Questions — OCR MEI (4333 questions)

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OCR MEI S3 2016 June Q4
18 marks Standard +0.3
4 An insurance company is investigating a new system designed to reduce the average time taken to process claim forms. The company has decided to use 10 experienced employees to process claims using the old system and the new system. Two procedures for comparing the systems are proposed.
Procedure \(A\) There are two sets of claim forms, set 1 and set 2. Each contains the same number of forms. Each employee processes set 1 on the old system and set 2 on the new system. The times taken are compared. Procedure \(B\) There is just one set of claim forms which each employee processes firstly on the old system and then on the new system. The times taken are compared.
  1. State one weakness of each of these procedures. In fact a third procedure which avoids these two weaknesses is adopted. In this procedure each employee is given a randomly selected set of claim forms. Each set contains the same number of forms. The employees each process their set of claim forms on both systems. The times taken, in minutes, are shown in the table.
    Employee12345678910
    Old system40.542.952.851.777.266.765.249.255.658.3
    New system39.240.750.650.771.470.571.147.752.155.5
  2. Carry out a paired \(t\) test at the \(5 \%\) level of significance to investigate whether the mean length of time taken to process a set of forms has reduced using the new system.
  3. State fully the usual conditions for a paired \(t\) test.
  4. Construct a \(99 \%\) confidence interval for the mean reduction in time taken to process a set of forms using the new system.
OCR MEI S4 Q4
12 marks Standard +0.8
4 An experiment is carried out to compare five industrial paints, A, B, C, D, E, that are intended to be used to protect exterior surfaces in polluted urban environments. Five different types of surface (I, II, III, IV, V) are to be used in the experiment, and five specimens of each type of surface are available. Five different external locations ( \(1,2,3,4,5\) ) are used in the experiment. The paints are applied to the specimens of the surfaces which are then left in the locations for a period of six months. At the end of this period, a "score" is given to indicate how effective the paint has been in protecting the surface.
  1. Name a suitable experimental design for this trial and give an example of an experimental layout. Initial analysis of the data indicates that any differences between the types of surface are negligible, as also are any differences between the locations. It is therefore decided to analyse the data by one-way analysis of variance.
  2. State the usual model, including the accompanying distributional assumptions, for the one-way analysis of variance. Interpret the terms in the model.
  3. The data for analysis are as follows. Higher scores indicate better performance. The underlying distributions of strengths are assumed to be Normal for both suppliers, with variances 2.45 for supplier A and 1.40 for supplier B.
  4. Test at the \(5 \%\) level of significance whether it is reasonable to assume that the mean strengths from the two suppliers are equal.
  5. Provide a two-sided 90\% confidence interval for the true mean difference.
  6. Show that the test procedure used in part (i), with samples of sizes 7 and 5 and a \(5 \%\) significance level, leads to acceptance of the null hypothesis of equal means if \(- 1.556 < \bar { x } - \bar { y } < 1.556\), where \(\bar { x }\) and \(\bar { y }\) are the observed sample means from suppliers A and B . Hence find the probability of a Type II error for this test procedure if in fact the true mean strength from supplier A is 2.0 units more than that from supplier B.
  7. A manager suggests that the Wilcoxon rank sum test should be used instead, comparing the median strengths for the samples of sizes 7 and 5 . Give one reason why this suggestion might be sensible and two why it might not.
OCR MEI S4 2009 June Q1
24 marks Challenging +1.2
1 An industrial process produces components. Some of the components contain faults. The number of faults in a component is modelled by the random variable \(X\) with probability function $$\mathrm { P } ( X = x ) = \theta ( 1 - \theta ) ^ { x } \quad \text { for } x = 0,1,2 , \ldots$$ where \(\theta\) is a parameter with \(0 < \theta < 1\). The numbers of faults in different components are independent.
A random sample of \(n\) components is inspected. \(n _ { 0 }\) are found to have no faults, \(n _ { 1 }\) to have one fault and the remainder \(\left( n - n _ { 0 } - n _ { 1 } \right)\) to have two or more faults.
