Questions (30179 questions)

Browse by module
All questions AEA AS Paper 1 AS Paper 2 AS Pure C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Extra Pure Further Mechanics Further Mechanics A AS Further Mechanics AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics Further Paper 4 Further Pure Core Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Pure with Technology Further Statistics Further Statistics A AS Further Statistics AS Further Statistics B AS Further Statistics Major Further Statistics Minor Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 H240/01 H240/02 H240/03 M1 M2 M3 M4 M5 Mechanics 1 P1 P2 P3 P4 PMT Mocks PURE Paper 1 Paper 2 Paper 3 Pure 1 S1 S2 S3 S4 Stats 1 Unit 1 Unit 2 Unit 3 Unit 4
Browse by board
AQA AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Paper 1 Further Paper 2 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics M1 M2 M3 Paper 1 Paper 2 Paper 3 S1 S2 S3 CAIE FP1 FP2 Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 4 M1 M2 P1 P2 P3 S1 S2 Edexcel AEA AS Paper 1 AS Paper 2 C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS M1 M2 M3 M4 M5 P1 P2 P3 P4 PMT Mocks Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 OCR AS Pure C1 C2 C3 C4 D1 D2 FD1 AS FM1 AS FP1 FP1 AS FP2 FP3 FS1 AS Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Mechanics Further Mechanics AS Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Statistics Further Statistics AS H240/01 H240/02 H240/03 M1 M2 M3 M4 Mechanics 1 PURE Pure 1 S1 S2 S3 S4 Stats 1 OCR MEI AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further Extra Pure Further Mechanics A AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Pure Core Further Pure Core AS Further Pure with Technology Further Statistics A AS Further Statistics B AS Further Statistics Major Further Statistics Minor M1 M2 M3 M4 Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 WJEC Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 Unit 1 Unit 2 Unit 3 Unit 4
Sort by: Default | Easiest first | Hardest first
CAIE M1 2016 June Q4
7 marks Standard +0.3
4 A particle of mass 15 kg is stationary on a rough plane inclined at an angle of \(20 ^ { \circ }\) to the horizontal. The coefficient of friction between the particle and the plane is 0.2 . A force of magnitude \(X \mathrm {~N}\) acting parallel to a line of greatest slope of the plane is used to keep the particle in equilibrium. Show that the least possible value of \(X\) is 23.1 , correct to 3 significant figures, and find the greatest possible value of \(X\).
CAIE M1 2016 June Q5
8 marks Moderate -0.3
5 The motion of a car of mass 1400 kg is resisted by a constant force of magnitude 650 N .
  1. Find the constant speed of the car on a horizontal road, assuming that the engine works at a rate of 20 kW .
  2. The car is travelling at a constant speed of \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) up a hill inclined at an angle of \(\theta\) to the horizontal, where \(\sin \theta = \frac { 1 } { 7 }\). Find the power of the car's engine.
  3. The car descends the same hill with the engine working at \(80 \%\) of the power found in part (ii). Find the acceleration of the car at an instant when the speed is \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
CAIE M1 2016 June Q6
10 marks Standard +0.8
6 Two particles of masses 1.3 kg and 0.7 kg are connected by a light inextensible string that passes over a fixed smooth pulley. The particles are held at the same vertical height with the string taut. The distance of each particle above a horizontal plane is 2 m , and the distance of each particle below the pulley is 4 m . The particles are released from rest.
  1. Find
    (a) the tension in the string before the particle of mass 1.3 kg reaches the plane,
    (b) the time taken for the particle of mass 1.3 kg to reach the plane.
  2. Find the greatest height of the particle of mass 0.7 kg above the plane.
CAIE M1 2016 June Q7
10 marks Standard +0.3
7 A particle \(P\) moves in a straight line. At time \(t \mathrm {~s}\), the displacement of \(P\) from \(O\) is \(s \mathrm {~m}\) and the acceleration of \(P\) is \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\), where \(a = 6 t - 2\). When \(t = 1 , s = 7\) and when \(t = 3 , s = 29\).
  1. Find the set of values of \(t\) for which the particle is decelerating.
  2. Find \(s\) in terms of \(t\).
  3. Find the time when the velocity of the particle is \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
CAIE M1 2017 June Q1
3 marks Moderate -0.8
1 A particle of mass 0.6 kg is dropped from a height of 8 m above the ground. The speed of the particle at the instant before hitting the ground is \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find the work done against air resistance.
