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OCR C1 2008 January Q1
3 marks Easy -1.2
1 Express \(\frac { 4 } { 3 - \sqrt { 7 } }\) in the form \(a + b \sqrt { 7 }\), where \(a\) and \(b\) are integers.
OCR C1 2008 January Q2
3 marks Easy -1.3
2
  1. Write down the equation of the circle with centre \(( 0,0 )\) and radius 7 .
  2. A circle with centre \(( 3,5 )\) has equation \(x ^ { 2 } + y ^ { 2 } - 6 x - 10 y - 30 = 0\). Find the radius of the circle.
OCR C1 2008 January Q3
4 marks Moderate -0.8
3 Given that \(3 x ^ { 2 } + b x + 10 = a ( x + 3 ) ^ { 2 } + c\) for all values of \(x\), find the values of the constants \(a , b\) and \(c\).
OCR C1 2008 January Q4
6 marks Easy -1.2
4 Solve the equations
  1. \(10 ^ { p } = 0.1\),
  2. \(\left( 25 k ^ { 2 } \right) ^ { \frac { 1 } { 2 } } = 15\),
  3. \(t ^ { - \frac { 1 } { 3 } } = \frac { 1 } { 2 }\).
OCR C1 2008 January Q5
7 marks Easy -1.2
5
  1. Sketch the curve \(y = x ^ { 3 } + 2\).
  2. Sketch the curve \(y = 2 \sqrt { x }\).
  3. Describe a transformation that transforms the curve \(y = 2 \sqrt { x }\) to the curve \(y = 3 \sqrt { x }\).
OCR C1 2008 January Q6
8 marks Moderate -0.3
6
  1. Solve the equation \(x ^ { 2 } + 8 x + 10 = 0\), giving your answers in simplified surd form.
  2. Sketch the curve \(y = x ^ { 2 } + 8 x + 10\), giving the coordinates of the point where the curve crosses the \(y\)-axis.
  3. Solve the inequality \(x ^ { 2 } + 8 x + 10 \geqslant 0\).
OCR C1 2008 January Q7
8 marks Moderate -0.8
7
  1. Find the gradient of the line \(l\) which has equation \(x + 2 y = 4\).
  2. Find the equation of the line parallel to \(l\) which passes through the point ( 6,5 ), giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.
  3. Solve the simultaneous equations $$y = x ^ { 2 } + x + 1 \quad \text { and } \quad x + 2 y = 4$$
OCR C1 2008 January Q8
11 marks Moderate -0.8
8
  1. Find the coordinates of the stationary points on the curve \(y = x ^ { 3 } + x ^ { 2 } - x + 3\).
  2. Determine whether each stationary point is a maximum point or a minimum point.
  3. For what values of \(x\) does \(x ^ { 3 } + x ^ { 2 } - x + 3\) decrease as \(x\) increases?
OCR C1 2008 January Q9
12 marks Moderate -0.3
9 The points \(A\) and \(B\) have coordinates \(( - 5 , - 2 )\) and \(( 3,1 )\) respectively.
  1. Find the equation of the line \(A B\), giving your answer in the form \(a x + b y + c = 0\).
  2. Find the coordinates of the mid-point of \(A B\). The point \(C\) has coordinates (-3,4).
  3. Calculate the length of \(A C\), giving your answer in simplified surd form.
  4. Determine whether the line \(A C\) is perpendicular to the line \(B C\), showing all your working.
OCR C1 2008 January Q10
10 marks Standard +0.3
10 Given that \(\mathrm { f } ( x ) = 8 x ^ { 3 } + \frac { 1 } { x ^ { 3 } }\),
  1. find \(\mathrm { f } ^ { \prime \prime } ( x )\),
  2. solve the equation \(\mathrm { f } ( x ) = - 9\).
OCR MEI S1 Q1
8 marks Easy -1.2
1 An amateur weather forecaster describes each day as either sunny, cloudy or wet. He keeps a record each day of his forecast and of the actual weather. His results for one particular year are given in the table,
\multirow{2}{*}{}Weather Forecast\multirow{2}{*}{Total}
SunnyCloudyWet
\multirow{3}{*}{Actual Weather}Sunny5512774
Cloudy1712829174
Wet33381117
Total75173117365
A day is selected at random from that year.
