Questions - OCR MEI AS Paper 2

OCR MEI AS Paper 2 2022 June Q7

7

On the pair of axes in the Printed Answer Booklet, sketch the graphs of
- $y = 2 x + 4$
- $\mathrm { y } = \frac { 2 } { \mathrm { x } }$
- Determine the $x$-coordinates of the points of intersection of the line $y = 2 x + 4$ and the curve $\mathrm { y } = \frac { 2 } { \mathrm { x } }$, giving your answers in an exact form.

OCR MEI AS Paper 2 2022 June Q8

8 In 2018 research showed that 81\% of young adults in England had never donated blood.
Following an advertising campaign in 2021, it is believed that the percentage of young adults in England who had never donated blood in 2021 is less than $81 \%$. Ling decides to carry out a hypothesis test at the 5\% level.
Ling collects data from a random sample of 400 young adults in England.

State the null and alternative hypotheses for the test, defining the parameter used.
Write down the probability that the null hypothesis is rejected when it should in fact be accepted.
Assuming the null hypothesis is correct, calculate the expected number of young adults in the sample who had never donated blood.
Calculate the probability that there were no more than 308 young adults who had never donated blood in the sample.
Determine the critical region for the test. In fact, the sample contained 314 young adults who had never donated blood.
Carry out the test, giving the conclusion in the context of the question.

OCR MEI AS Paper 2 2022 June Q9

9 The equation of a curve is $y = 12 x - 4 x ^ { \frac { 3 } { 2 } }$.

State the coordinates of the intersection of the curve with the $y$-axis.
Find the value of $y$ when $x = 9$.
Determine the coordinates of the stationary point.
Sketch the curve, giving the coordinates of the stationary point and of any intercepts with the axes.

OCR MEI AS Paper 2 2022 June Q10

10 In this question you must show detailed reasoning.
The equation of a curve is $y = 12 x ^ { 3 } - 24 x ^ { 2 } - 60 x + 72$.
Determine the magnitude of the total area bounded by the curve and the $x$-axis.

OCR MEI AS Paper 2 2022 June Q11

11 The pre-release material contains information about the Median Income of Taxpayers and the Percentage of Pupils Achieving at Least 5 A*- C grades, including English and Maths, at the end of KS4 in different areas of London. Alex is investigating whether there is a relationship between median income and the percentage of pupils achieving at least 5 A* - C grades, including English and Maths, at the end of KS4. Alex decides to use the first 12 rows of data for 2014-5 from the pre-release data as a sample. The sample is shown in Fig. 11.1. \begin{table}[h]

Area	Median Income of Taxpayers	Percentage of Pupils Achieving at Least 5 A*- C grades including English and Maths
City of London	61100	\#N/A
Barking and Dagenham	21800	54.0
Barnet	27100	70.1
Bexley	24400	55.0
Brent	22700	60.0
Bromley	28100	68.0
Camden	33100	56.4
Croydon	25100	59.6
Ealing	24600	62.1
Enfield	25300	54.5
Greenwich	24600	57.7
Hackney	26000	60.4

\captionsetup{labelformat=empty} \caption{Fig. 11.1}

\end{table}

Explain whether the data in Fig. 11.1 is a simple random sample of the data for 2014-5.
The City of London is included in Alex's sample. Explain why Alex is not able to use the data for the City of London in this investigation. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Fig. 11.2 shows a scatter diagram showing Percentage of Pupils against Median Income for all of the areas of London for which data is available.} \includegraphics[alt={},max width=\textwidth]{e0b502a8-c742-4d78-993c-8c0c7329ec9c-09_716_1378_356_244}
\end{figure} Fig. 11.2 Alex identifies some outliers.
On the copy of Fig. 11.2 in the Printed Answer Booklet, ring three of these outliers. Alex then discards all the outliers and uses the LINEST function on a spreadsheet to obtain the following model.
$\mathrm { P } = 0.0009049 \mathrm { M } + 37.38$,
where $P =$ percentage of pupils and $M =$ median income.
Show that the model is a good fit for the data for Hackney.
Use the model to find an estimate of the value of $P$ for City of London.
Give two reasons why this estimate may not be reliable. Alex states that more than 50\% of the pupils in London achieved at least a grade C at the end of KS4 in English and Maths in 2014-5.
Use the information in Fig. 11.2 together with your knowledge of the pre-release material to explain whether there is evidence to support this statement.

