Questions — OCR MEI (4455 questions)

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OCR MEI S1 Q4
4 marks Easy -1.2
25% of the plants of a particular species have red flowers. A random sample of 6 plants is selected.
  1. Find the probability that there are no plants with red flowers in the sample. [2]
  2. If 50 random samples of 6 plants are selected, find the expected number of samples in which there are no plants with red flowers. [2]
OCR MEI S1 Q5
8 marks Moderate -0.8
In a recent survey, a large number of working people were asked whether they worked full-time or part-time, with part-time being defined as less than 25 hours per week. One of the respondents is selected at random. • \(W\) is the event that this person works part-time. • \(F\) is the event that this person is female. You are given that \(\text{P}(W) = 0.14\), \(\text{P}(F) = 0.41\) and \(\text{P}(W \cap F) = 0.11\).
  1. Draw a Venn diagram showing the events \(W\) and \(F\), and fill in the probability corresponding to each of the four regions of your diagram. [3]
  2. Determine whether the events \(W\) and \(F\) are independent. [2]
  3. Find \(\text{P}(W|F)\) and explain what this probability represents. [3]
OCR MEI S1 Q6
4 marks Moderate -0.8
The table shows all the possible products of the scores on two fair four-sided dice. \includegraphics{figure_6}
  1. Find the probability that the product of the two scores is less than 10. [1]
  2. Show that the events 'the score on the first die is even' and 'the product of the scores on the two dice is less than 10' are not independent. [3]
OCR MEI S1 Q7
8 marks Moderate -0.8
Andy can walk to work, travel by bike or travel by bus. The tree diagram shows the probabilities of any day being dry or wet and the corresponding probabilities for each of Andy's methods of travel. \includegraphics{figure_7} A day is selected at random. Find the probability that
  1. the weather is wet and Andy travels by bus, [2]
  2. Andy walks or travels by bike, [3]
  3. the weather is dry given that Andy walks or travels by bike. [3]
OCR MEI S2 2007 January Q1
18 marks Moderate -0.8
In a science investigation into energy conservation in the home, a student is collecting data on the time taken for an electric kettle to boil as the volume of water in the kettle is varied. The student's data are shown in the table below, where \(v\) litres is the volume of water in the kettle and \(t\) seconds is the time taken for the kettle to boil (starting with the water at room temperature in each case). Also shown are summary statistics and a scatter diagram on which the regression line of \(t\) on \(v\) is drawn.
\(v\)0.20.40.60.81.0
\(t\)4478114156172
\(n = 5\), \(\Sigma v = 3.0\), \(\Sigma t = 564\), \(\Sigma v^2 = 2.20\), \(\Sigma vt = 405.2\). \includegraphics{figure_1}
  1. Calculate the equation of the regression line of \(t\) on \(v\), giving your answer in the form \(t = a + bv\). [5]
  2. Use this equation to predict the time taken for the kettle to boil when the amount of water which it contains is
    1. 0.5 litres,
    2. 1.5 litres.
    Comment on the reliability of each of these predictions. [4]
  3. In the equation of the regression line found in part (i), explain the role of the coefficient of \(v\) in the relationship between time taken and volume of water. [2]
  4. Calculate the values of the residuals for \(v = 0.8\) and \(v = 1.0\). [4]
  5. Explain how, on a scatter diagram with the regression line drawn accurately on it, a residual could be measured and its sign determined. [3]
OCR MEI S2 2007 January Q2
18 marks Moderate -0.3
  1. A farmer grows Brussels sprouts. The diameter of sprouts in a particular batch, measured in mm, is Normally distributed with mean 28 and variance 16. Sprouts that are between 24 mm and 33 mm in diameter are sold to a supermarket.
    1. Find the probability that the diameter of a randomly selected sprout will be within this range. [4]
    2. The farmer sells the sprouts in this range to the supermarket for 10 pence per kilogram. The farmer sells sprouts under 24 mm in diameter to a frozen food factory for 5 pence per kilogram. Sprouts over 33 mm in diameter are thrown away. Estimate the total income received by the farmer for the batch, which weighs 25 500 kg. [3]
    3. By harvesting sprouts earlier, the mean diameter for another batch can be reduced to \(k\) mm. Find the value of \(k\) for which only 5\% of the sprouts will be above 33 mm in diameter. You may assume that the variance is still 16. [3]
  2. The farmer also grows onions. The weight in kilograms of the onions is Normally distributed with mean 0.155 and variance 0.005. He is trying out a new variety, which he hopes will yield a higher mean weight. In order to test this, he takes a random sample of 25 onions of the new variety and finds that their total weight is 4.77 kg. You should assume that the weight in kilograms of the new variety is Normally distributed with variance 0.005.
