Questions — CAIE (7646 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 M1 2014 November Q2
5 marks Moderate -0.8
2 \includegraphics[max width=\textwidth, alt={}, center]{c7133fc4-9a14-43fd-b5ed-788da72291cd-2_666_953_662_596} Three coplanar forces act at a point. The magnitudes of the forces are \(20 \mathrm {~N} , 25 \mathrm {~N}\) and 30 N , and the directions in which the forces act are as shown in the diagram, where \(\sin \alpha = 0.28\) and \(\cos \alpha = 0.96\), and \(\sin \beta = 0.6\) and \(\cos \beta = 0.8\).
  1. Show that the resultant of the three forces has a zero component in the \(x\)-direction.
  2. Find the magnitude and direction of the resultant of the three forces.
  3. The force of magnitude 20 N is replaced by another force. The effect is that the resultant force is unchanged in magnitude but reversed in direction. State the magnitude and direction of the replacement force.
CAIE M1 2014 November Q3
5 marks Standard +0.3
3 A train of mass 200000 kg moves on a horizontal straight track. It passes through a point \(A\) with speed \(28 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and later it passes through a point \(B\). The power of the train's engine at \(B\) is 1.2 times the power of the train's engine at \(A\). The driving force of the train's engine at \(B\) is 0.96 times the driving force of the train's engine at \(A\).
  1. Show that the speed of the train at \(B\) is \(35 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  2. For the motion from \(A\) to \(B\), find the work done by the train's engine given that the work done against the resistance to the train's motion is \(2.3 \times 10 ^ { 6 } \mathrm {~J}\).
CAIE M1 2014 November Q4
7 marks Moderate -0.3
4 \includegraphics[max width=\textwidth, alt={}, center]{c7133fc4-9a14-43fd-b5ed-788da72291cd-3_383_791_262_678} Forces of magnitude \(X \mathrm {~N}\) and 40 N act on a block \(B\) of mass 15 kg , which is in equilibrium in contact with a horizontal surface between points \(A\) and \(C\) on the surface. The forces act in the same vertical plane and in the directions shown in the diagram.
  1. Given that the surface is smooth, find the value of \(X\).
  2. It is given instead that the surface is rough and that the block is in limiting equilibrium. The frictional force acting on the block has magnitude 10 N in the direction towards \(A\). Find the coefficient of friction between the block and the surface.
CAIE M1 2014 November Q5
8 marks Standard +0.3
5 \includegraphics[max width=\textwidth, alt={}, center]{c7133fc4-9a14-43fd-b5ed-788da72291cd-3_289_567_1233_788} Particles \(A\) and \(B\), each of mass 0.3 kg , are connected by a light inextensible string. The string passes over a small smooth pulley fixed at the edge of a rough horizontal surface. Particle \(A\) hangs freely and particle \(B\) is held at rest in contact with the surface (see diagram). The coefficient of friction between \(B\) and the surface is 0.7 . Particle \(B\) is released and moves on the surface without reaching the pulley.
  1. Find, for the first 0.9 m of \(B\) 's motion,
CAIE M1 2014 November Q7
12 marks Standard +0.3
7 \includegraphics[max width=\textwidth, alt={}, center]{c7133fc4-9a14-43fd-b5ed-788da72291cd-4_512_1351_998_397} The diagram shows the velocity-time graph for the motion of a particle \(P\) which moves on a straight line \(B A C\). It starts at \(A\) and travels to \(B\) taking 5 s. It then reverses direction and travels from \(B\) to \(C\) taking 10 s . For the first 3 s of \(P\) 's motion its acceleration is constant. For the remaining 12 s the velocity of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at time \(t \mathrm {~s}\) after leaving \(A\), where $$v = - 0.2 t ^ { 2 } + 4 t - 15 \text { for } 3 \leqslant t \leqslant 15$$
  1. Find the value of \(v\) when \(t = 3\) and the magnitude of the acceleration of \(P\) for the first 3 s of its motion.
  2. Find the maximum velocity of \(P\) while it is moving from \(B\) to \(C\).
  3. Find the average speed of \(P\),
    1. while moving from \(A\) to \(B\),
    2. for the whole journey.
CAIE M1 2014 November Q2
Easy -1.2
2 \includegraphics[max width=\textwidth, alt={}, center]{9c7e8624-c4cd-4a8e-83d9-f92d0bd6f95b-2_262_1004_760_575} The tops of each of two smooth inclined planes \(A\) and \(B\) meet at a right angle. Plane \(A\) is inclined at angle \(\alpha\) to the horizontal and plane \(B\) is inclined at angle \(\beta\) to the horizontal, where \(\sin \alpha = \frac { 63 } { 65 }\) and \(\sin \beta = \frac { 16 } { 65 }\). A small smooth pulley is fixed at the top of the planes and a light inextensible string passes over the pulley. Two particles \(P\) and \(Q\), each of mass 0.65 kg , are attached to the string, one at each end. Particle \(Q\) is held at rest at a point of the same line of greatest slope of the plane \(B\) as the pulley. Particle \(P\) rests freely below the pulley in contact with plane \(A\) (see diagram). Particle \(Q\) is released and the particles start to move with the string taut. Find the tension in the string.
