OXFORD CAMBRIDGE AND RSA EXAMINATIONS Advanced …...Advanced Subsidiary General Certificate of Education Advanced General Certificate of Education MEI STRUCTURED MATHEMATICS 4756
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OXFORD CAMBRIDGE AND RSA EXAMINATIONS
Advanced Subsidiary General Certificate of EducationAdvanced General Certificate of Education
MEI STRUCTURED MATHEMATICS 4756Further Methods for Advanced Mathematics (FP2)
1 (a) A curve has polar equation for where a is a positive constant.
(i) Sketch the curve, using a continuous line for sections where and a broken line forsections where [3]
(ii) Find the area enclosed by one of the loops. [5]
(b) Find the exact value of [5]
(c) Use a trigonometric substitution to find [5]
2 In this question, q is a real number with and
(i) State the modulus and argument of each of the complex numbers
Illustrate these three complex numbers on an Argand diagram. [6]
(ii) Show that [4]
Infinite series C and S are defined by
(iii) Show that and find a similar expression for S. [8]C �4 cos 2q � 2 cos q
5 � 4 cos 3q,
S � sin 2q � 12 sin 5q � 14 sin 8q � 1
8 sin 11q � ... .
C � cos 2q � 12 cos 5q � 14 cos 8q � 1
8 cos 11q � ... ,
(1 � w) (1 � w*) � 54 � cos 3q.
w, w* and jw.
w � 12 e3jq.0 � q � 1
6 p ,
1
1 3 20
1
32+( )
ÛıÙ x
xd .
1
3 4 20
34
-ÛıÙ x
xd .
r � 0.r � 0
– 12 p � q � 12 p,r � acos 3q
2
4756 January 2006
3 The matrix
(i) Show that the characteristic equation for M is [3]
(ii) Show that –1 is an eigenvalue of M, and find the other two eigenvalues. [3]
(iii) Find an eigenvector corresponding to the eigenvalue –1. [3]
(iv) Verify that are eigenvectors of M. [3]
(v) Write down a matrix P, and a diagonal matrix D, such that [3]
(vi) Use the Cayley-Hamilton theorem to express in the form [3]
Section B (18 marks)
Answer one question
Option 1: Hyperbolic functions
4 (a) Solve the equation
giving the answers in an exact logarithmic form. [6]
(b) Find the exact value of [4]
(c) (i) Differentiate with respect to x. [2]
(ii) Use integration by parts to show that [6]�2
0arsinh ( 23 x) dx � 2 ln 3 � 1.
arsinh ( 23 x)
�2
0e x sinh x dx .
sinh x � 4 cosh x � 8,
aM2 � bM � cI .M–1
M3 � PDP–1.
301
032
Ê
ËÁÁ
ˆ
¯˜˜ -
Ê
ËÁÁ
ˆ
¯˜˜
and
l 3 � 6l 2 � 9l � 14 � 0.
M = - --
Ê
ËÁÁ
ˆ
¯˜˜
1 2 32 3 62 2 4
.
3
4756 January 2006 [Turn over
Option 2: Investigation of curves
This question requires the use of a graphical calculator.
5 A curve has equation where k is a positive constant and
(i) Find the equations of the three asymptotes. [3]
(ii) Use your graphical calculator to obtain rough sketches of the curve in the two separate casesand [4]
(iii) In the case , your sketch may not show clearly the shape of the curve near . Usecalculus to show that the curve has a minimum point when [5]
(iv) In the case , your sketch may not show clearly how the curve approaches its asymptoteas Show algebraically that the curve crosses this asymptote. [2]
(v) Use the results of parts (iii) and (iv) to produce more accurate sketches of the curve in the twoseparate cases and These sketches should indicate where the curve crosses theaxes, and should show clearly how the curve approaches its asymptotes. The presence ofstationary points should be clearly shown, but there is no need to find their coordinates.
[4]
k � 2.k � 2
x Æ ��.k � 2
x � 0.x � 0k � 2
k � 2.k � 2
k � 2.y �x3 � k3
x2 � 4,
4
4756 January 2006
OXFORD CAMBRIDGE AND RSA EXAMINATIONS
Advanced Subsidiary General Certificate of EducationAdvanced General Certificate of Education
MEI STRUCTURED MATHEMATICS 4756Further Methods for Advanced Mathematics (FP2)
This question requires the use of a graphical calculator.
