OCR Further Additional Pure (Further Additional Pure) 2023 June

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. \(y = 1\)

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.

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.

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.

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
State, with justification, to which of these two groups \(G\) is isomorphic.
3. 1. List the odd quadratic residues modulo 32.
   2. Given that \(n\) is an odd integer, prove that \(n ^ { 6 } + 3 n ^ { 4 } + 7 n ^ { 2 } \equiv 11 ( \bmod 32 )\).

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 \(\beta + \tan \beta = 0\).

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 \(n \geqslant 1\).

8 Let \(f ( n )\) denote the base- \(n\) number \(2121 _ { n }\) where \(n \geqslant 3\).

1. 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 \(h = 5\).

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.
   If OCR has unwittingly failed to correctly acknowledge or clear any third-party content in this assessment material, OCR will be happy to correct its mistake at the earliest possible opportunity.
   For queries or further information please contact The OCR Copyright Team, The Triangle Building, Shaftesbury Road, Cambridge CB2 8EA.
   OCR is part of Cambridge University Press \& Assessment, which is itself a department of the University of Cambridge.