Past Exam Questions
โ Back9709 P32 - Jun 2019 - Q4
Find the exact coordinates of the point on the curve \(y = \frac{x}{1 + \ln x}\) at which the gradient of the tangent is equal to \(\frac{1}{4}\).
9709 P33 - Jun 2017 - Q5
A curve has equation \(y = \frac{2}{3} \ln(1 + 3 \cos^2 x)\) for \(0 \leq x \leq \frac{1}{2} \pi\).
(i) Express \(\frac{dy}{dx}\) in terms of \(\tan x\).
(ii) Hence find the \(x\)-coordinate of the point on the curve where the gradient is \(-1\). Give your answer correct to 3 significant figures.
9709 P32 - Mar 2017 - Q5
The curve with equation \(y = e^{-ax} \tan x\), where \(a\) is a positive constant, has only one point in the interval \(0 < x < \frac{1}{2}\pi\) at which the tangent is parallel to the \(x\)-axis. Find the value of \(a\) and state the exact value of the \(x\)-coordinate of this point.
9709 P31 - Nov 2015 - Q5
The equation of a curve is \(y = e^{-2x} \tan x\), for \(0 \leq x < \frac{1}{2} \pi\).
(i) Obtain an expression for \(\frac{dy}{dx}\) and show that it can be written in the form \(e^{-2x}(a + b \tan x)^2\), where \(a\) and \(b\) are constants.
(ii) Explain why the gradient of the curve is never negative.
(iii) Find the value of \(x\) for which the gradient is least.
9709 P32 - Jun 2014 - Q5
(i) The polynomial \(f(x)\) is of the form \((x - 2)^2 g(x)\), where \(g(x)\) is another polynomial. Show that \((x - 2)\) is a factor of \(f'(x)\).
(ii) The polynomial \(x^5 + ax^4 + 3x^3 + bx^2 + a\), where \(a\) and \(b\) are constants, has a factor \((x - 2)^2\). Using the factor theorem and the result of part (i), or otherwise, find the values of \(a\) and \(b\).
9709 P31 - Jun 2014 - Q10
The diagram shows the curve \(y = 10e^{-\frac{1}{2}x} \sin 4x\) for \(x \geq 0\). The stationary points are labelled \(T_1, T_2, T_3, \ldots\) as shown.
(i) Find the \(x\)-coordinates of \(T_1\) and \(T_2\), giving each \(x\)-coordinate correct to 3 decimal places.
(ii) It is given that the \(x\)-coordinate of \(T_n\) is greater than 25. Find the least possible value of \(n\).
9709 P41 - Nov 2023 - Q5
The diagram shows a particle A, of mass 1.2 kg, which lies on a plane inclined at an angle of 40ยฐ to the horizontal and a particle B, of mass 1.6 kg, which lies on a plane inclined at an angle of 50ยฐ to the horizontal. The particles are connected by a light inextensible string which passes over a small smooth pulley P fixed at the top of the planes. The parts AP and BP of the string are taut and parallel to lines of greatest slope of the respective planes. The two planes are rough, with the same coefficient of friction, ฮผ, between the particles and the planes.
Find the value of ฮผ for which the system is in limiting equilibrium.
9709 P43 - Jun 2010 - Q4
The diagram shows a vertical cross-section of a triangular prism which is fixed so that two of its faces are inclined at 60ยฐ to the horizontal. One of these faces is smooth and one is rough. Particles A and B, of masses 0.36 kg and 0.24 kg respectively, are attached to the ends of a light inextensible string which passes over a small smooth pulley fixed at the highest point of the cross-section. B is held at rest at a point of the cross-section on the rough face and A hangs freely in contact with the smooth face (see diagram). B is released and starts to move up the face with acceleration 0.25 m s-2.
- By considering the motion of A, show that the tension in the string is 3.03 N, correct to 3 significant figures.
- Find the coefficient of friction between B and the rough face, correct to 2 significant figures.
9709 P41 - Jun 2023 - Q6 - 8 marks
Two particles P and Q, of masses 0.2 kg and 0.1 kg respectively, are attached to the ends of a light inextensible string. The string passes over a fixed smooth pulley at B which is attached to two inclined planes. Particle P lies on a smooth plane AB which is inclined at 60ยฐ to the horizontal. Particle Q lies on a plane BC which is inclined at an angle of ฮธยฐ to the horizontal. The string is taut and the particles can move on lines of greatest slope of the two planes (see diagram).
