Questions - CAIE M1 - A-Level Maths

(i)
P $\theta$
$P \quad \underline { \theta }$
-
$\underline { \theta }$
$5 \theta$ $$\begin{gathered}
\theta \end{gathered} \quad P$$
$P$

CAIE M1 2013 June Q6

6 $\begin{array} { c c c c c c c c } & P & & P & & & \theta &
\theta & P & & A & \theta & \theta & P &
\text { (i) } & & P & & & \theta &
\text { (ii) } & P & & & P & & &
\text { (iii) } & & & & & & \theta
& & & & & & \theta \end{array}$ 7
\includegraphics[max width=\textwidth, alt={}, center]{bc436b32-01f9-41dc-b2f7-ce49e18d3e6c-3_512_1095_1439_580}

$B$
$\begin{array} { l l l } A & P & A \end{array}$ AN PA

CAIE M1 2013 June Q1

1 A straight ice track of length 50 m is inclined at $14 ^ { \circ }$ to the horizontal. A man starts at the top of the track, on a sledge, with speed $8 \mathrm {~m} \mathrm {~s} ^ { - 1 }$. He travels on the sledge to the bottom of the track. The coefficient of friction between the sledge and the track is 0.02 . Find the speed of the sledge and the man when they reach the bottom of the track.

CAIE M1 2013 June Q2

2
\includegraphics[max width=\textwidth, alt={}, center]{ceb367ee-4e12-4cb2-9020-078ea5724d6e-2_529_691_529_726} Particle $A$ of mass 1.6 kg and particle $B$ of mass 2 kg are attached to opposite ends of a light inextensible string. The string passes over a small smooth pulley fixed at the top of a smooth plane, which is inclined at angle $\theta$, where $\sin \theta = 0.8$. Particle $A$ is held at rest at the bottom of the plane and $B$ hangs at a height of 3.24 m above the level of the bottom of the plane (see diagram). $A$ is released from rest and the particles start to move.

Show that the loss of potential energy of the system, when $B$ reaches the level of the bottom of the plane, is 23.328 J .
Hence find the speed of the particles when $B$ reaches the level of the bottom of the plane.

CAIE M1 2013 June Q3

3 A car has mass 800 kg . The engine of the car generates constant power $P \mathrm {~kW}$ as the car moves along a straight horizontal road. The resistance to motion is constant and equal to $R \mathrm {~N}$. When the car's speed is $14 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ its acceleration is $1.4 \mathrm {~m} \mathrm {~s} ^ { - 2 }$, and when the car's speed is $25 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ its acceleration is $0.33 \mathrm {~m} \mathrm {~s} ^ { - 2 }$. Find the values of $P$ and $R$.

CAIE M1 2013 June Q4

4 An aeroplane moves along a straight horizontal runway before taking off. It starts from rest at $O$ and has speed $90 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ at the instant it takes off. While the aeroplane is on the runway at time $t$ seconds after leaving $O$, its acceleration is $( 1.5 + 0.012 t ) \mathrm { m } \mathrm { s } ^ { - 2 }$. Find

the value of $t$ at the instant the aeroplane takes off,
the distance travelled by the aeroplane on the runway.

CAIE M1 2013 June Q5

5 A particle $P$ is projected vertically upwards from a point on the ground with speed $17 \mathrm {~m} \mathrm {~s} ^ { - 1 }$. Another particle $Q$ is projected vertically upwards from the same point with speed $7 \mathrm {~ms} ^ { - 1 }$. Particle $Q$ is projected $T$ seconds later than particle $P$.

Given that the particles reach the ground at the same instant, find the value of $T$.
At a certain instant when both $P$ and $Q$ are in motion, $P$ is 5 m higher than $Q$. Find the magnitude and direction of the velocity of each of the particles at this instant.

