Questions Paper 2 - A-Level Maths

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.

OCR MEI Paper 2 2021 November Q14

14 The equation of a curve is
\(y = x ^ { 2 } ( x - 2 ) ^ { 3 }\).

Find \(\frac { \mathrm { dy } } { \mathrm { dx } }\), giving your answer in factorised form.
Determine the coordinates of the stationary points on the curve. In part (c) you may use the result \(\frac { d ^ { 2 } y } { d x ^ { 2 } } = 4 ( x - 2 ) \left( 5 x ^ { 2 } - 8 x + 2 \right)\).
Determine the nature of the stationary points on the curve.
Sketch the curve.

OCR MEI Paper 2 2021 November Q15

Show that \(\sum _ { r = 1 } ^ { \infty } 0.99 ^ { r - 1 } \times 0.01 = 1\). Kofi is a very good table tennis player. Layla is determined to beat him.
Every week they play one match of table tennis against each other. They will stop playing when Layla wins the match for the first time.
\(X\) is the discrete random variable "the number of matches they play in total". Kofi models the situation using the probability function
\(\mathrm { P } ( \mathrm { X } = \mathrm { r } ) = 0.99 ^ { \mathrm { r } - 1 } \times 0.01 \quad r = 1,2,3,4 , \ldots\) Kofi states that he is \(95 \%\) certain that Layla will not beat him within 6 years.
Determine whether Kofi's statement is consistent with his model. In between matches, Layla practises, but Kofi does not.
Explain why Layla might disagree with Kofi's model. Layla models the situation using the probability function
\(\mathrm { P } ( \mathrm { X } = \mathrm { r } ) = \mathrm { kr } ^ { 2 } \quad r = 1,2,3,4,5,6,7,8\).
Explain how Layla's model takes into account the fact that she practises between matches, but Kofi’s does not. Layla states that she is \(95 \%\) certain that she will beat Kofi within the first 6 matches.
Determine whether Layla’s statement is consistent with her model.

OCR MEI Paper 2 2021 November Q16

16 In this question you must show detailed reasoning.
Find \(\int \frac { x } { 1 + \sqrt { x } } d x\). END OF QUESTION PAPER

OCR MEI Paper 2 Specimen Q2

2 Given that \(\mathrm { f } ( x ) = x ^ { 3 }\) and \(\mathrm { g } ( x ) = 2 x ^ { 3 } - 1\), describe a sequence of two transformations which maps the curve \(y = \mathrm { f } ( x )\) onto the curve \(y = \mathrm { g } ( x )\).

OCR MEI Paper 2 Specimen Q3

3 Evaluate \(\int _ { 0 } ^ { \frac { \pi } { 12 } } \cos 3 x \mathrm {~d} x\), giving your answer in exact form.

OCR MEI Paper 2 Specimen Q4

4 The function \(\mathrm { f } ( x )\) is defined by \(\mathrm { f } ( x ) = x ^ { 3 } - 4\) for \(- 1 \leq x \leq 2\).
For \(\mathrm { f } ^ { - 1 } ( x )\), determine

the domain
the range.

OCR MEI Paper 2 Specimen Q5

5 In a particular country, \(8 \%\) of the population has blue eyes. A random sample of 20 people is selected from this population.
Find the probability that exactly two of these people have blue eyes.

OCR MEI Paper 2 Specimen Q6

6 Each day, for many years, the maximum temperature in degrees Celsius at a particular location is recorded. The maximum temperatures for days in October can be modelled by a Normal distribution. The appropriate Normal curve is shown in Fig. 6. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{e9f3a5f3-210b-453d-9ff5-8518340f5689-05_761_1397_484_246} \captionsetup{labelformat=empty} \caption{Fig. 6}

\end{figure}

1. Use the model to write down the mean of the maximum temperatures.
2. Explain why the curve indicates that the standard deviation is approximately 3 degrees Celsius. Temperatures can be converted from Celsius to Fahrenheit using the formula \(F = 1.8 C + 32\), where \(F\) is the temperature in degrees Fahrenheit and \(C\) is the temperature in degrees Celsius.
For maximum temperature in October in degrees Fahrenheit, estimate
- the mean
- the standard deviation.
\begin{displayquote} Answer all the questions.
Section B (77 marks) \end{displayquote} \(7 \quad\) Two events \(A\) and \(B\) are such that \(\mathrm { P } ( A ) = 0.6 , \mathrm { P } ( B ) = 0.5\) and \(\mathrm { P } ( A \cup B ) = 0.85\). Find \(\mathrm { P } ( A \mid B )\).

