Questions - CAIE M1 - A-Level Maths

Find, in kW , the greatest possible power of the car's engine.
Find the greatest possible acceleration of the car at an instant when its speed on a straight level road is $32 \mathrm {~m} \mathrm {~s} ^ { - 1 }$.
The car travels down a hill inclined at an angle of $\theta ^ { \circ }$ to the horizontal at a constant speed of $50 \mathrm {~m} \mathrm {~s} ^ { - 1 }$. The power of the car's engine is 60 kW . Find the value of $\theta$.

CAIE M1 2018 June Q7

7 A particle $P$ moves in a straight line starting from a point $O$. The velocity $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$ of $P$ at time $t \mathrm {~s}$ is given by $$\begin{array} { l l } v = 12 t - 4 t ^ { 2 } & \text { for } 0 \leqslant t \leqslant 2 ,
v = 16 - 4 t & \text { for } 2 \leqslant t \leqslant 4 . \end{array}$$

Find the maximum velocity of $P$ during the first 2 s .
Determine, with justification, whether there is any instantaneous change in the acceleration of $P$ when $t = 2$.
Sketch the velocity-time graph for $0 \leqslant t \leqslant 4$.
\includegraphics[max width=\textwidth, alt={}, center]{03e325b9-171a-4f76-95cd-57dad3741caf-13_684_1054_351_584}
Find the distance travelled by $P$ in the interval $0 \leqslant t \leqslant 4$.
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 M1 2019 June Q1

1
\includegraphics[max width=\textwidth, alt={}, center]{555678d3-f37d-4822-a005-de8c6094dc50-03_563_503_262_820} Given that $\tan \alpha = \frac { 12 } { 5 }$ and $\tan \theta = \frac { 4 } { 3 }$, show that the coplanar forces shown in the diagram are in equilibrium.

CAIE M1 2019 June Q2

2 A particle $P$ is projected vertically upwards with speed $25 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ from a point 3 m above horizontal ground.

Find the time taken for $P$ to reach its greatest height.
Find the length of time for which $P$ is higher than 23 m above the ground.
$P$ is higher than $h \mathrm {~m}$ above the ground for 1 second. Find $h$.

CAIE M1 2019 June Q3

3 A lorry has mass 12000 kg .

The lorry moves at a constant speed of $5 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ up a hill inclined at an angle of $\theta$ to the horizontal, where $\sin \theta = 0.08$. At this speed, the magnitude of the resistance to motion on the lorry is 1500 N . Show that the power of the lorry's engine is 55.5 kW .
When the speed of the lorry is $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$ the magnitude of the resistance to motion is $k v ^ { 2 } \mathrm {~N}$, where $k$ is a constant.
Show that $k = 60$.
The lorry now moves at a constant speed on a straight level road. Given that its engine is still working at 55.5 kW , find the lorry's speed.

CAIE M1 2019 June Q4

4 A particle of mass 1.3 kg rests on a rough plane inclined at an angle $\theta$ to the horizontal, where $\tan \theta = \frac { 12 } { 5 }$. The coefficient of friction between the particle and the plane is $\mu$.

A force of magnitude 20 N parallel to a line of greatest slope of the plane is applied to the particle and the particle is on the point of moving up the plane. Show that $\mu = 1.6$.
The force of magnitude 20 N is now removed.
Find the acceleration of the particle.
Find the work done against friction during the first 2 s of motion.

CAIE M1 2019 June Q5

5 A particle $P$ moves in a straight line from a fixed point $O$. The velocity $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$ of $P$ at time $t \mathrm {~s}$ is given by $$v = t ^ { 2 } - 8 t + 12 \quad \text { for } 0 \leqslant t \leqslant 8$$

Find the minimum velocity of $P$.
Find the total distance travelled by $P$ in the interval $0 \leqslant t \leqslant 8$.
\includegraphics[max width=\textwidth, alt={}, center]{555678d3-f37d-4822-a005-de8c6094dc50-12_401_1102_260_520} Two particles $A$ and $B$, of masses 0.4 kg and 0.2 kg respectively, are connected by a light inextensible string. Particle $A$ is held on a smooth plane inclined at an angle of $\theta ^ { \circ }$ to the horizontal. The string passes over a small smooth pulley $P$ fixed at the top of the plane, and $B$ hangs freely 0.5 m above horizontal ground (see diagram). The particles are released from rest with both sections of the string taut.
Given that the system is in equilibrium, find $\theta$.
It is given instead that $\theta = 20$. In the subsequent motion particle $A$ does not reach $P$ and $B$ remains at rest after reaching the ground.
(a) Find the tension in the string and the acceleration of the system.
(b) Find the speed of $A$ at the instant $B$ reaches the ground.
(c) Use an energy method to find the total distance $A$ moves up the plane before coming to instantaneous rest.
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 M1 2019 June Q2

2 A car moves in a straight line with initial speed $u \mathrm {~m} \mathrm {~s} ^ { - 1 }$ and constant acceleration $a \mathrm {~m} \mathrm {~s} ^ { - 2 }$. The car takes 5 s to travel the first 80 m and it takes 8 s to travel the first 160 m . Find $a$ and $u$.

