Questions — OCR MEI AS Paper 2 (98 questions)

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OCR MEI AS Paper 2 2021 November Q11
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
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
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
1 Find \(\int \left( x ^ { 2 } + \frac { 1 } { x ^ { 2 } } \right) \mathrm { d } x\).
OCR MEI AS Paper 2 Specimen Q2
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
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
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
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
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 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
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
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
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
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 AS Paper 2 2024 June Q11
  1. Verify that the curve cuts the \(x\)-axis at \(x = 4\) and at \(x = 9\). The curve does not cut or touch the \(x\)-axis at any other points.
  2. Determine the exact area bounded by the curve and the \(x\)-axis.
OCR MEI AS Paper 2 2021 November Q10
  1. Show that PQ is perpendicular to QR . A circle passes through \(\mathrm { P } , \mathrm { Q }\) and R .
  2. Determine the coordinates of the centre of the circle.
OCR MEI AS Paper 2 2018 June Q8
8 In this question you must show detailed reasoning. The centre of a circle C is at the point \(( - 1,3 )\) and C passes through the point \(( 1 , - 1 )\). The straight line L passes through the points \(( 1,9 )\) and \(( 4,3 )\). Show that L is a tangent to C .
OCR MEI AS Paper 2 2019 June Q10
10 In this question you must show detailed reasoning. The equation of a curve is \(y = \frac { x ^ { 2 } } { 4 } + \frac { 2 } { x } + 1\). A tangent and a normal to the curve are drawn at the point where \(x = 2\). Calculate the area bounded by the tangent, the normal and the \(x\)-axis. \section*{END OF QUESTION PAPER}
OCR MEI AS Paper 2 2023 June Q14
14 In this question you must show detailed reasoning. The equation of a curve is \(y = 16 \sqrt { x } + \frac { 8 } { x }\).
Determine the equation of the tangent to the curve at the point where \(x = 4\).
OCR MEI AS Paper 2 2024 June Q8
8 In this question you must show detailed reasoning. Determine the coordinates of the point of intersection of the line with equation \(y = 2 x + 3\) and the curve with equation \(y ^ { 2 } - 4 x ^ { 2 } = 33\).
OCR MEI AS Paper 2 2020 November Q7
7 In this question you must show detailed reasoning. A circle has centre \(( 2 , - 1 )\) and radius 5. A straight line passes through the points \(( 1,1 )\) and \(( 9,5 )\).
Find the coordinates of the points of intersection of the line and the circle.
OCR MEI AS Paper 2 2021 November Q3
3 In this question you must show detailed reasoning. You are given that \(\tan 30 ^ { \circ } = \frac { 1 } { \sqrt { 3 } }\).
Explain why \(\tan 690 ^ { \circ } = - \frac { 1 } { \sqrt { 3 } }\).
OCR MEI AS Paper 2 Specimen Q11
11 In this question you must show detailed reasoning. Fig. 11 shows the curve \(y = \mathrm { f } ( x )\), where \(\mathrm { f } ( x )\) is a cubic function. Fig. 11 also shows the coordinates of the turning points and the points of intersection with the axes. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{05376a51-e768-4b45-9c18-c98255a4bd70-11_805_620_543_317} \captionsetup{labelformat=empty} \caption{Fig. 11}
\end{figure} Show that the tangent to \(y = \mathrm { f } ( x )\) at \(x = t\) is parallel to the tangent to \(y = \mathrm { f } ( x )\) at \(x = - t\) for all values of \(t\).