Questions - CAIE M2 - A-Level Maths

Calculate the tensions in the two parts of the string when the speed of $P$ is $1.2 \mathrm {~m} \mathrm {~s} ^ { - 1 }$. The angular speed of the rod is increased to $5 \mathrm { rad } \mathrm { s } ^ { - 1 }$, and it is given that the system now rotates with angle $O A P =$ angle $O B P = 60 ^ { \circ }$.
Show that the tension in the part $A P$ of the string is zero. \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. To avoid the issue of disclosure of answer-related information to candidates, all copyright acknowledgements are reproduced online in the Cambridge International Examinations Copyright Acknowledgements Booklet. This is produced for each series of examinations and is freely available to download at \href{http://www.cie.org.uk}{www.cie.org.uk} after the live examination series. 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 2016 June Q2

2
\includegraphics[max width=\textwidth, alt={}, center]{76a47bf6-1982-4cdb-bcaa-2cdf84cc4f37-2_515_463_484_842} A uniform wire has the shape of a semicircular arc, with diameter $A B$ of length 0.8 m . The wire is attached to a vertical wall by a smooth hinge at $A$. The wire is held in equilibrium with $A B$ inclined at $70 ^ { \circ }$ to the upward vertical by a light string attached to $B$. The other end of the string is attached to the point $C$ on the wall 0.8 m vertically above $A$. The tension in the string is 15 N (see diagram).

Show that the horizontal distance of the centre of mass of the wire from the wall is 0.463 m , correct to 3 significant figures.
Calculate the weight of the wire.

CAIE M2 2016 June Q3

3 A particle $P$ of mass 0.4 kg is released from rest at a point $O$ on a smooth plane inclined at $30 ^ { \circ }$ to the horizontal. When the displacement of $P$ from $O$ is $x \mathrm {~m}$ down the plane, the velocity of $P$ is $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$. A force of magnitude $0.8 \mathrm { e } ^ { - x } \mathrm {~N}$ acts on $P$ up the plane along the line of greatest slope through $O$.

Show that $v \frac { \mathrm {~d} v } { \mathrm {~d} x } = 5 - 2 \mathrm { e } ^ { - x }$.
Find $v$ when $x = 0.6$.
\includegraphics[max width=\textwidth, alt={}, center]{76a47bf6-1982-4cdb-bcaa-2cdf84cc4f37-3_905_604_251_769} A uniform solid cone has base radius 0.4 m and height 4.4 m . A uniform solid cylinder has radius 0.4 m and weight equal to the weight of the cone. An object is formed by attaching the cylinder to the cone so that the base of the cone and a circular face of the cylinder are in contact and their circumferences coincide. The object rests in equilibrium with its circular base on a plane inclined at an angle of $20 ^ { \circ }$ to the horizontal (see diagram).
Calculate the least possible value of the coefficient of friction between the plane and the object.
Calculate the greatest possible height of the cylinder.

CAIE M2 2016 June Q6

6
\includegraphics[max width=\textwidth, alt={}, center]{76a47bf6-1982-4cdb-bcaa-2cdf84cc4f37-4_503_805_260_671} A light inextensible string passes through a small smooth bead $B$ of mass 0.4 kg . One end of the string is attached to a fixed point $A 0.4 \mathrm {~m}$ above a fixed point $O$ on a smooth horizontal surface. The other end of the string is attached to a fixed point $C$ which is vertically below $A$ and 0.3 m above the surface. The bead moves with constant speed on the surface in a circle with centre $O$ and radius 0.3 m (see diagram).

Given that the tension in the string is 2 N , calculate
(a) the angular speed of the bead,
(b) the magnitude of the contact force exerted on the bead by the surface.
Given instead that the bead is about to lose contact with the surface, calculate the speed of the bead.

