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OCR MEI C3 2008 January Q4
7 marks Moderate -0.8
4 When the gas in a balloon is kept at a constant temperature, the pressure \(P\) in atmospheres and the volume \(V \mathrm {~m} ^ { 3 }\) are related by the equation $$P = \frac { k } { V }$$ where \(k\) is a constant. [This is known as Boyle's Law.]
When the volume is \(100 \mathrm {~m} ^ { 3 }\), the pressure is 5 atmospheres, and the volume is increasing at a rate of \(10 \mathrm {~m} ^ { 3 }\) per second.
  1. Show that \(k = 500\).
  2. Find \(\frac { \mathrm { d } P } { \mathrm {~d} V }\) in terms of \(V\).
  3. Find the rate at which the pressure is decreasing when \(V = 100\).
OCR MEI C3 2008 January Q5
4 marks Moderate -0.8
5
  1. Verify the following statement: $$\text { ' } 2 ^ { p } - 1 \text { is a prime number for all prime numbers } p \text { less than } 11 \text { '. }$$
  2. Calculate \(23 \times 89\), and hence disprove this statement: $$\text { ' } 2 ^ { p } - 1 \text { is a prime number for all prime numbers } p ^ { \prime } \text {. }$$
OCR MEI C3 2008 January Q6
8 marks Standard +0.8
6 Fig. 6 shows the curve \(\mathrm { e } ^ { 2 y } = x ^ { 2 } + y\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{a66c2da5-46b4-4467-933e-179be04b03b1-3_739_1339_349_404} \captionsetup{labelformat=empty} \caption{Fig. 6}
\end{figure}
  1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 2 x } { 2 \mathrm { e } ^ { 2 y } - 1 }\).
  2. Hence find to 3 significant figures the coordinates of the point P , shown in Fig. 6, where the curve has infinite gradient. Section B (36 marks)
OCR MEI C3 2008 January Q7
19 marks Standard +0.3
7 A curve is defined by the equation \(y = 2 x \ln ( 1 + x )\).
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and hence verify that the origin is a stationary point of the curve.
  2. Find \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\), and use this to verify that the origin is a minimum point.
  3. Using the substitution \(u = 1 + x\), show that \(\int \frac { x ^ { 2 } } { 1 + x } \mathrm {~d} x = \int \left( u - 2 + \frac { 1 } { u } \right) \mathrm { d } u\). Hence evaluate \(\int _ { 0 } ^ { 1 } \frac { x ^ { 2 } } { 1 + x } \mathrm {~d} x\), giving your answer in an exact form.
  4. Using integration by parts and your answer to part (iii), evaluate \(\int _ { 0 } ^ { 1 } 2 x \ln ( 1 + x ) \mathrm { d } x\).
OCR MEI C3 2008 January Q8
17 marks Standard +0.3
8 Fig. 8 shows the curve \(y = \mathrm { f } ( x )\), where \(\mathrm { f } ( x ) = 1 + \sin 2 x\) for \(- \frac { 1 } { 4 } \pi \leqslant x \leqslant \frac { 1 } { 4 } \pi\).
[diagram]
  1. State a sequence of two transformations that would map part of the curve \(y = \sin x\) onto the curve \(y = \mathrm { f } ( x )\).
  2. Find the area of the region enclosed by the curve \(y = \mathrm { f } ( x )\), the \(x\)-axis and the line \(x = \frac { 1 } { 4 } \pi\).
  3. Find the gradient of the curve \(y = \mathrm { f } ( x )\) at the point \(( 0,1 )\). Hence write down the gradient of the curve \(y = \mathrm { f } ^ { - 1 } ( x )\) at the point \(( 1,0 )\).
  4. State the domain of \(\mathrm { f } ^ { - 1 } ( x )\). Add a sketch of \(y = \mathrm { f } ^ { - 1 } ( x )\) to a copy of Fig. 8.
