Questions — CAIE (7646 questions)

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CAIE M1 2023 June Q5
9 marks Standard +0.3
\includegraphics{figure_5} Four coplanar forces act at a point. The magnitudes of the forces are \(F\) N, \(10\) N, \(50\) N and \(40\) N. The directions of the forces are as shown in the diagram.
  1. Given that the forces are in equilibrium, find the value of \(F\) and the value of \(\theta\). [6]
  2. Given instead that \(F = 10\sqrt{2}\) and \(\theta = 45\), find the direction and the exact magnitude the resultant force. [3]
CAIE M1 2023 June Q6
8 marks Standard +0.3
\includegraphics{figure_6} Two particles \(P\) and \(Q\), of masses \(0.2\) kg and \(0.1\) kg respectively, are attached to the ends of a light inextensible string. The string passes over a fixed smooth pulley \(B\) which is attached to two inclined planes. Particle \(P\) lies on a smooth plane \(AB\) which is inclined at \(60°\) to the horizontal. Particle \(Q\) lies on a plane \(BC\) which is inclined at an angle of \(\theta°\) to the horizontal. The string is taut and the particles can move on lines of greatest slope of the two planes (see diagram).
  1. It is given that \(\theta = 60\), the plane \(BC\) is rough and the coefficient of friction between \(Q\) and the plane \(BC\) is \(0.7\). The particles are released from rest. Determine whether the particles move. [4]
  2. It is given instead that the plane \(BC\) is smooth. The particles are released from rest and in the subsequent motion the tension in the string is \((\sqrt{3} - 1)\) N. Find the magnitude of the acceleration of \(P\) as it moves on the plane, and find the value of \(\theta\). [4]
CAIE M1 2023 June Q7
11 marks Standard +0.3
A car of mass \(1200\) kg is travelling along a straight horizontal road. The power of the car's engine is constant and is equal to \(16\) kW. There is a constant resistance to motion of magnitude \(500\) N.
  1. Find the acceleration of the car at an instant when its speed is \(20\) m s\(^{-1}\). [3]
  2. Assuming that the power and the resistance forces remain unchanged, find the steady speed at which the car can travel. [2]
The car comes to the bottom of a straight hill of length \(316\) m, inclined at an angle to the horizontal of \(\sin^{-1}(\frac{4}{65})\). The power remains constant at \(16\) kW, but the magnitude of the resistance force is no longer constant and changes such that the work done against the resistance force in ascending the hill is \(128400\) J. The time taken to ascend the hill is \(15\) s.
  1. Given that the car is travelling at a speed of \(20\) m s\(^{-1}\) at the bottom of the hill, find its speed at the top of the hill. [6]
CAIE M1 2024 June Q1
3 marks Standard +0.3
A cyclist and bicycle have a total mass of 72 kg. The cyclist rides along a horizontal road against a total resistance force of 28 N. Find the total work done by the cyclist to increase his speed from \(8\text{ ms}^{-1}\) to \(16\text{ ms}^{-1}\) while travelling a distance of 100 metres. [3]
CAIE M1 2024 June Q2
5 marks Moderate -0.3
A particle \(P\) moves in a straight line. At time \(t\) s after leaving a point \(O\) on the line, \(P\) has velocity \(v\text{ ms}^{-1}\), where \(v = 44t - 6t^2 - 36\).
  1. Find the set of values of \(t\) for which the acceleration of the particle is positive. [2]
  2. Find the two values of \(t\) at which \(P\) returns to \(O\). [3]
CAIE M1 2024 June Q3
6 marks Standard +0.3
\includegraphics{figure_3} Four coplanar forces of magnitude \(P\) N, 10 N, 16 N and 2 N act at a point in the directions shown in the diagram. It is given that the forces are in equilibrium. Find the values of \(\theta\) and \(P\). [6]
CAIE M1 2024 June Q4
7 marks Standard +0.3
A car has mass 1400 kg. When the speed of the car is \(v\text{ ms}^{-1}\) the magnitude of the resistance to motion is \(kv^2\) N where \(k\) is a constant.
  1. The car moves at a constant speed of \(24\text{ ms}^{-1}\) up a hill inclined at an angle of \(\alpha\) to the horizontal where \(\sin \alpha = 0.12\). At this speed the magnitude of the resistance to motion is 480 N.
