Questions — CAIE (7659 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 Further Paper 3 2024 June Q5
7 marks Standard +0.8
5 \includegraphics[max width=\textwidth, alt={}, center]{73f73a7a-79d0-40fc-8c6d-1e46dacda788-08_579_987_292_539} A uniform lamina is in the form of a triangle \(O B C\), with \(O C = 18 a , O B = 24 a\) and angle \(C O B = 90 ^ { \circ }\). The point \(A\) on \(O B\) is such that \(O A = x\) (see diagram). The triangle \(O A C\) is removed from the lamina.
  1. Find, in terms of \(a\) and \(x\), the distance of the centre of mass of the remaining object \(A B C\) from \(O C\). [3] \includegraphics[max width=\textwidth, alt={}, center]{73f73a7a-79d0-40fc-8c6d-1e46dacda788-08_2718_40_141_2010}
    The object \(A B C\) is suspended from \(C\) .In its equilibrium position,the side \(A B\) makes an angle \(\theta\) with the vertical,where \(\tan \theta = \frac { 6 } { 5 }\) .
  2. Find \(x\) in terms of \(a\) .
CAIE Further Paper 3 2024 June Q6
10 marks Standard +0.8
6 A particle \(P\) is projected with speed \(u \mathrm {~ms} ^ { - 1 }\) 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 \(\frac { 7 } { 16 } u ^ { 2 } - 100 u \sin \theta + 2500 = 0\). \includegraphics[max width=\textwidth, alt={}, center]{73f73a7a-79d0-40fc-8c6d-1e46dacda788-10_2713_31_145_2014}
  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 \(\sin \theta\).
CAIE Further Paper 3 2024 June Q7
11 marks Standard +0.3
7 A parachutist of mass \(m \mathrm {~kg}\) opens his parachute when he is moving vertically downwards with a speed of \(50 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). At time \(t \mathrm {~s}\) after opening his parachute, he has fallen a distance \(x \mathrm {~m}\) from the point where he opened his parachute, and his speed is \(v \mathrm {~ms} ^ { - 1 }\). The forces acting on him are his weight and a resistive force of magnitude \(m v \mathrm {~N}\).
  1. Find an expression for \(v\) in terms of \(t\). \includegraphics[max width=\textwidth, alt={}, center]{73f73a7a-79d0-40fc-8c6d-1e46dacda788-12_2715_40_144_2007}
  2. Find an expression for \(x\) in terms of \(t\).
  3. Find the distance that the parachutist has fallen, since opening his parachute, when his speed is \(15 \mathrm {~ms} ^ { - 1 }\).
    If you use the following page to complete the answer to any question, the question number must be clearly shown. \includegraphics[max width=\textwidth, alt={}, center]{73f73a7a-79d0-40fc-8c6d-1e46dacda788-14_2715_35_143_2012}
CAIE Further Paper 3 2020 November Q1
3 marks Standard +0.8
1 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 \(3 m g\), 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. \includegraphics[max width=\textwidth, alt={}, center]{0581e302-2fc5-46f0-b597-e5cae1f664a2-04_515_707_267_685} 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 \(\frac { 4 } { 5 } \sqrt { 5 a g }\) perpendicular to the string and just completes a vertical circle (see diagram). Find the value of \(\cos \theta\).
CAIE Further Paper 3 2020 November Q3
6 marks Standard +0.8
3 One end of a light elastic string, of natural length \(a\) and modulus of elasticity \(4 m g\), 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 \(\sqrt { \frac { \mathrm { g } } { \mathrm { a } } }\) with the string inclined at an angle \(\theta\) to the downward vertical through \(O\). The length of the string during this motion is \(( \mathrm { k } + 1 ) \mathrm { a }\).
  1. Find the value of \(k\).
  2. Find the value of \(\cos \theta\). \includegraphics[max width=\textwidth, alt={}, center]{0581e302-2fc5-46f0-b597-e5cae1f664a2-06_581_695_267_667} The diagram shows the cross-section \(A B C D\) of a uniform solid object which is formed by removing a cone with cross-section \(D C E\) from the top of a larger cone with cross-section \(A B E\). The perpendicular distance between \(A B\) and \(D C\) is \(h\), the diameter \(A B\) is \(6 r\) and the diameter \(D C\) is \(2 r\).
