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OCR Further Discrete 2021 November Q2
8 marks Standard +0.3
2 A simply connected semi-Eulerian graph G has 6 vertices and 8 arcs. Two of the vertex degrees are 3 and 4.
    1. Determine the minimum possible vertex degree.
    2. Determine the maximum possible vertex degree.
  2. Write down the two possible degree sequences (ordered lists of vertex degrees). The adjacency matrix for a digraph H is given below.
    \multirow{7}{*}{From}\multirow{2}{*}{}To
    JKLMN
    J01100
    K10100
    L10001
    M00211
    N01010
  3. Write down the indegree and the outdegree of each vertex of H .
    1. Use the indegrees and outdegrees to determine whether graph H is Eulerian.
    2. Use the adjacency matrix to determine whether graph H is simply connected.
OCR Further Discrete 2021 November Q3
8 marks Standard +0.3
3 Six people play a game with 150 cards. Each player has a stack of cards in front of them and the remainder of the cards are in another stack on the table.
  1. Use the pigeonhole principle to explain why at least one of the stacks must have at least 22 cards in it. The set of cards is numbered from 1 to 150 . Each digit '2', '3' and '5', whether as a units digit or a tens digit, is coloured red.
    So, for example
    • the card numbered 25 has two red digits,
    • the card numbered 26 has one red digit,
    • the card numbered 148 has no red digits.
    • By considering the cards with one digit, two digits and three digits, or otherwise, determine how many cards have no red digits.
    The cards are put into a Venn diagram with three intersecting sets:
    \(\mathrm { A } = \{\) cards with a number that is a multiple of \(2 \}\)
    \(\mathrm { B } = \{\) cards with a number that is a multiple of \(3 \}\)
    \(\mathrm { C } = \{\) cards with a number that is a multiple of \(5 \}\) For example
    • the card numbered 2 is in set A ,
    • the card numbered 15 is in sets B and C ,
    • the card numbered 23 is in none of the sets.
      \includegraphics[max width=\textwidth, alt={}, center]{133395d2-5020-4054-a229-70168f1d0f95-4_588_1150_1667_246}
    • By considering the cards with one digit, two digits and three digits, or otherwise, determine how many cards in set A have no red digits.
    • Given that, for the cards with no red digits, \(n ( B ) = 21 , n ( C ) = 9\) and \(n ( A \cap B ) = 12\), use the inclusion-exclusion principle to determine how many of the cards with no red digits are in none of the sets A, B or C.
OCR Further Discrete 2021 November Q4
13 marks Moderate -0.3
4 One of these graphs is isomorphic to \(\mathrm { K } _ { 2,3 }\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{133395d2-5020-4054-a229-70168f1d0f95-5_175_195_285_242} \captionsetup{labelformat=empty} \caption{Graph A}
\end{figure} \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{133395d2-5020-4054-a229-70168f1d0f95-5_170_195_285_635} \captionsetup{labelformat=empty} \caption{Graph B}
\end{figure} \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{133395d2-5020-4054-a229-70168f1d0f95-5_168_191_287_1027} \captionsetup{labelformat=empty} \caption{Graph C}
\end{figure}
  1. Explain how you know that each of the other graphs is not isomorphic to \(\mathrm { K } _ { 2,3 }\). The arcs of the complete graph \(\mathrm { K } _ { 5 }\) can be partitioned as the complete bipartite graph \(\mathrm { K } _ { 2,3 }\) and a graph G.
  2. Draw the graph G.
  3. Explain carefully how you know that the graph \(\mathrm { K } _ { 5 }\) has thickness 2 . The following colouring algorithm can be used to determine whether a connected graph is bipartite or not. The algorithm colours each vertex of a graph in one of two colours, \(\alpha\) and \(\beta\). STEP 1 Choose a vertex and assign it colour \(\alpha\).