OCR MEI AS Paper 2 2023 June Q1

1 A researcher collects data concerning the number of different social media platforms used by school pupils on a typical weekday. The frequency table for the data is shown below.

Number of different social media platforms	0	1	2	3	4	5	6	7
Frequency	2	5	9	15	8	5	4	1

The researcher uses software to represent the results in this diagram.
\includegraphics[max width=\textwidth, alt={}, center]{82438df0-6550-4ffd-92d8-3c67bec59a6b-04_961_1195_737_242}

Explain why this diagram is inappropriate.
Calculate the following for the number of social media platforms used:
1. the mean,
2. the standard deviation.

OCR MEI AS Paper 2 2023 June Q2

2

Express $x ^ { 2 } - 6 x + 1$ in the form $( \mathrm { x } - \mathrm { a } ) ^ { 2 } - \mathrm { b }$, where $a$ and $b$ are integers to be determined.
Hence state the coordinates of the turning point on the graph of $y = x ^ { 2 } - 6 x + 1$.

OCR MEI AS Paper 2 2023 June Q3

3 A student makes the following conjecture.
For all positive integers $n , 6 n - 1$ is always prime. Use a counter example to disprove this conjecture.

OCR MEI AS Paper 2 2023 June Q4

4 The equation of a curve is $\mathrm { y } = \frac { \mathrm { k } } { \mathrm { x } ^ { 2 } }$, where $k$ is a constant.
The curve passes through the point $( 2,1 )$.

Find the value of $k$.
Sketch the curve.

OCR MEI AS Paper 2 2023 June Q5

5 Show that the distance between the points $( 5,2 )$ and $( 11 , - 1 )$ is $a \sqrt { b }$, where $a$ and $b$ are integers to be determined.

OCR MEI AS Paper 2 2023 June Q6

6 An app on my new smartphone records the number of times in a day I use the phone. The data for each day since I bought the phone are shown in the stem and leaf diagram.

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

Key: 3|1 means 31

Explain whether these data are a sample or a population.
Describe the shape of the distribution.
Determine the interquartile range.
Use your answer to part (c) to determine whether there are any outliers in the lower tail.

OCR MEI AS Paper 2 2023 June Q7

7

Use the factor theorem to show that $( x - 2 )$ is a factor of $x ^ { 3 } + 6 x ^ { 2 } - x - 30$.
Factorise $x ^ { 3 } + 6 x ^ { 2 } - x - 30$ completely.

OCR MEI AS Paper 2 2023 June Q8

8 The pre-release material contains information on Pulse Rate and Body Mass Index (BMI). A student is investigating whether there is a relationship between pulse rate and BMI. A section of the available data is shown in the table.

Sex	Age	BMI	Pulse
Male	62	29.54	60
Female	20	23.68	\#N/A
Male	17	26.97	72
Male	35	24.7	64
Male	17	20.09	54
Male	85	23.86	54
Female	81	24.04	\#N/A

The student decides to draw a scatter diagram.

