Questions - CAIE M1 - A-Level Maths

The car and caravan move along a horizontal part of the road at a constant speed of $30 \mathrm {~m} \mathrm {~s} ^ { - 1 }$.
1. Calculate, in kW , the power developed by the engine of the car.
2. Given that this power is suddenly decreased by 8 kW , find the instantaneous deceleration of the car and caravan and the tension in the tow-bar.
The car and caravan now travel along a part of the road inclined at $\sin ^ { - 1 } 0.06$ to the horizontal. The car and caravan travel up the incline at constant speed with the engine of the car working at 28 kW .
1. Find this constant speed.
2. Find the increase in the potential energy of the caravan in one minute.

CAIE M1 2021 June Q6

6 A particle $A$ is projected vertically upwards from level ground with an initial speed of $30 \mathrm {~m} \mathrm {~s} ^ { - 1 }$. At the same instant a particle $B$ is released from rest 15 m vertically above $A$. The mass of one of the particles is twice the mass of the other particle. During the subsequent motion $A$ and $B$ collide and coalesce to form particle $C$. Find the difference between the two possible times at which $C$ hits the ground.
$7 \quad$ A particle $P$ moving in a straight line starts from rest at a point $O$ and comes to rest 16 s later. At time $t \mathrm {~s}$ after leaving $O$, the acceleration $a \mathrm {~m} \mathrm {~s} ^ { - 2 }$ of $P$ is given by $$\begin{array} { l l } a = 6 + 4 t & 0 \leqslant t < 2 ,
a = 14 & 2 \leqslant t < 4 ,
a = 16 - 2 t & 4 \leqslant t \leqslant 16 . \end{array}$$ There is no sudden change in velocity at any instant.

Find the values of $t$ when the velocity of $P$ is $55 \mathrm {~m} \mathrm {~s} ^ { - 1 }$.
Complete the sketch of the velocity-time diagram.
\includegraphics[max width=\textwidth, alt={}, center]{41e63d05-d109-47dc-80a6-927953e3e607-11_511_1054_351_584}
Find the distance travelled by $P$ when it is decelerating.
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 2021 June Q1

1 Particles $P$ of mass 0.4 kg and $Q$ of mass 0.5 kg are free to move on a smooth horizontal plane. $P$ and $Q$ are moving directly towards each other with speeds $2.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ and $1.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ respectively. After $P$ and $Q$ collide, the speed of $Q$ is twice the speed of $P$. Find the two possible values of the speed of $P$ after the collision.

CAIE M1 2021 June Q2

2 A cyclist is travelling along a straight horizontal road. She is working at a constant rate of 150 W . At an instant when her speed is $4 \mathrm {~m} \mathrm {~s} ^ { - 1 }$, her acceleration is $0.25 \mathrm {~m} \mathrm {~s} ^ { - 2 }$. The resistance to motion is 20 N .

Find the total mass of the cyclist and her bicycle.
The cyclist comes to a straight hill inclined at an angle $\theta$ above the horizontal. She ascends the hill at constant speed $3 \mathrm {~m} \mathrm {~s} ^ { - 1 }$. She continues to work at the same rate as before and the resistance force is unchanged.
Find the value of $\theta$.

CAIE M1 2021 June Q3

3
\includegraphics[max width=\textwidth, alt={}, center]{ba29ddb2-3558-4be1-a8a8-134e27a70149-04_456_767_260_689} Four coplanar forces act at a point. The magnitudes of the forces are $20 \mathrm {~N} , 30 \mathrm {~N} , 40 \mathrm {~N}$ and $F \mathrm {~N}$. The directions of the forces are as shown in the diagram, where $\sin \alpha ^ { \circ } = 0.28$ and $\sin \beta ^ { \circ } = 0.6$. Given that the forces are in equilibrium, find $F$ and $\theta$.

CAIE M1 2021 June Q4

4 A particle is projected vertically upwards with speed $u \mathrm {~m} \mathrm {~s} ^ { - 1 }$ from a point on horizontal ground. After 2 seconds, the height of the particle above the ground is 24 m .

Show that $u = 22$.
The height of the particle above the ground is more than $h \mathrm {~m}$ for a period of 3.6 s . Find $h$.

