Questions - Edexcel - A-Level Maths

Edexcel S4 2002 June Q5

5. The times, $x$ seconds, taken by the competitors in the 100 m freestyle events at a school swimming gala are recorded. The following statistics are obtained from the data.

	No. of competitors	Sample Mean $\bar { x }$	$\sum x ^ { 2 }$
Girls	8	83.10	55746
Boys	7	88.90	56130

Following the gala a proud parent claims that girls are faster swimmers than boys. Assuming that the times taken by the competitors are two independent random samples from normal distributions,

test, at the $10 \%$ level of significance, whether or not the variances of the two distributions are the same. State your hypotheses clearly.
Stating your hypotheses clearly, test the parent's claim. Use a $5 \%$ level of significance.

Edexcel S4 2002 June Q6

6. A nutritionist studied the levels of cholesterol, $X \mathrm { mg } / \mathrm { cm } ^ { 3 }$, of male students at a large college. She assumed that $X$ was distributed $\mathrm { N } \left( \mu , \sigma ^ { 2 } \right)$ and examined a random sample of 25 male students. Using this sample she obtained unbiased estimates of $\mu$ and $\sigma ^ { 2 }$ as $$\hat { \mu } = 1.68 , \quad \hat { \sigma } ^ { 2 } = 1.79 .$$

Find a 95\% confidence interval for $\mu$.
Obtain a $95 \%$ confidence interval for $\sigma ^ { 2 }$. A cholesterol reading of more than $2.5 \mathrm { mg } / \mathrm { cm } ^ { 3 }$ is regarded as high.
Use appropriate confidence limits from parts (a) and (b) to find the lowest estimate of the proportion of male students in the college with high cholesterol.

Edexcel S4 2002 June Q7

A proportion $p$ of the items produced by a factory is defective. A quality assurance manager selects a random sample of 5 items from each batch produced to check whether or not there is evidence that $p$ is greater than 0.10 . The criterion that the manager uses for rejecting the hypothesis that $p$ is 0.10 is that there are more than 2 defective items in the sample.
1. Find the size of the test.
  (2)
Table 1 gives some values, to 2 decimal places, of the power function of this test. \begin{table}[h]
\captionsetup{labelformat=empty} \caption{Table 1}
$p$ 0.15 0.20 0.25 0.30 0.35 0.40
Power 0.03 $r$ 0.10 0.16 0.24 0.32
\end{table}
Find the value of $r$. One day the manager is away and an assistant checks the production by random sample of 10 items from each batch produced. The hypothesis that $p = 0.10$ is rejected if more than 4 defectives are found in the sample.
Find P (Type I error) using the assistant's test. Table 2 gives some values, to 2 decimal places, of the power function for this test. \begin{table}[h]
\captionsetup{labelformat=empty} \caption{Table 2}
$p$ 0.15 0.20 0.25 0.30 0.35 0.40
Power 0.01 0.03 0.08 0.15 0.25 $s$
\end{table}
Find the value of $s$.
Using the same axes, draw the graphs of the power functions of these two tests.
1. State the value of $p$ where these graphs cross.
2. Explain the significance if $p$ is greater than this value. The manager studies the graphs in part ( $e$ ) but decides to carry on using the test based on a sample of size 5 .
Suggest 2 reasons why the manager might have made this decision.

Edexcel M4 Q1

1. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{cf941854-3a33-4d9d-9fa0-ce9a63227599-03_457_638_233_598} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} A fixed smooth plane is inclined to the horizontal at an angle of $45 ^ { \circ }$. A particle $P$ is moving horizontally and strikes the plane. Immediately before the impact, $P$ is moving in a vertical plane containing a line of greatest slope of the inclined plane. Immediately after the impact, $P$ is moving in a direction which makes an angle of $30 ^ { \circ }$ with the inclined plane, as shown in Figure 1. Find the fraction of the kinetic energy of $P$ which is lost in the impact.

Edexcel M4 Q2

2. At time $t = 0$, a particle $P$ of mass $m$ is projected vertically upwards with speed $\sqrt { \frac { g } { k } }$, where $k$ is a constant. At time $t$ the speed of $P$ is $v$. The particle $P$ moves against air resistance whose magnitude is modelled as being $m k v ^ { 2 }$ when the speed of $P$ is $v$. Find, in terms of $k$, the distance travelled by $P$ until its speed first becomes half of its initial speed.

Edexcel M4 Q3

At noon a motorboat $P$ is 2 km north-west of another motorboat $Q$. The motorboat $P$ is moving due south at $20 \mathrm {~m} \mathrm {~s} ^ { - 1 }$. The motorboat $Q$ is pursuing motorboat $P$ at a speed of $12 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ and sets a course in order to get as close to motorboat $P$ as possible.
1. Find the course set by $Q$, giving your answer as a bearing to the nearest degree.
2. Find the shortest distance between $P$ and $Q$.
3. Find the distance travelled by $Q$ from its position at noon to the point of closest approach.

Edexcel M4 Q4

4. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{cf941854-3a33-4d9d-9fa0-ce9a63227599-08_479_807_246_571} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} A light inextensible string of length $2 a$ has one end attached to a fixed point $A$. The other end of the string is attached to a particle $P$ of mass $m$. A second light inextensible string of length $L$, where $L > \frac { 12 a } { 5 }$, has one of its ends attached to $P$ and passes over a small smooth peg fixed at a point $B$. The line $A B$ is horizontal and $A B = 2 a$. The other end of the second string is attached to a particle of mass $\frac { 7 } { 20 } m$, which hangs vertically below $B$, as shown in Figure 2.

Show that the potential energy of the system, when the angle $P A B = 2 \theta$, is $$\frac { 1 } { 5 } m g a ( 7 \sin \theta - 10 \sin 2 \theta ) + \text { constant. }$$
Show that there is only one value of $\cos \theta$ for which the system is in equilibrium and find this value.
Determine the stability of the position of equilibrium.

Edexcel M4 Q5

Two small smooth spheres $A$ and $B$, of mass 2 kg and 1 kg respectively, are moving on a smooth horizontal plane when they collide. Immediately before the collision the velocity of $A$ is $( \mathbf { i } + 2 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }$ and the velocity of $B$ is $- 2 \mathbf { i } \mathrm {~m} \mathrm {~s} ^ { - 1 }$. Immediately after the collision the velocity of $A$ is $\mathbf { j } \mathrm { m } \mathrm { s } ^ { - 1 }$.
1. Show that the velocity of $B$ immediately after the collision is $2 \mathbf { j } \mathrm {~m} \mathrm {~s} ^ { - 1 }$.
2. Find the impulse of $B$ on $A$ in the collision, giving your answer as a vector, and hence show that the line of centres is parallel to $\mathbf { i } + \mathbf { j }$.
3. Find the coefficient of restitution between $A$ and $B$.