With reference to the table, explain which data should be cleaned before any analysis takes place. The student cleans the data for BMI and Pulse Rate in the pre-release material and draws a scatter diagram. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Scatter diagram of Pulse Rate against BMI} \includegraphics[alt={},max width=\textwidth]{82438df0-6550-4ffd-92d8-3c67bec59a6b-06_869_1575_1585_246}
\end{figure} The student identifies one outlier.
On the copy of the scatter diagram in the Printed Answer Booklet, circle this outlier. The student decides to remove this outlier from the data. They then use the LINEST function in the spreadsheet to obtain the following formula for the line of best fit.
$\mathrm { P } = 0.29 \mathrm { Q } + 64.2$,
where $P =$ PulseRate and $Q = \mathrm { BMI }$. They use this to estimate the Pulse Rate of a person with BMI 23.68.
They obtain a value of 71 correct to the nearest whole number.
With reference to the scatter diagram, explain whether it is appropriate to use the formula for the line of best fit. It is suggested that all pairs of values where the pulse rate is above 100 should also be cleaned from the data, as they must be incorrect.
Use your knowledge of the pre-release material to explain whether or not all pairs of values with a pulse rate of more than 100 should be cleaned from the data.

OCR MEI AS Paper 2 2023 June Q9

9 The table shows the probability distribution for the discrete random variable $X$.

$x$	1	2	3	4	5
$\mathrm { P } ( \mathrm { X } = \mathrm { x } )$	0.1	0.3	$q$	$2 q$	$3 q$

You are given that $q$ is a positive constant.

Determine the value of $q$.
Calculate $\mathrm { P } ( X \leqslant 4 )$. Two independent values of $X$ are taken.
Determine the probability that the sum of the two values is 3 . Fifty independent values of $X$ are taken.
Find the probability that a value of 2 occurs exactly 17 times.

OCR MEI AS Paper 2 2023 June Q10

10 In this question you must show detailed reasoning.
The diagram shows triangle ABC , where $\mathrm { AB } = 3.9 \mathrm {~cm} , \mathrm { BC } = 4.5 \mathrm {~cm}$ and $\mathrm { AC } = 3.5 \mathrm {~cm}$. Determine the area of triangle ABC .

OCR MEI AS Paper 2 2023 June Q11

11 In this question you must show detailed reasoning.
The equation of a curve is $y = 2 x ^ { 3 } + 9 x ^ { 2 } + 24 x - 8$.
Show that there are no stationary points on this curve.

OCR MEI AS Paper 2 2023 June Q12

12 Doctors are investigating the weights of adult males registered at their surgery. One week they collect a sample by noting the weight in kilograms of all the adult males who have an appointment at their surgery.

State the sampling method they use.
Explain why this method will not generate a simple random sample of all the adult males registered at their surgery. They represent the data using a histogram.
\includegraphics[max width=\textwidth, alt={}, center]{82438df0-6550-4ffd-92d8-3c67bec59a6b-09_1166_1243_726_233} An incomplete frequency table for the data is shown below.
Weight in kg $50 -$ $65 -$ $75 -$ $80 -$ $90 -$ $100 - 120$
Frequency 8
Complete the copy of the frequency table in the Printed Answer Booklet. One of these patients is selected at random.
Determine an estimate of the probability that he weighs either less than 60 kg or more than 110 kg .
Explain why your answer to part (d) is an estimate and not exact.

OCR MEI AS Paper 2 2023 June Q13

13 In a report published in October 2021 it is stated that $37 \%$ of adults in the United Kingdom never exercise or play sport. A researcher believes that the true percentage is less than this. They decide to carry out a hypothesis test at the $5 \%$ level to investigate the claim.

State the null and alternative hypotheses for their test.
Define the parameter for their test. In a random sample of 118 adults, they find that 35 of them never exercise or play sport.
Carry out the test.

OCR MEI AS Paper 2 2023 June Q15

15 A family is planning a holiday in Europe. They need to buy some euros before they go. The exchange rate, $y$, is the number of euros they can buy per pound. They believe that the exchange rate may be modelled by the formula
$y = a t ^ { 2 } + b t + c$,
where $t$ is the time in days from when they first check the exchange rate.
Initially, when $t = 0$, the exchange rate is 1.14 .

Write down the value of $c$. When $t = 2 , y = 1.20$ and when $t = 4 , y = 1.25$.
Calculate the values of $a$ and $b$. The family will only buy their euros when their model predicts an exchange rate of at least 1.29 .
Determine the range of values of $t$ for which, according to their model, they will buy their euros.
Explain why the family's model is not viable in the long run.