  5. Find an expression for \(\mathrm { f } ^ { - 1 } ( x )\).
OCR MEI S3 2006 June Q1
18 marks Standard +0.3
1 Design engineers are simulating the load on a particular part of a complex structure. They intend that the simulated load, measured in a convenient unit, should be given by the random variable \(X\) having probability density function $$f ( x ) = 12 x ^ { 3 } - 24 x ^ { 2 } + 12 x , \quad 0 \leqslant x \leqslant 1 .$$
  1. Find the mean and the mode of \(X\).
  2. Find the cumulative distribution function \(\mathrm { F } ( x )\) of \(X\). $$\text { Verify that } \mathrm { F } \left( \frac { 1 } { 4 } \right) = \frac { 67 } { 256 } , \mathrm {~F} \left( \frac { 1 } { 2 } \right) = \frac { 11 } { 16 } \text { and } \mathrm { F } \left( \frac { 3 } { 4 } \right) = \frac { 243 } { 256 } .$$ The engineers suspect that the process for generating simulated loads might not be working as intended. To investigate this, they generate a random sample of 512 loads. These are recorded in a frequency distribution as follows.
    Load \(x\)\(0 \leqslant x \leqslant \frac { 1 } { 4 }\)\(\frac { 1 } { 4 } < x \leqslant \frac { 1 } { 2 }\)\(\frac { 1 } { 2 } < x \leqslant \frac { 3 } { 4 }\)\(\frac { 3 } { 4 } < x \leqslant 1\)
    Frequency12620913146
  3. Use a suitable statistical procedure to assess the goodness of fit of \(X\) to these data. Discuss your conclusions briefly.
OCR MEI S3 2006 June Q2
18 marks Standard +0.3
2 A bus route runs from the centre of town A through the town's urban area to a point B on its boundary and then through the country to a small town C . Because of traffic congestion and general road conditions, delays occur on both the urban and the country sections. All delays may be considered independent. The scheduled time for the journey from A to B is 24 minutes. In fact, journey times over this section are given by the Normally distributed random variable \(X\) with mean 26 minutes and standard deviation 3 minutes. The scheduled time for the journey from B to C is 18 minutes. In fact, journey times over this section are given by the Normally distributed random variable \(Y\) with mean 15 minutes and standard deviation 2 minutes. Journey times on the two sections of route may be considered independent. The timetable published to the public does not show details of times at intermediate points; thus, if a bus is running early, it merely continues on its journey and is not required to wait.
  1. Find the probability that a journey from A to B is completed in less than the scheduled time of 24 minutes.
  2. Find the probability that a journey from A to C is completed in less than the scheduled time of 42 minutes.
  3. It is proposed to introduce a system of bus lanes in the urban area. It is believed that this would mean that the journey time from A to B would be given by the random variable \(0.85 X\). Assuming this to be the case, find the probability that a journey from A to B would be completed in less than the currently scheduled time of 24 minutes.
  4. An alternative proposal is to introduce an express service. This would leave out some bus stops on both sections of the route and its overall journey time from A to C would be given by the random variable \(0.9 X + 0.8 Y\). The scheduled time from A to C is to be given as a whole number of minutes. Find the least possible scheduled time such that, with probability 0.75 , buses would complete the journey on time or early.
  5. A programme of minor road improvements is undertaken on the country section. After their completion, it is thought that the random variable giving the journey time from B to C is still Normally distributed with standard deviation 2 minutes. A random sample of 15 journeys is found to have a sample mean journey time from B to C of 13.4 minutes. Provide a two-sided \(95 \%\) confidence interval for the population mean journey time from B to C .
OCR MEI S3 2006 June Q3
18 marks Moderate -0.3
3 An employer has commissioned an opinion polling organisation to undertake a survey of the attitudes of staff to proposed changes in the pension scheme. The staff are categorised as management, professional and administrative, and it is thought that there might be considerable differences of opinion between the categories. There are 60,140 and 300 staff respectively in the categories. The budget for the survey allows for a sample of 40 members of staff to be selected for in-depth interviews.
  1. Explain why it would be unwise to select a simple random sample from all the staff.
  2. Discuss whether it would be sensible to consider systematic sampling.
  3. What are the advantages of stratified sampling in this situation?
  4. State the sample sizes in each category if stratified sampling with as nearly as possible proportional allocation is used. The opinion polling organisation needs to estimate the average wealth of staff in the categories, in terms of property, savings, investments and so on. In a random sample of 11 professional staff, the sample mean is \(\pounds 345818\) and the sample standard deviation is \(\pounds 69241\).
