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OCR MEI Paper 2 2018 June Q17
17
  1. Express \(\frac { \left( x ^ { 2 } - 8 x + 9 \right) } { ( x + 1 ) ( x - 2 ) ^ { 2 } }\) in partial fractions.
  2. Express \(y\) in terms of \(x\) given that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { y \left( x ^ { 2 } - 8 x + 9 \right) } { ( x + 1 ) ( x - 2 ) ^ { 2 } } \text { and } y = 16 \text { when } x = 3 .$$ \section*{END OF QUESTION PAPER}
OCR MEI Paper 2 2019 June Q1
1 Fig. 1 shows the probability distribution of the discrete random variable \(X\). \begin{table}[h]
\(x\)12345
\(\mathrm { P } ( X = x )\)0.20.1\(k\)\(2 k\)\(4 k\)
\captionsetup{labelformat=empty} \caption{Fig. 1}
\end{table}
  1. Find the value of \(k\).
  2. Find \(\mathrm { P } ( X \neq 4 )\).
OCR MEI Paper 2 2019 June Q2
2 Given that \(y = \left( x ^ { 2 } + 5 \right) ^ { 12 }\),
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
  2. Hence find \(\int 48 x \left( x ^ { 2 } + 5 \right) ^ { 11 } \mathrm {~d} x\).
OCR MEI Paper 2 2019 June Q3
3 Fig. 3 shows the time Lorraine spent in hours, \(t\), answering e-mails during the working day. The data were collected over a number of months. \begin{table}[h]
Time in hours,
\(t\)
\(0 \leqslant t < 1\)\(1 \leqslant t < 2\)\(2 \leqslant t < 3\)\(3 \leqslant t < 4\)\(4 \leqslant t < 6\)\(6 \leqslant t < 8\)
Number of
days
283642312412
\captionsetup{labelformat=empty} \caption{Fig. 3}
\end{table}
  1. Calculate an estimate of the mean time per day that Lorraine spent answering e-mails over this period.
  2. Explain why your answer to part (a) is an estimate. When Lorraine accepted her job, she was told that the mean time per day spent answering e-mails would not be more than 3 hours.
  3. Determine whether, according to the data in Fig. 3, it is possible that the mean time per day Lorraine spends answering e-mails is in fact more than 3 hours.
OCR MEI Paper 2 2019 June Q4
4 Fig. 4 shows the graph of \(y = \sqrt { 1 + x ^ { 3 } }\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{95eb3bcc-6d3c-4f7e-9b27-5e046ab57ec5-05_544_639_338_248} \captionsetup{labelformat=empty} \caption{Fig. 4}
\end{figure}
  1. Use the trapezium rule with \(h = 0.5\) to find an estimate of \(\int _ { - 1 } ^ { 0 } \sqrt { 1 + x ^ { 3 } } \mathrm {~d} x\), giving your answer correct to 6 decimal places.
  2. State whether your answer to part (a) is an under-estimate or an over-estimate, justifying your answer.
OCR MEI Paper 2 2019 June Q5
5 Fig. 5 shows the number of times that students at a sixth form college visited a recreational mathematics website during the first week of the summer term. \begin{table}[h]
Number of visits to website012345
Number of students2438171242
\captionsetup{labelformat=empty} \caption{Fig. 5}
\end{table}
  1. State the value of the mid-range of the data.
  2. Describe the shape of the distribution.
  3. State the value of the mode.
OCR MEI Paper 2 2019 June Q6
6 Find \(\int \frac { 32 } { x ^ { 5 } } \ln x \mathrm {~d} x\). Answer all the questions
Section B (78 marks)
OCR MEI Paper 2 2019 June Q7
7 The area of a sector of a circle is \(36.288 \mathrm {~cm} ^ { 2 }\). The angle of the sector is \(\theta\) radians and the radius of the circle is \(r \mathrm {~cm}\).
  1. Find an expression for \(\theta\) in terms of \(r\). The perimeter of the sector is 24.48 cm .
  2. Show that \(\theta = \frac { 24.48 } { r } - 2\).
  3. Find the possible values of \(r\).
OCR MEI Paper 2 2019 June Q8
8 A team called "The Educated Guess" enter a weekly quiz. If they win the quiz in a particular week, the probability that they will win the following week is 0.4 , but if they do not win, the probability that they will win the following week is 0.2 . In week 4 The Educated Guess won the quiz.
