Questions — CAIE M2 (519 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 M2 2016 June Q1
5 marks Standard +0.3
1 A small ball \(B\) is projected with speed \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(30 ^ { \circ }\) above the horizontal from a point \(O\) on horizontal ground. At the instant 0.8 s after projection, \(B\) is 0.5 m vertically above the top of a vertical post.
  1. Calculate the height of the top of the post above the ground.
  2. Show that \(B\) is at its greatest height 0.2 s before passing over the post.
CAIE M2 2016 June Q2
8 marks Standard +0.3
2 One end of a light elastic string of natural length 0.4 m is attached to a fixed point \(O\). The other end of the string is attached to a particle of weight 5 N which hangs in equilibrium 0.6 m vertically below \(O\).
  1. Find the modulus of elasticity of the string. The particle is projected vertically upwards from the equilibrium position and comes to instantaneous rest after travelling 0.3 m upwards.
  2. Calculate the speed of projection of the particle.
  3. Calculate the greatest extension of the string in the subsequent motion.
CAIE M2 2016 June Q3
7 marks Standard +0.3
3 The point \(O\) is 8 m above a horizontal plane. A particle \(P\) is projected from \(O\). After projection, the horizontal and vertically upwards displacements of \(P\) from \(O\) are \(x \mathrm {~m}\) and \(y \mathrm {~m}\) respectively. The equation of the trajectory of \(P\) is $$y = 2 x - x ^ { 2 }$$
  1. Find the value of \(x\) for the point where \(P\) strikes the plane.
  2. Find the angle and speed of projection of \(P\).
  3. Calculate the speed of \(P\) immediately before it strikes the plane.
CAIE M2 2016 June Q4
8 marks Standard +0.3
4 \includegraphics[max width=\textwidth, alt={}, center]{f8633b64-b20c-4471-9641-ccc3e6854f2c-3_784_556_260_790} A uniform object is made by drilling a cylindrical hole through a rectangular block. The axis of the cylindrical hole is perpendicular to the cross-section \(A B C D\) through the centre of mass of the object. \(A B = C D = 0.7 \mathrm {~m} , B C = A D = 0.4 \mathrm {~m}\), and the centre of the hole is 0.1 m from \(A B\) and 0.2 m from \(A D\) (see diagram). The hole has a cross-section of area \(0.03 \mathrm {~m} ^ { 2 }\).
  1. Show that the distance of the centre of mass of the object from \(A B\) is 0.212 m , and calculate the distance of the centre of mass from \(A D\). The object has weight 70 N and is placed on a rough horizontal surface, with \(A D\) in contact with the surface. A vertically upwards force of magnitude \(F \mathrm {~N}\) acts on the object at \(C\). The object is on the point of toppling.
  2. Find the value of \(F\). The force acting at \(C\) is removed, and the object is placed on a rough plane inclined at an angle \(\theta ^ { \circ }\) to the horizontal. \(A D\) lies along a line of greatest slope, with \(A\) higher than \(D\). The plane is sufficiently rough to prevent sliding, and the object does not topple.
  3. Find the greatest possible value of \(\theta\).
CAIE M2 2016 June Q5
10 marks Standard +0.3
5 A particle \(P\) of mass 0.4 kg is placed at rest at a point \(A\) on a rough horizontal surface. A horizontal force, directed away from \(A\) and with magnitude \(0.6 t \mathrm {~N}\), acts on \(P\), where \(t \mathrm {~s}\) is the time after \(P\) is placed at \(A\). The coefficient of friction between \(P\) and the surface is 0.3 , and \(P\) has displacement from \(A\) of \(x \mathrm {~m}\) at time \(t \mathrm {~s}\).
  1. Show that \(P\) starts to move when \(t = 2\). Show also that when \(P\) is in motion it has acceleration \(( 1.5 t - 3 ) \mathrm { m } \mathrm { s } ^ { - 2 }\).
