LIBS TASK OIGMATHS 11 0580 21 2012 - WELCOME IGCSE · 5 The Tiger Sky Tower in Singapore has a...

This document consists of 12 printed pages.

IB12 11_0580_21/4RP © UCLES 2012 [Turn over

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UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS International General Certificate of Secondary Education

MATHEMATICS 0580/21

Paper 2 (Extended) October/November 2012

1 hour 30 minutes

Candidates answer on the Question Paper.

Additional Materials: Electronic calculator Geometrical instruments Mathematical tables (optional) Tracing paper (optional)

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 fluid.

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 specified in the question, and if the answer is not exact, give the answer to three significant figures. 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 70.

2

© UCLES 2012 0580/21/O/N/12

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Examiner's

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1 On a mountain, the temperature decreases by 6.5 °C for every 1000 metres increase in height. At 2000 metres the temperature is 10 °C. Find the temperature at 6000 metres. Answer °C [2]

2 Use your calculator to find the value of

6.28.12

4.36.28.1222

××

−+.

Answer [2]

3 (a) The diagram shows a cuboid.

How many planes of symmetry does this cuboid have? Answer(a) [1]

(b) Write down the order of rotational symmetry for the following diagram.

Answer(b) [1]

3

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Examiner's

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4 Write down all your working to show that the following statement is correct.

2

1

9

8

2

1

+

+

= 45

34

Answer

[2]

5 Simplify the expression.

(a2

1

O b2

1

)(a2

1

+ b2

1

) Answer [2]

6

C

D O

B

A

108°

NOT TOSCALE

A, B, C and D lie on a circle centre O. Angle ADC = 108°. Work out the obtuse angle AOC. Answer Angle AOC = [2]

4

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7 The train fare from Bangkok to Chiang Mai is 768 baht. The exchange rate is £1 = 48 baht. Calculate the train fare in pounds (£). Answer £ [2]

8 Acri invested $500 for 3 years at a rate of 2.8% per year compound interest. Calculate the final amount he has after 3 years. Answer $ [3]

9 Solve the inequality.

35

32 xx

−

−

Y=2

Answer [3]

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10 A large water bottle holds 25 litres of water correct to the nearest litre. A drinking glass holds 0.3 litres correct to the nearest 0.1 litre. Calculate the lower bound for the number of glasses of water which can be filled from the bottle. Answer [3]

11 The electrical resistance, R, of a length of cylindrical wire varies inversely as the square of the

diameter, d, of the wire. R = 10 when d = 2. Find R when d = 4.

Answer R = [3]

12

6 cmNOT TOSCALE

The diagram shows a circular disc with radius 6 cm. In the centre of the disc there is a circular hole with radius 0.5 cm. Calculate the area of the shaded section. Answer cm2 [3]

6

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13 Find the matrix which represents the combined transformation of a reflection in the x axis followed

by a reflection in the line y = x.

Answer

[3]

14

x°C

A B4 cm4 cm NOT TO

SCALE

ABC is a sector of a circle, radius 4 cm and centre C. The length of the arc AB is 8 cm and angle ACB = x°. Calculate the value of x . Answer x = [3]

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For

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15

0

14

20 40 60

Speed(metres per

second)

Time (seconds) The diagram shows the speed-time graph of a bus journey between two bus stops. Hamid runs at a constant speed of 4 m/s along the bus route. He passes the bus as it leaves the first bus stop. The bus arrives at the second bus stop after 60 seconds. How many metres from the bus is Hamid at this time? Answer m [3]

16 Rearrange the formula y = 4

2

−

+

x

x

to make x the subject.

Answer x = [4]

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17

A BC

AB is the diameter of a circle. C is a point on AB such that AC = 4 cm. (a) Using a straight edge and compasses only, construct (i) the locus of points which are equidistant from A and from B, [2] (ii) the locus of points which are 4 cm from C. [1] (b) Shade the region in the diagram which is

• nearer to B than to A and

• less than 4 cm from C. [1]

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18 Lauris records the mass and grade of 300 eggs. The table shows the results.

Mass (x grams)

30 I x Y 40 40 I x Y 50 50 I x Y 60 60 I x Y 70 70 I x Y 80 80 I x Y 90

Frequency 15 48 72 81 54 30

Grade small medium large very large

(a) Find the probability that an egg chosen at random is graded very large.

Answer(a) [1]

(b) The cumulative frequency diagram shows the results from the table.

300

250

200

150

100

50

0 30 40 50 60 70 80 90

Cumulativefrequency

Mass (x grams) Use the cumulative frequency diagram to find (i) the median, Answer(b)(i) g [1]

(ii) the lower quartile, Answer(b)(ii) g [1]

(iii) the inter-quartile range, Answer(b)(iii) g [1]

(iv) the number of eggs with a mass greater than 65 grams. Answer(b)(iv) [2]

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19

M =

−

32

45

Find (a) M

2 ,

Answer(a)

[2]

(b) 2M ,

Answer(b)

[1]

(c) |M| , the determinant of M, Answer(c) [1]

(d) MO1.

Answer(d)

[2]

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20 f(x) = 4(x + 1) g(x) = 2

3x

O 1

(a) Write down the value of x when f O1(x) = 2.

Answer(a) x = [1]

(b) Find fg(x). Give your answer in its simplest form. Answer(b) fg(x)= [2]

(c) Find gO1(x).

