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OCR Further Additional Pure AS 2019 June Q4
8 marks Challenging +1.2
4 The sequence \(\left\{ \mathrm { u } _ { \mathrm { n } } \right\}\) is defined by \(u _ { 1 } = 1\) and \(\mathrm { u } _ { \mathrm { n } + 1 } = 2 \mathrm { u } _ { \mathrm { n } } + \mathrm { n } ^ { 2 }\) for \(\mathrm { n } \geqslant 1\).
Determine \(u _ { n }\) as a function of \(n\).
OCR Further Additional Pure AS 2019 June Q5
8 marks Challenging +1.2
5 The tetrahedron \(T\), shown below, has vertices at \(O ( 0,0,0 ) , A ( 1,2,2 ) , B ( 2,1,2 )\) and \(C ( 2,2,1 )\). \includegraphics[max width=\textwidth, alt={}, center]{59fa1650-a296-471e-93b9-0988177cd89d-3_360_464_319_555} Diagram not drawn to scale Show that the surface area of \(T\) is \(\frac { 1 } { 2 } \sqrt { 3 } ( 1 + \sqrt { 51 } )\).
OCR Further Additional Pure AS 2019 June Q6
8 marks Standard +0.8
6
  1. Determine all values of \(x\) for which \(16 x \equiv 5 ( \bmod 101 )\).
  2. Solve
    1. \(95 x \equiv 6 ( \bmod 101 )\),
    2. \(95 x \equiv 5 ( \bmod 101 )\).
OCR Further Additional Pure AS 2019 June Q7
12 marks Standard +0.8
7 You are given the set \(S = \{ 1,5,7,11,13,17 \}\) together with \(\times _ { 18 }\), the operation of multiplication modulo 18.
  1. Complete the Cayley table for \(\left( S , \times _ { 18 } \right)\) given in the Printed Answer Booklet.
  2. Prove that ( \(S , \times _ { 18 }\) ) is a group. (You may assume that \(\times _ { 18 }\) is associative.)
  3. Write down the order of each element of the group.
  4. Show that \(\left( S , \times _ { 18 } \right)\) is a cyclic group.
    1. Give an example of a non-cyclic group of order 6 .
    2. Give one reason why your example is structurally different to \(\left( S , { } _ { 18 } \right)\).
OCR Further Additional Pure AS 2019 June Q8
13 marks Standard +0.8
8 The motion of two remote controlled helicopters \(P\) and \(Q\) is modelled as two points moving along straight lines. Helicopter \(P\) moves on the line \(\mathbf { r } = \left( \begin{array} { r } 2 + 4 p \\ - 3 + p \\ 1 + 3 p \end{array} \right)\) and helicopter \(Q\) moves on the line \(\mathbf { r } = \left( \begin{array} { l } 5 + 8 q \\ 2 + q \\ 5 + 4 q \end{array} \right)\).
The function \(z\) denotes \(( P Q ) ^ { 2 }\), the square of the distance between \(P\) and \(Q\).
  1. Show that \(z = 26 p ^ { 2 } + 81 q ^ { 2 } - 90 p q - 58 p + 90 q + 50\).
  2. Use partial differentiation to find the values of \(p\) and \(q\) for which \(z\) has a stationary point.
  3. With the aid of a diagram, explain why this stationary point must be a minimum point, rather than a maximum point or a saddle point.
  4. Hence find the shortest possible distance between the two helicopters. The model is now refined by modelling each helicopter as a sphere of radius 0.5 units.
  5. Explain how this will change your answer to part (d). \section*{END OF QUESTION PAPER}
OCR Further Additional Pure AS 2022 June Q1
6 marks Standard +0.8
1 The points \(A , B\) and \(C\) have position vectors \(\mathbf { a } = \left( \begin{array} { l } 1 \\ 1 \\ 3 \end{array} \right) , \mathbf { b } = \left( \begin{array} { r } 2 \\ 3 \\ - 1 \end{array} \right)\) and \(\mathbf { c } = \left( \begin{array} { r } - 5 \\ 1 \\ 2 \end{array} \right)\) respectively, relative to the origin \(O\).
  1. Calculate, in its simplest exact form, the area of triangle \(O A B\).
  2. Show that \(\mathbf { a } \times ( \mathbf { b } \times \mathbf { c } ) + \mathbf { b } \times ( \mathbf { c } \times \mathbf { a } ) + \mathbf { c } \times ( \mathbf { a } \times \mathbf { b } ) = \mathbf { 0 }\).
OCR Further Additional Pure AS 2022 June Q2
6 marks Challenging +1.2
2 The surface \(S\) has equation \(z = x ^ { 3 } + y ^ { 3 } - 2 x ^ { 2 } - 5 y ^ { 2 } + 3 x y\).
It is given that \(S\) has two stationary points; one at the origin, \(O\), and the other at the point \(A\).
