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 2014 November Q3
6 marks Moderate -0.3
A particle \(P\) of mass \(0.2\) kg is attached to one end of a light inextensible string. The other end of the string is attached to a fixed point \(O\). The particle moves in a horizontal circle of radius \(0.8\) m with the string making a constant angle of \(60°\) with the vertical. Calculate the speed of the particle and the tension in the string. [6]
CAIE M2 2014 November Q4
8 marks Standard +0.3
\includegraphics{figure_4} The diagram shows the cross-section of a uniform solid consisting of a cylinder of radius \(0.4\) m and height \(1.5\) m with a hemisphere of radius \(0.4\) m on top.
  1. Find the distance of the centre of mass above the base of the cylinder. [5]
  2. The solid can just rest in equilibrium on a plane inclined at angle \(\alpha\) to the horizontal. Find \(\alpha\). [3]
CAIE M2 2014 November Q5
7 marks Moderate -0.3
The position vector of a particle at time \(t\) is given by \(\mathbf{r} = t^2\mathbf{i} + (3t - 1)\mathbf{j}\), where \(\mathbf{i}\) and \(\mathbf{j}\) are perpendicular unit vectors. Find the velocity and acceleration of the particle when \(t = 2\).
  1. Hence find the angle between the velocity and acceleration vectors when \(t = 2\). [3]
  2. Find the value of \(t\) for which the velocity and acceleration vectors are perpendicular. [4]
CAIE M2 2014 November Q6
12 marks Standard +0.3
A particle of mass \(2\) kg moves under the action of a variable force. At time \(t\) seconds the force is \((6t - 3)\mathbf{i} + 4\mathbf{j}\) newtons, where \(\mathbf{i}\) and \(\mathbf{j}\) are perpendicular unit vectors. When \(t = 0\), the particle is at rest at the origin.
  1. Find the velocity of the particle when \(t = 4\). [4]
  2. Find the kinetic energy of the particle when \(t = 4\). [2]
  3. Find the distance of the particle from the origin when \(t = 2\). [6]
CAIE M2 2014 November Q7
8 marks Standard +0.3
\includegraphics{figure_7} A particle of mass \(0.4\) 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 \(O\). The particle moves in a vertical circle and passes through the lowest point of the circle with speed \(6\) m s\(^{-1}\).
  1. Find the tension in the string when the particle is at the lowest point. [2]
  2. Find the speed of the particle when the string makes an angle of \(60°\) with the downward vertical. [4]
  3. Hence find the tension in the string at this position. [2]
CAIE M2 2014 November Q1
4 marks Standard +0.8
A particle \(P\) is projected with speed \(V\) m s\(^{-1}\) at an angle of \(30°\) above the horizontal from a point \(O\) on horizontal ground. At the instant \(2\) s after projection, \(OP\) makes an angle of \(15°\) above the horizontal. Calculate \(V\). [4]
CAIE M2 2014 November Q2
4 marks Standard +0.8
\includegraphics{figure_2} A uniform solid cone with height \(0.8\) m and semi-vertical angle \(30°\) has weight \(20\) N. The cone rests in equilibrium with a single point \(P\) of its base in contact with a rough horizontal surface, and its vertex \(V\) vertically above \(P\). Equilibrium is maintained by a force of magnitude \(F\) N acting along the axis of symmetry of the cone and applied to \(V\) (see diagram).
  1. Show that the moment of the weight of the cone about \(P\) is \(6\) N m. [2]
  2. Hence find \(F\). [2]
CAIE M2 2014 November Q3
5 marks Standard +0.3
One end of a light elastic string of natural length \(1.6\) m and modulus of elasticity \(28\) N is attached to a fixed point \(O\). The other end of the string is attached to a particle \(P\) of mass \(0.35\) kg which hangs in equilibrium vertically below \(O\). The particle \(P\) is projected vertically upwards from the equilibrium position with speed \(1.8\) m s\(^{-1}\). Calculate the speed of \(P\) at the instant the string first becomes slack. [5]
CAIE M2 2014 November Q4
7 marks Standard +0.8
\includegraphics{figure_4} \(ABCDEF\) is the cross-section through the centre of mass of a uniform solid prism. \(ABCF\) is a rectangle in which \(AB = CF = 1.6\) m, and \(BC = AF = 0.4\) m. \(CDE\) is a triangle in which \(CD = 1.8\) m, \(CE = 0.4\) m, and angle \(DCE = 90°\). The prism stands on a rough horizontal surface. A horizontal force of magnitude \(T\) N acts at \(B\) in the direction \(CB\) (see diagram). The prism is in equilibrium.
