Questions - CAIE M2 - A-Level Maths

CAIE M2 2016 November Q2

2
\includegraphics[max width=\textwidth, alt={}, center]{f9d76b90-9786-4a35-8f94-ffa7b18622b6-2_318_488_484_826} A uniform wire is bent to form an object which has a semicircular arc with diameter $A B$ of length 1.2 m , with a smaller semicircular arc with diameter $B C$ of length 0.6 m . The end $C$ of the smaller arc is at the centre of the larger arc (see diagram). The two semicircular arcs of the wire are in the same plane.

Show that the distance of the centre of mass of the object from the line $A C B$ is 0.191 m , correct to 3 significant figures. The object is freely suspended at $A$ and hangs in equilibrium.
Find the angle between $A C B$ and the vertical.

CAIE M2 2016 November Q4

4
\includegraphics[max width=\textwidth, alt={}, center]{f9d76b90-9786-4a35-8f94-ffa7b18622b6-3_388_650_264_749} The diagram shows the cross-section $A B C D$ through the centre of mass of a uniform solid prism. $A B = 0.9 \mathrm {~m} , B C = 2 a \mathrm {~m} , A D = a \mathrm {~m}$ and angle $A B C =$ angle $B A D = 90 ^ { \circ }$.

Calculate the distance of the centre of mass of the prism from $A D$.
Express the distance of the centre of mass of the prism from $A B$ in terms of $a$. The prism has weight 18 N and rests in equilibrium on a rough horizontal surface, with $A D$ in contact with the surface. A horizontal force of magnitude 6 N is applied to the prism. This force acts through the centre of mass in the direction $B C$.
Given that the prism is on the point of toppling, calculate $a$.

CAIE M2 2016 November Q6

6
\includegraphics[max width=\textwidth, alt={}, center]{f9d76b90-9786-4a35-8f94-ffa7b18622b6-3_483_419_1800_863} The diagram shows a smooth narrow tube formed into a fixed vertical circle with centre $O$ and radius 0.9 m . A light elastic string with modulus of elasticity 8 N and natural length 1.2 m has one end attached to the highest point $A$ on the inside of the tube. The other end of the string is attached to a particle $P$ of mass 0.2 kg . The particle is released from rest at the lowest point on the inside of the tube. By considering energy, calculate

the speed of $P$ when it is at the same horizontal level as $O$,
the speed of $P$ at the instant when the string becomes slack.

CAIE M2 2017 November Q1

1
\includegraphics[max width=\textwidth, alt={}, center]{8a7016eb-4e76-4104-aa00-fbf09e1d739a-02_560_421_258_861} A hollow cylinder with a rough inner surface has radius 0.5 m . A particle $P$ of mass 0.4 kg is in contact with the inner surface of the cylinder. The particle and cylinder rotate together with angular speed $6 \mathrm { rad } \mathrm { s } ^ { - 1 }$ about the vertical axis of the cylinder, so that the particle moves in a horizontal circle (see diagram). Given that $P$ is about to slip downwards, find the coefficient of friction between $P$ and the surface of the cylinder.

CAIE M2 2017 November Q2

2 A small ball is projected from a point 1.5 m above horizontal ground. At a point 9 m above the ground the ball is travelling at $45 ^ { \circ }$ above the horizontal and its velocity is $4 \mathrm {~m} \mathrm {~s} ^ { - 1 }$. Find the angle of projection of the ball.

CAIE M2 2017 November Q3

3
\includegraphics[max width=\textwidth, alt={}, center]{8a7016eb-4e76-4104-aa00-fbf09e1d739a-04_305_510_264_813} One end of a light inextensible string of length 0.4 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 $60 ^ { \circ }$ with the horizontal (see diagram).

Given that $v = 0.5$, calculate the magnitude of the force that the surface exerts on $P$.
Find the greatest possible value of $v$ for which $P$ remains in contact with the surface.

CAIE M2 2017 November Q4

4 A particle $P$ is projected with speed $25 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ at an angle of $30 ^ { \circ }$ above the horizontal from a point $O$ on horizontal ground. 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.

Express $x$ and $y$ in terms of $t$ and hence show that the equation of the trajectory of $P$ is $$y = \frac { x } { \sqrt { 3 } } - \frac { 4 x ^ { 2 } } { 375 }$$
Find the horizontal distance between the two points at which $P$ is 5 m above the ground.