  1. Find \(\mathrm { P } ( X \geqslant 2 )\) and hence show that the likelihood is $$\mathrm { L } ( \theta ) = \theta ^ { n _ { 0 } + n _ { 1 } } ( 1 - \theta ) ^ { 2 n - 2 n _ { 0 } - n _ { 1 } }$$
  2. Find the maximum likelihood estimator \(\hat { \theta }\) of \(\theta\). You are not required to verify that any turning point you locate is a maximum.
  3. Show that \(\mathrm { E } ( X ) = \frac { 1 - \theta } { \theta }\). Deduce that another plausible estimator of \(\theta\) is \(\tilde { \theta } = \frac { 1 } { 1 + \bar { X } }\) where \(\bar { X }\) is the sample mean. What additional information is needed in order to calculate the value of this estimator?
  4. You are given that, in large samples, \(\tilde { \theta }\) may be taken as Normally distributed with mean \(\theta\) and variance \(\theta ^ { 2 } ( 1 - \theta ) / n\). Use this to obtain a \(95 \%\) confidence interval for \(\theta\) for the case when 100 components are inspected and it is found that 92 have no faults, 6 have one fault and the remaining 2 have exactly four faults each.
OCR MEI S4 2009 June Q2
24 marks Standard +0.3
2
  1. The random variable \(Z\) has the standard Normal distribution with probability density function $$\mathrm { f } ( z ) = \frac { 1 } { \sqrt { 2 \pi } } \mathrm { e } ^ { - z ^ { 2 } / 2 } , \quad - \infty < z < \infty$$ Obtain the moment generating function of \(Z\).
  2. Let \(\mathrm { M } _ { Y } ( t )\) denote the moment generating function of the random variable \(Y\). Show that the moment generating function of the random variable \(a Y + b\), where \(a\) and \(b\) are constants, is \(\mathrm { e } ^ { b t } \mathrm { M } _ { Y } ( a t )\).
  3. Use the results in parts (i) and (ii) to obtain the moment generating function \(\mathrm { M } _ { X } ( t )\) of the random variable \(X\) having the Normal distribution with parameters \(\mu\) and \(\sigma ^ { 2 }\).
  4. If \(W = \mathrm { e } ^ { X }\) where \(X\) is as in part (iii), \(W\) is said to have a lognormal distribution. Show that, for any positive integer \(k\), the expected value of \(W ^ { k }\) is \(\mathrm { M } _ { X } ( k )\). Use this result to find the expected value and variance of the lognormal distribution.
OCR MEI S4 2009 June Q3
24 marks Standard +0.3
3
  1. At a waste disposal station, two methods for incinerating some of the rubbish are being compared. Of interest is the amount of particulates in the exhaust, which can be measured over the working day in a convenient unit of concentration. It is assumed that the underlying distributions of concentrations of particulates are Normal. It is also assumed that the underlying variances are equal. During a period of several months, measurements are made for method A on a random sample of 10 working days and for method B on a separate random sample of 7 working days, with results, in the convenient unit, as follows.
    Method A124.8136.4116.6129.1140.7120.2124.6127.5111.8130.3
    Method B130.4136.2119.8150.6143.5126.1130.7
    Use a \(t\) test at the \(10 \%\) level of significance to examine whether either method is better in resulting, on the whole, in a lower concentration of particulates. State the null and alternative hypotheses under test.
  2. The company's statistician criticises the design of the trial in part (i) on the grounds that it is not paired. Summarise the arguments the statistician will have used. A new trial is set up with a paired design, measuring the concentrations of particulates on a random sample of 9 paired occasions. The results are as follows.
    PairIIIIIIIVVVIVIIVIIIIX
    Method A119.6127.6141.3139.5141.3124.1116.6136.2128.8
    Method B112.2128.8130.2134.0135.1120.4116.9134.4125.2
    Use a \(t\) test at the \(5 \%\) level of significance to examine the same hypotheses as in part (i). State the underlying distributional assumption that is needed in this case.
  3. State the names of procedures that could be used in the situations of parts (i) and (ii) if the underlying distributional assumptions could not be made. What hypotheses would be under test?
