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OCR H240/01 2023 June Q5
8 marks Standard +0.3
5
  1. The function \(\mathrm { f } ( x )\) is defined for all values of \(x\) as \(\mathrm { f } ( x ) = | a x - b |\), where \(a\) and \(b\) are positive constants.
    1. The graph of \(y = \mathrm { f } ( x ) + c\), where \(c\) is a constant, has a vertex at \(( 3,1 )\) and crosses the \(y\)-axis at \(( 0,7 )\). Find the values of \(a , b\) and \(c\).
    2. Explain why \(\mathrm { f } ^ { - 1 } ( x )\) does not exist.
  2. The function \(\mathrm { g } ( x )\) is defined for \(x \geqslant \frac { q } { p }\) as \(\mathrm { g } ( x ) = | p x - q |\), where \(p\) and \(q\) are positive constants.
    1. Find, in terms of \(p\) and \(q\), an expression for \(\mathrm { g } ^ { - 1 } ( x )\), stating the domain of \(\mathrm { g } ^ { - 1 } ( x )\).
    2. State the set of values of \(p\) for which the equation \(\mathrm { g } ( x ) = \mathrm { g } ^ { - 1 } ( x )\) has no solutions.
OCR H240/01 2023 June Q6
9 marks Standard +0.3
6 A curve has equation \(y = \mathrm { e } ^ { x ^ { 2 } + 3 x }\).
  1. Determine the \(x\)-coordinates of any stationary points on the curve.
  2. Show that the curve is convex for all values of \(x\).
OCR H240/01 2023 June Q7
9 marks Moderate -0.3
7
  1. Use the result \(\cos ( A + B ) = \cos A \cos B - \sin A \sin B\) to show that \(\cos ( A - B ) = \cos A \cos B + \sin A \sin B\). The function \(\mathrm { f } ( \theta )\) is defined as \(\cos \left( \theta + 30 ^ { \circ } \right) \cos \left( \theta - 30 ^ { \circ } \right)\), where \(\theta\) is in degrees.
  2. Show that \(f ( \theta ) = \cos ^ { 2 } \theta - \frac { 1 } { 4 }\).
    1. Determine the following.
      • The maximum value of \(\mathrm { f } ( \theta )\)
  4. The smallest positive value of \(\theta\) for which this maximum value occurs
    (ii) Determine the following.
  5. The minimum value of \(\mathrm { f } ( \theta )\)
  6. The smallest positive value of \(\theta\) for which this minimum value occurs
OCR H240/01 2023 June Q8
9 marks Standard +0.3
8
  1. Find the first three terms in the expansion of \(( 4 + 3 x ) ^ { \frac { 3 } { 2 } }\) in ascending powers of \(x\).
  2. State the range of values of \(x\) for which the expansion in part (a) is valid.
  3. In the expansion of \(( 4 + 3 x ) ^ { \frac { 3 } { 2 } } ( 1 + a x ) ^ { 2 }\) the coefficient of \(x ^ { 2 }\) is \(\frac { 107 } { 16 }\). Determine the possible values of the constant \(a\).
OCR H240/01 2023 June Q9
6 marks Moderate -0.3
9 Conservationists are studying how the number of bees in a wildflower meadow varies according to the number of wildflower plants. The study takes place over a series of weeks in the summer. A model is suggested for the number of bees, \(B\), and the number of wildflower plants, \(F\), at time \(t\) weeks after the start of the study. In the model \(B = 20 + 2 t + \cos 3 t\) and \(F = 50 \mathrm { e } ^ { 0.1 t }\). The model assumes that \(B\) and \(F\) can be treated as continuous variables.
  1. State the meaning of \(\frac { \mathrm { d } B } { \mathrm {~d} F }\).
  2. Determine \(\frac { \mathrm { d } B } { \mathrm {~d} F }\) when \(t = 4\).
  3. Suggest a reason why this model may not be valid for values of \(t\) greater than 12 .
OCR H240/01 2023 June Q10
9 marks Standard +0.3
10 \includegraphics[max width=\textwidth, alt={}, center]{9473b8f7-616a-485e-963b-696c6640ae6b-07_805_775_251_242} The diagram shows part of the curve \(\mathrm { f } ( x ) = \frac { \mathrm { e } ^ { x } } { 4 x ^ { 2 } - 1 } + 2\). The equation \(\mathrm { f } ( x ) = 0\) has a positive root \(\alpha\) close to \(x = 0.3\).