    STEP 2 If any vertex is adjacent to another vertex of the same colour, jump to STEP 5. Otherwise assign colour \(\beta\) to each vertex that is adjacent to a vertex with colour \(\alpha\). STEP 3 If any vertex is adjacent to another vertex of the same colour, jump to STEP 5. Otherwise assign colour \(\alpha\) to each vertex that is adjacent to a vertex with colour \(\beta\). STEP 4 Repeat STEP 2 and STEP 3 until all vertices are coloured.
    STEP 5 If there are no adjacent vertices of the same colour then the graph is bipartite. Otherwise the graph is not bipartite. STEP 6 Stop.
  4. Apply this algorithm to graph A, starting with the vertex in the top left corner, to determine whether graph A is bipartite or not. A measure of the efficiency of the colouring algorithm is given by the number of passes through STEP 4.
  5. Write down how many passes through STEP 4 are needed for the bipartite graph \(\mathrm { K } _ { 2,3 }\). The worst case is when the graph is a path that starts at one vertex and ends at another, having passed through each of the other vertices once.
  6. What can you deduce about the efficiency of the colouring algorithm in this worst case? The colouring algorithm is used on two graphs, X and Y . It takes 10 seconds to run for graph X and 60 seconds to run for graph Y. Graph X has 1000 vertices.
  7. Estimate the number of vertices in graph Y . A different algorithm has efficiency \(\mathrm { O } \left( 2 ^ { n } \right)\). This algorithm takes 10 seconds to run for graph X .
  8. Explain why you would expect this algorithm to take longer than 60 seconds to run for graph Y .
OCR Further Discrete 2021 November Q5
12 marks Challenging +1.8
5 Alex and Beth play a zero-sum game. Alex chooses a strategy P, Q or R and Beth chooses a strategy \(\mathrm { X } , \mathrm { Y }\) or Z . The table shows the number of points won by Alex for each combination of strategies. The entry for cell \(( \mathrm { P } , \mathrm { X } )\) is \(x\), where \(x\) is an integer. \begin{table}[h]
\captionsetup{labelformat=empty} \caption{Beth}
XYZ
\cline { 3 - 5 }P\(x\)32
\cline { 3 - 5 }Q40- 2
\cline { 3 - 5 }R- 3- 1- 3
\cline { 3 - 5 }
\cline { 3 - 5 }
\end{table} Suppose that P is a play-safe strategy.
    1. Determine the values of \(x\) for which the game is stable.
    2. Determine the values of \(x\) for which the game is unstable. The game can be reduced to a \(2 \times 3\) game using dominance.
  2. Write down the pay-off matrix for the reduced game. When the game is unstable, Alex plays strategy P with probability \(p\).
  3. Determine, as a function of \(x\), the value of \(p\) for the optimal mixed strategy for Alex. Suppose, instead, that P is not a play-safe strategy and the value of \(x\) is - 5 .
  4. Show how to set up a linear programming formulation that could be used to find the optimal mixed strategy for Alex.
OCR Further Discrete 2021 November Q6
11 marks Moderate -0.5
6 An initial simplex tableau is shown below.
\(P\)\(x\)\(y\)\(s\)\(t\)\(u\)RHS
12-50000
02110025.8
0-1301013.8
04-300118.8
The variables \(s , t\) and \(u\) are slack variables.
  1. For the LP problem that this tableau represents, write down the following, in terms of \(x\) and \(y\) only.
    • The objective function, \(P\), to be maximised.
    • The constraints as inequalities.
    The graph below shows the feasible region for the problem (as the unshaded region, and its boundaries), and objective lines \(P = 10\) and \(P = 20\) (shown as dotted lines).
    \includegraphics[max width=\textwidth, alt={}, center]{133395d2-5020-4054-a229-70168f1d0f95-7_883_1043_1272_244} The optimal solution is \(P = 23\), when \(x = 0\) and \(y = 4.6\).
