Questions — OCR MEI Paper 2 (129 questions)

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OCR MEI Paper 2 2022 June Q4
4 marks Easy -1.3
A survey of university students revealed that
  • 31\% have a part-time job but do not play competitive sport.
  • 23\% play competitive sport but do not have a part-time job.
  • 22\% do not play competitive sport and do not have a part-time job.
  1. Show this information on a Venn diagram. [2]
A student is selected at random.
  1. Determine the probability that the student plays competitive sport and has a part-time job. [2]
OCR MEI Paper 2 2022 June Q5
3 marks Standard +0.3
Tom conjectures that if \(n\) is an odd number greater than 1, then \(2^n - 1\) is prime. Find a counter example to disprove Tom's conjecture. [3]
OCR MEI Paper 2 2022 June Q6
2 marks Easy -1.8
\(X\) is a continuous random variable such that \(X \sim N(\mu, \sigma^2)\). On the sketch of this Normal distribution in the Printed Answer Booklet, shade the area bounded by the curve, the \(x\)-axis and the lines \(x = \mu \pm \sigma\). [2]
OCR MEI Paper 2 2022 June Q7
2 marks Easy -1.2
Kareem bought some tomatoes. He recorded the mass of each tomato and displayed the results in a histogram, which is shown below. \includegraphics{figure_7} Determine how many tomatoes Kareem bought. [2]
OCR MEI Paper 2 2022 June Q8
3 marks Easy -1.8
Ali conducted an investigation into the distances ridden by those members of a cycling club who rode at least 120 km in a training week. She grouped all the distances into intervals of length 10 km and then constructed a cumulative frequency diagram, which is shown below. \includegraphics{figure_8}
  1. Explain whether the data Ali used is a sample or a population. [1]
The club is taking part in a competition. Eight team members and one reserve are to be selected. The club captain decides that the team members should be those cyclists who rode the furthest during the training week, and that the reserve should be the cyclist who rode the next furthest.
  1. Use the graph to estimate the shortest distance cycled by a team member. [1]
The captain's best friend rode 156 km in the training week and was selected as reserve. Ali complained that this was unjustifiable.
  1. Explain whether there is sufficient evidence in the diagram to support Ali's complaint. [1]
OCR MEI Paper 2 2022 June Q9
9 marks Moderate -0.3
At the beginning of the academic year, all the pupils in year 12 at a college take part in an assessment. Summary statistics for the marks obtained by the 2021 cohort are given below. \(n = 205 \quad \sum x = 23042 \quad \sum x^2 = 2591716\) Marks may only be whole numbers, but the Head of Mathematics believes that the distribution of marks may be modelled by a Normal distribution.
  1. Calculate
    [2]
  2. Use your answers to part (a) to write down a possible Normal model for the distribution of marks. [2]
One candidate in the cohort scored less than 105.
  1. Determine whether the model found in part (b) is consistent with this information. [3]
  2. Use the model to calculate an estimate of the number of candidates who scored 115 marks. [2]
OCR MEI Paper 2 2022 June Q10
7 marks Moderate -0.8
The parametric equations of a curve are \(x = 2 + 5\cos\theta\) and \(y = 1 + 5\sin\theta\), where \(0 \leq \theta < 2\pi\).
  1. Determine the cartesian equation of the curve. [3]
  2. Hence or otherwise, find the equation of the tangent to the curve at the point \((5, -3)\), giving your answer in the form \(ax + by + c = 0\), where \(a\), \(b\) and \(c\) are integers to be determined. [4]
OCR MEI Paper 2 2022 June Q11
10 marks Standard +0.3
A die in the form of a dodecahedron has its faces numbered from 1 to 12. The die is biased so that the probability that a score of 12 is achieved is different from any other score. The probability distribution of \(X\), the score on the die, is given in the table in terms of \(p\) and \(k\), where \(0 < p < 1\) and \(k\) is a positive integer.
\(x\)123456789101112
P\((X = x)\)\(p\)\(p\)\(p\)\(p\)\(p\)\(p\)\(p\)\(p\)\(p\)\(p\)\(p\)\(kp\)
Sam rolls the die 30 times, Leo rolls the die 60 times and Nina rolls the die 120 times. They each plot their scores on bar line graphs.
