Questions — OCR MEI (4455 questions)

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OCR MEI C1 Q1
2 marks Easy -2.0
1
  1. Statement P is \(a + b = 4\).
    Statement Q is \(\quad a = 1\) and \(b = 3\).
    Which one of the following is correct? $$\mathrm { P } \Rightarrow \mathrm { Q } , \quad \mathrm { P } \Leftrightarrow \mathrm { Q } , \quad \mathrm { P } \Leftarrow \mathrm { Q }$$
  2. Statement R is \(\quad x = 2\). Statement S is \(\quad x ^ { 2 } = 4\). Which one of the following is correct? $$R \Rightarrow S , \quad R \Leftrightarrow S , \quad R \Leftarrow S$$
OCR MEI C1 Q2
3 marks Easy -1.2
2 Find the equation of the straight line which is parallel to the line \(y = 3 x + 5\) and which goes through the point \(( 2,12 )\).
OCR MEI C1 Q3
3 marks Moderate -0.8
3 Find the term which has the highest coefficient in the expansion of \(( 1 + x ) ^ { 8 }\).
OCR MEI C1 Q4
3 marks Easy -1.8
4 The surface area of the surface of a cylinder is given by the formula $$A = 2 \pi r ( r + h )$$ Rearrange this formula so that \(h\) is the subject.
OCR MEI C1 Q5
3 marks Easy -1.3
5 Solve the following equations.
  1. \(\quad 2 ^ { x } = \frac { 1 } { 8 }\).
  2. \(\quad x ^ { - \frac { 1 } { 2 } } = \frac { 1 } { 4 }\)
OCR MEI C1 Q6
3 marks Easy -1.2
6 Find the positive integer values of \(x\) for which $$\frac { 1 } { 2 } ( 26 - 2 x ) \geq 2 ( 3 + x )$$
OCR MEI C1 Q7
5 marks Moderate -0.3
7 The remainder when \(x ^ { 3 } - 2 x + 4\) is divided by ( \(x - 2\) ) is twice the remainder when \(x ^ { 2 } + x + k\) is divided by ( \(x + 1\) ).
Find the value of \(k\).
OCR MEI C1 Q8
5 marks Easy -1.2
8 Find the values of \(a\) and \(b\) for which \(\frac { 4 } { ( 2 \sqrt { 3 } - 1 ) } = a + b \sqrt { 3 }\).
OCR MEI C1 Q9
4 marks Moderate -0.8
9 Find the coordinates of the points where the curve \(y = x ^ { 2 } - 2 x - 8\) meets the line \(y = x + 2\).
OCR MEI C1 Q10
5 marks Easy -1.2
10 The diagram shows the graph of \(y = \mathrm { f } ( x )\). \includegraphics[max width=\textwidth, alt={}, center]{4c556b8e-1a19-4480-bf2a-0ef9e67f98b4-3_507_1085_933_383} A is the minimum point of the curve at \(( 3 , - 4 )\) and B is the point \(( 5,0 )\).
On separate diagrams on graph paper, draw the graphs of the following. In each case give the coordinates of the images of the points A and B .
  1. \(\quad y = \mathrm { f } ( x ) + 2\),
  2. \(y = \mathrm { f } ( x + 2 )\).
OCR MEI C1 Q11
12 marks Moderate -0.3
11 Fig. 11 shows the graph of \(y = a x ^ { 2 } + b x + c\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{4c556b8e-1a19-4480-bf2a-0ef9e67f98b4-4_572_1509_465_285} \captionsetup{labelformat=empty} \caption{Fig. 11}
\end{figure}
  1. Explain why a must be negative.
  2. State two factors of \(y = a x ^ { 2 } + b x + c\).
  3. Hence, or otherwise, find the values of \(a , b\) and \(c\). Another function is given by \(y = x ^ { 2 } - 4 x + 10\).
  4. Write this in completed square form.
  5. Explain why the graphs of these two functions never meet.
OCR MEI C1 Q12
12 marks Standard +0.3
12 The function \(\mathrm { f } ( x )\) is given by \(\mathrm { f } ( x ) = x ^ { 3 } + 6 x ^ { 2 } + 5 x - 12\).
  1. Show that \(( x + 3 )\) is a factor of \(\mathrm { f } ( x )\).
  2. Find the other factors of \(\mathrm { f } ( x )\).
  3. State the coordinates where the graph of \(y = \mathrm { f } ( x )\) cuts the \(x\) axis. Hence sketch the graph of \(y = \mathrm { f } ( x )\).
