Questions — CAIE (7646 questions)

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CAIE M1 2019 November Q1
4 marks Moderate -0.8
A particle moves in a straight line. The displacement of the particle at time \(t\) s is \(s\) m, where $$s = t^3 - 6t^2 + 4t.$$ Find the velocity of the particle at the instant when its acceleration is zero. [4]
CAIE M1 2019 November Q2
5 marks Moderate -0.8
\includegraphics{figure_2} The diagram shows a velocity-time graph which models the motion of a tractor. The graph consists of four straight line segments. The tractor passes a point \(O\) at time \(t = 0\) with speed \(U\) m s\(^{-1}\). The tractor accelerates to a speed of \(V\) m s\(^{-1}\) over a period of 5 s, and then travels at this speed for a further 25 s. The tractor then accelerates to a speed of 12 m s\(^{-1}\) over a period of 5 s. The tractor then decelerates to rest over a period of 15 s.
  1. Given that the acceleration of the tractor between \(t = 30\) and \(t = 35\) is 0.8 m s\(^{-2}\), find the value of \(V\). [2]
  2. Given also that the total distance covered by the tractor in the 50 seconds of motion is 375 m, find the value of \(U\). [3]
CAIE M1 2019 November Q3
5 marks Moderate -0.3
\includegraphics{figure_3} A particle \(P\) of mass 0.3 kg is held in equilibrium above a horizontal plane by a force of magnitude 5 N, acting vertically upwards. The particle is attached to two strings \(PA\) and \(PB\) of lengths 0.9 m and 1.2 m respectively. The points \(A\) and \(B\) lie on the plane and angle \(APB = 90°\) (see diagram). Find the tension in each of the strings. [5]
CAIE M1 2019 November Q4
7 marks Moderate -0.3
A lorry of mass 25 000 kg travels along a straight horizontal road. There is a constant force of 3000 N resisting the motion.
  1. Find the power required to maintain a constant speed of 30 m s\(^{-1}\). [2]
The lorry comes to a straight hill inclined at 2° to the horizontal. The driver switches off the engine of the lorry at the point \(A\) which is at the foot of the hill. Point \(B\) is further up the hill. The speeds of the lorry at \(A\) and \(B\) are 30 m s\(^{-1}\) and 25 m s\(^{-1}\) respectively. The resistance force is still 3000 N.
  1. Use an energy method to find the height of \(B\) above the level of \(A\). [5]
CAIE M1 2019 November Q5
7 marks Moderate -0.3
Two particles \(A\) and \(B\) move in the same vertical line. Particle \(A\) is projected vertically upwards from the ground with speed 20 m s\(^{-1}\). One second later particle \(B\) is dropped from rest from a height of 40 m.
  1. Find the height above the ground at which the two particles collide. [4]
  2. Find the difference in the speeds of the two particles at the instant when the collision occurs. [3]
CAIE M1 2019 November Q6
11 marks Standard +0.3
A block of mass 3 kg is initially at rest on a rough horizontal plane. A force of magnitude 6 N is applied to the block at an angle of \(\theta\) above the horizontal, where \(\cos \theta = \frac{24}{25}\). The force is applied for a period of 5 s, during which time the block moves a distance of 4.5 m.
  1. Find the magnitude of the frictional force on the block. [4]
  2. Show that the coefficient of friction between the block and the plane is 0.165, correct to 3 significant figures. [3]
  3. When the block has moved a distance of 4.5 m, the force of magnitude 6 N is removed and the block then decelerates to rest. Find the total time for which the block is in motion. [4]
CAIE M1 2019 November Q7
11 marks Standard +0.3
\includegraphics{figure_7} Two particles \(P\) and \(Q\), of masses 0.3 kg and 0.2 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 smooth plane. The plane is inclined at an angle \(\theta\) to the horizontal, where \(\sin \theta = \frac{3}{5}\). \(P\) lies on the plane and \(Q\) hangs vertically below the pulley at a height of 0.8 m above the floor (see diagram). The string between \(P\) and the pulley is parallel to a line of greatest slope of the plane. \(P\) is released from rest and \(Q\) moves vertically downwards.
  1. Find the tension in the string and the magnitude of the acceleration of the particles. [5]
\(Q\) hits the floor and does not bounce. It is given that \(P\) does not reach the pulley in the subsequent motion.
