Questions M2 - A-Level Maths

Edexcel M2 2002 January Q7

7. \begin{figure}[h]

\captionsetup{labelformat=empty} \caption{Figure 3} \includegraphics[alt={},max width=\textwidth]{d5250c24-138a-4fe3-91eb-d6f8dca766fb-6_837_1284_468_397}

\end{figure} A rocket $R$ of mass 100 kg is projected from a point $A$ with speed $80 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ at an angle of elevation of $60 ^ { \circ }$, as shown in Fig. 3. The point $A$ is 20 m vertically above a point $O$ which is on horizontal ground. The rocket $R$ moves freely under gravity. At $B$ the velocity of $R$ is horizontal. By modelling $R$ as a particle, find

the height in m of $B$ above the ground,
the time taken for $R$ to reach $B$ from $A$. When $R$ is at $B$, there is an internal explosion and $R$ breaks into two parts $P$ and $Q$ of masses 60 kg and 40 kg respectively. Immediately after the explosion the velocity of $P$ is $80 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ horizontally away from $A$. After the explosion the paths of $P$ and $Q$ remain in the plane $O A B$. Part $Q$ strikes the ground at $C$. By modelling $P$ and $Q$ as particles,
show that the speed of $Q$ immediately after the explosion is $20 \mathrm {~m} \mathrm {~s} ^ { - 1 }$,
find the distance $O C$.

Edexcel M2 2003 January Q1

Three particles of mass $3 m , 5 m$ and $\lambda m$ are placed at points with coordinates (4, 0), (0, -3) and $( 4,2 )$ respectively. The centre of mass of the system of three particles is at $( 2 , k )$.
1. Show that $\lambda = 2$.
2. Calculate the value of $k$.
3. A car of mass 1000 kg is moving along a straight horizontal road with a constant acceleration of $f \mathrm {~m} \mathrm {~s} ^ { - 2 }$. The resistance to motion is modelled as a constant force of magnitude 1200 N . When the car is travelling at $12 \mathrm {~m} \mathrm {~s} ^ { - 1 }$, the power generated by the engine of the car is 24 kW .
4. Calculate the value of $f$.
When the car is travelling at $14 \mathrm {~m} \mathrm {~s} ^ { - 1 }$, the engine is switched off and the car comes to rest, without braking, in a distance of $d$ metres. Assuming the same model for resistance,
use the work-energy principle to calculate the value of $d$.
Give a reason why the model used for the resistance to motion may not be realistic. \section*{3.} \section*{Figure 1}
\includegraphics[max width=\textwidth, alt={}]{19f831ad-5e32-470c-9974-beb82d5c9753-3_751_678_440_657}
A uniform ladder $A B$, of mass $m$ and length $2 a$, has one end $A$ on rough horizontal ground. The other end $B$ rests against a smooth vertical wall. The ladder is in a vertical plane perpendicular to the wall. The ladder makes an angle $\alpha$ with the horizontal, where $\tan \alpha = \frac { 4 } { 3 }$. A child of mass $2 m$ stands on the ladder at $C$ where $A C = \frac { 1 } { 2 } a$, as shown in Fig. 1. The ladder and the child are in equilibrium. By modelling the ladder as a rod and the child as a particle, calculate the least possible value of the coefficient of friction between the ladder and the ground.

Edexcel M2 2003 January Q4

4. Figure 2
\includegraphics[max width=\textwidth, alt={}, center]{19f831ad-5e32-470c-9974-beb82d5c9753-4_574_1159_395_429} Figure 2 shows a uniform lamina $A B C D E$ such that $A B D E$ is a rectangle, $B C = C D , A B = 8 a$ and $A E = 6 a$. The point $X$ is the mid-point of $B D$ and $X C = 4 a$. The centre of mass of the lamina is at $G$.

Show that $G X = \frac { 44 } { 15 } a$.
(6) The mass of the lamina is $M$. A particle of mass $\lambda M$ is attached to the lamina at $C$. The lamina is suspended from $B$ and hangs freely under gravity with $A B$ horizontal.
Find the value of $\lambda$.
(3)

Edexcel M2 2003 January Q5

5. A particle $P$ moves on the $x$-axis. The acceleration of $P$ at time $t$ seconds is $( 4 t - 8 ) \mathrm { m } \mathrm { s } ^ { - 2 }$, measured in the direction of $x$ increasing. The velocity of $P$ at time $t$ seconds is $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$. Given that $v = 6$ when $t = 0$, find

$v$ in terms of $t$,
the distance between the two points where $P$ is instantaneously at rest.