Answer(c) g

O1(x) = [3]

Question 21 is printed on the next page.

12

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.

© UCLES 2012 0580/21/O/N/12

For

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21 y

x

2

1

–1

–2

0–1 1 2 3–2–3

P Q

R

The triangle PQR has co-ordinates P(O1, 1), Q(1, 1) and R(1, 2). (a) Rotate triangle PQR by 90° clockwise about (0, 0). Label your image P'Q'R'. [2] (b) Reflect your triangle P'Q'R' in the line y = x− . Label your image P''Q''R''. [2] (c) Describe fully the single transformation which maps triangle PQR onto triangle P''Q''R''. Answer(c) [2]



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MATHEMATICS 0580/22


1 hour 30 minutes

















2

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Examiner's

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1 Write the following numbers correct to one significant figure.

(a) 7682

Answer(a) [1]

(b) 0.07682

Answer(b) [1]

2 Work out 11.3139 – 2.28 × 3 29 .

Give your answer correct to one decimal place.

Answer [2]

3 m = 4

1[3h2 + 8ah + 3a2]

Calculate the exact value of m when h = 20 and a = O5.

Answer m = [2]

3


For

Examiner's

Use

4

6°

6°

NOT TOSCALE

The diagram shows two of the exterior angles of a regular polygon with n sides.

Calculate n.

Answer n = [2]

5 The Tiger Sky Tower in Singapore has a viewing capsule which holds 72 people.

This number is 75% of the population of Singapore when it was founded in 1819.

What was the population of Singapore in 1819?

Answer [2]

6 In a traffic survey of 125 cars the number of people in each car was recorded.

Number of people in each car 1 2 3 4 5

Frequency 50 40 10 20 5

Find

(a) the range,

Answer(a) [1]

(b) the median,

Answer(b) [1]

(c) the mode.

Answer(c) [1]

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7 The number of spectators at the 2010 World Cup match between Argentina and Mexico was

82 000 correct to the nearest thousand.

If each spectator paid 2600 Rand (R) to attend the game, what is the lower bound for the total amount

paid?

Write your answer in standard form.

Answer R [3]

8

0.65 m

85 km

NOT TOSCALE

A water pipeline in Australia is a cylinder with radius 0.65 metres and length 85 kilometres.

Calculate the volume of water the pipeline contains when it is full.

Give your answer in cubic metres.

Answer m3 [3]

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Examiner's

Use

9 A shop is open during the following hours.

Monday to Friday Saturday Sunday

Opening time 06 45 07 30 08 45

Closing time 17 30 17 30 12 00

(a) Write the closing time on Saturday in the 12-hour clock time.

Answer(a) [1]

(b) Calculate the total number of hours the shop is open in one week.

Answer(b) h [2]

10 Solve the equation 4x O 12 = 2(11 – 3x).

Answer x = [3]

6

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11 List all the prime numbers which satisfy this inequality.

16 I 2x – 5 I 48

Answer [3]

12

A company sells cereals in boxes which measure 10 cm by 25 cm by 35 cm.

They make a special edition box which is mathematically similar to the original box.

The volume of the special edition box is 15 120 cm3.

Work out the dimensions of this box.

Answer cm by cm by cm [3]

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Examiner's

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13 The mass, m, of an object varies directly as the cube of its length, l .

m = 250 when l = 5 .

Find m when l = 7 .

Answer m = [3]

14 (a)

8

3 8

3

×

8

3 8

1

= p

q

Find the value of p and the value of q.

Answer(a) p =

q = [2]

(b) 5O3 + 5

O4 = k × 5

O4

Find the value of k.

Answer(b) k = [2]

8

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15

0 5 10 15 20 25 30 35

25

20

15

10

5

Speed(metres per

second)

Time (seconds)

The diagram shows the speed-time graph for the last 35 seconds of a car journey.

(a) Find the deceleration of the car as it came to a stop.

Answer(a) m/s2 [1]

(b) Calculate the total distance travelled by the car in the 35 seconds.

Answer(b) m [3]

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Examiner's

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16 A company sends out ten different questionnaires to its customers.

The table shows the number sent and replies received for each questionnaire.

Questionnaire A B C D E F G H I J

Number sent out 100 125 150 140 70 105 100 90 120 130

Number of replies 24 30 35 34 15 25 22 21 30 31

40

35

30

25

20

15

10

5

0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150

Num

ber o

f rep

lies

Number sent out

(a) Complete the scatter diagram for these results.

The first two points have been plotted for you. [2]

(b) Describe the correlation between the two sets of data.

Answer(b) [1]

(c) Draw the line of best fit. [1]

10

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17 y

x1 2 3 4 5 6 7 8 9

3

2

1

0

–1

–2

A

D

B A' B'

A″ B″

C D' C'

D″ C″

(a) Describe the single transformation which maps ABCD onto A' B' C' D'.

Answer(a) [3]

(b) A single transformation maps A' B' C' D' onto A" B" C" D".

Find the matrix which represents this transformation.

Answer(b)

[2]

18 A =

0

1

1

0

B =

−

−

01

10

On the grid on the next page, draw the image of PQRS after the transformation represented by BA.

