Questions - Edexcel M2 - A-Level Maths

Edexcel M2 Q3

11 marks Standard +0.8

3. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{ef2dd10c-5a3c-4868-af00-aaf7f2937d7e-2_421_474_1080_664} \captionsetup{labelformat=empty} \caption{Fig. 1}

\end{figure} Figure 1 shows a ladder of mass 20 kg and length 6 m leaning against a rough vertical wall with its lower end on smooth horizontal ground. The ladder is prevented from slipping along the ground by a light rope which is attached to the ladder 2 m from its bottom end and fastened to the wall so that the rope is horizontal and perpendicular to the wall. The ladder is at an angle $\theta$ to the horizontal where $\tan \theta = \frac { 5 } { 2 }$ and the coefficient of friction between the ladder and the wall is $\frac { 1 } { 3 }$.

Draw a diagram showing all the forces acting on the ladder.
Show that the magnitude of the tension in the rope is $5 g$. A man wishes to use the ladder but fears the rope will snap as he climbs the ladder.
Suggest, giving a reason for your answer, a more suitable position for the rope.
(2 marks)

Edexcel M2 Q4

12 marks Standard +0.3

4. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{ef2dd10c-5a3c-4868-af00-aaf7f2937d7e-3_222_350_242_788} \captionsetup{labelformat=empty} \caption{Fig. 2}

\end{figure} Figure 2 shows an earring consisting of a uniform wire $A B C D$ of length $6 a$ bent to form right angles at $B$ and $C$ such that $A B$ and $C D$ are of length $2 a$ and $a$ respectively.

Find, in terms of $a$, the distance of the centre of mass from
1. $\quad A B$,
2. $B C$. The earring is to be worn such that it hangs in equilibrium suspended from the point $A$.
Find, to the nearest degree, the angle made by $A B$ with the downward vertical.
(4 marks)

Edexcel M2 Q5

13 marks Moderate -0.3

5. A lorry of mass 40 tonnes moves up a straight road inclined at an angle $\alpha$ to the horizontal where $\sin \alpha = \frac { 1 } { 20 }$. The lorry moves at a constant speed of $20 \mathrm {~m} \mathrm {~s} ^ { - 1 }$. In a model of the motion of the lorry, the non-gravitational resistance to motion is assumed to be constant and of magnitude 4400 N .

Show that the engine of the lorry is working at a rate of 480 kW . The road becomes horizontal. The lorry's engine continues to work at the same rate and the resistance to motion is assumed to remain unchanged.
Find the initial acceleration of the lorry.
Find, correct to 3 significant figures, the maximum speed of the lorry.
Using your answer to part (c), comment on the suitability of the modelling assumption.

Edexcel M2 Q6

13 marks Moderate -0.3

6. Particle $S$ of mass $2 M$ is moving with speed $U \mathrm {~m} \mathrm {~s} ^ { - 1 }$ on a smooth horizontal plane when it collides directly with a particle $T$ of mass $5 M$ which is lying at rest on the plane. The coefficient of restitution between $S$ and $T$ is $\frac { 3 } { 4 }$. Given that the speed of $T$ after the collision is $4 \mathrm {~ms} ^ { - 1 }$,

find $U$. As a result of the collision, $T$ is projected horizontally from the top of a building of height 19.6 m and falls freely under gravity. $T$ strikes the ground at the point $X$ as shown in Figure 3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{ef2dd10c-5a3c-4868-af00-aaf7f2937d7e-4_663_928_740_523} \captionsetup{labelformat=empty} \caption{Fig. 3}
\end{figure}
Find the time taken for $T$ to reach $X$.
Show that the angle between the horizontal and the direction of motion of $T$, just before it strikes the ground at $X$, is $78.5 ^ { \circ }$ correct to 3 significant figures.
(4 marks)

Edexcel M2 Q7

16 marks Standard +0.3

7. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{ef2dd10c-5a3c-4868-af00-aaf7f2937d7e-5_495_604_214_580} \captionsetup{labelformat=empty} \caption{Fig. 4}

\end{figure} Figure 4 shows a particle $P$ projected from the point $A$ up the line of greatest slope of a rough plane which is inclined at an angle $\alpha$ to the horizontal where $\sin \alpha = \frac { 4 } { 5 } . P$ is projected with speed $5.6 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ and the coefficient of friction between $P$ and the plane is $\frac { 4 } { 7 }$. Given that $P$ first comes to rest at point $B$,

use the Work-Energy principle to show that the distance $A B$ is 1.4 m . The particle then slides back down the plane.
Find, correct to 2 significant figures, the speed of $P$ when it returns to $A$.

