Questions - OCR Further Additional Pure AS

OCR Further Additional Pure AS 2023 June Q4

4 The equation of line $l$ can be written in either of the following vector forms.

$\mathbf { r } = \mathbf { a } + \lambda \mathbf { b }$, where $\lambda \in \mathbb { R }$
$( \mathbf { r } - \mathbf { c } ) \times \mathbf { d } = \mathbf { 0 }$
1. Write down two equations involving the vectors $\mathbf { a , b , c }$, and d, giving reasons for your answers.
2. Determine the value of $\mathbf { a } \cdot ( \mathbf { c } \times \mathbf { d } )$.

OCR Further Additional Pure AS 2023 June Q5

5

Express as a decimal (base-10) number the base-23 number $7119 _ { 23 }$.
Solve the linear congruence $7 n + 11 \equiv 9 ( \bmod 23 )$.
Let $N = 10 a + b$ and $M = a + 7 b$, where $a$ and $b$ are integers and $0 \leqslant b \leqslant 9$.
1. By considering $3 N - 7 M$, prove that $23 \mid N$ if and only if $23 \mid M$.
2. Use a procedure based on this result to show that $N = 711965$ is a multiple of 23 .

OCR Further Additional Pure AS 2023 June Q6

6 When $10 ^ { 6 }$ of a certain type of bacteria are detected in a blood sample of an infected animal, a course of treatment is started. The long-term aim of the treatment is to reduce the number of bacteria in such a sample to under 10000 . At this level the animal's immune system can fight the infection for itself. Once treatment has started, if the number of bacteria in a sample is 10000 or more, then treatment either continues or restarts. The model suggested to predict the progress of the course of treatment is based on the recurrence system $P _ { n + 1 } = \frac { 2 P _ { n } } { n + 1 } + \frac { n } { P _ { n } }$ for $n \geqslant 0$, with $P _ { 0 } = 1000$, where $P _ { n }$ denotes the number of bacteria (in thousands) present in the animal's body $n$ days after the treatment was started. The table below shows the values of $P _ { n }$, for certain chosen values of $n$. Each value has been given correct to 2 decimal places (where appropriate).

$n$	0	1	2	3	4	5	6	7	8	9
$P _ { n }$	1000	2000	2000	1333.33	666.67	266.67		25.47	6.64	2.68

$n$	10	20	40	60	80	100	200	300	400
$P _ { n }$	3.89	4.67	6.45	7.84	9.03	10.08	14.20	17.36	20.04

Find the value of $P _ { 6 }$ correct to 2 decimal places.
Using the given values for $P _ { 0 }$ to $P _ { 9 }$, and assuming that the model is valid,
1. describe the effects of this course of treatment during the first 9 days,
2. state the number of days after treatment is started when the animal's own immune system is expected to be able to fight the infection for itself.
1. Using information from the above table, suggest a function f such that, for $n > 10 , \mathrm { f } ( n )$ is a suitable approximation for $P _ { n }$.
2. Use your suggested function to estimate the number of days after treatment is started when the animal may once again require medical intervention in order to help fight off this bacterial infection.
3. Using information from the above table and the recurrence relation, verify or correct the estimate which you found in part (c)(ii).
One criticism of the system $P _ { n + 1 } = \frac { 2 P _ { n } } { n + 1 } + \frac { n } { P _ { n } }$, with $P _ { 0 } = 1000$, is that it gives non-integer
values of $P$. values of $P _ { n }$. Suggest a modification that would correct this issue.

OCR Further Additional Pure AS 2023 June Q7

7 The group $G$, of order 12, consists of the set $\{ 1,2,4,5,8,10,13,16,17,19,20 , x \}$ under the operation of multiplication modulo 21 . The identity of $G$ is the element 1 . The element $x$ is an integer, $0 < x < 21$, distinct from the other elements in the set. An incomplete copy of the Cayley table for $G$ is shown below:

G	1	2	4	5	8	10	13	16	17	19	20	$x$
1	1	2	4	5	8	10	13	16	17	19	20
2	2	4	8	10	16	20	5	$x$	13	17	19
4	4	8	16	20	$x$	19	10	1	5	13	17
5	5	10	20	4	19	8	2	17	1	$x$	16
8	8	16	$x$	19	1	17	20	2	10	5	13
10	10	20	19	8	17	16	4	13	2	1	$x$
13	13	5	10	2	20	4	1	19	$x$	16	8
16	16	$x$	1	17	2	13	19	4	20	10	5
17	17	13	5	1	10	2	$x$	20	16	8	4
19	19	17	13	$x$	5	1	16	10	8	4	2
20	20	19	17	16	13	$x$	8	5	4	2	1
$x$

State, with justification, the value of $x$.
In the table given in the Printed Answer Booklet, list the order of each of the non-identity elements of $G$.
1. Write down all the subgroups of $G$ of order 3 .
2. Write down all the subgroups of $G$ of order 6 .
Determine all the subgroups of $G$ of order 4, and prove that there are no other subgroups of order 4.
State, with a reason, whether $G$ is a cyclic group.