Edexcel M4 Q6

6. A light elastic spring $A B$ has natural length $2 a$ and modulus of elasticity $2 m n ^ { 2 } a$, where $n$ is a constant. A particle $P$ of mass $m$ is attached to the end $A$ of the spring. At time $t = 0$, the spring, with $P$ attached, lies at rest and unstretched on a smooth horizontal plane. The other end $B$ of the spring is then pulled along the plane in the direction $A B$ with constant acceleration $f$. At time $t$ the extension of the spring is $x$.

Show that $$\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + n ^ { 2 } x = f .$$
Find $x$ in terms of $n , f$ and $t$. Hence find
the maximum extension of the spring,
the speed of $P$ when the spring first reaches its maximum extension.
Turn over
1. \hspace{0pt} [In this question $\mathbf { i }$ and $\mathbf { j }$ are unit vectors due east and due north respectively]
A man cycles at a constant speed $u \mathrm {~m} \mathrm {~s} ^ { - 1 }$ on level ground and finds that when his velocity is $u \mathbf { j } \mathrm {~m} \mathrm {~s} ^ { - 1 }$ the velocity of the wind appears to be $v ( 3 \mathbf { i } - 4 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }$, where $v$ is a positive constant. When the man cycles with velocity $\frac { 1 } { 5 } u ( - 3 \mathbf { i } + 4 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }$, the velocity of the wind appears to be $w \mathbf { i } \mathrm {~m} \mathrm {~s} ^ { - 1 }$, where $w$ is a positive constant. Find, in terms of $u$, the true velocity of the wind.
2. Two smooth uniform spheres $S$ and $T$ have equal radii. The mass of $S$ is 0.3 kg and the mass of $T$ is 0.6 kg . The spheres are moving on a smooth horizontal plane and collide obliquely. Immediately before the collision the velocity of $S$ is $\mathbf { u } _ { 1 } \mathrm {~m} \mathrm {~s} ^ { - 1 }$ and the velocity of $T$ is $\mathbf { u } _ { 2 } \mathrm {~m} \mathrm {~s} ^ { - 1 }$. The coefficient of restitution between the spheres is 0.5 . Immediately after the collision the velocity of $S$ is $( - \mathbf { i } + 2 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }$ and the velocity of $T$ is $( \mathbf { i } + \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }$. Given that when the spheres collide the line joining their centres is parallel to $\mathbf { i }$,
find
1. $\mathbf { u } _ { 1 }$,
2. $\mathbf { u } _ { 2 }$. After the collision, $T$ goes on to collide with a smooth vertical wall which is parallel to $\mathbf { j }$. Given that the coefficient of restitution between $T$ and the wall is also 0.5 , find
the angle through which the direction of motion of $T$ is deflected as a result of the collision with the wall,
the loss in kinetic energy of $T$ caused by the collision with the wall.
1. At 12 noon, $\operatorname { ship } A$ is 8 km due west of $\operatorname { ship } B$. Ship $A$ is moving due north at a constant speed of $10 \mathrm {~km} \mathrm {~h} ^ { - 1 }$. Ship $B$ is moving at a constant speed of $6 \mathrm {~km} \mathrm {~h} ^ { - 1 }$ on a bearing so that it passes as close to $A$ as possible.
2. Find the bearing on which ship $B$ moves.
3. Find the shortest distance between the two ships.
4. Find the time when the two ships are closest.
1. A particle of mass $m$ is projected vertically upwards, at time $t = 0$, with speed $U$. The particle is subject to air resistance of magnitude $\frac { m g v ^ { 2 } } { k ^ { 2 } }$, where $v$ is the speed of the particle at time $t$ and $k$ is a positive constant.
2. Show that the particle reaches its greatest height above the point of projection at time
$$\frac { k } { g } \tan ^ { - 1 } \left( \frac { U } { k } \right)$$
Find the greatest height above the point of projection attained by the particle. 5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{cf941854-3a33-4d9d-9fa0-ce9a63227599-22_419_1212_260_365} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} The end $A$ of a uniform rod $A B$, of length $2 a$ and mass $4 m$, is smoothly hinged to a fixed point. The end $B$ is attached to one end of a light inextensible string which passes over a small smooth pulley, fixed at the same level as $A$. The distance from $A$ to the pulley is $4 a$. The other end of the string carries a particle of mass $m$ which hangs freely, vertically below the pulley, with the string taut. The angle between the rod and the downward vertical is $\theta$, where $0 < \theta < \frac { \pi } { 2 }$, as shown in Figure 1.
Show that the potential energy of the system is $$2 m g a ( \sqrt { } ( 5 - 4 \sin \theta ) - 2 \cos \theta ) + \text { constant }$$
Hence, or otherwise, show that any value of $\theta$ which corresponds to a position of equilibrium of the system satisfies the equation $$4 \sin ^ { 3 } \theta - 6 \sin ^ { 2 } \theta + 1 = 0$$
Given that $\theta = \frac { \pi } { 6 }$ corresponds to a position of equilibrium, determine its stability.
1. Two points $A$ and $B$ lie on a smooth horizontal table with $A B = 4 a$. One end of a light elastic spring, of natural length $a$ and modulus of elasticity $2 m g$, is attached to $A$. The other end of the spring is attached to a particle $P$ of mass $m$. Another light elastic spring, of natural length $a$ and modulus of elasticity $m g$, has one end attached to $B$ and the other end attached to $P$. The particle $P$ is on the table at rest and in equilibrium.
2. Show that $A P = \frac { 5 a } { 3 }$.
The particle $P$ is now moved along the table from its equilibrium position through a distance $0.5 a$ towards $B$ and released from rest at time $t = 0$. At time $t , P$ is moving with speed $v$ and has displacement $x$ from its equilibrium position. There is a resistance to motion of magnitude $4 m \omega v$ where $\omega = \sqrt { } \left( \frac { g } { a } \right)$.
Show that $\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 4 \omega \frac { \mathrm {~d} x } { \mathrm {~d} t } + 3 \omega ^ { 2 } x = 0$.
Find the velocity, $\frac { \mathrm { d } x } { \mathrm {~d} t }$, of $P$ in terms of $a , \omega$ and $t$. Turn over
advancing learning, changing lives
1. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{cf941854-3a33-4d9d-9fa0-ce9a63227599-27_780_1022_228_488} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Two smooth uniform spheres $A$ and $B$ have masses $2 m \mathrm {~kg}$ and $3 m \mathrm {~kg}$ respectively and equal radii. The spheres are moving on a smooth horizontal surface. Initially, sphere $A$ has velocity $( 3 \mathbf { i } - 4 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }$ and sphere $B$ has velocity $( 2 \mathbf { i } - 3 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }$. When the spheres collide, the line joining their centres is parallel to $\mathbf { j }$, as shown in Figure 1. The coefficient of restitution between the spheres is $\frac { 3 } { 7 }$. Find, in terms of $m$, the total kinetic energy lost in the collision. 2. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{cf941854-3a33-4d9d-9fa0-ce9a63227599-29_680_853_285_543} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 2 represents part of the smooth rectangular floor of a sports hall. A ball is at $B$, 4 m from one wall of the hall and 5 m from an adjacent wall. These two walls are smooth and meet at the corner $C$. The ball is kicked so that it travels along the floor, bounces off the first wall at the point $X$ and hits the second wall at the point $Y$. The point $Y$ is 7.5 m from the corner $C$.
The coefficient of restitution between the ball and the first wall is $\frac { 3 } { 4 }$.
Modelling the ball as a particle, find the distance $C X$.
1. \hspace{0pt} [In this question the unit vectors $\mathbf { i }$ and $\mathbf { j }$ are due east and due north respectively.]
A coastguard patrol boat $C$ is moving with constant velocity $( 8 \mathbf { i } + u \mathbf { j } ) \mathrm { km } \mathrm { h } ^ { - 1 }$. Another ship $S$ is moving with constant velocity $( 12 \mathbf { i } + 16 \mathbf { j } ) \mathrm { km } \mathrm { h } ^ { - 1 }$.
Find, in terms of $u$, the velocity of $C$ relative to $S$. At noon, $S$ is 10 km due west of $C$.
If $C$ is to intercept $S$,
1. find the value of $u$.
2. Using this value of $u$, find the time at which $C$ would intercept $S$. If instead, at noon, $C$ is moving with velocity $( 8 \mathbf { i } + 8 \mathbf { j } ) \mathrm { km } \mathrm { h } ^ { - 1 }$ and continues at this constant velocity,
find the distance of closest approach of $C$ to $S$.
1. A hiker walking due east at a steady speed of $5 \mathrm {~km} \mathrm {~h} ^ { - 1 }$ notices that the wind appears to come from a direction with bearing 050. At the same time, another hiker moving on a bearing of 320 , and also walking at $5 \mathrm {~km} \mathrm {~h} ^ { - 1 }$, notices that the wind appears to come from due north.
Find
the direction from which the wind is blowing,
the wind speed.
5. A particle $Q$ of mass 6 kg is moving along the $x$-axis. At time $t$ seconds the displacement of $Q$ from the origin $O$ is $x$ metres and the speed of $Q$ is $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$. The particle moves under the action of a retarding force of magnitude ( $a + b v ^ { 2 }$ ) N, where $a$ and $b$ are positive constants. At time $t = 0 , Q$ is at $O$ and moving with speed $U \mathrm {~m} \mathrm {~s} ^ { - 1 }$ in the positive $x$-direction. The particle $Q$ comes to instantaneous rest at the point $X$.
Show that the distance $O X$ is $$\frac { 3 } { b } \ln \left( 1 + \frac { b U ^ { 2 } } { a } \right) \mathrm { m }$$ Given that $a = 12$ and $b = 3$,
find, in terms of $U$, the time taken to move from $O$ to $X$. 6. A particle $P$ of mass 4 kg moves along a horizontal straight line under the action of a force directed towards a fixed point $O$ on the line. At time $t$ seconds, $P$ is $x$ metres from $O$ and the force towards $O$ has magnitude $9 x$ newtons. The particle $P$ is also subject to air resistance, which has magnitude $12 v$ newtons when $P$ is moving with speed $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$.
Show that the equation of motion of $P$ is $$4 \frac { \mathrm {~d} ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 12 \frac { \mathrm {~d} x } { \mathrm {~d} t } + 9 x = 0$$ It is given that the solution of this differential equation is of the form $$x = \mathrm { e } ^ { - \lambda t } ( A t + B )$$ When $t = 0$ the particle is released from rest at the point $R$, where $O R = 4 \mathrm {~m}$. Find,
the values of the constants $\lambda , A$ and $B$,
the greatest speed of $P$ in the subsequent motion.