Determine the coordinates of \(A\).
OCR Further Additional Pure AS 2022 June Q3
7 marks Challenging +1.2
3 The sequence \(\left\{ U _ { n } \right\}\) is given by \(U _ { 1 } = 0 , U _ { 2 } = - 1\) and \(U _ { n + 2 } = U _ { n + 1 } + U _ { n } + n - 1\) for \(n \geqslant 1\).
  1. List the first seven terms of this sequence. The Fibonacci sequence \(\left\{ \mathrm { F } _ { \mathrm { n } } \right\}\) is given by \(\mathrm { F } _ { 1 } = 1 , \mathrm {~F} _ { 2 } = 1\) and \(\mathrm { F } _ { \mathrm { n } + 2 } = \mathrm { F } _ { \mathrm { n } + 1 } + \mathrm { F } _ { \mathrm { n } }\) for \(n \geqslant 1\).
    1. By comparing the two sequences, give the relationship between \(\mathrm { U } _ { \mathrm { n } }\) and \(\mathrm { F } _ { \mathrm { n } }\).
    2. Show that the relationship found in part (b)(i) holds for all \(n \geqslant 1\).
OCR Further Additional Pure AS 2022 June Q4
8 marks
4 Let \(\mathrm { N } = 10 \mathrm { a } + \mathrm { b }\) and \(\mathrm { M } = \mathrm { a } + 3 \mathrm {~b}\), where \(a\) and \(b\) are integers such that \(a \geqslant 1\) and \(0 \leqslant b \leqslant 9\).
  1. Prove that \(29 \mid N\) if and only if \(29 \mid M\).
  2. Use an iterative method based on the result of part (a) to show that 899364472 is a multiple of 29 .
OCR Further Additional Pure AS 2022 June Q5
6 marks Standard +0.8
5 A research student is using 3-D graph-plotting software to model a chain of volcanic islands in the Pacific Ocean. These islands appear above sea-level at regular intervals, (approximately) distributed along a straight line. Each island takes the form of a single peak; also, along the line of islands, the heights of these peaks decrease in size in an (approximately) regular fashion (see Fig. 1.1). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{06496165-0b83-4050-ae26-fa5a0614bd46-3_476_812_495_246} \captionsetup{labelformat=empty} \caption{Fig. 1.1}
\end{figure} The student's model uses the surface with equation \(\mathrm { z } = \sin \mathrm { x } + \sin \mathrm { y }\), a part of which is shown in Fig. 1.2 below. The surface of the sea is taken to be the plane \(z = 0\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{06496165-0b83-4050-ae26-fa5a0614bd46-3_789_951_1270_242} \captionsetup{labelformat=empty} \caption{Fig. 1.2}
\end{figure}
  1. - Describe two problems with this model.
    • Suggest revisions to this model so that each of these problems is addressed.
    • Still using their original model, the student examines the contour \(z = 2\) for their surface only to find that the software shows what appears to be an empty graph.
    Explain what has happened.
OCR Further Additional Pure AS 2022 June Q6
14 marks Challenging +1.2
6 The sequence \(\left\{ u _ { n } \right\}\) is such that \(u _ { 1 } = 7 , u _ { 2 } = 37 , u _ { 3 } = 337 , u _ { 4 } = 3337 , \ldots\).
  1. Write down a first-order recurrence system for \(\left\{ \mathrm { u } _ { \mathrm { n } } \right\}\).
  2. By solving the recurrence system of part (a), show that \(\mathrm { u } _ { \mathrm { n } } = \frac { 1 } { 3 } \left( 10 ^ { \mathrm { n } } + 11 \right)\).
  3. Prove that \(\left\{ \mathrm { u } _ { \mathrm { n } } \right\}\) contains infinitely many terms which are multiples of 37 .
OCR Further Additional Pure AS 2022 June Q7
13 marks Standard +0.8
7 The diagram below shows an equilateral triangle \(A B C\). The three lines of reflection symmetry of \(A B C\) (the lines \(a , b\) and \(c\) ) are shown as broken lines. The point of intersection of these three lines, \(O\), is the centre of rotational symmetry of the triangle. \includegraphics[max width=\textwidth, alt={}, center]{06496165-0b83-4050-ae26-fa5a0614bd46-4_533_538_884_246} The group \(D _ { 3 }\) is defined as the set of symmetries of \(A B C\) under the composition of the following transformations. \(i\) : the identity transformation \(a\) : reflection in line \(a\) \(b\) : reflection in line \(b\) \(c\) : reflection in line \(c\) \(p\) : an anticlockwise rotation about \(O\) through \(120 ^ { \circ }\) \(q\) : a clockwise rotation about \(O\) through \(120 ^ { \circ }\) Note that the lines \(a , b\) and \(c\) are unaffected by the transformations and remain fixed.
