Questions — OCR Further Additional Pure (85 questions)

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OCR Further Additional Pure 2024 June Q1
1
  1. The number \(N\) has the base-10 form \(\mathrm { N } = \operatorname { abba } a b b a \ldots a b b a\), consisting of blocks of four digits, as shown, where \(a\) and \(b\) are integers such that \(1 \leqslant a < 10\) and \(0 \leqslant b < 10\). Use a standard divisibility test to show that \(N\) is always divisible by 11 .
  2. The number \(M\) has the base- \(n\) form \(\mathrm { M } = \operatorname { cddc } c d d c \ldots c d d c\), where \(n > 11\) and \(c\) and \(d\) are integers such that \(1 \leqslant \mathrm { c } < \mathrm { n }\) and \(0 \leqslant \mathrm {~d} < \mathrm { n }\). Show that \(M\) is always divisible by a number of the form \(\mathrm { k } _ { 1 } \mathrm { n } + \mathrm { k } _ { 2 }\), where \(k _ { 1 }\) and \(k _ { 2 }\) are integers to be determined.
OCR Further Additional Pure 2024 June Q2
2 A surface \(S\) has equation \(\mathrm { z } = 4 \mathrm { x } \sqrt { \mathrm { y } } - \mathrm { y } \sqrt { \mathrm { x } } + \mathrm { y } ^ { 2 }\) for \(x , y \geqslant 0\). Determine the equation of the tangent plane to \(S\) at the point (1,4,20). Give 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 2024 June Q3
3 Determine all integers \(x\) for which \(x \equiv 1 ( \bmod 7 )\) and \(x \equiv 22 ( \bmod 37 )\) and \(x \equiv 7 ( \bmod 67 )\).
Give your answer in the form \(\mathrm { x } = \mathrm { qn } + \mathrm { r }\) for integers \(n , q , r\) with \(q > 0\) and \(0 \leqslant \mathrm { r } < \mathrm { q }\).
OCR Further Additional Pure 2024 June Q4
4 The vectors \(\mathbf { a }\) and \(\mathbf { b }\) are given by \(\mathbf { a } = \left( \begin{array} { c } p - 1
q + 2
2 r - 3 \end{array} \right)\) and \(\mathbf { b } = \left( \begin{array} { c } 2 p + 4
2 q - 5
r + 3 \end{array} \right)\), where \(p , q\) and \(r\) are real numbers.
  1. Given that \(\mathbf { b }\) is not a multiple of \(\mathbf { a }\) and that \(\mathbf { a } \times \mathbf { b } = \mathbf { 0 }\), determine all possible sets of values of \(p , q\) and \(r\).
  2. You are given instead that \(\mathbf { b } = \lambda \mathbf { a }\), where \(\lambda\) is an integer with \(| \lambda | > 1\). By writing each of \(p , q\) and \(r\) in terms of \(\lambda\), show that there is a unique value of \(\lambda\) for which \(p , q\) and \(r\) are all integers, stating this set of values of \(p , q\) and \(r\).
OCR Further Additional Pure 2024 June Q5
5 In a conservation project in a nature reserve, scientists are modelling the population of one species of animal. The initial population of the species, \(P _ { 0 }\), is 10000 . After \(n\) years, the population is \(P _ { n }\). The scientists believe that the year-on-year change in the population can be modelled by a recurrence relation of the form
\(\mathrm { P } _ { \mathrm { n } + 1 } = 2 \mathrm { P } _ { \mathrm { n } } \left( 1 - \mathrm { k } \mathrm { P } _ { \mathrm { n } } \right)\) for \(n \geqslant 0\), where \(k\) is a constant.
  1. The initial aim of the project is to ensure that the population remains constant. Show that this happens, according to this model, when \(k = 0.00005\).
  2. After a few years, with the population still at 10000 , the scientists suggest increasing the population. One way of achieving this is by adding 50 more of these animals into the nature reserve at the end of each year. In this scenario, the recurrence system modelling the population (using \(k = 0.00005\) ) is given by
    \(P _ { 0 } = 10000\) and \(\mathrm { P } _ { \mathrm { n } + 1 } = 2 \mathrm { P } _ { \mathrm { n } } \left( 1 - 0.00005 \mathrm { P } _ { \mathrm { n } } \right) + 50\) for \(n \geqslant 0\).
    Use your calculator to find the long-term behaviour of \(P _ { n }\) predicted by this recurrence system.
  3. However, the scientists decide not to add any animals at the end of each year. Also, further research predicts that certain factors will remove 2400 animals from the population each year.
