Questions — OCR MEI Paper 2 (129 questions)

Browse by module
All questions AEA AS Paper 1 AS Paper 2 AS Pure C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Extra Pure Further Mechanics Further Mechanics A AS Further Mechanics AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics Further Paper 4 Further Pure Core Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Pure with Technology Further Statistics Further Statistics A AS Further Statistics AS Further Statistics B AS Further Statistics Major Further Statistics Minor Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 H240/01 H240/02 H240/03 M1 M2 M3 M4 M5 P1 P2 P3 P4 PMT Mocks PURE Paper 1 Paper 2 Paper 3 Pre-U 9794/1 Pre-U 9794/2 Pre-U 9794/3 Pre-U 9795 Pre-U 9795/1 Pre-U 9795/2 S1 S2 S3 S4 Unit 1 Unit 2 Unit 3 Unit 4
Browse by board
AQA AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Paper 1 Further Paper 2 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics M1 M2 M3 Paper 1 Paper 2 Paper 3 S1 S2 S3 CAIE FP1 FP2 Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 4 M1 M2 P1 P2 P3 S1 S2 Edexcel AEA AS Paper 1 AS Paper 2 C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS M1 M2 M3 M4 M5 P1 P2 P3 P4 PMT Mocks PURE Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 OCR AS Pure C1 C2 C3 C4 D1 D2 FD1 AS FM1 AS FP1 FP1 AS FP2 FP3 FS1 AS Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Mechanics Further Mechanics AS Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Statistics Further Statistics AS H240/01 H240/02 H240/03 M1 M2 M3 M4 PURE S1 S2 S3 S4 OCR MEI AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further Extra Pure Further Mechanics A AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Pure Core Further Pure Core AS Further Pure with Technology Further Statistics A AS Further Statistics B AS Further Statistics Major Further Statistics Minor M1 M2 M3 M4 Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 Pre-U Pre-U 9794/1 Pre-U 9794/2 Pre-U 9794/3 Pre-U 9795 Pre-U 9795/1 Pre-U 9795/2 WJEC Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 Unit 1 Unit 2 Unit 3 Unit 4
Sort by: Default | Easiest first | Hardest first
OCR MEI Paper 2 2021 November Q4
3 marks Easy -1.8
4 Sketch the graph of \(y = | 2 x - 3 |\).
OCR MEI Paper 2 2021 November Q5
3 marks Moderate -0.8
5 It is known that 40\% of people in Britain carry a certain gene.
A random sample of 32 people is collected.
  1. Calculate the probability that exactly 12 people carry the gene.
  2. 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
5 marks Moderate -0.8
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 \(\mathbf { b } = \binom { - 1 } { 6 }\).
  1. Determine the magnitude of \(\mathbf { v }\).
  2. Determine the angle between \(\mathbf { v }\) and the vector \(\binom { 1 } { 0 }\).
OCR MEI Paper 2 2021 November Q7
4 marks Easy -1.2
7 The parametric equations of a circle are \(x = 7 + 5 \cos \theta , \quad y = 5 \sin \theta - 3\), for \(0 \leqslant \theta \leqslant 2 \pi\).
  1. Find a cartesian equation of the circle.
  2. State the coordinates of the centre of the circle. Answer all the questions.
    Section B (77 marks)
OCR MEI Paper 2 2021 November Q8
4 marks Standard +0.3
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
10 marks Standard +0.8
9 Labrador puppies may be black, yellow or chocolate in colour. Some information about a litter of 9 puppies is given in the table.
malefemale
black13
yellow21
chocolate11
Four puppies are chosen at random to train as guide dogs.