CAIE M1 2014 November Q3
Moderate -0.5
3 \includegraphics[max width=\textwidth, alt={}, center]{9c7e8624-c4cd-4a8e-83d9-f92d0bd6f95b-2_487_696_1537_721} Each of three light inextensible strings has a particle attached to one of its ends. The other ends of the strings are tied together at a point \(O\). Two of the strings pass over fixed smooth pegs and the particles hang freely in equilibrium. The weights of the particles and the angles between the sloping parts of the strings and the vertical are as shown in the diagram. It is given that \(\sin \beta = 0.8\) and \(\cos \beta = 0.6\).
  1. Show that \(W \cos \alpha = 3.8\) and find the value of \(W \sin \alpha\).
  2. Hence find the values of \(W\) and \(\alpha\).
CAIE M1 2014 November Q4
Moderate -0.5
4 A particle \(P\) starts from rest and moves in a straight line for 18 seconds. For the first 8 seconds of the motion \(P\) has constant acceleration \(0.25 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). Subsequently \(P\) 's velocity, \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at time \(t\) seconds after the motion started, is given by $$v = - 0.1 t ^ { 2 } + 2.4 t - k ,$$ where \(8 \leqslant t \leqslant 18\) and \(k\) is a constant.
  1. Find the value of \(v\) when \(t = 8\) and hence find the value of \(k\).
  2. Find the maximum velocity of \(P\).
  3. Find the displacement of \(P\) from its initial position when \(t = 18\).
CAIE M1 2014 November Q5
Moderate -0.5
5 A box of mass 8 kg is on a rough plane inclined at \(5 ^ { \circ }\) to the horizontal. A force of magnitude \(P \mathrm {~N}\) acts on the box in a direction upwards and parallel to a line of greatest slope of the plane. When \(P = 7 X\) the box moves up the line of greatest slope with acceleration \(0.15 \mathrm {~ms} ^ { - 2 }\) and when \(P = 8 X\) the box moves up the line of greatest slope with acceleration \(1.15 \mathrm {~ms} ^ { - 2 }\). Find the value of \(X\) and the coefficient of friction between the box and the plane.
CAIE M1 2014 November Q6
Easy -1.2
6 \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{9c7e8624-c4cd-4a8e-83d9-f92d0bd6f95b-3_462_218_1343_287} \captionsetup{labelformat=empty} \caption{Fig. 1}
\end{figure} \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{9c7e8624-c4cd-4a8e-83d9-f92d0bd6f95b-3_563_1143_1238_712} \captionsetup{labelformat=empty} \caption{Fig. 2}
\end{figure} Particles \(P\) and \(Q\) have a total mass of 1 kg . The particles are attached to opposite ends of a light inextensible string which passes over a smooth fixed pulley. \(P\) is held at rest and \(Q\) hangs freely, with both straight parts of the string vertical. Both particles are at a height of \(h \mathrm {~m}\) above the floor (see Fig. 1). \(P\) is released from rest and the particles start to move with the string taut. Fig. 2 shows the velocity-time graphs for \(P\) 's motion and for \(Q\) 's motion, where the positive direction for velocity is vertically upwards. Find
  1. the magnitude of the acceleration with which the particles start to move and the mass of each of the particles,
  2. the value of \(h\),
  3. the greatest height above the floor reached by particle \(P\).
CAIE M1 2014 November Q7
Moderate -0.5
7 \includegraphics[max width=\textwidth, alt={}, center]{9c7e8624-c4cd-4a8e-83d9-f92d0bd6f95b-4_668_848_260_653} A small block of mass 3 kg is initially at rest at the bottom \(O\) of a rough plane inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = 0.6\) and \(\cos \alpha = 0.8\). A force of magnitude 35 N acts on the block at an angle \(\beta\) above the plane, where \(\sin \beta = 0.28\) and \(\cos \beta = 0.96\). The block starts to move up a line of greatest slope of the plane and passes through a point \(A\) with speed \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The distance \(O A\) is 12.5 m (see diagram).
  1. For the motion of the block from \(O\) to \(A\), find the work done against the frictional force acting on the block.
  2. Find the coefficient of friction between the block and the plane. At the instant that the block passes through \(A\) the force of magnitude 35 N ceases to act.