Edexcel M2 Q1

5 marks Moderate -0.3

An ice hockey puck of mass 0.5 kg is moving with velocity $( 5 \mathbf { i } - 8 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }$, where $\mathbf { i }$ and $\mathbf { j }$ are perpendicular horizontal unit vectors, when it is struck by a stick. After the impact, the puck travels with velocity $( 13 \mathbf { i } + 7 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }$.

Find the magnitude of the impulse exerted by the stick on the puck.
(5 marks)

Edexcel M2 Q2

8 marks Moderate -0.3

2. A car of mass 1 tonne is climbing a hill inclined at an angle $\theta$ to the horizontal where $\sin \theta = \frac { 1 } { 7 }$. When the car passes a point $X$ on the hill, it is travelling at $20 \mathrm {~m} \mathrm {~s} ^ { - 1 }$. When the car passes the point $Y , 200 \mathrm {~m}$ further up the hill, it has speed $10 \mathrm {~ms} ^ { - 1 }$. In a preliminary model of the situation, the car engine is assumed only to be doing work against gravity. Using this model,

find the change in the total mechanical energy of the car as it moves from $X$ to $Y$.
(6 marks)
In a more sophisticated model, the car engine is also assumed to work against other forces.
Write down two other forces which this model might include.
(2 marks)

Edexcel M2 Q3

8 marks Moderate -0.8

3. A particle moves along a straight horizontal track such that its displacement, $s$ metres, from a fixed point $O$ on the line after $t$ seconds is given by $$s = 2 t ^ { 3 } - 13 t ^ { 2 } + 20 t$$

Find the values of $t$ for which the particle is at $O$.
Find the values of $t$ at which the particle comes instantaneously to rest.

Edexcel M2 Q4

9 marks Standard +0.3

4. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{0ea2267e-6c46-4a4f-9a38-c242de57901d-3_378_730_196_609} \captionsetup{labelformat=empty} \caption{Fig. 1}

\end{figure} Figure 1 shows a uniform rod $A B$ of length 2 m and mass 6 kg inclined at an angle of $30 ^ { \circ }$ to the horizontal with $A$ on smooth horizontal ground and $B$ supported by a rough peg. The rod is in limiting equilibrium and the coefficient of friction between $B$ and the peg is $\mu$.

Find, in terms of $g$, the magnitude of the reactions at $A$ and $B$.
Show that $\mu = \frac { 1 } { \sqrt { 3 } }$.

Edexcel M2 Q5

12 marks Standard +0.3

5. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{0ea2267e-6c46-4a4f-9a38-c242de57901d-3_405_718_1169_555} \captionsetup{labelformat=empty} \caption{Fig. 2}

\end{figure} During a cricket match, a batsman hits the ball giving it an initial velocity of $22 \mathrm {~ms} ^ { - 1 }$ at an angle $\alpha$ to the horizontal where $\sin \alpha = \frac { 7 } { 8 }$. When the batsman strikes the ball it is 1.6 metres above the ground, as shown in Figure 2, and it subsequently moves freely under gravity.

Find, correct to 3 significant figures, the maximum height above the ground reached by the ball. The ball is caught by a fielder when it is 0.2 metres above the ground.
Find the length of time for which the ball is in the air. Assuming that the fielder who caught the ball ran at a constant speed of $6 \mathrm {~m} \mathrm {~s} ^ { - 1 }$,
find, correct to 3 significant figures, the maximum distance that the fielder could have been from the ball when it was struck.

Edexcel M2 Q6

16 marks Standard +0.3

6. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{0ea2267e-6c46-4a4f-9a38-c242de57901d-4_433_282_196_726} \captionsetup{labelformat=empty} \caption{Fig. 3}

\end{figure} Figure 3 shows a uniform rectangular lamina $A B C D$ of mass $8 m$ in which the sides $A B$ and $B C$ are of length $a$ and $2 a$ respectively. Particles of mass $2 m , 6 m$ and $4 m$ are fixed to the lamina at the points $A , B$ and $D$ respectively.