OCR Further Additional Pure AS 2023 June Q8

8 A surface, $C$, is given by the equation $z = \mathrm { f } ( x , y )$ for all real values of $x$ and $y$. You are given that $C$ has the following properties.

The surface is continuous for all $x$ and $y$.
The contour $z = - 1$ is a single point on the $z$-axis.
For $- 1 < a < 1$, the contour $z = a$ is a pair of circles with different radiuses but each having the same centre $( 0,0 , a )$.
The contour $z = 1$ consists of the circle, centre $( 0,0,1 )$ and radius 1 .

Sketch a possible section of $C$ corresponding to $y = 0$.

OCR Further Additional Pure AS 2020 November Q1

1

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

OCR Further Additional Pure AS 2020 November Q2

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 $A \mathrm {~cm} ^ { 2 }$.

Show that $\mathrm { A } = \mathrm { xy } + 2000 \left( \frac { 1 } { \mathrm { x } } + \frac { 1 } { \mathrm { y } } \right)$.
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 $A$.

OCR Further Additional Pure AS 2020 November Q3

3 In this question, $N$ is the number 26132652.

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

OCR Further Additional Pure AS 2020 November Q4

4

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.
Show that ( $S , \times _ { 14 }$ ) forms a group, $G$. (You may assume that $\times _ { 14 }$ is associative.)
1. Write down all the proper subgroups of $G$.
2. Given that $G$ is cyclic, write down all the possible generators of $G$.

OCR Further Additional Pure AS 2020 November Q5

5

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

OCR Further Additional Pure AS 2020 November Q6

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.

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 $O A B$.
Determine the two possible position vectors of $C$.

OCR Further Additional Pure AS 2020 November Q7

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. 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.
The frog population will survive provided there are at least thirty female frogs alive sixteen weeks after the start of this project.
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.
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}

OCR Further Additional Pure AS Specimen Q1

1 The sequence $\left\{ u _ { n } \right\}$ is defined by $u _ { 1 } = 2$ and $u _ { n + 1 } = \frac { 12 } { 1 + u _ { n } }$ for $n \geq 1$.
Given that the sequence converges, with limit $\alpha$, determine the value of $\alpha$.

OCR Further Additional Pure AS Specimen Q2

2 The points $A ( 1,2,2 ) , B ( 8,2,5 ) , C ( - 3,6,5 )$ and $D ( - 10,6,2 )$ are the vertices of parallelogram $A B C D$. Determine the area of $A B C D$.

OCR Further Additional Pure AS Specimen Q3

3 A non-commutative group $G$ consists of the six elements $\left\{ e , a , a ^ { 2 } , b , a b , b a \right\}$ where $e$ is the identity element, $a$ is an element of order 3 and $b$ is an element of order 2 .
By considering the row in $G$ 's group table in which each of the above elements is pre-multiplied by $b$, show that $b a ^ { 2 } = a b$.

OCR Further Additional Pure AS Specimen Q4

4 Let $S$ be the set $\{ 16,36,56,76,96 \}$ and $\times _ { H }$ the operation of multiplication modulo 100 .

Given that $a$ and $b$ are odd positive integers, show that $( 10 a + 6 ) ( 10 b + 6 )$ can also be written in the form $10 n + 6$ for some odd positive integer $n$.
Construct the Cayley table for $\left( S , \times _ { H } \right)$
Show that $\left( S , \times _ { H } \right)$ is a group.
[0pt] [You may use the result that $\times _ { H }$ is associative on $S$.]
Write down all generators of $\left( S , \times _ { H } \right)$.

OCR Further Additional Pure AS Specimen Q5

5 Let $\mathrm { f } ( x , y ) = x ^ { 3 } + y ^ { 3 } - 2 x y + 1$. The surface $S$ has equation $z = \mathrm { f } ( x , y )$.

(a) Find $f _ { x }$.
(b) Find $\mathrm { f } _ { y }$.
(c) Show that $S$ has a stationary point at ( $0,0,1$ ).
(d) Find the coordinates of the second stationary point of $S$.
The section $z = \mathrm { f } ( a , y )$, where $a$ is a constant, has exactly one stationary point. Determine the equation of the section. A customer takes out a loan of $\pounds P$ from a bank at an annual interest rate of $4.9 \%$. Interest is charged monthly at an equivalent monthly interest rate. This interest is added to the outstanding amount of the loan at the end of each month, and then the customer makes a fixed monthly payment of $\pounds M$ in order to reduce the outstanding amount of the loan. Let $L _ { n }$ denote the outstanding amount of the loan at the end of month $n$ after the fixed payment has been made, with $L _ { 0 } = P$.
Explain how the outstanding amount of the loan from one month to the next is modelled by the recurrence relation $$L _ { n + 1 } = 1.004 L _ { n } - M$$ with $L _ { 0 } = P , n \geq 0$.
Solve, in terms of $n , M$ and $P$, the first order recurrence relation given in part (i).
The loan amount is $\pounds 100000$ and will be fully repaid after 10 years. Find, to the nearest pound, the value of the monthly repayment.
The bank's procedures only allow for calculations using integer amounts of pounds. When each monthly amount of the outstanding $\operatorname { debt } \left( L _ { n } \right)$ is calculated it is always rounded up to the nearest pound before the monthly repayment ( $M$ ) is subtracted.
Rewrite (*) to take this into account.
Let $N = 10 a + b$ and $M = a - 5 b$ where $a$ and $b$ are integers such that $a \geq 1$ and $0 \leq b \leq 9$. $N$ is to be tested for divisibility by 17 .
(a) Prove that $17 \mid N$ if and only if $17 \mid M$.
(b) Demonstrate step-by-step how an algorithm based on these forms can be used to show that $17 \mid 4097$.
(a) Show that, for $n \geq 2$, any number of the form $1001 _ { n }$ is composite.
(b) Given that $n$ is a positive even number, provide a counter-example to show that the statement "any number of the form $10001 _ { n }$ is prime" is false. \section*{END OF QUESTION PAPER} OCR is committed to seeking permission to reproduce all third-party content that it uses in the 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 Copyright Team, First Floor, 9 Hills Road, Cambridge CB2 1GE.
OCR is part of the Cambridge Assessment Group; Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge.