CAIE M2 2017 June Q1

1 A particle is projected with speed $20 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ at an angle of $60 ^ { \circ }$ above the horizontal. Calculate the time after projection when the particle is descending at an angle of $40 ^ { \circ }$ below the horizontal.
\begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{7800deca-98e8-4eb4-9176-288bb1f44fec-04_376_713_260_715} \captionsetup{labelformat=empty} \caption{Fig. 1}

\end{figure} One end of a light inextensible string is attached to a fixed point $A$. The other end of the string is attached to a particle $P$ of mass $m \mathrm {~kg}$ which hangs vertically below $A$. The particle is also attached to one end of a light elastic string of natural length 0.25 m . The other end of this string is attached to a point $B$ which is 0.6 m from $P$ and on the same horizontal level as $P$. Equilibrium is maintained by a horizontal force of magnitude 7 N applied to $P$ (see Fig. 1).

Calculate the modulus of elasticity of the elastic string.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{7800deca-98e8-4eb4-9176-288bb1f44fec-05_348_488_262_826} \captionsetup{labelformat=empty} \caption{Fig. 2}
\end{figure} $P$ is released from rest by removing the 7 N force. In its subsequent motion $P$ first comes to instantaneous rest at a point where $B P = 0.3 \mathrm {~m}$ and the elastic string makes an angle of $30 ^ { \circ }$ with the horizontal (see Fig. 2).
Find the value of $m$.

CAIE M2 2017 June Q3

3
\includegraphics[max width=\textwidth, alt={}, center]{7800deca-98e8-4eb4-9176-288bb1f44fec-06_351_607_269_769} An object is made from a uniform solid hemisphere of radius 0.56 m and centre $O$ by removing a hemisphere of radius 0.28 m and centre $O$. The diagram shows a cross-section through $O$ of the object.

Calculate the distance of the centre of mass of the object from $O$.
[0pt] [The volume of a hemisphere is $\frac { 2 } { 3 } \pi r ^ { 3 }$.]
The object has weight 24 N . A uniform hemisphere $H$ of radius 0.28 m is placed in the hollow part of the object to create a non-uniform hemisphere with centre $O$. The centre of mass of the non-uniform hemisphere is 0.15 m from $O$.
Calculate the weight of $H$.

CAIE M2 2017 June Q4

4 A particle is projected from a point $O$ on horizontal ground. The initial components of the velocity of the particle are $10 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ horizontally and $15 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ vertically. At time $t \mathrm {~s}$ after projection, the horizontal and vertically upwards displacements of the particle from $O$ are $x \mathrm {~m}$ and $y \mathrm {~m}$ respectively.

Express $x$ and $y$ in terms of $t$, and hence find the equation of the trajectory of the particle.
\includegraphics[max width=\textwidth, alt={}, center]{7800deca-98e8-4eb4-9176-288bb1f44fec-08_63_1569_488_328}
The horizontal ground is at the top of a vertical cliff. The point $O$ is at a distance $d \mathrm {~m}$ from the edge of the cliff. The particle is projected towards the edge of the cliff and does not strike the ground before it passes over the edge of the cliff.
Show that $d$ is less than 30 .
Find the value of $x$ when the particle is 14 m below the level of $O$.
\includegraphics[max width=\textwidth, alt={}, center]{7800deca-98e8-4eb4-9176-288bb1f44fec-10_501_614_258_762} A uniform semicircular lamina of radius 0.7 m and weight 14 N has diameter $A B$. The lamina is in a vertical plane with $A$ freely pivoted at a fixed point. The straight edge $A B$ rests against a small smooth peg $P$ above the level of $A$. The angle between $A B$ and the horizontal is $30 ^ { \circ }$ and $A P = 0.9 \mathrm {~m}$ (see diagram).
Show that the magnitude of the force exerted by the peg on the lamina is 7.12 N , correct to 3 significant figures.
Find the angle with the horizontal of the force exerted by the pivot on the lamina at $A$.