  2. Complete the first three rows of branch-and-bound in the Printed Answer Booklet, branching on \(y\) first, to determine an optimal solution when \(x\) and \(y\) are constrained to take integer values. In your working, you should show non-integer values to \(\mathbf { 2 }\) decimal places. The tableau entry 18.8 is reduced to 0 .
  3. Describe carefully what changes, if any, this makes to the following.
    • The graph of the feasible region.
    • The optimal integer valued solution.
OCR Further Discrete 2021 November Q7
15 marks Moderate -0.8
7 A network is formed by weighting the graph below using the listed arc weights.
\includegraphics[max width=\textwidth, alt={}, center]{133395d2-5020-4054-a229-70168f1d0f95-8_168_190_310_258}
\(\begin{array} { l l l l l l l l } 2.9 & 0.9 & 1.5 & 3.5 & 4.2 & 5.3 & 4.7 & 2.3 \end{array}\)
    1. Show the result after the first pass and after the second pass, when the list of weights is sorted into increasing order using bubble sort.
    2. Show the result after the first pass and after the second pass, when the list of weights is sorted into increasing order using shuttle sort. In the remaining passes of bubble sort another 14 comparisons are made.
      In the remaining passes of shuttle sort another 11 comparisons are made.
      The total number of swaps needed is the same for both sorting methods.
  2. Use the total number of comparisons and the total number of swaps to compare the efficiency of bubble sort and shuttle sort for sorting this list of weights. The sorted list of arc weights for the network is as follows.
    \(\begin{array} { l l l l l l l l } 0.9 & 1.5 & 2.3 & 2.9 & 3.5 & 4.2 & 4.7 & 5.3 \end{array}\) These weights can be given to the arcs of the graph in several ways to form different networks.
    1. What is the smallest weight that does not have to appear in a minimum spanning tree for any of these networks? You must explain your reasoning.
    2. Show a way of weighting the arcs, using the weights in the list, that results in the largest possible total for a minimum spanning tree. You should state the total weight of your minimum spanning tree.
    3. Determine the total weight of an optimal solution of the route inspection problem for the network found in part (c)(ii). \section*{END OF QUESTION PAPER}
OCR Further Discrete Specimen Q1
13 marks Moderate -0.3
1 Fiona is a mobile hairdresser. One day she needs to visit five clients, A to E, starting and finishing at her own house at F . She wants to find a suitable route that does not involve her driving too far.
  1. Which standard network problem does Fiona need to solve? The shortest distances between clients, in km, are given in the matrix below.
    ABCDE
    A-12864
    B12-10810
    C810-1310
    D6813-10
    E4101010-
  2. Use the copy of the matrix in the Printed Answer Booklet to construct a minimum spanning tree for these five client locations.
    State the algorithm you have used, show the order in which you build your tree and give its total weight. Draw your minimum spanning tree. The distance from Fiona's house to each client, in km, is given in the table below.
    ABCDE
    F211975
  3. Use this information together with your answer to part (ii) to find a lower bound for the length of Fiona's route.
  4. (a) Find all the cycles that result from using the nearest neighbour method, starting at F .
    (b) Use these to find an upper bound for the length of Fiona's route.
  5. Fiona wants to drive less than 35 km . Using the information in your answers to parts (iii) and (iv) explain whether or not a route exists which is less than 35 km in length.
OCR Further Discrete Specimen Q2
13 marks Standard +0.3
2 Kirstie has bought a house that she is planning to renovate. She has broken the project into a list of activities and constructed an activity network, using activity on arc.
Activity
\(A\)Structural survey
\(B\)Replace damp course
\(C\)Scaffolding
\(D\)Repair brickwork
\(E\)Repair roof
\(F\)Check electrics
\(G\)Replaster walls
Activity
\(H\)Planning
\(I\)Build extension
\(J\)Remodel internal layout
\(K\)Kitchens and bathrooms
\(L\)Decoration and furnishing
\(M\)Landscape garden
\includegraphics[max width=\textwidth, alt={}, center]{0c9513fe-a471-427e-ba30-b18df11271e3-3_887_1751_1030_207}
  1. Construct a cascade chart for the project, showing the float for each non-critical activity.
  2. Calculate the float for remodelling the internal layout stating how much of this is independent float and how much is interfering float. Kirstie needs to supervise the project. This means that she cannot allow more than three activities to happen on any day.
