math may 2007 exam c4

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Edexcel GCECore Mathematics C4

Advanced

Monday 18 June 2007 Morning

Time: 1 hour 30 minutes

Materials required for examination Items included with question papers

Mathematical Formulae (Green) Nil

Candidates may use any calculator allowed by the regulations of the JointCouncil for Qualifications. Calculators must not have the facility for symbolicalgebra manipulation, differentiation and integration, or have retrievablemathematical formulas stored in them.

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.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 24 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.

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This publication may be reproduced only in accordance withEdexcel Limited copyright policy.

2007 Edexcel Limited.

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1.

Find the binomial expansion of f(x), in ascending powers ofx, as far as the term inx3.

Give each coefficient as a simplified fraction.

(5)

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f ( )x = 3(3 2 )x + , 32x < .

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Question 1 continued___________________________________________________________________________

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___________________________________________________________________________ Q1

(Total 5 marks)

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2. Use the substitution u = to find the exact value of

(6)

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1

20

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(2 1)

x

xdx

+ .

2x

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(Total 6 marks)

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3. (a) Find .

(4)

(b) Hence, using the identity , deduce2cosx x dx .

(3)

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cos2x x dx

2cos 2 2cos 1x x=

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___________________________________________________________________________Q3

(Total 7 marks)

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4.

(a) Find the values of the constantsA, B and C.

(4)

(b) Hence show that the exact value of , giving the

value of the constant k.

(6)

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2

2( 4 1) .(2 1)(2 1) (2 1) (2 1)

x B C A x x x x

+ + ++ +

2 2

1

2(4 1)

(2 1)(2 1)

xdx

x x

+

+ is 2 + ln k

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(Total 10 marks)

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5.

The line l1 has equation r =

The line l2 has equation r =

(a) Show that l1 and l2 do not meet.

(4)

The pointA is on l1 where = 1, and the pointB is on l2 where = 2.

(b) Find the cosine of the acute angle betweenAB and l1.(6)

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10

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(Total 10 marks)

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6. A curve has parametric equations

(a) Find an expression for in terms oft. You need not simplify your answer.

(3)

(b) Find an equation of the tangent to the curve at the point where .

Give your answer in the form , where a and b are constants to be

determined.

(5)

(c) Find a cartesian equation of the curve in the form .

(4)

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2

d

d

y

x

t =4

y = ax + b

2 f ( )y x=

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(Total 12 marks)

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7.

Figure 1

Figure 1 shows part of the curve with equation . The finite region R,

which is bounded by the curve, the x-axis and the line , is shown shaded in

Figure 1.

(a) Given that , complete the table with the values ofy corresponding to

,giving your answers to 5 decimal places.x 0

y 0 1

(3)

(b) Use the trapezium rule with all the values ofy in the completed table to obtain anestimate for the area of the shaded regionR, giving your answer to 4 decimal places.

(4)

The region R is rotated through 2 radians around the x-axis to generate a solid of

revolution.

(c) Use integration to find an exact value for the volume of the solid generated.

(4)

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y

xO 4

R

y = (tanx)

x =4

x =16 16

and,

8

3

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y = (tanx)

8

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*N26110A01824*


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(Total 11 marks)

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8. A population growth is modelled by the differential equation

wherePis the population, tis the time measured in days and kis a positive constant.

Given that the initial population isP0,

(a) solve the differential equation, givingPin terms ofP0, kand t.

(4)

Given also that k= 2.5,

(b) find the time taken, to the nearest minute, for the population to reach 2P0.

(3)

In an improved model the differential equation is given as

wherePis the population, tis the time measured in days and is a positive constant.

Given, again, that the initial population isP0 and that time is measured in days,

(c) solve the second differential equation, givingPin terms ofP0, and t.

(4)

Given also that = 2.5,

(d) find the time taken, to the nearest minute, for the population to reach 2P0 for the first

time, using the improved model.

(3)

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dPkP,

dt=

cosdP

P t,dt

=

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*N26110A02224*


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*N26110A02324*


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TOTAL FOR PAPER: 75 MARKS

END

Q8

(Total 14 marks)

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