Questions M1 - A-Level Maths

\hspace{0pt} [In this question $\mathbf { i }$ and $\mathbf { j }$ are horizontal unit vectors due east and due north respectively and position vectors are given relative to a fixed origin $O$ ]

A particle $P$ is moving with velocity $( \mathbf { i } - 2 \mathbf { j } ) \mathrm { km } \mathrm { h } ^ { - 1 }$. At time $t = 0$ hours, the position vector of $P$ is $( - 5 \mathbf { i } + 9 \mathbf { j } ) \mathrm { km }$. At time $t$ hours, the position vector of $P$ is $\mathbf { p } \mathrm { km }$.

Find an expression for $\mathbf { p }$ in terms of $t$. The point $A$ has position vector ( $3 \mathbf { i } + 2 \mathbf { j }$ ) km.
Find the position vector of $P$ when $P$ is due west of $A$. Another particle $Q$ is moving with velocity $[ ( 2 b - 1 ) \mathbf { i } + ( 5 - 2 b ) \mathbf { j } ] \mathrm { km } \mathrm { h } ^ { - 1 }$ where $b$ is a constant. Given that the particles are moving along parallel lines,
find the value of $b$.

Edexcel M1 2016 October Q5

9 marks Standard +0.3

5. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{6978be48-561b-49a0-a297-c8886ca66c19-10_419_933_123_525} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} A particle $P$ of mass 0.5 kg is at rest on a rough plane which is inclined to the horizontal at $30 ^ { \circ }$. The particle is held in equilibrium by a force of magnitude 8 N , acting at an angle of $40 ^ { \circ }$ to the plane, as shown in Figure 2. The line of action of the force lies in the vertical plane containing $P$ and a line of greatest slope of the plane. The coefficient of friction between $P$ and the plane is $\mu$. Given that $P$ is on the point of sliding up the plane, find the value of $\mu$.

Edexcel M1 2016 October Q6

9 marks Standard +0.3

6. Two cars $A$ and $B$ are moving in the same direction along a straight horizontal road. Car $A$ is moving with uniform acceleration $0.4 \mathrm {~ms} ^ { - 2 }$ and car $B$ is moving with uniform acceleration $0.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }$. At the instant when $B$ is 200 m behind $A$, the speed of $A$ is $35 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ and the speed of $B$ is $44 \mathrm {~ms} ^ { - 1 }$. Find the speed of $B$ when it overtakes $A$.
(9)

Edexcel M1 2016 October Q7

11 marks Standard +0.3

7. A train moves on a straight horizontal track between two stations $A$ and $B$. The train starts from rest at $A$ and moves with constant acceleration $1 \mathrm {~ms} ^ { - 2 }$ until it reaches a speed of $V \mathrm {~m} \mathrm {~s} ^ { - 1 }$. The train maintains this speed of $V \mathrm {~m} \mathrm {~s} ^ { - 1 }$ for the next $T$ seconds before slowing down with constant deceleration $0.5 \mathrm {~ms} ^ { - 2 }$, coming to rest at $B$. The journey from $A$ to $B$ takes 180 s and the distance between the stations is 4800 m .

Sketch a speed-time graph for the motion of the train from $A$ to $B$.
Show that $T = 180 - 3 V$.
Find the value of $V$.

Edexcel M1 2016 October Q8

13 marks Standard +0.3

8. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{6978be48-561b-49a0-a297-c8886ca66c19-20_312_1068_230_438} \captionsetup{labelformat=empty} \caption{Figure 3}

\end{figure} Two particles $P$ and $Q$ have masses 2 kg and 3 kg respectively. The particles are attached to the ends of a light inextensible string. The string passes over a smooth light pulley which is fixed at the top of a rough plane. The plane is inclined to horizontal ground at an angle $\alpha$, where tan $\alpha = \frac { 3 } { 4 }$. Initially $P$ is held at rest on the inclined plane with the part of the string from $P$ to the pulley parallel to a line of greatest slope of the plane. The particle $Q$ hangs freely below the pulley at a height of 0.5 m above the ground, as shown in Figure 3. The coefficient of friction between $P$ and the plane is $\mu$. The system is released from rest, with the string taut, and $Q$ strikes the ground before $P$ reaches the pulley. The speed of $Q$ at the instant when it strikes the ground is $1.4 \mathrm {~m} \mathrm {~s} ^ { - 1 }$.

