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OCR MEI AS Paper 2 2021 November Q9
5 marks Moderate -0.5
9 Arun, Beth and Charlie are investigating whether there is any association between death rate per 1000 and physician density per 1000. They each collect a random sample of size 10. Arun’s sample is shown in Fig.9.1. \begin{table}[h]
death rate per 1000physician density per 1000
Canberra7.23.62
Dhaka5.30.49
Brasilia6.82.23
Yaounde9.30.08
Zagreb12.53.08
Tehran5.41.16
Rome10.74.14
Tripoli3.82.09
Oslo7.94.51
Abuja9.70.35
\captionsetup{labelformat=empty} \caption{Fig. 9.1}
\end{table}
  1. Explain whether or not Arun collected his data from the pre-release material, or whether it is not possible to say. Beth and Charlie collected their samples from the pre-release material. Each of them drew a scatter diagram for their samples. The samples and scatter diagrams are shown in Figs. 9.2 and 9.3.
    Beth's sampledeath rate per 1000physician density per 1000
    Sudan6.70.41
    Cambodia7.40.17
    Gabon6.20.36
    Seychelles70.95
    Mexico5.42.25
    Kuwait2.32.58
    Haiti7.50.23
    Maldives41.04
    Nauru5.91.24
    Jordan3.42.34
    \includegraphics[max width=\textwidth, alt={}]{2b9ce212-84e2-4817-be94-98e2adff12a3-08_545_1024_340_918}
    \begin{table}[h]
    Charlie's sampledeath rate per 1000physician density per 1000
    Vanuata40.17
    Solomon Islands3.80.2
    N. Mariana Islands4.90.36
    Nauru5.91.24
    United Kingdom9.42.81
    Portugal10.63.34
    North Macedonia9.62.87
    Faroe Islands8.82.62
    Bulgaria14.53.99
    St. Kitts and Nevis7.22.52
    \captionsetup{labelformat=empty} \caption{Fig. 9.3}
    \end{table} \begin{figure}[h]
    \captionsetup{labelformat=empty} \caption{Fig. 9.2} \includegraphics[alt={},max width=\textwidth]{2b9ce212-84e2-4817-be94-98e2adff12a3-08_572_899_1400_1041}
    \end{figure} Arun states that Charlie's sample and Beth's sample cannot both be random for the following reasons.
    • Both samples include Nauru - there should not be any common values.
    • Beth's diagram suggests a negative association between death rate and physician density, whereas Charlie's diagram suggests a positive association. If both samples are random the same relationship would be suggested.
    • - Explain whether Arun’s reasons are valid.
    • State whether or not Arun is correct, or whether it is not possible to say.
    Kofi collects a sample of 10 African countries and 10 European countries. The scatter diagram for his results is shown in Fig. 9.4. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{2b9ce212-84e2-4817-be94-98e2adff12a3-09_485_903_902_260} \captionsetup{labelformat=empty} \caption{Fig. 9.4}
    \end{figure}
  2. On the copy of Fig. 9.4 in the Printed Answer Booklet, use your knowledge of the pre-release material to identify the points representing the 10 European countries, justifying your choice.
OCR MEI AS Paper 2 2021 November Q11
6 marks Easy -1.2
11 James is investigating the amount of time retired people spend each day using social media. He collects a sample by advertising in a local newspaper for people to complete an online survey.
  1. State
    • the name of the sampling technique he is using,
    • one disadvantage of using this technique.
    James processes his data in order to draw a histogram. His table of results is shown below.
    Time spent using social media in minutes\(0 -\)\(15 -\)\(30 -\)\(60 -\)\(120 - 240\)
    Number of people per minute12.214.08.47.33.1
  2. Show that the size of the sample is 1455 .
  3. Calculate an estimate of the probability that a retired person spends more than an hour per day using social media.
OCR MEI AS Paper 2 2021 November Q12
8 marks Moderate -0.3
12 A manufacturer of steel rods checks the length of each rod in randomly selected batches of 10 rods. 100 batches of 10 rods are checked and \(x\), the number of rods in each batch which are too long, is recorded. Summary statistics are as follows. \(n = 100\) $$\sum x = 210 \quad \sum x ^ { 2 } = 604$$
  1. Calculate
    • the mean number of rods in a batch which are too long,
    • the variance of the number of rods in a batch which are too long.
    Layla decides to use a binomial distribution to model the number of rods which are too long in a batch of 10 .
