Questions — CAIE (7646 questions)

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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]
CAIE M1 2019 June Q7
11 marks Moderate -0.3
Particles \(P\) and \(Q\) leave a fixed point \(A\) at the same time and travel in the same straight line. The velocity of \(P\) after \(t\) seconds is \(6(t - 3)\) m s\(^{-1}\) and the velocity of \(Q\) after \(t\) seconds is \((10 - 2t)\) m s\(^{-1}\).
  1. Sketch, on the same axes, velocity-time graphs for \(P\) and \(Q\) for \(0 \leq t \leq 5\). [3]
  2. Verify that \(P\) and \(Q\) meet after 5 seconds. [4]
  3. Find the greatest distance between \(P\) and \(Q\) for \(0 \leq t \leq 5\). [4]
CAIE M1 2017 March Q1
4 marks Moderate -0.8
A particle of mass \(0.4\) kg is projected with a speed of \(12\) m s\(^{-1}\) up a line of greatest slope of a smooth plane inclined at \(30°\) to the horizontal.
  1. Find the initial kinetic energy of the particle. [1]
  2. Use an energy method to find the distance the particle moves up the plane before coming to instantaneous rest. [3]
CAIE M1 2017 March Q2
6 marks Moderate -0.8
\includegraphics{figure_2} A particle \(P\) of mass \(1.6\) kg is suspended in equilibrium by two light inextensible strings attached to points \(A\) and \(B\). The strings make angles of \(20°\) and \(40°\) respectively with the horizontal (see diagram). Find the tensions in the two strings. [6]
CAIE M1 2017 March Q3
6 marks Standard +0.3
\includegraphics{figure_3} A particle of mass \(0.6\) kg is placed on a rough plane which is inclined at an angle of \(21°\) to the horizontal. The particle is kept in equilibrium by a force of magnitude \(P\) N acting parallel to a line of greatest slope of the plane, as shown in the diagram. The coefficient of friction between the particle and the plane is \(0.3\). Show that the least possible value of \(P\) is \(0.470\), correct to \(3\) significant figures, and find the greatest possible value of \(P\). [6]
CAIE M1 2017 March Q4
10 marks Standard +0.3
A car of mass \(900\) kg is moving on a straight horizontal road \(ABCD\). There is a constant resistance of magnitude \(800\) N in the sections \(AB\) and \(BC\), and a constant resistance of magnitude \(R\) N in the section \(CD\). The power of the car's engine is a constant \(36\) kW.
  1. The car moves from \(A\) to \(B\) at a constant speed in \(120\) s. Find the speed of the car and the distance \(AB\). [3]
  2. The distance \(BC\) is \(450\) m. Find the speed of the car at \(C\). [3]
  3. The car comes to rest at \(D\). The distance \(AD\) is \(6637.5\) m. Find the deceleration of the car and the value of \(R\). [4]
The car's engine is switched off at \(B\).
CAIE M1 2017 March Q5
12 marks Standard +0.3
A particle \(P\) moves in a straight line starting from a point \(O\) and comes to rest \(35\) s later. At time \(t\) s after leaving \(O\), the velocity \(v\) m s\(^{-1}\) of \(P\) is given by $$v = \frac{4}{5}t^2 \quad 0 \leq t \leq 5,$$ $$v = 2t + 10 \quad 5 \leq t \leq 15,$$ $$v = a + bt^2 \quad 15 \leq t \leq 35,$$ where \(a\) and \(b\) are constants such that \(a > 0\) and \(b < 0\).
  1. Show that the values of \(a\) and \(b\) are \(49\) and \(-0.04\) respectively. [3]
  2. Sketch the velocity-time graph. [4]
  3. Find the total distance travelled by \(P\) during the \(35\) s. [5]
CAIE M1 2017 March Q6
12 marks Standard +0.3
\includegraphics{figure_6} Two particles of masses \(1.2\) kg and \(0.8\) kg are connected by a light inextensible string that passes over a fixed smooth pulley. The particles hang vertically. The system is released from rest with both particles \(0.64\) m above the floor (see diagram). In the subsequent motion the \(0.8\) kg particle does not reach the pulley.
  1. Show that the acceleration of the particles is \(2\) m s\(^{-2}\) and find the tension in the string. [4]
  2. Find the total distance travelled by the \(0.8\) kg particle during the first second after the particles are released. [8]
CAIE M1 2019 March Q1
4 marks Moderate -0.3
\includegraphics{figure_1} A small ring \(P\) of mass \(0.03\) kg is threaded on a rough vertical rod. A light inextensible string is attached to the ring and is pulled upwards at an angle of \(15°\) to the horizontal. The tension in the string is \(2.5\) N (see diagram). The ring is in limiting equilibrium and on the point of sliding up the rod. Find the coefficient of friction between the ring and the rod. [4]
CAIE M1 2019 March Q2
6 marks Easy -1.2
A particle is projected vertically upwards with speed \(30\) m s\(^{-1}\) from a point on horizontal ground.
