Questions - OCR MEI - A-Level Maths

OCR MEI C4 2008 June Q2

2 Fig. 2 shows the curve $y = \sqrt { 1 + \mathrm { e } ^ { 2 x } }$. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{8ad99e2a-4cef-40b3-af8d-673b97536227-02_432_873_587_635} \captionsetup{labelformat=empty} \caption{Fig. 2}

\end{figure} The region bounded by the curve, the $x$-axis, the $y$-axis and the line $x = 1$ is rotated through $360 ^ { \circ }$ about the $x$-axis. Show that the volume of the solid of revolution produced is $\frac { 1 } { 2 } \pi \left( 1 + \mathrm { e } ^ { 2 } \right)$.

OCR MEI C4 2008 June Q3

3 Solve the equation $\cos 2 \theta = \sin \theta$ for $0 \leqslant \theta \leqslant 2 \pi$, giving your answers in terms of $\pi$.

OCR MEI C4 2008 June Q4

4 Given that $x = 2 \sec \theta$ and $y = 3 \tan \theta$, show that $\frac { x ^ { 2 } } { 4 } - \frac { y ^ { 2 } } { 9 } = 1$.

OCR MEI C4 2008 June Q5

5 A curve has parametric equations $x = 1 + u ^ { 2 } , y = 2 u ^ { 3 }$.

Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$ in terms of $u$.
Hence find the gradient of the curve at the point with coordinates $( 5,16 )$.

OCR MEI C4 2008 June Q6

6

Find the first three non-zero terms of the binomial series expansion of $\frac { 1 } { \sqrt { 1 + 4 x ^ { 2 } } }$, and state the set of values of $x$ for which the expansion is valid.
Hence find the first three non-zero terms of the series expansion of $\frac { 1 - x ^ { 2 } } { \sqrt { 1 + 4 x ^ { 2 } } }$.

OCR MEI C4 2008 June Q7

7 Express $\sqrt { 3 } \sin x - \cos x$ in the form $R \sin ( x - \alpha )$, where $R > 0$ and $0 < \alpha < \frac { 1 } { 2 } \pi$. Express $\alpha$ in the form $k \pi$. Find the exact coordinates of the maximum point of the curve $y = \sqrt { 3 } \sin x - \cos x$ for which $0 < x < 2 \pi$.

OCR MEI C4 2008 June Q8

8 The upper and lower surfaces of a coal seam are modelled as planes ABC and DEF, as shown in Fig. 8. All dimensions are metres. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{8ad99e2a-4cef-40b3-af8d-673b97536227-03_1004_1397_493_374} \captionsetup{labelformat=empty} \caption{Fig. 8}

\end{figure} Relative to axes $\mathrm { O } x$ (due east), $\mathrm { O } y$ (due north) and $\mathrm { O } z$ (vertically upwards), the coordinates of the points are as follows.
A: (0, 0, -15)
B: (100, 0, -30)
C: (0, 100, -25)
D: (0, 0, -40)
E: (100, 0, -50)
F: (0, 100, -35)

Verify that the cartesian equation of the plane ABC is $3 x + 2 y + 20 z + 300 = 0$.
Find the vectors $\overrightarrow { \mathrm { DE } }$ and $\overrightarrow { \mathrm { DF } }$. Show that the vector $2 \mathbf { i } - \mathbf { j } + 20 \mathbf { k }$ is perpendicular to each of these vectors. Hence find the cartesian equation of the plane DEF .
By calculating the angle between their normal vectors, find the angle between the planes ABC and DEF. It is decided to drill down to the seam from a point $\mathrm { R } ( 15,34,0 )$ in a line perpendicular to the upper surface of the seam. This line meets the plane ABC at the point S .
Write down a vector equation of the line RS. Calculate the coordinates of S.

OCR MEI C4 2008 June Q9

9 A skydiver drops from a helicopter. Before she opens her parachute, her speed $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$ after time $t$ seconds is modelled by the differential equation $$\frac { \mathrm { d } v } { \mathrm {~d} t } = 10 \mathrm { e } ^ { - \frac { 1 } { 2 } t }$$ When $t = 0 , v = 0$.

