Questions M1 - A-Level Maths

Sketch, in the space below, a speed-time graph to show the motion of the car from $A$ to $B$.
Calculate the deceleration of the car as it decelerates from $25 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ to $10 \mathrm {~m} \mathrm {~s} ^ { - 1 }$. Question 4 continued $\_\_\_\_$

Edexcel M1 2008 June Q5

9 marks Standard +0.3

5. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{9dbbbc01-fb66-460d-a42e-2c37ec8b451a-07_357_968_274_484} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} Two forces $\mathbf { P }$ and $\mathbf { Q }$ act on a particle at a point $O$. The force $\mathbf { P }$ has magnitude 15 N and the force $\mathbf { Q }$ has magnitude $X$ newtons. The angle between $\mathbf { P }$ and $\mathbf { Q }$ is $150 ^ { \circ }$, as shown in Figure 1. The resultant of $\mathbf { P }$ and $\mathbf { Q }$ is $\mathbf { R }$. Given that the angle between $\mathbf { R }$ and $\mathbf { Q }$ is $50 ^ { \circ }$, find

the magnitude of $\mathbf { R }$,
the value of $X$.

Edexcel M1 2008 June Q6

10 marks Standard +0.3

6. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{9dbbbc01-fb66-460d-a42e-2c37ec8b451a-08_392_678_260_614} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} A plank $A B$ has mass 12 kg and length 2.4 m . A load of mass 8 kg is attached to the plank at the point $C$, where $A C = 0.8 \mathrm {~m}$. The loaded plank is held in equilibrium, with $A B$ horizontal, by two vertical ropes, one attached at $A$ and the other attached at $B$, as shown in Figure 2. The plank is modelled as a uniform rod, the load as a particle and the ropes as light inextensible strings.

Find the tension in the rope attached at $B$. The plank is now modelled as a non-uniform rod. With the new model, the tension in the rope attached at $A$ is 10 N greater than the tension in the rope attached at $B$.
Find the distance of the centre of mass of the plank from $A$.

Edexcel M1 2008 June Q7

11 marks Standard +0.3

7. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{9dbbbc01-fb66-460d-a42e-2c37ec8b451a-10_291_726_265_607} \captionsetup{labelformat=empty} \caption{Figure 3}

\end{figure} A package of mass 4 kg lies on a rough plane inclined at $30 ^ { \circ }$ to the horizontal. The package is held in equilibrium by a force of magnitude 45 N acting at an angle of $50 ^ { \circ }$ to the plane, as shown in Figure 3. The force is acting in a vertical plane through a line of greatest slope of the plane. The package is in equilibrium on the point of moving up the plane. The package is modelled as a particle. Find

the magnitude of the normal reaction of the plane on the package,
the coefficient of friction between the plane and the package.

Edexcel M1 2008 June Q8

15 marks Standard +0.3

8. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{9dbbbc01-fb66-460d-a42e-2c37ec8b451a-12_131_940_269_498} \captionsetup{labelformat=empty} \caption{Figure 4}

\end{figure} Two particles $P$ and $Q$, of mass 2 kg and 3 kg respectively, are joined by a light inextensible string. Initially the particles are at rest on a rough horizontal plane with the string taut. A constant force $\mathbf { F }$ of magnitude 30 N is applied to $Q$ in the direction $P Q$, as shown in Figure 4. The force is applied for 3 s and during this time $Q$ travels a distance of 6 m . The coefficient of friction between each particle and the plane is $\mu$. Find

the acceleration of $Q$,
the value of $\mu$,
the tension in the string.
State how in your calculation you have used the information that the string is inextensible. When the particles have moved for 3 s , the force $\mathbf { F }$ is removed.
Find the time between the instant that the force is removed and the instant that $Q$ comes to rest.

