Questions — OCR MEI (4455 questions)

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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]
OCR MEI Paper 2 Specimen Q13
6 marks Challenging +1.2
Evaluate \(\int_0^1 \frac{1}{1 + \sqrt{x}} \, dx\), giving your answer in the form \(a + b \ln c\), where \(a\), \(b\) and \(c\) are integers. [6]
OCR MEI Paper 2 Specimen Q14
12 marks Standard +0.3
In a chemical reaction, the mass \(m\) grams of a chemical at time \(t\) minutes is modelled by the differential equation $$\frac{dm}{dt} = \frac{m}{t(1 + 2t)}.$$ At time 1 minute, the mass of the chemical is 1 gram.
  1. Solve the differential equation to show that \(m = \frac{3t}{(1 + 2t)}\). [8]
  2. Hence
    1. find the time when the mass is 1.25 grams, [2]
    2. show what happens to the mass of the chemical as \(t\) becomes large. [2]
OCR MEI Paper 2 Specimen Q15
15 marks Standard +0.3
A quality control department checks the lifetimes of batteries produced by a company. The lifetimes, \(x\) minutes, for a random sample of 80 'Superstrength' batteries are shown in the table below.
Lifetime\(160 \leq x < 165\)\(165 \leq x < 168\)\(168 \leq x < 170\)\(170 \leq x < 172\)\(172 \leq x < 175\)\(175 \leq x < 180\)
Frequency5142021164
  1. Estimate the proportion of these batteries which have a lifetime of at least 174.0 minutes. [2]
  2. Use the data in the table to estimate
    [3]
The data in the table on the previous page are represented in the following histogram, Fig 15. \includegraphics{figure_15} A quality control manager models the data by a Normal distribution with the mean and standard deviation you calculated in part (b).
  1. Comment briefly on whether the histogram supports this choice of model. [2]
    1. Use this model to estimate the probability that a randomly selected battery will have a lifetime of more than 174.0 minutes.
    2. Compare your answer with your answer to part (a). [3]
The company also manufactures 'Ultrapower' batteries, which are stated to have a mean lifetime of 210 minutes.
  1. A random sample of 8 Ultrapower batteries is selected. The mean lifetime of these batteries is 207.3 minutes. Carry out a hypothesis test at the 5% level to investigate whether the mean lifetime is as high as stated. You should use the following hypotheses \(\text{H}_0 : \mu = 210\), \(\text{H}_1 : \mu < 210\), where \(\mu\) represents the population mean for Ultrapower batteries. You should assume that the population is Normally distributed with standard deviation 3.4. [5]
OCR MEI Paper 2 Specimen Q16
20 marks Easy -1.8
Fig. 16.1, Fig. 16.2 and Fig. 16.3 show some data about life expectancy, including some from the pre-release data set. \includegraphics{figure_16_1} \includegraphics{figure_16_2} \includegraphics{figure_16_3}
  1. Comment on the shapes of the distributions of life expectancy at birth in 2014 and 1974. [2]
    1. The minimum value shown in the box plot is negative. What does a negative value indicate? [1]
    2. What feature of Fig 16.3 suggests that a Normal distribution would not be an appropriate model for increase in life expectancy from one year to another year? [1]
    3. Software has been used to obtain the values in the table in Fig. 16.3. Decide whether the level of accuracy is appropriate. Justify your answer. [1]
    4. John claims that for half the people in the world their life expectancy has improved by 10 years or more. Explain why Fig. 16.3 does not provide conclusive evidence for John's claim. [1]
  3. Decide whether the maximum increase in life expectancy from 1974 to 2014 is an outlier. Justify your answer. [3]
Here is some further information from the pre-release data set.
CountryLife expectancy at birth in 2014
Ethiopia60.8
Sweden81.9
    1. Estimate the change in life expectancy at birth for Ethiopia between 1974 and 2014.
    2. Estimate the change in life expectancy at birth for Sweden between 1974 and 2014.
    3. Give one possible reason why the answers to parts (i) and (ii) are so different. [4]
Fig. 16.4 shows the relationship between life expectancy at birth in 2014 and 1974. \includegraphics{figure_16_4} A spreadsheet gives the following linear model for all the data in Fig 16.4. (Life expectancy at birth 2014) = 30.98 + 0.67 × (Life expectancy at birth 1974) The life expectancy at birth in 1974 for the region that now constitutes the country of South Sudan was 37.4 years. The value for this country in 2014 is not available.
    1. Use the linear model to estimate the life expectancy at birth in 2014 for South Sudan. [2]
    2. Give two reasons why your answer to part (i) is not likely to be an accurate estimate for the life expectancy at birth in 2014 for South Sudan. You should refer to both information from Fig 16.4 and your knowledge of the large data set. [2]
  2. In how many of the countries represented in Fig. 16.4 did life expectancy drop between 1974 and 2014? Justify your answer. [3]
OCR MEI Further Pure Core AS 2018 June Q1
4 marks Moderate -0.8
The matrices \(\mathbf{A}\), \(\mathbf{B}\) and \(\mathbf{C}\) are defined as follows: $$\mathbf{A} = \begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix}, \quad \mathbf{B} = \begin{pmatrix} 2 & 0 & 3 \\ 1 & -1 & 3 \end{pmatrix}, \quad \mathbf{C} = \begin{pmatrix} 1 & 3 \end{pmatrix}.$$ Calculate all possible products formed from two of these three matrices. [4]
OCR MEI Further Pure Core AS 2018 June Q2
3 marks Moderate -0.8
Find, to the nearest degree, the angle between the vectors \(\begin{pmatrix} 1 \\ 0 \\ -2 \end{pmatrix}\) and \(\begin{pmatrix} -2 \\ 3 \\ -3 \end{pmatrix}\). [3]
OCR MEI Further Pure Core AS 2018 June Q3
5 marks Moderate -0.8
Find real numbers \(a\) and \(b\) such that \((a - 3i)(5 - i) = b - 17i\). [5]
OCR MEI Further Pure Core AS 2018 June Q4
5 marks Moderate -0.3
Find a cubic equation with real coefficients, two of whose roots are \(2 - i\) and \(3\). [5]
OCR MEI Further Pure Core AS 2018 June Q5
7 marks Standard +0.3
A transformation of the \(x\)-\(y\) plane is represented by the matrix \(\begin{pmatrix} \cos \theta & 2 \sin \theta \\ 2 \sin \theta & -\cos \theta \end{pmatrix}\), where \(\theta\) is a positive acute angle.
  1. Write down the image of the point \((2, 3)\) under this transformation. [2]
  2. You are given that this image is the point \((a, 0)\). Find the value of \(a\). [5]
OCR MEI Further Pure Core AS 2018 June Q6
4 marks Standard +0.3
Find the invariant line of the transformation of the \(x\)-\(y\) plane represented by the matrix \(\begin{pmatrix} 2 & 0 \\ 4 & -1 \end{pmatrix}\). [4]