the centre of mass of the lamina from \(B D\). The lamina has weight \(W \mathrm {~N}\) and is in equilibrium, suspended by a vertical string attached to the lamina at \(B\). The lamina rests on a vertical support at \(C\). The lamina is in a vertical plane with \(A B\) and \(D C\) horizontal.
  2. Find, in terms of \(W\), the tension in the string and the magnitude of the force exerted on the lamina at \(C\).
CAIE M2 2005 November Q4
7 marks Standard +0.3
4 A particle is projected from horizontal ground with speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(\theta ^ { \circ }\) above the horizontal. The greatest height reached by the particle is 10 m and the particle hits the ground at a distance of 40 m from the point of projection. In either order,
  1. find the values of \(u\) and \(\theta\),
  2. find the equation of the trajectory, in the form \(y = a x - b x ^ { 2 }\), where \(x \mathrm {~m}\) and \(y \mathrm {~m}\) are the horizontal and vertical displacements of the particle from the point of projection.
CAIE M2 2005 November Q5
8 marks Standard +0.8
5 \includegraphics[max width=\textwidth, alt={}, center]{a20a6641-d771-4c89-b40f-168a0c61f99d-3_577_693_1740_724} A particle \(P\) of mass 0.2 kg is attached to the mid-point of a light elastic string of natural length 5.5 m and modulus of elasticity \(\lambda \mathrm { N }\). The ends of the string are attached to fixed points \(A\) and \(B\) which are at the same horizontal level and 6 m apart. \(P\) is held at rest at a point 1.25 m vertically above the mid-point of \(A B\) and then released. \(P\) travels a distance 5.25 m downwards before coming to instantaneous rest (see diagram). By considering the changes in gravitational potential energy and elastic potential energy as \(P\) travels downwards, find the value of \(\lambda\).
CAIE M2 2005 November Q6
10 marks Challenging +1.2
6 \includegraphics[max width=\textwidth, alt={}, center]{a20a6641-d771-4c89-b40f-168a0c61f99d-4_673_773_269_685} A horizontal circular disc of radius 4 m is free to rotate about a vertical axis through its centre \(O\). One end of a light inextensible rope of length 5 m is attached to a point \(A\) of the circumference of the disc, and an object \(P\) of mass 24 kg is attached to the other end of the rope. When the disc rotates with constant angular speed \(\omega\) rad s \(^ { - 1 }\), the rope makes an angle of \(\theta\) radians with the vertical and the tension in the rope is \(T \mathrm {~N}\) (see diagram). You may assume that the rope is always in the same vertical plane as the radius \(O A\) of the disc.
  1. Given that \(\cos \theta = \frac { 24 } { 25 }\), find the value of \(\omega\).
  2. Given instead that the speed of \(P\) is twice the speed of the point \(A\), find
    (a) the value of \(T\),
    (b) the speed of \(P\).