Questions — CAIE (7646 questions)

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CAIE M1 2007 November Q4
6 marks Moderate -0.8
\includegraphics{figure_4} The diagram shows the vertical cross-section of a surface. \(A\) and \(B\) are two points on the cross-section, and \(A\) is 5 m higher than \(B\). A particle of mass \(0.35\) kg passes through \(A\) with speed \(7 \text{ m s}^{-1}\), moving on the surface towards \(B\).
  1. Assuming that there is no resistance to motion, find the speed with which the particle reaches \(B\). [3]
  2. Assuming instead that there is a resistance to motion, and that the particle reaches \(B\) with speed \(11 \text{ m s}^{-1}\), find the work done against this resistance as the particle moves from \(A\) to \(B\). [3]
CAIE M1 2007 November Q5
7 marks Moderate -0.3
\includegraphics{figure_5} A ring of mass 4 kg is threaded on a fixed rough vertical rod. A light string is attached to the ring, and is pulled with a force of magnitude \(T\) N acting at an angle of \(60°\) to the downward vertical (see diagram). The ring is in equilibrium.
  1. The normal and frictional components of the contact force exerted on the ring by the rod are \(R\) N and \(F\) N respectively. Find \(R\) and \(F\) in terms of \(T\). [4]
  2. The coefficient of friction between the rod and the ring is 0.7. Find the value of \(T\) for which the ring is about to slip. [3]
CAIE M1 2007 November Q6
11 marks Standard +0.3
  1. A man walks in a straight line from \(A\) to \(B\) with constant acceleration \(0.004 \text{ m s}^{-2}\). His speed at \(A\) is \(1.8 \text{ m s}^{-1}\) and his speed at \(B\) is \(2.2 \text{ m s}^{-1}\). Find the time taken for the man to walk from \(A\) to \(B\), and find the distance \(AB\). [3]
  2. A woman cyclist leaves \(A\) at the same instant as the man. She starts from rest and travels in a straight line to \(B\), reaching \(B\) at the same instant as the man. At time \(t\) s after leaving \(A\) the cyclist's speed is \(k(200t - t^2) \text{ m s}^{-1}\), where \(k\) is a constant. Find
    1. the value of \(k\), [4]
    2. the cyclist's speed at \(B\). [1]
  3. Sketch, using the same axes, the velocity-time graphs for the man's motion and the woman's motion from \(A\) to \(B\). [3]
CAIE M1 2007 November Q7
11 marks Standard +0.3
\includegraphics{figure_7} A rough inclined plane of length 65 cm is fixed with one end at a height of 16 cm above the other end. Particles \(P\) and \(Q\), of masses \(0.13\) kg and \(0.11\) kg respectively, are attached to the ends of a light inextensible string which passes over a small smooth pulley at the top of the plane. Particle \(P\) is held at rest on the plane and particle \(Q\) hangs vertically below the pulley (see diagram). The system is released from rest and \(P\) starts to move up the plane.
  1. Draw a diagram showing the forces acting on \(P\) during its motion up the plane. [1]
  2. Show that \(T - F > 0.32\), where \(T\) N is the tension in the string and \(F\) N is the magnitude of the frictional force on \(P\). [4]
The coefficient of friction between \(P\) and the plane is 0.6.
  1. Find the acceleration of \(P\). [6]
CAIE M1 2017 November Q1
5 marks Moderate -0.8
A particle of mass 0.2 kg is resting in equilibrium on a rough plane inclined at \(20°\) to the horizontal.
  1. Show that the friction force acting on the particle is 0.684 N, correct to 3 significant figures. [1]
The coefficient of friction between the particle and the plane is 0.6. A force of magnitude 0.9 N is applied to the particle down a line of greatest slope of the plane. The particle accelerates down the plane.
  1. Find this acceleration. [4]
CAIE M1 2017 November Q2
6 marks Moderate -0.3
\includegraphics{figure_2} A block of mass 15 kg hangs in equilibrium below a horizontal ceiling attached to two strings as shown in the diagram. One of the strings is inclined at \(45°\) to the horizontal and the tension in this string is 120 N. The other string is inclined at \(θ°\) to the horizontal and the tension in this string is \(T\) N. Find the values of \(T\) and \(θ\). [6]
CAIE M1 2017 November Q3
6 marks Standard +0.3
A car travels along a straight road with constant acceleration. It passes through points \(A\), \(B\) and \(C\). The car passes point \(A\) with velocity 14 m s\(^{-1}\). The two sections \(AB\) and \(BC\) are of equal length. The times taken to travel along \(AB\) and \(BC\) are 5 s and 3 s respectively.