Write down the distance of the centre of mass from $A D$.
Show that the distance of the centre of mass from $A B$ is $\frac { 4 } { 5 } a$. Another particle of mass $k m$ is attached to the lamina at the point $B$.
Show that the distance of the centre of mass from $A D$ is now given by $\frac { ( 10 + k ) a } { 20 + k }$.
(4 marks)
Given that when the lamina is suspended freely from the point $A$ the side $A B$ makes an angle of $45 ^ { \circ }$ with the vertical,
find the value of $k$.

Edexcel M2 Q7

17 marks Standard +0.3

7. Particle $A$ of mass 7 kg is moving with speed $u _ { 1 }$ on a smooth horizontal surface when it collides directly with particle $B$ of mass 4 kg moving in the same direction as $A$ with speed $u _ { 2 }$. After the impact, $A$ continues to move in the same direction but its speed has been halved. Given that the coefficient of restitution between the particles is $e$,

show that $8 u _ { 2 } ( e + 1 ) = u _ { 1 } ( 8 e - 3 )$. Given also that $u _ { 1 } = 14 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ and $u _ { 2 } = 3 \mathrm {~m} \mathrm {~s} ^ { - 1 }$,
find $e$,
show that the percentage of the kinetic energy of the particles lost as a result of the impact is $9.6 \%$, correct to 2 significant figures.

Edexcel M2 Q3

11 marks Moderate -0.8

3. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{90893903-4f36-4974-8eaa-0f462f35f442-02_650_1043_367_317} \captionsetup{labelformat=empty} \caption{Fig. 2}

\end{figure} The points $A ( 3,0 )$ and $B ( 0,4 )$ are two vertices of the rectangle $A B C D$, as shown in Fig. 2.

Write down the gradient of $A B$ and hence the gradient of $B C$. The point $C$ has coordinates $( 8 , k )$, where $k$ is a positive constant.
Find the length of $B C$ in terms of $k$. Given that the length of $B C$ is 10 and using your answer to part (b),
find the value of $k$,
find the coordinates of $D$.

Edexcel M2 Q4

5 marks Moderate -0.8

4. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{90893903-4f36-4974-8eaa-0f462f35f442-03_725_560_310_571} \captionsetup{labelformat=empty} \caption{Fig. 4}

\end{figure} A manufacturer produces cartons for fruit juice. Each carton is in the shape of a closed cuboid with base dimensions $2 x \mathrm {~cm}$ by $x \mathrm {~cm}$ and height $h \mathrm {~cm}$, as shown in Fig. 4. Given that the capacity of a carton has to be $1030 \mathrm {~cm} ^ { 3 }$,

express $h$ in terms of $x$,
show that the surface area, $A \mathrm {~cm} ^ { 2 }$, of a carton is given by $$A = 4 x ^ { 2 } + \frac { 3090 } { x } .$$

Edexcel M2 Q16

13 marks Moderate -0.8

16. \section*{Figure 3}

\includegraphics[max width=\textwidth, alt={}]{90893903-4f36-4974-8eaa-0f462f35f442-08_581_575_395_609}

The points $A ( - 3 , - 2 )$ and $B ( 8,4 )$ are at the ends of a diameter of the circle shown in Fig. 3.

Find the coordinates of the centre of the circle.
Find an equation of the diameter $A B$, giving your answer in the form $a x + b y + c = 0$, where $a , b$ and $c$ are integers.
Find an equation of tangent to the circle at $B$. The line $l$ passes through $A$ and the origin.
Find the coordinates of the point at which $l$ intersects the tangent to the circle at $B$, giving your answer as exact fractions.

Edexcel M2 Q20

14 marks Moderate -0.5

20. The curve $C$ has equation $y = \mathrm { f } ( x )$. Given that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = 3 x ^ { 2 } - 20 x + 29$$ and that $C$ passes through the point $P ( 2,6 )$,