11


For

Examiner's

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P Q

RS

y

x10 2 3 4 5 6 7–7 –6 –5 –4 –3 –2 –1

4

3

2

1

–1

–2

–3

–4 [5]

19 f(x) = x2 + 1 g(x) = 3

2+x

(a) Work out ff(O1).

Answer(a) [2]

(b) Find gf(3x), simplifying your answer as far as possible.

Answer(b) gf(3x) = [3]

(c) Find gO1(x).

Answer(c) gO1(x) = [2]


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20 (a) The two lines y = 2x + 8 and y = 2x – 12 intersect the x-axis at P and Q.

Work out the distance PQ.

Answer(a) PQ = [2]

(b) Write down the equation of the line with gradient O4 passing through (0, 5).

Answer(b) [2]

(c) Find the equation of the line parallel to the line in part (b) passing through (5, 4).

Answer(c) [3]



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MATHEMATICS 0580/23


1 hour 30 minutes

















2

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Examiner's

Use

1 Samantha invests $600 at a rate of 2% per year simple interest. Calculate the interest Samantha earns in 8 years. Answer $ [2]

2 Show that 10

1

2

+ 5

2

2

= 0.17.

Write down all the steps in your working. Answer

[2]

3 Jamie needs 300 g of flour to make 20 cakes. How much flour does he need to make 12 cakes? Answer g [2]

4 Expand the brackets.

y(3 O y3) Answer [2]

3


For

Examiner's

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5 Maria pays $84 rent. The rent is increased by 5%. Calculate Maria’s new rent. Answer $ [2]

6

Using a straight edge and compasses only, construct the locus of points which are equidistant from

R and from T. [2]

7 Find the value of 10.9511.8

7.2

−

.

Give your answer correct to 4 significant figures. Answer [2]

8 A carton contains 250 ml of juice, correct to the nearest millilitre. Complete the statement about the amount of juice, j ml, in the carton. Answer Y j I [2]

T

R

4

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9 Shade the required region in each of the Venn diagrams.

A B

A' (P ∩ R) ∪ Q

P Q

R

[2]

10 Without using a calculator, show that 2

3_

16

49

=

343

64.

Write down all the steps in your working. Answer

[2]

11 Simplify (256w256)4

1

.

Answer [2]

5


For

Examiner's

Use

12

Mass of parcel (m kilograms)

0 I m Y 0.5 0.5 I m Y 1.5 1.5 I m Y 3

Frequency 20 18 9

The table above shows information about parcels in a delivery van. John wants to draw a histogram using this information. Complete the table below.

Mass of parcel (m kilograms)

0 I m Y 0.5 0.5 I m Y 1.5 1.5 I m Y 3

Frequency density 18

[2]

13 Write the following as a single fraction in its simplest form.

3

2+x

_ 4

12 −x

+ 1

Answer [3]

6

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14 y varies inversely as the square root of x. When x = 9, y = 6. Find y when x = 36. Answer y = [3]

15 A model of a ship is made to a scale of 1 : 200. The surface area of the model is 7500 cm2. Calculate the surface area of the ship, giving your answer in square metres. Answer m2 [3]

16 Make y the subject of the formula.

A = πx2 O πy2 Answer y = [3]

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For

Examiner's

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17

O

B

A

4r

5rNOT TOSCALE

The diagram shows a sector of a circle, centre O, radius 5r. The length of the arc AB is 4r. Find the area of the sector in terms of r, giving your answer in its simplest form. Answer [3]

18

A

B

C

23°

13 cm6 cm

NOT TOSCALE

In triangle ABC, AB = 6 cm, BC = 13 cm and angle ACB = 23°. Calculate angle BAC, which is obtuse. Answer Angle BAC = [4]

8

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19

2 4 6 8 10 12 14 16 18

5

4

3

2

1

0

Time (seconds)

Speed(metres per

second)

The diagram shows the speed-time graph for the last 18 seconds of Roman’s cycle journey. (a) Calculate the deceleration. Answer(a) m/s2 [1]

(b) Calculate the total distance Roman travels during the 18 seconds. Answer(b) m [3]

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For

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20

O C

D

Ed

c

NOT TOSCALE

In the diagram, O is the origin.

= c and = d. E is on CD so that CE = 2ED. Find, in terms of c and d, in their simplest forms,

(a) , Answer(a) = [2]

(b) the position vector of E. Answer(b) [2]

10

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21 Simplify the following.

25

20

2

2

−

−−

h

hh

Answer [4]

22 (a) M =

− 11

23

Find M–1, the inverse of M.

Answer(a)

[2]

(b) D, E and X are 2 × 2 matrices. I is the identity 2 × 2 matrix. (i) Simplify DI. Answer(b)(i) [1]

(ii) DX = E Write X in terms of D and E. Answer(b)(ii) X = [1]

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23 f(x) = 3x + 5 g(x) = 4x O 1 (a) Find the value of gg(3). Answer(a) [2]

(b) Find fg(x), giving your answer in its simplest form. Answer(b)fg(x) = [2]

(c) Solve the equation. f

–1(x) = 11 Answer(c) x = [1]


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24 Q P

B

AD

C

10 cm

6 cm

5 cm

NOT TOSCALE

The diagram shows a triangular prism. ABCD is a horizontal rectangle with DA = 10 cm and AB = 5 cm. BCQP is a vertical rectangle and BP = 6 cm. Calculate (a) the length of DP, Answer(a) DP = cm [3]

(b) the angle between DP and the horizontal rectangle ABCD. Answer(b) [3]



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MATHEMATICS 0580/41


2 hours 30 minutes













For π use either your calculator value or 3.142.




