Questions — OCR MEI (4456 questions)

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OCR MEI C4 2006 June Q5
11 marks Standard +0.3
5
  1. Show that \(\int x \mathrm { e } ^ { - 2 x } \mathrm {~d} x = - \frac { 1 } { 4 } \mathrm { e } ^ { - 2 x } ( 1 + 2 x ) + c\). A vase is made in the shape of the volume of revolution of the curve \(y = x ^ { 1 / 2 } \mathrm { e } ^ { - x }\) about the \(x\)-axis between \(x = 0\) and \(x = 2\) (see Fig. 5). \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{c64062c4-4cbd-41b2-9b4d-60a43dceb700-3_716_741_1233_662} \captionsetup{labelformat=empty} \caption{Fig. 5}
    \end{figure}
  2. Show that this volume of revolution is \(\frac { 1 } { 4 } \pi \left( 1 - \frac { 5 } { \mathrm { e } ^ { 4 } } \right)\). Fig. 6 shows the arch ABCD of a bridge. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{c64062c4-4cbd-41b2-9b4d-60a43dceb700-4_378_1630_461_214} \captionsetup{labelformat=empty} \caption{Fig. 6}
    \end{figure} The section from B to C is part of the curve OBCE with parametric equations $$x = a ( \theta - \sin \theta ) , y = a ( 1 - \cos \theta ) \text { for } 0 \leqslant \theta \leqslant 2 \pi$$ where \(a\) is a constant.
  3. Find, in terms of \(a\),
    (A) the length of the straight line OE,
    (B) the maximum height of the arch.
  4. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(\theta\). The straight line sections AB and CD are inclined at \(30 ^ { \circ }\) to the horizontal, and are tangents to the curve at B and C respectively. BC is parallel to the \(x\)-axis. BF is parallel to the \(y\)-axis.
  5. Show that at the point B the parameter \(\theta\) satisfies the equation $$\sin \theta = \frac { 1 } { \sqrt { 3 } } ( 1 - \cos \theta )$$ Verify that \(\theta = \frac { 2 } { 3 } \pi\) is a solution of this equation.
    Hence show that \(\mathrm { BF } = \frac { 3 } { 2 } a\), and find OF in terms of \(a\), giving your answer exactly.
  6. Find BC and AF in terms of \(a\). Given that the straight line distance AD is 20 metres, calculate the value of \(a\). \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{c64062c4-4cbd-41b2-9b4d-60a43dceb700-5_748_1306_319_367} \captionsetup{labelformat=empty} \caption{Fig. 7}
    \end{figure} Fig. 7 illustrates a house. All units are in metres. The coordinates of A, B, C and E are as shown. BD is horizontal and parallel to AE .
  7. Find the length AE .
  8. Find a vector equation of the line BD . Given that the length of BD is 15 metres, find the coordinates of D.
  9. Verify that the equation of the plane ABC is $$- 3 x + 4 y + 5 z = 30$$ Write down a vector normal to this plane.
  10. Show that the vector \(\left( \begin{array} { l } 4 \\ 3 \\ 5 \end{array} \right)\) is normal to the plane ABDE . Hence find the equation of the plane ABDE .
  11. Find the angle between the planes ABC and ABDE . RECOGNISING ACHIEVEMENT \section*{OXFORD CAMBRIDGE AND RSA EXAMINATIONS} Advanced Subsidiary General Certificate of Education Advanced General Certificate of Education \section*{MEI STRUCTURED MATHEMATICS} Applications of Advanced Mathematics (C4) \section*{Paper B: Comprehension} Monday 12 JUNE 2006 Afternoon Up to 1 hour Additional materials:
    Rough paper
    MEI Examination Formulae and Tables (MF2) TIME Up to 1 hour
    For Examiner's Use
    Qu.Mark
    1
    2
    3
    4
    5
    6
    Total
    1 The marathon is 26 miles and 385 yards long ( 1 mile is 1760 yards). There are now several men who can run 2 miles in 8 minutes. Imagine that an athlete maintains this average speed for a whole marathon. How long does the athlete take?
    2 According to the linear model, in which calendar year would the record for the men's mile first become negative?
    3 Explain the statement in line 93 "According to this model the 2-hour marathon will never be run."
    4 Explain how the equation in line 49, $$R = L + ( U - L ) \mathrm { e } ^ { - k t } ,$$ is consistent with Fig. 2
  12. initially,
  13. for large values of \(t\).