  3. Find an expression, in terms of \(h\), for the distance of the centre of mass of the solid object from \(A B\).
    The object is freely suspended from the point \(B\) and hangs in equilibrium. The angle between \(A B\) and the downward vertical through \(B\) is \(\theta\).
  4. Given that \(h = \frac { 13 } { 4 } r\), find the value of \(\tan \theta\).
CAIE Further Paper 3 2020 November Q5
10 marks Standard +0.3
5 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 $$\mathrm { y } = \mathrm { x } \tan \alpha - \frac { \mathrm { gx } ^ { 2 } } { 2 \mathrm { u } ^ { 2 } } \sec ^ { 2 } \alpha$$ The point \(Q\) is the highest point on the trajectory of \(P\) in the case where \(\alpha = 45 ^ { \circ }\).
  2. Show that the \(x\)-coordinate of \(Q\) is \(\frac { \mathrm { u } ^ { 2 } } { 2 \mathrm {~g} }\).
  3. Find the other value of \(\alpha\) for which \(P\) would pass through the point \(Q\).
CAIE Further Paper 3 2020 November Q6
10 marks Challenging +1.2
6 Two smooth spheres \(A\) and \(B\) have equal radii and masses \(m\) and \(2 m\) 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.
    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 \(A\) is \(\frac { 5 } { 32 }\) of the kinetic energy of \(B\).
  2. Find the possible values of \(e\).
CAIE Further Paper 3 2020 November Q7
10 marks Challenging +1.8
7 A particle \(P\) moving in a straight line has displacement \(x \mathrm {~m}\) from a fixed point \(O\) on the line at time \(t \mathrm {~s}\). The acceleration of \(P\), in \(\mathrm { ms } ^ { - 2 }\), is given by \(\frac { 200 } { x ^ { 2 } } - \frac { 100 } { x ^ { 3 } }\) for \(x > 0\). When \(t = 0 , x = 1\) and \(P\) has velocity \(10 \mathrm {~ms} ^ { - 1 }\) directed towards \(O\).
  1. Show that the velocity \(v \mathrm {~ms} ^ { - 1 }\) of \(P\) is given by \(\mathrm { v } = \frac { 10 ( 1 - 2 \mathrm { x } ) } { \mathrm { x } }\).
  2. Show that \(x\) and \(t\) are related by the equation \(\mathrm { e } ^ { - 40 \mathrm { t } } = ( 2 \mathrm { x } - 1 ) \mathrm { e } ^ { 2 \mathrm { x } - 2 }\) and deduce what happens to \(x\) as \(t\) becomes large.
    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 M2 2010 November Q1
3 marks Moderate -0.8
1 A horizontal circular disc rotates with constant angular speed \(9 \mathrm { rad } \mathrm { s } ^ { - 1 }\) about its centre \(O\). A particle of mass 0.05 kg is placed on the disc at a distance 0.4 m from \(O\). The particle moves with the disc and no sliding takes place. Calculate the magnitude of the resultant force exerted on the particle by the disc.
CAIE M2 2010 November Q2
6 marks Standard +0.3
2 \includegraphics[max width=\textwidth, alt={}, center]{af7d1fc8-5552-48b8-a359-895b2b5d3d6c-2_673_401_525_872} A bow consists of a uniform curved portion \(A B\) of mass 1.4 kg , and a uniform taut string of mass \(m \mathrm {~kg}\) which joins \(A\) and \(B\). The curved portion \(A B\) is an arc of a circle centre \(O\) and radius 0.8 m . Angle \(A O B\) is \(\frac { 2 } { 3 } \pi\) radians (see diagram). The centre of mass of the bow (including the string) is 0.65 m from \(O\). Calculate \(m\).
CAIE M2 2010 November Q3
7 marks Standard +0.3
3 \includegraphics[max width=\textwidth, alt={}, center]{af7d1fc8-5552-48b8-a359-895b2b5d3d6c-2_279_905_1560_621} One end of a light inextensible string of length 0.2 m is attached to a fixed point \(A\) which is above a smooth horizontal surface. A particle \(P\) of mass 0.6 kg is attached to the other end of the string. \(P\) moves in a circle on the surface with constant speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), with the string taut and making an angle of \(30 ^ { \circ }\) to the horizontal (see diagram).
