Question 1:
1
y = g | (x) | y =
y = | cos
1
0
– | (cid:155)1 | (cid:652) | 3 | (cid:652) 2 | 2 | (cid:652)
(cid:652) | (cid:652)
2 | 2
–
1
PMT
3
6 Oil is leaking into the sea from a pipeline, creating a circular oil slick. The radius r metres of the oil slick
t hours after the start of the leak is modelled by the equation
r = 20(1 − e−0.2t ).
(i) Find the radius of the slick when t = 2, and the rate at which the radius is increasing at this time. [4]
(ii) Find the rate at which the area of the slick is increasing when t = 2. [4]
7 Fig. 7 shows the curve x3 + y3 = 3xy. The point P is a turning point of the curve.
y
P
x
Fig. 7
dy y − x2
(i) Show that = . [4]
dx y2 − x
(ii) Hence find the exact x-coordinate of P. [4]
Turn over
© OCR 2012 4753/01 Jan12
PMT
4
Section B (36 marks)
x
8 Fig. 8 shows the curve y = , together with the lines y = x and x = 11.
x – 2
The curve meets these lines at P and Q respectively. R is the point (11, 11).
y
y = x
R(11, 11)
P Q
x
x = 11
Fig. 8
(i) Verify that the x-coordinate of P is 3. [2]
dy x − 4
(ii) Show that, for the curve, = .
dx 2(x − 2) 3 2
Hence find the gradient of the curve at P. Use the result to show that the curve is not symmetrical about
y = x. [7]
(cid:2)
11
x
(iii) Using the substitution u = x − 2, show that dx = 251.
x − 2 3
3
Hence find the area of the region PQR bounded by the curve and the lines y = x and x = 11. [9]
© OCR 2012 4753/01 Jan12
Thursday 21 June 2012 – Afternoon
A2 GCE MATHEMATICS (MEI)
4753/01 Methods for Advanced Mathematics (C3)
QUESTION PAPER
Candidates answer on the Printed Answer Book.
OCR supplied materials:
(cid:129) Printed Answer Book 4753/01
(cid:129) MEI Examination Formulae and Tables (MF2)
Other materials required:
(cid:129) Scientific or graphical calculator
INSTRUCTIONS TO CANDIDATES
These instructions are the same on the Printed Answer Book and the Question Paper.
(cid:129) The Question Paper will be found in the centre of the Printed Answer Book.
(cid:129) Write your name, centre number and candidate number in the spaces provided on the
Printed Answer Book. Please write clearly and in capital letters.
(cid:129) 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).
(cid:129) Use black ink. HB pencil may be used for graphs and diagrams only.
(cid:129) Read each question carefully. Make sure you know what you have to do before starting
your answer.
(cid:129) Answer all the questions.
(cid:129) Do not write in the bar codes.
(cid:129) You are permitted to use a scientific or graphical calculator in this paper.
(cid:129) 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.
(cid:129) The number of marks is given in brackets [ ] at the end of each question or part question
on the Question Paper.
(cid:129) 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.
(cid:129) The total number of marks for this paper is 72.
(cid:129) The Printed Answer Book consists of 16 pages. The Question Paper consists of 8 pages.
Any blank pages are indicated.
INSTRUCTION TO EXAMS OFFICER / INVIGILATOR
(cid:129) 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 2012 [M/102/2652] OCR is an exempt Charity
DC (NH) 43916/3 Turn over
*4715660612*
PMT
Duration: 1 hour 30 minutes
PMT
2
Section A (36 marks)
(cid:2)
2
1
1 Show that dx = 2. [5]
3x − 2 3
1
2 Solve the inequality (cid:3)2x + 1(cid:3) > 4. [3]
3 Find the gradient at the point (0, ln 2) on the curve with equation e2y = 5 − e−x. [4]
4 Fig. 4 shows the curve y = f(x), where f(x) = 1 − 9x2 , −a x a.
y
y = f(x)
x
–a a
Fig. 4
(i) Find the value of a. [2]
(ii) Write down the range of f(x). [1]
(iii) Sketch the curve y = f(1x) − 1. [3]
3
5 A termites’ nest has a population of P million. P is modelled by the equation P = 7 − 2e−kt, where t is in
years, and k is a positive constant.
(i) Calculate the population when t = 0, and the long-term population, given by this model. [3]
(ii) Given that the population when t = 1 is estimated to be 5.5 million, calculate the value of k. [3]
© OCR 2012 4753/01 Jun12
PMT
3
6 Fig. 6 shows the curve y = f(x), where f(x) = 2arcsin x, −1 x 1.
Fig. 6 also shows the curve y = g(x), where g(x) is the inverse function of f(x).
P is the point on the curve y = f(x) with x-coordinate 1.
2
y
y = f(x)
(cid:652)
P
y = g(x)
Q
x
–1 1
–(cid:652)
Fig. 6
(i) Find the y-coordinate of P, giving your answer in terms of π. [2]
The point Q is the reflection of P in y = x.