  1. Show that the distance of the centre of mass of the prism from \(AB\) is \(0.488\) m. [4]
  2. Given that the weight of the prism is \(100\) N, find the greatest and least possible values of \(T\). [3]
CAIE M2 2014 November Q5
9 marks Standard +0.3
The equation of the trajectory of a small ball \(B\) projected from a fixed point \(O\) is $$y = -0.05x^2,$$ where \(x\) and \(y\) are, respectively, the displacements in metres of \(B\) from \(O\) in the horizontal and vertically upwards directions.
  1. Show that \(B\) is projected horizontally, and find its speed of projection. [3]
  2. Find the value of \(y\) when the direction of motion of \(B\) is \(60°\) below the horizontal, and find the corresponding speed of \(B\). [6]
CAIE M2 2014 November Q6
9 marks Challenging +1.2
\(O\), \(A\) and \(B\) are three points in a straight line on a smooth horizontal surface. A particle \(P\) of mass \(0.6\) kg moves along the line. At time \(t\) s the particle has displacement \(x\) m from \(O\) and speed \(v\) m s\(^{-1}\). The only horizontal force acting on \(P\) has magnitude \(0.4v^{\frac{1}{2}}\) N and acts in the direction \(OA\). Initially the particle is at \(A\), where \(x = 1\) and \(v = 1\).
  1. Show that \(3v^{\frac{1}{2}}\frac{dv}{dx} = 2\). [2]
  2. Express \(v\) in terms of \(x\). [4]
  3. Given that \(AB = 7\) m, find the value of \(t\) when \(P\) passes through \(B\). [3]
CAIE M2 2014 November Q7
12 marks Standard +0.8
\includegraphics{figure_7} One end of a light elastic string with modulus of elasticity \(15\) N is attached to a fixed point \(A\) which is \(2\) m vertically above a fixed small smooth ring \(R\). The string has natural length \(2\) m and it passes through \(R\). The other end of the string is attached to a particle \(P\) of mass \(m\) kg which moves with constant angular speed \(\omega\) rad s\(^{-1}\) in a horizontal circle which has its centre \(0.4\) m vertically below the ring. \(PR\) makes an acute angle \(\theta\) with the vertical (see diagram).
  1. Show that the tension in the string is \(\frac{3}{\cos\theta}\) N and hence find the value of \(m\). [4]
  2. Show that the value of \(\omega\) does not depend on \(\theta\). [4]
It is given that for one value of \(\theta\) the elastic potential energy stored in the string is twice the kinetic energy of \(P\).
  1. Find this value of \(\theta\). [4]
CAIE M2 2015 November Q1
4 marks Moderate -0.3
A particle \(P\) moves in a straight line and passes through a point \(O\) of the line with velocity \(2\text{ m s}^{-1}\). At time \(t\) s after passing through \(O\), the velocity of \(P\) is \(v\text{ m s}^{-1}\) and the acceleration of \(P\) is given by \(\text{e}^{-0.5t}\text{ m s}^{-2}\). Calculate the velocity of \(P\) when \(t = 1.2\). [4]
CAIE M2 2015 November Q2
5 marks Challenging +1.2
\includegraphics{figure_2} A uniform rigid rod \(AB\) of length \(1.2\text{ m}\) and weight \(8\text{ N}\) has a particle of weight \(2\text{ N}\) attached at the end \(B\). The end \(A\) of the rod is freely hinged to a fixed point. One end of a light elastic string of natural length \(0.8\text{ m}\) and modulus of elasticity \(20\text{ N}\) is attached to the hinge. The string passes over a small smooth pulley \(P\) fixed \(0.8\text{ m}\) vertically above the hinge. The other end of the string is attached to a small light smooth ring \(R\) which can slide on the rod. The system is in equilibrium with the rod inclined at an angle \(\theta°\) to the vertical (see diagram).
  1. Show that the tension in the string is \(20\sin\theta\text{ N}\). [1]
  2. Explain why the part of the string attached to the ring is perpendicular to the rod. [1]
  3. Find \(\theta\). [3]
CAIE M2 2015 November Q3
5 marks Standard +0.3
A particle \(P\) of mass \(0.3\text{ kg}\) moves in a straight line on a smooth horizontal surface. \(P\) passes through a fixed point \(O\) of the line with velocity \(8\text{ m s}^{-1}\). A force of magnitude \(2x\text{ N}\) acts on \(P\) in the direction \(PO\), where \(x\text{ m}\) is the displacement of \(P\) from \(O\).