CAIE M1 2021 June Q5

5 A car of mass 1400 kg is towing a trailer of mass 500 kg down a straight hill inclined at an angle of $5 ^ { \circ }$ to the horizontal. The car and trailer are connected by a light rigid tow-bar. At the top of the hill the speed of the car and trailer is $20 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ and at the bottom of the hill their speed is $30 \mathrm {~m} \mathrm {~s} ^ { - 1 }$.

It is given that as the car and trailer descend the hill, the engine of the car does 150000 J of work, and there are no resistance forces. Find the length of the hill.
It is given instead that there is a resistance force of 100 N on the trailer, the length of the hill is 200 m , and the acceleration of the car and trailer is constant. Find the tension in the tow-bar between the car and trailer.

CAIE M1 2021 June Q6

6 A particle moves in a straight line and passes through the point $A$ at time $t = 0$. The velocity of the particle at time $t \mathrm {~s}$ after leaving $A$ is $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$, where $$v = 2 t ^ { 2 } - 5 t + 3$$

Find the times at which the particle is instantaneously at rest. Hence or otherwise find the minimum velocity of the particle.
Sketch the velocity-time graph for the first 3 seconds of motion.
Find the distance travelled between the two times when the particle is instantaneously at rest.

CAIE M1 2021 June Q7

7
\includegraphics[max width=\textwidth, alt={}, center]{ba29ddb2-3558-4be1-a8a8-134e27a70149-10_220_609_260_769} A particle $P$ of mass 0.3 kg rests on a rough plane inclined at an angle $\theta$ to the horizontal, where $\sin \theta = \frac { 7 } { 25 }$. A horizontal force of magnitude 4 N , acting in the vertical plane containing a line of greatest slope of the plane, is applied to $P$ (see diagram). The particle is on the point of sliding up the plane.

Show that the coefficient of friction between the particle and the plane is $\frac { 3 } { 4 }$.
The force acting horizontally is replaced by a force of magnitude 4 N acting up the plane parallel to a line of greatest slope.
Find the acceleration of $P$.
Starting with $P$ at rest, the force of 4 N parallel to the plane acts for 3 seconds and is then removed. Find the total distance travelled until $P$ comes 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 2022 June Q1

1 A car starts from rest and moves in a straight line with constant acceleration for a distance of 200 m , reaching a speed of $25 \mathrm {~m} \mathrm {~s} ^ { - 1 }$. The car then travels at this speed for 400 m , before decelerating uniformly to rest over a period of 5 s .

Find the time for which the car is accelerating.
Sketch the velocity-time graph for the motion of the car, showing the key points.
Find the average speed of the car during its motion.

CAIE M1 2022 June Q2

2 Two particles $P$ and $Q$, of masses 0.5 kg and 0.3 kg respectively, are connected by a light inextensible string. The string is taut and $P$ is vertically above $Q$. A force of magnitude 10 N is applied to $P$ vertically upwards. Find the acceleration of the particles and the tension in the string connecting them.

CAIE M1 2022 June Q3

3 A crate of mass 300 kg is at rest on rough horizontal ground. The coefficient of friction between the crate and the ground is 0.5 . A force of magnitude $X \mathrm {~N}$, acting at an angle $\alpha$ above the horizontal, is applied to the crate, where $\sin \alpha = 0.28$. Find the greatest value of $X$ for which the crate remains at rest.
\includegraphics[max width=\textwidth, alt={}, center]{213e26a8-3e4e-4dd4-b287-02e5925f3f47-06_849_807_255_669} Three coplanar forces of magnitudes $20 \mathrm {~N} , 100 \mathrm {~N}$ and $F \mathrm {~N}$ act at a point. The directions of these forces are shown in the diagram. Given that the three forces are in equilibrium, find $F$ and $\alpha$.

CAIE M1 2022 June Q5

5 Two racing cars $A$ and $B$ are at rest alongside each other at a point $O$ on a straight horizontal test track. The mass of $A$ is 1200 kg . The engine of $A$ produces a constant driving force of 4500 N . When $A$ arrives at a point $P$ its speed is $25 \mathrm {~ms} ^ { - 1 }$. The distance $O P$ is $d \mathrm {~m}$. The work done against the resistance force experienced by $A$ between $O$ and $P$ is 75000 J .

Show that $d = 100$.
Car $B$ starts off at the same instant as car $A$. The two cars arrive at $P$ simultaneously and with the same speed. The engine of $B$ produces a driving force of 3200 N and the car experiences a constant resistance to motion of 1200 N .
Find the mass of $B$.
Find the steady speed which $B$ can maintain when its engine is working at the same rate as it is at $P$.

