Questions Further Paper 3 (180 questions)

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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]
CAIE Further Paper 3 2023 June Q6
10 marks Challenging +1.2
A particle of mass \(m\) kg falls vertically under gravity, from rest. At time \(t\) s, \(P\) has fallen \(x\) m and has velocity \(v\) m s\(^{-1}\). The only forces acting on \(P\) are its weight and a resistance of magnitude \(kmgv\) N, where \(k\) is a constant.
  1. Find an expression for \(v\) in terms of \(t\), \(g\) and \(k\). [5]
  2. Given that \(k = 0.05\), find, in metres, how far \(P\) has fallen when its speed is \(12\) m s\(^{-1}\). [5]
CAIE Further Paper 3 2023 June Q7
9 marks Challenging +1.2
The points \(O\) and \(P\) are on a horizontal plane, a distance \(8\) m apart. A ball is thrown from \(O\) with speed \(u\) m s\(^{-1}\) at an angle \(\theta\) above the horizontal, where \(\tan \theta = \frac{3}{4}\). At the same instant, a model aircraft is launched with speed \(5\) m s\(^{-1}\) parallel to the horizontal plane from a point \(4\) m vertically above \(P\). The model aircraft moves in the same vertical plane as the ball and in the same horizontal direction as the ball. The model aircraft moves horizontally with a constant speed of \(5\) m s\(^{-1}\). After \(T\) s, the ball and the model aircraft collide.
  1. Find the value of \(T\). [6]
  2. Find the direction in which the ball is moving immediately before the collision. [3]
CAIE Further Paper 3 2024 June Q1
6 marks Challenging +1.8
Two smooth uniform spheres \(A\) and \(B\) of equal radii have masses \(m\) and \(5m\) respectively. Sphere \(A\) is moving on a smooth horizontal surface with speed \(u\) when it collides with sphere \(B\) which is at rest on the surface. Immediately before the collision, \(A\)'s direction of motion makes an angle of \(\theta\) with the line of centres. After the collision, the kinetic energies of \(A\) and \(B\) are equal. The coefficient of restitution between the spheres is \(\frac{1}{3}\). \includegraphics{figure_1} Find the value of \(\tan\theta\). [6]
CAIE Further Paper 3 2024 June Q2
7 marks Challenging +1.2
The points \(A\) and \(B\) are at the same horizontal level a distance \(4a\) apart. The ends of a light elastic string, of natural length \(4a\) and modulus of elasticity \(\lambda\), are attached to \(A\) and \(B\). A particle \(P\) of mass \(m\) is attached to the midpoint of the string. The system is in equilibrium with \(P\) at a distance \(\frac{5}{8}a\) below \(M\), the midpoint of \(AB\).
  1. Find \(\lambda\) in terms of \(m\) and \(g\). [3]
The particle \(P\) is pulled down vertically and released from rest at a distance \(\frac{8}{5}a\) below \(M\).
  1. Find, in terms of \(a\) and \(g\), the speed of \(P\) as it passes through \(M\) in the subsequent motion. [4]
CAIE Further Paper 3 2024 June Q3
5 marks Challenging +1.2
At time \(t = 0\) seconds, a particle \(P\) is projected with speed \(u\) m s\(^{-1}\) at an angle \(60°\) above the horizontal from a point \(O\). In the subsequent motion \(P\) moves freely under gravity. The direction of motion of \(P\) when \(t = 5\) is perpendicular to its direction of motion when \(t = 15\). Find the value of \(u\). [5]
CAIE Further Paper 3 2024 June Q4
7 marks Challenging +1.8
A ring of weight \(W\), with radius \(a\) and centre \(O\), is at rest on a rough surface that is inclined to the horizontal at an angle \(\alpha\) where \(\tan\alpha = \frac{1}{3}\). The plane of the ring is perpendicular to the inclined surface and parallel to a line of greatest slope of the surface. The point \(P\) on the circumference of the ring is such that \(OP\) is parallel to the surface. A light inextensible string is attached to \(P\) and to the point \(Q\), which is on the surface, such that \(PQ\) is horizontal (see diagram). The points \(O\), \(P\) and \(Q\) are in the same vertical plane. The system is in limiting equilibrium and the coefficient of friction between the ring and the surface is \(\mu\). \includegraphics{figure_4}
  1. Find, in terms of \(W\), the tension in the string \(PQ\). [4]
  2. Find the value of \(\mu\). [3]
CAIE Further Paper 3 2024 June Q5
7 marks Challenging +1.2
Two particles \(A\) and \(B\) of masses \(m\) and \(km\) respectively are connected by a light inextensible string of length \(a\). The particles are placed on a rough horizontal circular turntable with the string taut and lying along a radius of the turntable. Particle \(A\) is at a distance \(a\) from the centre of the turntable and particle \(B\) is at a distance \(2a\) from the centre of the turntable. The coefficient of friction between each particle and the turntable is \(\frac{1}{3}\). When the turntable is made to rotate with angular speed \(\frac{2}{5}\sqrt{\frac{g}{a}}\), the system is in limiting equilibrium.
