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Math C3
Exam.EDEXEL AL
0181843818 للكمبيوتر MIC داخل معھد – بجوار مسجد الصحابه – ش خالد امين خلف الشبراوى 20رقم : فرع الھرم 0181847737 ش اسماعيل القبانى خلف الكازيتا فى مدخل سنتر فريندز مدينة نصر المنطقه االولىعز الدين طه موازى لــ ش 7رقم : فرع مدينة نصر
0127711092 الدور االول داخل سنتر نبراس – العماره اللى فيھا احمد محمود لالحذيه – ش الخليفه المأمون بجانب الشبراوى 41رقم : فرع مصر الجديده 0127746409: ت موبيل
Index of Math C3
EDEXEL . AL Exams
Moke exam ……………………………………………………………………....1 Specimen exam…………………………………………………………………...5.
June 2005………………………………………………………………………...19
Jan 2006………………………………………………………………………….39
June 2006……………………………………………………………...………….59
Jan 2007…………………………………………………………...……………..83
June 2007……………………………………………………..…………….......107
Jan 2008………………………………………………...………………………131
June 2008 ……………………..…………………………………...…………...155
Jan 2009 ……………………………………………...………………...............179.
June 2009 .. ….………………………………………..……………...………...207
Jan 2010 ….…. ……………………………………………………...…………235
June 2010……………………………………………………………………….263.
Jan 2011………………………………………………………………………..271.
6665Edexcel GCEPure Mathematics C3
Advanced LevelSpecimen Paper
Time: 1 hour 30 minutes
Materials required for examination Items included with question papersAnswer Book (AB16) NilMathematical Formulae (Lilac)Graph Paper (ASG2)
Candidates may use any calculator EXCEPT those with the facility forsymbolic algebra, differentiation and/or integration. Thus candidates mayNOT use calculators such as the Texas Instruments TI-89, TI-92,Casio cfx 9970G, Hewlett Packard HP 48G.
Instructions to Candidates
In the boxes on the answer book, write the name of the examining body (Edexcel), your centrenumber, candidate number, the unit title (Pure Mathematics C3), the paper reference (6665), yoursurname, other name and signature.When a calculator is used, the answer should be given to an appropriate degree of accuracy.
Information for Candidates
A booklet ‘Mathematical Formulae and Statistical Tables’ is provided.Full marks may be obtained for answers to ALL questions.This paper has seven questions.
Advice to Candidates
You must ensure that your answers to parts of questions are clearly labelled.You must show sufficient working to make your methods clear to the Examiner. Answers withoutworking may gain no credit.
© 2003 EdexcelThis publication may only be reproduced in accordance with Edexcel copyright policy.
32
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1. The function f is defined by
f: x � | x – 2 | – 3, x � ℝ.
(a) Solve the equation f(x) = 1. (3)
The function g is defined by
g: x � x2 – 4x + 11, x � 0.
(b) Find the range of g. (3)
(c) Find gf(–1).(2)
2. f(x) = x3 – 2x – 5.
(a) Show that there is a root � of f(x) = 0 for x in the interval [2, 3]. (2)
The root � is to be estimated using the iterative formula
xn + 1 = ,52 ���
����
��
nxx0 = 2.
(b) Calculate the values of x1, x2, x3 and x4, giving your answers to 4 significant figures. (3)
(c) Prove that, to 5 significant figures, � is 2.0946. (3)
3. (a) Using the identity for cos (A + B), prove that cos � � 1 – 2 sin2 ( 21 � ).
(3)
(b) Prove that 1 + sin � – cos � � 2 sin ( 21 � )[cos ( 2
1 � ) + sin ( 21 � )].
(3)
(c) Hence, or otherwise, solve the equation
���� 20,0cossin1 ����� . (4)
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34
4. f(x) = x + 1
3�x
– 32
122
�� xx, x ∈ ℝ, x > 1.
(a) Show that f(x) = 3
332
�
��
xxx .
(5)
(b) Solve the equation f � (x) = 2522 .
(5)
5. Figure 1
y
(0, q)
(p, 0) O x
Figure 1 shows part of the curve with equation y = f(x), x � ℝ. The curve meets the x-axis atP (p, 0) and meets the y-axis at Q (0, q).
(a) On separate diagrams, sketch the curve with equation
(i) y = |f(x)|,
(ii) y = 3f( 21 x).
In each case show, in terms of p or q, the coordinates of points at which the curve meets theaxes.
(5)
Given that f(x) = 3 ln(2x + 3),
(b) state the exact value of q,(1)
(c) find the value of p,(2)
(d) find an equation for the tangent to the curve at P.(4)
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6. As a substance cools its temperature, T �C, is related to the time (t minutes) for which it has beencooling. The relationship is given by the equation
T = 20 + 60e–0.1t, t � 0.
(a) Find the value of T when the substance started to cool.(1)
(b) Explain why the temperature of the substance is always above 20�C.(1)
(c) Sketch the graph of T against t.(2)
(d) Find the value, to 2 significant figures, of t at the instant T = 60.(4)
(e) Find tT
dd .
(2)
(f) Hence find the value of T at which the temperature is decreasing at a rate of 1.8 �C perminute.
(3)
7. (i) Given that y = tan x + 2 cos x, find the exact value of xy
dd at x =
4� .
(3)
(ii) Given that x = tan 21 y, prove that
xy
dd = 21
2x�
.
(4)
(iii) Given that y = e–x sin 2x, show that xy
dd can be expressed in the form R e–x cos (2x + �). Find,
to 3 significant figures, the values of R and α, where 0 < α < 2� .
(7)
END
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This publication may be reproduced only in accordance with
Edexcel Limited copyright policy.
©2005 Edexcel Limited.
Printer’s Log. No.
N23494BW850/R6665/57570 7/7/3/3/3
Paper Reference(s)
6665/01
Edexcel GCECore Mathematics C3
Advanced Level
Monday 20 June 2005 – Morning
Time: 1 hour 30 minutes
Materials required for examination Items included with question papers
Mathematical Formulae (Green) Nil
Candidates may use any calculator EXCEPT those with the facility forsymbolic algebra, differentiation and/or integration. Thus candidates mayNOT use calculators such as the Texas Instruments TI 89, TI 92, Casio CFX 9970G, Hewlett Packard HP 48G.
Instructions to Candidates
In the boxes above, write your centre number, candidate number, your surname, initial(s) andsignature.Check that you have the correct question paper.You must write your answer for each question in the space following the question.When a calculator is used, the answer should be given to an appropriate degree of accuracy.
Information for Candidates
A booklet ‘Mathematical Formulae and Statistical Tables’ is provided.Full marks may be obtained for answers to ALL questions.The marks for individual questions and the parts of questions are shown in round brackets: e.g. (2).There are 7 questions in this question paper. The total mark for this paper is 75. There are 20 pages in this question paper. Any blank pages are indicated.
Advice to Candidates
You must ensure that your answers to parts of questions are clearly labelled.You must show sufficient working to make your methods clear to the Examiner. Answers withoutworking may gain no credit.
Turn over
Examiner’s use only
Team Leader’s use only
Question LeaveNumber Blank
1
2
3
4
5
6
7
Total
CentreNo.
Candidate No.
Surname Initial(s)
Signature
Paper Reference
6 6 6 5 0 1
*N23494B0120*
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1. (a) Given that sin2θ + cos2θ ≡ 1, show that 1 + tan2θ ≡ sec2θ.
