Questions - Edexcel M1 - A-Level Maths

Sketch, on the same axes, the speed-time graph for the motion of the car and the speed-time graph for the motion of the motorcyclist, from time $t = 0$ to the instant when the motorcyclist catches up with the car. At the instant when $t = t _ { 1 }$ seconds, the car and the motorcyclist are moving at the same speed.
Find the value of $t _ { 1 }$
Show that $T ^ { 2 } + k T - 300 = 0$, where $k$ is a constant to be found. DO NOT WRITEIN THIS AREA

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Edexcel M1 2020 January Q6

6. A force $\mathbf { F }$ is given by $\mathbf { F } = ( 10 \mathbf { i } + \mathbf { j } ) \mathrm { N }$.

Find the exact value of the magnitude of $\mathbf { F }$.
Find, in degrees, the size of the angle between the direction of $\mathbf { F }$ and the direction of the vector $( \mathbf { i } + \mathbf { j } )$. The resultant of the force $\mathbf { F }$ and the force $( - 15 \mathbf { i } + a \mathbf { j } ) \mathrm { N }$, where $a$ is a constant, is parallel to, but in the opposite direction to, the vector $( 2 \mathbf { i } - 3 \mathbf { j } )$.
Find the value of $a$. \includegraphics[max width=\textwidth, alt={}, center]{916543cb-14f7-486c-ba3c-eda9be134045-19_104_59_2613_1886}

Edexcel M1 2020 January Q7

7. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{916543cb-14f7-486c-ba3c-eda9be134045-20_663_1290_260_335} \captionsetup{labelformat=empty} \caption{Figure 3}

\end{figure} A particle $A$ of mass 4 kg is held at rest on a rough horizontal table. Particle $A$ is attached to one end of a string that passes over a pulley $P$. The pulley is fixed the the of the table. The other end of the string is attached to a particle $B$, of mass 3 kg , which hangs freely below $P$. The part of the string from $A$ to $P$ is perpendicular to the edge of the table and $A , P$ and $B$ all lie in the same vertical plane. The string is modelled as being light and inextensible and the pulley is modelled as being small, smooth and light. The system is released from rest with the string taut. At the instant of release, $A$ is 2 m from the edge of the table and $B$ is 1.4 m above a horizontal floor, as shown in Figure 3. After descending with constant acceleration for 2 seconds, $B$ hits the floor and does not rebound.

Show that the acceleration of $A$ before $B$ hits the floor is $0.7 \mathrm {~ms} ^ { - 2 }$
State which of the modelling assumptions you have used in order to answer part (a).
Find the magnitude of the resultant force exerted on the pulley by the string. The coefficient of friction between $A$ and the table is $\mu$.
Find the value of $\mu$.
Determine, by calculation, whether or not $A$ reaches the pulley. DO NOT WRITEIN THIS AREA

VIXV SIHIANI III IM IONOO VIAV SIHI NI JYHAM ION OO VI4V SIHI NI JLIYM ION OO
\includegraphics[max width=\textwidth, alt={}, center]{916543cb-14f7-486c-ba3c-eda9be134045-23_2255_50_314_34}

Edexcel M1 2021 January Q1

A small stone is projected vertically upwards with speed $20 \mathrm {~ms} ^ { - 1 }$ from a point $O$ which is 5 m above horizontal ground. The stone is modelled as a particle moving freely under gravity.

Find

the speed of the stone at the instant when it is 2 m above the ground,
the total time between the instant when the stone is projected from $O$ and the instant when it first strikes the ground.

Edexcel M1 2021 January Q2

2. Two particles, $P$ and $Q$, have masses $2 m$ and $m$ respectively. The particles are moving towards each other in opposite directions along the same straight line on a smooth horizontal plane. The particles collide directly. Immediately before the collision, the speed of $P$ is $3 u$ and the speed of $Q$ is $2 u$. The magnitude of the impulse exerted on $Q$ by $P$ in the collision is 5mu. Find

the speed of $P$ immediately after the collision,
the speed of $Q$ immediately after the collision.

Edexcel M1 2021 January Q3

3. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{ca445c1e-078c-4a57-94df-de90f30f8efd-06_156_1009_255_470} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} A parcel of mass 20 kg is at rest on a rough horizontal floor. The coefficient of friction between the parcel and the floor is 0.3 Two forces, both acting in the same vertical plane, of magnitudes 200 N and $T \mathrm {~N}$ are applied to the parcel. The line of action of the 200 N force makes an angle of $15 ^ { \circ }$ with the horizontal and the line of action of the $T \mathrm {~N}$ force makes an angle of $25 ^ { \circ }$ with the horizontal, as shown in Figure 1. The parcel is modelled as a particle $P$. Find the smallest value of $T$ for which $P$ remains in equilibrium.

