Questions M1 - A-Level Maths

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Edexcel M1 2016 June Q3

7 marks Standard +0.3

3. A particle $P$ of mass 0.4 kg is moving on rough horizontal ground when it hits a fixed vertical plane wall. Immediately before hitting the wall, $P$ is moving with speed $4 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ in a direction perpendicular to the wall. The particle rebounds from the wall and comes to rest at a distance of 5 m from the wall. The coefficient of friction between $P$ and the ground is $\frac { 1 } { 8 }$. Find the magnitude of the impulse exerted on $P$ by the wall.

Edexcel M1 2016 June Q4

12 marks Moderate -0.3

4. Two trains $M$ and $N$ are moving in the same direction along parallel straight horizontal tracks. At time $t = 0 , M$ overtakes $N$ whilst they are travelling with speeds $40 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ and $30 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ respectively. Train $M$ overtakes train $N$ as they pass a point $X$ at the side of the tracks. After overtaking $N$, train $M$ maintains its speed of $40 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ for $T$ seconds and then decelerates uniformly, coming to rest next to a point $Y$ at the side of the tracks. After being overtaken, train $N$ maintains its speed of $30 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ for 25 s and then decelerates uniformly, also coming to rest next to the point $Y$. The times taken by the trains to travel between $X$ and $Y$ are the same.

Sketch, on the same diagram, the speed-time graphs for the motions of the two trains between $X$ and $Y$. Given that $X Y = 975 \mathrm {~m}$,
find the value of $T$.

Edexcel M1 2016 June Q5

10 marks Standard +0.3

5. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{d408dd83-c5b2-4e55-b5c1-3e7f3faadbcb-08_321_917_285_518} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} A particle $P$ of mass 2 kg is held at rest in equilibrium on a rough plane by a constant force of magnitude 40 N . The direction of the force is inclined to the plane at an angle of $30 ^ { \circ }$. The plane is inclined to the horizontal at an angle of $20 ^ { \circ }$, 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 June Q6

7 marks Standard +0.3

6. A non-uniform plank $A B$ has length 6 m and mass 30 kg . The plank rests in equilibrium in a horizontal position on supports at the points $S$ and $T$ of the plank where $A S = 0.5 \mathrm {~m}$ and $T B = 2 \mathrm {~m}$. When a block of mass $M \mathrm {~kg}$ is placed on the plank at $A$, the plank remains horizontal and in equilibrium and the plank is on the point of tilting about $S$. When the block is moved to $B$, the plank remains horizontal and in equilibrium and the plank is on the point of tilting about $T$. The distance of the centre of mass of the plank from $A$ is $d$ metres. The block is modelled as a particle and the plank is modelled as a non-uniform rod. Find

the value of $d$,
the value of $M$.

VIIIV SIHI NI JIIUM ION OC VIIIV SIHI NI JIIIM ION OC VI4V SIHI NI JIIYM ION OO

Edexcel M1 2016 June Q7

11 marks Moderate -0.3

7. Two forces $\mathbf { F } _ { 1 }$ and $\mathbf { F } _ { 2 }$ act on a particle $P$. The force $\mathbf { F } _ { 1 }$ is given by $\mathbf { F } _ { 1 } = ( - \mathbf { i } + 2 \mathbf { j } ) \mathrm { N }$ and $\mathbf { F } _ { 2 }$ acts in the direction of the vector $( \mathbf { i } + \mathbf { j } )$.
Given that the resultant of $\mathbf { F } _ { 1 }$ and $\mathbf { F } _ { 2 }$ acts in the direction of the vector ( $\mathbf { i } + 3 \mathbf { j }$ ),

find $\mathbf { F } _ { 2 }$ (7) The acceleration of $P$ is $( 3 \mathbf { i } + 9 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 2 }$. At time $t = 0$, the velocity of $P$ is $( 3 \mathbf { i } - 22 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }$
Find the speed of $P$ when $t = 3$ seconds.

Edexcel M1 2016 June Q8

12 marks Standard +0.3

8. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{d408dd83-c5b2-4e55-b5c1-3e7f3faadbcb-14_460_981_274_475} \captionsetup{labelformat=empty} \caption{Figure 3}

\end{figure} Two particles $P$ and $Q$ have masses 1.5 kg and 3 kg respectively. The particles are attached to the ends of a light inextensible string. Particle $P$ is held at rest on a fixed rough horizontal table. The coefficient of friction between $P$ and the table is $\frac { 1 } { 5 }$. The string is parallel to the table and passes over a small smooth light pulley which is fixed at the edge of the table. Particle $Q$ hangs freely at rest vertically below the pulley, as shown in Figure 3. Particle $P$ is released from rest with the string taut and slides along the table. Assuming that $P$ has not reached the pulley, find

the tension in the string during the motion,
the magnitude and direction of the resultant force exerted on the pulley by the string.