For the motion before $Q$ strikes the ground, find the tension in the string.
Find the value of $\mu$.
END

Edexcel M1 2017 October Q1

7 marks Moderate -0.3

A suitcase of mass 40 kg is being dragged in a straight line along a rough horizontal floor at constant speed using a thin strap. The strap is inclined at $20 ^ { \circ }$ above the horizontal. The coefficient of friction between the suitcase and the floor is $\frac { 3 } { 4 }$. The strap is modelled as a light inextensible string and the suitcase is modelled as a particle. Find the tension in the strap.

\begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{0fccbe27-ff7a-4c63-bb08-770b138696b7-03_442_1296_114_324} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} A metal girder $A B$, of weight 1080 N and length 6 m , rests in equilibrium in a horizontal position on two supports, one at $C$ and one at $D$, where $A C = 0.5 \mathrm {~m}$ and $B D = 2 \mathrm {~m}$, as shown in Figure 1. A boy of weight 400 N stands on the girder at $B$ and the girder remains horizontal and in equilibrium. The boy is modelled as a particle and the girder is modelled as a uniform rod.

Find
1. the magnitude of the reaction on the girder at $C$,
2. the magnitude of the reaction on the girder at $D$.
  (6) The boy now stands at a point $E$ on the girder, where $A E = x$ metres, and the girder remains horizontal and in equilibrium. Given that the magnitude of the reaction on the girder at $D$ is now 520 N greater than the magnitude of the reaction on the girder at $C$,
find the value of $x$.

Edexcel M1 2017 October Q3

6 marks Moderate -0.3

Two particles $P$ and $Q$ have masses $4 m$ and $m$ respectively. They are moving in opposite directions towards each other along the same straight line on a smooth horizontal plane and collide directly. Immediately before the collision the speed of $P$ is $2 u$ and the speed of $Q$ is $4 u$. In the collision, the particles join together to form a single particle.

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

Edexcel M1 2017 October Q4

9 marks Moderate -0.3

4. Two forces $\mathbf { F } _ { 1 }$ and $\mathbf { F } _ { 2 }$ act on a particle. The force $\mathbf { F } _ { 1 }$ has magnitude 8 N and acts due east. The resultant of $\mathbf { F } _ { 1 }$ and $\mathbf { F } _ { 2 }$ is a force of magnitude 14 N acting in a direction whose bearing is $120 ^ { \circ }$. Find

the magnitude of $\mathbf { F } _ { 2 }$,
the direction of $\mathbf { F } _ { 2 }$, giving your answer as a bearing to the nearest degree.

Edexcel M1 2017 October Q5

11 marks Moderate -0.8

A small ball is projected vertically upwards from a point $O$ with speed $14.7 \mathrm {~m} \mathrm {~s} ^ { - 1 }$. The point $O$ is 2.5 m above the ground. The motion of the ball is modelled as that of a particle moving freely under gravity.

Find

the maximum height above the ground reached by the ball,
the time taken for the ball to first reach a height of 1 m above the ground,
the speed of the ball at the instant before it strikes the ground for the first time.

Edexcel M1 2017 October Q6

14 marks Standard +0.3

An athlete goes for a run along a straight horizontal road. Starting from rest, she accelerates at $0.6 \mathrm {~m} \mathrm {~s} ^ { - 2 }$ up to a speed of $V \mathrm {~m} \mathrm {~s} ^ { - 1 }$. She then maintains this constant speed of $V \mathrm {~m} \mathrm {~s} ^ { - 1 }$ before finally decelerating at $0.2 \mathrm {~m} \mathrm {~s} ^ { - 2 }$ back to rest. She covers a total distance of 1500 m in 270 s .
1. Sketch a speed-time graph to represent the athlete's run.
2. Show that she accelerates for $\frac { 5 V } { 3 }$ seconds.
3. Show that $V ^ { 2 } - k V + 450 = 0$, where $k$ is a constant to be found.
4. Find the value of $V$, justifying your answer.

