Questions - CAIE - A-Level Maths

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 2022 November Q6

8 marks Challenging +1.8

6 \includegraphics[max width=\textwidth, alt={}, center]{7febbd80-4cbb-4b2e-b022-d6a20e7e13aa-10_426_1191_267_438} Two uniform smooth spheres $A$ and $B$ of equal radii have masses $m$ and $k m$ respectively. The two spheres are moving on a horizontal surface with speeds $u$ and $\frac { 5 } { 8 } u$ respectively. Immediately before the spheres collide, $A$ is travelling along the line of centres, and $B$ 's direction of motion makes an angle $\alpha$ with the line of centres (see diagram). The coefficient of restitution between the spheres is $\frac { 2 } { 3 }$ and $\tan \alpha = \frac { 3 } { 4 }$. After the collision, the direction of motion of $B$ is perpendicular to the line of centres.

Find the value of $k$.
Find the loss in the total kinetic energy as a result of the collision.

CAIE Further Paper 3 2022 November Q7

10 marks Challenging +1.2

7 A particle $P$ is projected with speed $\mathrm { Vms } ^ { - 1 }$ at an angle $75 ^ { \circ }$ above the horizontal from a point $O$ on a horizontal plane. It then moves freely under gravity.

Show that the total time of flight, in seconds, is $\frac { 2 \mathrm {~V} } { \mathrm {~g} } \sin 75 ^ { \circ }$.
A smooth vertical barrier is now inserted with its lower end on the plane at a distance 15 m from $O$. The particle is projected as before but now strikes the barrier, rebounds and returns to $O$. The coefficient of restitution between the barrier and the particle is $\frac { 3 } { 5 }$.
Explain why the total time of flight is unchanged.
Find an expression for $V$ in terms of $g$.
If you use the following page to complete the answer to any question, the question number must be clearly shown.

Find an expression for $v$ in terms of $x$.
State the value that the speed approaches for large values of $x$.

CAIE Further Paper 3 2022 November Q5

9 marks Standard +0.8

5 A particle $P$ is projected with speed $u \mathrm {~ms} ^ { - 1 }$ at an angle of $\theta$ 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 $t \mathrm {~s}$ are denoted by $x \mathrm {~m}$ and $y \mathrm {~m}$ respectively.

Show that the equation of the trajectory is given by $$\mathrm { y } = \mathrm { x } \tan \theta - \frac { \mathrm { gx } ^ { 2 } } { 2 \mathrm { u } ^ { 2 } } \left( 1 + \tan ^ { 2 } \theta \right)$$ In the subsequent motion $P$ passes through the point with coordinates $( 30,20 )$.
Given that one possible value of $\tan \theta$ is $\frac { 4 } { 3 }$, find the other possible value of $\tan \theta$. \includegraphics[max width=\textwidth, alt={}, center]{9b3f3add-17fd-4597-bd5d-27e3abb527be-10_451_1339_258_404} A light inextensible string is threaded through a fixed smooth ring $R$ which is at a height $h$ above a smooth horizontal surface. One end of the string is attached to a particle $A$ of mass $m$. The other end of the string is attached to a particle $B$ of mass $\frac { 6 } { 7 } m$. The particle $A$ moves in a horizontal circle on the surface. The particle $B$ hangs in equilibrium below the ring and above the surface (see diagram). When $A$ has constant angular speed $\omega$, the angle between $A R$ and $B R$ is $\theta$ and the normal reaction between $A$ and the surface is $N$. When $A$ has constant angular speed $\frac { 3 } { 2 } \omega$, the angle between $A R$ and $B R$ is $\alpha$ and the normal reaction between $A$ and the surface is $\frac { 1 } { 2 } N$.
Show that $\cos \theta = \frac { 4 } { 9 } \cos \alpha$.
Find $N$ in terms of $m$ and $g$ and find the value of $\cos \alpha$. \includegraphics[max width=\textwidth, alt={}, center]{9b3f3add-17fd-4597-bd5d-27e3abb527be-12_413_974_255_587} Two uniform smooth spheres $A$ and $B$ of equal radii have masses $m$ and $\frac { 1 } { 2 } m$ respectively. The two spheres are moving on a horizontal surface when they collide. Immediately before the collision, sphere $A$ is travelling with speed $u$ and its direction of motion makes an angle $\alpha$ with the line of centres. Sphere $B$ is travelling with speed $2 u$ and its direction of motion makes an angle $\beta$ with the line of centres (see diagram). The coefficient of restitution between the spheres is $\frac { 5 } { 8 }$ and $\alpha + \beta = 90 ^ { \circ }$.
Find the component of the velocity of $B$ parallel to the line of centres after the collision, giving your answer in terms of $u$ and $\alpha$.
The direction of motion of $B$ after the collision is parallel to the direction of motion of $A$ before the collision.
Find the value of $\tan \alpha$.
If you use the following page to complete the answer to any question, the question number must be clearly shown.