5 Cartesian coordinates and polar coordinates are set up in the usual way, with the poleat the origin and the initial line along the positive x-axis, so that and .
A curve has polar equation where k is a constant with
(i) Use your graphical calculator to obtain sketches of the curve in the three cases
[5]
(ii) Name the special feature which the curve has when [1]
(iii) For each of the three cases, state the number of points on the curve at which the tangent isparallel to the y-axis. [2]
(iv) Express x in terms of k and q , and find Hence find the range of values of k for which there
are just two points on the curve where the tangent is parallel to the y-axis. [4]
The distance between the point on the curve and the point on the x-axis is d.
(v) Use the cosine rule to express in terms of k and q , and deduce that [4]
(vi) Hence show that, when k is large, the shape of the curve is very nearly circular. [2]
Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Everyreasonable effort has been made by the publisher (OCR) to trace copyright holders, but if any items requiring clearance have unwittingly been included, thepublisher will be pleased to make amends at the earliest possible opportunity.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.
INSTRUCTIONS TO CANDIDATES
• Write your name, centre number and candidate number in the spaces provided on the answer booklet.
• Answer all the questions in Section A and one question from Section B.
• You are permitted to use a graphical calculator in this paper.
• Final answers should be given to a degree of accuracy appropriate to the context.
INFORMATION FOR CANDIDATES
• The number of marks is given in brackets [ ] at the end of each question or part question.
• The total number of marks for this paper is 72.
ADVICE TO CANDIDATES
• Read each question carefully and make sure you know what you have to do before starting youranswer.
• You are advised that an answer may receive no marks unless you show sufficient detail of theworking to indicate that a correct method is being used.
1 (a) A curve has polar equation , where a is a positive constant.
(i) Sketch the curve. [2]
(ii) Find the area of the region enclosed by the section of the curve for which
and the line [6]
(b) Use a trigonometric substitution to show that [4]
(c) In this part of the question,
(i) Find [2]
(ii) Use a standard series to expand and hence find the series for in ascendingpowers of x, up to the term in [4]
2 (a) Use de Moivre’s theorem to show that [5]
(b) (i) Find the cube roots of in the form where and [6]
These cube roots are represented by points A, B and C in the Argand diagram, with A in thefirst quadrant and ABC going anticlockwise. The midpoint of AB is M, and M represents thecomplex number w.
(ii) Draw an Argand diagram, showing the points A, B, C and M. [2]
This question requires the use of a graphical calculator.
5 The curve with equation is to be investigated for different values of k.
(i) Use your graphical calculator to obtain rough sketches of the curve in the cases and [6]
(ii) Show that the equation of the curve may be written as
Hence find the two values of k for which the curve is a straight line. [4]
(iii) When the curve is not a straight line, it is a conic.
(A) Name the type of conic. [1]
(B) Write down the equations of the asymptotes. [2]
(iv) Draw a sketch to show the shape of the curve when . This sketch should showwhere the curve crosses the axes and how it approaches its asymptotes. Indicate the points Aand B on the curve where and respectively. [5]x � kx � 1
Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Everyreasonable effort has been made by the publisher (OCR) to trace copyright holders, but if any items requiring clearance have unwittingly been included, thepublisher will be pleased to make amends at the earliest possible opportunity.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.
This question requires the use of a graphical calculator.
5 A curve has parametric equations x = t2
1 + t2, y = t3 − λ t, where λ is a constant.
(i) Use your calculator to obtain a sketch of the curve in each of the cases
λ = −1, λ = 0 and λ = 1.
Name any special features of these curves. [5]
(ii) By considering the value of x when t is large, write down the equation of the asymptote. [1]
For the remainder of this question, assume that λ is positive.
(iii) Find, in terms of λ , the coordinates of the point where the curve intersects itself. [3]
(iv) Show that the two points on the curve where the tangent is parallel to the x-axis have coordinates
( λ3 + λ
, ±√
4λ3
27). [4]
Fig. 5 shows a curve which intersects itself at the point (2, 0) and has asymptote x = 8. The stationarypoints A and B have y-coordinates 2 and −2.
