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OCR H240/02 Q6
12 marks Moderate -0.3
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
6 marks Moderate -0.3
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
7 marks Moderate -0.8
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
4 marks Moderate -0.8
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
7 marks Moderate -0.3
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
8 marks Moderate -0.3
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
    5 marks Challenging +1.2
    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
    5 marks Moderate -0.5
    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.
    OCR H240/02 Q14
    8 marks Standard +0.8
    14 A random variable \(X\) has probability distribution given by \(\mathrm { P } ( X = x ) = \frac { 1 } { 860 } ( 1 + x )\) for \(x = 1,2,3 , \ldots , 40\).
    1. Find \(\mathrm { P } ( X > 39 )\).
    2. Given that \(x\) is even, determine \(\mathrm { P } ( X < 10 )\). \section*{END OF QUESTION PAPER}
    OCR H240/03 2018 June Q1
    3 marks Easy -1.2
    1 A circle with centre \(C\) has equation \(x ^ { 2 } + y ^ { 2 } + 8 x - 2 y - 7 = 0\).
    Find
    1. the coordinates of \(C\),
    2. the radius of the circle.
    OCR H240/03 2018 June Q2
    3 marks Moderate -0.8
    2 Solve the equation \(| 2 x - 1 | = | x + 3 |\).
    OCR H240/03 2018 June Q4
    8 marks Moderate -0.3
    4 In this question you must show detailed reasoning.
    The functions f and g are defined for all real values of \(x\) by $$\mathrm { f } ( x ) = x ^ { 3 } \quad \text { and } \quad \mathrm { g } ( x ) = x ^ { 2 } + 2 .$$
    1. Write down expressions for
      (a) \(\mathrm { fg } ( x )\),
      (b) \(\operatorname { gf } ( x )\).
    2. Hence find the values of \(x\) for which \(\mathrm { fg } ( x ) - \mathrm { gf } ( x ) = 24\).
    OCR H240/03 2018 June Q5
    13 marks Standard +0.3
    5
    1. Use the trapezium rule, with two strips of equal width, to show that $$\int _ { 0 } ^ { 4 } \frac { 1 } { 2 + \sqrt { x } } \mathrm {~d} x \approx \frac { 11 } { 4 } - \sqrt { 2 }$$
    2. Use the substitution \(x = u ^ { 2 }\) to find the exact value of $$\int _ { 0 } ^ { 4 } \frac { 1 } { 2 + \sqrt { x } } \mathrm {~d} x$$
    3. Using your answers to parts (i) and (ii), show that $$\ln 2 \approx k + \frac { \sqrt { 2 } } { 4 }$$ where \(k\) is a rational number to be determined.
    OCR H240/03 2018 June Q6
    8 marks Standard +0.3
    6 It is given that the angle \(\theta\) satisfies the equation \(\sin \left( 2 \theta + \frac { 1 } { 4 } \pi \right) = 3 \cos \left( 2 \theta + \frac { 1 } { 4 } \pi \right)\).
    1. Show that \(\tan 2 \theta = \frac { 1 } { 2 }\).
    2. Hence find, in surd form, the exact value of \(\tan \theta\), given that \(\theta\) is an obtuse angle.
    OCR H240/03 2018 June Q7
    9 marks Standard +0.3
    7 The gradient of the curve \(y = \mathrm { f } ( x )\) is given by the differential equation $$( 2 x - 1 ) ^ { 3 } \frac { \mathrm {~d} y } { \mathrm {~d} x } + 4 y ^ { 2 } = 0$$ and the curve passes through the point \(( 1,1 )\). By solving this differential equation show that $$f ( x ) = \frac { a x ^ { 2 } - a x + 1 } { b x ^ { 2 } - b x + 1 }$$ where \(a\) and \(b\) are integers to be determined.
    OCR H240/03 2018 June Q8
    6 marks Moderate -0.8
    8 In this question \(\binom { 1 } { 0 }\) and \(\binom { 0 } { 1 }\) denote unit vectors which are horizontal and vertically upwards respectively.
