paper reference(s) edexcel gce - pearson … 1 shows the finite region r bounded by the x-axis, the...

32
Examiner’s use only Team Leader’s use only Surname Initial(s) Signature Centre No. Candidate No. Turn over Question Leave Number Blank 1 2 3 4 5 6 7 8 Total Paper Reference(s) 6666/01 Edexcel GCE Core Mathematics C4 Advanced Tuesday 18 June 2013 – Morning Time: 1 hour 30 minutes Materials required for examination Items included with question papers Mathematical Formulae (Pink) 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 or symbolic differentiation/integration, or have retrievable mathematical formulae stored in them. This publication may be reproduced only in accordance with Pearson Education Ltd copyright policy. ©2013 Pearson Education Ltd. Printer’s Log. No. P43137A W850/R6666/57570 5/5/5/5/5 *P43137A0132* Instructions to Candidates In 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 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 8 questions in this question paper. The total mark for this paper is 75. There are 32 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 should show sufficient working to make your methods clear to the Examiner. Answers without working may not gain full credit. Paper Reference 6666 01

Upload: nguyenhanh

Post on 10-May-2018

229 views

Category:

Documents


1 download

TRANSCRIPT

Page 1: Paper Reference(s) Edexcel GCE - Pearson … 1 shows the finite region R bounded by the x-axis, the y-axis, the line 2 π x = and the curve with equation 1 sec , 0 22 π yx x ⎛⎞

Examiner’s use only

Team Leader’s use only

Surname Initial(s)

Signature

Centre No.

Candidate No.

Turn over

Question Leave Number Blank

1

2

3

4

5

6

7

8

Total

Paper Reference(s)

6666/01Edexcel GCECore Mathematics C4Advanced Tuesday 18 June 2013 – MorningTime: 1 hour 30 minutes

Materials required for examination Items included with question papersMathematical Formulae (Pink) 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 or symbolic differentiation/integration, or have retrievable mathematical formulae stored in them.

This publication may be reproduced only in accordance with Pearson Education Ltd copyright policy. ©2013 Pearson Education Ltd.

Printer’s Log. No.

P43137AW850/R6666/57570 5/5/5/5/5

*P43137A0132*

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 32 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 6 0 1

Page 2: Paper Reference(s) Edexcel GCE - Pearson … 1 shows the finite region R bounded by the x-axis, the y-axis, the line 2 π x = and the curve with equation 1 sec , 0 22 π yx x ⎛⎞

Leave blank

2

*P43137A0232*

1. (a) Find x xx2e d .∫(5)

(b) Hence find the exact value of x xx2

0

1e d .∫

(2)

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

Page 3: Paper Reference(s) Edexcel GCE - Pearson … 1 shows the finite region R bounded by the x-axis, the y-axis, the line 2 π x = and the curve with equation 1 sec , 0 22 π yx x ⎛⎞

Leave blank

3

*P43137A0332* Turn over

Question 1 continued

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________ Q1

(Total 7 marks)

Page 4: Paper Reference(s) Edexcel GCE - Pearson … 1 shows the finite region R bounded by the x-axis, the y-axis, the line 2 π x = and the curve with equation 1 sec , 0 22 π yx x ⎛⎞

4

*P43137A0432*

Leave blank

2. (a) Use the binomial expansion to show that

21 11 , | | 11 2

x x x xx

+⎛ ⎞ ≈ + + <⎜ ⎟⎝ ⎠− (6)

(b) Substitute x = 126

into

21 x x= + +xx

+⎛ ⎞⎜ ⎟⎝ ⎠−

11

12

� � ��������������������������� �

Give your answer in the form ab

where a and b are integers.(3)

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

Page 5: Paper Reference(s) Edexcel GCE - Pearson … 1 shows the finite region R bounded by the x-axis, the y-axis, the line 2 π x = and the curve with equation 1 sec , 0 22 π yx x ⎛⎞

Leave blank

5

*P43137A0532* Turn over

Question 2 continued

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

Page 6: Paper Reference(s) Edexcel GCE - Pearson … 1 shows the finite region R bounded by the x-axis, the y-axis, the line 2 π x = and the curve with equation 1 sec , 0 22 π yx x ⎛⎞

Leave blank

6

*P43137A0632*

Question 2 continued

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

Page 7: Paper Reference(s) Edexcel GCE - Pearson … 1 shows the finite region R bounded by the x-axis, the y-axis, the line 2 π x = and the curve with equation 1 sec , 0 22 π yx x ⎛⎞

Leave blank

7

*P43137A0732* Turn over

Question 2 continued

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________ Q2

(Total 9 marks)

Page 8: Paper Reference(s) Edexcel GCE - Pearson … 1 shows the finite region R bounded by the x-axis, the y-axis, the line 2 π x = and the curve with equation 1 sec , 0 22 π yx x ⎛⎞

Leave blank

8

*P43137A0832*

3.