CAIE M1 2022 June Q6

6 A particle starts from a point $O$ and moves in a straight line. The velocity $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$ of the particle at time $t \mathrm {~s}$ after leaving $O$ is given by $$v = k \left( 3 t ^ { 2 } - 2 t ^ { 3 } \right)$$ where $k$ is a constant.

Verify that the particle returns to $O$ when $t = 2$.
It is given that the acceleration of the particle is $- 13.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }$ for the positive value of $t$ at which $v = 0$. Find $k$ and hence find the total distance travelled in the first two seconds of motion.

CAIE M1 2022 June Q7

7 Two particles $A$ and $B$, of masses 0.4 kg and 0.2 kg respectively, are moving down the same line of greatest slope of a smooth plane. The plane is inclined at $30 ^ { \circ }$ to the horizontal, and $A$ is higher up the plane than $B$. When the particles collide, the speeds of $A$ and $B$ are $3 \mathrm {~ms} ^ { - 1 }$ and $2 \mathrm {~ms} ^ { - 1 }$ respectively. In the collision between the particles, the speed of $A$ is reduced to $2.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }$.

Find the speed of $B$ immediately after the collision.
After the collision, when $B$ has moved 1.6 m down the plane from the point of collision, it hits a barrier and returns back up the same line of greatest slope. $B$ hits the barrier 0.4 s after the collision, and when it hits the barrier, its speed is reduced by $90 \%$. The two particles collide again 0.44 s after their previous collision, and they then coalesce on impact.
Show that the speed of $B$ immediately after it hits the barrier is $0.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }$. Hence find the speed of the combined particle immediately after the second collision between $A$ and $B$.
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 2022 June Q1

1 Small smooth spheres $A$ and $B$, of equal radii and of masses 5 kg and 3 kg respectively, lie on a smooth horizontal plane. Initially $B$ is at rest and $A$ is moving towards $B$ with speed $8.5 \mathrm {~ms} ^ { - 1 }$. The spheres collide and after the collision $A$ continues to move in the same direction but with a quarter of the speed of $B$.

Find the speed of $B$ after the collision.
Find the loss of kinetic energy of the system due to the collision.

CAIE M1 2022 June Q2

2
\includegraphics[max width=\textwidth, alt={}, center]{4ee2568e-5902-442f-9af1-3663fa1d59c7-03_680_636_255_756} Coplanar forces of magnitudes $60 \mathrm {~N} , 20 \mathrm {~N} , 16 \mathrm {~N}$ and 14 N act at a point in the directions shown in the diagram. Find the magnitude and direction of the resultant force.

CAIE M1 2022 June Q3

3 Two particles $A$ and $B$, of masses 2.4 kg and 1.2 kg respectively, are connected by a light inextensible string which passes over a fixed smooth pulley. $A$ is held at a distance of 2.1 m above a horizontal plane and $B$ is 1.5 m above the plane. The particles hang vertically and are released from rest. In the subsequent motion $A$ reaches the plane and does not rebound and $B$ does not reach the pulley.

Show that the tension in the string before $A$ reaches the plane is 16 N and find the magnitude of the acceleration of the particles before $A$ reaches the plane.
Find the greatest height of $B$ above the plane.

CAIE M1 2022 June Q4

4 A particle $A$, moving along a straight horizontal track with constant speed $8 \mathrm {~m} \mathrm {~s} ^ { - 1 }$, passes a fixed point $O$. Four seconds later, another particle $B$ passes $O$, moving along a parallel track in the same direction as $A$. Particle $B$ has speed $20 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ when it passes $O$ and has a constant deceleration of $2 \mathrm {~m} \mathrm {~s} ^ { - 2 } . B$ comes to rest when it returns to $O$.

Find expressions, in terms of $t$, for the displacement from $O$ of each particle $t$ seconds after $B$ passes $O$.
Find the values of $t$ when the particles are the same distance from $O$.
On the given axes, sketch the displacement-time graphs for both particles, for values of $t$ from 0 to 20 .
\includegraphics[max width=\textwidth, alt={}, center]{4ee2568e-5902-442f-9af1-3663fa1d59c7-07_805_1259_1672_484}
\includegraphics[max width=\textwidth, alt={}, center]{4ee2568e-5902-442f-9af1-3663fa1d59c7-08_467_583_255_781} A block of mass 12 kg is placed on a plane which is inclined at an angle of $24 ^ { \circ }$ to the horizontal. A light string, making an angle of $36 ^ { \circ }$ above a line of greatest slope, is attached to the block. The tension in the string is 65 N (see diagram). The coefficient of friction between the block and plane is $\mu$. The block is in limiting equilibrium and is on the point of sliding up the plane. Find $\mu$.

