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OCR H240/02 2018 March Q8
9 marks Standard +0.3
8 The masses, \(X\) grams, of tomatoes are normally distributed. Half of the tomatoes have masses greater than 56.0 g and \(70 \%\) of the tomatoes have masses greater than 53.0 g .
  1. Find the percentage of tomatoes with masses greater than 59.0 g .
  2. Find the percentage of tomatoes with masses greater than 65.0 g .
  3. Given that \(\mathrm { P } ( a < X < 50 ) = 0.1\), find \(a\).
OCR H240/02 2018 March Q9
10 marks Standard +0.3
9 A bag contains 100 black discs and 200 white discs. Paula takes five discs at random, without replacement. She notes the number \(X\) of these discs that are black.
  1. Find \(\mathrm { P } ( X = 3 )\). Paula decides to use the binomial distribution as a model for the distribution of \(X\).
  2. Explain why this model will give probabilities that are approximately, but not exactly, correct.
  3. Paula uses the binomial model to find an approximate value for \(\mathrm { P } ( X = 3 )\). Calculate the percentage by which her answer will differ from the answer in part (ii). Paula now assumes that the binomial distribution is a good model for \(X\). She uses a computer simulation to generate 1000 values of \(X\). The number of times that \(X = 3\) occurs is denoted by \(Y\).
  4. Calculate estimates of the limits between which two thirds of the values of \(Y\) will lie.
OCR H240/02 2018 March Q10
12 marks Moderate -0.8
10 A researcher is investigating the actual lengths of time that patients spend at their appointments with the doctors at a certain clinic. There are 12 doctors at the clinic, and each doctor has 24 appointments per day. The researcher plans to choose a sample of 24 appointments on a particular day.
  1. The researcher considers the following two methods for choosing the sample. Method A: Choose a random sample of 24 appointments from the 288 on that day.
    Method B: Choose one doctor's 1st and 2nd appointments. Choose another doctor's 3rd and 4th appointments and so on until the last doctor's 23rd and 24th appointments. For each of A and B state a disadvantage of using this method. Appointments are scheduled to last 10 minutes. The researcher suspects that the actual times that patients spend are more than 10 minutes on average. To test this suspicion, he uses method A , and takes a random sample of 24 appointments. He notes the actual time spent for each appointment and carries out a hypothesis test at the \(1 \%\) significance level.
  2. Explain why a 1-tail test is appropriate. The population mean of the actual times that patients spend at their appointments is denoted by \(\mu\) minutes.
  3. Assuming that \(\mu = 10\), state the probability that the conclusion of the test will be that \(\mu\) is not greater than 10 . The actual lengths of time, in minutes, that patients spend for their appointments may be assumed to have a normal distribution with standard deviation 3.4.
    [0pt]
  4. Given that the total length of time spent for the 24 appointments is 285 minutes, carry out the test. [7]
  5. In part (iv) it was necessary to use the fact that the sample mean is normally distributed. Give a reason why you know that this is true in this case.
OCR H240/02 2018 March Q11
7 marks Easy -1.2
11 The scatter diagram shows data, taken from the pre-release data set, for several Local Authorities in one region of the UK in 2011. The diagram shows, for each Local Authority, the number of workers who drove to work, and the number of workers who walked to work. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{2011} \includegraphics[alt={},max width=\textwidth]{6a6316e4-7b2d-4533-988a-4863d79ce668-08_483_956_479_557}
\end{figure}
  1. Four students calculated the value of Pearson's product-moment correlation coefficient for the data in the diagram. Their answers were \(0.913,0.124 , - 0.913\) and - 0.124 . One of these values is correct. Without calculation state, with a reason, which is the correct value.
  2. Sanjay makes the following statement.
    "The diagram shows that, in any Local Authority, if there are a large number of people who drive to work there will be a large number who walk to work." Give a reason why this statement is incorrect.
  3. Rosie makes the following statement.
    "The diagram must be wrong because it shows good positive correlation. If there are more people driving to work, there will be fewer people walking to work, so there would be negative correlation." Explain briefly why Rosie's statement is incorrect.
  4. The diagram shows a fairly close relationship between the two variables. One point on the diagram represents a Local Authority where this relationship is less strong than for the others. On the diagram in the Printed Answer Booklet, label this point A.
  5. Given that the point A represents a metropolitan borough, suggest a reason why the relationship is less strong for this Local Authority than for the others in the region. The scatter diagram below shows the corresponding data for the same region in 2001. \begin{figure}[h]
    \captionsetup{labelformat=empty} \caption{2001} \includegraphics[alt={},max width=\textwidth]{6a6316e4-7b2d-4533-988a-4863d79ce668-09_481_885_388_591}
    \end{figure}
  6. (a) State a change that has taken place in the metropolitan borough represented by the point A between 2001 and 2011.