Ox
y
Fig. 5
A
B
82
2
–2
(v) For the curve sketched in Fig. 5, find parametric equations of the form x = at2
1 + t2, y = b(t3 − λ t),
where a, λ and b are to be determined. [5]
Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonableeffort has been made by the publisher (OCR) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will bepleased to make amends at the earliest possible opportunity.
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.
This question requires the use of a graphical calculator.
5 A curve has parametric equations x = λ cos θ − 1λ
sin θ , y = cos θ + sin θ , where λ is a positiveconstant.
(i) Use your calculator to obtain a sketch of the curve in each of the cases
λ = 0.5, λ = 3 and λ = 5. [3](ii) Given that the curve is a conic, name the type of conic. [1]
(iii) Show that y has a maximum value of√
2 when θ = 14π. [2]
(iv) Show that x2 + y2 = (1 + λ 2) + ( 1
λ 2− λ 2) sin2 θ, and deduce that the distance from the origin of
any point on the curve is between
√1 + 1
λ 2and
√1 + λ 2. [6]
(v) For the case λ = 1, show that the curve is a circle, and find its radius. [2]
(vi) For the case λ = 2, draw a sketch of the curve, and label the points A, B, C, D, E, F, G, H on thecurve corresponding to θ = 0, 1
4π, 1
2π, 3
4π, π, 5
4π, 3
2π, 7
4π respectively. You should make clear what
is special about each of these points. [4]
Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonableeffort has been made by the publisher (OCR) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will bepleased to make amends at the earliest possible opportunity.
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.
MATHEMATICS (MEI) 4756Further Methods for Advanced Mathematics (FP2)
Candidates answer on the Answer Booklet
OCR Supplied Materials:• 8 page Answer Booklet• Graph paper
• MEI Examination Formulae and Tables (MF2)
Other Materials Required:None
Friday 9 January 2009
Morning
Duration: 1 hour 30 minutes
**
44
77
55
66
**
INSTRUCTIONS TO CANDIDATES
• Write your name clearly in capital letters, your Centre Number and Candidate Number in the spaces providedon the Answer Booklet.
• Use black ink. Pencil may be used for graphs and diagrams only.
• Read each question carefully and make sure that you know what you have to do before starting your answer.• Answer all the questions in Section A and one question from Section B.• Do not write in the bar codes.• You are permitted to use a graphical calculator in this paper.
• Final answers should be given to a degree of accuracy appropriate to the context.
INFORMATION FOR CANDIDATES
• The number of marks is given in brackets [ ] at the end of each question or part question.• 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.• This document consists of 4 pages. Any blank pages are indicated.
This question requires the use of a graphical calculator.
5 The limacon of Pascal has polar equation r = 1 + 2a cos θ, where a is a constant.
(i) Use your calculator to sketch the curve when a = 1. (You need not distinguish between parts of
the curve where r is positive and negative.) [3]
(ii) By using your calculator to investigate the shape of the curve for different values of a, positive
and negative,
(A) state the set of values of a for which the curve has a loop within a loop,
(B) state, with a reason, the shape of the curve when a = 0,
(C) state what happens to the shape of the curve as a → ±∞,
(D) name the feature of the curve that is evident when a = 0.5, and find another value of a for
which the curve has this feature. [7]
(iii) Given that a > 0 and that a is such that the curve has a loop within a loop, write down an equation
for the values of θ at which r = 0. Hence show that the angle at which the curve crosses itself is
2 arccos( 1
2a).
Obtain the cartesian equations of the tangents at the point where the curve crosses itself. Explain
briefly how these equations relate to the answer to part (ii)(A). [8]
Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable
effort has been made by the publisher (OCR) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will be
pleased to make amends at the earliest possible opportunity.
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.
MATHEMATICS (MEI) 4756Further Methods for Advanced Mathematics (FP2)
Candidates answer on the Answer Booklet
OCR Supplied Materials:• 8 page Answer Booklet• Graph paper
• MEI Examination Formulae and Tables (MF2)
Other Materials Required:None
Friday 5 June 2009
Afternoon
Duration: 1 hour 30 minutes
**
44
77
55
66
**
INSTRUCTIONS TO CANDIDATES
• Write your name clearly in capital letters, your Centre Number and Candidate Number in the spaces providedon the Answer Booklet.