  1. Show that the probability that the forecast is correct is \(\frac { 264 } { 365 }\). Find the probability that
  2. the forecast is correct, given that the forecast is sunny,
  3. the forecast is correct, given that the weather is wet,
  4. the weather is cloudy, given that the forecast is correct.
OCR MEI S1 Q2
17 marks Standard +0.3
2 A drug for treating a particular minor illness cures, on average, 78\% of patients. Twenty people with this minor illness are selected at random and treated with the drug.
  1. (A) Find the probability that exactly 19 patients are cured.
    (B) Find the probability that at most 18 patients are cured.
    (C) Find the expected number of patients who are cured.
  2. A pharmaceutical company is trialling a new drug to treat this illness. Researchers at the company hope that a higher percentage of patients will be cured when given this new drug. Twenty patients are selected at random, and given the new drug. Of these, 19 are cured. Carry out a hypothesis test at the \(1 \%\) significance level to investigate whether there is any evidence to suggest that the new drug is more effective than the old one.
  3. If the researchers had chosen to carry out the hypothesis test at the \(5 \%\) significance level, what would the result have been? Justify your answer.
OCR MEI S1 Q3
18 marks Standard +0.3
3 A coffee shop provides free internet access for its customers. It is known that the probability that a randomly selected customer is accessing the internet is 0.35 , independently of all other customers.
  1. 10 customers are selected at random.
    (A) Find the probability that exactly 5 of them are accessing the internet.
    (B) Find the probability that at least 5 of them are accessing the internet.
    (C) Find the expected number of these customers who are accessing the internet. Another coffee shop also provides free internet access. It is suspected that the probability that a randomly selected customer at this coffee shop is accessing the internet may be different from 0.35 . A random sample of 20 customers at this coffee shop is selected. Of these, 10 are accessing the internet.
  2. Carry out a hypothesis test at the \(5 \%\) significance level to investigate whether the probability for this coffee shop is different from 0.35. Give a reason for your choice of alternative hypothesis.
  3. To get a more reliable result, a much larger random sample of 200 customers is selected over a period of time, and another hypothesis test is carried out. You are given that 90 of the 200 customers were accessing the internet. You are also given that, if \(X\) has the binomial distribution with parameters \(n = 200\) and \(p = 0.35\), then \(\mathrm { P } ( X \geqslant 90 ) = 0.0022\). Using the same hypotheses and significance level which you used in part (ii), complete this test.
OCR MEI S1 Q1
18 marks Standard +0.3
1 A manufacturer produces tiles. On average 10\% of the tiles produced are faulty. Faulty tiles occur randomly and independently. A random sample of 18 tiles is selected.
  1. (A) Find the probability that there are exactly 2 faulty tiles in the sample.
    (B) Find the probability that there are more than 2 faulty tiles in the sample.
    (C) Find the expected number of faulty tiles in the sample. A cheaper way of producing the tiles is introduced. The manufacturer believes that this may increase the proportion of faulty tiles. In order to check this, a random sample of 18 tiles produced using the cheaper process is selected and a hypothesis test is carried out.
  2. (A) Write down suitable null and alternative hypotheses for the test.
    (B) Explain why the alternative hypothesis has the form that it does.
  3. Find the critical region for the test at the \(5 \%\) level, showing all of your calculations.
  4. In fact there are 4 faulty tiles in the sample. Complete the test, stating your conclusion clearly.
OCR MEI S1 Q2
5 marks Moderate -0.8
2 In a multiple-choice test there are 30 questions. For each question, there is a \(60 \%\) chance that a randomly selected student answers correctly, independently of all other questions.
  1. Find the probability that a randomly selected student gets a total of exactly 20 questions correct.
  2. If 100 randomly selected students take the test, find the expected number of students who get exactly 20 questions correct.