  5. Assuming the underlying population is Normally distributed, test at the \(5 \%\) level of significance the null hypothesis that the population mean is \(\pounds 300000\) against the alternative hypothesis that it is greater than \(\pounds 300000\). Provide also a two-sided \(95 \%\) confidence interval for the population mean.
    [0pt] [10]
OCR MEI S3 2006 June Q4
18 marks Moderate -0.3
4 A company has many factories. It is concerned about incidents of trespassing and, in the hope of reducing if not eliminating these, has embarked on a programme of installing new fencing.
  1. Records for a random sample of 9 factories of the numbers of trespass incidents in typical weeks before and after installation of the new fencing are as follows.
    FactoryABCDEFGHI
    Number before installation81264142241314
    Number after installation6110118101154
    Use a Wilcoxon test to examine at the \(5 \%\) level of significance whether it appears that, on the whole, the number of trespass incidents per week is lower after the installation of the new fencing than before.
  2. Records are also available of the costs of damage from typical trespass incidents before and after the introduction of the new fencing for a random sample of 7 factories, as follows (in £).
    FactoryTUVWXYZ
    Cost before installation1215955464672356236550
    Cost after installation12681105784802417318620
    Stating carefully the required distributional assumption, provide a two-sided \(99 \%\) confidence interval based on a \(t\) distribution for the population mean difference between costs of damage before and after installation of the new fencing. Explain why this confidence interval should not be based on the Normal distribution.
OCR MEI S3 2007 June Q1
18 marks Standard +0.3
1 A manufacturer of fireworks is investigating the lengths of time for which the fireworks burn. For a particular type of firework this length of time, in minutes, is modelled by the random variable \(T\) with probability density function $$\mathrm { f } ( t ) = k t ^ { 3 } ( 2 - t ) \quad \text { for } 0 < t \leqslant 2$$ where \(k\) is a constant.
  1. Show that \(k = \frac { 5 } { 8 }\).
  2. Find the modal time.
  3. Find \(\mathrm { E } ( T )\) and show that \(\operatorname { Var } ( T ) = \frac { 8 } { 63 }\).
  4. A large random sample of \(n\) fireworks of this type is tested. Write down in terms of \(n\) the approximate distribution of \(\bar { T }\), the sample mean time.
  5. For a random sample of 100 such fireworks the times are summarised as follows. $$\Sigma t = 145.2 \quad \Sigma t ^ { 2 } = 223.41$$ Find a 95\% confidence interval for the mean time for this type of firework and hence comment on the appropriateness of the model.
OCR MEI S3 2007 June Q2
18 marks Standard +0.3
2 The operator of a section of motorway toll road records its weekly takings according to the types of vehicles using the motorway. For purposes of charging, there are three types of vehicle: cars, coaches, lorries. The weekly takings (in thousands of pounds) for each type are assumed to be Normally distributed. These distributions are independent of each other and are summarised in the table.
Vehicle typeMeanStandard deviation
Cars60.25.2
Coaches33.96.3
Lorries52.44.9
  1. Find the probability that the weekly takings for coaches are less than \(\pounds 40000\).
  2. Find the probability that the weekly takings for lorries exceed the weekly takings for cars.
  3. Find the probability that over a 4 -week period the total takings for cars exceed \(\pounds 225000\). What assumption must be made about the four weeks?
  4. Each week the operator allocates part of the takings for repairs. This is determined for each type of vehicle according to estimates of the long-term damage caused. It is calculated as follows: \(5 \%\) of takings for cars, \(10 \%\) for coaches and \(20 \%\) for lorries. Find the probability that in any given week the total amount allocated for repairs will exceed \(\pounds 20000\).
OCR MEI S3 2007 June Q3
18 marks Standard +0.3
3 The management of a large chain of shops aims to reduce the level of absenteeism among its workforce by means of an incentive bonus scheme. In order to evaluate the effectiveness of the scheme, the management measures the percentage of working days lost before and after its introduction for each of a random sample of 11 shops. The results are shown below.
