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. Position vectors are given relative to a fixed origin $O$.]

A boy $B$ is running in a field with constant velocity ( $3 \mathbf { i } - 2 \mathbf { j }$ ) $\mathrm { m } \mathrm { s } ^ { - 1 }$. At time $t = 0 , B$ is at the point with position vector 10j m . Find

the speed of $B$,
the direction in which $B$ is running, giving your answer as a bearing. At time $t = 0$, a girl $G$ is at the point with position vector $( 4 \mathbf { i } - 2 \mathbf { j } ) \mathrm { m }$. The girl is running with constant velocity $\left( \frac { 5 } { 3 } \mathbf { i } + 2 \mathbf { j } \right) \mathrm { m } \mathrm { s } ^ { - 1 }$ and meets $B$ at the point $P$.
Find
1. the value of $t$ when they meet,
2. the position vector of $P$.

Edexcel M1 2014 June Q6

13 marks Moderate -0.3

A car starts from rest at a point $A$ and moves along a straight horizontal road. The car moves with constant acceleration $1.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }$ for the first 8 s . The car then moves with constant acceleration $0.8 \mathrm {~m} \mathrm {~s} ^ { - 2 }$ for the next 20 s . It then moves with constant speed for $T$ seconds before slowing down with constant deceleration $2.8 \mathrm {~m} \mathrm {~s} ^ { - 2 }$ until it stops at a point $B$.
1. Find the speed of the car 28 s after leaving $A$.
2. Sketch, in the space provided, a speed-time graph to illustrate the motion of the car as it travels from $A$ to $B$.
3. Find the distance travelled by the car during the first 28 s of its journey from $A$.
The distance from $A$ to $B$ is 2 km .
Find the value of $T$.

Edexcel M1 2014 June Q7

16 marks Standard +0.3

7. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{edcc4603-f006-4c4f-a4e5-063cab41da98-12_486_1257_230_347} \captionsetup{labelformat=empty} \caption{Figure 3}

\end{figure} Two particles $P$ and $Q$, of mass 2 kg and 3 kg respectively, are connected by a light inextensible string. Initially $P$ is held at rest on a fixed smooth plane inclined at $30 ^ { \circ }$ to the horizontal. The string passes over a small smooth fixed pulley at the top of the plane. The particle $Q$ hangs freely below the pulley and 0.6 m above the ground, as shown in Figure 3. The part of the string from $P$ to the pulley is parallel to a line of greatest slope of the plane. The system is released from rest with the string taut. For the motion before $Q$ hits the ground,

1. show that the acceleration of $Q$ is $\frac { 2 g } { 5 }$,
2. find the tension in the string. On hitting the ground $Q$ is immediately brought to rest by the impact.
Find the speed of $P$ at the instant when $Q$ hits the ground. In its subsequent motion $P$ does not reach the pulley.
Find the total distance moved up the plane by $P$ before it comes to instantaneous rest.
Find the length of time between $Q$ hitting the ground and $P$ first coming to instantaneous rest.

Edexcel M1 2015 June Q1

6 marks Moderate -0.8

Three forces $\mathbf { F } _ { 1 } , \mathbf { F } _ { 2 }$ and $\mathbf { F } _ { 3 }$ act on a particle $P$.

$$\mathbf { F } _ { 1 } = ( 2 \mathbf { i } + 3 a \mathbf { j } ) \mathrm { N } ; \quad \mathbf { F } _ { 2 } = ( 2 a \mathbf { i } + b \mathbf { j } ) \mathrm { N } ; \quad \mathbf { F } _ { 3 } = ( b \mathbf { i } + 4 \mathbf { j } ) \mathrm { N } .$$ The particle $P$ is in equilibrium under the action of these forces.
Find the value of $a$ and the value of $b$.

Edexcel M1 2015 June Q2

9 marks Standard +0.3

2. Particle $A$ of mass $2 m$ and particle $B$ of mass $k m$, where $k$ is a positive constant, 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 $A$ is $u$ and the speed of $B$ is $3 u$. The direction of motion of each particle is reversed by the collision. Immediately after the collision the speed of $A$ is $\frac { 1 } { 2 } u$.

Show that $k < 1$
Find, in terms of $m$ and $u$, the magnitude of the impulse exerted on $B$ by $A$ in the collision.