2

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1

A or A*

x°

E, F or G

(x + 18)°

B, C or DA or A*

E, F or GB, C or D

72°60°

Girls Boys

NOT TOSCALE

The pie charts show information on the grades achieved in mathematics by the girls and boys at a

school.

(a) For the Girls’ pie chart, calculate

(i) x,

Answer(a)(i) x = [2]

(ii) the angle for grades B, C or D.

Answer(a)(ii) [1]

(b) Calculate the percentage of the Boys who achieved grades E, F or G.

Answer(b) % [2]

(c) There were 140 girls and 180 boys.

(i) Calculate the percentage of students (girls and boys) who achieved grades A or A*.

Answer(c)(i) % [3]

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(ii) How many more boys than girls achieved grades B, C or D?

Answer(c)(ii) [2]

(d) The table shows information about the times, t minutes, taken by 80 of the girls to complete

their mathematics examination.

Time taken (t minutes) 40 I t Y 60 60 I t Y 80 80 I t Y 120 120 I t Y 150

Frequency 5 14 29 32

(i) Calculate an estimate of the mean time taken by these 80 girls to complete the examination.

Answer(d)(i) min [4]

(ii) On a histogram, the height of the column for the interval 60 I t Y 80 is 2.8 cm.

Calculate the heights of the other three columns.

Do not draw the histogram.

Answer(d)(ii) 40 I t Y 60 column height = cm

80 I t Y 120 column height = cm

120 I t Y 150 column height = cm [4]

4

© UCLES 2012 0580/41/O/N/12

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2 (a) (i) Complete the table of values for y = 2

1x3 + x2 – 7x.

x –5 –4 –3 –2 –1 0 1 2 3 4

y –2.5 12 16.5 7.5 0 –6 1.5

[3]

(ii) On the grid, draw the graph of y = 2

1x3 + x2 – 7x for –5 Y x Y 4 .

y

x

22

20

18

16

14

12

10

8

6

4

2

–2

–4

–6

–8

0 1 2 3 4–1–2–3–4–5

[4]

(b) Use your graph to solve the equation 2

1x3 + x2 – 7x = 2 .

Answer(b) x = or x = or x = [3]

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For

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(c) By drawing a suitable tangent, calculate an estimate of the gradient of the graph where x = O4 .

Answer(c) [3]

(d) (i) On the grid draw the line y = 10 – 5x for O2 Y x Y 3 . [3]

(ii) Use your graphs to solve the equation 2

1x3 + x2 – 7x = 10 – 5x.

Answer(d)(ii) x = [1]

6

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3 90 students are asked which school clubs they attend.

D = {students who attend drama club}

M = {students who attend music club}

S = { students who attend sports club}

39 students attend music club.

26 students attend exactly two clubs.

35 students attend drama club.

10

5

23

13........

........ ........

........

D M

S

(a) Write the four missing values in the Venn diagram. [4]

(b) How many students attend

(i) all three clubs,

Answer(b)(i) [1]

(ii) one club only?

Answer(b)(ii) [1]

(c) Find

(i) n(D ∩ M ),

Answer(c)(i) [1]

(ii) n((D ∩ M ) ∩ S' ).

Answer(c)(ii) [1]

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For

Examiner's

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(d) One of the 90 students is chosen at random.

Find the probability that the student

(i) only attends music club,

Answer(d)(i) [1]

(ii) attends both music and drama clubs.

Answer(d)(ii) [1]

(e) Two of the 90 students are chosen at random without replacement.

Find the probability that

(i) they both attend all three clubs,

Answer(e)(i) [2]

(ii) one of them attends sports club only and the other attends music club only.

Answer(e)(ii) [3]

8

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4 (a) Solve the equations.

(i) 4x – 7 = 8 – 2x

Answer(a)(i) x = [2]

(ii) 23

7=

−x

Answer(a)(ii) x = [2]

(b) Simplify the expressions.

(i) (3xy

4)3

Answer(b)(i) [2]

(ii) (16a6b2) 2

1

Answer(b)(ii) [2]

(iii)

64

87

2

2

−

−−

x

xx

Answer(b)(iii) [4]

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For

Examiner's

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5 (a)

20 cm

46 cm24 cm

NOT TOSCALE

Jose has a fish tank in the shape of a cuboid measuring 46 cm by 24 cm by 20 cm.

Calculate the length of the diagonal shown in the diagram.

Answer(a) cm [3]

(b) Maria has a fish tank with a volume of 20 000 cm3.

Write the volume of Maria’s fish tank as a percentage of the volume of Jose’s fish tank.

Answer(b) % [3]

(c) Lorenzo’s fish tank is mathematically similar to Jose’s and double the volume.

Calculate the dimensions of Lorenzo’s fish tank.

Answer(c) cm by cm by cm [3]

(d) A sphere has a volume of 20 000 cm3. Calculate its radius.

[The volume, V, of a sphere with radius r is V = 3

4πr3.]

Answer(d) cm [3]

10

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Examiner's

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6 (a) a =

−

3

2 b =

− 7

2 c =

−

21

10

(i) Find 2a + b.

