OCR MEI Paper 2 (Paper 2) 2020 November

1 Fig. 1 shows triangle \(A B C\). Fig. 1 Calculate the area of triangle \(A B C\), giving your answer correct to 3 significant figures.

2 Fig. 2 shows a sector of a circle of radius 8 cm . The angle of the sector is 2.1 radians. \begin{figure}[h] \end{figure}

1. Calculate the length of the arc \(L\).
2. Calculate the area of the sector.

3 You are given that \(y = 4 x + \sin 8 x\).

1. Find \(\frac { \mathrm { dy } } { \mathrm { dx } }\).
2. Find the smallest positive value of \(x\) for which \(\frac { \mathrm { dy } } { \mathrm { dx } } = 0\), giving your answer in an exact form.

4 Fig. 4 shows a cumulative frequency diagram for the time spent revising mathematics by year 11 students at a certain school during a week in the summer term. \begin{figure}[h] \end{figure}

1. Use the diagram to estimate the median time spent revising mathematics by these students. [1] A teacher comments that \(90 \%\) of the students spent less than an hour revising mathematics during this week.
2. Determine whether the information in the diagram supports this comment.

5 The first \(n\) terms of an arithmetic series are
\(17 + 28 + 39 + \ldots + 281 + 292\).

1. Find the value of \(n\).
2. Find the sum of these \(n\) terms.

6

1. Find the first three terms in ascending powers of \(x\) of the binomial expansion of \(( 1 + 4 x ) ^ { \frac { 1 } { 2 } }\).
2. State the range of values of \(x\) for which this expansion is valid.

7 You are given that \(P ( A ) = 0.6 , P ( B ) = 0.5\) and \(P ( A \cup B ) ^ { \prime } = 0.2\).

1. Find \(\mathrm { P } ( \mathrm { A } \cap \mathrm { B } )\).
2. Find \(\mathrm { P } ( \mathrm { A } \mid \mathrm { B } )\).
3. State, with a reason, whether \(A\) and \(B\) are independent.

8 Rosella is carrying out an investigation into the age at which adults retire from work in the city where she lives. She collects a sample of size 50 , ensuring this comprises of 25 randomly selected retired men and 25 randomly selected retired women.

1. State the name of the sampling method she uses. Fig. 8.1 shows the data she obtains in a frequency table and Fig. 8.2 shows these data displayed in a histogram. \begin{table}[h]

   Age in years at retirement	\(45 -\)	\(50 -\)	\(55 -\)	\(60 -\)	\(65 -\)	\(70 -\)	\(75 - 80\)
   Frequency density	0.4	1.8	2.4	2.2	1.8	1.2	0.2

   \captionsetup{labelformat=empty} \caption{Fig. 8.1}
   \end{table} \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{cea67565-8074-4703-8e1a-09b98e380baf-08_805_1006_1160_244} \captionsetup{labelformat=empty} \caption{Fig. 8.2}
   \end{figure}
2. How many people in the sample are aged between 50 and 55? Rosella obtains a list of the names of all 4960 people who have retired in the city during the previous month.
3. Describe how Rosella could collect a sample of size 200 from her list using
   • systematic sampling such that every item on the list could be selected,
   • simple random sampling.
   Rosella collects two simple random samples, one of size 200 and one of size 500, from her list. The histograms in Fig. 8.3 show the data from the sample of size 200 on the left and the data from the sample of 500 on the right. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{cea67565-8074-4703-8e1a-09b98e380baf-09_659_1909_388_77} \captionsetup{labelformat=empty} \caption{Fig. 8.3}
   \end{figure}
4. With reference to the histograms shown in Fig. 8.2 and Fig. 8.3, explain why it appears reasonable to model the age of retirement in this city using the Normal distribution. Summary statistics for the sample of 500 are shown in Fig. 8.4. \begin{table}[h]

   Statistics
   n	500
   Mean	60.0515
   \(\sigma\)	6.5717
   s	6.5783
   \(\Sigma x\)	30025.7601
   \(\Sigma \mathrm { x } ^ { 2 }\)	1824686.322
   Min	36.0793
   Q1	55.2573
   Median	59.9202
   Q3	64.4239
   Max	81.742

   \captionsetup{labelformat=empty} \caption{Fig. 8.4}
   \end{table}
5. Use an appropriate Normal model based on the information in Fig. 8.4 to estimate the number of people aged over 65 who retired in the city in the previous month.
6. Identify a limitation in using this model to predict the number of people aged over 65 retiring in the following month.

