Paper Reference(s) 6674/01 Edexcel GCEeiewebvip.edexcel.org.uk/Reports/Confidential Documents/0701/6674... · Edexcel GCE Further Pure Mathematics FP1 Advanced/Advanced Subsidiary

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Paper Reference(s)

6674/01Edexcel GCEFurther Pure Mathematics FP1Advanced/Advanced SubsidiaryFriday 26 January 2007 – AfternoonTime: 1 hour 30 minutes

Materials required for examination Items included with question papersMathematical Formulae (Green) Nil

Candidates may use any calculator EXCEPT those with the facility forsymbolic algebra, differentiation and/or integration. Thus candidates mayNOT use calculators such as the Texas Instruments TI 89, TI 92, Casio CFX 9970G, Hewlett Packard HP 48G.

Instructions to CandidatesIn the boxes above, write your centre number, candidate number, your surname, initial(s) and signature.Check that you have the correct question paper.You must write your answer for each question in the space following the question.When a calculator is used, the answer should be given to an appropriate degree of accuracy.

Information for CandidatesA booklet ‘Mathematical Formulae and Statistical Tables’ is provided.Full marks may be obtained for answers to ALL questions.The marks for individual questions and the parts of questions are shown in round brackets: e.g. (2).There are 8 questions in this question paper. The total mark for this paper is 75. There are 24 pages in this question paper. Any blank pages are indicated.

Advice to CandidatesYou must ensure that your answers to parts of questions are clearly labelled.You must show sufficient working to make your methods clear to the examiner. Answers withoutworking may gain no credit.

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Question LeaveNumber Blank

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Candidate No.

Surname Initial(s)

Signature

Paper Reference

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1. (a) Find the roots of the equation

z2 + 2z + 17 = 0,

giving your answers in the form a + ib, where a and b are integers.(3)

(b) Show these roots on an Argand diagram.(1)

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Q1

(Total 4 marks)

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2. Obtain the general solution of the differential equation

giving your answer in the form y = f(x).(8)

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d 2 cos , 0,dyx y x xx+ = >

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Question 2 continued

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Q2

(Total 8 marks)

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3. The complex numbers z1 and z2 are given by

z1 = 5 + 3i,

z2 = 1 + pi,where p is an integer.

(a) Find in the form a + ib, where a and b are expressed in terms of p.(3)

Given that arg

(b) find the value of p.(2)

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2

1

,4

zz

π =

2

1

zz

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Q3

(Total 5 marks)

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4. (a) Show that

(3)

(b) Find expressing your answer as a single fraction in its simplest form.(6)

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3

1

1,( 1)

n

r

r rr r=

− ++∑

3 1 1 11 , for 0, 1.( 1) 1

r r r rr r r r− +

≡ − + − ≠ −+ +

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Q4

(Total 9 marks)

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5. Figure 1

Figure 1 shows a sketch of the curve with equation

The curve crosses the x-axis at x = 1 and x = –1 and the line x = –2 is an asymptote of thecurve.

(a) Use algebra to solve the equation

(6)

(b) Hence, or otherwise, find the set of values of x for which

(3)

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2 1 3(1 ).2

x xx−

< −+

2 1 3(1 ).2

x xx−

= −+

2 1 , 2.2

xy xx−

= ≠ −+

*N23568A01024*

y

x10–2 –1

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Q5

(Total 9 marks)

*N23568A01324*

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6. f(x) = ln x + x – 3, x > 0.

(a) Find f(2.0) and f(2.5), each to 4 decimal places, and show that the root α of theequation f(x) = 0 satisfies 2.0 < α < 2.5.

(3)

(b) Use linear interpolation with your values of f(2.0) and f(2.5) to estimate α, givingyour answer to 3 decimal places.

(2)

(c) Taking 2.25 as a first approximation to α, apply the Newton-Raphson process onceto f(x) to obtain a second approximation to α, giving your answer to 3 decimal places.

(5)

(d) Show that your answer in part (c) gives α correct to 3 decimal places.(2)

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Q6

(Total 12 marks)

*N23568A01524*

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7. A scientist is modelling the amount of a chemical in the human bloodstream. The amountx of the chemical, measured in mg l –1, at time t hours satisfies the differential equation

(a) Show that the substitution transforms this differential equation into

(5)

(b) Find the general solution of differential equation .

(4)

Given that at time t = 0, and

(c) find an expression for x in terms of t,(4)

(d) write down the maximum value of x as t varies.(1)

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d 0,dxt=

12

x =

I

I2

2

d 3.d

y yt

+ =

2

1yx

=

222 4

2

d d2 6 3 , 0.d d

x xx x x xt t

− = − >

*N23568A01624*

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Q7

(Total 14 marks)

*N23568A01924*

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8. Figure 2

Figure 2 shows a sketch of the curve C with polar equation

The tangent to C at the point P is perpendicular to the initial line.

(a) Show that P has polar coordinates

(6)

The point Q on C has polar coordinates

The shaded region R is bounded by OP, OQ and C, as shown in Figure 2.

(b) Show that the area of R is given by

(3)

(c) Hence, or otherwise, find the area of R, giving your answer in the form a + bπ, wherea and b are rational numbers.

(5)

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24

6

1 1sin 2 cos 2 cos 4 d .2 2

π

πθ θ θ θ + −

∫

2, .4π √

3 , .2 6π

24sin cos , 0 .2

r πθ θ θ= <

*N23568A02024*

Q

RC

O

P

Initial line

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TOTAL FOR PAPER: 75 MARKS

END

Q8

(Total 14 marks)

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