  1. Write down an expression for the distance \(AB\) in terms of the acceleration of the car. Write down a similar expression for the distance \(AC\). Hence show that the acceleration of the car is 4 m s\(^{-2}\). [4]
  2. Find the speed of the car as it passes point \(C\). [2]
CAIE M1 2017 November Q4
6 marks Standard +0.3
A particle \(P\) is projected vertically upwards from horizontal ground with speed 12 m s\(^{-1}\).
  1. Find the time taken for \(P\) to return to the ground. [2]
The time in seconds after \(P\) is projected is denoted by \(t\). When \(t = 1\), a second particle \(Q\) is projected vertically upwards with speed 10 m s\(^{-1}\) from a point which is 5 m above the ground. Particles \(P\) and \(Q\) move in different vertical lines.
  1. Find the set of values of \(t\) for which the two particles are moving in the same direction. [4]
CAIE M1 2017 November Q5
8 marks Standard +0.3
A cyclist is riding up a straight hill inclined at an angle \(α\) to the horizontal, where \(\sin α = 0.04\). The total mass of the bicycle and rider is 80 kg. The cyclist is riding at a constant speed of 4 m s\(^{-1}\). There is a force resisting the motion. The work done by the cyclist against this resistance force over a distance of 25 m is 600 J.
  1. Find the power output of the cyclist. [4]
The cyclist reaches the top of the hill, where the road becomes horizontal, with speed 4 m s\(^{-1}\). The cyclist continues to work at the same rate on the horizontal part of the road.
  1. Find the speed of the cyclist 10 seconds after reaching the top of the hill, given that the work done by the cyclist during this period against the resistance force is 1200 J. [4]
CAIE M1 2017 November Q6
10 marks Standard +0.3
\includegraphics{figure_6} Two particles \(P\) and \(Q\), each of mass \(m\) kg, are attached to the ends of a light inextensible string. The string passes over a fixed smooth pulley which is attached to the edge of a rough plane. The plane is inclined at an angle \(α\) to the horizontal, where \(\tan α = \frac{4}{3}\). Particle \(P\) rests on the plane and particle \(Q\) hangs vertically, as shown in the diagram. The string between \(P\) and the pulley is parallel to a line of greatest slope of the plane. The system is in limiting equilibrium.
  1. Show that the coefficient of friction between \(P\) and the plane is \(\frac{4}{3}\). [5]
A force of magnitude 10 N is applied to \(P\), acting up a line of greatest slope of the plane, and \(P\) accelerates at 2.5 m s\(^{-2}\).
  1. Find the value of \(m\). [5]
CAIE M1 2017 November Q7
9 marks Standard +0.3
A particle starts from rest and moves in a straight line. The velocity of the particle at time \(t\) s after the start is \(v\) m s\(^{-1}\), where $$v = -0.01t^3 + 0.22t^2 - 0.4t.$$
  1. Find the two positive values of \(t\) for which the particle is instantaneously at rest. [2]
  2. Find the time at which the acceleration of the particle is greatest. [3]
  3. Find the distance travelled by the particle while its velocity is positive. [4]
CAIE M1 2018 November Q1
4 marks Standard +0.3
A smooth ring \(R\) of mass \(m\) kg is threaded on a light inextensible string \(ARB\). The ends of the string are attached to fixed points \(A\) and \(B\) with \(A\) vertically above \(B\). The string is taut and angle \(ARB = 90°\). The angle between the part \(AR\) of the string and the vertical is \(45°\). The ring is held in equilibrium in this position by a force of magnitude \(2.5\) N, acting on the ring in the direction \(BR\) (see diagram). Calculate the tension in the string and the mass of the ring. [4] \includegraphics{figure_1}
CAIE M1 2018 November Q2
4 marks Moderate -0.8
A block of mass \(5\) kg is being pulled by a rope up a rough plane inclined at \(6°\) to the horizontal. The rope is parallel to a line of greatest slope of the plane and the block is moving at constant speed. The coefficient of friction between the block and the plane is \(0.3\). Find the tension in the rope. [4]
CAIE M1 2018 November Q3
7 marks Moderate -0.8
\includegraphics{figure_3} The velocity of a particle moving in a straight line is \(v\) m s\(^{-1}\) at time \(t\) seconds. The diagram shows a velocity-time graph which models the motion of the particle from \(t = 0\) to \(t = T\). The graph consists of four straight line segments. The particle reaches its maximum velocity \(V\) m s\(^{-1}\) at \(t = 10\).