  2. Express the velocity of \(P\) in terms of \(t\), for \(t \geqslant 2\).
  3. Express \(x\) in terms of \(t\), for \(t \geqslant 2\).
CAIE M2 2016 June Q6
12 marks Standard +0.8
6 \includegraphics[max width=\textwidth, alt={}, center]{f8633b64-b20c-4471-9641-ccc3e6854f2c-4_479_499_255_824} \(O A\) is a rod which rotates in a horizontal circle about a vertical axis through \(O\). A particle \(P\) of mass 0.2 kg is attached to the mid-point of a light inextensible string. One end of the string is attached to the \(\operatorname { rod }\) at \(A\) and the other end of the string is attached to a point \(B\) on the axis. It is given that \(O A = O B\), angle \(O A P =\) angle \(O B P = 30 ^ { \circ }\), and \(P\) is 0.4 m from the axis. The rod and the particle rotate together about the axis with \(P\) in the plane \(O A B\) (see diagram).
  1. Calculate the tensions in the two parts of the string when the speed of \(P\) is \(1.2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The angular speed of the rod is increased to \(5 \mathrm { rad } \mathrm { s } ^ { - 1 }\), and it is given that the system now rotates with angle \(O A P =\) angle \(O B P = 60 ^ { \circ }\).
  2. Show that the tension in the part \(A P\) of the string is zero. {www.cie.org.uk} after the live examination series. }
CAIE M2 2017 June Q1
5 marks Moderate -0.5
1 A particle \(P\) of mass 0.2 kg moves with speed \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and angular speed \(5 \mathrm { rad } \mathrm { s } ^ { - 1 }\) in a horizontal circle on a smooth surface. \(P\) is attached to one end of a light elastic string of natural length 0.6 m . The other end of the string is attached to the point on the surface which is the centre of the circular motion of \(P\).
  1. Find the radius of this circle.
  2. Find the modulus of elasticity of the string.
CAIE M2 2017 June Q2
5 marks Standard +0.3
2
[diagram]
The ends of two light inextensible strings of length 0.7 m are attached to a particle \(P\). The other ends of the strings are attached to two fixed points \(A\) and \(B\) which lie in the same vertical line with \(A\) above \(B\). The particle \(P\) moves in a horizontal circle which has its centre at the mid-point of \(A B\). Both strings are inclined at \(60 ^ { \circ }\) to the vertical. The tension in the string attached to \(A\) is 6 N and the tension in the string attached to \(B\) is 4 N (see diagram).
  1. Find the mass of \(P\).
  2. Calculate the speed of \(P\).
CAIE M2 2017 June Q3
7 marks Challenging +1.2
3 An open box in the shape of a cube with edges of length 0.2 m is placed with its base horizontal and its four sides vertical. The four sides and base are uniform laminas, each with weight 3 N .
  1. Calculate the height of the centre of mass of the box above its base.
    The box is now fitted with a thin uniform square lid of weight 3 N and with edges of length 0.2 m . The lid is attached to the box by a hinge of length 0.2 m and weight 2 N . The lid of the box is held partly open.
  2. Find the angle which the lid makes with the horizontal when the centre of mass of the box (including the lid and hinge) is 0.12 m above the base of the box.
CAIE M2 2017 June Q4
8 marks Standard +0.8
4 A small object of mass 0.4 kg is released from rest at a point 8 m above the ground. The object descends vertically and when its downwards displacement from its initial position is \(x \mathrm {~m}\) the object has velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). While the object is moving, a force of magnitude \(0.2 v ^ { 2 } \mathrm {~N}\) opposes the motion.
  1. Show that \(v \frac { \mathrm {~d} v } { \mathrm {~d} x } = 10 - 0.5 v ^ { 2 }\).
  2. Express \(v\) in terms of \(x\).
  3. Find the increase in the value of \(v\) during the final 4 m of the descent of the object.
CAIE M2 2017 June Q5
8 marks Standard +0.3
5 A particle of mass 0.3 kg is attached to one end of a light elastic string of natural length 0.8 m and modulus of elasticity 6 N . The other end of the string is attached to a fixed point \(O\). The particle is projected vertically downwards from \(O\) with initial speed \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  1. Calculate the greatest speed of the particle during its descent.