  4. On the same graph sketch also \(y = x ( x - 1 ) ( x - 2 )\) Label the two points of intersection of the two curves A and B .
  5. By equating the two curves, show that the \(x\) coordinates of A and B satisfy the equation \(3 x ^ { 2 } + x - 4 = 0\).
    Solve this equation to find the \(x\)-coordinates of A and B .
OCR MEI C1 Q13
12 marks Standard +0.3
13 In Fig.13, XP and XQ are the perpendicular bisectors of AC and BC respectively. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{4c556b8e-1a19-4480-bf2a-0ef9e67f98b4-5_409_768_383_604} \captionsetup{labelformat=empty} \caption{Fig. 13}
\end{figure}
  1. Find the coordinates of X .
  2. Hence show that \(\mathrm { AX } = \mathrm { BX } = \mathrm { CX }\).
  3. The circumcircle of a triangle is the circle which passes through the vertices of the triangle.
    Write down the equation of the circumcircle of the triangle ABC .
  4. Find the coordinates of the points where the circle cuts the \(x\) axis.
OCR MEI S2 2006 January Q1
18 marks Moderate -0.8
1 A roller-coaster ride has a safety system to detect faults on the track.
  1. State conditions for a Poisson distribution to be a suitable model for the number of faults occurring on a randomly selected day. Faults are detected at an average rate of 0.15 per day. You may assume that a Poisson distribution is a suitable model.
  2. Find the probability that on a randomly chosen day there are
    (A) no faults,
    (B) at least 2 faults.
  3. Find the probability that, in a randomly chosen period of 30 days, there are at most 3 faults. There is also a separate safety system to detect faults on the roller-coaster train itself. Faults are detected by this system at an average rate of 0.05 per day, independently of the faults detected on the track. You may assume that a Poisson distribution is also suitable for modelling the number of faults detected on the train.
  4. State the distribution of the total number of faults detected by the two systems in a period of 10 days. Find the probability that a total of 5 faults is detected in a period of 10 days.
    [0pt]
  5. The roller-coaster is operational for 200 days each year. Use a suitable approximating distribution to find the probability that a total of at least 50 faults is detected in 200 days. [5]
OCR MEI S2 2006 January Q2
18 marks Standard +0.3
2 The drug EPO (erythropoetin) is taken by some athletes to improve their performance. This drug is in fact banned and blood samples taken from athletes are tested to measure their 'hematocrit level'. If the level is over 50 it is considered that the athlete is likely to have taken EPO and the result is described as 'positive'. The measured hematocrit level of each athlete varies over time, even if EPO has not been taken.
  1. For each athlete in a large population of innocent athletes, the variation in measured hematocrit level is described by the Normal distribution with mean 42.0 and standard deviation 3.0.
    (A) Show that the probability that such an athlete tests positive for EPO in a randomly chosen test is 0.0038 .
    (B) Find the probability that such an athlete tests positive on at least 1 of the 7 occasions during the year when hematocrit level is measured. (These occasions are spread at random through the year and all test results are assumed to be independent.)
    (C) It is standard policy to apply a penalty after testing positive. Comment briefly on this policy in the light of your answer to part (i)(B).
  2. Suppose that 1000 tests are carried out on innocent athletes whose variation in measured hematocrit level is as described in part (i). It may be assumed that the probability of a positive result in each test is 0.0038 , independently of all other test results.
    (A) State the exact distribution of the number of positive tests.
    (B) Use a suitable approximating distribution to find the probability that at least 10 tests are positive.
  3. Because of genetic factors, a particular innocent athlete has an abnormally high natural hematocrit level. This athlete's measured level is Normally distributed with mean 48.0 and standard deviation 2.0. The usual limit of 50 for a positive test is to be altered for this athlete to a higher value \(h\). Find the value of \(h\) for which this athlete would test positive on average just once in 200 occasions.