  1. On the diagrams provided in the Printed Answer Booklet, show each of the six elements of \(D _ { 3 }\) obtained when the above transformations are applied to triangle \(A B C\).
  2. Complete the Cayley table given in the Printed Answer Booklet.
  3. List all the proper subgroups of \(D _ { 3 }\).
  4. State, with justification, whether \(D _ { 3 }\) is
    1. cyclic,
    2. abelian.
  5. The group \(H\), also of order 6, is the set of rotational symmetries of the regular hexagon. Describe two structural differences between \(D _ { 3 }\) and \(H\). \section*{END OF QUESTION PAPER}
OCR Further Additional Pure AS 2023 June Q1
3 marks Moderate -0.8
1
  1. Express 205 in the form \(7 q + r\) for positive integers \(q\) and \(r\), with \(0 \leqslant r < 7\).
  2. Given that \(7 \mid ( 205 \times 8666 )\), use the result of part (a) to justify that \(7 \mid 8666\).
OCR Further Additional Pure AS 2023 June Q2
4 marks Standard +0.8
2 For all positive integers \(n\), the terms of the sequence \(\left\{ u _ { n } \right\}\) are given by the formula \(u _ { n } = 3 n ^ { 2 } + 3 n + 7 ( \bmod 10 )\).
  1. Show that \(u _ { n + 5 } = u _ { n }\) for all positive integers \(n\).
  2. Hence describe the behaviour of the sequence, justifying your answer.
OCR Further Additional Pure AS 2023 June Q3
6 marks Challenging +1.2
3 A surface has equation \(z = x ^ { 2 } y ^ { 2 } - 3 x y + 2 x + y\) for all real values of \(x\) and \(y\). Determine the coordinates of all stationary points of this surface.
OCR Further Additional Pure AS 2023 June Q4
7 marks Challenging +1.2
4 The equation of line \(l\) can be written in either of the following vector forms.
  • \(\mathbf { r } = \mathbf { a } + \lambda \mathbf { b }\), where \(\lambda \in \mathbb { R }\)
  • \(( \mathbf { r } - \mathbf { c } ) \times \mathbf { d } = \mathbf { 0 }\)
    1. Write down two equations involving the vectors \(\mathbf { a , b , c }\), and d, giving reasons for your answers.
    2. Determine the value of \(\mathbf { a } \cdot ( \mathbf { c } \times \mathbf { d } )\).
OCR Further Additional Pure AS 2023 June Q5
11 marks Standard +0.8
5
  1. Express as a decimal (base-10) number the base-23 number \(7119 _ { 23 }\).
  2. Solve the linear congruence \(7 n + 11 \equiv 9 ( \bmod 23 )\).
  3. Let \(N = 10 a + b\) and \(M = a + 7 b\), where \(a\) and \(b\) are integers and \(0 \leqslant b \leqslant 9\).
    1. By considering \(3 N - 7 M\), prove that \(23 \mid N\) if and only if \(23 \mid M\).
    2. Use a procedure based on this result to show that \(N = 711965\) is a multiple of 23 .
OCR Further Additional Pure AS 2023 June Q6
9 marks Standard +0.8
6 When \(10 ^ { 6 }\) of a certain type of bacteria are detected in a blood sample of an infected animal, a course of treatment is started. The long-term aim of the treatment is to reduce the number of bacteria in such a sample to under 10000 . At this level the animal's immune system can fight the infection for itself. Once treatment has started, if the number of bacteria in a sample is 10000 or more, then treatment either continues or restarts. The model suggested to predict the progress of the course of treatment is based on the recurrence system \(P _ { n + 1 } = \frac { 2 P _ { n } } { n + 1 } + \frac { n } { P _ { n } }\) for \(n \geqslant 0\), with \(P _ { 0 } = 1000\), where \(P _ { n }\) denotes the number of bacteria (in thousands) present in the animal's body \(n\) days after the treatment was started. The table below shows the values of \(P _ { n }\), for certain chosen values of \(n\). Each value has been given correct to 2 decimal places (where appropriate).
\(n\)0123456789
\(P _ { n }\)1000200020001333.33666.67266.6725.476.642.68
\(n\)1020406080100200300400
\(P _ { n }\)3.894.676.457.849.0310.0814.2017.3620.04
  1. Find the value of \(P _ { 6 }\) correct to 2 decimal places.
  2. Using the given values for \(P _ { 0 }\) to \(P _ { 9 }\), and assuming that the model is valid,
    1. describe the effects of this course of treatment during the first 9 days,
    2. state the number of days after treatment is started when the animal's own immune system is expected to be able to fight the infection for itself.
    1. Using information from the above table, suggest a function f such that, for \(n > 10 , \mathrm { f } ( n )\) is a suitable approximation for \(P _ { n }\).
