Questions - OCR MEI Paper 2

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

OCR MEI Paper 2 2020 November Q3

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

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

OCR MEI Paper 2 2020 November Q4

1 marks

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]

\includegraphics[alt={},max width=\textwidth]{cea67565-8074-4703-8e1a-09b98e380baf-05_554_1070_737_242} \captionsetup{labelformat=empty} \caption{Fig. 4}

\end{figure}

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.
Determine whether the information in the diagram supports this comment.

OCR MEI Paper 2 2020 November Q5

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

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

OCR MEI Paper 2 2020 November Q6

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

OCR MEI Paper 2 2020 November Q7

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

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

OCR MEI Paper 2 2020 November Q8

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.

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}
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.
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}
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}
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.
Identify a limitation in using this model to predict the number of people aged over 65 retiring in the following month.

OCR MEI Paper 2 2020 November Q9

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.

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.

OCR MEI Paper 2 2020 November Q11

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]

\includegraphics[alt={},max width=\textwidth]{cea67565-8074-4703-8e1a-09b98e380baf-12_572_1751_447_159} \captionsetup{labelformat=empty} \caption{Fig. 11.1}

\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.

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.
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.
Find the equation Professor Jackson uses in her model.
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}
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.

OCR MEI Paper 2 2020 November Q13

2 marks

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]

\includegraphics[alt={},max width=\textwidth]{cea67565-8074-4703-8e1a-09b98e380baf-14_629_1424_397_242} \captionsetup{labelformat=empty} \caption{Fig. 13.1}

\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 \(0.37154 > 0.05\).

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}
Use the information in Fig. 13.3 and Fig. 13.2 to show that there are at least two outliers.
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.

OCR MEI Paper 2 2020 November Q15

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
\(\mathrm { h } ( x ) = \mathrm { fg } ( x )\).

Find \(\mathrm { h } ( x )\) in terms of \(x\) and state its domain.
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}
Determine the gradient of \(\mathrm { y } = \mathrm { h } ^ { - 1 } ( \mathrm { x } )\) at the point where \(y = 3\).

OCR MEI Paper 2 2021 November Q1

1 The equation of a curve is \(y = 4 x ^ { 2 } + 8 x + 1\).
The curve is stretched parallel to the \(x\)-axis with scale factor 2 .
Find the equation of the new curve, giving your answer in the form \(\mathrm { y } = a \mathrm { x } ^ { 2 } + b \mathrm { x } + c\), where \(a , b\) and \(c\) are integers to be determined.

OCR MEI Paper 2 2021 November Q2

Write \(65 ^ { \circ }\) in radians, giving your answer in the form \(k \pi\), where \(k\) is a fraction in its lowest terms.
Write 0.211 radians in degrees, giving your answer correct to \(\mathbf { 1 }\) decimal place.

OCR MEI Paper 2 2021 November Q3

3 Draw a number line to show the values of \(x\) which belong to the set \(\{ x : x \geqslant 2 \} \cap \{ x : x < 7 \}\).

OCR MEI Paper 2 2021 November Q4

4 Sketch the graph of \(y = | 2 x - 3 |\).

OCR MEI Paper 2 2021 November Q5

5 It is known that 40\% of people in Britain carry a certain gene.
A random sample of 32 people is collected.

Calculate the probability that exactly 12 people carry the gene.
Calculate the probability that at least 8 people carry the gene, giving your answer correct to \(\mathbf { 3 }\) decimal places.

OCR MEI Paper 2 2021 November Q6

6 You are given that \(\mathbf { v } = 2 \mathbf { a } + 3 \mathbf { b }\), where \(\mathbf { a }\) and \(\mathbf { b }\) are the position vectors
\(\mathbf { a } = \binom { 5 } { 3 }\) and \(\mathbf { b } = \binom { - 1 } { 6 }\).

Determine the magnitude of \(\mathbf { v }\).
Determine the angle between \(\mathbf { v }\) and the vector \(\binom { 1 } { 0 }\).

OCR MEI Paper 2 2021 November Q7

7 The parametric equations of a circle are
\(x = 7 + 5 \cos \theta , \quad y = 5 \sin \theta - 3\), for \(0 \leqslant \theta \leqslant 2 \pi\).

Find a cartesian equation of the circle.
State the coordinates of the centre of the circle. Answer all the questions.
Section B (77 marks)

OCR MEI Paper 2 2021 November Q8

8 The Normal variable \(X\) is transformed to the Normal variable \(Y\).
The transformation is \(\mathrm { y } = \mathrm { a } + \mathrm { bx }\), where \(a\) and \(b\) are positive constants.
You are given that \(X \sim N ( 42,6.8 )\) and \(Y \sim N ( 57.2,11.492 )\).
Determine the values of \(a\) and \(b\).

OCR MEI Paper 2 2021 November Q9

9 Labrador puppies may be black, yellow or chocolate in colour. Some information about a litter of 9 puppies is given in the table.

	male	female
black	1	3
yellow	2	1
chocolate	1	1

Four puppies are chosen at random to train as guide dogs.

Determine the probability that exactly 3 females are chosen.
Determine the probability that at least 3 black puppies are chosen.
Determine the probability that exactly 3 females are chosen given that at least 3 black puppies are chosen.
Explain whether the 2 events ‘choosing exactly 3 females’ and ‘choosing at least 3 black puppies’ are independent events.