Edexcel M4 Q7

7. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{cf941854-3a33-4d9d-9fa0-ce9a63227599-38_451_1077_315_370} \captionsetup{labelformat=empty} \caption{Figure 3}

\end{figure} Figure 3 shows a framework $A B C$, consisting of two uniform rods rigidly joined together at $B$ so that $\angle A B C = 90 ^ { \circ }$. The $\operatorname { rod } A B$ has length $2 a$ and mass $4 m$, and the $\operatorname { rod } B C$ has length $a$ and mass $2 m$. The framework is smoothly hinged at $A$ to a fixed point, so that the framework can rotate in a fixed vertical plane. One end of a light elastic string, of natural length $2 a$ and modulus of elasticity $3 m g$, is attached to $A$. The string passes through a small smooth ring $R$ fixed at a distance $2 a$ from $A$, on the same horizontal level as $A$ and in the same vertical plane as the framework. The other end of the string is attached to $B$. The angle $A R B$ is $\theta$, where $0 < \theta < \frac { \pi } { 2 }$.

Show that the potential energy $V$ of the system is given by $$V = 8 a m g \sin 2 \theta + 5 a m g \cos 2 \theta + \text { constant }$$
Find the value of $\theta$ for which the system is in equilibrium.
Determine the stability of this position of equilibrium. Turn over
1. A smooth uniform sphere $S$, of mass $m$, is moving on a smooth horizontal plane when it collides obliquely with another smooth uniform sphere $T$, of the same radius as $S$ but of mass $2 m$, which is at rest on the plane. Immediately before the collision the velocity of $S$ makes an angle $\alpha$, where $\tan \alpha = \frac { 3 } { 4 }$, with the line joining the centres of the spheres. Immediately after the collision the speed of $T$ is $V$. The coefficient of restitution between the spheres is $\frac { 3 } { 4 }$.
2. Find, in terms of $V$, the speed of $S$
  1. immediately before the collision,
  2. immediately after the collision.
3. Find the angle through which the direction of motion of $S$ is deflected as a result of the collision.
1. A $\operatorname { ship } A$ is moving at a constant speed of $8 \mathrm {~km} \mathrm {~h} \mathrm {~h} ^ { - 1 }$ on a bearing of $150 ^ { \circ }$. At noon a second ship $B$ is 6 km from $A$, on a bearing of $210 ^ { \circ }$. Ship $B$ is moving due east at a constant speed. At a later time, $B$ is $2 \sqrt { 3 } \mathrm {~km}$ due south of $A$.
Find
(i) the time at which $B$ will be due east of $A$,
(ii) the distance between the ships at that time.
1. Two particles, of masses $m$ and $2 m$, are connected to the ends of a long light inextensible string. The string passes over a small smooth fixed pulley and hangs vertically on either side. The particles are released from rest with the string taut. Each particle is subject to air resistance of magnitude $k v ^ { 2 }$, where $v$ is the speed of each particle after it has moved a distance $x$ from rest and $k$ is a positive constant.
2. Show that $\frac { \mathrm { d } } { \mathrm { d } x } \left( v ^ { 2 } \right) + \frac { 4 k } { 3 m } v ^ { 2 } = \frac { 2 g } { 3 }$
3. Find $v ^ { 2 }$ in terms of $x$.
4. Deduce that the tension in the string, $T$, satisfies
$$\frac { 4 m g } { 3 } \leqslant T < \frac { 3 m g } { 2 }$$
1. A rescue boat, whose maximum speed is $20 \mathrm {~km} \mathrm {~h} ^ { - 1 }$, receives a signal which indicates that a yacht is in distress near a fixed point $P$. The rescue boat is 15 km south-west of $P$. There is a constant current of $5 \mathrm {~km} \mathrm {~h} ^ { - 1 }$ flowing uniformly from west to east. The rescue boat sets the course needed to get to $P$ as quickly as possible. Find
2. the course the rescue boat sets,
3. the time, to the nearest minute, to get to $P$.
When the rescue boat arrives at $P$, the yacht is just visible 4 km due north of $P$ and is drifting with the current. Find
the course that the rescue boat should set to get to the yacht as quickly as possible,
the time taken by the rescue boat to reach the yacht from $P$. 5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{cf941854-3a33-4d9d-9fa0-ce9a63227599-49_977_1228_205_356} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} A uniform rod $A B$, of length $4 a$ and weight $W$, is free to rotate in a vertical plane about a fixed smooth horizontal axis which passes through the point $C$ of the rod, where $A C = 3 a$. One end of a light inextensible string of length $L$, where $L > 10 a$, is attached to the end $A$ of the rod and passes over a small smooth fixed peg at $P$ and another small smooth fixed peg at $Q$. The point $Q$ lies in the same vertical plane as $P , A$ and $B$. The point $P$ is at a distance $3 a$ vertically above $C$ and $P Q$ is horizontal with $P Q = 4 a$. A particle of weight $\frac { 1 } { 2 } W$ is attached to the other end of the string and hangs vertically below $Q$. The rod is inclined at an angle $2 \theta$ to the vertical, where $- \pi < 2 \theta < \pi$, as shown in Figure 1.
Show that the potential energy of the system is $$W a ( 3 \cos \theta - \cos 2 \theta ) + \text { constant }$$
Find the positions of equilibrium and determine their stability.
1. Two points $A$ and $B$ are in a vertical line, with $A$ above $B$ and $A B = 4 a$. One end of a light elastic spring, of natural length $a$ and modulus of elasticity $3 m g$, is attached to $A$. The other end of the spring is attached to a particle $P$ of mass $m$. Another light elastic spring, of natural length $a$ and modulus of elasticity $m g$, has one end attached to $B$ and the other end attached to $P$. The particle $P$ hangs at rest in equilibrium.
2. Show that $A P = \frac { 7 a } { 4 }$
The particle $P$ is now pulled down vertically from its equilibrium position towards $B$ and at time $t = 0$ it is released from rest. At time $t$, the particle $P$ is moving with speed $v$ and has displacement $x$ from its equilibrium position. The particle $P$ is subject to air resistance of magnitude $m k v$, where $k$ is a positive constant.
Show that $$\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + k \frac { \mathrm {~d} x } { \mathrm {~d} t } + \frac { 4 g } { a } x = 0$$
Find the range of values of $k$ which would result in the motion of $P$ being a damped oscillation.

Edexcel M4 2002 January Q1

A river of width 40 m flows with uniform and constant speed between straight banks. A swimmer crosses as quickly as possible and takes 30 s to reach the other side. She is carried 25 m downstream.

Find

the speed of the river,
the speed of the swimmer relative to the water.

Edexcel M4 2002 January Q2

2. A ball of mass $m$ is thrown vertically upwards from the ground with an initial speed $u$. When the speed of the ball is $v$, the magnitude of the air resistance is $m k v$, where $k$ is a positive constant. By modelling the ball as a particle, find, in terms of $u , k$ and $g$, the time taken for the ball to reach its greatest height.

Edexcel M4 2002 January Q3

3. A smooth uniform sphere $P$ of mass $m$ is falling vertically and strikes a fixed smooth inclined plane with speed $u$. The plane is inclined at an angle $\theta , \theta < 45 ^ { \circ }$, to the horizontal. The coefficient of restitution between $P$ and the inclined plane is $e$. Immediately after $P$ strikes the plane, $P$ moves horizontally.

Show that $e = \tan ^ { 2 } \theta$.
Show that the magnitude of the impulse exerted by $P$ on the plane is $m u \sec \theta$.

Edexcel M4 2002 January Q4

4. A pilot flying an aircraft at a constant speed of $2000 \mathrm { kmh } ^ { - 1 }$ detects an enemy aircraft 100 km away on a bearing of $045 ^ { \circ }$. The enemy aircraft is flying at a constant velocity of $1500 \mathrm { kmh } ^ { - 1 }$ due west. Find

the course, as a bearing to the nearest degree, that the pilot should set up in order to intercept the enemy aircraft,
the time, to the nearest s , that the pilot will take to reach the enemy aircraft.

Edexcel M4 2002 January Q5

5. \section*{Figure 1}

\includegraphics[max width=\textwidth, alt={}]{f70e9177-fbda-409d-8f80-d900a33a6481-3_394_1000_425_519}

A smooth uniform sphere $S$ of mass $m$ is moving on a smooth horizontal table. The sphere $S$ collides with another smooth uniform sphere $T$, of the same radius as $S$ but of mass $k m , k > 1$, which is at rest on the table. The coefficient of restitution between the spheres is $e$. Immediately before the spheres collide the direction of motion of $S$ makes an angle $\theta$ with the line joing their centres, as shown in Fig. 1. Immediately after the collision the directions of motion of $S$ and $T$ are perpendicular.

Show that $e = \frac { 1 } { k }$.
(6) Given that $k = 2$ and that the kinetic energy lost in the collision is one quarter of the initial kinetic energy,
find the value of $\theta$.
(6)

Edexcel M4 2002 January Q7

7. \begin{figure}[h]

\captionsetup{labelformat=empty} \caption{Figure 3} \includegraphics[alt={},max width=\textwidth]{f70e9177-fbda-409d-8f80-d900a33a6481-5_622_506_395_803}

\end{figure} A uniform rod $A B$, of mass $m$ and length $2 a$, can rotate freely in a vertical plane about a fixed smooth horizontal axis through $A$. The fixed point $C$ is vertically above $A$ and $A C = 4 a$. A light elastic string, of natural length $2 a$ and modulus of elasticity $\frac { 1 } { 2 } m g$, joins $B$ to $C$. The rod $A B$ makes an angle $\theta$ with the upward vertical at $A$, as shown in Fig. 3.

