Paper Reference(s) 6672/01 Edexcel GCE - Past Paperspastpapers.org/Edexcel/c4/june 05 pure2.pdf · Figure 3 shows part of the graph of y = f(x), x ∈R. The graph consists of two

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N21140AW850/R6672/57570 7/7/3/3/3/3/47,300

Paper Reference(s)

6672/01Edexcel GCEPure Mathematics P2Advanced/Advanced SubsidiaryMonday 20 June 2005 – MorningTime: 1 hour 30 minutes

Materials required for examination Items included with question papersMathematical Formulae (Lilac) 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) andsignature.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 question 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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1. Solve

(a) 5x = 8, giving your answer to 3 significant figures,(3)

(b) log2(x + 1) – log2x = log2 7.(3)

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Q1

(Total 6 marks)

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2. (a) Write down the first three terms, in ascending powers of x, of the binomial expansionof (1 + px)12, where p is a non-zero constant.

(2)

Given that, in the expansion of (1 + px)12, the coefficient of x is (–q) and the coefficientof x2 is 11q,

(b) find the value of p and the value of q.(4)

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Q2

(Total 6 marks)

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3. A river, running between parallel banks, is 20 m wide. The depth, y metres, of the rivermeasured at a point x metres from one bank, is given by the formula

, 0 x 20.

(a) Complete the table below, giving values of y to 3 decimal places.

(2)

(b) Use the trapezium rule with all the values in the table to estimate the cross-sectionalarea of the river.

(4)

Given that the cross-sectional area is constant and that the river is flowing uniformly at 2 m s–1,

(c) estimate, in m3, the volume of water flowing per minute, giving your answer to 3 significant figures.

(2)

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1 (20 )10

y x x= −√

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x 0 4 8 12 16 20

y 0 2.771 0

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Q3

(Total 8 marks)

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4. The function f is defined by

(a) Show that (4)

(b) Find f –1(x).(3)

The function g is defined by

g : x→ x2 + 5, x ∈ R.

(c) Solve fg(x) = .(3)

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14

2f ( ) , 1.1

x xx

= >−

2

5 1 3f : , 1.2 2

xx xx x x

+→ − >

+ − +

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Q4

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5. Figure 1

Figure 1 shows part of the curve C with equation

The finite region enclosed by C, the lines x = 1, x = 3, and the x-axis is rotated through360° about the x-axis to generate a solid S.

(a) Using integration, find the exact volume of S.(7)

Figure 2

The tangent T to C at the point (1, 2) meets the x-axis at the point (3, 0). The shadedregion R is bounded by C, the line x = 3 and T, as shown in Figure 2.

(b) Using your answer to part (a), find the exact volume generated by R when it is rotatedthrough 360° about the x-axis.

(3)

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1 , 0.xy xx+

= >

y

xO

C

1 3

(1, 2)

1xyx+

=

y

xO

CT

1 3

(1, 2)

1xyx+

=R

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Question 5 continued

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Q5

(Total 10 marks)

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6. f(x) = 3ex – ln x – 2, x > 0.

(a) Differentiate to find f´(x).(3)

The curve with equation y = f(x) has a turning point P. The x-coordinate of P is α.

(b) Show that α = e–α.(2)

The iterative formula

is used to find an approximate value for α.

(c) Calculate the values of x1, x2, x3 and x4, giving your answers to 4 decimal places.(2)

(d) By considering the change of sign of f ´(x) in a suitable interval, prove that α = 0.1443correct to 4 decimal places.

(2)

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11 06 e , 1,nx

nx x−+ = =

16

12

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Q6

(Total 9 marks)

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7. Figure 3

Figure 3 shows part of the graph of y = f(x), x ∈ R. The graph consists of two linesegments that meet at the point (1, a), a < 0. One line meets the x-axis at (3, 0). The otherline meets the x-axis at (–1, 0) and the y-axis at (0, b), b < 0.

In separate diagrams, sketch the graph with equation

(a) y = f(x + 1),(2)

(b) y = f( |x | ).(3)

Indicate clearly on each sketch the coordinates of any points of intersection with the axes.

Given that f(x) = |x – 1| – 2, find

(c) the value of a and the value of b,(2)

(d) the value of x for which f(x) = 5x.(4)

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y

xb

–1 O 3

(1, a)

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Question 7 continued

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8. (a) Given that 2 sin(θ + 30)° = cos(θ + 60)°, find the exact value of tan θ °.(5)

(b) (i) Using the identity cos(A + B) ≡ cosA cosB – sinA sinB, prove that

cos 2A ≡ 1 – 2 sin2A.(2)

(ii) Hence solve, for 0 x < 2π,

cos 2x = sin x,

giving your answers in terms of π.(5)

(iii) Show that sin 2y tan y + cos 2y ≡ 1, for 0 y < π.(3)

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Question 8 continued

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TOTAL FOR PAPER: 75 MARKS

END

Q8

(Total 15 marks)

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paper reference(s) 6672/01 edexcel gce - past paperspastpapers.org/edexcel/c4/june 05 pure2.pdf ·...

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