Answer(a)(i)

[1]

(ii) Find ö=b ö.

Answer(a)(ii) [2]

(iii) ma + nb = c

Find the values of m and n.

Show all your working.

Answer(a)(iii) m =

n = [6]

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For

Examiner's

Use

(b)

O

P

Q

Y

XNOT TOSCALE

In the diagram, OX : XP = 3 : 2 and OY : YQ = 3 : 2 .

= p and = q.

(i) Write in terms of p and q.

Answer(b)(i) = [1]

(ii) Write in terms of p and q.

Answer(b)(ii) = [1]

(iii) Complete the following sentences.

The lines XY and PQ are

The triangles OXY and OPQ are

The ratio of the area of triangle OXY to the area of triangle OPQ is : [3]

12

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7

O

AW X

Z Y

BE

D C

7 cm

NOT TOSCALE

The vertices A, B, C, D and E of a regular pentagon lie on the circumference of a circle, centre O,

radius 7 cm.

They also lie on the sides of a rectangle WXYZ.

(a) Show that

(i) angle DOC = 72°,

Answer(a)(i)

[1]

(ii) angle DCB = 108°,

Answer(a)(ii)

[2]

(iii) angle CBY = 18°.

Answer(a)(iii)

[1]

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(b) Show that the length CD of one side of the pentagon is 8.23 cm correct to three significant

figures.

Answer(b)

[3]

(c) Calculate

(i) the area of the triangle DOC,

Answer(c)(i) cm2 [2]

(ii) the area of the pentagon ABCDE,

Answer(c)(ii) cm2 [1]

(iii) the area of the sector ODC,

Answer(c)(iii) cm2 [2]

(iv) the length XY.

Answer(c)(iv) cm [2]

(d) Calculate the ratio

area of the pentagon ABCDE : area of the rectangle WXYZ.

Give your answer in the form 1 : n.

Answer(d) 1 : [5]

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8 A rectangular piece of card has a square of side 2 cm removed from each corner.

NOT TOSCALE

2 cm

(2x + 3) cm

(x + 5) cm

2 cm

(a) Write expressions, in terms of x, for the dimensions of the rectangular card before the squares

are removed from the corners.

Answer(a) cm by cm [2]

(b) The diagram shows a net for an open box.

Show that the volume, V cm3, of the open box is given by the formula V = 4x2 + 26x + 30 .

Answer(b)

[3]

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(c) (i) Calculate the values of x when V = 75.

Show all your working and give your answers correct to two decimal places.

Answer(c)(i) x = or x = [5]

(ii) Write down the length of the longest edge of the box.

Answer(c)(ii) cm [1]


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9 Distances from the Sun can be measured in astronomical units, AU.

Earth is a distance of 1 AU from the Sun.

One AU is approximately 1.496 × 108 km.

The table shows distances from the Sun.

Name Distance from the Sun in AU Distance from the Sun in kilometres

Earth 1 1.496 × 108

Mercury 0.387

Jupiter

7.79 × 108

Pluto

5.91 × 109

(a) Complete the table. [3]

(b) Light travels at approximately 300 000 kilometres per second.

(i) How long does it take light to travel from the Sun to Earth?

Give your answer in seconds.

Answer(b)(i) s [2]

(ii) How long does it take light to travel from the Sun to Pluto?

Give your answer in minutes.

Answer(b)(ii) min [2]

(c) One light year is the distance that light travels in one year (365 days).

How far is one light year in kilometres?

Give your answer in standard form.

Answer(c) km [3]

(d) How many astronomical units (AU) are equal to one light year?

Answer(d) AU [2]



*4849274249*


MATHEMATICS 0580/42


2 hours 30 minutes

















2

© UCLES 2012 0580/42/O/N/12

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1 A factory produces bird food made with sunflower seed, millet and maize. (a) The amounts of sunflower seed, millet and maize are in the ratio sunflower seed : millet : maize = 5 : 3 : 1 . (i) How much millet is there in 15 kg of bird food? Answer(a)(i) kg [2]

(ii) In a small bag of bird food there is 60 g of sunflower seed. What is the mass of bird food in a small bag? Answer(a)(ii) g [2]

(b) Sunflower seeds cost $204.50 for 30 kg from Jon’s farm or €96.40 for 20 kg from Ann’s farm. The exchange rate is $1 = €0.718. Which farm has the cheapest price per kilogram? You must show clearly all your working. Answer(b) [4]

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(c) Bags are filled with bird food at a rate of 420 grams per second. How many 20 kg bags can be completely filled in 4 hours? Answer(c) [3]

(d) Brian buys bags of bird food from the factory and sells them in his shop for $15.30 each. He makes 12.5% profit on each bag. How much does Brian pay for each bag of bird food? Answer(d) $ [3]

(e) Brian orders 600 bags of bird food.

The probability that a bag is damaged is 50

1.