CAIE M2 2017 November Q5

5 One end of a light elastic string of natural length 0.8 m and modulus of elasticity 24 N is attached to a fixed point $O$. The other end of the string is attached to a particle $P$ of mass $0.3 \mathrm {~kg} . P$ is projected vertically upwards with speed $4 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ from a position 1.2 m vertically below $O$.

Calculate the speed of the particle at the position where it is moving with zero acceleration. [5
Show that the particle moves 1.2 m while moving upwards with constant deceleration.

CAIE M2 2017 November Q6

6 A solid object consists of a uniform hemisphere of radius 0.4 m attached to a uniform cylinder of radius 0.4 m so that the circumferences of their circular faces coincide. The hemisphere and cylinder each have weight 20 N . The centre of mass of the object lies at the centre $O$ of their common circular face.

Show that the height of the cylinder is 0.3 m .
A new object is made by cutting the cylinder in half and removing the half not attached to the hemisphere. The cut is perpendicular to the axis of symmetry, so the new object consists of a hemisphere and a cylinder half the height of the original cylinder.
Find the distance of the centre of mass of the new object from $O$.
The new object is placed with its hemispherical part on a rough horizontal surface. The new object is held in equilibrium by a force of magnitude $P \mathrm {~N}$ acting along its axis of symmetry, which is inclined at $30 ^ { \circ }$ to the horizontal.
Find $P$.

CAIE M2 2017 November Q7

7 A particle $P$ of mass 0.2 kg is released from rest at a point $O$ on a rough plane inclined at $60 ^ { \circ }$ to the horizontal, and travels down a line of greatest slope. The coefficient of friction between $P$ and the plane is 0.3 . A force of magnitude $0.6 x \mathrm {~N}$ acts on $P$ in the direction $P O$, where $x \mathrm {~m}$ is the displacement of $P$ from $O$.

Show that $v \frac { \mathrm {~d} v } { \mathrm {~d} x } = 5 \sqrt { } 3 - 1.5 - 3 x$, where $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$ is the velocity of $P$ at a displacement $x \mathrm {~m}$ from $O$.
Find the value of $x$ for which $P$ reaches its maximum velocity, and calculate this maximum velocity.
Calculate the magnitude of the acceleration of $P$ immediately after it has first come to instantaneous rest.

CAIE M2 2017 November Q1

1 A particle $P$ of mass 0.2 kg is released from rest at a point $O$ on a smooth horizontal surface. A horizontal force of magnitude $t \mathrm { e } ^ { - v } \mathrm {~N}$ directed away from $O$ acts on $P$, where $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$ is the velocity of $P$ at time $t \mathrm {~s}$ after release. Find the velocity of $P$ when $t = 2$.

CAIE M2 2017 November Q2

2 A uniform solid cone has height 0.6 m and base radius 0.2 m . A uniform hollow cylinder, open at both ends, has the same dimensions. An object is made by putting the cone inside the cylinder so that the base of the cone coincides with one end of the cylinder (see diagram, which shows a cross-section). The total weight of the object is 60 N and its centre of mass is 0.25 m from the base of the cone. Calculate the weight of the cone.

CAIE M2 2017 November 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. $P$ moves down the line of greatest slope through $O$. The velocity of $P$ is $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$ when its displacement from $O$ is $x \mathrm {~m}$. A retarding force of magnitude $0.2 v ^ { 2 } \mathrm {~N}$ acts on $P$ in the direction $P O$.

Show that $v \frac { \mathrm {~d} v } { \mathrm {~d} x } = 5 - 0.5 v ^ { 2 }$.
Express $v$ in terms of $x$.

CAIE M2 2017 November Q4

4
\includegraphics[max width=\textwidth, alt={}, center]{6b220343-1d64-4dbc-a42d-77967eef9c6d-06_264_839_260_653} A light elastic string has natural length 2 m and modulus of elasticity 39 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 $m \mathrm {~kg}$ is attached to the mid-point of the string and hangs in equilibrium at a point 0.5 m below $A B$ (see diagram).

Show that $m = 0.9$.
$P$ is projected vertically downwards from the equilibrium position, and comes to instantaneous rest at a point 1.6 m below $A B$.
Calculate the speed of projection of $P$.

CAIE M2 2017 November Q5

5
\includegraphics[max width=\textwidth, alt={}, center]{6b220343-1d64-4dbc-a42d-77967eef9c6d-08_449_890_262_630}
$O A B$ is a uniform lamina in the shape of a quadrant of a circle with centre $O$ and radius 0.8 m which has its centre of mass at $G$. The lamina is smoothly hinged at $A$ to a fixed point and is free to rotate in a vertical plane. A horizontal force of magnitude 12 N acting in the plane of the lamina is applied to the lamina at $B$. The lamina is in equilibrium with $A G$ horizontal (see diagram).

