Questions M2 - A-Level Maths

Calculate the height, above the horizontal surface, of the centre of mass of the cone with the particle.
$P$ is set in motion, and moves with angular speed $4 \mathrm { rad } \mathrm { s } ^ { - 1 }$ in a circular path on the surface of the cone.
Show that the tension in the string is 1.96 N , and calculate the magnitude of the force exerted on $P$ by the cone.
Find the speed of $P$.

CAIE M2 2013 June Q7

7 A particle $P$ of mass 0.5 kg moves in a straight line on a smooth horizontal surface. The velocity of $P$ is $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$ when the displacement of $P$ from $O$ is $x \mathrm {~m}$. A single horizontal force of magnitude $0.16 \mathrm { e } ^ { x } \mathrm {~N}$ acts on $P$ in the direction $O P$. The velocity of $P$ when it is at $O$ is $0.8 \mathrm {~m} \mathrm {~s} ^ { - 1 }$.

Show that $v = 0.8 \mathrm { e } ^ { \frac { 1 } { 2 } x }$.
Find the time taken by $P$ to travel 1.4 m from $O$.

CAIE M2 2013 June Q4

4
\includegraphics[max width=\textwidth, alt={}, center]{3607beed-0b98-46aa-8d95-403b05446a63-2_622_496_1121_804} A ll l $l \quad x \quad x v l \quad O \quad l$
xl $l \quad x$
$l$
$k v l$ \section*{$x$} $l$
$l$

l l
$l$
Cll
$x$
$l v$

CAIE M2 2013 June Q5

5 Oll
$x$
vll $l$

$l x l$
vl $l$
vll $l$
$x l$
(a)
(b)
(c)
$x l$
Cll

CAIE M2 2013 June Q6

6
6
$\begin{array} { l l l l } A & l & & v l
\text { lll } & l & l \end{array}$
l l l $l$
l ll l

v $$l v \quad l$$ l ll

\multirow[t]{3}{*}{A ll ll $l$}

\multirow{3}{*}{}

\multirow[t]{2}{*}{$k v$}

\multirow[t]{5}{*}{(i)}

\multirow[b]{5}{*}{

}




(ii)	$x$ vl Cll	$v$	$l l$	$l$

CAIE M2 2014 June Q2

2
\includegraphics[max width=\textwidth, alt={}, center]{edb816ee-741a-4d2d-8d60-f2a801005b1e-2_85_86_228_255}
\includegraphics[max width=\textwidth, alt={}, center]{edb816ee-741a-4d2d-8d60-f2a801005b1e-2_560_483_434_831}
$\theta \quad \theta \quad \theta$

$\theta$
$\theta$
$\theta$
$\theta$

CAIE M2 2014 June Q5

5
\includegraphics[max width=\textwidth, alt={}, center]{edb816ee-741a-4d2d-8d60-f2a801005b1e-3_350_611_276_767}
$\theta$
$\theta$

$\theta$

CAIE M2 2014 June Q6

$\theta$
$\theta$
$\theta$
[0pt] [uestion 7 is printed on the next page.]

CAIE M2 2014 June Q7

7
\includegraphics[max width=\textwidth, alt={}, center]{edb816ee-741a-4d2d-8d60-f2a801005b1e-4_342_732_278_683}

\multirow[t]{3}{*}{P m} m m v
C m v \multirow{5}{*}{
m
n
} \multirow{5}{*}{}
m m m
C m m C m A m C m A m \multirow{3}{*}{} \multirow[t]{3}{*}{} \multirow[t]{2}{*}{C m}

CAIE M2 2014 June Q2

2
\includegraphics[max width=\textwidth, alt={}, center]{3788852a-cd1a-49ae-be79-f365894bfa71-2_85_86_228_255}
\includegraphics[max width=\textwidth, alt={}, center]{3788852a-cd1a-49ae-be79-f365894bfa71-2_560_483_434_831}
$\theta \quad \theta \quad \theta$

$\theta$
$\theta$

CAIE M2 2014 June Q5

5
\includegraphics[max width=\textwidth, alt={}, center]{3788852a-cd1a-49ae-be79-f365894bfa71-3_350_611_276_767}
$\theta$
$\theta$