find $y$ in terms of $x$.
Verify that $C$ passes through the point ( 4,0 ).
Find an equation of the tangent to $C$ at $P$. The tangent to $C$ at the point $Q$ is parallel to the tangent at $P$.
Calculate the exact $x$-coordinate of $Q$.
21. $$y = 7 + 10 x ^ { \frac { 3 } { 2 } }$$
Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$.
Find $\int y \mathrm {~d} x$.
22. (a) Given that $3 ^ { x } = 9 ^ { y - 1 }$, show that $x = 2 y - 2$.
Solve the simultaneous equations $$\begin{gathered} x = 2 y - 2 \\ x ^ { 2 } = y ^ { 2 } + 7 \end{gathered}$$
1. The straight line $l _ { 1 }$ with equation $y = \frac { 3 } { 2 } x - 2$ crosses the $y$-axis at the point $P$. The point $Q$ has coordinates $( 5 , - 3 )$.
The straight line $l _ { 2 }$ is perpendicular to $l _ { 1 }$ and passes through $Q$.
Calculate the coordinates of the mid-point of $P Q$.
Find an equation for $l _ { 2 }$ in the form $a x + b y = c$, where $a$, b and $c$ are integer constants. The lines $l _ { 1 }$ and $l _ { 2 }$ intersect at the point $R$.
Calculate the exact coordinates of $R$.
24. $$\frac { \mathrm { d } y } { \mathrm {~d} x } = 5 + \frac { 1 } { x ^ { 2 } } .$$
Use integration to find $y$ in terms of $x$.
Given that $y = 7$ when $x = 1$, find the value of $y$ at $x = 2$.
25. Find the set of values for $x$ for which
$6 x - 7 < 2 x + 3$,
$2 x ^ { 2 } - 11 x + 5 < 0$,
both $6 x - 7 < 2 x + 3$ and $2 x ^ { 2 } - 11 x + 5 < 0$.
[0pt] [P1 June 2003 Question 2]
26. In the first month after opening, a mobile phone shop sold 280 phones. A model for future trading assumes that sales will increase by $x$ phones per month for the next 35 months, so that $( 280 + x )$ phones will be sold in the second month, $( 280 + 2 x )$ in the third month, and so on. Using this model with $x = 5$, calculate
1. the number of phones sold in the 36th month,
2. the total number of phones sold over the 36 months. The shop sets a sales target of 17000 phones to be sold over the 36 months.
  Using the same model,
find the least value of $x$ required to achieve this target.
[0pt] [P1 June 2003 Question 3]
27. The points $A$ and $B$ have coordinates $( 4,6 )$ and $( 12,2 )$ respectively. The straight line $l _ { 1 }$ passes through $A$ and $B$.
Find an equation for $l _ { 1 }$ in the form $a x + b y = c$, where $a$, b and $c$ are integers. The straight line $l _ { 2 }$ passes through the origin and has gradient - 4 .
Write down an equation for $l _ { 2 }$. The lines $l _ { 1 }$ and $l _ { 2 }$ intercept at the point $C$.
Find the exact coordinates of the mid-point of $A C$.
28. For the curve $C$ with equation $y = x ^ { 4 } - 8 x ^ { 2 } + 3$,
find $\frac { \mathrm { d } y } { \mathrm {~d} x }$, The point $A$, on the curve $C$, has $x$-coordinate 1 .
Find an equation for the normal to $C$ at $A$, giving your answer in the form $a x + b y + c = 0$, where $a , b$ and $c$ are integers.
[0pt] [P1 June 2003 Question 8*]
29. The sum of an arithmetic series is $$\sum _ { r = 1 } ^ { n } ( 80 - 3 r )$$
Write down the first two terms of the series.
Find the common difference of the series. Given that $n = 50$,
find the sum of the series.
30. (a) Solve the equation $4 x ^ { 2 } + 12 x = 0$. $$f ( x ) = 4 x ^ { 2 } + 12 x + c$$ where $c$ is a constant.
Given that $\mathrm { f } ( x ) = 0$ has equal roots, find the value of $c$ and hence solve $\mathrm { f } ( x ) = 0$.
31. Solve the simultaneous equations $$\begin{aligned} & x - 3 y + 1 = 0 \\ & x ^ { 2 } - 3 x y + y ^ { 2 } = 11 \end{aligned}$$
1. A container made from thin metal is in the shape of a right circular cylinder with height $h \mathrm {~cm}$ and base radius $r \mathrm {~cm}$. The container has no lid. When full of water, the container holds $500 \mathrm {~cm} ^ { 3 }$ of water.
Show that the exterior surface area, $A \mathrm {~cm} ^ { 2 }$, of the container is given by $$A = \pi r ^ { 2 } + \frac { 1000 } { r } .$$ 33. \section*{Figure 1}
