Questions — OCR H240/02 (94 questions)

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OCR H240/02 2021 November Q2
2 The diagram shows part of the graph of \(y = \mathrm { f } ( x )\), where \(\mathrm { f } ( x )\) is a cubic polynomial in \(x\).
\includegraphics[max width=\textwidth, alt={}, center]{7298e7b9-ad52-480c-bc2b-8289aeab9ebb-04_437_620_909_274} Explain why one of the roots of the equation \(\mathrm { f } ( x ) = 0\) cannot be found by the sign change method.
OCR H240/02 2021 November Q3
3 The 15th term of an arithmetic sequence is 88. The sum of the first 10 terms is 310 .
Determine the first term and the common difference.
OCR H240/02 2021 November Q4
4 The size, \(P\), of a population of a certain species of insect at time \(t\) months is modelled by the following formula.
\(P = 5000 - 1000 \cos ( 30 t ) ^ { \circ }\)
  1. Write down the maximum size of the population.
  2. Write down the difference between the largest and smallest values of \(P\).
  3. Without giving any numerical values, describe briefly the behaviour of the population over time.
  4. Find the time taken for the population to return to its initial size for the first time.
  5. Determine the time on the second occasion when \(P = 4500\). A scientist observes the population over a period of time. He notices that, although the population varies in a way similar to the way predicted by the model, the variations become smaller and smaller over time, and \(P\) converges to 5000 .
  6. Suggest a change to the model that will take account of this observation.
OCR H240/02 2021 November Q6
6 Alex is investigating the area, \(A\), under the graph of \(y = x ^ { 2 }\) between \(x = 1\) and \(x = 1.5\). They draw the graph, together with rectangles of width \(\delta x = 0.1\), and varying heights \(y\).
\includegraphics[max width=\textwidth, alt={}, center]{7298e7b9-ad52-480c-bc2b-8289aeab9ebb-06_531_714_356_251}
  1. Use the rectangles in the diagram to show that lower and upper bounds for the area \(A\) are 0.73 and 0.855 respectively.
  2. Alex finds lower and upper bounds for the area \(A\), using widths \(\delta x\) of decreasing size. The results are shown in the table. Where relevant, values are given correct to 3 significant figures.
    Width \(\delta x\)0.10.050.0250.0125
    Lower bound for area \(A\)0.730.7610.7760.784
    Upper bound for area \(A\)0.8550.8230.8070.799
    Use Alex's results to estimate the value of \(A\) correct to \(\mathbf { 2 }\) significant figures. Give a brief justification for your estimate.
  3. Write down an expression, in terms of \(y\) and \(\delta x\), for the exact value of the area \(A\).
OCR H240/02 2021 November Q7
7 Differentiate \(\cos x\) with respect to \(x\), from first principles.
OCR H240/02 2021 November Q8
8 The number \(K\) is defined by \(K = n ^ { 3 } + 1\), where \(n\) is an integer greater than 2 .
  1. Given that \(n ^ { 3 } + 1 \equiv ( n + 1 ) \left( n ^ { 2 } + b n + c \right)\), find the constants \(b\) and \(c\).
  2. Prove that \(K\) has at least two distinct factors other than 1 and \(K\).
OCR H240/02 2021 November Q9
9 Points \(A , B\) and \(C\) have position vectors \(\mathbf { a } , \mathbf { b }\) and \(\mathbf { c }\) relative to an origin \(O\) in 3-dimensional space. Rectangles \(O A D C\) and \(B E F G\) are the base and top surface of a cuboid.
\includegraphics[max width=\textwidth, alt={}, center]{7298e7b9-ad52-480c-bc2b-8289aeab9ebb-07_522_812_952_280}
  • The point \(M\) is the midpoint of \(B C\).
  • The point \(X\) lies on \(A M\) such that \(A X = 2 X M\).
    1. Find \(\overrightarrow { O X }\) in terms of \(\mathbf { a } , \mathbf { b }\) and \(\mathbf { c }\), simplifying your answer.
    2. Hence show that the lines \(O F\) and \(A M\) intersect.
OCR H240/02 2021 November Q10
10 A researcher plans to carry out a statistical investigation to test whether there is linear correlation between the time ( \(T\) weeks) from conception to birth, and the birth weight ( \(W\) grams) of new-born babies.
  1. Explain why a 1-tail test is appropriate in this context. The researcher records the values of \(T\) and \(W\) for a random sample of 11 babies. They calculate Pearson's product-moment correlation coefficient for the sample and find that the value is 0.722 .