OCR MEI S2 2006 January Q3
18 marks Standard +0.3
3 A researcher is investigating the relationship between temperature and levels of the air pollutant nitrous oxide at a particular site. The researcher believes that there will be a positive correlation between the daily maximum temperature, \(x\), and nitrous oxide level, \(y\). Data are collected for 10 randomly selected days. The data, measured in suitable units, are given in the table and illustrated on the scatter diagram.
\(x\)13.317.216.918.718.419.323.115.020.614.4
\(y\)911142643255215107
[diagram]
  1. Calculate the value of Spearman's rank correlation coefficient for these data.
  2. Perform a hypothesis test at the \(5 \%\) level to check the researcher's belief, stating your hypotheses clearly.
  3. It is suggested that it would be preferable to carry out a test based on the product moment correlation coefficient. State the distributional assumption required for such a test to be valid. Explain how a scatter diagram can be used to check whether the distributional assumption is likely to be valid and comment on the validity in this case.
  4. A statistician investigates data over a much longer period and finds that the assumptions for the use of the product moment correlation coefficient are in fact valid. Give the critical region for the test at the \(1 \%\) level, based on a sample of 60 days.
  5. In a different research project, into the correlation between daily temperature and ozone pollution levels, a positive correlation is found. It is argued that this shows that high temperatures cause increased ozone levels. Comment on this claim.
OCR MEI S2 2006 January Q4
18 marks Standard +0.3
4 The table summarises the usual method of travelling to school for 200 randomly selected pupils from primary and secondary schools in a city.
PrimarySecondary
\multirow{3}{*}{
Method of
travel
}		Bus2149
\cline { 2 - 4 }Car6515
\cline { 2 - 4 }Cycle or Walk3416
  1. Write down null and alternative hypotheses for a test to examine whether there is any association between method of travel and type of school.
  2. Calculate the expected frequency for primary school bus users. Calculate also the corresponding contribution to the test statistic for the usual \(\chi ^ { 2 }\) test.
  3. Given that the value of the test statistic for the usual \(\chi ^ { 2 }\) test is 42.64 , carry out the test at the \(5 \%\) level of significance, stating your conclusion clearly. The mean travel time for pupils who travel by bus is known to be 18.3 minutes. A survey is carried out to determine whether the mean travel time to school by car is different from 18.3 minutes. In the survey, 20 pupils who travel by car are selected at random. Their mean travel time is found to be 22.4 minutes.
  4. Assuming that car travel times are Normally distributed with standard deviation 8.0 minutes, carry out a test at the \(10 \%\) level, stating your hypotheses and conclusion clearly.
  5. Comment on the suggestion that pupils should use a bus if they want to get to school quickly.
OCR MEI S2 2008 January Q1
18 marks Moderate -0.3
1 A biology student is carrying out an experiment to study the effect of a hormone on the growth of plant shoots. The student applies the hormone at various concentrations to a random sample of twelve shoots and measures the growth of each shoot. The data are illustrated on the scatter diagram below, together with the summary statistics for these data. The variables \(x\) and \(y\), measured in suitable units, represent concentration and growth respectively. \includegraphics[max width=\textwidth, alt={}, center]{20fc4222-95c6-4b59-8e89-913dd988eb44-2_693_897_534_625} $$n = 12 , \Sigma x = 30 , \Sigma y = 967.6 , \Sigma x ^ { 2 } = 90 , \Sigma y ^ { 2 } = 78926 , \Sigma x y = 2530.3 .$$
  1. State which of the two variables \(x\) and \(y\) is the independent variable and which is the dependent variable. Briefly explain your answers.
  2. Calculate the equation of the regression line of \(y\) on \(x\).
  3. Use the equation of the regression line to calculate estimates of shoot growth for concentrations of
    (A) 1.2,
    (B) 4.3. Comment on the reliability of each of these estimates.
  4. Calculate the value of the residual for the data point where \(x = 3\) and \(y = 80\).
  5. In further experiments, the student finds that using concentration \(x = 6\) results in shoot growths of around \(y = 20\). In the light of all the available information, what can be said about the relationship between \(x\) and \(y\) ?
