Questions — OCR MEI AS Paper 2 (99 questions)

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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,
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 AS Paper 2 2024 June Q11
6 marks Moderate -0.8
  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
6 marks Standard +0.3
  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 2019 June Q10
10 marks Standard +0.8
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
7 marks Moderate -0.8
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
4 marks Moderate -0.3
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
8 marks Moderate -0.3
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 marks Moderate -0.8
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
6 marks Standard +0.8
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 \(t\).
OCR MEI AS Paper 2 2018 June Q1
2 marks Easy -2.0
Write down the value of (A) \(\log_a (a^4)\), [1] (B) \(\log_a \left(\frac{1}{a}\right)\). [1]
OCR MEI AS Paper 2 2018 June Q2
3 marks Easy -1.3
Doug has a list of times taken by competitors in a 'fun run'. He has grouped the data and calculated the frequency densities in order to draw a histogram to represent the information. Some of the data are presented in Fig. 2.
Time in minutes\(15-\)\(20-\)\(25-\)\(35-\)\(45-60\)
Number of runners12235971
Frequency density2.45.97.11.4
Fig. 2
  1. Write down the missing values in the copy of Fig. 2 in the Printed Answer Booklet. [2]
  2. Doug labels the horizontal axis on the histogram 'time in minutes' and the vertical axis 'number of minutes per runner'. State which one of these labels is incorrect and write down a correct version. [1]
OCR MEI AS Paper 2 2018 June Q3
3 marks Moderate -0.8
\(P\) and \(Q\) are consecutive odd positive integers such that \(P > Q\). Prove that \(P^2 - Q^2\) is a multiple of 8. [3]
OCR MEI AS Paper 2 2018 June Q4
5 marks Moderate -0.8
The probability distribution of the discrete random variable \(X\) is given in Fig. 4.
\(r\)01234
P\((X = r)\)0.20.150.3\(k\)0.25
Fig. 4
  1. Find the value of \(k\). [2]
\(X_1\) and \(X_2\) are two independent values of \(X\).
  1. Find P\((X_1 + X_2 = 6)\). [3]
OCR MEI AS Paper 2 2018 June Q5
3 marks Moderate -0.8
Find the set of values of \(a\) for which the equation $$ax^2 + 8x + 2 = 0$$ has no real roots. [3]
OCR MEI AS Paper 2 2018 June Q6
4 marks Moderate -0.8
Show that \(\int_0^9 (3 + 4\sqrt{x})dx = 99\). [4]
OCR MEI AS Paper 2 2018 June Q7
8 marks Moderate -0.8
Rose and Emma each wear a device that records the number of steps they take in a day. All the results for a 7-day period are given in Fig. 7.
Day1234567
Rose10014112621014993619708992110369
Emma9204991387411001510261739110856
Fig. 7 The 7-day mean is the mean number of steps taken in the last 7 days. The 7-day mean for Rose is 10112.
  1. Calculate the 7-day mean for Emma. [1]
At the end of day 8 a new 7-day mean is calculated by including the number of steps taken on day 8 and omitting the number of steps taken on day 1. On day 8 Rose takes 10259 steps.
  1. Determine the number of steps Emma must take on day 8 so that her 7-day mean at the end of day 8 is the same as for Rose. [4]
In fact, over a long period of time, the mean of the number of steps per day that Emma takes is 10341 and the standard deviation is 948.
  1. Determine whether the number of steps Emma needs to take on day 8 so that her 7-day mean is the same as that for Rose in part (ii) is unusually high. [3]
OCR MEI AS Paper 2 2018 June Q8
7 marks Standard +0.3
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. [7]
OCR MEI AS Paper 2 2018 June Q9
7 marks Standard +0.3
In this question you must show detailed reasoning. Research showed that in May 2017 62% of adults over 65 years of age in the UK used a certain online social media platform. Later in 2017 it was believed that this proportion had increased. In December 2017 a random sample of 59 adults over 65 years of age in the UK was collected. It was found that 46 of the 59 adults used this online social media platform. Use a suitable hypothesis test to determine whether there is evidence at the 1% level to suggest that the proportion of adults over 65 in the UK who used this online social media platform had increased from May 2017 to December 2017. [7]
OCR MEI AS Paper 2 2018 June Q10
9 marks Moderate -0.8
  1. A curve has equation \(y = 16x + \frac{1}{x}\). Find
    1. \(\frac{dy}{dx}\), [2]
    2. \(\frac{d^2y}{dx^2}\). [2]
  2. Hence
OCR MEI AS Paper 2 2018 June Q11
9 marks Easy -1.8
The pre-release material contains data concerning the death rate per thousand people and the birth rate per thousand people in all the countries of the world. The diagram in Fig. 11.1 was generated using a spreadsheet and summarises the birth rates for all the countries in Africa. \includegraphics{figure_11_1} Fig. 11.1
  1. Identify two respects in which the presentation of the data is incorrect. [2]
Fig. 11.2 shows a scatter diagram of death rate, \(y\), against birth rate, \(x\), for a sample of 55 countries, all of which are in Africa. A line of best fit has also been drawn. \includegraphics{figure_11_2} Fig. 11.2 The equation of the line of best fit is \(y = 0.15x + 4.72\).
    1. What does the diagram suggest about the relationship between death rate and birth rate? [1]
    2. The birth rate in Togo is recorded as 34.13 per thousand, but the data on death rate has been lost. Use the equation of the line of best fit to estimate the death rate in Togo. [1]
    3. Explain why it would not be sensible to use the equation of the line of best fit to estimate the death rate in a country where the birth rate is 5.5 per thousand. [1]
    4. Explain why it would not be sensible to use the equation of the line of best fit to estimate the death rate in a Caribbean country where the birth rate is known. [1]
    5. Explain why it is unlikely that the sample is random. [1]
Including Togo there were 56 items available for selection.
  1. Describe how a sample of size 14 from this data could be generated for further analysis using systematic sampling. [2]
OCR MEI AS Paper 2 2018 June Q12
10 marks Moderate -0.8
In an experiment 500 fruit flies were released into a controlled environment. After 10 days there were 650 fruit flies present. Munirah believes that \(N\), the number of fruit flies present at time \(t\) days after the fruit flies are released, will increase at the rate of 4.4% per day. She proposes that the situation is modelled by the formula \(N = Ak^t\).
  1. Write down the values of \(A\) and \(k\). [2]
  2. Determine whether the model is consistent with the value of \(N\) at \(t = 10\). [2]
  3. What does the model suggest about the number of fruit flies in the long run? [1]
Subsequently it is found that for large values of \(t\) the number of fruit flies in the controlled environment oscillates about 750. It is also found that as \(t\) increases the oscillations decrease in magnitude. Munirah proposes a second model in the light of this new information. $$N = 750 - 250 \times e^{-0.092t}$$
  1. Identify three ways in which this second model is consistent with the known data. [3]
    1. Identify one feature which is not accounted for by the second model. [1]
    2. Give an example of a mathematical function which needs to be incorporated in the model to account for this feature. [1]