    1. Write down suitable null and alternative hypotheses for the test in terms of \(\mu\). State the meaning of \(\mu\) in this case. [2]
    2. Carry out the test at the 1\% level. [6]
OCR MEI S2 2007 January Q3
18 marks Standard +0.3
An electrical retailer gives customers extended guarantees on washing machines. Under this guarantee all repairs in the first 3 years are free. The retailer records the numbers of free repairs made to 80 machines.
Number of repairs0123\(>3\)
Frequency5320610
  1. Show that the sample mean is 0.4375. [1]
  2. The sample standard deviation \(s\) is 0.6907. Explain why this supports a suggestion that a Poisson distribution may be a suitable model for the distribution of the number of free repairs required by a randomly chosen washing machine. [2]
The random variable \(X\) denotes the number of free repairs required by a randomly chosen washing machine. For the remainder of this question you should assume that \(X\) may be modelled by a Poisson distribution with mean 0.4375.
  1. Find P\((X = 1)\). Comment on your answer in relation to the data in the table. [4]
  2. The manager decides to monitor 8 washing machines sold on one day. Find the probability that there are at least 12 free repairs in total on these 8 machines. You may assume that the 8 machines form an independent random sample. [3]
  3. A launderette with 8 washing machines has needed 12 free repairs. Why does your answer to part (iv) suggest that the Poisson model with mean 0.4375 is unlikely to be a suitable model for free repairs on the machines in the launderette? Give a reason why the model may not be appropriate for the launderette. [3]
The retailer also sells tumble driers with the same guarantee. The number of free repairs on a tumble drier in three years can be modelled by a Poisson distribution with mean 0.15. A customer buys a tumble drier and a washing machine.
  1. Assuming that free repairs are required independently, find the probability that
    1. the two appliances need a total of 3 free repairs between them,
    2. each appliance needs exactly one free repair.
    [5]
OCR MEI S2 2007 January Q4
18 marks Standard +0.3
Two educational researchers are investigating the relationship between personal ambitions and home location of students. The researchers classify students into those whose main personal ambition is good academic results and those who have some other ambition. A random sample of 480 students is selected.
  1. One researcher summarises the data as follows.
    \multirow{2}{*}{Observed}Home location
    \cline{2-3}CityNon-city
    \multirow{2}{*}{Ambition}Good results102147
    \cline{2-3}Other75156
    Carry out a test at the 5\% significance level to examine whether there is any association between home location and ambition. State carefully your null and alternative hypotheses. Your working should include a table showing the contributions of each cell to the test statistic. [9]
  2. The other researcher summarises the same data in a different way as follows.
    \multirow{2}{*}{Observed}Home location
    \cline{2-4}CityTownCountry
    \multirow{2}{*}{Ambition}Good results1028364
    \cline{2-4}Other756492
    1. Calculate the expected frequencies for both 'Country' cells. [2]
    2. The test statistic for these data is 10.94. Carry out a test at the 5\% level based on this table, using the same hypotheses as in part (i). [3]
    3. The table below gives the contribution of each cell to the test statistic. Discuss briefly how personal ambitions are related to home location. [2]
      \multirow{2}{*}{
      Contribution to the
      test statistic
      }      		Home location
      \cline{2-4}CityTownCountry
      \multirow{2}{*}{Ambition}Good results1.1290.5963.540
      \cline{2-4}Other1.2170.6433.816
  3. Comment briefly on whether the analysis in part (ii) means that the conclusion in part (i) is invalid. [2]
OCR MEI S3 2006 January Q1
18 marks Standard +0.3
A railway company is investigating operations at a junction where delays often occur. Delays (in minutes) are modelled by the random variable \(T\) with the following cumulative distribution function. $$F(t) = \begin{cases} 0 & t \leq 0 \\ 1 - e^{-\frac{1}{t}} & t > 0 \end{cases}$$
  1. Find the median delay and the 90th percentile delay. [5]
  2. Derive the probability density function of \(T\). Hence use calculus to find the mean delay. [5]
  3. Find the probability that a delay lasts longer than the mean delay. [2]
You are given that the variance of \(T\) is 9.