  2. Find the greatest distance of the particle below \(O\). \includegraphics[max width=\textwidth, alt={}, center]{2b0425b2-2f8f-491a-996c-3d3b589bd7df-12_558_554_260_794} The end \(A\) of a non-uniform rod \(A B\) of length 0.6 m and weight 8 N rests on a rough horizontal plane, with \(A B\) inclined at \(60 ^ { \circ }\) to the horizontal. Equilibrium is maintained by a force of magnitude 3 N applied to the rod at \(B\). This force acts at \(30 ^ { \circ }\) above the horizontal in the vertical plane containing the rod (see diagram).
  3. Find the distance of the centre of mass of the rod from \(A\).
    The 3 N force is removed, and the rod is held in equilibrium by a force of magnitude \(P \mathrm {~N}\) applied at \(B\), acting in the vertical plane containing the rod, at an angle of \(30 ^ { \circ }\) below the horizontal.
  4. Calculate \(P\).
    In one of the two situations described, the \(\operatorname { rod } A B\) is in limiting equilibrium.
  5. Find the coefficient of friction at \(A\). \(7 \quad\) A particle \(P\) is projected from a point \(O\) with speed \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\). At time \(t \mathrm {~s}\) after projection the horizontal and vertically upwards displacements of \(P\) from \(O\) are \(x \mathrm {~m}\) and \(y \mathrm {~m}\) respectively. The equation of the trajectory of \(P\) is \(y = 2 x - \frac { 25 x ^ { 2 } } { V ^ { 2 } }\).
  6. Write down the value of \(\tan \theta\), where \(\theta\) is the angle of projection of \(P\).
    When \(t = 4 , P\) passes through the point \(A\) where \(x = y = a\).
  7. Calculate \(V\) and \(a\).
  8. Find the direction of motion of \(P\) when it passes through \(A\).
CAIE M2 2018 June Q6
9 marks Standard +0.8
6 \includegraphics[max width=\textwidth, alt={}, center]{8dda6c21-7cb5-43b6-9a34-485bdf4042c4-10_262_732_264_705} A particle \(P\) of mass 0.2 kg is attached to one end of a light inextensible string of length 0.6 m . The other end of the string is attached to a particle \(Q\) of mass 0.3 kg . The string passes through a small hole \(H\) in a smooth horizontal surface. A light elastic string of natural length 0.3 m and modulus of elasticity 15 N joins \(Q\) to a fixed point \(A\) which is 0.4 m vertically below \(H\). The particle \(P\) moves on the surface in a horizontal circle with centre \(H\) (see diagram).
  1. Calculate the greatest possible speed of \(P\) for which the elastic string is not extended.
  2. Find the distance \(H P\) given that the angular speed of \(P\) is \(8 \mathrm { rad } \mathrm { s } ^ { - 1 }\).
CAIE M2 2018 June Q7
10 marks Standard +0.8
7 \includegraphics[max width=\textwidth, alt={}, center]{8dda6c21-7cb5-43b6-9a34-485bdf4042c4-12_732_581_260_774} A uniform solid cone has height 1.2 m and base radius 0.5 m . A uniform object is made by drilling a cylindrical hole of radius 0.2 m through the cone along the axis of symmetry (see diagram).
  1. Show that the height of the object is 0.72 m and that the volume of the cone removed by the drilling is \(0.0352 \pi \mathrm {~m} ^ { 3 }\).
    [0pt] [The volume of a cone is \(\frac { 1 } { 3 } \pi r ^ { 2 } h\).]
  2. Find the distance of the centre of mass of the object from its base.
    If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.