  15. \(\_\_\_\_\) 5 A model for an athletics record has the form $$R = A - ( A - B ) \mathrm { e } ^ { - k t } \text { where } A > B > 0 \text { and } k > 0 .$$
  16. Sketch the graph of \(R\) against \(t\), showing \(A\) and \(B\) on your graph.
  17. Name one event for which this might be an appropriate model.
  18. \includegraphics[max width=\textwidth, alt={}, center]{c64062c4-4cbd-41b2-9b4d-60a43dceb700-9_803_808_721_575}
  19. \(\_\_\_\_\)
OCR MEI C4 2006 June Q6
21 marks Moderate -0.8
6 A number of cases of the general exponential model for the marathon are given in Table 6. One of these is $$R = 115 + ( 175 - 115 ) \mathrm { e } ^ { - 0.0467 t ^ { 0.797 } }$$
  1. What is the value of \(t\) for the year 2012?
  2. What record time does this model predict for the year 2012?
  1. \(\_\_\_\_\)
  2. \(\_\_\_\_\)
OCR MEI C4 2008 June Q1
3 marks Easy -1.2
1 Express \(\frac { x } { x ^ { 2 } - 4 } + \frac { 2 } { x + 2 }\) as a single fraction, simplifying your answer.
OCR MEI C4 2008 June Q2
4 marks Standard +0.3
2 Fig. 2 shows the curve \(y = \sqrt { 1 + \mathrm { e } ^ { 2 x } }\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{8ad99e2a-4cef-40b3-af8d-673b97536227-02_432_873_587_635} \captionsetup{labelformat=empty} \caption{Fig. 2}
\end{figure} The region bounded by the curve, the \(x\)-axis, the \(y\)-axis and the line \(x = 1\) is rotated through \(360 ^ { \circ }\) about the \(x\)-axis. Show that the volume of the solid of revolution produced is \(\frac { 1 } { 2 } \pi \left( 1 + \mathrm { e } ^ { 2 } \right)\).
OCR MEI C4 2008 June Q3
7 marks Moderate -0.3
3 Solve the equation \(\cos 2 \theta = \sin \theta\) for \(0 \leqslant \theta \leqslant 2 \pi\), giving your answers in terms of \(\pi\).
OCR MEI C4 2008 June Q4
3 marks Moderate -0.8
4 Given that \(x = 2 \sec \theta\) and \(y = 3 \tan \theta\), show that \(\frac { x ^ { 2 } } { 4 } - \frac { y ^ { 2 } } { 9 } = 1\).
OCR MEI C4 2008 June Q5
5 marks Moderate -0.8
5 A curve has parametric equations \(x = 1 + u ^ { 2 } , y = 2 u ^ { 3 }\).
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(u\).
  2. Hence find the gradient of the curve at the point with coordinates \(( 5,16 )\).
OCR MEI C4 2008 June Q6
8 marks Standard +0.3
6
  1. Find the first three non-zero terms of the binomial series expansion of \(\frac { 1 } { \sqrt { 1 + 4 x ^ { 2 } } }\), and state the set of values of \(x\) for which the expansion is valid.
  2. Hence find the first three non-zero terms of the series expansion of \(\frac { 1 - x ^ { 2 } } { \sqrt { 1 + 4 x ^ { 2 } } }\).
OCR MEI C4 2008 June Q7
6 marks Standard +0.3
7 Express \(\sqrt { 3 } \sin x - \cos x\) in the form \(R \sin ( x - \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { 1 } { 2 } \pi\). Express \(\alpha\) in the form \(k \pi\). Find the exact coordinates of the maximum point of the curve \(y = \sqrt { 3 } \sin x - \cos x\) for which \(0 < x < 2 \pi\).
OCR MEI C4 2008 June Q8
18 marks Standard +0.3
8 The upper and lower surfaces of a coal seam are modelled as planes ABC and DEF, as shown in Fig. 8. All dimensions are metres. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{8ad99e2a-4cef-40b3-af8d-673b97536227-03_1004_1397_493_374} \captionsetup{labelformat=empty} \caption{Fig. 8}
\end{figure} Relative to axes \(\mathrm { O } x\) (due east), \(\mathrm { O } y\) (due north) and \(\mathrm { O } z\) (vertically upwards), the coordinates of the points are as follows.