OCR MEI S1 Q3
16 marks Moderate -0.3
3 The birth weights in grams of a random sample of 1000 babies are displayed in the cumulative frequency diagram below. \includegraphics[max width=\textwidth, alt={}, center]{dfb0acd8-d84b-4291-a811-a68f4942794b-2_1266_1546_487_335}
  1. Use the diagram to estimate the median and interquartile range of the data.
  2. Use your answers to part (i) to estimate the number of outliers in the sample.
  3. Should these outliers be excluded from any further analysis? Briefly explain your answer.
  4. Any baby whose weight is below the 10th percentile is selected for careful monitoring. Use the diagram to determine the range of weights of the babies who are selected. \(12 \%\) of new-born babies require some form of special care. A maternity unit has 17 new-born babies. You may assume that these 17 babies form an independent random sample.
  5. Find the probability that
    (A) exactly 2 of these 17 babies require special care,
    (B) more than 2 of the 17 babies require special care.
  6. On 100 independent occasions the unit has 17 babies. Find the expected number of occasions on which there would be more than 2 babies who require special care.
OCR MEI S1 Q4
18 marks Moderate -0.3
4 When onion seeds are sown outdoors, on average two-thirds of them germinate. A gardener sows seeds in pairs, in the hope that at least one will germinate.
  1. Assuming that germination of one of the seeds in a pair is independent of germination of the other seed, find the probability that, if a pair of seeds is selected at random,
    (A) both seeds germinate,
    (B) just one seed germinates,
    (C) neither seed germinates.
  2. Explain why the assumption of independence is necessary in order to calculate the above probabilities. Comment on whether the assumption is likely to be valid.
  3. A pair of seeds is sown. Find the expectation and variance of the number of seeds in the pair which germinate.
  4. The gardener plants 200 pairs of seeds. If both seeds in a pair germinate, the gardener destroys one of the two plants so that only one is left to grow. Of the plants that remain after this, only \(85 \%\) successfully grow to form an onion. Find the expected number of onions grown from the 200 pairs of seeds. If the seeds are sown in a greenhouse, the germination rate is higher. The seed manufacturing company claims that the germination rate is \(90 \%\). The gardener suspects that the rate will not be as high as this, and carries out a trial to investigate. 18 randomly selected seeds are sown in the greenhouse and it is found that 14 germinate.
  5. Write down suitable hypotheses and carry out a test at the \(5 \%\) level to determine whether there is any evidence to support the gardener's suspicions.
OCR MEI S1 Q3
18 marks Moderate -0.3
3 The birth weights in grams of a random sample of 1000 babies are displayed in the cumulative frequency diagram below. \includegraphics[max width=\textwidth, alt={}, center]{79f1015b-7c3d-4576-8d5b-e9fc89d8a49e-2_1266_1546_487_335}
  1. Use the diagram to estimate the median and interquartile range of the data.
  2. Use your answers to part (i) to estimate the number of outliers in the sample.
  3. Should these outliers be excluded from any further analysis? Briefly explain your answer.
  4. Any baby whose weight is below the 10th percentile is selected for careful monitoring. Use the diagram to determine the range of weights of the babies who are selected. \(12 \%\) of new-born babies require some form of special care. A maternity unit has 17 new-born babies. You may assume that these 17 babies form an independent random sample.
  5. Find the probability that
    (A) exactly 2 of these 17 babies require special care,
    (B) more than 2 of the 17 babies require special care.
  6. On 100 independent occasions the unit has 17 babies. Find the expected number of occasions on which there would be more than 2 babies who require special care.
OCR MEI S1 Q1
8 marks Easy -1.2
1 The stem and leaf diagram illustrates the heights in metres of 25 young oak trees.
3467899
402234689
501358
6245
746
81
Key: 4 |2 represents 4.2
  1. State the type of skewness of the distribution.
  2. Use your calculator to find the mean and standard deviation of these data.
  3. Determine whether there are any outliers.
OCR MEI S1 Q2
7 marks Easy -1.8
2 The mean daily maximum temperatures at a research station over a 12 -month period, measured to the nearest degree Celsius, are given below.