  1. Find the time, from the instant at which \(P\) is released, for \(Q\) to reach the floor. [2]
  2. When \(Q\) hits the floor the string becomes slack. Find the time, from the instant at which \(P\) is released, for the string to become taut again. [4]
CAIE M1 Specimen Q1
4 marks Easy -1.2
A weightlifter performs an exercise in which he raises a mass of 200 kg from rest vertically through a distance of 0.7 m and holds it at that height.
  1. Find the work done by the weightlifter. [2]
  2. Given that the time taken to raise the mass is 1.2 s, find the average power developed by the weightlifter. [2]
CAIE M1 Specimen Q2
6 marks Moderate -0.8
A particle of mass 0.5 kg starts from rest and slides down a line of greatest slope of a smooth plane. The plane is inclined at an angle of 30° to the horizontal.
  1. Find the time taken for the particle to reach a speed of 2.5 m s\(^{-1}\). [3]
  2. Find the distance that the particle travels along the ground before it comes to rest. [3]
When the particle has travelled 3 m down the slope from its starting point, it reaches rough horizontal ground at the bottom of the slope. The frictional force acting on the particle is 1 N.
CAIE M1 Specimen Q3
6 marks Standard +0.3
A lorry of mass 24 000 kg is travelling up a hill which is inclined at 3° to the horizontal. The power developed by the lorry's engine is constant, and there is a constant resistance to motion of 3200 N.
  1. When the speed of the lorry is 25 m s\(^{-1}\), its acceleration is 0.2 m s\(^{-2}\). Find the power developed by the lorry's engine. [4]
  2. Find the steady speed at which the lorry moves up the hill if the power is 500 kW and the resistance remains 3200 N. [2]
CAIE M1 Specimen Q4
6 marks Standard +0.3
\includegraphics{figure_4} Blocks \(P\) and \(Q\), of mass \(m\) kg and 5 kg respectively, are attached to the ends of a light inextensible string. The string passes over a small smooth pulley which is fixed at the top of a rough plane inclined at 35° to the horizontal. Block \(P\) is at rest on the plane and block \(Q\) hangs vertically below the pulley (see diagram). The coefficient of friction between block \(P\) and the plane is 0.2. Find the set of values of \(m\) for which the two blocks remain at rest. [6]
CAIE M1 Specimen Q5
8 marks Standard +0.3
\includegraphics{figure_5} A small bead \(Q\) can move freely along a smooth horizontal straight wire \(AB\) of length 3 m. Three horizontal forces of magnitudes \(F\) N, 10 N and 20 N act on the bead in the directions shown in the diagram. The magnitude of the resultant of the three forces is \(R\) N in the direction shown in the diagram.
  1. Find the values of \(F\) and \(R\). [5]
  2. Initially the bead is at rest at \(A\). It reaches \(B\) with a speed of 11.7 m s\(^{-1}\). Find the mass of the bead. [3]
CAIE M1 Specimen Q6
10 marks Standard +0.3
A particle \(P\) moves in a straight line, starting from a point \(O\). The velocity of \(P\), measured in m s\(^{-1}\), at time \(t\) s after leaving \(O\) is given by $$v = 0.6t - 0.03t^2.$$
  1. Verify that, when \(t = 5\), the particle is 6.25 m from \(O\). Find the acceleration of the particle at this time. [4]
  2. Find the values of \(t\) at which the particle is travelling at half of its maximum velocity. [6]
CAIE M1 Specimen Q7
10 marks Standard +0.3
A cyclist starts from rest at point \(A\) and moves in a straight line with acceleration 0.5 m s\(^{-2}\) for a distance of 36 m. The cyclist then travels at constant speed for 25 s before slowing down, with constant deceleration, to come to rest at point \(B\). The distance \(AB\) is 210 m.
  1. Find the total time that the cyclist takes to travel from \(A\) to \(B\). [5]
  2. Find the time that it takes from when the cyclist starts until the car overtakes her. [5]
24 s after the cyclist leaves point \(A\), a car starts from rest from point \(A\), with constant acceleration 4 m s\(^{-2}\) towards \(B\). It is given that the car overtakes the cyclist while the cyclist is moving with constant speed.
