0RQGD\-XQH²$IWHUQRRQ
A2 GCE MATHEMATICS (MEI)
4798/01
Further Pure Mathematics with Technology (FPT)
QUESTION PAPER
* 4 7 3 9 3 8 0 6 1 3 *
Candidates answer on the Printed Answer Book.
OCR supplied materials:
•
Printed Answer Book 4798/01
•
MEI Examination Formulae and Tables (MF2)
Duration: Up to 2 hours
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.
INSTRUCTION TO EXAMS OFFICER / INVIGILATOR
•
Do not send this Question Paper for marking; it should be retained in the centre or
recycled. Please contact OCR Copyright should you wish to re-use this document.
© OCR 2013 [R/503/9789]
DC (SLM) 75975/2
OCR is an exempt Charity
Turn over
7KLVTXHVWLRQFRQFHUQVFXUYHVZLWKSDUDPHWULFHTXDWLRQV
x = a cos t + cos at , y = a sin t - sin at
ZKHUHDLVDSRVLWLYHLQWHJHUDQG 0 G W 1 2r L 6NHWFKWKHFXUYHVIRUWKHFDVHV D = 2, D = 3DQG D = DQGGHVFULEHWZRRIWKHFRPPRQIHDWXUHVRI
WKHVHWKUHHFXUYHV
>@
LL )RUWKHFDVHD = ILQGWKHYDOXHVRIWIRUWKHSRLQWVZKHUHWKHFXUYHLQWHUVHFWVWKHD[HVDQGKHQFHILQG
WKHFRRUGLQDWHVRIWKHSRLQWVRILQWHUVHFWLRQZLWKWKHD[HV
>@
2r
E\LQYHVWLJDWLQJWKH
LLL )RUWKHFDVH D = FRQILUPWKHIHDWXUHRIWKHFXUYHDWWKHSRLQWZKHUH W =
3
2r
JUDGLHQWDVW "
>@
3
LY 6NHWFKWKHFXUYH
x = k cos 3 t y = k sin 3 t
ZKHUHNLVSRVLWLYHDQG 0 G W 1 2r <RXDUHJLYHQWKDWIRUWKHFDVHD = WKHFXUYH
x = a cos t + cos at y = a sin t - sin at
FDQEHZULWWHQLQWKHIRUP
x = k cos 3 t y = k sin 3 t
© OCR 2013
IRUDSDUWLFXODUSRVLWLYHYDOXHRINDQG 0 G W 1 2r )LQGWKLVYDOXHRINDQGREWDLQDFDUWHVLDQHTXDWLRQ
IRUWKHFXUYHLQWKLVFDVH
>@
4798/01 Jun13
7KLVTXHVWLRQFRQFHUQVWKHIXQFWLRQ f (]) = sin ] IRU ] ! C ZLWKGHULYDWLYH f l (]) = cos ] L)LQG ZLWK UHDO DQG LPDJLQDU\ SDUWV JLYHQ WR GHFLPDO SODFHV WKH YDOXHV RI ] ] DQG ] ZKHUH
] 1 = f (3 + 2i) ] 2 = f (3.1 + 2i) DQG ] 3 = f l (3 + 2i) 3ORWWKHVHSRLQWVRQDQ$UJDQGGLDJUDP
([SUHVV
]2 - ]1
DQG]LQWKHIRUPUe ii DQGH[SODLQZK\WKH\DUHDSSUR[LPDWHO\HTXDO
0.1
>@
LL &RQVWUXFWDVSUHDGVKHHWWRGHPRQVWUDWHWKDW