Edexcel M1 2021 January Q4

4. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{ca445c1e-078c-4a57-94df-de90f30f8efd-08_426_1428_118_258} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} \begin{verbatim} A metal girder $A B$ has weight $W$ newtons and length 6 m . The girder rests in a horizontal position on two supports $C$ and $D$ where $A C = D B = 1 \mathrm {~m}$, as shown in Figure 2. When a force of magnitude 900 N is applied vertically upwards to the girder at $A$, the girder is about to tilt about $D$. When a force of magnitude 1500 N is applied vertically upwards to the girder at $B$, the girder is about to tilt about $C$. The girder is modelled as a non-uniform rod whose centre of mass is a distance $x$ metres from $A$. Find the value of $x$. A metal girder AB has weight When a force of magnitude 1500 N is applied vertically upwards to the girder at $B$, the girder is about to tilt about $C$. The girder is modelled as a non-uniform rod whose centre of mass is a distance $x$ metres from $A$. Find the value of $x$. \end{verbatim}

Edexcel M1 2021 January Q5

5. A particle is acted upon by two forces $\mathbf { F }$ and $\mathbf { G }$. The force $\mathbf { F }$ has magnitude 8 N and acts in a direction with a bearing of $240 ^ { \circ }$. The force $\mathbf { G }$ has magnitude 10 N and acts due South. Given that $\mathbf { R } = \mathbf { F } + \mathbf { G }$, find

the magnitude of $\mathbf { R }$,
the direction of $\mathbf { R }$, giving your answer as a bearing to the nearest degree. in a direction with a bearing of $240 ^ { \circ }$. The force $\mathbf { G }$ has magnitude 10 N and acts due South. Given that $\mathbf { R } = \mathbf { F } + \mathbf { G }$, find

Edexcel M1 2021 January Q6

6. Two girls, Agatha and Brionie, are roller skating inside a large empty building. The girls are modelled as particles. At time $t = 0$, Agatha is at the point with position vector $( 11 \mathbf { i } + 11 \mathbf { j } ) \mathrm { m }$ and Brionie is at the point with position vector $( 7 \mathbf { i } + 16 \mathbf { j } ) \mathrm { m }$. The position vectors are given relative to the door, $O$, and $\mathbf { i }$ and $\mathbf { j }$ are horizontal perpendicular unit vectors. Agatha skates with constant velocity ( $3 \mathbf { i } - \mathbf { j }$ ) $\mathrm { m } \mathrm { s } ^ { - 1 }$
Brionie skates with constant velocity ( $4 \mathbf { i } - 2 \mathbf { j }$ ) $\mathrm { m } \mathrm { s } ^ { - 1 }$

Find the position vector of Agatha at time $t$ seconds. At time $t = 6$ seconds, Agatha passes through the point $P$.
Show that Brionie also passes through $P$ and find the value of $t$ when this occurs. At time $t$ seconds, Agatha is at the point $A$ and Brionie is at the point $B$.
Show that $\overrightarrow { A B } = [ ( t - 4 ) \mathbf { i } + ( 5 - t ) \mathbf { j } ] \mathrm { m }$
Find the distance between the two girls when they are closest together. \includegraphics[max width=\textwidth, alt={}, center]{ca445c1e-078c-4a57-94df-de90f30f8efd-13_2255_50_314_34}

VIXV SIHIANI III IM IONOO VIAV SIHI NI JYHAM ION OO VI4V SIHI NI JLIYM ION OO

Edexcel M1 2021 January Q7

7. A helicopter is hovering at rest above horizontal ground at the point $H$. A parachutist steps out of the helicopter and immediately falls vertically and freely under gravity from rest for 2.5 s . His parachute then opens and causes him to immediately decelerate at a constant rate of $3.9 \mathrm {~m} \mathrm {~s} ^ { - 2 }$ for $T$ seconds ( $T < 6$ ), until his speed is reduced to $V \mathrm {~m} \mathrm {~s} ^ { - 1 }$. He then moves with this constant speed $V \mathrm {~m} \mathrm {~s} ^ { - 1 }$ until he hits the ground. While he is decelerating, he falls a distance of 73.75 m . The total time between the instant when he leaves $H$ and the instant when he hits the ground is 20 s . The parachutist is modelled as a particle.