\includegraphics[max width=\textwidth, alt={}]{90893903-4f36-4974-8eaa-0f462f35f442-15_668_748_358_699}
The points $A$ and $B$ have coordinates $( 2 , - 3 )$ and $( 8,5 )$ respectively, and $A B$ is a chord of a circle with centre $C$, as shown in Fig. 1.
Find the gradient of $A B$. The point $M$ is the mid-point of $A B$.
Find an equation for the line through $C$ and $M$. Given that the $x$-coordinate of $C$ is 4 ,
find the $y$-coordinate of $C$,
show that the radius of the circle is $\frac { 5 \sqrt { } 17 } { 4 }$.
34. The first three terms of an arithmetic series are $p , 5 p - 8$, and $3 p + 8$ respectively.
Show that $p = 4$.
Find the value of the 40th term of this series.
Prove that the sum of the first $n$ terms of the series is a perfect square.
35. $$\mathrm { f } ( x ) = x ^ { 2 } - k x + 9 , \text { where } k \text { is a constant. }$$
Find the set of values of $k$ for which the equation $\mathrm { f } ( x ) = 0$ has no real solutions. Given that $k = 4$,
express $\mathrm { f } ( x )$ in the form $( x - p ) ^ { 2 } + q$, where $p$ and $q$ are constants to be found,
36. The curve $C$ with equation $y = \mathrm { f } ( x )$ is such that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = 3 \sqrt { } x + \frac { 12 } { \sqrt { } x } , \quad x > 0 .$$
Show that, when $x = 8$, the exact value of $\frac { \mathrm { d } y } { \mathrm {~d} x }$ is $9 \sqrt { } 2$. The curve $C$ passes through the point $( 4,30 )$.
Using integration, find $\mathrm { f } ( x )$.
37. \section*{Figure 2}
\includegraphics[max width=\textwidth, alt={}]{90893903-4f36-4974-8eaa-0f462f35f442-17_687_1074_351_539}
Figure 2 shows the curve with equation $y ^ { 2 } = 4 ( x - 2 )$ and the line with equation $2 x - 3 y = 12$.
The curve crosses the $x$-axis at the point $A$, and the line intersects the curve at the points $P$ and $Q$.
Write down the coordinates of $A$.
Find, using algebra, the coordinates of $P$ and $Q$.
Show that $\angle P A Q$ is a right angle.
38. A sequence is defined by the recurrence relation $$u _ { n + 1 } = \sqrt { \left( \frac { u _ { n } } { 2 } + \frac { a } { u _ { n } } \right) } , \quad n = 1,2,3 , \ldots ,$$ where $a$ is a constant.
Given that $a = 20$ and $u _ { 1 } = 3$, find the values of $u _ { 2 } , u _ { 3 }$ and $u _ { 4 }$, giving your answers to 2 decimal places.
Given instead that $u _ { 1 } = u _ { 2 } = 3$,
1. calculate the value of $a$,
2. write down the value of $u _ { 5 }$.
  [0pt] [P2 January 2004 Question 2]
  39. The points $A$ and $B$ have coordinates $( 1,2 )$ and $( 5,8 )$ respectively.
Find the coordinates of the mid-point of $A B$.
Find, in the form $y = m x + c$, an equation for the straight line through $A$ and $B$.
40. Giving your answers in the form $a + b \sqrt { 2 }$, where $a$ and $b$ are rational numbers, find
$( 3 - \sqrt { } 8 ) ^ { 2 }$,
$\frac { 1 } { 4 - \sqrt { 8 } }$.
41. The width of a rectangular sports pitch is $x$ metres, $x > 0$. The length of the pitch is 20 m more than its width. Given that the perimeter of the pitch must be less than 300 m ,
form a linear inequality in $x$. Given that the area of the pitch must be greater than $4800 \mathrm {~m} ^ { 2 }$,
form a quadratic inequality in $x$.
by solving your inequalities, find the set of possible values of $x$.
42. The curve $C$ has equation $y = x ^ { 2 } - 4$ and the straight line $l$ has equation $y + 3 x = 0$.
In the space below, sketch $C$ and $l$ on the same axes.
Write down the coordinates of the points at which $C$ meets the coordinate axes.
Using algebra, find the coordinates of the points at which $l$ intersects $C$.
43. $$f ( x ) = \frac { \left( x ^ { 2 } - 3 \right) ^ { 2 } } { x ^ { 3 } } , x \neq 0$$
Show that $\mathrm { f } ( x ) \equiv x - 6 x ^ { - 1 } + 9 x ^ { - 3 }$.
Hence, or otherwise, differentiate $\mathrm { f } ( x )$ with respect to $x$.

