Questions — OCR MEI Paper 2 (127 questions)

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OCR MEI Paper 2 2018 June Q1
1 Show that \(\sqrt { 27 } + \sqrt { 192 } = a \sqrt { b }\), where \(a\) and \(b\) are prime numbers to be determined.
OCR MEI Paper 2 2018 June Q2
2 Solve the inequality \(| 2 x + 1 | < 5\).
OCR MEI Paper 2 2018 June Q3
3 The probability that Chipping FC win a league football match is \(\mathrm { P } ( W ) = 0.4\).
  1. Calculate the probability that Chipping FC fail to win each of their next two league football matches. The probability that Chipping FC lose a league football match is \(\mathrm { P } ( L ) = 0.3\).
  2. Explain why \(\mathrm { P } ( W ) + \mathrm { P } ( L ) \neq 1\).
OCR MEI Paper 2 2018 June Q4
4 A survey of the number of cars per household in a certain village generated the data in Fig. 4. \begin{table}[h]
Number of cars01234
Number of households82231277
\captionsetup{labelformat=empty} \caption{Fig. 4}
\end{table}
  1. Calculate the mean number of cars per household.
  2. Calculate the standard deviation of the number of cars per household.
OCR MEI Paper 2 2018 June Q5
5
  1. (A) Sketch the graph of \(y = 3 ^ { x }\).
    (B) Give the coordinates of any intercepts. The curve \(y = \mathrm { f } ( x )\) is the reflection of the curve \(y = 3 ^ { x }\) in the line \(y = x\).
  2. Find \(\mathrm { f } ( x )\).
OCR MEI Paper 2 2018 June Q6
6
  1. Express \(7 \cos x - 24 \sin x\) in the form \(R \cos ( x + \alpha )\), where \(0 < \alpha < \frac { \pi } { 2 }\).
  2. Write down the range of the function $$f ( x ) = 12 + 7 \cos x - 24 \sin x , \quad 0 \leqslant x \leqslant 2 \pi .$$
OCR MEI Paper 2 2018 June Q7
7 Find \(\int \left( 4 \sqrt { x } - \frac { 6 } { x ^ { 3 } } \right) \mathrm { d } x\). Answer all the questions
Section B (79 marks)
OCR MEI Paper 2 2018 June Q8
8 Every morning before breakfast Laura and Mike play a game of chess. The probability that Laura wins is 0.7 . The outcome of any particular game is independent of the outcome of other games. Calculate the probability that, in the next 20 games,
  1. Laura wins exactly 14 games,
  2. Laura wins at least 14 games.
OCR MEI Paper 2 2018 June Q9
9 At the end of each school term at North End College all the science classes in year 10 are given a test. The marks out of 100 achieved by members of set 1 are shown in Fig. 9. \begin{table}[h]
35
409
5236
601356
701256899
83466889
955567
\captionsetup{labelformat=empty} \caption{Fig. 9}
\end{table} Key \(5 \quad\) 2 represents a mark of 52
  1. Describe the shape of the distribution.
  2. The teacher for set 1 claimed that a typical student in his class achieved a mark of 95. How did he justify this statement?
  3. Another teacher said that the average mark in set 1 is 76 . How did she justify this statement? Benson's mark in the test is 35 . If the mark achieved by any student is an outlier in the lower tail of the distribution, the student is moved down to set 2 .
  4. Determine whether Benson is moved down to set 2 .
OCR MEI Paper 2 2018 June Q10
10 The screenshot in Fig. 10 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]{d8ff9511-aff7-45ea-ba55-e6667e8ba760-06_515_1009_338_529} \captionsetup{labelformat=empty} \caption{Fig. 10}
\end{figure} The area of each of the unshaded regions under the curve is 0.025 . The lower boundary of the shaded region is at 16.452 and the upper boundary of the shaded region is at 21.548 .
