Questions — CAIE (7646 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 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 \(P\).
CAIE M2 2019 June Q6
8 marks Standard +0.3
6 A particle is projected with speed \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(\theta ^ { \circ }\) above the horizontal. At the instant 4 s after projection the speed of the particle is \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  1. Find \(\theta\).
  2. Show that at the instant 4 s after projection the particle is 33.75 m below the level of the point of projection and find the direction of motion at this instant.
    \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{f3a35846-075d-4e03-ba6b-82774ef0e4f8-12_259_609_255_769} \captionsetup{labelformat=empty} \caption{Fig. 1}
    \end{figure} Fig. 1 shows an object made from a uniform wire of length 0.8 m . The object consists of a straight part \(A B\), and a semicircular part \(B C\) such that \(A , B\) and \(C\) lie in the same straight line. The radius of the semicircle is \(r \mathrm {~m}\) and the centre of mass of the object is 0.1 m from line \(A B C\).
  1. Show that \(r = 0.2\).
    \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{f3a35846-075d-4e03-ba6b-82774ef0e4f8-13_615_383_260_881} \captionsetup{labelformat=empty} \caption{Fig. 2}
    \end{figure} The object is freely suspended at \(A\) and a horizontal force of magnitude 7 N is applied to the object at \(C\) so that the object is in equilibrium with \(A B C\) vertical (see Fig. 2).
  2. Calculate the weight of the object.
    The 7 N force is removed and the object hangs in equilibrium with \(A B C\) at an angle of \(\theta ^ { \circ }\) with the vertical.
  3. Find \(\theta\).
    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 Standard +0.3
1 \includegraphics[max width=\textwidth, alt={}, center]{111bcbf6-daaf-4d8d-9299-d591ac7369f1-03_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 Q3
5 marks Standard +0.3
3 \includegraphics[max width=\textwidth, alt={}, center]{111bcbf6-daaf-4d8d-9299-d591ac7369f1-05_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 2016 March Q1
5 marks Standard +0.3
1 A particle is projected from a point on horizontal ground. At the instant 2 s after projection, the particle has travelled a horizontal distance of 30 m and is at its greatest height above the ground. Find the initial speed and the angle of projection of the particle.
CAIE M2 2016 March Q2
5 marks Challenging +1.2
2 \includegraphics[max width=\textwidth, alt={}, center]{334b4bdf-6d9c-4208-9032-572eb7c5f9ee-2_295_805_484_671} A uniform solid hemisphere of weight 60 N and radius 0.8 m rests in limiting equilibrium with its curved surface on a rough horizontal plane. The axis of symmetry of the hemisphere is inclined at an angle of \(\theta\) to the horizontal, where \(\cos \theta = 0.28\). Equilibrium is maintained by a horizontal force of magnitude \(P\) N applied to the lowest point of the circular rim of the hemisphere (see diagram).
  1. Show that \(P = 8.75\).
  2. Find the coefficient of friction between the hemisphere and the plane.
CAIE M2 2016 March Q3
5 marks Standard +0.8
3 A stone is thrown with speed \(9 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(60 ^ { \circ }\) above the horizontal from a point on horizontal ground. Find the distance between the two points at which the path of the stone makes an angle of \(45 ^ { \circ }\) with the horizontal.
CAIE M2 2016 March Q4
7 marks Standard +0.3
4 \includegraphics[max width=\textwidth, alt={}, center]{334b4bdf-6d9c-4208-9032-572eb7c5f9ee-2_549_579_1505_781} A uniform lamina is made by joining a rectangle \(A B C D\), in which \(A B = C D = 0.56 \mathrm {~m}\) and \(B C = A D = 2 \mathrm {~m}\), and a square \(E F G A\) of side 1.2 m . The vertex \(E\) of the square lies on the edge \(A D\) of the rectangle (see diagram). The centre of mass of the lamina is a distance \(h \mathrm {~m}\) from \(B C\) and a distance \(v \mathrm {~m}\) from \(B A G\).
  1. Find the value of \(h\) and show that \(v = h\). The lamina is freely suspended at the point \(B\) and hangs in equilibrium.
  2. State the angle which the edge \(B C\) makes with the horizontal. Instead, the lamina is now freely suspended at the point \(F\) and hangs in equilibrium.
  3. Calculate the angle between \(F G\) and the vertical.
CAIE M2 2016 March Q5
9 marks Standard +0.3
5 A particle \(P\) of mass 0.6 kg is attached to one end of a light elastic string of natural length 0.8 m and modulus of elasticity 24 N . The other end of the string is attached to a fixed point \(A\), and \(P\) hangs in equilibrium.
  1. Calculate the extension of the string. \(P\) is projected vertically downwards from the equilibrium position with speed \(4.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  2. Find the distance \(A P\) when the speed of \(P\) is \(3.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(P\) is below the equilibrium position.
  3. Calculate the speed of \(P\) when it is 0.5 m above the equilibrium position.
CAIE M2 2016 March Q6
9 marks Challenging +1.8
6 A particle \(P\) of mass 0.2 kg is released from rest at a point \(O\) on a plane inclined at \(30 ^ { \circ }\) to the horizontal. At time \(t \mathrm {~s}\) after its release, \(P\) has velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and displacement \(x \mathrm {~m}\) down the plane from \(O\). The coefficient of friction between \(P\) and the plane increases as \(P\) moves down the plane, and equals \(0.1 x ^ { 2 }\).