  1. Explain why using the sign change method with \(x = 0\) and \(x = 1\) will fail to locate \(\alpha\).
  2. Show that the equation \(\mathrm { f } ( x ) = 0\) can be written as \(x = \frac { 1 } { 4 } \sqrt { \left( 4 - 2 \mathrm { e } ^ { x } \right) }\).
  3. Use the iterative formula \(x _ { n + 1 } = \frac { 1 } { 4 } \sqrt { \left( 4 - 2 \mathrm { e } ^ { x _ { n } } \right) }\) with a starting value of \(x _ { 1 } = 0.3\) to find the value of \(\alpha\) correct to \(\mathbf { 4 }\) significant figures, showing the result of each iteration.
  4. An alternative iterative formula is \(x _ { n + 1 } = \mathrm { F } \left( x _ { n } \right)\), where \(\mathrm { F } \left( x _ { n } \right) = \ln \left( 2 - 8 x _ { n } ^ { 2 } \right)\). By considering \(\mathrm { F } ^ { \prime } ( 0.3 )\) explain why this iterative formula will not find \(\alpha\).
OCR H240/01 2023 June Q11
12 marks Moderate -0.3
11 The owners of an online shop believe that their sales can be modelled by \(S = a b ^ { t }\), where \(a\) and \(b\) are both positive constants, \(S\) is the number of items sold in a month and \(t\) is the number of complete months since starting their online shop. The sales for the first six months are recorded, and the values of \(\log _ { 10 } S\) are plotted against \(t\) in the graph below. The graph is repeated in the Printed Answer Booklet. \includegraphics[max width=\textwidth, alt={}, center]{9473b8f7-616a-485e-963b-696c6640ae6b-08_1203_1408_552_244}
  1. Explain why the graph suggests that the given model is appropriate. The owners believe that \(a = 120\) and \(b = 1.15\) are good estimates for the parameters in the model.
  2. Show that the graph supports these estimates for the parameters.
  3. Use the model \(S = 120 \times 1.15 ^ { t }\) to predict the number of items sold in the seventh month after opening.
    1. Use the model \(S = 120 \times 1.15 ^ { t }\) to predict the number of months after opening when the total number of items sold after opening will first exceed 70000 .
    2. Comment on how reliable this prediction may be.
OCR H240/01 2023 June Q12
10 marks Challenging +1.2
12
  1. Use the substitution \(u = \mathrm { e } ^ { x } - 2\) to show that $$\int \frac { 7 \mathrm { e } ^ { x } - 8 } { \left( \mathrm { e } ^ { x } - 2 \right) ^ { 2 } } \mathrm {~d} x = \int \frac { 7 u + 6 } { u ^ { 2 } ( u + 2 ) } \mathrm { d } u$$
  2. Hence show that $$\int _ { \ln 4 } ^ { \ln 6 } \frac { 7 \mathrm { e } ^ { x } - 8 } { \left( \mathrm { e } ^ { x } - 2 \right) ^ { 2 } } \mathrm {~d} x = a + \ln b$$ where \(a\) and \(b\) are rational numbers to be determined. \section*{END OF QUESTION PAPER}
OCR H240/02 2018 June Q1
7 marks Moderate -0.8
1
  1. Express \(2 x ^ { 2 } - 12 x + 23\) in the form \(a ( x + b ) ^ { 2 } + c\).
  2. Use your result to show that the equation \(2 x ^ { 2 } - 12 x + 23 = 0\) has no real roots.
  3. Given that the equation \(2 x ^ { 2 } - 12 x + k = 0\) has repeated roots, find the value of the constant \(k\).
OCR H240/02 2018 June Q2
6 marks Moderate -0.8
2 The points \(A\) and \(B\) have position vectors \(\left( \begin{array} { c } 1 \\ - 2 \\ 5 \end{array} \right)\) and \(\left( \begin{array} { c } - 3 \\ - 1 \\ 2 \end{array} \right)\) respectively.
