Questions — CAIE (7646 questions)

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CAIE M1 2018 November Q6
10 marks Moderate -0.3
6 A van of mass 3200 kg travels along a horizontal road. The power of the van's engine is constant and equal to 36 kW , and there is a constant resistance to motion acting on the van.
  1. When the speed of the van is \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), its acceleration is \(0.2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). Find the resistance force.
    When the van is travelling at \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), it begins to ascend a hill inclined at \(1.5 ^ { \circ }\) to the horizontal. The power is increased and the resistance force is still equal to the value found in part (i).
  2. Find the power required to maintain this speed of \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  3. The engine is now stopped, with the van still travelling at \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), and the van decelerates to rest. Find the distance the van moves up the hill from the point at which the engine is stopped until it comes to rest.
CAIE M1 2018 November Q7
10 marks Standard +0.3
7 A particle moves in a straight line. The particle is initially at rest at a point \(O\) on the line. At time \(t \mathrm {~s}\) after leaving \(O\), the acceleration \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\) of the particle is given by \(a = 25 - t ^ { 2 }\) for \(0 \leqslant t \leqslant 9\).
  1. Find the maximum velocity of the particle in this time period.
  2. Find the total distance travelled until the maximum velocity is reached.
    The acceleration of the particle for \(t > 9\) is given by \(a = - 3 t ^ { - \frac { 1 } { 2 } }\).
  3. Find the velocity of the particle when \(t = 25\).
    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 M1 2019 November Q1
3 marks Easy -1.2
1 A crate of mass 500 kg is being pulled along rough horizontal ground by a horizontal rope attached to a winch. The winch produces a constant pulling force of 2500 N and the crate is moving at constant speed. Find the coefficient of friction between the crate and the ground.
CAIE M1 2019 November Q2
5 marks Standard +0.3
2 A train of mass 150000 kg ascends a straight slope inclined at \(\alpha ^ { \circ }\) to the horizontal with a constant driving force of 16000 N . At a point \(A\) on the slope the speed of the train is \(45 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Point \(B\) on the slope is 500 m beyond \(A\). At \(B\) the speed of the train is \(42 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). There is a resistance force acting on the train and the train does \(4 \times 10 ^ { 6 } \mathrm {~J}\) of work against this resistance force between \(A\) and \(B\). Find the value of \(\alpha\).
CAIE M1 2019 November Q3
6 marks Moderate -0.3
3 \includegraphics[max width=\textwidth, alt={}, center]{60a41d3b-62a0-40d9-a30d-0560903429af-05_479_647_264_749} Three coplanar forces of magnitudes \(50 \mathrm {~N} , 60 \mathrm {~N}\) and 100 N act at a point. The resultant of the forces has magnitude \(R \mathrm {~N}\). The directions of these forces are shown in the diagram. Find the values of \(R\) and \(\alpha\).
CAIE M1 2019 November Q4
6 marks Standard +0.3
4 A car travels along a straight road with constant acceleration. It passes through points \(P , Q , R\) and \(S\). The times taken for the car to travel from \(P\) to \(Q , Q\) to \(R\) and \(R\) to \(S\) are each equal to 10 s . The distance \(Q R\) is 1.5 times the distance \(P Q\). At point \(Q\) the speed of the car is \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  1. Show that the acceleration of the car is \(0.8 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
  2. Find the distance \(Q S\) and hence find the average speed of the car between \(Q\) and \(S\).
CAIE M1 2019 November Q5
8 marks Moderate -0.3
5 A cyclist is travelling along a straight horizontal road. The total mass of the cyclist and his bicycle is 80 kg . His power output is a constant 240 W . His acceleration when he is travelling at \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is \(0.3 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
  1. Show that the resistance to the cyclist's motion is 16 N .
  2. Find the steady speed that the cyclist can maintain if his power output and the resistance force are both unchanged.
  3. The cyclist later ascends a straight hill inclined at \(3 ^ { \circ }\) to the horizontal. His power output and the resistance force are still both unchanged. Find his acceleration when he is travelling at \(4 \mathrm {~ms} ^ { - 1 }\).