A: (0, 0, -15)
B: (100, 0, -30)
C: (0, 100, -25)
D: (0, 0, -40)
E: (100, 0, -50)
F: (0, 100, -35)
  1. Verify that the cartesian equation of the plane ABC is \(3 x + 2 y + 20 z + 300 = 0\).
  2. Find the vectors \(\overrightarrow { \mathrm { DE } }\) and \(\overrightarrow { \mathrm { DF } }\). Show that the vector \(2 \mathbf { i } - \mathbf { j } + 20 \mathbf { k }\) is perpendicular to each of these vectors. Hence find the cartesian equation of the plane DEF .
  3. By calculating the angle between their normal vectors, find the angle between the planes ABC and DEF. It is decided to drill down to the seam from a point \(\mathrm { R } ( 15,34,0 )\) in a line perpendicular to the upper surface of the seam. This line meets the plane ABC at the point S .
  4. Write down a vector equation of the line RS. Calculate the coordinates of S.
OCR MEI C4 2008 June Q9
18 marks Standard +0.3
9 A skydiver drops from a helicopter. Before she opens her parachute, her speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) after time \(t\) seconds is modelled by the differential equation $$\frac { \mathrm { d } v } { \mathrm {~d} t } = 10 \mathrm { e } ^ { - \frac { 1 } { 2 } t }$$ When \(t = 0 , v = 0\).
  1. Find \(v\) in terms of \(t\).
  2. According to this model, what is the speed of the skydiver in the long term? She opens her parachute when her speed is \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Her speed \(t\) seconds after this is \(w \mathrm {~m} \mathrm {~s} ^ { - 1 }\), and is modelled by the differential equation $$\frac { \mathrm { d } w } { \mathrm {~d} t } = - \frac { 1 } { 2 } ( w - 4 ) ( w + 5 )$$
  3. Express \(\frac { 1 } { ( w - 4 ) ( w + 5 ) }\) in partial fractions.
  4. Using this result, show that \(\frac { w - 4 } { w + 5 } = 0.4 \mathrm { e } ^ { - 4.5 t }\).
  5. According to this model, what is the speed of the skydiver in the long term? RECOGNISING ACHIEVEMENT \section*{ADVANCED GCE} \section*{4754/01B} \section*{MATHEMATICS (MEI)} Applications of Advanced Mathematics (C4) Paper B: Comprehension
    WEDNESDAY 21 MAY 2008
    Afternoon
    Time: Up to 1 hour
    Additional materials: Rough paper
    MEI Examination Formulae and Tables (MF 2) \section*{Candidate Forename}
    \includegraphics[max width=\textwidth, alt={}]{8ad99e2a-4cef-40b3-af8d-673b97536227-05_125_547_986_516}
    This document consists of \(\mathbf { 6 }\) printed pages, \(\mathbf { 2 }\) blank pages and an insert. 1 Complete these Latin square puzzles.
  1. 213
    3
  2. \includegraphics[max width=\textwidth, alt={}, center]{8ad99e2a-4cef-40b3-af8d-673b97536227-06_391_419_836_854} 2 In line 51, the text says that the Latin square
    1234
    3142
    2413
    4321
    could not be the solution to a Sudoku puzzle.
    Explain this briefly.
    3 On lines 114 and 115 the text says "It turns out that there are 16 different ways of filling in the remaining cells while keeping to the Sudoku rules. One of these ways is shown in Fig.10." Complete the grid below with a solution different from that given in Fig. 10.
    1234
    4 Lines 154 and 155 of the article read "There are three other embedded Latin squares in Fig. 14; one of them is illustrated in Fig. 16." Indicate one of the other two embedded Latin squares on this copy of Fig. 14.
    4231
    24
    42
    2413
    5 The number of \(9 \times 9\) Sudokus is given in line 121 .
    Without doing any calculations, explain why you would expect 9! to be a factor of this number.
    6 In the table below, \(M\) represents the maximum number of givens for which a Sudoku puzzle may have no unique solution (Investigation 3 in the article). \(s\) is the side length of the Sudoku grid and \(b\) is the side length of its blocks.