CAIE M1 2019 June Q3

3 A particle of mass 13 kg is on a rough plane inclined at an angle of $\theta$ to the horizontal, where $\tan \theta = \frac { 5 } { 12 }$. The coefficient of friction between the particle and the plane is 0.3 . A force of magnitude $T \mathrm {~N}$, acting parallel to a line of greatest slope, moves the particle a distance of 2.5 m up the plane at a constant speed. Find the work done by this force.

CAIE M1 2019 June Q4

4 A constant resistance to motion of magnitude 350 N acts on a car of mass 1250 kg . The engine of the car exerts a constant driving force of 1200 N . The car travels along a road inclined at an angle of $\theta$ to the horizontal, where $\sin \theta = 0.05$. Find the speed of the car when it has moved 100 m from rest in each of the following cases.

The car is moving up the hill.
The car is moving down the hill.
\includegraphics[max width=\textwidth, alt={}, center]{bc7a0101-e433-48d3-8fc5-4a93343203b0-08_567_511_258_817}

Two particles $A$ and $B$, of masses 0.4 kg and 0.2 kg respectively, are connected by a light inextensible string which passes over a fixed smooth pulley. Both $A$ and $B$ are 0.5 m above the ground. The particles hang vertically (see diagram). The particles are released from rest. In the subsequent motion $B$ does not reach the pulley and $A$ remains at rest after reaching the ground.

For the motion before $A$ reaches the ground, show that the magnitude of the acceleration of each particle is $\frac { 10 } { 3 } \mathrm {~m} \mathrm {~s} ^ { - 2 }$ and find the tension in the string.
Find the maximum height of $B$ above the ground.

CAIE M1 2019 June Q6

6 A car has mass 1000 kg . When the car is travelling at a steady speed of $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$, where $v > 2$, the resistance to motion of the car is $( A v + B ) \mathrm { N }$, where $A$ and $B$ are constants. The car can travel along a horizontal road at a steady speed of $18 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ when its engine is working at 36 kW . The car can travel up a hill inclined at an angle of $\theta$ to the horizontal, where $\sin \theta = 0.05$, at a steady speed of $12 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ when its engine is working at 21 kW . Find $A$ and $B$.

CAIE M1 2019 June Q7

7 Particles $P$ and $Q$ leave a fixed point $A$ at the same time and travel in the same straight line. The velocity of $P$ after $t$ seconds is $6 t ( t - 3 ) \mathrm { m } \mathrm { s } ^ { - 1 }$ and the velocity of $Q$ after $t$ seconds is $( 10 - 2 t ) \mathrm { m } \mathrm { s } ^ { - 1 }$.

Sketch, on the same axes, velocity-time graphs for $P$ and $Q$ for $0 \leqslant t \leqslant 5$.
Verify that $P$ and $Q$ meet after 5 seconds.
Find the greatest distance between $P$ and $Q$ for $0 \leqslant t \leqslant 5$.
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 M1 2019 June Q1

1 A bus moves in a straight line between two bus stops. The bus starts from rest and accelerates at $2.1 \mathrm {~m} \mathrm {~s} ^ { - 2 }$ for 5 s . The bus then travels for 24 s at constant speed and finally slows down, with a constant deceleration, stopping in a further 6 s . Sketch a velocity-time graph for the motion and hence find the distance between the two bus stops.

CAIE M1 2019 June Q2

2
\includegraphics[max width=\textwidth, alt={}, center]{539be201-7bfc-4ba0-8378-c7aec4473ac7-03_577_691_262_724} Coplanar forces of magnitudes $12 \mathrm {~N} , 24 \mathrm {~N}$ and 30 N act at a point in the directions shown in the diagram.

Find the components of the resultant of the three forces in the $x$-direction and in the $y$-direction. Component in $x$-direction $\_\_\_\_$
Component in $y$-direction. $\_\_\_\_$
Hence find the direction of the resultant.

CAIE M1 2019 June Q3

3 A car of mass 1400 kg is travelling up a hill inclined at an angle of $4 ^ { \circ }$ to the horizontal. There is a constant resistance to motion of magnitude 1550 N acting on the car.

Given that the engine of the car is working at 30 kW , find the speed of the car at an instant when its acceleration is $0.4 \mathrm {~m} \mathrm {~s} ^ { - 2 }$.
The greatest possible constant speed at which the car can travel up the hill is $40 \mathrm {~m} \mathrm {~s} ^ { - 1 }$. Find the maximum possible power of the engine.
\includegraphics[max width=\textwidth, alt={}, center]{539be201-7bfc-4ba0-8378-c7aec4473ac7-06_643_419_255_863} Two particles $A$ and $B$, of masses 1.3 kg and 0.7 kg respectively, are connected by a light inextensible string which passes over a smooth fixed pulley. Particle $A$ is 1.75 m above the floor and particle $B$ is 1 m above the floor (see diagram). The system is released from rest with the string taut, and the particles move vertically. When the particles are at the same height the string breaks.