  1. Given that \(v = 1.5\), calculate the magnitude of the force that the surface exerts on \(P\).
  2. Given instead that \(P\) moves with its greatest possible speed while remaining in contact with the surface, find \(v\).
CAIE M2 2010 November Q4
7 marks Standard +0.3
4 \includegraphics[max width=\textwidth, alt={}, center]{af7d1fc8-5552-48b8-a359-895b2b5d3d6c-3_560_894_258_628} A uniform beam \(A B\) has length 2 m and weight 70 N . The beam is hinged at \(A\) to a fixed point on a vertical wall, and is held in equilibrium by a light inextensible rope. One end of the rope is attached to the wall at a point 1.7 m vertically above the hinge. The other end of the rope is attached to the beam at a point 0.8 m from \(A\). The rope is at right angles to \(A B\). The beam carries a load of weight 220 N at \(B\) (see diagram).
  1. Find the tension in the rope.
  2. Find the direction of the force exerted on the beam at \(A\).
CAIE M2 2010 November Q5
7 marks Standard +0.8
5 A particle \(P\) of mass 0.28 kg is attached to the mid-point of a light elastic string of natural length 4 m . The ends of the string are attached to fixed points \(A\) and \(B\) which are at the same horizontal level and 4.8 m apart. \(P\) is released from rest at the mid-point of \(A B\). In the subsequent motion, the acceleration of \(P\) is zero when \(P\) is at a distance 0.7 m below \(A B\).
  1. Show that the modulus of elasticity of the string is 20 N .
  2. Calculate the maximum speed of \(P\).
CAIE M2 2010 November Q6
10 marks Standard +0.3
6 A cyclist and his bicycle have a total mass of 81 kg . The cyclist starts from rest and rides in a straight line. The cyclist exerts a constant force of 135 N and the motion is opposed by a resistance of magnitude \(9 v \mathrm {~N}\), where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the cyclist's speed at time \(t \mathrm {~s}\) after starting.
  1. Show that \(\frac { 9 } { 15 - v } \frac { \mathrm {~d} v } { \mathrm {~d} t } = 1\).
  2. Solve this differential equation to show that \(v = 15 \left( 1 - \mathrm { e } ^ { - \frac { 1 } { 9 } t } \right)\).
  3. Find the distance travelled by the cyclist in the first 9 s of the motion.
CAIE M2 2010 November Q7
10 marks Standard +0.3
7 \includegraphics[max width=\textwidth, alt={}, center]{af7d1fc8-5552-48b8-a359-895b2b5d3d6c-4_433_841_255_653} A particle \(P\) is projected from a point \(O\) with initial speed \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(45 ^ { \circ }\) above the horizontal. \(P\) subsequently passes through the point \(A\) which is at an angle of elevation of \(30 ^ { \circ }\) from \(O\) (see diagram). At time \(t \mathrm {~s}\) after projection the horizontal and vertically upward displacements of \(P\) from \(O\) are \(x \mathrm {~m}\) and \(y \mathrm {~m}\) respectively.
  1. Write down expressions for \(x\) and \(y\) in terms of \(t\), and hence obtain the equation of the trajectory of \(P\).
  2. Calculate the value of \(x\) when \(P\) is at \(A\).
  3. Find the angle the trajectory makes with the horizontal when \(P\) is at \(A\). \footnotetext{Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable effort has been made by the publisher (UCLES) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will be pleased to make amends at the earliest possible opportunity.
    University of Cambridge International Examinations is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge. }
CAIE M2 2010 November Q2
6 marks Standard +0.3
2 \includegraphics[max width=\textwidth, alt={}, center]{8425223a-1924-43ef-bd7c-e9b424fdc311-2_673_401_525_872} A bow consists of a uniform curved portion \(A B\) of mass 1.4 kg , and a uniform taut string of mass \(m \mathrm {~kg}\) which joins \(A\) and \(B\). The curved portion \(A B\) is an arc of a circle centre \(O\) and radius 0.8 m . Angle \(A O B\) is \(\frac { 2 } { 3 } \pi\) radians (see diagram). The centre of mass of the bow (including the string) is 0.65 m from \(O\). Calculate \(m\).