CAIE M1 2019 November Q6
9 marks Standard +0.3
6 Particle \(P\) travels in a straight line from \(A\) to \(B\). The velocity of \(P\) at time \(t \mathrm {~s}\) after leaving \(A\) is denoted by \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), where $$v = 0.04 t ^ { 3 } + c t ^ { 2 } + k t$$ \(P\) takes 5 s to travel from \(A\) to \(B\) and it reaches \(B\) with speed \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The distance \(A B\) is 25 m .
  1. Find the values of the constants \(c\) and \(k\).
  2. Show that the acceleration of \(P\) is a minimum when \(t = 2.5\).
CAIE M1 2019 November Q7
13 marks Standard +0.3
7 \includegraphics[max width=\textwidth, alt={}, center]{60a41d3b-62a0-40d9-a30d-0560903429af-12_565_511_260_817} Two particles \(A\) and \(B\) have masses \(m \mathrm {~kg}\) and \(k m \mathrm {~kg}\) respectively, where \(k > 1\). The particles are attached to the ends of a light inextensible string. The string passes over a fixed smooth pulley and the particles hang vertically below it. Both particles are at a height of 0.81 m above horizontal ground (see diagram). The system is released from rest and particle \(B\) reaches the ground 0.9 s later. The particle \(A\) does not reach the pulley in its subsequent motion.
  1. Find the value of \(k\) and show that the tension in the string before \(B\) reaches the ground is equal to \(12 m \mathrm {~N}\).
    At the instant when \(B\) reaches the ground, the string breaks.
  2. Show that the speed of \(A\) when it reaches the ground is \(5.97 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), correct to 3 significant figures, and find the time taken, after the string breaks, for \(A\) to reach the ground.
  3. Sketch a velocity-time graph for the motion of particle \(A\) from the instant when the system is released until \(A\) reaches the ground. 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 June Q1
5 marks Standard +0.3
1 One end of a light elastic string of natural length 1.6 m and modulus of elasticity 25 N is attached to a fixed point \(A\). A particle \(P\) of mass 0.15 kg is attached to the other end of the string. \(P\) is held at rest at a point 2 m vertically below \(A\) and is then released.
  1. For the motion from the instant of release until the string becomes slack, find the loss of elastic potential energy and the gain in gravitational potential energy.
  2. Hence find the speed of \(P\) at the instant the string becomes slack.
CAIE M2 2002 June Q2
4 marks Standard +0.3
2 \includegraphics[max width=\textwidth, alt={}, center]{3e7472a8-df1e-45c4-81fb-e4397bddf5ad-2_316_1065_712_541} Two identical uniform heavy triangular prisms, each of base width 10 cm , are arranged as shown at the ends of a smooth horizontal shelf of length 1 m . Some books, each of width 5 cm , are placed on the shelf between the prisms.
  1. Find how far the base of a prism can project beyond an end of the shelf without the prism toppling.
  2. Find the greatest number of books that can be stored on the shelf without either of the prisms toppling.
CAIE M2 2002 June Q3
5 marks Standard +0.3
3 \includegraphics[max width=\textwidth, alt={}, center]{3e7472a8-df1e-45c4-81fb-e4397bddf5ad-2_202_972_1619_584} A light elastic string has natural length 0.8 m and modulus of elasticity 12 N . The ends of the string are attached to fixed points \(A\) and \(B\), which are at the same horizontal level and 0.96 m apart. A particle of weight \(W \mathrm {~N}\) is attached to the mid-point of the string and hangs in equilibrium at a point 0.14 m below \(A B\) (see diagram). Find \(W\).
CAIE M2 2002 June Q4
8 marks Standard +0.3
4 \includegraphics[max width=\textwidth, alt={}, center]{3e7472a8-df1e-45c4-81fb-e4397bddf5ad-3_576_826_258_662} A hollow cone with semi-vertical angle \(45 ^ { \circ }\) is fixed with its axis vertical and its vertex \(O\) downwards. A particle \(P\) of mass 0.3 kg moves in a horizontal circle on the inner surface of the cone, which is smooth. \(P\) is attached to one end of a light inextensible string of length 1.2 m . The other end of the string is attached to the cone at \(O\) (see diagram). The string is taut and rotates at a constant angular speed of \(4 \mathrm { rad } \mathrm { s } ^ { - 1 }\).