OCR MEI S2 2008 January Q2
18 marks Standard +0.3
2 A large hotel has 90 bedrooms. Sometimes a guest makes a booking for a room, but then does not arrive. This is called a 'no-show'. On average \(10 \%\) of bookings are no-shows. The hotel manager accepts up to 94 bookings before saying that the hotel is full. If at least 4 of these bookings are no-shows then there will be enough rooms for all of the guests. 94 bookings have been made for each night in August. You should assume that all bookings are independent.
  1. State the distribution of the number of no-shows on one night in August.
  2. State the conditions under which the use of a Poisson distribution is appropriate as an approximation to a binomial distribution.
  3. Use a Poisson approximating distribution to find the probability that, on one night in August,
    (A) there are exactly 4 no-shows,
    (B) there are enough rooms for all of the guests who do arrive.
  4. Find the probability that, on all of the 31 nights in August, there are enough rooms for all of the guests who arrive.
  5. (A) In August there are \(31 \times 94 = 2914\) bookings altogether. State the exact distribution of the total number of no-shows during August.
    (B) Use a suitable approximating distribution to find the probability that there are at most 300 no-shows altogether during August.
OCR MEI S2 2008 January Q3
17 marks Standard +0.3
3 In a large population, the diastolic blood pressure (DBP) of 5-year-old children is Normally distributed with mean 56 and standard deviation 6.5.
  1. Find the probability that the DBP of a randomly selected 5-year-old child is between 52.5 and 57.5. The DBP of young adults is Normally distributed with mean 68 and standard deviation 10.
  2. A 5-year-old child and a young adult are selected at random. Find the probability that the DBP of one of them is over 62 and the other is under 62.
  3. Sketch both distributions on a single diagram.
  4. For another age group, the DBP is Normally distributed with mean 82. The DBP of \(12 \%\) of people in this age group is below 62. Find the standard deviation for this age group.
OCR MEI S2 2008 January Q4
19 marks Standard +0.3
4
  1. A researcher believes that there may be some association between a student's sex and choice of certain subjects at A-level. A random sample of 250 A -level students is selected. The table below shows, for each sex, how many study either or both of the two subjects, Mathematics and English.
    Mathematics onlyEnglish onlyBothNeitherRow totals
    Male381963295
    Female4255949155
    Column totals80741581250
    Carry out a test at the \(5 \%\) significance level to examine whether there is any association between a student's sex and choice of subjects. State carefully your null and alternative hypotheses. Your working should include a table showing the contributions of each cell to the test statistic. [12]
  2. Over a long period it has been determined that the mean score of students in a particular English module is 67.4 and the standard deviation is 8.9. A new teaching method is introduced with the aim of improving the results. A random sample of 12 students taught by the new method is selected. Their mean score is found to be 68.3. Carry out a test at the \(10 \%\) level to investigate whether the new method appears to have been successful. State carefully your null and alternative hypotheses. You should assume that the scores are Normally distributed and that the standard deviation is unchanged.
OCR MEI S2 2005 June Q1
19 marks Standard +0.3
1 A student is collecting data on traffic arriving at a motorway service station during weekday lunchtimes. The random variable \(X\) denotes the number of cars arriving in a randomly chosen period of ten seconds.
  1. State two assumptions necessary if a Poisson distribution is to provide a suitable model for the distribution of \(X\). Comment briefly on whether these assumptions are likely to be valid. The student counts the number of arrivals, \(x\), in each of 100 ten-second periods. The data are shown in the table below.
    \(x\)012345\(> 5\)
    Frequency, \(f\)18392012830
  2. Show that the sample mean is 1.62 and calculate the sample variance.
  3. Do your calculations in part (ii) support the suggestion that a Poisson distribution is a suitable model for the distribution of \(X\) ? Explain your answer. For the remainder of this question you should assume that \(X\) may be modelled by a Poisson distribution with mean 1.62 .
  4. Find \(\mathrm { P } ( X = 2 )\). Comment on your answer in relation to the data in the table.
  5. Find the probability that at least ten cars arrive in a period of 50 seconds during weekday lunchtimes.
  6. Use a suitable approximating distribution to find the probability that no more than 550 cars arrive in a randomly chosen period of one hour during weekday lunchtimes.