  1. Find the acceleration of the particle during the first \(2\) seconds. [1]
  2. Find the value of \(V\). [2]
At \(t = 6\), the particle is instantaneously at rest at the point \(A\). At \(t = T\), the particle comes to rest at the point \(B\). At \(t = 0\) the particle starts from rest at a point one third of the way from \(A\) to \(B\).
  1. Find the distance \(AB\) and hence find the value of \(T\). [4]
CAIE M1 2018 November Q4
8 marks Standard +0.3
\includegraphics{figure_4} Two particles \(P\) and \(Q\), of masses \(0.4\) kg and \(0.7\) kg respectively, are attached to the ends of a light inextensible string. The string passes over a fixed smooth pulley which is attached to the edge of a rough plane. The coefficient of friction between \(P\) and the plane is \(0.5\). The plane is inclined at an angle \(\alpha\) to the horizontal, where \(\tan \alpha = \frac{3}{4}\). Particle \(P\) lies on the plane and particle \(Q\) hangs vertically. The string between \(P\) and the pulley is parallel to a line of greatest slope of the plane (see diagram). A force of magnitude \(X\) N, acting directly down the plane, is applied to \(P\).
  1. Show that the greatest value of \(X\) for which \(P\) remains stationary is \(6.2\). [4]
  2. Given instead that \(X = 0.8\), find the acceleration of \(P\). [4]
CAIE M1 2018 November Q5
8 marks Standard +0.3
A particle moves in a straight line starting from a point \(O\) with initial velocity \(1\) m s\(^{-1}\). The acceleration of the particle at time \(t\) s after leaving \(O\) is \(a\) m s\(^{-2}\), where $$a = 1.2t^{\frac{1}{2}} - 0.6t.$$
  1. At time \(T\) s after leaving \(O\) the particle reaches its maximum velocity. Find the value of \(T\). [2]
  2. Find the velocity of the particle when its acceleration is maximum (you do not need to verify that the acceleration is a maximum rather than a minimum). [6]
CAIE M1 2018 November Q6
8 marks Moderate -0.3
A car of mass \(1200\) kg is driving along a straight horizontal road at a constant speed of \(15\) m s\(^{-1}\). There is a constant resistance to motion of \(350\) N.
  1. Find the power of the car's engine. [1]
The car comes to a hill inclined at \(1°\) to the horizontal, still travelling at \(15\) m s\(^{-1}\).
  1. The car starts to descend the hill with reduced power and with an acceleration of \(0.12\) m s\(^{-2}\). Given that there is no change in the resistance force, find the new power of the car's engine at the instant when it starts to descend the hill. [3]
  2. When the car is travelling at \(20\) m s\(^{-1}\) down the hill, the power is cut off and the car gradually slows down. Assuming that the resistance force remains \(350\) N, find the distance travelled from the moment when the power is cut off until the speed of the car is reduced to \(18\) m s\(^{-1}\). [4]
CAIE M1 2018 November Q7
11 marks Standard +0.3
A particle of mass \(0.3\) kg is released from rest above a tank containing water. The particle falls vertically, taking \(0.8\) s to reach the water surface. There is no instantaneous change of speed when the particle enters the water. The depth of water in the tank is \(1.25\) m. The water exerts a force on the particle resisting its motion. The work done against this resistance force from the instant that the particle enters the water until it reaches the bottom of the tank is \(1.2\) J.
  1. Use an energy method to find the speed of the particle when it reaches the bottom of the tank. [4]
When the particle reaches the bottom of the tank, it bounces back vertically upwards with initial speed \(7\) m s\(^{-1}\). As the particle rises through the water, it experiences a constant resistance force of \(1.8\) N. The particle comes to instantaneous rest \(t\) seconds after it bounces on the bottom of the tank.