  1. Let \(\overline{T}\) denote the mean of a random sample of 30 delays. Write down an approximation to the distribution of \(\overline{T}\). [3]
  2. A random sample of 30 delays is found to have mean 4.2 minutes. Does this cast any doubt on the modelling? [3]
OCR MEI S3 2006 January Q2
18 marks Standard +0.3
Geoffrey is a university lecturer. He has to prepare five questions for an examination. He knows by experience that it takes about 3 hours to prepare a question, and he models the time (in minutes) taken to prepare one by the Normally distributed random variable \(X\) with mean 180 and standard deviation 12, independently for all questions.
  1. One morning, Geoffrey has a gap of 2 hours 50 minutes (170 minutes) between other activities. Find the probability that he can prepare a question in this time. [3]
  2. One weekend, Geoffrey can devote 14 hours to preparing the complete examination paper. Find the probability that he can prepare all five questions in this time. [3]
A colleague, Helen, has to check the questions.
  1. She models the time (in minutes) to check a question by the Normally distributed random variable \(Y\) with mean 50 and standard deviation 6, independently for all questions and independently of \(X\). Find the probability that the total time for Geoffrey to prepare a question and Helen to check it exceeds 4 hours. [3]
  2. When working under pressure of deadlines, Helen models the time to check a question in a different way. She uses the Normally distributed random variable \(\frac{1}{2}X\), where \(X\) is as above. Find the length of time, as given by this model, which Helen needs to ensure that, with probability 0.9, she has time to check a question. [4]
Ian, an educational researcher, suggests that a better model for the time taken to prepare a question would be a constant \(k\) representing "thinking time" plus a random variable \(T\) representing the time required to write the question itself, independently for all questions.
  1. Taking \(k\) as 45 and \(T\) as Normally distributed with mean 120 and standard deviation 10 (all units are minutes), find the probability according to Ian's model that a question can be prepared in less than 2 hours 30 minutes. [2]
Juliet, an administrator, proposes that the examination should be reduced in time and shorter questions should be used.
  1. Juliet suggests that Ian's model should be used for the time taken to prepare such shorter questions but with \(k = 30\) and \(T\) replaced by \(\frac{2}{3}T\). Find the probability as given by this model that a question can be prepared in less than \(1\frac{1}{4}\) hours. [3]
OCR MEI S3 2006 January Q3
18 marks Standard +0.3
A production line has two machines, A and B, for delivering liquid soap into bottles. Each machine is set to deliver a nominal amount of 250 ml, but it is not expected that they will work to a high level of accuracy. In particular, it is known that the ambient temperature affects the rate of flow of the liquid and leads to variation in the amounts delivered. The operators think that machine B tends to deliver a somewhat greater amount than machine A, no matter what the ambient temperature. This is being investigated by an experiment. A random sample of 10 results from the experiment is shown below. Each column of data is for a different ambient temperature.
Ambient temperature\(T_1\)\(T_2\)\(T_3\)\(T_4\)\(T_5\)\(T_6\)\(T_7\)\(T_8\)\(T_9\)\(T_{10}\)
Amount delivered by machine A246.2251.6252.0246.6258.4251.0247.5247.1248.1253.4
Amount delivered by machine B248.3252.6252.8247.2258.8250.0247.2247.9249.0254.5
  1. Use an appropriate \(t\) test to examine, at the 5\% level of significance, whether the mean amount delivered by machine B may be taken as being greater than that delivered by machine A, stating carefully your null and alternative hypotheses and the required distributional assumption. [11]
  2. Using the data for machine A in the table above, provide a two-sided 95\% confidence interval for the mean amount delivered by this machine, stating the required distributional assumption. Explain whether you would conclude that the machine appears to be working correctly in terms of the nominal amount as set. [7]
OCR MEI S3 2006 January Q4
18 marks Standard +0.3
Quality control inspectors in a factory are investigating the lengths of glass tubes that will be used to make laboratory equipment.
  1. Data on the observed lengths of a random sample of 200 glass tubes from one batch are available in the form of a frequency distribution as follows.