    1. Write down a modified form of the recurrence relation given in part (b), that will model the population of these animals in the nature reserve when 2400 animals are removed each year and no additional animals are added.
    2. Use your calculator to find the behaviour of \(P _ { n }\) predicted by this modified form of the recurrence relation over the course of the next ten years.
    3. Show algebraically that this modified form of the recurrence relation also gives a constant value of \(P _ { n }\) in the long term, which should be stated.
    4. Determine what constant value should replace 0.00005 in this modified form of the recurrence relation to ensure that the value of \(P _ { n }\) remains constant at 10000 .
OCR Further Additional Pure 2024 June Q6
6 The surface \(C\) is given by the equation \(z = x ^ { 2 } + y ^ { 3 } + a x y\) for all real \(x\) and \(y\), where \(a\) is a non-zero real number.
  1. Show that \(C\) has two stationary points, one of which is at the origin, and give the coordinates of the second in terms of \(a\).
  2. Determine the nature of these stationary points of \(C\).
  3. Explain what can be said about the location and nature of the stationary point(s) of the surface given by the equation \(z = x ^ { 2 } + y ^ { 3 }\) for all real \(x\) and \(y\).
OCR Further Additional Pure 2024 June Q7
7 Let \(\mathrm { I } _ { \mathrm { n } } = \int _ { 0 } ^ { 2 } \frac { \mathrm { x } ^ { \mathrm { n } } } { \sqrt { \mathrm { x } ^ { 3 } + 1 } } \mathrm { dx }\) for integers \(n > 0\).
  1. By considering the derivative of \(\sqrt { x ^ { 3 } + 1 }\) with respect to \(x\), determine the exact value of \(I _ { 2 }\).
  2. Given that \(n > 3\), show that \(\left. ( 2 n - 1 ) \right| _ { n } = 3 \times 2 ^ { n - 1 } - \left. 2 ( n - 2 ) \right| _ { n - 3 }\).
  3. Hence determine the exact value of \(\int _ { 0 } ^ { 2 } x ^ { 5 } \sqrt { x ^ { 3 } + 1 } \mathrm {~d} x\).
OCR Further Additional Pure 2024 June Q8
8 The group \(G\) is cyclic and of order 12.
    1. State the possible orders of all the proper subgroups of \(G\). You must justify your answers.
    2. List all the elements of each of these subgroups.
    3. Explain why \(G\) must be abelian. The group \(\mathbb { Z } _ { k }\) is the cyclic group of order \(k\), consisting of the elements \(\{ 0,1,2 , \ldots , k - 1 \}\) under the operation \(+ _ { k }\) of addition modulo \(k\). The coordinate group \(\mathrm { C } _ { \mathrm { mn } }\) is the group which consists of elements of the form \(( x , y )\), where \(\mathrm { x } \in \mathbb { Z } _ { \mathrm { m } }\) and \(\mathrm { y } \in \mathbb { Z } _ { \mathrm { n } }\), under the operation \(\oplus\) given by \(\left( \mathrm { x } _ { 1 } , \mathrm { y } _ { 1 } \right) \oplus \left( \mathrm { x } _ { 2 } , \mathrm { y } _ { 2 } \right) = \left( \mathrm { x } _ { 1 } + { } _ { \mathrm { m } } \mathrm { x } _ { 2 } , \mathrm { y } _ { 1 } + { } _ { \mathrm { n } } \mathrm { y } _ { 2 } \right)\). For example, for \(m = 5\) and \(n = 2 , ( 3,0 ) \oplus ( 4,1 ) = ( 2,1 )\).
    1. List all the elements of \(\mathrm { J } = \mathrm { C } _ { 34 }\).
    2. Show that \(G\) and \(J\) are isomorphic. There is a second coordinate group of order 12; that is, \(\mathrm { K } = \mathrm { C } _ { \mathrm { mn } }\), where \(1 < \mathrm { m } < \mathrm { n } < 12\) but neither \(m\) nor \(n\) is equal to 3 or 4 .
    1. State the values of \(m\) and \(n\) which give \(K\).
    2. Hence list all of the elements of \(K\).
    3. Explain why \(K\) must be abelian.
  4. Show that \(G\) and \(K\) are not isomorphic. \section*{END OF QUESTION PAPER}
OCR Further Additional Pure 2020 November Q1
1 The following Cayley table is for a set \(\{ a , b , c , d \}\) under a suitable binary operation.
\(a\)\(b\)\(c\)\(d\)
\(a\)\(b\)\(a\)
\(b\)
\(c\)\(c\)
\(d\)\(d\)\(a\)
  1. Given that the Latin square property holds for this Cayley table, complete it using the table supplied in the Printed Answer Booklet.
