Questions — OCR MEI (4456 questions)

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OCR MEI C1 2007 January Q11
12 marks Moderate -0.3
11 There is an insert for use in this question. The graph of \(y = x + \frac { 1 } { x }\) is shown on the insert. The lowest point on one branch is \(( 1,2 )\). The highest point on the other branch is \(( - 1 , - 2 )\).
  1. Use the graph to solve the following equations, showing your method clearly. $$\text { (A) } x + \frac { 1 } { x } = 4$$ $$\text { (B) } 2 x + \frac { 1 } { x } = 4$$
  2. The equation \(( x - 1 ) ^ { 2 } + y ^ { 2 } = 4\) represents a circle. Find in exact form the coordinates of the points of intersection of this circle with the \(y\)-axis.
  3. State the radius and the coordinates of the centre of this circle. Explain how these can be used to deduce from the graph that this circle touches one branch of the curve \(y = x + \frac { 1 } { x }\) but does not intersect with the other.
OCR MEI C1 2009 January Q13
11 marks Moderate -0.3
13 Answer part (i) of this question on the insert provided. The insert shows the graph of \(y = \frac { 1 } { x }\).
  1. On the insert, on the same axes, plot the graph of \(y = x ^ { 2 } - 5 x + 5\) for \(0 \leqslant x \leqslant 5\).
  2. Show algebraically that the \(x\)-coordinates of the points of intersection of the curves \(y = \frac { 1 } { x }\) and \(y = x ^ { 2 } - 5 x + 5\) satisfy the equation \(x ^ { 3 } - 5 x ^ { 2 } + 5 x - 1 = 0\).
  3. Given that \(x = 1\) at one of the points of intersection of the curves, factorise \(x ^ { 3 } - 5 x ^ { 2 } + 5 x - 1\) into a linear and a quadratic factor. Show that only one of the three roots of \(x ^ { 3 } - 5 x ^ { 2 } + 5 x - 1 = 0\) is rational.
OCR MEI C1 Q4
12 marks Moderate -0.8
4 There is an insert for use in this question. The graph of \(y = x + \frac { 1 } { x }\) is shown on the insert. The lowest point on one branch is \(( 1,2 )\). The highest point on the other branch is \(( - 1 , - 2 )\).
  1. Use the graph to solve the following equations, showing your method clearly.
    (A) \(x + \frac { 1 } { x } = 4\) (B) \(2 x + \frac { 1 } { x } = 4\)
  2. The equation \(( x - 1 ) ^ { 2 } + y ^ { 2 } = 4\) represents a circle. Find in exact form the coordinates of the points of intersection of this circle with the \(y\)-axis.
  3. State the radius and the coordinates of the centre of this circle. Explain how these can be used to deduce from the graph that this circle touches one branch of the curve \(y = x + \frac { 1 } { x }\) but does not intersect with the other.
OCR MEI C2 2007 January Q13
12 marks Moderate -0.3
13 Answer part (ii) of this question on the insert provided. The table gives a firm's monthly profits for the first few months after the start of its business, rounded to the nearest \(\pounds 100\).
Number of months after start-up \(( x )\)123456
Profit for this month \(( \pounds y )\)5008001200190030004800
The firm's profits, \(\pounds y\), for the \(x\) th month after start-up are modelled by $$y = k \times 10 ^ { a x }$$ where \(a\) and \(k\) are constants.
  1. Show that, according to this model, a graph of \(\log _ { 10 } y\) against \(x\) gives a straight line of gradient \(a\) and intercept \(\log _ { 10 } k\).