    Block side
    length, \(b\)
    Sudoku,
    \(s \times s\)
    		\(M\)
    1\(1 \times 1\)-
    2\(4 \times 4\)12
    3\(9 \times 9\)
    4\(16 \times 16\)
    5
  1. Complete the table.
  2. Give a formula for \(M\) in terms of \(b\).
    7 A man is setting a Sudoku puzzle and starts with this solution.
    123456789
    456897312
    789312564
    231564897
    564978123
    897123645
    312645978
    645789231
    978231456
    He then removes some of the numbers to give the puzzles in parts (i) and (ii). In each case explain briefly, and without trying to solve the puzzle, why it does not have a unique solution.
    [0pt] [2,2]
  1. 12469
    4891
    86
    2147
    647812
    8924
    16497
    64791
    982146
  2. 123456789
    456897312
    789564
    231564897
    564978123
    897645
    312645978
    645789231
    978456
  1. \(\_\_\_\_\)
  2. \(\_\_\_\_\)
OCR MEI S3 Q2
20 marks Standard +0.3
2 Geoffrey is a university lecturer. He has to prepare five questions for an examination. He knows by experience that it takes about 3 hours to prepare a question, and he models the time (in minutes) taken to prepare one by the Normally distributed random variable \(X\) with mean 180 and standard deviation 12, independently for all questions.
  1. One morning, Geoffrey has a gap of 2 hours 50 minutes ( 170 minutes) between other activities. Find the probability that he can prepare a question in this time.
  2. One weekend, Geoffrey can devote 14 hours to preparing the complete examination paper. Find the probability that he can prepare all five questions in this time. A colleague, Helen, has to check the questions.
  3. She models the time (in minutes) to check a question by the Normally distributed random variable \(Y\) with mean 50 and standard deviation 6, independently for all questions and independently of \(X\). Find the probability that the total time for Geoffrey to prepare a question and Helen to check it exceeds 4 hours.
  4. When working under pressure of deadlines, Helen models the time to check a question in a different way. She uses the Normally distributed random variable \(\frac { 1 } { 4 } X\), where \(X\) is as above. Find the length of time, as given by this model, which Helen needs to ensure that, with probability 0.9 , she has time to check a question. Ian, an educational researcher, suggests that a better model for the time taken to prepare a question would be a constant \(k\) representing "thinking time" plus a random variable \(T\) representing the time required to write the question itself, independently for all questions.
  5. Taking \(k\) as 45 and \(T\) as Normally distributed with mean 120 and standard deviation 10 (all units are minutes), find the probability according to Ian's model that a question can be prepared in less than 2 hours 30 minutes. Juliet, an administrator, proposes that the examination should be reduced in time and shorter questions should be used.
  6. Juliet suggests that Ian's model should be used for the time taken to prepare such shorter questions but with \(k = 30\) and \(T\) replaced by \(\frac { 3 } { 5 } T\). Find the probability as given by this model that a question can be prepared in less than \(1 \frac { 3 } { 4 }\) hours.
OCR MEI FP3 2015 June Q1
24 marks Standard +0.8
1 The point A has coordinates \(( 2,5,4 )\) and the line BC has equation $$\mathbf { r } = \left( \begin{array} { c } 8 \\ 25 \\ 43 \end{array} \right) + \lambda \left( \begin{array} { c } 4 \\ 15 \\ 25 \end{array} \right)$$ You are given that \(\mathrm { AB } = \mathrm { AC } = 15\).
  1. Show that the coordinates of one of the points B and C are (4, 10, 18). Find the coordinates of the other point. These points are B and C respectively.
  2. Find the equation of the plane ABC in cartesian form.
  3. Show that the plane containing the line BC and perpendicular to the plane ABC has equation \(5 y - 3 z + 4 = 0\). The point D has coordinates \(( 1,1,3 )\).
  4. Show that \(| \overrightarrow { B C } \times \overrightarrow { A D } | = \sqrt { 7667 }\) and hence find the shortest distance between the lines \(B C\) and \(A D\).
  5. Find the volume of the tetrahedron ABCD .
OCR MEI FP3 2015 June Q2
24 marks Challenging +1.2
2 A surface has equation \(z = 3 x ^ { 2 } - 12 x y + 2 y ^ { 3 } + 60\).
  1. Show that the point \(\mathrm { A } ( 8,4 , - 4 )\) is a stationary point on the surface. Find the coordinates of the other stationary point, B , on this surface.
  2. A point P with coordinates \(( 8 + h , 4 + k , p )\) lies on the surface.
    (A) Show that \(p = - 4 + 3 ( h - 2 k ) ^ { 2 } + 2 k ^ { 2 } ( 6 + k )\).