OCR MEI Paper 2 Specimen Q8

8 Alison selects 10 of her male friends. For each one she measures the distance between his eyes. The distances, measured in mm , are as follows:

The mean of these data is 61.4 . The sample standard deviation is 5.232 , correct to 3 decimal places. One of the friends decides he does not want his measurement to be used. Alison replaces his measurement with the measurement from another male friend. This increases the mean to 62.0 and reduces the standard deviation. Give a possible value for the measurement which has been removed and find the measurement which has replaced it.

OCR MEI Paper 2 Specimen Q9

9 A geyser is a hot spring which erupts from time to time. For two geysers, the duration of each eruption, \(x\) minutes, and the waiting time until the next eruption, \(y\) minutes, are recorded.

For a random sample of 50 eruptions of the first geyser, the correlation coefficient between \(x\) and \(y\) is 0.758 .
The critical value for a 2 -tailed hypothesis test for correlation at the \(5 \%\) level is 0.279 . Explain whether or not there is evidence of correlation in the population of eruptions. The scatter diagram in Fig. 9 shows the data from a random sample of 50 eruptions of the second geyser. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{e9f3a5f3-210b-453d-9ff5-8518340f5689-07_794_1298_383_251} \captionsetup{labelformat=empty} \caption{Fig. 9}
\end{figure}
Stella claims the scatter diagram shows evidence of correlation between duration of eruption and waiting time. Make two comments about Stella's claim.

OCR MEI Paper 2 Specimen Q10

10 A researcher wants to find out how many adults in a large town use the internet at least once a week. The researcher has formulated a suitable question to ask. For each of the following methods of taking a sample of the adults in the town, give a reason why the method may be biased. Method A: Ask people walking along a particular street between 9 am and 5 pm on one Monday.
Method B: Put the question through every letter box in the town and ask people to send back answers.
Method C: Put the question on the local council website for people to answer online.

OCR MEI Paper 2 Specimen Q11

11 Suppose \(x\) is an irrational number, and \(y\) is a rational number, so that \(y = \frac { m } { n }\), where \(m\) and \(n\) are integers and \(n \neq 0\).
Prove by contradiction that \(x + y\) is not rational.

OCR MEI Paper 2 Specimen Q12

12 Fig. 12 shows the curve \(2 x ^ { 3 } + y ^ { 3 } = 5 y\).
\includegraphics[max width=\textwidth, alt={}, center]{e9f3a5f3-210b-453d-9ff5-8518340f5689-08_841_606_900_212}

Find the gradient of the curve \(2 x ^ { 3 } + y ^ { 3 } = 5 y\) at the point \(( 1,2 )\), giving your answer in exact form.
Show that all the stationary points of the curve lie on the \(y\)-axis.

OCR MEI Paper 2 Specimen Q13

13 Evaluate \(\int _ { 0 } ^ { 1 } \frac { 1 } { 1 + \sqrt { x } } \mathrm {~d} x\), giving your answer in the form \(a + b \ln c\), where \(a , b\) and \(c\) are integers.

OCR MEI Paper 2 Specimen Q14

14 In a chemical reaction, the mass \(m\) grams of a chemical at time \(t\) minutes is modelled by the differential equation
\(\frac { \mathrm { d } m } { \mathrm {~d} t } = \frac { m } { t ( 1 + 2 t ) }\).
At time 1 minute, the mass of the chemical is 1 gram.

Solve the differential equation to show that \(m = \frac { 3 t } { ( 1 + 2 t ) }\).
Hence
1. find the time when the mass is 1.25 grams,
2. show what happens to the mass of the chemical as \(t\) becomes large.

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

Questions Paper 2 (402 questions)

Browse by module

Browse by board

OCR MEI Paper 2 2021 November Q4

OCR MEI Paper 2 2021 November Q5

OCR MEI Paper 2 2021 November Q6

OCR MEI Paper 2 2021 November Q7

OCR MEI Paper 2 2021 November Q8

OCR MEI Paper 2 2021 November Q9

OCR MEI Paper 2 2021 November Q10

OCR MEI Paper 2 2021 November Q11

OCR MEI Paper 2 2021 November Q12

OCR MEI Paper 2 2021 November Q13

OCR MEI Paper 2 2021 November Q14

OCR MEI Paper 2 2021 November Q15

OCR MEI Paper 2 2021 November Q16

OCR MEI Paper 2 Specimen Q2

OCR MEI Paper 2 Specimen Q3

OCR MEI Paper 2 Specimen Q4

OCR MEI Paper 2 Specimen Q5

OCR MEI Paper 2 Specimen Q6

OCR MEI Paper 2 Specimen Q8

OCR MEI Paper 2 Specimen Q9

OCR MEI Paper 2 Specimen Q10

OCR MEI Paper 2 Specimen Q11

OCR MEI Paper 2 Specimen Q12

OCR MEI Paper 2 Specimen Q13

OCR MEI Paper 2 Specimen Q14