Figure 1

Figure 1 shows the finite region R bounded by the x-axis, the y-axis, the line 2πx = and

the curve with equation

1sec , 02 2

πy x x⎛ ⎞= ⎜ ⎟⎝ ⎠� �

The table shows corresponding values of x and y for y = sec 12

x⎛⎝⎜

⎞⎠⎟

.

x 0 6π

y 1 1.035276 1.414214

(a) Complete the table above giving the missing value of y to 6 decimal places.(1)

(b) Using the trapezium rule, with all of the values of y from the completed table, find an approximation for the area of R, giving your answer to 4 decimal places.

(3)

Region R is rotated through 2� radians about the x-axis.

(c) Use calculus to find the exact volume of the solid formed.(4)

y

y x= ⎛⎝⎜

⎞⎠⎟

sec 12

2π x

R

O

Page 9: Paper Reference(s) Edexcel GCE - Pearson … 1 shows the finite region R bounded by the x-axis, the y-axis, the line 2 π x = and the curve with equation 1 sec , 0 22 π yx x ⎛⎞

Leave blank

9

*P43137A0932* Turn over

Question 3 continued

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

Page 10: Paper Reference(s) Edexcel GCE - Pearson … 1 shows the finite region R bounded by the x-axis, the y-axis, the line 2 π x = and the curve with equation 1 sec , 0 22 π yx x ⎛⎞

Leave blank

10

*P43137A01032*

Question 3 continued

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

Page 11: Paper Reference(s) Edexcel GCE - Pearson … 1 shows the finite region R bounded by the x-axis, the y-axis, the line 2 π x = and the curve with equation 1 sec , 0 22 π yx x ⎛⎞

Leave blank

11

*P43137A01132* Turn over

Question 3 continued

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________ Q3

(Total 8 marks)

Page 12: Paper Reference(s) Edexcel GCE - Pearson … 1 shows the finite region R bounded by the x-axis, the y-axis, the line 2 π x = and the curve with equation 1 sec , 0 22 π yx x ⎛⎞

Leave blank

12

*P43137A01232*

4. A curve C has parametric equations

x = 2sin t, y = 1 – cos 2t, –2π

� t ��2π

(a) Find dd

yx at the point where t =

(4)

(b) Find a cartesian equation for C in the form

y = f(x), – k � x � k,

stating the value of the constant k.(3)

(c) Write down the range of f(x).(2)

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

Page 13: Paper Reference(s) Edexcel GCE - Pearson … 1 shows the finite region R bounded by the x-axis, the y-axis, the line 2 π x = and the curve with equation 1 sec , 0 22 π yx x ⎛⎞

Leave blank

13

*P43137A01332* Turn over

Question 4 continued

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

Page 14: Paper Reference(s) Edexcel GCE - Pearson … 1 shows the finite region R bounded by the x-axis, the y-axis, the line 2 π x = and the curve with equation 1 sec , 0 22 π yx x ⎛⎞

Leave blank

14

*P43137A01432*

Question 4 continued

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

Page 15: Paper Reference(s) Edexcel GCE - Pearson … 1 shows the finite region R bounded by the x-axis, the y-axis, the line 2 π x = and the curve with equation 1 sec , 0 22 π yx x ⎛⎞

Leave blank

15

*P43137A01532* Turn over

Question 4 continued

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________ Q4

(Total 9 marks)

Page 16: Paper Reference(s) Edexcel GCE - Pearson … 1 shows the finite region R bounded by the x-axis, the y-axis, the line 2 π x = and the curve with equation 1 sec , 0 22 π yx x ⎛⎞

Leave blank

16

*P43137A01632*

5. (a) Use the substitution x = u2, u > 0, to show that

2d =(2 1)

duxu u −

1(2x x − 1)∫ ∫

(3)

(b) Hence show that9

1

1 d = 2ln axb

⎛ ⎞⎜ ⎟⎝ ⎠∫ (2x x − 1)

where a and b are integers to be determined.(7)

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

Page 17: Paper Reference(s) Edexcel GCE - Pearson … 1 shows the finite region R bounded by the x-axis, the y-axis, the line 2 π x = and the curve with equation 1 sec , 0 22 π yx x ⎛⎞

Leave blank

17

*P43137A01732* Turn over

Question 5 continued

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

Page 18: Paper Reference(s) Edexcel GCE - Pearson … 1 shows the finite region R bounded by the x-axis, the y-axis, the line 2 π x = and the curve with equation 1 sec , 0 22 π yx x ⎛⎞

Leave blank

18

*P43137A01832*

Question 5 continued

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

Page 19: Paper Reference(s) Edexcel GCE - Pearson … 1 shows the finite region R bounded by the x-axis, the y-axis, the line 2 π x = and the curve with equation 1 sec , 0 22 π yx x ⎛⎞

Leave blank

19

*P43137A01932* Turn over

Question 5 continued

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________ Q5

(Total 10 marks)

Page 20: Paper Reference(s) Edexcel GCE - Pearson … 1 shows the finite region R bounded by the x-axis, the y-axis, the line 2 π x = and the curve with equation 1 sec , 0 22 π yx x ⎛⎞

Leave blank

20

*P43137A02032*

6. Water is being heated in a kettle. At time t seconds, the temperature of the water is � °C.

The rate of increase of the temperature of the water at any time t is modelled by the differential equation

d (120 )dθ λ θt

= − , � � 100

where � is a positive constant.