9 A company supplies computers to businesses. In the past the company has found that computers are kept by businesses for a mean time of 5 years before being replaced. Claud, the manager of the company, thinks that the mean time before replacing computers is now different.

1. Describe how Claud could obtain a cluster sample of 120 computers used by businesses the company supplies. Claud decides to conduct a hypothesis test at the \(5 \%\) level to test whether there is evidence to suggest that the mean time that businesses keep computers is not 5 years. He takes a random sample of 120 computers. Summary statistics for the length of time computers in this sample are kept are shown in Fig. 9. \begin{table}[h]

   Statistics
   \(n\)	120
   Mean	4.8855
   \(\sigma\)	2.6941
   \(s\)	2.7054
   \(\Sigma x\)	586.2566
   \(\Sigma x ^ { 2 }\)	3735.1475
   Min	0.1213
   Q1	2.5472
   Median	4.8692
   Q3	7.0349
   Max	9.9856

   \captionsetup{labelformat=empty} \caption{Fig. 9}
   \end{table} \section*{(b) In this question you must show detailed reasoning.}
   • State the hypotheses for this test, explaining why the alternative hypothesis takes the form it does.
   • Use a suitable distribution to carry out the test.

10 In this question you must show detailed reasoning. The equation of a curve is $$y = \frac { \sin 2 x - x } { x \sin x }$$

1. Use the small angle approximation given in the list of formulae on pages 2-3 of this question paper to show that $$\int _ { 0.01 } ^ { 0.05 } \mathrm { ydx } \approx \ln 5$$
2. Use the same small angle approximation to show that $$\frac { d y } { d x } \approx - 10000 \text { at the point where } x = 0.01 \text {. }$$ The equation \(y = 0\) has a root near \(x = 1\). Joan uses the Newton-Raphson method to find this root. The output from the spreadsheet she uses is shown in Fig. 10.1. \begin{table}[h]

   \(n\)	0	1	2	3	4	5	6	7
   \(\mathrm { x } _ { \mathrm { n } }\)	1	0.958509	0.950084	0.948261	0.94786	0.947772	0.947753	0.947748

   \captionsetup{labelformat=empty} \caption{Fig. 10.1}
   \end{table} Joan carries out some analysis of this output. The results are shown in Fig. 10.2. \begin{table}[h]

   \(x\)	\(y\)
   0.9477475	\(- 7.79967 \mathrm { E } - 07\)
   0.9477485	\(- 2.90821 \mathrm { E } - 06\)
   \(x\)	\(y\)
   0.947745	\(4.54066 \mathrm { E } - 06\)
   0.947755	\(- 1.67417 \mathrm { E } - 05\)

   \captionsetup{labelformat=empty} \caption{Fig. 10.2}
   \end{table}
3. Consider the information in Fig. 10.1 and Fig. 10.2.
   • Write 4.54066E-06 in standard mathematical notation.
   • State the value of the root as accurately as you can, justifying your answer.

11 The pre-release material contains information concerning median house prices over the period 2004-2015. A spreadsheet has been used to generate a time series graph for two areas: the London borough of "Barking and Dagenham" and "North West". This is shown together with the raw data in Fig. 11.1. \begin{figure}[h] \end{figure} Dr Procter suggests that it is unusual for median house prices in a London borough to be consistently higher than those in other parts of the country.

1. Use your knowledge of the large data set to comment on Dr Procter's suggestion. Dr Procter wishes to predict the median house price in Barking and Dagenham in 2016. She uses the spreadsheet function LINEST to find the equation of the line of best fit for the given data. She obtains the equation
   \(P = 4897 Y - 9657847\), where \(P\) is the median house price in pounds and \(Y\) is the calendar year, for example 2015.
2. Use Dr Procter's equation to predict the median house price in Barking and Dagenham in
   • 2016
   • 2017.
   Professor Jackson uses a simpler model by using the data from 2014 and 2015 only to form a straight-line model.
3. Find the equation Professor Jackson uses in her model.
4. Use Professor Jackson’s equation to predict the median house price in Barking and Dagenham in
   • 2016
   • 2017.
   Professor Jackson carries out some research online. She finds some information about median house prices in Barking and Dagenham, which is shown in Fig. 11.2. \begin{table}[h]

   2016	2017
   \(\pounds 290000\)	\(\pounds 300000\)

   \captionsetup{labelformat=empty} \caption{Fig. 11.2}
   \end{table}
5. Comment on how well
   • Dr Procter’s model fits the data,
   • Professor Jackson’s model fits the data.
   • Explain which, if any, of the models is likely to be more reliable for predicting median house prices in Barking and Dagenham in 2020.