    (b) Suggest a possible reason for this change.
OCR H240/02 2018 March Q12
12 marks Standard +0.3
12 Rob has two six-sided dice, each with sides numbered 1, 2, 3, 4, 5, 6.
One dice is fair. The other dice is biased, with probabilities as shown in the table.
Biased die
\(y\)123456
\(\mathrm { P } ( Y = y )\)0.30.250.20.140.10.01
Rob throws each dice once and notes the two scores, \(X\) on the fair dice and \(Y\) on the biased dice. He then calculates the value of the variable \(S\) which is defined as follows.
  • If \(X \leqslant 3\), then \(S = X + 2 Y\).
  • If \(X > 3\), then \(S = X + Y\).
    1. (a) Draw up a sample space diagram showing all the possible outcomes and the corresponding values of \(S\).
      (b) On your diagram, circle the four cells where the value \(S = 10\) occurs.
    2. Explain the mistake in the following calculation.
$$\mathrm { P } ( S = 10 ) = \frac { \text { Number of outcomes giving } S = 10 } { \text { Total number of outcomes } } = \frac { 4 } { 36 } = \frac { 1 } { 9 } .$$
  • Find the correct value of \(\mathrm { P } ( S = 10 )\).
  • Given that \(S = 10\), find the probability that the score on one of the dice is 4 .
  • The events " \(X = 1\) or 2 " and " \(S = n\) " are mutually exclusive. Given that \(\mathrm { P } ( S = n ) \neq 0\), find the value of \(n\).
    •
    OCR H240/01 2018 September Q1
    5 marks Easy -1.8
    1 Solve the following inequalities.
    1. \(- 5 < 3 x + 1 < 14\)
    2. \(4 x ^ { 2 } + 3 > 28\)
    OCR H240/01 2018 September Q2
    6 marks Easy -1.2
    2 Vector \(\mathbf { v } = a \mathbf { i } + 0.6 \mathbf { j }\), where \(a\) is a constant.
    1. Given that the direction of \(\mathbf { v }\) is \(45 ^ { \circ }\), state the value of \(a\).
    2. Given instead that \(\mathbf { v }\) is parallel to \(8 \mathbf { i } + 3 \mathbf { j }\), find the value of \(a\).
    3. Given instead that \(\mathbf { v }\) is a unit vector, find the possible values of \(a\).
    OCR H240/01 2018 September Q3
    6 marks Moderate -0.3
    3
    1. The diagram below shows the graphs of \(y = | 3 x - 2 |\) and \(y = | 2 x + 1 |\). \includegraphics[max width=\textwidth, alt={}, center]{e3942549-bfc0-432a-bf49-7d01d44af01a-4_423_682_1110_694} On the diagram in your Printed Answer Booklet, give the coordinates of the points of intersection of the graphs with the coordinate axes.
    2. Solve the equation \(| 2 x + 1 | = | 3 x - 2 |\).
    OCR H240/01 2018 September Q4
    7 marks Standard +0.3
    4 \includegraphics[max width=\textwidth, alt={}, center]{e3942549-bfc0-432a-bf49-7d01d44af01a-5_487_789_251_639} The diagram shows the triangle \(A O B\), in which angle \(A O B = 0.8\) radians, \(O A = 7 \mathrm {~cm}\) and \(O B = 10 \mathrm {~cm}\). \(C D\) is the arc of a circle with centre \(O\) and radius \(O C\). The area of the triangle \(A O B\) is twice the area of the sector COD
    1. Find the length \(O C\).
    2. Find the perimeter of the region \(A B C D\).
    OCR H240/01 2018 September Q5
    6 marks Moderate -0.3
    5 A student was asked to solve the equation \(2 \left( \log _ { 3 } x \right) ^ { 2 } - 3 \log _ { 3 } x - 2 = 0\). The student's attempt is written out below. $$\begin{aligned} & 2 \left( \log _ { 3 } x \right) ^ { 2 } - 3 \log _ { 3 } x - 2 = 0 \\ & 4 \log _ { 3 } x - 3 \log _ { 3 } x - 2 = 0 \\ & \log _ { 3 } x - 2 = 0 \\ & \log _ { 3 } x = 2 \\ & x = 8 \end{aligned}$$
    1. Identify the two mistakes that the student has made.
    2. Solve the equation \(2 \left( \log _ { 3 } x \right) ^ { 2 } - 3 \log _ { 3 } x - 2 = 0\), giving your answers in an exact form.