OCR MEI AS Paper 2 2024 June Q1

1 Express $2 x ( x + 3 ) + 5 x ^ { 2 } - 2 ( x - 3 )$ in the form $a x ^ { 2 } + b x + c$, where $a , b$ and $c$ are integers to be determined.

OCR MEI AS Paper 2 2024 June Q2

2

Find the discriminant of the equation $3 x ^ { 2 } - 2 x + 5 = 0$.
Use your answer to part (a) to find the number of real roots of the equation $3 x ^ { 2 } - 2 x + 5 = 0$.

OCR MEI AS Paper 2 2024 June Q3

3 A student conducts an investigation into the number of hours spent cooking per week by people who live in village A. The student represents the data in the cumulative frequency diagram below. \section*{Hours spent cooking per week by people who live in village A} \includegraphics[max width=\textwidth, alt={}, center]{ce94c1ea-ffe5-42d0-8f8a-43c47105d6bf-3_796_1494_918_233}

How many people were involved in the investigation?
Use the copy of the diagram in the Printed Answer Booklet to determine an estimate for the interquartile range. The student conducts a similar investigation into the number of hours spent cooking per week by 200 people who live in village B. The interquartile range is found to be 3.9 hours.
Explain whether the evidence suggests that the number of hours spent cooking by people who live in village B is more variable, equally variable or less variable than the number of hours spent cooking by people who live in village A .

OCR MEI AS Paper 2 2024 June Q4

4 In this question you must show detailed reasoning.
Express $\frac { 1 + 4 \sqrt { 3 } } { 2 + \sqrt { 3 } }$ in the form $\mathrm { a } + \mathrm { b } \sqrt { 3 }$, where $a$ and $b$ are integers to be determined.

OCR MEI AS Paper 2 2024 June Q5

5 The pre-release material contains information for countries in the world concerning real GDP per capita in US\$ and mobile phone subscribers per 100 population. In an investigation into the relationship between these two variables, a student takes a sample of 20 countries in Africa. The student draws a scatter diagram for the data, which is shown in Fig. 5.1. \section*{Fig. 5.1} \section*{Africa 1st sample} \includegraphics[max width=\textwidth, alt={}, center]{ce94c1ea-ffe5-42d0-8f8a-43c47105d6bf-4_433_1043_842_244}

What does Fig. 5.1 suggest about the relationship between real GDP per capita and the number of mobile phone subscribers per 100 population? Another student collects a different sample of 20 countries from Africa, and draws a scatter diagram for the data, which is shown in Fig. 5.2. \section*{Fig. 5.2} \section*{Africa 2nd sample}
\includegraphics[max width=\textwidth, alt={}]{ce94c1ea-ffe5-42d0-8f8a-43c47105d6bf-4_273_1084_1818_244}
Mobile phone subscribers per 100 population
What does Fig. 5.2 suggest about the relationship between real GDP per capita and the number of mobile phone subscribers per 100 population?
Explain whether either of the two scatter diagrams is likely to be representative of the true relationship between real GDP per capita and the number of mobile phone subscribers per 100 population, for countries in Africa.

OCR MEI AS Paper 2 2024 June Q6

6 Determine the equation of the line which passes through the point $( 4 , - 1 )$ and is perpendicular to the line with equation $2 x + 3 y = 6$. Give your answer in the form $y = m x + c$, where $m$ is a fraction in its lowest terms and $c$ is an integer.

\(x\)	1	2	3	4	5
\(\mathrm { P } ( \mathrm { X } = \mathrm { x } )\)	0.1	0.3	\(q\)	\(2 q\)	\(3 q\)

Weight in kg	\(50 -\)	\(65 -\)	\(75 -\)	\(80 -\)	\(90 -\)	\(100 - 120\)
Frequency		8

Questions — OCR MEI AS Paper 2 (98 questions)

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