Edexcel M1 2009 June Q1

7 marks Moderate -0.3

Three posts $P , Q$ and $R$, are fixed in that order at the side of a straight horizontal road. The distance from $P$ to $Q$ is 45 m and the distance from $Q$ to $R$ is 120 m . A car is moving along the road with constant acceleration $a \mathrm {~m} \mathrm {~s} ^ { - 2 }$. The speed of the car, as it passes $P$, is $u \mathrm {~m} \mathrm {~s} ^ { - 1 }$. The car passes $Q$ two seconds after passing $P$, and the car passes $R$ four seconds after passing $Q$. Find
1. the value of $u$,
2. the value of $a$.
3. A particle is acted upon by two forces $\mathbf { F } _ { 1 }$ and $\mathbf { F } _ { 2 }$, given by
  $\mathbf { F } _ { 1 } = ( \mathbf { i } - 3 \mathbf { j } ) \mathrm { N }$,
  $\mathbf { F } _ { 2 } = ( p \mathbf { i } + 2 p \mathbf { j } ) \mathrm { N }$, where $p$ is a positive constant.
  (a) Find the angle between $\mathbf { F } _ { 2 }$ and $\mathbf { j }$.
The resultant of $\mathbf { F } _ { 1 }$ and $\mathbf { F } _ { 2 }$ is $\mathbf { R }$. Given that $\mathbf { R }$ is parallel to $\mathbf { i }$,
(b) find the value of $p$.

Edexcel M1 2009 June Q3

6 marks Moderate -0.3

3. Two particles $A$ and $B$ are moving on a smooth horizontal plane. The mass of $A$ is $2 m$ and the mass of $B$ is $m$. The particles are moving along the same straight line but in opposite directions and they collide directly. Immediately before they collide the speed of $A$ is $2 u$ and the speed of $B$ is $3 u$. The magnitude of the impulse received by each particle in the collision is $\frac { 7 m u } { 2 }$. Find

the speed of $A$ immediately after the collision,
the speed of $B$ immediately after the collision.

Edexcel M1 2009 June Q4

9 marks Moderate -0.3

4. A small brick of mass 0.5 kg is placed on a rough plane which is inclined to the horizontal at an angle $\theta$, where $\tan \theta = \frac { 4 } { 3 }$, and released from rest. The coefficient of friction between the brick and the plane is $\frac { 1 } { 3 }$. Find the acceleration of the brick.
(9)

Edexcel M1 2009 June Q5

9 marks Moderate -0.3

5. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{05fd9db2-2ff3-4b84-99c2-f348ff567ebd-06_332_780_292_585} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} A small box of mass 15 kg rests on a rough horizontal plane. The coefficient of friction between the box and the plane is 0.2 . Aforce of magnitude $P$ newtons is applied to the box at $50 ^ { \circ }$ to the horizontal, as shown in Figure 1. The box is on the point of sliding along the plane. Find the value of $P$, giving your answer to 2 significant figures.

Edexcel M1 2009 June Q6

13 marks Moderate -0.3

6. A car of mass 800 kg pulls a trailer of mass 200 kg along a straight horizontal road using a light towbar which is parallel to the road. The horizontal resistances to motion of the car and the trailer have magnitudes 400 N and 200 N respectively. The engine of the car produces a constant horizontal driving force on the car of magnitude 1200 N . Find

the acceleration of the car and trailer,
the magnitude of the tension in the towbar. The car is moving along the road when the driver sees a hazard ahead. He reduces the force produced by the engine to zero and applies the brakes. The brakes produce a force on the car of magnitude $F$ newtons and the car and trailer decelerate. Given that the resistances to motion are unchanged and the magnitude of the thrust in the towbar is 100 N ,
find the value of $F$.

Edexcel M1 2009 June Q7

12 marks Standard +0.3

7. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{05fd9db2-2ff3-4b84-99c2-f348ff567ebd-09_337_1287_228_370} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} A beam $A B$ is supported by two vertical ropes, which are attached to the beam at points $P$ and $Q$, where $A P = 0.3 \mathrm {~m}$ and $B Q = 0.3 \mathrm {~m}$. The beam is modelled as a uniform rod, of length 2 m and mass 20 kg . The ropes are modelled as light inextensible strings. A gymnast of mass 50 kg hangs on the beam between $P$ and $Q$. The gymnast is modelled as a particle attached to the beam at the point $X$, where $P X = x \mathrm {~m} , 0 < x < 1.4$ as shown in Figure 2. The beam rests in equilibrium in a horizontal position.