Find the speed of the parachutist at the instant when his parachute opens.
Sketch a speed-time graph for the motion of the parachutist from the instant when he leaves $H$ to the instant when he hits the ground.
Find the value of $T$.
Find, to the nearest metre, the height of the point $H$ above the ground.
7. A helicopter is hovering at rest above horizontal ground at the point $H$. A parachutist steps

Edexcel M1 2021 January Q8

8. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{ca445c1e-078c-4a57-94df-de90f30f8efd-20_369_1264_248_342} \captionsetup{labelformat=empty} \caption{Figure 3}

\end{figure} Two particles, $A$ and $B$, have masses 2 kg and 4 kg respectively. The particles are connected by a light inextensible string. The string passes over a small smooth pulley which is fixed at the top of a rough plane. The plane is inclined to the horizontal ground at an angle $\alpha$ where $\tan \alpha = \frac { 3 } { 4 }$. The particle $A$ is held at rest on the plane at a distance $d$ metres from the pulley. The particle $B$ hangs freely at rest, vertically below the pulley, at a distance $h$ metres above the ground, as shown in Figure 3. The part of the string between $A$ and the pulley is parallel to a line of greatest slope of the plane. The coefficient of friction between $A$ and the plane is $\frac { 1 } { 4 }$ The system is released from rest with the string taut and $B$ descends.

Find the tension in the string as $B$ descends. On hitting the ground, $B$ immediately comes to rest. Given that $A$ comes to rest before reaching the pulley,
find, in terms of $h$, the range of possible values of $d$.
State one physical factor, other than air resistance, that could be taken into account to make the model described above more realistic.

Edexcel M1 2022 January Q1

1. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{f1bdc84b-c8a1-4e7c-a2ba-48b40c6a6d36-02_486_638_248_653} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} A particle $P$ of weight 5 N is attached to one end of a light inextensible string. The other end of the string is attached to a fixed point $O$. The particle $P$ is held in equilibrium by a force of magnitude $F$ newtons. The direction of this force is perpendicular to the string and $O P$ makes an angle of $60 ^ { \circ }$ with the vertical, as shown in Figure 1. Find

the value of $F$
the tension in the string.

Edexcel M1 2022 January Q2

2. A particle $P$ has mass $k m$ and a particle $Q$ has mass $m$. The particles are moving towards each other in opposite directions along the same straight line when they collide directly. Immediately before the collision, $P$ has speed $3 u$ and $Q$ has speed $u$.
As a result of the collision, the direction of motion of each particle is reversed and the speed of each particle is halved.

Find the value of $k$.
Find, in terms of $m$ and $u$, the magnitude of the impulse exerted on $Q$ in the collision.

Edexcel M1 2022 January Q3

3. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{f1bdc84b-c8a1-4e7c-a2ba-48b40c6a6d36-06_328_1356_244_296} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} A beam $A D C B$ has length 5 m . The beam lies on a horizontal step with the end $A$ on the step and the end $B$ projecting over the edge of the step. The edge of the step is at the point $D$ where $D B = 1.3 \mathrm {~m}$, as shown in Figure 2. When a small boy of mass 30 kg stands on the beam at $C$, where $C B = 0.5 \mathrm {~m}$, the beam is on the point of tilting. The boy is modelled as a particle and the beam is modelled as a uniform rod.

Find the mass of the beam. A block of mass $X \mathrm {~kg}$ is now placed on the beam at $A$.
The block is modelled as a particle.
Find the smallest value of $X$ that will enable the boy to stand on the beam at $B$ without the beam tilting.
State how you have used the modelling assumption that the block is a particle in your calculations.

Edexcel M1 2022 January Q4

4. At time $t = 0$, a small ball is projected vertically upwards from a point $A$ which is 24.5 m above the ground. The ball first comes to instantaneous rest at the point $B$, where $A B = 19.6 \mathrm {~m}$ and first hits the ground at time $t = T$ seconds. The ball is modelled as a particle moving freely under gravity.

Find the value of $T$.
Sketch a speed-time graph for the motion of the ball from $t = 0$ to $t = T$ seconds.
(No further calculations are needed in order to draw this sketch.)

Edexcel M1 2022 January Q5

5. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{f1bdc84b-c8a1-4e7c-a2ba-48b40c6a6d36-14_209_511_246_721} \captionsetup{labelformat=empty} \caption{Figure 3}

\end{figure} A particle of mass $m$ rests in equilibrium on a fixed rough plane under the action of a force of magnitude $X$. The force acts up a line of greatest slope of the plane, as shown in Figure 3. The plane is inclined at an angle $\alpha$ to the horizontal, where $\tan \alpha = \frac { 3 } { 4 }$
The coefficient of friction between the particle and the plane is $\mu$.