  1. Find the exact length of \(A B\).
  2. Find the position vector of the midpoint of \(A B\). The points \(P\) and \(Q\) have position vectors \(\left( \begin{array} { l } 1 \\ 2 \\ 0 \end{array} \right)\) and \(\left( \begin{array} { l } 5 \\ 1 \\ 3 \end{array} \right)\) respectively.
  3. Show that \(A B P Q\) is a parallelogram.
OCR H240/02 2018 June Q3
4 marks Easy -1.2
3 Ayesha, Bob, Chloe and Dave are discussing the relationship between the time, \(t\) hours, they might spend revising for an examination, and the mark, \(m\), they would expect to gain. Each of them draws a graph to model this relationship for himself or herself. \includegraphics[max width=\textwidth, alt={}, center]{65d9d34c-8c78-45fe-b9f0-dab071ae56bb-04_437_423_1576_187} \includegraphics[max width=\textwidth, alt={}, center]{65d9d34c-8c78-45fe-b9f0-dab071ae56bb-04_439_426_1576_609} \includegraphics[max width=\textwidth, alt={}, center]{65d9d34c-8c78-45fe-b9f0-dab071ae56bb-04_439_428_1576_1032} \includegraphics[max width=\textwidth, alt={}, center]{65d9d34c-8c78-45fe-b9f0-dab071ae56bb-04_437_419_1576_1462}
  1. Assuming Ayesha's model is correct, how long would you recommend that she spends revising?
  2. State one feature of Dave's model that is likely to be unrealistic.
  3. Suggest a reason for the shape of Bob's graph as compared with Ayesha's graph.
  4. What does Chloe's model suggest about her attitude to revision?
OCR H240/02 2018 June Q4
4 marks Moderate -0.3
4 Prove that \(\sin ^ { 2 } ( \theta + 45 ) ^ { \circ } - \cos ^ { 2 } ( \theta + 45 ) ^ { \circ } \equiv \sin 2 \theta ^ { \circ }\).
OCR H240/02 2018 June Q5
6 marks Standard +0.3
5 Charlie claims to have proved the following statement.
"The sum of a square number and a prime number cannot be a square number."
  1. Give an example to show that Charlie's statement is not true. Charlie's attempt at a proof is below.
    Assume that the statement is not true.
    ⇒ There exist integers \(n\) and \(m\) and a prime \(p\) such that \(n ^ { 2 } + p = m ^ { 2 }\). \(\Rightarrow p = m ^ { 2 } - n ^ { 2 }\) \(\Rightarrow p = ( m - n ) ( m + n )\) \(\Rightarrow p\) is the product of two integers. \(\Rightarrow p\) is not prime, which is a contradiction.
    ⇒ Charlie's statement is true.
  2. Explain the error that Charlie has made.
  3. Given that 853 is a prime number, find the square number \(S\) such that \(S + 853\) is also a square number.
OCR H240/02 2018 June Q7
7 marks Standard +0.3
7 The diagram shows a part \(A B C\) of the curve \(y = 3 - 2 x ^ { 2 }\), together with its reflections in the lines \(y = x\), \(y = - x\) and \(y = 0\). \includegraphics[max width=\textwidth, alt={}, center]{65d9d34c-8c78-45fe-b9f0-dab071ae56bb-05_691_673_1957_678}
OCR H240/02 2018 June Q8
8 marks Standard +0.3
8
  1. The variable \(X\) has the distribution \(\mathrm { N } ( 20,9 )\).
    (a) Find \(\mathrm { P } ( X > 25 )\).
    (b) Given that \(\mathrm { P } ( X > a ) = 0.2\), find \(a\).
    (c) Find \(b\) such that \(\mathrm { P } ( 20 - b < X < 20 + b ) = 0.5\).
  2. The variable \(Y\) has the distribution \(\mathrm { N } \left( \mu , \frac { \mu ^ { 2 } } { 9 } \right)\). Find \(\mathrm { P } ( Y > 1.5 \mu )\).
OCR H240/02 2018 June Q9
7 marks Standard +0.3
9 Briony suspects that a particular 6-sided dice is biased in favour of 2. She plans to throw the dice 35 times and note the number of times that it shows a 2 . She will then carry out a test at the \(4 \%\) significance level. Find the rejection region for the test.