  3. Find the distance the block travels up the plane after passing through \(A\). \end{document}
CAIE M1 2015 November Q1
4 marks Easy -1.2
1 A weightlifter performs an exercise in which he raises a mass of 200 kg from rest vertically through a distance of 0.7 m and holds it at that height.
  1. Find the work done by the weightlifter.
  2. Given that the time taken to raise the mass is 1.2 s , find the average power developed by the weightlifter.
CAIE M1 2015 November Q2
6 marks Moderate -0.3
2 A particle of mass 0.5 kg starts from rest and slides down a line of greatest slope of a smooth plane. The plane is inclined at an angle of \(30 ^ { \circ }\) to the horizontal.
  1. Find the time taken for the particle to reach a speed of \(2.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). When the particle has travelled 3 m down the slope from its starting point, it reaches rough horizontal ground at the bottom of the slope. The frictional force acting on the particle is 1 N .
  2. Find the distance that the particle travels along the ground before it comes to rest.
CAIE M1 2015 November Q3
6 marks Moderate -0.3
3 A lorry of mass 24000 kg is travelling up a hill which is inclined at \(3 ^ { \circ }\) to the horizontal. The power developed by the lorry's engine is constant, and there is a constant resistance to motion of 3200 N .
  1. When the speed of the lorry is \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), its acceleration is \(0.2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). Find the power developed by the lorry's engine.
  2. Find the steady speed at which the lorry moves up the hill if the power is 500 kW and the resistance remains 3200 N .
CAIE M1 2015 November Q4
6 marks Standard +0.3
4 \includegraphics[max width=\textwidth, alt={}, center]{2a91fb7a-0eaf-4c50-8a2c-4755c0b44c17-2_499_784_1617_685} Blocks \(P\) and \(Q\), of mass \(m \mathrm {~kg}\) and 5 kg respectively, are attached to the ends of a light inextensible string. The string passes over a small smooth pulley which is fixed at the top of a rough plane inclined at \(35 ^ { \circ }\) to the horizontal. Block \(P\) is at rest on the plane and block \(Q\) hangs vertically below the pulley (see diagram). The coefficient of friction between block \(P\) and the plane is 0.2 . Find the set of values of \(m\) for which the two blocks remain at rest.
CAIE M1 2015 November Q5
8 marks Moderate -0.3
5 \includegraphics[max width=\textwidth, alt={}, center]{2a91fb7a-0eaf-4c50-8a2c-4755c0b44c17-3_355_1048_255_552} A small bead \(Q\) can move freely along a smooth horizontal straight wire \(A B\) of length 3 m . Three horizontal forces of magnitudes \(F \mathrm {~N} , 10 \mathrm {~N}\) and 20 N act on the bead in the directions shown in the diagram. The magnitude of the resultant of the three forces is \(R \mathrm {~N}\) in the direction shown in the diagram.
  1. Find the values of \(F\) and \(R\).
  2. Initially the bead is at rest at \(A\). It reaches \(B\) with a speed of \(11.7 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find the mass of the bead.
CAIE M1 2015 November Q6
10 marks Standard +0.3
6 A particle \(P\) moves in a straight line, starting from a point \(O\). The velocity of \(P\), measured in \(\mathrm { m } \mathrm { s } ^ { - 1 }\), at time \(t \mathrm {~s}\) after leaving \(O\) is given by $$v = 0.6 t - 0.03 t ^ { 2 }$$
  1. Verify that, when \(t = 5\), the particle is 6.25 m from \(O\). Find the acceleration of the particle at this time.
  2. Find the values of \(t\) at which the particle is travelling at half of its maximum velocity.
CAIE M1 2015 November Q7
10 marks Standard +0.3
7 A cyclist starts from rest at point \(A\) and moves in a straight line with acceleration \(0.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) for a distance of 36 m . The cyclist then travels at constant speed for 25 s before slowing down, with constant deceleration, to come to rest at point \(B\). The distance \(A B\) is 210 m .
  1. Find the total time that the cyclist takes to travel from \(A\) to \(B\). 24 s after the cyclist leaves point \(A\), a car starts from rest from point \(A\), with constant acceleration \(4 \mathrm {~m} \mathrm {~s} ^ { - 2 }\), towards \(B\). It is given that the car overtakes the cyclist while the cyclist is moving with constant speed.
  2. Find the time that it takes from when the cyclist starts until the car overtakes her.
CAIE M1 2015 November Q1
5 marks Moderate -0.3
1 \includegraphics[max width=\textwidth, alt={}, center]{48f66bd5-33c1-4ce9-85f9-69faf10e871c-2_558_529_258_808} Four horizontal forces act at a point \(O\) and are in equilibrium. The magnitudes of the forces are \(F \mathrm {~N}\), \(G \mathrm {~N} , 15 \mathrm {~N}\) and \(F \mathrm {~N}\), and the forces act in directions as shown in the diagram.
