Questions M1 (2067 questions)

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CAIE M1 2017 June Q2
1 marks Easy -1.2
\includegraphics{figure_2} The diagram shows a wire \(ABCD\) consisting of a straight part \(AB\) of length \(5\) m and a part \(BCD\) in the shape of a semicircle of radius \(6\) m and centre \(O\). The diameter \(BD\) of the semicircle is horizontal and \(AB\) is vertical. A small ring is threaded onto the wire and slides along the wire. The ring starts from rest at \(A\). The part \(AB\) of the wire is rough, and the ring accelerates at a constant rate of \(2.5\) m s\(^{-2}\) between \(A\) and \(B\).
  1. Show that the speed of the ring as it reaches \(B\) is \(5\) m s\(^{-1}\). [1]
CAIE M1 2017 June Q3
9 marks Standard +0.3
A particle \(A\) moves in a straight line with constant speed \(10\) m s\(^{-1}\). Two seconds after \(A\) passes a point \(O\) on the line, a particle \(B\) passes through \(O\), moving along the line in the same direction as \(A\). Particle \(B\) has speed \(16\) m s\(^{-1}\) at \(O\) and has a constant deceleration of \(2\) m s\(^{-2}\).
  1. Find expressions, in terms of \(t\), for the displacement from \(O\) of each particle \(t\) s after \(B\) passes through \(O\). [3]
  2. Find the distance between the particles when \(B\) comes to instantaneous rest. [3]
  3. Find the minimum distance between the particles. [3]
CAIE M1 2017 June Q4
9 marks Moderate -0.3
A car of mass \(1200\) kg is moving on a straight road against a constant force of \(850\) N resisting the motion.
  1. On a part of the road that is horizontal, the car moves with a constant speed of \(42\) m s\(^{-1}\).
    1. Calculate, in kW, the power developed by the engine of the car. [2]
    2. Given that this power is suddenly increased by \(6\) kW, find the instantaneous acceleration of the car. [3]
  2. On a part of the road that is inclined at \(\theta°\) to the horizontal, the car moves up the hill at a constant speed of \(24\) m s\(^{-1}\), with the engine working at \(80\) kW. Find \(\theta\). [4]
CAIE M1 2017 June Q5
8 marks Challenging +1.2
\includegraphics{figure_5} A particle of mass \(0.12\) kg is placed on a plane which is inclined at an angle of \(40°\) to the horizontal. The particle is kept in equilibrium by a force of magnitude \(P\) N acting up the plane at an angle of \(30°\) above a line of greatest slope, as shown in the diagram. The coefficient of friction between the particle and the plane is \(0.32\). Find the set of possible values of \(P\). [8]
CAIE M1 2017 June Q6
14 marks Standard +0.3
\includegraphics{figure_6} The diagram shows a fixed block with a horizontal top surface and a surface which is inclined at an angle of \(\theta°\) to the horizontal, where \(\sin \theta = \frac{3}{5}\). A particle \(A\) of mass \(0.3\) kg rests on the horizontal surface and is attached to one end of a light inextensible string. The string passes over a small smooth pulley \(P\) fixed at the edge of the block. The other end of the string is attached to a particle \(B\) of mass \(1.5\) kg which rests on the sloping surface of the block. The system is released from rest with the string taut.