Edexcel M1 2015 June Q3

10 marks Standard +0.3

3. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{3280fdf1-d81a-4729-b065-e84dece6a220-05_325_947_267_493} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} A particle $P$ of mass 2 kg is pushed by a constant horizontal force of magnitude 30 N up a line of greatest slope of a rough plane. The plane is inclined to the horizontal at an angle $\alpha$, where $\tan \alpha = \frac { 3 } { 4 }$, as shown in Figure 1. The line of action of the force lies in the vertical plane containing $P$ and the line of greatest slope of the plane. The particle $P$ starts from rest. The coefficient of friction between $P$ and the plane is $\mu$. After 2 seconds, $P$ has travelled a distance of 5.5 m up the plane.

Find the acceleration of $P$ up the plane.
Find the value of $\mu$.

Edexcel M1 2015 June Q4

7 marks Standard +0.3

A small stone is released from rest from a point $A$ which is at height $h$ metres above horizontal ground. Exactly one second later another small stone is projected with speed $19.6 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ vertically downwards from a point $B$, which is also at height $h$ metres above the horizontal ground. The motion of each stone is modelled as that of a particle moving freely under gravity. The two stones hit the ground at the same time.

Find the value of $h$.

Edexcel M1 2015 June Q5

10 marks Standard +0.3

5. A car travelling along a straight horizontal road takes 170 s to travel between two sets of traffic lights at $A$ and $B$ which are 2125 m apart. The car starts from rest at $A$ and moves with constant acceleration until it reaches a speed of $17 \mathrm {~m} \mathrm {~s} ^ { - 1 }$. The car then maintains this speed before moving with constant deceleration, coming to rest at $B$. The magnitude of the deceleration is twice the magnitude of the acceleration.

Sketch, in the space below, a speed-time graph for the motion of the car between $A$ and $B$.
Find the deceleration of the car.

Edexcel M1 2015 June Q6

12 marks Standard +0.3

6. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{3280fdf1-d81a-4729-b065-e84dece6a220-10_238_1258_267_342} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} A plank $A B$ has length 4 m and mass 6 kg . The plank rests in a horizontal position on two supports, one at $B$ and one at $C$, where $A C = 1.5 \mathrm {~m}$. A load of mass 15 kg is placed on the plank at the point $X$, as shown in Figure 2, and the plank remains horizontal and in equilibrium. The plank is modelled as a uniform rod and the load is modelled as a particle. The magnitude of the reaction on the plank at $C$ is twice the magnitude of the reaction on the plank at $B$.

Find the magnitude of the reaction on the plank at $C$.
Find the distance $A X$. The load is now moved along the plank to a point $Y$, between $A$ and $C$. Given that the plank is on the point of tipping about $C$,
find the distance $A Y$.

Edexcel M1 2015 June Q7

5 marks Moderate -0.3

A particle $P$ moves from point $A$ to point $B$ with constant acceleration $( c \mathbf { i } + d \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 2 }$, where $c$ and $d$ are positive constants. The velocity of $P$ at $A$ is $( - 3 \mathbf { i } - 3 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }$ and the velocity of $P$ at $B$ is $( 2 \mathbf { i } + 9 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }$. The magnitude of the acceleration of $P$ is $2.6 \mathrm {~m} \mathrm {~s} ^ { - 2 }$.

Find the value of $c$ and the value of $d$.

Edexcel M1 2015 June Q8

16 marks Standard +0.3

8. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{3280fdf1-d81a-4729-b065-e84dece6a220-13_648_1280_271_331} \captionsetup{labelformat=empty} \caption{Figure 3}

\end{figure} Two particles $P$ and $Q$ have masses $m$ and $4 m$ respectively. The particles are attached to the ends of a light inextensible string. Particle $P$ is held at rest on a rough horizontal table. The string lies along the table and passes over a small smooth light pulley which is fixed at the edge of the table. Particle $Q$ hangs at rest vertically below the pulley, at a height $h$ above a horizontal plane, as shown in Figure 3. The coefficient of friction between $P$ and the table is 0.5 . Particle $P$ is released from rest with the string taut and slides along the table.

Find, in terms of $m g$, the tension in the string while both particles are moving. The particle $P$ does not reach the pulley before $Q$ hits the plane.
Show that the speed of $Q$ immediately before it hits the plane is $\sqrt { 1.4 g h }$ When $Q$ hits the plane, $Q$ does not rebound and $P$ continues to slide along the table. Given that $P$ comes to rest before it reaches the pulley,
show that the total length of the string must be greater than 2.4 h

Edexcel M1 2016 June Q1

7 marks Standard +0.3

A car is moving along a straight horizontal road with constant acceleration $a \mathrm {~m} \mathrm {~s} ^ { - 2 } ( a > 0 )$. At time $t = 0$ the car passes the point $P$ moving with speed $u \mathrm {~m} \mathrm {~s} ^ { - 1 }$. In the next 4 s , the car travels 76 m and then in the following 6 s it travels a further 219 m .

