Questions Paper 2 - A-Level Maths

$x$	0	1	2	3
$\mathrm { P } ( \mathrm { X } = \mathrm { x } )$	0.2	$a$	$3 a$	0.4

Calculate the value of the constant $a$.
A single value of $X$ is chosen at random. Find the probability that the value is an odd number.
Two independent values of $X$ are chosen at random. Calculate the probability that the total of the two values is 3 .

OCR MEI Paper 2 2024 June Q7

7 A sequence is defined by the recurrence relation $\mathrm { u } _ { \mathrm { k } + 1 } = \mathrm { u } _ { \mathrm { k } } + 5$ with $\mathrm { u } _ { 1 } = - 2$.

Write down the values of $u _ { 2 } , u _ { 3 }$, and $u _ { 4 }$.
Explain whether this sequence is divergent or convergent.
Determine the value of $u _ { 30 }$.
Determine the value of $\sum _ { \mathrm { k } = 1 } ^ { 30 } \mathrm { u } _ { \mathrm { k } }$.

OCR MEI Paper 2 2024 June Q8

8 The equation of a curve is $y = 2 x ^ { 3 } + 3 m x ^ { 2 } - 9 m x + 4$. Determine the range of values of $m$ for which the curve has no stationary values.

OCR MEI Paper 2 2024 June Q9

9 A teacher is investigating how pupils travel to and from school each day. Pupils can either travel by bus, train, car, bicycle or walk. The teacher decides to collect a sample of size 60 for the investigation.

The teacher lives in a village 10 miles away from the school. Explain how collecting a sample which just consists of pupils who live in the same village as the teacher might introduce bias. The table below shows how many students there are in each year.
Year 7 Year 8 Year 9 Year 10 Year 11
86 105 107 101 101
The teacher decides to use the method of proportional stratified sampling. Calculate the number of pupils in the sample who are in Year 9. The teacher generates a sample of 10 pupils from the 86 in Year 7 by listing them in alphabetical order and selecting the first name on the list and every ninth name thereafter.
Explain whether this method will generate a simple random sample of the pupils who travel in Year 7.

OCR MEI Paper 2 2024 June Q10

Determine the first three terms in ascending powers of $x$ of the binomial expansion of $( 8 + 3 x ) ^ { \frac { 1 } { 3 } }$.
State the range of values of $x$ for which this expansion is valid.

OCR MEI Paper 2 2024 June Q11

11 A householder is investigating whether there is any relationship between his monthly cost of gas and his monthly cost of electricity, both measured in pounds ( $\pounds$ ). The householder collects a random sample of monthly costs and presents them in the scatter diagram below.
\includegraphics[max width=\textwidth, alt={}, center]{8e48bbd3-2166-49e7-8906-833261f331ca-08_604_1452_392_244} One of the points on the diagram represents the energy costs in a month when the householder was away on holiday for three weeks. The other points represent the energy costs in months when the householder did not go away on holiday.

On the copy of the diagram in the Printed Answer Booklet, circle the point which represents the month when the householder was most likely to have been away on holiday for three weeks.
With reference to the diagram, describe the relationship between the cost of gas and the cost of electricity. The householder decides to test whether there is evidence to suggest that there is any association between the monthly cost of gas and the monthly cost of electricity. The value of Spearman's rank correlation coefficient for this sample is 0.4359 and the associated $p$-value is 0.09195 .
Determine whether there is any evidence to suggest, at the $5 \%$ level, that there is any association between the monthly cost of gas and the monthly cost of electricity.

OCR MEI Paper 2 2024 June Q12

12 A survey conducted in 2021 showed that 10\% of British adults were vegetarians. A dietitian believes that the proportion of British adults who are vegetarians may have changed, so decides to conduct a hypothesis test at the $5 \%$ level of significance. In a random sample of 112 adults, the dietitian finds that there are 19 vegetarians. Carry out the hypothesis test to determine whether there is any evidence to support the dietitian’s belief.

