Questions — CAIE Further Paper 3 (180 questions)

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CAIE Further Paper 3 2024 June Q3
7 marks Standard +0.8
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 \(O\). When the particle is hanging vertically below \(O\), it is projected horizontally with speed \(u\) so that it begins to move along a circular path. When \(P\) is at the lowest point of its motion, the tension in the string is \(T\). When \(OP\) makes an angle \(\theta\) with the upward vertical, the tension in the string is \(S\).
  1. Show that \(S = T - 3mg(1 + \cos\theta)\). [5]
  2. Given that \(u = \sqrt{4ag}\), find the value of \(\cos\theta\) when the string goes slack. [2]
CAIE Further Paper 3 2024 June Q4
7 marks Challenging +1.2
\includegraphics{figure_4} A light spring of natural length \(a\) and modulus of elasticity \(kmg\) is attached to a fixed point \(O\) on a smooth plane inclined to the horizontal at an angle \(\theta\), where \(\sin\theta = \frac{1}{4}\). A particle of mass \(m\) is attached to the lower end of the spring and is held at the point \(A\) on the plane, where \(OA = 2a\) and \(OA\) is along a line of greatest slope of the plane (see diagram). The particle is released from rest and is moving with speed \(V\) when it passes through the point \(B\) on the plane, where \(OB = \frac{3}{2}a\). The speed of the particle is \(\frac{1}{3}V\) when it passes through the point \(C\) on the plane, where \(OC = \frac{3}{4}a\). Find the value of \(k\). [7]
CAIE Further Paper 3 2024 June Q5
7 marks Standard +0.8
\includegraphics{figure_5} A uniform lamina is in the form of a triangle \(OBC\), with \(OC = 18a\), \(OB = 24a\) and angle \(COB = 90°\). The point \(A\) on \(OB\) is such that \(OA = x\) (see diagram). The triangle \(OAC\) is removed from the lamina.
  1. Find, in terms of \(a\) and \(x\), the distance of the centre of mass of the remaining object \(ABC\) from \(OC\). [3]
The object \(ABC\) is suspended from \(C\). In its equilibrium position, the side \(AB\) makes an angle \(\theta\) with the vertical, where \(\tan\theta = \frac{8}{5}\).
  1. Find \(x\) in terms of \(a\). [4]
CAIE Further Paper 3 2024 June Q6
8 marks Standard +0.8
A particle \(P\) is projected with speed \(u\text{ ms}^{-1}\) at an angle \(\theta\) above the horizontal from a point \(O\) and moves freely under gravity. After 5 seconds the speed of \(P\) is \(\frac{3}{4}u\).
  1. Show that \(\frac{7}{16}u^2 - 100u\sin\theta + 2500 = 0\). [3]
  2. It is given that the velocity of \(P\) after 5 seconds is perpendicular to the initial velocity. Find, in either order, the value of \(u\) and the value of \(\sin\theta\). [5]
CAIE Further Paper 3 2024 June Q7
11 marks Standard +0.8
A parachutist of mass \(m\) kg opens his parachute when he is moving vertically downwards with a speed of \(50\text{ ms}^{-1}\). At time \(t\) s after opening his parachute, he has fallen a distance \(x\) m from the point where he opened his parachute, and his speed is \(v\text{ ms}^{-1}\). The forces acting on him are his weight and a resistive force of magnitude \(mv\) N.
  1. Find an expression for \(v\) in terms of \(t\). [6]
  2. Find an expression for \(x\) in terms of \(t\). [3]
  3. Find the distance that the parachutist has fallen, since opening his parachute, when his speed is \(15\text{ ms}^{-1}\). [2]
CAIE Further Paper 3 2020 November Q1
3 marks Standard +0.3
A particle \(P\) of mass \(m\) is placed on a fixed smooth plane which is inclined at an angle \(\theta\) to the horizontal. A light spring, of natural length \(a\) and modulus of elasticity \(3mg\), has one end attached to \(P\) and the other end attached to a fixed point \(O\) at the top of the plane. The spring lies along a line of greatest slope of the plane. The system is released from rest with the spring at its natural length. Find, in terms of \(a\) and \(\theta\), an expression for the greatest extension of the spring in the subsequent motion. [3]
CAIE Further Paper 3 2020 November Q2
5 marks Challenging +1.2
\includegraphics{figure_2} A particle \(P\) 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 with the string taut and making an angle \(\theta\) with the downward vertical. The particle \(P\) is then projected with speed \(\frac{1}{2}\sqrt{5ag}\) perpendicular to the string and just completes a vertical circle (see diagram). Find the value of \(\cos\theta\). [5]
CAIE Further Paper 3 2020 November Q3
6 marks Challenging +1.2
One end of a light elastic string, of natural length \(a\) and modulus of elasticity \(4mg\), is attached to a fixed point \(O\). The other end of the string is attached to a particle of mass \(m\). The particle moves in a horizontal circle with a constant angular speed \(\sqrt{\frac{g}{a}}\) with the string inclined at an angle \(\theta\) to the downward vertical through \(O\). The length of the string during this motion is \((k+1)a\).
  1. Find the value of \(k\). [4]
  2. Find the value of \(\cos\theta\). [2]
CAIE Further Paper 3 2020 November Q4
6 marks Standard +0.8
\includegraphics{figure_4} The diagram shows the cross-section \(ABCD\) of a uniform solid object which is formed by removing a cone with cross-section \(DCE\) from the top of a larger cone with cross-section \(ABE\). The perpendicular distance between \(AB\) and \(DC\) is \(h\), the diameter \(AB\) is \(6r\) and the diameter \(DC\) is \(2r\).
  1. Find an expression, in terms of \(h\), for the distance of the centre of mass of the solid object from \(AB\). [4]
The object is freely suspended from the point \(B\) and hangs in equilibrium. The angle between \(AB\) and the downward vertical through \(B\) is \(\theta\).
  1. Given that \(h = \frac{13}{4}r\), find the value of \(\tan\theta\). [2]
CAIE Further Paper 3 2020 November Q5
10 marks Standard +0.3
A particle \(P\) is projected with speed \(u\) at an angle \(\alpha\) 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. Derive the equation of the trajectory of \(P\) in the form $$y = x\tan\alpha - \frac{gx^2}{2u^2}\sec^2\alpha.$$ [3]
The point \(Q\) is the highest point on the trajectory of \(P\) in the case where \(\alpha = 45°\).
  1. Show that the \(x\)-coordinate of \(Q\) is \(\frac{u^2}{2g}\). [3]
  2. Find the other value of \(\alpha\) for which \(P\) would pass through the point \(Q\). [4]
CAIE Further Paper 3 2020 November Q6
10 marks Challenging +1.8
Two smooth spheres \(A\) and \(B\) have equal radii and masses \(m\) and \(2m\) respectively. Sphere \(B\) is at rest on a smooth horizontal floor. Sphere \(A\) is moving on the floor with velocity \(u\) and collides directly with \(B\). The coefficient of restitution between the spheres is \(e\).
  1. Find, in terms of \(u\) and \(e\), the velocities of \(A\) and \(B\) after the collision. [3]
Subsequently, \(B\) collides with a fixed vertical wall which makes an angle \(\theta\) with the direction of motion of \(B\), where \(\tan\theta = \frac{3}{4}\). The coefficient of restitution between \(B\) and the wall is \(\frac{2}{3}\). Immediately after \(B\) collides with the wall, the kinetic energy of \(B\) is \(\frac{5}{27}\) of the kinetic energy of \(B\).
  1. Find the possible values of \(e\). [7]
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]