• Use black ink. Pencil may be used for graphs and diagrams only.
• Read each question carefully and make sure that you know what you have to do before starting your answer.• Answer all the questions in Section A and one question from Section B.• Do not write in the bar codes.• You are permitted to use a graphical calculator in this paper.
• Final answers should be given to a degree of accuracy appropriate to the context.
INFORMATION FOR CANDIDATES
• The number of marks is given in brackets [ ] at the end of each question or part question.• 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.• This document consists of 4 pages. Any blank pages are indicated.
This question requires the use of a graphical calculator.
5 Fig. 5 shows a circle with centre C (a, 0) and radius a. B is the point (0, 1). The line BC intersects
the circle at P and Q; P is above the x-axis and Q is below.
B
O
P
Q
C ( , 0)a
Fig. 5
(i) Show that, in the case a = 1, P has coordinates (1 − 1√2
,1√2). Write down the coordinates of Q.
[3]
(ii) Show that, for all positive values of a, the coordinates of P are
x = a(1 − a√a2 + 1
), y = a√a2 + 1
. (∗)Write down the coordinates of Q in a similar form. [4]
Now let the variable point P be defined by the parametric equations (∗) for all values of the parameter
a, positive, zero and negative. Let Q be defined for all a by your answer in part (ii).
(iii) Using your calculator, sketch the locus of P as a varies. State what happens to P as a → ∞ and
as a → −∞.
Show algebraically that this locus has an asymptote at y = −1.
On the same axes, sketch, as a dotted line, the locus of Q as a varies. [8]
(The single curve made up of these two loci and including the point B is called a right strophoid.)
(iv) State, with a reason, the size of the angle POQ in Fig. 5. What does this indicate about the angle
at which a right strophoid crosses itself? [3]
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, is given to all schools that receive assessment material 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 1PB.
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
MATHEMATICS (MEI) 4756Further Methods for Advanced Mathematics (FP2)
Candidates answer on the Answer Booklet
OCR Supplied Materials:
• 8 page Answer Booklet• MEI Examination Formulae and Tables (MF2)
Other Materials Required:
None
Monday 11 January 2010
Morning
Duration: 1 hour 30 minutes
**
44
77
55
66
**
INSTRUCTIONS TO CANDIDATES
• Write your name clearly in capital letters, your Centre Number and Candidate Number in the spaces provided
on the Answer Booklet.• Use black ink. Pencil may be used for graphs and diagrams only.• Read each question carefully and make sure that you know what you have to do before starting your answer.• Answer all the questions in Section A and one question from Section B.
• Do not write in the bar codes.• You are permitted to use a graphical calculator in this paper.• Final answers should be given to a degree of accuracy appropriate to the context.
INFORMATION FOR CANDIDATES
• The number of marks is given in brackets [ ] at the end of each question or part question.
• You are advised that an answer may receive no marks unless you show sufficient detail of the working toindicate that a correct method is being used.
• The total number of marks for this paper is 72.• This document consists of 4 pages. Any blank pages are indicated.
This question requires the use of a graphical calculator.
5 A line PQ is of length k (where k > 1) and it passes through the point (1, 0). PQ is inclined at angle θto the positive x-axis. The end Q moves along the y-axis. See Fig. 5. The end P traces out a locus.
y
x
O
Q
P
1
q
Fig. 5
(i) Show that the locus of P may be expressed parametrically as follows. [3]
x = k cos θ y = k sin θ − tan θ
You are now required to investigate curves with these parametric equations, where k may take any
non-zero value and −12π < θ < 1
2π.
(ii) Use your calculator to sketch the curve in each of the cases k = 2, k = 1, k = 12
and k = −1. [4]
(iii) For what value(s) of k does the curve have
(A) an asymptote (you should state what the asymptote is),
(B) a cusp,
(C) a loop? [3]
(iv) For the case k = 2, find the angle at which the curve crosses itself. [2]
(v) For the case k = 8, find in an exact form the coordinates of the highest point on the loop. [3]
(vi) Verify that the cartesian equation of the curve is
y2 = (x − 1)2
x2(k2 − x2). [3]
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, is given to all schools that receive assessment material 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
MATHEMATICS (MEI) 4756Further Methods for Advanced Mathematics (FP2)
Candidates answer on the Answer Booklet
OCR Supplied Materials:• 8 page Answer Booklet• MEI Examination Formulae and Tables (MF2)
Other Materials Required:• Scientific or graphical calculator
Friday 11 June 2010
Morning
Duration: 1 hour 30 minutes
**
44
77
55
66
**
INSTRUCTIONS TO CANDIDATES
• Write your name clearly in capital letters, your Centre Number and Candidate Number in the spaces providedon the Answer Booklet.