CAIE Further Paper 3 2022 November Q2

6 marks Standard +0.3

2 A light elastic string has natural length $a$ and modulus of elasticity 4 mg . One end of the string is fixed to a point $O$ 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 along the surface in the direction $O P$. When the length of the string is $\frac { 5 } { 4 } a$, the speed of $P$ is $v$. When the length of the string is $\frac { 3 } { 2 } a$, the speed of $P$ is $\frac { 1 } { 2 } v$.

Find an expression for $v$ in terms of $a$ and $g$.
Find, in terms of $g$, the acceleration of $P$ when the stretched length of the string is $\frac { 3 } { 2 } a$. \includegraphics[max width=\textwidth, alt={}, center]{5e95e0c9-d47d-4f2b-89da-ab949b9661f4-04_552_1059_264_502} A smooth cylinder is fixed to a rough horizontal surface with its axis of symmetry horizontal. A uniform rod $A B$, of length $4 a$ and weight $W$, rests against the surface of the cylinder. The end $A$ of the rod is in contact with the horizontal surface. The vertical plane containing the rod $A B$ is perpendicular to the axis of the cylinder. The point of contact between the rod and the cylinder is $C$, where $A C = 3 a$. The angle between the rod and the horizontal surface is $\theta$ where $\tan \theta = \frac { 3 } { 4 }$ (see diagram). The coefficient of friction between the rod and the horizontal surface is $\frac { 6 } { 7 }$. A particle of weight $k W$ is attached to the rod at $B$. The rod is about to slip. The normal reaction between the rod and the cylinder is $N$.

CAIE Further Paper 3 2022 November Q6

8 marks Challenging +1.8

6 \includegraphics[max width=\textwidth, alt={}, center]{5e95e0c9-d47d-4f2b-89da-ab949b9661f4-10_426_1191_267_438} Two uniform smooth spheres $A$ and $B$ of equal radii have masses $m$ and $k m$ respectively. The two spheres are moving on a horizontal surface with speeds $u$ and $\frac { 5 } { 8 } u$ respectively. Immediately before the spheres collide, $A$ is travelling along the line of centres, and $B$ 's direction of motion makes an angle $\alpha$ with the line of centres (see diagram). The coefficient of restitution between the spheres is $\frac { 2 } { 3 }$ and $\tan \alpha = \frac { 3 } { 4 }$. After the collision, the direction of motion of $B$ is perpendicular to the line of centres.

Find the value of $k$.
Find the loss in the total kinetic energy as a result of the collision.

CAIE Further Paper 3 2023 November Q2

7 marks Challenging +1.2

2 A ball of mass 2 kg is projected vertically downwards with speed $5 \mathrm {~ms} ^ { - 1 }$ through a liquid. At time $t \mathrm {~s}$ after projection, the velocity of the ball is $v \mathrm {~ms} ^ { - 1 }$ and its displacement from its starting point is $x \mathrm {~m}$. The forces acting on the ball are its weight and a resistive force of magnitude $0.2 v ^ { 2 } \mathrm {~N}$.