  3. Describe how Kirstie should organise the activities so that the project is completed in the minimum project completion time and no more than three activities happen on any day.
OCR Further Discrete Specimen Q3
9 marks Standard +0.8
3 Bob has been given a pile of five letters addressed to five different people. He has also been given a pile of five envelopes addressed to the same five people. Bob puts one letter in each envelope at random.
  1. How many different ways are there to pair the letters with the envelopes?
  2. Find the number of arrangements with exactly three letters in the correct envelopes.
  3. (a) Show that there are two derangements of the three symbols A , B and C .
    (b) Hence find the number of arrangements with exactly two letters in the correct envelopes. Let \(\mathrm { D } _ { n }\) represent the number of derangements of \(n\) symbols.
  4. Explain why \(\mathrm { D } _ { n } = ( n - 1 ) \times \left( \mathrm { D } _ { n - 1 } + \mathrm { D } _ { n - 2 } \right)\).
  5. Find the number of ways in which all five letters are in the wrong envelopes.
OCR Further Discrete Specimen Q4
11 marks Standard +0.8
4 The table shows the pay-off matrix for player \(A\) in a two-person zero-sum game between \(A\) and \(B\). Player \(A\)
Player \(B\)
Strategy \(X\)Strategy \(Y\)Strategy \(Z\)
Strategy \(P\)45- 4
Strategy \(Q\)3- 12
Strategy \(R\)402
  1. Find the play-safe strategy for player \(A\) and the play-safe strategy for player \(B\). Use the values of the play-safe strategies to determine whether the game is stable or unstable.
  2. If player \(B\) knows that player \(A\) will use their play-safe strategy, which strategy should player \(B\) use?
  3. Suppose that the value in the cell where both players use their play-safe strategies can be changed, but all other entries are unchanged. Show that there is no way to change this value that would make the game stable.
  4. Suppose, instead, that the value in one cell can be changed, but all other entries are unchanged, so that the game becomes stable. Identify a suitable cell and write down a new pay-off value for that cell which would make the game stable.
  5. Show that the zero-sum game in the table above has a Nash equilibrium and explain what this means for the players.
OCR Further Discrete Specimen Q5
13 marks Standard +0.3
5 A garden centre sells tulip bulbs in mixed packs. The cost of each pack and the number of tulips of each colour are given in the table.
Cost \(( \pounds )\)RedWhiteYellowPink
Pack A5025252525
Pack B484030300
Pack C5320304010
Dirk is designing a floral display in which he will need the number of red tulips to be at most 50 more than the number of white tulips, and the number of white tulips to be less than or equal to twice the number of pink tulips. He has a budget of \(\pounds 240\) and wants to find out which packs to buy to maximise the total number of bulbs. Dirk uses the variables \(x , y\) and \(z\) to represent, respectively, how many of pack A , pack B and pack C he buys. He sets up his problem as an initial simplex tableau, which is shown below. Initial tableau
Row 1
Row 2
Row 3
Row 4
\(P\)\(x\)\(y\)\(z\)\(s\)\(t\)\(u\)RHS
1- 1- 1- 10000
001- 11005
0- 5620100
0504853001240
  1. Show how the constraint on the number of red tulips leads to one of the rows of the tableau. The tableau that results after the first iteration is shown below.
    After first iteration
    Row 5
    Row 6
    Row 7
    Row 8
    \(P\)\(x\)\(y\)\(z\)\(s\)\(t\)\(u\)RHS
    10- 0.040.06000.024.8
    001- 11005
    0010.87.3010.124
    010.961.06000.024.8
  2. Which cell was used as the pivot?
  3. Explain why row 2 and row 6 are the same.
  4. (a) Read off the values of \(x , y\) and \(z\) after the first iteration.