  2. Using your completed Cayley table, explain why the set does not form a group under the binary operation.
OCR Further Additional Pure 2020 November Q2
2 For \(x , y \in \mathbb { R }\), the function f is given by \(\mathrm { f } ( x , y ) = 2 x ^ { 2 } \mathrm { y } ^ { 7 } + 3 x ^ { 5 } y ^ { 4 } - 5 x ^ { 8 } y\).
  1. Prove that \(\mathrm { xf } _ { \mathrm { x } } + \mathrm { yf } _ { \mathrm { y } } = \mathrm { nf }\), where \(n\) is a positive integer to be determined.
  2. Show that \(\mathrm { xf } _ { \mathrm { xx } } + \mathrm { yf } _ { \mathrm { xy } } = ( \mathrm { n } - 1 ) \mathrm { f } _ { \mathrm { x } }\).
OCR Further Additional Pure 2020 November Q3
3 For integers \(n \geqslant 0 , \mathrm { I } _ { \mathrm { n } } = \int _ { 0 } ^ { 1 } \frac { \mathrm { x } ^ { \mathrm { n } } } { 1 + \mathrm { x } ^ { 2 } } \mathrm { dx }\).
  1. For integers \(n \geqslant 2\), show that \(I _ { n } + I _ { n - 2 } = \frac { 1 } { n - 1 }\).
    1. Determine the exact value of \(I _ { 10 }\).
    2. Deduce that \(\pi < 3 \frac { 107 } { 315 }\).
OCR Further Additional Pure 2020 November Q4
4 Points \(A , B\) and \(C\) have position vectors \(\mathbf { a } , \mathbf { b }\) and \(\mathbf { c }\) respectively, relative to origin \(O\). It is given that \(\mathbf { b } \times \mathbf { c } = \mathbf { a }\) and that \(| \mathbf { a } | = 3\).
  1. Determine each of the following as either a single vector or a scalar quantity.
    1. \(\mathbf { c } \times \mathbf { b }\)
    2. \(\mathbf { a } \times ( \mathbf { b } \times \mathbf { c } )\)
    3. \(\mathbf { a } \cdot ( \mathbf { b } \times \mathbf { c } )\)
  2. Describe a geometrical relationship between the points \(O , A , B\) and \(C\) which can be deduced from
    1. the statement \(\mathbf { b } \times \mathbf { c } = \mathbf { a }\),
    2. the result of (a)(iii).
OCR Further Additional Pure 2020 November Q5
5 A designer intends to manufacture a product using a 3-D printer. The product will take the form of a surface \(S\) which must meet a number of design specifications. The designer chooses to model \(S\) with the equation \(\mathrm { Z } = \mathrm { y } \cosh \mathrm { x }\) for \(- \ln 20 \leqslant x \leqslant \ln 20 , - 2 \leqslant y \leqslant 2\).
    1. In the Printed Answer Booklet, on the axes provided, sketch the section of \(S\) given by \(y = 1\).
    2. One of the design specifications of the product is that this section should have a length no greater than 20 units. Determine whether the product meets this requirement according to the model.
    1. In the Printed Answer Booklet, on the axes provided, sketch the contour of \(S\) given by \(z = 1\).
    2. When this contour is rotated through \(2 \pi\) radians about the \(x\)-axis, the surface \(T\) is generated. The surface area of \(T\) is denoted by \(A\). Show that \(A\) can be written in the form \(\mathrm { k } \pi \int _ { 0 } ^ { \ln 20 } \frac { 1 } { \cosh ^ { 3 } \mathrm { x } } \sqrt { \cosh ^ { 4 } \mathrm { x } + \cosh ^ { 2 } \mathrm { x } - 1 } \mathrm { dx }\) for some
      integer \(k\) to be determined. integer \(k\) to be determined.
    3. A second design specification is that the surface area of \(T\) must not be greater than 20 square units. Use your calculator to decide whether the product meets this requirement according to the model.
OCR Further Additional Pure 2020 November Q6
6 The group \(G\) consists of the set \(\{ 3,6,9,12,15,18,21,24,27,30,33,36 \}\) under \(\times _ { 39 }\), the operation of multiplication modulo 39.
