Questions H240/02 (94 questions)

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OCR H240/02 Q14
92 marks
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 H240/02 2018 June Q6
  1. Find the \(x\)-coordinate of the point where the curve crosses the \(x\) axis.
  2. The points \(A\) and \(B\) lie on the curve and have \(x\) coordinates 2 and 4. Show that the line \(A B\) is parallel to the \(x\)-axis.
  3. Find the coordinates of the turning point on the curve.
  4. Determine whether this turning point is a maximum or a minimum.
OCR H240/02 2018 June Q13
13 In this question you must show detailed reasoning. The probability that Paul's train to work is late on any day is 0.15 , independently of other days.
  1. The number of days on which Paul's train to work is late during a 450-day period is denoted by the random variable \(Y\). Find a value of \(a\) such that \(\mathrm { P } ( Y > a ) \approx \frac { 1 } { 6 }\). In the expansion of \(( 0.15 + 0.85 ) ^ { 50 }\), the terms involving \(0.15 ^ { r }\) and \(0.15 ^ { r + 1 }\) are denoted by \(T _ { r }\) and \(T _ { r + 1 }\) respectively.
  2. Show that \(\frac { T _ { r } } { T _ { r + 1 } } = \frac { 17 ( r + 1 ) } { 3 ( 50 - r ) }\).
  3. The number of days on which Paul's train to work is late during a 50-day period is modelled by the random variable \(X\).
    (a) Find the values of \(r\) for which \(\mathrm { P } ( X = r ) \leqslant \mathrm { P } ( X = r + 1 )\).
    (b) Hence find the most likely number of days on which the train will be late during a 50-day period.
OCR H240/02 2021 November Q5
5 In this question you must show detailed reasoning. Points \(A , B\) and \(C\) have coordinates \(( 0,6 ) , ( 7,5 )\) and \(( 6 , - 2 )\) respectively.
  1. Find an equation of the perpendicular bisector of \(A B\).
  2. Hence, or otherwise, find an equation of the circle that passes through points \(A , B\) and \(C\).
OCR H240/02 2020 November Q1
1
  1. Differentiate the following with respect to \(x\).
    1. \(( 2 x + 3 ) ^ { 7 }\)
    2. \(x ^ { 3 } \ln x\)
  2. Find \(\int \cos 5 x \mathrm {~d} x\).
  3. Find the equation of the curve through \(( 1,3 )\) for which \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 6 x - 5\).
OCR H240/02 2020 November Q2
2 Simplify fully \(\frac { 2 x ^ { 3 } + x ^ { 2 } - 7 x - 6 } { x ^ { 2 } - x - 2 }\).
OCR H240/02 2020 November Q3
3 In this question you should assume that \(- 1 < x < 1\).
  1. For the binomial expansion of \(( 1 - x ) ^ { - 2 }\)
    1. find and simplify the first four terms,
    2. write down the term in \(x ^ { n }\).
  2. Write down the sum to infinity of the series \(1 + x + x ^ { 2 } + x ^ { 3 } + \ldots\).
  3. Hence or otherwise find and simplify an expression for \(2 + 3 x + 4 x ^ { 2 } + 5 x ^ { 3 } + \ldots\) in the form \(\frac { a - x } { ( b - x ) ^ { 2 } }\) where \(a\) and \(b\) are constants to be determined.
OCR H240/02 2020 November Q4
4 In this question you must show detailed reasoning.
Solve the equation \(3 \sin ^ { 4 } \phi + \sin ^ { 2 } \phi = 4\), for \(0 \leqslant \phi < 2 \pi\), where \(\phi\) is measured in radians.
OCR H240/02 2020 November Q5
5
  1. Determine the set of values of \(n\) for which \(\frac { n ^ { 2 } - 1 } { 2 }\) and \(\frac { n ^ { 2 } + 1 } { 2 }\) are positive integers. A 'Pythagorean triple' is a set of three positive integers \(a , b\) and \(c\) such that \(a ^ { 2 } + b ^ { 2 } = c ^ { 2 }\).
  2. Prove that, for the set of values of \(n\) found in part (a), the numbers \(n , \frac { n ^ { 2 } - 1 } { 2 }\) and \(\frac { n ^ { 2 } + 1 } { 2 }\) form a Pythagorean triple.
OCR H240/02 2020 November Q6
6 Prove that \(\sqrt { 2 } \cos \left( 2 \theta + 45 ^ { \circ } \right) \equiv \cos ^ { 2 } \theta - 2 \sin \theta \cos \theta - \sin ^ { 2 } \theta\), where \(\theta\) is measured in degrees.