    1. Find the value of \(k\). [1]
    2. Find the power of the car's engine. [3]
  2. The car now moves at a constant speed on a straight level road. Given that its engine is working at 54 kW, find this speed. [3]
CAIE M1 2024 June Q5
8 marks Standard +0.8
\includegraphics{figure_5} A particle of mass 0.8 kg lies on a rough plane which is inclined at an angle of \(28°\) to the horizontal. The particle is kept in equilibrium by a force of magnitude \(T\) N. This force acts at an angle of \(35°\) above a line of greatest slope of the plane (see diagram). The coefficient of friction between the particle and the plane is 0.2. Find the least and greatest possible values of \(T\). [8]
CAIE M1 2024 June Q6
11 marks Standard +0.3
Three particles \(A\), \(B\) and \(C\) of masses 5 kg, 1 kg and 2 kg respectively lie at rest in that order on a straight smooth horizontal track \(XYZ\). Initially \(A\) is at \(X\), \(B\) is at \(Y\) and \(C\) is at \(Z\). Particle \(A\) is projected towards \(B\) with a speed of \(6\text{ ms}^{-1}\) and at the same instant \(C\) is projected towards \(B\) with a speed of \(v\text{ ms}^{-1}\). In the subsequent motion, \(A\) collides and coalesces with \(B\) to form particle \(D\). Particle \(D\) then collides and coalesces with \(C\) to form particle \(E\) and \(E\) moves towards \(Z\).
  1. Show that after the second collision the speed of \(E\) is \(\frac{15-v}{4}\text{ ms}^{-1}\). [3]
  2. The total loss of kinetic energy of the system due to the two collisions is 63 J. Use the result from (a) to show that \(v = 3\). [3]
  3. It is given that the distance \(XY\) is 36 m and the distance \(YZ\) is 98 m.
    1. Find the time between the two collisions. [4]
    2. Find the time between the instant that \(A\) is projected from \(X\) and the instant that \(E\) reaches \(Z\). [1]
CAIE M1 2024 June Q7
10 marks Standard +0.3
\includegraphics{figure_7} Two particles \(P\) and \(Q\) of masses 2.5 kg and 0.5 kg respectively are connected by a light inextensible string that passes over a small smooth pulley fixed at the top of a plane inclined at an angle of \(30°\) to the horizontal. Particle \(P\) is on the plane and \(Q\) hangs below the pulley such that the level of \(Q\) is 2 m below the level of \(P\) (see diagram). Particle \(P\) is released from rest with the string taut and slides down the plane. The plane is rough with coefficient of friction 0.2 between the plane and \(P\).
  1. Find the acceleration of \(P\). [5]
  2. Use an energy method to find the speed of the particles at the instant when they are at the same vertical height. [5]
CAIE M1 2023 March Q1
5 marks Moderate -0.8
A crate of mass 200 kg is being pulled at constant speed along horizontal ground by a horizontal rope attached to a winch. The winch is working at a constant rate of 4.5 kW and there is a constant resistance to the motion of the crate of magnitude 600 N.
  1. Find the time that it takes for the crate to move a distance of 15 m. [2] The rope breaks after the crate has moved 15 m.
  2. Find the time taken, after the rope breaks, for the crate to come to rest. [3]
CAIE M1 2023 March Q2
5 marks Moderate -0.8
A particle \(P\) is projected vertically upwards from horizontal ground with speed \(15\,\text{m}\,\text{s}^{-1}\).
  1. Find the speed of \(P\) when it is 10 m above the ground. [2] At the same instant that \(P\) is projected, a second particle \(Q\) is dropped from a height of 18 m above the ground in the same vertical line as \(P\).
  2. Find the height above the ground at which the two particles collide. [3]
CAIE M1 2023 March Q3
5 marks Standard +0.3
A particle moves in a straight line starting from rest from a point \(O\). The acceleration of the particle at time \(t\) s after leaving \(O\) is \(a\,\text{m}\,\text{s}^{-2}\), where \(a = 4t^2\).
  1. Find the speed of the particle when \(t = 9\). [2]
  2. Find the time after leaving \(O\) at which the speed (in metres per second) and the distance travelled (in metres) are numerically equal. [3]
CAIE M1 2023 March Q4
7 marks Moderate -0.3
A toy railway locomotive of mass 0.8 kg is towing a truck of mass 0.4 kg on a straight horizontal track at a constant speed of \(2\,\text{m}\,\text{s}^{-1}\). There is a constant resistance force of magnitude 0.2 N on the locomotive, but no resistance force on the truck. There is a light rigid horizontal coupling connecting the locomotive and the truck.
