Questions — CAIE (7646 questions)

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CAIE Further Paper 3 2021 June Q5
6 marks Standard +0.8
A particle \(P\) of mass \(m\) kg is projected vertically upwards from a point \(O\), with speed \(20\) m s\(^{-1}\), and moves under gravity. There is a resistive force of magnitude \(2mv\) N, where \(v\) m s\(^{-1}\) is the speed of \(P\) at time \(t\) s after projection.
  1. Find an expression for \(v\) in terms of \(t\), while \(P\) is moving upwards. [6]
CAIE Further Paper 3 2021 June Q5
4 marks Standard +0.8
The displacement of \(P\) from \(O\) is \(x\) m at time \(t\) s.
  1. Find an expression for \(x\) in terms of \(t\), while \(P\) is moving upwards. [2]
  2. Find, correct to 3 significant figures, the greatest height above \(O\) reached by \(P\). [2]
CAIE Further Paper 3 2021 June Q6
3 marks Standard +0.8
\includegraphics{figure_6} Two uniform smooth spheres \(A\) and \(B\) of equal radii have masses \(m\) and \(km\) respectively. Sphere \(A\) is moving with speed \(u\) on a smooth horizontal surface when it collides with sphere \(B\) which is at rest. Immediately before the collision, \(A\)'s direction of motion makes an angle \(\theta\) with the line of centres (see diagram). The coefficient of restitution between the spheres is \(\frac{1}{3}\).
  1. Show that the speed of \(B\) after the collision is \(\frac{4u \cos \theta}{3(1 + k)}\). [3]
CAIE Further Paper 3 2021 June Q6
6 marks Challenging +1.8
70% of the total kinetic energy of the spheres is lost as a result of the collision.
  1. Given that \(\tan \theta = \frac{1}{3}\), find the value of \(k\). [6]
CAIE Further Paper 3 2021 June Q7
4 marks Easy -1.2
A particle \(P\) is projected with speed \(u\) at an angle \(\theta\) 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 \(t\) are denoted by \(x\) and \(y\) respectively.
  1. Use the equation of the trajectory given in the List of formulae (MF19), together with the condition \(y = 0\), to establish an expression for the range \(R\) in terms of \(u\), \(\theta\) and \(g\). [2]
  2. Deduce an expression for the maximum height \(H\), in terms of \(u\), \(\theta\) and \(g\). [2]
CAIE Further Paper 3 2021 June Q7
5 marks Challenging +1.2
It is given that \(R = \frac{4H}{\sqrt{3}}\).
  1. Show that \(\theta = 60°\). [1]
It is given also that \(u = \sqrt{40}\) m s\(^{-1}\).
  1. Find, by differentiating the equation of the trajectory or otherwise, the set of values of \(x\) for which the direction of motion makes an angle of less than \(45°\) with the horizontal. [4]
CAIE Further Paper 3 2022 June Q1
5 marks Standard +0.3
\includegraphics{figure_1} A particle of weight 10 N is attached to one end of a light elastic string. The other end of the string is attached to a fixed point \(A\) on a horizontal ceiling. A horizontal force of 7.5 N acts on the particle. In the equilibrium position, the string makes an angle \(\theta\) with the ceiling (see diagram). The string has natural length 0.8 m and modulus of elasticity 50 N.