Find $v$ in terms of $t$.
According to this model, what is the speed of the skydiver in the long term? She opens her parachute when her speed is $10 \mathrm {~m} \mathrm {~s} ^ { - 1 }$. Her speed $t$ seconds after this is $w \mathrm {~m} \mathrm {~s} ^ { - 1 }$, and is modelled by the differential equation $$\frac { \mathrm { d } w } { \mathrm {~d} t } = - \frac { 1 } { 2 } ( w - 4 ) ( w + 5 )$$
Express $\frac { 1 } { ( w - 4 ) ( w + 5 ) }$ in partial fractions.
Using this result, show that $\frac { w - 4 } { w + 5 } = 0.4 \mathrm { e } ^ { - 4.5 t }$.
According to this model, what is the speed of the skydiver in the long term? RECOGNISING ACHIEVEMENT \section*{ADVANCED GCE} \section*{4754/01B} \section*{MATHEMATICS (MEI)} Applications of Advanced Mathematics (C4) Paper B: Comprehension
WEDNESDAY 21 MAY 2008
Afternoon
Time: Up to 1 hour
Additional materials: Rough paper
MEI Examination Formulae and Tables (MF 2) \section*{Candidate Forename}
\includegraphics[max width=\textwidth, alt={}]{8ad99e2a-4cef-40b3-af8d-673b97536227-05_125_547_986_516}
This document consists of $\mathbf { 6 }$ printed pages, $\mathbf { 2 }$ blank pages and an insert. 1 Complete these Latin square puzzles.
2 1 3
3
\includegraphics[max width=\textwidth, alt={}, center]{8ad99e2a-4cef-40b3-af8d-673b97536227-06_391_419_836_854} 2 In line 51, the text says that the Latin square
1 2 3 4
3 1 4 2
2 4 1 3
4 3 2 1
could not be the solution to a Sudoku puzzle.
Explain this briefly.
3 On lines 114 and 115 the text says "It turns out that there are 16 different ways of filling in the remaining cells while keeping to the Sudoku rules. One of these ways is shown in Fig.10." Complete the grid below with a solution different from that given in Fig. 10.
1 2 3 4
4 Lines 154 and 155 of the article read "There are three other embedded Latin squares in Fig. 14; one of them is illustrated in Fig. 16." Indicate one of the other two embedded Latin squares on this copy of Fig. 14.
4 2 3 1
2 4
4 2
2 4 1 3
5 The number of $9 \times 9$ Sudokus is given in line 121 .
Without doing any calculations, explain why you would expect 9! to be a factor of this number.
6 In the table below, $M$ represents the maximum number of givens for which a Sudoku puzzle may have no unique solution (Investigation 3 in the article). $s$ is the side length of the Sudoku grid and $b$ is the side length of its blocks.
Block side
length, $b$
Sudoku,
$s \times s$
$M$
1 $1 \times 1$ -
2 $4 \times 4$ 12
3 $9 \times 9$
4 $16 \times 16$
5
Complete the table.
Give a formula for $M$ in terms of $b$.
7 A man is setting a Sudoku puzzle and starts with this solution.
1 2 3 4 5 6 7 8 9
4 5 6 8 9 7 3 1 2
7 8 9 3 1 2 5 6 4
2 3 1 5 6 4 8 9 7
5 6 4 9 7 8 1 2 3
8 9 7 1 2 3 6 4 5
3 1 2 6 4 5 9 7 8
6 4 5 7 8 9 2 3 1
9 7 8 2 3 1 4 5 6
He then removes some of the numbers to give the puzzles in parts (i) and (ii). In each case explain briefly, and without trying to solve the puzzle, why it does not have a unique solution.
[0pt] [2,2]
1 2 4 6 9
4 8 9 1
8 6
2 1 4 7
6 4 7 8 1 2
8 9 2 4
1 6 4 9 7
6 4 7 9 1
9 8 2 1 4 6
1 2 3 4 5 6 7 8 9
4 5 6 8 9 7 3 1 2
7 8 9 5 6 4
2 3 1 5 6 4 8 9 7
5 6 4 9 7 8 1 2 3
8 9 7 6 4 5
3 1 2 6 4 5 9 7 8
6 4 5 7 8 9 2 3 1
9 7 8 4 5 6
$\_\_\_\_$
$\_\_\_\_$

OCR MEI S3 Q2

2 Geoffrey is a university lecturer. He has to prepare five questions for an examination. He knows by experience that it takes about 3 hours to prepare a question, and he models the time (in minutes) taken to prepare one by the Normally distributed random variable $X$ with mean 180 and standard deviation 12, independently for all questions.

One morning, Geoffrey has a gap of 2 hours 50 minutes ( 170 minutes) between other activities. Find the probability that he can prepare a question in this time.
One weekend, Geoffrey can devote 14 hours to preparing the complete examination paper. Find the probability that he can prepare all five questions in this time. A colleague, Helen, has to check the questions.
She models the time (in minutes) to check a question by the Normally distributed random variable $Y$ with mean 50 and standard deviation 6, independently for all questions and independently of $X$. Find the probability that the total time for Geoffrey to prepare a question and Helen to check it exceeds 4 hours.
When working under pressure of deadlines, Helen models the time to check a question in a different way. She uses the Normally distributed random variable $\frac { 1 } { 4 } X$, where $X$ is as above. Find the length of time, as given by this model, which Helen needs to ensure that, with probability 0.9 , she has time to check a question. Ian, an educational researcher, suggests that a better model for the time taken to prepare a question would be a constant $k$ representing "thinking time" plus a random variable $T$ representing the time required to write the question itself, independently for all questions.
Taking $k$ as 45 and $T$ as Normally distributed with mean 120 and standard deviation 10 (all units are minutes), find the probability according to Ian's model that a question can be prepared in less than 2 hours 30 minutes. Juliet, an administrator, proposes that the examination should be reduced in time and shorter questions should be used.
Juliet suggests that Ian's model should be used for the time taken to prepare such shorter questions but with $k = 30$ and $T$ replaced by $\frac { 3 } { 5 } T$. Find the probability as given by this model that a question can be prepared in less than $1 \frac { 3 } { 4 }$ hours.

OCR MEI S3 Q4

10 marks

4 Quality control inspectors in a factory are investigating the lengths of glass tubes that will be used to make laboratory equipment.