  1. State the tension in the coupling. [1]
  2. Find the power produced by the locomotive's engine. [1] The power produced by the locomotive's engine is now changed to 1.2 W.
  3. Find the magnitude of the tension in the coupling at the instant that the locomotive begins to accelerate. [5]
CAIE M1 2023 March Q5
6 marks Standard +0.3
\includegraphics{figure_5} The diagram shows a block \(D\) of mass 100 kg supported by two sloping struts \(AD\) and \(BD\), each attached at an angle of \(45°\) to fixed points \(A\) and \(B\) respectively on a horizontal floor. The block is also held in place by a vertical rope \(CD\) attached to a fixed point \(C\) on a horizontal ceiling. The tension in the rope \(CD\) is 500 N and the block rests in equilibrium.
  1. Find the magnitude of the force in each of the struts \(AD\) and \(BD\). [3] A horizontal force of magnitude \(F\) N is applied to the block in a direction parallel to \(AB\).
  2. Find the value of \(F\) for which the magnitude of the force in the strut \(AD\) is zero. [3]
CAIE M1 2023 March Q6
9 marks Standard +0.3
\includegraphics{figure_6} A block \(B\), of mass 2 kg, lies on a rough inclined plane sloping at \(30°\) to the horizontal. A light rope, inclined at an angle of \(20°\) above a line of greatest slope, is attached to \(B\). The tension in the rope is \(T\) N. There is a friction force of \(F\) N acting on \(B\) (see diagram). The coefficient of friction between \(B\) and the plane is \(\mu\).
  1. It is given that \(F = 5\) and that the acceleration of \(B\) up the plane is \(1.2\,\text{m}\,\text{s}^{-2}\).
    1. Find the value of \(T\). [3]
    2. Find the value of \(\mu\). [3]
  2. It is given instead that \(\mu = 0.8\) and \(T = 15\). Determine whether \(B\) will move up the plane. [3]
CAIE M1 2023 March Q7
13 marks Standard +0.3
\includegraphics{figure_7} The diagram shows a smooth track which lies in a vertical plane. The section \(AB\) is a quarter circle of radius 1.8 m with centre \(O\). The section \(BC\) is a horizontal straight line of length 7.0 m and \(OB\) is perpendicular to \(BC\). The section \(CFE\) is a straight line inclined at an angle of \(\theta°\) above the horizontal. A particle \(P\) of mass 0.5 kg is released from rest at \(A\). Particle \(P\) collides with a particle \(Q\) of mass 0.1 kg which is at rest at \(B\). Immediately after the collision, the speed of \(P\) is \(4\,\text{m}\,\text{s}^{-1}\) in the direction \(BC\). You should assume that \(P\) is moving horizontally when it collides with \(Q\).
  1. Show that the speed of \(Q\) immediately after the collision is \(10\,\text{m}\,\text{s}^{-1}\). [4] When \(Q\) reaches \(C\), it collides with a particle \(R\) of mass 0.4 kg which is at rest at \(C\). The two particles coalesce. The combined particle comes instantaneously to rest at \(F\). You should assume that there is no instantaneous change in speed as the combined particle leaves \(C\), nor when it passes through \(C\) again as it returns down the slope.
  2. Given that the distance \(CF\) is 0.4 m, find the value of \(\theta\). [4]
  3. Find the distance from \(B\) at which \(P\) collides with the combined particle. [5]
CAIE M1 2024 March Q1
5 marks Easy -1.3
\includegraphics{figure_1} The displacement of a particle at time \(t\) s after leaving a fixed point \(O\) is \(s\) m. The diagram shows a displacement-time graph which models the motion of the particle. The graph consists of 4 straight line segments. The particle travels 50 m in the first 10 s, then travels at \(2\) m s\(^{-1}\) for a period of 10 s. The particle then comes to rest for a period of 20 s, before returning to its starting point when \(t = 60\).