(2)
(b) Solve, for 0 ��θ < 360°, the equation
2 tan2θ + secθ = 1,
giving your answers to 1 decimal place.
(6)
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2 *n23494B0220*
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3
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Question 1 continued
Q1
(Total 8 marks)
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2. (a) Differentiate with respect to x
(i) 3 sin2x + sec2x,
(3)
(ii) {x + ln(2x)}3.
(3)
Given that
(b) show that
(6)
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3
d 8.
d ( 1)
yx x
= −−
2
2
5 10 9, 1,
( 1)
x xy xx− +
= ≠−
4 *n23494B0420*
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Question 2 continued
Q2
(Total 12 marks)
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3. The function f is defined by
(a) Show that
(4)
(b) Find f –1(x).
(3)
The function g is defined by
g : x → x2 + 5, x ∈ R.
(c) Solve fg(x) = .
(3)
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14
2f ( ) , 1.
1x x
x= >
−
2
5 1 3f : , 1.
2 2
xx xx x x
+→ − >
+ − +
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Question 3 continued
Q3
(Total 10 marks)
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4. f(x) = 3ex – ln x – 2, x > 0.
(a) Differentiate to find f ´(x).
(3)
The curve with equation y = f(x) has a turning point at P. The x-coordinate of P is α.
(b) Show that α = e–α.
(2)
The iterative formula
is used to find an approximate value for α.
(c) Calculate the values of x1, x2, x3 and x4, giving your answers to 4 decimal places.
(2)
(d) By considering the change of sign of f ´(x) in a suitable interval, prove that α = 0.1443
correct to 4 decimal places.
(2)
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11 06
e , 1,nxnx x−
+ = =
16
12
8 *n23494B0820*
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Q4
(Total 9 marks)
Question 4 continued
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5. (a) Using the identity cos(A + B) ≡ cosA cosB – sin A sinB, prove that
cos 2A ≡ 1 – 2 sin2A.
(2)
(b) Show that
2 sin 2θ – 3 cos 2θ – 3 sin θ + 3 ≡ sin θ (4 cos θ + 6 sin θ – 3).
(4)
(c) Express 4 cos θ + 6 sin θ in the form R sin(θ + α), where R > 0 and 0 < α < π.
(4)
(d) Hence, for 0 � θ < π, solve
2 sin 2θ = 3(cos 2θ + sin θ – 1),
giving your answers in radians to 3 significant figures, where appropriate.
(5)
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11
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Question 5 continued
Q5
(Total 15 marks)
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6. Figure 1
Figure 1 shows part of the graph of y = f(x), x ∈ R. The graph consists of two line
segments that meet at the point (1, a), a < 0. One line meets the x-axis at (3, 0). The other
line meets the x-axis at (–1, 0) and the y-axis at (0, b), b < 0.
In separate diagrams, sketch the graph with equation
(a) y = f(x + 1),
(2)
(b) y = f( |x | ).
(3)
Indicate clearly on each sketch the coordinates of any points of intersection with the axes.
Given that f(x) = |x – 1| – 2, find
(c) the value of a and the value of b,
(2)
(d) the value of x for which f(x) = 5x.
(4)
14 *n23494B01420*
y
xb
–1 O 3
(1, a)
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Q6
(Total 11 marks)
Question 6 continued
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7. A particular species of orchid is being studied. The population p at time t years after the
study started is assumed to be
where a is a constant.
Given that there were 300 orchids when the study started,
(a) show that a = 0.12,
(3)
(b) use the equation with a = 0.12 to predict the number of years before the population
of orchids reaches 1850.
(4)
(c) Show that
(1)
(d) Hence show that the population cannot exceed 2800.
(2)
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0.2
336.
0.12 e tp −=+
0.2
0.2
2800 e,
1 e
t
t
apa
=+
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TOTAL FOR PAPER: 75 MARKS
END
20 *n23494B02020*
Question 7 continued
Q7
(Total 10 marks)
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Paper Reference(s)
6665/01Edexcel GCECore Mathematics C3Advanced LevelMonday 23 January 2006 – AfternoonTime: 1 hour 30 minutes
Materials required for examination Items included with question papersMathematical Formulae (Green) Nil
Candidates may use any calculator EXCEPT those with the facility forsymbolic algebra, differentiation and/or integration. Thus candidates mayNOT use calculators such as the Texas Instruments TI 89, TI 92, Casio CFX 9970G, Hewlett Packard HP 48G.
Instructions to CandidatesIn the boxes above, write your centre number, candidate number, your surname, initial(s) andsignature.Check that you have the correct question paper.You must write your answer for each question in the space following the question.When a calculator is used, the answer should be given to an appropriate degree of accuracy.
Information for CandidatesA booklet ‘Mathematical Formulae and Statistical Tables’ is provided.Full marks may be obtained for answers to ALL questions.The marks for individual questions and the parts of questions are shown in round brackets: e.g. (2).There are 8 questions in this question paper. The total mark for this paper is 75. There are 20 pages in this question paper. Any blank pages are indicated.
Advice to CandidatesYou must ensure that your answers to parts of questions are clearly labelled.You must show sufficient working to make your methods clear to the examiner. Answers withoutworking may gain no credit.
Examiner’s use only
Team Leader’s use only
Question LeaveNumber Blank
1
2
3
4
5
6
7
8
Total
CentreNo.
Candidate No.
Surname Initial(s)
Signature
Paper Reference
6 6 6 5 0 1
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1. Figure 1
Figure 1 shows the graph of y = f(x), –5 x 5.The point M (2, 4) is the maximum turning point of the graph.
Sketch, on separate diagrams, the graphs of
(a) y = f(x) + 3,(2)
(b) y = | f(x) | ,(2)
(c) y = f( |x | ).(3)
Show on each graph the coordinates of any maximum turning points.
2 *N23495A0220*
y
x–5 O 5
M (2, 4)
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Q1
(Total 7 marks)
3
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2. Express
as a single fraction in its simplest form.(7)
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2
2
2 3 6(2 3)( 2) 2
x xx x x x
+ −+ − − −
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(Total 7 marks)
5
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3. The point P lies on the curve with equation The x-coordinate of P is 3.
Find an equation of the normal to the curve at the point P in the form y = ax + b, wherea and b are constants.
(5)
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1ln .3
y x⎛ ⎞= ⎜ ⎟⎝ ⎠
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Q3
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4. (a) Differentiate with respect to x
(i) x2e3x + 2,(4)
(ii) .(4)
(b) Given that x = 4 sin(2y + 6), find in terms of x.
(5)
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ddyx
3cos(2 )3
xx
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(Total 13 marks)
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5. f(x) = 2x3 – x – 4.
(a) Show that the equation f(x) = 0 can be written as
x = √(3)
The equation 2x3 – x – 4 = 0 has a root between 1.35 and 1.4.
(b) Use the iteration formula
xn + 1 = √with x0 = 1.35, to find, to 2 decimal places, the values of x1, x2 and x3.
(3)
The only real root of f(x) = 0 is α.
(c) By choosing a suitable interval, prove that α = 1.392, to 3 decimal places.(3)
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2 1 ,2nx
⎛ ⎞+⎜ ⎟
⎝ ⎠
2 1 .2x
⎛ ⎞+⎜ ⎟⎝ ⎠
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(Total 9 marks)
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6. f(x) = 12 cos x – 4 sin x.