Find

the value of $u$,
the value of $a$.

Edexcel M1 2016 June Q2

6 marks Standard +0.2

Two particles $P$ and $Q$ are moving in opposite directions along the same horizontal straight line. Particle $P$ has mass $m$ and particle $Q$ has mass $k m$. The particles collide directly. Immediately before the collision, the speed of $P$ is $u$ and the speed of $Q$ is $2 u$. As a result of the collision, the direction of motion of each particle is reversed and the speed of each particle is halved.
1. Find the value of $k$.
  (4)
2. Find, in terms of $m$ and $u$ only, the magnitude of the impulse exerted on $Q$ by $P$ in the collision.
  (2)
3. A block $A$ of mass 9 kg is released from rest from a point $P$ which is a height $h$ metres above horizontal soft ground. The block falls and strikes another block $B$ of mass 1.5 kg which is on the ground vertically below $P$. The speed of $A$ immediately before it strikes $B$ is $7 \mathrm {~m} \mathrm {~s} ^ { - 1 }$. The blocks are modelled as particles.
4. Find the value of $h$.
  (2)
Immediately after the impact the blocks move downwards together with the same speed and both come to rest after sinking a vertical distance of 12 cm into the ground. Assuming that the resistance offered by the ground has constant magnitude $R$ newtons,
find the value of $R$.
(8)
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Edexcel M1 2016 June Q4

10 marks Moderate -0.3

4. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{55024b00-68f0-4f8b-9b08-58479bb291fd-06_244_1168_264_388} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} A diving board $A B$ consists of a wooden plank of length 4 m and mass 30 kg . The plank is held at rest in a horizontal position by two supports at the points $A$ and $C$, where $A C = 0.6 \mathrm {~m}$, as shown in Figure 1. The force on the plank at $A$ acts vertically downwards and the force on the plank at $C$ acts vertically upwards. A diver of mass 50 kg is standing on the board at the end $B$. The diver is modelled as a particle and the plank is modelled as a uniform rod. The plank is in equilibrium.

Find
1. the magnitude of the force acting on the plank at $A$,
2. the magnitude of the force acting on the plank at $C$. The support at $A$ will break if subjected to a force whose magnitude is greater than 5000 N .
Find, in kg, the greatest integer mass of a diver who can stand on the board at $B$ without breaking the support at $A$.
Explain how you have used the fact that the diver is modelled as a particle.

Edexcel M1 2016 June Q5

10 marks Moderate -0.8

5. Two forces, $\mathbf { F } _ { 1 }$ and $\mathbf { F } _ { 2 }$, act on a particle $A$.
$\mathbf { F } _ { 1 } = ( 2 \mathbf { i } - 3 \mathbf { j } ) \mathrm { N }$ and $\mathbf { F } _ { 2 } = ( p \mathbf { i } + q \mathbf { j } ) \mathrm { N }$, where $p$ and $q$ are constants.
Given that the resultant of $\mathbf { F } _ { 1 }$ and $\mathbf { F } _ { 2 }$ is parallel to ( $\mathbf { i } + 2 \mathbf { j }$ ),

show that $2 p - q + 7 = 0$ Given that $q = 11$ and that the mass of $A$ is 2 kg , and that $\mathbf { F } _ { 1 }$ and $\mathbf { F } _ { 2 }$ are the only forces acting on $A$,
find the magnitude of the acceleration of $A$.

Edexcel M1 2016 June Q6

17 marks Moderate -0.3

6. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{55024b00-68f0-4f8b-9b08-58479bb291fd-09_264_997_269_461} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} Two cars, $A$ and $B$, move on parallel straight horizontal tracks. Initially $A$ and $B$ are both at rest with $A$ at the point $P$ and $B$ at the point $Q$, as shown in Figure 2. At time $t = 0$ seconds, $A$ starts to move with constant acceleration $a \mathrm {~m} \mathrm {~s} ^ { - 2 }$ for 3.5 s , reaching a speed of $14 \mathrm {~m} \mathrm {~s} ^ { - 1 }$. Car $A$ then moves with constant speed $14 \mathrm {~m} \mathrm {~s} ^ { - 1 }$.