  1. Find the velocity of the particle during the last 20 s of its motion. [2]
  2. Sketch a velocity-time graph for the motion of the particle from \(t = 0\) to \(t = 60\). [3]
CAIE M1 2024 March Q2
4 marks Moderate -0.8
A particle is projected vertically upwards from horizontal ground. The speed of the particle 2 seconds after it is projected is \(5\) m s\(^{-1}\) and it is travelling downwards.
  1. Find the speed of projection of the particle. [2]
  2. Find the distance travelled by the particle between the two times at which its speed is \(10\) m s\(^{-1}\). [2]
CAIE M1 2024 March Q3
5 marks Standard +0.3
A crate of mass 600 kg is being pulled up a line of greatest slope of a rough plane at a constant speed of \(2\) m s\(^{-1}\) by a rope attached to a winch. The plane is inclined at an angle of \(30°\) to the horizontal and the rope is parallel to the plane. The winch is working at a constant rate of 8 kW. Find the coefficient of friction between the crate and the plane. [5]
CAIE M1 2024 March Q4
6 marks Moderate -0.3
\includegraphics{figure_4} Four coplanar forces act at a point. The magnitudes of the forces are \(F\) N, \(2F\) N, \(3F\) N and \(30\) N. The directions of the forces are as shown in the diagram. Given that the forces are in equilibrium, find the value of \(F\) and the value of \(\theta\). [6]
CAIE M1 2024 March Q5
8 marks Standard +0.3
A particle moves in a straight line starting from a point \(O\). The velocity \(v\) m s\(^{-1}\) of the particle \(t\) s after leaving \(O\) is given by $$v = t^3 - \frac{9}{2}t^2 + 1 \text{ for } 0 \leqslant t \leqslant 4.$$ You may assume that the velocity of the particle is positive for \(t < \frac{1}{2}\), is zero at \(t = \frac{1}{2}\) and is negative for \(t > \frac{1}{2}\).
  1. Find the distance travelled between \(t = 0\) and \(t = \frac{1}{2}\). [4]
  2. Find the positive value of \(t\) at which the acceleration is zero. Hence find the total distance travelled between \(t = 0\) and this instant. [4]
CAIE M1 2024 March Q6
10 marks Standard +0.3
A car of mass 1800 kg is towing a trailer of mass 300 kg up a straight road inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = 0.05\). The car and trailer are connected by a tow-bar which is light and rigid and is parallel to the road. There is a resistance force of 800 N acting on the car and a resistance force of \(F\) N acting on the trailer. The driving force of the car's engine is 3000 N.
  1. It is given that \(F = 100\). Find the acceleration of the car and the tension in the tow-bar. [5]
  2. It is given instead that the total work done against \(F\) in moving a distance of 50 m up the road is 6000 J. The speed of the car at the start of the 50 m is \(20\) m s\(^{-1}\). Use an energy method to find the speed of the car at the end of the 50 m. [5]
CAIE M1 2024 March Q7
12 marks Standard +0.8
\includegraphics{figure_7} The diagram shows two particles \(P\) and \(Q\) which lie on a line of greatest slope of a plane \(ABC\). Particles \(P\) and \(Q\) are each of mass \(m\) kg. The plane is inclined at an angle \(\theta\) to the horizontal, where \(\sin \theta = 0.6\). The length of \(AB\) is 0.75 m and the length of \(BC\) is 3.25 m. The section \(AB\) of the plane is smooth and the section \(BC\) is rough. The coefficient of friction between each particle and the section \(BC\) is 0.25. Particle \(P\) is released from rest at \(A\). At the same instant, particle \(Q\) is released from rest at \(B\).
  1. Verify that particle \(P\) reaches \(B\) 0.5 s after it is released, with speed \(3\) m s\(^{-1}\). [3]
  2. Find the time that it takes from the instant the two particles are released until they collide. [4]
The two particles coalesce when they collide. The coefficient of friction between the combined particle and the plane is still 0.25.
  1. Find the time that it takes from the instant the particles collide until the combined particle reaches \(C\). [5]
CAIE M1 2020 November Q1
5 marks Moderate -0.8
A particle \(B\) of mass 5 kg is at rest on a smooth horizontal table. A particle \(A\) of mass 2.5 kg moves on the table with a speed of \(6 \text{ m s}^{-1}\) and collides directly with \(B\). In the collision the two particles coalesce.
  1. Find the speed of the combined particle after the collision. [2]
  2. Find the loss of kinetic energy of the system due to the collision. [3]