Questions — CAIE (7646 questions)

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CAIE Further Paper 3 2020 November Q7
10 marks Challenging +1.8
A particle \(P\) moving in a straight line has displacement \(x\) m from a fixed point \(O\) on the line at time \(t\) s. The acceleration of \(P\), in m s\(^{-2}\), is given by \(\frac{200}{x^2} - \frac{100}{x^3}\) for \(x > 0\). When \(t = 0\), \(x = 1\) and \(P\) has velocity \(10\) m s\(^{-1}\) directed towards \(O\).
  1. Show that the velocity \(v\) m s\(^{-1}\) of \(P\) is given by \(v = \frac{10(1-2x)}{x}\). [5]
  2. Show that \(x\) and \(t\) are related by the equation \(e^{-40t} = (2x-1)e^{2x-2}\) and deduce what happens to \(x\) as \(t\) becomes large. [5]
CAIE Further Paper 3 2021 November Q1
4 marks Standard +0.3
One end of a light elastic string, of natural length \(a\) and modulus of elasticity \(3mg\), is attached to a fixed point \(O\) on a smooth horizontal plane. A particle \(P\) of mass \(m\) is attached to the other end of the string and moves in a horizontal circle with centre \(O\). The speed of \(P\) is \(\sqrt{\frac{1}{4}ga}\). Find the extension of the string. [4]
CAIE Further Paper 3 2021 November Q2
6 marks Challenging +1.2
A particle \(P\) of mass \(m\) kg moves along a horizontal straight line with acceleration \(a\) ms\(^{-2}\) given by $$a = \frac{v(1-2t^2)}{t},$$ where \(v\) ms\(^{-1}\) is the velocity of \(P\) at time \(t\) s.
  1. Find an expression for \(v\) in terms of \(t\) and an arbitrary constant. [3]
  2. Given that \(a = 5\) when \(t = 1\), find an expression, in terms of \(m\) and \(t\), for the horizontal force acting on \(P\) at time \(t\). [3]
CAIE Further Paper 3 2021 November Q3
6 marks Challenging +1.2
A light elastic string has natural length \(a\) and modulus of elasticity \(12mg\). One end of the string is attached to a fixed point \(O\). The other end of the string is attached to a particle of mass \(m\). The particle hangs in equilibrium vertically below \(O\). The particle is pulled vertically down and released from rest with the extension of the string equal to \(e\), where \(e > \frac{1}{4}a\). In the subsequent motion the particle has speed \(\sqrt{2ga}\) when it has ascended a distance \(\frac{1}{4}a\). Find \(e\) in terms of \(a\). [6]
CAIE Further Paper 3 2021 November Q4
8 marks Standard +0.8
\includegraphics{figure_4} A uniform lamina \(AECF\) is formed by removing two identical triangles \(BCE\) and \(CDF\) from a square lamina \(ABCD\). The square has side \(3a\) and \(EB = DF = h\) (see diagram).
  1. Find the distance of the centre of mass of the lamina \(AECF\) from \(AD\) and from \(AB\), giving your answers in terms of \(a\) and \(h\). [5]
The lamina \(AECF\) is placed vertically on its edge \(AE\) on a horizontal plane.
  1. Find, in terms of \(a\), the set of values of \(h\) for which the lamina remains in equilibrium. [3]
CAIE Further Paper 3 2021 November Q5
7 marks Challenging +1.8
A particle \(P\) is projected from a point \(O\) on a horizontal plane and moves freely under gravity. Its initial speed is \(u\) ms\(^{-1}\) and its angle of projection is \(\sin^{-1}(\frac{3}{5})\) above the horizontal. At time 8 s after projection, \(P\) is at the point \(A\). At time 32 s after projection, \(P\) is at the point \(B\). The direction of motion of \(P\) at \(B\) is perpendicular to its direction of motion at \(A\). Find the value of \(u\). [7]
CAIE Further Paper 3 2021 November Q6
8 marks Challenging +1.8
A particle \(P\), of mass \(m\), is attached to one end of a light inextensible string of length \(a\). The other end of the string is attached to a fixed point \(O\). The particle \(P\) moves in complete vertical circles about \(O\) with the string taut. The points \(A\) and \(B\) are on the path of \(P\) with \(AB\) a diameter of the circle. \(OA\) makes an angle \(\theta\) with the downward vertical through \(O\) and \(OB\) makes an angle \(\theta\) with the upward vertical through \(O\). The speed of \(P\) when it is at \(A\) is \(\sqrt{5ag}\). The ratio of the tension in the string when \(P\) is at \(A\) to the tension in the string when \(P\) is at \(B\) is \(9 : 5\).