  2. Write down the parameters that Layla should use in her model.
  3. Use Layla's model to determine the expected number of batches out of 100 in which there are exactly 2 rods which are too long.
OCR MEI AS Paper 2 2021 November Q13
9 marks Standard +0.3
13 In this question you must show detailed reasoning.
The equation of a curve is \(y = 3 x + \frac { 7 } { x } - \frac { 3 } { x ^ { 2 } }\).
Determine the coordinates of the points on the curve where the curve is parallel to the line \(y = 2 x\).
[0pt] [9] END OF QUESTION PAPER
OCR MEI AS Paper 2 Specimen Q1
3 marks Easy -1.2
1 Find \(\int \left( x ^ { 2 } + \frac { 1 } { x ^ { 2 } } \right) \mathrm { d } x\).
OCR MEI AS Paper 2 Specimen Q2
4 marks Moderate -0.8
2
  1. Express \(2 \log _ { 3 } x + \log _ { 3 } a\) as a single logarithm.
  2. Given that \(2 \log _ { 3 } x + \log _ { 3 } a = 2\), express \(x\) in terms of \(a\).
OCR MEI AS Paper 2 Specimen Q3
5 marks Moderate -0.8
3 Show that the area of the region bounded by the curve \(y = 3 x ^ { - \frac { 3 } { 2 } }\), the lines \(x = 1 , x = 3\) and the \(x\)-axis is \(6 - 2 \sqrt { 3 }\).
OCR MEI AS Paper 2 Specimen Q4
5 marks Moderate -0.8
4 There are four human blood groups; these are called \(\mathrm { O } , \mathrm { A } , \mathrm { B }\) and AB . Each person has one of these blood groups. The table below shows the distribution of blood groups in a large country.
Blood group
Proportion of
population
O\(49 \%\)
A\(38 \%\)
B\(10 \%\)
AB\(3 \%\)
Two people are selected at random from this country.
  1. Find the probability that at least one of these two people has blood group O .
  2. Find the probability that each of these two people has a different blood group.
OCR MEI AS Paper 2 Specimen Q5
6 marks Moderate -0.3
5 A triangular field has sides of length \(100 \mathrm {~m} , 120 \mathrm {~m}\) and 135 m .
  1. Find the area of the field.
  2. Explain why it would not be reasonable to expect your answer in (a) to be accurate to the nearest square metre.
OCR MEI AS Paper 2 Specimen Q6
8 marks Challenging +1.2
6
  1. The graph of \(y = 3 \sin ^ { 2 } \theta\) for \(0 ^ { \circ } \leq \theta \leq 360 ^ { \circ }\) is shown in Fig. 6.
    On the copy of Fig. 6 in the Printed Answer Booklet, sketch the graph of \(y = 2 \cos \theta\) for \(0 ^ { \circ } \leq \theta \leq 360 ^ { \circ }\). \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{05376a51-e768-4b45-9c18-c98255a4bd70-05_818_1507_571_351} \captionsetup{labelformat=empty} \caption{Fig. 6}
    \end{figure}
  2. In this question you must show detailed reasoning. Determine the values of \(\theta , 0 ^ { \circ } \leq \theta \leq 360 ^ { \circ }\), for which the two graphs cross.
OCR MEI AS Paper 2 Specimen Q7
7 marks Easy -1.2
7 A farmer has 200 apple trees. She is investigating the masses of the crops of apples from individual trees. She decides to select a sample of these trees and find the mass of the crop for each tree.
  1. Explain how she can select a random sample of 10 different trees from the 200 trees. The masses of the crops from the 10 trees, measured in kg, are recorded as follows. \(\begin{array} { l l l l l l l l l l } 23.5 & 27.4 & 26.2 & 29.0 & 25.1 & 27.4 & 26.2 & 28.3 & 38.1 & 24.9 \end{array}\)
  2. For these data find
    • the mean,
    • the sample standard deviation.
    • Show that there is one outlier at the upper end of the data. How should the farmer decide whether to use this outlier in any further analysis of the data?
OCR MEI AS Paper 2 Specimen Q8
7 marks Moderate -0.8
8 In an experiment, the temperature of a hot liquid is measured every minute.
The difference between the temperature of the hot liquid and room temperature is \(D ^ { \circ } \mathrm { C }\) at time \(t\) minutes. Fig. 8 shows the experimental data. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{05376a51-e768-4b45-9c18-c98255a4bd70-07_1144_1541_497_276} \captionsetup{labelformat=empty} \caption{Fig. 8}
\end{figure} It is thought that the model \(D = 70 \mathrm { e } ^ { - 0.03 t }\) might fit the data.