    (b) Interpret this solution in terms of the original problem.
  5. Identify the variable that has become non-basic. Use the pivot row of the initial tableau to eliminate \(x\) algebraically from the equation represented by Row 1 of the initial tableau. The feasible region can be represented graphically in three dimensions, with the variables \(x , y\) and \(z\) corresponding to the \(x\)-axis, \(y\)-axis and \(z\)-axis respectively. The boundaries of the feasible region are planes. Pairs of these planes intersect in lines and at the vertices of the feasible region these lines intersect.
  6. The planes defined by each of the new basic variables being set equal to 0 intersect at a point. Show how the equations from part (v) are used to find the values \(P\) and \(x\) at this point. A planar graph \(G\) is described by the adjacency matrix below. \(\quad\)
    \(A\)
    \(B\)
    \(C\)
    \(D\)
    \(E\)
    \(F\) \(\left( \begin{array} { c c c c c c } A & B & C & D & E & F \\ 0 & 1 & 0 & 0 & 1 & 1 \\ 1 & 0 & 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 & 1 & 1 \\ 1 & 1 & 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 1 & 0 & 0 \end{array} \right)\)
  7. Draw the graph \(G\).
  8. Use Euler's formula to verify that there are four regions. Identify each region by listing the vertices that define it.
  9. Explain why graph \(G\) cannot have a Hamiltonian cycle that includes the edge \(A B\). Deduce how many Hamiltonian cycles graph \(G\) has. A colouring algorithm is given below. STEP 1: Choose a vertex, colour this vertex using colour 1. STEP 2: If all vertices are coloured, STOP. Otherwise use colour 2 to colour all uncoloured vertices for which there is an edge that joins that vertex to a vertex of colour 1 . STEP 3: If all vertices are coloured, STOP. Otherwise use colour 1 to colour all uncoloured vertices for which there is an edge that joins that vertex to a vertex of colour 2 . STEP 4: Go back to STEP 2.
  10. Apply this algorithm to graph \(G\), starting at \(E\). Explain how the colouring shows you that graph \(G\) is not bipartite. By removing just one edge from graph \(G\) it is possible to make a bipartite graph.
  11. Identify which edge needs to be removed and write down the two sets of vertices that form the bipartite graph. Graph \(G\) is augmented by the addition of a vertex \(X\) joined to each of \(A , B , C , D , E\) and \(F\).
  12. Apply Kuratowski's theorem to a contraction of the augmented graph to explain how you know that the augmented graph has thickness 2.
OCR Further Additional Pure 2019 June Q1
4 marks Challenging +1.8
1 The sequence \(\left\{ u _ { n } \right\}\) is defined by \(u _ { 0 } = 2 , u _ { 1 } = 5\) and \(u _ { n } = \frac { 1 + u _ { n - 1 } } { u _ { n - 2 } }\) for \(n \geqslant 2\).
Prove that the sequence is periodic with period 5.
OCR Further Additional Pure 2019 June Q2
11 marks Standard +0.8
2 A surface has equation \(z = \mathrm { f } ( x , y )\) where \(\mathrm { f } ( x , y ) = x ^ { 2 } \sin y + 2 y \cos x\).
  1. Determine \(\mathrm { f } _ { x } , \mathrm { f } _ { y } , \mathrm { f } _ { x x } , \mathrm { f } _ { y y } , \mathrm { f } _ { x y }\) and \(\mathrm { f } _ { y x }\).
    1. Verify that \(z\) has a stationary point at \(\left( \frac { 1 } { 2 } \pi , \frac { 1 } { 2 } \pi , \frac { 1 } { 4 } \pi ^ { 2 } \right)\).