OCR MEI Paper 2 2021 November Q10

10 Ben has an interest in birdwatching. For many years he has identified, at the start of the year, 32 days on which he will spend an hour counting the number of birds he sees in his garden. He divides the year into four using the Meteorological Office definition of seasons. Each year he uses stratified sampling to identify the 32 days on which he will count the birds in his garden, drawn equally from the four seasons. Ben’s data for 2019 are shown in the stem and leaf diagram in Fig. 10.1. \begin{table}[h]

0	3	5	9	9	9
1	0	0	1	1	2	4	5	6	7	8	9
2	0	1	4	6	7	8	9
3	0	0	2	3
4	0	3	6
5	1
6	0

\captionsetup{labelformat=empty} \caption{Fig. 10.1}

\end{table}

Suggest a reason why Ben chose to use stratified sampling instead of simple random sampling.
Describe the shape of the distribution.
Explain why the mode is not a useful measure of central tendency in this case.
For Ben's sample, determine
- the median,
- the interquartile range.
Ben found a boxplot for the sample of size 32 he collected using stratified sampling in 2015. The boxplot is shown in Fig. 10.2. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{c9d14a4d-a1c8-42ad-9c0b-42cef6b3612f-06_483_1163_1982_242} \captionsetup{labelformat=empty} \caption{Fig. 10.2}
\end{figure} In 2016 Ben replaced his hedge with a garden fence.
Ben now believes that
- he sees fewer birds in his garden,
- the number of birds he sees in his garden is more variable.
- With reference to Fig. 10.2 and your answer to part (d), comment on whether there is any evidence to support Ben’s belief.
Jane says she can tell that the data for 2015 is definitely uniformly distributed by looking at the boxplot.
Explain why Jane is wrong.

OCR MEI Paper 2 2021 November Q11

11 In 2010 the heights of adult women in the UK were found to have mean \(\mu = 161.6 \mathrm {~cm}\) and variance \(\sigma ^ { 2 } = 1.96 \mathrm {~cm} ^ { 2 }\). It is believed that the mean height of adult women in 2020 in the UK is greater than in 2010. In 2020 a researcher collected a random sample of the heights of 200 adult women in the UK. The researcher calculated the sample mean height and carried out a hypothesis test at the \(5 \%\) level to investigate whether there was any evidence to suggest that the mean height of adult women in the UK had increased. The researcher assumed that the variance was unaltered.

- State suitable hypotheses for the test, defining any variables you use.
- Explain whether the researcher conducted a 1-tail or a 2-tail test.
- Determine the critical region for the test.
The researcher found that the sample mean was 161.9 cm and made the following statements.
- The sample mean is in the critical region.
- The null hypothesis is accepted.
- This proves that the mean height of adult women in the UK is unaltered at 161.6 cm .
- Explain whether each of these statements is correct.

OCR MEI Paper 2 2021 November Q12

12 Fig. 12.1 shows an excerpt from the pre-release material. \begin{table}[h]

	A	B	C	D	E	F	G	H
1	Sex	Age	Marital	Weight	Height	BMI	Waist	Pulse
2	Female	34	Married	60.3	173.4	20.05	82.5	74
3	Female	85	Widowed	64.7	161.2	24.9	\#N/A	\#N/A
4	Female	48	Divorced	100.6	171.4	34.24	105.6	92
5	Male	61	Married	70.9	169.5	24.68	92.2	70
6	Male	68	Divorced	96.8	181.6	29.35	112.9	68

\captionsetup{labelformat=empty} \caption{Fig. 12.1}

\end{table} There was no data available for cell H3.

Explain why \#N/A is used when no data is available. Fig. 12.2 shows a scatter diagram of pulse rate against BMI (Body Mass Index) for females. All the available data was used. Pulse rate against BMI for females \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{c9d14a4d-a1c8-42ad-9c0b-42cef6b3612f-08_659_1552_1363_233} \captionsetup{labelformat=empty} \caption{Fig. 12.2}
\end{figure} There are two outliers on the diagram.
On the copy of Fig. 12.2 in the Printed Answer Booklet, ring these outliers.
Use your knowledge of the pre-release material to explain whether either of these outliers should be removed.
State whether the diagram suggests there is any correlation between pulse rate and BMI. The product moment correlation coefficient between waist measurement, \(w\), in cm and BMI, \(b\), for females was found to be 0.912 . All the available data was used.
Explain why a model of the form \(\mathrm { w } = \mathrm { mb } + \mathrm { c }\) for the relationship between waist measurement and BMI is likely to be appropriate. The LINEST function on a spreadsheet gives \(m = 2.16\) and \(c = 33.0\).
Calculate an estimate of the value for cell G3 in Fig. 12.1.

OCR MEI Paper 2 2021 November Q13

13 At a certain factory Christmas tree decorations are packed in boxes of 10 . The quality control manager collects a random sample of 100 boxes of decorations and records the number of decorations in each box which are damaged. His results are displayed in Fig. 13.1. \begin{table}[h]

Number of damaged decorations	0	1	2	3	4	5 or more
Number of boxes	19	35	28	13	5	0

\captionsetup{labelformat=empty} \caption{Fig. 13.1}

\end{table}

Calculate
- the mean number of damaged decorations per box,
- the standard deviation of the number of damaged decorations per box.
It is believed that the number of damaged decorations in a box of 10, \(X\), may be modelled by a binomial distribution such that \(\mathrm { X } \sim \mathrm { B } ( \mathrm { n } , \mathrm { p } )\).
State suitable values for \(n\) and \(p\).
Use the binomial model to complete the copy of Fig. 13.2 in the Printed Answer Booklet, giving your answers correct to \(\mathbf { 1 }\) decimal place. \begin{table}[h]
Number of damaged decorations 0 1 2 3 4 5 or more
Observed number of boxes 19 35 28 13 5 0
Expected number of boxes
\captionsetup{labelformat=empty} \caption{Fig. 13.2}
\end{table}
Explain whether the model is a good fit for these data.

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

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

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

Number of damaged decorations	0	1	2	3	4	5 or more
Observed number of boxes	19	35	28	13	5	0
Expected number of boxes

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

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