  1. Write down the derivative of \(\mathrm { e } ^ { - 0.03 t }\).
  2. Explain how you know that \(70 \mathrm { e } ^ { - 0.03 t }\) is a decreasing function of \(t\).
  3. Calculate the value of \(70 \mathrm { e } ^ { - 0.03 t }\) when
    1. \(\quad t = 0\),
    2. \(t = 20\).
  4. Using your answers to parts (b) and (c), discuss how well the model \(D = 70 \mathrm { e } ^ { - 0.03 t }\) fits the data.
OCR MEI AS Paper 2 Specimen Q9
7 marks Easy -1.3
9 Fig. 9.1 shows box and whisker diagrams which summarise the birth rates per 1000 people for all the countries in three of the regions as given in the pre-release data set.
The diagrams were drawn as part of an investigation comparing birth rates in different regions of the world. Africa (Sub-Saharan) \includegraphics[max width=\textwidth, alt={}, center]{05376a51-e768-4b45-9c18-c98255a4bd70-08_104_991_557_730} East and South East Asia \includegraphics[max width=\textwidth, alt={}, center]{05376a51-e768-4b45-9c18-c98255a4bd70-08_109_757_744_671} Caribbean \includegraphics[max width=\textwidth, alt={}, center]{05376a51-e768-4b45-9c18-c98255a4bd70-08_99_369_982_730} \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{05376a51-e768-4b45-9c18-c98255a4bd70-08_202_1595_1153_299} \captionsetup{labelformat=empty} \caption{Fig. 9.1}
\end{figure}
  1. Discuss the distributions of birth rates in these regions of the world. Make three different statements. You should refer to both information from the box and whisker diagrams and your knowledge of the large data set.
  2. The birth rates for all the countries in Australasia are shown below.
    CountryBirth rate per 1000
    Australia12.19
    New Zealand13.4
    Papua New Guinea24.89
    1. Explain why the calculation below is not a correct method for finding the birth rate per 1000 for Australasia as a whole. $$\frac { 12.19 + 13.4 + 24.89 } { 3 } \approx 16.83$$
    2. Without doing any calculations, explain whether the birth rate per 1000 for Australasia as a whole is higher or lower than 16.83 . The scatter diagram in Fig. 9.2 shows birth rate per 1000 and physicians/ 1000 population for all the countries in the pre-release data set. \begin{figure}[h]
      \includegraphics[alt={},max width=\textwidth]{05376a51-e768-4b45-9c18-c98255a4bd70-09_898_1698_386_274} \captionsetup{labelformat=empty} \caption{Fig. 9.2}
      \end{figure}
  3. Describe the correlation in the scatter diagram.
  4. Discuss briefly whether the scatter diagram shows that high birth rates would be reduced by increasing the number of physicians in a country.
OCR MEI AS Paper 2 Specimen Q10
9 marks Moderate -0.3
10 A company operates trains. The company claims that \(92 \%\) of its trains arrive on time. You should assume that in a random sample of trains, they arrive on time independently of each other.
  1. Assuming that \(92 \%\) of the company's trains arrive on time, find the probability that in a random sample of 30 trains operated by this company
    1. exactly 28 trains arrive on time,
    2. more than 27 trains arrive on time. A journalist believes that the percentage of trains operated by this company which arrive on time is lower than \(92 \%\).
  2. To investigate the journalist's belief a hypothesis test will be carried out at the \(1 \%\) significance level. A random sample of 18 trains is selected. For this hypothesis test,
    • state the hypotheses,
    • find the critical region.
OCR MEI AS Paper 2 Specimen Q12
3 marks Standard +0.3
12 Given that \(\arcsin x = \arccos y\), prove that \(x ^ { 2 } + y ^ { 2 } = 1\). [Hint: Let \(\arcsin x = \theta\) ] \section*{END OF QUESTION PAPER}
OCR MEI Paper 1 2018 June Q1
3 marks Easy -1.8
1 Show that ( \(x - 2\) ) is a factor of \(3 x ^ { 3 } - 8 x ^ { 2 } + 3 x + 2\).
OCR MEI Paper 1 2018 June Q2
2 marks Moderate -0.8
2 By considering a change of sign, show that the equation \(\mathrm { e } ^ { x } - 5 x ^ { 3 } = 0\) has a root between 0 and 1 .
