Questions - OCR Further Additional Pure AS

OCR Further Additional Pure AS 2018 March Q1

1 Use standard divisibility tests to show that the number $$N = 91039173588$$

is divisible by 9
is divisible by 11
is not divisible by 8 .

OCR Further Additional Pure AS 2018 March Q2

2 The surface $S$ has equation $z = x ^ { 2 } y - 8 x y ^ { 2 } + \frac { x } { y }$ for $y \neq 0$.

(a) Find the following.
- $\frac { \partial z } { \partial x }$
- $\frac { \partial z } { \partial y }$
  (b) Find the coordinates of all stationary points of $S$.
- Find all four second partial derivatives of $z$ with respect to $x$ and/or $y$.

OCR Further Additional Pure AS 2018 March Q3

3 In this question you must show detailed reasoning.

The sequence $\left\{ A _ { n } \right\}$ is given by $A _ { 1 } = \sqrt { 3 }$ and $A _ { n + 1 } = ( \sqrt { 3 } + 1 ) A _ { n }$ for $n \geqslant 1$. Find an expression for $A _ { n }$ in terms of $n$.
The sequence $\left\{ B _ { n } \right\}$ is given by the formula $$B _ { n } = \frac { 1 } { \sqrt { 3 } } \left( ( \sqrt { 3 } + 1 ) ^ { n } - ( \sqrt { 3 } - 1 ) ^ { n } \right) \text { for } n \geqslant 1 .$$ Explain why $B _ { n } \rightarrow \frac { 1 } { \sqrt { 3 } } ( \sqrt { 3 } + 1 ) ^ { n }$ as $n \rightarrow \infty$.
The sequence $\left\{ C _ { n } \right\}$ converges and is defined by $C _ { n } = \frac { A _ { n } } { B _ { n } }$ for $n \geqslant 1$. Identify the limit of $C n$ as $n \rightarrow \infty$.

OCR Further Additional Pure AS 2018 March Q4

4 The group $G$ consists of the symmetries of the equilateral triangle $A B C$ under the operation of composition of transformations (which may be assumed to be associative). Three elements of $G$ are

$\boldsymbol { i }$, the identity
$\boldsymbol { j }$, the reflection in the vertical line of symmetry of the triangle
$\boldsymbol { k }$, the anticlockwise rotation of $120 ^ { \circ }$ about the centre of the triangle.

These are shown in the diagram below.
\includegraphics[max width=\textwidth, alt={}, center]{0b4458dc-4f82-40e4-adcf-cbffca088389-3_204_531_735_772}
\includegraphics[max width=\textwidth, alt={}, center]{0b4458dc-4f82-40e4-adcf-cbffca088389-3_211_543_975_762}
\includegraphics[max width=\textwidth, alt={}, center]{0b4458dc-4f82-40e4-adcf-cbffca088389-3_216_543_1215_762}

Explain why the order of $G$ is 6 .
Determine
- the order of $\boldsymbol { j }$,
- the order of $\boldsymbol { k }$.
- - Express, in terms of $\boldsymbol { j }$ and/or $\boldsymbol { k }$, each of the remaining three elements of $G$.
- Draw a diagram for each of these elements.
- Is the operation of composition of transformations on $G$ commutative? Justify your answer.
- List all the proper subgroups of $G$.

OCR Further Additional Pure AS 2018 March Q5

5 The points $A$ and $B$ have position vectors $\mathbf { a }$ and $\mathbf { b }$ respectively, relative to a fixed origin $O$.

(a) Prove that $\mathbf { a } \times ( \mathbf { b } - \mathbf { a } ) = \mathbf { a } \times \mathbf { b }$.
(b) Determine the relationship between $\mathbf { a } \times ( \mathbf { b } - \mathbf { a } )$ and $\mathbf { b } \times ( \mathbf { b } - \mathbf { a } )$.
The point $D$ is on the line $A B$. $O D$ is perpendicular to $A B$. By considering the area of triangle $O A B$, show
that $| O D | = \frac { | \mathbf { a } \times \mathbf { b } | } { | \mathbf { b } - \mathbf { a } | }$.