  2. On the insert, complete the table and plot \(\log _ { 10 } y\) against \(x\), drawing by eye a line of best fit.
  3. Use your graph to find an equation for \(y\) in terms of \(x\) for this model.
  4. For which month after start-up does this model predict profits of about \(\pounds 75000\) ?
  5. State one way in which this model is unrealistic.
OCR MEI C2 2009 January Q5
4 marks Moderate -0.8
5 Answer this question on the insert provided. Fig. 5 shows the graph of \(y = \mathrm { f } ( x )\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{838d6b15-69a9-4e67-bc36-5bf60254a767-3_979_1077_422_536} \captionsetup{labelformat=empty} \caption{Fig. 5}
\end{figure} On the insert, draw the graph of
  1. \(y = \mathrm { f } ( x - 2 )\),
  2. \(y = 3 \mathrm { f } ( x )\).
OCR MEI C2 2009 January Q12
12 marks Moderate -0.3
12 Answer part (ii) of this question on the insert provided. The proposal for a major building project was accepted, but actual construction was delayed. Each year a new estimate of the cost was made. The table shows the estimated cost, \(\pounds y\) million, of the project \(t\) years after the project was first accepted.
Years after proposal accepted \(( t )\)12345
Cost \(( \pounds y\) million \()\)250300360440530
The relationship between \(y\) and \(t\) is modelled by \(y = a b ^ { t }\), where \(a\) and \(b\) are constants.
  1. Show that \(y = a b ^ { t }\) may be written as $$\log _ { 10 } y = \log _ { 10 } a + t \log _ { 10 } b$$
  2. On the insert, complete the table and plot \(\log _ { 10 } y\) against \(t\), drawing by eye a line of best fit.
  3. Use your graph and the results of part (i) to find the values of \(\log _ { 10 } a\) and \(\log _ { 10 } b\) and hence \(a\) and \(b\).
  4. According to this model, what was the estimated cost of the project when it was first accepted?
  5. Find the value of \(t\) given by this model when the estimated cost is \(\pounds 1000\) million. Give your answer rounded to 1 decimal place.
OCR MEI C3 Q9
18 marks Standard +0.3
9 Answer parts (i) and (iii) on the insert provided. Fig. 9 shows a sketch graph of \(y = \mathrm { f } ( x )\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{3f8be5ab-d241-4027-af26-c49da9394adc-4_401_799_488_593} \captionsetup{labelformat=empty} \caption{Fig. 9}
\end{figure}
  1. On the Insert sketch graphs of
    (A) \(y = 2 \mathrm { f } ( x )\),
    (B) \(y = \mathrm { f } ( - x )\),
    (C) \(y = \mathrm { f } ( x - 2 )\) In each case describe the transformations.
  2. Explain why the function \(y = \mathrm { f } ( x )\) does not have an inverse function.
  3. The function \(\mathrm { g } ( x )\) is defined as follows: $$\mathrm { g } ( x ) = \mathrm { f } ( x ) \text { for } x \geq 0$$ On the Insert sketch the graph of \(y = \mathrm { g } ^ { - 1 } ( x )\).
  4. You are given that \(\mathrm { f } ( x ) = x ^ { 2 } ( x + 2 )\). Calculate the gradient of the curve \(y = \mathrm { f } ( x )\) at the point \(( 1,3 )\).
    Deduce the gradient of the function \(\mathrm { g } ^ { - 1 } ( x )\) at the point where \(x = 3\).
