Questions Further Paper 3 (180 questions)

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CAIE Further Paper 3 2024 November Q4
4 marks Standard +0.3
\includegraphics{figure_4} An object is formed by removing a cylinder of radius \(\frac{2}{3}a\) and height \(kh\) (\(k < 1\)) from a uniform solid cylinder of radius \(a\) and height \(h\). The vertical axes of symmetry of the two cylinders coincide. The upper faces of the two cylinders are in the same plane as each other. The points \(A\) and \(B\) are the opposite ends of a diameter of the upper face of the object (see diagram).
  1. Find, in terms of \(h\) and \(k\), the distance of the centre of mass of the object from \(AB\). [4]
CAIE Further Paper 3 2024 November Q4
3 marks Standard +0.8
When the object is suspended from \(A\), the angle between \(AB\) and the vertical is \(\theta\), where \(\tan\theta = \frac{1}{2}\).
  1. Given that \(h = \frac{8}{3}a\), find the possible values of \(k\). [3]
CAIE Further Paper 3 2024 November Q5
4 marks Challenging +1.2
A particle \(P\) of mass \(2\text{kg}\) moving on a horizontal straight line has displacement \(x\text{m}\) from a fixed point \(O\) on the line and velocity \(v\text{ms}^{-1}\) at time \(t\). The only horizontal force acting on \(P\) is a variable force \(F\text{N}\) which can be expressed as a function of \(t\). It is given that $$\frac{v}{x} = \frac{3-t}{1+t}$$ and when \(t = 0\), \(x = 5\).
  1. Find an expression for \(x\) in terms of \(t\). [4]
CAIE Further Paper 3 2024 November Q5
3 marks Moderate -0.5
  1. Find the magnitude of \(F\) when \(t = 3\). [3]
CAIE Further Paper 3 2024 November Q6
3 marks Standard +0.3
\includegraphics{figure_6} A particle \(P\) of mass \(0.05\text{kg}\) is attached to one end of a light inextensible string of length \(1\text{m}\). The other end of the string is attached to a fixed point \(O\). A particle \(Q\) of mass \(0.04\text{kg}\) is attached to one end of a second light inextensible string. The other end of this string is attached to \(P\). The particle \(P\) moves in a horizontal circle of radius \(0.8\text{m}\) with angular speed \(\omega\text{rads}^{-1}\). The particle \(Q\) moves in a horizontal circle of radius \(1.4\text{m}\) also with angular speed \(\omega\text{rads}^{-1}\). The centres of the circles are vertically below \(O\), and \(O\), \(P\) and \(Q\) are always in the same vertical plane. The strings \(OP\) and \(PQ\) remain at constant angles \(\alpha\) and \(\beta\) respectively to the vertical (see diagram).
  1. Find the tension in the string \(OP\). [3]
CAIE Further Paper 3 2024 November Q6
3 marks Moderate -0.5
  1. Find the value of \(\omega\). [3]
CAIE Further Paper 3 2024 November Q6
2 marks Moderate -1.0
  1. Find the value of \(\beta\). [2]
CAIE Further Paper 3 2024 November Q7
4 marks Challenging +1.8
A particle \(P\) is projected with speed \(u\) at an angle \(\tan^{-1}\left(\frac{4}{3}\right)\) above the horizontal from a point \(O\) on a horizontal plane and moves freely under gravity. When \(P\) is moving horizontally, it strikes a smooth inclined plane at the point \(A\). This plane is inclined to the horizontal at an angle \(\alpha\), and the line of greatest slope through \(A\) lies in the vertical plane through \(O\) and \(A\). As a result of the impact, \(P\) moves vertically upwards. The coefficient of restitution between \(P\) and the inclined plane is \(e\).
  1. Show that \(e\tan^2\alpha = 1\). [4]
CAIE Further Paper 3 2024 November Q7
6 marks Challenging +1.2
In its subsequent motion, the greatest height reached by \(P\) above \(A\) is \(\frac{3}{10}\) of the vertical height of \(A\) above the horizontal plane.