Edexcel M2 2024 October Q1

Standard +0.3

In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.

At time $t$ seconds, $t \geqslant 0$, a particle $P$ is moving with velocity $\mathbf { v } \mathrm { ms } ^ { - 1 }$, where $$\mathbf { v } = 3 ( t + 2 ) ^ { 2 } \mathbf { i } + 5 t ( t + 2 ) \mathbf { j }$$ Position vectors are given relative to the fixed point $O$ At time $t = 0 , P$ is at the point with position vector $( - 30 \mathbf { i } - 45 \mathbf { j } ) \mathrm { m }$.

Find the position vector of $P$ when $t = 3$
Find the magnitude of the acceleration of $P$ when $t = 3$ At time $T$ seconds, $P$ is moving in the direction of the vector $2 \mathbf { i } + \mathbf { j }$
Find the value of $T$

Edexcel M2 2024 October Q2

Standard +0.3

A particle $Q$ of mass 3 kg is moving on a smooth horizontal surface.

Particle $Q$ is moving with velocity $5 \mathbf { i } \mathrm {~ms} ^ { - 1 }$ when it receives a horizontal impulse of magnitude $3 \sqrt { 82 } \mathrm { Ns }$. Immediately after receiving the impulse, the velocity of $Q$ is $( x \mathbf { i } + y \mathbf { j } ) \mathrm { ms } ^ { - 1 }$, where $x$ and $y$ are positive constants. The kinetic energy gained by $Q$ as a result of receiving the impulse is 138 J .
Find, in terms of $\mathbf { i }$ and $\mathbf { j }$, the velocity of $Q$ immediately after receiving the impulse.

Edexcel M2 2024 October Q3

Standard +0.3

3. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{3e78f951-041d-4227-aa4b-e67a6ab5b4cd-06_275_1143_303_461} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} A van of mass 900 kg is moving up a straight road inclined at an angle $\alpha$ to the horizontal, where $\sin \alpha = \frac { 1 } { 25 }$. The van is towing a trailer of mass 200 kg . The trailer is attached to the van by a rigid towbar which is parallel to the direction of motion of the van and the trailer, as shown in Figure 1. The resistance to the motion of the van from non-gravitational forces is modelled as a constant force of magnitude 400 N .
The resistance to the motion of the trailer from non-gravitational forces is modelled as a constant force of magnitude 240 N . The towbar is modelled as a light rod.
The engine of the van is working at a constant rate of 15 kW .

Find the acceleration of the van at the instant when the speed of the van is $12 \mathrm {~ms} ^ { - 1 }$ At the instant when the speed of the van is $14 \mathrm {~ms} ^ { - 1 }$, the trailer is passing the point $A$ on the slope and the towbar breaks. The trailer continues to move up the slope until it comes to rest at the point $B$.
The resistance to the motion of the trailer from non-gravitational forces is still modelled as a constant force of magnitude 240 N .
Use the work-energy principle to find the distance $A B$.

Edexcel M2 2024 October Q4

Standard +0.3

4. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{3e78f951-041d-4227-aa4b-e67a6ab5b4cd-10_301_871_319_598} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} The uniform lamina $A B C D$ shown in Figure 2 is in the shape of an isosceles trapezium.

$B C$ is parallel to $A D$ and angle $B A D$ is equal to angle $A D C$
$B C = 5 a$ and $A D = 7 a$
the perpendicular distance between $B C$ and $A D$ is $3 a$
the distance of the centre of mass of $A B C D$ from $A D$ is $d$
1. Show that $d = \frac { 17 } { 12 } a$

The uniform lamina $P Q R S$ is a rectangle with $P Q = 5 a$ and $Q R = 9 a$.
The lamina $A B C D$ in Figure 2 is used to cut a hole in $P Q R S$ to form the template shown shaded in Figure 3. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{3e78f951-041d-4227-aa4b-e67a6ab5b4cd-10_364_876_1567_593} \captionsetup{labelformat=empty} \caption{Figure 3}

\end{figure}

The template is freely suspended from $P$ and hangs in equilibrium with $P S$ at an angle of $\theta ^ { \circ }$ to the downward vertical.

Find the value of $\theta$

Edexcel M2 2024 October Q5

Standard +0.3

The fixed points $X$ and $Y$ lie on horizontal ground.