CAIE M1 2013 June Q6

6
\includegraphics[max width=\textwidth, alt={}, center]{ceb367ee-4e12-4cb2-9020-078ea5724d6e-3_703_700_255_721} A small box of mass 40 kg is moved along a rough horizontal floor by three men. Two of the men apply horizontal forces of magnitudes 100 N and 120 N , making angles of $30 ^ { \circ }$ and $60 ^ { \circ }$ respectively with the positive $x$-direction. The third man applies a horizontal force of magnitude $F \mathrm {~N}$ making an angle of $\alpha ^ { \circ }$ with the negative $x$-direction (see diagram). The resultant of the three horizontal forces acting on the box is in the positive $x$-direction and has magnitude 136 N .

Find the values of $F$ and $\alpha$.
Given that the box is moving with constant speed, state the magnitude of the frictional force acting on the box and hence find the coefficient of friction between the box and the floor.

CAIE M1 2013 June Q7

7
\includegraphics[max width=\textwidth, alt={}, center]{ceb367ee-4e12-4cb2-9020-078ea5724d6e-3_430_860_1585_641} Particle $A$ of mass 1.26 kg and particle $B$ of mass 0.9 kg are attached to the ends of a light inextensible string. The string passes over a small smooth pulley $P$ which is fixed at the edge of a rough horizontal table. $A$ is held at rest at a point 0.48 m from $P$, and $B$ hangs vertically below $P$, at a height of 0.45 m above the floor (see diagram). The coefficient of friction between $A$ and the table is $\frac { 2 } { 7 } . A$ is released and the particles start to move.

Show that the magnitude of the acceleration of the particles is $2.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }$ and find the tension in the string.
Find the speed with which $B$ reaches the floor.
Find the speed with which $A$ reaches the pulley.

CAIE M1 2014 June Q1

1 A ta v at tat th ta $k a$ th tta ta
a a hta taht tak $v$ that th $t$ th tat $k$ th va
$2 A h a$ at a a th $a$ Th $a$

th va
th at va th
$t$ th hta $A$ a at at th $a t t$

CAIE M1 2014 June Q3

3
\includegraphics[max width=\textwidth, alt={}, center]{9950eb77-25f9-4275-88ab-8dce9fb07cec-2_524_572_806_749}
aa at at a
$t$ Th at th $a \quad a$ th
$t$ hh th at $a h$ th aa that a $t$ th
tat th

CAIE M1 2014 June Q4

4 A at $t$ vta a th
Th at $v$ avt
a $t$ av hta
That $v$ avt
$t$ tht th
$t$
$h$ at $t$
th tat $t$ th at $h$
$t$ th th tta
ta tav a
th tta t tak

CAIE M1 2014 June Q5

5
\includegraphics[max width=\textwidth, alt={}, center]{9950eb77-25f9-4275-88ab-8dce9fb07cec-2_135_1351_1928_415} A a a k tat tat
th $a$ th hta th
$t \quad$ taht $a$ hta aa $h$
$a$ ath tat th $a t \quad T h \quad$ th $a$
a ha tav a ta

th a $t$
t a tta $t$
$h$ that Th $t$
at a hht av thv
tha $v$ th $v$ th
$t \quad t t a k t \quad t$
th va hh
th va hh
taht
Th
$h t h$
$a$
$a$
$h h$

CAIE M1 2014 June Q6

6
\includegraphics[max width=\textwidth, alt={}, center]{9950eb77-25f9-4275-88ab-8dce9fb07cec-3_492_556_260_792} Pat aka akaattahtt a ht xt thh $a v a x$ th Tht $h$ at $t$ th th $t$ tat a $t$ taht at vta Bt $h$ ataatahht $\quad$ avth $T h t a$ t $a$ at $h$ th at $a$ th $t$ ak Th at $t$ ah th th $t$ t

th aat
a th ta tav
th t ak
\includegraphics[max width=\textwidth, alt={}, center]{9950eb77-25f9-4275-88ab-8dce9fb07cec-3_668_1086_1336_518} Th vtt ah $h$ th $t$ at
t t ht th Th
a th t tak
$t$ ah $t$ att
$a a h$