  5. Show that \(\mathrm { g } ( x )\) and \(\mathrm { g } ^ { - 1 } ( x )\) cross where \(x = - 1 + \sqrt { 2 }\). \section*{Insert for question 9.}
  6. (A) On the axes below sketch the graph of \(y = 2 \mathrm { f } ( x )\). Describe the transformation. \includegraphics[max width=\textwidth, alt={}, center]{3f8be5ab-d241-4027-af26-c49da9394adc-5_563_1102_484_395} Description:
  7. (B) On the axes below sketch the graph of \(y = \mathrm { f } ( - x )\). Describe the transformation. \includegraphics[max width=\textwidth, alt={}, center]{3f8be5ab-d241-4027-af26-c49da9394adc-5_588_1154_1576_404} Description:
  8. (C) On the axes below sketch the graph of \(y = \mathrm { f } ( x - 2 )\). Describe the transformation. \includegraphics[max width=\textwidth, alt={}, center]{3f8be5ab-d241-4027-af26-c49da9394adc-6_615_1230_402_406} Description:
  9. The function \(\mathrm { g } ( x )\) is defined as follows: $$\mathrm { g } ( x ) = \mathrm { f } ( x ) \text { for } x \geq 0$$ On the axes below sketch the graph of \(y = g ^ { - 1 } ( x )\). \includegraphics[max width=\textwidth, alt={}, center]{3f8be5ab-d241-4027-af26-c49da9394adc-6_677_1356_1567_312}
OCR MEI C3 Q9
18 marks Moderate -0.3
9 Answer parts (ii) and (iii) of this question on the Insert provided. The bat population of a colony is being investigated and data are collected of the estimated number of bats in the colony at the beginning of each year. It is thought that the population may be modelled by the formula $$P = P _ { 0 } \mathrm { e } ^ { k t }$$ where \(P _ { 0 }\) and \(k\) are constants, \(P\) is the number of bats and \(t\) is the number of years after the start of the collection of data.
  1. Explain why a graph of \(\ln P\) against \(t\) should give a straight line. State the gradient and intercept of this line.
  2. The data collected are as follows.
    Time \(( t\) years \()\)01234
    Number of bats, \(P\)100170300340360
    Using the first three pairs of data in the table, plot \(\ln P\) against \(t\) on the axes given on the Insert, and hence estimate values for \(P _ { 0 }\) and \(k\).
    (Work to three significant figures.) This model assumes exponential growth, and assumes that once born a bat does not die, continuing to reproduce. This is unrealistic and so a second model is proposed with formula $$P = 150 \arctan ( t - 1 ) + 170$$ (You are reminded that arctan values should be given in radians.)
  3. Plot on a single graph on the Insert the curves \(P = P _ { 0 } \mathrm { e } ^ { k t }\) for your values of \(P _ { 0 }\) and \(k\) and \(P = 150 \arctan ( t - 1 ) + 170\). The data pairs in the table above have been plotted for you.
  4. Using the second model calculate an estimate of the number of years it is before the bat population exceeds 375. \section*{Insert for question 3.}
  1. Sketch the graph of \(y = 2 \mathrm { f } ( x )\) \includegraphics[max width=\textwidth, alt={}, center]{3853d1e7-ae1f-4eca-93c7-96f03b6d31c3-6_641_1431_541_354}
  2. Sketch the graph of \(y = \mathrm { f } ( 2 x )\). \includegraphics[max width=\textwidth, alt={}, center]{3853d1e7-ae1f-4eca-93c7-96f03b6d31c3-6_691_1539_1468_374} \section*{Insert for question 9.} (ii) Plot \(\ln P\) against \(t\). \includegraphics[max width=\textwidth, alt={}, center]{3853d1e7-ae1f-4eca-93c7-96f03b6d31c3-7_704_1442_443_338}
    (iii) Plot the curves \(P = P _ { 0 } \mathrm { e } ^ { k t }\) and \(P = 150 \arctan ( t - 1 ) + 170\) for your values of \(P _ { 0 }\) and \(k\). The data pairs are plotted on the graph. \includegraphics[max width=\textwidth, alt={}, center]{3853d1e7-ae1f-4eca-93c7-96f03b6d31c3-7_780_1399_1546_333}
OCR MEI C4 Q6
4 marks Moderate -0.8
6 Use the Insert provided for this question. The graph of \(y = \tan x\) is given on the Insert.
On this graph sketch the graph of \(y = \operatorname { cotx }\).
Show clearly where your graph crosses the graph of \(y = \tan x\) and indicate the asymptotes.