  1. Calculate the probability that The Educated Guess will win the quiz in week 6. Every week the same 20 quiz teams, each with 6 members, take part in a quiz. Every member of every team buys a raffle ticket. Five winning tickets are drawn randomly, without replacement. Alf, who is a member of one of the teams, takes part every week.
  2. Calculate the probability that, in a randomly chosen week, Alf wins a raffle prize.
  3. Find the smallest number of weeks after which it will be \(95 \%\) certain that Alf has won at least one raffle prize.
OCR MEI Paper 2 2019 June Q9
9 You are given that
\(\mathrm { f } ( x ) = 2 x + 3 \quad\) for \(x < 0 \quad\) and
\(\mathrm { g } ( x ) = x ^ { 2 } - 2 x + 1\) for \(x > 1\).
  1. Find \(\mathrm { gf } ( x )\), stating the domain.
  2. State the range of \(\mathrm { gf } ( x )\).
  3. Find (gf) \({ } ^ { - 1 } ( x )\).
OCR MEI Paper 2 2019 June Q10
2 marks
10 Club 65-80 Holidays fly jets between Liverpool and Magaluf. Over a long period of time records show that half of the flights from Liverpool to Magaluf take less than 153 minutes and \(5 \%\) of the flights take more than 183 minutes. An operations manager believes that flight times from Liverpool to Magaluf may be modelled by the Normal distribution.
  1. Use the information above to write down the mean time the operations manager will use in his Normal model for flight times from Liverpool to Magaluf.
  2. Use the information above to find the standard deviation the operations manager will use in his Normal model for flight times from Liverpool to Magaluf, giving your answer correct to 1 decimal place.
  3. Data is available for 452 flights. A flight time of under 2 hours was recorded in 16 of these flights. Use your answers to parts (a) and (b) to determine whether the model is consistent with this data. The operations manager suspects that the mean time for the journey from Magaluf to Liverpool is less than from Liverpool to Magaluf. He collects a random sample of 24 flight times from Magaluf to Liverpool. He finds that the mean flight time is 143.6 minutes.
  4. Use the Normal model used in part (c) to conduct a hypothesis test to determine whether there is evidence at the \(1 \%\) level to suggest that the mean flight time from Magaluf to Liverpool is less than the mean flight time from Liverpool to Magaluf.
    [0pt]
  5. Identify two ways in which the Normal model for flight times from Liverpool to Magaluf might be adapted to provide a better model for the flight times from Magaluf to Liverpool. [2]
OCR MEI Paper 2 2019 June Q11
11 Fig. 11 shows the graph of \(y = x ^ { 2 } - 4 x + x \ln x\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{95eb3bcc-6d3c-4f7e-9b27-5e046ab57ec5-08_697_463_338_246} \captionsetup{labelformat=empty} \caption{Fig. 11}
\end{figure}
  1. Show that the \(x\)-coordinate of the stationary point on the curve may be found from the equation \(2 x - 3 + \ln x = 0\).
  2. Use an iterative method to find the \(x\)-coordinate of the stationary point on the curve \(y = x ^ { 2 } - 4 x + x \ln x\), giving your answer correct to 4 decimal places.
OCR MEI Paper 2 2019 June Q12
12 The jaguar is a species of big cat native to South America. Records show that 6\% of jaguars are born with black coats. Jaguars with black coats are known as black panthers. Due to deforestation a population of jaguars has become isolated in part of the Amazon basin. Researchers believe that the percentage of black panthers may not be \(6 \%\) in this population.
  1. Find the minimum sample size needed to conduct a two-tailed test to determine whether there is any evidence at the \(5 \%\) level to suggest that the percentage of black panthers is not \(6 \%\). A research team identifies 70 possible sites for monitoring the jaguars remotely. 30 of these sites are randomly selected and cameras are installed. 83 different jaguars are filmed during the evidence gathering period. The team finds that 10 of the jaguars are black panthers.
  2. Conduct a hypothesis test to determine whether the information gathered by the research team provides any evidence at the \(5 \%\) level to suggest that the percentage of black panthers in this population is not \(6 \%\).
