OCR Further Additional Pure (Further Additional Pure) 2019 June

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.

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 \(\mathrm { f } _ { y x }\).
2. 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.

3

1. Solve \(7 x \equiv 6 ( \bmod 19 )\).
2. Show that the following simultaneous linear congruences have no solution. $$x \equiv 3 ( \bmod 4 ) , x \equiv 4 ( \bmod 6 )$$

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 \(T _ { 20 }\).

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|}{\(\times _ { 80 }\)}	1	9	31	39
   1	1	9	31	39
   9	9	1	39	31
   31	31	39	1	9
   39	39	31	9	1

2. Use the two Cayley tables to explain why \(G\) and \(H\) are not isomorphic.
3. (i) List

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
   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 (*).

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.

8 In this question you must show detailed reasoning.

1. Prove that \(2 ( p - 2 ) ^ { p - 2 } \equiv - 1 ( \bmod p )\), where \(p\) is an odd prime.
2. Find two odd prime factors of the number \(N = 2 \times 34 ^ { 34 } - 2 ^ { 15 }\). \section*{END OF QUESTION PAPER} \section*{OCR
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