    2. Determine the nature of this stationary point.
OCR Further Additional Pure 2019 June Q3
4 marks Standard +0.3
3
  1. Solve \(7 x \equiv 6 ( \bmod 19 )\).
  2. Show that the following simultaneous linear congruences have no solution. $$x \equiv 3 ( \bmod 4 ) , x \equiv 4 ( \bmod 6 )$$
OCR Further Additional Pure 2019 June Q4
10 marks Challenging +1.2
4
  1. Solve the second-order recurrence relation \(T _ { n + 2 } + 2 T _ { n } = - 87\) given that \(T _ { 0 } = - 27\) and \(T _ { 1 } = 27\).
  2. Determine the value of \(T _ { 20 }\).
OCR Further Additional Pure 2019 June Q5
11 marks Challenging +1.2
5 The group \(G\) consists of a set \(S\) together with \(\times _ { 80 }\), the operation of multiplication modulo 80. It is given that \(S\) is the smallest set which contains the element 11 .
  1. By constructing the Cayley table for \(G\), determine all the elements of \(S\). The Cayley table for a second group, \(H\), also with the operation \(\times _ { 80 }\), is shown below.
    \cline { 2 - 5 } \multicolumn{1}{c|}{\(\times _ { 80 }\)}193139
    1193139
    9913931
    31313919
    39393191
  2. Use the two Cayley tables to explain why \(G\) and \(H\) are not isomorphic.
    1. List
      • all the proper subgroups of \(G\),
  4. all the proper subgroups of \(H\).
    (ii) Use your answers to (c) (i) to give another reason why \(G\) and \(H\) are not isomorphic.
OCR Further Additional Pure 2019 June Q6
12 marks Challenging +1.2
6
  1. For the vectors \(\mathbf { p } = \left( \begin{array} { l } 1 \\ 2 \\ 3 \end{array} \right) , \mathbf { q } = \left( \begin{array} { r } 3 \\ 1 \\ - 1 \end{array} \right)\) and \(\mathbf { r } = \left( \begin{array} { r } 2 \\ - 4 \\ 5 \end{array} \right)\), calculate
    • \(\mathbf { p } \cdot \mathbf { q } \times \mathbf { r }\),
    • \(\mathbf { p } \times ( \mathbf { q } \times \mathbf { r } )\),
    • \(( \mathbf { p } \times \mathbf { q } ) \times \mathbf { r }\).
    • State whether the vector product is associative for three-dimensional column vectors with real components. Justify your answer.
    It is given that \(\mathbf { a } , \mathbf { b }\) and \(\mathbf { c }\) are three-dimensional column vectors with real components.
  2. Explain geometrically why the vector \(\mathbf { a } \times ( \mathbf { b } \times \mathbf { c } )\) must be expressible in the form \(\lambda \mathbf { b } + \mu \mathbf { c }\), where \(\lambda\) and \(\mu\) are scalar constants. It is given that the following relationship holds for \(\mathbf { a } , \mathbf { b }\) and \(\mathbf { c }\).
    \(\mathbf { a } \times ( \mathbf { b } \times \mathbf { c } ) = ( \mathbf { a } \cdot \mathbf { c } ) \mathbf { b } - ( \mathbf { a } \cdot \mathbf { b } ) \mathbf { c }\)
  3. Find an expression for ( \(\mathbf { a } \times \mathbf { b ) } \times \mathbf { c }\) in the form of (*).
OCR Further Additional Pure 2019 June Q7
12 marks Challenging +1.8
7 The points \(P \left( \frac { 1 } { 2 } , \frac { 13 } { 24 } \right)\) and \(Q \left( \frac { 3 } { 2 } , \frac { 31 } { 24 } \right)\) lie on the curve \(y = \frac { 1 } { 3 } x ^ { 3 } + \frac { 1 } { 4 x }\).
The area of the surface generated when arc \(P Q\) is rotated completely about the \(x\)-axis is denoted by \(A\).
