Download MEI STRUCTURED MATHEMATICS

OXFORD CAMBRIDGE AND RSA EXAMINATIONS
Advanced Subsidiary General Certificate of Education
Advanced General Certificate of Education
MEI STRUCTURED MATHEMATICS
4753/1
Methods for Advanced Mathematics (C3)
Thursday
8 JUNE 2006
Morning
1 hour 30 minutes
Additional materials:
8 page answer booklet
Graph paper
MEI Examination Formulae and Tables (MF2)
TIME
1 hour 30 minutes
INSTRUCTIONS TO CANDIDATES
•
Write your name, centre number and candidate number in the spaces provided on the answer
booklet.
•
Answer all the questions.
•
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 question paper consists of 5 printed pages and 3 blank pages.
HN/2
© OCR 2006 [M/102/2652]
Registered Charity 1066969
[Turn over
2
Section A (36 marks)
1
Solve the equation 3x 2 x.
2
p
Û 6 x sin 2 x dx = 3 3 - p .
Show that Ù
24
ı0
3
Fig. 3 shows the curve defined by the equation y arcsin ( x 1 ) , for 0 x 2.
[3]
1
[6]
y
0
2
x
Fig. 3
(i) Find x in terms of y, and show that
dx
cos y.
dy
(ii) Hence find the exact gradient of the curve at the point where x 1.5.
4753/1 June 2006
[3]
[4]
3
4
Fig. 4 is a diagram of a garden pond.
hm
Fig. 4
The volume V m3 of water in the pond when the depth is h metres is given by
V 13 p h2 ( 3 h ) .
(i) Find
dV
.
dh
[2]
Water is poured into the pond at the rate of 0.02 m3 per minute.
(ii) Find the value of
5
dh
when h 0.4.
dt
[4]
Positive integers a, b and c are said to form a Pythagorean triple if a 2 b 2 c 2.
(i) Given that t is an integer greater than 1, show that 2t , t 2 1 and t 2 1 form a Pythagorean
triple.
[3]
(ii) The two smallest integers of a Pythagorean triple are 20 and 21. Find the third integer.
Use this triple to show that not all Pythagorean triples can be expressed in the form 2t , t 2 1
and t 2 1 .
[3]
6
The mass M kg of a radioactive material is modelled by the equation
M M 0 e k t,
where M 0 is the initial mass, t is the time in years, and k is a constant which measures the rate of
radioactive decay.
(i) Sketch the graph of M against t.
[2]
(ii) For Carbon 14, k 0.000 121. Verify that after 5730 years the mass M has reduced to
approximately half the initial mass.
[2]
The half-life of a radioactive material is the time taken for its mass to reduce to exactly half the
initial mass.
(iii) Show that, in general, the half-life T is given by T ln 2
.
k
[3]
(iv) Hence find the half-life of Plutonium 239, given that for this material k 2.88 10 5. [1]
4753/1 June 2006
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4
Section B (36 marks)
7
Fig. 7 shows the curve y x2 3
. It has a minimum at the point P. The line l is an asymptote to
x1
the curve.
y
P
O
x
l
Fig. 7
(i) Write down the equation of the asymptote l.
[1]
(ii) Find the coordinates of P.
[6]
(iii) Using the substitution u x 1, show that the area of the region enclosed by the x-axis, the
curve and the lines x 2 and x 3 is given by
2
4ˆ
Û Ê
Ù Ë u + 2 + ¯ du.
u
ı1
Evaluate this area exactly.
[7]
dy
x2 3
. Find
(iv) Another curve is defined by the equation
in terms of x and y by
x1
dx
differentiating implicitly. Hence find the gradient of this curve at the point where x 2.
[4]
ey
4753/1 June 2006
5
8
Fig. 8 shows part of the curve y f ( x ) , where f ( x ) = e
-1x
5 sin x,
for all x.
y
P
–2p
–p
0
p
2p
x
Fig. 8
(i) Sketch the graphs of
(A) y f ( 2x ) ,
(B) y f ( x p ) .
[4]
(ii) Show that the x-coordinate of the turning point P satisfies the equation tan x 5.
Hence find the coordinates of P.
(iii) Show that f ( x + p ) = - e
- 1p
5 f ( x ).
[6]
Hence, using the substitution u x p , show that
2p
p
Û f ( x ) dx = - e - 15 p Û f ( u) du.