Edexcel M2 2003 January Q6

6. A smooth sphere $P$ of mass $2 m$ is moving in a straight line with speed $u$ on a smooth horizontal table. Another smooth sphere $Q$ of mass $m$ is at rest on the table. The sphere $P$ collides directly with $Q$. The coefficient of restitution between $P$ and $Q$ is $\frac { 1 } { 3 }$. The spheres are modelled as particles.

Show that, immediately after the collision, the speeds of $P$ and $Q$ are $\frac { 5 } { 9 } u$ and $\frac { 8 } { 9 } u$ respectively. After the collision, $Q$ strikes a fixed vertical wall which is perpendicular to the direction of motion of $P$ and $Q$. The coefficient of restitution between $Q$ and the wall is $e$. When $P$ and $Q$ collide again, $P$ is brought to rest.
Find the value of $e$.
Explain why there must be a third collision between $P$ and $Q$.

Edexcel M2 2003 January Q7

7. \begin{figure}[h]

\captionsetup{labelformat=empty} \caption{Figure 3} \includegraphics[alt={},max width=\textwidth]{19f831ad-5e32-470c-9974-beb82d5c9753-6_636_1617_406_184}

\end{figure} A ball $B$ of mass 0.4 kg is struck by a bat at a point $O$ which is 1.2 m above horizontal ground. The unit vectors $\mathbf { i }$ and $\mathbf { j }$ are respectively horizontal and vertical. Immediately before being struck, $B$ has velocity $( - 20 \mathbf { i } + 4 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }$. Immediately after being struck it has velocity $( 15 \mathbf { i } + 16 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }$. After $B$ has been struck, it moves freely under gravity and strikes the ground at the point $A$, as shown in Fig. 3. The ball is modelled as a particle.

Calculate the magnitude of the impulse exerted by the bat on $B$.
By using the principle of conservation of energy, or otherwise, find the speed of $B$ when it reaches $A$.
Calculate the angle which the velocity of $B$ makes with the ground when $B$ reaches $A$.
State two additional physical factors which could be taken into account in a refinement of the model of the situation which would make it more realistic.

Edexcel M2 Q3

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

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

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

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 $x$.

AQA M2 Q1

1 A uniform beam, $A B$, has mass 20 kg and length 7 metres. A rope is attached to the beam at $A$. A second rope is attached to the beam at the point $C$, which is 2 metres from $B$. Both of the ropes are vertical. The beam is in equilibrium in a horizontal position, as shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{88aec6ab-af83-4d5e-84b6-5fd84c16a6c9-003_298_906_756_552} Find the tensions in the two ropes.

AQA M2 Q2

2 A particle, of mass 2 kg , is attached to one end of a light inextensible string. The other end is fixed to the point $O$. The particle is set into motion, so that it describes a horizontal circle of radius 0.6 metres, with the string at an angle of $30 ^ { \circ }$ to the vertical. The centre of the circle is vertically below $O$.
\includegraphics[max width=\textwidth, alt={}, center]{88aec6ab-af83-4d5e-84b6-5fd84c16a6c9-003_346_340_1580_842}

Show that the tension in the string is 22.6 N , correct to three significant figures.
Find the speed of the particle.

AQA M2 Q3

3 A particle moves in a straight line and at time $t$ has velocity $v$, where $$v = 2 t - 12 \mathrm { e } ^ { - t } , \quad t \geqslant 0$$

1. Find an expression for the acceleration of the particle at time $t$.
2. State the range of values of the acceleration of the particle.
When $t = 0$, the particle is at the origin. Find an expression for the displacement of the particle from the origin at time $t$.
(4 marks)

AQA M2 Q4

4 A car has a maximum speed of $42 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ when it is moving on a horizontal road. When the speed of the car is $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$, it experiences a resistance force of magnitude $30 v$ newtons.

Show that the maximum power of the car is 52920 W .
The car has mass 1200 kg . It travels, from rest, up a slope inclined at $5 ^ { \circ }$ to the horizontal.
1. Show that, when the car is travelling at its maximum speed $\mathrm { V } \mathrm { m } \mathrm { s } ^ { - 1 }$ up the slope, $$V ^ { 2 } + 392 \sin 5 ^ { \circ } V - 1764 = 0$$
2. Hence find $V$.

