OCR Further Additional Pure AS (Further Additional Pure AS) 2019 June

1 In decimal (base 10) form, the number \(N\) is 15260.

1. Express \(N\) in binary (base 2) form.
2. Using the binary form of \(N\), show that \(N\) is divisible by 7 .

2

1. The convergent sequence \(\left\{ \mathrm { a } _ { \mathrm { n } } \right\}\) is defined by \(a _ { 0 } = 1\) and \(\mathrm { a } _ { \mathrm { n } + 1 } = \sqrt { \mathrm { a } _ { \mathrm { n } } } + \frac { 4 } { \sqrt { \mathrm { a } _ { \mathrm { n } } } }\) for \(n \geqslant 0\). Calculate the limit of the sequence.
2. The convergent sequence \(\left\{ \mathrm { b } _ { \mathrm { n } } \right\}\) is defined by \(\mathrm { b } _ { 0 } = 1\) and \(\mathrm { b } _ { \mathrm { n } + 1 } = \sqrt { \mathrm { b } _ { \mathrm { n } } } + \frac { \mathrm { k } } { \sqrt { \mathrm { b } _ { \mathrm { n } } } }\) for \(n \geqslant 0\), where \(k\) is a constant. Determine the value of \(k\) for which the limit of the sequence is 9 .

3 The non-zero vectors \(\mathbf { x }\) and \(\mathbf { y }\) are such that \(\mathbf { x } \times \mathbf { y } = \mathbf { 0 }\).

1. Explain the geometrical significance of this statement.
2. Use your answer to part (a) to explain how the line equation \(\mathbf { r } = \mathbf { a } + t \mathbf { d }\) can be written in the form \(( \mathbf { r } - \mathbf { a } ) \times \mathbf { d } = \mathbf { 0 }\).

4 The sequence \(\left\{ \mathrm { u } _ { \mathrm { n } } \right\}\) is defined by \(u _ { 1 } = 1\) and \(\mathrm { u } _ { \mathrm { n } + 1 } = 2 \mathrm { u } _ { \mathrm { n } } + \mathrm { n } ^ { 2 }\) for \(\mathrm { n } \geqslant 1\).
Determine \(u _ { n }\) as a function of \(n\).

5 The tetrahedron \(T\), shown below, has vertices at \(O ( 0,0,0 ) , A ( 1,2,2 ) , B ( 2,1,2 )\) and \(C ( 2,2,1 )\).
\includegraphics[max width=\textwidth, alt={}, center]{59fa1650-a296-471e-93b9-0988177cd89d-3_360_464_319_555} Diagram not drawn to scale Show that the surface area of \(T\) is \(\frac { 1 } { 2 } \sqrt { 3 } ( 1 + \sqrt { 51 } )\).

6

1. Determine all values of \(x\) for which \(16 x \equiv 5 ( \bmod 101 )\).
2. Solve
   1. \(95 x \equiv 6 ( \bmod 101 )\),
   2. \(95 x \equiv 5 ( \bmod 101 )\).

7 You are given the set \(S = \{ 1,5,7,11,13,17 \}\) together with \(\times _ { 18 }\), the operation of multiplication modulo 18.

1. Complete the Cayley table for \(\left( S , \times _ { 18 } \right)\) given in the Printed Answer Booklet.
2. Prove that ( \(S , \times _ { 18 }\) ) is a group. (You may assume that \(\times _ { 18 }\) is associative.)
3. Write down the order of each element of the group.
4. Show that \(\left( S , \times _ { 18 } \right)\) is a cyclic group.
5. 1. Give an example of a non-cyclic group of order 6 .
   2. Give one reason why your example is structurally different to \(\left( S , { } _ { 18 } \right)\).

8 The motion of two remote controlled helicopters \(P\) and \(Q\) is modelled as two points moving along straight lines. Helicopter \(P\) moves on the line \(\mathbf { r } = \left( \begin{array} { r } 2 + 4 p
- 3 + p
1 + 3 p \end{array} \right)\) and helicopter \(Q\) moves on the line \(\mathbf { r } = \left( \begin{array} { l } 5 + 8 q
2 + q
5 + 4 q \end{array} \right)\).
The function \(z\) denotes \(( P Q ) ^ { 2 }\), the square of the distance between \(P\) and \(Q\).

1. Show that \(z = 26 p ^ { 2 } + 81 q ^ { 2 } - 90 p q - 58 p + 90 q + 50\).
2. Use partial differentiation to find the values of \(p\) and \(q\) for which \(z\) has a stationary point.
3. With the aid of a diagram, explain why this stationary point must be a minimum point, rather than a maximum point or a saddle point.
4. Hence find the shortest possible distance between the two helicopters. The model is now refined by modelling each helicopter as a sphere of radius 0.5 units.
5. Explain how this will change your answer to part (d). \section*{END OF QUESTION PAPER}