Show that the tension in the rope attached to the beam at $P$ is $( 588 - 350 x ) \mathrm { N }$.
Find, in terms of $x$, the tension in the rope attached to the beam at $Q$.
Hence find, justifying your answer carefully, the range of values of the tension which could occur in each rope. Given that the tension in the rope attached at $Q$ is three times the tension in the rope attached at $P$,
find the value of $x$. \section*{LU
$\_\_\_\_$}

Edexcel M1 2009 June Q8

13 marks Moderate -0.8

\hspace{0pt} [In this question $\mathbf { i }$ and $\mathbf { j }$ are horizontal unit vectors due east and due north respectively.] A hiker $H$ is walking with constant velocity $( 1.2 \mathbf { i } - 0.9 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }$.
1. Find the speed of $H$.
  (2)
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{05fd9db2-2ff3-4b84-99c2-f348ff567ebd-11_599_1057_521_445} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} A horizontal field $O A B C$ is rectangular with $O A$ due east and $O C$ due north, as shown in Figure 3. At twelve noon hiker $H$ is at the point $Y$ with position vector $100 \mathbf { j } \mathrm {~m}$, relative to the fixed origin $O$.
Write down the position vector of $H$ at time $t$ seconds after noon. At noon, another hiker $K$ is at the point with position vector $( 9 \mathbf { i } + 46 \mathbf { j } )$ m. Hiker $K$ is moving with constant velocity $( 0.75 \mathbf { i } + 1.8 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }$.
Show that, at time $t$ seconds after noon, $$\overrightarrow { H K } = [ ( 9 - 0.45 t ) \mathbf { i } + ( 2.7 t - 54 ) \mathbf { j } ] \text { metres. }$$ Hence,
show that the two hikers meet and find the position vector of the point where they meet.

Edexcel M1 2010 June Q1

5 marks Moderate -0.8

A particle $P$ is moving with constant velocity $( - 3 \mathbf { i } + 2 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }$. At time $t = 6 \mathrm {~s} P$ is at the point with position vector $( - 4 \mathbf { i } - 7 \mathbf { j } ) \mathrm { m }$. Find the distance of $P$ from the origin at time $t = 2 \mathrm {~s}$.
(5)
Particle $P$ has mass $m \mathrm {~kg}$ and particle $Q$ has mass $3 m \mathrm {~kg}$. The particles are moving in opposite directions along a smooth horizontal plane when they collide directly. Immediately before the collision $P$ has speed $4 u \mathrm {~m} \mathrm {~s} ^ { - 1 }$ and $Q$ has speed $k u \mathrm {~m} \mathrm {~s} ^ { - 1 }$, where $k$ is a constant. 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 $k$.
2. Find, in terms of $m$ and $u$, the magnitude of the impulse exerted on $P$ by $Q$.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{25300ba0-1e54-4242-8db4-a593f5d5a80e-04_195_579_260_507} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} A small box is pushed along a floor. The floor is modelled as a rough horizontal plane and the box is modelled as a particle. The coefficient of friction between the box and the floor is $\frac { 1 } { 2 }$. The box is pushed by a force of magnitude 100 N which acts at an angle of $30 ^ { \circ }$ with the floor, as shown in Figure 1. Given that the box moves with constant speed, find the mass of the box.

Edexcel M1 2010 June Q4

7 marks Standard +0.3

4. A beam $A B$ has length 6 m and weight 200 N . The beam rests in a horizontal position on two supports at the points $C$ and $D$, where $A C = 1 \mathrm {~m}$ and $D B = 1 \mathrm {~m}$. Two children, Sophie and Tom, each of weight 500 N , stand on the beam with Sophie standing twice as far from the end $B$ as Tom. The beam remains horizontal and in equilibrium and the magnitude of the reaction at $D$ is three times the magnitude of the reaction at $C$. By modelling the beam as a uniform rod and the two children as particles, find how far Tom is standing from the end $B$.

Edexcel M1 2010 June Q5

12 marks Standard +0.3

5. Two cars $P$ and $Q$ are moving in the same direction along the same straight horizontal road. Car $P$ is moving with constant speed $25 \mathrm {~m} \mathrm {~s} ^ { - 1 }$. At time $t = 0 , P$ overtakes $Q$ which is moving with constant speed $20 \mathrm {~m} \mathrm {~s} ^ { - 1 }$. From $t = T$ seconds, P decelerates uniformly, coming to rest at a point $X$ which is 800 m from the point where $P$ overtook $Q$. From $t = 25 \mathrm {~s}$, $Q$ decelerates uniformly, coming to rest at the same point $X$ at the same instant as $P$.

Sketch, on the same axes, the speed-time graphs of the two cars for the period from $t = 0$ to the time when they both come to rest at the point $X$.
Find the value of $T$.