Given that f(x) = R cos(x + α), where R 0 and 0 α 90°,
(a) find the value of R and the value of α.(4)
(b) Hence solve the equation
12 cos x – 4 sin x = 7
for 0 x < 360°, giving your answers to one decimal place.(5)
(c) (i) Write down the minimum value of 12 cos x – 4 sin x.(1)
(ii) Find, to 2 decimal places, the smallest positive value of x for which thisminimum value occurs.
(2)
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(Total 12 marks)
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7. (a) Show that
(i) x ≠ (n – )π, n ∈ Z,(2)
(ii) (cos 2x – sin 2x) ≡ cos2 x – cos x sin x – .(3)
(b) Hence, or otherwise, show that the equation
can be written as
sin 2θ = cos 2θ.(3)
(c) Solve, for 0 θ < 2π,
sin 2θ = cos 2θ,
giving your answers in terms of π.(4)
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cos 2 1coscos sin 2
θθθ θ
⎛ ⎞ =⎜ ⎟+⎝ ⎠
12
12
14
cos 2 cos sin ,cos sin
x x xx x
≡ −+
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(Total 12 marks)
17
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8. The functions f and g are defined by
f:x → 2x + ln 2, x ∈ R,
g :x → e2x, x ∈ R.
(a) Prove that the composite function gf is
gf :x → 4e4x, x ∈ R.(4)
(b) In the space provided on page 19, sketch the curve with equation y = gf(x), and showthe coordinates of the point where the curve cuts the y-axis.
(1)
(c) Write down the range of gf.(1)
(d) Find the value of x for which , giving your answer to 3 significant figures.
(4)
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d [gf ( )] 3d
xx
=
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TOTAL FOR PAPER: 75 MARKS
END
Q8
(Total 10 marks)
20 *N23495A02020*
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This publication may be reproduced only in accordance with Edexcel Limited copyright policy. ©2006 Edexcel Limited.
Printer’s Log. No.
N23581AW850/R6665/57570 3/3/3/3/3/36,300
Paper Reference(s)
6665/01Edexcel GCECore Mathematics C3Advanced LevelMonday 12 June 2006 – AfternoonTime: 1 hour 30 minutes
Materials required for examination Items included with question papersMathematical Formulae (Green) Nil
Candidates may use any calculator EXCEPT those with the facility forsymbolic algebra, differentiation and/or integration. Thus candidates mayNOT use calculators such as the Texas Instruments TI 89, TI 92, CasioCFX 9970G, Hewlett Packard HP 48G.
Instructions to CandidatesIn the boxes above, write your centre number, candidate number, your surname, initial(s) andsignature.Check that you have the correct question paper.When a calculator is used, the answer should be given to an appropriate degree of accuracy.You must write your answer for each question in the space following the question.If you need more space to complete your answer to any question, use additional answer sheets.
Information for CandidatesA booklet ‘Mathematical Formulae and Statistical Tables’ is provided.The marks for individual questions and the parts of questions are shown in round brackets: e.g. (2).Full marks may be obtained for answers to ALL questions.There are 8 questions in this question paper. The total mark for this paper is 75.There are 24 pages in this question paper. Any blank pages are indicated.
Advice to CandidatesYou must ensure that your answers to parts of questions are clearly labelled.You must show sufficient working to make your methods clear to the Examiner. Answers withoutworking may gain no credit.
Turn over
Examiner’s use only
Team Leader’s use only
Question LeaveNumber Blank
1
2
3
4
5
6
7
8
Total
CentreNo.
Candidate No.
Surname Initial(s)
Signature
Paper Reference
6 6 6 5 0 1
*N23581A0124*
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2
1. (a) Simplify .(3)
(b) Hence, or otherwise, express as a single fraction in its simplest
form.(3)
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2
2
3 2 11 ( 1)
x xx x x− −
−− +
2
2
3 21
x xx− −
−
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3
Question 1 continued
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(Total 6 marks)
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4
2. Differentiate, with respect to x,
(a) e3x + ln 2x,(3)
(b) .(3)
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322(5 )x+
Q2
(Total 6 marks)
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5
3. Figure 1
Figure 1 shows part of the curve with equation y = f(x), x ∈R, where f is an increasingfunction of x. The curve passes through the points P(0, –2) and Q(3, 0) as shown.
In separate diagrams, sketch the curve with equation
(a) y = |f (x) | ,(3)
(b) y = f –1(x),(3)
(c) y = f(3x).(3)
Indicate clearly on each sketch the coordinates of the points at which the curve crosses ormeets the axes.
12
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y
xO
P
Q (3, 0)
y = f (x)
(0, –2)
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7
Question 3 continued
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Q3
(Total 9 marks)
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8
4. A heated metal ball is dropped into a liquid. As the ball cools, its temperature, T °C, t minutes after it enters the liquid, is given by
T = 400 e–0.05t + 25, t 0.
(a) Find the temperature of the ball as it enters the liquid.(1)
(b) Find the value of t for which T = 300, giving your answer to 3 significant figures.(4)
(c) Find the rate at which the temperature of the ball is decreasing at the instant when t = 50. Give your answer in °C per minute to 3 significant figures.
(3)
(d) From the equation for temperature T in terms of t, given above, explain why thetemperature of the ball can never fall to 20 °C.
(1)
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(Total 9 marks)
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10
5. Figure 2
Figure 2 shows part of the curve with equation
The curve has a minimum at the point P. The x-coordinate of P is k.
(a) Show that k satisfies the equation
4k + sin 4k – 2 = 0.(6)
The iterative formula
is used to find an approximate value for k.
(b) Calculate the values of x1, x2, x3 and x4, giving your answers to 4 decimal places.(3)
(c) Show that k = 0.277, correct to 3 significant figures.(2)
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1 01 (2 sin 4 ), 0.3,4n nx x x+ = − =
(2 1) tan 2 , 0 .4
y x x x π= − <
*N23581A01024*
y
O
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x4π
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Turn over
Q5
(Total 11 marks)
*N23581A01324*
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6. (a) Using sin2θ + cos2θ ≡ 1, show that cosec2θ – cot2θ ≡ 1.(2)
(b) Hence, or otherwise, prove that
cosec4θ – cot4θ ≡ cosec2θ + cot2θ.(2)
(c) Solve, for 90° < θ < 180°,
cosec4θ – cot4θ = 2 – cot θ.(6)
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Q6
(Total 10 marks)
*N23581A01724*
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7. For the constant k, where k > 1, the functions f and g are defined by
f: x ln (x + k), x > –k,g: x |2x – k |, x ∈ R.
(a) On separate axes, sketch the graph of f and the graph of g.
On each sketch state, in terms of k, the coordinates of points where the graph meets thecoordinate axes.
(5)
(b) Write down the range of f.(1)
(c) Find in terms of k, giving your answer in its simplest form.(2)
The curve C has equation y = f(x). The tangent to C at the point with x-coordinate 3 isparallel to the line with equation 9y = 2x + 1.
(d) Find the value of k.(4)
fg4k
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Question 7 continued
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Q7
(Total 12 marks)
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8. (a) Given that where 270° < A < 360°, find the exact value of sin 2A.(5)
(b) (i) Show that (3)
Given that
(ii) show that (4)
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d sin 2 .dy xx
=
23sin cos 2 cos 2 ,3 3
y x x xπ π = + + + −
cos 2 cos 2 cos 2 .3 3
x x xπ π + + − ≡
3cos ,4
A =
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Question 8 continued
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Question 8 continued
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TOTAL FOR PAPER: 75 MARKS
END
*N23581A02424*
Q8
(Total 12 marks)
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This publication may be reproduced only in accordance with Edexcel Limited copyright policy. ©2007 Edexcel Limited.