Edexcel M1 2017 June Q1

6 marks Moderate -0.8

Three forces, $( 15 \mathbf { i } + \mathbf { j } ) \mathrm { N } , ( 5 q \mathbf { i } - p \mathbf { j } ) \mathrm { N }$ and $( - 3 p \mathbf { i } - q \mathbf { j } ) \mathrm { N }$, where $p$ and $q$ are constants, act on a particle. Given that the particle is in equilibrium, find the value of $p$ and the value of $q$.
(6)

Edexcel M1 2017 June Q2

7 marks Moderate -0.8

2. Two particles, $P$ and $Q$, have masses $2 m$ and $3 m$ respectively. They are moving towards each other in opposite directions on a smooth horizontal plane when they collide directly. Immediately before they collide the speed of $P$ is $4 u$ and the speed of $Q$ is $3 u$. As a result of the collision, $Q$ has its direction of motion reversed and is moving with speed $u$.

Find the speed of $P$ immediately after the collision.
State whether or not the direction of motion of $P$ has been reversed by the collision.
Find the magnitude of the impulse exerted on $P$ by $Q$ in the collision.

Edexcel M1 2017 June Q3

9 marks Standard +0.3

3. A plank $A B$ has length 6 m and mass 30 kg . The point $C$ is on the plank with $C B = 2 \mathrm {~m}$. The plank rests in equilibrium in a horizontal position on supports at $A$ and $C$. Two people, each of mass 75 kg , stand on the plank. One person stands at the point $P$ of the plank, where $A P = x$ metres, and the other person stands at the point $Q$ of the plank, where $A Q = 2 x$ metres. The plank remains horizontal and in equilibrium with the magnitude of the reaction at $C$ five times the magnitude of the reaction at $A$. The plank is modelled as a uniform rod and each person is modelled as a particle.

Find the value of $x$.
State two ways in which you have used the assumptions made in modelling the plank as a uniform rod.

the acceleration of the cyclist as she moves from $A$ to $B$,
the time it takes her to travel from $A$ to the midpoint of $A B$.

Edexcel M1 2017 June Q7

14 marks Standard +0.3

7. [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 $O$.] Two ships, $P$ and $Q$, are moving with constant velocities.
The velocity of $P$ is $( 9 \mathbf { i } - 2 \mathbf { j } ) \mathrm { km } \mathrm { h } ^ { - 1 }$ and the velocity of $Q$ is $( 4 \mathbf { i } + 8 \mathbf { j } ) \mathrm { km } \mathrm { h } ^ { - 1 }$

Find the direction of motion of $P$, giving your answer as a bearing to the nearest degree. When $t = 0$, the position vector of $P$ is $( 9 \mathbf { i } + 10 \mathbf { j } ) \mathrm { km }$ and the position vector of $Q$ is $( \mathbf { i } + 4 \mathbf { j } ) \mathrm { km }$. At time $t$ hours, the position vectors of $P$ and $Q$ are $\mathbf { p } \mathrm { km }$ and $\mathbf { q } \mathrm { km }$ respectively.
Find an expression for
1. $\mathbf { p }$ in terms of $t$,
2. $\mathbf { q }$ in terms of $t$.
Hence show that, at time $t$ hours, $$\overrightarrow { Q P } = ( 8 + 5 t ) \mathbf { i } + ( 6 - 10 t ) \mathbf { j }$$
Find the values of $t$ when the ships are 10 km apart.

Edexcel M1 2017 June Q8

15 marks Standard +0.3

8. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{c809d34e-83db-4a16-a831-001f9f36b1c3-24_638_951_242_500} \captionsetup{labelformat=empty} \caption{Figure 3}

\end{figure} Two particles, $A$ and $B$, have masses $2 m$ and $m$ respectively. The particles are attached to the ends of a light inextensible string. Particle $A$ is held at rest on a fixed rough horizontal table at a distance $d$ from a small smooth light pulley which is fixed at the edge of the table at the point $P$. The coefficient of friction between $A$ and the table is $\mu$, where $\mu < \frac { 1 } { 2 }$. The string is parallel to the table from $A$ to $P$ and passes over the pulley. Particle $B$ hangs freely at rest vertically below $P$ with the string taut and at a height $h$, ( $h < d$ ), above a horizontal floor, as shown in Figure 3. Particle $A$ is released from rest with the string taut and slides along the table.