AQA M2 Q5

5 A car, of mass 1600 kg , is travelling along a straight horizontal road at a speed of $20 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ when the driving force is removed. The car then freewheels and experiences a resistance force. The resistance force has magnitude $40 v$ newtons, where $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$ is the speed of the car after it has been freewheeling for $t$ seconds. Find an expression for $v$ in terms of $t$.

AQA M2 Q6

6 A particle $P$, of mass $m \mathrm {~kg}$, is placed at the point $Q$ on the top of a smooth upturned hemisphere of radius 3 metres and centre $O$. The plane face of the hemisphere is fixed to a horizontal table. The particle is set into motion with an initial horizontal velocity of $2 \mathrm {~ms} ^ { - 1 }$. When the particle is on the surface of the hemisphere, the angle between $O P$ and $O Q$ is $\theta$ and the particle has speed $v \mathrm {~ms} ^ { - 1 }$.
\includegraphics[max width=\textwidth, alt={}, center]{88aec6ab-af83-4d5e-84b6-5fd84c16a6c9-005_419_1013_607_511}

Show that $v ^ { 2 } = 4 + 6 g ( 1 - \cos \theta )$.
Find the value of $\theta$ when the particle leaves the hemisphere.

AQA M2 Q7

7 A particle, of mass 10 kg , is attached to one end of a light elastic string of natural length 0.4 metres and modulus of elasticity 100 N . The other end of the string is fixed to the point $O$.

Find the length of the elastic string when the particle hangs in equilibrium directly below $O$.
The particle is pulled down and held at a point $P$, which is 1 metre vertically below $O$. Show that the elastic potential energy of the string when the particle is in this position is 45 J .
The particle is released from rest at the point $P$. In the subsequent motion, the particle has speed $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$ when it is $x$ metres below $\boldsymbol { O }$.
1. Show that, while the string is taut, $$v ^ { 2 } = 39.6 x - 25 x ^ { 2 } - 14.6$$
2. Find the value of $x$ when the particle comes to rest for the first time after being released, given that the string is still taut.

AQA M2 Q8

8 Two small blocks, $A$ and $B$, of masses 0.8 kg and 1.2 kg respectively, are stuck together. A spring has natural length 0.5 metres and modulus of elasticity 49 N . One end of the spring is attached to the top of the block $A$ and the other end of the spring is attached to a fixed point $O$.

The system hangs in equilibrium with the blocks stuck together, as shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{88aec6ab-af83-4d5e-84b6-5fd84c16a6c9-017_385_239_669_881} Find the extension of the spring.
Show that the elastic potential energy of the spring when the system is in equilibrium is 1.96 J .
The system is hanging in this equilibrium position when block $B$ falls off and block $A$ begins to move vertically upwards. Block $A$ next comes to rest when the spring is compressed by $x$ metres.
1. Show that $x$ satisfies the equation $$x ^ { 2 } + 0.16 x - 0.008 = 0$$
2. Find the value of $x$.

AQA M2 2007 January Q1

1 A child, of mass 35 kg , slides down a slide in a water park. The child, starting from rest, slides from the point $A$ to the point $B$, which is 10 metres vertically below the level of $A$, as shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{480a817d-074f-440d-829e-c8f8a9746151-2_259_595_685_705}

In a simple model, all resistance forces are ignored. Use an energy method to find the speed of the child at $B$.
State one resistance force that has been ignored in answering part (a).
In fact, when the child slides down the slide, she reaches $B$ with a speed of $12 \mathrm {~m} \mathrm {~s} ^ { - 1 }$. Given that the slide is 20 metres long and the sum of the resistance forces has a constant magnitude of $F$ newtons, use an energy method to find the value of $F$.
(4 marks)

AQA M2 2007 January Q2

2 A hotel sign consists of a uniform rectangular lamina of weight $W$. The sign is suspended in equilibrium in a vertical plane by two vertical light chains attached to the sign at the points $A$ and $B$, as shown in the diagram. The edge containing $A$ and $B$ is horizontal.
\includegraphics[max width=\textwidth, alt={}, center]{480a817d-074f-440d-829e-c8f8a9746151-2_289_529_1859_726} The tensions in the chains attached at $A$ and $B$ are $T _ { A }$ and $T _ { B }$ respectively.

Draw a diagram to show the forces acting on the sign.
Find $T _ { A }$ and $T _ { B }$ in terms of $W$.
Explain how you have used the fact that the lamina is uniform in answering part (b).