  1. Find the acceleration of \(P\).
  2. Find the tension in the string and the force exerted on \(P\) by the cone.
CAIE M2 2002 June Q5
9 marks Standard +0.3
5 \includegraphics[max width=\textwidth, alt={}, center]{3e7472a8-df1e-45c4-81fb-e4397bddf5ad-3_590_754_1425_699} A uniform lamina of weight 9 N has dimensions as shown in the diagram. The lamina is freely hinged to a fixed point at \(A\). A light inextensible string has one end attached to \(B\), and the other end attached to a fixed point \(C\), which is in the same vertical plane as the lamina. The lamina is in equilibrium with \(A B\) horizontal and angle \(A B C = 150 ^ { \circ }\).
  1. Show that the tension in the string is 12.2 N .
  2. Find the magnitude of the force acting on the lamina at \(A\).
CAIE M2 2002 June Q6
10 marks Standard +0.3
6 \includegraphics[max width=\textwidth, alt={}, center]{3e7472a8-df1e-45c4-81fb-e4397bddf5ad-4_182_844_264_653} A particle \(P\) of mass 0.4 kg travels on a horizontal surface along the line \(O A\) in the direction from \(O\) to \(A\). Air resistance of magnitude \(0.1 v \mathrm {~N}\) opposes the motion, where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the speed of \(P\) at time \(t \mathrm {~s}\) after it passes through the fixed point \(O\) (see diagram). The speed of \(P\) at \(O\) is \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  1. Assume that the horizontal surface is smooth. Show that \(\frac { \mathrm { d } v } { \mathrm {~d} x } = - \frac { 1 } { 4 }\), where \(x \mathrm {~m}\) is the distance of \(P\) from \(O\) at time \(t \mathrm {~s}\), and hence find the distance from \(O\) at which the speed of \(P\) is zero.
  2. Assume instead that the horizontal surface is not smooth and that the coefficient of friction between \(P\) and the surface is \(\frac { 3 } { 40 }\).
    1. Show that \(4 \frac { \mathrm {~d} v } { \mathrm {~d} t } = - ( v + 3 )\).
    2. Hence find the value of \(t\) for which the speed of \(P\) is zero.
CAIE M2 2002 June Q7
9 marks Standard +0.3
7 A ball is projected from a point \(O\) with speed \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\), at an angle of \(30 ^ { \circ }\) above the horizontal. At time \(T\) s after projection, the ball passes through the point \(A\), whose horizontal and vertically upward displacements from \(O\) are 10 m and 2 m respectively.
  1. By using the equation of the trajectory, or otherwise, find the value of \(V\).
  2. Find the value of \(T\).
  3. Find the angle that the direction of motion of the ball at \(A\) makes with the horizontal.
CAIE M2 2003 June Q1
4 marks Standard +0.3
1 \includegraphics[max width=\textwidth, alt={}, center]{7f8646df-a7d8-4ca1-a6ee-3ceab6bb83af-2_533_497_269_824} A frame consists of a uniform circular ring of radius 25 cm and mass 1.5 kg , and a uniform rod of length 48 cm and mass 0.6 kg . The ends \(A\) and \(B\) of the rod are attached to points on the circumference of the ring, as shown in the diagram. Find the distance of the centre of mass of the frame from the centre of the ring.
CAIE M2 2003 June Q2
5 marks Standard +0.3
2 \includegraphics[max width=\textwidth, alt={}, center]{7f8646df-a7d8-4ca1-a6ee-3ceab6bb83af-2_439_608_1181_772} A uniform solid hemisphere, with centre \(O\) and radius 4 cm , is held so that a point \(P\) of its rim is in contact with a horizontal surface. The plane face of the hemisphere makes an angle of \(70 ^ { \circ }\) with the horizontal. \(Q\) is the point on the axis of symmetry of the hemisphere which is vertically above \(P\). The diagram shows the vertical cross-section of the hemisphere which contains \(O , P\) and \(Q\).
  1. Determine whether or not the centre of mass of the hemisphere is between \(O\) and \(Q\). The hemisphere is now released.
  2. State whether or not the hemisphere falls on to its plane face, giving a reason for your answer.
CAIE M2 2003 June Q3
7 marks Standard +0.3
3 \includegraphics[max width=\textwidth, alt={}, center]{7f8646df-a7d8-4ca1-a6ee-3ceab6bb83af-3_464_439_274_854} A uniform beam \(A B\) has length 6 m and mass 45 kg . One end of a light inextensible rope is attached to the beam at the point 2.5 m from \(A\). The other end of the rope is attached to a fixed point \(P\) on a vertical wall. The beam is in equilibrium with \(A\) in contact with the wall at a point 5 m below \(P\). The rope is taut and at right angles to \(A B\) (see diagram). Find
  1. the tension in the rope,
  2. the horizontal and vertical components of the force exerted by the wall on the beam at \(A\).