Printer’s Log. No.
N23583AW850/R6665/57570 3/3/3/3/3/3/21,900
Paper Reference(s)
6665/01Edexcel GCECore Mathematics C3Advanced LevelThursday 18 January 2007 – AfternoonTime: 1 hour 30 minutes
Materials required for examination Items included with question papersMathematical Formulae (Green) Nil
Candidates may use any calculator EXCEPT those with the facility forsymbolic algebra, differentiation and/or integration. Thus candidates mayNOT use calculators such as the Texas Instruments TI 89, TI 92, Casio CFX 9970G, Hewlett Packard HP 48G.
Instructions to CandidatesIn the boxes above, write your centre number, candidate number, your surname, initial(s) andsignature.Check that you have the correct question paper.You must write your answer for each question in the space following the question.When a calculator is used, the answer should be given to an appropriate degree of accuracy.
Information for CandidatesA booklet ‘Mathematical Formulae and Statistical Tables’ is provided.Full marks may be obtained for answers to ALL questions.The marks for individual questions and the parts of questions are shown in round brackets: e.g. (2).There are 8 questions in this question paper. The total mark for this paper is 75.There are 24 pages in this question paper. Any blank pages are indicated.
Advice to CandidatesYou must ensure that your answers to parts of questions are clearly labelled.You must show sufficient working to make your methods clear to the Examiner. Answers withoutworking may gain no credit.
Turn over
Examiner’s use only
Team Leader’s use only
Question LeaveNumber Blank
1
2
3
4
5
6
7
8
Total
CentreNo.
Candidate No.
Surname Initial(s)
Signature
Paper Reference
6 6 6 5 0 1
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1. (a) By writing sin 3θ as sin (2θ +θ), show that
(5)
(b) Given that sin , find the exact value of sin3θ .(2)
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34
θ√
=
3sin 3 3sin 4sin .θ θ θ= −
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Question 1 continued
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Q1
(Total 7 marks)
*N23583A0324*
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2.
(a) Show that (4)
(b) Show that x2 + x + 1 > 0 for all values of x.(3)
(c) Show that f (x) > 0 for all values of x, x ≠ –2.(1)
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2
2
1f ( ) , 2.( 2)x xx x
x+ +
= ≠ −+
2
3 3f ( ) 1 , 2.2 ( 2)
x xx x
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Question 2 continued
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Q2
(Total 8 marks)
*N23583A0524*
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3. The curve C has equation
x = 2 sin y.
(a) Show that the point lies on C.(1)
(b) Show that at P.(4)
(c) Find an equation of the normal to C at P. Give your answer in the form y = mx + c,where m and c are exact constants.
(4)
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d 1dyx
=√ 2
,4
P π √ 2
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Question 3 continued
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Q3
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*N23583A0724*
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4. (i) The curve C has equation
Use calculus to find the coordinates of the turning points of C.(6)
(ii) Given that
find the value of .(5)
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d 1at ln 3d 2y xx
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32 2(1 e ) ,xy = +
2 .9
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Question 4 continued
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Q4
(Total 11 marks)
*N23583A01124*
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5. Figure 1
Figure 1 shows an oscilloscope screen.
The curve shown on the screen satisfies the equation
(a) Express the equation of the curve in the form y = Rsin(x + α), where R and α areconstants,
(4)
(b) Find the values of x, , for which y = 1.(4)
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0 2x π<
0 and 0 .2
R πα> < <
cos sin .y x x= √ 3 +
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Question 5 continued
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Q5
(Total 8 marks)
*N23583A01324*
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6. The function f is defined by
(a) Show that the inverse function of f is defined by
and write down the domain of f –1.(4)
(b) Write down the range of f –1.(1)
(c) In the space provided on page 16, sketch the graph of y = f –1(x). State the coordinatesof the points of intersection with the x and y axes.
(4)
The graph of y = x + 2 crosses the graph of y = f –1(x) at x = k.
The iterative formula
is used to find an approximate value for k.
(d) Calculate the values of x1 and x2, giving your answers to 4 decimal places.(2)
(e) Find the value of k to 3 decimal places.(2)
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1 01 e , 0.32
nxnx x+ = − = −
1 1f : 2 e2
xx− −
f : ln (4 2 ), 2 and .x x x x− < ∈
*N23583A01424*
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Question 6 continued
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*N23583A01624*
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Turn over
Q6
(Total 13 marks)
*N23583A01724*
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7.
(a) Show that there is a root of f (x) = 0 in the interval [–2, –1].(3)
(b) Find the coordinates of the turning point on the graph of y = f (x).(3)
(c) Given that f (x) = (x – 2)(x3 + ax2 + bx + c), find the values of the constants, a, b and c.(3)
(d) In the space provided on page 21, sketch the graph of y = f (x).(3)
(e) Hence sketch the graph of y = |f (x)|.(1)
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4f ( ) 4 8.x x x= − −
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Question 7 continued
Turn over
Q7
(Total 13 marks)
*N23583A02124*
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8. (i) Prove that
(3)
(ii) Given that
(a) express arcsin x in terms of y.(2)
(b) Hence evaluate arccos x + arcsin x. Give your answer in terms of π.(1)
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arccos , 1 1 and 0 ,y x x y π= −
2 2 2 2sec cosec tan cot .x x x x− ≡ −
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Question 8 continued
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Question 8 continued
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TOTAL FOR PAPER: 75 MARKS
END
*N23583A02424*
Q8
(Total 6 marks)
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Paper Reference(s)
6665/01Edexcel GCECore Mathematics C3Advanced LevelThursday 14 June 2007 – AfternoonTime: 1 hour 30 minutes
Materials required for examination Items included with question papersMathematical Formulae (Green) Nil
Candidates may use any calculator allowed by the regulations of the Joint Council for Qualifications. Calculators must not have the facility for symbolic algebra manipulation, differentiation and integration, or have retrievable mathematical formulas stored in them.
Instructions to CandidatesIn the boxes above, write your centre number , candidate number, your surname, initial(s) and signature.Check that you have the correct question paper.You must write your answer for each question in the space following the question.When a calculator is used, the answer should be given to an appropriate degree of accuracy.
Information for CandidatesA booklet ‘Mathematical Formulae and Statistical Tables’ is provided.Full marks may be obtained for answers to ALL questions.The marks for individual questions and the parts of questions are shown in round brackets: e.g. (2).There are 8 questions in this question paper. The total mark for this paper is 75.There are 24 pages in this question paper. Any blank pages are indicated.
Advice to CandidatesYou must ensure that your answers to parts of questions are clearly labelled.You should show sufficient working to make your methods clear to the Examiner .Answers without working may not gain full credit.
Examiner’s use only
Team Leader’s use only
Question Leave Number Blank
1
2
3
4
5
6
7
8
Total
Surname Initial(s)
Signature
Centre No.
*N26109A0124*Turn over
Candidate No.
Paper Reference
6 6 6 5 0 1
This publication may be reproduced only in accordance with Edexcel Limited copyright policy. ©2007 Edexcel Limited.
Printer’s Log. No.