\(7 \quad A\) and \(B\) are fixed points in the \(x - y\) plane. The position vectors of \(A\) and \(B\) are \(\mathbf { a }\) and \(\mathbf { b }\) respectively. State, with reference to points \(A\) and \(B\), the geometrical significance of
  1. the quantity \(| \mathbf { a } - \mathbf { b } |\),
  2. the vector \(\frac { 1 } { 2 } ( \mathbf { a } + \mathbf { b } )\). The circle \(P\) is the set of points with position vector \(\mathbf { p }\) in the \(x - y\) plane which satisfy \(\left| \mathbf { p } - \frac { 1 } { 2 } ( \mathbf { a } + \mathbf { b } ) \right| = \frac { 1 } { 2 } | \mathbf { a } - \mathbf { b } |\).
  3. State, in terms of \(\mathbf { a }\) and \(\mathbf { b }\),
    1. the position vector of the centre of \(P\),
    2. the radius of \(P\). It is now given that \(\mathbf { a } = \binom { 2 } { - 1 } , \mathbf { b } = \binom { 4 } { 5 }\) and \(\mathbf { p } = \binom { x } { y }\).
  4. Find a cartesian equation of \(P\).
OCR H240/02 2020 November Q8
8 The rate of change of a certain population \(P\) at time \(t\) is modelled by the equation \(\frac { \mathrm { d } P } { \mathrm {~d} t } = ( 100 - P )\). Initially \(P = 2000\).
  1. Determine an expression for \(P\) in terms of \(t\).
  2. Describe how the population changes over time.
OCR H240/02 2020 November Q9
9 The histogram shows information about the numbers of cars in five different price ranges, sold in one year at a car showroom.
\includegraphics[max width=\textwidth, alt={}, center]{4ba60d6b-c987-4f6b-8c55-8b594c90c854-06_922_1413_495_244} It is given that 66 cars in the price range \(\pounds 10000\) to \(\pounds 20000\) were sold.
  1. Find the number of cars sold in the price range \(\pounds 50000\) to \(\pounds 90000\).
  2. State the units of the frequency density.
  3. Suggest one change that the management could make to the diagram so that it would provide more information.
  4. Estimate the number of cars sold in the price range \(\pounds 50000\) to \(\pounds 60000\).
OCR H240/02 2020 November Q10
10 Pierre is a chef. He claims that \(90 \%\) of his customers are satisfied with his cooking. Yvette suspects that Pierre is over-confident about the level of satisfaction amongst his customers. She talks to a random sample of 15 of Pierre's customers, and finds that 11 customers say that they are satisfied. She then performs a hypothesis test. Carry out the test at the 5\% significance level.
OCR H240/02 2020 November Q11
11 As part of a research project, the masses, \(m\) grams, of a random sample of 1000 pebbles from a certain beach were recorded. The results are summarised in the table.
Mass \(( \mathrm { g } )\)\(50 \leqslant m < 150\)\(150 \leqslant m < 200\)\(200 \leqslant m < 250\)\(250 \leqslant m < 350\)
Frequency162318355165
  1. Calculate estimates of the mean and standard deviation of these masses. The masses, \(x\) grams, of a random sample of 1000 pebbles on a different beach were also found. It was proposed that the distribution of these masses should be modelled by the random variable \(X \sim \mathrm {~N} ( 200,3600 )\).
  2. Use the model to find \(\mathrm { P } ( 150 < X < 210 )\).
  3. Use the model to determine \(x _ { 1 }\) such that \(\mathrm { P } \left( 160 < X < x _ { 1 } \right) = 0.6\), giving your answer correct to five significant figures. It was found that the smallest and largest masses of the pebbles in this second sample were 112 g and 288 g respectively.
  4. Use these results to show that the model may not be appropriate.
  5. Suggest a different value of a parameter of the model in the light of these results.
OCR H240/02 2020 November Q12
12 In the past, the time for Jeff's journey to work had mean 45.7 minutes and standard deviation 5.6 minutes. This year he is trying a new route. In order to test whether the new route has reduced his journey time, Jeff finds the mean time for a random sample of 30 journeys using the new route. He carries out a hypothesis test at the 2.5\% significance level. Jeff assumes that, for the new route, the journey time has a normal distribution with standard deviation 5.6 minutes.
  1. State appropriate null and alternative hypotheses for the test.
  2. Determine the rejection region for the test.
OCR H240/02 2020 November Q13
13 Andy and Bev are playing a game.
  • The game consists of three points.