Data on the observed lengths of a random sample of 200 glass tubes from one batch are available in the form of a frequency distribution as follows.
Use a suitable statistical procedure to assess the goodness of fit of $X$ to these data. Discuss your conclusions briefly. 2 A bus route runs from the centre of town A through the town's urban area to a point B on its boundary and then through the country to a small town C . Because of traffic congestion and general road conditions, delays occur on both the urban and the country sections. All delays may be considered independent. The scheduled time for the journey from A to B is 24 minutes. In fact, journey times over this section are given by the Normally distributed random variable $X$ with mean 26 minutes and standard deviation 3 minutes. The scheduled time for the journey from B to C is 18 minutes. In fact, journey times over this section are given by the Normally distributed random variable $Y$ with mean 15 minutes and standard deviation 2 minutes. Journey times on the two sections of route may be considered independent. The timetable published to the public does not show details of times at intermediate points; thus, if a bus is running early, it merely continues on its journey and is not required to wait.
Find the probability that a journey from A to B is completed in less than the scheduled time of 24 minutes.
Find the probability that a journey from A to C is completed in less than the scheduled time of 42 minutes.
It is proposed to introduce a system of bus lanes in the urban area. It is believed that this would mean that the journey time from A to B would be given by the random variable $0.85 X$. Assuming this to be the case, find the probability that a journey from A to B would be completed in less than the currently scheduled time of 24 minutes.
An alternative proposal is to introduce an express service. This would leave out some bus stops on both sections of the route and its overall journey time from A to C would be given by the random variable $0.9 X + 0.8 Y$. The scheduled time from A to C is to be given as a whole number of minutes. Find the least possible scheduled time such that, with probability 0.75 , buses would complete the journey on time or early.
A programme of minor road improvements is undertaken on the country section. After their completion, it is thought that the random variable giving the journey time from B to C is still Normally distributed with standard deviation 2 minutes. A random sample of 15 journeys is found to have a sample mean journey time from B to C of 13.4 minutes. Provide a two-sided $95 \%$ confidence interval for the population mean journey time from B to C . 3 An employer has commissioned an opinion polling organisation to undertake a survey of the attitudes of staff to proposed changes in the pension scheme. The staff are categorised as management, professional and administrative, and it is thought that there might be considerable differences of opinion between the categories. There are 60,140 and 300 staff respectively in the categories. The budget for the survey allows for a sample of 40 members of staff to be selected for in-depth interviews.
Explain why it would be unwise to select a simple random sample from all the staff.
Discuss whether it would be sensible to consider systematic sampling.
What are the advantages of stratified sampling in this situation?
State the sample sizes in each category if stratified sampling with as nearly as possible proportional allocation is used. The opinion polling organisation needs to estimate the average wealth of staff in the categories, in terms of property, savings, investments and so on. In a random sample of 11 professional staff, the sample mean is $\pounds 345818$ and the sample standard deviation is $\pounds 69241$.
Assuming the underlying population is Normally distributed, test at the $5 \%$ level of significance the null hypothesis that the population mean is $\pounds 300000$ against the alternative hypothesis that it is greater than $\pounds 300000$. Provide also a two-sided $95 \%$ confidence interval for the population mean.
[0pt] [10] 4 A company has many factories. It is concerned about incidents of trespassing and, in the hope of reducing if not eliminating these, has embarked on a programme of installing new fencing.
Records for a random sample of 9 factories of the numbers of trespass incidents in typical weeks before and after installation of the new fencing are as follows.
Find the probability that, on a randomly chosen visit, it takes less than 50 minutes to mow the lawns.
Find the probability that, on a randomly chosen visit, the total time for hoeing and pruning is less than 50 minutes.
If Bill mows the lawns while Ben does the hoeing and pruning, find the probability that, on a randomly chosen visit, Ben finishes first. Bill and Ben do my gardening twice a month and send me an invoice at the end of the month.
Write down the mean and variance of the total time (in minutes) they spend on mowing, hoeing and pruning per month.
The company charges for the total time spent at 15 pence per minute. There is also a fixed charge of $\pounds 10$ per month. Find the probability that the total charge for a month does not exceed $\pounds 40$. 4 (a) An amateur weather forecaster has been keeping records of air pressure, measured in atmospheres. She takes the measurement at the same time every day using a barometer situated in her garden. A random sample of 100 of her observations is summarised in the table below. The corresponding expected frequencies for a Normal distribution, with its two parameters estimated by sample statistics, are also shown in the table.
Find the probability that the weekly takings for coaches are less than $\pounds 40000$.
Find the probability that the weekly takings for lorries exceed the weekly takings for cars.
Find the probability that over a 4 -week period the total takings for cars exceed $\pounds 225000$. What assumption must be made about the four weeks?
Each week the operator allocates part of the takings for repairs. This is determined for each type of vehicle according to estimates of the long-term damage caused. It is calculated as follows: $5 \%$ of takings for cars, $10 \%$ for coaches and $20 \%$ for lorries. Find the probability that in any given week the total amount allocated for repairs will exceed $\pounds 20000$. 3 The management of a large chain of shops aims to reduce the level of absenteeism among its workforce by means of an incentive bonus scheme. In order to evaluate the effectiveness of the scheme, the management measures the percentage of working days lost before and after its introduction for each of a random sample of 11 shops. The results are shown below.