  1. Given that the block is smooth, find the acceleration of particle \(A\) and the tension in the string. [5]
  2. It is given instead that the block is rough. The coefficient of friction between \(A\) and the block is \(\mu\) and the coefficient of friction between \(B\) and the block is also \(\mu\). In the first \(3\) seconds of the motion, \(A\) does not reach \(P\) and \(B\) does not reach the bottom of the sloping surface. The speed of the particles after \(3\) s is \(5\) m s\(^{-1}\). Find the acceleration of particle \(A\) and the value of \(\mu\). [9]
CAIE M1 2018 June Q1
3 marks Easy -1.2
A particle \(P\) is projected vertically upwards with speed \(24 \text{ m s}^{-1}\) from a point \(5 \text{ m}\) above ground level. Find the time from projection until \(P\) reaches the ground. [3]
CAIE M1 2018 June Q2
4 marks Standard +0.3
\includegraphics{figure_2} The diagram shows three coplanar forces acting at the point \(O\). The magnitudes of the forces are \(6 \text{ N}\), \(8 \text{ N}\) and \(10 \text{ N}\). The angle between the \(6 \text{ N}\) force and the \(8 \text{ N}\) force is \(90°\). The forces are in equilibrium. Find the other angles between the forces. [4]
CAIE M1 2018 June Q3
6 marks Standard +0.3
\includegraphics{figure_3} A particle \(P\) of mass \(8 \text{ kg}\) is on a smooth plane inclined at an angle of \(30°\) to the horizontal. A force of magnitude \(100 \text{ N}\), making an angle of \(\theta°\) with a line of greatest slope and lying in the vertical plane containing the line of greatest slope, acts on \(P\) (see diagram).
  1. Given that \(P\) is in equilibrium, show that \(\theta = 66.4\), correct to \(1\) decimal place, and find the normal reaction between the plane and \(P\). [4]
  2. Given instead that \(\theta = 30\), find the acceleration of \(P\). [2]
CAIE M1 2018 June Q4
7 marks Moderate -0.8
A particle \(P\) moves in a straight line starting from a point \(O\). At time \(t \text{ s}\) after leaving \(O\), the displacement \(s \text{ m}\) from \(O\) is given by \(s = t^3 - 4t^2 + 4t\) and the velocity is \(v \text{ m s}^{-1}\).
  1. Find an expression for \(v\) in terms of \(t\). [2]
  2. Find the two values of \(t\) for which \(P\) is at instantaneous rest. [2]
  3. Find the minimum velocity of \(P\). [3]
CAIE M1 2018 June Q5
8 marks Moderate -0.8
A sprinter runs a race of \(200 \text{ m}\). His total time for running the race is \(20 \text{ s}\). He starts from rest and accelerates uniformly for \(6 \text{ s}\), reaching a speed of \(12 \text{ m s}^{-1}\). He maintains this speed for the next \(10 \text{ s}\), before decelerating uniformly to cross the finishing line with speed \(V \text{ m s}^{-1}\).
  1. Find the distance travelled by the sprinter in the first \(16 \text{ s}\) of the race. Hence sketch a displacement-time graph for the \(20 \text{ s}\) of the sprinter's race. [6]
  2. Find the value of \(V\). [2]
CAIE M1 2018 June Q6
10 marks Standard +0.3
A car has mass \(1250 \text{ kg}\).
  1. The car is moving along a straight level road at a constant speed of \(36 \text{ m s}^{-1}\) and is subject to a constant resistance of magnitude \(850 \text{ N}\). Find, in kW, the rate at which the engine of the car is working. [2]
  2. The car travels at a constant speed up a hill and is subject to the same resistance as in part (i). The hill is inclined at an angle of \(\theta°\) to the horizontal, where \(\sin \theta° = 0.1\), and the engine is working at \(63 \text{ kW}\). Find the speed of the car. [3]
  3. The car descends the same hill with the engine of the car working at a constant rate of \(20 \text{ kW}\). The resistance is not constant. The initial speed of the car is \(20 \text{ m s}^{-1}\). Eight seconds later the car has speed \(24 \text{ m s}^{-1}\) and has moved \(176 \text{ m}\) down the hill. Use an energy method to find the total work done against the resistance during the eight seconds. [5]
CAIE M1 2018 June Q7
12 marks Standard +0.3
\includegraphics{figure_7} The diagram shows a triangular block with sloping faces inclined to the horizontal at \(45°\) and \(30°\). Particle \(A\) of mass \(0.8 \text{ kg}\) lies on the face inclined at \(45°\) and particle \(B\) of mass \(1.2 \text{ kg}\) lies on the face inclined at \(30°\). The particles are connected by a light inextensible string which passes over a small smooth pulley \(P\) fixed at the top of the faces. The parts \(AP\) and \(BP\) of the string are parallel to lines of greatest slope of the respective faces. The particles are released from rest with both parts of the string taut. In the subsequent motion neither particle reaches the pulley and neither particle reaches the bottom of a face.