At time $t = 0$, a particle $P$ is projected from $X$ with speed $u \mathrm {~ms} ^ { - 1 }$ at angle $\theta$ to the ground. Particle $P$ moves freely under gravity and first hits the ground at $Y$.

Show that $X Y = \frac { u ^ { 2 } \sin 2 \theta } { g }$ The points $A$ and $B$ lie on horizontal ground. A vertical pole $C D$ has length 5 m .
The end $C$ is fixed to the ground between $A$ and $B$, where $A C = 12 \mathrm {~m}$.
At time $t = 0$, a particle $Q$ is projected from $A$ with speed $20 \mathrm {~ms} ^ { - 1 }$ at $60 ^ { \circ }$ to the ground.
Particle $Q$ moves freely under gravity, passes over the pole and first hits the ground at $B$, as shown in Figure 4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{3e78f951-041d-4227-aa4b-e67a6ab5b4cd-14_335_1179_1032_443} \captionsetup{labelformat=empty} \caption{Figure 4}
\end{figure}
Find the distance $C B$.
Find the height of $Q$ above $D$ at the instant when $Q$ passes over the pole.

Edexcel M2 2024 October Q6

Standard +0.3

6. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{3e78f951-041d-4227-aa4b-e67a6ab5b4cd-18_419_1307_315_379} \captionsetup{labelformat=empty} \caption{Figure 5}

\end{figure} A uniform beam $A B$, of weight $5 W$ and length $12 a$, rests with end $A$ on rough horizontal ground.
A package of weight $W$ is attached to the beam at $B$.
The beam rests in equilibrium on a smooth horizontal peg at $C$, with $A C = 9 a$, as shown in Figure 5.
The beam is inclined at an angle $\theta$ to the ground, where $\tan \theta = \frac { 5 } { 12 }$ The beam is modelled as a rod that lies in a vertical plane perpendicular to the peg. The package is modelled as a particle. The normal reaction between the beam and the peg at $C$ has magnitude $k W$ Using the model,

show that $k = \frac { 56 } { 13 }$ The coefficient of friction between $A$ and the ground is $\mu$ Given that the beam is resting in limiting equilibrium,
find the value of $\mu$

Edexcel M2 2024 October Q7

Standard +0.3

A particle $P$ has mass $5 m$ and a particle $Q$ has mass $2 m$.

The particles are moving in opposite directions along the same straight line on a smooth horizontal surface.
Particle $P$ collides directly with particle $Q$.
Immediately before the collision, the speed of $P$ is $2 u$ and the speed of $Q$ is $3 u$. Immediately after the collision, the speed of $P$ is $x$ and the speed of $Q$ is $y$.
The direction of motion of $Q$ is reversed as a result of the collision.
The coefficient of restitution between $P$ and $Q$ is $e$.

Find the set of values of $e$ for which the direction of motion of $P$ is unchanged as a result of the collision. In the collision, $Q$ receives an impulse of magnitude $\frac { 60 } { 7 } m u$
Show that $e = \frac { 1 } { 5 }$ After the collision, $Q$ hits a smooth fixed vertical wall that is perpendicular to the direction of motion of $Q$. Particle $Q$ rebounds and there is a second collision between $P$ and $Q$.
The coefficient of restitution between $Q$ and the wall is $\frac { 1 } { 3 }$
Find, in terms of $m$ and $u$, the magnitude of the impulse received by $Q$ in the second collision between $P$ and $Q$.

Edexcel M2 2014 January Q1

8 marks Moderate -0.3

A particle $P$ of mass 2 kg is moving with velocity $(3\mathbf{i} + 4\mathbf{j})$ m s$^{-1}$ when it receives an impulse. Immediately after the impulse is applied, $P$ has velocity $(2\mathbf{i} - 3\mathbf{j})$ m s$^{-1}$.

Find the magnitude of the impulse. [5]
Find the angle between the direction of the impulse and the direction of motion of $P$ immediately before the impulse is applied. [3]

Edexcel M2 2014 January Q2

11 marks Standard +0.3

A particle $P$ moves on the $x$-axis. At time $t$ seconds the velocity of $P$ is $v$ m s$^{-1}$ in the direction of $x$ increasing, where $$v = (t - 2)(3t - 10), \quad t \geq 0$$ When $t = 0$, $P$ is at the origin $O$.

Find the acceleration of $P$ when $t = 3$ [3]
Find the total distance travelled by $P$ in the first 3 seconds of its motion. [6]
Show that $P$ never returns to $O$. [2]

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