• Use black ink. Pencil may be used for graphs and diagrams only.
• Read each question carefully and make sure that you know what you have to do before starting your answer.• Answer all the questions in Section A and one question from Section B.• Do not write in the bar codes.• You are permitted to use a graphical calculator in this paper.
• Final answers should be given to a degree of accuracy appropriate to the context.
INFORMATION FOR CANDIDATES
• The number of marks is given in brackets [ ] at the end of each question or part question.• 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.• This document consists of 4 pages. Any blank pages are indicated.
This question requires the use of a graphical calculator.
5 In parts (i), (ii), (iii) of this question you are required to investigate curves with the equation
xk + yk = 1
for various positive values of k.
(i) Firstly consider cases in which k is a positive even integer.
(A) State the shape of the curve when k = 2.
(B) Sketch, on the same axes, the curves for k = 2 and k = 4.
(C) Describe the shape that the curve tends to as k becomes very large.
(D) State the range of possible values of x and y. [6]
(ii) Now consider cases in which k is a positive odd integer.
(A) Explain why x and y may take any value.
(B) State the shape of the curve when k = 1.
(C) Sketch the curve for k = 3. State the equation of the asymptote of this curve.
(D) Sketch the shape that the curve tends to as k becomes very large. [6]
(iii) Now let k = 12.
Sketch the curve, indicating the range of possible values of x and y. [2]
(iv) Now consider the modified equation |x |k + |y |k = 1.
(A) Sketch the curve for k = 12.
(B) Investigate the shape of the curve for k = 1
nas the positive integer n becomes very large.
[4]
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, is given to all schools that receive assessment material 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
MATHEMATICS (MEI) 4756Further Methods for Advanced Mathematics (FP2)
Candidates answer on the answer booklet.
OCR supplied materials:
• 8 page answer booklet(sent with general stationery)
• MEI Examination Formulae and Tables (MF2)
Other materials required:• Scientific or graphical calculator
Monday 10 January 2011
Morning
Duration: 1 hour 30 minutes
**
44
77
55
66
**
INSTRUCTIONS TO CANDIDATES
• Write your name, centre number and candidate number in the spaces provided on theanswer booklet. Please write clearly and in capital letters.
• Use black ink. 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 in Section A and one question from Section B.• 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
• The number of marks is given in brackets [ ] at the end of each question or part question.• 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.• This document consists of 4 pages. Any blank pages are indicated.
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
MATHEMATICS (MEI) 4756Further Methods for Advanced Mathematics (FP2)
Candidates answer on the answer booklet.
OCR supplied materials:
• 8 page answer booklet(sent with general stationery)
• MEI Examination Formulae and Tables (MF2)
Other materials required:• Scientific or graphical calculator
Monday 20 June 2011
Morning
Duration: 1 hour 30 minutes
**
44
77
55
66
**
INSTRUCTIONS TO CANDIDATES
• Write your name, centre number and candidate number in the spaces provided on theanswer booklet. Please write clearly and in capital letters.
• Use black ink. 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 in Section A and one question from Section B.• 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
• The number of marks is given in brackets [ ] at the end of each question or part question.• 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.• This document consists of 4 pages. Any blank pages are indicated.
This question requires the use of a graphical calculator.
5 In this question, you are required to investigate the curve with equation
y = xm(1 − x)n, 0 ≤ x ≤ 1,
for various positive values of m and n.
(i) On separate diagrams, sketch the curve in each of the following cases.
(A) m = 1, n = 1,
(B) m = 2, n = 2,
(C) m = 2, n = 4,
(D) m = 4, n = 2. [4]
(ii) What feature does the curve have when m = n?