Find an expression for $v$ in terms of $t$.
Deduce what happens to $v$ for large values of $t$. \includegraphics[max width=\textwidth, alt={}, center]{6964a0b0-8fc8-4fa8-b3fe-f51dafdaaeec-06_803_652_251_703} A uniform square lamina of side $2 a$ and weight $W$ is suspended from a light inextensible string attached to the midpoint $E$ of the side $A B$. The other end of the string is attached to a fixed point $P$ on a rough vertical wall. The vertex $B$ of the lamina is in contact with the wall. The string $E P$ is perpendicular to the side $A B$ and makes an angle $\theta$ with the wall (see diagram). The string and the lamina are in a vertical plane perpendicular to the wall. The coefficient of friction between the wall and the lamina is $\frac { 1 } { 2 }$. Given that the vertex $B$ is about to slip up the wall, find the value of $\tan \theta$. \includegraphics[max width=\textwidth, alt={}, center]{6964a0b0-8fc8-4fa8-b3fe-f51dafdaaeec-08_581_576_269_731} A light elastic string has natural length $8 a$ and modulus of elasticity $5 m g$. A particle $P$ of mass $m$ is attached to the midpoint of the string. The ends of the string are attached to points $A$ and $B$ which are a distance $12 a$ apart on a smooth horizontal table. The particle $P$ is held on the table so that $A P = B P = L$ (see diagram). The particle $P$ is released from rest. When $P$ is at the midpoint of $A B$ it has speed $\sqrt { 80 a g }$.
Find $L$ in terms of $a$.
Find the initial acceleration of $P$ in terms of $g$.

CAIE Further Paper 3 2023 November Q5

9 marks Challenging +1.2

5 A particle $P$ is projected with speed $u \mathrm {~ms} ^ { - 1 }$ at an angle $\theta$ above the horizontal from a point $O$ on a horizontal plane and moves freely under gravity. During its flight $P$ passes through the point which is a horizontal distance $3 a$ from $O$ and a vertical distance $\frac { 3 } { 8 } a$ above the horizontal plane. It is given that $\tan \theta = \frac { 1 } { 3 }$.

Show that $\mathrm { u } ^ { 2 } = 8 \mathrm { ag }$.
A particle $Q$ is projected with speed $V \mathrm {~ms} ^ { - 1 }$ at an angle $\alpha$ above the horizontal from $O$ at the instant when $P$ is at its highest point. Particles $P$ and $Q$ both land at the same point on the horizontal plane at the same time.
Find $V$ in terms of $a$ and $g$.

CAIE Further Paper 3 2023 November Q6

11 marks Challenging +1.8

6 A particle $P$ of mass $m$ is attached to one end of a light inextensible rod of length $3 a$. An identical particle $Q$ is attached to the other end of the rod. The rod is smoothly pivoted at a point $O$ on the rod, where $\mathrm { OQ } = \mathrm { x }$. The system, of rod and particles, rotates about $O$ in a vertical plane. At an instant when the rod is vertical, with $P$ above $Q$, the particle $P$ is moving horizontally with speed $u$. When the rod has turned through an angle of $60 ^ { \circ }$ from the vertical, the speed of $P$ is $2 \sqrt { \mathrm { ag } }$, and the tensions in the two parts of the rod, $O P$ and $O Q$, have equal magnitudes.

Show that the speed of $Q$ when the rod has turned through an angle of $60 ^ { \circ }$ from the vertical is $\frac { 2 x } { 3 a - x } \sqrt { a g }$.
Find $x$ in terms of $a$.
Find $u$ in terms of $a$ and $g$.
If you use the following page to complete the answer to any question, the question number must be clearly shown.

CAIE Further Paper 3 2023 November Q1

4 marks Moderate -0.5

1 One end of a light inextensible string of length $a$ is attached to a fixed point $O$. The other end of the string is attached to a particle of mass $m$. The string is taut and makes an angle $\theta$ with the downward vertical through $O$, where $\cos \theta = \frac { 2 } { 3 }$. The particle moves in a horizontal circle with speed $v$. Find $v$ in terms of $a$ and $g$.
..................................................................................................................................................