  1. Explain whose graph is most likely to give the best representation of the theoretical probability distribution for the score on the die. [1]
  2. Find \(p\) in terms of \(k\). [2]
  3. Determine, in terms of \(k\), the expected number of times Nina rolls a 12. [3]
  4. Given that Nina rolls a 12 on 32 occasions, calculate an estimate of the value of \(k\). [2]
Nina rolls the die a further 30 times.
  1. Use your answer to part (d) to calculate an estimate for the probability that she obtains a 12 exactly 8 times in these 30 rolls. [2]
OCR MEI Paper 2 2022 June Q12
8 marks Moderate -0.8
A retailer sells bags of flour which are advertised as containing 1.5 kg of flour. A trading standards officer is investigating whether there is enough flour in each bag. He collects a random sample and uses software to carry out a hypothesis test at the 5\% level. The analysis is shown in the software printout below. \includegraphics{figure_12}
  1. State the hypotheses the officer uses in the test, defining any parameters used. [2]
  2. State the distribution used in the analysis. [3]
  3. Carry out the hypothesis test, giving your conclusion in context. [3]
OCR MEI Paper 2 2022 June Q13
8 marks Moderate -0.3
Records from the 1950s showed that 35\% of human babies were born without wisdom teeth. It is believed that as part of the evolutionary process more babies are now born without wisdom teeth. In a random sample of 140 babies, collected in 2020, a researcher found that 61 were born without wisdom teeth. The researcher made the following statement. ``This shows that the percentage of babies born without wisdom teeth has increased from 35\%.''
  1. Explain whether this statement can be fully justified. [1]
  2. Conduct a hypothesis test at the 5\% level to determine whether there is any evidence to suggest that more than 35\% of babies are now born without wisdom teeth. [7]
OCR MEI Paper 2 2022 June Q14
8 marks Standard +0.3
Fig. 14.1 shows the curve with equation \(y = \frac{1}{1 + x^2}\), together with 5 rectangles of equal width. \includegraphics{figure_14_1} Fig. 14.2 shows the coordinates of the points A, B, C, D, E and F. \includegraphics{figure_14_2}
  1. Use the 5 rectangles shown in Fig. 14.1 and the information in Fig. 14.2 to show that a lower bound for \(\int_0^1 \frac{1}{1 + x^2}\,dx\) is 0.7337, correct to 4 decimal places. [2]
  2. Use the 5 rectangles shown in Fig. 14.1 and the information in Fig. 14.2 to calculate an upper bound for \(\int_0^1 \frac{1}{1 + x^2}\,dx\) correct to 4 decimal places. [2]
  3. Hence find the length of the interval in which your answers to parts (a) and (b) indicate the value of \(\int_0^1 \frac{1}{1 + x^2}\,dx\) lies. [1]
Amit uses \(n\) rectangles, each of width \(\frac{1}{n}\), to calculate upper and lower bounds for \(\int_0^1 \frac{1}{1 + x^2}\,dx\), using different values of \(n\). His results are shown in Fig. 14.3. \includegraphics{figure_14_3}
  1. Find the length of the smallest interval in which Amit now knows \(\int_0^1 \frac{1}{1 + x^2}\,dx\) lies. [2]
  2. Without doing any calculation, explain how Amit could find a smaller interval which contains the value of \(\int_0^1 \frac{1}{1 + x^2}\,dx\). [1]
OCR MEI Paper 2 2022 June Q15
9 marks Easy -2.0
The pre-release material includes information on life expectancy at birth in countries of the world. Fig. 15.1 shows the data for Liberia, which is in Africa, together with a time series graph. \includegraphics{figure_15_1} Sundip uses the LINEST function on a spreadsheet to model life expectancy as a function of calendar year by a straight line. The equation of this line is \(L = 0.473y - 892\), where \(L\) is life expectancy at birth and \(y\) is calendar year.
  1. Use this model to find an estimate of the life expectancy at birth in Liberia in 1995. [1]
According to the model, the life expectancy at birth in Liberia in 2025 is estimated to be 65.83 years.