7		$a$	tav a a taht a Bth t a thh				\multirow[t]{4}{*}{$a$ av tat th}
	$T t$ ta at				tat ta at at av	$C t t h a h$
a a th th tat aat
	(i)	$h$ that th ta	$a$	at
	$C t$		$t$	at $t$	at av		h
	$a$
	(ii)		$a$ th $a x$		h ha ah
	$C t$ tav		$t \quad$ a ta at th h ha ah at $t$ th tat aat					Ct
	(iii)		th t tak th t t tav		$t$

CAIE M1 2014 June Q1

1 A a a tav a a tat $t$ tat a a $t$ a a T tat tat

$t$ va $v$ tat $t$ a aat at a tat $t a$ at at a at
tat ta tavta tat 2 a att a aat ta taa A at
tat t ta t tat aat At at v
t ta t tat a at at t tat tat tat t
v At t at t tat at $a$
$t v a$
$t$ at $t$

CAIE M1 2014 June Q3

3
\includegraphics[max width=\textwidth, alt={}, center]{cb3c4728-fd7f-48c0-bbd0-7ea97c6d50f2-2_522_633_1135_740}

$a$		vta av	A at	$a$
atta $t$		a atta $t$		a txt
$t \quad t$	ata at a ta	$t a a$
ta at at		aa Bt $t$ a tat a $t$
at ta	atta tt tt

\multirow[t]{3}{*}{4}

A at

$v$ a tat tat tat a $t$

\multirow{2}{*}{}

$t$ Ttat

\multirow{2}{*}{

}
	$t v$	a tatva		t tat aat			-
	tat	at $t$		At a $t$	$t v t$		- ${ } ^ { - }$	\multirow{4}{*}{}	\multirow{4}{*}{}
\multirow{3}{*}{}	(i) taat	\multirow{2}{*}{}	\multirow{2}{*}{atattat atat}		\multirow{2}{*}{}	tat t aat
	$a$
	(ii) tav

\includegraphics[max width=\textwidth, alt={}]{cb3c4728-fd7f-48c0-bbd0-7ea97c6d50f2-3_334_689_258_726}

\multirow{2}{*}{At xt a a atta at $t t T a v a t a$ at a a}	$a$ atta at $a a$		\multirow[t]{2}{*}{$a$}
	$x$ att t a	at $t$ at $t$ tt $t a$
$a$	aa Tttt			a $t$
$a$	a tatt tat $v$ a $t$
a v a ta	a a

t tat tt t
$t$
(a) $t$ avtata tta $t t$
(b) $t$ aat $t$ ta
tat

CAIE M1 2014 June Q6

6 Aat a a tatat avtat a ta attaatt $T$ a taa $t$ tata a $a t$ at a ta $t$ a $t T \quad$ tt a tat ta tt at

tat tattatva Tt $t$ ta a $t$ xt $t$
a $t$ tat aat $T$ ta $t t$
a ta at av t tt t ta
tt tt
ta t ta a atta $t$
a at t tt t ta a t vta
t t at T at

CAIE M1 2014 June Q7

7
\includegraphics[max width=\textwidth, alt={}, center]{cb3c4728-fd7f-48c0-bbd0-7ea97c6d50f2-4_323_1008_274_566}

A $t$ xt $t \quad t \quad a$ at tv atta $t$ t At at tt Tat

\multirow[b]{4}{*}{$a$}

\multirow{6}{*}{$a$}

\multirow{4}{*}{at $t$ vta}

a a t t tat $t v t$ tat aat at

CAIE M1 2014 June Q1

1
\includegraphics[max width=\textwidth, alt={}, center]{139371b7-e142-4ed6-bff3-3ca4c32b9c6b-2_426_424_258_863} A block $B$ of mass 7 kg is at rest on rough horizontal ground. A force of magnitude $X \mathrm {~N}$ acts on $B$ at an angle of $15 ^ { \circ }$ to the upward vertical (see diagram).

Given that $B$ is in equilibrium find, in terms of $X$, the normal component of the force exerted on $B$ by the ground.
The coefficient of friction between $B$ and the ground is 0.4 . Find the value of $X$ for which $B$ is in limiting equilibrium.