OCR MEI S4 2009 June Q4
24 marks Standard +0.3
4
  1. Describe, with the aid of a specific example, an experimental situation for which a Latin square design is appropriate, indicating carefully the features which show that a completely randomised or randomised blocks design would be inappropriate.
  2. The model for the one-way analysis of variance may be written, in a customary notation, as $$x _ { i j } = \mu + \alpha _ { i } + e _ { i j }$$ State the distributional assumptions underlying \(e _ { i j }\) in this model. What is the interpretation of the term \(\alpha _ { i }\) ?
  3. An experiment for comparing 5 treatments is carried out, with a total of 20 observations. A partial one-way analysis of variance table for the analysis of the results is as follows.
    Source of variationSums of squaresDegrees of freedomMean squaresMean square ratio
    Between treatments
    Residual68.76
    Total161.06
    Copy and complete the table, and carry out the appropriate test using a \(1 \%\) significance level.
OCR MEI S4 2011 June Q1
24 marks Standard +0.8
1 The random variable \(X\) has the Normal distribution with mean 0 and variance \(\theta\), so that its probability density function is $$\mathrm { f } ( x ) = \frac { 1 } { \sqrt { 2 \pi \theta } } \mathrm { e } ^ { - x ^ { 2 } / 2 \theta } , \quad - \infty < x < \infty$$ where \(\theta ( \theta > 0 )\) is unknown. A random sample of \(n\) observations from \(X\) is denoted by \(X _ { 1 } , X _ { 2 } , \ldots , X _ { n }\).
  1. Find \(\hat { \theta }\), the maximum likelihood estimator of \(\theta\).
  2. Show that \(\hat { \theta }\) is an unbiased estimator of \(\theta\).
  3. In large samples, the variance of \(\hat { \theta }\) may be estimated by \(\frac { 2 \hat { \theta } ^ { 2 } } { n }\). Use this and the results of parts (i) and (ii) to find an approximate \(95 \%\) confidence interval for \(\theta\) in the case when \(n = 100\) and \(\Sigma X _ { i } ^ { 2 } = 1000\).
OCR MEI S4 2011 June Q2
24 marks Standard +0.8
2 The random variable \(X\) has the \(\chi _ { n } ^ { 2 }\) distribution. This distribution has moment generating function \(\mathrm { M } ( \theta ) = ( 1 - 2 \theta ) ^ { - \frac { 1 } { 2 } n }\), where \(\theta < \frac { 1 } { 2 }\).
  1. Verify the expression for \(\mathrm { M } ( \theta )\) quoted above for the cases \(n = 2\) and \(n = 4\), given that the probability density functions of \(X\) in these cases are as follows. $$\begin{array} { l l } n = 2 : & \mathrm { f } ( x ) = \frac { 1 } { 2 } \mathrm { e } ^ { - \frac { 1 } { 2 } x } \quad ( x > 0 ) \\ n = 4 : & \mathrm { f } ( x ) = \frac { 1 } { 4 } x \mathrm { e } ^ { - \frac { 1 } { 2 } x } \quad ( x > 0 ) \end{array}$$
  2. For the general case, use \(\mathrm { M } ( \theta )\) to find the mean and variance of \(X\) in terms of \(n\).
  3. \(Y _ { 1 } , Y _ { 2 } , \ldots , Y _ { k }\) are independent random variables, each with the \(\chi _ { 1 } ^ { 2 }\) distribution. Show that \(W = \sum _ { i = 1 } ^ { k } Y _ { i }\) has the \(\chi _ { k } ^ { 2 }\) distribution.
  4. Use the Central Limit Theorem to find an approximation for \(\mathrm { P } ( W < 118.5 )\) for the case \(k = 100\).
OCR MEI S4 2011 June Q3
24 marks Challenging +1.2
3
  1. Explain the meaning of the following terms in the context of hypothesis testing: Type I error, Type II error, operating characteristic, power.