CAIE M2 2019 June Q1
5 marks Moderate -0.3
1 \includegraphics[max width=\textwidth, alt={}, center]{bba68fb2-88c6-4883-931b-f738cda2dce3-03_231_970_258_591} A particle \(P\) of mass 0.3 kg is attached to a fixed point \(A\) by a light inextensible string of length 0.8 m . The fixed point \(O\) is 0.15 m vertically below \(A\). The particle \(P\) moves with constant speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in a horizontal circle with centre \(O\) (see diagram).
  1. Show that the tension in the string is 16 N .
  2. Find the value of \(v\).
CAIE M2 2019 June Q2
5 marks Standard +0.3
2 A particle is projected with speed \(\mathrm { V } \mathrm { m } \mathrm { s } ^ { - 1 }\) at an angle of \(\theta ^ { \circ }\) above the horizontal. At the instant 4 s after projection the speed of the particle is \(16 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and its direction of motion is \(30 ^ { \circ }\) above the horizontal. Find \(V\) and \(\theta\).
CAIE M2 2019 June Q3
5 marks Standard +0.3
3 \includegraphics[max width=\textwidth, alt={}, center]{bba68fb2-88c6-4883-931b-f738cda2dce3-05_448_802_258_676} The diagram shows the cross-section through the centre of mass of a uniform solid object. The object is a cylinder of radius 0.2 m and length 0.7 m , from which a hemisphere of radius 0.2 m has been removed at one end. The point \(A\) is the centre of the plane face at the other end of the object. Find the distance of the centre of mass of the object from \(A\).
[0pt] [The volume of a hemisphere is \(\frac { 2 } { 3 } \pi r ^ { 3 }\).]
CAIE M2 2019 June Q4
8 marks Moderate -0.3
4 A small ball is projected with speed \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(30 ^ { \circ }\) above the horizontal from a point \(O\) on horizontal ground. At time \(t \mathrm {~s}\) after projection the horizontal and vertically upwards displacements of the ball from \(O\) are \(x \mathrm {~m}\) and \(y \mathrm {~m}\) respectively.
  1. Express \(x\) and \(y\) in terms of \(t\) and hence find the equation of the trajectory of the ball.
  2. Find \(x\) for the position of the ball when its path makes an angle of \(15 ^ { \circ }\) below the horizontal. [4]
CAIE M2 2019 June Q5
8 marks Challenging +1.2
5 A particle \(P\) of mass 0.4 kg is attached to one end of a light elastic string of natural length 0.5 m and modulus of elasticity 6 N . The other end of the string is attached to a fixed point \(O\). The particle \(P\) is released from rest at the point \(( 0.5 + x ) \mathrm { m }\) vertically below \(O\). The particle \(P\) comes to instantaneous rest at \(O\).
  1. Find \(x\).
  2. Find the greatest speed of \(P\).
CAIE M2 2019 June Q6
7 marks Standard +0.3
6 \(A B C\) is a uniform lamina in the form of a triangle with \(A B = 0.3 \mathrm {~m} , B C = 0.6 \mathrm {~m}\) and a right angle at \(B\) (see diagram).
  1. State the distances of the centre of mass of the lamina from \(A B\) and from \(B C\). Distance from \(A B\) Distance from \(B C\) \(\_\_\_\_\) The lamina is freely suspended at \(B\) and hangs in equilibrium.
  2. Find the angle between \(A B\) and the horizontal.
    A force of magnitude 12 N is applied along the edge \(A C\) of the lamina in the direction from \(A\) towards \(C\). The lamina, still suspended at \(B\), is now in equilibrium with \(A B\) vertical.
  3. Calculate the weight of the lamina.
CAIE M2 2019 June Q7
12 marks Challenging +1.8
7 A particle \(P\) of mass 0.5 kg is attached to a fixed point \(O\) by a light elastic string of natural length 1 m and modulus of elasticity 16 N . The particle \(P\) is projected vertically upwards from \(O\) with speed \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). A resisting force of magnitude \(0.1 x ^ { 2 } \mathrm {~N}\) acts on \(P\) when \(P\) has displacement \(x \mathrm {~m}\) above \(O\). After projection the upwards velocity of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  1. Show that, before the string becomes taut, \(v \frac { \mathrm {~d} v } { \mathrm {~d} x } = - 10 - 0.2 x ^ { 2 }\).