(a) It is given that ฮธ = 60, the plane BC is rough and the coefficient of friction between Q and the plane BC is 0.7. The particles are released from rest. Determine whether the particles move.
(b) It is given instead that the plane BC is smooth. The particles are released from rest and in the subsequent motion the tension in the string is \\(\sqrt{3} - 1 \\\) N. Find the magnitude of the acceleration of P as it moves on the plane, and find the value of ฮธ.
9709 P43 - Jun 2022 - Q6
Two particles P and Q, of masses 0.3 kg and 0.2 kg respectively, are attached to the ends of a light inextensible string. The string passes over a fixed smooth pulley at B which is attached to two inclined planes. P lies on a smooth plane AB which is inclined at 60ยฐ to the horizontal. Q lies on a plane BC which is inclined at 30ยฐ to the horizontal. The string is taut and the particles can move on lines of greatest slope of the two planes (see diagram).
(a) It is given that the plane BC is smooth and that the particles are released from rest. Find the tension in the string and the magnitude of the acceleration of the particles. [5]
(b) It is given instead that the plane BC is rough. A force of magnitude 3 N is applied to Q directly up the plane along a line of greatest slope of the plane. Find the least value of the coefficient of friction between Q and the plane BC for which the particles remain at rest. [5]
9709 P43 - Nov 2020 - Q7
As shown in the diagram, particles A and B of masses 2 kg and 3 kg respectively are attached to the ends of a light inextensible string. The string passes over a small fixed smooth pulley which is attached to the top of two inclined planes. Particle A is on plane P, which is inclined at an angle of 10ยฐ to the horizontal. Particle B is on plane Q, which is inclined at an angle of 20ยฐ to the horizontal. The string is taut, and the two parts of the string are parallel to lines of greatest slope of their respective planes.
(a) It is given that plane P is smooth, plane Q is rough, and the particles are in limiting equilibrium. Find the coefficient of friction between particle B and plane Q.
(b) It is given instead that both planes are smooth and that the particles are released from rest at the same horizontal level. Find the time taken until the difference in the vertical height of the particles is 1 m. [You should assume that this occurs before A reaches the pulley or B reaches the bottom of plane Q.]
9709 P41 - Jun 2018 - Q7
The diagram shows a triangular block with sloping faces inclined to the horizontal at 45ยฐ and 30ยฐ. Particle A of mass 0.8 kg lies on the face inclined at 45ยฐ and particle B of mass 1.2 kg lies on the face inclined at 30ยฐ. The particles are connected by a light inextensible string which passes over a small smooth pulley P fixed at the top of the faces. The parts AP and BP of the string are parallel to lines of greatest slope of the respective faces. The particles are released from rest with both parts of the string taut. In the subsequent motion neither particle reaches the pulley and neither particle reaches the bottom of a face.
- Given that both faces are smooth, find the speed of A after each particle has travelled a distance of 0.4 m.
- It is given instead that both faces are rough. The coefficient of friction between each particle and a face of the block is \(\mu\). Find the value of \(\mu\) for which the system is in limiting equilibrium.
9709 P41 - Nov 2017 - Q7
Two particles A and B of masses 0.9 kg and 0.4 kg respectively are attached to the ends of a light inextensible string. The string passes over a fixed smooth pulley which is attached to the top of two inclined planes. The particles are initially at rest with A on a smooth plane inclined at angle ฮธยฐ to the horizontal and B on a plane inclined at angle 25ยฐ to the horizontal. The string is taut and the particles can move on lines of greatest slope of the two planes. A force of magnitude 2.5 N is applied to B acting down the plane (see diagram).
- For the case where ฮธ = 15 and the plane on which B rests is smooth, find the acceleration of B.
- For a different value of ฮธ, the plane on which B rests is rough with coefficient of friction between the plane and B of 0.8. The system is in limiting equilibrium with B on the point of moving in the direction of the 2.5 N force. Find the value of ฮธ.
9709 P41 - Jun 2017 - Q7
As shown in the diagram, a particle A of mass 0.8 kg lies on a plane inclined at an angle of 30ยฐ to the horizontal and a particle B of mass 1.2 kg lies on a plane inclined at an angle of 60ยฐ to the horizontal. The particles are connected by a light inextensible string which passes over a small smooth pulley P fixed at the top of the planes. The parts AP and BP of the string are parallel to lines of greatest slope of the respective planes. The particles are released from rest with both parts of the string taut.
