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2013年CIE剑桥国际考试IGCSE数学真题(43)


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UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS International General Certi?cate of Secondary Education

*9687546924*

MATHEMATICS Paper 4 (Extended)

0580/43
May/June 2013 2 hours 30 minutes

Candidates answer on the Question Paper. Additional Materials: Electronic calculator Tracing paper (optional) Geometrical instruments

READ THESE INSTRUCTIONS FIRST Write your Centre number, candidate number and name on all the work you hand in. Write in dark blue or black pen. You may use a pencil for any diagrams or graphs. Do not use staples, paper clips, highlighters, glue or correction ?uid. DO NOT WRITE IN ANY BARCODES. Answer all questions. If working is needed for any question it must be shown below that question. Electronic calculators should be used. If the degree of accuracy is not speci?ed in the question, and if the answer is not exact, give the answer to three signi?cant ?gures. Give answers in degrees to one decimal place. For π, use either your calculator value or 3.142. At the end of the examination, fasten all your work securely together. The number of marks is given in brackets [ ] at the end of each question or part question. The total of the marks for this paper is 130.

This document consists of 19 printed pages and 1 blank page.
IB13 06_0580_43/5RP ? UCLES 2013

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2 1 (a) Ali and Ben receive a sum of money. They share it in the ratio 5 : 1. Ali receives $2345. Calculate the total amount.
For Examiner′s Use

Answer(a) $ ............................................... [2] (b) Ali uses 11% of his $2345 to buy a television. Calculate the cost of the television.

Answer(b) $ ............................................... [2] (c) A different television costs $330. (i) Ben buys one in a sale when this cost is reduced by 15%. How much does Ben pay?

Answer(c)(i) $ ............................................... [2] (ii) $330 is 12% less than the cost last year. Calculate the cost last year.

Answer(c)(ii) $ ............................................... [3]

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畅享冠军纯正特产?快来绵羊国正品代购!

3 (d) Ali invests $1500 of his share in a bank account. The account pays compound interest at a rate of 2.3% per year. Calculate the total amount in the account at the end of 3 years.
For Examiner′s Use

Answer(d) $ ............................................... [3] (e) Ali also buys a computer for $325. He later sells this computer for $250. Calculate Ali’s percentage loss.

Answer(e) ........................................... % [3] _____________________________________________________________________________________

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4 2 (a) In this question show all your construction arcs and use only a ruler and compasses to draw the boundaries of your region. This scale drawing shows the positions of four towns, P, Q, R and S, on a map where 1 cm represents 10 km.
For Examiner′s Use

North

P

Q

Scale: 1 cm to 10 km

S

R

A nature reserve lies in the quadrilateral PQRS. The boundaries of the nature reserve are: ● ● ● ● equidistant from Q and from R equidistant from PS and from PQ 60 km from R along QR. [7]

(i) Shade the region which represents the nature reserve. (ii) Measure the bearing of S from P.

Answer(a)(ii) ............................................... [1]
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畅享冠军纯正特产?快来绵羊国正品代购!

5 (b) A circular lake in the nature reserve has a radius of 45 m. (i) Calculate the area of the lake.
For Examiner′s Use

Answer(b)(i) .......................................... m2 [2] (ii) NOT TO SCALE

A fence is placed along part of the circumference of the lake. This arc subtends an angle of 210° at the centre of the circle. Calculate the length of the fence.

Answer(b)(ii) ........................................... m [2] _____________________________________________________________________________________

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6 3 (a) Luk wants to buy x goats and y sheep. (i) He wants to buy at least 5 goats. Write down an inequality in x to represent this condition. Answer(a)(i) ............................................... [1] (ii) He wants to buy at least 11 sheep. Write down an inequality in y to represent this condition. Answer(a)(ii) ............................................... [1] (iii) He wants to buy at least 20 animals. Write down an inequality in x and y to represent this condition. Answer(a)(iii) ............................................... [1] (b) Goats cost $4 and sheep cost $8. The maximum Luk can spend is $160. Write down an inequality in x and y and show that it simpli?es to x + 2y Y 40 . Answer(b)
For Examiner′s Use

[1]

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畅享冠军纯正特产?快来绵羊国正品代购!

