OCR Further Additional Pure (Further Additional Pure) 2021 November

1 In this question you must show detailed reasoning. Express the number \(\mathbf { 4 1 7 2 3 } _ { 10 }\) in hexadecimal (base 16).

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 \(b\).

1. List all the proper subgroups of \(G\).
2. State another group of order 6 to which \(G\) is isomorphic.

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 .

4 Solve the simultaneous linear congruences \(x \equiv 1 ( \bmod 3 ) , x \equiv 5 ( \bmod 11 ) , 2 x \equiv 5 ( \bmod 17 )\).

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.

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.

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 \(\int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \cos ^ { 6 } x \sin ^ { 2 } x d x\).

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 \(n \geqslant 0\).
2. 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.

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 .

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 \(y = 0\).
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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}