ShopABCDEFGHIJK
\% days lost before3.55.03.53.24.54.94.16.06.88.16.0
\% days lost after1.84.32.94.54.45.83.56.76.45.45.1
  1. The management decides to carry out a \(t\) test to investigate whether there has been a reduction in absenteeism.
    1. State clearly the hypotheses that should be used together with any necessary assumptions.
    2. Carry out the test using a \(5 \%\) significance level.
  2. Find a 95\% confidence interval for the true mean percentage of days lost after the introduction of the incentive scheme and state any assumption needed. The management has set a target that the mean percentage should be 3.5. Do you think this has been achieved? Explain your answer.
OCR MEI S3 2007 June Q4
18 marks Standard +0.3
4 A machine produces plastic strip in a continuous process. Occasionally there is a flaw at some point along the strip. The length of strip (in hundreds of metres) between successive flaws is modelled by a continuous random variable \(X\) with probability density function \(\mathrm { f } ( x ) = \frac { 18 } { ( 3 + x ) ^ { 3 } }\) for \(x > 0\). The table below gives the frequencies for 100 randomly chosen observations of \(X\). It also gives the probabilities for the class intervals using the model.
Length \(x\) (hundreds of metres)Observed frequencyProbability
\(0 < x \leqslant 0.5\)210.2653
\(0.5 < x \leqslant 1\)240.1722
\(1 < x \leqslant 2\)120.2025
\(2 < x \leqslant 3\)150.1100
\(3 < x \leqslant 5\)130.1094
\(5 < x \leqslant 10\)90.0874
\(x > 10\)60.0532
  1. Examine the fit of this model to the data at the \(5 \%\) level of significance. You are given that the median length between successive flaws is 124 metres. At a later date the following random sample of ten lengths (in metres) between flaws is obtained. $$\begin{array} { l l l l l l l l l l } 239 & 77 & 179 & 221 & 100 & 312 & 52 & 129 & 236 & 42 \end{array}$$
  2. Test at the \(10 \%\) level of significance whether the median length may still be assumed to be 124 metres.
OCR MEI C3 2005 June Q1
3 marks Easy -1.2
1 Solve the equation \(| 3 x + 2 | = 1\).
OCR MEI C3 2005 June Q2
3 marks Easy -1.2
2 Given that \(\arcsin x = \frac { 1 } { 6 } \pi\), find \(x\). Find \(\arccos x\) in terms of \(\pi\).
OCR MEI C3 2005 June Q3
3 marks Moderate -0.8
3 The functions \(\mathrm { f } ( x )\) and \(\mathrm { g } ( x )\) are defined for the domain \(x > 0\) as follows: $$\mathrm { f } ( x ) = \ln x , \quad \mathrm {~g} ( x ) = x ^ { 3 } .$$ Express the composite function \(\mathrm { fg } ( x )\) in terms of \(\ln x\).
State the transformation which maps the curve \(y = \mathrm { f } ( x )\) onto the curve \(y = \mathrm { fg } ( x )\).
OCR MEI C3 2005 June Q4
6 marks Moderate -0.8
4 The temperature \(T ^ { \circ } \mathrm { C }\) of a liquid at time \(t\) minutes is given by the equation $$T = 30 + 20 \mathrm { e } ^ { - 0.05 t } , \quad \text { for } t \geqslant 0 .$$ Write down the initial temperature of the liquid, and find the initial rate of change of temperature.
Find the time at which the temperature is \(40 ^ { \circ } \mathrm { C }\).
OCR MEI C3 2005 June Q5
6 marks Standard +0.3
5 Using the substitution \(u = 2 x + 1\), show that \(\int _ { 0 } ^ { 1 } \frac { x } { 2 x + 1 } \mathrm {~d} x = \frac { 1 } { 4 } ( 2 - \ln 3 )\).
OCR MEI C3 2005 June Q6
7 marks Standard +0.3
6 A curve has equation \(y = \frac { x } { 2 + 3 \ln x }\). Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\). Hence find the exact coordinates of the stationary point of the curve.
OCR MEI C3 2005 June Q7
8 marks Standard +0.3
7 Fig. 7 shows the curve defined implicitly by the equation $$y ^ { 2 } + y = x ^ { 3 } + 2 x ,$$ together with the line \(x = 2\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{3efea8db-9fa1-47a8-89b8-e4888f87a313-3_465_378_534_808} \captionsetup{labelformat=empty} \caption{Not to scale}
\end{figure} Fig. 7 Find the coordinates of the points of intersection of the line and the curve.
Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\) and \(y\). Hence find the gradient of the curve at each of these two points.
OCR MEI C3 2005 June Q8
17 marks Standard +0.3
8 Fig. 8 shows part of the curve \(y = x \sin 3 x\). It crosses the \(x\)-axis at P . The point on the curve with \(x\)-coordinate \(\frac { 1 } { 6 } \pi\) is Q . \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{3efea8db-9fa1-47a8-89b8-e4888f87a313-3_421_789_1748_610} \captionsetup{labelformat=empty} \caption{Fig. 8}
\end{figure}
  1. Find the \(x\)-coordinate of P .
  2. Show that Q lies on the line \(y = x\).
  3. Differentiate \(x \sin 3 x\). Hence prove that the line \(y = x\) touches the curve at Q .
  4. Show that the area of the region bounded by the curve and the line \(y = x\) is \(\frac { 1 } { 72 } \left( \pi ^ { 2 } - 8 \right)\).
OCR MEI C3 2005 June Q9
19 marks Standard +0.3
9 The function \(\mathrm { f } ( x ) = \ln \left( 1 + x ^ { 2 } \right)\) has domain \(- 3 \leqslant x \leqslant 3\).
Fig. 9 shows the graph of \(y = \mathrm { f } ( x )\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{3efea8db-9fa1-47a8-89b8-e4888f87a313-4_540_943_477_550} \captionsetup{labelformat=empty} \caption{Fig. 9}
\end{figure}
  1. Show algebraically that the function is even. State how this property relates to the shape of the curve.
  2. Find the gradient of the curve at the point \(\mathrm { P } ( 2 , \ln 5 )\).
  3. Explain why the function does not have an inverse for the domain \(- 3 \leqslant x \leqslant 3\). The domain of \(\mathrm { f } ( x )\) is now restricted to \(0 \leqslant x \leqslant 3\). The inverse of \(\mathrm { f } ( x )\) is the function \(\mathrm { g } ( x )\).
  4. Sketch the curves \(y = \mathrm { f } ( x )\) and \(y = \mathrm { g } ( x )\) on the same axes. State the domain of the function \(\mathrm { g } ( x )\). Show that \(\mathrm { g } ( x ) = \sqrt { \mathrm { e } ^ { x } - 1 }\).
  5. Differentiate \(\mathrm { g } ( x )\). Hence verify that \(\mathrm { g } ^ { \prime } ( \ln 5 ) = 1 \frac { 1 } { 4 }\). Explain the connection between this result and your answer to part (ii).
OCR S4 2007 June Q1
6 marks Standard +0.3
1 For the events \(A\) and \(B , \mathrm { P } ( A ) = 0.3 , \mathrm { P } ( B ) = 0.6\) and \(\mathrm { P } \left( A ^ { \prime } \cap B ^ { \prime } \right) = c\), where \(c \neq 0\).
  1. Find \(\mathrm { P } ( A \cap B )\) in terms of \(c\).
  2. Find \(\mathrm { P } ( B \mid A )\) and deduce that \(0.1 \leqslant c \leqslant 0.4\).
OCR S4 2007 June Q2
7 marks Standard +0.3
2 Of 9 randomly chosen students attending a lecture, 4 were found to be smokers and 5 were nonsmokers. During the lecture their pulse-rates were measured, with the following results in beats per minute.
Smokers77859098
Non-smokers5964688088
It may be assumed that these two groups of students were random samples from the student populations of smokers and non-smokers. Using a suitable Wilcoxon test at the \(10 \%\) significance level, test whether there is a difference in the median pulse-rates of the two populations.
OCR S4 2007 June Q3
7 marks Standard +0.3
3 The discrete random variables \(X\) and \(Y\) have the joint probability distribution given in the following table.
\(X\)
\cline { 2 - 5 } \multicolumn{1}{l}{}- 101
10.240.220.04
20.260.180.06
  1. Show that \(\operatorname { Cov } ( X , Y ) = 0\).
  2. Find the conditional distribution of \(X\) given that \(Y = 2\).