CAIE M2 2017 June Q6

6 A particle $P$ of mass 0.15 kg is attached to one end of a light elastic string of natural length 0.4 m and modulus of elasticity 12 N . The other end of the string is attached to a fixed point $A$. The particle $P$ moves in a horizontal circle which has its centre vertically below $A$, with the string inclined at $\theta ^ { \circ }$ to the vertical and $A P = 0.5 \mathrm {~m}$.

Find the angular speed of $P$ and the value of $\theta$.
Calculate the difference between the elastic potential energy stored in the string and the kinetic energy of $P$.
$7 \quad$ A particle $P$ of mass 0.5 kg is at rest at a point $O$ on a rough horizontal surface. At time $t = 0$, where $t$ is in seconds, a horizontal force acting in a fixed direction is applied to $P$. At time $t \mathrm {~s}$ the magnitude of the force is $0.6 t ^ { 2 } \mathrm {~N}$ and the velocity of $P$ away from $O$ is $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$. It is given that $P$ remains at rest at $O$ until $t = 0.5$.
Calculate the coefficient of friction between $P$ and the surface, and show that $$\frac { \mathrm { d } v } { \mathrm {~d} t } = 1.2 t ^ { 2 } - 0.3 \quad \text { for } t > 0.5$$
Express $v$ in terms of $t$ for $t > 0.5$.
Find the displacement of $P$ from $O$ when $t = 1.2$.

CAIE M2 2017 June Q1

1 A particle $P$ of mass 0.2 kg moves with speed $4 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ and angular speed $5 \mathrm { rad } \mathrm { s } ^ { - 1 }$ in a horizontal circle on a smooth surface. $P$ is attached to one end of a light elastic string of natural length 0.6 m . The other end of the string is attached to the point on the surface which is the centre of the circular motion of $P$.

Find the radius of this circle.
Find the modulus of elasticity of the string.

CAIE M2 2017 June Q2

2
\includegraphics[max width=\textwidth, alt={}, center]{2b0425b2-2f8f-491a-996c-3d3b589bd7df-04_444_455_260_845} The ends of two light inextensible strings of length 0.7 m are attached to a particle $P$. The other ends of the strings are attached to two fixed points $A$ and $B$ which lie in the same vertical line with $A$ above $B$. The particle $P$ moves in a horizontal circle which has its centre at the mid-point of $A B$. Both strings are inclined at $60 ^ { \circ }$ to the vertical. The tension in the string attached to $A$ is 6 N and the tension in the string attached to $B$ is 4 N (see diagram).

Find the mass of $P$.
Calculate the speed of $P$.

CAIE M2 2017 June Q3

3 An open box in the shape of a cube with edges of length 0.2 m is placed with its base horizontal and its four sides vertical. The four sides and base are uniform laminas, each with weight 3 N .

Calculate the height of the centre of mass of the box above its base.
The box is now fitted with a thin uniform square lid of weight 3 N and with edges of length 0.2 m . The lid is attached to the box by a hinge of length 0.2 m and weight 2 N . The lid of the box is held partly open.
Find the angle which the lid makes with the horizontal when the centre of mass of the box (including the lid and hinge) is 0.12 m above the base of the box.

CAIE M2 2017 June Q4

4 A small object of mass 0.4 kg is released from rest at a point 8 m above the ground. The object descends vertically and when its downwards displacement from its initial position is $x \mathrm {~m}$ the object has velocity $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$. While the object is moving, a force of magnitude $0.2 v ^ { 2 } \mathrm {~N}$ opposes the motion.

Show that $v \frac { \mathrm {~d} v } { \mathrm {~d} x } = 10 - 0.5 v ^ { 2 }$.
Express $v$ in terms of $x$.
Find the increase in the value of $v$ during the final 4 m of the descent of the object.