  1. Calculate the value of \(\mu\).
  2. Calculate the value of \(\sigma ^ { 2 }\).
  3. \(Y\) is the random variable given by \(Y = 4 X + 5\).
    (A) Write down the distribution of \(Y\).
    (B) Find \(\mathrm { P } ( \mathrm { Y } > 90 )\).
OCR MEI Paper 2 2018 June Q11
11 The discrete random variable \(X\) takes the values \(0,1,2,3,4\) and 5 with probabilities given by the formula $$\mathrm { P } ( X = x ) = k ( x + 1 ) ( 6 - x ) .$$
  1. Find the value of \(k\). In one half-term Ben attends school on 40 days. The probability distribution above is used to model \(X\), the number of lessons per day in which Ben receives a gold star for excellent work.
  2. Find the probability that Ben receives no gold stars on each of the first 3 days of the half-term and two gold stars on each of the next 2 days.
  3. Find the expected number of days in the half-term on which Ben receives no gold stars.
OCR MEI Paper 2 2018 June Q13
13 Each weekday Keira drives to work with her son Kaito. She always sets off at 8.00 a.m. She models her journey time, \(x\) minutes, by the distribution \(X \sim \mathrm {~N} ( 15,4 )\). Over a long period of time she notes that her journey takes less than 14 minutes on \(7 \%\) of the journeys, and takes more than 18 minutes on \(31 \%\) of the journeys.
  1. Investigate whether Keira's model is a good fit for the data. Kaito believes that Keira’s value for the variance is correct, but realises that the mean is not correct.
  2. Find, correct to two significant figures, the value of the mean that Keira should use in a refined model which does fit the data. Keira buys a new car. After driving to work in it each day for several weeks, she randomly selects the journey times for \(n\) of these days. Her mean journey time for these \(n\) days is 16 minutes. Using the refined model she conducts a hypothesis test to see if her mean journey time has changed, and finds that the result is significant at the \(5 \%\) level.
  3. Determine the smallest possible value of \(n\).
OCR MEI Paper 2 2018 June Q14
14 The pre-release material includes data on unemployment rates in different countries. A sample from this material has been taken. All the countries in the sample are in Europe. The data have been grouped and are shown in Fig 14.1. \begin{table}[h]
Unemployment rate\(0 -\)\(5 -\)\(10 -\)\(15 -\)\(20 -\)\(35 - 50\)
Frequency15215522
\captionsetup{labelformat=empty} \caption{Fig. 14.1}
\end{table} A cumulative frequency curve has been generated for the sample data using a spreadsheet. This is shown in Fig. 14.2. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{d8ff9511-aff7-45ea-ba55-e6667e8ba760-08_639_1081_808_466} \captionsetup{labelformat=empty} \caption{Fig. 14.2}
\end{figure} Hodge used Fig. 14.2 to estimate the median unemployment rate in Europe. He obtained the answer 5.0. The correct value for this sample is 6.9.
  1. (A) There is a systematic error in the diagram.
    • Identify this error.
    • State how this error affects Hodge’s estimate.
      (B) There is another factor which has affected Hodge’s estimate.
    • Identify this factor.
    • State how this factor affects Hodge’s estimate.
    • Use your knowledge of the pre-release material to give another reason why any estimation of the median unemployment rate in Europe may be unreliable.
    • Use your knowledge of the pre-release material to explain why it is very unlikely that the sample has been randomly selected from the pre-release material.
    The scatter diagram shown in Fig. 14.3 shows the unemployment rate and life expectancy at birth for the 47 countries in the sample for which this information is available. \begin{figure}[h]
    \captionsetup{labelformat=empty} \caption{Scatter diagram to show life expectancy at birth against unemployment rate} \includegraphics[alt={},max width=\textwidth]{d8ff9511-aff7-45ea-ba55-e6667e8ba760-09_627_1281_456_367}
    \end{figure} Fig. 14.3 The product moment correlation coefficient for the 47 items in the sample is - 0.2607 .