$\theta$

CAIE M2 2014 June Q7

7
\includegraphics[max width=\textwidth, alt={}, center]{3788852a-cd1a-49ae-be79-f365894bfa71-4_342_732_278_683}

\multirow[t]{3}{*}{P m} m m v
C m v \multirow{5}{*}{
m
n
} \multirow{5}{*}{}
m m m
C m m C m A m C m A m \multirow{3}{*}{} \multirow[t]{3}{*}{} \multirow[t]{2}{*}{C m}

CAIE M2 2014 June Q1

1 A light elastic string has modulus of elasticity 5 N and natural length 1.5 m . One end of the string is attached to a fixed point $O$ and a particle $P$ of mass 0.1 kg is attached to the other end of the string. $P$ is released from rest at the point 2.4 m vertically below $O$. Calculate the speed of $P$ at the instant the string first becomes slack.

CAIE M2 2014 June Q2

2
\includegraphics[max width=\textwidth, alt={}, center]{9c82b387-8e5e-48b9-973d-5337b4e56a66-2_536_905_520_621} A uniform lamina $A B C$ in the shape of an isosceles triangle has weight 24 N . The perpendicular distance from $A$ to $B C$ is 12 cm . The lamina rests in a vertical plane in equilibrium, with the vertex $A$ in contact with a horizontal surface. Angle $B A C = 100 ^ { \circ }$ and $A B$ makes an angle of $10 ^ { \circ }$ with the horizontal. Equilibrium is maintained by a force of magnitude $F \mathrm {~N}$ acting along $B C$ (see diagram). Show that $F = 8$.

CAIE M2 2014 June Q3

3 A small block $B$ of mass 0.2 kg is placed at a fixed point $O$ on a smooth horizontal surface. A horizontal force of magnitude 0.42 N is applied to $B$. At time $t \mathrm {~s}$ after the force is first applied, the velocity of $B$ away from $O$ is $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$.

Find the value of $v$ when $t = 1$. For $t > 1$ an additional force, of magnitude $0.32 t \mathrm {~N}$ and directed towards $O$, is applied to $B$. The force of magnitude 0.42 N continues to act as before.
Find the value of $v$ when $t = 2$. For $t > 2$ a third force, of magnitude $0.06 t ^ { 2 } \mathrm {~N}$ and directed away from $O$, is applied to $B$. The other two forces continue to act as before.
Show that the velocity of $B$ is the same when $t = 2$ and when $t = 3$.

CAIE M2 2014 June Q4

4 One end of a light inextensible string of length 2.4 m is attached to a fixed point $A$. The other end of the string is attached to a particle $P$ of mass $0.2 \mathrm {~kg} . P$ moves with constant speed in a horizontal circle which has its centre vertically below $A$, with the string taut and making an angle of $60 ^ { \circ }$ with the vertical.

Find the speed of $P$. The string of length 2.4 m is removed, and $P$ is now connected to $A$ by a light inextensible string of length 1.2 m . The particle $P$ moves with angular speed $4 \mathrm { rad } \mathrm { s } ^ { - 1 }$ in a horizontal circle with its centre vertically below $A$.
Calculate the angle between the string and the vertical.

CAIE M2 2014 June Q5

5 A small ball is thrown horizontally with speed $5 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ from a point $O$ on the roof of a building. At time $t \mathrm {~s}$ after projection, the horizontal and vertically downwards displacements of the ball 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 the ball is $y = 0.2 x ^ { 2 }$. The ball strikes the horizontal ground which surrounds the building at a point $A$.
Given that $O A = 18 \mathrm {~m}$, calculate the value of $x$ at $A$, and the speed of the ball immediately before it strikes the ground at $A$.