OCR MEI Paper 1 2018 June Q3
4 marks Standard +0.3
3 In this question you must show detailed reasoning.
Solve the equation \(\sec ^ { 2 } \theta + 2 \tan \theta = 4\) for \(0 ^ { \circ } \leqslant \theta < 360 ^ { \circ }\).
OCR MEI Paper 1 2018 June Q4
4 marks Easy -1.2
4 Rory pushes a box of mass 2.8 kg across a rough horizontal floor against a resistance of 19 N . Rory applies a constant horizontal force. The box accelerates from rest to \(1.2 \mathrm {~ms} ^ { - 1 }\) as it travels 1.8 m .
  1. Calculate the acceleration of the box.
  2. Find the magnitude of the force that Rory applies.
OCR MEI Paper 1 2018 June Q5
4 marks Moderate -0.3
5 The position vector \(\mathbf { r }\) metres of a particle at time \(t\) seconds is given by $$\mathbf { r } = \left( 1 + 12 t - 2 t ^ { 2 } \right) \mathbf { i } + \left( t ^ { 2 } - 6 t \right) \mathbf { j }$$
  1. Find an expression for the velocity of the particle at time \(t\).
  2. Determine whether the particle is ever stationary.
OCR MEI Paper 1 2018 June Q6
6 marks Moderate -0.8
6 Aleela and Baraka are saving to buy a car. Aleela saves \(\pounds 50\) in the first month. She increases the amount she saves by \(\pounds 20\) each month.
  1. Calculate how much she saves in two years. Baraka also saves \(\pounds 50\) in the first month. The amount he saves each month is \(12 \%\) more than the amount he saved in the previous month.
  2. Explain why the amounts Baraka saves each month form a geometric sequence.
  3. Determine whether Baraka saves more in two years than Aleela. Answer all the questions
    Section B (77 marks)
OCR MEI Paper 1 2018 June Q7
3 marks Moderate -0.3
7 A rod of length 2 m hangs vertically in equilibrium. Parallel horizontal forces of 30 N and 50 N are applied to the top and bottom and the rod is held in place by a horizontal force \(F \mathrm {~N}\) applied \(x \mathrm {~m}\) below the top of the rod as shown in Fig. 7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{904025c9-6d68-4344-bd41-8c0fccfcf92f-05_445_390_609_824} \captionsetup{labelformat=empty} \caption{Fig. 7}
\end{figure}
  1. Find the value of \(F\).
  2. Find the value of \(x\).
OCR MEI Paper 1 2018 June Q8
6 marks Standard +0.3
8
  1. Show that \(8 \sin ^ { 2 } x \cos ^ { 2 } x\) can be written as \(1 - \cos 4 x\).
  2. Hence find \(\int \sin ^ { 2 } x \cos ^ { 2 } x \mathrm {~d} x\).
OCR MEI Paper 1 2018 June Q9
10 marks Standard +0.3
9 A pebble is thrown horizontally at \(14 \mathrm {~ms} ^ { - 1 }\) from a window which is 5 m above horizontal ground. The pebble goes over a fence 2 m high \(d \mathrm {~m}\) away from the window as shown in Fig. 9. The origin is on the ground directly below the window with the \(x\)-axis horizontal in the direction in which the pebble is thrown and the \(y\)-axis vertically upwards. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{904025c9-6d68-4344-bd41-8c0fccfcf92f-06_538_1082_452_488} \captionsetup{labelformat=empty} \caption{Fig. 9}
\end{figure}
  1. Find the time the pebble takes to reach the ground.
  2. Find the cartesian equation of the trajectory of the pebble.
  3. Find the range of possible values for \(d\).
OCR MEI Paper 1 2018 June Q10
8 marks Challenging +1.2
10 Fig. 10 shows the graph of \(y = ( k - x ) \ln x\) where \(k\) is a constant ( \(k > 1\) ). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{904025c9-6d68-4344-bd41-8c0fccfcf92f-06_454_1266_1564_395} \captionsetup{labelformat=empty} \caption{Fig. 10}
\end{figure} Find, in terms of \(k\), the area of the finite region between the curve and the \(x\)-axis.