  1. Determine the probability that exactly 3 females are chosen.
  2. Determine the probability that at least 3 black puppies are chosen.
  3. Determine the probability that exactly 3 females are chosen given that at least 3 black puppies are chosen.
  4. 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
9 marks Moderate -0.8
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]
035999
100112456789
20146789
30023
4036
51
60
\captionsetup{labelformat=empty} \caption{Fig. 10.1}
\end{table}
  1. Suggest a reason why Ben chose to use stratified sampling instead of simple random sampling.
  2. Describe the shape of the distribution.
  3. Explain why the mode is not a useful measure of central tendency in this case.
  4. For Ben's sample, determine
    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
    Jane says she can tell that the data for 2015 is definitely uniformly distributed by looking at the boxplot.
  5. Explain why Jane is wrong.
OCR MEI Paper 2 2021 November Q11
8 marks Standard +0.3
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.
  1. - State suitable hypotheses for the test, defining any variables you use.
    The researcher found that the sample mean was 161.9 cm and made the following statements.
OCR MEI Paper 2 2021 November Q12
5 marks Moderate -0.5
12 Fig. 12.1 shows an excerpt from the pre-release material. \begin{table}[h]
ABCDEFGH
1SexAgeMaritalWeightHeightBMIWaistPulse
2Female34Married60.3173.420.0582.574
3Female85Widowed64.7161.224.9\#N/A\#N/A
4Female48Divorced100.6171.434.24105.692
5Male61Married70.9169.524.6892.270
6Male68Divorced96.8181.629.35112.968
\captionsetup{labelformat=empty} \caption{Fig. 12.1}
\end{table} There was no data available for cell H3.
  1. 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.
  2. On the copy of Fig. 12.2 in the Printed Answer Booklet, ring these outliers.
  3. Use your knowledge of the pre-release material to explain whether either of these outliers should be removed.
  4. 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.
  5. 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 \(c = 33.0\).
  6. Calculate an estimate of the value for cell G3 in Fig. 12.1.
OCR MEI Paper 2 2021 November Q13
7 marks Moderate -0.3
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 decorations012345 or more
Number of boxes1935281350
\captionsetup{labelformat=empty} \caption{Fig. 13.1}
\end{table}
  1. Calculate
    It is believed that the number of damaged decorations in a box of 10, \(X\), may be modelled by a binomial distribution such that \(\mathrm { X } \sim \mathrm { B } ( \mathrm { n } , \mathrm { p } )\).
  2. State suitable values for \(n\) and \(p\).
  3. 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 decorations012345 or more
    Observed number of boxes1935281350
    Expected number of boxes
    \captionsetup{labelformat=empty} \caption{Fig. 13.2}
    \end{table}
  4. Explain whether the model is a good fit for these data.
OCR MEI Paper 2 2021 November Q14
13 marks Moderate -0.3
14 The equation of a curve is \(y = x ^ { 2 } ( x - 2 ) ^ { 3 }\).
  1. Find \(\frac { \mathrm { dy } } { \mathrm { dx } }\), giving your answer in factorised form.
  2. Determine the coordinates of the stationary points on the curve. In part (c) you may use the result \(\frac { d ^ { 2 } y } { d x ^ { 2 } } = 4 ( x - 2 ) \left( 5 x ^ { 2 } - 8 x + 2 \right)\).
  3. Determine the nature of the stationary points on the curve.
  4. Sketch the curve.
OCR MEI Paper 2 2021 November Q15
11 marks Moderate -0.8
15
  1. 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.
  2. Determine whether Kofi's statement is consistent with his model. In between matches, Layla practises, but Kofi does not.
  3. Explain why Layla might disagree with Kofi's model. Layla models the situation using the probability function \(\mathrm { P } ( \mathrm { X } = \mathrm { r } ) = \mathrm { kr } ^ { 2 } \quad r = 1,2,3,4,5,6,7,8\).
  4. 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.
  5. Determine whether Layla's statement is consistent with her model.
OCR MEI Paper 2 2021 November Q16
8 marks Standard +0.8
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 2024 June Q5
4 marks Easy -1.2
  1. In the Printed Answer Booklet, complete the copy of the two-way table.
  2. Calculate the probability that an A-level student selected at random does not study chemistry given that they do not study mathematics.