N26109AW850/R6665/57570 3/3/3/3/3/3/37,500
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*N26109A0224*
1. Find the exact solutions to the equations
(a) ln x + ln 3 = ln 6,(2)
(b) ex + 3e–x = 4.(4)
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(Total 6 marks)
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*N26109A0424*
2. f ( ) ,x xx
xx x
x=++
−++ −
>2 3
29 2
2 3 2122 .
(a) Show that f ( ) .x xx
=−−
4 62 1 (7)
(b) Hence, or otherwise, find f ′(x) in its simplest form.(3)
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Question 2 continued
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_________________________________________________________________________________________________________________________________ Q2
(Total 10 marks)
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*N26109A0624*
3. A curve C has equationy = x2ex.
(a) Find ddyx
, using the product rule for dif ferentiation.(3)
(b) Hence find the coordinates of the turning points of C.(3)
(c) Find dd
2
2
yx
.(2)
(d) Determine the nature of each turning point of the curve C.(2)
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(Total 10 marks)
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*N26109A0824*
4. f( x) = – x3 + 3x2 – 1.
(a) Show that the equation f(x) = 0 can be rewritten as
xx
=−
13
.(2)
(b) Starting with x1 = 0.6, use the iteration
xxn
n+ =
−
1
13
to calculate the values of x2, x3 and x4, giving all your answers to 4 decimal places.(2)
(c) Show that x = 0.653 is a root of f( x) = 0 correct to 3 decimal places.(3)
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(Total 7 marks)
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*N26109A01024*
5. The functions f and g are defined by
f : ln( ),x x2 1− x x∈ >, ,12
g : ,xx
23−
x x∈ ≠, .3
(a) Find the exact value of fg(4).(2)
(b) Find the inverse function f –1(x), stating its domain.(4)
(c) Sketch the graph of y x= | | .g( ) Indicate clearly the equation of the vertical asymptote and the coordinates of the point at which the graph crosses the y-axis.
(3)
(d) Find the exact values of x for which 23x −
= 3.(3)
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Question 5 continued
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*N26109A01224*
Question 5 continued
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(Total 12 marks)
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*N26109A01424*
6. (a) Express 3 sin x + 2 cos x in the form R sin(x + α) where R > 0 and 0 < α < π2
.(4)
(b) Hence find the greatest value of (3 sin x + 2 cos x)4.(2)
(c) Solve, for 0 < x < 2π, the equation
3 sin x + 2 cos x = 1,
giving your answers to 3 decimal places.(5)
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Question 6 continued
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*N26109A01624*
Question 6 continued
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Question 6 continued
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_________________________________________________________________________________________________________________________________ Q6
(Total 11 marks)
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*N26109A01824*
7. (a) Prove that sin cos 2cosec 2 , 90 .cos sinθ θ θ θ nθ θ+ = ≠ °
(4)
(b) On the axes on page 20, sketch the graph of y = 2 cosec 2θ for 0° < θ < 360°.(2)
(c) Solve, for 0° < θ < 360°, the equation
sincos
cossin
,θθ
θθ
+ = 3
giving your answers to 1 decimal place.(6)
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Question 7 continued
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y
O 90° 180° 270° 360° θ
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Question 7 continued
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(Total 12 marks)
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*N26109A02224*
8. The amount of a certain type of drug in the bloodstream t hours after it has been taken is given by the formula
x Dt
=−
e18 ,
where x is the amount of the drug in the bloodstream in milligrams and D is the dose given in milligrams.
A dose of 10 mg of the drug is given.
(a) Find the amount of the drug in the bloodstream 5 hours after the dose is given. Give your answer in mg to 3 decimal places.
(2)
A second dose of 10 mg is given after 5 hours.
(b) Show that the amount of the drug in the bloodstream 1 hour after the second dose is 13.549 mg to 3 decimal places.
(2)
No more doses of the drug are given. At time T hours after the second dose is given, the amount of the drug in the bloodstream is 3 mg.
(c) Find the value of T.(3)
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Question 8 continued
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TOTAL FOR PAPER: 75 MARKS
END
Q8
(Total 7 marks)
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Paper Reference(s)
6665/01Edexcel GCECore Mathematics C3AdvancedThursday 17 January 2008 – AfternoonTime: 1 hour 30 minutes
Materials required for examination Items included with question papersMathematical Formulae (Green) Nil
Candidates may use any calculator allowed by the regulations of the Joint Council for Qualifications. Calculators must not have the facility for symbolic algebra manipulation, differentiation and integration, or have retrievable mathematical formulae stored in them.
Examiner’s use only
Team Leader’s use only
Question Leave Number Blank
1
2
3
4
5
6
7
8
9
10
Total
Surname Initial(s)
Signature
Turn over
Paper Reference
6 6 6 5 0 1
Instructions to CandidatesIn the boxes above, write your centre number, candidate number, your surname, initials and signature. Check that you have the correct question paper.Answer ALL the questions. Write your answers in the spaces provided in this question paper .When a calculator is used, the answer should be given to an appropriate degree of accuracy .
Information for CandidatesA booklet ‘Mathematical Formulae and Statistical Tables’ is provided.Full marks may be obtained for answers to ALL questions.The marks for individual questions and the parts of questions are shown in round brackets: e.g. (2).There are 8 questions in this question paper. The total mark for this paper is 75. There are 24 pages in this question paper. Any blank pages are indicated.
Advice to CandidatesYou must ensure that your answers to parts of questions are clearly labelled.You should show sufficient working to make your methods clear to the Examiner . Answers without working may not gain full credit.
Examiner’s use only
Team Leader’s use only
Question Leave Number Blank
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3
4
5
6
7
8
Total
Surname Initial(s)
Signature
Centre No.
*H26315RB0124*Turn over
Candidate No.
This publication may be reproduced only in accordance with Edexcel Limited copyright policy. ©2008 Edexcel Limited.
Printer’s Log. No.
H26315RBW850/R6665/57570 3/3/3/3/3/3/3/2/2/
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1. Given that
2 3 11 1
4 2
22
2
x x xx
ax bx c dx ex
− + +−
≡ + + ++−( )
( )( )
,
find the values of the constants a, b, c, d and e.(4)
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Question 1 continued
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(Total 4 marks)
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2. A curve C has equation
y x x nx= ≠ +e2 2 12
tan , ( )π .
(a) Show that the turning points on C occur where tan 1x = − .(6)
(b) Find an equation of the tangent to C at the point where 0x = .(2)
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Question 2 continued
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(Total 8 marks)
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3. f ln( ) ( ) , ,x x x x x= + − + > − ∈2 1 2 .
(a) Show that there is a root of f( ) 0x = in the interval 2 3x< < .(2)
(b) Use the iterative formula
to calculate the values of 1 2 3, and x x x giving your answers to 5 decimal places.(3)
(c) Show that 2.505x = is a root of f( ) 0x = correct to 3 decimal places.(2)
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(Total 7 marks)
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4.
Figure 1
Figure 1 shows a sketch of the curve with equation f ( )y x= . The curve passes through the origin O and the points A(5, 4) and B(– 5, – 4).
In separate diagrams, sketch the graph with equation
(a) f ( )y x= ,(3)
(b) f ( )y x= ,(3)
(c) 2f ( 1)y x= + .(4)
On each sketch, show the coordinates of the points corresponding to A and B.
y
x
A (5, 4)
B (– 5, – 4)
O
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Question 4 continued
Q4
(Total 10 marks)
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5. The radioactive decay of a substance is given by
R ct= −1000e , t 0.
where R is the number of atoms at time t years and c is a positive constant.