CAIE Further Paper 3 2023 November Q2

6 marks Challenging +1.2

2 A particle $P$ of mass 0.5 kg moves in a straight line. At time $t \mathrm {~s}$ the velocity of $P$ is $v \mathrm {~ms} ^ { - 1 }$ and its displacement from a fixed point $O$ on the line is $x \mathrm {~m}$. The only forces acting on $P$ are a force of magnitude $\frac { 150 } { ( x + 1 ) ^ { 2 } } \mathrm {~N}$ in the direction of increasing displacement and a resistive force of magnitude $\frac { 450 } { ( x + 1 ) ^ { 3 } } \mathrm {~N}$. When $t = 0 , x = 0$ and $v = 20$.
Find $v$ in terms of $x$, giving your answer in the form $v = \frac { A x + B } { ( x + 1 ) }$, where $A$ and $B$ are constants to be determined. \includegraphics[max width=\textwidth, alt={}, center]{0270d51a-e252-46d3-8c97-7f71ba91fa65-04_451_812_255_625} A uniform lamina is in the form of an isosceles triangle $A B C$ in which $A C = 2 \mathrm { a }$ and angle $A B C = 90 ^ { \circ }$. The point $D$ on $A B$ is such that the ratio $D B : A B = 1 : k$. The point $E$ on $C B$ is such that $D E$ is parallel to $A C$. The triangle $D B E$ is removed from the lamina (see diagram).

Find, in terms of $k$, the distance of the centre of mass of the remaining lamina $A D E C$ from the midpoint of $A C$.
When the lamina $A D E C$ is freely suspended from the vertex $A$, the edge $A C$ makes an angle $\theta$ with the downward vertical, where $\tan \theta = \frac { 5 } { 18 }$.
Find the value of $k$. \includegraphics[max width=\textwidth, alt={}, center]{0270d51a-e252-46d3-8c97-7f71ba91fa65-06_604_798_251_635} Two smooth vertical walls meet at right angles. The smooth sphere $A$, with mass $m$, is at rest on a smooth horizontal surface and is at a distance $d$ from each wall. An identical smooth sphere $B$ is moving on the horizontal surface with speed $u$ at an angle $\theta$ with the line of centres when the spheres collide (see diagram). After the collision, the spheres take the same time to reach a wall. The coefficient of restitution between the spheres is $\frac { 1 } { 2 }$.

CAIE Further Paper 3 2023 November Q6

9 marks Challenging +1.2

6 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.

Derive the equation of the trajectory of $P$ in the form $$\mathrm { y } = \mathrm { x } \tan \alpha - \frac { \mathrm { gx } ^ { 2 } } { 2 \mathrm { u } ^ { 2 } } \sec ^ { 2 } \alpha$$ During its flight, $P$ must clear an obstacle of height $h \mathrm {~m}$ that is at a horizontal distance of 32 m from the point of projection. When $u = 40 \sqrt { 2 } \mathrm {~ms} ^ { - 1 } , P$ just clears the obstacle. When $u = 40 \mathrm {~ms} ^ { - 1 } , P$ only achieves $80 \%$ of the height required to clear the obstacle.
Find the two possible values of $h$.

CAIE Further Paper 3 2023 November Q7

9 marks Challenging +1.8

7 \includegraphics[max width=\textwidth, alt={}, center]{0270d51a-e252-46d3-8c97-7f71ba91fa65-12_618_835_255_616} A particle $P$ of mass $m$ is attached to one end of a light rod of length $3 a$. The other end of the rod is able to pivot smoothly about the fixed point $A$. The particle is also attached to one end of a light spring of natural length $a$ and modulus of elasticity $k m g$. The other end of the spring is attached to a fixed point $B$. The points $A$ and $B$ are in a horizontal line, a distance $5 a$ apart, and these two points and the rod are in a vertical plane. Initially, $P$ is held in equilibrium by a vertical force $F$ with the stretched length of the spring equal to $4 a$ (see diagram). The particle is released from rest in this position and has a speed of $\frac { 6 } { 5 } \sqrt { 2 a g }$ when the rod becomes horizontal.

Find the value of $k$.
Find $F$ in terms of $m$ and $g$.
Find, in terms of $m$ and $g$, the tension in the rod immediately before it is released.
If you use the following page to complete the answer to any question, the question number must be clearly shown.