Calculate the length $A G$.
Find the weight of the lamina.

CAIE M2 2017 November Q6

6 One end of a light elastic string of natural length 0.4 m and modulus of elasticity 8 N is attached to a fixed point $O$ on a smooth horizontal plane. The other end of the string is attached to a particle $P$ of mass 0.2 kg which moves on the plane in a circular path with centre $O$. The speed of $P$ is $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$ and the extension of the string is $x \mathrm {~m}$.

Given that $v = 2.5$, find $x$.
It is given instead that the kinetic energy of $P$ is twice the elastic potential energy stored in the string.
Form two simultaneous equations and hence find $x$ and $v$.

CAIE M2 2017 November Q7

7 A small ball $B$ is projected from a point $O$ which is $h \mathrm {~m}$ above a horizontal plane. At time 2 s after projection $B$ has speed $18 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ and is moving in the direction $30 ^ { \circ }$ above the horizontal.

Find the initial speed and the angle of projection of $B$.
$B$ has speed $38 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ immediately before it strikes the plane.
Calculate $h$.
$B$ bounces when it strikes the plane, and leaves the plane with speed $20 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ but with its horizontal component of velocity unchanged.
Find the total time which elapses between the initial projection of $B$ and the instant when it strikes the plane for the second time.

CAIE M2 2017 November Q3

3
\includegraphics[max width=\textwidth, alt={}, center]{2e4e6e32-eafc-4196-aaa8-42909cc2078e-04_305_510_264_813} One end of a light inextensible string of length 0.4 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 $60 ^ { \circ }$ with the horizontal (see diagram).

Given that $v = 0.5$, calculate the magnitude of the force that the surface exerts on $P$.
Find the greatest possible value of $v$ for which $P$ remains in contact with the surface.

CAIE M2 2018 November Q1

4 marks

1 A small ball $B$ is projected with speed $30 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ at an angle of $60 ^ { \circ }$ to the horizontal from a point on horizontal ground. Find the time after projection when the speed of $B$ is $25 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ for the second time.
[0pt] [4]
\includegraphics[max width=\textwidth, alt={}, center]{ef38fda2-230b-431a-8064-82e4a3bff393-04_620_668_255_742} A uniform object is made by attaching the base of a solid hemisphere to the base of a solid cone so that the object has an axis of symmetry. The base of the cone has radius 0.3 m , and the hemisphere has radius 0.2 m . The object is placed on a horizontal plane with a point $A$ on the curved surface of the hemisphere and a point $B$ on the circumference of the cone in contact with the plane (see diagram).

Given that the object is on the point of toppling about $B$, find the distance of the centre of mass of the object from the base of the cone.
Given instead that the object is on the point of toppling about $A$, calculate the height of the cone.
[0pt] [The volume of a cone is $\frac { 1 } { 3 } \pi r ^ { 2 } h$. The volume of a hemisphere is $\frac { 2 } { 3 } \pi r ^ { 3 }$.]

CAIE M2 2018 November Q3

3 A particle $P$ of mass 0.4 kg is attached to a fixed point $O$ by a light elastic string of natural length 0.5 m and modulus of elasticity 20 N . The particle $P$ is released from rest at $O$.

Find the greatest speed of $P$ in the subsequent motion.
Find the distance below $O$ of the point at which $P$ comes to instantaneous rest.

CAIE M2 2018 November Q4

4
\includegraphics[max width=\textwidth, alt={}, center]{ef38fda2-230b-431a-8064-82e4a3bff393-08_152_885_262_630} A particle $P$ of mass 0.5 kg is projected along a smooth horizontal surface towards a fixed point $A$. Initially $P$ is at a point $O$ on the surface, and after projection, $P$ has a displacement from $O$ of $x \mathrm {~m}$ and velocity $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$. The particle $P$ is connected to $A$ by a light elastic string of natural length 0.8 m and modulus of elasticity 16 N . The distance $O A$ is 1.6 m (see diagram). The motion of $P$ is resisted by a force of magnitude $24 x ^ { 2 } \mathrm {~N}$.

Show that $v \frac { \mathrm {~d} v } { \mathrm {~d} x } = 32 - 40 x - 48 x ^ { 2 }$ while $P$ is in motion and the string is stretched.
The maximum value of $v$ is 4.5 .
Find the initial value of $v$.