OCR MEI Paper 2 2019 June Q13
13 The population of Melchester is 185207. During a nationwide flu epidemic the number of new cases in Melchester are recorded each day. The results from the first three days are shown in Fig. 13. \begin{table}[h]
Day123
Number of new cases82472
\captionsetup{labelformat=empty} \caption{Fig. 13}
\end{table} A doctor notices that the numbers of new cases on successive days are in geometric progression.
  1. Find the common ratio for this geometric progression. The doctor uses this geometric progression to model the number of new cases of flu in Melchester.
  2. According to the model, how many new cases will there be on day 5?
  3. Find a formula for the total number of cases from day 1 to day \(n\) inclusive according to this model, simplifying your answer.
  4. Determine the maximum number of days for which the model could be viable in Melchester.
  5. State, with a reason, whether it is likely that the model will be viable for the number of days found in part (d).
OCR MEI Paper 2 2019 June Q14
14 The pre-release material includes data concerning crude death rates in different countries of the world. Fig. 14.1 shows some information concerning crude death rates in countries in Europe and in Africa. \begin{table}[h]
EuropeAfrica
\(n\)4856
minimum6.283.58
lower quartile8.507.31
median9.538.71
upper quartile11.4111.93
maximum14.4614.89
\captionsetup{labelformat=empty} \caption{Fig. 14.1}
\end{table}
  1. Use your knowledge of the large data set to suggest a reason why the statistics in Fig. 14.1 refer to only 48 of the 51 European countries.
  2. Use the information in Fig. 14.1 to show that there are no outliers in either data set. The crude death rate in Libya is recorded as 3.58 and the population of Libya is recorded as 6411776.
  3. Calculate an estimate of the number of deaths in Libya in a year. The median age in Germany is 46.5 and the crude death rate is 11.42. The median age in Cyprus is 36.1 and the crude death rate is 6.62 .
  4. Explain why a country like Germany, with a higher median age than Cyprus, might also be expected to have a higher crude death rate than Cyprus. Fig. 14.2 shows a scatter diagram of median age against crude death rate for countries in Africa and Fig. 14.3 shows a scatter diagram of median age against crude death rate for countries in Europe. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{95eb3bcc-6d3c-4f7e-9b27-5e046ab57ec5-10_678_1221_1975_248} \captionsetup{labelformat=empty} \caption{Fig. 14.2}
    \end{figure} \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{95eb3bcc-6d3c-4f7e-9b27-5e046ab57ec5-11_588_1248_223_228} \captionsetup{labelformat=empty} \caption{Fig. 14.3}
    \end{figure} The rank correlation coefficient for the data shown in Fig. 14.2 is - 0.281206 .
    The rank correlation coefficient for the data shown in Fig. 14.3 is 0.335215 .
  5. Compare and contrast what may be inferred about the relationship between median age and crude death rate in countries in Africa and in countries in Europe.
OCR MEI Paper 2 2022 June Q1
1 Express \(\cos \theta + \sqrt { 3 } \sin \theta\) in the form \(R \cos ( \theta - \alpha )\), where \(R\) and \(\alpha\) are exact values to be determined.
OCR MEI Paper 2 2022 June Q2
2 Find the sum of the infinite series \(50 + 25 + 12.5 + 6.25 + \ldots\).
OCR MEI Paper 2 2022 June Q3
3
  1. On the axes in the Printed Answer Booklet, sketch the curve with equation \(\mathrm { y } = 3 \times 0.4 ^ { \mathrm { x } }\).
  2. Given that \(3 \times 0.4 ^ { x } = 0.8\), determine the value of \(x\) correct to 3 significant figures.
OCR MEI Paper 2 2022 June Q4
4 A survey of university students revealed that
  • \(31 \%\) have a part-time job but do not play competitive sport.
  • \(23 \%\) play competitive sport but do not have a part-time job.
  • \(22 \%\) do not play competitive sport and do not have a part-time job.
    1. Show this information on a Venn diagram.
A student is selected at random.
  • Determine the probability that the student plays competitive sport and has a part-time job.
    •
    OCR MEI Paper 2 2022 June Q5
    5 Tom conjectures that if \(n\) is an odd number greater than 1 , then \(2 ^ { n } - 1\) is prime.
    Find a counter example to disprove Tom's conjecture.