OCR Further Additional Pure AS 2017 December Q1

1 Solve $12 x \equiv 3 ( \bmod 99 )$.

OCR Further Additional Pure AS 2017 December Q2

2

For non-zero vectors $\mathbf { x }$ and $\mathbf { y }$, explain the geometrical significance of the statement $\mathbf { x } \times \mathbf { y } = \mathbf { 0 }$.
The points $P$ and $Q$ have position vectors $\mathbf { p } = 2 \mathbf { i } + 7 \mathbf { j } - \mathbf { k }$ and $\mathbf { q } = \mathbf { i } + 3 \mathbf { j } + 4 \mathbf { k }$ respectively. Find, in the form $( \mathbf { r } - \mathbf { a } ) \times \mathbf { d } = \mathbf { 0 }$, the equation of line $P Q$.

OCR Further Additional Pure AS 2017 December Q3

3 The surface with equation $z = x ^ { 3 } + y ^ { 3 } - 6 x y$ has two stationary points; one at the origin and the second at the point $A$. Determine the coordinates of $A$.

OCR Further Additional Pure AS 2017 December Q4

4

The binary operation is defined on $\mathbb { Z }$ by $a$ b $b = a + b - a b$ for all $a , b \in \mathbb { Z }$. Prove that is associative on $\mathbb { Z }$. The operation ∘ is defined on the set $A = \{ 0,2,3,4,5,6 \}$ by $a \circ b = a + b - a b ( \bmod 7 )$ for all $a , b \in A$.
Complete the Cayley table for $\left( A , { } ^ { \circ } \right)$ given in the Printed Answer Booklet.
Prove that $( A , \circ )$ is a group. You may assume that the operation is associative.
List all the subgroups of $( A , \circ )$.

OCR Further Additional Pure AS 2017 December Q5

5 Given that $n$ is a positive integer greater than 2 , prove that

$\quad 10201 _ { n }$ is a square number.
$\quad 1221 _ { n }$ is a composite number.

OCR Further Additional Pure AS 2017 December Q6

6 For real constants $a$ and $b$, the sequence $U _ { 1 } , U _ { 2 } , U _ { 3 } , \ldots$ is given by $$U _ { 1 } = a \text { and } U _ { n } = \left( U _ { n - 1 } \right) ^ { 2 } - b \text { for } n \geqslant 2 .$$

Determine the behaviour of the sequence in the case where $a = 1$ and $b = 3$.
In the case where $b = 6$, find the values of $a$ for which the sequence is constant.
In the case where $a = - 1$ and $b = 8$, prove that $U _ { n }$ is divisible by 7 for all even values of $n$.

OCR Further Additional Pure AS 2017 December Q7

7 The points $A ( 0,35,120 ) , B ( 28,21,120 )$ and $C ( 96,35 , - 72 )$ lie on the sphere $S$, with centre $O$ and radius 125 . Triangle $A B C$ is denoted by $\triangle$.

Find, in simplest surd form, the area of $\Delta$. The points $A , B$ and $C$ also form a spherical triangle, $T$, on the surface of $S$. Each 'side' of $T$ is the shorter arc of the circle, centre $O$ and radius 125, which passes through two of the given vertices of $T$. In order to find an approximation to the area of the spherical triangle, $T$ is being modelled by $\Delta$.
(a) State the assumption being made in using this model.
(b) Say, giving a reason, whether the model gives an under-estimate or an over-estimate of the area of $T$.

OCR Further Additional Pure AS 2017 December Q8

8 A surface $S$ has equation $z = 8 y ^ { 3 } - 6 x ^ { 2 } y + 60 x y - 15 x ^ { 2 } + 186 x - 150 y - 100$.

(a) Find any stationary points of the section of $S$ given by $y = - 3$.
(b) Find any stationary points of the section of $S$ given by $x = - 1$.
Show that the surface $S$ has a least one saddle point. \section*{OCR} Oxford Cambridge and RSA

Questions — OCR Further Additional Pure AS (73 questions)

Browse by module

Browse by board