CAIE Further Paper 3 2023 November Q2

7 marks Challenging +1.2

Find an expression for $v$ in terms of $t$.
Deduce what happens to $v$ for large values of $t$. \includegraphics[max width=\textwidth, alt={}, center]{e7091f6c-af72-49f3-b825-cdce9fb2c06f-06_803_652_251_703} A uniform square lamina of side $2 a$ and weight $W$ is suspended from a light inextensible string attached to the midpoint $E$ of the side $A B$. The other end of the string is attached to a fixed point $P$ on a rough vertical wall. The vertex $B$ of the lamina is in contact with the wall. The string $E P$ is perpendicular to the side $A B$ and makes an angle $\theta$ with the wall (see diagram). The string and the lamina are in a vertical plane perpendicular to the wall. The coefficient of friction between the wall and the lamina is $\frac { 1 } { 2 }$. Given that the vertex $B$ is about to slip up the wall, find the value of $\tan \theta$. \includegraphics[max width=\textwidth, alt={}, center]{e7091f6c-af72-49f3-b825-cdce9fb2c06f-08_581_576_269_731} A light elastic string has natural length $8 a$ and modulus of elasticity $5 m g$. A particle $P$ of mass $m$ is attached to the midpoint of the string. The ends of the string are attached to points $A$ and $B$ which are a distance $12 a$ apart on a smooth horizontal table. The particle $P$ is held on the table so that $A P = B P = L$ (see diagram). The particle $P$ is released from rest. When $P$ is at the midpoint of $A B$ it has speed $\sqrt { 80 a g }$.
Find $L$ in terms of $a$.
Find the initial acceleration of $P$ in terms of $g$.

CAIE Further Paper 3 2024 November Q1

5 marks Challenging +1.2

1 A particle $P$ is projected with speed $u \mathrm {~m} \mathrm {~s} ^ { - 1 }$ 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 m horizontally from $O$, it is at a vertical height $H \mathrm {~m}$ above the plane. When $P$ has travelled a distance 84 m horizontally from $O$, it is at a vertical height $\frac { 1 } { 2 } H \mathrm {~m}$ above the plane. Find, in either order, the value of $u$ and the value of $H$. \includegraphics[max width=\textwidth, alt={}, center]{1eb2ef87-e858-41c1-8e96-75fb2222b57a-02_2718_38_106_2009}

CAIE Further Paper 3 2024 November Q2

5 marks Challenging +1.2

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 $O A$ makes an angle $\theta$ with the downward vertical through $O$, where $\tan \theta = \frac { 3 } { 4 }$. The particle $P$ is projected perpendicular to $O A$ in an upwards direction with speed $\sqrt { 5 a g }$, and it starts to move along a circular path in a vertical plane. When $P$ is at the point $B$, where angle $A O B$ is a right angle, the tension in the string is $T$. Find $T$ in terms of $m$ and $g$.

CAIE Further Paper 3 2024 November Q3

8 marks Challenging +1.2

3 A particle $P$ of mass $m \mathrm {~kg}$ is attached to one end of a light elastic string of natural length 2 m and modulus of elasticity $2 m g \mathrm {~N}$. 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 $d$ m below its equilibrium position and released from rest.

Given that the particle just reaches $O$ in the subsequent motion, find the value of $d$. \includegraphics[max width=\textwidth, alt={}, center]{1eb2ef87-e858-41c1-8e96-75fb2222b57a-04_2713_31_111_2017} \includegraphics[max width=\textwidth, alt={}, center]{1eb2ef87-e858-41c1-8e96-75fb2222b57a-05_2725_35_99_20}
Hence find the speed of $P$ when it is 2 m below $O$. \includegraphics[max width=\textwidth, alt={}, center]{1eb2ef87-e858-41c1-8e96-75fb2222b57a-06_785_729_255_708} An object is formed by removing a cylinder of radius $\frac { 2 } { 3 } a$ and height $k h ( k < 1 )$ 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).
Find, in terms of $h$ and $k$, the distance of the centre of mass of the object from $A B$.

