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OCR Further Additional Pure 2022 June Q8
10 marks Challenging +1.8
8
  1. Explain why all groups of even order must contain at least one self-inverse element (that is, an element of order 2).
  2. Prove that any group, in which every (non-identity) element is self-inverse, is abelian.
  3. A student believes that, if \(x\) and \(y\) are two distinct, non-identity, self-inverse elements of a group, then the element \(x y\) is also self-inverse. The table shown here is the Cayley table for the non-cyclic group of order 6, having elements \(i , a , b , c , d\) and \(e\), where \(i\) is the identity.
    \(i\)\(a\)\(b\)\(c\)\(d\)\(e\)
    \(i\)\(i\)\(a\)\(b\)\(c\)\(d\)\(e\)
    \(a\)\(a\)\(i\)\(d\)\(e\)\(b\)\(c\)
    \(b\)\(b\)\(e\)\(i\)\(d\)\(c\)\(a\)
    \(c\)\(c\)\(d\)\(e\)\(i\)\(a\)\(b\)
    \(d\)\(d\)\(c\)\(a\)\(b\)\(e\)\(i\)
    \(e\)\(e\)\(b\)\(c\)\(a\)\(i\)\(d\)
    By considering the elements of this group, produce a counter-example which proves that this student is wrong.
  4. A group \(G\) has order \(4 n + 2\), for some positive integer \(n\), and \(i\) is the identity element of \(G\). Let \(x\) and \(y\) be two distinct, non-identity, self-inverse elements of \(G\). By considering the set \(\mathrm { H } = \{ \mathrm { i } , \mathrm { x } , \mathrm { y } , \mathrm { xy } \}\), prove by contradiction that not all elements of \(G\) are self-inverse.
OCR Further Additional Pure 2022 June Q9
9 marks Challenging +1.2
9 For all real values of \(x\) and \(y\) the surface \(S\) has equation \(z = 4 x ^ { 2 } + 4 x y + y ^ { 2 } + 6 x + 3 y + k\), where \(k\) is a constant and an integer.
  1. Find \(\frac { \partial z } { \partial x }\) and \(\frac { \partial z } { \partial y }\).
  2. Determine the smallest value of the integer \(k\) for which the whole of \(S\) lies above the \(x - y\) plane.
OCR Further Additional Pure 2023 June Q1
4 marks Standard +0.8
1 The surface \(S\) is defined for all real \(x\) and \(y\) by the equation \(z = x ^ { 2 } + 2 x y\). The intersection of \(S\) with the plane \(\Pi\) gives a section of the surface. On the axes provided in the Printed Answer Booklet, sketch this section when the equation of \(\Pi\) is each of the following.
  1. \(x = 1\)
  2. \(y = 1\)
OCR Further Additional Pure 2023 June Q2
6 marks Challenging +1.2
2 A curve has equation \(\mathrm { y } = \sqrt { 1 + \mathrm { x } ^ { 2 } }\), for \(0 \leqslant x \leqslant 1\), where both the \(x\) - and \(y\)-units are in cm. The area of the surface generated when this curve is rotated fully about the \(x\)-axis is \(A \mathrm {~cm} ^ { 2 }\).
  1. Show that \(\mathrm { A } = 2 \pi \int _ { 0 } ^ { 1 } \sqrt { 1 + \mathrm { kx } ^ { 2 } } \mathrm { dx }\) for some integer \(k\) to be determined. A small component for a car is produced in the shape of this surface. The curved surface area of the component must be \(8 \mathrm {~cm} ^ { 2 }\), accurate to within one percent. The engineering process produces such components with a curved surface area accurate to within one half of one percent.
  2. Determine whether all components produced will be suitable for use in the car.
OCR Further Additional Pure 2023 June Q3
7 marks Challenging +1.2
3 The points \(A\) and \(B\) have position vectors \(\mathbf { a } = \mathbf { i } + \mathrm { pj } + \mathrm { q } \mathbf { k }\) and \(\mathbf { b } = 2 \mathbf { i } + 3 \mathbf { j } + 2 \mathbf { k }\) respectively, relative to the origin \(O\).
  1. Determine the value of \(p\) and the value of \(q\) for which \(\mathbf { a } \times \mathbf { b } = 2 \mathbf { i } + 6 \mathbf { j } - 1 \mathbf { 1 } \mathbf { k }\).
  2. The point \(C\) has coordinates ( \(d , e , f\) ) and the tetrahedron \(O A B C\) has volume 7.
    1. Using the values of \(p\) and \(q\) found in part (a), find the possible relationships between \(d , e\) and \(f\).
    2. Explain the geometrical significance of these relationships.
OCR Further Additional Pure 2023 June Q4
7 marks Hard +2.3
4 The sequence \(\left\{ A _ { n } \right\}\) is given for all integers \(n \geqslant 0\) by \(A _ { n } = \frac { I _ { n + 2 } } { I _ { n } }\), where \(I _ { n } = \int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \cos ^ { n } x d x\).
  • Show that \(\left\{ A _ { n } \right\}\) increases monotonically.
  • Show that \(\left\{ \mathrm { A } _ { \mathrm { n } } \right\}\) converges to a limit, \(A\), whose exact value should be stated.
OCR Further Additional Pure 2023 June Q5
10 marks Challenging +1.2
5
  1. The group \(G\) consists of the set \(S = \{ 1,9,17,25 \}\) under \(\times _ { 32 }\), the operation of multiplication modulo 32.