    OCR H240/01 2018 September Q6
    7 marks Standard +0.3
    6 In this question you must show detailed reasoning. A curve has equation \(y = \frac { 2 x } { 3 x - 1 } + \sqrt { 5 x + 1 }\). Show that the equation of the tangent to the curve at the point where \(x = 3\) is \(19 x - 32 y + 95 = 0\).
    OCR H240/01 2018 September Q7
    11 marks Moderate -0.3
    7 A line has equation \(y = 2 x\) and a circle has equation \(x ^ { 2 } + y ^ { 2 } + 2 x - 16 y + 56 = 0\).
    1. Show that the line does not meet the circle.
    2. (a) Find the equation of the line through the centre of the circle that is perpendicular to the line \(y = 2 x\).
      (b) Hence find the shortest distance between the line \(y = 2 x\) and the circle, giving your answer in an exact form.
    OCR H240/01 2018 September Q8
    9 marks Standard +0.3
    8 \includegraphics[max width=\textwidth, alt={}, center]{e3942549-bfc0-432a-bf49-7d01d44af01a-6_533_524_246_772} The diagram shows a container which consists of a cylinder with a solid base and a hemispherical top. The radius of the cylinder is \(r \mathrm {~cm}\) and the height is \(h \mathrm {~cm}\). The container is to be made of thin plastic. The volume of the container is \(45 \pi \mathrm {~cm} ^ { 3 }\).
    1. Show that the surface area of the container, \(A \mathrm {~cm} ^ { 2 }\), is given by $$A = \frac { 5 } { 3 } \pi r ^ { 2 } + \frac { 90 \pi } { r } .$$ [The volume of a sphere is \(V = \frac { 4 } { 3 } \pi r ^ { 3 }\) and the surface area of a sphere is \(S = 4 \pi r ^ { 2 }\).]
    2. Use calculus to find the minimum surface area of the container, justifying that it is a minimum.
    3. Suggest a reason why the manufacturer would wish to minimise the surface area.
    OCR H240/01 2018 September Q9
    8 marks Standard +0.3
    9 An analyst believes that the sales of a particular electronic device are growing exponentially. In 2015 the sales were 3.1 million devices and the rate of increase in the annual sales is 0.8 million devices per year.
    1. Find a model to represent the annual sales, defining any variables used.
    2. In 2017 the sales were 5.2 million devices. Determine whether this is consistent with the model in part (i).
    3. The analyst uses the model in part (i) to predict the sales for 2025. Comment on the reliability of this prediction.
    OCR H240/01 2018 September Q10
    13 marks Standard +0.3
    10 \includegraphics[max width=\textwidth, alt={}, center]{e3942549-bfc0-432a-bf49-7d01d44af01a-7_579_764_255_651} The diagram shows the graph of \(\mathrm { f } ( x ) = \ln ( 3 x + 1 ) - x\), which has a stationary point at \(x = \alpha\). A student wishes to find the non-zero root \(\beta\) of the equation \(\ln ( 3 x + 1 ) - x = 0\) using the Newton-Raphson method.
    1. (a) Determine the value of \(\alpha\).
      (b) Explain why the Newton-Raphson method will fail if \(\alpha\) is used as the initial value.
    2. Show that the Newton-Raphson iterative formula for finding \(\beta\) can be written as $$x _ { n + 1 } = \frac { 3 x _ { n } - \left( 3 x _ { n } + 1 \right) \ln \left( 3 x _ { n } + 1 \right) } { 2 - 3 x _ { n } } .$$
    3. Apply the iterative formula in part (ii) with initial value \(x _ { 1 } = 1\) to find the value of \(\beta\) correct to 5 significant figures. You should show the result of each iteration.
    4. Use a change of sign method to verify that the value of \(\beta\) found in part (iii) is correct to 5 significant figures.
    OCR H240/01 2018 September Q11
    12 marks Challenging +1.2
    11 In this question you must show detailed reasoning. The \(n\)th term of a geometric progression is denoted by \(g _ { n }\) and the \(n\)th term of an arithmetic progression is denoted by \(a _ { n }\). It is given that \(g _ { 1 } = a _ { 1 } = 1 + \sqrt { 5 } , g _ { 3 } = a _ { 2 }\) and \(g _ { 4 } + a _ { 3 } = 0\). Given also that the geometric progression is convergent, show that its sum to infinity is \(4 + 2 \sqrt { 5 }\).
    OCR H240/01 2018 September Q12
    10 marks Challenging +1.2
    12 The gradient function of a curve is given by \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { x ^ { 2 } \sin 2 x } { 2 \cos ^ { 2 } 4 y - 1 }\).