  1. Find the tension in the string. [2]
  2. Find the vertical distance between the particle and the ceiling. [3]
CAIE Further Paper 3 2022 June Q2
5 marks Challenging +1.2
One end of a light inextensible string of length \(a\) is attached to a fixed point \(O\). A particle of mass \(m\) is attached to the other end of the string. The particle is held at the point \(A\) with the string taut. The angle between \(OA\) and the downward vertical is equal to \(\alpha\), where \(\cos \alpha = \frac{4}{5}\). The particle is projected from \(A\), perpendicular to the string in an upwards direction, with a speed \(\sqrt{3ga}\). It then moves along a circular path in a vertical plane. The string first goes slack when it makes an angle \(\theta\) with the upward vertical through \(O\). Find the value of \(\cos \theta\). [5]
CAIE Further Paper 3 2022 June Q3
5 marks Challenging +1.2
A particle \(P\) is moving in a horizontal straight line. Initially \(P\) is at the point \(O\) on the line and is moving with velocity \(25 \text{ m s}^{-1}\). At time \(t\) s after passing through \(O\), the acceleration of \(P\) is \(-\frac{4000}{(5t + 4)^3} \text{ m s}^{-2}\) in the direction \(PO\). The displacement of \(P\) from \(O\) at time \(t\) is \(x\) m. Find an expression for \(x\) in terms of \(t\). [5]
CAIE Further Paper 3 2022 June Q4
8 marks Challenging +1.2
\includegraphics{figure_4} An object is composed of a hemispherical shell of radius \(2a\) attached to a closed hollow circular cylinder of height \(h\) and base radius \(a\). The hemispherical shell and the hollow cylinder are made of the same uniform material. The axes of symmetry of the shell and the cylinder coincide. \(AB\) is a diameter of the lower end of the cylinder (see diagram).
  1. Find, in terms of \(a\) and \(h\), an expression for the distance of the centre of mass of the object from \(AB\). [4]
The object is placed on a rough plane which is inclined to the horizontal at an angle \(\theta\), where \(\tan \theta = \frac{2}{5}\). The object is in equilibrium with \(AB\) in contact with the plane and lying along a line of greatest slope of the plane.
  1. Find the set of possible values of \(h\), in terms of \(a\). [4]
CAIE Further Paper 3 2022 June Q5
7 marks Challenging +1.8
\includegraphics{figure_5} A light inextensible string \(AB\) passes through two small holes \(C\) and \(D\) in a smooth horizontal table where \(AC = 3a\) and \(DB = a\). A particle of mass \(m\) is attached at the end \(A\) and moves in a horizontal circle with angular velocity \(\omega\). A particle of mass \(\frac{3}{4}m\) is attached to the end \(B\) and moves in a horizontal circle with angular velocity \(k\omega\). \(AC\) makes an angle \(\theta\) with the downward vertical and \(DB\) makes an angle \(\theta\) with the horizontal (see diagram). Find the value of \(k\). [7]
CAIE Further Paper 3 2022 June Q6
9 marks Challenging +1.2
Two uniform smooth spheres \(A\) and \(B\) of equal radii have masses \(m\) and \(km\) respectively. The two spheres are on a horizontal surface. Sphere \(A\) is travelling with speed \(u\) towards sphere \(B\) which is at rest. The spheres collide. Immediately before the collision, the direction of motion of \(A\) makes an angle \(\alpha\) with the line of centres. The coefficient of restitution between the spheres is \(\frac{1}{2}\).
  1. Show that the speed of \(B\) after the collision is \(\frac{3u \cos \alpha}{2(1 + k)}\) and find also an expression for the speed of \(A\) along the line of centres after the collision, in terms of \(k\), \(u\) and \(\alpha\). [4]
After the collision, the kinetic energy of \(A\) is equal to the kinetic energy of \(B\).
  1. Given that \(\tan \alpha = \frac{2}{3}\), find the possible values of \(k\). [5]
CAIE Further Paper 3 2022 June Q7
11 marks Challenging +1.2
Particles \(P\) and \(Q\) are projected in the same vertical plane from a point \(O\) at the top of a cliff. The height of the cliff exceeds 50 m. Both particles move freely under gravity. Particle \(P\) is projected with speed \(\frac{35}{2} \text{ m s}^{-1}\) at an angle \(\alpha\) above the horizontal, where \(\tan \alpha = \frac{4}{3}\). Particle \(Q\) is projected with speed \(u \text{ m s}^{-1}\) at an angle \(\beta\) above the horizontal, where \(\tan \beta = \frac{1}{2}\). Particle \(Q\) is projected one second after the projection of particle \(P\). The particles collide \(T\) s after the projection of particle \(Q\).
  1. Write down expressions, in terms of \(T\), for the horizontal displacements of \(P\) and \(Q\) from \(O\) when they collide and hence show that \(4uT = 21\sqrt{5(T + 1)}\). [4]
  2. Find the value of \(T\). [4]
  3. Find the horizontal and vertical displacements of the particles from \(O\) when they collide. [3]
CAIE Further Paper 3 2023 June Q1
5 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 \(O\). The other end of the string is attached to a particle \(P\) of mass \(m\). The string hangs with \(P\) vertically below \(O\). The particle \(P\) is pulled vertically downwards so that the extension of the string is \(2a\). The particle \(P\) is then released from rest.
