OCR Further Additional Pure (Further Additional Pure) 2020 November

1 The following Cayley table is for a set \(\{ a , b , c , d \}\) under a suitable binary operation.

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.

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 \(\mathrm { xf } _ { \mathrm { xx } } + \mathrm { yf } _ { \mathrm { xy } } = ( \mathrm { n } - 1 ) \mathrm { f } _ { \mathrm { x } }\).

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 \(I _ { n } + I _ { n - 2 } = \frac { 1 } { n - 1 }\).
2. 1. Determine the exact value of \(I _ { 10 }\).
   2. Deduce that \(\pi < 3 \frac { 107 } { 315 }\).

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).

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. 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.
2. 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.

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.

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 \(n ^ { 5 } \equiv n ( \bmod 10 )\).
3. 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.

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}