(a) Find the number of atoms when the substance started to decay .(1)
It takes 5730 years for half of the substance to decay .
(b) Find the value of c to 3 significant figures.(4)
(c) Calculate the number of atoms that will be left when 22 920t = .(2)
(d) In the space provided on page 13, sketch the graph of R against t .(2)
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6. (a) Use the double angle formulae and the identity
cos( ) cos cos sin sinA B A B A B+ ≡ −
to obtain an expression for cos 3x in terms of powers of cos x only.(4)
(b) (i) Prove that
cossin
sincos
sec , ( )xx
xx
x x n1
1 2 2 12+
++
≡ ≠ +π .
(4)
(ii) Hence find, for 0 x< < 2π , all the solutions of
cossin
sincos
xx
xx1
1 4+
++
= .
(3)
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7. A curve C has equation
y x x= +3 2 4 2sin cos , -π x π .
The point A(0, 4) lies on C.
(a) Find an equation of the normal to the curve C at A.(5)
(b) Express y in the form R xsin( ),2 +α where R > 0 and 02
< <απ .
Give the value of α to 3 significant figures.(4)
(c) Find the coordinates of the points of intersection of the curve C with the x-axis. Give your answers to 2 decimal places.
(4)
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Question 7 continued
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Question 7 continued
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(Total 13 marks)
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*H26315RB02224*
8. The functions f and g are defined by
f : 1 2
g :
x x x
xx
x x
− ∈
− > ∈
3
3 4 0
,
, ,
(a) Find the inverse function -1f .(2)
(b) Show that the composite function gf is
gf : x xx
8 11 2
3
3
−−
.
(4)
(c) Solve gf ( ) 0x = .(2)
(d) Use calculus to find the coordinates of the stationary point on the graph of y = gf(x).(5)
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TOTAL FOR PAPER: 75 MARKSEND
Q8
(Total 13 marks)
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Examiner’s use only
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Surname Initial(s)
Signature
Centre No.
Turn over
Candidate No.
Question Leave Number Blank
1
2
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5
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7
Total
Paper Reference(s)
6665/01Edexcel GCECore Mathematics C3Advanced Friday 6 June 2008 – AfternoonTime: 1 hour 30 minutes
Materials required for examination Items included with question papersMathematical Formulae (Green) Nil
Candidates may use any calculator allowed by the regulations of the Joint Council for Qualifications. Calculators must not have the facility for symbolic algebra manipulation, differentiation and integration, or have retrievable mathematical formulae stored in them.
Instructions to CandidatesIn the boxes above, write your centre number, candidate number, your surname, initials and signature. Check that you have the correct question paper.Answer ALL the questions.You must write your answer for each question in the space following the question.When a calculator is used, the answer should be given to an appropriate degree of accuracy.
Information for CandidatesA booklet ‘Mathematical Formulae and Statistical Tables’ is provided.Full marks may be obtained for answers to ALL questions.The marks for individual questions and the parts of questions are shown in round brackets: e.g. (2).There are 7 questions in this question paper. The total mark for this paper is 75. There are 24 pages in this question paper. Any blank pages are indicated.
Advice to CandidatesYou must ensure that your answers to parts of questions are clearly labelled.You should show sufficient working to make your methods clear to the Examiner . Answers without working may not gain full credit.
Paper Reference
6 6 6 5 0 1
This publication may be reproduced only in accordance with Edexcel Limited copyright policy. ©2008 Edexcel Limited.
Printer’s Log. No.
N30745AW850/R6665/57570 3/3/3/3
*N30745A0124*
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1. The point P lies on the curve with equation
The y-coordinate of P is 8.
(a) Find, in terms of ln 2, the x-coordinate of P.(2)
(b) Find the equation of the tangent to the curve at the point P in the form y = ax + b, where a and b are exact constants to be found.
(4)
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y e x= +4 2 1.
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2.
Given that where R > 0 and
(a) find the value of R and the value of α to 3 decimal places.(4)
(b) Hence solve the equation
for 0 x < 2π.(5)
(c) (i) Write down the maximum value of (1)
(ii) Find the smallest positive value of x for which this maximum value occurs.(2)
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f ( ) cos sinx x x= +5 12
f ( ) cos( ),x R x= −α 0 < <α π2
,
5 12 6cos sinx x+ =
5 12cos sin .x x+
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(Total 12 marks)
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3.
Figure 1
Figure 1 shows the graph of The graph consists of two line segments that meet at the point P. The graph cuts the y-axis at the point Q and the x-axis at the points (–3, 0) and R. Sketch, on separate diagrams, the graphs of
(a) (2)
(b) y = f (– x).(2)
Given that
(c) find the coordinates of the points P, Q and R,(3)
(d) solve (5)
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xR
P
Q
y
–3
y x x= ∈f ( ), .
y x= f ( ) ,
f ( ) ,x x= − +2 1
f ( ) .x x=12
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4. The function f is defined by
(a) Show that(4)
(b) Find the range of f.(2)
(c) Find f –1 (x). State the domain of this inverse function.(3)
The function g is defined by
(d) Solve (3)
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f : ( ) , .x xx x x
x2 12 3
13
32
−− −
−−
>
f ( ) , .xx
x=+
>1
13
g : , .x x x2 32 − ∈
fg( ) .x =18
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5. (a) Given that sin2 θ + cos2 θ ≡ 1, show that 1 + cot 2 θ ≡ cosec2 θ .(2)
(b) Solve, for 0 θ < 180°, the equation
2 cot2 θ – 9 cosec θ = 3,
giving your answers to 1 decimal place.(6)
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6. (a) Differentiate with respect to x,
(i) (3)
(ii) (3)
Given that
(b) show that (5)
(c) Hence find and the real values of x for which
(3)
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e3 2x x x(sin cos ) ,+
y x xx
x=+ −+
≠ −3 6 7
11
2
2( ), ,
ddyx x=
+20
1 3( ).
dd
2 yx2
dd
2 yx2
154
= − .
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7.
(a) Show that f (x) = 0 has a root, α, between x = 1.4 and x = 1.45(2)
(b) Show that the equation f (x) = 0 can be written as
(3)
(c) Starting with x0=1.43, use the iteration
xn+1 xn
2 23( )= +
to calculate the values of x1, x2 and x3, giving your answers to 4 decimal places.(3)
(d) By choosing a suitable interval, show that α = 1.435 is correct to 3 decimal places.(3)
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f ( )x x x= − −3 2 63
xx
x ≠2 23
0, .( )= +
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TOTAL FOR PAPER: 75 MARKSEND
Q7
(Total 11 marks)
© Science Exam Paperswww.XtremePapers.netSAT & IG Bookstore 0127746409 page178
Examiner’s use only
Team Leader’s use only
Surname Initial(s)
Signature
Centre No.
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Candidate No.
Question Leave Number Blank
1
2
3
4
5
6
7
8
Total
Paper Reference(s)
6665/01Edexcel GCECore Mathematics C3AdvancedThursday 15 January 2009 – MorningTime: 1 hour 30 minutes
Materials required for examination Items included with question papersMathematical Formulae (Green) Nil
Candidates may use any calculator allowed by the regulations of the Joint Council for Qualifications. Calculators must not have the facility for symbolic algebra manipulation, differentiation and integration, or have retrievable mathematical formulae stored in them.
