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6665/01Edexcel GCECore Mathematics C3

Advanced

Thursday 14 June 2012 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 JointCouncil for Qualifications. Calculators must not have the facility for symbolicalgebra manipulation or symbolic differentiation/integration, or haveretrievable mathematical formulae 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.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.

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No.

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No.

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1. Express

2 3 2

9 4

2

3 12( )x

x x

+

+

as a single fraction in its simplest form.

(4)

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Question 1 continued

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(Total 4 marks)

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2. f ( )x x x x= + + 3 23 4 12

(a) Show that the equation f ( )x = 0 can be written as

xx

x=

+

4 3

3

( )

( ), x 3

(3)

The equation x x x3 23 4 12 0+ + = has a single root which is between 1 and 2

(b) Use the iteration formula

xx

xnn

n+ =

+

1

4 3

3

( )

( ) , n. 0with x

01= to find, to 2 decimal places, the value ofx x

1 2, and x

3.

(3)

The root of f ( )x = 0 is .

(c) By choosing a suitable interval, prove that = 1 272. to 3 decimal places.(3)

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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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(Total 9 marks)

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3.

Figure 1

Figure 1 shows a sketch of the curve Cwhich has equation

y xx

= e3

3sin , 3 3 x

(a) Find thex coordinate of the turning pointPon C, for which x 0

Give your answer as a multiple of.

(6)

(b) Find an equation of the normal to Cat the point where x = 0(3)

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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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(Total 9 marks)

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4.

Figure 2

Figure 2 shows part of the curve with equation y x= f ( )

The curve passes through the points P( . , )1 5 0 and Q( , )0 5 as shown.On separate diagrams, sketch the curve with equation

(a) y x= f ( )(2)

(b) y x= f ( )(2)

(c) y x= 2 3f ( )

(3)

Indicate clearly on each sketch the coordinates of the points at which the curve crosses or

meets the axes.

O

P

Q

(1.5, 0)

(0, 5)

y

x

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*P40686RA01432*

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Question 4 continued

Q4

(Total 7 marks)

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*P40686RA01632*

5. (a) Express 4 2 2cosec cosec2 in terms of sinR and cosR.

(2)

(b) Hence show that

4 22

cosec cosec2

2

sec=

(4)

(c) Hence or otherwise solve, for 0 < R

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Question 5 continued

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*P40686RA01832*

Question 5 continued

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Question 5 continued

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(Total 9 marks)

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*P40686RA02032*

6. The functions f and g are defined by

f e:x x6 + 2 , x \

g : lnx x6 , x 0

(a) State the range of f.(1)

(b) Find fg( )x , giving your answer in its simplest form.

(2)

(c) Find the exact value ofx for which f ( )2 3 6x + =(4)

(d) Find f1 , the inverse function of f, stating its domain.

(3)

(e) On the same axes sketch the curves with equation y x= f ( ) and y x= f 1( ) , giving thecoordinates of all the points where the curves cross the axes.

(4)

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Question 6 continued

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*P40686RA02232*

Question 6 continued

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Question 6 continued

Q6

(Total 14 marks)

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*P40686RA02432*

7. (a) Differentiate with respect tox,

(i) x x1

2 3ln( )

(ii)1 10

2 1 5

x

x( ), giving your answer in its simplest form.

(6)

(b) Given that x y= 3 2tan findd

d

y

xin terms ofx.

(5)

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Question 7 continued

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*P40686RA02632*

Question 7 continued

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Question 7 continued

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(Total 11 marks)

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*P40686RA02832*

8. f ( ) cos sinx x x= 7 2 24 2

Given that f ( ) cos( )x R x= +2 , where R 0 and 0 90< < ,

(a) find the value ofR and the value of .

(3)

(b) Hence solve the equation

7 2 24 2 12 5cos sin .x x =

for 0 180-x D , giving your answers to 1 decimal place.

(5)

(c) Express 14 482cos sin cosx x x in the form a x b x ccos sin2 2+ + ,where a, b, and c are constants to be found.

(2)

(d) Hence, using your answers to parts (a) and (c), deduce the maximum value of

14 482cos sin cosx x x(2)

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Question 8 continued

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*P40686RA03032*

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 12 marks)