OCR H240/02 2018 June Q10
9 marks Standard +0.3
10 A certain forest contains only trees of a particular species. Dipak wished to take a random sample of 5 trees from the forest. He numbered the trees from 1 to 784. Then, using his calculator, he generated the random digits 14781049 . Using these digits, Dipak formed 5 three-digit numbers. He took the first, second and third digits, followed by the second, third and fourth digits and so on. In this way he obtained the following list of numbers for his sample. $$\begin{array} { l l l l l } 147 & 478 & 781 & 104 & 49 \end{array}$$
  1. Explain why Dipak omitted the number 810 from his list.
  2. Explain why Dipak's sample is not random. The mean height of all trees of this species is known to be 4.2 m . Dipak wishes to test whether the mean height of trees in the forest is less than 4.2 m . He now uses a correct method to choose a random sample of 50 trees and finds that their mean height is 4.0 m . It is given that the standard deviation of trees in the forest is 0.8 m .
  3. Carry out the test at the \(2 \%\) significance level.
OCR H240/02 2018 June Q11
6 marks Moderate -0.8
11 Christa used Pearson's product-moment correlation coefficient, \(r\), to compare the use of public transport with the use of private vehicles for travel to work in the UK.
  1. Using the pre-release data set for all 348 UK Local Authorities, she considered the following four variables.
    Number of employees using
    public transport
    		\(x\)
    Number of employees using
    private vehicles
    		\(y\)
    Proportion of employees using
    public transport
    		\(a\)
    Proportion of employees using
    private vehicles
    		\(b\)
    (a) Explain, in context, why you would expect strong, positive correlation between \(x\) and \(y\).
    (b) Explain, in context, what kind of correlation you would expect between \(a\) and \(b\).
  2. Christa also considered the data for the 33 London boroughs alone and she generated the following scatter diagram. \begin{figure}[h]
    \captionsetup{labelformat=empty} \caption{London} \includegraphics[alt={},max width=\textwidth]{65d9d34c-8c78-45fe-b9f0-dab071ae56bb-07_467_707_1366_653}
    \end{figure} One London Borough is represented by an outlier in the diagram.
    (a) Suggest what effect this outlier is likely to have on the value of \(r\) for the 32 London Boroughs.
    (b) Suggest what effect this outlier is likely to have on the value of \(r\) for the whole country.
    (c) What can you deduce about the area of the London Borough represented by the outlier? Explain your answer.
OCR H240/02 2018 June Q12
11 marks Standard +0.3
12 The discrete random variable \(X\) takes values 1, 2, 3, 4 and 5, and its probability distribution is defined as follows. $$\mathrm { P } ( X = x ) = \begin{cases} a & x = 1 , \\ \frac { 1 } { 2 } \mathrm { P } ( X = x - 1 ) & x = 2,3,4,5 , \\ 0 & \text { otherwise } , \end{cases}$$ where \(a\) is a constant.
  1. Show that \(a = \frac { 16 } { 31 }\). The discrete probability distribution for \(X\) is given in the table.
    \(x\)12345
    \(\mathrm { P } ( X = x )\)\(\frac { 16 } { 31 }\)\(\frac { 8 } { 31 }\)\(\frac { 4 } { 31 }\)\(\frac { 2 } { 31 }\)\(\frac { 1 } { 31 }\)
  2. Find the probability that \(X\) is odd. Two independent values of \(X\) are chosen, and their sum \(S\) is found.
  3. Find the probability that \(S\) is odd.
  4. Find the probability that \(S\) is greater than 8 , given that \(S\) is odd. Sheila sometimes needs several attempts to start her car in the morning. She models the number of attempts she needs by the discrete random variable \(Y\) defined as follows. $$\mathrm { P } ( Y = y + 1 ) = \frac { 1 } { 2 } \mathrm { P } ( Y = y ) \quad \text { for all positive integers } y .$$
  5. Find \(\mathrm { P } ( Y = 1 )\).
  6. Give a reason why one of the variables, \(X\) or \(Y\), might be more appropriate as a model for the number of attempts that Sheila needs to start her car.