1. Write down an equation of motion for $A$.
2. Write down an equation of motion for $B$.
Hence show that, until $B$ hits the floor, the acceleration of $A$ is $\frac { g } { 3 } ( 1 - 2 \mu )$.
Find, in terms of $g , h$ and $\mu$, the speed of $A$ at the instant when $B$ hits the floor. After $B$ hits the floor, $A$ continues to slide along the table. Given that $\mu = \frac { 1 } { 3 }$ and that $A$ comes to rest at $P$,
find $d$ in terms of $h$.
Describe what would happen if $\mu = \frac { 1 } { 2 }$
(Total 15 marks)
Leave blank
Q8

Edexcel M1 2018 June Q1

6 marks Moderate -0.8

Two particles, $P$ and $Q$, have masses $3 m$ 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. The speeds of $P$ and $Q$ immediately before the collision are $2 u$ and $4 u$ respectively. The magnitude of the impulse received by each particle in the collision is $\frac { 21 m u } { 4 }$.

Find the speed of $P$ after the collision.
Find the speed of $Q$ after the collision.

Edexcel M1 2018 June Q2

10 marks Standard +0.8

2. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{4fd21e83-0bdf-4bb1-8a3f-76beada511ae-04_333_976_287_550} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} A particle of mass 2 kg lies on a rough plane. The plane is inclined to the horizontal at $30 ^ { \circ }$. The coefficient of friction between the particle and the plane is $\frac { 1 } { 4 }$. The particle is held in equilibrium by a force of magnitude $P$ newtons. The force makes an angle of $20 ^ { \circ }$ with the horizontal and acts in a vertical plane containing a line of greatest slope of the plane, as shown in Figure 1. Find the least possible value of $P$.

Edexcel M1 2018 June Q3

7 marks Standard +0.3

3. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{4fd21e83-0bdf-4bb1-8a3f-76beada511ae-08_426_1226_221_360} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} A wooden beam $A B$, of mass 150 kg and length 9 m , rests in a horizontal position supported by two vertical ropes. The ropes are attached to the beam at $C$ and $D$, where $A C = 1.5 \mathrm {~m}$ and $B D = 3.5 \mathrm {~m}$. A gymnast of mass 60 kg stands on the beam at the point $P$, where $A P = 3 \mathrm {~m}$, as shown in Figure 2. The beam remains horizontal and in equilibrium. By modelling the gymnast as a particle, the beam as a uniform rod and the ropes as light inextensible strings,

find the tension in the rope attached to the beam at $C$. The gymnast at $P$ remains on the beam at $P$ and another gymnast, who is also modelled as a particle, stands on the beam at $B$. The beam remains horizontal and in equilibrium. The mass of the gymnast at $B$ is the largest possible for which the beam remains horizontal and in equilibrium.
Find the tension in the rope attached to the beam at $D$.

Edexcel M1 2018 June Q4

13 marks Standard +0.3

4. A ball of mass 0.2 kg is projected vertically downwards with speed $U \mathrm {~m} \mathrm {~s} ^ { - 1 }$ from a point A which is 2.5 m above horizontal ground. The ball hits the ground. Immediately after hitting the ground, the ball rebounds vertically with a speed of $10 \mathrm {~m} \mathrm {~s} ^ { - 1 }$. The ball receives an impulse of magnitude 7 Ns in its impact with the ground. By modelling the ball as a particle and ignoring air resistance, find

the value of $U$. After hitting the ground, the ball moves vertically upwards and passes through a point $B$ which is 1 m above the ground.
Find the time between the instant when the ball hits the ground and the instant when the ball first passes through $B$.
Sketch a velocity-time graph for the motion of the ball from when it was projected from $A$ to when it first passes through $B$. (You need not make any further calculations to draw this sketch.)

Edexcel M1 2018 June Q5

10 marks Moderate -0.3

5. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{4fd21e83-0bdf-4bb1-8a3f-76beada511ae-16_359_298_233_824} \captionsetup{labelformat=empty} \caption{Figure 3}

\end{figure} A lift of mass 250 kg is being raised by a vertical cable attached to the top of the lift. A woman of mass 60 kg stands on the horizontal floor inside the lift, as shown in Figure 3. The lift ascends vertically with constant acceleration $2 \mathrm {~m} \mathrm {~s} ^ { - 2 }$. There is a constant downwards resistance of magnitude 100 N on the lift. By modelling the woman as a particle,

find the magnitude of the normal reaction exerted by the floor of the lift on the woman. The tension in the cable must not exceed 10000 N for safety reasons, and the maximum upward acceleration of the lift is $3 \mathrm {~m} \mathrm {~s} ^ { - 2 }$. A typical occupant of the lift is modelled as a particle of mass 75 kg and the cable is modelled as a light inextensible string. There is still a constant downwards resistance of magnitude 100 N on the lift.
Find the maximum number of typical occupants that can be safely carried in the lift when it is ascending with an acceleration of $3 \mathrm {~m} \mathrm {~s} ^ { - 2 }$.