  1. Find the speed of \(P\) when it is at a distance \(\frac{3}{4}a\) below \(O\). [3]
  2. Find the initial acceleration of \(P\) when it is released from rest. [2]
CAIE Further Paper 3 2023 June Q2
5 marks Challenging +1.2
\includegraphics{figure_2} A particle \(P\) of mass \(m\) is moving with speed \(u\) on a fixed smooth horizontal surface. It collides at an angle \(\alpha\) with a fixed smooth vertical barrier. After the collision, \(P\) moves at an angle \(\theta\) with the barrier, where \(\tan\theta = \frac{1}{3}\) (see diagram). The coefficient of restitution between \(P\) and the barrier is \(e\). The particle \(P\) loses 20% of its kinetic energy as a result of the collision. Find the value of \(e\). [5]
CAIE Further Paper 3 2023 June Q3
7 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 particle \(P\) is held at the point \(A\), where \(OA\) makes an angle \(\theta\) with the downward vertical through \(O\), and with the string taut. The particle \(P\) is projected perpendicular to \(OA\) in an upwards direction with speed \(u\). It then starts to move along a circular path in a vertical plane. The string goes slack when \(P\) is at \(B\), where angle \(AOB\) is \(90°\) and the speed of \(P\) is \(\sqrt{\frac{1}{3}ag}\).
  1. Find the value of \(\sin\theta\). [2]
  2. Find, in terms of \(m\) and \(g\), the tension in the string when \(P\) is at \(A\). [5]
CAIE Further Paper 3 2023 June Q4
8 marks Challenging +1.2
\includegraphics{figure_4} An object is formed from a solid hemisphere, of radius \(2a\), and a solid cylinder, of radius \(a\) and height \(d\). The hemisphere and the cylinder are made of the same material. The cylinder is attached to the plane face of the hemisphere. The line \(OC\) forms a diameter of the base of the cylinder, where \(C\) is the centre of the plane face of the hemisphere and \(O\) is common to both circumferences (see diagram). Relative to axes through \(O\), parallel and perpendicular to \(OC\) as shown, the centre of mass of the object is \((\bar{x}, \bar{y})\).
  1. Show that \(\bar{x} = \frac{32a^2 + 3ad}{16a + 3d}\) and find an expression, in terms of \(a\) and \(d\), for \(\bar{y}\). [5]
The object is placed on a rough plane which is inclined to the horizontal at an angle \(\theta\) where \(\sin\theta = \frac{1}{6}\). The object is in equilibrium with \(CO\) horizontal, where \(CO\) lies in a vertical plane through a line of greatest slope.
  1. Find \(d\) in terms of \(a\). [3]
CAIE Further Paper 3 2023 June Q5
7 marks Standard +0.8
A light elastic string of natural length \(a\) and modulus of elasticity \(\lambda mg\) has one end attached to a fixed point \(O\) on a smooth horizontal surface. When a particle of mass \(m\) is attached to the free end of the string, it moves with speed \(v\) in a horizontal circle with centre \(O\) and radius \(x\). When, instead, a particle of mass \(2m\) is attached to the free end of the string, this particle moves with speed \(\frac{1}{2}v\) in a horizontal circle with centre \(O\) and radius \(\frac{4}{3}x\).
  1. Find \(x\) in terms of \(a\). [5]
  2. Given that \(v = \sqrt{12ag}\), find the value of \(\lambda\). [2]
CAIE Further Paper 3 2023 June Q6
9 marks Challenging +1.8
A particle \(P\) moving in a straight line has displacement \(x\)m from a fixed point \(O\) on the line and velocity \(v\)m s\(^{-1}\) at time \(t\)s. The acceleration of \(P\), in m s\(^{-2}\), is given by \(6\sqrt{v + 9}\). When \(t = 0\), \(x = 2\) and \(v = 72\).
