Questions — OCR MEI AS Paper 2 (99 questions)

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OCR MEI AS Paper 2 2020 November Q4
4 marks Moderate -0.3
4 In a certain country it is known that 11\% of people are left-handed.
  1. Calculate the probability that, in a random sample of 98 people from this country, 5 or fewer are found to be left-handed, giving your answer correct to 3 significant figures. An anthropologist believes that the proportion of left-handed people is lower in a particular ethnic group. The anthropologist collects a random sample of 98 people from this particular ethnic group in order to test the hypothesis that the proportion of left-handed people is less than \(11 \%\). The anthropologist carries out the test at the \(1 \%\) level.
  2. Determine the critical region for this test.
OCR MEI AS Paper 2 2020 November Q5
6 marks Moderate -0.8
5 A company needs to appoint 3 new assistants. 8 candidates are invited for interview; each candidate has a different surname. The candidates are to be interviewed one after another. The personnel officer randomly selects the order in which the candidates are to be interviewed by drawing their names out of a hat. One of the candidates is called Mr Browne and another is called Mrs Green.
  1. Calculate the probability that Mr Browne is interviewed first and Mrs Green is interviewed last. 5 of the 8 candidates invited for interview are women and the other 3 are men. The chief executive can't make up his mind who to appoint so he randomly selects 3 candidates by drawing their names out of a hat.
  2. Determine the probability that more women than men are selected.
OCR MEI AS Paper 2 2020 November Q6
6 marks Moderate -0.8
6 Use integration to show that the area bounded by the \(x\)-axis and the curve with equation \(y = ( x - 1 ) ^ { 2 } ( x - 3 )\) is \(\frac { 4 } { 3 }\) square units.
OCR MEI AS Paper 2 2020 November Q8
6 marks Standard +0.3
8 In this question you must show detailed reasoning.
Solve the equation \(3 \cos \theta + 8 \tan \theta = 0\) for \(0 ^ { \circ } < \theta < 360 ^ { \circ }\), giving your answers correct to the nearest degree.
OCR MEI AS Paper 2 2020 November Q9
8 marks Moderate -0.8
9 The equation of a curve is \(y = 24 \sqrt { x } - 8 x ^ { \frac { 3 } { 2 } } + 16\).
  1. Find \(\frac { \mathrm { dy } } { \mathrm { dx } }\).
  2. Find the coordinates of the turning point.
  3. Determine the nature of the turning point.
OCR MEI AS Paper 2 2020 November Q10
9 marks Moderate -0.8
10 Fig. 10.1 shows a sample collected from the large data set. BMI is defined as \(\frac { \text { mass of person in kilograms } } { \text { square of person's height in metres } }\). \begin{table}[h]
SexAge in yearsMass in kgHeight in cmBMI
Male3877.6164.828.57
Male1763.5170.321.89
Male1868.0172.322.91
Male1857.2172.219.29
Male1977.6191.221.23
Male2472.7177.023.21
Male2592.5177.929.23
Male2670.4159.427.71
Male3177.5174.025.60
Male34132.4182.239.88
Male38115.0186.433.10
Male40112.1171.738.02
\captionsetup{labelformat=empty} \caption{Fig. 10.1}
\end{table}
  1. Calculate the mass in kg of a person with a BMI of 23.56 and a height of 181.6 cm , giving your answer correct to 1 decimal place. Fig. 10.2 shows a scatter diagram of BMI against age for the data in the table. A line of best fit has also been drawn. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{c08a2212-3104-425e-8aee-7f2d46f23924-09_682_1212_351_248} \captionsetup{labelformat=empty} \caption{Fig. 10.2}
    \end{figure}
  2. Describe the correlation between age and BMI.
  3. Use the line of best fit to estimate the BMI of a 30-year-old man.
  4. Explain why it would not be sensible to use the line of best fit to estimate the BMI of a 60-year-old man.
  5. Use your knowledge of the large data set to suggest two reasons why the sample data in the table may not be representative of the population.
  6. Once the data in the large data set had been cleaned there were 196 values available for selection. Describe how a sample of size 12 could be generated using systematic sampling so that each of the 196 values could be selected in the sample.
OCR MEI AS Paper 2 2020 November Q11
10 marks Standard +0.3
11 A car is travelling along a stretch of road at a steady speed of \(11 \mathrm {~ms} ^ { - 1 }\).
The driver accelerates, and \(t\) seconds after starting to accelerate the speed of the car, \(V\), is modelled by the formula \(\mathrm { V } = \mathrm { A } + \mathrm { B } \left( 1 - \mathrm { e } ^ { - 0.17 \mathrm { t } } \right)\).
When \(t = 3 , V = 13.8\).
  1. Find the values of \(A\) and \(B\), giving your answers correct to 2 significant figures. When \(t = 4 , V = 14.5\) and when \(t = 5 , V = 14.9\).
  2. Determine whether the model is a good fit for these data.
  3. Determine the acceleration of the car according to the model when \(t = 5\), giving your answer correct to 3 decimal places. The car continues to accelerate until it reaches its maximum speed.