When the object is suspended from $A$, the angle between $A B$ and the vertical is $\theta$, where $\tan \theta = \frac { 3 } { 2 }$.
Given that $h = \frac { 8 } { 3 } a$, find the possible values of $k$.

CAIE Further Paper 3 2024 November Q5

7 marks Challenging +1.2

5 A particle $P$ of mass 2 kg moving on a horizontal straight line has displacement $x \mathrm {~m}$ from a fixed point $O$ on the line and velocity $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$ at time $t$ s. The only horizontal force acting on $P$ is a variable force $F \mathrm {~N}$ which can be expressed as a function of $t$. It is given that $$\frac { v } { x } = \frac { 3 - t } { 1 + t }$$ and when $t = 0 , x = 5$.

Find an expression for $x$ in terms of $t$. \includegraphics[max width=\textwidth, alt={}, center]{1eb2ef87-e858-41c1-8e96-75fb2222b57a-08_2718_38_106_2009} \includegraphics[max width=\textwidth, alt={}, center]{1eb2ef87-e858-41c1-8e96-75fb2222b57a-09_2723_33_99_21}
Find the magnitude of $F$ when $t = 3$. \includegraphics[max width=\textwidth, alt={}, center]{1eb2ef87-e858-41c1-8e96-75fb2222b57a-10_559_1257_255_445} A particle $P$ of mass 0.05 kg is attached to one end of a light inextensible string of length 1 m . The other end of the string is attached to a fixed point $O$. A particle $Q$ of mass 0.04 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 m with angular speed $\omega \operatorname { rad~s } ^ { - 1 }$. The particle $Q$ moves in a horizontal circle of radius 1.4 m also with angular speed $\omega \mathrm { rad } \mathrm { s } ^ { - 1 }$. The centres of the circles are vertically below $O$, and $O , P$ and $Q$ are always in the same vertical plane. The strings $O P$ and $P Q$ remain at constant angles $\alpha$ and $\beta$ respectively to the vertical (see diagram).
Find the tension in the string $O P$. \includegraphics[max width=\textwidth, alt={}, center]{1eb2ef87-e858-41c1-8e96-75fb2222b57a-10_2716_38_109_2012} \includegraphics[max width=\textwidth, alt={}, center]{1eb2ef87-e858-41c1-8e96-75fb2222b57a-11_2725_35_99_20}
Find the value of $\omega$.
Find the value of $\beta$.

CAIE Further Paper 3 2024 November Q7

10 marks Challenging +1.8

7 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 $e$.

Show that $e \tan ^ { 2 } \alpha = 1$. \includegraphics[max width=\textwidth, alt={}, center]{1eb2ef87-e858-41c1-8e96-75fb2222b57a-12_2716_36_106_2014} In its subsequent motion, the greatest height reached by $P$ above $A$ is $\frac { 3 } { 16 }$ of the vertical height of $A$ above the horizontal plane.
Find the value of $e$.
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]{1eb2ef87-e858-41c1-8e96-75fb2222b57a-14_2715_33_109_2012}

CAIE Further Paper 3 2024 November Q1

3 marks Standard +0.3

1 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. \includegraphics[max width=\textwidth, alt={}, center]{123017e8-8536-4716-aa01-5e9367770575-02_2717_33_109_2014}

CAIE Further Paper 3 2024 November Q2

4 marks Challenging +1.2

2 A particle $P$ of mass $m$ is attached to one end of a light elastic spring of natural length $a$ and modulus of elasticity 5 mg . 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 $\frac { 3 } { 2 } a$. Find the distance of $P$ below $O$ when $P$ first comes to instantaneous rest.

CAIE Further Paper 3 2024 November Q3

7 marks Challenging +1.2

3 \includegraphics[max width=\textwidth, alt={}, center]{123017e8-8536-4716-aa01-5e9367770575-04_348_828_251_621} \includegraphics[max width=\textwidth, alt={}, center]{123017e8-8536-4716-aa01-5e9367770575-04_2717_35_110_2012} 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 $2 u$ and $3 u$ 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 $\frac { 1 } { 3 }$.

