Questions — CAIE M1 (786 questions)

Browse by module
All questions AEA AS Paper 1 AS Paper 2 AS Pure C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Extra Pure Further Mechanics Further Mechanics A AS Further Mechanics AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics Further Paper 4 Further Pure Core Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Pure with Technology Further Statistics Further Statistics A AS Further Statistics AS Further Statistics B AS Further Statistics Major Further Statistics Minor Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 H240/01 H240/02 H240/03 M1 M2 M3 M4 M5 P1 P2 P3 P4 PMT Mocks PURE Paper 1 Paper 2 Paper 3 Pre-U 9794/1 Pre-U 9794/2 Pre-U 9794/3 Pre-U 9795 Pre-U 9795/1 Pre-U 9795/2 S1 S2 S3 S4 Unit 1 Unit 2 Unit 3 Unit 4
Browse by board
AQA AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Paper 1 Further Paper 2 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics M1 M2 M3 Paper 1 Paper 2 Paper 3 S1 S2 S3 CAIE FP1 FP2 Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 4 M1 M2 P1 P2 P3 S1 S2 Edexcel AEA AS Paper 1 AS Paper 2 C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS M1 M2 M3 M4 M5 P1 P2 P3 P4 PMT Mocks PURE Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 OCR AS Pure C1 C2 C3 C4 D1 D2 FD1 AS FM1 AS FP1 FP1 AS FP2 FP3 FS1 AS Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Mechanics Further Mechanics AS Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Statistics Further Statistics AS H240/01 H240/02 H240/03 M1 M2 M3 M4 PURE S1 S2 S3 S4 OCR MEI AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further Extra Pure Further Mechanics A AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Pure Core Further Pure Core AS Further Pure with Technology Further Statistics A AS Further Statistics B AS Further Statistics Major Further Statistics Minor M1 M2 M3 M4 Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 Pre-U Pre-U 9794/1 Pre-U 9794/2 Pre-U 9794/3 Pre-U 9795 Pre-U 9795/1 Pre-U 9795/2 WJEC Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 Unit 1 Unit 2 Unit 3 Unit 4
Sort by: Default | Easiest first | Hardest first
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]
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]