CAIE M1 2014 June Q2

2 A car of mass 1250 kg travels up a straight hill inclined at an angle $\alpha$ to the horizontal, where $\sin \alpha = 0.02$. The power provided by the car's engine is 23 kW . The resistance to motion is constant and equal to 600 N . Find the speed of the car at an instant when its acceleration is $0.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }$.

CAIE M1 2014 June Q3

6 marks

3
\includegraphics[max width=\textwidth, alt={}, center]{139371b7-e142-4ed6-bff3-3ca4c32b9c6b-2_657_913_1450_616} A particle $P$ of weight 1.4 N is attached to one end of a light inextensible string $S _ { 1 }$ of length 1.5 m , and to one end of another light inextensible string $S _ { 2 }$ of length 1.3 m . The other end of $S _ { 1 }$ is attached to a wall at the point 0.9 m vertically above a point $O$ of the wall. The other end of $S _ { 2 }$ is attached to the wall at the point 0.5 m vertically below $O$. The particle is held in equilibrium, at the same horizontal level as $O$, by a horizontal force of magnitude 2.24 N acting away from the wall and perpendicular to it (see diagram). Find the tensions in the strings.
[0pt] [6]

CAIE M1 2014 June Q4

4 A small ball of mass 0.4 kg is released from rest at a point 5 m above horizontal ground. At the instant the ball hits the ground it loses 12.8 J of kinetic energy and starts to move upwards.

Show that the greatest height above the ground that the ball reaches after hitting the ground is 1.8 m .
Find the time taken for the ball's motion from its release until reaching this greatest height.

CAIE M1 2014 June Q5

5 A lorry of mass 16000 kg travels at constant speed from the bottom, $O$, to the top, $A$, of a straight hill. The distance $O A$ is 1200 m and $A$ is 18 m above the level of $O$. The driving force of the lorry is constant and equal to 4500 N .

Find the work done against the resistance to the motion of the lorry. On reaching $A$ the lorry continues along a straight horizontal road against a constant resistance of 2000 N . The driving force of the lorry is not now constant, and the speed of the lorry increases from $9 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ at $A$ to $21 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ at the point $B$ on the road. The distance $A B$ is 2400 m .
Use an energy method to find $F$, where $F \mathrm {~N}$ is the average value of the driving force of the lorry while moving from $A$ to $B$.
Given that the driving force at $A$ is 1280 N greater than $F \mathrm {~N}$ and that the driving force at $B$ is 1280 N less than $F \mathrm {~N}$, show that the power developed by the lorry's engine is the same at $B$ as it is at $A$.

CAIE M1 2014 June Q6

6 A particle starts from rest at a point $O$ and moves in a horizontal straight line. The velocity of the particle is $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$ at time $t \mathrm {~s}$ after leaving $O$. For $0 \leqslant t < 60$, the velocity is given by $$v = 0.05 t - 0.0005 t ^ { 2 }$$ The particle hits a wall at the instant when $t = 60$, and reverses the direction of its motion. The particle subsequently comes to rest at the point $A$ when $t = 100$, and for $60 < t \leqslant 100$ the velocity is given by $$v = 0.025 t - 2.5$$

Find the velocity of the particle immediately before it hits the wall, and its velocity immediately after its hits the wall.
Find the total distance travelled by the particle.
Find the maximum speed of the particle and sketch the particle's velocity-time graph for $0 \leqslant t \leqslant 100$, showing the value of $t$ for which the speed is greatest. \section*{[Question 7 is printed on the next page.]}

7		\(a\)	tav a a taht a Bth t a thh				\multirow[t]{4}{*}{\(a\) av tat th}
	\(T t\) ta at				tat ta at at av	\(C t t h a h\)
a a th th tat aat
	(i)	\(h\) that th ta	\(a\)	at
	\(C t\)		\(t\)	at \(t\)	at av		h
	\(a\)
	(ii)		\(a\) th \(a x\)		h ha ah
	\(C t\) tav		\(t \quad\) a ta at th h ha ah at \(t\) th tat aat					Ct
	(iii)		th t tak th t t tav		\(t\)

tat	t ta	t tat aat		At at	v
t ta	t tat		a at	at t tat tat	tat t
v At t	at t tat at	\(a\)

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