Show that the potential energy of the system is $$- m g a [ \cos \theta + \sqrt { } ( 5 - 4 \cos \theta ) ] + \text { constant. }$$
Hence determine the values of $\theta$ for which the system is in equilibrium. END

Edexcel M5 Q3

A spaceship is moving in a straight line in deep space and needs to increase its speed. This is done by ejecting fuel backwards from the spaceship at a constant speed $c$ relative to the spaceship. When the speed of the spaceship is $v$, its mass is $m$.
1. Show that, while the spaceship is ejecting fuel,
$$\frac { \mathrm { d } v } { \mathrm {~d} m } = - \frac { c } { m }$$ The initial mass of the spaceship is $m _ { 0 }$ and at time $t$ the mass of the spaceship is given by $m = m _ { 0 } ( 1 - k t )$, where $k$ is a positive constant.
Find the acceleration of the spaceship at time $t$.

Edexcel M5 Q4

4. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{19c5a621-c175-4d58-9002-4bcdefd02b71-08_515_417_210_758} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} A uniform lamina of mass $M$ is in the shape of a right-angled triangle $O A B$. The angle $O A B$ is $90 ^ { \circ } , O A = a$ and $A B = 2 a$, as shown in Figure 1.

Prove, using integration, that the moment of inertia of the lamina $O A B$ about the edge $O A$ is $\frac { 2 } { 3 } M a ^ { 2 }$.
(You may assume without proof that the moment of inertia of a uniform rod of mass $m$ and length $2 l$ about an axis through one end and perpendicular to the rod is $\frac { 4 } { 3 } m l ^ { 2 }$.) The lamina $O A B$ is free to rotate about a fixed smooth horizontal axis along the edge $O A$ and hangs at rest with $B$ vertically below $A$. The lamina is then given a horizontal impulse of magnitude $J$. The impulse is applied to the lamina at the point $B$, in a direction which is perpendicular to the plane of the lamina. Given that the lamina first comes to instantaneous rest after rotating through an angle of $120 ^ { \circ }$,
find an expression for $J$, in terms of $M , a$ and $g$.

