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Physics & Maths TutorTypewritten TextEdexcel Maths C3

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Physics & Maths TutorTypewritten Text

Physics & Maths TutorTypewritten Text2005-2013

This publication may be reproduced only in accordance with

Edexcel Limited copyright policy.

2005 Edexcel Limited.

Printers 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

Examiners use only

Team Leaders 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 sin2 + cos2 1, show that 1 + tan2 sec2.(2)

(b) Solve, for 0 < 360, the equation

2 tan2 + sec = 1,

giving your answers to 1 decimal place.

(6)

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2 *n23494B0220*

physicsandmathstutor.com June 2005

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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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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

+ >

+ +

6 *n23494B0620*


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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 .

(b) Show that = e.(2)

The iterative formula

is used to find an approximate value for .

(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 = 0.1443correct to 4 decimal places.

(2)

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11 06

e , 1,nx

nx x

+ = =

16

12

8 *n23494B0820*


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5. (a) Using the identity cos(A + B) cosA cosB sinA sinB, prove that

cos 2A 1 2 sin2A.(2)

(b) Show that

2 sin 2 3 cos 2 3 sin + 3 sin (4 cos + 6 sin 3).(4)

(c) Express 4 cos + 6 sin in the form R sin( + ), where R > 0 and 0 < < .(4)

(d) Hence, for 0 < , solve

2 sin 2 = 3(cos 2 + sin 1),

giving your answers in radians to 3 significant figures, where appropriate.

(5)

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12

10 *n23494B01020*


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11

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Question 5 continued


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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 linesegments that meet at the point (1, a), a < 0. One line meets the x-axis at (3, 0). The otherline 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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15

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7. A particular species of orchid is being studied. The population p at time t years after thestudy 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 populationof 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

=+

18 *n23494B01820*


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TOTAL FOR PAPER: 75 MARKS

END

20 *n23494B02020*


Q7

(Total 10 marks)


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.

Examiners use only



1

2

3

4

5

6

7

8

Total

CentreNo.

Candidate No.

Surname Initial(s)

Signature

Paper Reference

6 6 6 5 0 1

This publication may be reproduced only in accordance with Edexcel Limited copyright policy. 2006 Edexcel Limited.

Printers Log. No.

N23495AW850/R6665/57570 7/7/3/3/3/3/3

Turn over

*N23495A0120*


Leaveblank

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

x5 O 5

M (2, 4)

physicsandmathstutor.com January 2006

Leaveblank


Q1

(Total 7 marks)

3

Turn over*N23495A0320*


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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

+

+

4 *N23495A0420*


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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 =

6 *N23495A0620*


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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

8 *N23495A0820*


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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 .

(c) By choosing a suitable interval, prove that = 1.392, to 3 decimal places.(3)

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2 1 ,2nx

+

2 1 .2x

+

10 *N23495A01020*


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6. f(x) = 12 cos x 4 sin x.

Given that f(x) = R cos(x + ), where R . 0 and 0 - - 90,(a) find the value of R and the value of .

(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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12 *N23495A01220*


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__________________________________________________________________________ Q6

(Total 12 marks)

13



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7. (a) Show that

(i) x (n ), 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 2 = cos 2.(3)

(c) Solve, for 0 - < 2,sin 2 = cos 2,

giving your answers in terms of .(4)

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cos 2 1coscos sin 2

= +

12

12

14

cos 2 cos sin ,cos sin

x x xx x

+

14 *N23495A01420*


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15



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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

=

18 *N23495A01820*


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19



This publication may be reproduced only in accordance with Edexcel Limited copyright policy. 2006 Edexcel Limited.

Printers 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

Examiners use only



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*


Leaveblank

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

*N23581A0224*


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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)

*N23581A0424*


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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


y

xO

P

Q (3, 0)

y = f (x)

(0, 2)


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6


*N23581A0624*


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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 e0.05t + 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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*N23581A0824*


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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 n

x x x+ = =

(2 1) tan 2 , 0 .4

y x x x =

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11


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6. (a) Using sin2 + cos2 1, show that cosec2 cot2 1.(2)

(b) Hence, or otherwise, prove that

cosec4 cot4 cosec2 + cot2.(2)

(c) Solve, for 90 < < 180,cosec4 cot4 = 2 cot .

(6)

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7. For the constant k, where k > 1, the functions f and g are defined by

f: x6 ln (x + k), x > k,g: x6 |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

*N23581A01824*


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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 x = + + +

cos 2 cos 2 cos 2 .3 3

x x x + +

3cos ,4

A =

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Documents

Combined QP - C3 Edexcel