Edexcel M1 2010 June Q6

10 marks Moderate -0.3

6. A ball is projected vertically upwards with a speed of $14.7 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ from a point which is 49 m above horizontal ground. Modelling the ball as a particle moving freely under gravity, find

the greatest height, above the ground, reached by the ball,
the speed with which the ball first strikes the ground,
the total time from when the ball is projected to when it first strikes the ground.

Edexcel M1 2010 June Q7

10 marks Standard +0.3

7. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{25300ba0-1e54-4242-8db4-a593f5d5a80e-10_275_712_269_612} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} A particle of mass 0.4 kg is held at rest on a fixed rough plane by a horizontal force of magnitude $P$ newtons. The force acts in the vertical plane containing the line of greatest slope of the inclined plane which passes through the particle. The plane is inclined to the horizontal at an angle $\alpha$, where tan $\alpha = \frac { 3 } { 4 }$, as shown in Figure 2. The coefficient of friction between the particle and the plane is $\frac { 1 } { 3 }$.
Given that the particle is on the point of sliding up the plane, find

the magnitude of the normal reaction between the particle and the plane,
the value of $P$.

Edexcel M1 2010 June Q8

17 marks Standard +0.3

8. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{25300ba0-1e54-4242-8db4-a593f5d5a80e-12_890_428_237_754} \captionsetup{labelformat=empty} \caption{Figure 3}

\end{figure} Two particles $A$ and $B$ have mass 0.4 kg and 0.3 kg respectively. The particles are attached to the ends of a light inextensible string. The string passes over a small smooth pulley which is fixed above a horizontal floor. Both particles are held, with the string taut, at a height of 1 m above the floor, as shown in Figure 3. The particles are released from rest and in the subsequent motion $B$ does not reach the pulley.

Find the tension in the string immediately after the particles are released.
Find the acceleration of $A$ immediately after the particles are released. When the particles have been moving for 0.5 s , the string breaks.
Find the further time that elapses until $B$ hits the floor.

Edexcel M1 2011 June Q1

8 marks Moderate -0.3

At time $t = 0$ a ball is projected vertically upwards from a point $O$ and rises to a maximum height of 40 m above $O$. The ball is modelled as a particle moving freely under gravity.
1. Show that the speed of projection is $28 \mathrm {~m} \mathrm {~s} ^ { - 1 }$.
2. Find the times, in seconds, when the ball is 33.6 m above $O$.
3. Particle $P$ has mass 3 kg and particle $Q$ has mass 2 kg . The particles are moving in opposite directions on a smooth horizontal plane when they collide directly. Immediately before the collision, $P$ has speed $3 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ and $Q$ has speed $2 \mathrm {~m} \mathrm {~s} ^ { - 1 }$. Immediately after the collision, both particles move in the same direction and the difference in their speeds is $1 \mathrm {~m} \mathrm {~s} ^ { - 1 }$.
4. Find the speed of each particle after the collision.
5. Find the magnitude of the impulse exerted on $P$ by $Q$.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{9a1ffe48-cea7-49aa-9b6f-f781568d0600-04_344_771_221_589} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} A particle of weight $W$ newtons is held in equilibrium on a rough inclined plane by a horizontal force of magnitude 4 N . The force acts in a vertical plane containing a line of greatest slope of the inclined plane. The plane is inclined to the horizontal at an angle $\alpha$, where $\tan \alpha = \frac { 3 } { 4 }$, as shown in Figure 1.
The coefficient of friction between the particle and the plane is $\frac { 1 } { 2 }$.
Given that the particle is on the point of sliding down the plane,
(i) show that the magnitude of the normal reaction between the particle and the plane is 20 N ,
(ii) find the value of $W$.

Edexcel M1 2011 June Q4

12 marks Moderate -0.8

A girl runs a 400 m race in a time of 84 s . In a model of this race, it is assumed that, starting from rest, she moves with constant acceleration for 4 s , reaching a speed of $5 \mathrm {~m} \mathrm {~s} ^ { - 1 }$. She maintains this speed for 60 s and then moves with constant deceleration for 20 s , crossing the finishing line with a speed of $V \mathrm {~m} \mathrm {~s} ^ { - 1 }$.
1. Sketch, in the space below, a speed-time graph for the motion of the girl during the whole race.
2. Find the distance run by the girl in the first 64 s of the race.
3. Find the value of $V$.
4. Find the deceleration of the girl in the final 20 s of her race.

Edexcel M1 2011 June Q5

11 marks Moderate -0.3

A plank $P Q R$, of length 8 m and mass 20 kg , is in equilibrium in a horizontal position on two supports at $P$ and $Q$, where $P Q = 6 \mathrm {~m}$.