Edexcel M1 2017 October Q7

17 marks Standard +0.3

7. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{0fccbe27-ff7a-4c63-bb08-770b138696b7-13_349_1347_248_303} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} Figure 2 shows two particles $A$ and $B$, of masses $3 m$ and $4 m$ respectively, attached to the ends of a light inextensible string. Initially $A$ is held at rest 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 coefficient of friction between $A$ and the plane is $\frac { 1 } { 4 }$. The string passes over a small smooth light pulley $P$ which is fixed at the top of the plane. The part of the string from $A$ to $P$ is parallel to a line of greatest slope of the plane. The particle $B$ hangs freely and is vertically below $P$. The system is released from rest with the string taut and with $B$ at a height of 1.75 m above the ground. In the subsequent motion, $A$ does not hit the pulley. For the period before $B$ hits the ground,

write down an equation of motion for each particle.
Hence show that the acceleration of $B$ is $\frac { 8 } { 35 } \mathrm {~g}$.
Explain how you have used the fact that the string is inextensible in your calculation. When $B$ hits the ground, $B$ does not rebound and comes immediately to rest.
Find the distance travelled by $A$ from the instant when the system is released to the instant when $A$ first comes to rest.

Edexcel M1 2018 October Q1

6 marks Moderate -0.8

A particle $P$ of mass 0.8 kg is moving along a straight horizontal line on a smooth hoizontal surface with speed $4 \mathrm {~ms} ^ { - 1 }$. A second particle $Q$ of mass 2 kg is moving, in the opposite direction to $P$, along the same straight line with speed $2 \mathrm {~ms} ^ { - 1 }$. The particles collide directly. Immediately after the collision the direction of motion of each particle is reversed and the speed of $P$ is $2.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }$.
1. Find the speed of $Q$ immediately after the collision.
2. Find the magnitude of the impulse exerted by $Q$ on $P$ in the collision, stating the units of your answer.
VILU SIHI NI III M I ION OC VIIV 5141 NI 311814 ION OC VI4V SIHI NI JIIYM ION OC
Figure 1 A non-uniform plank $A B$ has weight 60 N and length 5 m . The plank rests horizontally in equilibrium on two smooth supports at $A$ and $C$, where $A C = 3 \mathrm {~m}$, as shown in Figure 1. A parcel of weight 12 N is placed on the plank at $B$ and the plank remains horizontal and in equilibrium. The magnitude of the reaction of the support at $A$ on the plank is half the magnitude of the reaction of the support at $C$ on the plank. By modelling the plank as a non-uniform rod and the parcel as a particle,
find the distance of the centre of mass of the plank from $A$.
State briefly how you have used the modelling assumption
1. that the parcel is a particle,
2. that the plank is a rod.

Edexcel M1 2018 October Q2

8 marks Easy -1.2

2.
\includegraphics[max width=\textwidth, alt={}, center]{5f2d38d9-b719-4205-8cb0-caa959afc46f-04_269_1175_296_375}

Edexcel M1 2018 October Q3

7 marks Standard +0.3

At time $t = 0$, a stone is thrown vertically upwards with speed $19.6 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ from a point $A$ which is $h$ metres above horizontal ground. At time $t = 3 \mathrm {~s}$, another stone is released from rest from a point $B$ which is also $h$ metres above the same horizontal ground. Both stones hit the ground at time $t = T$ seconds. The motion of each stone is modelled as that of a particle moving freely under gravity.

Find

the value of $T$,
the value of $h$.

VILU SIHI NI III M I ION OC VIIV 5141 NI JINAM ION OC VI4V SIHI NI JIIYM ION OO

Edexcel M1 2018 October Q4

7 marks Moderate -0.3

4. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{5f2d38d9-b719-4205-8cb0-caa959afc46f-12_540_584_294_680} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} A particle $P$ of mass $m \mathrm {~kg}$ is attached to one end of a light inextensible string of length 2.5 m . The other end of the string is attached to a fixed point $A$ on a vertical wall. The tension in the string is 16 N . The particle is held in equilibrium by a force of magnitude $F$ newtons, acting in the vertical plane which is perpendicular to the wall and contains the string. This force acts in a direction perpendicular to the string, as shown in Figure 2. Given that the horizontal distance of $P$ from the wall is 1.5 m , find

the value of $F$,
the value of $m$.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{5f2d38d9-b719-4205-8cb0-caa959afc46f-16_186_830_292_557} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} Two posts, $A$ and $B$, are fixed at the side of a straight horizontal road and are 816 m apart, as shown in Figure 3. A car and a van are at rest side by side on the road and level with $A$. The car and the van start to move at the same time in the direction $A B$. The car accelerates from rest with constant acceleration until it reaches a speed of $24 \mathrm {~m} \mathrm {~s} ^ { - 1 }$. The car then moves at a constant speed of $24 \mathrm {~m} \mathrm {~s} ^ { - 1 }$. The van accelerates from rest with constant acceleration for 12 s until it reaches a speed of $V \mathrm {~m} \mathrm {~s} ^ { - 1 }$. The van then moves at a constant speed of $V \mathrm {~m} \mathrm {~s} ^ { - 1 }$. When the car has been moving at $24 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ for 30 s , the van draws level with the car at $B$, and each vehicle has then travelled a distance of 816 m .
(a) Sketch, on the same diagram, a speed-time graph for the motion of each vehicle from $A$ to $B$.
(b) Find the time for which the car is accelerating.
(c) Find the value of $V$.