How many bags would Brian expect to be damaged? Answer(e) [1]

4

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2

B

A

DC

64 m43 m

32 m

NOT TOSCALE

The diagram represents a field in the shape of a quadrilateral ABCD. AB = 32 m, BC = 43 m and AC = 64 m. (a) (i) Show clearly that angle CAB = 37.0° correct to one decimal place. Answer(a)(i) [4] (ii) Calculate the area of the triangle ABC. Answer(a)(ii) m2 [2]

(b) CD = 70 m and angle DAC = 55°. Calculate the perimeter of the whole field ABCD. Answer(b) m [6]

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3 (a) (i) Factorise completely the expression 4x2 O 18x O 10 . Answer(a)(i) [3]

(ii) Solve 4x2 O 18x O 10 = 0. Answer(a)(ii) x = or x = [1]

(b) Solve the equation 2x2 O 7x O 10 = 0 . Show all your working and give your answers correct to two decimal places. Answer(b) x = or x = [4]

(c) Write 13

6

−x

O=

2

2

−x

as a single fraction in its simplest form.

Answer(c) [3]

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4 (a)

32°B

A

D O

C

143°

NOT TOSCALE

Points A, C and D lie on a circle centre O. BA and BC are tangents to the circle. Angle ABC = 32° and angle DAB = 143°. (i) Calculate angle AOC in quadrilateral AOCB. Answer(a)(i) Angle AOC = [2]

(ii) Calculate angle ADC. Answer(a)(ii) Angle ADC = [1]

(iii) Calculate angle OCD. Answer(a)(iii) Angle OCD = [2]

(iv) OA = 6 cm. Calculate the length of AB. Answer(a)(iv) AB = cm [3]

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(b) B

C

A

D

O

X

17°

39°

NOT TOSCALE

A, B, C and D are on the circumference of the circle centre O. AC is a diameter. Angle CAB = 39° and angle ABD = 17°. (i) Calculate angle ACB. Answer(b)(i) Angle ACB = [2]

(ii) Calculate angle BXC. Answer(b)(ii) Angle BXC = [2]

(iii) Give the reason why angle DOA is 34°. Answer(b)(iii) [1]

(iv) Calculate angle BDO. Answer(b)(iv) Angle BDO = [1]

(v) The radius of the circle is 12 cm. Calculate the length of major arc ABCD. Answer(b)(v) Arc ABCD = cm [3]

8

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5 (a) A farmer takes a sample of 158 potatoes from his crop. He records the mass of each potato and the results are shown in the table.

Mass (m grams) Frequency

0 I m Y 40 6

40 I m Y 80 10

80 I m Y 120 28

120 I m Y 160 76

160 I m Y 200 22

200 I m Y 240 16

Calculate an estimate of the mean mass. Show all your working. Answer(a) g [4]

(b) A new frequency table is made from the results shown in the table in part (a).

Mass (m grams) Frequency

0 I m Y 80

80 I m Y 200

200 I m Y 240 16

(i) Complete the table above. [2] (ii) On the grid opposite, complete the histogram to show the information in this new table.

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1.2

1.0

0.8

0.6

0.4

0.2

040 80 120

Mass (grams)160 200 240

Frequencydensity

m

[3] (c) A bag contains 15 potatoes which have a mean mass of 136 g. The farmer puts 3 potatoes which have a mean mass of 130 g into the bag. Calculate the mean mass of all the potatoes in the bag. Answer(c) g [3]

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6 (a) Calculate the magnitude of the vector

− 5

3.

Answer(a) [2]

(b)

16

14

12

10

8

6

4

2

04 8 12 16 182 6 10 14

x

y

R P

(i) The points P and R are marked on the grid above.

=

− 5

3. Draw the vector on the grid above. [1]

(ii) Draw the image of vector after rotation by 90° anticlockwise about R. [2]

(c) = 2a + b and = 3b O a.

Find in terms of a and b. Write your answer in its simplest form. Answer(c) = [2]

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(d) =

−

5

2 and =

−1

5.

Write as a column vector.

Answer(d) =

[2]

(e)

A

B

CX

MNOT TOSCALE

= b and = c.

(i) Find in terms of b and c. Answer(e)(i) = [1]

(ii) X divides CB in the ratio 1 : 3 . M is the midpoint of AB.

Find in terms of b and c. Show all your working and write your answer in its simplest form. Answer(e)(ii) = [4]

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7 Jay makes wooden boxes in two sizes. He makes x small boxes and y large boxes. He makes at least 5 small boxes. The greatest number of large boxes he can make is 8. The greatest total number of boxes is 14. The number of large boxes is at least half the number of small boxes.

(a) (i) Write down four inequalities in x and y to show this information. Answer(a)(i)

[4]

(ii) Draw four lines on the grid and write the letter R in the region which represents these inequalities.

15

14

13

12

11

10

9

8

7

6

5

4

3

2

1

02 4 6 8 10 12 13 14 151 3 5 7 9 11

y

x

[5]

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(b) The price of the small box is $20 and the price of the large box is $45. (i) What is the greatest amount of money he receives when he sells all the boxes he has made? Answer(b)(i) $ [2]

(ii) For this amount of money, how many boxes of each size did he make? Answer(b)(ii) small boxes and large boxes [1]

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8 The graph of y = f(x) is drawn on the grid for 0 Y x Y 3.2 .

y

x0 1 2

5

4

3

2

1

3

y = f(x)

(a) (i) Draw the tangent to the curve y = f(x) at x = 2.5 . [1] (ii) Use your tangent to estimate the gradient of the curve at x = 2.5 . Answer(a)(ii) [2]

(b) Use the graph to solve f(x) = 2, for 0 Y x Y 3.2 . Answer(b) x = or x = [2]

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(c) g(x) = 2

2

2 x

x

+ x ≠ 0 .