Ù
Ù
ıp
ı0
Interpret this result graphically. [You should not attempt to integrate f ( x ) .]
4753/1 June 2006
[8]
6
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4753/1 June 2006
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BLANK PAGE
4753/1 June 2006
4753/01
ADVANCED GCE
MATHEMATICS (MEI)
Methods for Advanced Mathematics (C3)
FRIDAY 11 JANUARY 2008
Morning
Time: 1 hour 30 minutes
Additional materials:
Answer Booklet (8 pages)
Graph paper
MEI Examination Formulae and Tables (MF2)
INSTRUCTIONS TO CANDIDATES
•
Write your name in capital letters, your Centre Number and Candidate Number in the spaces
provided on the Answer Booklet.
•
Read each question carefully and make sure you know what you have to do before starting
your answer.
Answer all the questions.
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.
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.
This document consists of 4 printed pages.
© OCR 2008 [M/102/2652]
OCR is an exempt Charity
[Turn over
2
Section A (36 marks)
3
1 + 6x2 .
1
Differentiate
[4]
2
The functions f(x) and g(x) are defined for all real numbers x by
f(x) = x2 ,
3
g(x) = x − 2.
(i) Find the composite functions fg(x) and gf(x).
[3]
(ii) Sketch the curves y = f(x), y = fg(x) and y = gf(x), indicating clearly which is which.
[2]
The profit £P made by a company in its nth year is modelled by the exponential function
P = Aebn .
In the first year (when n = 1), the profit was £10 000. In the second year, the profit was £16 000.
4
(i) Show that eb = 1.6, and find b and A.
[6]
(ii) What does this model predict the profit to be in the 20th year?
[2]
When the gas in a balloon is kept at a constant temperature, the pressure P in atmospheres and the
volume V m3 are related by the equation
P=
k
,
V
where k is a constant. [This is known as Boyle’s Law.]
When the volume is 100 m3 , the pressure is 5 atmospheres, and the volume is increasing at a rate of
10 m3 per second.
(i) Show that k = 500.
(ii) Find
[1]
dP
in terms of V .
dV
[2]
(iii) Find the rate at which the pressure is decreasing when V = 100.
5
[4]
(i) Verify the following statement:
‘2p − 1 is a prime number for all prime numbers p less than 11’.
[2]
(ii) Calculate 23 × 89, and hence disprove this statement:
‘2p − 1 is a prime number for all prime numbers p’.
© OCR 2008
4753/01 Jan08
[2]
3
6
Fig. 6 shows the curve e2y = x2 + y.
y
O
x
P
Fig. 6
(i) Show that
2x
dy
= 2y
.
dx 2e − 1
[4]
(ii) Hence find to 3 significant figures the coordinates of the point P, shown in Fig. 6, where the curve
has infinite gradient.
[4]
Section B (36 marks)
7
A curve is defined by the equation y = 2x ln(1 + x).
(i) Find
dy
and hence verify that the origin is a stationary point of the curve.
dx
[4]
(ii) Find
d2 y
, and use this to verify that the origin is a minimum point.
dx2
[5]
(iii) Using the substitution u = 1 + x, show that 1
Hence evaluate 0
x2
1
dx = u − 2 + du.
1+x
u
x2
dx, giving your answer in an exact form.
1+x
[6]
1
(iv) Using integration by parts and your answer to part (iii), evaluate 2x ln(1 + x) dx.
[4]
0
© OCR 2008
4753/01 Jan08
[Turn over
4
8
Fig. 8 shows the curve y = f(x), where f(x) = 1 + sin 2x for − 14 π ≤ x ≤ 14 π .
y
2
– 41 p
O
1p
4
x
Fig. 8
(i) State a sequence of two transformations that would map part of the curve y = sin x onto the curve
y = f(x).
[4]
(ii) Find the area of the region enclosed by the curve y = f(x), the x-axis and the line x = 14 π .
[4]
(iii) Find the gradient of the curve y = f(x) at the point (0, 1). Hence write down the gradient of the
[4]
curve y = f −1 (x) at the point (1, 0).
(iv) State the domain of f −1 (x). Add a sketch of y = f −1 (x) to a copy of Fig. 8.
[3]
(v) Find an expression for f −1 (x).
[2]
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.
© OCR 2008
4753/01 Jan08