Give three reasons why a $t$ test would be appropriate.
Carry out the test using a $5 \%$ significance level. State your hypotheses and conclusion carefully.
Find a 95\% confidence interval for the true mean temperature in the reaction chamber.
Describe briefly one advantage and one disadvantage of having a 99\% confidence interval instead of a 95\% confidence interval. 4 (a) In Germany, towards the end of the nineteenth century, a study was undertaken into the distribution of the sexes in families of various sizes. The table shows some data about the numbers of girls in 500 families, each with 5 children. It is thought that the binomial distribution $\mathrm { B } ( 5 , p )$ should model these data.
The grower intends to perform a $t$ test to examine whether there is any difference in the mean yield of the two types of plant. State the hypotheses he should use and also any necessary assumption.
Carry out the test using a $5 \%$ significance level.
(b) The tea grower deals with many types of tea and employs tasters to rate them. The tasters do this by giving each tea a score out of 100. The tea grower wishes to compare the scores given by two of the tasters. Their scores for a random selection of 10 teas are as follows. A Wilcoxon signed rank test is to be used to decide whether there is any evidence of a preference for one of the uniforms.
Explain why this test is appropriate in these circumstances and state the hypotheses that should be used.
Carry out the test at the $5 \%$ significance level. 4 A random variable $X$ has probability density function $\mathrm { f } ( x ) = \frac { 2 x } { \lambda ^ { 2 } }$ for $0 < x < \lambda$, where $\lambda$ is a positive constant.
Show that, for any value of $\lambda , \mathrm { f } ( x )$ is a valid probability density function.
Find $\mu$, the mean value of $X$, in terms of $\lambda$ and show that $\mathrm { P } ( X < \mu )$ does not depend on $\lambda$.
Given that $\mathrm { E } \left( X ^ { 2 } \right) = \frac { \lambda ^ { 2 } } { 2 }$, find $\sigma ^ { 2 }$, the variance of $X$, in terms of $\lambda$. The random variable $X$ is used to model the depth of the space left by the filling machine at the top of a jar of jam. The model gives the following probabilities for $X$ (whatever the value of $\lambda$ ).
Initially it is assumed that the value of $p$ is $\frac { 1 } { 2 }$. Test at the $5 \%$ level of significance whether it is reasonable to suppose that the model applies with $p = \frac { 1 } { 2 }$.
The model is refined by estimating $p$ from the data. Find the mean of the observed data and hence an estimate of $p$.
Using the estimated value of $p$, the value of the test statistic $X ^ { 2 }$ turns out to be 2.3857 . Is it reasonable to suppose, at the $5 \%$ level of significance, that this refined model applies?
Discuss the reasons for the different outcomes of the tests in parts (i) and (iii). 2 (a) A continuous random variable, $X$, has probability density function $$f ( x ) = \begin{cases} \frac { 1 } { 72 } \left( 8 x - x ^ { 2 } \right) & 2 \leqslant x \leqslant 8
0 & \text { otherwise } \end{cases}$$
Find $\mathrm { F } ( x )$, the cumulative distribution function of $X$.
Sketch $\mathrm { F } ( x )$.
The median of $X$ is $m$. Show that $m$ satisfies the equation $m ^ { 3 } - 12 m ^ { 2 } + 148 = 0$. Verify that $m \approx 4.42$.
(b) The random variable in part (a) is thought to model the weights, in kilograms, of lambs at birth. The birth weights, in kilograms, of a random sample of 12 lambs, given in ascending order, are as follows. $$\begin{array} { l l l l l l l l l l l l } 3.16 & 3.62 & 3.80 & 3.90 & 4.02 & 4.72 & 5.14 & 6.36 & 6.50 & 6.58 & 6.68 & 6.78 \end{array}$$ Test at the 5\% level of significance whether a median of 4.42 is consistent with these data. 3 Cholesterol is a lipid (fat) which is manufactured by the liver from the fatty foods that we eat. It plays a vital part in allowing the body to function normally. However, when high levels of cholesterol are present in the blood there is a risk of arterial disease. Among the factors believed to assist with achieving and maintaining low cholesterol levels are weight loss and exercise. A doctor wishes to test the effectiveness of exercise in lowering cholesterol levels. For a random sample of 12 of her patients, she measures their cholesterol levels before and after they have followed a programme of exercise. The measurements obtained are as follows. This sample is to be tested to see whether the campaign appears to have been successful in raising the percentage receiving the booster.
Explain why the use of paired data is appropriate in this context.
Carry out an appropriate Wilcoxon signed rank test using these data, at the $5 \%$ significance level.
(b) Benford's Law predicts the following probability distribution for the first significant digit in some large data sets.
Digit 1 2 3 4 5 6 7 8 9
Probability 0.301 0.176 0.125 0.097 0.079 0.067 0.058 0.051 0.046
On one particular day, the first significant digits of the stock market prices of the shares of a random sample of 200 companies gave the following results.
Digit 1 2 3 4 5 6 7 8 9
Frequency 55 34 27 16 15 17 12 15 9
Test at the $10 \%$ level of significance whether Benford's Law provides a reasonable model in the context of share prices. 4 A random variable $X$ has an exponential distribution with probability density function $\mathrm { f } ( x ) = \lambda \mathrm { e } ^ { - \lambda x }$ for $x \geqslant 0$, where $\lambda$ is a positive constant.
Verify that $\int _ { 0 } ^ { \infty } \mathrm { f } ( x ) \mathrm { d } x = 1$ and sketch $\mathrm { f } ( x )$.
In this part of the question you may use the following result. $$\int _ { 0 } ^ { \infty } x ^ { r } \mathrm { e } ^ { - \lambda x } \mathrm {~d} x = \frac { r ! } { \lambda ^ { r + 1 } } \quad \text { for } r = 0,1,2 , \ldots$$ Derive the mean and variance of $X$ in terms of $\lambda$. The random variable $X$ is used to model the lifetime, in years, of a particular type of domestic appliance. The manufacturer of the appliance states that, based on past experience, the mean lifetime is 6 years.
Let $\bar { X }$ denote the mean lifetime, in years, of a random sample of 50 appliances. Write down an approximate distribution for $\bar { X }$.
A random sample of 50 appliances is found to have a mean lifetime of 7.8 years. Does this cast any doubt on the model?