What is the effect on the curve of interchanging m and n when m ≠ n? [2]
(iii) Describe how the x-coordinate of the maximum on the curve varies as m and n vary. Use calculus
to determine the x-coordinate of the maximum. [6]
(iv) Find the condition on m for the gradient to be zero when x = 0. State a corresponding result for
the gradient to be zero when x = 1. [2]
(v) Use your calculator to investigate the shape of the curve for large values of m and n. Hence
conjecture what happens to the value of the integral ã 1
0
xm(1 − x)n dx as m and n tend to infinity.
[2]
(vi) Use your calculator to investigate the shape of the curve for small values of m and n. Hence
conjecture what happens to the shape of the curve as m and n tend to zero. [2]
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Friday 13 January 2012 – MorningA2 GCE MATHEMATICS (MEI)
4756 Further Methods for Advanced Mathematics (FP2)
QUESTION PAPER
INSTRUCTIONS TO CANDIDATESThese 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.• Answer all the questions in Section A and one question from Section B.• Read each question carefully. Make sure you know what you have to do before starting
your answer.• 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 CANDIDATESThis 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 16 pages. The Question Paper consists of 4 pages.
Any blank pages are indicated.
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.
4 (i) Define tanh t in terms of exponential functions. Sketch the graph of tanh t. [3]
(ii) Show that artanh x = 12ln
1 + x1 − x . State the set of values of x for which this equation is valid. [5]
(iii) Differentiate the equation tanh y = x with respect to x and hence show that the derivative of artanh x is
11 − x2 .
Show that this result may also be obtained by differentiating the equation in part (ii). [5]
(iv) By considering artanh x as 1 × artanh x and using integration by parts, show that
�12
0
artanh x dx = 14ln
2716 . [5]
Option 2: Investigation of curves
This question requires the use of a graphical calculator.
5 The points A(−1, 0), B(1, 0) and P(x, y) are such that the product of the distances PA and PB is 1. You are given that the cartesian equation of the locus of P is
((x + 1)2 + y2)((x − 1)2 + y2) = 1.
(i) Show that this equation may be written in polar form as
r4 + 2r2 = 4r2 cos2 θ.
Show that the polar equation simplifies to
r2 = 2 cos 2θ. [4]
(ii) Give a sketch of the curve, stating the values of θ for which the curve is defined. [4]
(iii) The equation in part (i) is now to be generalised to
r2 = 2 cos 2θ + k,
where k is a constant.
(A) Give sketches of the curve in the cases k = 1, k = 2. Describe how these two curves differ at the pole.
(B) Give a sketch of the curve in the case k = 4. What happens to the shape of the curve as k tends to infinity? [7]
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.
Thursday 21 June 2012 – AfternoonA2 GCE MATHEMATICS (MEI)
4756 Further Methods for Advanced Mathematics (FP2)
QUESTION PAPER
INSTRUCTIONS TO CANDIDATESThese 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 in Section A and one question from Section B.• 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 CANDIDATESThis 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 16 pages. The Question Paper consists of 4 pages.
Any blank pages are indicated.
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.
2 (a) (i) Given that z = cos θ + j sin θ, express zn + 1zn and zn − 1
zn in simplified trigonometric form. [2]
(ii) Beginning with an expression for �z + 1z�
4, find the constants A, B, C in the identity
cos4θ ≡ A + B cos 2θ + C cos 4θ. [4]
(iii) Use the identity in part (ii) to obtain an expression for cos 4θ as a polynomial in cos θ. [2]
(b) (i) Given that z = 4e jπ/3 and that w2 = z, write down the possible values of w in the form r e jθ, where r > 0. Show z and the possible values of w in an Argand diagram. [5]
(ii) Find the least positive integer n for which zn is real.
Show that there is no positive integer n for which zn is imaginary.
For each possible value of w, find the value of w3 in the form a + jb where a and b are real. [5]
3 (i) Find the value of a for which the matrix
M = 1−1
3
2a
−2
342
does not have an inverse.
Assuming that a does not have this value, find the inverse of M in terms of a. [7]
(ii) Hence solve the following system of equations.
x + 2y + 3z = 1 −x + 4z = −2 3x – 2y + 2z = 1 [4]
(iii) Find the value of b for which the following system of equations has a solution.