CAIE M1 2017 June Q2
6 marks Moderate -0.3
2 A particle of mass 0.8 kg is projected with a speed of \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) up a line of greatest slope of a rough plane inclined at an angle of \(10 ^ { \circ }\) to the horizontal. The coefficient of friction between the particle and the plane is 0.4 .
  1. Find the acceleration of the particle.
  2. Find the distance the particle moves up the plane before coming to rest.
CAIE M1 2017 June Q3
6 marks Moderate -0.3
3 \includegraphics[max width=\textwidth, alt={}, center]{4941e074-2f93-4a0c-80ba-0ca96a48389e-04_442_584_255_778} Two light inextensible strings are attached to a particle of weight 25 N . The strings pass over two smooth fixed pulleys and have particles of weights \(A \mathrm {~N}\) and \(B \mathrm {~N}\) hanging vertically at their ends. The sloping parts of the strings make angles of \(30 ^ { \circ }\) and \(40 ^ { \circ }\) respectively with the vertical (see diagram). The system is in equilibrium. Find the values of \(A\) and \(B\).
CAIE M1 2017 June Q4
6 marks Moderate -0.3
4 A car of mass 800 kg is moving up a hill inclined at \(\theta ^ { \circ }\) to the horizontal, where \(\sin \theta = 0.15\). The initial speed of the car is \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Twelve seconds later the car has travelled 120 m up the hill and has speed \(14 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  1. Find the change in the kinetic energy and the change in gravitational potential energy of the car.
  2. The engine of the car is working at a constant rate of 32 kW . Find the total work done against the resistive forces during the twelve seconds.
CAIE M1 2017 June Q5
7 marks Moderate -0.3
5 A particle \(P\) moves in a straight line \(A B C D\) with constant deceleration. The velocities of \(P\) at \(A , B\) and \(C\) are \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 } , 12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) respectively.
  1. Find the ratio of distances \(A B : B C\).
  2. The particle comes to rest at \(D\). Given that the distance \(A D\) is 80 m , find the distance \(B C\).
CAIE M1 2017 June Q6
10 marks Standard +0.3
6 A particle \(P\) moves in a straight line passing through a point \(O\). At time \(t \mathrm {~s}\), the velocity of \(P , v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), is given by \(v = q t + r t ^ { 2 }\), where \(q\) and \(r\) are constants. The particle has velocity \(4 \mathrm {~ms} ^ { - 1 }\) when \(t = 1\) and when \(t = 2\).
  1. Show that, when \(t = 0.5\), the acceleration of \(P\) is \(4 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
    …………………………………………………………………………………………......................... .
  2. Find the values of \(t\) when \(P\) is at instantaneous rest.
  3. The particle is at \(O\) when \(t = 3\). Find the distance of \(P\) from \(O\) when \(t = 0\).
CAIE M1 2017 June Q7
12 marks Standard +0.3
7 \includegraphics[max width=\textwidth, alt={}, center]{4941e074-2f93-4a0c-80ba-0ca96a48389e-10_374_762_259_688} As shown in the diagram, a particle \(A\) of mass 0.8 kg lies on a plane inclined at an angle of \(30 ^ { \circ }\) to the horizontal and a particle \(B\) of mass 1.2 kg lies on a plane inclined at an angle of \(60 ^ { \circ }\) 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 \(A P\) and \(B P\) 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.
  1. Given that both planes are smooth, find the acceleration of \(A\) and the tension in the string.
  2. It is given instead that both planes are rough, with the same coefficient of friction, \(\mu\), for both particles. Find the value of \(\mu\) for which the system is in limiting equilibrium.