(ii) Find g(x) and its derivative g′(x). Hence determine the exact gradient of the curve y = g(x) at the
point Q.
Write down the exact gradient of y = f(x) at the point P. [6]
7 You are given that f(x) and g(x) are odd functions, defined for x ∈ (cid:2).
(i) Given that s(x) = f(x) + g(x), prove that s(x) is an odd function. [2]
(ii) Given that p(x) = f(x)g(x), determine whether p(x) is odd, even or neither. [2]
Turn over
© OCR 2012 4753/01 Jun12
PMT
4
Section B (36 marks)
8 Fig. 8 shows a sketch of part of the curve y = x sin 2x, where x is in radians.
The curve crosses the x-axis at the point P. The tangent to the curve at P crosses the y-axis at Q.
y
Q
x
O P
y = x sin 2x
Fig. 8
dy
(i) Find . Hence show that the x-coordinates of the turning points of the curve satisfy the equation
dx
tan 2x + 2x = 0. [4]
(ii) Find, in terms of π, the x-coordinate of the point P.
Show that the tangent PQ has equation 2πx + 2y = π2.
Find the exact coordinates of Q. [7]
(iii) Show that the exact value of the area shaded in Fig. 8 is 1 π(π2 − 2). [7]
8
© OCR 2012 4753/01 Jun12
PMT
5
9 Fig. 9 shows the curve y = f(x), which has a y-intercept at P(0, 3), a minimum point at Q(1, 2), and an
asymptote x = −1.
y
y = f(x)
x = –1
P (0, 3)
Q (1, 2)
x
Fig. 9
(i) Find the coordinates of the images of the points P and Q when the curve y = f(x) is transformed to
(A) y = 2f(x),
(B) y = f(x + 1) + 2. [4]
x2 + 3
You are now given that f(x) = , x ≠ −1.
x + 1
(ii) Find f ′(x), and hence find the coordinates of the other turning point on the curve y = f(x). [6]
4
(iii) Show that f(x − 1) = x − 2 + . [3]
x
(cid:2)
b
(cid:4) 4(cid:5)
(iv) Find x − 2 + dx in terms of a and b.
x
a
Hence, by choosing suitable values for a and b, find the exact area enclosed by the curve y = f(x), the
x-axis, the y-axis and the line x = 1. [5]
© OCR 2012 4753/01 Jun12
Wednesday 23 January 2013 – Morning
A2 GCE MATHEMATICS (MEI)
4753/01 Methods for Advanced Mathematics (C3)
QUESTION PAPER
*4733990113*
PMT
Candidates answer on the Printed Answer Book.
Duration: 1 hour 30 minutes
OCR supplied materials:
• Printed Answer Book 4753/01
• MEI Examination Formulae and Tables (MF2)
Other materials required:
• Scientific or graphical calculator
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 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.
© OCR 2013 [M/102/2652] OCR is an exempt Charity
DC (AC/SW) 63784/5 Turn over
PMT
2
Section A (36 marks)
dy
1 (i) Given that y= e -x sin2x, find . [3]
dx
(ii) Hence show that the curve y= e -x sin2x has a stationary point when x= 1 arctan2. [3]
2
2 A curve has equation x 2+ 2y 2 = 4x.
dy
(i) By differentiating implicitly, find in terms of x and y. [3]
dx
(ii) Hence find the exact coordinates of the stationary points of the curve. [You need not determine their
nature.] [3]
3 Express 1 1x13 in the form x- a 1b, where a and b are to be determined. [2]
4 The temperature i °C of water in a container after t minutes is modelled by the equation
i= a- be-kt,
where a, b and k are positive constants.
The initial and long-term temperatures of the water are 15 °C and 100 °C respectively. After 1 minute, the
temperature is 30 °C.
(i) Find a, b and k. [6]
(ii) Find how long it takes for the temperature to reach 80 °C. [2]
5 The driving force F newtons and velocity vkms -1 of a car at time t seconds are related by the
25
equation F = .
v
dF
(i) Find . [2]
dv
dF dv
(ii) Find when v = 50 and = 1.5. [3]
dt dt
6 Evaluate y 3 x(x+ 1) - 2 1 dx, giving your answer as an exact fraction. [5]
0
7 (i) Disprove the following statement:
3 n+ 2 is prime for all integers n H 0. [2]
(ii) Prove that no number of the form 3n (where n is a positive integer) has 5 as its final digit. [2]
© OCR 2013 4753/01 Jan13
PMT
3
Section B (36 marks)
8 Fig. 8 shows parts of the curves y= f(x) and y= g(x), where f(x)= tanx and g(x)= 1+ f(x- 1 r).
4
y
y = f(x) y = g(x)
x
-1
r
1
r
1
r
4 4 2
Fig. 8
(i) Describe a sequence of two transformations which maps the curve y= f(x) to the curve y= g(x). [4]
2sinx
It can be shown that g(x)= .