(i) Complete the table for values of g(x), correct to 1 decimal place.

x 0.7 1 1.5 2 2.5 3

g(x) 1.6 1.6 1.7

[2]

(ii) On the grid opposite, draw the graph of y = g(x) for 0.7 Y x Y 3 . [3]

(iii) Solve f(x) = g(x) for 0.7 Y x Y 3. Answer(c) (iii) x = or x = or x = [3]

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9 (a) = {25 students in a class}

F = {students who study French} S = {students who study Spanish} 16 students study French and 18 students study Spanish. 2 students study neither of these. (i) Complete the Venn diagram to show this information.

F S

..... ..... ..... .....

[2] (ii) Find n(F ' ). Answer(a)(ii) [1]

(iii) Find n(F ∩ S)'. Answer(a)(iii) [1]

(iv) One student is chosen at random. Find the probability that this student studies both French and Spanish. Answer(a)(iv) [1]

(v) Two students are chosen at random without replacement. Find the probability that they both study only Spanish. Answer(a)(v) [2]

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(b) In another class the students all study at least one language from French, German and Spanish. No student studies all three languages. The set of students who study German is a proper subset of the set of students who study

French. 4 students study both French and German. 12 students study Spanish but not French. 9 students study French but not Spanish. A total of 16 students study French. (i) Draw a Venn diagram to represent this information. [4] (ii) Find the total number of students in this class. Answer(b)(ii) [1]

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10 Consecutive integers are set out in rows in a grid. (a) This grid has 5 columns.

1 2 3 4 5

6 7 8 9 10 a b

11 12 13 14 15 n

16 17 18 19 20 c d

21 22 23 24 25

26 27 28 29 30

31 32 33 34 35

The shape drawn encloses five numbers 7, 9, 13, 17 and 19. This is the n = 13 shape. In this shape, a = 7, b = 9, c = 17 and d = 19.

(i) Calculate bc O ad for the n = 13 shape. Answer(a)(i) [1]

(ii) For the 5 column grid, a = n O 6. Write down b, c and d in terms of n for this grid. Answer(a)(ii) b =

c =

d = [2]

(iii) Write down bc O ad in terms of n. Show clearly that it simplifies to 20. Answer(a)(iii) [2]

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(b) This grid has 6 columns. The shape is drawn for n = 10.

1 2 3 4 5 6 a b

7 8 9 10 11 12 n

13 14 15 16 17 18 c d

19 20 21 22 23 24

25 26 27 28 29 30

31 32 33 34 35 36

(i) Calculate the value of bc O ad for n = 10. Answer(b)(i) [1]

(ii) Without simplifying, write down bc O ad in terms of n for this grid. Answer(b)(ii) [2]

(c) This grid has 7 columns.

1 2 3 4 5 6 7 a b

8 9 10 11 12 13 14 n

15 16 17 18 19 20 21 c d

22 23 24 25 26 27 28

29 30 31 32 33 34 35

Show clearly that bc O ad = 28 for n = 17. Answer(c)

[1] Question 10 continues on the next page.

20



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(d) Write down the value of bc O ad when there are t columns in the grid. Answer(d) [1]

(e) Find the values of c, d and bc O ad for this shape.

2 3 4

16

c d

Answer (e) c =

d =

bc O ad = [2]

This document consists of 19 printed pages and 1 blank page.


*5306291564*


MATHEMATICS 0580/43


2 hours 30 minutes

















2

© UCLES 2012 0580/43/O/N/12

For

Examiner's

Use

1 (a) The Martinez family travels by car to Seatown. The distance is 92 km and the journey takes 1 hour 25 minutes. (i) The family leaves home at 07 50. Write down the time they arrive at Seatown. Answer(a)(i) [1]

(ii) Calculate the average speed for the journey. Answer(a)(ii) km/h [2]

(iii) During the journey, the family stops for 10 minutes. Calculate 10 minutes as a percentage of 1 hour 25 minutes. Answer(a)(iii) % [1]

(iv) 92 km is 15% more than the distance from Seatown to Deecity. Calculate the distance from Seatown to Deecity. Answer(a)(iv) km [3]

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(b) The Martinez family spends $150 in the ratio fuel : meals : gifts = 11 : 16 : 3 . (i) Show that $15 is spent on gifts. Answer (b)(i) [2] (ii) The family buys two gifts. The first gift costs $8.25. Find the ratio cost of first gift : cost of second gift. Give your answer in its simplest form. Answer(b)(ii) : [2]

4

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2 (a) y

x

8

7

6

5

4

3

2

1

–1

–2

–3

–4

–5

–6

–7

–8

–9

0–1 1 2 3 4 5 6 7 8–2–3–4–5–6–7–8

X

Y

(i) Draw the translation of triangle X by the vector

−

−

1

11. [2]

(ii) Draw the enlargement of triangle Y with centre (–6, – 4) and scale factor 2

1. [2]

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(b)

X

y

x

87654321

–1–2–3–4–5–6–7–8–9

0 1–1–2–3–4–5–6–7–8 2 3 4 5 6 7 8

W

Z

Y

Describe fully the single transformation that maps (i) triangle X onto triangle Z, Answer(b)(i) [2]

(ii) triangle X onto triangle Y, Answer(b)(ii) [3]

(iii) triangle X onto triangle W. Answer(b)(iii) [3]

(c) Find the matrix that represents the transformation in part (b)(iii).