    A particle of mass 5 kg , initially at rest at the point with position vector \(\binom { 2 } { 45 } \mathrm {~m}\), is acted on by gravity and also by two forces \(\binom { 15 } { - 8 } \mathrm {~N}\) and \(\binom { - 7 } { - 2 } \mathrm {~N}\).
    1. Find the acceleration vector of the particle.
    2. Find the position vector of the particle after 10 seconds.
    OCR H240/03 2018 June Q9
    9 marks Standard +0.3
    9 A uniform plank \(A B\) has weight 100 N and length 4 m . The plank rests horizontally in equilibrium on two smooth supports \(C\) and \(D\), where \(A C = x \mathrm {~m}\) and \(C D = 0.5 \mathrm {~m}\) (see diagram). \includegraphics[max width=\textwidth, alt={}, center]{d5ab20c8-afd5-473e-8238-96762bd3786d-6_181_1271_1101_395} The magnitude of the reaction of the support on the plank at \(C\) is 75 N . Modelling the plank as a rigid rod, find
    1. the magnitude of the reaction of the support on the plank at \(D\),
    2. the value of \(x\). A stone block, which is modelled as a particle, is now placed at the end of the plank at \(B\) and the plank is on the point of tilting about \(D\).
    3. Find the weight of the stone block.
    4. Explain the limitation of modelling
      (a) the stone block as a particle,
      (b) the plank as a rigid rod.
    OCR H240/03 2018 June Q10
    11 marks Standard +0.3
    10 Three forces, of magnitudes \(4 \mathrm {~N} , 6 \mathrm {~N}\) and \(P \mathrm {~N}\), act at a point in the directions shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{d5ab20c8-afd5-473e-8238-96762bd3786d-7_604_601_306_724} The forces are in equilibrium.
    1. Show that \(\theta = 41.4 ^ { \circ }\), correct to 3 significant figures.
    2. Hence find the value of \(P\). The force of magnitude 4 N is now removed and the force of magnitude 6 N is replaced by a force of magnitude 3 N acting in the same direction.
    3. Find
      (a) the magnitude of the resultant of the two remaining forces,
      (b) the direction of the resultant of the two remaining forces.
    OCR H240/03 2018 June Q11
    10 marks Moderate -0.3
    11 The velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) of a car at time \(t \mathrm {~s}\), during the first 20 s of its journey, is given by \(v = k t + 0.03 t ^ { 2 }\), where \(k\) is a constant. When \(t = 20\) the acceleration of the car is \(1.3 \mathrm {~ms} ^ { - 2 }\). For \(t > 20\) the car continues its journey with constant acceleration \(1.3 \mathrm {~ms} ^ { - 2 }\) until its speed reaches \(25 \mathrm {~ms} ^ { - 1 }\).
    1. Find the value of \(k\).
    2. Find the total distance the car has travelled when its speed reaches \(25 \mathrm {~ms} ^ { - 1 }\).
    OCR H240/03 2018 June Q12
    14 marks Standard +0.3
    12 One end of a light inextensible string is attached to a particle \(A\) of mass \(m \mathrm {~kg}\). The other end of the string is attached to a second particle \(B\) of mass \(\lambda m \mathrm {~kg}\), where \(\lambda\) is a constant. Particle \(A\) is in contact with a rough plane inclined at \(30 ^ { \circ }\) to the horizontal. The string is taut and passes over a small smooth pulley \(P\) at the top of the plane. The part of the string from \(A\) to \(P\) is parallel to a line of greatest slope of the plane. The particle \(B\) hangs freely below \(P\) (see diagram). \includegraphics[max width=\textwidth, alt={}, center]{d5ab20c8-afd5-473e-8238-96762bd3786d-8_405_670_493_685} The coefficient of friction between \(A\) and the plane is \(\mu\).