CAIE M2 2017 June Q5

5 A particle of mass 0.3 kg is attached to one end of a light elastic string of natural length 0.8 m and modulus of elasticity 6 N . The other end of the string is attached to a fixed point $O$. The particle is projected vertically downwards from $O$ with initial speed $2 \mathrm {~m} \mathrm {~s} ^ { - 1 }$.

Calculate the greatest speed of the particle during its descent.
Find the greatest distance of the particle below $O$.
\includegraphics[max width=\textwidth, alt={}, center]{2b0425b2-2f8f-491a-996c-3d3b589bd7df-12_558_554_260_794} The end $A$ of a non-uniform rod $A B$ of length 0.6 m and weight 8 N rests on a rough horizontal plane, with $A B$ inclined at $60 ^ { \circ }$ to the horizontal. Equilibrium is maintained by a force of magnitude 3 N applied to the rod at $B$. This force acts at $30 ^ { \circ }$ above the horizontal in the vertical plane containing the rod (see diagram).
Find the distance of the centre of mass of the rod from $A$.
The 3 N force is removed, and the rod is held in equilibrium by a force of magnitude $P \mathrm {~N}$ applied at $B$, acting in the vertical plane containing the rod, at an angle of $30 ^ { \circ }$ below the horizontal.
Calculate $P$.
In one of the two situations described, the $\operatorname { rod } A B$ is in limiting equilibrium.
Find the coefficient of friction at $A$.
$7 \quad$ A particle $P$ is projected from a point $O$ with speed $V \mathrm {~m} \mathrm {~s} ^ { - 1 }$. At time $t \mathrm {~s}$ after projection the horizontal and vertically upwards displacements of $P$ from $O$ are $x \mathrm {~m}$ and $y \mathrm {~m}$ respectively. The equation of the trajectory of $P$ is $y = 2 x - \frac { 25 x ^ { 2 } } { V ^ { 2 } }$.
Write down the value of $\tan \theta$, where $\theta$ is the angle of projection of $P$.
When $t = 4 , P$ passes through the point $A$ where $x = y = a$.
Calculate $V$ and $a$.
Find the direction of motion of $P$ when it passes through $A$.

CAIE M2 2017 June Q1

\includegraphics[alt={},max width=\textwidth]{37a752de-04aa-4b65-bc44-a5f28d769902-04_376_713_260_715} \captionsetup{labelformat=empty} \caption{Fig. 1}

Calculate the modulus of elasticity of the elastic string.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{37a752de-04aa-4b65-bc44-a5f28d769902-05_348_488_262_826} \captionsetup{labelformat=empty} \caption{Fig. 2}
\end{figure} $P$ is released from rest by removing the 7 N force. In its subsequent motion $P$ first comes to instantaneous rest at a point where $B P = 0.3 \mathrm {~m}$ and the elastic string makes an angle of $30 ^ { \circ }$ with the horizontal (see Fig. 2).
Find the value of $m$.

CAIE M2 2017 June Q3

3
\includegraphics[max width=\textwidth, alt={}, center]{37a752de-04aa-4b65-bc44-a5f28d769902-06_351_607_269_769} An object is made from a uniform solid hemisphere of radius 0.56 m and centre $O$ by removing a hemisphere of radius 0.28 m and centre $O$. The diagram shows a cross-section through $O$ of the object.

Calculate the distance of the centre of mass of the object from $O$.
[0pt] [The volume of a hemisphere is $\frac { 2 } { 3 } \pi r ^ { 3 }$.]
The object has weight 24 N . A uniform hemisphere $H$ of radius 0.28 m is placed in the hollow part of the object to create a non-uniform hemisphere with centre $O$. The centre of mass of the non-uniform hemisphere is 0.15 m from $O$.
Calculate the weight of $H$.