Instructions to CandidatesIn the boxes above, write your centre number, candidate number, your surname, initials and signature. Check that you have the correct question paper.Answer ALL the questions.You must write your answer for each question in the space following the question.When a calculator is used, the answer should be given to an appropriate degree of accuracy.
Information for CandidatesA booklet ‘Mathematical Formulae and Statistical Tables’ is provided.Full marks may be obtained for answers to ALL questions.The marks for individual questions and the parts of questions are shown in round brackets: e.g. (2).There are 8 questions in this question paper. The total mark for this paper is 75. There are 28 pages in this question paper. Any blank pages are indicated.
Advice to CandidatesYou must ensure that your answers to parts of questions are clearly labelled.You should show sufficient working to make your methods clear to the Examiner . Answers without working may not gain full credit.
Paper Reference
6 6 6 5 0 1
This publication may be reproduced only in accordance with Edexcel Limited copyright policy. ©2009 Edexcel Limited.
Printer’s Log. No.
H31123AW850/R6665/57570 3/3/3/3
*H31123A0128*
www.xtremepapers.netwww.XtremePapers.netSAT & IG Bookstore 0127746409 page179
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2
*H31123A0228*
1. (a) Find the value of at the point where x = 2 on the curve with equation
y = x2√(5x – 1).(6)
(b) Differentiate with respect to x.(4)
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ddyx
sin 22
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(Total 10 marks)
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2.
(a) Express f (x) as a single fraction in its simplest form.(4)
(b) Hence show that
(3)
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f ( )x xx x
xx
=+
− −−
+−
2 22 3
132
′ =−
f ( )( )
xx
23 2
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*H31123A0728* Turn over
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3.
Figure 1
Figure 1 shows the graph of y = f (x), 1 < x < 9. The points T(3, 5) and S(7, 2) are turning points on the graph.
Sketch, on separate diagrams, the graphs of
(a) y = 2f (x) – 4,(3)
(b) .(3)
Indicate on each diagram the coordinates of any turning points on your sketch.
y x= f ( )
5T (3, 5)
S (7, 2)
y
2
O 3 7 x
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Q3
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4. Find the equation of the tangent to the curve
Give your answer in the form y = ax + b, where a and b are constants to be found.(6)
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x y= + ⎛⎝⎜
⎞⎠⎟cos( ) , .2 0
4π
π at
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*H31123A01128* Turn over
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5. The functions f and g are defined by
(a) Write down the range of g.(1)
(b) Show that the composite function fg is defined by
(2)
(c) Write down the range of fg.(1)
(d) Solve the equation (6)
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fg e ,: .x x xx� 2 32
+ ∈�
)+dd
fgx
x x xex( ) ( .⎡⎣ ⎤⎦ =2
2
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6. (a) (i) By writing 3θ = (2θ + θ), show thatsin 3θ = 3 sin θ – 4 sin3θ.
(4)
(ii) Hence, or otherwise, for solve
8 sin3θ – 6 sin θ + 1 = 0.
Give your answers in terms of π.(5)
(b) Using or otherwise, show that
(4)
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0 < <θ π3
,
sin( ) sin cos cos sin ,θ α θ α θ α− = −
sin15 14
° = (√6 −√ 2).
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*H31123A01928* Turn over
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*H31123A02028*
7.
The curve with equation y = f (x) has a turning point P.
(a) Find the exact coordinates of P.(5)
The equation f (x) = 0 has a root between x = 0.25 and x = 0.3
(b) Use the iterative formula
with x0 = 0.25 to find, to 4 decimal places, the values of x1, x2 and x3.(3)
(c) By choosing a suitable interval, show that a root of f (x) = 0 is x = 0.2576 correct to4 decimal places.
(3)
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f ( )x xex= −3 1
xnxn
+−=1
13
e
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*H31123A02228*
Question 7 continued
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*H31123A02428*
8. (a) Express 3 cos θ + 4 sin θ in the form R cos(θ – α), where R and α are constants, R > 0 and 0 < α < 90°.
(4)
(b) Hence find the maximum value of 3 cos θ + 4 sin θ and the smallest positive value of θ for which this maximum occurs.
(3)
The temperature, f (t), of a warehouse is modelled using the equation
f (t) = 10 + 3 cos(15t)° + 4 sin(15t)°,
where t is the time in hours from midday and 0 t < 24.
(c) Calculate the minimum temperature of the warehouse as given by this model.(2)
(d) Find the value of t when this minimum temperature occurs.(3)
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*H31123A02528* Turn over
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Question 8 continued
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Question 8 continued
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TOTAL FOR PAPER: 75 MARKSEND
Q8
(Total 12 marks)
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BLANK PAGE
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Examiner’s use only
Team Leader’s use only
Surname Initial(s)
Signature
Centre No.
Turn over
Candidate No.
Question Leave Number Blank
1
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Paper Reference(s)
6665/01Edexcel GCECore Mathematics C3Advanced Thursday 11 June 2009 – MorningTime: 1 hour 30 minutes
Materials required for examination Items included with question papersMathematical Formulae (Orange or Nil Green)
Candidates may use any calculator allowed by the regulations of the Joint Council for Qualifications. Calculators must not have the facility for symbolic algebra manipulation, differentiation and integration, or have retrievable mathematical formulae stored in them.
Instructions to CandidatesIn the boxes above, write your centre number, candidate number, your surname, initials and signature. Check that you have the correct question paper.Answer ALL the questions.You must write your answer for each question in the space following the question.When a calculator is used, the answer should be given to an appropriate degree of accuracy.
Information for CandidatesA booklet ‘Mathematical Formulae and Statistical Tables’ is provided.Full marks may be obtained for answers to ALL questions.The marks for individual questions and the parts of questions are shown in round brackets: e.g. (2).There are 8 questions in this question paper. The total mark for this paper is 75. There are 28 pages in this question paper. Any blank pages are indicated.
Advice to CandidatesYou must ensure that your answers to parts of questions are clearly labelled.You should show sufficient working to make your methods clear to the Examiner . Answers without working may not gain full credit.
Paper Reference
6 6 6 5 0 1
This publication may be reproduced only in accordance with Edexcel Limited copyright policy. ©2009 Edexcel Limited.
Printer’s Log. No.
H34264AW850/R6665/57570 4/5/5/3
*H34264A0128*
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*H34264A0228*
1.
Figure 1
Figure 1 shows part of the curve with equation 3 22 2,y x x= − + + which intersects the x-axis at the point A where x = α.
To find an approximation to α, the iterative formula
1 2
2 2( )n
n
xx+ = +
is used.
(a) Taking x0 = 2.5, find the values of x1, x2, x3 and x4. Give your answers to 3 decimal places where appropriate.
(3)
(b) Show that α = 2.359 correct to 3 decimal places.(3)
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–5
O
5
10
15
–2 –1 1 2 3A
y
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(Total 6 marks)
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2. (a) Use the identity cos2 θ + sin2 θ = 1 to prove that tan 2 θ = sec2 θ – 1.(2)
(b) Solve, for 0 θ < 360°, the equation
2 tan2 θ + 4 sec θ + sec2 θ = 2 (6)
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(Total 8 marks)
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3. Rabbits were introduced onto an island. The number of rabbits, P, t years after they were introduced is modelled by the equation
(a) Write down the number of rabbits that were introduced to the island.(1)
(b) Find the number of years it would take for the number of rabbits to first exceed 1000.