    Length \(x\) (mm)Observed frequency
    \(x \leq 298\)1
    \(298 < x \leq 300\)30
    \(300 < x \leq 301\)62
    \(301 < x \leq 302\)70
    \(302 < x \leq 304\)34
    \(x > 304\)3
    The sample mean and standard deviation are 301.08 and 1.2655 respectively. The corresponding expected frequencies for the Normal distribution with parameters estimated by the sample statistics are
    Length \(x\) (mm)Expected frequency
    \(x \leq 298\)1.49
    \(298 < x \leq 300\)37.85
    \(300 < x \leq 301\)55.62
    \(301 < x \leq 302\)58.32
    \(302 < x \leq 304\)44.62
    \(x > 304\)2.10
    Examine the goodness of fit of a Normal distribution, using a 5\% significance level. [7]
  2. It is thought that the lengths of tubes in another batch have an underlying distribution similar to that for the batch in part (i) but possibly with different location and dispersion parameters. A random sample of 10 tubes from this batch gives the following lengths (in mm). 301.3 \quad 301.4 \quad 299.6 \quad 302.2 \quad 300.3 \quad 303.2 \quad 302.6 \quad 301.8 \quad 300.9 \quad 300.8
    1. Discuss briefly whether it would be appropriate to use a \(t\) test to examine a hypothesis about the population mean length for this batch. [2]
    2. Use a Wilcoxon test to examine at the 10\% significance level whether the population median length for this batch is 301 mm. [9]
OCR MEI S3 2008 June Q1
19 marks Moderate -0.8
  1. Sarah travels home from work each evening by bus; there is a bus every 20 minutes. The time at which Sarah arrives at the bus stop varies randomly in such a way that the probability density function of \(X\), the length of time in minutes she has to wait for the next bus, is given by $$f(x) = k(20-x) \text{ for } 0 \leq x \leq 20, \text{ where } k \text{ is a constant.}$$
    1. Find \(k\). Sketch the graph of \(f(x)\) and use its shape to explain what can be deduced about how long Sarah has to wait. [5]
    2. Find the cumulative distribution function of \(X\) and hence, or otherwise, find the probability that Sarah has to wait more than 10 minutes for the bus. [4]
    3. Find the median length of time that Sarah has to wait. [3]
    1. Define the term 'simple random sample'. [2]
    2. Explain briefly how to carry out cluster sampling. [3]
    3. A researcher wishes to investigate the attitudes of secondary school pupils to pollution. Explain why he might prefer to collect his data using a cluster sample rather than a simple random sample. [2]
OCR MEI S3 2008 June Q2
18 marks Standard +0.3
An electronics company purchases two types of resistor from a manufacturer. The resistances of the resistors (in ohms) are known to be Normally distributed. Type A have a mean of 100 ohms and standard deviation of 1.9 ohms. Type B have a mean of 50 ohms and standard deviation of 1.3 ohms.
  1. Find the probability that the resistance of a randomly chosen resistor of type A is less than 103 ohms. [3]
  2. Three resistors of type A are chosen at random. Find the probability that their total resistance is more than 306 ohms. [3]
  3. One resistor of type A and one resistor of type B are chosen at random. Find the probability that their total resistance is more than 147 ohms. [3]
  4. Find the probability that the total resistance of two randomly chosen type B resistors is within 3 ohms of one randomly chosen type A resistor. [5]
  5. The manufacturer now offers type C resistors which are specified as having a mean resistance of 300 ohms. The resistances of a random sample of 100 resistors from the first batch supplied have sample mean 302.3 ohms and sample standard deviation 3.7 ohms. Find a 95\% confidence interval for the true mean resistance of the resistors in the batch. Hence explain whether the batch appears to be as specified. [4]
OCR MEI S3 2008 June Q3
18 marks Standard +0.3
  1. A tea grower is testing two types of plant for the weight of tea they produce. A trial is set up in which each type of plant is grown at each of 8 sites. The total weight, in grams, of tea leaves harvested from each plant is measured and shown below.
    SiteABCDEFGH
    Type I225.2268.9303.6244.1230.6202.7242.1247.5
    Type II215.2242.1260.9241.7245.5204.7225.8236.0
    1. The grower intends to perform a \(t\) test to examine whether there is any difference in the mean yield of the two types of plant. State the hypotheses he should use and also any necessary assumption. [3]
    2. Carry out the test using a 5\% significance level. [7]
  2. The tea grower deals with many types of tea and employs tasters to rate them. The tasters do this by giving each tea a score out of 100. The tea grower wishes to compare the scores given by two of the tasters. Their scores for a random selection of 10 teas are as follows.