  1. Find the value of \(\cos \theta\). [6]
  2. Find, in terms of \(a\) and \(g\), the greatest speed of \(P\) during its motion. [2]
CAIE Further Paper 3 2021 November Q7
11 marks Challenging +1.8
\includegraphics{figure_7} The smooth vertical walls \(AB\) and \(CB\) are at right angles to each other. A particle \(P\) is moving with speed \(u\) on a smooth horizontal floor and strikes the wall \(CB\) at an angle \(\alpha\). It rebounds at an angle \(\beta\) to the wall \(CB\). The particle then strikes the wall \(AB\) and rebounds at an angle \(\gamma\) to that wall (see diagram). The coefficient of restitution between each wall and \(P\) is \(e\).
  1. Show that \(\tan \beta = e \tan \alpha\). [3]
  2. Express \(\gamma\) in terms of \(\alpha\) and explain what this result means about the final direction of motion of \(P\). [4]
As a result of the two impacts the particle loses \(\frac{8}{9}\) of its initial kinetic energy.
  1. Given that \(\alpha + \beta = 90°\), find the value of \(e\) and the value of \(\tan \alpha\). [4]
CAIE Further Paper 3 2021 November Q1
5 marks Moderate -0.8
A particle is projected with speed \(u\) at an angle \(\alpha\) above the horizontal from a point \(O\) on a horizontal plane. The particle moves freely under gravity.
  1. Write down the horizontal and vertical components of the velocity of the particle at time \(T\) after projection. [2] At time \(T\) after projection, the direction of motion of the particle is perpendicular to the direction of projection.
  2. Express \(T\) in terms of \(u\), \(g\) and \(\alpha\). [2]
  3. Deduce that \(T > \frac{u}{g}\). [1]
CAIE Further Paper 3 2021 November Q2
6 marks Challenging +1.2
A light spring \(AB\) has natural length \(a\) and modulus of elasticity \(5mg\). The end \(A\) of the spring is attached to a fixed point on a smooth horizontal surface. A particle \(P\) of mass \(m\) is attached to the end \(B\) of the spring. The spring and particle \(P\) are at rest on the surface. Another particle \(Q\) of mass \(km\) is moving with speed \(\sqrt{4ga}\) along the horizontal surface towards \(P\) in the direction \(BA\). The particles \(P\) and \(Q\) collide directly and coalesce. In the subsequent motion the greatest amount by which the spring is compressed is \(\frac{2}{3}a\). Find the value of \(k\). [6]
CAIE Further Paper 3 2021 November Q3
6 marks Challenging +1.2
\includegraphics{figure_3} Particles \(A\) and \(B\), of masses \(m\) and \(3m\) respectively, are connected by a light inextensible string of length \(a\) that passes through a fixed smooth ring \(R\). Particle \(B\) hangs in equilibrium vertically below the ring. Particle \(A\) moves in horizontal circles with speed \(v\). Particles \(A\) and \(B\) are at the same horizontal level. The angle between \(AR\) and \(BR\) is \(\theta\) (see diagram).
  1. Show that \(\cos\theta = \frac{1}{3}\). [2]
  2. Find an expression for \(v\) in terms of \(a\) and \(g\). [4]
CAIE Further Paper 3 2021 November Q4
7 marks Challenging +1.2
\includegraphics{figure_4} An object is formed by removing a solid cylinder, of height \(ka\) and radius \(\frac{1}{2}a\), from a uniform solid hemisphere of radius \(a\). The axes of symmetry of the hemisphere and the cylinder coincide and one circular face of the cylinder coincides with the plane face of the hemisphere. \(AB\) is a diameter of the circular face of the hemisphere (see diagram).
  1. Show that the distance of the centre of mass of the object from \(AB\) is \(\frac{3a(2-k^2)}{2(8-3k)}\). [4] When the object is freely suspended from the point \(A\), the line \(AB\) makes an angle \(\theta\) with the downward vertical, where \(\tan\theta = \frac{7}{18}\).
  2. Find the possible values of \(k\). [3]
CAIE Further Paper 3 2021 November Q5
9 marks Challenging +1.8
\includegraphics{figure_5} Two uniform smooth spheres \(A\) and \(B\) of equal radii have masses \(m\) and \(\frac{2}{3}m\) respectively. The two spheres are each moving with speed \(u\) on a horizontal surface when they collide. Immediately before the collision \(A\)'s direction of motion is along the line of centres, and \(B\)'s direction of motion makes an angle of \(60°\) with the line of centres (see diagram). The coefficient of restitution between the spheres is \(\frac{2}{3}\).