  1. Find the value of \(t\). [7]
CAIE M1 2019 November Q1
2 marks Easy -1.2
A crane is lifting a load of 1250 kg vertically at a constant speed \(V\) m s\(^{-1}\). Given that the power of the crane is a constant 20 kW, find the value of \(V\). [2]
CAIE M1 2019 November Q2
5 marks Standard +0.3
The total mass of a cyclist and her bicycle is 75 kg. The cyclist ascends a straight hill of length 0.7 km inclined at 1.5° to the horizontal. Her speed at the bottom of the hill is 10 m s\(^{-1}\) and at the top it is 5 m s\(^{-1}\). There is a resistance to motion, and the work done against this resistance as the cyclist ascends the hill is 2000 J. The cyclist exerts a constant force of magnitude \(F\) N in the direction of motion. Find \(F\). [5]
CAIE M1 2019 November Q3
7 marks Moderate -0.3
A block of mass 3 kg is at rest on a rough plane inclined at 60° to the horizontal. A force of magnitude 15 N acting up a line of greatest slope of the plane is just sufficient to prevent the block from sliding down the plane.
  1. Find the coefficient of friction between the block and the plane. [5]
The force of magnitude 15 N is now replaced by a force of magnitude \(X\) N acting up the line of greatest slope.
  1. Find the greatest value of \(X\) for which the block does not move. [2]
CAIE M1 2019 November Q4
7 marks Standard +0.3
\includegraphics{figure_4} Two blocks \(A\) and \(B\) of masses 4 kg and 5 kg respectively are joined by a light inextensible string. The blocks rest on a smooth plane inclined at an angle \(\alpha\) to the horizontal, where tan \(\alpha = \frac{7}{24}\). The string is parallel to a line of greatest slope of the plane with \(B\) above \(A\). A force of magnitude 36 N acts on \(B\), parallel to a line of greatest slope of the plane (see diagram).
  1. Find the acceleration of the blocks and the tension in the string. [5]
  1. At a particular instant, the speed of the blocks is 1 m s\(^{-1}\). Find the time, after this instant, that it takes for the blocks to travel 0.65 m. [2]
CAIE M1 2019 November Q5
8 marks Moderate -0.3
\includegraphics{figure_5} A small ring \(P\) is threaded on a fixed smooth horizontal rod \(AB\). Three horizontal forces of magnitudes 4.5 N, 7.5 N and \(F\) N act on \(P\) (see diagram).
  1. Given that these three forces are in equilibrium, find the values of \(F\) and \(\theta\). [6]
  1. It is given instead that the values of \(F\) and \(\theta\) are 9.5 and 30 respectively, and the acceleration of the ring is 1.5 m s\(^{-2}\). Find the mass of the ring. [2]
CAIE M1 2019 November Q6
9 marks Moderate -0.3
A particle of mass 0.4 kg is released from rest at a height of 1.8 m above the surface of the water in a tank. There is no instantaneous change of speed when the particle enters the water. The water exerts an upward force of 5.6 N on the particle when it is in the water.
  1. Find the velocity of the particle at the instant when it reaches the surface of the water. [2]
  1. Find the time that it takes from the instant when the particle enters the water until it comes to instantaneous rest in the water. You may assume that the tank is deep enough so that the particle does not reach the bottom of the tank. [4]
  1. Sketch a velocity-time graph for the motion of the particle from the instant at which it is released until it comes to instantaneous rest in the water. [3]
CAIE M1 2019 November Q7
12 marks Standard +0.3
A particle moves in a straight line, starting from rest at a point \(O\), and comes to instantaneous rest at a point \(P\). The velocity of the particle at time \(t\) s after leaving \(O\) is \(v\) m s\(^{-1}\), where $$v = 0.6t^2 - 0.12t^3.$$
  1. Show that the distance \(OP\) is 6.25 m. [5]
On another occasion, the particle also moves in the same straight line. On this occasion, the displacement of the particle at time \(t\) s after leaving \(O\) is \(s\) m, where $$s = kt^3 + ct^5.$$ It is given that the particle passes point \(P\) with velocity 1.25 m s\(^{-1}\) at time \(t = 5\).
  1. Find the values of the constants \(k\) and \(c\). [5]
  1. Find the acceleration of the particle at time \(t = 5\). [2]