    TeaQRSTUVWXYZ
    Taster 169798563816585868977
    Taster 274759966756496949686
    Use a Wilcoxon test to examine, at the 5\% level of significance, whether it appears that, on the whole, the scores given to teas by these two tasters differ. [8]
OCR MEI S3 2008 June Q4
17 marks Standard +0.3
  1. A researcher is investigating the feeding habits of bees. She sets up a feeding station some distance from a beehive and, over a long period of time, records the numbers of bees arriving each minute. For a random sample of 100 one-minute intervals she obtains the following results.
    Number of bees01234567\(\geq 8\)
    Number of intervals61619181714640
    1. Show that the sample mean is 3.1 and find the sample variance. Do these values support the possibility of a Poisson model for the number of bees arriving each minute? Explain your answer. [3]
    2. Use the mean in part (i) to carry out a test of the goodness of fit of a Poisson model to the data. [10]
  2. The researcher notes the length of time, in minutes, that each bee spends at the feeding station. The times spent are assumed to be Normally distributed. For a random sample of 10 bees, the mean is found to be 1.465 minutes and the standard deviation is 0.3288 minutes. Find a 95\% confidence interval for the overall mean time. [4]
OCR MEI S3 2010 June Q1
18 marks Moderate -0.8
  1. The manager of a company that employs 250 travelling sales representatives wishes to carry out a detailed analysis of the expenses claimed by the representatives. He has an alphabetical (by surname) list of the representatives. He chooses a sample of representatives by selecting the 10th, 20th, 30th and so on. Name the type of sampling the manager is attempting to use. Describe a weakness in his method of using it, and explain how he might overcome this weakness. [3]
The representatives each use their own cars to drive to meetings with customers. The total distance, in miles, travelled by a representative in a month is Normally distributed with mean 2018 and standard deviation 96.
  1. Find the probability that, in a randomly chosen month, a randomly chosen representative travels more than 2100 miles. [3]
  2. Find the probability that, in a randomly chosen 3-month period, a randomly chosen representative travels less than 6000 miles. What assumption is needed here? Give a reason why it may not be realistic. [5]
  3. Each month every representative submits a claim for travelling expenses plus commission. Travelling expenses are paid at the rate of 45 pence per mile. The commission is 10\% of the value of sales in that month. The value, in £, of the monthly sales has the distribution N(21200, 1100²). Find the probability that a randomly chosen claim lies between £3000 and £3300. [7]
OCR MEI S3 2010 June Q2
18 marks Standard +0.3
William Sealy, a biochemistry student, is doing work experience at a brewery. One of his tasks is to monitor the specific gravity of the brewing mixture during the brewing process. For one particular recipe, an initial specific gravity of 1.040 is required. A random sample of 9 measurements of the specific gravity at the start of the process gave the following results. 1.046 \quad 1.048 \quad 1.039 \quad 1.055 \quad 1.038 \quad 1.054 \quad 1.038 \quad 1.051 \quad 1.038
  1. William has to test whether the specific gravity of the mixture meets the requirement. Why might a \(t\) test be used for these data and what assumption must be made? [3]
  2. Carry out the test using a significance level of 10\%. [9]
  3. Find a 95\% confidence interval for the true mean specific gravity of the mixture and explain what is meant by a 95\% confidence interval. [6]
OCR MEI S3 2010 June Q3
18 marks Standard +0.3
  1. In order to prevent and/or control the spread of infectious diseases, the Government has various vaccination programmes. One such programme requires people to receive a booster injection at the age of 18. It is felt that the proportion of people receiving this booster could be increased and a publicity campaign is undertaken for this purpose. In order to assess the effectiveness of this campaign, health authorities across the country are asked to report the percentage of 18-year-olds receiving the booster before and after the campaign. The results for a randomly chosen sample of 9 authorities are as follows.
    AuthorityABCDEFGHI
    Before769888818684839380
    After829793778395919589
    This sample is to be tested to see whether the campaign appears to have been successful in raising the percentage receiving the booster.
    1. Explain why the use of paired data is appropriate in this context. [1]
    2. Carry out an appropriate Wilcoxon signed rank test using these data, at the 5\% significance level. [10]
  2. Benford's Law predicts the following probability distribution for the first significant digit in some large data sets.