Find the value of $\tan \theta$. \includegraphics[max width=\textwidth, alt={}, center]{123017e8-8536-4716-aa01-5e9367770575-05_2723_33_99_21}
Find the total loss of kinetic energy as a result of the collision.
Find, in degrees, the angle through which the direction of motion of $A$ is deflected as a result of the collision. \includegraphics[max width=\textwidth, alt={}, center]{123017e8-8536-4716-aa01-5e9367770575-06_776_785_255_680} The end $A$ of a uniform rod $A B$ of length $6 a$ and weight $W$ is in contact with a rough vertical wall. One end of a light inextensible string of length $3 a$ 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 $\tan \theta = \frac { 3 } { 2 }$. A particle of weight $W$ is attached to the point $E$ on the rod, where the distance $A E$ is equal to $k a ( 3 < k < 6 )$ (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.
Find the value of $k$.
Find, in terms of $W$, the magnitude of the frictional force between the rod and the wall.

CAIE Further Paper 3 2024 November Q5

8 marks Challenging +1.2

5 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 $\left( 3 a , \frac { 4 } { 5 } a \right)$ relative to horizontal and vertical axes through $O$ in the plane of the motion.

Use the equation of the trajectory to show that $u ^ { 2 } = 25 a g$. \includegraphics[max width=\textwidth, alt={}, center]{123017e8-8536-4716-aa01-5e9367770575-09_2725_35_99_20} 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.
Express $V ^ { 2 }$ in the form $k a g$, where $k$ is a rational number. \includegraphics[max width=\textwidth, alt={}, center]{123017e8-8536-4716-aa01-5e9367770575-10_506_803_255_630} 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 $P$ is projected at right angles to the string with speed $\frac { 1 } { 3 } \sqrt { 10 a g }$ 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 $\frac { 5 } { 9 } a$.
Find the value of $\cos \theta$. \includegraphics[max width=\textwidth, alt={}, center]{123017e8-8536-4716-aa01-5e9367770575-11_2725_35_99_20}
Find the ratio of the tensions in the string immediately before and immediately after it strikes the peg. $7 \quad$ A particle $P$ of mass $m \mathrm {~kg}$ is held at rest at a point $O$ and released so that it moves vertically under gravity against a resistive force of magnitude $0.1 m v ^ { 2 } \mathrm {~N}$, where $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$ is the velocity of $P$ at time $t \mathrm {~s}$.
Find an expression for $v$ in terms of $t$.
The displacement of $P$ from $O$ at time $t \mathrm {~s}$ is $x \mathrm {~m}$.
Find an expression for $v ^ { 2 }$ in terms of $x$.
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]{123017e8-8536-4716-aa01-5e9367770575-14_2716_31_106_2016}

Questions — CAIE (7659 questions)

Browse by module

Browse by board

CAIE Further Paper 3 2022 November Q6

CAIE Further Paper 3 2022 November Q7

CAIE Further Paper 3 2022 November Q1

CAIE Further Paper 3 2022 November Q3

CAIE Further Paper 3 2022 November Q4

CAIE Further Paper 3 2022 November Q5

CAIE Further Paper 3 2022 November Q2

CAIE Further Paper 3 2022 November Q6

CAIE Further Paper 3 2023 November Q2

CAIE Further Paper 3 2023 November Q5

CAIE Further Paper 3 2023 November Q6

CAIE Further Paper 3 2023 November Q1

CAIE Further Paper 3 2023 November Q2

CAIE Further Paper 3 2023 November Q6

CAIE Further Paper 3 2023 November Q7

CAIE Further Paper 3 2023 November Q2

CAIE Further Paper 3 2024 November Q1

CAIE Further Paper 3 2024 November Q2

CAIE Further Paper 3 2024 November Q3

CAIE Further Paper 3 2024 November Q5

CAIE Further Paper 3 2024 November Q7

CAIE Further Paper 3 2024 November Q1

CAIE Further Paper 3 2024 November Q2

CAIE Further Paper 3 2024 November Q3

CAIE Further Paper 3 2024 November Q5