  1. Show that the maximum height above the ground reached by the particle is \(45\) m. [2]
  2. Find the time that it takes for the particle to reach a height of \(33.75\) m above the ground for the first time. Find also the speed of the particle at this time. [4]
CAIE M1 2019 March Q3
6 marks Moderate -0.3
\includegraphics{figure_3} Four coplanar forces of magnitudes \(F\) N, \(5\) N, \(25\) N and \(15\) N are acting at a point \(P\) in the directions shown in the diagram. Given that the forces are in equilibrium, find the values of \(F\) and \(α\). [6]
CAIE M1 2019 March Q4
7 marks Moderate -0.3
A car of mass \(1500\) kg is pulling a trailer of mass \(300\) kg along a straight horizontal road at a constant speed of \(20\) m s\(^{-1}\). The system of the car and trailer is modelled as two particles, connected by a light rigid horizontal rod. The power of the car's engine is \(6000\) W. There are constant resistances to motion of \(R\) N on the car and \(80\) N on the trailer.
  1. Find the value of \(R\). [2]
  2. The power of the car's engine is increased to \(12\,500\) W. The resistance forces do not change. Find the acceleration of the car and trailer and the tension in the rod at an instant when the speed of the car is \(25\) m s\(^{-1}\). [5]
CAIE M1 2019 March Q5
7 marks Moderate -0.8
\includegraphics{figure_5} The velocity of a particle moving in a straight line is \(v\) m s\(^{-1}\) at time \(t\) seconds after leaving a fixed point \(O\). The diagram shows a velocity-time graph which models the motion of the particle from \(t = 0\) to \(t = 16\). The graph consists of five straight line segments. The acceleration of the particle from \(t = 0\) to \(t = 3\) is \(\frac{7}{3}\) m s\(^{-2}\). The velocity of the particle at \(t = 5\) is \(7\) m s\(^{-1}\) and it comes to instantaneous rest at \(t = 8\). The particle then comes to rest again at \(t = 16\). The minimum velocity of the particle is \(V\) m s\(^{-1}\).
  1. Find the distance travelled by the particle in the first \(8\) s of its motion. [3]
  2. Given that when the particle comes to rest at \(t = 16\) its displacement from \(O\) is \(32\) m, find the value of \(V\). [4]
CAIE M1 2019 March Q6
9 marks Standard +0.3
A particle moves in a straight line. It starts from rest at a fixed point \(O\) on the line. Its acceleration at time \(t\) s after leaving \(O\) is \(a\) m s\(^{-2}\), where \(a = 0.4t^3 - 4.8t^2\).
  1. Show that, in the subsequent motion, the acceleration of the particle when it comes to instantaneous rest is \(16\) m s\(^{-2}\). [6]
  2. Find the displacement of the particle from \(O\) at \(t = 5\). [3]
CAIE M1 2019 March Q7
11 marks Standard +0.3
\includegraphics{figure_7} The diagram shows the vertical cross-section \(PQR\) of a slide. The part \(PQ\) is a straight line of length \(8\) m inclined at angle \(α\) to the horizontal, where \(\sin α = 0.8\). The straight part \(PQ\) is tangential to the curved part \(QR\) at \(Q\), and \(R\) is \(h\) m above the level of \(P\). The straight part \(PQ\) of the slide is rough and the curved part \(QR\) is smooth. A particle of mass \(0.25\) kg is projected with speed \(15\) m s\(^{-1}\) from \(P\) towards \(Q\) and comes to rest at \(R\). The coefficient of friction between the particle and \(PQ\) is \(0.5\).
  1. Find the work done by the friction force during the motion of the particle from \(P\) to \(Q\). [4]
  2. Hence find the speed of the particle at \(Q\). [4]
  3. Find the value of \(h\). [3]
CAIE M1 2007 November Q1
4 marks Moderate -0.3
A car of mass 900 kg travels along a horizontal straight road with its engine working at a constant rate of \(P\) kW. The resistance to motion of the car is 550 N. Given that the acceleration of the car is \(0.2 \text{ m s}^{-2}\) at an instant when its speed is \(30 \text{ m s}^{-1}\), find the value of \(P\). [4]
CAIE M1 2007 November Q2
5 marks Moderate -0.5
A particle is projected vertically upwards from a point \(O\) with initial speed \(12.5 \text{ m s}^{-1}\). At the same instant another particle is released from rest at a point 10 m vertically above \(O\). Find the height above \(O\) at which the particles meet. [5]
CAIE M1 2007 November Q3
6 marks Moderate -0.8
\includegraphics{figure_3} A particle is in equilibrium on a smooth horizontal table when acted on by the three horizontal forces shown in the diagram.
  1. Find the values of \(F\) and \(\theta\). [4]
  2. The force of magnitude 7 N is now removed. State the magnitude and direction of the resultant of the remaining two forces. [2]