Questions M2 - A-Level Maths

Find the value of $h$.
Given that $r = 10$, find the value of $\alpha$.

CAIE M2 2008 November Q3

3 A particle $P$ of mass 0.5 kg moves along the $x$-axis on a horizontal surface. When the displacement of $P$ from the origin $O$ is $x \mathrm {~m}$ the velocity of $P$ is $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$ in the positive $x$-direction. Two horizontal forces act on $P$; one force has magnitude $\left( 1 + 0.3 x ^ { 2 } \right) \mathrm { N }$ and acts in the positive $x$-direction, and the other force has magnitude $8 \mathrm { e } ^ { - x } \mathrm {~N}$ and acts in the negative $x$-direction.

Show that $v \frac { \mathrm {~d} v } { \mathrm {~d} x } = 2 + 0.6 x ^ { 2 } - 16 \mathrm { e } ^ { - x }$.
The velocity of $P$ as it passes through $O$ is $6 \mathrm {~ms} ^ { - 1 }$. Find the velocity of $P$ when $x = 3$.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{5109244c-3062-4f5f-9277-fc6b5b28f2d4-3_259_745_278_740} \captionsetup{labelformat=empty} \caption{Fig. 1}
\end{figure} A small sphere $A$ of mass 0.15 kg is moving inside a fixed smooth hollow cylinder whose axis is vertical. $A$ moves with constant speed $1.2 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ in a horizontal circle of radius 0.35 m , and is continuously in contact with both the plane base and the curved surface of the cylinder. Fig. 1 shows a vertical cross-section of the cylinder through its axis. Find the magnitude of the force exerted on $A$ by

CAIE M2 2008 November Q5

5
\includegraphics[max width=\textwidth, alt={}, center]{5109244c-3062-4f5f-9277-fc6b5b28f2d4-4_495_1405_264_370}
$A B C D$ is a central cross-section of a uniform rectangular block of mass 35 kg . The lengths of $A B$ and $B C$ are 1.2 m and 0.8 m respectively. The block is held in equilibrium by a rope, one end of which is attached to the point $E$ of a rough horizontal floor. The other end of the rope is attached to the block at $A$. The rope is in the same vertical plane as $A B C D$, and $E A B$ is a straight line making an angle of $20 ^ { \circ }$ with the horizontal (see diagram).

Show that the tension in the rope is 187 N , correct to the nearest whole number.
The block is on the point of slipping. Find the coefficient of friction between the block and the floor.

CAIE M2 2008 November Q6

6 A light elastic string has natural length 4 m and modulus of elasticity 2 N . One end of the string is attached to a fixed point $O$ of a smooth plane which is inclined at $30 ^ { \circ }$ to the horizontal. The other end of the string is attached to a particle $P$ of mass $0.1 \mathrm {~kg} . P$ is held at rest at $O$ and then released. The speed of $P$ is $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$ when the extension of the string is $x \mathrm {~m}$.

Show that $v ^ { 2 } = 45 - 5 ( x - 1 ) ^ { 2 }$. Hence find
the distance of $P$ from $O$ when $P$ is at its lowest point,
the maximum speed of $P$.

CAIE M2 2008 November Q7

7 A particle $P$ is projected from a point $O$ on horizontal ground with speed $V \mathrm {~m} \mathrm {~s} ^ { - 1 }$ and direction $60 ^ { \circ }$ upwards from the horizontal. At time $t \mathrm {~s}$ later the horizontal and vertical displacements of $P$ from $O$ are $x \mathrm {~m}$ and $y \mathrm {~m}$ respectively.

Write down expressions for $x$ and $y$ in terms of $V$ and $t$ and hence show that the equation of the trajectory of $P$ is $$y = ( \sqrt { } 3 ) x - \frac { 20 x ^ { 2 } } { V ^ { 2 } }$$ $P$ passes through the point $A$ at which $x = 70$ and $y = 10$. Find
the value of $V$,
the direction of motion of $P$ at the instant it passes through $A$.

CAIE M2 2009 November Q1

1
\includegraphics[max width=\textwidth, alt={}, center]{68acf474-5da2-4949-b3b2-fc42cd73bd4a-2_113_787_264_680} A light elastic spring of natural length 0.25 m and modulus of elasticity 100 N is held horizontally between two parallel plates. The axis of the spring is at right angles to each of the plates. The horizontal force exerted on the spring by each of the plates is 20 N (see diagram). Find the amount by which the spring is compressed and hence write down the distance between the plates.