1. Given that \(q < 2 p\), determine the values of \(p\) and \(q\).
2. The spinner is spun 10 times. Calculate the probability that exactly one 5 is obtained. Elaine's teacher believes that the probability that the spinner shows a 1 is greater than 0.2 . The spinner is spun 100 times and gives a score of 1 on 28 occasions.
3. Conduct a hypothesis test at the \(5 \%\) level to determine whether there is any evidence to suggest that the probability of obtaining a score of 1 is greater than 0.2 .

13 The pre-release material contains information concerning median house prices, recycling rates and employment rates. Fig. 13.1 shows a scatter diagram of recycling rate against employment rate for a random sample of 33 regions. \begin{figure}[h] \end{figure} The product moment correlation coefficient for this sample is 0.37154 and the associated \(p\)-value is 0.033. Lee conducts a hypothesis test at the \(5 \%\) level to test whether there is any evidence to suggest there is positive correlation between recycling rate and employment rate. He concludes that there is no evidence to suggest positive correlation because \(0.033 \approx 0\) and \(0.37154 > 0.05\).

1. Explain whether Lee's reasoning is correct. Fig. 13.2 shows a scatter diagram of recycling rate against median house price for a random sample of 33 regions. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{cea67565-8074-4703-8e1a-09b98e380baf-14_648_1474_1758_242} \captionsetup{labelformat=empty} \caption{Fig. 13.2}
   \end{figure} The product moment correlation coefficient for this sample is - 0.33278 and the associated \(p\)-value is 0.058 . Fig. 13.3 shows summary statistics for the median house prices for the data in this sample. \begin{table}[h]

   Statistics
   \(n\)	33
   Mean	465467.9697
   \(\sigma\)	201236.1345
   \(s\)	204356.2606
   \(\Sigma x\)	15360443
   \(\Sigma x ^ { 2 }\)	8486161617387
   Min	243500
   Q1	342500
   Median	410000
   Q3	521000
   Max	1200000

   \captionsetup{labelformat=empty} \caption{Fig. 13.3}
   \end{table}
2. Use the information in Fig. 13.3 and Fig. 13.2 to show that there are at least two outliers.
3. Describe the effect of removing the outliers on
   • the product moment correlation coefficient between recycling rate and median house price,
   • the \(p\)-value associated with this correlation coefficient,
     in each case explaining your answer.
     [0pt] [2]
     All 33 items in the sample are areas in London. A student suggests that it is very unlikely that only areas in London would be selected in a random sample.
   • Use your knowledge of the pre-release material to explain whether you think the student's suggestion is reasonable.

14 In this question you must show detailed reasoning. Fig. 14 shows the graphs of \(y = \sin x \cos 2 x\) and \(y = \frac { 1 } { 2 } - \sin 2 x \cos x\). \begin{figure}[h] \end{figure} Use integration to find the area between the two curves, giving your answer in an exact form.

15 Functions \(\mathrm { f } ( x )\) and \(\mathrm { g } ( x )\) are defined as follows.
\(\mathrm { f } ( x ) = \sqrt { x }\) for \(x > 0\) and \(\mathrm { g } ( x ) = x ^ { 3 } - x - 6\) for \(x > 2\). The function \(\mathrm { h } ( x )\) is defined as
\(\mathrm { h } ( x ) = \mathrm { fg } ( x )\).

1. Find \(\mathrm { h } ( x )\) in terms of \(x\) and state its domain.
2. Find \(\mathrm { h } ( 3 )\). Fig. 15 shows \(\mathrm { h } ( x )\) and \(\mathrm { h } ^ { - 1 } ( x )\), together with the straight line \(y = x\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{cea67565-8074-4703-8e1a-09b98e380baf-17_780_796_895_242} \captionsetup{labelformat=empty} \caption{Fig. 15}
   \end{figure}
3. Determine the gradient of \(\mathrm { y } = \mathrm { h } ^ { - 1 } ( \mathrm { x } )\) at the point where \(y = 3\).