    Digit123456789
    Probability0.3010.1760.1250.0970.0790.0670.0580.0510.046
    On one particular day, the first significant digits of the stock market prices of the shares of a random sample of 200 companies gave the following results.
    Digit123456789
    Frequency55342716151712159
    Test at the 10\% level of significance whether Benford's Law provides a reasonable model in the context of share prices. [7]
OCR MEI S3 2010 June Q4
18 marks Moderate -0.3
A random variable \(X\) has an exponential distribution with probability density function \(f(x) = \lambda e^{-\lambda x}\) for \(x \geq 0\), where \(\lambda\) is a positive constant.
  1. Verify that \(\int_0^{\infty} f(x) \, dx = 1\) and sketch \(f(x)\). [5]
  2. In this part of the question you may use the following result. $$\int_0^{\infty} x^r e^{-\lambda x} \, dx = \frac{r!}{\lambda^{r+1}} \text{ for } r = 0, 1, 2, \ldots$$ Derive the mean and variance of \(X\) in terms of \(\lambda\). [6]
The random variable \(X\) is used to model the lifetime, in years, of a particular type of domestic appliance. The manufacturer of the appliance states that, based on past experience, the mean lifetime is 6 years.
  1. Let \(\overline{X}\) denote the mean lifetime, in years, of a random sample of 50 appliances. Write down an approximate distribution for \(\overline{X}\). [4]
  2. A random sample of 50 appliances is found to have a mean lifetime of 7.8 years. Does this cast any doubt on the model? [3]
OCR MEI M1 2008 January Q1
6 marks Easy -1.3
A cyclist starts from rest and takes 10 seconds to accelerate at a constant rate up to a speed of 15 m s\(^{-1}\). After travelling at this speed for 20 seconds, the cyclist then decelerates to rest at a constant rate over the next 5 seconds.
  1. Sketch a velocity-time graph for the motion. [3]
  2. Calculate the distance travelled by the cyclist. [3]
OCR MEI M1 2008 January Q2
7 marks Moderate -0.8
The force acting on a particle of mass 1.5 kg is given by the vector \(\begin{pmatrix} 6 \\ 9 \end{pmatrix}\) N.
  1. Give the acceleration of the particle as a vector. [2]
  2. Calculate the angle that the acceleration makes with the direction \(\begin{pmatrix} 1 \\ 0 \end{pmatrix}\). [2]
  3. At a certain point of its motion, the particle has a velocity of \(\begin{pmatrix} -2 \\ 3 \end{pmatrix}\) m s\(^{-1}\). Calculate the displacement of the particle over the next two seconds. [3]
OCR MEI M1 2008 January Q3
8 marks Moderate -0.8
\includegraphics{figure_3} Fig. 3 shows a block of mass 15 kg on a rough, horizontal plane. A light string is fixed to the block at A, passes over a smooth, fixed pulley B and is attached at C to a sphere. The section of the string between the block and the pulley is inclined at 40° to the horizontal and the section between the pulley and the sphere is vertical. The system is in equilibrium and the tension in the string is 58.8 N.
  1. The sphere has a mass of \(m\) kg. Calculate the value of \(m\). [2]
  2. Calculate the frictional force acting on the block. [3]
  3. Calculate the normal reaction of the plane on the block. [3]
OCR MEI M1 2008 January Q4
7 marks Easy -1.2
Force \(\mathbf{F}\) is \(\begin{pmatrix} 4 \\ 1 \\ 2 \end{pmatrix}\) N and force \(\mathbf{G}\) is \(\begin{pmatrix} -6 \\ 2 \\ 4 \end{pmatrix}\) N.
  1. Find the resultant of \(\mathbf{F}\) and \(\mathbf{G}\) and calculate its magnitude. [4]
  2. Forces \(\mathbf{F}\), \(2\mathbf{G}\) and \(\mathbf{H}\) act on a particle which is in equilibrium. Find \(\mathbf{H}\). [3]
OCR MEI M1 2008 January Q5
8 marks Standard +0.3
\includegraphics{figure_5} A toy car is moving along the straight line \(Ox\), where O is the origin. The time \(t\) is in seconds. At time \(t = 0\) the car is at A, 3 m from O as shown in Fig. 5. The velocity of the car, \(v\) m s\(^{-1}\), is given by $$v = 2 + 12t - 3t^2.$$ Calculate the distance of the car from O when its acceleration is zero. [8]