CAIE M2 2014 June Q6

6
\includegraphics[max width=\textwidth, alt={}, center]{9c82b387-8e5e-48b9-973d-5337b4e56a66-3_652_618_849_762} A particle $P$ of mass 0.6 kg is attached to one end of a light elastic string of natural length 1.5 m and modulus of elasticity 9 N . The string passes through a small smooth ring $R$ fixed at a height of 0.4 m above a rough horizontal surface. The other end of the string is attached to a fixed point $O$ which is 1.5 m vertically above $R$. The points $A$ and $B$ are on the horizontal surface, and $B$ is vertically below $R$. When $P$ is on the surface between $A$ and $B , R P$ makes an acute angle $\theta ^ { \circ }$ with the horizontal (see diagram).

Show that the normal force exerted on $P$ by the surface has magnitude 3.6 N , for all values of $\theta$.
$P$ is projected with speed $2.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ towards $B$ from its initial position at $A$ where $\theta = 30$. The speed of $P$ when it passes through $B$ is $3 \mathrm {~m} \mathrm {~s} ^ { - 1 }$.
Find the work done against friction as $P$ moves from $A$ to $B$.
Calculate the value of the coefficient of friction between $P$ and the surface.

CAIE M2 2014 June Q7

7
\includegraphics[max width=\textwidth, alt={}, center]{9c82b387-8e5e-48b9-973d-5337b4e56a66-4_553_630_258_753} The diagram shows a container which consists of a bowl of weight 14 N and a handle of weight 8 N . The bowl of the container is in the form of a uniform hemispherical shell with centre $O$ and radius 0.3 m . The handle is in the form of a uniform semicircular arc of radius 0.3 m and is freely hinged to the bowl at $A$ and $B$, where $A B$ is a diameter of the bowl.

Calculate the distance of the centre of mass of the container from $O$ for the position indicated in the diagram, where the handle is perpendicular to the rim of the bowl.
Show that the distance of the centre of mass of the container from $O$ when the handle lies on the rim of the bowl is 0.118 m , correct to 3 significant figures. In the case when the handle lies on the rim of the bowl, the container rests in equilibrium with the curved surface of the bowl on a horizontal table.
Find the angle which the plane containing the rim of the bowl makes with the horizontal.

CAIE M2 2015 June Q1

1 One end of a light elastic string of natural length 0.7 m is attached to a fixed point $A$ on a smooth horizontal surface. The other end of the string is attached to a particle $P$ of mass 0.3 kg which is held at a point $B$ on the horizontal surface, where $A B = 1.2 \mathrm {~m}$. It is given that $P$ is released from rest at $B$ and that when $A P = 0.9 \mathrm {~m}$, the particle has speed $4 \mathrm {~m} \mathrm {~s} ^ { - 1 }$. Calculate the modulus of elasticity of the string.

CAIE M2 2015 June Q2

2 A stone is projected from a point $O$ on horizontal ground. The equation of the trajectory of the stone is $$y = 1.2 x - 0.15 x ^ { 2 }$$ where $x \mathrm {~m}$ and $y \mathrm {~m}$ are respectively the horizontal and vertically upwards displacements of the stone from $O$. Find

the greatest height of the stone,
the distance from $O$ of the point where the stone strikes the ground.

CAIE M2 2015 June Q3

3
\includegraphics[max width=\textwidth, alt={}, center]{7958ad9d-1a28-439d-8de0-5039f2dba770-2_255_926_1073_614} One end of a light inextensible string is attached to a fixed point $A$ and the other end of the string is attached to a particle $P$. The particle $P$ moves with constant angular speed $5 \mathrm { rad } \mathrm { s } ^ { - 1 }$ in a horizontal circle which has its centre $O$ vertically below $A$. The string makes an angle $\theta$ with the vertical (see diagram). The tension in the string is three times the weight of $P$.

Show that the length of the string is 1.2 m .
Find the speed of $P$.
\includegraphics[max width=\textwidth, alt={}, center]{7958ad9d-1a28-439d-8de0-5039f2dba770-3_521_1004_258_575} A small ball $B$ is projected from a point $O$ above horizontal ground, with initial speed $15 \mathrm {~ms} ^ { - 1 }$ at an angle of projection of $30 ^ { \circ }$ above the horizontal (see diagram). The ball strikes the ground 3 s after projection.
Calculate the speed and direction of motion of the ball immediately before it strikes the ground.
Find the height of $O$ above the ground.