Given that � = 20 when t = 0,

(a) solve this differential equation to show that

� = 120 – 100e–��

(8)

When the temperature of the water reaches 100 °C, the kettle switches off.

(b) Given that � = 0.01, find the time, to the nearest second, when the kettle switches off.(3)

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

Page 21: Paper Reference(s) Edexcel GCE - Pearson … 1 shows the finite region R bounded by the x-axis, the y-axis, the line 2 π x = and the curve with equation 1 sec , 0 22 π yx x ⎛⎞

Leave blank

21

*P43137A02132* Turn over

Question 6 continued

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

Page 22: Paper Reference(s) Edexcel GCE - Pearson … 1 shows the finite region R bounded by the x-axis, the y-axis, the line 2 π x = and the curve with equation 1 sec , 0 22 π yx x ⎛⎞

Leave blank

22

*P43137A02232*

Question 6 continued

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

Page 23: Paper Reference(s) Edexcel GCE - Pearson … 1 shows the finite region R bounded by the x-axis, the y-axis, the line 2 π x = and the curve with equation 1 sec , 0 22 π yx x ⎛⎞

Leave blank

23

*P43137A02332* Turn over

Question 6 continued

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________ Q6

(Total 11 marks)

Page 24: Paper Reference(s) Edexcel GCE - Pearson … 1 shows the finite region R bounded by the x-axis, the y-axis, the line 2 π x = and the curve with equation 1 sec , 0 22 π yx x ⎛⎞

Leave blank

24

*P43137A02432*

7. A curve is described by the equation

x2 + 4xy + y2 + 27 = 0

(a) Find dd

yx

in terms of x and y.(5)

A point Q lies on the curve.

The tangent to the curve at Q is parallel to the y-axis.

Given that the x coordinate of Q is negative,

(b) use your answer to part (a) to find the coordinates of Q.(7)

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

Page 25: Paper Reference(s) Edexcel GCE - Pearson … 1 shows the finite region R bounded by the x-axis, the y-axis, the line 2 π x = and the curve with equation 1 sec , 0 22 π yx x ⎛⎞

Leave blank

25

*P43137A02532* Turn over

Question 7 continued

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

Page 26: Paper Reference(s) Edexcel GCE - Pearson … 1 shows the finite region R bounded by the x-axis, the y-axis, the line 2 π x = and the curve with equation 1 sec , 0 22 π yx x ⎛⎞

Leave blank

26

*P43137A02632*

Question 7 continued

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

Page 27: Paper Reference(s) Edexcel GCE - Pearson … 1 shows the finite region R bounded by the x-axis, the y-axis, the line 2 π x = and the curve with equation 1 sec , 0 22 π yx x ⎛⎞

Leave blank

27

*P43137A02732* Turn over

Question 7 continued

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________ Q7

(Total 12 marks)

Page 28: Paper Reference(s) Edexcel GCE - Pearson … 1 shows the finite region R bounded by the x-axis, the y-axis, the line 2 π x = and the curve with equation 1 sec , 0 22 π yx x ⎛⎞

Leave blank

28

*P43137A02832*

8. With respect to a fixed origin O, the line l has equation

13 28 21 1

λ⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟= +⎜ ⎟ ⎜ ⎟

−⎝ ⎠ ⎝ ⎠r

13 28 21 1

λ⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟= +⎜ ⎟ ⎜ ⎟

−⎝ ⎠ ⎝ ⎠r , where � is a scalar parameter.

The point A lies on l and has coordinates (3, – 2, 6).

The point P has position vector (–p i + 2p k) relative to O, where p is a constant.

Given that vector PA

is perpendicular to l,

(a) find the value of p.(4)

Given also that B is a point on l such that �BPA = 45°,

(b) find the coordinates of the two possible positions of B.(5)

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

Page 29: Paper Reference(s) Edexcel GCE - Pearson … 1 shows the finite region R bounded by the x-axis, the y-axis, the line 2 π x = and the curve with equation 1 sec , 0 22 π yx x ⎛⎞

Leave blank

29

*P43137A02932* Turn over

Question 8 continued

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

Page 30: Paper Reference(s) Edexcel GCE - Pearson … 1 shows the finite region R bounded by the x-axis, the y-axis, the line 2 π x = and the curve with equation 1 sec , 0 22 π yx x ⎛⎞

Leave blank

30

*P43137A03032*

Question 8 continued

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

Page 31: Paper Reference(s) Edexcel GCE - Pearson … 1 shows the finite region R bounded by the x-axis, the y-axis, the line 2 π x = and the curve with equation 1 sec , 0 22 π yx x ⎛⎞

Leave blank

31

*P43137A03132* Turn over

Question 8 continued

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

Page 32: Paper Reference(s) Edexcel GCE - Pearson … 1 shows the finite region R bounded by the x-axis, the y-axis, the line 2 π x = and the curve with equation 1 sec , 0 22 π yx x ⎛⎞

32

*P43137A03232*

Question 8 continued

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

TOTAL FOR PAPER: 75 MARKS

END

Q8

(Total 9 marks)

Leave blank