OCR H240/02 2019 June Q1
11 marks Moderate -0.3
1
  1. Differentiate the following.
    1. \(\frac { x ^ { 2 } } { 2 x + 1 }\)
    2. \(\tan \left( x ^ { 2 } - 3 x \right)\)
  2. Use the substitution \(u = \sqrt { x } - 1\) to integrate \(\frac { 1 } { \sqrt { x } - 1 }\).
  3. Integrate \(\frac { x - 2 } { 2 x ^ { 2 } - 8 x - 1 }\).
OCR H240/02 2019 June Q2
8 marks Moderate -0.3
2
  1. Find the coefficient of \(x ^ { 5 }\) in the expansion of \(( 3 - 2 x ) ^ { 8 }\).
    1. Expand \(( 1 + 3 x ) ^ { 0.5 }\) as far as the term in \(x ^ { 3 }\).
    2. State the range of values of \(x\) for which your expansion is valid. A student suggests the following check to determine whether the expansion obtained in part (b)(i) may be correct.
      "Use the expansion to find an estimate for \(\sqrt { 103 }\), correct to five decimal places, and compare this with the value of \(\sqrt { 103 }\) given by your calculator."
    3. Showing your working, carry out this check on your expansion from part (b)(i).
OCR H240/02 2019 June Q3
9 marks Standard +0.3
3
  1. A circle is defined by the parametric equations \(x = 3 + 2 \cos \theta , y = - 4 + 2 \sin \theta\).
    1. Find a cartesian equation of the circle.
    2. Write down the centre and radius of the circle.
  2. In this question you must show detailed reasoning. The curve \(S\) is defined by the parametric equations \(x = 4 \cos t , y = 2 \sin t\). The line \(L\) is a tangent to \(S\) at the point given by \(t = \frac { 1 } { 6 } \pi\). Find where the line \(L\) cuts the \(x\)-axis.
OCR H240/02 2019 June Q4
5 marks Moderate -0.5
4 A species of animal is to be introduced onto a remote island. Their food will consist only of various plants that grow on the island. A zoologist proposes two possible models for estimating the population \(P\) after \(t\) years. The diagrams show these models as they apply to the first 20 years. \includegraphics[max width=\textwidth, alt={}, center]{1a0e0afb-81be-45d1-8c86-f98e508e9a49-05_725_606_406_242} \includegraphics[max width=\textwidth, alt={}, center]{1a0e0afb-81be-45d1-8c86-f98e508e9a49-05_714_593_413_968}
  1. Without calculation, describe briefly how the rate of growth of \(P\) will vary for the first 20 years, according to each of these two models. The equation of the curve for model A is \(P = 20 + 1000 \mathrm { e } ^ { - \frac { ( t - 20 ) ^ { 2 } } { 100 } }\).
    The equation of the curve for model B is \(P = 20 + 1000 \left( 1 - \mathrm { e } ^ { - \frac { t } { 5 } } \right)\).
  2. Describe the behaviour of \(P\) that is predicted for \(t > 20\)
    1. using model A,
    2. using model B . There is only a limited amount of food available on the island, and the zoologist assumes that the size of the population depends on the amount of food available and on no other external factors.
  3. State what is suggested about the long-term food supply by
    1. model A,
    2. model B.
OCR H240/02 2019 June Q5
9 marks Standard +0.8
5 \includegraphics[max width=\textwidth, alt={}, center]{1a0e0afb-81be-45d1-8c86-f98e508e9a49-06_371_339_255_251} For a cone with base radius \(r\), height \(h\) and slant height \(l\), the following formulae are given.
Curved surface area, \(S = \pi r l\) Volume, \(V = \frac { 1 } { 3 } \pi r ^ { 2 } h\) A container is to be designed in the shape of an inverted cone with no lid. The base radius is \(r \mathrm {~m}\) and the volume is \(V \mathrm {~m} ^ { 3 }\). The area of the material to be used for the cone is \(4 \pi \mathrm {~m} ^ { 2 }\).
  1. Show that \(V = \frac { 1 } { 3 } \pi \sqrt { 16 r ^ { 2 } - r ^ { 6 } }\).
  2. In this question you must show detailed reasoning. It is given that \(V\) has a maximum value for a certain value of \(r\).
    Find the maximum value of \(V\), giving your answer correct to 3 significant figures.
OCR H240/02 2019 June Q6
4 marks Standard +0.8
6 Shona makes the following claim.
" \(n\) is an even positive integer greater than \(2 \Rightarrow 2 ^ { n } - 1\) is not prime"
Prove that Shona's claim is true.