CAIE M1 2022 June Q6

6 A car of mass 900 kg is moving up a hill inclined at $\sin ^ { - 1 } 0.12$ to the horizontal. The initial speed of the car is $11 \mathrm {~m} \mathrm {~s} ^ { - 1 }$. After 12 s, the car has travelled 150 m up the hill and has speed $16 \mathrm {~ms} ^ { - 1 }$. The engine of the car is working at a constant rate of 24 kW .

Find the work done against the resistive forces during the 12 s .
The car then travels along a straight horizontal road. There is a resistance to the motion of the car of $( 1520 + 4 v ) \mathrm { N }$ when the speed of the car is $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$. The car travels at a constant speed with the engine working at a constant rate of 32 kW .
Find this speed.

CAIE M1 2022 June Q7

7 A particle $P$ moves in a straight line. The velocity $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$ at time $t$ seconds is given by $$\begin{array} { l l } v = 0.5 t & \text { for } 0 \leqslant t \leqslant 10
v = 0.25 t ^ { 2 } - 8 t + 60 & \text { for } 10 \leqslant t \leqslant 20 \end{array}$$

Show that there is an instantaneous change in the acceleration of the particle at $t = 10$.
Find the total distance covered by $P$ in the interval $0 \leqslant t \leqslant 20$.
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 2022 June Q1

1 Two particles $P$ and $Q$, of masses 0.3 kg and 0.2 kg respectively, are at rest on a smooth horizontal plane. $P$ is projected at a speed of $4 \mathrm {~ms} ^ { - 1 }$ directly towards $Q$. After $P$ and $Q$ collide, $Q$ begins to move with a speed of $3 \mathrm {~m} \mathrm {~s} ^ { - 1 }$.

Find the speed of $P$ after the collision.
After the collision, $Q$ moves directly towards a third particle $R$, of mass $m \mathrm {~kg}$, which is at rest on the plane. The two particles $Q$ and $R$ coalesce on impact and move with a speed of $2 \mathrm {~ms} ^ { - 1 }$.
Find $m$.

CAIE M1 2022 June Q2

2 A particle $P$ is projected vertically upwards from horizontal ground. $P$ reaches a maximum height of 45 m . After reaching the ground, $P$ comes to rest without rebounding.

Find the speed at which $P$ was projected.
Find the total time for which the speed of $P$ is at least $10 \mathrm {~m} \mathrm {~s} ^ { - 1 }$.

CAIE M1 2022 June Q3

3
\includegraphics[max width=\textwidth, alt={}, center]{4e555003-16f1-4453-ab25-c50929d4b5b3-04_824_1636_264_258} The displacement of a particle moving in a straight line is $s$ metres at time $t$ seconds after leaving a fixed point $O$. The particle starts from rest and passes through points $P , Q$ and $R$, at times $t = 5 , t = 10$ and $t = 15$ respectively, and returns to $O$ at time $t = 20$. The distances $O P , O Q$ and $O R$ are 50 m , 150 m and 200 m respectively. The diagram shows a displacement-time graph which models the motion of the particle from $t = 0$ to $t = 20$. The graph consists of two curved segments $A B$ and $C D$ and two straight line segments $B C$ and $D E$.

Find the speed of the particle between $t = 5$ and $t = 10$.
Find the acceleration of the particle between $t = 0$ and $t = 5$, given that it is constant.
Find the average speed of the particle during its motion.
\includegraphics[max width=\textwidth, alt={}, center]{4e555003-16f1-4453-ab25-c50929d4b5b3-06_483_880_258_630} The diagram shows a block of mass 10 kg suspended below a horizontal ceiling by two strings $A C$ and $B C$, of lengths 0.8 m and 0.6 m respectively, attached to fixed points on the ceiling. Angle $A C B = 90 ^ { \circ }$. There is a horizontal force of magnitude $F \mathrm {~N}$ acting on the block. The block is in equilibrium.

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