  1. List the possible orders of proper subgroups of \(G\), justifying your answer.
  2. List the elements of the subset of \(G\) generated by the element 3 .
  3. State the identity element of \(G\).
  4. Determine the order of the element 18 .
  5. Find the two elements \(g _ { 1 }\) and \(g _ { 2 }\) in \(G\) which satisfy \(g \times { } _ { 39 } g = 3\). The group \(H\) consists of the set \(\{ 1,2,3,4,5,6,7,8,9,10,11,12 \}\) under \(\times _ { 13 }\), the operation of multiplication modulo 13. You are given that \(G\) is isomorphic to \(H\). A student states that \(G\) is isomorphic to \(H\) because each element \(3 x\) in \(G\) maps directly to the element \(x\) in \(H\) (for \(x = 1,2,3,4,5,6,7,8,9,10,11,12\) ).
  6. Explain why this student is incorrect.
OCR Further Additional Pure 2020 November Q7
7 Throughout this question, \(n\) is a positive integer.
  1. Explain why \(n ^ { 5 } \equiv n ( \bmod 5 )\).
  2. By proving that \(n ^ { 5 } \equiv n ( \bmod 2 )\), show that \(n ^ { 5 } \equiv n ( \bmod 10 )\).
    1. Prove that \(n ^ { 5 } - n\) is divisible by 30 for all positive integers \(n\).
    2. Is there an integer \(N\), greater than 30 , such that \(n ^ { 5 } - n\) is divisible by \(N\) for all positive integers \(n\) ? Justify your answer.
OCR Further Additional Pure 2020 November Q8
8 The sequence \(\left\{ u _ { n } \right\}\) of positive real numbers is defined by \(u _ { 1 } = 1\) and \(u _ { n + 1 } = \frac { 2 u _ { n } + 3 } { u _ { n } + 2 }\) for \(n \geqslant 1\).
  1. Prove by induction that \(u _ { n } ^ { 2 } - 3 < 0\) for all positive integers \(n\).
  2. By considering \(u _ { n + 1 } - u _ { n }\), use the result of part (a) to show that \(u _ { n + 1 } > u _ { n }\) for all positive integers \(n\). The sequence \(\left\{ u _ { n } \right\}\) has a limit for \(n \rightarrow \infty\).
  3. Find the limit of the sequence \(\left\{ u _ { n } \right\}\) as \(n \rightarrow \infty\).
  4. Describe as fully as possible the behaviour of the sequence \(\left\{ u _ { n } \right\}\). \section*{END OF QUESTION PAPER}
OCR Further Additional Pure 2021 November Q2
2 The following Cayley table is for \(G\), a group of order 6. The identity element is \(e\) and the group is generated by the elements \(a\) and \(b\).
G\(e\)\(а\)\(a ^ { 2 }\)\(b\)\(a b\)\(\mathrm { a } ^ { 2 } \mathrm {~b}\)
\(e\)\(e\)\(а\)\(a ^ { 2 }\)\(b\)\(a b\)\(\mathrm { a } ^ { 2 } \mathrm {~b}\)
\(a\)\(а\)\(a ^ { 2 }\)\(e\)\(a b\)\(\mathrm { a } ^ { 2 } \mathrm {~b}\)\(b\)
\(a ^ { 2 }\)\(a ^ { 2 }\)\(e\)\(a\)\(\mathrm { a } ^ { 2 } \mathrm {~b}\)\(b\)\(a b\)
\(b\)b\(\mathrm { a } ^ { 2 } \mathrm {~b}\)\(a b\)\(e\)\(a ^ { 2 }\)\(a\)
\(a b\)\(a b\)b\(\mathrm { a } ^ { 2 } \mathrm {~b}\)\(a\)\(e\)\(a ^ { 2 }\)
\(\mathrm { a } ^ { 2 } \mathrm {~b}\)\(\mathrm { a } ^ { 2 } \mathrm {~b}\)\(a b\)b\(a ^ { 2 }\)\(a\)\(e\)
  1. List all the proper subgroups of \(G\).
  2. State another group of order 6 to which \(G\) is isomorphic.
OCR Further Additional Pure 2021 November Q3
3 The points \(P , Q\) and \(R\) have position vectors \(\mathbf { p } = 2 \mathbf { i } + \mathbf { j } + 5 \mathbf { k } , \mathbf { q } = \mathbf { i } - \mathbf { j } + \mathbf { k }\) and \(\mathbf { r } = 2 \mathbf { i } + \mathbf { j } + t \mathbf { k }\) respectively, relative to the origin \(O\). Determine the value(s) of \(t\) in each of the following cases.
  1. The line \(O R\) is parallel to \(\mathbf { p } \times \mathbf { q }\).
  2. The volume of tetrahedron \(O P Q R\) is 13 .
OCR Further Additional Pure 2021 November Q4
4 Solve the simultaneous linear congruences \(x \equiv 1 ( \bmod 3 ) , x \equiv 5 ( \bmod 11 ) , 2 x \equiv 5 ( \bmod 17 )\).