  1. Find the angle through which the direction of motion of \(B\) is deflected by the collision. [6]
  2. Find the loss in the total kinetic energy of the system as a result of the collision. [3]
CAIE Further Paper 3 2021 November Q6
9 marks Challenging +1.8
A particle \(P\) of mass \(2\) kg moves along a horizontal straight line. The point \(O\) is a fixed point on this line. At time \(t\) s the velocity of \(P\) is \(v\) m s\(^{-1}\) and the displacement of \(P\) from \(O\) is \(x\) m. A force of magnitude \(\left(8x - \frac{128}{x^3}\right)\) N acts on \(P\) in the direction \(OP\). When \(t = 0\), \(x = 8\) and \(v = -15\).
  1. Show that \(v = -\frac{2}{3}(x^2 - 4)\). [5]
  2. Find an expression for \(x\) in terms of \(t\). [4]
CAIE Further Paper 3 2021 November Q7
8 marks Challenging +1.8
One end of a light inextensible string of length \(a\) is attached to a fixed point \(O\). The other end of the string is attached to a particle \(P\) of mass \(m\). The particle \(P\) is held vertically below \(O\) with the string taut and then projected horizontally. When the string makes an angle of \(60°\) with the upward vertical, \(P\) becomes detached from the string. In its subsequent motion, \(P\) passes through the point \(A\) which is a distance \(a\) vertically above \(O\).
  1. The speed of \(P\) when it becomes detached from the string is \(V\). Use the equation of the trajectory of a projectile to find \(V\) in terms of \(a\) and \(g\). [4]
  2. Find, in terms of \(m\) and \(g\), the tension in the string immediately after \(P\) is initially projected horizontally. [4]
CAIE Further Paper 3 2022 November Q1
4 marks Challenging +1.2
A particle \(P\) of mass \(m\) is attached to one end of a light inextensible string of length \(a\). The other end of the string is attached to a fixed point \(O\). The string is held taut with \(OP\) making an angle \(\alpha\) with the downward vertical, where \(\cos \alpha = \frac{2}{3}\). The particle \(P\) is projected perpendicular to \(OP\) in an upwards direction with speed \(\sqrt{3ag}\). It then starts to move along a circular path in a vertical plane. Find the cosine of the angle between the string and the upward vertical when the string first becomes slack. [4]
CAIE Further Paper 3 2022 November Q2
6 marks Challenging +1.2
\includegraphics{figure_2} A uniform lamina is in the form of a triangle \(ABC\) in which angle \(B\) is a right angle, \(AB = 9a\) and \(BC = 6a\). The point \(D\) is on \(BC\) such that \(BD = x\) (see diagram). The region \(ABD\) is removed from the lamina. The resulting shape \(ADC\) is placed with the edge \(DC\) on a horizontal surface and the plane \(ADC\) is vertical. Find the set of values of \(x\), in terms of \(a\), for which the shape is in equilibrium. [6]
CAIE Further Paper 3 2022 November Q3
6 marks Challenging +1.2
One end of a light elastic string, of natural length \(a\) and modulus of elasticity \(\frac{16}{9}Mg\), is attached to a fixed point \(O\). A particle \(P\) of mass \(4M\) is attached to the other end of the string and hangs vertically in equilibrium. Another particle of mass \(2M\) is attached to \(P\) and the combined particle is then released from rest. The speed of the combined particle when it has descended a distance \(\frac{1}{4}a\) is \(v\). Find an expression for \(v\) in terms of \(g\) and \(a\). [6]
CAIE Further Paper 3 2022 November Q4
7 marks Challenging +1.8
A particle \(P\) of mass \(5\) kg moves along a horizontal straight line. At time \(t\) s, the velocity of \(P\) is \(v\) m s\(^{-1}\) and its displacement from a fixed point \(O\) on the line is \(x\) m. The forces acting on \(P\) are a force of magnitude \(\frac{500}{v}\) N in the direction \(OP\) and a resistive force of magnitude \(\frac{1}{2}v^2\) N. When \(t = 0\), \(x = 0\) and \(v = 5\).
  1. Find an expression for \(v\) in terms of \(x\). [6]
  2. State the value that the speed approaches for large values of \(x\). [1]
CAIE Further Paper 3 2022 November Q5
9 marks Standard +0.3
A particle \(P\) is projected with speed \(u\) m s\(^{-1}\) at an angle of \(\theta\) above the horizontal from a point \(O\) on a horizontal plane and moves freely under gravity. The horizontal and vertical displacements of \(P\) from \(O\) at a subsequent time \(t\) s are denoted by \(x\) m and \(y\) m respectively.
  1. Show that the equation of the trajectory is given by $$y = x \tan \theta - \frac{gx^2}{2u^2}(1 + \tan^2 \theta).$$ [4]
In the subsequent motion \(P\) passes through the point with coordinates \((30, 20)\).