CAIE M2 2009 November Q2

2 A particle of mass 0.2 kg is attached to one end of a light elastic string of natural length 0.6 m and modulus of elasticity 4 N . The other end of the string is attached to a fixed point $O$. The particle is held at a point which is $( 0.6 + x ) \mathrm { m }$ vertically below $O$. The particle is released from rest. In the subsequent motion the speed of the particle is $3 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ when the string becomes slack. By considering energy, find the value of $x$.

CAIE M2 2009 November Q3

3 \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{68acf474-5da2-4949-b3b2-fc42cd73bd4a-2_408_291_1027_927} \captionsetup{labelformat=empty} \caption{Fig. 1}

\end{figure} A uniform solid cylinder has mass 8 kg and height 16 cm . A uniform solid cone, whose base radius is the same as the radius of the cylinder, has mass 2 kg and height 12 cm . A composite solid is formed by joining the cylinder and cone so that the base of the cone coincides with one end of the cylinder (see Fig. 1).

Show that the centre of mass of the composite solid is 10.2 cm from its base. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{68acf474-5da2-4949-b3b2-fc42cd73bd4a-2_401_444_1877_849} \captionsetup{labelformat=empty} \caption{Fig. 2}
\end{figure} The composite solid is held with a point on the circumference of its base in contact with a horizontal table. The base makes an angle $\theta ^ { \circ }$ with the table (see Fig. 2, which shows a cross-section). When the cone is released it moves towards the equilibrium position in which its base is in contact with the table.
Given that the radius of the base is 4 cm , find the greatest possible value of $\theta$, correct to 1 decimal place.

CAIE M2 2009 November Q4

4 A particle is projected from a point $O$ with speed $V \mathrm {~m} \mathrm {~s} ^ { - 1 }$ at an angle $\theta$ above the horizontal. After 0.3 s the particle is moving with speed $25 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ at an angle $\tan ^ { - 1 } \left( \frac { 7 } { 24 } \right)$ above the horizontal.

Show that $V \cos \theta = 24$.
Find the value of $V \sin \theta$, and hence find $V$ and $\theta$.

CAIE M2 2009 November Q5

5
\includegraphics[max width=\textwidth, alt={}, center]{68acf474-5da2-4949-b3b2-fc42cd73bd4a-3_405_545_630_799} A uniform lamina $A O B$ is in the shape of a sector of a circle with centre $O$ and radius 0.5 m , and has angle $A O B = \frac { 1 } { 3 } \pi$ radians and weight 3 N . The lamina is freely hinged at $O$ to a fixed point and is held in equilibrium with $A O$ vertical by a force of magnitude $F \mathrm {~N}$ acting at $B$. The direction of this force is at right angles to $O B$ (see diagram). Find

the value of $F$,
the magnitude of the force acting on the lamina at $O$.

CAIE M2 2009 November Q6

6
\includegraphics[max width=\textwidth, alt={}, center]{68acf474-5da2-4949-b3b2-fc42cd73bd4a-3_504_878_1557_632} One end of a light inextensible string of length 0.7 m is attached to a fixed point $A$. The other end of the string is attached to a particle $P$ of mass 0.25 kg . The particle $P$ moves in a circle on a smooth horizontal table with constant speed $1.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }$. The string is taut and makes an angle of $40 ^ { \circ }$ with the vertical (see diagram). Find

the tension in the string,
the force exerted on $P$ by the table.
$P$ now moves in the same horizontal circle with constant angular speed $\omega \operatorname { rad~s } ^ { - 1 }$.
Find the maximum value of $\omega$ for which $P$ remains on the table.

CAIE M2 2009 November Q7

7 A particle $P$ of mass 0.1 kg is projected vertically upwards from a point $O$ with speed $20 \mathrm {~m} \mathrm {~s} ^ { - 1 }$. Air resistance of magnitude $0.1 v \mathrm {~N}$ opposes the motion, where $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$ is the speed of $P$ at time $t \mathrm {~s}$ after projection.

Show that, while $P$ is moving upwards, $\frac { 1 } { v + 10 } \frac { \mathrm {~d} v } { \mathrm {~d} t } = - 1$.
Hence find an expression for $v$ in terms of $t$, and explain why it is valid only for $0 \leqslant t \leqslant \ln 3$.
Find the initial acceleration of $P$. \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 2009 November Q3

3
\includegraphics[max width=\textwidth, alt={}, center]{fe5c198d-5d05-4241-98f5-894ba92f7afe-3_408_1164_248_493} A particle $P$ is released from rest at a point $A$ which is 7 m above horizontal ground. At the same instant that $P$ is released a particle $Q$ is projected from a point $O$ on the ground. The horizontal distance of $O$ from $A$ is 24 m . Particle $Q$ moves in the vertical plane containing $O$ and $A$, with initial speed $50 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ and initial direction making an angle $\theta$ above the horizontal, where $\tan \theta = \frac { 7 } { 24 }$ (see diagram). Show that the particles collide.