  1. Find the exact value of \(A\). Give your answer as a rational multiple of \(\pi\). Student X finds an approximation to \(A\) by modelling the arc \(P Q\) as the straight line segment \(P Q\), then rotating this line segment completely about the \(x\)-axis to form a surface.
  2. Find the approximation to \(A\) obtained by student X . Give your answer as a rational multiple of \(\pi\). Student Y finds a second approximation to \(A\) by modelling the original curve as the line \(y = M\), where \(M\) is the mean value of the function \(\mathrm { f } ( x ) = \frac { 1 } { 3 } x ^ { 3 } + \frac { 1 } { 4 x }\), then rotating this line completely about the \(x\)-axis to form a surface.
  3. Find the approximation to \(A\) obtained by student Y . Give your answer correct to four decimal places.
OCR Further Additional Pure 2022 June Q1
6 marks Standard +0.8
1 The surface \(E\) has equation \(z = \sqrt { 500 - 3 x ^ { 2 } - 2 y ^ { 2 } }\).
  1. Determine the values of \(\frac { \partial z } { \partial x }\) and \(\frac { \partial z } { \partial y }\) at the point \(P\) on \(E\) with coordinates \(( 11 , - 8,3 )\).
  2. Find the equation of the tangent plane to \(E\) at \(P\), giving your answer in the form \(\mathrm { ax } + \mathrm { by } + \mathrm { cz } = \mathrm { d }\) where \(a , b , c\) and \(d\) are integers.
OCR Further Additional Pure 2022 June Q2
5 marks Challenging +1.2
2 Consider the integers \(a\) and \(b\), where, for each integer \(n , \mathrm { a } = 7 \mathrm { n } + 4\) and \(\mathrm { b } = 8 \mathrm { n } + 5\). Let \(\mathrm { h } = \mathrm { hcf } ( \mathrm { a } , \mathrm { b } )\).
  1. Determine all possible values of \(h\).
  2. Find all values of \(n\) for which \(a\) and \(b\) are not co-prime.
OCR Further Additional Pure 2022 June Q3
6 marks Challenging +1.2
3 The irrational number \(\phi = \frac { 1 } { 2 } ( 1 + \sqrt { 5 } )\) plays a significant role in the sequence of Fibonacci numbers given by \(\mathrm { F } _ { 0 } = 0 , \mathrm {~F} _ { 1 } = 1\) and \(\mathrm { F } _ { \mathrm { n } + 1 } = \mathrm { F } _ { \mathrm { n } } + \mathrm { F } _ { \mathrm { n } - 1 }\) for \(n \geqslant 1\). Prove by induction that, for each positive integer \(n , \phi ^ { n } = \mathrm { F } _ { \mathrm { n } } \times \phi + \mathrm { F } _ { \mathrm { n } - 1 }\).
OCR Further Additional Pure 2022 June Q4
9 marks Challenging +1.2
4 Let \(N\) be the number 15824578 .
    1. Use a standard divisibility test to show that \(N\) is a multiple of 11 .
    2. A student uses the following test for divisibility by 7 . \begin{displayquote} 'Throw away' multiples of 7 that appear either individually or within a pair of consecutive digits of the test number.
      Stop when the number obtained is \(0,1,2,3,4,5\) or 6 .
      The test number is only divisible by 7 if that obtained number is 0 . \end{displayquote} For example, for the number \(N\), they first 'throw away' the " 7 " in the tens column, leaving the number \(N _ { 1 } = 15824508\). At the second stage, they 'throw away' the " 14 " from the left-hand pair of digits of \(N _ { 1 }\), leaving \(N _ { 2 } = 01824508\); and so on, until a number is obtained which is \(0,1,2,3,4,5\) or 6 .
      • Justify the validity of this process.
  2. Continue the student's test to show that \(7 \mid N\).
    (iii) Given that \(N = 11 \times 1438598\), explain why 7| 1438598 .
