Questions — CAIE M1 (786 questions)

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CAIE M1 2013 June Q6
9 marks Standard +0.3
6 \includegraphics[max width=\textwidth, alt={}, center]{2c628138-0729-4160-a95c-d6ab0f199cc5-3_639_939_1260_603} A particle \(P\) of mass 0.5 kg lies on a smooth horizontal plane. Horizontal forces of magnitudes \(F \mathrm {~N}\), 2.5 N and 2.6 N act on \(P\). The directions of the forces are as shown in the diagram, where \(\tan \alpha = \frac { 12 } { 5 }\) and \(\tan \beta = \frac { 7 } { 24 }\).
  1. Given that \(P\) is in equilibrium, find the values of \(F\) and \(\tan \theta\).
  2. The force of magnitude \(F \mathrm {~N}\) is removed. Find the magnitude and direction of the acceleration with which \(P\) starts to move.
CAIE M1 2013 June Q7
11 marks Standard +0.3
7 A car driver makes a journey in a straight line from \(A\) to \(B\), starting from rest. The speed of the car increases to a maximum, then decreases until the car is at rest at \(B\). The distance travelled by the car \(t\) seconds after leaving \(A\) is \(0.0000117 \left( 400 t ^ { 3 } - 3 t ^ { 4 } \right)\) metres.
  1. Find the distance \(A B\).
  2. Find the maximum speed of the car.
  3. Find the acceleration of the car
    1. as it starts from \(A\),
    2. as it arrives at \(B\).
    3. Sketch the velocity-time graph for the journey.
CAIE M1 2013 June Q3
Moderate -0.5
3 \includegraphics[max width=\textwidth, alt={}, center]{bc436b32-01f9-41dc-b2f7-ce49e18d3e6c-2_314_1193_1366_276} \({ } ^ { P A } { } _ { P A } ^ { P B }\) \(P\) \(P\) \begin{verbatim} " \end{verbatim}
CAIE M1 2013 June Q4
Standard +0.3
4 \(A\) B \(A \quad B\) \(\begin{array} { l l } B & \\ A & B \end{array}\) \(P B\) $$P \theta$$ \(\theta\) P \(\theta\) L \(P\) \(P\)
  1. (i)
    P \(\theta\) \(P \quad \underline { \theta }\)
  2. - \(\underline { \theta }\) \(5 \theta\) $$\begin{gathered} \\ \theta \end{gathered} \quad P$$
  3. \(P\)
CAIE M1 2013 June Q6
Standard +0.3
6 \(\begin{array} { c c c c c c c c } & P & & P & & & \theta & \\ \theta & P & & A & \theta & \theta & P & \\ \text { (i) } & & P & & & \theta & \\ \text { (ii) } & P & & & P & & & \\ \text { (iii) } & & & & & & \theta \\ & & & & & & \theta \end{array}\) 7 \includegraphics[max width=\textwidth, alt={}, center]{bc436b32-01f9-41dc-b2f7-ce49e18d3e6c-3_512_1095_1439_580}
  2. \(B\) \(\begin{array} { l l l } A & P & A \end{array}\) AN PA
CAIE M1 2013 June Q1
4 marks Moderate -0.3
1 A straight ice track of length 50 m is inclined at \(14 ^ { \circ }\) to the horizontal. A man starts at the top of the track, on a sledge, with speed \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). He travels on the sledge to the bottom of the track. The coefficient of friction between the sledge and the track is 0.02 . Find the speed of the sledge and the man when they reach the bottom of the track.
CAIE M1 2013 June Q2
5 marks Standard +0.3
2 \includegraphics[max width=\textwidth, alt={}, center]{ceb367ee-4e12-4cb2-9020-078ea5724d6e-2_529_691_529_726} Particle \(A\) of mass 1.6 kg and particle \(B\) of mass 2 kg are attached to opposite ends of a light inextensible string. The string passes over a small smooth pulley fixed at the top of a smooth plane, which is inclined at angle \(\theta\), where \(\sin \theta = 0.8\). Particle \(A\) is held at rest at the bottom of the plane and \(B\) hangs at a height of 3.24 m above the level of the bottom of the plane (see diagram). \(A\) is released from rest and the particles start to move.
  1. Show that the loss of potential energy of the system, when \(B\) reaches the level of the bottom of the plane, is 23.328 J .
  2. Hence find the speed of the particles when \(B\) reaches the level of the bottom of the plane.
CAIE M1 2013 June Q3
6 marks Standard +0.3
3 A car has mass 800 kg . The engine of the car generates constant power \(P \mathrm {~kW}\) as the car moves along a straight horizontal road. The resistance to motion is constant and equal to \(R \mathrm {~N}\). When the car's speed is \(14 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) its acceleration is \(1.4 \mathrm {~m} \mathrm {~s} ^ { - 2 }\), and when the car's speed is \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) its acceleration is \(0.33 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). Find the values of \(P\) and \(R\).