  1. Find the value of \(e\). [6]
CAIE Further Paper 3 2024 November Q1
3 marks Standard +0.3
A particle of mass \(2\) kg is attached to one end of a light elastic string of natural length \(0.8\) m and modulus of elasticity \(100\) N. The other end of the string is attached to a fixed point \(O\) on a smooth horizontal surface. The particle is moving in a horizontal circle about \(O\) with the string taut and with constant angular speed \(5\) radians per second. Find the extension of the string. [3]
CAIE Further Paper 3 2024 November Q2
4 marks Standard +0.3
A particle \(P\) of mass \(m\) is attached to one end of a light elastic spring of natural length \(a\) and modulus of elasticity \(5mg\). The other end of the spring is attached to a fixed point \(O\). The spring hangs vertically with \(P\) below \(O\). The particle \(P\) is pulled down vertically and released from rest when the length of the spring is \(\frac{7}{5}a\). Find the distance of \(P\) below \(O\) when \(P\) first comes to instantaneous rest. [4]
CAIE Further Paper 3 2024 November Q3
7 marks Challenging +1.8
\includegraphics{figure_3} The diagram shows two identical smooth uniform spheres \(A\) and \(B\) of equal radii and each of mass \(m\). The two spheres are moving on a smooth horizontal surface when they collide with speeds \(2u\) and \(3u\) respectively. Immediately before the collision, \(A\)'s direction of motion makes an angle \(\theta\) with the line of centres and \(B\)'s direction of motion is perpendicular to that of \(A\). After the collision, \(B\) moves perpendicular to the line of centres. The coefficient of restitution between the spheres is \(\frac{1}{3}\).
  1. Find the value of \(\tan \theta\). [3]
  2. Find the total loss of kinetic energy as a result of the collision. [2]
  3. Find, in degrees, the angle through which the direction of motion of \(A\) is deflected as a result of the collision. [2]
CAIE Further Paper 3 2024 November Q4
7 marks Challenging +1.8
\includegraphics{figure_4} The end \(A\) of a uniform rod \(AB\) of length \(6a\) and weight \(W\) is in contact with a rough vertical wall. One end of a light inextensible string of length \(3a\) is attached to the midpoint \(C\) of the rod. The other end of the string is attached to a point \(D\) on the wall, vertically above \(A\). The rod is in equilibrium when the angle between the rod and the wall is \(\theta\), where \(\tan \theta = \frac{3}{4}\). A particle of weight \(W\) is attached to the point \(E\) on the rod, where the distance \(AE\) is equal to \(ka\) (\(3 < k < 6\)) (see diagram). The rod and the string are in a vertical plane perpendicular to the wall. The coefficient of friction between the rod and the wall is \(\frac{1}{3}\). The rod is about to slip down the wall.
  1. Find the value of \(k\). [5]
  2. Find, in terms of \(W\), the magnitude of the frictional force between the rod and the wall. [2]
CAIE Further Paper 3 2024 November Q5
8 marks Challenging +1.2
A particle \(P\) is projected from a point \(O\) on horizontal ground with speed \(u\) at an angle \(\theta\) above the horizontal, where \(\tan \theta = \frac{1}{3}\). The particle \(P\) moves freely under gravity and passes through the point with coordinates \((3a, \frac{4}{5}a)\) relative to horizontal and vertical axes through \(O\) in the plane of the motion.
  1. Use the equation of the trajectory to show that \(u^2 = 25ag\). [2]
  2. Express \(V^2\) in the form \(kag\), where \(k\) is a rational number. [6]
At the instant when \(P\) is moving horizontally, a particle \(Q\) is projected from \(O\) with speed \(V\) at an angle \(\alpha\) above the horizontal. The particles \(P\) and \(Q\) reach the ground at the same point and at the same time.
CAIE Further Paper 3 2024 November Q6
10 marks Challenging +1.8
\includegraphics{figure_6} 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 with the string taut and the string makes an angle \(\theta\) with the downward vertical through \(O\). The particle is projected at right angles to the string with speed \(\frac{1}{3}\sqrt{10ag}\) and begins to move downwards along a circular path. When the string is vertical, it strikes a small smooth peg at the point \(A\) which is vertically below \(O\). The circular path and the point \(A\) are in the same vertical plane. After the string strikes the peg, the particle \(P\) begins to move in a vertical circle with centre \(A\). When the string makes an angle \(\theta\) with the upward vertical through \(A\) the string becomes slack (see diagram). The distance of \(A\) below \(O\) is \(\frac{5}{6}a\).
  1. Find the value of \(\cos \theta\). [6]
  2. Find the ratio of the tensions in the string immediately before and immediately after it strikes the peg. [4]
CAIE Further Paper 3 2024 November Q7
11 marks Challenging +1.2
A particle \(P\) of mass \(m\) kg is held at rest at a point \(O\) and released so that it moves vertically under gravity against a resistive force of magnitude \(0.1mv^2\) N, where \(v\) m s\(^{-1}\) is the velocity of \(P\) at time \(t\) s.
  1. Find an expression for \(v\) in terms of \(t\). [6]
  2. Find an expression for \(v^2\) in terms of \(x\). [5]
The displacement of \(P\) from \(O\) at time \(t\) s is \(x\) m.