CAIE M2 2017 June Q4

Express $x$ and $y$ in terms of $t$, and hence find the equation of the trajectory of the particle.
\includegraphics[max width=\textwidth, alt={}, center]{37a752de-04aa-4b65-bc44-a5f28d769902-08_63_1569_488_328}
The horizontal ground is at the top of a vertical cliff. The point $O$ is at a distance $d \mathrm {~m}$ from the edge of the cliff. The particle is projected towards the edge of the cliff and does not strike the ground before it passes over the edge of the cliff.
Show that $d$ is less than 30 .
Find the value of $x$ when the particle is 14 m below the level of $O$.
\includegraphics[max width=\textwidth, alt={}, center]{37a752de-04aa-4b65-bc44-a5f28d769902-10_501_614_258_762} A uniform semicircular lamina of radius 0.7 m and weight 14 N has diameter $A B$. The lamina is in a vertical plane with $A$ freely pivoted at a fixed point. The straight edge $A B$ rests against a small smooth peg $P$ above the level of $A$. The angle between $A B$ and the horizontal is $30 ^ { \circ }$ and $A P = 0.9 \mathrm {~m}$ (see diagram).
Show that the magnitude of the force exerted by the peg on the lamina is 7.12 N , correct to 3 significant figures.
Find the angle with the horizontal of the force exerted by the pivot on the lamina at $A$.

CAIE M2 2018 June Q1

1 A small ball $B$ is projected from a point $O$ on horizontal ground. The initial velocity of $B$ has horizontal and vertically upwards components of $18 \mathrm {~ms} ^ { - 1 }$ and $25 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ respectively. For the instant 4 s after projection, find the speed and direction of motion of $B$.
A non-uniform rod $A B$ of length 0.5 m and weight 8 N is freely hinged to a fixed point at $A$. The rod makes an angle of $30 ^ { \circ }$ with the horizontal with $B$ above the level of $A$. The rod is held in equilibrium by a force of magnitude 12 N acting in the vertical plane containing the rod at an angle of $30 ^ { \circ }$ to $A B$ applied at $B$ (see diagram). Find the distance of the centre of mass of the rod from $A$.

CAIE M2 2018 June Q3

3 A particle $P$ of mass 0.4 kg is projected horizontally along a smooth horizontal plane from a point $O$. At time $t \mathrm {~s}$ after projection the velocity of $P$ is $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$. A force of magnitude $0.8 t \mathrm {~N}$ directed away from $O$ acts on $P$ and a force of magnitude $2 \mathrm { e } ^ { - t } \mathrm {~N}$ opposes the motion of $P$.

Show that $\frac { \mathrm { d } v } { \mathrm {~d} t } = 2 t - 5 \mathrm { e } ^ { - t }$.
Given that $v = 8$ when $t = 1$, express $v$ in terms of $t$.
Find the speed of projection of $P$.

CAIE M2 2018 June Q4

4 A small object is projected from a point $O$ with speed $V \mathrm {~m} \mathrm {~s} ^ { - 1 }$ at an angle of $45 ^ { \circ }$ above the horizontal. At time $t \mathrm {~s}$ after projection, the horizontal and vertically upwards displacements of the object from $O$ are $x \mathrm {~m}$ and $y \mathrm {~m}$ respectively.

Express $x$ and $y$ in terms of $t$, and hence find the equation of the path.
The object passes through the point with coordinates $( 24,18 )$.
Find $V$.
The object passes through two points which are 22.5 m above the level of $O$. Find the values of $x$ for these points.

CAIE M2 2018 June Q5

5 A particle $P$ of mass 0.7 kg is attached by a light elastic string to a fixed point $O$ on a smooth plane inclined at an angle of $30 ^ { \circ }$ to the horizontal. The natural length of the string is 0.5 m and the modulus of elasticity is 20 N . The particle $P$ is projected up the line of greatest slope through $O$ from a point $A$ below the level of $O$. The initial kinetic energy of $P$ is 1.8 J and the initial elastic potential energy in the string is also 1.8 J .

Find the distance $O A$.
....................................................................................................................................
Find the greatest speed of $P$ in the motion.