    (B) Deduce that the stationary point A is a local minimum.
    (C) By considering sections of the surface near to B in each of the planes \(x = 0\) and \(y = 0\), investigate the nature of the stationary point B .
  3. The point Q with coordinates \(( 1,1,53 )\) lies on the surface. Show that the equation of the tangent plane at Q is $$6 x + 6 y + z = 65$$
  4. The tangent plane at the point R has equation \(6 x + 6 y + z = \lambda\) where \(\lambda \neq 65\). Find the coordinates of R .
OCR MEI FP3 2015 June Q3
24 marks Challenging +1.8
3 Fig. 3 shows an ellipse with parametric equations \(x = a \cos \theta , y = b \sin \theta\), for \(0 \leqslant \theta \leqslant 2 \pi\), where \(0 < b \leqslant a\).
The curve meets the positive \(x\)-axis at A and the positive \(y\)-axis at B . \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{0e032f23-0549-4adc-bfae-59333108fab5-4_668_1255_477_404} \captionsetup{labelformat=empty} \caption{Fig. 3}
\end{figure}
  1. Show that the radius of curvature at A is \(\frac { b ^ { 2 } } { a }\) and find the corresponding centre of curvature.
  2. Write down the radius of curvature and the centre of curvature at B .
  3. Find the relationship between \(a\) and \(b\) if the radius of curvature at B is equal to the radius of curvature at A . What does this mean geometrically?
  4. Show that the arc length from A to B can be expressed as $$b \int _ { 0 } ^ { \frac { \pi } { 2 } } \sqrt { 1 + \lambda ^ { 2 } \sin ^ { 2 } \theta } d \theta$$ where \(\lambda ^ { 2 }\) is to be determined in terms of \(a\) and \(b\).
    Evaluate this integral in the case \(a = b\) and comment on your answer.
  5. Find the cartesian equation of the evolute of the ellipse.
OCR MEI FP3 2015 June Q4
24 marks Challenging +1.8
4 M is the set of all \(2 \times 2\) matrices \(\mathrm { m } ( a , b )\) where \(a\) and \(b\) are rational numbers and $$\mathrm { m } ( a , b ) = \left( \begin{array} { l l } a & b \\ 0 & \frac { 1 } { a } \end{array} \right) , a \neq 0$$
  1. Show that under matrix multiplication M is a group. You may assume associativity of matrix multiplication.
  2. Determine whether the group is commutative. The set \(\mathrm { N } _ { k }\) consists of all \(2 \times 2\) matrices \(\mathrm { m } ( k , b )\) where \(k\) is a fixed positive integer and \(b\) can take any integer value.
  3. Prove that \(\mathrm { N } _ { k }\) is closed under matrix multiplication if and only if \(k = 1\). Now consider the set P consisting of the matrices \(\mathrm { m } ( 1,0 ) , \mathrm { m } ( 1,1 ) , \mathrm { m } ( 1,2 )\) and \(\mathrm { m } ( 1,3 )\). The elements of P are combined using matrix multiplication but with arithmetic carried out modulo 4 .
  4. Show that \(( \mathrm { m } ( 1,1 ) ) ^ { 2 } = \mathrm { m } ( 1,2 )\).
  5. Construct the group combination table for P . The group R consists of the set \(\{ e , a , b , c \}\) combined under the operation *. The identity element is \(e\), and elements \(a , b\) and \(c\) are such that $$a ^ { * } a = b ^ { * } b = c ^ { * } c \quad \text { and } \quad a ^ { * } c = c ^ { * } a = b$$
  6. Determine whether R is isomorphic to P . Option 5: Markov chains \section*{This question requires the use of a calculator with the ability to handle matrices.}
OCR MEI FP3 2015 June Q5
24 marks Standard +0.8
5 An inspector has three factories, A, B, C, to check. He spends each day in one of the factories. He chooses the factory to visit on a particular day according to the following rules.
  • If he is in A one day, then the next day he will never choose A but he is equally likely to choose B or C .
  • If he is in B one day, then the next day he is equally likely to choose \(\mathrm { A } , \mathrm { B }\) or C .
  • If he is in C one day, then the next day he will never choose A but he is equally likely to choose B or C .
    1. Write down the transition matrix, \(\mathbf { P }\).
    2. On Day 1 the inspector chooses A.
      (A) Find the probability that he will choose A on Day 4.