    The \(p\)-value associated with \(r = - 0.2607\) and \(n = 47\) is 0.0383 .
  2. Does this information suggest that there is an association between unemployment rate and life expectancy at birth in countries in Europe? Hodge uses the spreadsheet tools to obtain the equation of a line of best fit for this data.
  3. The unemployment rate in Kosovo is 35.3 , but there is no data available on life expectancy. Is it reasonable to use Hodge’s line of best fit to estimate life expectancy at birth in Kosovo?
OCR MEI Paper 2 2018 June Q15
15 You must show detailed reasoning in this question. The equation of a curve is $$y ^ { 3 } - x y + 4 \sqrt { x } = 4 .$$ Find the gradient of the curve at each of the points where \(y = 1\).
OCR MEI Paper 2 2018 June Q16
16 In the first year of a course, an A-level student, Aaishah, has a mathematics test each week. The night before each test she revises for \(t\) hours. Over the course of the year she realises that her percentage mark for a test, \(p\), may be modelled by the following formula, where \(A , B\) and \(C\) are constants. $$p = A - B ( t - C ) ^ { 2 }$$
  • Aaishah finds that, however much she revises, her maximum mark is achieved when she does 2 hours revision. This maximum mark is 62 .
  • Aaishah had a mark of 22 when she didn't spend any time revising.
    1. Find the values of \(A , B\) and \(C\).
    2. According to the model, if Aaishah revises for 45 minutes on the night before the test, what mark will she achieve?
    3. What is the maximum amount of time that Aaishah could have spent revising for the model to work?
In an attempt to improve her marks Aaishah now works through problems for a total of \(t\) hours over the three nights before the test. After taking a number of tests, she proposes the following new formula for \(p\). $$p = 22 + 68 \left( 1 - \mathrm { e } ^ { - 0.8 t } \right)$$ For the next three tests she recorded the data in Fig. 16. \begin{table}[h]
\(t\)135
\(p\)598489
\captionsetup{labelformat=empty} \caption{Fig. 16}
\end{table}
  • Verify that the data is consistent with the new formula.
  • Aaishah's tutor advises her to spend a minimum of twelve hours working through problems in future. Determine whether or not this is good advice.
    •
    OCR MEI Paper 2 2018 June Q17
    17
    1. Express \(\frac { \left( x ^ { 2 } - 8 x + 9 \right) } { ( x + 1 ) ( x - 2 ) ^ { 2 } }\) in partial fractions.
    2. Express \(y\) in terms of \(x\) given that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { y \left( x ^ { 2 } - 8 x + 9 \right) } { ( x + 1 ) ( x - 2 ) ^ { 2 } } \text { and } y = 16 \text { when } x = 3 .$$ \section*{END OF QUESTION PAPER}
    OCR MEI Paper 2 2019 June Q1
    1 Fig. 1 shows the probability distribution of the discrete random variable \(X\). \begin{table}[h]
    \(x\)12345
    \(\mathrm { P } ( X = x )\)0.20.1\(k\)\(2 k\)\(4 k\)
    \captionsetup{labelformat=empty} \caption{Fig. 1}
    \end{table}
    1. Find the value of \(k\).
    2. Find \(\mathrm { P } ( X \neq 4 )\).
    OCR MEI Paper 2 2019 June Q2
    2 Given that \(y = \left( x ^ { 2 } + 5 \right) ^ { 12 }\),
    1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
    2. Hence find \(\int 48 x \left( x ^ { 2 } + 5 \right) ^ { 11 } \mathrm {~d} x\).
    OCR MEI Paper 2 2019 June Q3
    3 Fig. 3 shows the time Lorraine spent in hours, \(t\), answering e-mails during the working day. The data were collected over a number of months. \begin{table}[h]
    Time in hours,
    \(t\)
    		\(0 \leqslant t < 1\)\(1 \leqslant t < 2\)\(2 \leqslant t < 3\)\(3 \leqslant t < 4\)\(4 \leqslant t < 6\)\(6 \leqslant t < 8\)
    Number of
    days
    		283642312412
    \captionsetup{labelformat=empty} \caption{Fig. 3}
    \end{table}
    1. Calculate an estimate of the mean time per day that Lorraine spent answering e-mails over this period.
    2. Explain why your answer to part (a) is an estimate. When Lorraine accepted her job, she was told that the mean time per day spent answering e-mails would not be more than 3 hours.