When $X = 2 P$, the particle is on the point of sliding up the plane.
When $X = P$, the particle is on the point of sliding down the plane.

Find the value of $\mu$.

Edexcel M1 2022 January Q6

\hspace{0pt} [In this question $\mathbf { i }$ and $\mathbf { j }$ are horizontal unit vectors.]

A particle $P$ of mass 2 kg moves under the action of two forces, $( p \mathbf { i } + q \mathbf { j } ) \mathrm { N }$ and $( 2 q \mathbf { i } + p \mathbf { j } ) \mathrm { N }$, where $p$ and $q$ are constants. Given that the acceleration of $P$ is $( \mathbf { i } - \mathbf { j } ) \mathrm { ms } ^ { - 2 }$

find the value of $p$ and the value of $q$.
Find the size of the angle between the direction of the acceleration and the vector $\mathbf { j }$. At time $t = 0$, the velocity of $P$ is $( 3 \mathbf { i } - 4 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }$
At $t = T$ seconds, $P$ is moving in the direction of the vector $( 11 \mathbf { i } - 13 \mathbf { j } )$.
Find the value of $T$.

Edexcel M1 2022 January Q7

7. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{f1bdc84b-c8a1-4e7c-a2ba-48b40c6a6d36-22_342_1203_246_374} \captionsetup{labelformat=empty} \caption{Figure 4}

\end{figure} A particle $P$ of mass $4 m$ lies on the surface of a fixed rough inclined plane.
The plane is inclined to the horizontal at an angle $\alpha$ where $\tan \alpha = \frac { 3 } { 4 }$
The particle $P$ is attached to one end of a light inextensible string.
The string passes over a small smooth pulley that is fixed at the top of the plane. The other end of the string is attached to a particle $Q$ of mass $m$ which lies on a smooth horizontal plane. The string lies along the horizontal plane and in the vertical plane that contains the pulley and a line of greatest slope of the inclined plane. The system is released from rest with the string taut, as shown in Figure 4, and $P$ moves down the plane. The coefficient of friction between $P$ and the plane is $\frac { 1 } { 4 }$
For the motion before $Q$ reaches the pulley

write down an equation of motion for $Q$,
find, in terms of $m$ and $g$, the tension in the string,
find the magnitude of the force exerted on the pulley by the string.
State where in your working you have used the information that the string is light.

Edexcel M1 2022 January Q8

8. [In this question $\mathbf { i }$ and $\mathbf { j }$ are horizontal unit vectors directed due east and due north respectively and position vectors are given relative to a fixed origin.] A ship $A$ moves with constant velocity $( 3 \mathbf { i } - 10 \mathbf { j } ) \mathrm { kmh } ^ { - 1 }$
At time $t$ hours, the position vector of $A$ is $\mathbf { r } \mathrm { km }$.
At time $t = 0 , A$ is at the point with position vector $( 13 \mathbf { i } + 5 \mathbf { j } ) \mathrm { km }$.

Find $\mathbf { r }$ in terms of $t$. Another ship $B$ moves with constant velocity $( 15 \mathbf { i } + 14 \mathbf { j } ) \mathrm { km } \mathrm { h } ^ { - 1 }$
At time $t = 0 , B$ is at the point with position vector $( 3 \mathbf { i } - 5 \mathbf { j } ) \mathrm { km }$.
Show that, at time $t$ hours, $$\overrightarrow { A B } = [ ( 12 t - 10 ) \mathbf { i } + ( 24 t - 10 ) \mathbf { j } ] \mathrm { km }$$ Given that the two ships do not change course,
find the shortest distance between the two ships,
find the bearing of ship $B$ from ship $A$ when the ships are closest.

\includegraphics[max width=\textwidth, alt={}]{f1bdc84b-c8a1-4e7c-a2ba-48b40c6a6d36-28_2820_1967_102_100}

Edexcel M1 2023 January Q1

A train travels along a straight horizontal track between two stations $A$ and $B$.

The train starts from rest at station $A$ and accelerates uniformly for $T$ seconds until it reaches a speed of $20 \mathrm {~ms} ^ { - 1 }$ The train then travels at a constant speed of $20 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ for 3 minutes before decelerating uniformly until it comes to rest at station $B$. The magnitude of the acceleration of the train is twice the magnitude of the deceleration.