CAIE M1 2013 June Q4
7 marks Standard +0.3
4 An aeroplane moves along a straight horizontal runway before taking off. It starts from rest at \(O\) and has speed \(90 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at the instant it takes off. While the aeroplane is on the runway at time \(t\) seconds after leaving \(O\), its acceleration is \(( 1.5 + 0.012 t ) \mathrm { m } \mathrm { s } ^ { - 2 }\). Find
  1. the value of \(t\) at the instant the aeroplane takes off,
  2. the distance travelled by the aeroplane on the runway.
CAIE M1 2013 June Q5
8 marks Standard +0.3
5 A particle \(P\) is projected vertically upwards from a point on the ground with speed \(17 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Another particle \(Q\) is projected vertically upwards from the same point with speed \(7 \mathrm {~ms} ^ { - 1 }\). Particle \(Q\) is projected \(T\) seconds later than particle \(P\).
  1. Given that the particles reach the ground at the same instant, find the value of \(T\).
  2. At a certain instant when both \(P\) and \(Q\) are in motion, \(P\) is 5 m higher than \(Q\). Find the magnitude and direction of the velocity of each of the particles at this instant.
CAIE M1 2013 June Q6
9 marks Moderate -0.3
6 \includegraphics[max width=\textwidth, alt={}, center]{ceb367ee-4e12-4cb2-9020-078ea5724d6e-3_703_700_255_721} A small box of mass 40 kg is moved along a rough horizontal floor by three men. Two of the men apply horizontal forces of magnitudes 100 N and 120 N , making angles of \(30 ^ { \circ }\) and \(60 ^ { \circ }\) respectively with the positive \(x\)-direction. The third man applies a horizontal force of magnitude \(F \mathrm {~N}\) making an angle of \(\alpha ^ { \circ }\) with the negative \(x\)-direction (see diagram). The resultant of the three horizontal forces acting on the box is in the positive \(x\)-direction and has magnitude 136 N .
  1. Find the values of \(F\) and \(\alpha\).
  2. Given that the box is moving with constant speed, state the magnitude of the frictional force acting on the box and hence find the coefficient of friction between the box and the floor.
CAIE M1 2013 June Q7
11 marks Standard +0.3
7 \includegraphics[max width=\textwidth, alt={}, center]{ceb367ee-4e12-4cb2-9020-078ea5724d6e-3_430_860_1585_641} Particle \(A\) of mass 1.26 kg and particle \(B\) of mass 0.9 kg are attached to the ends of a light inextensible string. The string passes over a small smooth pulley \(P\) which is fixed at the edge of a rough horizontal table. \(A\) is held at rest at a point 0.48 m from \(P\), and \(B\) hangs vertically below \(P\), at a height of 0.45 m above the floor (see diagram). The coefficient of friction between \(A\) and the table is \(\frac { 2 } { 7 } . A\) is released and the particles start to move.
  1. Show that the magnitude of the acceleration of the particles is \(2.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) and find the tension in the string.
  2. Find the speed with which \(B\) reaches the floor.
  3. Find the speed with which \(A\) reaches the pulley.
CAIE M1 2014 June Q1
Moderate -1.0
1 hour 15 minutes
Additional Materials:
Answer Booklet/Paper
Graph Paper
List of Formulae (MF9) \section*{READ THESE INSTRUCTIONS FIRST} If you have been given an Answer Booklet, follow the instructions on the front cover of the Booklet.
Write your Centre number, candidate number and name on all the work you hand in.
Write in dark blue or black pen.
You may use an HB pencil for any diagrams or graphs.
Do not use staples, paper clips, glue or correction fluid.
DO NOT WRITE IN ANY BARCODES. Answer all the questions.
Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place in the case of angles in degrees, unless a different level of accuracy is specified in the question.
Where a numerical value for the acceleration due to gravity is needed, use \(10 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
The use of an electronic calculator is expected, where appropriate.
You are reminded of the need for clear presentation in your answers.
At the end of the examination, fasten all your work securely together.
[0pt] The number of marks is given in brackets [ ] at the end of each question or part question.
The total number of marks for this paper is 50.
Questions carrying smaller numbers of marks are printed earlier in the paper, and questions carrying larger numbers of marks later in the paper.