sinx+ cosx
(ii) Show that gl(x)= 2 . Hence verify that the gradient of y= g(x) at the point ( 1 r,1) is
(sinx+ cosx) 2 4
the same as that of y= f(x) at the origin. [7]
1r 1
sin x y4 y 1
(iii) By writing tanx= and using the substitution u = cosx, show that f(x)dx= du.
cos x u
0 1
2
Evaluate this integral exactly. [4]
(iv) Hence find the exact area of the region enclosed by the curve y= g(x), the x-axis and the lines
x= 1 r and x= 1 r . [2]
4 2
© OCR 2013 4753/01 Jan13 Turn over
PMT
4
9 Fig. 9 shows the line y= x and the curve y= f(x), where f(x)= 1 (e x- 1). The line and the curve intersect
2
at the origin and at the point P(a, a).
y
y = f(x)
y = x
P(a, a)
O
x
Fig. 9
(i) Show that e a= 1+ 2a. [1]
(ii) Show that the area of the region enclosed by the curve, the x-axis and the line x= a is 1 a. Hence find,
2
in terms of a, the area enclosed by the curve and the line y= x. [6]
(iii) Show that the inverse function of f(x) is g(x), where g(x)= ln(1+ 2x). Add a sketch of y= g(x) to
the copy of Fig. 9. [5]
1
(iv) Find the derivatives of f(x) and g(x). Hence verify that gl(a)= .
fl(a)
Give a geometrical interpretation of this result. [7]
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 4753/01 Jan13
Tuesday 18 June 2013 – Morning
A2 GCE MATHEMATICS (MEI)
4753/01 Methods for Advanced Mathematics (C3)
QUESTION PAPER
*4715660613*
PMT
Candidates answer on the Printed Answer Book.
Duration: 1 hour 30 minutes
OCR supplied materials:
• Printed Answer Book 4753/01
• MEI Examination Formulae and Tables (MF2)
Other materials required:
• Scientific or graphical calculator
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 16 pages. The Question Paper consists of 8 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.
© OCR 2013 [M/102/2652] OCR is an exempt Charity
DC (RW/SW) 65527/3 Turn over
PMT
2
Section A (36 marks)
1 Fig. 1 shows the graphs of y= x and y= a x+ b , where a and b are constants. The intercepts of
y= a x+ b with the x- and y-axes are _ -1,0i and a0, 2 1 k respectively.
y
y= x
y= a x+b
2
x
–1 O 2
Fig. 1
(i) Find a and b. [2]
(ii) Find the coordinates of the two points of intersection of the graphs. [4]
2 (i) Factorise fully n 3- n. [2]
(ii) Hence prove that, if n is an integer, n 3- n is divisible by 6. [2]
© OCR 2013 4753/01 Jun13
h
PMT
4
2x
7 (i) Show algebraically that the function f(x)= is odd. [2]
1- x 2
Fig. 7 shows the curve y= f(x) for 0 G xG 4, together with the asymptote x= 1.
y
x
O 1 4
Fig. 7
(ii) Use the copy of Fig. 7 to complete the curve for -4 G xG 4. [2]
© OCR 2013 4753/01 Jun13
PMT
5
Section B (36 marks)
8 Fig. 8 shows the curve y= f(x), where f(x)= _1- xie 2x, with its turning point P.
y
P
y = f(x)
x
O
Fig. 8
(i) Write down the coordinates of the intercepts of y= f(x) with the x- and y-axes. [2]
(ii) Find the exact coordinates of the turning point P. [6]
(iii) Show that the exact area of the region enclosed by the curve and the x- and y-axes is 1 _e 2- 3i. [5]
4
The function g(x) is defined by g(x)= 3fa2 1 xk .
(iv) Express g(x) in terms of x.
Sketch the curve y= g(x) on the copy of Fig. 8, indicating the coordinates of its intercepts with the
x- and y-axes and of its turning point. [4]
(v) Write down the exact area of the region enclosed by the curve y= g(x) and the x- and y-axes. [1]
© OCR 2013 4753/01 Jun13 Turn over
PMT
6
3
9 Fig. 9 shows the curve with equation y 3 = x . It has an asymptote x= a and turning point P.
2x- 1
y
P
x
O x = a
Fig. 9
(i) Write down the value of a. [1]
dy 4x 3- 3x 2
(ii) Show that = .
dx 3y 2 _2x- 1i 2
Hence find the coordinates of the turning point P, giving the y-coordinate to 3 significant figures. [9]
(iii) Show that the substitution u = 2x- 1 transforms y x dx to 1y _u 2 3+ u - 3 1 idu.
3 2x- 1 4
3
Hence find the exact area of the region enclosed by the curve y 3 = x , the x-axis and the lines
2x- 1
x= 1 and x= 4.5. [8]
© OCR 2013 4753/01 Jun13
PMT
8
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 4753/01 Jun13