Edexcel M1 2018 October Q5

9 marks Moderate -0.8

[In this question the unit vectors $\mathbf { i }$ and $\mathbf { j }$ are horizontal vectors due east and due north respectively and position vectors are given relative to a fixed origin.]

Edexcel M1 2018 October Q6

11 marks Moderate -0.3

6. The point $A$ on a horizontal playground has position vector $( 3 \mathbf { i } - 2 \mathbf { j } ) \mathrm { m }$. At time $t = 0$, a girl kicks a ball from $A$. The ball moves horizontally along the playground with constant velocity $( 4 \mathbf { i } + 5 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }$. Modelling the ball as a particle, find

the speed of the ball,
the position vector of the ball at time $t$ seconds. The point $B$ on the playground has position vector $( \mathbf { i } + 6 \mathbf { j } ) \mathrm { m }$. At time $t = T$ seconds, the ball is due east of $B$.
Find the value of $T$. A boy is running due east with constant speed $\nu \mathrm { ms } ^ { - 1 }$. At the instant when the girl kicks the ball from $A$, the boy is at $B$. Given that the boy intercepts the ball,
find the value of $v$. \includegraphics[max width=\textwidth, alt={}, center]{5f2d38d9-b719-4205-8cb0-caa959afc46f-23_68_47_2617_1886}

Edexcel M1 2018 October Q7

10 marks Moderate -0.3

7. A truck of mass 1600 kg is towing a car of mass 960 kg along a straight horizontal road. The truck and the car are joined by a light rigid tow bar. The tow bar is horizontal and is parallel to the direction of motion. The truck and the car experience constant resistances to motion of magnitude 640 N and $R$ newtons respectively. The truck's engine produces a constant driving force of magnitude 2100 N . The magnitude of the acceleration of the truck and the car is $0.4 \mathrm {~ms} ^ { - 2 }$.

Show that $R = 436$
Find the tension in the tow bar. The two vehicles come to a hill inclined at an angle $\alpha$ to the horizontal where $\sin \alpha = \frac { 1 } { 15 }$. The truck and the car move down a line of greatest slope of the hill with the tow bar parallel to the direction of motion. The truck's engine produces a constant driving force of magnitude 2100 N . The magnitudes of the resistances to motion on the truck and the car are 640 N and 436 N respectively.
Find the magnitude of the acceleration of the truck and the car as they move down the hill.

\includegraphics[max width=\textwidth, alt={}, center]{5f2d38d9-b719-4205-8cb0-caa959afc46f-27_67_59_2654_1886}

Edexcel M1 2018 October Q8

17 marks Standard +0.3

8. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{5f2d38d9-b719-4205-8cb0-caa959afc46f-28_268_634_292_657} \captionsetup{labelformat=empty} \caption{Figure 4}

\end{figure} A rough plane is inclined at $30 ^ { \circ }$ to the horizontal. A particle $P$ of mass 0.5 kg is held at rest on the plane by a horizontal force of magnitude 5 N , as shown in Figure 4. The force acts in a vertical plane containing a line of greatest slope of the inclined plane. The particle is on the point of moving up the plane.

Find the magnitude of the normal reaction of the plane on $P$.
Find the coefficient of friction between $P$ and the plane. The force of magnitude 5 N is now removed and $P$ accelerates from rest down the plane.
Find the speed of $P$ after it has travelled 3 m down the plane.

Edexcel M1 2021 October Q1

6 marks Standard +0.3

1. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{151d9232-5a78-4bc1-a57e-6c9cae80e473-02_298_1288_264_328} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} A non-uniform rod $A B$ has length 9 m and mass $M \mathrm {~kg}$.
The rod rests in equilibrium in a horizontal position on two supports, one at $C$ where $A C = 2.5 \mathrm {~m}$ and the other at $D$ where $D B = 2 \mathrm {~m}$, as shown in Figure 1 . The magnitude of the force acting on the rod at $D$ is twice the magnitude of the force acting on the $\operatorname { rod }$ at $C$. The centre of mass of the rod is $d$ metres from $A$.
Find the value of $d$.