CAIE M1 2014 June Q3
Moderate -0.5
3 \includegraphics[max width=\textwidth, alt={}, center]{77976dad-c055-45fd-93fe-e37fa8e9ae22-2_520_719_1137_712} \(A\) and \(B\) are fixed points of a vertical wall with \(A\) vertically above \(B\). A particle \(P\) of mass 0.7 kg is attached to \(A\) by a light inextensible string of length \(3 \mathrm {~m} . P\) is also attached to \(B\) by a light inextensible string of length \(2.5 \mathrm {~m} . P\) is maintained in equilibrium at a distance of 2.4 m from the wall by a horizontal force of magnitude 10 N acting on \(P\) (see diagram). Both strings are taut, and the 10 N force acts in the plane \(A P B\) which is perpendicular to the wall. Find the tensions in the strings. [6]
CAIE M1 2014 June Q6
Moderate -0.5
6 A particle \(P\) of mass 0.2 kg is released from rest at a point 7.2 m above the surface of the liquid in a container. \(P\) falls through the air and into the liquid. There is no air resistance and there is no instantaneous change of speed as \(P\) enters the liquid. When \(P\) is at a distance of 0.8 m below the surface of the liquid, \(P\) 's speed is \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The only force on \(P\) due to the liquid is a constant resistance to motion of magnitude \(R \mathrm {~N}\).
  1. Find the deceleration of \(P\) while it is falling through the liquid, and hence find the value of \(R\). The depth of the liquid in the container is \(3.6 \mathrm {~m} . P\) is taken from the container and attached to one end of a light inextensible string. \(P\) is placed at the bottom of the container and then pulled vertically upwards with constant acceleration. The resistance to motion of \(R \mathrm {~N}\) continues to act. The particle reaches the surface 4 s after leaving the bottom of the container.
  2. Find the tension in the string.
CAIE M1 2014 June Q7
Easy -1.2
7 \includegraphics[max width=\textwidth, alt={}, center]{77976dad-c055-45fd-93fe-e37fa8e9ae22-4_333_1001_264_573} A light inextensible string of length 5.28 m has particles \(A\) and \(B\), of masses 0.25 kg and 0.75 kg respectively, attached to its ends. Another particle \(P\), of mass 0.5 kg , is attached to the mid-point of the string. Two small smooth pulleys \(P _ { 1 }\) and \(P _ { 2 }\) are fixed at opposite ends of a rough horizontal table of length 4 m and height 1 m . The string passes over \(P _ { 1 }\) and \(P _ { 2 }\) with particle \(A\) held at rest vertically below \(P _ { 1 }\), the string taut and \(B\) hanging freely below \(P _ { 2 }\). Particle \(P\) is in contact with the table halfway between \(P _ { 1 }\) and \(P _ { 2 }\) (see diagram). The coefficient of friction between \(P\) and the table is 0.4 . Particle \(A\) is released and the system starts to move with constant acceleration of magnitude \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\). The tension in the part \(A P\) of the string is \(T _ { A } \mathrm {~N}\) and the tension in the part \(P B\) of the string is \(T _ { B } \mathrm {~N}\).
  1. Find \(T _ { A }\) and \(T _ { B }\) in terms of \(a\).
  2. Show by considering the motion of \(P\) that \(a = 2\).
  3. Find the speed of the particles immediately before \(B\) reaches the floor.
  4. Find the deceleration of \(P\) immediately after \(B\) reaches the floor. \end{document}
CAIE M1 2014 June Q1
5 marks Moderate -0.8
1 \includegraphics[max width=\textwidth, alt={}, center]{139371b7-e142-4ed6-bff3-3ca4c32b9c6b-2_426_424_258_863} A block \(B\) of mass 7 kg is at rest on rough horizontal ground. A force of magnitude \(X \mathrm {~N}\) acts on \(B\) at an angle of \(15 ^ { \circ }\) to the upward vertical (see diagram).
  1. Given that \(B\) is in equilibrium find, in terms of \(X\), the normal component of the force exerted on \(B\) by the ground.
  2. The coefficient of friction between \(B\) and the ground is 0.4 . Find the value of \(X\) for which \(B\) is in limiting equilibrium.
CAIE M1 2014 June Q2
5 marks Standard +0.3
2 A car of mass 1250 kg travels up a straight hill inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = 0.02\). The power provided by the car's engine is 23 kW . The resistance to motion is constant and equal to 600 N . Find the speed of the car at an instant when its acceleration is \(0.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
CAIE M1 2014 June Q3
6 marks Standard +0.3
3 \includegraphics[max width=\textwidth, alt={}, center]{139371b7-e142-4ed6-bff3-3ca4c32b9c6b-2_657_913_1450_616} A particle \(P\) of weight 1.4 N is attached to one end of a light inextensible string \(S _ { 1 }\) of length 1.5 m , and to one end of another light inextensible string \(S _ { 2 }\) of length 1.3 m . The other end of \(S _ { 1 }\) is attached to a wall at the point 0.9 m vertically above a point \(O\) of the wall. The other end of \(S _ { 2 }\) is attached to the wall at the point 0.5 m vertically below \(O\). The particle is held in equilibrium, at the same horizontal level as \(O\), by a horizontal force of magnitude 2.24 N acting away from the wall and perpendicular to it (see diagram). Find the tensions in the strings.