OCR Further Additional Pure AS 2018 March Q6

6 You are given that $n$ is an integer.

(a) Show that $\operatorname { hcf } ( 2 n + 1,3 n + 2 ) = 1$.
(b) Hence prove that, if $( 2 n + 1 )$ divides $\left( 36 n ^ { 2 } + 3 n - 14 \right)$, then $( 2 n + 1 )$ divides $( 12 n - 7 )$.
Use the results of part (i) to find all integers $n$ for which $\frac { 36 n ^ { 2 } + 3 n - 14 } { 2 n + 1 }$ is also an integer.

OCR Further Additional Pure AS 2018 March Q7

7 Irrational numbers can be modelled by sequences $\left\{ u _ { n } \right\}$ of rational numbers of the form $$u _ { 0 } = 1 \text { and } u _ { n + 1 } = a + \frac { 1 } { b + u _ { n } } \text { for } n \geqslant 0 \text {, }$$ where $a$ and $b$ are non-negative integer constants.

(a) The constants $a = 1$ and $b = 0$ produce the irrational number $\omega$. State the value of $\omega$ correct to six decimal places.
(b) By setting $u _ { n + 1 }$ and $u _ { n }$ equal to $\omega$, determine the exact value of $\omega$.
Use the method of part (i) (b) to find the exact value of the irrational number produced by taking $a = 0$ and $b = 2$.
Find positive integers $a$ and $b$ which would produce the irrational number $2 + \sqrt { 10 }$. \section*{END OF QUESTION PAPER}

OCR Further Additional Pure AS 2024 June Q1

1 In this question you must show detailed reasoning. The number $N$ is written as 28 A 3 B in base-12 form. Express $N$ in decimal (base-10) form.

OCR Further Additional Pure AS 2024 June Q2

2 The points $A$ and $B$ have position vectors $\mathbf { a }$ and $\mathbf { b }$ relative to the origin $O$. It is given that $\mathbf { a } = \left( \begin{array} { c } 2
4
3 \lambda \end{array} \right)$ and $\mathbf { b } = \left( \begin{array} { r } \lambda
- 4
6 \end{array} \right)$, where $\lambda$ is a real parameter.

In the case when $\lambda = 3$, determine the area of triangle $O A B$.
Determine the value of $\lambda$ for which $\mathbf { a } \times \mathbf { b } = \mathbf { 0 }$.

OCR Further Additional Pure AS 2024 June Q3

3 The surface $S$ has equation $z = f ( x , y )$, where $f ( x , y ) = 4 x ^ { 2 } y - 6 x y ^ { 2 } - \frac { 1 } { 12 } x ^ { 4 }$ for all real values of $x$ and $y$. You are given that $S$ has a stationary point at the origin, $O$, and a second stationary point at the point $P ( a , b , c )$, where $\mathrm { c } = \mathrm { f } ( \mathrm { a } , \mathrm { b } )$.

Determine the values of $a , b$ and $c$.
Throughout this part, take the values of $a$ and $b$ to be those found in part (a).
1. Evaluate $\mathrm { f } _ { x }$ at the points $\mathrm { U } _ { 1 } ( \mathrm { a } - 0.1 , \mathrm {~b} , \mathrm { f } ( \mathrm { a } - 0.1 , \mathrm {~b} ) )$ and $\mathrm { U } _ { 2 } ( \mathrm { a } + 0.1 , \mathrm {~b} , \mathrm { f } ( \mathrm { a } + 0.1 , \mathrm {~b} ) )$.
2. Evaluate $\mathrm { f } _ { y }$ at the points $\mathrm { V } _ { 1 } ( \mathrm { a } , \mathrm { b } - 0.1 , \mathrm { f } ( \mathrm { a } , \mathrm { b } - 0.1 ) )$ and $\mathrm { V } _ { 2 } ( \mathrm { a } , \mathrm { b } + 0.1 , \mathrm { f } ( \mathrm { a } , \mathrm { b } + 0.1 ) )$.
3. Use the answers to parts (b)(i) and (b)(ii) to sketch the portions of the sections of $S$, given by
  - $z = f ( x , b )$, for $| x - a | \leqslant 0.1$,
$z = f ( a , y )$, for $| y - b | \leqslant 0.1$.

OCR Further Additional Pure AS 2024 June Q4

4 The first five terms of the Fibonacci sequence, $\left\{ \mathrm { F } _ { \mathrm { n } } \right\}$, where $n \geqslant 1$, are $F _ { 1 } = 1 , F _ { 2 } = 1 , F _ { 3 } = 2 , F _ { 4 } = 3$ and $F _ { 5 } = 5$.