OCR Further Additional Pure 2021 November Q5
5 The surface \(S\) has equation \(x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = x y z - 1\).
  1. Show that \(( 2 z - x y ) \left( x \frac { \partial z } { \partial x } + y \frac { \partial z } { \partial y } \right) = 2 \left( 1 + z ^ { 2 } \right)\).
  2. Deduce that \(S\) has no stationary point.
OCR Further Additional Pure 2021 November Q6
6 The binary operation ◇ is defined on the set \(\mathbb { C }\) of complex numbers by
\(( a + i b ) \diamond ( c + i d ) = a c + i ( b + a d )\)
where \(a , b , c\) and \(d\) are real numbers.
  1. Is \(\mathbb { C }\) closed under △ ? Justify your answer.
  2. Prove that ◇ is associative on \(\mathbb { C }\).
  3. Determine the identity element of \(\mathbb { C }\) under \(\diamond\).
  4. Determine the largest subset S of \(\mathbb { C }\) such that \(( \mathrm { S } , \diamond )\) is a group.
OCR Further Additional Pure 2021 November Q7
7 Let \(\mathrm { I } _ { \mathrm { n } } = \int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \cos ^ { \mathrm { n } } \mathrm { xdx }\) for integers \(n \geqslant 0\).
  1. Show that, for \(n \geqslant 2 , \quad \mathrm { nl } _ { \mathrm { n } } = ( \mathrm { n } - 1 ) \mathrm { I } _ { \mathrm { n } - 2 }\).
  2. Use this reduction formula to deduce the exact value of \(I _ { 8 }\).
  3. Use the results of parts (a) and (b) to determine the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \cos ^ { 6 } x \sin ^ { 2 } x d x\).
OCR Further Additional Pure 2021 November Q8
8
  1. Solve the second-order recurrence system \(\mathrm { H } _ { \mathrm { n } + 2 } = 5 \mathrm { H } _ { \mathrm { n } + 1 } - 4 \mathrm { H } _ { \mathrm { n } }\) with \(H _ { 0 } = 3 , H _ { 1 } = 7\) for \(n \geqslant 0\).
    1. Write down the quadratic residues modulo 10 .
    2. By considering the sequence \(\left\{ \mathrm { H } _ { \mathrm { n } } \right\}\) modulo 10, prove that \(\mathrm { H } _ { \mathrm { n } }\) is never a perfect square.
OCR Further Additional Pure 2021 November Q9
9 For each value of \(k\) the sequence of real numbers \(\left\{ u _ { n } \right\}\) is given by \(u _ { 1 } = 2\) and \(u _ { n + 1 } = \frac { k } { 6 + u _ { n } }\). For each of the following cases, either determine a value of \(k\) or prove that one does not exist.
  1. \(\left\{ \mathrm { u } _ { n } \right\}\) is constant.
  2. \(\left\{ \mathrm { u } _ { \mathrm { n } } \right\}\) is periodic, with period 2 .
  3. \(\left\{ \mathrm { u } _ { \mathrm { n } } \right\}\) is periodic, with period 4 .
OCR Further Additional Pure 2021 November Q10
10
\includegraphics[max width=\textwidth, alt={}, center]{df94bc38-5187-4349-9005-f9b72691c70d-4_519_770_251_242} A student wishes to model the saddle of a horse. They use a surface described by a function of the form \(\mathrm { z } = \mathrm { f } ( \mathrm { x } , \mathrm { y } )\) with a saddle point at the origin \(O\). The z -axis is vertically upwards. The \(x\) - and \(y\)-axes lie in a horizontal plane, with the \(x\)-axis across the horse and the \(y\)-axis along the length of the horse (see diagram). The arc \(A O B\) is part of a parabola which lies in the \(y z\)-plane. The arc \(C O D\) is part of a parabola which lies in the \(x z\)-plane. The saddle is symmetric in both the \(x z\)-plane and \(y z\)-plane. The length of the saddle, the distance \(A B\), is to be 0.6 m with both \(A\) and \(B\) at a height of 0.27 m above \(O\). The width of the saddle, the distance \(C D\), is to be 0.5 m with both \(C\) and \(D\) at a depth of 0.4 m below \(O\).
  1. On separate diagrams, sketch the sections \(x = 0\) and \(y = 0\).
    [0pt]
  2. Determine a function f that describes the saddle. [You do not need to state the domain of function f .] \section*{END OF QUESTION PAPER} \section*{OCR
    Oxford Cambridge and RSA}