  2. A market research organisation is designing a sample survey to investigate whether expenditure on everyday food items has increased in 2011 compared with 2010. For one of the populations being studied, the random variable \(X\) is used to model weekly expenditure, in \(\pounds\), on these items in 2011, where \(X\) is Normally distributed with mean \(\mu\) and variance \(\sigma ^ { 2 }\). As the corresponding mean value in 2010 was 94 , the hypotheses to be examined are $$\begin{aligned} & \mathrm { H } _ { 0 } : \mu = 94 \\ & \mathrm { H } _ { 1 } : \mu > 94 \end{aligned}$$ By comparison with the corresponding 2010 value, \(\sigma ^ { 2 }\) is assumed to be 25 .
    The following criteria for the survey are laid down.
    • If in fact \(\mu = 94\), the probability of concluding that \(\mu > 94\) must be only \(2 \%\)
    • If in fact \(\mu = 97\), the probability of concluding that \(\mu > 94\) must be \(95 \%\)
    A random sample of size \(n\) is to be taken and the usual Normal test based on \(\bar { X }\) is to be used, with a critical value of \(c\) such that \(\mathrm { H } _ { 0 }\) is rejected if the value of \(\bar { X }\) exceeds \(c\). Find \(c\) and the smallest value of \(n\) that is required.
  3. Sketch the power function of an ideal test for examining the hypotheses in part (ii).
OCR MEI S4 2011 June Q4
24 marks Standard +0.3
4
  1. Provide an example of an experimental situation where there is one factor of primary interest and where a suitable experimental design would be
    1. randomised blocks,
    2. a Latin square. In each case, explain carefully why the design is suitable and why the other design would not be appropriate.
  2. An industrial experiment to compare four treatments for increasing the tensile strength of steel is carried out according to a completely randomised design. For various reasons, it is not possible to use the same number of replicates for each treatment. The increases, in a suitable unit of tensile strength, are as follows.
    Treatment
    A
    Treatment
    B
    Treatment
    C
    Treatment
    D
    10.121.19.222.6
    21.220.38.817.4
    11.616.015.223.1
    13.615.019.2
    [The sum of these data items is 256.8 and the sum of their squares is 4471.92 .] Construct the usual one-way analysis of variance table. Carry out the appropriate test, using a \(5 \%\) significance level. RECOGNISING ACHIEVEMENT
OCR MEI S4 2013 June Q1
24 marks Challenging +1.2
1 Traffic engineers are studying the flow of vehicles along a road. At an initial stage of the investigation, they assume that the average flow remains the same throughout the working day. An automatic counter records the number of vehicles passing a certain point per minute during the working day. A random sample of these records is selected; the sample values are denoted by \(x _ { 1 } , x _ { 2 } , \ldots , x _ { n }\).
  1. The engineers model the underlying random variable \(X\) by a Poisson distribution with unknown parameter \(\theta\). Obtain the likelihood of \(x _ { 1 } , x _ { 2 } , \ldots , x _ { n }\) and hence find the maximum likelihood estimate of \(\theta\).
  2. Write down the maximum likelihood estimate of the probability that no vehicles pass during a minute.
  3. The engineers note that, in a sample of size 1000 with sample mean \(\bar { x } = 5\), there are no observations of zero. Suggest why this might cast some doubt on the investigation.
  4. On checking the automatic counter, the engineers find that, due to a fault, no record at all is made if no vehicle passes in a minute. They therefore model \(X\) as a Poisson random variable, again with an unknown parameter \(\theta\), except that the value \(x = 0\) cannot occur. Show that, under this model, $$\mathrm { P } ( X = x ) = \frac { \theta ^ { x } } { \left( \mathrm { e } ^ { \theta } - 1 \right) x ! } , \quad x = 1,2 , \ldots$$ and hence show that the maximum likelihood estimate of \(\theta\) satisfies the equation $$\frac { \theta \mathrm { e } ^ { \theta } } { \mathrm { e } ^ { \theta } - 1 } = \bar { x }$$
OCR MEI S4 2013 June Q2
24 marks Challenging +1.8
2 The random variable \(X\) takes values \(- 2,0\) and 2 , each with probability \(\frac { 1 } { 3 }\).
  1. Write down the values of
    (A) \(\mu\), the mean of \(X\),
    (B) \(\mathrm { E } \left( X ^ { 2 } \right)\),
    (C) \(\sigma ^ { 2 }\), the variance of \(X\).