Edexcel M5 Q6

A pendulum consists of a uniform rod $A B$, of length $4 a$ and mass $2 m$, whose end $A$ is rigidly attached to the centre $O$ of a uniform square lamina $P Q R S$, of mass $4 m$ and side $a$. The $\operatorname { rod } A B$ is perpendicular to the plane of the lamina. The pendulum is free to rotate about a fixed smooth horizontal axis $L$ which passes through $B$. The axis $L$ is perpendicular to $A B$ and parallel to the edge $P Q$ of the square.
1. Show that the moment of inertia of the pendulum about $L$ is $75 m a ^ { 2 }$.
The pendulum is released from rest when $B A$ makes an angle $\alpha$ with the downward vertical through $B$, where $\tan \alpha = \frac { 7 } { 24 }$. When $B A$ makes an angle $\theta$ with the downward vertical through $B$, the magnitude of the component, in the direction $A B$, of the force exerted by the axis $L$ on the pendulum is $X$.
Find an expression for $X$ in terms of $m , g$ and $\theta$. Using the approximation $\theta \approx \sin \theta$,
find an estimate of the time for the pendulum to rotate through an angle $\alpha$ from its initial rest position. Turn over
1. At time $t = 0$, the position vector of a particle $P$ is $- 3 \mathbf { j } \mathrm {~m}$. At time $t$ seconds, the position vector of $P$ is $\mathbf { r }$ metres and the velocity of $P$ is $\mathbf { v } \mathrm { m } \mathrm { s } ^ { - 1 }$. Given that
$$\mathbf { v } - 2 \mathbf { r } = 4 \mathrm { e } ^ { t } \mathbf { j }$$ find the time when $P$ passes through the origin.
2. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{19c5a621-c175-4d58-9002-4bcdefd02b71-16_504_586_267_671} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} A uniform circular disc has mass $4 m$, centre $O$ and radius $4 a$. The line $P O Q$ is a diameter of the disc. A circular hole of radius $2 a$ is made in the disc with the centre of the hole at the point $R$ on $P Q$ where $Q R = 5 a$, as shown in Figure 1. The resulting lamina is free to rotate about a fixed smooth horizontal axis $L$ which passes through $Q$ and is perpendicular to the plane of the lamina.
Show that the moment of inertia of the lamina about $L$ is $69 m a ^ { 2 }$. The lamina is hanging at rest with $P$ vertically below $Q$ when it is given an angular velocity $\Omega$. Given that the lamina turns through an angle $\frac { 2 \pi } { 3 }$ before it first comes to instantaneous rest,
find $\Omega$ in terms of $g$ and $a$.
1. A uniform lamina $A B C$ of mass $m$ is in the shape of an isosceles triangle with $A B = A C = 5 a$ and $B C = 8 a$.
2. Show, using integration, that the moment of inertia of the lamina about an axis through $A$, parallel to $B C$, is $\frac { 9 } { 2 } m a ^ { 2 }$.
The foot of the perpendicular from $A$ to $B C$ is $D$. The lamina is free to rotate in a vertical plane about a fixed smooth horizontal axis which passes through $D$ and is perpendicular to the plane of the lamina. The lamina is released from rest when $D A$ makes an angle $\alpha$ with the downward vertical. It is given that the moment of inertia of the lamina about an axis through $A$, perpendicular to $B C$ and in the plane of the lamina, is $\frac { 8 } { 3 } m a ^ { 2 }$.
Find the angular acceleration of the lamina when $D A$ makes an angle $\theta$ with the downward vertical. Given that $\alpha$ is small,
find an approximate value for the period of oscillation of the lamina about the vertical.
1. Two forces $\mathbf { F } _ { 1 } = ( \mathbf { i } + 2 \mathbf { j } + 3 \mathbf { k } ) \mathrm { N }$ and $\mathbf { F } _ { 2 } = ( 3 \mathbf { i } + \mathbf { j } + 2 \mathbf { k } ) \mathrm { N }$ act on a rigid body.
The force $\mathbf { F } _ { 1 }$ acts through the point with position vector ( $2 \mathbf { i } + \mathbf { k }$ ) m and the force $\mathbf { F } _ { 2 }$ acts through the point with position vector $( \mathbf { j } + 2 \mathbf { k } ) \mathrm { m }$.
If the two forces are equivalent to a single force $\mathbf { R }$, find
1. $\mathbf { R }$,
2. a vector equation of the line of action of $\mathbf { R }$, in the form $\mathbf { r } = \mathbf { a } + \lambda \mathbf { b }$.
If the two forces are equivalent to a single force acting through the point with position vector $( \mathbf { i } + 2 \mathbf { j } + \mathbf { k } ) \mathrm { m }$ together with a couple of moment $\mathbf { G }$, find the magnitude of $\mathbf { G }$.
1. A raindrop falls vertically under gravity through a cloud. In a model of the motion the raindrop is assumed to be spherical at all times and the cloud is assumed to consist of stationary water particles. At time $t = 0$, the raindrop is at rest and has radius $a$. As the raindrop falls, water particles from the cloud condense onto it and the radius of the raindrop is assumed to increase at a constant rate $\lambda$. A time $t$ the speed of the raindrop is $v$.
2. Show that
$$\frac { \mathrm { d } v } { \mathrm {~d} t } + \frac { 3 \lambda v } { ( \lambda t + a ) } = g$$
Find the speed of the raindrop when its radius is $3 a$.
1. A uniform circular disc has mass $m$, centre $O$ and radius $2 a$. It is free to rotate about a fixed smooth horizontal axis $L$ which lies in the same plane as the disc and which is tangential to the disc at the point $A$. The disc is hanging at rest in equilibrium with $O$ vertically below $A$ when it is struck at $O$ by a particle of mass $m$. Immediately before the impact the particle is moving perpendicular to the plane of the disc with speed $3 \sqrt { } ( a g )$. The particle adheres to the disc at $O$.