    2. Use your suggested function to estimate the number of days after treatment is started when the animal may once again require medical intervention in order to help fight off this bacterial infection.
    3. Using information from the above table and the recurrence relation, verify or correct the estimate which you found in part (c)(ii).
  4. One criticism of the system \(P _ { n + 1 } = \frac { 2 P _ { n } } { n + 1 } + \frac { n } { P _ { n } }\), with \(P _ { 0 } = 1000\), is that it gives non-integer
    values of \(P\). values of \(P _ { n }\). Suggest a modification that would correct this issue.
OCR Further Additional Pure AS 2023 June Q7
14 marks Challenging +1.2
7 The group \(G\), of order 12, consists of the set \(\{ 1,2,4,5,8,10,13,16,17,19,20 , x \}\) under the operation of multiplication modulo 21 . The identity of \(G\) is the element 1 . The element \(x\) is an integer, \(0 < x < 21\), distinct from the other elements in the set. An incomplete copy of the Cayley table for \(G\) is shown below:
G12458101316171920\(x\)
112458101316171920
22481016205\(x\)131719
4481620\(x\)1910151317
55102041982171\(x\)16
8816\(x\)1911720210513
101020198171641321\(x\)
13135102204119\(x\)168
1616\(x\)11721319420105
17171351102\(x\)201684
19191713\(x\)511610842
202019171613\(x\)85421
\(x\)
  1. State, with justification, the value of \(x\).
  2. In the table given in the Printed Answer Booklet, list the order of each of the non-identity elements of \(G\).
    1. Write down all the subgroups of \(G\) of order 3 .
    2. Write down all the subgroups of \(G\) of order 6 .
  4. Determine all the subgroups of \(G\) of order 4, and prove that there are no other subgroups of order 4.
  5. State, with a reason, whether \(G\) is a cyclic group.
OCR Further Additional Pure AS 2023 June Q8
6 marks Challenging +1.8
8 A surface, \(C\), is given by the equation \(z = \mathrm { f } ( x , y )\) for all real values of \(x\) and \(y\). You are given that \(C\) has the following properties.
  • The surface is continuous for all \(x\) and \(y\).
  • The contour \(z = - 1\) is a single point on the \(z\)-axis.
  • For \(- 1 < a < 1\), the contour \(z = a\) is a pair of circles with different radiuses but each having the same centre \(( 0,0 , a )\).
  • The contour \(z = 1\) consists of the circle, centre \(( 0,0,1 )\) and radius 1 .
Sketch a possible section of \(C\) corresponding to \(y = 0\).
OCR Further Additional Pure AS 2020 November Q1
4 marks Moderate -0.8
1
  1. Evaluate \(13 \times 19\) modulo 31 .
  2. Solve the linear congruence \(13 x \equiv 9 ( \bmod 31 )\).
OCR Further Additional Pure AS 2020 November Q2
11 marks Challenging +1.2
2 An open-topped rectangular box is to be manufactured with a fixed volume of \(1000 \mathrm {~cm} ^ { 3 }\). The dimensions of the base of the box are \(x \mathrm {~cm}\) by \(y \mathrm {~cm}\). The surface area of the box is \(A \mathrm {~cm} ^ { 2 }\).
  1. Show that \(\mathrm { A } = \mathrm { xy } + 2000 \left( \frac { 1 } { \mathrm { x } } + \frac { 1 } { \mathrm { y } } \right)\).
    1. Use partial differentiation to determine, in exact form, the values of \(x\) and \(y\) for which \(A\) has a stationary value.
    2. Find the stationary value of \(A\).
OCR Further Additional Pure AS 2020 November Q3
6 marks
3 In this question, \(N\) is the number 26132652.
  1. Without dividing \(N\) by 13, explain why 13 is a factor of \(N\).
  2. Use standard divisibility tests to show that 36 is a factor of \(N\). It is given that \(N = 36 \times 725907\).
  3. Use the results of parts (a) and (b) to deduce that 13 is a factor of 725907.
OCR Further Additional Pure AS 2020 November Q4
12 marks Standard +0.8
4
  1. For the set \(S = \{ 2,4,6,8,10,12 \}\), under the operation \(\times _ { 14 }\) of multiplication modulo 14, complete the Cayley table given in the Printed Answer Booklet.
  2. Show that ( \(S , \times _ { 14 }\) ) forms a group, \(G\). (You may assume that \(\times _ { 14 }\) is associative.)
    1. Write down all the proper subgroups of \(G\).
    2. Given that \(G\) is cyclic, write down all the possible generators of \(G\).
OCR Further Additional Pure AS 2020 November Q5
8 marks Standard +0.8
5
  1. Determine the general solution of the first-order recurrence relation \(V _ { n + 1 } = 2 V _ { n } + n\).
  2. Given that \(V _ { 1 } = 8\), find the exact value of \(V _ { 20 }\).