OCR MEI Paper 2 2024 June Q13

13 Determine the coordinates of the turning points on the curve with equation $$y ^ { 2 } + x y + x ^ { 2 } - x = 1 .$$

OCR MEI Paper 2 2024 June Q14

14 The pre-release material contains medical data for 103 women and 97 men.
The boxplot represents the weights in kg of 101 of the women from the pre-release material.
\includegraphics[max width=\textwidth, alt={}, center]{8e48bbd3-2166-49e7-8906-833261f331ca-09_421_1232_735_244}

Use your knowledge of the pre-release material to give a reason why the weights of all 103 women were not included in the diagram.
Determine the range of values in which any outliers lie.
Use your knowledge of the pre-release material to explain whether these outliers should be removed from any further analysis of the data.
The median weight of men in the sample was found to be 79.9 kg . Explain what may be inferred by comparing the median weight of men with the median weight of women. Further analysis of the weights of both men and women is carried out. The table shows some of the results.
mean standard deviation
men 82.69 kg 19.98 kg
women 72.5 kg 19.95 kg
Use the information in the table to make two inferences about the distribution of the weights of men compared with the distribution of the weights of women.

OCR MEI Paper 2 2024 June Q15

15 Bottles of Fizzipop nominally contain 330 ml of drink. A consumer affairs researcher collects a random sample of 55 bottles of Fizzipop and records the volume of drink in each bottle. Summary statistics for the researcher's sample are shown in the table.

$n$	55
$\sum x$	18535
$\sum x ^ { 2 }$	6247066.6

1. Calculate the mean volume of drink in a bottle of Fizzipop.
2. Show that the standard deviation of the volume of drink in a bottle of Fizzipop is 3.78 ml . The researcher uses software to produce a histogram with equal class intervals, which is shown below.
  \includegraphics[max width=\textwidth, alt={}, center]{8e48bbd3-2166-49e7-8906-833261f331ca-10_533_759_1181_251}
Explain why the researcher decides that the Normal distribution is a suitable model for the volume of drink in a bottle of Fizzipop.
Use your answers to parts (a) and (b) to determine the expected number of bottles which contain less than 330 ml in a random sample of 100 bottles. In order to comply with new regulations, no more than 1\% of bottles of Fizzipop should contain less than 330 ml . The manufacturer decides to meet the new regulations by adjusting the manufacturing process so that the mean volume of drink in a bottle of Fizzipop is increased. The standard deviation is unaltered.
Determine the minimum mean volume of drink in a bottle of Fizzipop which should ensure that the new regulations are met. Give your answer to $\mathbf { 3 }$ significant figures. The mean volume of drink in a bottle of Fizzipop is set to 340 ml . After several weeks the quality control manager suspects the mean volume may have reduced. She collects a random sample of 100 bottles of Fizzipop. The mean volume of drink in a bottle in the sample is found to be 339.37 ml .
Assuming the standard deviation is unaltered, conduct a hypothesis test at the $5 \%$ level to determine whether there is any evidence to suggest that the mean volume of drink in a bottle of Fizzipop is less than 340 ml .

OCR MEI Paper 2 2020 November Q1

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.

OCR MEI Paper 2 2020 November Q2

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]

\includegraphics[alt={},max width=\textwidth]{cea67565-8074-4703-8e1a-09b98e380baf-04_423_296_1366_246} \captionsetup{labelformat=empty} \caption{Fig. 2}

\end{figure}

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 $\{ x : x \geqslant 2 \} \cap \{ x : x < 7 \}$.

\(x\)	0	1	2	3
\(\mathrm { P } ( \mathrm { X } = \mathrm { x } )\)	0.2	\(a\)	\(3 a\)	0.4

	mean	standard deviation
men	82.69 kg	19.98 kg
women	72.5 kg	19.95 kg

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

Year 7	Year 8	Year 9	Year 10	Year 11
86	105	107	101	101

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

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