  3. Let \(\mathrm { M } = \mathrm { N } ^ { 2 }\).
    1. Express \(N\) in the unique form 101a + b for positive integers \(a\) and \(b\), with \(0 \leqslant b < 101\).
    2. Hence write \(M\) in the form \(\mathrm { M } \equiv \mathrm { r } ( \bmod 101 )\), where \(0 < r < 101\).
    3. Deduce the order of \(N\) modulo 101.
OCR Further Additional Pure 2022 June Q5
8 marks Standard +0.8
5 You are given the variable point \(A ( 3 , - 8 , t )\), where \(t\) is a real parameter, and the fixed point \(B ( 1,2 , - 2 )\).
  1. Using only the geometrical properties of the vector product, explain why the statement " \(\overrightarrow { \mathrm { OA } } \times \overrightarrow { \mathrm { OB } } = \mathbf { 0 }\) " is false for all values of \(t\).
    1. Use the vector product to find an expression, in terms of \(t\), for the area of triangle \(O A B\).
    2. Hence determine the value of \(t\) for which the area of triangle \(O A B\) is a minimum.
OCR Further Additional Pure 2022 June Q6
12 marks Standard +0.3
6 In a national park, the number of adults of a given species is carefully monitored and controlled. The number of adults, \(n\) months after the start of this project, is \(A _ { n }\). Initially, there are 1000 adults. It is predicted that this number will have declined to 960 after one month. The first model for the number of adults is that, from one month to the next, a fixed proportion of adults is lost. In order to maintain a fixed number of adults, the park managers "top up" the numbers by adding a constant number of adults from other parks at the end of each month.
  1. Use this model to express the number of adults as a first-order recurrence system. Instead, it is found that, the proportion of adults lost each month is double the predicted amount, with no change being made to the constant number of adults added each month.
    1. Show that the revised recurrence system for \(A _ { n }\) is \(A _ { 0 } = 1000 , A _ { n + 1 } = 0.92 A _ { n } + 40\). [1]
    2. Solve this revised recurrence system.
    3. Describe the long-term behaviour of the sequence \(\left\{ A _ { n } \right\}\) in this case. A more refined model for the number of adults uses the second-order recurrence system \(\mathrm { A } _ { \mathrm { n } + 1 } = 0.9 \mathrm {~A} _ { \mathrm { n } } - 0.1 \mathrm {~A} _ { \mathrm { n } - 1 } + 50\), for \(n \geqslant 1\), with \(A _ { 0 } = 1000\) and \(A _ { 1 } = 920\).
    1. Determine the long-term behaviour of the sequence \(\left\{ A _ { n } \right\}\) for this more refined model.
    2. A criticism of this more refined model is that it does not take account of the fact that the number of adults must be an integer at all times. State a modified form of the second-order recurrence relation for this more refined model that will satisfy this requirement.
OCR Further Additional Pure 2022 June Q7
10 marks Challenging +1.8
7
  1. Differentiate \(\left( 16 + t ^ { 2 } \right) ^ { \frac { 3 } { 2 } }\) with respect to \(t\). Let \(I _ { n } = \int _ { 0 } ^ { 3 } t ^ { n } \sqrt { 16 + t ^ { 2 } } d t\) for integers \(n \geqslant 1\).
  2. Show that, for \(n \geqslant 3 , \left. ( n + 2 ) \right| _ { n } = 125 \times 3 ^ { n - 1 } - \left. 16 ( n - 1 ) \right| _ { n - 2 }\).
  3. The curve \(C\) is defined parametrically by \(\mathrm { x } = \mathrm { t } ^ { 4 } \cos \mathrm { t }\), \(\mathrm { y } = \mathrm { t } ^ { 4 } \sin \mathrm { t }\), for \(0 \leqslant t \leqslant 3\). The length of \(C\) is denoted by \(L\). Show that \(\mathrm { L } = \mathrm { I } _ { 3 }\). (You are not required to evaluate this integral.)