QUESTION PAPER
Candidates answer on the Printed Answer Book.
OCR supplied materials:
• Printed Answer Book 4797
• MEI Examination Formulae and Tables (MF2)
Other materials required:
• Scientific or graphical calculator
• Computer with appropriate software
•
•
•
•
•
•
•
•
INSTRUCTIONS TO CANDIDATES
These instructions are the same on the Printed Answer Book and the Question Paper.
•
The Question Paper will be found in the centre of the Printed Answer Book.
Write your name, centre number and candidate number in the spaces provided on the Printed
Answer Book. Please write clearly and in capital letters.
Write your answer to each question in the space provided in the Printed Answer Book.
Additional paper may be used if necessary but you must clearly show your candidate number, centre number and question number(s).
Use black ink. HB pencil may be used for graphs and diagrams only.
Read each question carefully. Make sure you know what you have to do before starting your answer.
Answer all the questions.
Do not write in the bar codes.
You are permitted to use a scientific or graphical calculator in this paper.
Final answers should be given to a degree of accuracy appropriate to the context.
•
•
INFORMATION FOR CANDIDATES
This information is the same on the Printed Answer Book and the Question Paper.
•
•
The number of marks is given in brackets [ ] at the end of each question or part question on the
Question Paper.
You are advised that an answer may receive no marks unless you show sufficient detail of the working to indicate that a correct method is being used.
The total number of marks for this paper is 72 .
The Printed Answer Book consists of 12 pages. The Question Paper consists of 4 pages. Any blank pages are indicated.
COMPUTING RESOURCES
• Candidates will require access to a computer with a computer algebra system, a spreadsheet, a programming language and graph-plotting software throughout the examination.
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2
1 This question concerns curves with parametric equations x = t − k sin t , y = 1 − cos t , where k is positive and t takes all real values.
(i) Investigate the curves for values of k where 0 < k < 1. Describe the common features of these curves and sketch, for − 2 π ≤ t ≤ 4 π , a typical example.
[5]
(ii) Prove that, if 0 < k < 1, the curve is never parallel to the y -axis.
[3]
(iii) Sketch the curve for k = 1. Describe the main feature by which this curve differs from the curves where 0 < k < 1.
[3]
(iv) For the case k = 1, confirm the behaviour of the curve at the point where t = 0 by investigating the gradient as t → 0.
[5]
(v) Sketch the curve for k = 2. Show that the width of each loop, measured parallel to the x -axis, is
2 p
3 −
2
3
π .
[6]
(vi) Form an equation to find the value of k where the loops will just touch each other. Find the value of k to 4 decimal places.
[3]
2 (i) The function f is defined by f (ß) = ß
3
− ( 3 − 3i )ß
2 plot the roots as points on an Argand diagram.
− 6i ß + 2 + i. Solve the equation f (ß) = 0 and
[6] Show that these points form an equilateral triangle.
(ii) Find f
`
(ß) and show that the equation f
`
(ß) = 0 has a repeated root.
Plot this root on the Argand diagram drawn in part (i) .
[4]
The function f has the following properties:
(∗) the roots of the equation f (ß) = 0 form an equilateral triangle when plotted in the Argand diagram;
(∗∗) the equation f
` (ß) = 0 has a single repeated root.
The rest of this question explores the relationship between (∗) and (∗∗) for other cubic functions.
(iii) Let g (ß) = ß
3
+ b ß
2
+ c ß + d . Find the relationship between b and c when there is a single repeated root of the equation g
`
(ß) = 0.
[3]
(iv) ( A ) Obtain another cubic, with distinct roots, for which (∗∗) is true. Show that (∗) is also true for your cubic.
( B ) Obtain a cubic for which (∗) and (∗∗) are both false.
[3]
The three points representing 0, ß
1
, ß
2 form an equilateral triangle on the Argand diagram.
(v) Given that ß
2
= β ß
1
, β ∈ ; , find the two possible values of β .
[3]
(vi) Use one of your values of β to write down a cubic with roots 0, ß
1 for this cubic.
and ß
2
. Show that (∗∗) is true
[5]
© OCR 2012 DRAFT SPECIMEN

3
3
(i) Create a program to find all the positive integer solutions to x 2 − 3 y 2 = 1 with x ≤ 100, y ≤ 100.
Write out your program in full and list the solutions it gives.
[10]
(ii) Show how the other solutions can be derived from the solution with the smallest x -value.
Use each solution to give a rational approximation to p
3.
[5]
(iii) Edit your program so that it will find solutions to x
2 − ny
2
Write out the lines of your program that you have changed.
= 1, where n is a positive integer.
Use the edited program to find a rational approximation to p
5 that is accurate to within 0.1%.
[6]
(iv) Explain why the edited program will not give any results if n is a square number.
[2]
© OCR 2012 DRAFT SPECIMEN

4
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Copyright Information
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 (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 2012 DRAFT SPECIMEN

PRINTED ANSWER BOOK
Candidates answer on this Printed Answer Book.
OCR supplied materials:
• Question Paper 4797 (inserted)
• MEI Examination Formulae and Tables (MF2)
Other materials required:
• Scientific or graphical calculator
• Computer with appropriate software
*
*
4
4
7
7
9
9
7
7
*
*
Candidate forename
Candidate surname
Centre number Candidate number
•
•
•
•
•
•
•
•
INSTRUCTIONS TO CANDIDATES
These instructions are the same on the Printed Answer Book and the Question Paper.
•
The Question Paper will be found in the centre of the Printed Answer Book.
Write your name, centre number and candidate number in the spaces provided on the Printed
Answer Book. Please write clearly and in capital letters.
Write your answer to each question in the space provided in the Printed Answer Book.
Additional paper may be used if necessary but you must clearly show your candidate number, centre number and question number(s).
Use black ink. HB pencil may be used for graphs and diagrams only.
Read each question carefully. Make sure you know what you have to do before starting your answer.
Answer all the questions.
Do not write in the bar codes.
You are permitted to use a scientific or graphical calculator in this paper.
Final answers should be given to a degree of accuracy appropriate to the context.
•
•
INFORMATION FOR CANDIDATES
This information is the same on the Printed Answer Book and the Question Paper.
•
•
The number of marks is given in brackets [ ] at the end of each question or part question on the
Question Paper.
You are advised that an answer may receive no marks unless you show sufficient detail of the working to indicate that a correct method is being used.
The total number of marks for this paper is 72 .
The Printed Answer Book consists of 12 pages. The Question Paper consists of 4 pages. Any blank pages are indicated.
© OCR 2012 [x/xxx/xxxx] OCR is an exempt Charity Turn over

1 (i)
2
1 (ii)
© OCR 2012

1 (iii)
3
1 (iv)
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1 (v)
4
1 (vi)
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2 (i)
5
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2 (ii)
6
2 (iii)
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2 (iv) ( A )
7
2 (iv) ( B )
2 (v)
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2 (vi)
8
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3 (i)
9
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3 (ii)
10
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3 (iii)
11
3 (iv)
© OCR 2012

12
PLEASE DO NOT WRITE ON THIS PAGE
Copyright Information
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 (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 2012