CAIE M2 2015 June Q5

5 A particle $P$ of mass 0.3 kg is attached to one end of a light elastic string of natural length 0.9 m and modulus of elasticity 18 N . The other end of the string is attached to a fixed point $O$ which is 3 m above the ground.

Find the extension of the string when $P$ is in the equilibrium position.
$P$ is projected vertically downwards from the equilibrium position with initial speed $6 \mathrm {~ms} ^ { - 1 }$. At the instant when the tension in the string is 12 N the string breaks. $P$ continues to descend vertically.
(a) Calculate the height of $P$ above the ground at the instant when the string breaks.
(b) Find the speed of $P$ immediately before it strikes the ground.

CAIE M2 2015 June Q6

6 A particle $P$ of mass 0.1 kg moves with decreasing speed in a straight line on a smooth horizontal surface. A horizontal resisting force of magnitude $0.2 \mathrm { e } ^ { - x } \mathrm {~N}$ acts on $P$, where $x \mathrm {~m}$ is the displacement of $P$ from a fixed point $O$ on the line. The velocity of $P$ is $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$ when its displacement from $O$ is $x \mathrm {~m}$.

Show that $$v \frac { \mathrm {~d} v } { \mathrm {~d} x } = k \mathrm { e } ^ { - x }$$ where $k$ is a constant to be found.
$P$ passes through $O$ with velocity $2.2 \mathrm {~m} \mathrm {~s} ^ { - 1 }$.
Calculate the value of $x$ at the instant when the velocity of $P$ is $2 \mathrm {~m} \mathrm {~s} ^ { - 1 }$.
Show that the speed of $P$ does not fall below $0.917 \mathrm {~m} \mathrm {~s} ^ { - 1 }$, correct to 3 significant figures.
[0pt] [Question 7 is printed on the next page.]

CAIE M2 2015 June Q7

7
\includegraphics[max width=\textwidth, alt={}, center]{7958ad9d-1a28-439d-8de0-5039f2dba770-4_454_588_262_776} The diagram shows the cross-section $O A B C D E$ through the centre of mass of a uniform prism on a rough inclined plane. The portion $A D E O$ is a rectangle in which $A D = O E = 0.6 \mathrm {~m}$ and $D E = A O = 0.8 \mathrm {~m}$; the portion $B C D$ is an isosceles triangle in which angle $B C D$ is a right angle, and $A$ is the mid-point of $B D$. The plane is inclined at $45 ^ { \circ }$ to the horizontal, $B C$ lies along a line of greatest slope of the plane and $D E$ is horizontal.

Calculate the distance of the centre of mass of the prism from $B D$. The weight of the prism is 21 N , and it is held in equilibrium by a horizontal force of magnitude $P \mathrm {~N}$ acting along $E D$.
(a) Find the smallest value of $P$ for which the prism does not topple.
(b) It is given that the prism is about to slip for this smallest value of $P$. Calculate the coefficient of friction between the prism and the plane. The value of $P$ is gradually increased until the prism ceases to be in equilibrium.
Show that the prism topples before it begins to slide, stating the value of $P$ at which equilibrium is broken. \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. }

Questions M2 (1391 questions)

Browse by module

Browse by board

CAIE M2 2013 June Q6

CAIE M2 2013 June Q7

CAIE M2 2013 June Q4

CAIE M2 2013 June Q5

CAIE M2 2013 June Q6

CAIE M2 2014 June Q2

CAIE M2 2014 June Q5

CAIE M2 2014 June Q6

CAIE M2 2014 June Q7

CAIE M2 2014 June Q2

CAIE M2 2014 June Q5

CAIE M2 2014 June Q7

CAIE M2 2014 June Q1

CAIE M2 2014 June Q2

CAIE M2 2014 June Q3

CAIE M2 2014 June Q4

CAIE M2 2014 June Q5

CAIE M2 2014 June Q6

CAIE M2 2014 June Q7

CAIE M2 2015 June Q1

CAIE M2 2015 June Q2

CAIE M2 2015 June Q3

CAIE M2 2015 June Q5

CAIE M2 2015 June Q6

CAIE M2 2015 June Q7