(PDF) Edexcel GCE core 3 mathematics C3 6665/01 advanced subsidiary jun 2006 question paper

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N23581AW850/R6665/57570 3/3/3/3/3/36,300

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6665/01Edexcel GCECore Mathematics C3Advanced LevelMonday 12 June 2006 – 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, CasioCFX 9970G, Hewlett Packard HP 48G.

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

Information for CandidatesA booklet ‘Mathematical Formulae and Statistical Tables’ is provided.The marks for individual questions and the parts of questions are shown in round brackets: e.g. (2).Full marks may be obtained for answers to ALL questions.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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6 6 6 5 0 1

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1. (a) Simplify .(3)

(b) Hence, or otherwise, express as a single fraction in its simplest

form.(3)

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2

3 2 11 ( 1)

x xx x x− −

−− +

2

3 21

x xx− −

−

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Q1

(Total 6 marks)

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2. Differentiate, with respect to x,

(a) e3x + ln 2x,(3)

(b) .(3)

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322(5 )x+

Q2

(Total 6 marks)

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3. Figure 1

Figure 1 shows part of the curve with equation y = f(x), x ∈R, where f is an increasingfunction of x. The curve passes through the points P(0, –2) and Q(3, 0) as shown.

In separate diagrams, sketch the curve with equation

(a) y = |f (x) | ,(3)

(b) y = f –1(x),(3)

(c) y = f(3x).(3)

Indicate clearly on each sketch the coordinates of the points at which the curve crosses ormeets the axes.

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y

xO

P

Q (3, 0)

y = f (x)

(0, –2)

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Question 3 continued

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Q3

(Total 9 marks)

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4. A heated metal ball is dropped into a liquid. As the ball cools, its temperature, T °C, t minutes after it enters the liquid, is given by

T = 400 e–0.05t + 25, t 0.

(a) Find the temperature of the ball as it enters the liquid.(1)

(b) Find the value of t for which T = 300, giving your answer to 3 significant figures.(4)

(c) Find the rate at which the temperature of the ball is decreasing at the instant when t = 50. Give your answer in °C per minute to 3 significant figures.

(3)

(d) From the equation for temperature T in terms of t, given above, explain why thetemperature of the ball can never fall to 20 °C.

(1)

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Q4

(Total 9 marks)

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5. Figure 2

Figure 2 shows part of the curve with equation

The curve has a minimum at the point P. The x-coordinate of P is k.

(a) Show that k satisfies the equation

4k + sin 4k – 2 = 0.(6)

The iterative formula

is used to find an approximate value for k.

(b) Calculate the values of x1, x2, x3 and x4, giving your answers to 4 decimal places.(3)

(c) Show that k = 0.277, correct to 3 significant figures.(2)

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1 01 (2 sin 4 ), 0.3,4n nx x x+ = − =

(2 1) tan 2 , 0 .4

y x x x π= − <

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y

O

P

x4π

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Question 5 continued

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Q5

(Total 11 marks)

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6. (a) Using sin2θ + cos2θ ≡ 1, show that cosec2θ – cot2θ ≡ 1.(2)

(b) Hence, or otherwise, prove that

cosec4θ – cot4θ ≡ cosec2θ + cot2θ.(2)

(c) Solve, for 90° < θ < 180°,

cosec4θ – cot4θ = 2 – cot θ.(6)

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Q6

(Total 10 marks)

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7. For the constant k, where k > 1, the functions f and g are defined by

f: x ln (x + k), x > –k,g: x |2x – k |, x ∈ R.

(a) On separate axes, sketch the graph of f and the graph of g.

On each sketch state, in terms of k, the coordinates of points where the graph meets thecoordinate axes.

(5)

(b) Write down the range of f.(1)

(c) Find in terms of k, giving your answer in its simplest form.(2)

The curve C has equation y = f(x). The tangent to C at the point with x-coordinate 3 isparallel to the line with equation 9y = 2x + 1.

(d) Find the value of k.(4)

fg4k

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Question 7 continued

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Q7

(Total 12 marks)

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8. (a) Given that where 270° < A < 360°, find the exact value of sin 2A.(5)

(b) (i) Show that (3)

Given that

(ii) show that (4)

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d sin 2 .dy xx

=

23sin cos 2 cos 2 ,3 3

y x x xπ π = + + + −

cos 2 cos 2 cos 2 .3 3

x x xπ π + + − ≡

3cos ,4

A =

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Question 8 continued

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TOTAL FOR PAPER: 75 MARKS

END

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Q8

(Total 12 marks)

Edexcel GCE core 3 mathematics C3 6665/01 advanced subsidiary jun 2006 question paper

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