  1. Show that \(2 v \frac { \mathrm {~d} v } { \mathrm {~d} x } = 10 - ( \sqrt { } 3 ) x ^ { 2 }\).
  2. Calculate the maximum speed of \(P\).
  3. Find the value of \(x\) at the point where \(P\) comes to rest.
CAIE M2 2016 March Q7
10 marks Standard +0.3
7 \includegraphics[max width=\textwidth, alt={}, center]{334b4bdf-6d9c-4208-9032-572eb7c5f9ee-3_451_432_1434_852} One end of a light inextensible string is attached to the highest point \(A\) of a solid fixed sphere with centre \(O\) and radius 0.6 m . The other end of the string is attached to a particle \(P\) of mass 0.2 kg which rests in contact with the smooth surface of the sphere. The angle \(A O P = 60 ^ { \circ }\) (see diagram). The sphere exerts a contact force of magnitude \(R \mathrm {~N}\) on \(P\) and the tension in the string is \(T \mathrm {~N}\).
  1. By resolving vertically, show that \(R + ( \sqrt { } 3 ) T = 4\). \(P\) is now set in motion, and moves with angular speed \(\omega \mathrm { rad } \mathrm { s } ^ { - 1 }\) in a horizontal circle on the surface of the sphere.
  2. Find an equation involving \(R , T\) and \(\omega\).
  3. Hence
    1. calculate \(R\) when \(\omega = 2\),
    2. find the greatest possible value of \(T\) and the corresponding speed of \(P\).
CAIE M2 2019 March Q1
5 marks Moderate -0.5
1 A particle is projected with speed \(24 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(30 ^ { \circ }\) above the horizontal. Find the speed and direction of motion of the particle at the instant 4 s after projection.
CAIE M2 2019 March Q2
6 marks Standard +0.3
2 \includegraphics[max width=\textwidth, alt={}, center]{b8e52188-f9a6-46fc-90bf-97965c6dd324-04_606_376_260_881} A uniform object is made by joining together three solid cubes with edges \(3 \mathrm {~m} , 2 \mathrm {~m}\) and 1 m . The object has an axis of symmetry, with the cubes stacked vertically and the cube of edge 2 m between the other two cubes (see diagram).
  1. Calculate the distance of the centre of mass of the object above the base of the largest cube.
    The smallest cube is now removed from the object. It is replaced by a heavier uniform cube with 1 m edges which is made of a different material. The centre of mass of the object is now at the base of the 2 m cube.
  2. Find the ratio of the masses of the two cubes of edge 1 m .
CAIE M2 2019 March Q3
6 marks Moderate -0.8
3 A small ball is projected 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, where \(x = 4 t\) and \(y = 6 t - 5 t ^ { 2 }\).
  1. Find the equation of the trajectory of the ball.
  2. Hence or otherwise calculate the angle of projection of the ball and its initial speed.
CAIE M2 2019 March Q4
6 marks Standard +0.3
4 \includegraphics[max width=\textwidth, alt={}, center]{b8e52188-f9a6-46fc-90bf-97965c6dd324-07_337_526_262_726} \includegraphics[max width=\textwidth, alt={}, center]{b8e52188-f9a6-46fc-90bf-97965c6dd324-07_111_116_486_1308} A particle \(P\) of mass 0.3 kg is attached to a fixed point \(A\) by a light elastic string of natural length 0.8 m and modulus of elasticity 16 N . The particle \(P\) moves in a horizontal circle which has centre \(O\). It is given that \(A O\) is vertical and that angle \(O A P\) is \(60 ^ { \circ }\) (see diagram). Calculate the speed of \(P\). [6]
CAIE M2 2019 March Q5
8 marks Standard +0.3
5 A particle \(P\) of mass 0.3 kg is attached to one end of a light elastic string of natural length 0.6 m and modulus of elasticity 24 N . The other end of the string is attached to a fixed point \(O\). The particle \(P\) is released from rest at the point 0.4 m vertically below \(O\).
  1. Find the greatest speed of \(P\).
  2. Calculate the greatest distance of \(P\) below \(O\).
    \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{b8e52188-f9a6-46fc-90bf-97965c6dd324-10_608_611_258_767} \captionsetup{labelformat=empty} \caption{Fig. 1}
    \end{figure} Fig. 1 shows the cross-section of a solid cylinder through which a cylindrical hole has been drilled to make a uniform prism. The radius of the cylinder is \(5 r\) and the radius of the hole is \(r\). The centre of the hole is a distance \(2 r\) from the centre of the cylinder.
CAIE M2 2019 March Q7
11 marks Challenging +1.2
7 A particle \(P\) is projected horizontally from a point \(O\) on a rough horizontal surface. The coefficient of friction between the particle and the surface is 0.2 . A horizontal force of magnitude \(0.06 t \mathrm {~N}\) directed away from \(O\) acts on \(P\), where \(t \mathrm {~s}\) is the time after projection. \(P\) comes to rest when \(t = 4\).
  1. The particle begins to move again when \(t = 8\). Show that the mass of \(P\) is 0.24 kg .
  2. Show that, for \(0 \leqslant t \leqslant 4 , \frac { \mathrm {~d} v } { \mathrm {~d} t } = 0.25 t - 2\), and find the speed of projection of \(P\).
  3. Find the distance from \(O\) at which \(P\) comes to rest.
    If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.