CAIE M2 2010 November Q3
7 marks Standard +0.3
3 \includegraphics[max width=\textwidth, alt={}, center]{8425223a-1924-43ef-bd7c-e9b424fdc311-2_279_905_1560_621} One end of a light inextensible string of length 0.2 m is attached to a fixed point \(A\) which is above a smooth horizontal surface. A particle \(P\) of mass 0.6 kg is attached to the other end of the string. \(P\) moves in a circle on the surface with constant speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), with the string taut and making an angle of \(30 ^ { \circ }\) to the horizontal (see diagram).
  1. Given that \(v = 1.5\), calculate the magnitude of the force that the surface exerts on \(P\).
  2. Given instead that \(P\) moves with its greatest possible speed while remaining in contact with the surface, find \(v\).
CAIE M2 2010 November Q4
7 marks Standard +0.3
4 \includegraphics[max width=\textwidth, alt={}, center]{8425223a-1924-43ef-bd7c-e9b424fdc311-3_560_894_258_628} A uniform beam \(A B\) has length 2 m and weight 70 N . The beam is hinged at \(A\) to a fixed point on a vertical wall, and is held in equilibrium by a light inextensible rope. One end of the rope is attached to the wall at a point 1.7 m vertically above the hinge. The other end of the rope is attached to the beam at a point 0.8 m from \(A\). The rope is at right angles to \(A B\). The beam carries a load of weight 220 N at \(B\) (see diagram).
  1. Find the tension in the rope.
  2. Find the direction of the force exerted on the beam at \(A\).
CAIE M2 2010 November Q7
10 marks Standard +0.3
7 \includegraphics[max width=\textwidth, alt={}, center]{8425223a-1924-43ef-bd7c-e9b424fdc311-4_433_841_255_653} A particle \(P\) is projected from a point \(O\) with initial speed \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(45 ^ { \circ }\) above the horizontal. \(P\) subsequently passes through the point \(A\) which is at an angle of elevation of \(30 ^ { \circ }\) from \(O\) (see diagram). At time \(t \mathrm {~s}\) after projection the horizontal and vertically upward displacements of \(P\) from \(O\) are \(x \mathrm {~m}\) and \(y \mathrm {~m}\) respectively.
  1. Write down expressions for \(x\) and \(y\) in terms of \(t\), and hence obtain the equation of the trajectory of \(P\).
  2. Calculate the value of \(x\) when \(P\) is at \(A\).
  3. Find the angle the trajectory makes with the horizontal when \(P\) is at \(A\). \footnotetext{Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable effort has been made by the publisher (UCLES) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will be pleased to make amends at the earliest possible opportunity.
    University of Cambridge International Examinations is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge. }
CAIE M2 2010 November Q1
6 marks Standard +0.3
1 \(A B C D\) is a uniform lamina with \(A B = 1.8 \mathrm {~m} , A D = D C = 0.9 \mathrm {~m}\), and \(A D\) perpendicular to \(A B\) and \(D C\) (see diagram).
  1. Find the distance of the centre of mass of the lamina from \(A B\) and the distance from \(A D\). The lamina is freely suspended at \(A\) and hangs in equilibrium.
  2. Calculate the angle between \(A B\) and the vertical.
CAIE M2 2010 November Q2
7 marks Standard +0.3
2 A particle \(P\) is projected with speed \(26 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(30 ^ { \circ }\) above the horizontal from a point \(O\) on a horizontal plane.
  1. For the instant when the vertical component of the velocity of \(P\) is \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) downwards, find the direction of motion of \(P\) and the height of \(P\) above the plane.
  2. \(P\) strikes the plane at the point \(A\). Calculate the time taken by \(P\) to travel from \(O\) to \(A\) and the distance \(O A\). \includegraphics[max width=\textwidth, alt={}, center]{022776e2-80bb-4e73-a309-8cd972ae7377-2_679_455_1544_845} Particles \(P\) and \(Q\) have masses 0.8 kg and 0.4 kg respectively. \(P\) is attached to a fixed point \(A\) by a light inextensible string which is inclined at an angle \(\alpha ^ { \circ }\) to the vertical. \(Q\) is attached to a fixed point \(B\), which is vertically below \(A\), by a light inextensible string of length 0.3 m . The string \(B Q\) is horizontal. \(P\) and \(Q\) are joined to each other by a light inextensible string which is vertical. The particles rotate in horizontal circles of radius 0.3 m about the axis through \(A\) and \(B\) with constant angular speed \(5 \mathrm { rad } \mathrm { s } ^ { - 1 }\) (see diagram).