  • On each point, \(\mathrm { P } (\) Andy wins \() = 0.4\) and \(\mathrm { P } (\) Bev wins \() = 0.6\).
  • If one player wins two consecutive points, then they win the game, otherwise neither player wins.
    1. Determine the probability of the following events.
      1. Andy wins the game.
      2. Neither player wins the game.
Andy and Bev now decide to play a match which consists of a series of games.
  • In each game, if a player wins the game then they win the match.
  • If neither player wins the game then the players play another game.
  • Determine the probability that Andy wins the match.
OCR H240/02 2020 November Q14
14 Table 1 shows the numbers of usual residents in the age range 0 to 4 in 15 Local Authorities (LAs) in 2001 and 2011. The table also shows the increase in the numbers in this age group, and the same increase as a percentage. \begin{table}[h]
20012011Increase\% Increase
Bolton1677918765198611.84\%
Bury1111712235111810.06\%
Knowsley94549121-333-3.52\%
Liverpool248402609912595.07\%
Manchester24693364131172047.46\%
Oldham151961649112958.52\%
Rochdale13771147549837.14\%
Salford1252916255372629.74\%
Sefton1489614601-295-1.98\%
St. Helens10083102691861.84\%
Stockport16457173428855.38\%
Tameside1280314439163612.78\%
Trafford1197114870289924.22\%
Wigan1756119681212012.07\%
Wirral174751851410395.95\%
\captionsetup{labelformat=empty} \caption{Table 1}
\end{table} Fig. 2 shows the increase in each LA in raw numbers, and Fig. 3 shows the percentage increase in each LA. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{4ba60d6b-c987-4f6b-8c55-8b594c90c854-10_792_1691_1838_187} \captionsetup{labelformat=empty} \caption{Fig. 2}
\end{figure} \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{4ba60d6b-c987-4f6b-8c55-8b594c90c854-11_707_1700_214_185} \captionsetup{labelformat=empty} \caption{Fig. 3}
\end{figure}
  1. The Education Committees in these LAs need to plan for the provision of schools for pupils in their districts.
    1. Explain why, in this context, the increase is more important than the actual numbers.
    2. In which of the following LAs was there likely to have been the greatest need for extra teachers in the years following 2011: Bolton, Sefton, Tameside or Wigan? Give a reason for your answer.
    3. State an assumption about the populations needed to make your answer in part (ii) valid.
  2. In two of the 15 LAs the proportion of young families is greater than in the other 13 LAs. Suggest, using only data from Fig. 2 and Fig. 3 and/or Table 1, which two LAs these are most likely to be.
OCR H240/02 2020 November Q15
15 In this question you must show detailed reasoning. The random variable \(X\) has probability distribution defined as follows.
\(\mathrm { P } ( X = x ) = \begin{cases} \frac { 15 } { 64 } \times \frac { 2 ^ { x } } { x ! } & x = 2,3,4,5 ,
0 & \text { otherwise. } \end{cases}\)
  1. Show that \(\mathrm { P } ( X = 2 ) = \frac { 15 } { 32 }\). The values of three independent observations of \(X\) are denoted by \(X _ { 1 } , X _ { 2 }\) and \(X _ { 3 }\).
  2. Given that \(X _ { 1 } + X _ { 2 } + X _ { 3 } = 9\), determine the probability that at least one of these three values is equal to 2 . Freda chooses values of \(X\) at random until she has obtained \(X = 2\) exactly three times. She then stops.
  3. Determine the probability that she chooses exactly 10 values of \(X\). \section*{END OF QUESTION PAPER} \section*{OCR} Oxford Cambridge and RSA
OCR H240/02 2022 June Q1
1 In this question you must show detailed reasoning. Solve the following equations.
  1. \(\frac { x } { x + 1 } - \frac { x - 1 } { x + 2 } = 0\)
  2. \(\frac { 8 } { x ^ { 6 } } - \frac { 7 } { x ^ { 3 } } - 1 = 0\)
  3. \(3 ^ { x ^ { 2 } - 7 } = \frac { 1 } { 243 }\)
OCR H240/02 2022 June Q2
2 The points \(A\) and \(B\) have position vectors \(3 \mathbf { i } + 2 \mathbf { j }\) and \(4 \mathbf { i } + 2 \mathbf { j } - 5 \mathbf { k }\) respectively.