CAIE M1 2017 June Q1
3 marks Moderate -0.5
1 One end of a light inextensible string is attached to a block. The string makes an angle of \(\theta ^ { \circ }\) with the horizontal. The tension in the string is 20 N . The string pulls the block along a horizontal surface at a constant speed of \(1.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) for 12 s . The work done by the tension in the string is 50 J . Find \(\theta\). [3]
CAIE M1 2017 June Q2
7 marks Standard +0.3
2 \includegraphics[max width=\textwidth, alt={}, center]{39f94377-0dd0-4d1b-98de-815bd6e2c409-02_438_565_1347_790} The diagram shows a wire \(A B C D\) consisting of a straight part \(A B\) of length 5 m and a part \(B C D\) in the shape of a semicircle of radius 6 m and centre \(O\). The diameter \(B D\) of the semicircle is horizontal and \(A B\) is vertical. A small ring is threaded onto the wire and slides along the wire. The ring starts from rest at \(A\). The part \(A B\) of the wire is rough, and the ring accelerates at a constant rate of \(2.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) between \(A\) and \(B\).
  1. Show that the speed of the ring as it reaches \(B\) is \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
    The part \(B C D\) of the wire is smooth. The mass of the ring is 0.2 kg .
  2. (a) Find the speed of the ring at \(C\), where angle \(B O C = 30 ^ { \circ }\).
    (b) Find the greatest speed of the ring.
CAIE M1 2017 June Q3
9 marks Standard +0.3
3 A particle \(A\) moves in a straight line with constant speed \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Two seconds after \(A\) passes a point \(O\) on the line, a particle \(B\) passes through \(O\), moving along the line in the same direction as \(A\). Particle \(B\) has speed \(16 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at \(O\) and has a constant deceleration of \(2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
  1. Find expressions, in terms of \(t\), for the displacement from \(O\) of each particle \(t \mathrm {~s}\) after \(B\) passes through \(O\).
  2. Find the distance between the particles when \(B\) comes to instantaneous rest.
  3. Find the minimum distance between the particles.
CAIE M1 2017 June Q4
7 marks Moderate -0.3
4 A car of mass 1200 kg is moving on a straight road against a constant force of 850 N resisting the motion.
  1. On a part of the road that is horizontal, the car moves with a constant speed of \(42 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
    (a) Calculate, in kW , the power developed by the engine of the car.
    (b) Given that this power is suddenly increased by 6 kW , find the instantaneous acceleration of the car.
  2. On a part of the road that is inclined at \(\theta ^ { \circ }\) to the horizontal, the car moves up the hill at a constant speed of \(24 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), with the engine working at 80 kW . Find \(\theta\). \includegraphics[max width=\textwidth, alt={}, center]{39f94377-0dd0-4d1b-98de-815bd6e2c409-08_406_401_264_872} A particle of mass 0.12 kg is placed on a plane which is inclined at an angle of \(40 ^ { \circ }\) to the horizontal. The particle is kept in equilibrium by a force of magnitude \(P \mathrm {~N}\) acting up the plane at an angle of \(30 ^ { \circ }\) above a line of greatest slope, as shown in the diagram. The coefficient of friction between the particle and the plane is 0.32 . Find the set of possible values of \(P\).
CAIE M1 2017 June Q6
14 marks Standard +0.8
6 \includegraphics[max width=\textwidth, alt={}, center]{39f94377-0dd0-4d1b-98de-815bd6e2c409-10_330_946_260_612} The diagram shows a fixed block with a horizontal top surface and a surface which is inclined at an angle of \(\theta ^ { \circ }\) to the horizontal, where \(\sin \theta = \frac { 3 } { 5 }\). A particle \(A\) of mass 0.3 kg rests on the horizontal surface and is attached to one end of a light inextensible string. The string passes over a small smooth pulley \(P\) fixed at the edge of the block. The other end of the string is attached to a particle \(B\) of mass 1.5 kg which rests on the sloping surface of the block. The system is released from rest with the string taut.
  1. Given that the block is smooth, find the acceleration of particle \(A\) and the tension in the string.
  2. It is given instead that the block is rough. The coefficient of friction between \(A\) and the block is \(\mu\) and the coefficient of friction between \(B\) and the block is also \(\mu\). In the first 3 seconds of the motion, \(A\) does not reach \(P\) and \(B\) does not reach the bottom of the sloping surface. The speed of the particles after 3 s is \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find the acceleration of particle \(A\) and the value of \(\mu\).
CAIE Further Paper 2 2020 June Q1
6 marks Standard +0.3
1 Find the solution of the differential equation $$\frac { d y } { d x } + 5 y = e ^ { - 7 x }$$ for which \(y = 0\) when \(x = 0\). Give your answer in the form \(y = f ( x )\).