OCR MEI D1 2006 January Q5
16 marks Moderate -0.8
5 Answer this question on the insert provided. Table 5 specifies a road network connecting 7 towns, A, B, \(\ldots\), G. The entries in Table 5 give the distances in miles between towns which are connected directly by roads. \begin{table}[h]
ABCDEFG
A-10---1215
B10-1520--8
C-15-7--11
D-207-20-13
E---20-179
F12---17-13
G1581113913-
\captionsetup{labelformat=empty} \caption{Table 5}
\end{table}
  1. Using the copy of Table 5 in the insert, apply the tabular form of Prim's algorithm to the network, starting at vertex A. Show the order in which you connect the vertices. Draw the resulting tree, give its total length and describe a practical application.
  2. The network in the insert shows the information in Table 5. Apply Dijkstra's algorithm to find the shortest route from A to E. Give your route and its length.
  3. A tunnel is built through a hill between A and B , shortening the distance between A and B to 6 miles. How does this affect your answers to parts (i) and (ii)?
OCR MEI D1 2006 January Q6
16 marks Easy -1.2
6 Answer part (iv) of this question on the insert provided. There are two types of customer who use the shop at a service station. \(70 \%\) buy fuel, the other \(30 \%\) do not. There is only one till in operation.
  1. Give an efficient rule for using one-digit random numbers to simulate the type of customer arriving at the service station. Table 6.1 shows the distribution of time taken at the till by customers who are buying fuel.
    Time taken (mins)11.522.5
    Probability\(\frac { 3 } { 10 }\)\(\frac { 2 } { 5 }\)\(\frac { 1 } { 5 }\)\(\frac { 1 } { 10 }\)
    \section*{Table 6.1}
  2. Specify an efficient rule for using one-digit random numbers to simulate the time taken at the till by customers purchasing fuel. Table 6.2 shows the distribution of time taken at the till by customers who are not buying fuel.
    Time taken (mins)11.522.53
    Probability\(\frac { 1 } { 7 }\)\(\frac { 2 } { 7 }\)\(\frac { 2 } { 7 }\)\(\frac { 1 } { 7 }\)\(\frac { 1 } { 7 }\)
    \section*{Table 6.2}
  3. Specify an efficient rule for using two-digit random numbers to simulate the time taken at the till by customers not buying fuel. What is the advantage in using two-digit random numbers instead of one-digit random numbers in this part of the question? The table in the insert shows a partially completed simulation study of 10 customers arriving at the till.
  4. Complete the table using the random numbers which are provided.
  5. Calculate the mean total time spent queuing and paying.
OCR MEI S1 2005 June Q6
15 marks Standard +0.3
6 Answer part (i) of this question on the insert provided. Mancaster Hockey Club invite prospective new players to take part in a series of three trial games. At the end of each game the performance of each player is assessed as pass or fail. Players who achieve a pass in all three games are invited to join the first team squad. Players who achieve a pass in two games are invited to join the second team squad. Players who fail in two games are asked to leave. This may happen after two games.
  • The probability of passing the first game is 0.9
  • Players who pass any game have probability 0.9 of passing the next game
  • Players who fail any game have probability 0.5 of failing the next game
    1. On the insert, complete the tree diagram which illustrates the information above. \includegraphics[max width=\textwidth, alt={}, center]{668963b4-994d-475a-a1c8-c3e3a252e4e6-4_691_1329_978_397}
    2. Find the probability that a randomly selected player
      (A) is invited to join the first team squad,
      (B) is invited to join the second team squad.
    3. Hence write down the probability that a randomly selected player is asked to leave.
    4. Find the probability that a randomly selected player is asked to leave after two games, given that the player is asked to leave.
Angela, Bryony and Shareen attend the trials at the same time. Assuming their performances are independent, find the probability that
  • at least one of the three is asked to leave,
  • they pass a total of 7 games between them.
    •
    OCR MEI S1 Q3
    8 marks Easy -1.3
    3 Answer part (i) of this question on the insert provided. A taxi driver operates from a taxi rank at a main railway station in London. During one particular week he makes 120 journeys, the lengths of which are summarised in the table.