  1. Given that both faces are smooth, find the speed of \(A\) after each particle has travelled a distance of \(0.4 \text{ m}\). [6]
  2. It is given instead that both faces are rough. The coefficient of friction between each particle and a face of the block is \(\mu\). Find the value of \(\mu\) for which the system is in limiting equilibrium. [6]
CAIE M1 2018 June Q1
5 marks Easy -1.2
\includegraphics{figure_1} The diagram shows the velocity-time graph for a train which travels from rest at one station to rest at the next station. The graph consists of three straight line segments. The distance between the two stations is 9040 m.
  1. Find the acceleration of the train during the first 40 s. [1]
  2. Find the length of time for which the train is travelling at constant speed. [2]
  3. Find the distance travelled by the train while it is decelerating. [2]
CAIE M1 2018 June Q2
5 marks Moderate -0.8
A small ball is projected vertically downwards with speed \(5\text{ m s}^{-1}\) from a point \(A\) at a height of \(7.2\text{ m}\) above horizontal ground. The ball hits the ground with speed \(V\text{ m s}^{-1}\) and rebounds vertically upwards with speed \(\frac{1}{2}V\text{ m s}^{-1}\). The highest point the ball reaches after rebounding is \(B\). Find \(V\) and hence find the total time taken for the ball to reach the ground from \(A\) and rebound to \(B\). [5]
CAIE M1 2018 June Q3
6 marks Moderate -0.3
\includegraphics{figure_3} Coplanar forces of magnitudes 8 N, 12 N and 18 N act at a point in the directions shown in the diagram. Find the magnitude and direction of the single additional force acting at the same point which will produce equilibrium. [6]
CAIE M1 2018 June Q4
6 marks Standard +0.3
\includegraphics{figure_4} Two particles \(A\) and \(B\), of masses \(0.8\text{ kg}\) and \(1.6\text{ kg}\) respectively, are connected by a light inextensible string. Particle \(A\) is placed on a smooth plane inclined at an angle \(\theta\) to the horizontal, where \(\sin \theta = \frac{3}{5}\). The string passes over a small smooth pulley \(P\) fixed at the top of the plane, and \(B\) hangs freely (see diagram). The section \(AP\) of the string is parallel to a line of greatest slope of the plane. The particles are released from rest with both sections of the string taut. Use an energy method to find the speed of the particles after each particle has moved a distance of \(0.5\text{ m}\), assuming that \(A\) has not yet reached the pulley. [6]
CAIE M1 2018 June Q5
6 marks Standard +0.3
A particle of mass \(3\text{ kg}\) is on a rough plane inclined at an angle of \(20°\) to the horizontal. A force of magnitude \(P\text{ N}\) acting parallel to a line of greatest slope of the plane is used to keep the particle in equilibrium. The coefficient of friction between the particle and the plane is \(0.35\). Show that the least possible value of \(P\) is \(0.394\), correct to 3 significant figures, and find the greatest possible value of \(P\). [6]
CAIE M1 2018 June Q6
9 marks Standard +0.3
A car of mass \(1400\text{ kg}\) travelling at a speed of \(v\text{ m s}^{-1}\) experiences a resistive force of magnitude \(40v\text{ N}\). The greatest possible constant speed of the car along a straight level road is \(56\text{ m s}^{-1}\).