Edexcel M1 2018 June Q6

13 marks Moderate -0.3

6. [In this question $\mathbf { i }$ and $\mathbf { j }$ are horizontal unit vectors due east and due north respectively] Two forces $\mathbf { F } _ { 1 }$ and $\mathbf { F } _ { 2 }$ act on a particle $P$ of mass 0.5 kg . $\mathbf { F } _ { 1 } = ( 4 \mathbf { i } - 6 \mathbf { j } ) \mathrm { N }$ and $\mathbf { F } _ { 2 } = ( p \mathbf { i } + q \mathbf { j } ) \mathrm { N }$.
Given that the resultant force of $\mathbf { F } _ { 1 }$ and $\mathbf { F } _ { 2 }$ is in the same direction as $- 2 \mathbf { i } - \mathbf { j }$,

show that $p - 2 q = - 16$ Given that $q = 3$
find the magnitude of the acceleration of $P$,
find the direction of the acceleration of $P$, giving your answer as a bearing to the nearest degree. XXXXXXXXXXIXITEINTIIS AREA XX女X女X女X女X DO NOT WIRIE IN THS AREA.

Edexcel M1 2018 June Q7

16 marks Standard +0.3

7. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{4fd21e83-0bdf-4bb1-8a3f-76beada511ae-24_391_917_251_516} \captionsetup{labelformat=empty} \caption{Figure 4}

\end{figure} A particle $P$ of mass $4 m$ is held at rest at the point $X$ on the surface of a rough inclined plane which is fixed to horizontal ground. The point $X$ is a distance $h$ from the bottom of the inclined plane. The plane is inclined to the horizontal at an angle $\alpha$ where $\tan \alpha = \frac { 3 } { 4 }$. The coefficient of friction between $P$ and the plane is $\frac { 1 } { 4 }$. The particle $P$ is attached to one end of a light inextensible string. The string passes over a small smooth pulley which is fixed at the top of the plane. The other end of the string is attached to a particle $Q$ of mass $m$ which hangs freely at a distance $d$, where $d > h$, below the pulley, as shown in Figure 4. The string lies in a vertical plane through a line of greatest slope of the inclined plane. The system is released from rest with the string taut and $P$ moves down the plane. For the motion of the particles before $P$ hits the ground,

state which of the information given above implies that the magnitudes of the accelerations of the two particles are the same,
write down an equation of motion for each particle,
find the acceleration of each particle. When $P$ hits the ground, it immediately comes to rest. Given that $Q$ comes to instantaneous rest before reaching the pulley,
show that $d > \frac { 28 h } { 25 }$.

\includegraphics[max width=\textwidth, alt={}, center]{4fd21e83-0bdf-4bb1-8a3f-76beada511ae-27_56_20_109_1950}

END

Edexcel M1 2002 November Q1

6 marks Moderate -0.8

1. \begin{figure}[h]

\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{14703bfa-abd8-4a8d-bc18-20d66eea409e-2_671_829_294_663}

\end{figure} A particle $P$ of weight 6 N is attached to one end of a light inextensible string. The other end of the string is attached to a fixed point $O$. A horizontal force of magnitude $F$ newtons is applied to $P$. The particle $P$ is in equilibrium under gravity with the string making an angle of $30 ^ { \circ }$ with the vertical, as shown in Fig. 1. Find, to 3 significant figures,

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

Edexcel M1 2002 November Q2

7 marks Moderate -0.8

2. A particle $P$ of mass 1.5 kg is moving under the action of a constant force ( $3 \mathbf { i } - 7.5 \mathbf { j }$ ) N. Initially $P$ has velocity $( 2 \mathbf { i } + 3 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }$. Find

the magnitude of the acceleration of $P$,
the velocity of $P$, in terms of $\mathbf { i }$ and $\mathbf { j }$, when $P$ has been moving for 4 seconds.

Edexcel M1 2002 November Q3

7 marks Moderate -0.3

3. A car accelerates uniformly from rest to a speed of $20 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ in $T$ seconds. The car then travels at a constant speed of $20 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ for $4 T$ seconds and finally decelerates uniformly to rest in a further 50 s .

Sketch a speed-time graph to show the motion of the car. The total distance travelled by the car is 1220 m . Find
the value of $T$,
the initial acceleration of the car.

Edexcel M1 2002 November Q4

9 marks Standard +0.2

[diagram]

A uniform plank $A B$ has weight 80 N and length $x$ metres. The plank rests in equilibrium horizontally on two smooth supports at $A$ and $C$, where $A C = 2 \mathrm {~m}$, as shown in Fig. 2. A rock of weight 20 N is placed at $B$ and the plank remains in equilibrium. The reaction on the plank at $C$ has magnitude 90 N . The plank is modelled as a rod and the rock as a particle.

Find the value of $x$.
State how you have used the model of the rock as a particle. The support at $A$ is now moved to a point $D$ on the plank and the plank remains in equilibrium with the rock at $B$. The reaction on the plank at $C$ is now three times the reaction at $D$.
Find the distance $A D$.

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Edexcel M1 2016 June Q3

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