    Length
    \(( x\) miles \()\)
    		\(0 < x \leqslant 1\)\(1 < x \leqslant 2\)\(2 < x \leqslant 3\)\(3 < x \leqslant 4\)\(4 < x \leqslant 6\)\(6 < x \leqslant 10\)
    Number of
    journeys
    		3830211498
    1. On the insert, draw a cumulative frequency diagram to illustrate the data.
    2. Use your graph to estimate the median length of journey and the quartiles. Hence find the interquartile range.
    3. State the type of skewness of the distribution of the data.
    OCR MEI S1 Q4
    18 marks Standard +0.8
    4 Answer part (i) of this question on the insert provided. Mancaster Hockey Club invite prospective new players to take part in a series of three trial games. At the end of each game the performance of each player is assessed as pass or fail. Players who achieve a pass in all three games are invited to join the first team squad. Players who achieve a pass in two games are invited to join the second team squad. Players who fail in two games are asked to leave. This may happen after two games.
    • The probability of passing the first game is 0.9
    • Players who pass any game have probability 0.9 of passing the next game
    • Players who fail any game have probability 0.5 of failing the next game
      1. On the insert, complete the tree diagram which illustrates the information above. \includegraphics[max width=\textwidth, alt={}, center]{64f25a40-d3bf-4212-b92e-655f980c702b-4_643_1239_942_417}
      2. Find the probability that a randomly selected player
        (A) is invited to join the first team squad,
        (B) is invited to join the second team squad.
      3. Hence write down the probability that a randomly selected player is asked to leave.
      4. Find the probability that a randomly selected player is asked to leave after two games, given that the player is asked to leave.
    Angela, Bryony and Shareen attend the trials at the same time. Assuming their performances are independent, find the probability that
  • at least one of the three is asked to leave,
  • they pass a total of 7 games between them.
    •
    OCR MEI S1 Q4
    9 marks Easy -1.8
    4 Answer part (i) of this question on the insert provided. A taxi driver operates from a taxi rank at a main railway station in London. During one particular week he makes 120 journeys, the lengths of which are summarised in the table.
    Length
    \(( x\) miles \()\)
    		\(0 < x \leqslant 1\)\(1 < x \leqslant 2\)\(2 < x \leqslant 3\)\(3 < x \leqslant 4\)\(4 < x \leqslant 6\)\(6 < x \leqslant 10\)
    Number of
    journeys
    		3830211498
    1. On the insert, draw a cumulative frequency diagram to illustrate the data.
    2. Use your graph to estimate the median length of journey and the quartiles. Hence find the interquartile range.
    3. State the type of skewness of the distribution of the data.
    OCR MEI AS Paper 1 2018 June Q8
    8 marks Moderate -0.8
    8 In this question you must show detailed reasoning. Fig. 8 shows the graph of a quadratic function. The graph crosses the axes at the points \(( - 1,0 ) , ( 0 , - 4 )\) and \(( 2,0 )\). \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{1513048a-d53b-4b85-82f4-c86e0d81f8f8-4_689_606_1114_731} \captionsetup{labelformat=empty} \caption{Fig. 8}
    \end{figure} Find the area of the finite region bounded by the curve and the \(x\)-axis.
    OCR MEI AS Paper 1 2019 June Q1
    3 marks Easy -1.2
    1 In this question you must show detailed reasoning. Show that the equation \(x = 7 + 2 x ^ { 2 }\) has no real roots.
    OCR MEI AS Paper 1 2019 June Q2
    3 marks Standard +0.3
    2 In this question you must show detailed reasoning. Fig. 2 shows the graphs of \(y = 4 \sin x ^ { \circ }\) and \(y = 3 \cos x ^ { \circ }\) for \(0 \leqslant x \leqslant 360\). \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{0b1c272a-f0f4-4931-be89-5d045804a7af-3_549_768_813_258} \captionsetup{labelformat=empty} \caption{Fig. 2}
    \end{figure} Find the \(x\)-coordinates of the two points of intersection, giving your answers correct to 1 decimal place.