  1. Show that \(F = 41.0\), correct to 3 significant figures.
  2. Find the value of \(G\).
CAIE M1 2015 November Q2
5 marks Moderate -0.8
2 A particle is released from rest at a point \(H \mathrm {~m}\) above horizontal ground and falls vertically. The particle passes through a point 35 m above the ground with a speed of \(( V - 10 ) \mathrm { m } \mathrm { s } ^ { - 1 }\) and reaches the ground with a speed of \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find
  1. the value of \(V\),
  2. the value of \(H\).
CAIE M1 2015 November Q3
6 marks Standard +0.3
3 A particle \(P\) moves along a straight line for 100 s . It starts at a point \(O\) and at time \(t\) seconds after leaving \(O\) the velocity of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), where $$v = 0.00004 t ^ { 3 } - 0.006 t ^ { 2 } + 0.288 t$$
  1. Find the values of \(t\) at which the acceleration of \(P\) is zero.
  2. Find the displacement of \(P\) from \(O\) when \(t = 100\).
CAIE M1 2015 November Q4
6 marks Standard +0.3
4 \includegraphics[max width=\textwidth, alt={}, center]{48f66bd5-33c1-4ce9-85f9-69faf10e871c-3_574_483_260_829} The diagram shows a vertical cross-section \(A B C\) of a surface. The part of the surface containing \(A B\) is smooth and \(A\) is 2.5 m above the level of \(B\). The part of the surface containing \(B C\) is rough and is at \(45 ^ { \circ }\) to the horizontal. The distance \(B C\) is 4 m (see diagram). A particle \(P\) of mass 0.2 kg is released from rest at \(A\) and moves in contact with the curve \(A B\) and then with the straight line \(B C\). The coefficient of friction between \(P\) and the part of the surface containing \(B C\) is 0.4 . Find the speed with which \(P\) reaches \(C\).
CAIE M1 2015 November Q5
8 marks Standard +0.3
5 \includegraphics[max width=\textwidth, alt={}, center]{48f66bd5-33c1-4ce9-85f9-69faf10e871c-3_259_828_1288_660} A smooth inclined plane of length 2.5 m is fixed with one end on the horizontal floor and the other end at a height of 0.7 m above the floor. Particles \(P\) and \(Q\), of masses 0.5 kg and 0.1 kg respectively, are attached to the ends of a light inextensible string which passes over a small smooth pulley fixed at the top of the plane. Particle \(Q\) is held at rest on the floor vertically below the pulley. The string is taut and \(P\) is at rest on the plane (see diagram). \(Q\) is released and starts to move vertically upwards towards the pulley and \(P\) moves down the plane.
  1. Find the tension in the string and the magnitude of the acceleration of the particles before \(Q\) reaches the pulley. At the instant just before \(Q\) reaches the pulley the string breaks; \(P\) continues to move down the plane and reaches the floor with a speed of \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  2. Find the length of the string.
CAIE M1 2015 November Q6
10 marks Standard +0.3
6 \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{48f66bd5-33c1-4ce9-85f9-69faf10e871c-4_149_410_306_518} \captionsetup{labelformat=empty} \caption{Fig. 1}
\end{figure} \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{48f66bd5-33c1-4ce9-85f9-69faf10e871c-4_133_406_260_1210} \captionsetup{labelformat=empty} \caption{Fig. 2}
\end{figure} A small ring of mass 0.024 kg is threaded on a fixed rough horizontal rod. A light inextensible string is attached to the ring and the string is pulled with a force of magnitude 0.195 N at an angle of \(\theta\) with the horizontal, where \(\sin \theta = \frac { 5 } { 13 }\). When the angle \(\theta\) is below the horizontal (see Fig. 1) the ring is in limiting equilibrium.
  1. Find the coefficient of friction between the ring and the rod. When the angle \(\theta\) is above the horizontal (see Fig. 2) the ring moves.
  2. Find the acceleration of the ring.
CAIE M1 2015 November Q7
10 marks Moderate -0.8
7 A car of mass 1600 kg moves with constant power 14 kW as it travels along a straight horizontal road. The car takes 25 s to travel between two points \(A\) and \(B\) on the road.
  1. Find the work done by the car's engine while the car travels from \(A\) to \(B\). The resistance to the car's motion is constant and equal to 235 N . The car has accelerations at \(A\) and \(B\) of \(0.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) and \(0.25 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) respectively. Find
  2. the gain in kinetic energy by the car in moving from \(A\) to \(B\),
  3. the distance \(A B\). {www.cie.org.uk} after the live examination series. }