OCR Further Additional Pure AS (Further Additional Pure AS) 2020 November

1

1. Evaluate \(13 \times 19\) modulo 31 .
2. Solve the linear congruence \(13 x \equiv 9 ( \bmod 31 )\).

2 An open-topped rectangular box is to be manufactured with a fixed volume of \(1000 \mathrm {~cm} ^ { 3 }\). The dimensions of the base of the box are \(x \mathrm {~cm}\) by \(y \mathrm {~cm}\). The surface area of the box is \(A \mathrm {~cm} ^ { 2 }\).

1. Show that \(\mathrm { A } = \mathrm { xy } + 2000 \left( \frac { 1 } { \mathrm { x } } + \frac { 1 } { \mathrm { y } } \right)\).
2. 1. Use partial differentiation to determine, in exact form, the values of \(x\) and \(y\) for which \(A\) has a stationary value.
   2. Find the stationary value of \(A\).

3 In this question, \(N\) is the number 26132652.

1. Without dividing \(N\) by 13, explain why 13 is a factor of \(N\).
2. Use standard divisibility tests to show that 36 is a factor of \(N\). It is given that \(N = 36 \times 725907\).
3. Use the results of parts (a) and (b) to deduce that 13 is a factor of 725907.

4

1. For the set \(S = \{ 2,4,6,8,10,12 \}\), under the operation \(\times _ { 14 }\) of multiplication modulo 14, complete the Cayley table given in the Printed Answer Booklet.
2. Show that ( \(S , \times _ { 14 }\) ) forms a group, \(G\). (You may assume that \(\times _ { 14 }\) is associative.)
3. 1. Write down all the proper subgroups of \(G\).
   2. Given that \(G\) is cyclic, write down all the possible generators of \(G\).

5

1. Determine the general solution of the first-order recurrence relation \(V _ { n + 1 } = 2 V _ { n } + n\).
2. Given that \(V _ { 1 } = 8\), find the exact value of \(V _ { 20 }\).

6 The points \(A\) and \(B\) have position vectors \(\mathbf { a } = \mathbf { i } + 2 \mathbf { j } + \mathbf { k }\) and \(\mathbf { b } = - 3 \mathbf { i } + 4 \mathbf { j } - 5 \mathbf { k }\) respectively.

1. Determine the area of triangle \(O A B\), giving your answer in an exact form. The point \(C\) lies on the line \(( \mathbf { r } - \mathbf { a } ) \times ( \mathbf { b } - \mathbf { a } ) = \mathbf { O }\) such that the area of triangle \(O A C\) is half the area of triangle \(O A B\).
2. Determine the two possible position vectors of \(C\).

7 In a conservation project, a batch of 100000 tadpoles which have just hatched from eggs is introduced into an environment which has no frog population. Previous research suggests that for every 1 million tadpoles hatched only 3550 will live to maturity at 12 weeks, when they become adult frogs. It is assumed that the steady decline in the population of tadpoles, from all causes, can be explained by a weekly death-rate factor, \(r\), which is constant across each week of this twelve-week period. Let \(\mathrm { T } _ { \mathrm { k } }\) denote the total number of tadpoles alive at the end of \(k\) weeks after the start of this project.

1. 1. Explain why a recurrence system for \(\mathrm { T } _ { \mathrm { k } }\) is given by \(T _ { 0 } = 100000\) and \(\mathrm { T } _ { \mathrm { k } + 1 } = ( 1 - \mathrm { r } ) \mathrm { T } _ { \mathrm { k } }\) for \(0 \leqslant k \leqslant 12\).
   2. Show that \(r = 0.375\), correct to 3 significant figures. The proportion of females within each batch of tadpoles is \(p\), where \(0 < p < 1\). In a simple model of the frog population the following assumptions are made.
      • The death rate factor for adult frogs is also \(r\) and is the same for males and females.
2. The frog population will survive provided there are at least thirty female frogs alive sixteen weeks after the start of this project.
3. 1. Find the smallest value of \(p\) for which the frog population will survive according to the model.
   2. Write down one assumption that has been made in order to obtain this result.
Each surviving female will then lay a batch of eggs from which 2500 tadpoles are hatched.
4. By considering the total number of tadpoles hatched, give one criticism of the assumption that the frog population will survive provided there are at least thirty female frogs alive sixteen weeks after the start of this project. \section*{END OF QUESTION PAPER}