  1. Find the tension in the string, in terms of \(m\) and \(g\). [4]
  2. Find the value of \(k\). [3]
CAIE Further Paper 3 2024 June Q6
9 marks Challenging +1.8
A particle \(P\) of mass \(2\) kg moving on a horizontal 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 only horizontal force acting on \(P\) has magnitude \(\frac{1}{10}(2v - 1)^2e^{-t}\) N and acts towards \(O\). When \(t = 0\), \(x = 1\) and \(v = 3\).
  1. Find an expression for \(v\) in terms of \(t\). [5]
  2. Find an expression for \(x\) in terms of \(t\). [4]
CAIE Further Paper 3 2024 June Q7
9 marks Challenging +1.2
A smooth sphere with centre \(O\) and of radius \(a\) is fixed to a horizontal plane. A particle \(P\) of mass \(m\) is projected horizontally from the highest point of the sphere with speed \(u\), so that it begins to move along the surface of the sphere. The particle \(P\) loses contact with the sphere at the point \(Q\) on the sphere, where \(OQ\) makes an angle \(\theta\) with the upward vertical through \(O\).
  1. Show that \(\cos\theta = \frac{u^2 + 2ag}{3ag}\). [4]
It is given that \(\cos\theta = \frac{5}{9}\).
  1. Find, in terms of \(a\) and \(g\), an expression for the vertical component of the velocity of \(P\) just before it hits the horizontal plane to which the sphere is fixed. [3]
  2. Find an expression for the time taken by \(P\) to fall from \(Q\) to the plane. Give your answer in the form \(k\sqrt{\frac{a}{g}}\), stating the value of \(k\) correct to 3 significant figures. [2]
CAIE Further Paper 3 2024 June Q1
6 marks Challenging +1.8
\includegraphics{figure_1} Two smooth uniform spheres \(A\) and \(B\) of equal radii have masses \(m\) and \(5m\) respectively. Sphere \(A\) is moving on a smooth horizontal surface with speed \(u\) when it collides with sphere \(B\) which is at rest on the surface. Immediately before the collision, \(A\)'s direction of motion makes an angle of \(\theta\) with the line of centres. After the collision, the kinetic energies of \(A\) and \(B\) are equal. The coefficient of restitution between the spheres is \(\frac{1}{3}\). Find the value of \(\tan\theta\). [6]
CAIE Further Paper 3 2024 June Q4
7 marks Challenging +1.8
\includegraphics{figure_4} A ring of weight \(W\), with radius \(a\) and centre \(O\), is at rest on a rough surface that is inclined to the horizontal at an angle \(\alpha\) where \(\tan\alpha = \frac{1}{3}\). The plane of the ring is perpendicular to the inclined surface and parallel to a line of greatest slope of the surface. The point \(P\) on the circumference of the ring is such that \(OP\) is parallel to the surface. A light inextensible string is attached to \(P\) and to the point \(Q\), which is on the surface, such that \(PQ\) is horizontal (see diagram). The points \(O\), \(P\) and \(Q\) are in the same vertical plane. The system is in limiting equilibrium and the coefficient of friction between the ring and the surface is \(\mu\).
  1. Find, in terms of \(W\), the tension in the string \(PQ\). [4]
  2. Find the value of \(\mu\). [3]
CAIE Further Paper 3 2024 June Q6
9 marks Challenging +1.8
A particle \(P\) of mass \(2\) kg moving on a horizontal 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 only horizontal force acting on \(P\) has magnitude \(\frac{1}{10}(2v - 1)^2 e^{-t}\) N and acts towards \(O\). When \(t = 0\), \(x = 1\) and \(v = 3\).
  1. Find an expression for \(v\) in terms of \(t\). [5]
  2. Find an expression for \(x\) in terms of \(t\). [4]
CAIE Further Paper 3 2024 June Q1
4 marks Challenging +1.2
\includegraphics{figure_1} Two smooth uniform spheres \(A\) and \(B\) of equal radii have masses \(m\) and \(2m\) respectively. The two spheres are moving on a smooth horizontal surface when they collide with speeds \(u\) and \(\frac{1}{2}u\) respectively. Immediately before the collision, \(A\)'s direction of motion is along the line of centres, and \(B\)'s direction of motion makes an angle \(\theta\) with the line of centres (see diagram). As a result of the collision, the direction of motion of \(A\) is reversed and its speed is reduced to \(\frac{1}{4}u\). The direction of motion of \(B\) again makes an angle \(\theta\) with the line of centres, but on the opposite side of the line of centres. The speed of \(B\) is unchanged. Find the value of the coefficient of restitution between the spheres. [4]
CAIE Further Paper 3 2024 June Q2
6 marks Standard +0.8
A particle \(P\) of mass \(m\) is attached to one end of a light elastic string of natural length \(a\) and modulus of elasticity \(2mg\). A particle \(Q\) of mass \(km\) is attached to the other end of the string. Particle \(P\) lies on a smooth horizontal table. The string passes through a small smooth hole \(H\) in the table and then passes through a small smooth hole \(H\) in the table. Particle \(P\) moves in a horizontal circle on the surface of the table with constant speed \(\sqrt{\frac{1}{3}ga}\). Particle \(Q\) hangs in equilibrium vertically below the hole with \(HQ = \frac{1}{4}a\).
  1. Find, in terms of \(a\), the extension in the string. [4]
  2. Find the value of \(k\). [2]