    1. Find an equation for the curve in the form \(\mathrm { f } ( y ) = g ( x )\). The curve passes through the point \(\left( \frac { 1 } { 4 } \pi , \frac { 1 } { 12 } \pi \right)\).
    2. Find the smallest positive value of \(y\) for which \(x = 0\). \section*{END OF QUESTION PAPER}
    OCR H240/02 2018 September Q1
    7 marks Moderate -0.8
    1
    1. Differentiate the following with respect to \(x\).
      1. \(\frac { 1 } { ( 3 x - 4 ) ^ { 2 } }\)
      2. \(\frac { \ln ( x + 2 ) } { x }\)
      3. Find \(\int \mathrm { e } ^ { ( 2 x + 3 ) } \mathrm { d } x\).
    OCR H240/02 2018 September Q2
    9 marks Easy -1.3
    2
    1. Ben saves his pocket money as follows.
      Each week he puts money into his piggy bank (which pays no interest). In the first week he puts in 10p. In the second week he puts in 12p. In the third week he puts in 14p, and so on. How much money does Ben have in his piggy bank after 25 weeks?
    2. On January 1st Shirley invests \(\pounds 500\) in a savings account that pays compound interest at \(3 \%\) per annum. She makes no further payments into this account. The interest is added on 31st December each year.
      1. Find the number of years after which her investment will first be worth more than \(\pounds 600\).
      2. State an assumption that you have made in answering part (ii)(a).
    OCR H240/02 2018 September Q3
    4 marks Standard +0.3
    3 Use small angle approximations to estimate the solution of the equation \(\frac { \cos \frac { 1 } { 2 } \theta } { 1 + \sin \theta } = 0.825\), if \(\theta\) is small enough to neglect terms in \(\theta ^ { 3 }\) or above.
    OCR H240/02 2018 September Q4
    4 marks Moderate -0.5
    4 Prove that the sum of the squares of any two consecutive integers is of the form \(4 k + 1\), where \(k\) is an integer.
    OCR H240/02 2018 September Q5
    3 marks Easy -1.2
    5 The diagram shows the graph of \(y = \sin x ^ { \circ }\) for \(0 \leqslant x \leqslant 360\). \includegraphics[max width=\textwidth, alt={}, center]{85de9a39-f8be-40ee-b0c8-e2e632be93d8-4_597_965_1909_539} Find an equation for the transformed curve when the curve \(y = \sin x ^ { \circ }\) is reflected in
    1. the \(x\)-axis,
    2. the line \(y = 0.5\).
    OCR H240/02 2018 September Q6
    9 marks Standard +0.3
    6
    1. Find the coefficient of \(x ^ { 4 }\) in the expansion of \(( 3 x - 2 ) ^ { 10 }\).
    2. In the expansion of \(( 1 + 2 x ) ^ { n }\), where \(n\) is a positive integer, the coefficients of \(x ^ { 7 }\) and \(x ^ { 8 }\) are equal. Find the value of \(n\).
    3. Find the coefficient of \(x ^ { 3 }\) in the expansion of \(\frac { 1 } { \sqrt { 4 + x } }\).
    OCR H240/02 2018 September Q7
    7 marks Standard +0.3
    7 The diagram shows part of the curve \(y = x ^ { 2 }\) for \(0 \leqslant x \leqslant p\), where \(p\) is a constant. \includegraphics[max width=\textwidth, alt={}, center]{85de9a39-f8be-40ee-b0c8-e2e632be93d8-5_736_543_669_762} The area \(A\) of the region enclosed by the curve, the \(x\)-axis and the line \(x = p\) is given approximately by the sum \(S\) of the areas of \(n\) rectangles, each of width \(h\), where \(h\) is small and \(n h = p\). The first three such rectangles are shown in the diagram.
    1. Find an expression for \(S\) in terms of \(n\) and \(h\).
    2. Use the identity \(\sum _ { r = 1 } ^ { n } r ^ { 2 } \equiv \frac { 1 } { 6 } n ( n + 1 ) ( 2 n + 1 )\) to show that \(S = \frac { 1 } { 6 } p ( p + h ) ( 2 p + h )\).
    3. Show how to use this result to find \(A\) in terms of \(p\).
    OCR H240/02 2018 September Q8
    9 marks Moderate -0.5
    8 The points \(A , B\) and \(C\) have position vectors \(\mathbf { a } , \mathbf { b }\) and \(\mathbf { c }\), relative to an origin \(O\), in three dimensions. The figure \(O A P B S C T U\) is a cuboid, with vertices labelled as in the following diagram. \(M\) is the midpoint of \(A U\). \includegraphics[max width=\textwidth, alt={}, center]{85de9a39-f8be-40ee-b0c8-e2e632be93d8-5_557_1221_2087_420}