  2. Find the velocity of \(P\) at the instant the string becomes taut.
  3. Find an expression for the acceleration of \(P\) while it is moving upwards after the string becomes taut.
  4. Verify that \(P\) comes to instantaneous rest before the extension of the string is 0.5 m .
    If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.
CAIE M2 2019 June Q1
5 marks Moderate -0.8
1 A small ball is projected from a point \(O\) on horizontal ground at an angle of \(30 ^ { \circ }\) above the horizontal. At time \(t \mathrm {~s}\) after projection the vertically upwards displacement of the ball from \(O\) is \(\left( 14 t - k t ^ { 2 } \right) \mathrm { m }\), where \(k\) is a constant.
  1. State the value of \(k\). \includegraphics[max width=\textwidth, alt={}, center]{f3a35846-075d-4e03-ba6b-82774ef0e4f8-03_56_1563_495_331}
  2. Show that the initial speed of the ball is \(28 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  3. Find the horizontal displacement of the ball from \(O\) when \(t = 3\).
CAIE M2 2019 June Q2
4 marks Standard +0.3
2 \includegraphics[max width=\textwidth, alt={}, center]{f3a35846-075d-4e03-ba6b-82774ef0e4f8-04_442_554_260_794} A uniform lamina \(A B C E F G\) is formed from a square \(A B D G\) by removing a smaller square \(C D F E\) from one corner. \(A B = 0.7 \mathrm {~m}\) and \(D F = 0.3 \mathrm {~m}\) (see diagram). Find the distance of the centre of mass of the lamina from \(A\).
CAIE M2 2019 June Q3
5 marks Moderate -0.3
3 A particle \(P\) of mass 0.4 kg is attached to a fixed point \(A\) by a light inextensible string of length 0.5 m . The point \(A\) is 0.3 m above a smooth horizontal surface. The particle \(P\) moves in a horizontal circle on the surface with constant angular speed \(5 \mathrm { rad } \mathrm { s } ^ { - 1 }\).
  1. Calculate the tension in the string. \includegraphics[max width=\textwidth, alt={}, center]{f3a35846-075d-4e03-ba6b-82774ef0e4f8-05_67_1569_486_328}
  2. Find the magnitude of the force exerted by the surface on \(P\).
CAIE M2 2019 June Q4
8 marks Challenging +1.8
4 A particle \(P\) of mass 0.5 kg is attached to one end of a light elastic string of natural length 0.8 m and modulus of elasticity 16 N . The other end of the string is attached to a fixed point \(O\). The particle \(P\) is released from rest at the point 0.8 m vertically below \(O\). When the extension of the string is \(x \mathrm {~m}\), the downwards velocity of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and a force of magnitude \(25 x ^ { 2 } \mathrm {~N}\) opposes the motion of \(P\).
  1. Show that, when \(P\) is moving downwards, \(v \frac { \mathrm {~d} v } { \mathrm {~d} x } = 10 - 40 x - 50 x ^ { 2 }\).
  2. For the instant when \(P\) has its greatest downwards speed, find the kinetic energy of \(P\) and the elastic potential energy stored in the string.
CAIE M2 2019 June Q5
8 marks Standard +0.3
5 A light elastic string has natural length \(a \mathrm {~m}\) and modulus of elasticity \(\lambda \mathrm { N }\). When the length of the string is 1.6 m the tension is 4 N . When the length of the string is 2 m the tension is 6 N .
  1. Find the values of \(a\) and \(\lambda\).
    One end of the string is attached to a fixed point \(O\) on a smooth horizontal surface. The other end of the string is attached to a particle \(P\) of mass 0.2 kg . The particle \(P\) moves with constant speed on the surface in a circle with centre \(O\) and radius 1.9 m .
  2. Find the speed of \(P\).