A child of mass 40 kg stands on the plank at a distance of 2 m from $P$ and a block of mass $M \mathrm {~kg}$ is placed on the plank at the end $R$. The plank remains horizontal and in equilibrium. The force exerted on the plank by the support at $P$ is equal to the force exerted on the plank by the support at $Q$. By modelling the plank as a uniform rod, and the child and the block as particles,

1. find the magnitude of the force exerted on the plank by the support at $P$,
2. find the value of $M$.
State how, in your calculations, you have used the fact that the child and the block can be modelled as particles.

Edexcel M1 2011 June Q6

16 marks Standard +0.3

6. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{9a1ffe48-cea7-49aa-9b6f-f781568d0600-10_369_954_214_497} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} Two particles $P$ and $Q$ have masses 0.3 kg and $m \mathrm {~kg}$ respectively. The particles are attached to the ends of a light inextensible string. The string passes over a small smooth pulley which is fixed at the top of a fixed rough 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 $\frac { 1 } { 2 }$. The string lies in a vertical plane through a line of greatest slope of the inclined plane. The particle $P$ is held at rest on the inclined plane and the particle $Q$ hangs freely below the pulley with the string taut, as shown in Figure 2. The system is released from rest and $Q$ accelerates vertically downwards at $1.4 \mathrm {~m} \mathrm {~s} ^ { - 2 }$. Find

the magnitude of the normal reaction of the inclined plane on $P$,
the value of $m$. When the particles have been moving for 0.5 s , the string breaks. Assuming that $P$ does not reach the pulley,
find the further time that elapses until $P$ comes to instantaneous rest.

Edexcel M1 2011 June Q7

11 marks Moderate -0.3

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

Two ships $P$ and $Q$ are moving with constant velocities. Ship $P$ moves with velocity $( 2 \mathbf { i } - 3 \mathbf { j } ) \mathrm { km } \mathrm { h } ^ { - 1 }$ and ship $Q$ moves with velocity $( 3 \mathbf { i } + 4 \mathbf { j } ) \mathrm { km } \mathrm { h } ^ { - 1 }$.

Find, to the nearest degree, the bearing on which $Q$ is moving. At 2 pm , ship $P$ is at the point with position vector $( \mathbf { i } + \mathbf { j } ) \mathrm { km }$ and $\operatorname { ship } Q$ is at the point with position vector $( - 2 \mathbf { j } ) \mathrm { km }$. At time $t$ hours after 2 pm , the position vector of $P$ is $\mathbf { p } \mathrm { km }$ and the position vector of $Q$ is $\mathbf { q } \mathrm { km }$.
Write down expressions, in terms of $t$, for
1. $\mathbf { p }$,
2. $\mathbf { q }$,
3. $\overrightarrow { P Q }$.
Find the time when
1. $Q$ is due north of $P$,
2. $Q$ is north-west of $P$.

Edexcel M1 2012 June Q1

6 marks Moderate -0.8

Two particles $A$ and $B$, of mass $5 m \mathrm {~kg}$ and $2 m \mathrm {~kg}$ respectively, are moving in opposite directions along the same straight horizontal line. The particles collide directly. Immediately before the collision, the speeds of $A$ and $B$ are $3 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ and $4 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ respectively. The direction of motion of $A$ is unchanged by the collision. Immediately after the collision, the speed of $A$ is $0.8 \mathrm {~m} \mathrm {~s} ^ { - 1 }$.
1. Find the speed of $B$ immediately after the collision.
In the collision, the magnitude of the impulse exerted on $A$ by $B$ is 3.3 N s .
Find the value of $m$.

Edexcel M1 2012 June Q2

7 marks Standard +0.3

2. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{5c908e75-73df-46be-93bb-09dba2cb3b7e-03_215_716_233_614} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} A non-uniform rod $A B$ has length 3 m and mass 4.5 kg . The rod rests in equilibrium, in a horizontal position, on two smooth supports at $P$ and at $Q$, where $A P = 0.8 \mathrm {~m}$ and $Q B = 0.6 \mathrm {~m}$, as shown in Figure 1. The centre of mass of the rod is at $G$. Given that the magnitude of the reaction of the support at $P$ on the rod is twice the magnitude of the reaction of the support at $Q$ on the rod, find

the magnitude of the reaction of the support at $Q$ on the rod,
the distance $A G$.

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