OCR MEI FP3 2015 June Q1

1 The point A has coordinates $( 2,5,4 )$ and the line BC has equation $$\mathbf { r } = \left( \begin{array} { c } 8
25
43 \end{array} \right) + \lambda \left( \begin{array} { c } 4
15
25 \end{array} \right)$$ You are given that $\mathrm { AB } = \mathrm { AC } = 15$.

Show that the coordinates of one of the points B and C are (4, 10, 18). Find the coordinates of the other point. These points are B and C respectively.
Find the equation of the plane ABC in cartesian form.
Show that the plane containing the line BC and perpendicular to the plane ABC has equation $5 y - 3 z + 4 = 0$. The point D has coordinates $( 1,1,3 )$.
Show that $| \overrightarrow { B C } \times \overrightarrow { A D } | = \sqrt { 7667 }$ and hence find the shortest distance between the lines $B C$ and $A D$.
Find the volume of the tetrahedron ABCD .

OCR MEI FP3 2015 June Q2

2 A surface has equation $z = 3 x ^ { 2 } - 12 x y + 2 y ^ { 3 } + 60$.

Show that the point $\mathrm { A } ( 8,4 , - 4 )$ is a stationary point on the surface. Find the coordinates of the other stationary point, B , on this surface.
A point P with coordinates $( 8 + h , 4 + k , p )$ lies on the surface.
(A) Show that $p = - 4 + 3 ( h - 2 k ) ^ { 2 } + 2 k ^ { 2 } ( 6 + k )$.
(B) Deduce that the stationary point A is a local minimum.
(C) By considering sections of the surface near to B in each of the planes $x = 0$ and $y = 0$, investigate the nature of the stationary point B .
The point Q with coordinates $( 1,1,53 )$ lies on the surface. Show that the equation of the tangent plane at Q is $$6 x + 6 y + z = 65$$
The tangent plane at the point R has equation $6 x + 6 y + z = \lambda$ where $\lambda \neq 65$. Find the coordinates of R .

OCR MEI FP3 2015 June Q3

3 Fig. 3 shows an ellipse with parametric equations $x = a \cos \theta , y = b \sin \theta$, for $0 \leqslant \theta \leqslant 2 \pi$, where $0 < b \leqslant a$.
The curve meets the positive $x$-axis at A and the positive $y$-axis at B . \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{0e032f23-0549-4adc-bfae-59333108fab5-4_668_1255_477_404} \captionsetup{labelformat=empty} \caption{Fig. 3}

\end{figure}

Show that the radius of curvature at A is $\frac { b ^ { 2 } } { a }$ and find the corresponding centre of curvature.
Write down the radius of curvature and the centre of curvature at B .
Find the relationship between $a$ and $b$ if the radius of curvature at B is equal to the radius of curvature at A . What does this mean geometrically?
Show that the arc length from A to B can be expressed as $$b \int _ { 0 } ^ { \frac { \pi } { 2 } } \sqrt { 1 + \lambda ^ { 2 } \sin ^ { 2 } \theta } d \theta$$ where $\lambda ^ { 2 }$ is to be determined in terms of $a$ and $b$.
Evaluate this integral in the case $a = b$ and comment on your answer.
Find the cartesian equation of the evolute of the ellipse.

OCR MEI FP3 2015 June Q4

4 M is the set of all $2 \times 2$ matrices $\mathrm { m } ( a , b )$ where $a$ and $b$ are rational numbers and $$\mathrm { m } ( a , b ) = \left( \begin{array} { l l } a & b
0 & \frac { 1 } { a } \end{array} \right) , a \neq 0$$

Show that under matrix multiplication M is a group. You may assume associativity of matrix multiplication.
Determine whether the group is commutative. The set $\mathrm { N } _ { k }$ consists of all $2 \times 2$ matrices $\mathrm { m } ( k , b )$ where $k$ is a fixed positive integer and $b$ can take any integer value.
Prove that $\mathrm { N } _ { k }$ is closed under matrix multiplication if and only if $k = 1$. Now consider the set P consisting of the matrices $\mathrm { m } ( 1,0 ) , \mathrm { m } ( 1,1 ) , \mathrm { m } ( 1,2 )$ and $\mathrm { m } ( 1,3 )$. The elements of P are combined using matrix multiplication but with arithmetic carried out modulo 4 .
Show that $( \mathrm { m } ( 1,1 ) ) ^ { 2 } = \mathrm { m } ( 1,2 )$.
Construct the group combination table for P . The group R consists of the set $\{ e , a , b , c \}$ combined under the operation *. The identity element is $e$, and elements $a , b$ and $c$ are such that $$a ^ { * } a = b ^ { * } b = c ^ { * } c \quad \text { and } \quad a ^ { * } c = c ^ { * } a = b$$
Determine whether R is isomorphic to P . Option 5: Markov chains \section*{This question requires the use of a calculator with the ability to handle matrices.}

OCR MEI FP3 2015 June Q5

5 An inspector has three factories, A, B, C, to check. He spends each day in one of the factories. He chooses the factory to visit on a particular day according to the following rules.

If he is in A one day, then the next day he will never choose A but he is equally likely to choose B or C .
If he is in B one day, then the next day he is equally likely to choose $\mathrm { A } , \mathrm { B }$ or C .
If he is in C one day, then the next day he will never choose A but he is equally likely to choose B or C .
1. Write down the transition matrix, $\mathbf { P }$.
2. On Day 1 the inspector chooses A.
  (A) Find the probability that he will choose A on Day 4.
  (B) Find the probability that the factory he chooses on Day 7 is the same factory that he chose on Day 2.
3. Find the equilibrium probabilities and explain what they mean.

The inspector is not satisfied with the number of times he visits A so he changes the rules as follows.

If he is in A one day, then the next day he will choose $\mathrm { A } , \mathrm { B } , \mathrm { C }$, with probabilities $0.8,0.1,0.1$, respectively.
If he is in B or C one day, then the probabilities for choosing the factory the next day remain as before.
Write down the new transition matrix, $\mathbf { Q }$, and find the new equilibrium probabilities.
On a particular day, the inspector visits factory A. Find the expected number of consecutive further days on which he will visit factory A.

Still not satisfied, the inspector changes the rules as follows.

If he is in A one day, then the next day he will choose $\mathrm { A } , \mathrm { B } , \mathrm { C }$, with probabilities $1,0,0$, respectively.
If he is in B or C one day, then the probabilities for choosing the factory the next day remain as before.

The new transition matrix is $\mathbf { R }$.

On Day 15 he visits C . Find the first subsequent day for which the probability that he visits B is less than 0.1.