Use the recurrence definition of the Fibonacci sequence, $\mathrm { F } _ { \mathrm { n } + 1 } = \mathrm { F } _ { \mathrm { n } } + \mathrm { F } _ { \mathrm { n } - 1 }$, to express $\mathrm { F } _ { \mathrm { n } + 4 }$ in terms of $\mathrm { F } _ { \mathrm { n } }$ and $\mathrm { F } _ { \mathrm { n } - 1 }$.
Hence prove by induction that $\mathrm { F } _ { \mathrm { n } }$ is a multiple of 3 when $n$ is a multiple of 4 .

OCR Further Additional Pure AS 2024 June Q5

5 The set $S$ consists of all $2 \times 2$ matrices having determinant 1 or - 1 . For instance, the matrices $\mathbf { P } = \left( \begin{array} { c c } \frac { 1 } { 2 } & \frac { \sqrt { 3 } } { 2 }
- \frac { \sqrt { 3 } } { 2 } & \frac { 1 } { 2 } \end{array} \right) , \mathbf { Q } = - \left( \begin{array} { c c } \frac { 1 } { 2 } & \frac { \sqrt { 3 } } { 2 }
- \frac { \sqrt { 3 } } { 2 } & \frac { 1 } { 2 } \end{array} \right)$ and $\mathbf { R } = \left( \begin{array} { r r } 1 & 0
0 & - 1 \end{array} \right)$ are elements of $S$. It is given that $\times _ { \mathbf { M } }$ is the operation of matrix multiplication.

State the identity element of $S$ under $\times _ { \mathbf { M } }$. The group $G$ is generated by $\mathbf { P }$, under $\times _ { \mathbf { M } }$.
Determine the order of $G$. The group $H$ is generated by $\mathbf { Q }$ and $\mathbf { R }$, also under $\times _ { \mathbf { M } }$.
1. By finding each element of $H$, determine the order of $H$.
2. List all the proper subgroups of $H$.
State whether each of the following statements is true or false. Give a reason for each of your answers.
- $G$ is abelian
- $G$ is cyclic
- $H$ is abelian
- $H$ is cyclic

OCR Further Additional Pure AS 2024 June Q6

6 For positive integers $n$, let $f ( n ) = 1 + 2 ^ { n } + 4 ^ { n }$.

1. Given that $n$ is a multiple of 3 , but not of 9 , use the division algorithm to write down the two possible forms that $n$ can take.
2. Show that when $n$ is a multiple of 3 , but not of 9 , $f ( n )$ is a multiple of 73 .
Determine the value of $\mathrm { f } ( n )$, modulo 73 , in the case when $n$ is a multiple of 9 .

OCR Further Additional Pure AS 2024 June Q7

7 In a long-running biochemical experiment, an initial amount of 1200 mg of an enzyme is placed into a mixture. The model for the amount of enzyme present in the mixture suggests that, at the end of each hour, one-eighth of the amount of enzyme that was present at the start of that hour is used up due to chemical reactions within the mixture. To compensate for this, at the end of each six-hour period of time, a further 500 mg of the enzyme is added to the mixture.

Let $n$ be the number of six-hour periods that have elapsed since the experiment began. Explain how the amount of enzyme, $\mathrm { E } _ { \mathrm { n } } \mathrm { mg }$, in the mixture is given by the recurrence system $E _ { 0 } = 1200$ and $E _ { n + 1 } = \left( \frac { 7 } { 8 } \right) ^ { 6 } E _ { n } + 500$ for $n \geqslant 0$.
Solve the recurrence system given in part (a) to obtain an exact expression for $\mathrm { E } _ { \mathrm { n } }$ in terms of $n$.
Hence determine, in the long term, the amount of enzyme in the mixture. Give your answer correct to $\mathbf { 3 }$ significant figures.
In this question you must show detailed reasoning. The long-running experiment is then repeated. This time a new requirement is added that the amount of enzyme present in the mixture must always be at least 500 mg . Show that the new requirement ceases to be satisfied before 12 hours have elapsed. \section*{END OF QUESTION PAPER} OCR is committed to seeking permission to reproduce all third-party content that it uses in its 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 OCR Copyright Team, The Triangle Building, Shaftesbury Road, Cambridge CB2 8EA.
OCR is part of Cambridge University Press \& Assessment, which is itself a department of the University of Cambridge.