OCR MEI S2 2005 June Q2
18 marks Standard +0.3
2 The fuel economy of a car varies from day to day according to weather and driving conditions. Fuel economy is measured in miles per gallon (mpg). The fuel economy of a particular petrol-fuelled type of car is known to be Normally distributed with mean 38.5 mpg and standard deviation 4.0 mpg .
  1. Find the probability that on a randomly selected day the fuel economy of a car of this type will be above 45.0 mpg .
  2. The manufacturer wishes to quote a fuel economy figure which will be exceeded on \(90 \%\) of days. What figure should be quoted? The daily fuel economy of a similar type of car which is diesel-fuelled is known to be Normally distributed with mean 51.2 mpg and unknown standard deviation \(\sigma \mathrm { mpg }\).
  3. Given that on 75\% of days the fuel economy of this type of car is below 55.0 mpg , show that \(\sigma = 5.63\).
  4. Draw a sketch to illustrate both distributions on a single diagram.
  5. Find the probability that the fuel economy of either the petrol or the diesel model (or both) will be above 45.0 mpg on a randomly selected day. You may assume that the fuel economies of the two models are independent.
OCR MEI S2 2005 June Q3
17 marks Standard +0.3
3 In a triathlon, competitors have to swim 600 metres, cycle 40 kilometres and run 10 kilometres. To improve her strength, a triathlete undertakes a training programme in which she carries weights in a rucksack whilst running. She runs a specific course and notes the total time taken for each run. Her coach is investigating the relationship between time taken and weight carried. The times taken with eight different weights are illustrated on the scatter diagram below, together with the summary statistics for these data. The variables \(x\) and \(y\) represent weight carried in kilograms and time taken in minutes respectively. \includegraphics[max width=\textwidth, alt={}, center]{be463718-caf7-4bc8-b838-143ab4681d6e-4_627_1536_630_281} Summary statistics: \(n = 8 , \Sigma x = 36 , \Sigma y = 214.8 , \Sigma x ^ { 2 } = 204 , \Sigma y ^ { 2 } = 5775.28 , \Sigma x y = 983.6\).
  1. Calculate the equation of the regression line of \(y\) on \(x\). On one of the eight runs, the triathlete was carrying 4 kilograms and took 27.5 minutes. On this run she was delayed when she tripped and fell over.
  2. Calculate the value of the residual for this weight.
  3. The coach decides to recalculate the equation of the regression line without the data for this run. Would it be preferable to use this recalculated equation or the equation found in part (i) to estimate the delay when the triathlete tripped and fell over? Explain your answer. The triathlete's coach claims that there is positive correlation between cycling and swimming times in triathlons. The product moment correlation coefficient of the times of twenty randomly selected competitors in these two sections is 0.209 .
  4. Carry out a hypothesis test at the \(5 \%\) level to examine the coach's claim, explaining your conclusions clearly.
  5. What distributional assumption is necessary for this test to be valid? How can you use a scatter diagram to decide whether this assumption is likely to be true?
OCR MEI S2 2005 June Q9
Easy -1.2
9 JUNE 2005
Morning
1 hour 30 minutes
Additional materials:
Answer booklet
Graph paper
MEI Examination Formulae and Tables (MF2) TIME 1 hour 30 minutes
  • Write your name, centre number and candidate number in the spaces provided on the answer booklet.
  • Answer all the questions.
  • You are permitted to use a graphical calculator in this paper.
  • The number of marks is given in brackets [ ] at the end of each question or part question.
  • You are advised that an answer may receive no marks unless you show sufficient detail of the working to indicate that a correct method is being used.
  • Final answers should be given to a degree of accuracy appropriate to the context.
  • The total number of marks for this paper is 72.
1 A student is collecting data on traffic arriving at a motorway service station during weekday lunchtimes. The random variable \(X\) denotes the number of cars arriving in a randomly chosen period of ten seconds.
  1. State two assumptions necessary if a Poisson distribution is to provide a suitable model for the distribution of \(X\). Comment briefly on whether these assumptions are likely to be valid. The student counts the number of arrivals, \(x\), in each of 100 ten-second periods. The data are shown in the table below. Carry out a test at the \(5 \%\) level of significance to examine whether there is any association between type of customer and type of drink. State carefully your null and alternative hypotheses.