  1. Explain whether each of these two estimates is likely to be reliable. [2]
  2. Use your knowledge of the pre-release material to explain whether this model could be used to obtain a reliable estimate of the life expectancy at birth in other countries in 1995. [1]
Fig. 15.2 shows the life expectancy at birth between 1960 and 2010 for Italy and South Africa. \includegraphics{figure_15_2}
  1. Use your knowledge of the pre-release material to
    [2]
Sundip is investigating whether there is an association between the wealth of a country and life expectancy at birth in that country. As part of her analysis she draws a scatter diagram of GDP per capita in US\$ and life expectancy at birth in 2010 for all the countries in Europe for which data is available. She accidentally includes the data for the Central African Republic. The diagram is shown in Fig. 15.3. \includegraphics{figure_15_3}
  1. On the copy of Fig. 15.3 in the Printed Answer Booklet, use your knowledge of the pre-release material to circle the point representing the data for the Central African Republic. [1]
Sundip states that as GDP per capita increases, life expectancy at birth increases.
  1. Explain to what extent the information in Fig. 15.3 supports Sundip's statement. [2]
OCR MEI Paper 2 2022 June Q16
15 marks Standard +0.3
The equation of a curve is $$y = 6x^4 + 8x^3 - 21x^2 + 12x - 6.$$
  1. In this question you must show detailed reasoning. Determine
    [12]
  2. On the axes in the Printed Answer Booklet, sketch the curve whose equation is $$y = 6x^4 + 8x^3 - 21x^2 + 12x - 6.$$ [3]
OCR MEI Paper 2 Specimen Q1
5 marks Moderate -0.8
In this question you must show detailed reasoning. Find the coordinates of the points of intersection of the curve \(y = x^2 + x\) and the line \(2x + y = 4\). [5]
OCR MEI Paper 2 Specimen Q2
4 marks Moderate -0.8
Given that \(\text{f}(x) = x^3\) and \(\text{g}(x) = 2x^3 - 1\), describe a sequence of two transformations which maps the curve \(y = \text{f}(x)\) onto the curve \(y = \text{g}(x)\). [4]
OCR MEI Paper 2 Specimen Q3
3 marks Easy -1.2
Evaluate \(\int_0^{\frac{\pi}{12}} \cos 3x \, dx\), giving your answer in exact form. [3]
OCR MEI Paper 2 Specimen Q4
5 marks Moderate -0.3
The function f(x) is defined by \(\text{f}(x) = x^3 - 4\) for \(-1 \leq x \leq 2\). For \(\text{f}^{-1}(x)\), determine
  • the domain
  • the range.
[5]
OCR MEI Paper 2 Specimen Q5
2 marks Moderate -0.8
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. [2]
OCR MEI Paper 2 Specimen Q6
4 marks Moderate -0.8
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. \includegraphics{figure_6}
    1. Use the model to write down the mean of the maximum temperatures. [1]
    2. Explain why the curve indicates that the standard deviation is approximately 3 degrees Celsius. [1]
Temperatures can be converted from Celsius to Fahrenheit using the formula \(F = 1.8C + 32\), where \(F\) is the temperature in degrees Fahrenheit and \(C\) is the temperature in degrees Celsius.
  1. For maximum temperature in October in degrees Fahrenheit, estimate
    [2]
OCR MEI Paper 2 Specimen Q7
4 marks Moderate -0.8
Two events \(A\) and \(B\) are such that \(\text{P}(A) = 0.6\), \(\text{P}(B) = 0.5\) and \(\text{P}(A \cup B) = 0.85\). Find \(\text{P}(A | B)\). [4]
OCR MEI Paper 2 Specimen Q8
3 marks Standard +0.8
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: 51 57 58 59 61 64 64 65 67 68 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. [3]
OCR MEI Paper 2 Specimen Q9
4 marks Moderate -0.8
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.
  1. 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. [2]
The scatter diagram in Fig. 9 shows the data from a random sample of 50 eruptions of the second geyser. \includegraphics{figure_9}
  1. Stella claims the scatter diagram shows evidence of correlation between duration of eruption and waiting time. Make two comments about Stella's claim. [2]
OCR MEI Paper 2 Specimen Q10
3 marks Easy -1.8
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. [3]
OCR MEI Paper 2 Specimen Q11
4 marks Moderate -0.5
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. [4]
OCR MEI Paper 2 Specimen Q12
6 marks Standard +0.8
Fig. 12 shows the curve \(2x^3 + y^3 = 5y\). \includegraphics{figure_12}
  1. Find the gradient of the curve \(2x^3 + y^3 = 5y\) at the point \((1,2)\), giving your answer in exact form. [4]
  2. Show that all the stationary points of the curve lie on the \(y\)-axis. [2]