  1. Find the length of \(A B\). Point \(P\) has position vector \(p \mathbf { i } - 3 \mathbf { k }\), where \(p\) is a constant. \(P\) lies on the circumference of a circle of which \(A B\) is a diameter.
  2. Find the two possible values of \(p\).
OCR H240/02 2022 June Q3
3
  1. Amaya and Ben integrated \(( 1 + x ) ^ { 2 }\), with respect to \(x\), using different methods, as follows. Amaya: \(\quad \int ( 1 + x ) ^ { 2 } \mathrm {~d} x = \frac { ( 1 + x ) ^ { 3 } } { 3 } + c \quad = \frac { 1 } { 3 } + x + x ^ { 2 } + \frac { 1 } { 3 } x ^ { 3 } + c\) Ben: \(\quad \int ( 1 + x ) ^ { 2 } \mathrm {~d} x = \int \left( 1 + 2 x + x ^ { 2 } \right) \mathrm { d } x = x + x ^ { 2 } + \frac { 1 } { 3 } x ^ { 3 } + c\) Charlie said that, because these answers are different, at least one of them must be wrong. Explain whether you agree with Charlie's statement.
  2. You are given that \(a\) is a constant greater than 1 .
    1. Find \(\int _ { 1 } ^ { a } \frac { 1 } { ( 1 + x ) ^ { 2 } } \mathrm {~d} x\), giving your answer as a single fraction in terms of the constant \(a\).
    2. You are given that the area enclosed by the curve \(y = \frac { 1 } { ( 1 + x ) ^ { 2 } }\), the \(x\)-axis and the lines \(x = 1\) and \(x = a\) is equal to \(\frac { 1 } { 3 }\). Determine the value of \(a\).
  3. In this question you must show detailed reasoning. Find the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 12 } \pi } \frac { \cos 2 x } { \sin 2 x + 2 } \mathrm {~d} x\), giving your answer in its simplest form.
OCR H240/02 2022 June Q4
4 An artist is creating a design for a large painting. The design includes a set of steps of varying heights. In the painting the lowest step has height 20 cm and the height of each other step is \(5 \%\) less than the height of the step immediately below it. In the painting the total height of the steps is 205 cm , correct to the nearest centimetre. Determine the number of steps in the design.
OCR H240/02 2022 June Q5
5 In this question you must show detailed reasoning. A curve has equation \(y = x ^ { 3 } - 3 x ^ { 2 } + 4 x\).
  1. Show that the curve has no stationary points.
  2. Show that the curve has exactly one point of inflection.
OCR H240/02 2022 June Q6
6
  1. The diagrams show five different graphs. In each case the whole of the graph is shown. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{cb83836f-753f-4b3a-99e8-a18aff0f49ff-06_376_382_310_306} \captionsetup{labelformat=empty} \caption{Fig. 1.1}
    \end{figure} \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{cb83836f-753f-4b3a-99e8-a18aff0f49ff-06_376_378_310_842} \captionsetup{labelformat=empty} \caption{Fig. 1.2}
    \end{figure} \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{cb83836f-753f-4b3a-99e8-a18aff0f49ff-06_378_378_310_1379} \captionsetup{labelformat=empty} \caption{Fig. 1.3}
    \end{figure} \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{cb83836f-753f-4b3a-99e8-a18aff0f49ff-06_378_382_872_306} \captionsetup{labelformat=empty} \caption{Fig. 1.4}
    \end{figure} \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{cb83836f-753f-4b3a-99e8-a18aff0f49ff-06_378_378_872_845} \captionsetup{labelformat=empty} \caption{Fig. 1.5}
    \end{figure} Place ticks in the boxes in the table in the Printed Answer Booklet to indicate, for each graph, whether it represents a one-one function, a many-one function, a function that is its own inverse or it does not represent a function. There may be more than one tick in some rows or columns of the table.
  2. A function f is defined by \(\mathrm { f } ( x ) = \frac { 1 } { x }\) for the domain \(\{ x : 0 < x \leqslant 2 \}\). State the range of f , giving your answer in set notation.
OCR H240/02 2022 June Q7
7 It is given that any integer can be expressed in the form \(3 m + r\), where \(m\) is an integer and \(r\) is 0,1 or 2 . Use this fact to answer the following.
  1. By considering the different values of \(r\), prove that the square of any integer cannot be expressed in the form \(3 n + 2\), where \(n\) is an integer.
  2. Three integers are chosen at random from the integers 1 to 99 inclusive. The three integers are not necessarily different. By considering the different values of \(r\), determine the probability that the sum of these three integers is divisible by 3 .