CAIE Further Paper 2 2020 June Q2
7 marks Standard +0.3
2 It is given that \(y = 2 ^ { x }\).
  1. By differentiating \(\ln y\) with respect to \(x\), show that \(\frac { \mathrm { dy } } { \mathrm { dx } } = 2 ^ { \mathrm { x } } \ln 2\).
  2. Write down \(\frac { d ^ { 2 } y } { d x ^ { 2 } }\).
  3. Hence find the first three terms in the Maclaurin's series for \(2 ^ { X }\).
CAIE Further Paper 2 2020 June Q3
8 marks Standard +0.8
3
  1. Find the roots of the equation \(z ^ { 3 } = - 1 - \mathrm { i }\), giving your answers in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), where \(r > 0\) and \(0 \leqslant \theta < 2 \pi\).
    Let \(\mathbf { w } = \mathbf { z } _ { 1 } ^ { 3 \mathrm { k } } + \mathbf { z } _ { 2 } ^ { 3 \mathrm { k } } + \mathbf { z } _ { 3 } ^ { 3 \mathrm { k } }\), where \(k\) is a positive integer and \(\mathrm { z } _ { 1 } , \mathrm { z } _ { 2 } , \mathrm { z } _ { 3 }\) are the roots of \(\mathrm { z } ^ { 3 } = - 1 - \mathrm { i }\).
  2. Express \(w\) in the form \(R \mathrm { e } ^ { \mathrm { i } \alpha }\), where \(R > 0\), giving \(R\) and \(\alpha\) in terms of \(k\). \includegraphics[max width=\textwidth, alt={}, center]{20e14db3-0eb0-4954-91cf-027e16f8bf14-06_889_824_267_616} The diagram shows the curve with equation \(\mathrm { y } = \mathrm { x } ^ { 2 }\) for \(0 \leqslant x \leqslant 1\), together with a set of \(n\) rectangles of width \(\frac { 1 } { n }\).
CAIE Further Paper 2 2020 June Q5
11 marks Challenging +1.2
5 The curves \(C _ { 1 } : y = \cosh x\) and \(C _ { 2 } : y = \sinh 2 x\) intersect at the point where \(x = a\).
  1. Find the exact value of \(a\), giving your answer in logarithmic form.
  2. Sketch \(C _ { 1 }\) and \(C _ { 2 }\) on the same diagram.
  3. Find the exact value of the length of the arc of \(C _ { 1 }\) from \(x = 0\) to \(\mathrm { x } = \mathrm { a }\).
CAIE Further Paper 2 2020 June Q6
10 marks Challenging +1.8
6 The integral \(\mathrm { I } _ { \mathrm { n } }\), where \(n\) is an integer, is defined by \(\mathrm { I } _ { \mathrm { n } } = \int _ { 0 } ^ { \frac { 1 } { 2 } } \left( 1 - \mathrm { x } ^ { 2 } \right) ^ { - \frac { 1 } { 2 } \mathrm { n } } \mathrm { dx }\).
  1. Find the exact value of \(I _ { 1 }\).
  2. By considering \(\frac { \mathrm { d } } { \mathrm { dx } } \left( \mathrm { x } \left( 1 - \mathrm { x } ^ { 2 } \right) ^ { - \frac { 1 } { 2 } \mathrm { n } } \right)\), or otherwise, show that $$\mathrm { nl } _ { \mathrm { n } + 2 } = 2 ^ { \mathrm { n } - 1 } 3 ^ { - \frac { 1 } { 2 } \mathrm { n } } + ( \mathrm { n } - 1 ) \mathrm { I } _ { \mathrm { n } } .$$
  3. Find the exact value of \(I _ { 5 }\) giving the answer in the form \(k \sqrt { 3 }\), where \(k\) is a rational number to be determined. \includegraphics[max width=\textwidth, alt={}, center]{20e14db3-0eb0-4954-91cf-027e16f8bf14-11_78_1576_336_321}
CAIE Further Paper 2 2020 June Q7
11 marks Challenging +1.8
7 It is given that \(x = t ^ { 3 } y\) and $$t ^ { 3 } \frac { d ^ { 2 } y } { d t ^ { 2 } } + \left( 4 t ^ { 3 } + 6 t ^ { 2 } \right) \frac { d y } { d t } + \left( 13 t ^ { 3 } + 12 t ^ { 2 } + 6 t \right) y = 61 e ^ { \frac { 1 } { 2 } t }$$