    1. Complete the Cayley table for \(G\) given in the Printed Answer Booklet.
    2. Up to isomorphisms, there are only two groups of order 4.
      • \(C _ { 4 }\), the cyclic group of order 4
  2. \(K _ { 4 }\), the non-cyclic (Klein) group of order 4
    3. State, with justification, to which of these two groups \(G\) is isomorphic.
    1. List the odd quadratic residues modulo 32.
    2. Given that \(n\) is an odd integer, prove that \(n ^ { 6 } + 3 n ^ { 4 } + 7 n ^ { 2 } \equiv 11 ( \bmod 32 )\).
OCR Further Additional Pure 2023 June Q6
11 marks Challenging +1.8
6 The surface \(S\) has equation \(z = x \sin y + \frac { y } { x }\) for \(x > 0\) and \(0 < y < \pi\).
  1. Determine, as a function of \(x\) and \(y\), the determinant of \(\mathbf { H }\), the Hessian matrix of \(S\).
  2. Given that \(S\) has just one stationary point, \(P\), use the answer to part (a) to deduce the nature of \(P\).
  3. The coordinates of \(P\) are \(( \alpha , \beta , \gamma )\). Show that \(\beta\) satisfies the equation \(\beta + \tan \beta = 0\).
OCR Further Additional Pure 2023 June Q7
10 marks Challenging +1.8
7 Binet's formula for the \(n\)th Fibonacci number is given by \(\mathrm { F } _ { \mathrm { n } } = \frac { 1 } { \sqrt { 5 } } \left( \alpha ^ { \mathrm { n } } - \beta ^ { \mathrm { n } } \right)\) for \(n \geqslant 0\), where \(\alpha\) and \(\beta\) (with \(\alpha > 0 > \beta\) ) are the roots of \(x ^ { 2 } - x - 1 = 0\).
  1. Write down the values of \(\alpha + \beta\) and \(\alpha \beta\).
  2. Consider the sequence \(\left\{ \mathrm { S } _ { \mathrm { n } } \right\}\), where \(\mathrm { S } _ { \mathrm { n } } = \alpha ^ { \mathrm { n } } + \beta ^ { \mathrm { n } }\) for \(n \geqslant 0\).
    1. Determine the values of \(S _ { 2 }\) and \(S _ { 3 }\).
    2. Show that \(S _ { n + 2 } = S _ { n + 1 } + S _ { n }\) for \(n \geqslant 0\).
    3. Deduce that \(S _ { n }\) is an integer for all \(n \geqslant 0\).
  3. A student models the terms of the sequence \(\left\{ \mathrm { S } _ { \mathrm { n } } \right\}\) using the formula \(\mathrm { T } _ { \mathrm { n } } = \alpha ^ { \mathrm { n } }\).
    1. Explain why this formula is unsuitable for every \(n \geqslant 1\).
    2. Considering the cases \(n\) even and \(n\) odd separately, state a modification of the formula \(\mathrm { T } _ { \mathrm { n } } = \alpha ^ { \mathrm { n } }\), other than \(\mathrm { T } _ { \mathrm { n } } = \alpha ^ { \mathrm { n } } + \beta ^ { \mathrm { n } }\), such that \(\mathrm { T } _ { \mathrm { n } } = \mathrm { S } _ { \mathrm { n } }\) for all \(n \geqslant 1\).
OCR Further Additional Pure 2023 June Q8
9 marks Challenging +1.8
8 Let \(f ( n )\) denote the base- \(n\) number \(2121 _ { n }\) where \(n \geqslant 3\).
    1. For each \(n \geqslant 3\), show that \(\mathrm { f } ( n )\) can be written as the product of two positive integers greater than \(1 , \mathrm { a } ( n )\) and \(\mathrm { b } ( n )\), each of which is a function of \(n\).
    2. Deduce that \(\mathrm { f } ( n )\) is always composite.
  2. Let \(h\) be the highest common factor of \(\mathrm { a } ( n )\) and \(\mathrm { b } ( n )\).
    1. Prove that \(h\) is either 1 or 5 .
    2. Find a value of \(n\) for which \(h = 5\).
OCR Further Additional Pure 2023 June Q9
11 marks Challenging +1.8
9 The set \(C\) consists of the set of all complex numbers excluding 1 and - 1 . The operation ⊕ is defined on the elements of \(C\) by \(\mathrm { a } \oplus \mathrm { b } = \frac { \mathrm { a } + \mathrm { b } } { \mathrm { ab } + 1 }\) where \(\mathrm { a } , \mathrm { b } \in \mathrm { C }\).
  1. Determine the identity element of \(C\) under ⊕.
  2. For each element \(x\) in \(C\) show that it has an inverse element in \(C\).
  3. Show that \(\oplus\) is associative on \(C\).
  4. Explain why \(( C , \oplus )\) is not a group.
  5. Find a subset, \(D\), of \(C\) such that \(( D , \oplus )\) is a group of order 3 . \section*{END OF QUESTION PAPER} OCR is committed to seeking permission to reproduce all third-party content that it uses in its assessment materials. OCR has attempted to identify and contact all copyright holders whose work is used in this paper. To avoid the issue of disclosure of answer-related information to candidates, all copyright acknowledgements are reproduced in the OCR Copyright Acknowledgements Booklet. This is produced for each series of examinations and is freely available to download from our public website (\href{http://www.ocr.org.uk}{www.ocr.org.uk}) after the live examination series.
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OCR Further Additional Pure 2024 June Q1
6 marks Challenging +1.2
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
5 marks Standard +0.8
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
6 marks Standard +0.8
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
10 marks Challenging +1.8
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
10 marks Standard +0.8
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
13 marks Standard +0.8
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
10 marks Challenging +1.8
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
15 marks Challenging +1.8
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
5 marks Standard +0.8
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
9 marks Challenging +1.2
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
9 marks Challenging +1.8
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
7 marks Challenging +1.2
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
13 marks Challenging +1.8
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
10 marks Challenging +1.8
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.