CAIE Further Paper 3 2024 November Q1
5 marks Challenging +1.2
A particle \(P\) is projected with speed \(u \text{ m s}^{-1}\) at an angle \(\tan^{-1} 2\) above the horizontal from a point \(O\) on a horizontal plane and moves freely under gravity. When \(P\) has travelled a distance \(56 \text{ m}\) horizontally from \(O\), it is at a vertical height \(H \text{ m}\) above the plane. When \(P\) has travelled a distance \(84 \text{ m}\) horizontally from \(O\), it is at a vertical height \(\frac{1}{2}H \text{ m}\) above the plane. Find, in either order, the value of \(u\) and the value of \(H\). [5]
CAIE Further Paper 3 2024 November Q2
5 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 \(A\) with the string taut. It is given that \(OA\) makes an angle \(\theta\) with the downward vertical through \(O\), where \(\tan \theta = \frac{3}{4}\). The particle \(P\) is projected perpendicular to \(OA\) in an upwards direction with speed \(\sqrt{5ag}\), and it starts to move along a circular path in a vertical plane. When \(P\) is at the point \(B\), where angle \(AOB\) is a right angle, the tension in the string is \(T\). Find \(T\) in terms of \(m\) and \(g\). [5]
CAIE Further Paper 3 2024 November Q3
6 marks Challenging +1.2
A particle \(P\) of mass \(m \text{ kg}\) is attached to one end of a light elastic string of natural length \(2 \text{ m}\) and modulus of elasticity \(2mg \text{ N}\). The other end of the string is attached to a fixed point \(O\). The particle \(P\) hangs in equilibrium vertically below \(O\). The particle \(P\) is pulled down vertically a distance \(d \text{ m}\) below its equilibrium position and released from rest.
  1. Given that the particle just reaches \(O\) in the subsequent motion, find the value of \(d\). [6]
CAIE Further Paper 3 2024 November Q3
2 marks Moderate -0.5
  1. Hence find the speed of \(P\) when it is \(2 \text{ m}\) below \(O\). [2]
CAIE Further Paper 3 2024 November Q4
3 marks Standard +0.8
When the object is suspended from \(A\), the angle between \(AB\) and the vertical is \(\theta\), where \(\tan \theta = \frac{1}{2}\).
  1. Given that \(h = \frac{8}{3}a\), find the possible values of \(k\). [3]
CAIE Further Paper 3 2024 November Q5
4 marks Challenging +1.2
A particle \(P\) of mass \(2 \text{ kg}\) moving on a horizontal straight line has displacement \(x \text{ m}\) from a fixed point \(O\) on the line and velocity \(v \text{ m s}^{-1}\) at time \(t \text{ s}\). The only horizontal force acting on \(P\) is a variable force \(F \text{ N}\) which can be expressed as a function of \(t\). It is given that $$\frac{v}{x} = \frac{3-t}{1+t}$$ and when \(t = 0\), \(x = 5\).
  1. Find an expression for \(x\) in terms of \(t\). [4]
CAIE Further Paper 3 2024 November Q6
3 marks Standard +0.8
\includegraphics{figure_6} A particle \(P\) of mass \(0.05 \text{ kg}\) is attached to one end of a light inextensible string of length \(1 \text{ m}\). The other end of the string is attached to a fixed point \(O\). A particle \(Q\) of mass \(0.04 \text{ kg}\) is attached to one end of a second light inextensible string. The other end of this string is attached to \(P\). The particle \(P\) moves in a horizontal circle of radius \(0.8 \text{ m}\) with angular speed \(\omega \text{ rad s}^{-1}\). The particle \(Q\) moves in a horizontal circle of radius \(1.4 \text{ m}\) also with angular speed \(\omega \text{ rad s}^{-1}\). The centres of the circles are vertically below \(O\), and \(O\), \(P\) and \(Q\) are always in the same vertical plane. The strings \(OP\) and \(PQ\) remain at constant angles \(\alpha\) and \(\beta\) respectively to the vertical (see diagram).
  1. Find the tension in the string \(OP\). [3]
CAIE Further Paper 3 2024 November Q7
4 marks Challenging +1.8
A particle \(P\) is projected with speed \(u\) at an angle \(\tan^{-1}\left(\frac{4}{3}\right)\) above the horizontal from a point \(O\) on a horizontal plane and moves freely under gravity. When \(P\) is moving horizontally, it strikes a smooth inclined plane at the point \(A\). This plane is inclined to the horizontal at an angle \(\alpha\), and the line of greatest slope through \(A\) lies in the vertical plane through \(O\) and \(A\). As a result of the impact, \(P\) moves vertically upwards. The coefficient of restitution between \(P\) and the inclined plane is \(e\).
  1. Show that \(e \tan^2 \alpha = 1\). [4]
CAIE Further Paper 3 2020 Specimen Q1
4 marks Standard +0.8
A child's toy consists of an iron disc of radius \(r\) and a vertical bead with \(3r\) at rail that is rigidly fixed to the disc so that the toy rocks as it rolls. The circumference of the disc is such that the disc and bead have the same material. Show that the centre of mass of the toy is at a distance \(\frac{27r}{10}\) from the centre of the disc. [4]