    The speed limit on this road is \(60 \mathrm { kmh } ^ { - 1 }\). All drivers who exceed this speed limit are recorded by a speed camera and automatically fined \(\pounds 100\).
  4. Determine whether, according to the model, the driver of this car is fined \(\pounds 100\).
OCR MEI AS Paper 2 2021 November Q1
3 marks Easy -1.2
1 Find the coefficient of \(x ^ { 4 }\) in the expansion of \(( 1 + 3 x ) ^ { 6 }\).
OCR MEI AS Paper 2 2021 November Q2
2 marks Easy -1.8
2 Mia rolls a six-sided die 24 times and records the scores. She displays her results in a vertical line chart. This is shown in Fig. 2.1. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{2b9ce212-84e2-4817-be94-98e2adff12a3-03_534_1168_648_242} \captionsetup{labelformat=empty} \caption{Fig. 2.1}
\end{figure}
  1. Describe the shape of the distribution. She repeats the experiment, but this time she rolls the die 50 times. Her results are displayed in Fig. 2.2. \begin{figure}[h]
    \captionsetup{labelformat=empty} \caption{Scores on a six-sided die} \includegraphics[alt={},max width=\textwidth]{2b9ce212-84e2-4817-be94-98e2adff12a3-03_476_1161_1617_242}
    \end{figure} Fig. 2.2 Her brother Kai rolls the same die 1000 times and displays his results in a similar diagram.
  2. Assuming the die is fair, describe the distribution you would expect to see in Kai's diagram.
OCR MEI AS Paper 2 2021 November Q4
4 marks Easy -1.3
4 Find \(\int \left( 9 x ^ { 2 } + \frac { 6 } { \sqrt { x } } \right) \mathrm { d } x\).
OCR MEI AS Paper 2 2021 November Q5
7 marks Moderate -0.8
5 In 2019 scientists developed a model for comparing the ages of humans and dogs.
According to the model, \(Y = A \ln X + B\) where \(X =\) dog age in years and \(Y =\) human age in years.
For the model, it is known that when \(X = 1 , Y = 31\) and when \(X = 12 , Y = 71\).
  1. Find the value of \(B\).
  2. Determine the value of \(A\), correct to the nearest whole number. Use the model, with the exact value of \(B\) and the value of \(A\) correct to the nearest whole number, to answer parts (c) and (d).
  3. Find the human age corresponding to a dog age of 20 years.
  4. Determine the dog age corresponding to a human age of 120 years.
OCR MEI AS Paper 2 2021 November Q6
6 marks Moderate -0.3
6 The probability distribution for the discrete random variable \(X\) is shown below.
\(x\)0123
\(\mathrm { P } ( X = x )\)\(3 p ^ { 2 }\)\(0.5 p ^ { 2 } + 2 p\)\(1.5 p\)\(1.5 p ^ { 2 } + 0.5 p\)
  1. Determine the value of \(p\).
  2. Determine the modal value of \(X\).
OCR MEI AS Paper 2 2021 November Q7
7 marks Easy -1.2
7 The pre-release material contains information about health expenditure. Fig. 7.1 shows an extract from the data. \begin{table}[h]
CountryHealth expenditure (\% of GDP)
Algeria7.2
Egypt5.6
Libya5
Morocco5.9
Sudan8.4
Tunisia7
Western Sahara\#N/A
Angola3.3
Benin4.6
Botswana5.4
Burkina Faso5
\captionsetup{labelformat=empty} \caption{Fig. 7.1}
\end{table}
  1. Explain how the data should be cleaned before any analysis takes place. Kareem uses all the available data to conduct an investigation into health expenditure as a percentage of GDP in different countries. He calculates the mean to be 6.79 and the standard deviation to be 2.78 . Fig. 7.2 shows the smallest values and the largest values of health expenditure as a percentage of GDP. \begin{table}[h]
    Smallest values of Health expenditure (\% of GDP)Largest values of Health expenditure (\% of GDP)
    1.511.7
    1.911.9
    2.113.7
    13.7
    16.5
    17.1
    17.1
    \captionsetup{labelformat=empty} \caption{Fig. 7.2}
    \end{table}
  2. Determine which of these values are outliers. Kareem removes the outliers from the data and finds that there are 187 values left. He decides to collect a sample of size 30 . He uses the following sampling procedure.
    Assign each value a number from 1 to 187. Generate a random number, \(n\), between 1 and 13 . Starting with the \(n\)th value, choose every 6th value after that until 30 values have been chosen.
  3. Explain whether Kareem is using simple random sampling.
OCR MEI AS Paper 2 2021 November Q8
4 marks Easy -1.2
8 With respect to an origin O , the position vectors of the points A and B are \(\overrightarrow { \mathrm { OA } } = \binom { - 3 } { 20 }\) and \(\overrightarrow { \mathrm { OB } } = \binom { 6 } { 8 }\).
  1. Determine whether \(| \overrightarrow { \mathrm { AB } } | > 200\). The point C is such that \(\overrightarrow { \mathrm { AC } } = \binom { 18 } { - 24 }\).
  2. Determine whether \(\mathrm { A } , \mathrm { B }\) and C are collinear.
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.
    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
    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
    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