  1. Show that \(v\frac{\text{d}v}{\text{d}x} = kx\) and state the value of the constant \(k\). [2]
  2. Find the value of \(x\) at the instant when \(P\) comes to instantaneous rest. [3]
CAIE M2 2015 November Q4
8 marks Standard +0.3
\includegraphics{figure_4} One end of a light inextensible string is attached to a fixed point \(A\). The string passes through a smooth bead \(B\) of mass \(0.3\text{ kg}\) and the other end of the string is attached to a fixed point \(C\) vertically below \(A\). The bead \(B\) moves with constant speed in a horizontal circle of radius \(0.6\text{ m}\) which has its centre between \(A\) and \(C\). The string makes an angle of \(30°\) with the vertical at \(A\) and an angle of \(45°\) with the vertical at \(C\) (see diagram).
  1. Calculate the speed of \(B\). [5]
The lower end of the string is detached from \(C\), and \(B\) is now attached to this end of the string. The other end of the string remains attached to \(A\). The bead is set in motion so that it moves with angular speed \(3\text{ rad s}^{-1}\) in a horizontal circle which has its centre vertically below \(A\).
  1. Calculate the tension in the string. [3]
CAIE M2 2015 November Q5
8 marks Standard +0.8
A particle \(P\) of mass \(0.2\text{ kg}\) is attached to one end of a light elastic string of natural length \(0.75\text{ m}\) and modulus of elasticity \(21\text{ N}\). The other end of the string is attached to a fixed point \(A\) which is \(0.8\text{ m}\) vertically above a smooth horizontal surface. \(P\) rests in equilibrium on the surface.
  1. Find the magnitude of the force exerted on \(P\) by the surface. [2]
\(P\) is now projected horizontally along the surface with speed \(3\text{ m s}^{-1}\).
  1. Calculate the extension of the string at the instant when \(P\) leaves the surface. [3]
  2. Hence find the speed of \(P\) at the instant when it leaves the surface. [3]
CAIE M2 2015 November Q6
9 marks Standard +0.3
[diagram]
A uniform circular disc has centre \(O\) and radius \(1.2\text{ m}\). The centre of the disc is at the origin of \(x\)- and \(y\)-axes. Two circular holes with centres at \(A\) and \(B\) are made in the disc (see diagram). The point \(A\) is on the negative \(x\)-axis with \(OA = 0.5\text{ m}\). The point \(B\) is on the negative \(y\)-axis with \(OB = 0.7\text{ m}\). The hole with centre \(A\) has radius \(0.3\text{ m}\) and the hole with centre \(B\) has radius \(0.4\text{ m}\). Find the distance of the centre of mass of the object from
  1. the \(x\)-axis, [4]
  2. the \(y\)-axis. [3]
The object can rotate freely in a vertical plane about a horizontal axis through \(O\).
  1. Calculate the angle which \(OA\) makes with the vertical when the object rests in equilibrium. [2]
CAIE M2 2015 November Q7
11 marks Challenging +1.2
A particle \(P\) is projected with speed \(V\text{ m s}^{-1}\) at an angle of \(60°\) above the horizontal from a point \(O\). At the instant \(1\text{ s}\) later a particle \(Q\) is projected from \(O\) with the same initial speed at an angle of \(45°\) above the horizontal. The two particles collide when \(Q\) has been in motion for \(t\text{ s}\).
  1. Show that \(t = 2.414\), correct to 3 decimal places. [3]
  2. Find the value of \(V\). [4]
The collision occurs after \(P\) has passed through the highest point of its trajectory.