Show that in this situation there is an absorbing state, explaining what this means. \section*{END OF QUESTION PAPER}

OCR MEI D1 2007 January Q1

1 Each of the following symbols consists of boundaries enclosing regions.
(0) 1

OCR MEI D1 2007 January Q3

3
$\triangle$
5
(5) 7
(8)
(O) The symbol representing zero has three regions, the outside, that between the two boundaries and the inside. To classify the symbols a graph is produced for each one. The graph has a vertex for each region, with arcs connecting regions which share a boundary. Thus the graph for
\includegraphics[max width=\textwidth, alt={}, center]{a56f79f4-3b71-4ae3-a88b-5a88b0197433-2_110_81_900_767}
is
\includegraphics[max width=\textwidth, alt={}, center]{a56f79f4-3b71-4ae3-a88b-5a88b0197433-2_49_350_959_959}

Produce the graph for the symbol
Give two symbols each having the graph
\includegraphics[max width=\textwidth, alt={}, center]{a56f79f4-3b71-4ae3-a88b-5a88b0197433-2_52_199_1187_1030}
Produce the graph for the symbol .
Produce a single graph for the composite symbol
\includegraphics[max width=\textwidth, alt={}, center]{a56f79f4-3b71-4ae3-a88b-5a88b0197433-2_112_158_1398_1165}
Give the name of a connected graph with $n$ nodes and $n - 1 \operatorname { arcs }$. 2 The following algorithm is a version of bubble sort.
Step 1 Store the values to be sorted in locations $\mathrm { L } ( 1 ) , \mathrm { L } ( 2 ) , \ldots , \mathrm { L } ( \mathrm { n } )$ and set i to be the number, n , of values to be sorted. Step $2 \quad$ Set $\mathrm { j } = 1$.
Step 3 Compare the values in locations $\mathrm { L } ( \mathrm { j } )$ and $\mathrm { L } ( \mathrm { j } + 1 )$ and swap them if that $\mathrm { in } \mathrm { L } ( \mathrm { j } )$ is larger than that in $\mathrm { L } ( \mathrm { j } + 1 )$. Step 4 Add 1 to j.
Step 5 If j is less than i then go to step 3.
Step 5 Write out the current list, $\mathrm { L } ( 1 ) , \mathrm { L } ( 2 ) , \ldots , \mathrm { L } ( \mathrm { n } )$.
Step 6 Subtract 1 from i .
Step 7 If i is larger than 1 then go to step 2.
Step 8 Stop.
Apply this algorithm to sort the following list. $$\begin{array} { l l l l l } 109 & 32 & 3 & 523 & 58 \end{array} .$$ Count the number of comparisons and the number of swaps which you make in applying the algorithm.
Put the five values into the order which maximises the number of swaps made in applying the algorithm, and give that number.
Bubble sort has quadratic complexity. Using bubble sort it takes a computer 1.5 seconds to sort a list of 1000 values. Approximately how long would it take to sort a list of 100000 values? (Give your answer in hours and minutes.) 3 A bag contains five pieces of paper labelled A, B, C, D and E. One piece is drawn at random from the bag. If the piece is labelled with a vowel (A or E) then the process stops. Otherwise the piece of paper is replaced, the bag is shaken, and the process is repeated. You are to simulate this process to estimate the mean number of draws needed to get a vowel.
Show how to use single digit random numbers to simulate the process efficiently. You need to describe exactly how your simulation will work.
Use the random numbers in your answer book to run your simulation 5 times, recording your results.
From your results compute an estimate of the mean number of draws needed to get a vowel.
State how you could produce a more accurate estimate. Section B (48 marks)

OCR MEI D1 2007 January Q5

5
(5) 7
(8)
(O) The symbol representing zero has three regions, the outside, that between the two boundaries and the inside. To classify the symbols a graph is produced for each one. The graph has a vertex for each region, with arcs connecting regions which share a boundary. Thus the graph for
\includegraphics[max width=\textwidth, alt={}, center]{a56f79f4-3b71-4ae3-a88b-5a88b0197433-2_110_81_900_767}
is
\includegraphics[max width=\textwidth, alt={}, center]{a56f79f4-3b71-4ae3-a88b-5a88b0197433-2_49_350_959_959}

Produce the graph for the symbol
Give two symbols each having the graph
\includegraphics[max width=\textwidth, alt={}, center]{a56f79f4-3b71-4ae3-a88b-5a88b0197433-2_52_199_1187_1030}
Produce the graph for the symbol .
Produce a single graph for the composite symbol
\includegraphics[max width=\textwidth, alt={}, center]{a56f79f4-3b71-4ae3-a88b-5a88b0197433-2_112_158_1398_1165}
Give the name of a connected graph with $n$ nodes and $n - 1 \operatorname { arcs }$. 2 The following algorithm is a version of bubble sort.
Step 1 Store the values to be sorted in locations $\mathrm { L } ( 1 ) , \mathrm { L } ( 2 ) , \ldots , \mathrm { L } ( \mathrm { n } )$ and set i to be the number, n , of values to be sorted. Step $2 \quad$ Set $\mathrm { j } = 1$.
Step 3 Compare the values in locations $\mathrm { L } ( \mathrm { j } )$ and $\mathrm { L } ( \mathrm { j } + 1 )$ and swap them if that $\mathrm { in } \mathrm { L } ( \mathrm { j } )$ is larger than that in $\mathrm { L } ( \mathrm { j } + 1 )$. Step 4 Add 1 to j.
Step 5 If j is less than i then go to step 3.
Step 5 Write out the current list, $\mathrm { L } ( 1 ) , \mathrm { L } ( 2 ) , \ldots , \mathrm { L } ( \mathrm { n } )$.
Step 6 Subtract 1 from i .
Step 7 If i is larger than 1 then go to step 2.
Step 8 Stop.
Apply this algorithm to sort the following list. $$\begin{array} { l l l l l } 109 & 32 & 3 & 523 & 58 \end{array} .$$ Count the number of comparisons and the number of swaps which you make in applying the algorithm.
Put the five values into the order which maximises the number of swaps made in applying the algorithm, and give that number.
Bubble sort has quadratic complexity. Using bubble sort it takes a computer 1.5 seconds to sort a list of 1000 values. Approximately how long would it take to sort a list of 100000 values? (Give your answer in hours and minutes.) 3 A bag contains five pieces of paper labelled A, B, C, D and E. One piece is drawn at random from the bag. If the piece is labelled with a vowel (A or E) then the process stops. Otherwise the piece of paper is replaced, the bag is shaken, and the process is repeated. You are to simulate this process to estimate the mean number of draws needed to get a vowel.
Show how to use single digit random numbers to simulate the process efficiently. You need to describe exactly how your simulation will work.
Use the random numbers in your answer book to run your simulation 5 times, recording your results.
From your results compute an estimate of the mean number of draws needed to get a vowel.