Find the value of $a$. Car $B$ also starts to move at time $t = 0$ seconds, in the same direction as car $A$. Car $B$ moves with a constant acceleration of $3 \mathrm {~m} \mathrm {~s} ^ { - 2 }$. At time $t = T$ seconds, $B$ overtakes $A$. At this instant $A$ is moving with constant speed.
On a diagram, sketch, on the same axes, a speed-time graph for the motion of $A$ for the interval $0 \leqslant t \leqslant T$ and a speed-time graph for the motion of $B$ for the interval $0 \leqslant t \leqslant T$.
Find the value of $T$.
Find the distance of car $B$ from the point $Q$ when $B$ overtakes $A$.
On a new diagram, sketch, on the same axes, an acceleration-time graph for the motion of $A$ for the interval $0 \leqslant t \leqslant T$ and an acceleration-time graph for the motion of $B$ for the interval $0 \leqslant t \leqslant T$.

Edexcel M1 2016 June Q7

15 marks Standard +0.8

7. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{55024b00-68f0-4f8b-9b08-58479bb291fd-11_570_1045_262_459} \captionsetup{labelformat=empty} \caption{Figure 3}

\end{figure} A particle $P$ of mass 4 kg is attached to one end of a light inextensible string. A particle $Q$ of mass $m \mathrm {~kg}$ is attached to the other end of the string. The string passes over a small smooth pulley which is fixed at a point on the intersection of two fixed inclined planes. The string lies in a vertical plane that contains a line of greatest slope of each of the two inclined planes. The first plane is inclined to the horizontal at an angle $\alpha$, where $\tan \alpha = \frac { 3 } { 4 }$ and the second plane is inclined to the horizontal at an angle $\beta$, where $\tan \beta = \frac { 4 } { 3 }$. Particle $P$ is on the first plane and particle $Q$ is on the second plane with the string taut, as shown in Figure 3. The first plane is rough and the coefficient of friction between $P$ and the plane is $\frac { 1 } { 4 }$. The second plane is smooth. The system is in limiting equilibrium. Given that $P$ is on the point of slipping down the first plane,

find the value of $m$,
find the magnitude of the force exerted on the pulley by the string,
find the direction of the force exerted on the pulley by the string.

Edexcel M1 2017 June Q1

7 marks Moderate -0.3

1. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{5c3869c7-008f-4131-b68d-8ecdd4da3377-02_346_499_251_721} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} A particle $P$ of weight 5 N is attached to one end of a light string. The other end of the string is attached to a fixed point $O$. A force of magnitude $F$ newtons is applied to $P$. The line of action of the force is inclined to the horizontal at $30 ^ { \circ }$ and lies in the same vertical plane as the string. The particle $P$ is in equilibrium with the string making an angle of $40 ^ { \circ }$ with the downward vertical, as shown in Figure 1. Find

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

Edexcel M1 2017 June Q2

9 marks Standard +0.3

2. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{5c3869c7-008f-4131-b68d-8ecdd4da3377-04_429_1298_255_324} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} A wooden beam $A B$ has weight 140 N and length $2 a$ metres. The beam rests horizontally in equilibrium on two supports at $C$ and $D$, where $A C = 2 \mathrm {~m}$ and $A D = 6 \mathrm {~m}$. A block of weight 30 N is placed on the beam at $B$ and the beam remains horizontal and in equilibrium, as shown in Figure 2. The reaction on the beam at $D$ has magnitude 120 N . The block is modelled as a particle and the beam is modelled as a uniform rod.

Find the value of $a$. The support at $D$ is now moved to a point $E$ on the beam and the beam remains horizontal and in equilibrium with the block at $B$. The magnitude of the reaction on the beam at $C$ is now equal to the magnitude of the reaction on the beam at $E$.
Find the distance $A E$.

Edexcel M1 2017 June Q3

7 marks Standard +0.3

3. Two particles, $P$ and $Q$, have masses 0.5 kg and $m \mathrm {~kg}$ 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 $4 \mathrm {~ms} ^ { - 1 }$ and the speed of $Q$ is $2 \mathrm {~m} \mathrm {~s} ^ { - 1 }$. The magnitude of the impulse exerted on $P$ by $Q$ in the collision is 4.2 N s . As a result of the collision the direction of motion of $P$ is reversed.

Find the speed of $P$ immediately after the collision. The speed of $Q$ immediately after the collision is $1 \mathrm {~m} \mathrm {~s} ^ { - 1 }$.
Find the two possible values of $m$.