  1. Find an expression for \(v\) in terms of \(x\). [4]
  2. Find an expression for \(x\) in terms of \(t\). [5]
CAIE Further Paper 3 2023 June Q7
9 marks Standard +0.8
At time \(t\)s, a particle \(P\) is projected with speed \(40\)m s\(^{-1}\) at an angle \(\theta\) above the horizontal from a point \(O\) on a horizontal plane and moves freely under gravity. The greatest height achieved by \(P\) during its flight is \(H\)m and the corresponding time is \(T\)s.
  1. Obtain expressions for \(H\) and \(T\) in terms of \(\theta\). [2]
During the time between \(t = T\) and \(t = 3\), \(P\) descends a distance \(\frac{1}{4}H\).
  1. Find the value of \(\theta\). [4]
  2. Find the speed of \(P\) when \(t = 3\). [3]
CAIE Further Paper 3 2023 June 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 particle \(P\) is held at the point \(A\), where \(OA\) makes an angle \(\alpha\) with the downward vertical through \(O\), and with the string taut. The particle \(P\) is projected perpendicular to \(OA\) in an upwards direction with speed \(\sqrt{3ag}\). It then starts to move along a circular path in a vertical plane. The string goes slack when \(P\) is at \(B\), where \(OB\) makes an angle \(\theta\) with the upward vertical. Given that \(\cos \alpha = \frac{3}{5}\), find the value of \(\cos \theta\). [4]
CAIE Further Paper 3 2023 June Q2
4 marks Standard +0.8
One end of a light elastic string, of natural length \(a\) and modulus of elasticity \(\lambda mg\), is attached to a fixed point \(O\). The string lies on a smooth horizontal surface. A particle \(P\) of mass \(m\) is attached to the other end of the string. The particle \(P\) is projected in the direction \(OP\). When the length of the string is \(\frac{4}{3}a\), the speed of \(P\) is \(\sqrt{2ag}\). When the length of the string is \(\frac{5}{3}a\), the speed of \(P\) is \(\frac{1}{2}\sqrt{2ag}\). Find the value of \(\lambda\). [4]
CAIE Further Paper 3 2023 June Q3
7 marks Standard +0.8
\includegraphics{figure_3} A uniform lamina is in the form of a triangle \(ABC\), with \(AC = 8a\), \(BC = 6a\) and angle \(ACB = 90°\). The point \(D\) on \(AC\) is such that \(AD = 3a\). The point \(E\) on \(CB\) is such that \(CE = x\) (see diagram). The triangle \(CDE\) is removed from the lamina.
  1. Find, in terms of \(a\) and \(x\), the distance of the centre of mass of the remaining object \(ADEB\) from \(AC\). [4]
The object \(ADEB\) is on the point of toppling about the point \(E\) when the object is in the vertical plane with its edge \(EB\) on a smooth horizontal surface.
  1. Find \(x\) in terms of \(a\). [3]
CAIE Further Paper 3 2023 June Q4
8 marks Challenging +1.2
\includegraphics{figure_4} Two identical smooth uniform spheres \(A\) and \(B\) each have mass \(m\). The two spheres are moving on a smooth horizontal surface when they collide with speeds \(u\) and \(2u\) respectively. Immediately before the collision, \(A\)'s direction of motion makes an angle of \(30°\) with the line of centres, and \(B\)'s direction of motion is perpendicular to the line of centres (see diagram). After the collision, \(A\) and \(B\) are moving in the same direction. The coefficient of restitution between the spheres is \(e\).
  1. Find the value of \(e\). [5]
  2. Find the loss in the total kinetic energy of the spheres as a result of the collision. [3]
CAIE Further Paper 3 2023 June Q5
8 marks Challenging +1.2
One end of a light elastic string, of natural length \(12a\) and modulus of elasticity \(kmg\), is attached to a fixed point \(O\). The other end of the string is attached to a particle of mass \(m\). The particle moves with constant speed \(\frac{2}{3}\sqrt{3ag}\) in a horizontal circle with centre at a distance \(12a\) below \(O\). The string is inclined at an angle \(\theta\) to the downward vertical through \(O\).
  1. Find, in terms of \(a\), the extension of the string. [5]
  2. Find the value of \(k\). [3]