CAIE M2 2010 June Q1
4 marks Moderate -0.3
\includegraphics{figure_1} A frame consists of a uniform semicircular wire of radius 20 cm and mass 2 kg, and a uniform straight wire of length 40 cm and mass 0.9 kg. The ends of the semicircular wire are attached to the ends of the straight wire (see diagram). Find the distance of the centre of mass of the frame from the straight wire. [4]
CAIE M2 2010 June Q2
5 marks Standard +0.3
\includegraphics{figure_2} A uniform solid cone has height 30 cm and base radius \(r\) cm. The cone is placed with its axis vertical on a rough horizontal plane. The plane is slowly tilted and the cone remains in equilibrium until the angle of inclination of the plane reaches \(35°\), when the cone topples. The diagram shows a cross-section of the cone.
  1. Find the value of \(r\). [3]
  2. Show that the coefficient of friction between the cone and the plane is greater than 0.7. [2]
CAIE M2 2010 June Q3
6 marks Standard +0.3
\includegraphics{figure_3} A particle of mass 0.24 kg is attached to one end of a light inextensible string of length 2 m. The other end of the string is attached to a fixed point. The particle moves with constant speed in a horizontal circle. The string makes an angle \(\theta\) with the vertical (see diagram), and the tension in the string is \(T\) N. The acceleration of the particle has magnitude \(7.5 \text{ m s}^{-2}\).
  1. Show that \(\tan \theta = 0.75\) and find the value of \(T\). [4]
  2. Find the speed of the particle. [2]
CAIE M2 2010 June Q4
5 marks Standard +0.3
\includegraphics{figure_4} A uniform lamina of weight 15 N is in the form of a trapezium \(ABCD\) with dimensions as shown in the diagram. The lamina is freely hinged at \(A\) to a fixed point. One end of a light inextensible string is attached to the lamina at \(B\). The lamina is in equilibrium with \(AB\) horizontal; the string is taut and in the same vertical plane as the lamina, and makes an angle of \(30°\) upwards from the horizontal (see diagram). Find the tension in the string. [5]
CAIE M2 2010 June Q5
9 marks Standard +0.3
A particle is projected from a point \(O\) on horizontal ground. The velocity of projection has magnitude \(20 \text{ m s}^{-1}\) and direction upwards at an angle \(\theta\) to the horizontal. The particle passes through the point which is 7 m above the ground and 16 m horizontally from \(O\), and hits the ground at the point \(A\).
  1. Using the equation of the particle's trajectory and the identity \(\sec^2 \theta = 1 + \tan^2 \theta\), show that the possible values of \(\tan \theta\) are \(\frac{4}{3}\) and \(\frac{1}{4}\). [4]
  2. Find the distance \(OA\) for each of the two possible values of \(\tan \theta\). [3]
  3. Sketch in the same diagram the two possible trajectories. [2]
CAIE M2 2010 June Q6
10 marks Standard +0.3
\includegraphics{figure_6} A particle \(P\) of mass 0.35 kg is attached to the mid-point of a light elastic string of natural length 4 m. The ends of the string are attached to fixed points \(A\) and \(B\) which are 4.8 m apart at the same horizontal level. \(P\) hangs in equilibrium at a point 0.7 m vertically below the mid-point \(M\) of \(AB\) (see diagram).
  1. Find the tension in the string and hence show that the modulus of elasticity of the string is 25 N. [4]
\(P\) is now held at rest at a point 1.8 m vertically below \(M\), and is then released.
  1. Find the speed with which \(P\) passes through \(M\). [6]
CAIE M2 2010 June Q7
11 marks Challenging +1.2
A particle \(P\) of mass 0.25 kg moves in a straight line on a smooth horizontal surface. \(P\) starts at the point \(O\) with speed \(10 \text{ m s}^{-1}\) and moves towards a fixed point \(A\) on the line. At time \(t\) s the displacement of \(P\) from \(O\) is \(x\) m and the velocity of \(P\) is \(v \text{ m s}^{-1}\). A resistive force of magnitude \((5 - x)\) N acts on \(P\) in the direction towards \(O\).
  1. Form a differential equation in \(v\) and \(x\). By solving this differential equation, show that \(v = 10 - 2x\). [6]
  2. Find \(x\) in terms of \(t\), and hence show that the particle is always less than 5 m from \(O\). [5]
CAIE M2 2014 June Q1
Standard +0.3
\includegraphics{figure_1}
CAIE M2 2014 June Q2
Standard +0.3
\includegraphics{figure_2}
CAIE M2 2014 June Q3
Standard +0.8
\includegraphics{figure_3}
CAIE M2 2014 June Q4
Standard +0.8
\includegraphics{figure_4}