AQA M2 2007 January Q3

3 A light inextensible string has length $2 a$. One end of the string is attached to a fixed point $O$ and a particle of mass $m$ is attached to the other end. Initially, the particle is held at the point $A$ with the string taut and horizontal. The particle is then released from rest and moves in a circular path. Subsequently, it passes through the point $B$, which is directly below $O$. The points $O , A$ and $B$ are as shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{480a817d-074f-440d-829e-c8f8a9746151-3_426_437_575_772}

Show that the speed of the particle at $B$ is $2 \sqrt { a g }$.
Find the tension in the string as the particle passes through $B$. Give your answer in terms of $m$ and $g$.

AQA M2 2007 January Q4

4 A uniform T-shaped lamina is formed by rigidly joining two rectangles $A B C H$ and $D E F G$, as shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{480a817d-074f-440d-829e-c8f8a9746151-4_748_652_456_644}

Show that the centre of mass of the lamina is 26 cm from the edge $A B$.
Explain why the centre of mass of the lamina is 5 cm from the edge $G F$.
The point $X$ is on the edge $A B$ and is 7 cm from $A$, as shown in the diagram below.
\includegraphics[max width=\textwidth, alt={}, center]{480a817d-074f-440d-829e-c8f8a9746151-4_697_534_1576_753} The lamina is freely suspended from $X$ and hangs in equilibrium.
Find the angle between the edge $A B$ and the vertical, giving your answer to the nearest degree.
(4 marks)

AQA M2 2007 January Q5

5 Tom is on a fairground ride.
Tom's position vector, $\mathbf { r }$ metres, at time $t$ seconds is given by $$\mathbf { r } = 2 \cos t \mathbf { i } + 2 \sin t \mathbf { j } + ( 10 - 0.4 t ) \mathbf { k }$$ The perpendicular unit vectors $\mathbf { i }$ and $\mathbf { j }$ are in the horizontal plane and the unit vector $\mathbf { k }$ is directed vertically upwards.

1. Find Tom's position vector when $t = 0$.
2. Find Tom's position vector when $t = 2 \pi$.
3. Write down the first two values of $t$ for which Tom is directly below his starting point.
Find an expression for Tom's velocity at time $t$.
Tom has mass 25 kg . Show that the resultant force acting on Tom during the motion has constant magnitude. State the magnitude of the resultant force.
(5 marks)

AQA M2 2007 January Q6

6 A particle is attached to one end of a light inextensible string. The other end of the string is attached to a fixed point $O$. The particle is set into motion, so that it describes a horizontal circle whose centre is vertically below $O$. The angle between the string and the vertical is $\theta$, as shown in the diagram.
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The particle completes 40 revolutions every minute. Show that the angular speed of the particle is $\frac { 4 \pi } { 3 }$ radians per second.
The radius of the circle is 0.2 metres. Find, in terms of $\pi$, the magnitude of the acceleration of the particle.
The mass of the particle is $m \mathrm {~kg}$ and the tension in the string is $T$ newtons.
1. Draw a diagram showing the forces acting on the particle.
2. Explain why $T \cos \theta = m g$.
3. Find the value of $\theta$, giving your answer to the nearest degree.

AQA M2 2007 January Q7

7 A motorcycle has a maximum power of 72 kilowatts. The motorcycle and its rider are travelling along a straight horizontal road. When they are moving at a speed of $\mathrm { V } \mathrm { m } \mathrm { s } ^ { - 1 }$, they experience a total resistance force of magnitude $k V$ newtons, where $k$ is a constant.

The maximum speed of the motorcycle and its rider is $60 \mathrm {~ms} ^ { - 1 }$. Show that $k = 20$.
When the motorcycle is travelling at $20 \mathrm {~m} \mathrm {~s} ^ { - 1 }$, the rider allows the motorcycle to freewheel so that the only horizontal force acting is the resistance force. When the motorcycle has been freewheeling for $t$ seconds, its speed is $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$ and the magnitude of the resistance force is $20 v$ newtons. The mass of the motorcycle and its rider is 500 kg .
1. Show that $\frac { \mathrm { d } v } { \mathrm {~d} t } = - \frac { v } { 25 }$.
2. Hence find the time that it takes for the speed of the motorcycle to reduce from $20 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ to $10 \mathrm {~m} \mathrm {~s} ^ { - 1 }$.
  (6 marks)

Questions M2 (1391 questions)

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