    \(6 X\) is a continuous random variable such that \(X \sim \mathrm {~N} \left( \mu , \sigma ^ { 2 } \right)\).
    On the sketch of this Normal distribution in the Printed Answer Booklet, shade the area bounded by the curve, the \(x\)-axis and the lines \(x = \mu \pm \sigma\).
    OCR MEI Paper 2 2022 June Q7
    2 marks
    7 Kareem bought some tomatoes. He recorded the mass of each tomato and displayed the results in a histogram, which is shown below.
    \includegraphics[max width=\textwidth, alt={}, center]{57007d39-abb0-475e-9ed8-03021fa1273b-05_1273_1849_363_109} Determine how many tomatoes Kareem bought.
    [0pt] [2] Answer all the questions.
    Section B (77 marks)
    OCR MEI Paper 2 2022 June Q8
    8 Ali conducted an investigation into the distances ridden by those members of a cycling club who rode at least 120 km in a training week. She grouped all the distances into intervals of length 10 km and then constructed a cumulative frequency diagram, which is shown below.
    \includegraphics[max width=\textwidth, alt={}, center]{57007d39-abb0-475e-9ed8-03021fa1273b-06_1086_1627_587_233}
    1. Explain whether the data Ali used is a sample or a population. The club is taking part in a competition. Eight team members and one reserve are to be selected. The club captain decides that the team members should be those cyclists who rode the furthest during the training week, and that the reserve should be the cyclist who rode the next furthest.
    2. Use the graph to estimate the shortest distance cycled by a team member. The captain's best friend rode 156 km in the training week and was selected as reserve. Ali complained that this was unjustifiable.
    3. Explain whether there is sufficient evidence in the diagram to support Ali's complaint.
    OCR MEI Paper 2 2022 June Q9
    9 At the beginning of the academic year, all the pupils in year 12 at a college take part in an assessment. Summary statistics for the marks obtained by the 2021 cohort are given below.
    \(n = 205 \sum x = 23042 \sum x ^ { 2 } = 2591716\) Marks may only be whole numbers, but the Head of Mathematics believes that the distribution of marks may be modelled by a Normal distribution.
    1. Calculate
      • The mean mark
      • The variance of the marks
      • Use your answers to part (a) to write down a possible Normal model for the distribution of marks.
      One candidate in the cohort scored less than 105.
    2. Determine whether the model found in part (b) is consistent with this information.
    3. Use the model to calculate an estimate of the number of candidates who scored 115 marks.
    OCR MEI Paper 2 2022 June Q10
    10 The parametric equations of a curve are \(x = 2 + 5 \cos \theta\) and \(y = 1 + 5 \sin \theta\), where \(0 \leqslant \theta \leqslant 2 \pi\).
    1. Determine the cartesian equation of the curve.
    2. Hence or otherwise, find the equation of the tangent to the curve at the point ( \(5 , - 3\) ), giving your answer in the form \(\mathrm { ax } + \mathrm { by } + \mathrm { c } = 0\), where \(a\), \(b\) and \(c\) are integers to be determined.
    OCR MEI Paper 2 2022 June Q11
    11 A die in the form of a dodecahedron has its faces numbered from 1 to 12 . The die is biased so that the probability that a score of 12 is achieved is different from any other score. The probability distribution of \(X\), the score on the die, is given in the table in terms of \(p\) and \(k\), where \(0 < p < 1\) and \(k\) is a positive integer.
    \(x\)123456789101112
    \(\mathrm { P } ( X = x )\)\(p\)\(p\)\(p\)\(p\)\(p\)\(p\)\(p\)\(p\)\(p\)\(p\)\(p\)\(k p\)
    Sam rolls the die 30 times, Leo rolls the die 60 times and Nina rolls the die 120 times. They each plot their scores on bar line graphs.
    1. Explain whose graph is most likely to give the best representation of the theoretical probability distribution for the score on the die.
    2. Find \(p\) in terms of \(k\).
    3. Determine, in terms of \(k\), the expected number of times Nina rolls a 12 .
    4. Given that Nina rolls a 12 on 32 occasions, calculate an estimate of the value of \(k\). Nina rolls the die a further 30 times.
    5. Use your answer to part (d) to calculate an estimate for the probability that she obtains a 12 exactly 8 times in these 30 rolls.