Questions — OCR MEI (4455 questions)

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OCR MEI AS Paper 2 2018 June Q1
2 marks Easy -2.0
Write down the value of (A) \(\log_a (a^4)\), [1] (B) \(\log_a \left(\frac{1}{a}\right)\). [1]
OCR MEI AS Paper 2 2018 June Q2
3 marks Easy -1.3
Doug has a list of times taken by competitors in a 'fun run'. He has grouped the data and calculated the frequency densities in order to draw a histogram to represent the information. Some of the data are presented in Fig. 2.
Time in minutes\(15-\)\(20-\)\(25-\)\(35-\)\(45-60\)
Number of runners12235971
Frequency density2.45.97.11.4
Fig. 2
  1. Write down the missing values in the copy of Fig. 2 in the Printed Answer Booklet. [2]
  2. Doug labels the horizontal axis on the histogram 'time in minutes' and the vertical axis 'number of minutes per runner'. State which one of these labels is incorrect and write down a correct version. [1]
OCR MEI AS Paper 2 2018 June Q3
3 marks Moderate -0.8
\(P\) and \(Q\) are consecutive odd positive integers such that \(P > Q\). Prove that \(P^2 - Q^2\) is a multiple of 8. [3]
OCR MEI AS Paper 2 2018 June Q4
5 marks Moderate -0.8
The probability distribution of the discrete random variable \(X\) is given in Fig. 4.
\(r\)01234
P\((X = r)\)0.20.150.3\(k\)0.25
Fig. 4
  1. Find the value of \(k\). [2]
\(X_1\) and \(X_2\) are two independent values of \(X\).
  1. Find P\((X_1 + X_2 = 6)\). [3]
OCR MEI AS Paper 2 2018 June Q5
3 marks Moderate -0.8
Find the set of values of \(a\) for which the equation $$ax^2 + 8x + 2 = 0$$ has no real roots. [3]
OCR MEI AS Paper 2 2018 June Q6
4 marks Moderate -0.8
Show that \(\int_0^9 (3 + 4\sqrt{x})dx = 99\). [4]
OCR MEI AS Paper 2 2018 June Q7
8 marks Moderate -0.8
Rose and Emma each wear a device that records the number of steps they take in a day. All the results for a 7-day period are given in Fig. 7.
Day1234567
Rose10014112621014993619708992110369
Emma9204991387411001510261739110856
Fig. 7 The 7-day mean is the mean number of steps taken in the last 7 days. The 7-day mean for Rose is 10112.
  1. Calculate the 7-day mean for Emma. [1]
At the end of day 8 a new 7-day mean is calculated by including the number of steps taken on day 8 and omitting the number of steps taken on day 1. On day 8 Rose takes 10259 steps.
  1. Determine the number of steps Emma must take on day 8 so that her 7-day mean at the end of day 8 is the same as for Rose. [4]
In fact, over a long period of time, the mean of the number of steps per day that Emma takes is 10341 and the standard deviation is 948.
  1. Determine whether the number of steps Emma needs to take on day 8 so that her 7-day mean is the same as that for Rose in part (ii) is unusually high. [3]
OCR MEI AS Paper 2 2018 June Q8
7 marks Standard +0.3
In this question you must show detailed reasoning. The centre of a circle C is at the point \((-1, 3)\) and C passes through the point \((1, -1)\). The straight line L passes through the points \((1, 9)\) and \((4, 3)\). Show that L is a tangent to C. [7]
OCR MEI AS Paper 2 2018 June Q9
7 marks Standard +0.3
In this question you must show detailed reasoning. Research showed that in May 2017 62% of adults over 65 years of age in the UK used a certain online social media platform. Later in 2017 it was believed that this proportion had increased. In December 2017 a random sample of 59 adults over 65 years of age in the UK was collected. It was found that 46 of the 59 adults used this online social media platform. Use a suitable hypothesis test to determine whether there is evidence at the 1% level to suggest that the proportion of adults over 65 in the UK who used this online social media platform had increased from May 2017 to December 2017. [7]
OCR MEI AS Paper 2 2018 June Q10
9 marks Moderate -0.8
  1. A curve has equation \(y = 16x + \frac{1}{x}\). Find
    1. \(\frac{dy}{dx}\), [2]
    2. \(\frac{d^2y}{dx^2}\). [2]
  2. Hence
OCR MEI AS Paper 2 2018 June Q11
9 marks Easy -1.8
The pre-release material contains data concerning the death rate per thousand people and the birth rate per thousand people in all the countries of the world. The diagram in Fig. 11.1 was generated using a spreadsheet and summarises the birth rates for all the countries in Africa. \includegraphics{figure_11_1} Fig. 11.1
  1. Identify two respects in which the presentation of the data is incorrect. [2]
Fig. 11.2 shows a scatter diagram of death rate, \(y\), against birth rate, \(x\), for a sample of 55 countries, all of which are in Africa. A line of best fit has also been drawn. \includegraphics{figure_11_2} Fig. 11.2 The equation of the line of best fit is \(y = 0.15x + 4.72\).
    1. What does the diagram suggest about the relationship between death rate and birth rate? [1]
    2. The birth rate in Togo is recorded as 34.13 per thousand, but the data on death rate has been lost. Use the equation of the line of best fit to estimate the death rate in Togo. [1]
    3. Explain why it would not be sensible to use the equation of the line of best fit to estimate the death rate in a country where the birth rate is 5.5 per thousand. [1]
    4. Explain why it would not be sensible to use the equation of the line of best fit to estimate the death rate in a Caribbean country where the birth rate is known. [1]
    5. Explain why it is unlikely that the sample is random. [1]
Including Togo there were 56 items available for selection.
  1. Describe how a sample of size 14 from this data could be generated for further analysis using systematic sampling. [2]
OCR MEI AS Paper 2 2018 June Q12
10 marks Moderate -0.8
In an experiment 500 fruit flies were released into a controlled environment. After 10 days there were 650 fruit flies present. Munirah believes that \(N\), the number of fruit flies present at time \(t\) days after the fruit flies are released, will increase at the rate of 4.4% per day. She proposes that the situation is modelled by the formula \(N = Ak^t\).
  1. Write down the values of \(A\) and \(k\). [2]
  2. Determine whether the model is consistent with the value of \(N\) at \(t = 10\). [2]
  3. What does the model suggest about the number of fruit flies in the long run? [1]
Subsequently it is found that for large values of \(t\) the number of fruit flies in the controlled environment oscillates about 750. It is also found that as \(t\) increases the oscillations decrease in magnitude. Munirah proposes a second model in the light of this new information. $$N = 750 - 250 \times e^{-0.092t}$$
  1. Identify three ways in which this second model is consistent with the known data. [3]
    1. Identify one feature which is not accounted for by the second model. [1]
    2. Give an example of a mathematical function which needs to be incorporated in the model to account for this feature. [1]
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]
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]