2. Find the angular speed of the disc immediately after the impact.
3. Find the magnitude of the force exerted on the disc by the axis immediately after the impact.
Turn over
advancing learning, changing lives
1. A particle moves from the point $A$ with position vector $( 3 \mathbf { i } - \mathbf { j } + 3 \mathbf { k } ) \mathrm { m }$ to the point $B$ with position vector $( \mathbf { i } - 2 \mathbf { j } - 4 \mathbf { k } ) \mathrm { m }$ under the action of the force $( 2 \mathbf { i } - 3 \mathbf { j } - \mathbf { k } ) \mathrm { N }$. Find the work done by the force.
2. A particle $P$ moves in the $x - y$ plane so that its position vector $\mathbf { r }$ metres at time $t$ seconds satisfies the differential equation
$$\frac { \mathrm { d } ^ { 2 } \mathbf { r } } { \mathrm {~d} t ^ { 2 } } - 4 \mathbf { r } = - 3 \mathrm { e } ^ { t } \mathbf { j }$$ When $t = 0$, the particle is at the origin and is moving with velocity $( 2 \mathbf { i } + \mathbf { j } ) \mathrm { ms } ^ { - 1 }$.
Find $\mathbf { r }$ in terms of $t$.
1. A rocket propels itself by its engine ejecting burnt fuel. Initially the rocket has total mass $M$, of which a mass $k M , k < 1$, is fuel. The rocket is at rest when its engine is started. The burnt fuel is ejected with constant speed $c$, relative to the rocket, in a direction opposite to that of the rocket's motion. Assuming that there are no external forces, find the speed of the rocket when all its fuel has been burnt.
2. Two forces $\mathbf { F } _ { 1 } = ( 3 \mathbf { j } + \mathbf { k } ) \mathrm { N }$ and $\mathbf { F } _ { 2 } = ( 4 \mathbf { i } + \mathbf { j } - \mathbf { k } ) \mathrm { N }$ act on a rigid body.
The force $\mathbf { F } _ { 1 }$ acts at the point with position vector ( $2 \mathbf { i } - \mathbf { j } + 3 \mathbf { k }$ ) m and the force $\mathbf { F } _ { 2 }$ acts at the point with position vector $( - 3 \mathbf { i } + 2 \mathbf { k } ) \mathrm { m }$.
The two forces are equivalent to a single force $\mathbf { R }$ acting at the point with position vector $( \mathbf { i } + 2 \mathbf { j } + \mathbf { k } ) \mathrm { m }$ together with a couple of moment $\mathbf { G }$. Find,
$\mathbf { R }$,
G. A third force $\mathbf { F } _ { 3 }$ is now added to the system. The force $\mathbf { F } _ { 3 }$ acts at the point with position vector $( 2 \mathbf { i } - \mathbf { k } ) \mathrm { m }$ and the three forces $\mathbf { F } _ { 1 } , \mathbf { F } _ { 2 }$ and $\mathbf { F } _ { 3 }$ are equivalent to a couple.
Find the magnitude of the couple.
5. A uniform rod $P Q$, of mass $m$ and length $2 a$, is made to rotate in a vertical plane with constant angular speed $\sqrt { } \left( \frac { g } { a } \right)$ about a fixed smooth horizontal axis through the end $P$ of the rod. Show that, when the rod is inclined at an angle $\theta$ to the downward vertical, the magnitude of the force exerted on the axis by the rod is $2 m g \left| \cos \left( \frac { 1 } { 2 } \theta \right) \right|$.
6. A uniform $\operatorname { rod } A B$ of mass $4 m$ is free to rotate in a vertical plane about a fixed smooth horizontal axis, $L$, through $A$. The rod is hanging vertically at rest when it is struck at its end $B$ by a particle of mass $m$. The particle is moving with speed $u$, in a direction which is horizontal and perpendicular to $L$, and after striking the rod it rebounds in the opposite direction with speed $v$. The coefficient of restitution between the particle and the rod is 1 . Show that $u = 7 v$.

Edexcel M5 Q7

7. Prove, using integration, that the moment of inertia of a uniform solid right circular cone, of mass $M$ and base radius $a$, about its axis is $\frac { 3 } { 10 } M a ^ { 2 }$.
[0pt] [You may assume, without proof, that the moment of inertia of a uniform circular disc, of mass $m$ and radius $r$, about an axis through its centre and perpendicular to its plane is $\frac { 1 } { 2 } m r ^ { 2 }$.]

Edexcel M5 Q8

8. A pendulum consists of a uniform rod $P Q$, of mass $3 m$ and length $2 a$, which is rigidly fixed at its end $Q$ to the centre of a uniform circular disc of mass $m$ and radius $a$. The rod is perpendicular to the plane of the disc. The pendulum is free to rotate about a fixed smooth horizontal axis $L$ which passes through the end $P$ of the rod and is perpendicular to the rod.

Show that the moment of inertia of the pendulum about $L$ is $\frac { 33 } { 4 } m a ^ { 2 }$. The pendulum is released from rest in the position where $P Q$ makes an angle $\alpha$ with the downward vertical. At time $t , P Q$ makes an angle $\theta$ with the downward vertical.
Show that the angular speed, $\dot { \theta }$, of the pendulum satisfies $$\dot { \theta } ^ { 2 } = \frac { 40 g ( \cos \theta - \cos \alpha ) } { 33 a }$$
Hence, or otherwise, find the angular acceleration of the pendulum. Given that $\alpha = \frac { \pi } { 20 }$ and that $P Q$ has length $\frac { 8 } { 33 } \mathrm {~m}$,
find, to 3 significant figures, an approximate value for the angular speed of the pendulum 0.2 s after it has been released from rest.