VIAV SIHI NI III IM IONOO

VIAV SIHI NI III IM I ON OO

VIAV SIHI NI III HM ION OC

Edexcel M1 2021 October Q2

10 marks Standard +0.3

2. A particle $P$ of mass $2 m$ is moving on a rough horizontal plane when it collides directly with a particle $Q$ of mass $4 m$ which is at rest on the plane. The speed of $P$ immediately before the collision is $3 u$. The speed of $Q$ immediately after the collision is $2 u$.

Find, in terms of $u$, the speed of $P$ immediately after the collision.
State clearly the direction of motion of $P$ immediately after the collision. Following the collision, $Q$ comes to rest after travelling a distance $\frac { 6 u ^ { 2 } } { g }$ along the plane. The coefficient of friction between $Q$ and the plane is $\mu$.
Find the value of $\mu$.

Edexcel M1 2021 October Q3

10 marks Moderate -0.8

3. A car is moving at a constant speed of $25 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ along a straight horizontal road. The car is modelled as a particle.
At time $t = 0$, the car is at the point $A$ and the driver sees a road sign 48 m ahead.
Let $t$ seconds be the time that elapses after the car passes $A$.
In a first model, the car is assumed to decelerate uniformly at $6 \mathrm {~ms} ^ { - 2 }$ from $A$ until the car reaches the road sign.

Use this first model to find the speed of the car as it reaches the sign. The road sign indicates that the speed limit immediately after the sign is $13 \mathrm {~m} \mathrm {~s} ^ { - 1 }$.
In a second model, the car is assumed to decelerate uniformly at $6 \mathrm {~m} \mathrm {~s} ^ { - 2 }$ from $A$ until it reaches a speed of $13 \mathrm {~ms} ^ { - 1 }$. The car then maintains this speed until it reaches the road sign.
Use this second model to find the value of $t$ at which the car reaches the sign. In a third model, the car is assumed to move with constant speed $25 \mathrm {~ms} ^ { - 1 }$ from $A$ until time $t = 0.2$, the car then decelerates uniformly at $6 \mathrm {~ms} ^ { - 2 }$ until it reaches a speed of $13 \mathrm {~m} \mathrm {~s} ^ { - 1 }$. The car then maintains this speed until it reaches the road sign.
Use this third model to find the value of $t$ at which the car reaches the sign.

Edexcel M1 2021 October Q4

8 marks Moderate -0.8

The position vector, $\mathbf { r }$ metres, of a particle $P$ at time $t$ seconds, relative to a fixed origin $O$, is given by

$$\mathbf { r } = ( t - 3 ) \mathbf { i } + ( 1 - 2 t ) \mathbf { j }$$

Find, to the nearest degree, the size of the angle between $\mathbf { r }$ and the vector $\mathbf { j }$, when $t = 2$
Find the values of $t$ for which the distance of $P$ from $O$ is 2.5 m .

Edexcel M1 2021 October Q5

10 marks Standard +0.3

5. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{151d9232-5a78-4bc1-a57e-6c9cae80e473-18_440_230_248_856} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} A small bead of mass 0.2 kg is attached to the end $P$ of a light rod $P Q$. The bead is threaded onto a fixed vertical rough wire. The bead is held in equilibrium with the $\operatorname { rod } P Q$ inclined to the wire at an angle $\alpha$, where $\tan \alpha = \frac { 4 } { 3 }$, as shown in Figure 2. The thrust in the rod is $T$ newtons.
The bead is modelled as a particle.

Find the magnitude and direction of the friction force acting on the bead when $T = 2.5$ The coefficient of friction between the bead and the wire is $\mu$.
Given that the greatest possible value of $T$ is 6.125
find the value of $\mu$.

Edexcel M1 2021 October Q6

5 marks Moderate -0.5

6. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{151d9232-5a78-4bc1-a57e-6c9cae80e473-22_428_993_251_479} \captionsetup{labelformat=empty} \caption{Figure 3}

\end{figure} A small ball is thrown vertically upwards at time $t = 0$ from a point $A$ which is above horizontal ground. The ball hits the ground 7 s later. The ball is modelled as a particle moving freely under gravity.
The velocity-time graph shown in Figure 3 represents the motion of the ball for $0 \leqslant t \leqslant 7$

Find the speed with which the ball is thrown.
Find the height of $A$ above the ground.

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Edexcel M1 2016 October Q4

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