OCR MEI Paper 2 2020 November Q12
15 marks Standard +0.3
  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 .
OCR MEI Paper 2 2018 June Q12
5 marks Standard +0.3
12 You must show detailed reasoning in this question. In the summer of 2017 in England a large number of candidates sat GCSE examinations in both mathematics and English. 56\% of these candidates achieved at least level 4 in mathematics and \(80 \%\) of these candidates achieved at least level 4 in English. 14\% of these candidates did not achieve at least level 4 in either mathematics or English. Determine whether achieving level 4 or above in English and achieving level 4 or above in mathematics were independent events.
OCR MEI Paper 2 2019 June Q15
6 marks Challenging +1.2
15 You must show detailed reasoning in this question. The screenshot in Fig. 15 shows the probability distribution for the continuous random variable \(X\), where \(X \sim \mathrm {~N} \left( \mu , \sigma ^ { 2 } \right)\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{95eb3bcc-6d3c-4f7e-9b27-5e046ab57ec5-11_387_954_1599_260} \captionsetup{labelformat=empty} \caption{Fig. 15}
\end{figure} The distribution is symmetrical about the line \(x = 35\) and there is a point of inflection at \(x = 31\).
Fifty independent readings of \(X\) are made. Show that the probability that at least 45 of these readings are between 30 and 40 is less than 0.05 .
OCR MEI Paper 2 2023 June Q15
7 marks Standard +0.3
15 In this question you must show detailed reasoning. The equation of a curve is $$\ln y + x ^ { 3 } y = 8$$ Find the equation of the normal to the curve at the point where \(y = 1\), giving your answer in the form \(\mathrm { ax } + \mathrm { by } + \mathrm { c } = 0\), where \(a , b\) and \(c\) are constants to be found.
OCR MEI Paper 2 2023 June Q17
6 marks Standard +0.8
17 In this question you must show detailed reasoning. Solve the equation \(2 \sin x + \sec x = 4 \cos x\), where \(- \pi < x < \pi\).
OCR MEI Paper 2 2024 June Q16
12 marks Standard +0.3
16 In this question you must show detailed reasoning. Find the particular solution of the differential equation $$\frac { d y } { d x } = \frac { 9 y } { ( x - 1 ) ( x + 2 ) }$$ given that \(x = 2\) when \(y = 16\). \section*{END OF QUESTION PAPER}
OCR MEI Paper 2 2020 November Q10
9 marks Standard +0.3
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]
    \(n\)01234567
    \(\mathrm { x } _ { \mathrm { n } }\)10.9585090.9500840.9482610.947860.9477720.9477530.947748
    \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.
OCR MEI Paper 2 2020 November Q14
8 marks Challenging +1.2
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]
\includegraphics[alt={},max width=\textwidth]{cea67565-8074-4703-8e1a-09b98e380baf-16_647_898_404_233} \captionsetup{labelformat=empty} \caption{Fig. 14}
\end{figure} Use integration to find the area between the two curves, giving your answer in an exact form.
OCR MEI Paper 2 2022 June Q1
4 marks Moderate -0.8
Express \(\cos\theta + \sqrt{3}\sin\theta\) in the form \(R\cos(\theta - \alpha)\), where \(R\) and \(\alpha\) are exact values to be determined. [4]
OCR MEI Paper 2 2022 June Q2
2 marks Easy -1.2
Find the sum of the infinite series \(50 + 25 + 12.5 + 6.25 + \ldots\). [2]
OCR MEI Paper 2 2022 June Q3
6 marks Moderate -0.8
  1. On the axes in the Printed Answer Booklet, sketch the curve with equation \(y = 3 \times 0.4^x\). [3]
  2. Given that \(3 \times 0.4^x = 0.8\), determine the value of \(x\) correct to 3 significant figures. [3]