On the axes below, sketch a speed-time graph to illustrate the motion of the train as it moves from station $A$ to station $B$.
\includegraphics[max width=\textwidth, alt={}, center]{84c0eead-0a87-4d87-b33d-794a94bb466c-02_670_1422_813_312} If you need to redraw your graph, use the axes on page 3 Stations $A$ and $B$ are 4.8 km apart.
Find the value of $T$
Find the acceleration of the train during the first $T$ seconds of its motion. Only use these axes if you need to redraw your graph. ${ } _ { O } ^ { \substack { \text { speed }
\left( \mathrm { ms } ^ { - 1 } \right) } }$

Edexcel M1 2023 January Q2

2. Two particles, $A$ and $B$, are moving in a straight line in opposite directions towards each other on a smooth horizontal surface when they collide directly. Particle $A$ has mass $3 m \mathrm {~kg}$ and particle $B$ has mass $m \mathrm {~kg}$.
Immediately before the collision, both particles have a speed of $1.5 \mathrm {~ms} ^ { - 1 }$
Immediately after the collision, the direction of motion of $A$ is unchanged and the difference between the speed of $A$ and speed of $B$ is $1 \mathrm {~ms} ^ { - 1 }$

Find (i) the speed of $A$ immediately after the collision,
(ii) the speed of $B$ immediately after the collision.
Find, in terms of $m$, the magnitude of the impulse exerted on $B$ in the collision.

Edexcel M1 2023 January Q3

A particle $P$ is moving with constant acceleration ( $- 4 \mathbf { i } + \mathbf { j }$ ) $\mathrm { ms } ^ { - 2 }$

At time $t = 0 , P$ has velocity $( 14 \mathbf { i } - 5 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }$

Find the speed of $P$ at time $t = 2$ seconds.
Find the size of the angle between the direction of $\mathbf { i }$ and the direction of motion of $P$ at time $t = 2$ seconds. At time $t = T$ seconds, $P$ is moving in the direction of vector ( $2 \mathbf { i } - 3 \mathbf { j }$ )
Find the value of $T$

Edexcel M1 2023 January Q4

4. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{84c0eead-0a87-4d87-b33d-794a94bb466c-10_419_1445_283_312} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} A branch $A B$, of length 1.5 m , rests horizontally in equilibrium on two supports.
The two supports are at the points $C$ and $D$, where $A C = 0.24 \mathrm {~m}$ and $D B = 0.36 \mathrm {~m}$, as shown in Figure 1. When a force of 150 N is applied vertically upwards at $B$, the branch is on the point of tilting about $C$. When a force of 225 N is applied vertically downwards at $B$, the branch is on the point of tilting about $D$. The branch is modelled as a non-uniform rod $A B$ of weight $W$ newtons.
The distance from the point $C$ to the centre of mass of the rod is $x$ metres.
Use the model to find

the value of $W$
the value of $x$

Edexcel M1 2023 January Q5

5. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{84c0eead-0a87-4d87-b33d-794a94bb466c-14_117_1393_328_337} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} Three points $P , Q$ and $R$ are on a horizontal road where $P Q R$ is a straight line.
The point $Q$ is between $P$ and $R$, with $P Q = 6 x$ metres and $Q R = 5 x$ metres, as shown in Figure 2. A vehicle moves along the road from $P$ to $Q$ with constant acceleration.
The vehicle is modelled as a particle.
At time $t = 0$, the vehicle passes $P$ with speed $u \mathrm {~ms} ^ { - 1 }$
At time $t = 12 \mathrm {~s}$, the vehicle passes $Q$ with speed $2 u \mathrm {~ms} ^ { - 1 }$
Using the model,

show that $x = 3 u$ As the vehicle passes $Q$, the acceleration of the vehicle changes instantaneously to $1.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }$ The vehicle continues to move with a constant acceleration of $1.5 \mathrm {~ms} ^ { - 2 }$ and passes $R$ with speed $3 u \mathrm {~ms} ^ { - 1 }$ Using the model,
find the value of $u$,
find the distance travelled by the vehicle during the first 14 seconds after passing $P$

Edexcel M1 2023 January Q6

6. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{84c0eead-0a87-4d87-b33d-794a94bb466c-18_502_1429_280_319} \captionsetup{labelformat=empty} \caption{Figure 3}

\end{figure} A boat is pulled along a river at a constant speed by two ropes.
The banks of the river are parallel and the boat travels horizontally in a straight line, parallel to the riverbanks.

The tension in the first rope is 500 N acting at an angle of $40 ^ { \circ }$ to the direction of motion, as shown in Figure 3.
The tension in the second rope is $P$ newtons, acting at an angle of $\alpha ^ { \circ }$ to the direction of motion, also shown in Figure 3.
The resistance to motion of the boat as it moves through the water is a constant force of magnitude 900 N

The boat is modelled as a particle. The ropes are modelled as being light and lying in a horizontal plane. Use the model to find

the value of $\alpha$
the value of $P$

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