Edexcel M1 2017 June Q4

8 marks Moderate -0.8

A small ball of mass 0.2 kg is moving vertically downwards when it hits a horizontal floor. Immediately before hitting the floor the ball has speed $10 \mathrm {~m} \mathrm {~s} ^ { - 1 }$. Immediately after hitting the floor the ball rebounds vertically with speed $7 \mathrm {~m} \mathrm {~s} ^ { - 1 }$.
1. Find the magnitude of the impulse exerted by the floor on the ball.
By modelling the motion of the ball as that of a particle moving freely under gravity,
find the maximum height above the floor reached by the ball after it has rebounded from the floor,
find the time between the instant when the ball first hits the floor and the instant when the ball is first 1 m above the floor and moving upwards.

Edexcel M1 2017 June Q5

13 marks Standard +0.3

Two trains, $P$ and $Q$, move on horizontal parallel straight tracks. Initially both are at rest in a station and level with each other. At time $t = 0 , P$ starts off and moves with constant acceleration for 10 s up to a speed of $25 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ and then moves at a constant speed of $25 \mathrm {~m} \mathrm {~s} ^ { - 1 }$. At time $t = 20$, where $t$ is measured in seconds, train $Q$ starts to move in the same direction as $P$. Train $Q$ accelerates with the same initial constant acceleration as $P$, up to a speed of $40 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ and then moves at a constant speed of $40 \mathrm {~m} \mathrm {~s} ^ { - 1 }$. Train $Q$ overtakes $P$ at time $t = T$, after both trains have reached their constant speeds.
1. Sketch, on the same axes, the speed-time graphs of both trains for $0 \leqslant t \leqslant T$.
2. Find the value of $t$ at the instant when both trains are moving at the same speed.
3. Find the value of $T$.

Edexcel M1 2017 June Q6

9 marks Moderate -0.3

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

A particle $P$ moves with constant acceleration $( - 2 \mathbf { i } + 3 \mathbf { j } ) \mathrm { ms } ^ { - 2 }$. At time $t$ seconds, the velocity of $P$ is $\mathbf { v m ~ s } ^ { - 1 }$. When $t = 0 , \mathbf { v } = 10 \mathbf { i } + 4 \mathbf { j }$.

Find the direction of motion of $P$ when $t = 6$, giving your answer as a bearing to the nearest degree.
Find the value of $t$ when $P$ is moving north east.

Edexcel M1 2017 June Q7

8 marks Moderate -0.8

7. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{5c3869c7-008f-4131-b68d-8ecdd4da3377-22_254_291_251_831} \captionsetup{labelformat=empty} \caption{Figure 3}

\end{figure} Two forces, $\mathbf { P }$ and $\mathbf { Q }$, act on a particle. The force $\mathbf { P }$ has magnitude 8 N and the force $\mathbf { Q }$ has magnitude 5 N . The angle between the directions of $\mathbf { P }$ and $\mathbf { Q }$ is $50 ^ { \circ }$, as shown in Figure 3. The resultant of $\mathbf { P }$ and $\mathbf { Q }$ is the force $\mathbf { R }$.

Find, to 3 significant figures, the magnitude of $\mathbf { R }$.
Find, to the nearest degree, the size of the angle between the direction of $\mathbf { P }$ and the direction of $\mathbf { R }$.

Edexcel M1 2017 June Q8

14 marks Standard +0.3

8. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{5c3869c7-008f-4131-b68d-8ecdd4da3377-24_369_1200_248_370} \captionsetup{labelformat=empty} \caption{Figure 4}

\end{figure} Two particles, $P$ and $Q$, with masses $2 m$ and $m$ respectively, are attached to the ends of a light inextensible string. The string passes over a small smooth pulley which is fixed at the edge of a rough horizontal table. Particle $Q$ is held at rest on the table and particle $P$ is on the surface of a smooth inclined plane. The top of the plane coincides with the edge of the table. The plane is inclined to the horizontal at an angle $\alpha$, where $\tan \alpha = \frac { 3 } { 4 }$, as shown in Figure 4. The string lies in a vertical plane containing the pulley and a line of greatest slope of the plane. The coefficient of friction between $Q$ and the table is $\frac { 1 } { 2 }$. Particle $Q$ is released from rest with the string taut and $P$ begins to slide down the plane.

By writing down an equation of motion for each particle,
1. find the initial acceleration of the system,
2. find the tension in the string. Suppose now that the coefficient of friction between $Q$ and the table is $\mu$ and when $Q$ is released it remains at rest.
Find the smallest possible value of $\mu$.
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