      (B) Find the probability that the factory he chooses on Day 7 is the same factory that he chose on Day 2.
    3. Find the equilibrium probabilities and explain what they mean.
The inspector is not satisfied with the number of times he visits A so he changes the rules as follows.
Still not satisfied, the inspector changes the rules as follows.
The new transition matrix is \(\mathbf { R }\).
  • On Day 15 he visits C . Find the first subsequent day for which the probability that he visits B is less than 0.1.
  • Show that in this situation there is an absorbing state, explaining what this means. \section*{END OF QUESTION PAPER}
    •
    OCR MEI D1 2013 June Q1
    8 marks Moderate -0.8
    1 The adjacency graph for a map has a vertex for each country. Two vertices are connected by an arc if the corresponding countries share a border.
    1. Draw the adjacency graph for the following map of four countries. The graph is planar and you should draw it with no arcs crossing. \includegraphics[max width=\textwidth, alt={}, center]{e528b905-7419-44b6-b700-4c04ad96c816-2_531_1486_561_292}
    2. Number the regions of your planar graph, including the outside region. Regarding the graph as a map, draw its adjacency graph.
    3. Repeat parts (i) and (ii) for the following map. \includegraphics[max width=\textwidth, alt={}, center]{e528b905-7419-44b6-b700-4c04ad96c816-2_533_1484_1361_294}
    OCR MEI D1 2013 June Q2
    8 marks Easy -1.8
    2 The instructions labelled 1 to 7 describe the steps of a sorting algorithm applied to a list of six numbers.
    1 Let \(i\) equal 1.
    2 Repeat lines 3 to 7, stopping when \(i\) becomes 6 .
    OCR MEI D1 2013 June Q3
    8 marks Easy -1.8
    3 Let \(j\) equal 1.
    OCR MEI D1 2013 June Q4
    16 marks Easy -1.8
    4 Repeat lines 5 and 6 , until \(j\) becomes \(7 - i\).
    OCR MEI D1 2013 June Q5
    16 marks Easy -1.8
    5 If the \(j\) th number in the list is bigger than the \(( j + 1 )\) th, then swap them.
    OCR MEI D1 2013 June Q6
    16 marks Easy -1.8
    6 Let the new value of \(j\) be \(j + 1\).
    OCR MEI D1 2013 June Q7
    Moderate -0.8
    7 Let the new value of \(i\) be \(i + 1\).
    1. Apply the sorting algorithm to the list of numbers shown below. Record in the table provided the state of the list after each pass. Record the number of comparisons and the number of swaps that you make in each pass. (The result of the first pass has already been recorded.) List: \(\begin{array} { l l l l l l } 9 & 11 & 7 & 3 & 13 & 5 \end{array}\)
    2. Suppose now that the list is split into two sublists, \(\{ 9,11,7 \}\) and \(\{ 3,13,5 \}\). The sorting algorithm is adapted to apply to three numbers, and is applied to each sublist separately. This gives \(\{ 7,9,11 \}\) and \(\{ 3,5,13 \}\). How many comparisons and swaps does this need?
    3. How many further swaps would the original algorithm need to sort the revised list \(\{ 7,9,11,3,5,13 \}\) into increasing order? 3 The network below represents a number of villages together with connecting roads. The numbers on the arcs represent distances (in miles). \includegraphics[max width=\textwidth, alt={}, center]{e528b905-7419-44b6-b700-4c04ad96c816-3_684_785_1612_625}
    1. Use Dijkstra's algorithm to find the shortest routes from A to each of the other villages. Give these shortest routes and the corresponding distances. Traffic in the area travels at 30 mph . An accident delays all traffic passing through C by 20 minutes.
    2. Describe how the network can be adapted and used to find the fastest journey time from A to F .
    OCR MEI Paper 3 2019 June Q1
    6 marks Moderate -0.8
    1 The function \(\mathrm { f } ( x )\) is defined for all real \(x\) by \(f ( x ) = 3 x - 2\).
    1. Find an expression for \(\mathrm { f } ^ { - 1 } ( x )\).
    2. Sketch the graphs of \(y = \mathrm { f } ( x )\) and \(y = \mathrm { f } ^ { - 1 } ( x )\) on the same diagram.
    3. Find the set of values of \(x\) for which \(\mathrm { f } ( x ) > \mathrm { f } ^ { - 1 } ( x )\).