CAIE M2 2018 June Q6

6
\includegraphics[max width=\textwidth, alt={}, center]{c403a227-586d-4c1f-a392-e475234fc0a0-10_262_732_264_705} A particle $P$ of mass 0.2 kg is attached to one end of a light inextensible string of length 0.6 m . The other end of the string is attached to a particle $Q$ of mass 0.3 kg . The string passes through a small hole $H$ in a smooth horizontal surface. A light elastic string of natural length 0.3 m and modulus of elasticity 15 N joins $Q$ to a fixed point $A$ which is 0.4 m vertically below $H$. The particle $P$ moves on the surface in a horizontal circle with centre $H$ (see diagram).

Calculate the greatest possible speed of $P$ for which the elastic string is not extended.
Find the distance $H P$ given that the angular speed of $P$ is $8 \mathrm { rad } \mathrm { s } ^ { - 1 }$.

CAIE M2 2018 June Q7

7
\includegraphics[max width=\textwidth, alt={}, center]{c403a227-586d-4c1f-a392-e475234fc0a0-12_732_581_260_774} A uniform solid cone has height 1.2 m and base radius 0.5 m . A uniform object is made by drilling a cylindrical hole of radius 0.2 m through the cone along the axis of symmetry (see diagram).

Show that the height of the object is 0.72 m and that the volume of the cone removed by the drilling is $0.0352 \pi \mathrm {~m} ^ { 3 }$.
[0pt] [The volume of a cone is $\frac { 1 } { 3 } \pi r ^ { 2 } h$.]
Find the distance of the centre of mass of the object from its base.
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 2018 June Q2

2 One end of a light elastic string is attached to a fixed point $O$. The other end of the string is attached to a particle $P$ of mass 0.4 kg . The string has natural length 0.6 m and modulus of elasticity 24 N . The particle is released from rest at $O$. Find the two possible values of the distance $O P$ for which the particle has speed $1.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }$.

CAIE M2 2018 June Q3

3
$A B C$ is an object made from a uniform wire consisting of two straight portions $A B$ and $B C$, in which $A B = a , B C = x$ and angle $A B C = 90 ^ { \circ }$. When the object is freely suspended from $A$ and in equilibrium, the angle between $A B$ and the horizontal is $\theta$ (see diagram).

Show that $x ^ { 2 } \tan \theta - 2 a x - a ^ { 2 } = 0$.
Given that $\tan \theta = 1.25$, calculate the length of the wire in terms of $a$.

CAIE M2 2018 June Q4

4 A particle $P$ is projected from a point $O$ on horizontal ground with initial speed $20 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ and angle of projection $30 ^ { \circ }$. At the instant $t s$ after projection, the horizontal and vertically upwards displacements of $P$ from $O$ are $x \mathrm {~m}$ and $y \mathrm {~m}$ respectively.

Express $x$ and $y$ in terms of $t$ and hence find the equation of the trajectory of $P$.
$P$ is at the same height above the ground at two points which are a horizontal distance apart of 15 m .
Calculate this height.
\includegraphics[max width=\textwidth, alt={}, center]{874622ab-4c75-4a32-bae5-eef780ed0cc0-08_607_1022_255_573} A uniform object is made by joining a solid cone of height 0.8 m and base radius 0.6 m and a cylinder. The cylinder has length 0.4 m and radius 0.5 m . The cylinder has a cylindrical hole of length 0.4 m and radius $x \mathrm {~m}$ drilled through it along the axis of symmetry. A plane face of the cylinder is attached to the base of the cone so that the object has an axis of symmetry perpendicular to its base and passing through the vertex of the cone. The object is placed with points on the base of the cone and the base of the cylinder in contact with a horizontal surface (see diagram). The object is on the point of toppling.
Show that the centre of mass of the object is 0.15 m from the base of the cone.
Find $x$.
[0pt] [The volume of a cone is $\frac { 1 } { 3 } \pi r ^ { 2 } h$.]

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