  2. Obtain the moment generating function (mgf) of \(X\). A random sample of \(n\) independent observations on \(X\) has sample mean \(\bar { X }\), and the standardised mean is denoted by \(Z\) where $$Z = \frac { \bar { X } - \mu } { \frac { \sigma } { \sqrt { n } } }$$
  3. Stating carefully the required general results for mgfs of sums and of linear transformations, show that the mgf of \(Z\) is $$M _ { Z } ( \theta ) = \left\{ \frac { 1 } { 3 } \left( 1 + e ^ { \frac { \theta \sqrt { 3 } } { \sqrt { 2 n } } } + e ^ { - \frac { \theta \sqrt { 3 } } { \sqrt { 2 n } } } \right) \right\} ^ { n } .$$
  4. By expanding the exponential functions in \(\mathrm { M } _ { Z } ( \theta )\), show that, for large \(n\), $$\mathrm { M } _ { Z } ( \theta ) \approx \left( 1 + \frac { \theta ^ { 2 } } { 2 n } \right) ^ { n }$$
  5. Use the result \(\mathrm { e } ^ { y } = \lim _ { n \rightarrow \infty } \left( 1 + \frac { y } { n } \right) ^ { n }\) to find the limit of \(\mathrm { M } _ { Z } ( \theta )\) as \(n \rightarrow \infty\), and deduce the approximate distribution of \(Z\) for large \(n\).
OCR MEI S4 2013 June Q3
24 marks Standard +0.3
3
  1. Explain the meaning of the following terms in the context of hypothesis testing: Type I error, Type II error, operating characteristic, power.
  2. A test is to be carried out concerning a parameter \(\theta\). The null hypothesis is that \(\theta\) has the particular value \(\theta _ { 0 }\). The alternative hypothesis is \(\theta \neq \theta _ { 0 }\). Draw a sketch of the operating characteristic for a perfect test that never makes an error.
  3. The random variable \(X\) is distributed as \(\mathrm { N } ( \mu , 9 )\). A random sample of size 25 is available. The null hypothesis \(\mu = 0\) is to be tested against the alternative hypothesis \(\mu \neq 0\). The null hypothesis will be accepted if \(- 1 < \bar { x } < 1\) where \(\bar { x }\) is the value of the sample mean, otherwise it will be rejected. Calculate the probability of a Type I error. Calculate the probability of a Type II error if in fact \(\mu = 0.5\); comment on the value of this probability.
  4. Without carrying out any further calculations, draw a sketch of the operating characteristic for the test in part (iii).
OCR MEI S4 2013 June Q4
24 marks Moderate -0.3
4
  1. Explain the advantages of randomisation and replication in a statistically designed experiment.
  2. The usual statistical model underlying the one-way analysis of variance is given, in the usual notation, by $$x _ { i j } = \mu + \alpha _ { i } + e _ { i j }$$ where \(x _ { i j }\) denotes the \(j\) th observation on the \(i\) th treatment. Define carefully all the terms in this model and state the properties of the term that represents experimental error.
  3. A trial of five fertilisers is carried out at an agricultural research station according to a completely randomised design in which each fertiliser is applied to four experimental plots of a crop (so that there are 20 experimental units altogether). The sums of squares in a one-way analysis of variance of the resulting data on yields of the crop are as follows.
    Source of variationSum of squares
    Between fertilisers219.2
    Residual304.5
    Total523.7
    State the customary null and alternative hypotheses that are tested. Provide the degrees of freedom for each sum of squares. Hence copy and complete the analysis of variance table and carry out the test at the 5\% level.
