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OCR MEI Paper 1 Specimen Q5
4 marks Moderate -0.3
5 A woman is pulling a loaded sledge along horizontal ground. The only resistance to motion of the sledge is due to friction between it and the ground. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{ff44367e-c992-4e79-b255-5a04e0b8e21e-05_314_1024_486_356} \captionsetup{labelformat=empty} \caption{Fig. 5}
\end{figure} At first, she pulls with a force of 100 N inclined at \(32 ^ { \circ }\) to the horizontal, as shown in Fig.5, but the sledge does not move.
  1. Determine the frictional force between the ground and the sledge. Give your answer correct to 3 significant figures.
  2. Next she pulls with a force of 100 N inclined at a smaller angle to the horizontal. The sledge still does not move. Compare the frictional force in this new situation with that in part (a), justifying your answer.
OCR MEI Paper 1 Specimen Q6
4 marks Moderate -0.8
6 Fig. 6 shows a partially completed spreadsheet.
This spreadsheet uses the trapezium rule with four strips to estimate \(\int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \sqrt { 1 + \sin x } \mathrm {~d} x\). \begin{table}[h]
ABCDE
1\(x\)\(\sin x\)\(y\)
200.00000.00001.00000.5000
30.1250.39270.38271.17591.1759
40.250.78540.70711.30661.3066
50.3751.17810.92391.38701.3870
60.51.57081.00001.41420.7071
75.0766
8
\captionsetup{labelformat=empty} \caption{Fig. 6}
\end{table}
  1. Show how the value in cell B3 is calculated.
  2. Show how the values in cells D2 to D6 are used to calculate the value in cell E7.
  3. Complete the calculation to estimate \(\int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \sqrt { 1 + \sin x } \mathrm {~d} x\). Give your answer to 3 significant figures.
OCR MEI Paper 1 Specimen Q7
10 marks Moderate -0.3
7 In this question take \(\boldsymbol { g } = \mathbf { 1 0 }\).
A small stone is projected from a point O with a speed of \(26 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle \(\theta\) above the horizontal. The initial velocity and part of the path of the stone are shown in Fig. 7.
You are given that \(\sin \theta = \frac { 12 } { 13 }\).
After \(t\) seconds the horizontal displacement of the stone from O is \(x\) metres and the vertical displacement is \(y\) metres. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{ff44367e-c992-4e79-b255-5a04e0b8e21e-07_419_479_904_248} \captionsetup{labelformat=empty} \caption{Fig. 7}
\end{figure}
  1. Using the standard model for projectile motion,
    The stone passes through a point A . Point A is 16 m above the level of O .
  2. Find the two possible horizontal distances of A from O . A toy balloon is projected from O with the same initial velocity as the small stone.
  3. Suggest two ways in which the standard model could be adapted.
OCR MEI Paper 1 Specimen Q8
7 marks Standard +0.3
8 Find \(\int x ^ { 2 } \mathrm { e } ^ { 2 x } \mathrm {~d} x\).
OCR MEI Paper 1 Specimen Q9
8 marks Standard +0.3
9 In an experiment, a small box is hit across a floor. After it has been hit, the box slides without rotation. The box passes a point A. The distance the box travels after passing A before coming to rest is \(S\) metres and the time this takes is \(T\) seconds. The only resistance to the box's motion is friction due to the floor. The mass of the box is \(m \mathrm {~kg}\) and the frictional force is a constant \(F\).
    1. Find the equation of motion for the box while it is sliding.
    2. Show that \(S = k T ^ { 2 }\) where \(k = \frac { F } { 2 m }\).
  2. Given that \(k = 1.4\), find the value of the coefficient of friction between the box and the floor.
OCR MEI Paper 1 Specimen Q10
15 marks Standard +0.3
10 In a certain region, the populations of grey squirrels, \(P _ { \mathrm { G } }\) and red squirrels \(P _ { \mathrm { R } }\), at time \(t\) years are modelled by the equations: \(P _ { \mathrm { G } } = 10000 \left( 1 - \mathrm { e } ^ { - k t } \right)\) \(P _ { \mathrm { R } } = 20000 \mathrm { e } ^ { - k t }\) where \(t \geq 0\) and \(k\) is a positive constant.
    1. On the axes in your Printed Answer Book, sketch the graphs of \(P _ { \mathrm { G } }\) and \(P _ { \mathrm { R } }\) on the same axes.