  1. Find, in kW, the greatest possible power of the car's engine. [2]
  2. Find the greatest possible acceleration of the car at an instant when its speed on a straight level road is \(32\text{ m s}^{-1}\). [3]
  3. The car travels down a hill inclined at an angle of \(\theta°\) to the horizontal at a constant speed of \(50\text{ m s}^{-1}\). The power of the car's engine is \(60\text{ kW}\). Find the value of \(\theta\). [4]
CAIE M1 2018 June Q7
13 marks Moderate -0.3
A particle \(P\) moves in a straight line starting from a point \(O\). The velocity \(v\text{ m s}^{-1}\) of \(P\) at time \(t\text{ s}\) is given by $$v = 12t - 4t^2 \quad \text{for } 0 \leqslant t \leqslant 2,$$ $$v = 16 - 4t \quad \text{for } 2 \leqslant t \leqslant 4.$$
  1. Find the maximum velocity of \(P\) during the first \(2\text{ s}\). [3]
  2. Determine, with justification, whether there is any instantaneous change in the acceleration of \(P\) when \(t = 2\). [2]
  3. Sketch the velocity-time graph for \(0 \leqslant t \leqslant 4\). [3]
  4. Find the distance travelled by \(P\) in the interval \(0 \leqslant t \leqslant 4\). [5]
CAIE M1 2019 June Q1
6 marks Standard +0.3
\includegraphics{figure_1} Coplanar forces of magnitudes 40 N, 32 N, \(P\) N and 17 N act at a point in the directions shown in the diagram. The system is in equilibrium. Find the values of \(P\) and \(\theta\). [6]
CAIE M1 2019 June Q2
6 marks Moderate -0.5
A car moves in a straight line with initial speed \(u\) m s\(^{-1}\) and constant acceleration \(a\) m s\(^{-2}\). The car takes 5 s to travel the first 80 m and it takes 8 s to travel the first 160 m. Find \(a\) and \(u\). [6]
CAIE M1 2019 June Q3
5 marks Moderate -0.3
A particle of mass 13 kg is on a rough plane inclined at an angle of \(\theta\) to the horizontal, where \(\tan \theta = \frac{5}{12}\). The coefficient of friction between the particle and the plane is 0.3. A force of magnitude \(T\) N, acting parallel to a line of greatest slope, moves the particle a distance of 2.5 m up the plane at a constant speed. Find the work done by this force. [5]
CAIE M1 2019 June Q4
7 marks Moderate -0.3
A constant resistance to motion of magnitude 350 N acts on a car of mass 1250 kg. The engine of the car exerts a constant driving force of 1200 N. The car travels along a road inclined at an angle of \(\theta\) to the horizontal, where \(\sin \theta = 0.05\). Find the speed of the car when it has moved 100 m from rest in each of the following cases. • The car is moving up the hill. • The car is moving down the hill. [7]
CAIE M1 2019 June Q5
8 marks Standard +0.3
\includegraphics{figure_5} Two particles \(A\) and \(B\), of masses 0.4 kg and 0.2 kg respectively, are connected by a light inextensible string which passes over a fixed smooth pulley. Both \(A\) and \(B\) are 0.5 m above the ground. The particles hang vertically (see diagram). The particles are released from rest. In the subsequent motion \(B\) does not reach the pulley and \(A\) remains at rest after reaching the ground.
  1. For the motion before \(A\) reaches the ground, show that the magnitude of the acceleration of each particle is \(\frac{10}{3}\) m s\(^{-2}\) and find the tension in the string. [4]
  2. Find the maximum height of \(B\) above the ground. [4]
CAIE M1 2019 June Q6
7 marks Standard +0.8
A car has mass 1000 kg. When the car is travelling at a steady speed of \(v\) m s\(^{-1}\), where \(v > 2\), the resistance to motion of the car is \((Av + B)\) N, where \(A\) and \(B\) are constants. The car can travel along a horizontal road at a steady speed of 18 m s\(^{-1}\) when its engine is working at 36 kW. The car can travel up a hill inclined at an angle of \(\theta\) to the horizontal, where \(\sin \theta = 0.05\), at a steady speed of 12 m s\(^{-1}\) when its engine is working at 21 kW. Find \(A\) and \(B\). [7]