State how you could produce a more accurate estimate. Section B (48 marks)
4 Cassi is managing the building of a house. The table shows the major activities that are involved, their durations and their precedences.

Activity		Duration (days)	Immediate predecessors
A	Build concrete frame	10	-
B	Lay bricks	7	A
C	Lay roof tiles	10	A
D	First fit electrics	5	B
E	First fit plumbing	4	B
F	Plastering	6	C, D, E
G	Second fit electrics	3	F
H	Second fit plumbing	2	F
I	Tiling	10	G, H
J	Fit sanitary ware	2	H
K	Fit windows and doors	5	I

Draw an activity-on-arc network to represent this information.
Find the early time and the late time for each event. Give the project duration and list the critical activities.
Calculate total and independent floats for each non-critical activity. Cassi's clients wish to take delivery in 42 days. Some durations can be reduced, at extra cost, to achieve this.
- The tiler will finish activity I in 9 days for an extra $\pounds 250$, or in 8 days for an extra $\pounds 500$.
- The bricklayer will cut his total of 7 days on activity B by up to 3 days at an extra cost of $\pounds 350$ per day.
- The electrician could be paid $\pounds 300$ more to cut a day off activity D, or $\pounds 600$ more to cut two days.
- What is the cheapest way in which Cassi can get the house built in 42 days?
5 Leone is designing her new garden. She wants to have at least $1000 \mathrm {~m} ^ { 2 }$, split between lawn and flower beds. Initial costs are $\pounds 0.80$ per $\mathrm { m } ^ { 2 }$ for lawn and $\pounds 0.40$ per $\mathrm { m } ^ { 2 }$ for flowerbeds. Leone's budget is $\pounds 500$.
Leone prefers flower beds to lawn, and she wants the area for flower beds to be at least twice the area for lawn. However, she wants to have at least $200 \mathrm {~m} ^ { 2 }$ of lawn. Maintenance costs each year are $\pounds 0.15$ per $\mathrm { m } ^ { 2 }$ for lawn and $\pounds 0.25$ per $\mathrm { m } ^ { 2 }$ for flower beds. Leone wants to minimize the maintenance costs of her garden.
Formulate Leone's problem as a linear programming problem.
Produce a graph to illustrate the inequalities.
Solve Leone's problem.
If Leone had more than $\pounds 500$ available initially, how much extra could she spend to minimize maintenance costs?

OCR MEI D1 2007 January Q7

7
(8)
(O) The symbol representing zero has three regions, the outside, that between the two boundaries and the inside. To classify the symbols a graph is produced for each one. The graph has a vertex for each region, with arcs connecting regions which share a boundary. Thus the graph for
\includegraphics[max width=\textwidth, alt={}, center]{a56f79f4-3b71-4ae3-a88b-5a88b0197433-2_110_81_900_767}
is
\includegraphics[max width=\textwidth, alt={}, center]{a56f79f4-3b71-4ae3-a88b-5a88b0197433-2_49_350_959_959}

Produce the graph for the symbol
Give two symbols each having the graph
\includegraphics[max width=\textwidth, alt={}, center]{a56f79f4-3b71-4ae3-a88b-5a88b0197433-2_52_199_1187_1030}
Produce the graph for the symbol .
Produce a single graph for the composite symbol
\includegraphics[max width=\textwidth, alt={}, center]{a56f79f4-3b71-4ae3-a88b-5a88b0197433-2_112_158_1398_1165}
Give the name of a connected graph with $n$ nodes and $n - 1 \operatorname { arcs }$. 2 The following algorithm is a version of bubble sort.
Step 1 Store the values to be sorted in locations $\mathrm { L } ( 1 ) , \mathrm { L } ( 2 ) , \ldots , \mathrm { L } ( \mathrm { n } )$ and set i to be the number, n , of values to be sorted. Step $2 \quad$ Set $\mathrm { j } = 1$.
Step 3 Compare the values in locations $\mathrm { L } ( \mathrm { j } )$ and $\mathrm { L } ( \mathrm { j } + 1 )$ and swap them if that $\mathrm { in } \mathrm { L } ( \mathrm { j } )$ is larger than that in $\mathrm { L } ( \mathrm { j } + 1 )$. Step 4 Add 1 to j.
Step 5 If j is less than i then go to step 3.
Step 5 Write out the current list, $\mathrm { L } ( 1 ) , \mathrm { L } ( 2 ) , \ldots , \mathrm { L } ( \mathrm { n } )$.
Step 6 Subtract 1 from i .
Step 7 If i is larger than 1 then go to step 2.
Step 8 Stop.
Apply this algorithm to sort the following list. $$\begin{array} { l l l l l } 109 & 32 & 3 & 523 & 58 \end{array} .$$ Count the number of comparisons and the number of swaps which you make in applying the algorithm.
Put the five values into the order which maximises the number of swaps made in applying the algorithm, and give that number.
Bubble sort has quadratic complexity. Using bubble sort it takes a computer 1.5 seconds to sort a list of 1000 values. Approximately how long would it take to sort a list of 100000 values? (Give your answer in hours and minutes.) 3 A bag contains five pieces of paper labelled A, B, C, D and E. One piece is drawn at random from the bag. If the piece is labelled with a vowel (A or E) then the process stops. Otherwise the piece of paper is replaced, the bag is shaken, and the process is repeated. You are to simulate this process to estimate the mean number of draws needed to get a vowel.
Show how to use single digit random numbers to simulate the process efficiently. You need to describe exactly how your simulation will work.
Use the random numbers in your answer book to run your simulation 5 times, recording your results.
From your results compute an estimate of the mean number of draws needed to get a vowel.