7 (c) (i) On the grid below, draw four lines to show the four inequalities and shade the unwanted regions. y 40
For Examiner′s Use

35

30

25

20

15

10

5

0

x 5 10 15 20 25 30 35 40 [7] (ii) Work out the maximum number of animals that Luk can buy.

Answer(c)(ii) ............................................... [2] _____________________________________________________________________________________
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畅享冠军纯正特产?快来绵羊国正品代购!

8 4 I NOT TO SCALE
For Examiner′s Use

H F 40 cm

J

7 cm

E

22 cm

G

EFGHIJ is a solid metal prism of length 40 cm. The cross section EFG is a right-angled triangle. EF = 7 cm and EG = 22 cm. (a) Calculate the volume of the prism.

Answer(a) ........................................ cm3 [2] (b) Calculate the length FJ.

Answer(b) FJ = ......................................... cm [4]

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畅享冠军纯正特产?快来绵羊国正品代购!

9 (c) Calculate the angle between FJ and the base EGJH of the prism.
For Examiner′s Use

Answer(c) ............................................... [3] (d) The prism is melted and made into spheres. Each sphere has a radius 1.5 cm. Work out the greatest number of spheres that can be made. 4 [The volume, V, of a sphere with radius r is V = πr3.] 3

Answer(d) ............................................... [3] (e) (i) A right-angled triangle is the cross section of another prism. This triangle has height 4.5 cm and base 11.0 cm. Both measurements are correct to 1 decimal place. Calculate the upper bound for the area of this triangle.

Answer(e)(i) ........................................ cm2 [2] (ii) Write your answer to part (e)(i) correct to 4 signi?cant ?gures. Answer(e)(ii) ........................................ cm2 [1] _____________________________________________________________________________________

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10 5
2 (a) Complete this table of values for the function f(x) = 1 x – x , x ? 0.

For Examiner′s Use

x f(x)

–3 –9.33

–2 –4.5

–1 –2

–0.5 –2.25

–0.2

0.2 4.96

0.5

1

2 –3.5

3 –8.67 [3]

2 (b) Draw the graph of f(x) = 1 x – x for –3 Y x Y –0.2 and 0.2 Y x Y 3.

y 5 4 3 2 1 –3 –2 –1 0 –1 –2 –3 –4 –5 –6 –7 –8 –9 –10 [5] 1 2 3 x

? UCLES 2013

0580/43/M/J/13

畅享冠军纯正特产?快来绵羊国正品代购!

11 (c) Use your graph to solve f(x) = –3.
For Examiner′s Use

Answer(c) x = ................ or x = ................ or x = ................ [3] (d) By drawing a suitable line on your graph, solve the equation f(x) = 2x – 2.

Answer(d) x = ................ or x = ................ or x = ................ [3] (e) By drawing a suitable tangent, work out an estimate of the gradient of the curve at the point where x = –2. You must show your working.

Answer(e) ............................................... [3] _____________________________________________________________________________________

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12 6 In a box there are 7 red cards and 3 blue cards. A card is drawn at random from the box and is not replaced. A second card is then drawn at random from the box. (a) Complete this tree diagram. First card ........ 7 10 Red ........ ........ Blue ........ Blue [3] (b) Work out the probability that the two cards are of different colours. Give your answer as a fraction. Blue Second card Red
For Examiner′s Use

Red

........

Answer(b) ............................................... [3] _____________________________________________________________________________________

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畅享冠军纯正特产?快来绵羊国正品代购!