Edexcel D2 Q1

1. \section*{Figure 1}

\includegraphics[max width=\textwidth, alt={}]{4f494f19-5690-4d9f-8c18-db03d41da203-01_435_682_383_374}

Figure 1 shows a network of roads connecting six villages $A , B , C , D , E$ and $F$. The lengths of the roads are given in km.

Complete the table in the answer booklet, in which the entries are the shortest distances between pairs of villages. You should do this by inspection.
(2) The table can now be taken to represent a complete network.
Use the nearest-neighbour algorithm, starting at $A$, on your completed table in part (a). Obtain an upper bound to the length of a tour in this complete network, which starts and finishes at $A$ and visits every village exactly once.
Interpret your answer in part (b) in terms of the original network of roads connecting the six villages.
By choosing a different vertex as your starting point, use the nearest-neighbour algorithm to obtain a shorter tour than that found in part (b). State the tour and its length.

Edexcel D2 Q2

2. A two-person zero-sum game is represented by the following pay-off matrix for player $A$.

		$B$
		I	II	III	IV
\cline { 2 - 6 }	I	- 4	- 5	- 2	4
A	II	- 1	1	- 1	2
	III	0	5	- 2	- 4
	IV	- 1	3	- 1	1

Determine the play-safe strategy for each player.
Verify that there is a stable solution and determine the saddle points.
State the value of the game to $B$. Figure 2 \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{3.} \includegraphics[alt={},max width=\textwidth]{4f494f19-5690-4d9f-8c18-db03d41da203-02_502_990_319_148}
\end{figure} The network in Fig. 2 shows possible routes that an aircraft can take from $S$ to $T$. The numbers on the directed arcs give the amount of fuel used on that part of the route, in appropriate units. The airline wishes to choose the route for which the maximum amount of fuel used on any part of the route is as small as possible. This is the rninimax route.
Complete the table in the answer booklet.
Hence obtain the minimax route from $S$ to $T$ and state the maximum amount of fuel used on any part of this route.
(2)
4. Andrew ( $A$ ) and Barbara ( $B$ ) play a zero-sum game. This game is represented by the following payoff matrix for Andrew. $$A \left( \begin{array} { c c c } & B &
3 & 5 & 4
1 & 4 & 2
6 & 3 & 7 \end{array} \right)$$
Explain why this matrix may be reduced to $$\left( \begin{array} { l l } 3 & 5
6 & 3 \end{array} \right) .$$
Hence find the best strategy for each player and the value of the game.
5. An engineering company has 4 machines available and 4 jobs to be completed. Each machine is to be assigned to one job. The time, in hours, required by each machine to complete each job is shown in the table below.
\cline { 2 - 5 } \multicolumn{1}{c|}{} Job 1 Job 2 Job 3 Job 4
Machine 1 14 5 8 7
Machine 2 2 12 6 5
Machine 3 7 8 3 9
Machine 4 2 4 6 10
Use the Hungarian algorithm, reducing rows first, to obtain the allocation of machines to jobs which minimises the total time required. State this minimum time.
6. The table below shows the distances, in km , between six towns $A , B , C , D , E$ and $F$.
\cline { 2 - 7 } \multicolumn{1}{c|}{} $A$ $B$ $C$ $D$ $E$ $F$
$A$ - 85 110 175 108 100
$B$ 85 - 38 175 160 93
$C$ 110 38 - 148 156 73
$D$ 175 175 148 - 110 84
$E$ 108 160 156 110 - 92
$F$ 100 93 73 84 92 -
Starting from $A$, use Prim's algorithm to find a minimum connector and draw the minimum spanning tree. You must make your method clear by stating the order in which the arcs are selected.
1. Using your answer to part (a) obtain an initial upper hound for the solution of the travelling salesman problem.
2. Use a short cut to reduce the upper bound to a value less than 680 .

Starting by deleting $F$, find a lower bound for the solution of the travelling salesman problem.
7. A steel manufacturer has 3 factories $F _ { 1 } , F _ { 2 }$ and $F _ { 3 }$ which can produce 35,25 and 15 kilotonnes of steel per year, respectively. Three businesses $B _ { 1 } , B _ { 2 }$ and $B _ { 3 }$ have annual requirements of 20,25 and 30 kilotonnes respectively. The table below shows the cost $C _ { i j }$ in appropriate units, of transporting one kilotonne of steel from factory $F _ { i }$ to business $B _ { j }$.

\cline { 3 - 5 } \multicolumn{2}{c\|}{}	Business
\cline { 3 - 5 } \multicolumn{2}{c\|}{}	$B _ { 1 }$	$B _ { 2 }$	$B _ { 3 }$
\multirow{3}{*}{Factory}	$F _ { 1 }$	10	4	11
\cline { 2 - 5 }	$F _ { 2 }$	12	5	8
\cline { 2 - 5 }	$F _ { 3 }$	9	6	7

The manufacturer wishes to transport the steel to the businesses at minimum total cost.

Write down the transportation pattern obtained by using the North-West corner rule.
Calculate all of the improvement indices $I _ { i j }$, and hence show that this pattern is not optimal.
Use the stepping-stone method to obtain an improved solution.
Show that the transportation pattern obtained in part (c) is optimal and find its cost. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 4} \includegraphics[alt={},max width=\textwidth]{4f494f19-5690-4d9f-8c18-db03d41da203-04_346_922_319_278}
\end{figure} The network in Fig. 4 models a drainage system. The number on each arc indicates the capacity of that arc, in litres per second.
Write down the source vertices. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 5} \includegraphics[alt={},max width=\textwidth]{4f494f19-5690-4d9f-8c18-db03d41da203-04_341_920_900_278}
\end{figure} Figure 5 shows a feasible flow through the same network.
State the value of the feasible flow shown in Fig. 5. Taking the flow in Fig. 5 as your initial flow pattern,
use the labelling procedure on Diagram 1 to find a maximum flow through this network. You should list each flow-augmenting route you use, together with its flow.
Show the maximal flow on Diagram 2 and state its value.
Prove that your flow is maximal.
9. T42 Co. Ltd produces three different blends of tea, Morning, Afternoon and Evening. The teas must be processed, blended and then packed for distribution. The table below shows the time taken, in hours, for each stage of the production of a tonne of tea. It also shows the profit, in hundreds of pounds, on each tonne.
\cline { 2 - 5 } \multicolumn{1}{c|}{} Processing Blending Packing Profit (£100)
Morning blend 3 1 2 4
Afternoon blend 2 3 4 5
Evening blend 4 2 3 3
The total times available each week for processing, blending and packing are 35, 20 and 24 hours respectively. T42 Co. Ltd wishes to maximise the weekly profit. Let $x , y$ and $z$ be the number of tonnes of Morning, Afternoon and Evening blend produced each week.
Formulate the above situation as a linear programming problem, listing clearly the objective function, and the constraints as inequalities. An initial Simplex tableau for the above situation is
Basic
variable
$x$ $y$ $z$ $r$ $s$ $t$ Value
$r$ 3 2 4 1 0 0 35
$s$ 1 3 2 0 1 0 20
$t$ 2 4 3 0 0 1 24
$P$ - 4 - 5 - 3 0 0 0 0
Solve this linear programming problem using the Simplex algorithm. Take the most negative number in the profit row to indicate the pivot column at each stage.
(11) T42 Co. Ltd wishes to increase its profit further and is prepared to increase the time available for processing or blending or packing or any two of these three.