  3. By considering the motion of \(Q\), find the tensions in the strings \(P Q\) and \(B Q\).
  4. Find the tension in the string \(A P\) and the value of \(\alpha\).
CAIE M2 2010 November Q4
8 marks Standard +0.3
4 \includegraphics[max width=\textwidth, alt={}, center]{022776e2-80bb-4e73-a309-8cd972ae7377-3_371_570_258_790} A uniform \(\operatorname { rod } A B\) has weight 15 N and length 1.2 m . The end \(A\) of the rod is in contact with a rough plane inclined at \(30 ^ { \circ }\) to the horizontal, and the rod is perpendicular to the plane. The rod is held in equilibrium in this position by means of a horizontal force applied at \(B\), acting in the vertical plane containing the rod (see diagram).
  1. Show that the magnitude of the force applied at \(B\) is 4.33 N , correct to 3 significant figures.
  2. Find the magnitude of the frictional force exerted by the plane on the rod.
  3. Given that the rod is in limiting equilibrium, calculate the coefficient of friction between the rod and the plane.
CAIE M2 2010 November Q5
9 marks Challenging +1.2
5 \includegraphics[max width=\textwidth, alt={}, center]{022776e2-80bb-4e73-a309-8cd972ae7377-3_287_1068_1306_536} A light elastic string has natural length 2 m and modulus of elasticity \(\lambda \mathrm { N }\). The ends of the string are attached to fixed points \(A\) and \(B\) which are at the same horizontal level and 2.4 m apart. A particle \(P\) of mass 0.6 kg is attached to the mid-point of the string and hangs in equilibrium at a point 0.5 m below \(A B\) (see diagram).
  1. Show that \(\lambda = 26\). \(P\) is projected vertically downwards from the equilibrium position, and comes to instantaneous rest at a point 0.9 m below \(A B\).
  2. Calculate the speed of projection of \(P\).
CAIE M2 2010 November Q6
12 marks Challenging +1.2
6 \includegraphics[max width=\textwidth, alt={}, center]{022776e2-80bb-4e73-a309-8cd972ae7377-4_341_572_258_790} A particle \(P\) of mass 0.2 kg is projected with velocity \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) upwards along a line of greatest slope on a plane inclined at \(30 ^ { \circ }\) to the horizontal (see diagram). Air resistance of magnitude \(0.5 v \mathrm {~N}\) opposes the motion of \(P\), where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the velocity of \(P\) at time \(t \mathrm {~s}\) after projection. The coefficient of friction between \(P\) and the plane is \(\frac { 1 } { 2 \sqrt { } 3 }\). The particle \(P\) reaches a position of instantaneous rest when \(t = T\).
  1. Show that, while \(P\) is moving up the plane, \(\frac { \mathrm { d } v } { \mathrm {~d} t } = - 2.5 ( 3 + v )\).
  2. Calculate \(T\).
  3. Calculate the speed of \(P\) when \(t = 2 T\). \footnotetext{Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable effort has been made by the publisher (UCLES) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will be pleased to make amends at the earliest possible opportunity. University of Cambridge International Examinations is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge. }
CAIE M2 2011 November Q1
5 marks Standard +0.3
1 \includegraphics[max width=\textwidth, alt={}, center]{d1f1f036-1676-443e-b733-ca1fe79972d4-2_334_679_258_731} A non-uniform \(\operatorname { rod } A B\), of length 0.6 m and weight 9 N , has its centre of mass 0.4 m from \(A\). The end \(A\) of the rod is in contact with a rough vertical wall. The rod is held in equilibrium, perpendicular to the wall, by means of a light string attached to \(B\). The string is inclined at \(30 ^ { \circ }\) to the horizontal. The tension in the string is \(T \mathrm {~N}\) (see diagram).
  1. Calculate \(T\).
  2. Find the least possible value of the coefficient of friction at \(A\).