    3. Determine whether, according to the data in Fig. 3, it is possible that the mean time per day Lorraine spends answering e-mails is in fact more than 3 hours.
    OCR MEI Paper 2 2019 June Q4
    4 Fig. 4 shows the graph of \(y = \sqrt { 1 + x ^ { 3 } }\). \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{95eb3bcc-6d3c-4f7e-9b27-5e046ab57ec5-05_544_639_338_248} \captionsetup{labelformat=empty} \caption{Fig. 4}
    \end{figure}
    1. Use the trapezium rule with \(h = 0.5\) to find an estimate of \(\int _ { - 1 } ^ { 0 } \sqrt { 1 + x ^ { 3 } } \mathrm {~d} x\), giving your answer correct to 6 decimal places.
    2. State whether your answer to part (a) is an under-estimate or an over-estimate, justifying your answer.
    OCR MEI Paper 2 2019 June Q5
    5 Fig. 5 shows the number of times that students at a sixth form college visited a recreational mathematics website during the first week of the summer term. \begin{table}[h]
    Number of visits to website012345
    Number of students2438171242
    \captionsetup{labelformat=empty} \caption{Fig. 5}
    \end{table}
    1. State the value of the mid-range of the data.
    2. Describe the shape of the distribution.
    3. State the value of the mode.
    OCR MEI Paper 2 2019 June Q6
    6 Find \(\int \frac { 32 } { x ^ { 5 } } \ln x \mathrm {~d} x\). Answer all the questions
    Section B (78 marks)
    OCR MEI Paper 2 2019 June Q7
    7 The area of a sector of a circle is \(36.288 \mathrm {~cm} ^ { 2 }\). The angle of the sector is \(\theta\) radians and the radius of the circle is \(r \mathrm {~cm}\).
    1. Find an expression for \(\theta\) in terms of \(r\). The perimeter of the sector is 24.48 cm .
    2. Show that \(\theta = \frac { 24.48 } { r } - 2\).
    3. Find the possible values of \(r\).
    OCR MEI Paper 2 2019 June Q8
    8 A team called "The Educated Guess" enter a weekly quiz. If they win the quiz in a particular week, the probability that they will win the following week is 0.4 , but if they do not win, the probability that they will win the following week is 0.2 . In week 4 The Educated Guess won the quiz.
    1. Calculate the probability that The Educated Guess will win the quiz in week 6. Every week the same 20 quiz teams, each with 6 members, take part in a quiz. Every member of every team buys a raffle ticket. Five winning tickets are drawn randomly, without replacement. Alf, who is a member of one of the teams, takes part every week.
    2. Calculate the probability that, in a randomly chosen week, Alf wins a raffle prize.
    3. Find the smallest number of weeks after which it will be \(95 \%\) certain that Alf has won at least one raffle prize.
    OCR MEI Paper 2 2019 June Q9
    9 You are given that
    \(\mathrm { f } ( x ) = 2 x + 3 \quad\) for \(x < 0 \quad\) and
    \(\mathrm { g } ( x ) = x ^ { 2 } - 2 x + 1\) for \(x > 1\).
    1. Find \(\mathrm { gf } ( x )\), stating the domain.
    2. State the range of \(\mathrm { gf } ( x )\).
    3. Find (gf) \({ } ^ { - 1 } ( x )\).