Use your answer to part (b) to advise the company as to which stage(s) it should increase the time available.
(2)
10. While solving a maximizing linear programming problem, the following tableau was obtained.

Basic

variable

$x$

$y$

$z$

$r$

$s$

$t$

Value

$r$

0

$1 \frac { 2 } { 3 }$

1

0

$- \frac { 1 } { 6 }$

$\frac { 2 } { 3 }$

$y$

0

1

$3 \frac { 1 } { 3 }$

0

1

$- \frac { 1 } { 3 }$

$\frac { 1 } { 3 }$

$x$

1

0

- 3

0

- 1

$\frac { 1 } { 2 }$

1

$P$

0

1

0

1

11

Explain why this is an optimal tableau.
Write down the optimal solution of this problem, stating the value of every variable.
Write down the profit equation from the tableau. Use it to explain why changing the value of any of the non-basic variables will decrease the value of $P$.
11. A company wishes to transport its products from 3 factories $F _ { 1 } , F _ { 2 }$ and $F _ { 3 }$ to a single retail outlet $R$. The capacities of the possible routes, in van loads per day, are shown in Fig. 5. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 5} \includegraphics[alt={},max width=\textwidth]{4f494f19-5690-4d9f-8c18-db03d41da203-05_470_766_447_1695}
\end{figure}
On Diagram 1 in the answer booklet add a supersource $S$ to obtain a capacitated network with a single source and a single sink. State the minimum capacity of each arc you have added.
1. State the maximum flow along $S F _ { 1 } A B R$ and $S F _ { 3 } C R$.
2. Show these maximum flows on Diagram 2 in the answer booklet, using numbers in circles. Taking your answer to part (b)(ii) as the initial flow pattern,
1. use the labelling procedure to find a maximum flow from $S$ to $R$. Your working should be shown on Diagram 3. List each flow-augmenting route you find together with its flow.
2. Prove that your final flow is maximal.
  12. \begin{figure}[h]
  \captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{4f494f19-5690-4d9f-8c18-db03d41da203-06_405_791_301_221}
  \end{figure} A company has 3 warehouses $W _ { 1 } , W _ { 2 }$, and $W _ { 3 }$. It needs to transport the goods stored there to 2 retail outlets $R _ { 1 }$ and $R _ { 2 }$. The capacities of the possible routes, in van loads per day, are shown in Fig 2. Warehouses $W _ { 1 } , W _ { 2 }$ and $W _ { 3 }$ have 14, 12 and 14 van loads respectively available per day and retail outlets $R _ { 1 }$ and $R _ { 2 }$ can accept 6 and 25 van loads respectively per day.
On Diagram 1 on the answer sheet add a supersource $W$, a supersink $R$ and the appropriate directed arcs to obtain a single-source, single-sink capacitated network. State the minimum capacity of each arc you have added.
State the maximum flow along
1. $W \quad W _ { 1 } \quad A \quad R _ { 1 } \quad R$,
2. $W W _ { 3 } \quad C \quad R _ { 2 } \quad R$.
Taking your answers to part (b) as the initial flow pattern, use the labelling procedure to obtain a maximum flow through the network from $W$ to $R$. Show your working on Diagram 2. List each flowaugmenting route you use, together with its flow.
From your final flow pattern, determine the number of van loads passing through $B$ each day. \section*{D2 2003 (adapted for new spec)}
1. A two person zero-sum game is represented by the following pay-off matrix for player $A$.
\cline { 2 - 4 } \multicolumn{1}{c|}{} B plays I $B$ plays II $B$ plays III
$A$ plays I - 3 2 5
$A$ plays II 4 - 1 - 4
Write down the pay off matrix for player $B$.
Formulate the game as a linear programming problem for player $B$, writing the constraints as equalities and stating your variables clearly.
2. (a) Explain the difference between the classical and practical travelling salesman problems.
\includegraphics[max width=\textwidth, alt={}, center]{4f494f19-5690-4d9f-8c18-db03d41da203-06_454_857_737_1736} The network in the diagram above shows the distances, in kilometres, between eight McBurger restaurants. An inspector from head office wishes to visit each restaurant. His route should start and finish at $A$, visit each restaurant at least once and cover a minimum distance.
Obtain a minimum spanning tree for the network using Kruskal's algorithm. You should draw your tree and state the order in which the arcs were added.
Use your answer to part (b) to determine an initial upper bound for the length of the route.
Starting from your initial upper bound and using an appropriate method, find an upper bound which is less than 135 km . State your tour.

Edexcel D2 Q4

4. Andrew ( $A$ ) and Barbara ( $B$ ) play a zero-sum game. This game is represented by the following payoff matrix for Andrew. $$A \left( \begin{array} { c c c } & B &
3 & 5 & 4
1 & 4 & 2
6 & 3 & 7 \end{array} \right)$$

Explain why this matrix may be reduced to $$\left( \begin{array} { l l } 3 & 5
6 & 3 \end{array} \right) .$$
Hence find the best strategy for each player and the value of the game.

Edexcel D2 Q6

6. The table below shows the distances, in km , between six towns $A , B , C , D , E$ and $F$.

\cline { 2 - 7 } \multicolumn{1}{c\|}{}	$A$	$B$	$C$	$D$	$E$	$F$
$A$	-	85	110	175	108	100
$B$	85	-	38	175	160	93
$C$	110	38	-	148	156	73
$D$	175	175	148	-	110	84
$E$	108	160	156	110	-	92
$F$	100	93	73	84	92	-

Starting from $A$, use Prim's algorithm to find a minimum connector and draw the minimum spanning tree. You must make your method clear by stating the order in which the arcs are selected.
1. Using your answer to part (a) obtain an initial upper hound for the solution of the travelling salesman problem.
2. Use a short cut to reduce the upper bound to a value less than 680 .
Starting by deleting $F$, find a lower bound for the solution of the travelling salesman problem.

\cline { 3 - 5 } \multicolumn{2}{c\|}{}	Business
\cline { 3 - 5 } \multicolumn{2}{c\|}{}	\(B _ { 1 }\)	\(B _ { 2 }\)	\(B _ { 3 }\)
\multirow{3}{*}{Factory}	\(F _ { 1 }\)	10	4	11
\cline { 2 - 5 }	\(F _ { 2 }\)	12	5	8
\cline { 2 - 5 }	\(F _ { 3 }\)	9	6	7

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