JanFebMarAprMayJunJulAugSepOctNovDec
8152529313134363426158
  1. Construct a sorted stem and leaf diagram to represent these data, taking stem values of \(0,10 , \ldots\).
  2. Write down the median of these data.
  3. The mean of these data is 24.3. Would the mean or the median be a better measure of central tendency of the data? Briefly explain your answer.
OCR MEI S1 Q3
3 marks Easy -1.2
3 The stem and leaf diagram shows the weights, rounded to the nearest 10 grams, of 25 female iguanas.
839
935666899
100223469
112478
12345
132
Key: 11 | 2 represents 1120 grams
  1. Find the mode and the median of the data.
  2. Identify the type of skewness of the distribution.
OCR MEI S1 Q4
8 marks Easy -1.2
4 A camera records the speeds in miles per hour of 15 vehicles on a motorway. The speeds are given below. $$\begin{array} { l l l l l l l l l l l l l l l } 73 & 67 & 75 & 64 & 52 & 63 & 75 & 81 & 77 & 72 & 68 & 74 & 79 & 72 & 71 \end{array}$$
  1. Construct a sorted stem and leaf diagram to represent these data, taking stem values of \(50,60 , \ldots\).
  2. Write down the median and midrange of the data.
  3. Which of the median and midrange would you recommend to measure the central tendency of the data? Briefly explain your answer.
OCR MEI S1 Q5
5 marks Easy -1.8
5 In a traffic survey, the number of people in each car passing the survey point is recorded. The results are given in the following frequency table.
Number of people1234
Frequency5031165
  1. Write down the median and mode of these data.
  2. Draw a vertical line diagram for these data.
  3. State the type of skewness of the distribution.
OCR MEI S1 Q6
7 marks Easy -1.2
6 A supermarket chain buys a batch of 10000 scratchcard draw tickets for sale in its stores. 50 of these tickets have a \(\pounds 10\) prize, 20 of them have a \(\pounds 100\) prize, one of them has a \(\pounds 5000\) prize and all of the rest have no prize. This information is summarised in the frequency table below.
Prize money\(\pounds 0\)\(\pounds 10\)\(\pounds 100\)\(\pounds 5000\)
Frequency992950201
  1. Find the mean and standard deviation of the prize money per ticket.
  2. I buy two of these tickets at random. Find the probability that I win either two \(\pounds 10\) prizes or two \(\pounds 100\) prizes.
OCR MEI S1 Q7
20 marks Moderate -0.8
7 The histogram shows the age distribution of people living in Inner London in 2001. \includegraphics[max width=\textwidth, alt={}, center]{aabf9d8b-5f91-4a3b-bcf8-e46cb45127c4-4_805_1372_392_401} Data sourced from he 2001 Census, \href{http://www.statistics.gov.uk}{www.statistics.gov.uk}
  1. State the type of skewness shown by the distribution.
  2. Use the histogram to estimate the number of people aged under 25.
  3. The table below shows the cumulative frequency distribution.
    Age2030405065100
    Cumulative frequency (thousands)66012401810\(a\)24902770
    (A) Use the histogram to find the value of \(a\).
    (B) Use the table to calculate an estimate of the median age of these people. The ages of people living in Outer London in 2001 are summarised below.
    Age ( \(x\) years)\(0 \leqslant x < 20\)\(20 \leqslant x < 30\)\(30 \leqslant x < 40\)\(40 \leqslant x < 50\)\(50 \leqslant x < 65\)\(65 \leqslant x < 100\)
    Frequency (thousands)1120650770590680610
  4. Illustrate these data by means of a histogram.
  5. Make two brief comments on the differences between the age distributions of the populations of Inner London and Outer London.
  6. The data given in the table for Outer London are used to calculate the following estimates. Mean 38.5, median 35.7, midrange 50, standard deviation 23.7, interquartile range 34.4.
    The final group in the table assumes that the maximum age of any resident is 100 years. These estimates are to be recalculated, based on a maximum age of 105, rather than 100. For each of the five estimates, state whether it would increase, decrease or be unchanged.
    [0pt] [4]