  2. Use the table below to carry out the test at the \(1 \%\) significance level. \section*{Critical values of Pearson's product-moment correlation coefficient.}
    \multirow{2}{*}{}1-tail test5\%2.5\%1\%0.5\%
    2-tail test10\%5\%2.5\%1\%
    \multirow{4}{*}{\(n\)}100.54940.63190.71550.7646
    110.52140.60210.68510.7348
    120.49730.57600.65810.7079
    130.47620.55290.63390.6835
OCR H240/02 2021 November Q11
11 Zac is planning to write a report on the music preferences of the students at his college. There is a large number of students at the college.
  1. State one reason why Zac might wish to obtain information from a sample of students, rather than from all the students.
  2. Amaya suggests that Zac should use a sample that is stratified by school year. Give one advantage of this method as compared with random sampling, in this context. Zac decides to take a random sample of 60 students from his college. He asks each student how many hours per week, on average, they spend listening to music during term. From his results he calculates the following statistics.
    Mean
    Standard
    deviation
    		Median
    Lower
    quartile
    Upper
    quartile
    21.04.2020.518.022.9
  3. Sundip tells Zac that, during term, she spends on average 30 hours per week listening to music. Discuss briefly whether this value should be considered an outlier.
  4. Layla claims that, during term, each student spends on average 20 hours per week listening to music. Zac believes that the true figure is higher than 20 hours. He uses his results to carry out a hypothesis test at the 5\% significance level. Assume that the time spent listening to music is normally distributed with standard deviation 4.20 hours. Carry out the test.
OCR H240/02 2021 November Q12
12 Anika and Beth are playing a game which consists of several points.
  • The probability that Anika will win any point is 0.7 .
  • The probability that Beth will win any point is 0.3 .
  • The outcome of each point is independent of the outcome of every other point.
The first player to win two points wins the game.
  1. Write down the probability that the game consists of more than three points.
  2. Complete the probability tree diagram in the Printed Answer Booklet showing all the possibilities for the game.
  3. Determine the probability that Beth wins the game.
  4. Determine the probability that the game consists of exactly three points.
  5. Given that Beth wins the game, determine the probability that the game consists of exactly three points.
OCR H240/02 2021 November Q13
13 The four pie charts illustrate the numbers of employees using different methods of travel in four Local Authorities in 2011.
\includegraphics[max width=\textwidth, alt={}, center]{7298e7b9-ad52-480c-bc2b-8289aeab9ebb-10_1131_1077_347_242}
\multirow[t]{4}{*}{Key:}\multirow{4}{*}{\includegraphics[max width=\textwidth, alt={}]{7298e7b9-ad52-480c-bc2b-8289aeab9ebb-10_105_142_1578_465} }Public transport
Private motorised transport
Bicycle
All other methods of travel
  1. State, with reasons, which of the four Local Authorities is most likely to be a rural area with many hills.
  2. Explain why pie charts are more suitable for answering part (a) than bar charts showing the same data.
  3. Two of the Local Authorities represent urban areas.
    1. State with a reason which two Local Authorities are likely to be urban.
    2. One urban Local Authority introduced a Park-and-Ride service in 2006. Users of this service drive to the edge of the urban area and then use buses to take them into the centre of the area. A student claims that a comparison of the corresponding pie charts for 2001 (not shown) and 2011 would enable them to identify which Local Authority this was. State with a reason whether you agree with the student.
OCR H240/02 2021 November Q14
14 The probability distribution of a random variable \(X\) is modelled as follows.
\(\mathrm { P } ( X = x ) = \begin{cases} \frac { k } { x } & x = 1,2,3,4 ,
0 & \text { otherwise, } \end{cases}\)
where \(k\) is a constant.
  1. Show that \(k = \frac { 12 } { 25 }\).
  2. Show in a table the values of \(X\) and their probabilities.
  3. The values of three independent observations of \(X\) are denoted by \(X _ { 1 } , X _ { 2 }\) and \(X _ { 3 }\). Find \(\mathrm { P } \left( X _ { 1 } > X _ { 2 } + X _ { 3 } \right)\). In a game, a player notes the values of successive independent observations of \(X\) and keeps a running total. The aim of the game is to reach a total of exactly 7 .
  4. Determine the probability that a total of exactly 7 is first reached on the 5th observation. \section*{OCR} Oxford Cambridge and RSA
OCR H240/02 Q1
1 Simplify fully.
  1. \(\sqrt { a ^ { 3 } } \times \sqrt { 16 a }\)
  2. \(\quad \left( 4 b ^ { 6 } \right) ^ { \frac { 5 } { 2 } }\)
OCR H240/02 Q2
2 A curve has equation \(y = x ^ { 5 } - 5 x ^ { 4 }\).
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).