13 7 10 9 8 7 6 5 4 3 2 A 1 0 x 1 2 3 4 5 6 7 8 [2] B y
For Examiner′s Use

(a) (i) Draw the image of shape A after a stretch, factor 3, x-axis invariant. (ii) Write down the matrix representing a stretch, factor 3, x-axis invariant. Answer(a)(ii)

e

o

[2]

(b) (i) Describe fully the single transformation which maps shape A onto shape B. Answer(b)(i) ...................................................................................................................... [3] (ii) Write down the matrix representing the transformation which maps shape A onto shape B. Answer(b)(ii)

_____________________________________________________________________________________

e

o

[2]

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14 8 (a) B 27° C A O NOT TO SCALE
For Examiner′s Use

E D

A, B, C, D and E are points on the circle centre O. Angle ABD = 27°. Find (i) angle ACD, Answer(a)(i) Angle ACD = ............................................... [1] (ii) angle AOD, Answer(a)(ii) Angle AOD = ............................................... [1] (iii) angle AED. Answer(a)(iii) Angle AED = ............................................... [1] (b) L 100° 45 cm 32 cm K 67° M NOT TO SCALE

N The diagram shows quadrilateral KLMN. KL = 45 cm, LN = 32 cm, angle KLN = 100° and angle NLM = 67°.

? UCLES 2013

0580/43/M/J/13

畅享冠军纯正特产?快来绵羊国正品代购!

15 (i) Calculate the length KN.
For Examiner′s Use

Answer(b)(i) KN = ......................................... cm [4] (ii) The area of triangle LMN is 324 cm2. Calculate the length LM.

Answer(b)(ii) LM = ......................................... cm [3] (iii) Another triangle XYZ is mathematically similar to triangle LMN. M L Y X NOT TO SCALE Z N XZ = 16 cm and the area of triangle LMN is 324 cm2. Calculate the area of triangle XYZ.

Answer(b)(iii) ........................................ cm2 [2] _____________________________________________________________________________________
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16 9 Sam asked 80 people how many minutes their journey to work took on one day. The cumulative frequency diagram shows the times taken (m minutes). 80
For Examiner′s Use

70

60

50

Cumulative 40 frequency

30

20

10

0

m 10 20 30 40 50 Time (minutes)

(a) Find (i) the median, Answer(a)(i) ........................................ min [1] (ii) the lower quartile, Answer(a)(ii) ........................................ min [1] (iii) the inter-quartile range. Answer(a)(iii) ........................................ min [1]

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畅享冠军纯正特产?快来绵羊国正品代购!

17 (b) One of the 80 people is chosen at random. Find the probability that their journey to work took more than 35 minutes. Give your answer as a fraction. Answer(b) ............................................... [2] (c) Use the cumulative frequency diagram to complete this frequency table. Time (m minutes) Frequency 0 < m Y 10 30 10 < m Y 15 15 < m Y 30 30 < m Y 40 40 < m Y 50 12 18 [2] (d) Using mid-interval values, calculate an estimate of the mean journey time for the 80 people.
For Examiner′s Use

Answer(d) ........................................ min [3] (e) Use the table in part (c) to complete the histogram to show the times taken by the 80 people. One column has already been completed for you. 4

3

Frequency density

2

1

0

m 10 20 30 Time (minutes) 40 50

[5] _____________________________________________________________________________________

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18 10 (a) (i) Solve 2(3x – 7) = 13.
For Examiner′s Use

Answer(a)(i) x = ............................................... [3] (ii) Solve by factorising x2 – 7x + 6 = 0.

Answer(a)(ii) x = ................. or x = ................. [3] (iii) Solve x 2 3x - 2 + + = 4. 5 10

Answer(a)(iii) x = ............................................... [4]

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19 (b) 12 12 + 22 12 + 22 + 32 12 + 22 + 32 + 42 =1 =5 = 14 = 30
For Examiner′s Use

12 + 22 + 32 + 42 + ..................... + n2 = an3 + bn2 + Work out the values of a and b.

n 6

Answer(b) a = ............................................... b = ............................................... [6] _____________________________________________________________________________________

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20 BLANK PAGE

Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable effort has been made by the publisher (UCLES) to trace copyright holders, but if any items requiring clearance have unwittingly been included the publisher will be pleased to make amends at the earliest possible opportunity. University of Cambridge International Examinations is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge.

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