State how you could produce a more accurate estimate. Section B (48 marks)
4 Cassi is managing the building of a house. The table shows the major activities that are involved, their durations and their precedences.

Activity		Duration (days)	Immediate predecessors
A	Build concrete frame	10	-
B	Lay bricks	7	A
C	Lay roof tiles	10	A
D	First fit electrics	5	B
E	First fit plumbing	4	B
F	Plastering	6	C, D, E
G	Second fit electrics	3	F
H	Second fit plumbing	2	F
I	Tiling	10	G, H
J	Fit sanitary ware	2	H
K	Fit windows and doors	5	I

Draw an activity-on-arc network to represent this information.
Find the early time and the late time for each event. Give the project duration and list the critical activities.
Calculate total and independent floats for each non-critical activity. Cassi's clients wish to take delivery in 42 days. Some durations can be reduced, at extra cost, to achieve this.
- The tiler will finish activity I in 9 days for an extra $\pounds 250$, or in 8 days for an extra $\pounds 500$.
- The bricklayer will cut his total of 7 days on activity B by up to 3 days at an extra cost of $\pounds 350$ per day.
- The electrician could be paid $\pounds 300$ more to cut a day off activity D, or $\pounds 600$ more to cut two days.
- What is the cheapest way in which Cassi can get the house built in 42 days?
5 Leone is designing her new garden. She wants to have at least $1000 \mathrm {~m} ^ { 2 }$, split between lawn and flower beds. Initial costs are $\pounds 0.80$ per $\mathrm { m } ^ { 2 }$ for lawn and $\pounds 0.40$ per $\mathrm { m } ^ { 2 }$ for flowerbeds. Leone's budget is $\pounds 500$.
Leone prefers flower beds to lawn, and she wants the area for flower beds to be at least twice the area for lawn. However, she wants to have at least $200 \mathrm {~m} ^ { 2 }$ of lawn. Maintenance costs each year are $\pounds 0.15$ per $\mathrm { m } ^ { 2 }$ for lawn and $\pounds 0.25$ per $\mathrm { m } ^ { 2 }$ for flower beds. Leone wants to minimize the maintenance costs of her garden.
Formulate Leone's problem as a linear programming problem.
Produce a graph to illustrate the inequalities.
Solve Leone's problem.
If Leone had more than $\pounds 500$ available initially, how much extra could she spend to minimize maintenance costs? 6 In a factory a network of pipes connects 6 vats, A, B, C, D, E and F. Two separate connectors need to be chosen from the network The table shows the lengths of pipes (metres) connecting the 6 vats.
A B C D E F
A - 7 - - 12 -
B 7 - 5 3 6 6
C - 5 - 8 4 7
D - 3 8 - 1 5
E 12 6 4 1 - 7
F - 6 7 5 7 -
Use Kruskal's algorithm to find a minimum connector. Show the order in which you select pipes, draw your connector and give its total length.
Produce a new table excluding the pipes which you selected in part (i). Use the tabular form of Prim's algorithm to find a second minimum connector from this reduced set of pipes. Show your working, draw your connector and give its total length.
The factory manager prefers the following pair of connectors:
$\{ \mathrm { AB } , \mathrm { BC } , \mathrm { BD } , \mathrm { BE } , \mathrm { BF } \}$ and $\{ \mathrm { AE } , \mathrm { BF } , \mathrm { CE } , \mathrm { DE } , \mathrm { DF } \}$.
Give two possible reasons for this preference.

OCR MEI D1 2013 June Q1

1 The adjacency graph for a map has a vertex for each country. Two vertices are connected by an arc if the corresponding countries share a border.

Draw the adjacency graph for the following map of four countries. The graph is planar and you should draw it with no arcs crossing.
\includegraphics[max width=\textwidth, alt={}, center]{e528b905-7419-44b6-b700-4c04ad96c816-2_531_1486_561_292}
Number the regions of your planar graph, including the outside region. Regarding the graph as a map, draw its adjacency graph.
Repeat parts (i) and (ii) for the following map.
\includegraphics[max width=\textwidth, alt={}, center]{e528b905-7419-44b6-b700-4c04ad96c816-2_533_1484_1361_294}

OCR MEI D1 2013 June Q2

2 The instructions labelled 1 to 7 describe the steps of a sorting algorithm applied to a list of six numbers.
1 Let $i$ equal 1.
2 Repeat lines 3 to 7, stopping when $i$ becomes 6 .

OCR MEI D1 2013 June Q3

3 Let $j$ equal 1.

OCR MEI D1 2013 June Q4

4 Repeat lines 5 and 6 , until $j$ becomes $7 - i$.

OCR MEI D1 2013 June Q5

5 If the $j$ th number in the list is bigger than the $( j + 1 )$ th, then swap them.

OCR MEI D1 2013 June Q6

6 Let the new value of $j$ be $j + 1$.

Questions — OCR MEI (4301 questions)

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