OCR Further Additional Pure AS 2021 November Q1

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. 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 $\mathbf { c }$.
Use a vector product method to calculate the area of triangle $A B C$.

OCR Further Additional Pure AS 2021 November Q2

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. 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.
Write down the equation of any one contour of $S$ which does not include the origin.

OCR Further Additional Pure AS 2021 November Q3

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 $\mathrm { F } _ { \mathrm { n } } = \mathrm { F } _ { \mathrm { n } - 1 } + \mathrm { F } _ { \mathrm { n } - 2 } ( \mathrm { n } \geqslant 3 )$.

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.
Prove by induction that $\mathrm { F } _ { 3 n }$ is even for all positive integers $n$.

OCR Further Additional Pure AS 2021 November Q4

4

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 $b$.
Use the fact that $358 \times 715 - 239 \times 1071 = 1$ to prove that 715 and 1071 are co-prime.

OCR Further Additional Pure AS 2021 November Q5

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 $y > 0$.

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.
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.)

OCR Further Additional Pure AS 2021 November Q6

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. 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 $( S , \bigcirc )$.
Write down the only proper subgroup of ( $S , \bigcirc$ ).
1. Explain why ( $S , \bigcirc$ ) is a cyclic group.
2. List all possible generators of $( S , \bigcirc )$.

OCR Further Additional Pure AS 2021 November Q7

7

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 $n \geqslant 0$.
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