CAIE M1 2019 June Q5

5 A particle of mass 18 kg is on a plane inclined at an angle of $30 ^ { \circ }$ to the horizontal. The particle is projected up a line of greatest slope of the plane with a speed of $20 \mathrm {~m} \mathrm {~s} ^ { - 1 }$.

Given that the plane is smooth, use an energy method to find the distance the particle moves up the plane before coming to instantaneous rest.
Given instead that the plane is rough and the coefficient of friction between the particle and the plane is 0.25 , find the speed of the particle as it returns to its starting point.

CAIE M1 2019 June Q6

6 A particle $P$ moves in a straight line. The acceleration $a \mathrm {~m} \mathrm {~s} ^ { - 2 }$ of $P$ at time $t \mathrm {~s}$ is given by $a = 6 t - 12$. The displacement of $P$ from a fixed point $O$ on the line is $s \mathrm {~m}$. It is given that $s = 5$ when $t = 1$ and $s = 1$ when $t = 3$.

Show that $s = t ^ { 3 } - 6 t ^ { 2 } + p t + q$, where $p$ and $q$ are constants to be found.
Find the values of $t$ when $P$ is at instantaneous rest.
Find the total distance travelled by $P$ in the interval $0 \leqslant t \leqslant 4$.
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 M1 2016 March Q1

1 A cyclist has mass 85 kg and rides a bicycle of mass 20 kg . The cyclist rides along a horizontal road against a total resistance force of 40 N . Find the total work done by the cyclist in increasing his speed from $5 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ to $10 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ while travelling a distance of 50 m .

CAIE M1 2016 March Q2

2 A constant resistance of magnitude 1350 N acts on a car of mass 1200 kg .

The car is moving along a straight level road at a constant speed of $32 \mathrm {~m} \mathrm {~s} ^ { - 1 }$. Find, in kW , the rate at which the engine of the car is working.
The car travels at a constant speed up a hill inclined at an angle of $\theta$ to the horizontal, where $\sin \theta = 0.1$, with the engine working at 76.5 kW . Find this speed.

CAIE M1 2016 March Q3

3
\includegraphics[max width=\textwidth, alt={}, center]{9a99a969-db40-4d29-bb37-ea7ac15cdc2d-2_476_659_897_742} Coplanar forces of magnitudes $50 \mathrm {~N} , 40 \mathrm {~N}$ and 30 N act at a point $O$ in the directions shown in the diagram, where $\tan \alpha = \frac { 7 } { 24 }$.

Find the magnitude and direction of the resultant of the three forces.
The force of magnitude 50 N is replaced by a force of magnitude $P \mathrm {~N}$ acting in the same direction. The resultant of the three forces now acts in the positive $x$-direction. Find the value of $P$.

CAIE M1 2016 March Q4

4 A particle $P$ of mass 0.8 kg is placed on a rough horizontal table. The coefficient of friction between $P$ and the table is $\mu$. A force of magnitude 5 N , acting upwards at an angle $\alpha$ above the horizontal, where $\tan \alpha = \frac { 3 } { 4 }$, is applied to $P$. The particle is on the point of sliding on the table.

Find the value of $\mu$.
The magnitude of the force acting on $P$ is increased to 10 N , with the direction of the force remaining the same. Find the acceleration of $P$.

CAIE M1 2016 March Q5

5 A car of mass 1200 kg is pulling a trailer of mass 800 kg up a hill inclined at an angle $\alpha$ to the horizontal, where $\sin \alpha = 0.1$. The system of the car and the trailer is modelled as two particles connected by a light inextensible cable. The driving force of the car's engine is 2500 N and the resistances to the car and trailer are 100 N and 150 N respectively.

Find the acceleration of the system and the tension in the cable.
When the car and trailer are travelling at a speed of $30 \mathrm {~m} \mathrm {~s} ^ { - 1 }$, the driving force becomes zero. The cable remains taut. Find the time, in seconds, before the system comes to rest.

CAIE M1 2016 March Q6

6 Two particles $A$ and $B$, of masses 0.8 kg and 0.2 kg respectively, are connected by a light inextensible string. Particle $A$ is placed on a horizontal surface. The string passes over a small smooth pulley $P$ fixed at the edge of the surface, and $B$ hangs freely. The horizontal section of the string, $A P$, is of length 2.5 m . The particles are released from rest with both sections of the string taut.

Given that the surface is smooth, find the time taken for $A$ to reach the pulley.
Given instead that the surface is rough and the coefficient of friction between $A$ and the surface is 0.1 , find the speed of $A$ immediately before it reaches the pulley.
$7 \quad$ A particle $P$ moves in a straight line. The velocity $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$ at time $t \mathrm {~s}$ is given by $$\begin{array} { l l } v = 5 t ( t - 2 ) & \text { for } 0 \leqslant t \leqslant 4
v = k & \text { for } 4 \leqslant t \leqslant 14
v = 68 - 2 t & \text { for } 14 \leqslant t \leqslant 20 \end{array}$$ where $k$ is a constant.
Find $k$.
Sketch the velocity-time graph for $0 \leqslant t \leqslant 20$.
Find the set of values of $t$ for which the acceleration of $P$ is positive.
Find the total distance travelled by $P$ in the interval $0 \leqslant t \leqslant 20$.

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