Jf (z + h) - f (z)N
OO = cos z IRU ] = 3 + 2i DQGK ! R lim KK
h"0
h
L
P
6WDWHZKLFKYDOXHVRIK\RXKDYHXVHGDQGWKHH[SUHVVLRQV\RXKDYHHYDOXDWHG4XRWLQJVXIILFLHQW
YDOXHVIURP\RXUVSUHDGVKHHWH[SODLQKRZWKHUHVXOWLVGHPRQVWUDWHG
)LQGFRUUHFWWRVLJQLILFDQWILJXUHWKHODUJHVWYDOXHRIKK ! R VXFKWKDW
f ((3 + 2i) + K) - f (3 + 2i)
- cos (3 + 2i) 1 0.01
K
>@
LLL)LQGFRUUHFWWRVLJQLILFDQWILJXUHWKHODUJHVWYDOXHRIKK ! R VXFKWKDW
f ((3 + 2i) + Ki) - f (3 + 2i)
- cos (3 + 2i) 1 0.01
Ki
>@
LY8VH\RXUVRIWZDUHWRILQGWKHURRWVRIWKHHTXDWLRQ cos ] = 0 ZKHUH - 2r 1 Re (]) 1 2r 3ORWWKHVH
URRWVRQDQ$UJDQGGLDJUDP
8VH WKH UHDO DQG LPDJLQDU\ SDUWV RI cos ] WR VKRZ DOJHEUDLFDOO\ WKDW DOO WKH URRWV RI WKH HTXDWLRQ
>@
cos ] = 0 DUHUHDO
© OCR 2013
4798/01 Jun13
7XUQRYHU
7KLV TXHVWLRQ FRQFHUQV DULWKPHWLF PRGXOR 7KH XQNQRZQV [ DQG \ DUH LQWHJHUV VXFK WKDW
0 G [ 1 17 0 G \ 1 17 L &UHDWHDSURJUDPWRILQGDOOWKHVROXWLRQV[DQG\WRWKHFRQJUXHQFH
ax + by / c PRG
ZKHUHDEDQGFDUHSRVLWLYHLQWHJHUV<RXVKRXOGZULWHRXW\RXUSURJUDPLQIXOO
)LQGWKHQXPEHURIVROXWLRQVWRWKHFRQJUXHQFH
x + 5y / 13PRG
6WDWHWKHVROXWLRQIRUZKLFKWKHVXP x + y LVODUJHVW
>@
LL (GLW\RXUSURJUDPWRILQGWKHVROXWLRQV[DQG\WRWKHVLPXOWDQHRXVFRQJUXHQFHV
ax + by / c PRG
dx + ey / f PRG
ZKHUHDEFGHDQGIDUHSRVLWLYHLQWHJHUV,QGLFDWHFOHDUO\DOOWKHFKDQJHVWR\RXUSURJUDP
8VHWKHHGLWHGSURJUDPWRVROYHWKHVLPXOWDQHRXVFRQJUXHQFHV
3x + 5y / 7 PRG
2x + 7y / 1PRG
DQGVWDWHWKHVROXWLRQ
&KHFNWKHVROXWLRQE\FDOFXODWLQJWKHYDOXHVRI [ \ DQG [ \ >@
LLL([SODLQKRZ\RXZRXOGLQYHVWLJDWHWKHQXPEHURIVROXWLRQV[DQG\WRWKHVLPXOWDQHRXVFRQJUXHQFHV
kx + 5y / 7 PRG
2x + 7y / 1PRG
IRUGLIIHUHQWLQWHJHUYDOXHVRINZKHUH 0 G N 1 17 6WDWHWKHYDOXHRINIRUZKLFKWKHVLPXOWDQHRXVFRQJUXHQFHVGRQRWKDYHDVROXWLRQ([SODLQZK\WKH
FRQJUXHQFHVGRQRWKDYHDVROXWLRQIRUWKLVYDOXHRIN
>@
LY)LQGWKHQXPEHURIVROXWLRQV[DQG\WRWKHVLPXOWDQHRXVFRQJUXHQFHV
([SODLQ\RXUUHVXOW
7x + y / 6 PRG
x + 5y / 13 PRG
>@
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 2013
4798/01 Jun13