  2. Verify that the curve has a stationary point when \(x = 4\).
  3. Determine the nature of this stationary point.
OCR H240/02 Q3
3 A publisher has to choose the price at which to sell a certain new book. The total profit, \(\pounds t\), that the publisher will make depends on the price, \(\pounds p\). He decides to use a model that includes the following assumptions.
  • If the price is low, many copies will be sold, but the profit on each copy sold will be small, and the total profit will be small.
  • If the price is high, the profit on each copy sold will be high, but few copies will be sold, and the total profit will be small.
The graphs below show two possible models. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{f2f45d6c-cfdc-455b-ab08-597b06a69f36-05_346_465_1027_374} \captionsetup{labelformat=empty} \caption{Model A}
\end{figure} \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{f2f45d6c-cfdc-455b-ab08-597b06a69f36-05_348_476_1025_1080} \captionsetup{labelformat=empty} \caption{Model B}
\end{figure}
  1. Explain how model A is inconsistent with one of the assumptions given above.
  2. Given that the equation of the curve in model B is quadratic, show that this equation is of the form \(t = k \left( 12 p - p ^ { 2 } \right)\), and find the value of the constant \(k\).
  3. The publisher needs to make a total profit of at least \(\pounds 6400\). Use the equation found in part (b) to find the range of values within which model B suggests that the price of the book must lie.
  4. Comment briefly on how realistic model B may be in the following cases.
    • \(p = 0\)
    • \(p = 12.1\)
OCR H240/02 Q4
2 marks
4
  1. Express \(\frac { 1 } { ( x - 1 ) ( x + 2 ) }\) in partial fractions
    [0pt] [2]
  2. In this question you must show detailed reasoning. Hence find \(\int _ { 2 } ^ { 3 } \frac { 1 } { ( x - 1 ) ( x + 2 ) } \mathrm { d } x\).
    Give your answer in its simplest form.
OCR H240/02 Q5
8 marks
5 The diagram shows the circle with centre O and radius 2, and the parabola \(y = \frac { 1 } { \sqrt { 3 } } \left( 4 - x ^ { 2 } \right)\).
\includegraphics[max width=\textwidth, alt={}, center]{f2f45d6c-cfdc-455b-ab08-597b06a69f36-06_838_970_1059_280} The circle meets the parabola at points \(P\) and \(Q\), as shown in the diagram.
  1. Verify that the coordinates of \(Q\) are \(( 1 , \sqrt { 3 } )\).
  2. Find the exact area of the shaded region enclosed by the \(\operatorname { arc } P Q\) of the circle and the parabola.
    [0pt] [8]
OCR H240/02 Q6
6 Helga invests \(\pounds 4000\) in a savings account.
After \(t\) days, her investment is worth \(\pounds y\).
The rate of increase of \(y\) is \(k y\), where \(k\) is a constant.
  1. Write down a differential equation in terms of \(t , y\) and \(k\).
  2. Solve your differential equation to find the value of Helga's investment after \(t\) days. Give your answer in terms of \(k\) and \(t\). It is given that \(k = \frac { 1 } { 365 } \ln \left( 1 + \frac { r } { 100 } \right)\) where \(r \%\) is the rate of interest per annum. During the first year the rate of interest is \(6 \%\) per annum.
  3. Find the value of Helga's investment after 90 days. After one year (365 days), the rate of interest drops to 5\% per annum.
  4. Find the total time that it will take for Helga's investment to double in value.
OCR H240/02 Q7
7
  1. The heights of English men aged 25 to 34 are normally distributed with mean 178 cm and standard deviation 8 cm .
    Three English men aged 25 to 34 are chosen at random. Find the probability that all three men have a height less than 194 cm .
  2. The diagram shows the distribution of heights of Scottish women aged 25 to 34.
    \includegraphics[max width=\textwidth, alt={}, center]{f2f45d6c-cfdc-455b-ab08-597b06a69f36-08_585_1477_909_342} The distribution is approximately normal. Use the diagram in the Printed Answer Booklet to estimate the standard deviation of these heights, explaining your method.
OCR H240/02 Q8
8 A market gardener records the masses of a random sample of 100 of this year's crop of plums. The table shows his results.
Mass,
\(m\) grams
\(m < 25\)\(25 \leq m < 35\)\(35 \leq m < 45\)\(45 \leq m < 55\)\(55 \leq m < 65\)\(65 \leq m < 75\)\(m \geq 75\)
Number
of plums
0329363020
  1. Explain why the normal distribution might be a reasonable model for this distribution. The market gardener models the distribution of masses by \(\mathrm { N } \left( 47.5,10 ^ { 2 } \right)\).