OCR MEI M1 2009 January Q1
8 marks Easy -1.2
1 A particle is travelling in a straight line. Its velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at time \(t\) seconds is given by $$v = 6 + 4 t \quad \text { for } 0 \leqslant t \leqslant 5$$
  1. Write down the initial velocity of the particle and find the acceleration for \(0 \leqslant t \leqslant 5\).
  2. Write down the velocity of the particle when \(t = 5\). Find the distance travelled in the first 5 seconds. For \(5 \leqslant t \leqslant 15\), the acceleration of the particle is \(3 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
  3. Find the total distance travelled by the particle during the 15 seconds.
OCR MEI M1 2009 January Q2
4 marks Moderate -0.3
2 Fig. 2 shows an acceleration-time graph modelling the motion of a particle. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{93a5d409-ade4-418b-9c09-620d97df97de-2_684_1070_1064_536} \captionsetup{labelformat=empty} \caption{Fig. 2}
\end{figure} At \(t = 0\) the particle has a velocity of \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in the positive direction.
  1. Find the velocity of the particle when \(t = 2\).
  2. At what time is the particle travelling in the negative direction with a speed of \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) ?
OCR MEI M1 2009 January Q3
6 marks Moderate -0.3
3 The resultant of the force \(\binom { - 4 } { 8 } \mathrm {~N}\) and the force \(\mathbf { F }\) gives an object of mass 6 kg an acceleration of \(\binom { 2 } { 3 } \mathrm {~ms} ^ { - 2 }\).
  1. Calculate \(\mathbf { F }\).
  2. Calculate the angle between \(\mathbf { F }\) and the vector \(\binom { 0 } { 1 }\).
OCR MEI M1 2009 January Q4
6 marks Moderate -0.3
4 Sandy is throwing a stone at a plum tree. The stone is thrown from a point O at a speed of \(35 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(\alpha\) to the horizontal, where \(\cos \alpha = 0.96\). You are given that, \(t\) seconds after being thrown, the stone is \(\left( 9.8 t - 4.9 t ^ { 2 } \right) \mathrm { m }\) higher than O . When descending, the stone hits a plum which is 3.675 m higher than O . Air resistance should be neglected. Calculate the horizontal distance of the plum from O .
OCR MEI M1 2009 January Q5
5 marks Moderate -0.3
5 A man of mass 75 kg is standing in a lift. He is holding a parcel of mass 5 kg by means of a light inextensible string, as shown in Fig. 5. The tension in the string is 55 N . \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{93a5d409-ade4-418b-9c09-620d97df97de-3_456_476_833_833} \captionsetup{labelformat=empty} \caption{Fig. 5}
\end{figure}
  1. Find the upward acceleration.
  2. Find the reaction on the man of the lift floor.
OCR MEI M1 2009 January Q6
7 marks Standard +0.3
6 Small stones A and B are initially in the positions shown in Fig. 6 with B a height \(H \mathrm {~m}\) directly above A. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{93a5d409-ade4-418b-9c09-620d97df97de-3_318_271_1800_938} \captionsetup{labelformat=empty} \caption{Fig. 6}
\end{figure} At the instant when B is released from rest, A is projected vertically upwards with a speed of \(29.4 \mathrm {~m} \mathrm {~s} \mathrm {~s} ^ { - 1 }\). Air resistance may be neglected. The stones collide \(T\) seconds after they begin to move. At this instant they have the same speed, \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\), and A is still rising. By considering when the speed of A upwards is the same as the speed of B downwards, or otherwise, show that \(T = 1.5\) and find the values of \(V\) and \(H\). Section B (36 marks)
OCR MEI M1 2009 January Q7
17 marks Moderate -0.3
7 An explorer is trying to pull a loaded sledge of total mass 100 kg along horizontal ground using a light rope. The only resistance to motion of the sledge is from friction between it and the ground. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{93a5d409-ade4-418b-9c09-620d97df97de-4_327_1013_482_566} \captionsetup{labelformat=empty} \caption{Fig. 7}
\end{figure} Initially she pulls with a force of 121 N on the rope inclined at \(34 ^ { \circ }\) to the horizontal, as shown in Fig. 7, but the sledge does not move.
  1. Draw a diagram showing all the forces acting on the sledge. Show that the frictional force between the ground and the sledge is 100 N , correct to 3 significant figures. Calculate the normal reaction of the ground on the sledge. The sledge is given a small push to set it moving at \(0.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The explorer continues to pull on the rope with the same force and the same angle as before. The frictional force is also unchanged.