- Given that both planes are smooth, find the acceleration of A and the tension in the string. [6]
- It is given instead that both planes are rough, with the same coefficient of friction, ฮผ, for both particles. Find the value of ฮผ for which the system is in limiting equilibrium. [6]
9709 P43 - Nov 2014 - Q2
The tops of each of two smooth inclined planes A and B meet at a right angle. Plane A is inclined at angle \(\alpha\) to the horizontal and plane B is inclined at angle \(\beta\) to the horizontal, where \(\sin \alpha = \frac{63}{65}\) and \(\sin \beta = \frac{16}{65}\). A small smooth pulley is fixed at the top of the planes and a light inextensible string passes over the pulley. Two particles P and Q, each of mass 0.65 kg, are attached to the string, one at each end. Particle Q is held at rest at a point of the same line of greatest slope of the plane B as the pulley. Particle P rests freely below the pulley in contact with plane A (see diagram). Particle Q is released and the particles start to move with the string taut. Find the tension in the string.
9709 P41 - Jun 2012 - Q6
Particles P and Q, of masses 0.6 kg and 0.4 kg respectively, are attached to the ends of a light inextensible string. The string passes over a small smooth pulley which is fixed at the top of a vertical cross-section of a triangular prism. The base of the prism is fixed on horizontal ground and each of the sloping sides is smooth. Each sloping side makes an angle ฮธ with the ground, where \\sin ฮธ = 0.8\\. Initially the particles are held at rest on the sloping sides, with the string taut (see diagram). The particles are released and move along lines of greatest slope.
- Find the tension in the string and the acceleration of the particles while both are moving. [5]
The speed of P when it reaches the ground is 2 m s-1. On reaching the ground P comes to rest and remains at rest. Q continues to move up the slope but does not reach the pulley.
- Find the time taken from the instant that the particles are released until Q reaches its greatest height above the ground. [4]
9709 P43 - Nov 2010 - Q3
A small smooth pulley is fixed at the highest point A of a cross-section ABC of a triangular prism. Angle \(\angle ABC = 90^\circ\) and angle \(\angle BCA = 30^\circ\). The prism is fixed with the face containing BC in contact with a horizontal surface. Particles P and Q are attached to opposite ends of a light inextensible string, which passes over the pulley. The particles are in equilibrium with P hanging vertically below the pulley and Q in contact with AC. The resultant force exerted on the pulley by the string is \(3\sqrt{3} \text{ N}\) (see diagram).
(i) Show that the tension in the string is 3 N.
The coefficient of friction between Q and the prism is 0.75.
(ii) Given that Q is in limiting equilibrium and on the point of moving upwards, find its mass.
9709 P4 - Jun 2006 - Q5
Particles P and Q are attached to opposite ends of a light inextensible string. P is at rest on a rough horizontal table. The string passes over a small smooth pulley which is fixed at the edge of the table. Q hangs vertically below the pulley (see diagram). The force exerted on the string by the pulley has magnitude \(4\sqrt{2}\) N. The coefficient of friction between P and the table is 0.8.
- Show that the tension in the string is 4 N and state the mass of Q.
- Given that P is on the point of slipping, find its mass.
- A particle of mass 0.1 kg is now attached to Q and the system starts to move. Find the tension in the string while the particles are in motion.
9709 P43 - Nov 2021 - Q6
The diagram shows a particle of mass 5 kg on a rough horizontal table, and two light inextensible strings attached to it passing over smooth pulleys fixed at the edges of the table. Particles of masses 4 kg and 6 kg hang freely at the ends of the strings. The particle of mass 6 kg is 0.5 m above the ground. The system is in limiting equilibrium.
(a) Show that the coefficient of friction between the 5 kg particle and the table is 0.4.
The 6 kg particle is now replaced by a particle of mass 8 kg and the system is released from rest.
(b) Find the acceleration of the 4 kg particle and the tensions in the strings.
(c) In the subsequent motion the 8 kg particle hits the ground and does not rebound. Find the time that elapses after the 8 kg particle hits the ground before the other two particles come to instantaneous rest. (You may assume this occurs before either particle reaches a pulley.)
9709 P43 - Nov 2016 - Q3
Particles P and Q, of masses 7 kg and 3 kg respectively, are attached to the two ends of a light inextensible string. The string passes over two small smooth pulleys attached to the two ends of a horizontal table. The two particles hang vertically below the two pulleys. The two particles are both initially at rest, 0.5 m below the level of the table, and 0.4 m above the horizontal floor (see diagram).
(i) Find the acceleration of the particles and the speed of P immediately before it reaches the floor.
(ii) Determine whether Q comes to instantaneous rest before it reaches the pulley directly above it.