    OCR MEI AS Paper 1 2019 June Q9
    9 marks Moderate -0.3
    9 In this question you must show detailed reasoning. A car accelerates from rest along a straight level road. The velocity of the car after 8 s is \(25.6 \mathrm {~ms} ^ { - 1 }\).
    In one model for the motion, the velocity \(v \mathrm {~ms} ^ { - 1 }\) at time \(t\) seconds is given by \(v = 1.2 t ^ { 2 } - k t ^ { 3 }\), where \(k\) is a constant and \(0 \leqslant t \leqslant 8\).
    1. The model gives the correct velocity of \(25.6 \mathrm {~ms} ^ { - 1 }\) at time 8 s . Show that \(k = 0.1\). A second model for the motion uses constant acceleration.
    2. Find the value of the acceleration which gives the correct velocity of \(25.6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at time 8 s .
    3. Show that these two models give the same value for the displacement in the first 8 s .
    OCR MEI AS Paper 1 2020 November Q12
    12 marks Standard +0.3
    12 In this question you must show detailed reasoning. Fig. 12 shows part of the graph of \(y = x ^ { 2 } + \frac { 1 } { x ^ { 2 } }\). \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{a1b6c827-7d74-4527-9b60-58872e3d5ef7-7_574_574_402_233} \captionsetup{labelformat=empty} \caption{Fig. 12}
    \end{figure} The tangent to the curve \(\mathrm { y } = \mathrm { x } ^ { 2 } + \frac { 1 } { \mathrm { x } ^ { 2 } }\) at the point \(\left( 2 , \frac { 17 } { 4 } \right)\) meets the \(x\)-axis at A and meets the \(y\)-axis at B . O is the origin.
    1. Find the exact area of the triangle OAB .
    2. Use calculus to prove that the complete curve has two minimum points and no maximum point. \section*{END OF QUESTION PAPER}
    OCR MEI AS Paper 1 2021 November Q8
    12 marks Moderate -0.8
    8 In this question you must show detailed reasoning.
    1. Use differentiation to find the coordinates of the stationary point on the curve with equation \(y = 2 x ^ { 2 } - 3 x - 2\).
    2. Use the second derivative to determine the nature of the stationary point.
    3. Show by shading on a sketch the region defined by the inequality \(y \geqslant 2 x ^ { 2 } - 3 x - 2\), indicating clearly whether the boundary is included or not.
    4. Solve the inequality \(2 x ^ { 2 } - 3 x - 2 > 0\) using set notation for your answer.
    OCR MEI AS Paper 2 2019 June Q10
    10 marks Standard +0.8
    10 In this question you must show detailed reasoning. The equation of a curve is \(y = \frac { x ^ { 2 } } { 4 } + \frac { 2 } { x } + 1\). A tangent and a normal to the curve are drawn at the point where \(x = 2\). Calculate the area bounded by the tangent, the normal and the \(x\)-axis. \section*{END OF QUESTION PAPER}
    OCR MEI AS Paper 2 2023 June Q14
    7 marks Moderate -0.8
    14 In this question you must show detailed reasoning. The equation of a curve is \(y = 16 \sqrt { x } + \frac { 8 } { x }\).
    Determine the equation of the tangent to the curve at the point where \(x = 4\).
    OCR MEI AS Paper 2 2024 June Q8
    4 marks Moderate -0.3
    8 In this question you must show detailed reasoning. Determine the coordinates of the point of intersection of the line with equation \(y = 2 x + 3\) and the curve with equation \(y ^ { 2 } - 4 x ^ { 2 } = 33\).
    OCR MEI AS Paper 2 2020 November Q7
    8 marks Moderate -0.3
    7 In this question you must show detailed reasoning. A circle has centre \(( 2 , - 1 )\) and radius 5. A straight line passes through the points \(( 1,1 )\) and \(( 9,5 )\).
    Find the coordinates of the points of intersection of the line and the circle.