    2. Give the equations of any asymptotes.
  2. What does the model predict about the long term population of
    Grey squirrels and red squirrels compete for food and space. Grey squirrels are larger and more successful than red squirrels.
  3. Comment on the validity of the model given by the equations, giving a reason for your answer.
  4. Show that, according to the model, the rate of decrease of the population of red squirrels is always double the rate of increase of the population of grey squirrels.
  5. When \(t = 3\), the numbers of grey and red squirrels are equal. Find the value of \(k\).
OCR MEI Paper 1 Specimen Q11
9 marks Standard +0.3
11 Fig. 11 shows the curve with parametric equations $$x = 2 \cos \theta , y = \sin \theta , 0 \leq \theta \leq 2 \pi .$$ The point P has parameter \(\frac { 1 } { 4 } \pi\). The tangent at P to the curve meets the axes at A and B . \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{ff44367e-c992-4e79-b255-5a04e0b8e21e-10_668_1075_543_255} \captionsetup{labelformat=empty} \caption{Fig. 11}
\end{figure}
  1. Show that the equation of the line AB is \(x + 2 y = 2 \sqrt { 2 }\).
  2. Determine the area of the triangle AOB .
OCR MEI Paper 1 Specimen Q12
9 marks Standard +0.3
12 A model boat has velocity \(\mathbf { v } = ( ( 2 t - 2 ) \mathbf { i } + ( 2 t + 2 ) \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) for \(t \geq 0\), where \(t\) is the time in seconds. \(\mathbf { i }\) is the unit vector east and \(\mathbf { j }\) is the unit vector north.
When \(t = 3\), the position vector of the boat is \(( 3 \mathbf { i } + 14 \mathbf { j } ) \mathrm { m }\).
  1. Show that the boat is never instantaneously at rest.
  2. Determine any times at which the boat is moving directly northwards.
  3. Determine any times at which the boat is north-east of the origin.
OCR MEI Paper 1 Specimen Q14
9 marks Moderate -0.3
14 Blocks A and B are connected by a light rigid horizontal bar and are sliding on a rough horizontal surface. A light horizontal string exerts a force of 40 N on B .
This situation is shown in Fig. 14, which also shows the direction of motion, the mass of each of the blocks and the resistances to their motion. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{ff44367e-c992-4e79-b255-5a04e0b8e21e-11_266_1283_664_255} \captionsetup{labelformat=empty} \caption{Fig. 14}
\end{figure}
  1. Calculate the tension in the bar. The string breaks while the blocks are sliding. The resistances to motion are unchanged.
  2. Determine
OCR MEI Paper 1 Specimen Q15
6 marks Standard +0.3
15 Fig. 15 shows a uniform shelf AB of weight \(W \mathrm {~N}\).
The shelf is 180 cm long and rests on supports at points C and D . Point C is 30 cm from A and point D is 60 cm from B .
side view \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{ff44367e-c992-4e79-b255-5a04e0b8e21e-11_284_1169_1987_383} \captionsetup{labelformat=empty} \caption{Fig. 15}
\end{figure} Determine the range of positions a point load of \(3 W\) could be placed on the shelf without the shelf tipping. \section*{Copyright Information:} }{www.ocr.org.uk}) after the live examination series. If OCR has unwittingly failed to correctly acknowledge or clear any third-party content in this assessment material, OCR will be happy to correct its mistake at the earliest possible opportunity. For queries or further information please contact the Copyright Team, The Triangle Building, Shaftesbury Road, Cambridge CB2 8EA.
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OCR MEI Paper 2 2018 June Q1
2 marks Easy -1.8
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
3 marks Easy -1.8
2 Solve the inequality \(| 2 x + 1 | < 5\).
OCR MEI Paper 2 2018 June Q3
2 marks Easy -1.8
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
2 marks Easy -1.3
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
3 marks Easy -1.2
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
5 marks Moderate -0.3
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
4 marks Easy -1.2
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
4 marks Moderate -0.8
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
5 marks Easy -1.8
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
8 marks Standard +0.3
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
6 marks Moderate -0.8
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
10 marks Challenging +1.2
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
9 marks Moderate -0.8
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.
    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
9 marks Standard +0.8
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
11 marks Standard +0.3
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.
    •