  1. Calculate the vertical distance of \(P\) below its greatest height when \(P\) and \(Q\) collide. [4]
CAIE M2 2015 November Q1
4 marks Moderate -0.3
A particle \(P\) moves in a straight line and passes through a point \(O\) of the line with velocity \(2\,\text{m s}^{-1}\). At time \(t\) s after passing through \(O\), the velocity of \(P\) is \(v\,\text{m s}^{-1}\) and the acceleration of \(P\) is given by \(e^{-0.5t}\,\text{m s}^{-2}\). Calculate the velocity of \(P\) when \(t = 1.2\). [4]
CAIE M2 2015 November Q2
5 marks Standard +0.8
\includegraphics{figure_2} A uniform rigid rod \(AB\) of length \(1.2\,\text{m}\) and weight \(8\,\text{N}\) has a particle of weight \(2\,\text{N}\) attached at the end \(B\). The end \(A\) of the rod is freely hinged to a fixed point. One end of a light elastic string of natural length \(0.8\,\text{m}\) and modulus of elasticity \(20\,\text{N}\) is attached to the hinge. The string passes over a small smooth pulley \(P\) fixed \(0.8\,\text{m}\) vertically above the hinge. The other end of the string is attached to a small light smooth ring \(R\) which can slide on the rod. The system is in equilibrium with the rod inclined at an angle \(\theta°\) to the vertical (see diagram).
  1. Show that the tension in the string is \(20\sin\theta\,\text{N}\). [1]
  2. Explain why the part of the string attached to the ring is perpendicular to the rod. [1]
  3. Find \(\theta\). [3]
CAIE M2 2015 November Q3
5 marks Standard +0.3
A particle \(P\) of mass \(0.3\,\text{kg}\) moves in a straight line on a smooth horizontal surface. \(P\) passes through a fixed point \(O\) of the line with velocity \(8\,\text{m s}^{-1}\). A force of magnitude \(2x\,\text{N}\) acts on \(P\) in the direction \(PO\), where \(x\,\text{m}\) is the displacement of \(P\) from \(O\).
  1. Show that \(v\frac{dv}{dx} = kx\) and state the value of the constant \(k\). [2]
  2. Find the value of \(x\) at the instant when \(P\) comes to instantaneous rest. [3]
CAIE M2 2015 November Q4
8 marks Standard +0.3
\includegraphics{figure_4} One end of a light inextensible string is attached to a fixed point \(A\). The string passes through a smooth bead \(B\) of mass \(0.3\,\text{kg}\) and the other end of the string is attached to a fixed point \(C\) vertically below \(A\). The bead \(B\) moves with constant speed in a horizontal circle of radius \(0.6\,\text{m}\) which has its centre between \(A\) and \(C\). The string makes an angle of \(30°\) with the vertical at \(A\) and an angle of \(45°\) with the vertical at \(C\) (see diagram).
  1. Calculate the speed of \(B\). [5]
The lower end of the string is detached from \(C\), and \(B\) is now attached to this end of the string. The other end of the string remains attached to \(A\). The bead is set in motion so that it moves with angular speed \(3\,\text{rad s}^{-1}\) in a horizontal circle which has its centre vertically below \(A\).
  1. Calculate the tension in the string. [3]
CAIE M2 2015 November Q5
8 marks Standard +0.3
A particle \(P\) of mass \(0.2\,\text{kg}\) is attached to one end of a light elastic string of natural length \(0.75\,\text{m}\) and modulus of elasticity \(21\,\text{N}\). The other end of the string is attached to a fixed point \(A\) which is \(0.8\,\text{m}\) vertically above a smooth horizontal surface. \(P\) rests in equilibrium on the surface.
  1. Find the magnitude of the force exerted on \(P\) by the surface. [2]
\(P\) is now projected horizontally along the surface with speed \(3\,\text{m s}^{-1}\).
  1. Calculate the extension of the string at the instant when \(P\) leaves the surface. [3]
  2. Hence find the speed of \(P\) at the instant when it leaves the surface. [3]
CAIE M2 2015 November Q6
9 marks Standard +0.3
\includegraphics{figure_6} A uniform circular disc has centre \(O\) and radius \(1.2\,\text{m}\). The centre of the disc is at the origin of \(x\)- and \(y\)-axes. Two circular holes with centres at \(A\) and \(B\) are made in the disc (see diagram). The point \(A\) is on the negative \(x\)-axis with \(OA = 0.5\,\text{m}\). The point \(B\) is on the negative \(y\)-axis with \(OB = 0.7\,\text{m}\). The hole with centre \(A\) has radius \(0.3\,\text{m}\) and the hole with centre \(B\) has radius \(0.4\,\text{m}\). Find the distance of the centre of mass of the object from
  1. the \(x\)-axis, [4]
  2. the \(y\)-axis. [3]
The object can rotate freely in a vertical plane about a horizontal axis through \(O\).
  1. Calculate the angle which \(OA\) makes with the vertical when the object rests in equilibrium. [2]