CAIE M2 2009 November Q4

4 One end of a light elastic string of natural length 3 m and modulus of elasticity 15 mN is attached to a fixed point $O$. A particle $P$ of mass $m \mathrm {~kg}$ is attached to the other end of the string. $P$ is released from rest at $O$ and moves vertically downwards. When the extension of the string is $x \mathrm {~m}$ the velocity of $P$ is $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$.

Show that $v ^ { 2 } = 5 \left( 12 + 4 x - x ^ { 2 } \right)$.
Find the magnitude of the acceleration of $P$ when it is at its lowest point, and state the direction of this acceleration.

CAIE M2 2009 November Q5

5
\includegraphics[max width=\textwidth, alt={}, center]{fe5c198d-5d05-4241-98f5-894ba92f7afe-3_593_828_1530_660} A horizontal disc of radius 0.5 m is rotating with constant angular speed $\omega \mathrm { rad } \mathrm { s } ^ { - 1 }$ about a fixed vertical axis through its centre $O$. One end of a light inextensible string of length 0.8 m is attached to a point $A$ of the circumference of the disc. A particle $P$ of mass 0.4 kg is attached to the other end of the string. The string is taut and the system rotates so that the string is always in the same vertical plane as the radius $O A$ of the disc. The string makes a constant angle $\theta$ with the vertical (see diagram). The speed of $P$ is 1.6 times the speed of $A$.

Show that $\sin \theta = \frac { 3 } { 8 }$.
Find the tension in the string.
Find the value of $\omega$.

CAIE M2 2009 November Q6

6
\includegraphics[max width=\textwidth, alt={}, center]{fe5c198d-5d05-4241-98f5-894ba92f7afe-4_447_736_269_701}
$P$ is the vertex of a uniform solid cone of mass 5 kg , and $O$ is the centre of its base. Strings are attached to the cone at $P$ and at $O$. The cone hangs in equilibrium with $P O$ horizontal and the strings taut. The strings attached at $P$ and $O$ make angles of $\theta ^ { \circ }$ and $20 ^ { \circ }$, respectively, with the vertical (see diagram, which shows a cross-section).

By taking moments about $P$ for the cone, find the tension in the string attached at $O$.
Find the value of $\theta$ and the tension in the string attached at $P$.

CAIE M2 2009 November Q7

7 A particle $P$ of mass 0.3 kg is projected vertically upwards from the ground with an initial speed of $20 \mathrm {~m} \mathrm {~s} ^ { - 1 }$. When $P$ is at height $x \mathrm {~m}$ above the ground, its upward speed is $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$. It is given that $$3 v - 90 \ln ( v + 30 ) + x = A ,$$ where $A$ is a constant.

Differentiate this equation with respect to $x$ and hence show that the acceleration of the particle is $- \frac { 1 } { 3 } ( v + 30 ) \mathrm { m } \mathrm { s } ^ { - 2 }$.
Find, in terms of $v$, the resisting force acting on the particle.
Find the time taken for $P$ to reach its maximum height. \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

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

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

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).

Given that $v = 1.5$, calculate the magnitude of the force that the surface exerts on $P$.
Given instead that $P$ moves with its greatest possible speed while remaining in contact with the surface, find $v$.

CAIE M2 2010 November Q4

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).

Find the tension in the rope.
Find the direction of the force exerted on the beam at $A$.

CAIE M2 2010 November Q5

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 $A B$.

Show that the modulus of elasticity of the string is 20 N .
Calculate the maximum speed of $P$.

CAIE M2 2010 November Q6

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.

Show that $\frac { 9 } { 15 - v } \frac { \mathrm {~d} v } { \mathrm {~d} t } = 1$.
Solve this differential equation to show that $v = 15 \left( 1 - \mathrm { e } ^ { - \frac { 1 } { 9 } t } \right)$.
Find the distance travelled by the cyclist in the first 9 s of the motion.

CAIE M2 2010 November Q7

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.

Write down expressions for $x$ and $y$ in terms of $t$, and hence obtain the equation of the trajectory of $P$.
Calculate the value of $x$ when $P$ is at $A$.
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. }

Questions M2 (1391 questions)

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