  2. Find the number of plums in the sample that this model would predict to have masses in the range:
    1. \(35 \leq m < 45\)
    2. \(m < 25\).
  3. Use your answers to parts (b)(i) and (b)(ii) to comment on the suitability of this model. The market gardener plans to use this model to predict the distribution of the masses of next year's crop of plums.
  4. Comment on this plan.
OCR H240/02 Q9
9 The diagram below shows some "Cycle to work" data taken from the 2001 and 2011 UK censuses. The diagram shows the percentages, by age group, of male and female workers in England and Wales, excluding London, who cycled to work in 2001 and 2011.
\includegraphics[max width=\textwidth, alt={}, center]{f2f45d6c-cfdc-455b-ab08-597b06a69f36-10_951_1635_559_207} The following questions refer to the workers represented by the graphs in the diagram.
  1. A researcher is going to take a sample of men and a sample of women and ask them whether or not they cycle to work. Why would it be more important to stratify the sample of men? A research project followed a randomly chosen large sample of the group of male workers who were aged 30-34 in 2001.
  2. Does the diagram suggest that the proportion of this group who cycled to work has increased or decreased from 2001 to 2011?
    Justify your answer.
  3. Write down one assumption that you have to make about these workers in order to draw this conclusion.
OCR H240/02 Q10
10 In the past, the time spent in minutes, by customers in a certain library had mean 32.5 and standard deviation 8.2. Following a change of layout in the library, the mean time spent in the library by a random sample of 50 customers is found to be 34.5 minutes. Assuming that the standard deviation remains at 8.2 , test at the \(5 \%\) significance level whether the mean time spent by customers in the library has changed.
OCR H240/02 Q11
11 Each of the 30 students in a class plays at least one of squash, hockey and tennis.
  • 18 students play squash
  • 19 students play hockey
  • 17 students play tennis
  • 8 students play squash and hockey
  • 9 students play hockey and tennis
  • 11 students play squash and tennis
    1. Find the number of students who play all three sports.
A student is picked at random from the class.
  • Given that this student plays squash, find the probability that this student does not play hockey. Two different students are picked at random from the class, one after the other, without replacement.
  • Given that the first student plays squash, find the probability that the second student plays hockey.
    •
    OCR H240/02 Q12
    12 The table shows information for England and Wales, taken from the UK 2011 census.
    Total populationNumber of children aged 5-17
    560759128473617
    A random sample of 10000 people in another country was chosen in 2011 , and the number, \(m\), of children aged 5-17 was noted.
    It was found that there was evidence at the \(2.5 \%\) level that the proportion of children aged 5-17 in the same year was higher than in the UK.
    Unfortunately, when the results were recorded the value of \(m\) was omitted. Use an appropriate normal distribution to find an estimate of the smallest possible value of \(m\). TURN OVER FOR THE NEXT QUESTION
    OCR H240/02 Q13
    13 The table and the four scatter diagrams below show data taken from the 2011 UK census for four regions. On the scatter diagrams the names have been replaced by letters.
    The table shows, for each region, the mean and standard deviation of the proportion of workers in each Local Authority who travel to work by driving a car or van and the proportion of workers in each Local Authority who travel to work as a passenger in a car or van.
    Each scatter diagram shows, for each of the Local Authorities in a particular region, the proportion of workers who travel to work by driving a car or van and the proportion of workers who travel to work as a passenger in a car or van.
    Driving a car or vanPassenger in a car or van
    MeanStandard deviationMeanStandard deviation
    London0.2570.1330.0170.008
    South East0.5780.0640.0450.010
    South West0.5800.0840.0490.007
    Wales0.6440.0450.0680.015
    Region A
    \includegraphics[max width=\textwidth, alt={}, center]{f2f45d6c-cfdc-455b-ab08-597b06a69f36-14_634_1116_1308_299} Region B
    \includegraphics[max width=\textwidth, alt={}, center]{f2f45d6c-cfdc-455b-ab08-597b06a69f36-14_636_1109_2049_301}
    \includegraphics[max width=\textwidth, alt={}, center]{f2f45d6c-cfdc-455b-ab08-597b06a69f36-15_737_1183_237_240}
    \includegraphics[max width=\textwidth, alt={}, center]{f2f45d6c-cfdc-455b-ab08-597b06a69f36-15_723_1169_1046_246}
    1. Using the values given in the table, match each region to its corresponding scatter diagram, explaining your reasoning.
    2. Steven claims that the outlier in the scatter diagram for Region C consists of a group of small islands. Explain whether or not the data given above support his claim.
    3. One of the Local Authorities in Region B consists of a single large island. Explain whether or not you would expect this Local Authority to appear as an outlier in the scatter diagram for Region B.