  1. Show that $$\frac { d ^ { 2 } x } { d t ^ { 2 } } + 4 \frac { d x } { d t } + 13 x = 61 e ^ { \frac { 1 } { 2 } t }$$
  2. Find the general solution for \(y\) in terms of \(t\).
CAIE Further Paper 2 2020 June Q8
14 marks Standard +0.8
8
  1. Find the values of \(a\) for which the system of equations $$\begin{aligned} 3 x + y + z & = 0 \\ a x + 6 y - z & = 0 \\ a y - 2 z & = 0 \end{aligned}$$ does not have a unique solution.
    The matrix \(\mathbf { A }\) is given by $$\mathbf { A } = \left( \begin{array} { r r r } 3 & 1 & 1 \\ 0 & 6 & - 1 \\ 0 & 0 & - 2 \end{array} \right) .$$
  2. Use the characteristic equation of \(\mathbf { A }\) to find the inverse of \(\mathbf { A } ^ { 2 }\).
  3. Find a matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that \(\mathbf { A } ^ { 5 } = \mathbf { P D P } ^ { - 1 }\).
    If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.
CAIE Further Paper 2 2020 June Q3
8 marks
3
  1. Find the roots of the equation \(z ^ { 3 } = - 1 - \mathrm { i }\), giving your answers in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), where \(r > 0\) and \(0 \leqslant \theta < 2 \pi\).
    Let \(\mathbf { w } = \mathbf { z } _ { 1 } ^ { 3 \mathrm { k } } + \mathbf { z } _ { 2 } ^ { 3 \mathrm { k } } + \mathbf { z } _ { 3 } ^ { 3 \mathrm { k } }\), where \(k\) is a positive integer and \(\mathrm { z } _ { 1 } , \mathrm { z } _ { 2 } , \mathrm { z } _ { 3 }\) are the roots of \(\mathrm { z } ^ { 3 } = - 1 - \mathrm { i }\).
  2. Express \(w\) in the form \(R \mathrm { e } ^ { \mathrm { i } \alpha }\), where \(R > 0\), giving \(R\) and \(\alpha\) in terms of \(k\). \includegraphics[max width=\textwidth, alt={}, center]{1de67949-6262-4ade-b986-02b6563ae404-06_889_824_267_616} The diagram shows the curve with equation \(\mathrm { y } = \mathrm { x } ^ { 2 }\) for \(0 \leqslant x \leqslant 1\), together with a set of \(n\) rectangles of width \(\frac { 1 } { n }\).
CAIE Further Paper 2 2020 June Q6
10 marks Challenging +1.8
6 The integral \(\mathrm { I } _ { \mathrm { n } }\), where \(n\) is an integer, is defined by \(\mathrm { I } _ { \mathrm { n } } = \int _ { 0 } ^ { \frac { 1 } { 2 } } \left( 1 - \mathrm { x } ^ { 2 } \right) ^ { - \frac { 1 } { 2 } \mathrm { n } } \mathrm { dx }\).
  1. Find the exact value of \(I _ { 1 }\).
  2. By considering \(\frac { \mathrm { d } } { \mathrm { dx } } \left( \mathrm { x } \left( 1 - \mathrm { x } ^ { 2 } \right) ^ { - \frac { 1 } { 2 } \mathrm { n } } \right)\), or otherwise, show that $$\mathrm { nl } _ { \mathrm { n } + 2 } = 2 ^ { \mathrm { n } - 1 } 3 ^ { - \frac { 1 } { 2 } \mathrm { n } } + ( \mathrm { n } - 1 ) \mathrm { I } _ { \mathrm { n } } .$$
  3. Find the exact value of \(I _ { 5 }\) giving the answer in the form \(k \sqrt { 3 }\), where \(k\) is a rational number to be determined. \includegraphics[max width=\textwidth, alt={}, center]{1de67949-6262-4ade-b986-02b6563ae404-11_78_1576_336_321}