CAIE M2 2003 June Q4
7 marks Standard +0.8
4 A particle of mass 0.2 kg moves in a straight line on a smooth horizontal surface. When its displacement from a fixed point on the surface is \(x \mathrm {~m}\), its velocity is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The motion is opposed by a force of magnitude \(\frac { 1 } { 3 v } \mathrm {~N}\).
  1. Show that \(3 v ^ { 2 } \frac { \mathrm {~d} v } { \mathrm {~d} x } = - 5\).
  2. Find the value of \(v\) when \(x = 7.4\), given that \(v = 4\) when \(x = 0\).
CAIE M2 2003 June Q5
7 marks Moderate -0.3
5
[diagram]
A toy aircraft of mass 0.5 kg is attached to one end of a light inextensible string of length 9 m . The other end of the string is attached to a fixed point \(O\). The aircraft moves with constant speed in a horizontal circle. The string is taut, and makes an angle of \(60 ^ { \circ }\) with the upward vertical at \(O\) (see diagram). In a simplified model of the motion, the aircraft is treated as a particle and the force of the air on the aircraft is taken to act vertically upwards with magnitude 8 N . Find
  1. the tension in the string,
  2. the speed of the aircraft.
CAIE M2 2003 June Q6
9 marks Moderate -0.3
6 A particle is projected with speed \(60 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a point on horizontal ground. The angle of projection is \(\alpha ^ { \circ }\) above the horizontal. The particle reaches the ground again after 10 s .
  1. Find the value of \(\alpha\).
  2. Find the greatest height reached by the particle.
  3. At time \(T\) s after the instant of projection the direction of motion of the particle is at an angle of \(45 ^ { \circ }\) above the horizontal. Find the value of \(T\).
CAIE M2 2003 June Q7
11 marks Standard +0.3
7 \includegraphics[max width=\textwidth, alt={}, center]{7f8646df-a7d8-4ca1-a6ee-3ceab6bb83af-4_232_905_762_621} A light elastic string has natural length 10 m and modulus of elasticity 130 N . The ends of the string are attached to fixed points \(A\) and \(B\), which are at the same horizontal level. A small stone is attached to the mid-point of the string and hangs in equilibrium at a point 2.5 m below \(A B\), as shown in the diagram. With the stone in this position the length of the string is 13 m .
  1. Find the tension in the string.
  2. Show that the mass of the stone is 3 kg . The stone is now held at rest at a point 8 m vertically below the mid-point of \(A B\).
  3. Find the elastic potential energy of the string in this position.
  4. The stone is now released. Find the speed with which it passes through the mid-point of \(A B\).
CAIE M2 2004 June Q1
4 marks Standard +0.3
1 \includegraphics[max width=\textwidth, alt={}, center]{835616aa-0b2b-4e8c-bbbf-60b72dc5ea3e-2_182_843_264_651} A uniform rigid plank has mass 10 kg and length 4 m . The plank has 0.9 m of its length in contact with a horizontal platform. A man \(M\) of mass 75 kg stands on the end of the plank which is in contact with the platform. A child \(C\) of mass 25 kg walks on to the overhanging part of the plank (see diagram). Find the distance between the man and the child when the plank is on the point of tilting.
CAIE M2 2004 June Q2
6 marks Standard +0.3
2 \includegraphics[max width=\textwidth, alt={}, center]{835616aa-0b2b-4e8c-bbbf-60b72dc5ea3e-2_291_732_822_708} A uniform lamina \(A B C D E\) consists of a rectangular part with sides 5 cm and 10 cm , and a part in the form of a quarter of a circle of radius 5 cm , as shown in the diagram.
  1. Show that the distance of the centre of mass of the part \(C D E\) of the lamina is \(\frac { 20 } { 3 \pi } \mathrm {~cm}\) from \(C E\).
  2. Find the distance of the centre of mass of the lamina \(A B C D E\) from the edge \(A B\).