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CandidateNo.

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Paper Reference(s)

6666/01Edexcel GCECore Mathematics C4

Advanced

Tuesday 18 June 2013 MorningTime: 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 JointCouncil for Qualifications. Calculators must not have the facility for symbolicalgebra manipulation or symbolic differentiation/integration, or haveretrievable mathematical formulae stored in them.

This publication may be reproduced only in accordance with

Pearson Education Ltd copyright policy.2013 Pearson Education Ltd.

Printers Log. No.

P43137AW850/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

6 6 6 6 0 1

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1. (a) Find x xx2e d .(5)

(b) Hence find the exact value of x xx20

1e d .

(2)

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Question 1 continued

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___________________________________________________________________________ Q1

(Total 7 marks)

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

21 11 , | | 11 2

xx x x

x

+ + +

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Question 2 continued

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Question 2 continued

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Question 2 continued

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___________________________________________________________________________ Q2

(Total 9 marks)

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3.

Figure 1

Figure 1 shows the finite regionRbounded by thex-axis, they-axis, the line2

x= and

the curve with equation

1sec , 0

2 2

y x x

=

The table shows corresponding values ofx andy for y= sec 1

2

x

.

x 06

3

2

y 1 1.035276 1.414214

(a) Complete the table above giving the missing value ofyto 6 decimal places.(1)

(b) Using the trapezium rule, with all of the values ofyfrom the completed table, find anapproximation for the area ofR, giving your answer to 4 decimal places.

(3)

RegionRis rotated through 2radians about thex-axis.

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

y

y x

=

sec

1

2

2

x

R

O

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Question 3 continued

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Question 3 continued

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Question 3 continued

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___________________________________________________________________________ Q3

(Total 8 marks)

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4. A curve Chas parametric equations

x = 2sint, y= 1 cos 2t, 2

t

2

(a) Findd

d

y

x

at the point where t=

6

(4)

(b) Find a cartesian equation for Cin the form

y= f(x), kx k,

stating the value of the constant k.(3)

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

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Question 4 continued

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Question 4 continued

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Question 4 continued

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(Total 9 marks)

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

2d =

(2 1)dux

u u1

(2x x 1) (3)

(b) Hence show that9

1

1d = 2ln

ax

b

(2x x 1)

where aand bare integers to be determined.(7)

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Question 5 continued

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Question 5 continued

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Question 5 continued

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(Total 10 marks)

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6. Water is being heated in a kettle. At time tseconds, the temperature of the water is C.

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

d(120 )

d

t = , 100

whereis 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 thetime, to the nearest second, when the kettle switches off.(3)

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Question 6 continued

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Question 6 continued

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Question 6 continued

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(Total 11 marks)

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7. A curve is described by the equation

x2+ 4xy+y2+ 27 = 0

(a) Findddy

xin terms ofxandy.

(5)

A point Qlies on the curve.

The tangent to the curve at Qis parallel to they-axis.

Given that thexcoordinate of Qis negative,

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

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Question 7 continued

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Question 7 continued

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Question 7 continued

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(Total 12 marks)

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

13 2

8 2

1 1

= +

r

13 2

8 2

1 1

= +

r , whereis a scalar parameter.

The pointAlies on land has coordinates (3, 2, 6).

The pointPhas position vector (p i+ 2pk)relative to O, wherepis a constant.

Given that vectorPA

is perpendicular to l,

(a) find the value ofp.(4)

Given also thatB is a point on lsuch that BPA= 45,

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

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Question 8 continued

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Question 8 continued

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Question 8 continued

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Question 8 continued

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TOTAL FOR PAPER: 75 MARKS

Q8

(Total 9 marks)

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