  2. Describe the subsequent motion of the sledge. The explorer now pulls the rope, still at an angle of \(34 ^ { \circ }\) to the horizontal, so that the tension in it is 155 N . The frictional force is now 95 N .
  3. Calculate the acceleration of the sledge. In a new situation, there is no rope and the sledge slides down a uniformly rough slope inclined at \(26 ^ { \circ }\) to the horizontal. The sledge starts from rest and reaches a speed of \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in 2 seconds.
  4. Calculate the frictional force between the slope and the sledge.
OCR MEI M1 2009 January Q8
19 marks Moderate -0.3
8 A toy boat moves in a horizontal plane with position vector \(\mathbf { r } = x \mathbf { i } + y \mathbf { j }\), where \(\mathbf { i }\) and \(\mathbf { j }\) are the standard unit vectors east and north respectively. The origin of the position vectors is at O . The displacements \(x\) and \(y\) are in metres. First consider only the motion of the boat parallel to the \(x\)-axis. For this motion $$x = 8 t - 2 t ^ { 2 }$$ The velocity of the boat in the \(x\)-direction is \(v _ { x } \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  1. Find an expression in terms of \(t\) for \(v _ { x }\) and determine when the boat instantaneously has zero speed in the \(x\)-direction. Now consider only the motion of the boat parallel to the \(y\)-axis. For this motion $$v _ { y } = ( t - 2 ) ( 3 t - 2 )$$ where \(v _ { y } \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the velocity of the boat in the \(y\)-direction at time \(t\) seconds.
  2. Given that \(y = 3\) when \(t = 1\), use integration to show that \(y = t ^ { 3 } - 4 t ^ { 2 } + 4 t + 2\). The position vector of the boat is given in terms of \(t\) by \(\mathbf { r } = \left( 8 t - 2 t ^ { 2 } \right) \mathbf { i } + \left( t ^ { 3 } - 4 t ^ { 2 } + 4 t + 2 \right) \mathbf { j }\).
  3. Find the time(s) when the boat is due north of O and also the distance of the boat from O at any such times.
  4. Find the time(s) when the boat is instantaneously at rest. Find the distance of the boat from O at any such times.
  5. Plot a graph of the path of the boat for \(0 \leqslant t \leqslant 2\).
OCR MEI M1 2010 June Q1
3 marks Easy -1.2
1 An egg falls from rest a distance of 75 cm to the floor.
Neglecting air resistance, at what speed does it hit the floor?
OCR MEI M1 2010 June Q2
4 marks Moderate -0.8
2 Fig. 2 shows a sack of rice of weight 250 N hanging in equilibrium supported by a light rope AB . End A of the rope is attached to the sack. The rope passes over a small smooth fixed pulley. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{6cca1e5e-82b0-487d-8048-b9db7745dea6-2_458_479_705_833} \captionsetup{labelformat=empty} \caption{Fig. 2}
\end{figure} Initially, end B of the rope is attached to a vertical wall as shown in Fig. 2.
  1. Calculate the horizontal and the vertical forces acting on the wall due to the rope. End B of the rope is now detached from the wall and attached instead to the top of the sack. The sack is in equilibrium with both sections of the rope vertical.
  2. Calculate the tension in the rope.
OCR MEI M1 2010 June Q3
8 marks Moderate -0.3
3 The three forces \(\left( \begin{array} { r } - 1 \\ 14 \\ - 8 \end{array} \right) \mathrm { N } , \left( \begin{array} { r } 3 \\ - 9 \\ 10 \end{array} \right) \mathrm { N }\) and \(\mathbf { F } \mathrm { N }\) act on a body of mass 4 kg in deep space and give it an acceleration of \(\left( \begin{array} { r } - 1 \\ 2 \\ 4 \end{array} \right) \mathrm { m } \mathrm { s } ^ { - 2 }\).
  1. Calculate \(\mathbf { F }\). At one instant the velocity of the body is \(\left( \begin{array} { r } - 3 \\ 3 \\ 6 \end{array} \right) \mathrm { m } \mathrm { s } ^ { - 1 }\).
  2. Calculate the velocity and also the speed of the body 3 seconds later.