    1. It is given that \(A\) is on the point of moving down the plane.
      (a) Find the exact value of \(\mu\) when \(\lambda = \frac { 1 } { 4 }\).
      (b) Show that the value of \(\lambda\) must be less than \(\frac { 1 } { 2 }\).
    2. Given instead that \(\lambda = 2\) and that the acceleration of \(A\) is \(\frac { 1 } { 4 } g \mathrm {~ms} ^ { - 2 }\), find the exact value of \(\mu\). \section*{END OF QUESTION PAPER}
    OCR H240/03 2019 June Q1
    2 marks Easy -1.2
    1 \includegraphics[max width=\textwidth, alt={}, center]{7d1b7598-8f97-43a0-8366-efa8192d549e-04_239_867_504_255} The diagram shows triangle \(A B C\), with \(A C = 13.5 \mathrm {~cm} , B C = 8.3 \mathrm {~cm}\) and angle \(A B C = 32 ^ { \circ }\).
    Find angle \(C A B\).
    OCR H240/03 2019 June Q2
    8 marks Standard +0.3
    2 A circle with centre \(C\) has equation \(x ^ { 2 } + y ^ { 2 } - 6 x + 4 y + 4 = 0\).
    1. Find
      1. the coordinates of \(C\),
      2. the radius of the circle.
    2. Determine the set of values of \(k\) for which the line \(y = k x - 3\) does not intersect or touch the circle.
    OCR H240/03 2019 June Q3
    7 marks Moderate -0.3
    3
    1. In this question you must show detailed reasoning.
      Solve the inequality \(| x - 2 | \leqslant | 2 x - 6 |\).
    2. Give full details of a sequence of two transformations needed to transform the graph of \(y = | x - 2 |\) to the graph of \(y = | 2 x - 6 |\).
    OCR H240/03 2019 June Q4
    14 marks Standard +0.3
    4 \includegraphics[max width=\textwidth, alt={}, center]{7d1b7598-8f97-43a0-8366-efa8192d549e-05_456_634_260_251} The diagram shows the part of the curve \(y = 3 x \sin 2 x\) for which \(0 \leqslant x \leqslant \frac { 1 } { 2 } \pi\).
    The maximum point on the curve is denoted by \(P\).
    1. Show that the \(x\)-coordinate of \(P\) satisfies the equation \(\tan 2 x + 2 x = 0\).
    2. Use the Newton-Raphson method, with a suitable initial value, to find the \(x\)-coordinate of \(P\), giving your answer correct to 4 decimal places. Show the result of each iteration.
    3. The trapezium rule, with four strips of equal width, is used to find an approximation to \(\int _ { 0 } ^ { \frac { 1 } { 2 } \pi } 3 x \sin 2 x \mathrm {~d} x\). Show that the result can be expressed as \(k \pi ^ { 2 } ( \sqrt { 2 } + 1 )\), where \(k\) is a rational number to be determined.
      1. Evaluate \(\int _ { 0 } ^ { \frac { 1 } { 2 } \pi } 3 x \sin 2 x \mathrm {~d} x\).
      2. Hence determine whether using the trapezium rule with four strips of equal width gives an under- or over-estimate for the area of the region enclosed by the curve \(y = 3 x \sin 2 x\) and the \(x\)-axis for \(0 \leqslant x \leqslant \frac { 1 } { 2 } \pi\).
      3. Explain briefly why it is not easy to tell from the diagram alone whether the trapezium rule with four strips of equal width gives an under- or over-estimate for the area of the region in this case.
    OCR H240/03 2019 June Q7
    4 marks Easy -1.2
    7 A cyclist starting from rest accelerates uniformly at \(1.5 \mathrm {~ms} ^ { - 2 }\) for 4 s and then travels at constant speed.
    1. Sketch a velocity-time graph to represent the first 10 seconds of the cyclist's motion.
    2. Calculate the distance travelled by the cyclist in the first 10 seconds.