[0pt] [6]
CAIE M1 2014 June Q4
7 marks Moderate -0.3
4 A small ball of mass 0.4 kg is released from rest at a point 5 m above horizontal ground. At the instant the ball hits the ground it loses 12.8 J of kinetic energy and starts to move upwards.
  1. Show that the greatest height above the ground that the ball reaches after hitting the ground is 1.8 m .
  2. Find the time taken for the ball's motion from its release until reaching this greatest height.
CAIE M1 2014 June Q5
8 marks Standard +0.3
5 A lorry of mass 16000 kg travels at constant speed from the bottom, \(O\), to the top, \(A\), of a straight hill. The distance \(O A\) is 1200 m and \(A\) is 18 m above the level of \(O\). The driving force of the lorry is constant and equal to 4500 N .
  1. Find the work done against the resistance to the motion of the lorry. On reaching \(A\) the lorry continues along a straight horizontal road against a constant resistance of 2000 N . The driving force of the lorry is not now constant, and the speed of the lorry increases from \(9 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at \(A\) to \(21 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at the point \(B\) on the road. The distance \(A B\) is 2400 m .
  2. Use an energy method to find \(F\), where \(F \mathrm {~N}\) is the average value of the driving force of the lorry while moving from \(A\) to \(B\).
  3. Given that the driving force at \(A\) is 1280 N greater than \(F \mathrm {~N}\) and that the driving force at \(B\) is 1280 N less than \(F \mathrm {~N}\), show that the power developed by the lorry's engine is the same at \(B\) as it is at \(A\).
CAIE M1 2014 June Q6
10 marks Moderate -0.3
6 A particle starts from rest at a point \(O\) and moves in a horizontal straight line. The velocity of the particle is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at time \(t \mathrm {~s}\) after leaving \(O\). For \(0 \leqslant t < 60\), the velocity is given by $$v = 0.05 t - 0.0005 t ^ { 2 }$$ The particle hits a wall at the instant when \(t = 60\), and reverses the direction of its motion. The particle subsequently comes to rest at the point \(A\) when \(t = 100\), and for \(60 < t \leqslant 100\) the velocity is given by $$v = 0.025 t - 2.5$$
  1. Find the velocity of the particle immediately before it hits the wall, and its velocity immediately after its hits the wall.
  2. Find the total distance travelled by the particle.
  3. Find the maximum speed of the particle and sketch the particle's velocity-time graph for \(0 \leqslant t \leqslant 100\), showing the value of \(t\) for which the speed is greatest. \section*{[Question 7 is printed on the next page.]}
CAIE M1 2014 June Q7
9 marks Standard +0.3
7 \includegraphics[max width=\textwidth, alt={}, center]{139371b7-e142-4ed6-bff3-3ca4c32b9c6b-4_342_1257_255_445} A smooth inclined plane of length 160 cm is fixed with one end at a height of 40 cm above the other end, which is on horizontal ground. Particles \(P\) and \(Q\), of masses 0.76 kg and 0.49 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 \(P\) is held at rest on the same line of greatest slope as the pulley and \(Q\) hangs vertically below the pulley at a height of 30 cm above the ground (see diagram). \(P\) is released from rest. It starts to move up the plane and does not reach the pulley. Find
  1. the acceleration of the particles and the tension in the string before \(Q\) reaches the ground,
  2. the speed with which \(Q\) reaches the ground,
  3. the total distance travelled by \(P\) before it comes to instantaneous rest.
CAIE M1 2015 June Q1
4 marks Moderate -0.8
1 A block \(B\) of mass 2.7 kg is pulled at constant speed along a straight line on a rough horizontal floor. The pulling force has magnitude 25 N and acts at an angle of \(\theta\) above the horizontal. The normal component of the contact force acting on \(B\) has magnitude 20 N .
  1. Show that \(\sin \theta = 0.28\).
  2. Find the work done by the pulling force in moving the block a distance of 5 m .
CAIE M1 2015 June Q2
5 marks Moderate -0.3
2 \includegraphics[max width=\textwidth, alt={}, center]{f4f2996b-5382-4b0d-9804-b5f5945946b3-2_636_519_664_813} Three horizontal forces of magnitudes \(F \mathrm {~N} , 63 \mathrm {~N}\) and 25 N act at \(O\), the origin of the \(x\)-axis and \(y\)-axis. The forces are in equilibrium. The force of magnitude \(F \mathrm {~N}\) makes an angle \(\theta\) anticlockwise with the positive \(x\)-axis. The force of magnitude 63 N acts along the negative \(y\)-axis. The force of magnitude 25 N acts at \(\tan ^ { - 1 } 0.75\) clockwise from the negative \(x\)-axis (see diagram). Find the value of \(F\) and the value of \(\tan \theta\).