OCR Further Additional Pure AS 2021 November Q8

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}

OCR Further Additional Pure AS 2017 Specimen Q4

60 marks

4 Let $S$ be the set $\{ 16,36,57,76,96 \}$ 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 multiplication table for $\left( S , \times _ { H } \right)$ and show that it is a group. [You may use the result that $\times _ { H }$ is associative on $S$.]
Write down all generators of $\left( S , \times _ { H } \right)$. 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 $\mathrm { f } _ { x }$ and $\mathrm { f } _ { y }$.
(b) Show that f has a stationary point at $P ( 0,0,1 )$ and find the coordinates of the second stationary point, $Q$, of $z$.
(a) State the equation of the plane of symmetry of the surface $S$.
(b) By considering the intersections of $S$ at $P$ with the planes $y = x$ and $y = - x$, show that $P$ is a saddle-point of $S$. A customer takes out a loan of $\pounds P$ from a bank at an annual interest rate of $6.32 \%$. 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 $\pounds 960$ 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 $$L _ { n + 1 } = 1.00512 L _ { n } - 960 ,$$ with $L _ { 0 } = P , n \geq 0$.
Solve, in terms of $n$ and $P$, the first order recurrence relation given in part (i).
Given that the loan is fully repaid after 10 years, evaluate $P$ (to the nearest whole number).
Comment on the practicality of the model.
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*{Copyright Information:} 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. \section*{...day June 20XX - Morning/Afternoon} AS Level Further Mathematics A
Unit Y535 Additional Pure Mathematics SAMPLE MARK SCHEME Duration: 1 hour 15 minutes MAXIMUM MARK 60 \section*{DRAFT} \section*{Text Instructions} \section*{1. Annotations and abbreviations} \section*{2. Subject-specific Marking Instructions for AS Level Further Mathematics A} Annotations should be used whenever appropriate during your marking. The A, M and B annotations must be used on your standardisation scripts for responses that are not awarded either 0 or full marks. It is vital that you annotate standardisation scripts fully to show how the marks have been awarded. For subsequent marking you must make it clear how you have arrived at the mark you have awarded. An element of professional judgement is required in the marking of any written paper. Remember that the mark scheme is designed to assist in marking incorrect solutions. Correct solutions leading to correct answers are awarded full marks but work must not be judged on the answer alone, and answers that are given in the question, especially, must be validly obtained; key steps in the working must always be looked at and anything unfamiliar must be investigated thoroughly. Correct but unfamiliar or unexpected methods are often signalled by a correct result following an apparently incorrect method. Such work must be carefully assessed. When a candidate adopts a method which does not correspond to the mark scheme, escalate the question to your Team Leader who will decide on a course of action with the Principal Examiner.
If you are in any doubt whatsoever you should contact your Team Leader.
The following types of marks are available. \section*{M} A suitable method has been selected and applied in a manner which shows that the method is essentially understood. Method marks are not usually lost for numerical errors, algebraic slips or errors in units. However, it is not usually sufficient for a candidate just to indicate an intention of using some method or just to quote a formula; the formula or idea must be applied to the specific problem in hand, e.g. by substituting the relevant quantities into the formula. In some cases the nature of the errors allowed for the award of an M mark may be specified. \section*{A} Accuracy mark, awarded for a correct answer or intermediate step correctly obtained. Accuracy marks cannot be given unless the associated Method mark is earned (or implied). Therefore M0 A1 cannot ever be awarded. \section*{B} Mark for a correct result or statement independent of Method marks. \section*{E} Mark for explaining a result or establishing a given result. This usually requires more working or explanation than the establishment of an unknown result.
Unless otherwise indicated, marks once gained cannot subsequently be lost, e.g. wrong working following a correct form of answer is ignored. Sometimes this is reinforced in the mark scheme by the abbreviation isw. However, this would not apply to a case where a candidate passes through the correct answer as part of a wrong argument.
d When a part of a question has two or more 'method' steps, the M marks are in principle independent unless the scheme specifically says otherwise; and similarly where there are several B marks allocated. (The notation 'dep*' is used to indicate that a particular mark is dependent on an earlier, asterisked, mark in the scheme.) Of course, in practice it may happen that when a candidate has once gone wrong in a part of a question, the work from there on is worthless so that no more marks can sensibly be given. On the other hand, when two or more steps are successfully run together by the candidate, the earlier marks are implied and full credit must be given.
e The abbreviation FT implies that the A or B mark indicated is allowed for work correctly following on from previously incorrect results. Otherwise, A and B marks are given for correct work only - differences in notation are of course permitted. A (accuracy) marks are not given for answers obtained from incorrect working. When A or B marks are awarded for work at an intermediate stage of a solution, there may be various alternatives that are equally acceptable. In such cases, what is acceptable will be detailed in the mark scheme. If this is not the case please, escalate the question to your Team Leader who will decide on a course of action with the Principal Examiner.
Sometimes the answer to one part of a question is used in a later part of the same question. In this case, A marks will often be 'follow through'. In such cases you must ensure that you refer back to the answer of the previous part question even if this is not shown within the image zone. You may find it easier to mark follow through questions candidate-by-candidate rather than question-by-question.
f Unless units are specifically requested, there is no penalty for wrong or missing units as long as the answer is numerically correct and expressed either in SI or in the units of the question (e.g. lengths will be assumed to be in metres unless in a particular question all the lengths are in km , when this would be assumed to be the unspecified unit.) We are usually quite flexible about the accuracy to which the final answer is expressed; over-specification is usually only penalised where the scheme explicitly says so. When a value is given in the paper only accept an answer correct to at least as many significant figures as the given value. This rule should be applied to each case. When a value is not given in the paper accept any answer that agrees with the correct value to 2 s.f. Follow through should be used so that only one mark is lost for each distinct accuracy error, except for errors due to premature approximation which should be penalised only once in the examination. There is no penalty for using a wrong value for $g$. E marks will be lost except when results agree to the accuracy required in the question. Rules for replaced work: if a candidate attempts a question more than once, and indicates which attempt he/she wishes to be marked, then examiners should do as the candidate requests; if there are two or more attempts at a question which have not been crossed out, examiners should mark what appears to be the last (complete) attempt and ignore the others. NB Follow these maths-specific instructions rather than those in the assessor handbook.
h For a genuine misreading (of numbers or symbols) which is such that the object and the difficulty of the question remain unaltered, mark according to the scheme but following through from the candidate's data. A penalty is then applied; 1 mark is generally appropriate, though this may differ for some units. This is achieved by withholding one A mark in the question. Marks designated as cao may be awarded as long as there are no other errors. E marks are lost unless, by chance, the given results are established by equivalent working. 'Fresh starts' will not affect an earlier decision about a misread. Note that a miscopy of the candidate's own working is not a misread but an accuracy error.
i If a calculator is used, some answers may be obtained with little or no working visible. Allow full marks for correct answers (provided, of course, that there is nothing in the wording of the question specifying that analytical methods are required). Where an answer is wrong but there is some evidence of method, allow appropriate method marks. Wrong answers with no supporting method score zero. If in doubt, consult your Team Leader.
j If in any case the scheme operates with considerable unfairness consult your Team Leader. \end{table} PS = Problem Solving
M = Modelling

Questions — OCR Further Additional Pure AS (73 questions)

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