4 (i) Prove, from definitions involving exponential functions, that
cosh 2u = 2 sinh2 u + 1. [3]
(ii) Prove that, if y 0 and cosh y = u, then y = ln (u + √(u2 − 1)). [4]
(iii) Using the substitution 2x = cosh u, show that
� 4x2 – 1dx = ax 4x2 – 1 − b arcosh 2x + c,
where a and b are constants to be determined and c is an arbitrary constant. [7]
(iv) Find �12
1
4x2 – 1dx, expressing your answer in an exact form involving logarithms. [4]
Option 2: Investigation of curves
This question requires the use of a graphical calculator.
5 This question concerns curves with polar equation r = sec θ + a, where a is a constant.
(i) State the set of values of θ between 0 and 2π for which r is undefined. [2]
For the rest of the question you should assume that θ takes all values between 0 and 2π for which r is defined.
(ii) Use your graphical calculator to obtain a sketch of the curve in the case a = 0. Confirm the shape of the curve by writing the equation in cartesian form. [3]
(iii) Sketch the curve in the case a = 1.
Now consider the curve in the case a = −1. What do you notice?
By considering both curves for 0 < θ < π and π < θ < 2π separately, describe the relationship between the cases a = 1 and a = −1. [6]
(iv) What feature does the curve exhibit for values of a greater than 1?
Sketch a typical case. [3]
(v) Show that a cartesian equation of the curve r = sec θ + a is (x2 + y2)(x −1)2 = a2x2. [4]
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.
Monday 14 January 2013 – MorningA2 GCE MATHEMATICS (MEI)
4756/01 Further Methods for Advanced Mathematics (FP2)
QUESTION PAPER
*4734080113*
INSTRUCTIONS TO CANDIDATESThese 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 in Section A and one question from Section B.• 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 CANDIDATESThis 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 16 pages. The Question Paper consists of 4 pages.
Any blank pages are indicated.
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.
1 (a) (i) Differentiate with respect to x the equation tana y x= (where a is a constant), and hence show that the derivative of arctan a
x is a xa
2 2+. [3]
(ii) By first expressing x x4 82 - + in completed square form, evaluate the integral x x
x4 81 d
20
4
- += ,
giving your answer exactly. [4]
(iii) Use integration by parts to find arctan x xdy . [4]
(b) (i) A curve has polar equation 2cosr i= , for 21
21G Gr i r- . Show, by considering its cartesian
equation, that the curve is a circle. State the centre and radius of the circle. [5]
(ii) Another circle has radius 2 and its centre, in cartesian coordinates, is (0, 2). Find the polar equation of this circle. [2]
2 (a) (i) Show that
)sini1 2 (cos cose j2j i i+ = +i . [2]
(ii) The series C and S are defined as follows.
cos cos cosCn n
n1 1 2 2 4 2fi i i= + + + +J
LKK
J
LKK
N
POO
N
POO
S = sinn1 2i +J
LKKN
POO sin
n2 4iJ
LKKN
POO + sin n2f i+
By considering C Sj+ , show that
2 cos cosC nn ni i= ,
and find a corresponding expression for S. [7]
(b) (i) Express e 2 /3j r in the form x yj+ , where the real numbers x and y should be given exactly. [1]
(ii) An equilateral triangle in the Argand diagram has its centre at the origin. One vertex of the triangle is at the point representing 2 4j+ . Obtain the complex numbers representing the other two vertices, giving your answers in the form x yj+ , where the real numbers x and y should be given exactly. [6]
(iii) Show that the length of a side of the triangle is 2 15 . [2]
(ii) Find the eigenvalues and corresponding eigenvectors of M. [12]
(iii) Write down a matrix P and a diagonal matrix D such that
M PDPn 1= - .
(You are not required to calculate P 1- .) [3]
Section B (18 marks)
Answer one question
Option 1: Hyperbolic functions
4 (i) Show that the curve with equation
sinh coshy x x3 2= -
has no turning points.
Show that the curve crosses the x-axis at 5lnx 21= . Show that this is also the point at which the
gradient of the curve has a stationary value. [7]
(ii) Sketch the curve. [2]
(iii) Express ( )sinh coshx x3 2 2- in terms of sinh x2 and cosh x2 .
Hence or otherwise, show that the volume of the solid of revolution formed by rotating the region bounded by the curve and the axes through 360° about the x-axis is