  1. Given that one possible value of \(\tan \theta\) is \(\frac{4}{3}\), find the other possible value of \(\tan \theta\). [5]
CAIE Further Paper 3 2022 November Q6
9 marks Challenging +1.8
\includegraphics{figure_6} A light inextensible string is threaded through a fixed smooth ring \(R\) which is at a height \(h\) above a smooth horizontal surface. One end of the string is attached to a particle \(A\) of mass \(m\). The other end of the string is attached to a particle \(B\) of mass \(\frac{1}{2}m\). The particle \(A\) moves in a horizontal circle on the surface. The particle \(B\) hangs in equilibrium below the ring and above the surface (see diagram). When \(A\) has constant angular speed \(\omega\), the angle between \(AR\) and \(BR\) is \(\theta\) and the normal reaction between \(A\) and the surface is \(N\). When \(A\) has constant angular speed \(\frac{3}{2}\omega\), the angle between \(AR\) and \(BR\) is \(\alpha\) and the normal reaction between \(A\) and the surface is \(\frac{1}{2}N\).
  1. Show that \(\cos \theta = \frac{4}{9}\cos \alpha\). [5]
  2. Find \(N\) in terms of \(m\) and \(g\) and find the value of \(\cos \alpha\). [4]
CAIE Further Paper 3 2022 November Q7
9 marks Challenging +1.8
\includegraphics{figure_7} Two uniform smooth spheres \(A\) and \(B\) of equal radii have masses \(m\) and \(\frac{1}{2}m\) respectively. The two spheres are moving on a horizontal surface when they collide. Immediately before the collision, sphere \(A\) is travelling with speed \(u\) and its direction of motion makes an angle \(\alpha\) with the line of centres. Sphere \(B\) is travelling with speed \(2u\) and its direction of motion makes an angle \(\beta\) with the line of centres (see diagram). The coefficient of restitution between the spheres is \(\frac{5}{8}\) and \(\alpha + \beta = 90°\).
  1. Find the component of the velocity of \(B\) parallel to the line of centres after the collision, giving your answer in terms of \(u\) and \(\alpha\). [4]
The direction of motion of \(B\) after the collision is parallel to the direction of motion of \(A\) before the collision.
  1. Find the value of \(\tan \alpha\). [5]
CAIE Further Paper 3 2023 November Q1
7 marks Challenging +1.8
\includegraphics{figure_1} Two uniform smooth spheres \(A\) and \(B\) of equal radii have masses \(m\) and \(2m\) respectively. The two spheres are moving with equal speeds \(u\) on a smooth horizontal surface when they collide. Immediately before the collision, \(A\)'s direction of motion makes an angle of \(60°\) with the line of centres, and \(B\)'s direction of motion makes an angle \(\theta\) with the line of centres (see diagram). The coefficient of restitution between the spheres is \(e\). After the collision, the component of the velocity of \(A\) along the line of centres is \(v\) and \(B\) moves perpendicular to the line of centres. Sphere \(A\) now has twice as much kinetic energy as sphere \(B\).
  1. Show that \(v = \frac{1}{2}u(4\cos\theta - 1)\). [1]
  2. Find the value of \(\cos\theta\). [4]
  3. Find the value of \(e\). [2]
CAIE Further Paper 3 2023 November Q2
7 marks Challenging +1.2
A ball of mass \(2\) kg is projected vertically downwards with speed \(5\text{ ms}^{-1}\) through a liquid. At time \(t\) s after projection, the velocity of the ball is \(v\text{ ms}^{-1}\) and its displacement from its starting point is \(x\) m. The forces acting on the ball are its weight and a resistive force of magnitude \(0.2v^2\) N.
  1. Find an expression for \(v\) in terms of \(t\). [6]
  2. Deduce what happens to \(v\) for large values of \(t\). [1]
CAIE Further Paper 3 2023 November Q3
8 marks Challenging +1.8
\includegraphics{figure_3} A uniform square lamina of side \(2a\) and weight \(W\) is suspended from a light inextensible string attached to the midpoint \(E\) of the side \(AB\). The other end of the string is attached to a fixed point \(P\) on a rough vertical wall. The vertex \(B\) of the lamina is in contact with the wall. The string \(EP\) is perpendicular to the side \(AB\) and makes an angle \(\theta\) with the wall (see diagram). The string and the lamina are in a vertical plane perpendicular to the wall. The coefficient of friction between the wall and the lamina is \(\frac{1}{2}\). Given that the vertex \(B\) is about to slip up the wall, find the value of \(\tan\theta\). [8]