CAIE M2 2018 November Q5

5 A particle $P$ of mass 0.1 kg is attached to one end of a light inextensible string of length 0.5 m . The other end of the string is attached to a fixed point $A$. The particle $P$ moves in a circle which has its centre $O$ on a smooth horizontal surface 0.3 m below $A$. The tension in the string has magnitude $T \mathrm {~N}$ and the magnitude of the force exerted on $P$ by the surface is $R \mathrm {~N}$.

Given that the speed of $P$ is $1.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }$, calculate $T$ and $R$.
Given instead that $T = R$, calculate the angular speed of $P$.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{ef38fda2-230b-431a-8064-82e4a3bff393-12_449_621_260_762} \captionsetup{labelformat=empty} \caption{Fig. 1}
\end{figure} Fig. 1 shows the cross-section $A B C D E$ through the centre of mass $G$ of a uniform prism. The crosssection consists of a rectangle $A B C F$ from which a triangle $D E F$ has been removed; $A B = 0.6 \mathrm {~m}$, $B C = 0.7 \mathrm {~m}$ and $D F = E F = 0.3 \mathrm {~m}$.
Show that the distance of $G$ from $B C$ is 0.276 m , and find the distance of $G$ from $A B$.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{ef38fda2-230b-431a-8064-82e4a3bff393-13_494_583_258_781} \captionsetup{labelformat=empty} \caption{Fig. 2}
\end{figure} The prism is placed with $C D$ on a rough horizontal surface. A force of magnitude 2 N acting in the plane of the cross-section is applied to the prism. The line of action of the force passes through $G$ and is perpendicular to $D E$ (see Fig. 2). The prism is on the point of toppling about the edge through $D$.
Calculate the weight of the prism.
\includegraphics[max width=\textwidth, alt={}, center]{ef38fda2-230b-431a-8064-82e4a3bff393-14_512_520_258_817} A small object is projected with speed $24 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ from a point $O$ at the foot of a plane inclined at $45 ^ { \circ }$ to the horizontal. The angle of projection of the object is $15 ^ { \circ }$ above a line of greatest slope of the plane (see diagram). 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 value of $t$ for the instant when the object strikes the plane.
Express the vertical height of the object above the plane in terms of $t$ and hence find the greatest vertical height of the object above the plane.
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CAIE M2 2018 November Q1

1 A small ball $B$ is projected with speed $38 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ at an angle of $30 ^ { \circ }$ to the horizontal from a point on horizontal ground. Find the speed of $B$ when the path of $B$ makes an angle of $20 ^ { \circ }$ above the horizontal.

CAIE M2 2018 November Q2

2 A uniform solid object is made by attaching a cone to a cylinder so that the circumferences of the base of the cone and a plane face of the cylinder coincide. The cone and the cylinder each have radius 0.3 m and height 0.4 m .

Calculate the distance of the centre of mass of the object from the vertex of the cone.
[0pt] [The volume of a cone is $\frac { 1 } { 3 } \pi r ^ { 2 } h$.]
The object has weight $W \mathrm {~N}$ and is placed with its plane circular face on a rough horizontal surface. A force of magnitude $k W \mathrm {~N}$ acting at $30 ^ { \circ }$ to the upward vertical is applied to the vertex of the cone. The object does not slip.
Find the greatest possible value of $k$ for which the object does not topple.

CAIE M2 2018 November Q3

3 A particle $P$ of mass 0.4 kg is projected horizontally along a smooth horizontal plane from a point $O$. After projection the velocity of $P$ is $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$ and its displacement from $O$ is $x \mathrm {~m}$. A force of magnitude $8 x \mathrm {~N}$ directed away from $O$ acts on $P$ and a force of magnitude ( $2 \mathrm { e } ^ { - x } + 4$ ) N opposes the motion of $P$. One end of a light elastic string of natural length 0.5 m is attached to $O$ and the other end of the string is attached to $P$.

Show that $v \frac { \mathrm {~d} v } { \mathrm {~d} x } = 20 x - 10 - 5 \mathrm { e } ^ { - x }$ before the elastic string becomes taut.
Given that the initial velocity of $P$ is $6 \mathrm {~m} \mathrm {~s} ^ { - 1 }$, find $v$ when the string first becomes taut.
When the string is taut, the acceleration of $P$ is proportional to $\mathrm { e } ^ { - x }$.
Find the modulus of elasticity of the string.

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