(2)
(c) Find d .dP
t (2)
(d) Find P when d 50.dP
t=
(3)
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t ∈ , t 0580e1
P = t,
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(Total 8 marks)
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4. (i) Differentiate with respect to x
(a) 2 cos3x x(3)
(b) 2
2
ln( 1)1
x
x
++ (4)
(ii) A curve C has the equation
The point P on the curve has x-coordinate 2. Find an equation of the tangent to C at P in the form 0,ax by c+ + = where a, b and c are integers.
(6)
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144 1 , x > – –, y >y= √( x+ ) 0
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(Total 13 marks)
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5.
Figure 2
Figure 2 shows a sketch of part of the curve with equation y = f(x), x ∈ . The curve meets the coordinate axes at the points A(0,1– k ) and 1
2 ln ,0B ( – k ), where k is a constant and 1,k > as shown in Figure 2.
On separate diagrams, sketch the curve with equation
(a) f ( ) ,y x=(3)
(b) 1f ( ).y x−=(2)
Show on each sketch the coordinates, in terms of k, of each point at which the curve meets or cuts the axes.
Given that 2f ( ) e ,xx k= −
(c) state the range of f ,(1)
(d) find 1f ( ) ,x−
(3)
(e) write down the domain of 1f .−
(1)
B x
y
O
A
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Question 5 continued
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6. (a) Use the identity cos( ) cos cos sin sin ,A B A B A B+ = − to show that
cos sin2 1 2 2A A= −(2)
The curves C1 and C2 have equations
C1: 3sin 2y x=
C2: y x= − 2 cos 2x4 2sin
(b) Show that the x-coordinates of the points where C1 and C2 intersect satisfy the equation
4cos 2 3sin 2 2x x+ =(3)
(c) Express 4cos 2 3sin 2x x+ in the form R cos (2x – α), where R > 0 and 0 < α < 90°, giving the value of α to 2 decimal places.
(3)
(d) Hence find, for 0 180x < °, all the solutions of
4cos 2 3sin 2 2x x+ =
giving your answers to 1 decimal place.(4)
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Question 6 continued
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7. The function f is defined by
2 8f( ) 1 ,( 4) ( 2)( 4)
xx
x x x
−= − ++ − +
x ∈ 4 , 2x x≠ − ≠
(a) Show that 3f ( )2
xxx−
=− (5)
The function g is defined by
e 3g( ) ,e 2
x
xx −=
− x ∈ , ln 2x ≠
(b) Differentiate g( )x to show that 2
eg ( )(e 2)
x
xx′ =
− (3)
(c) Find the exact values of x for which g ( ) 1x′ =(4)
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(Total 12 marks)
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8. (a) Write down sin 2x in terms of sin x and cos .x(1)
(b) Find, for 0 < x < π, all the solutions of the equation
cosec 8cos 0x x− =
giving your answers to 2 decimal places.(5)
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Question 8 continued
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TOTAL FOR PAPER: 75 MARKSEND
Q8
(Total 6 marks)
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Examiner’s use only
Team Leader’s use only
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Centre No.
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Paper Reference(s)
6665/01Edexcel GCECore Mathematics C3Advanced Wednesday 20 January 2010 – AfternoonTime: 1 hour 30 minutes
Materials required for examination Items included with question papersMathematical Formulae (Pink or NilGreen)
Candidates may use any calculator allowed by the regulations of the Joint Council for Qualifications. Calculators must not have the facility for symbolic algebra manipulation, differentiation and integration, or have retrievable mathematical formulae stored in them.
Instructions to CandidatesIn the boxes above, write your centre number, candidate number, your surname, initials and signature. Check that you have the correct question paper.Answer ALL the questions.You must write your answer to each question in the space following the question.When a calculator is used, the answer should be given to an appropriate degree of accuracy.
Information for CandidatesA booklet ‘Mathematical Formulae and Statistical Tables’ is provided.Full marks may be obtained for answers to ALL questions.The marks for individual questions and the parts of questions are shown in round brackets: e.g. (2).There are 9 questions in this question paper. The total mark for this paper is 75. There are 28 pages in this question paper. Any blank pages are indicated.
Advice to CandidatesYou must ensure that your answers to parts of questions are clearly labelled.You should show sufficient working to make your methods clear to the Examiner . Answers without working may not gain full credit.
Paper Reference
6 6 6 5 0 1
This publication may be reproduced only in accordance with Edexcel Limited copyright policy. ©2010 Edexcel Limited.
Printer’s Log. No.
N35381AW850/R6665/57570 5/5/5/4/3/3
*N35381A0128*
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*N35381A0228*
1. Express
xx x+−
−+
13 3
13 12
as a single fraction in its simplest form.(4)
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(Total 4 marks)
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2.f( )x x x x= + − −3 22 3 11
(a) Show that f(x) = 0 can be rearranged as
x xx
x=++
⎛⎝⎜
⎞⎠⎟ ≠ −
3 112
2, .
(2)
The equation f(x) = 0 has one positive root α.
The iterative formula x xxn
n
n+ =
++
⎛
⎝⎜
⎞
⎠⎟1
3 112
is used to find an approximation to α .
(b) Taking x1 = 0, find, to 3 decimal places, the values of x2, x3 and x4.(3)
(c) Show that α = 2.057 correct to 3 decimal places.(3)
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(Total 8 marks)
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3. (a) Express 5 cos x – 3 sin x in the form R cos(x + α), where R > 0 and 0 < α < π12 .
(4)
(b) Hence, or otherwise, solve the equation
5 cos x – 3 sin x = 4
for 0 x < 2 , giving your answers to 2 decimal places.(5)
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(Total 9 marks)
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4. (i) Given that y xx
=+ln ( )2 1 , find d
dyx
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(4)
(ii) Given that x = tan y, show that ddyx x
=+1
1 2 .
(5)
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(Total 9 marks)
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5. Sketch the graph of y x= ln , stating the coordinates of any points of intersection with the axes.
(3)
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Q5
(Total 3 marks)
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6.
Figure 1
Figure 1 shows a sketch of the graph of y = f (x).
The graph intersects the y-axis at the point (0, 1) and the point A(2, 3) is the maximum turning point.
Sketch, on separate axes, the graphs of
(i) y = f(–x) + 1, (ii) y = f(x + 2) + 3, (iii) y = 2f(2x) .
On each sketch, show the coordinates of the point at which your graph intersects the y-axis and the coordinates of the point to which A is transformed.
(9)
yA(2, 3)
1
O x
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Q6
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7. (a) By writing seccos
xx
as 1 , show that d xdx
x x(sec ) sec tan= .
(3)
Given that y = e2x sec 3x,
(b) find ddyx
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(4)
The curve with equation y = e2x sec 3x, − < <π6
π6x , has a minimum turning point at
(a, b).
(c) Find the values of the constants a and b, giving your answers to 3 significant figures.
(4)
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Question 7 continued
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(Total 11 marks)
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8. Solvecosec2 2x – cot 2x = 1
for 0 x 180°.(7)
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(Total 7 marks)
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9. (i) Find the exact solutions to the equations
(a) ln (3x – 7) = 5(3)
(b) 3xe7x + 2 = 15(5)
(ii) The functions f and g are defined by
f (x) = e2x + 3, x ∈
g(x) = ln (x – 1), x ∈ , x > 1
(a) Find f –1 and state its domain.(4)
(b) Find fg and state its range.(3)
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Question 9 continued
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TOTAL FOR PAPER: 75 MARKS
END
Q9
(Total 15 marks)
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