OCR Further Additional Pure AS (Further Additional Pure AS) 2021 November

1 The points \(A , B\) and \(C\) have position vectors \(\mathbf { a } = \left( \begin{array} { l } 3
0
0 \end{array} \right) , \mathbf { b } = \left( \begin{array} { l } 0
4
0 \end{array} \right)\) and \(\mathbf { c } = \left( \begin{array} { l } 0
0
1 \end{array} \right)\) respectively, relative to the origin \(O\).

1. 1. Calculate \(\mathbf { a } \times \mathbf { b }\), giving your answer as a multiple of \(\mathbf { c }\).
   2. Explain, geometrically, why \(\mathbf { a } \times \mathbf { b }\) must be a multiple of \(\mathbf { c }\).
2. Use a vector product method to calculate the area of triangle \(A B C\).

2 The surface \(S\) is given by \(z = x ^ { 2 } + 4 x y\) for \(- 6 \leqslant x \leqslant 6\) and \(- 2 \leqslant y \leqslant 2\).

1. 1. Write down the equation of any one section of \(S\) which is parallel to the \(x\)-z plane
   2. Sketch the section of (a)(i) on the axes provided in the Printed Answer Booklet.
2. Write down the equation of any one contour of \(S\) which does not include the origin.

3 For positive integers \(n\), the sequence of Fibonacci numbers, \(\left\{ \mathrm { F } _ { \mathrm { n } } \right\}\), starts with the terms \(F _ { 1 } = 1 , F _ { 2 } = 1 , F _ { 3 } = 2 , \ldots\) and is given by the recurrence relation \(\mathrm { F } _ { \mathrm { n } } = \mathrm { F } _ { \mathrm { n } - 1 } + \mathrm { F } _ { \mathrm { n } - 2 } ( \mathrm { n } \geqslant 3 )\).

1. Show that \(\mathrm { F } _ { 3 \mathrm { k } + 3 } = 2 \mathrm {~F} _ { 3 \mathrm { k } + 1 } + \mathrm { F } _ { 3 \mathrm { k } }\), where \(k\) is a positive integer.
2. Prove by induction that \(\mathrm { F } _ { 3 n }\) is even for all positive integers \(n\).

4

1. Let \(a = 1071\) and \(b = 67\).
   1. Find the unique integers \(q\) and \(r\) such that \(\mathrm { a } = \mathrm { bq } + \mathrm { r }\), where \(q > 0\) and \(0 \leqslant r < b\).
   2. Hence express the answer to (a)(i) in the form of a linear congruence modulo \(b\).
2. Use the fact that \(358 \times 715 - 239 \times 1071 = 1\) to prove that 715 and 1071 are co-prime.

5 A trading company deals in two goods. The formula used to estimate \(z\), the total weekly cost to the company of trading the two goods, in tens of thousands of pounds, is
\(z = 0.9 x + \frac { 0.096 y } { x } - x ^ { 2 } y ^ { 2 }\),
where \(x\) and \(y\) are the masses, in thousands of tonnes, of the two goods. You are given that \(x > 0\) and \(y > 0\).

1. In the first week of trading, it was found that the values of \(x\) and \(y\) corresponded to the stationary value of \(z\). Determine the total cost to the company for this week.
2. For the second week, the company intends to make a small change in either \(x\) or \(y\) in order to reduce the total weekly cost. Determine whether the company should change \(x\) or \(y\). (You are not expected to say by how much the company should reduce its costs.)

6 The set \(S\) consists of the following four complex numbers.
\(\begin{array} { l l l l } \sqrt { 3 } + \mathrm { i } & - \sqrt { 3 } - \mathrm { i } & 1 - \mathrm { i } \sqrt { 3 } & - 1 + \mathrm { i } \sqrt { 3 } \end{array}\)
For \(z _ { 1 } , z _ { 2 } \in S\), the binary operation \(\bigcirc\) is defined by \(z _ { 1 } \bigcirc z _ { 2 } = \frac { 1 } { 4 } ( 1 + i \sqrt { 3 } ) z _ { 1 } z _ { 2 }\).

1. 1. Complete the Cayley table for \(( S , \bigcirc )\) given in the Printed Answer Booklet.
   2. Verify that ( \(S , \bigcirc\) ) is a group.
   3. State the order of each element of \(( S , \bigcirc )\).
2. Write down the only proper subgroup of ( \(S , \bigcirc\) ).
3. 1. Explain why ( \(S , \bigcirc\) ) is a cyclic group.
   2. List all possible generators of \(( S , \bigcirc )\).

7

1. Let \(f ( n ) = 2 ^ { 4 n + 3 } + 3 ^ { 3 n + 1 }\). Use arithmetic modulo 11 to prove that \(\mathrm { f } ( n ) \equiv 0 ( \bmod 11 )\) for all integers \(n \geqslant 0\).
2. Use the standard test for divisibility by 11 to prove the following statements.
   1. \(10 ^ { 33 } + 1\) is divisible by 11
   2. \(10 ^ { 33 } + 1\) is divisible by 121

8 A sequence \(\left\{ \mathrm { u } _ { \mathrm { n } } \right\}\) is defined by the recurrence system
\(u _ { 1 } = 1\) and \(\mathrm { u } _ { \mathrm { n } + 1 } = \mathrm { a } - \frac { \mathrm { a } ^ { 2 } } { 2 \mathrm { u } _ { \mathrm { n } } }\) for \(n \geqslant 1\), where \(a\) is a positive constant.
Determine with justification the behaviour of the sequence for all possible values of \(a\). \section*{END OF QUESTION PAPER}