Answer(c)

[2]

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3 A metal cuboid has a volume of 1080 cm3 and a mass of 8 kg. (a) Calculate the mass of one cubic centimetre of the metal. Give your answer in grams. Answer(a) g [1]

(b) The base of the cuboid measures 12 cm by 10 cm. Calculate the height of the cuboid. Answer(b) cm [2]

(c) The cuboid is melted down and made into a sphere with radius r cm. (i) Calculate the value of r.

[The volume, V, of a sphere with radius r is V = 3

4πr

3.]

Answer(c)(i) r = [3]

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(ii) Calculate the surface area of the sphere. [The surface area, A, of a sphere with radius r is A = 4πr

2.] Answer(c)(ii) cm2 [2]

(d) A larger sphere has a radius R cm. The surface area of this sphere is double the surface area of the sphere with radius r cm in

part (c).

Find the value of r

R.

Answer(d) [2]

8

© UCLES 2012 0580/43/O/N/12

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Use

4 f(x) = 2

2

x

O 3x, x ≠ 0

(a) Complete the table.

x O3 O2.5 O2 O1.5 O1 O0.5 0.5 1 1.5 2 2.5 3

f(x) 9.2 7.8 6.5 5.4 9.5 6.5 O3.6 O5.5 O7.2 O8.8

[2]

(b) On the grid, draw the graph of y = f(x), for O3 Y x Y O0.5 and 0.5 Y x Y 3 .

y

x

10

9

8

7

6

5

4

3

2

1

–1

–2

–3

–4

–5

–6

–7

–8

–9

0–1–2–3 321

[5]

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(c) Use your graph to solve the equations. (i) f(x) = 4 Answer(c)(i) x = [1]

(ii) f(x) = 3x Answer(c)(ii) x = [2]

(d) The equation f(x) = 3x can be written as x3 = k. Find the value of k. Answer(d) k = [2]

(e) (i) Draw the straight line through the points (–1, 5) and (3, –9). [1] (ii) Find the equation of this line. Answer(e)(ii) [3]

(iii) Complete the statement.

The straight line in part (e)(ii) is a to the graph of y = f(x). [1]

10

© UCLES 2012 0580/43/O/N/12

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5 (a) Marcos buys 2 bottles of water and 3 bottles of lemonade. The total cost is $3.60. The cost of one bottle of lemonade is $0.25 more than the cost of one bottle of water. Find the cost of one bottle of water. Answer(a) $ [4]

(b)

5 cm2

x cm

y cm 6 cm2

(x + 2) cm

Y cm NOT TOSCALE

The diagram shows two rectangles. The first rectangle measures x cm by y cm and has an area of 5 cm2. The second rectangle measures (x + 2) cm by Y cm and has an area of 6 cm2.

(i) When y + Y = 1, show that x2 O 9x O 10 = 0 . Answer (b)(i) [4]

(ii) Factorise x2 O 9x O 10 . Answer(b)(ii) [2]

(iii) Calculate the perimeter of the first rectangle. Answer(b)(iii) cm [2]

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(c)

(2x + 3) cm

(x + 3) cm

5 cm NOT TOSCALE

The diagram shows a right-angled triangle with sides of length 5 cm, (x + 3) cm and (2x + 3) cm.

(i) Show that 3x2 + 6x O 25 = 0 . Answer (c)(i) [4]

(ii) Solve the equation 3x2 + 6x O 25 = 0 . Show all your working and give your answers correct to 2 decimal places. Answer(c)(ii) x = or x = [4]

(iii) Calculate the area of the triangle. Answer(c)(iii) cm2 [2]

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6 A

B C

16 cm

25 cm

NOT TOSCALE

The area of triangle ABC is 130 cm2. AB = 16 cm and BC = 25 cm. (a) Show clearly that angle ABC = 40.5°, correct to one decimal place. Answer (a)

[3] (b) Calculate the length of AC. Answer(b) AC = cm [4]

(c) Calculate the shortest distance from A to BC. Answer(c) cm [2]

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7 (a)

1 2 2 3 4

Two discs are chosen at random without replacement from the five discs shown in the diagram. (i) Find the probability that both discs are numbered 2 . Answer(a)(i) [2]

(ii) Find the probability that the numbers on the two discs have a total of 5 . Answer(a)(ii) [3]

(iii) Find the probability that the numbers on the two discs do not have a total of 5. Answer(a)(iii) [1]

(b) A group of international students take part in a survey on the nationality of their parents. E = {students with an English parent} F = {students with a French parent}

n( ) = 50, n(E) = 15, n(F ) = 9 and n(E ∪ F )' = 33 .

(i) Find n(E ∩ F ). Answer(b)(i) [1]

(ii) Find n(E' ∪ F ). Answer(b)(ii) [1]

(iii) A student is chosen at random. Find the probability that this student has an English parent and a French parent. Answer(b)(iii) [1]

(iv) A student who has a French parent is chosen at random. Find the probability that this student also has an English parent. Answer(b)(iv) [1]

E F

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8 (a) D

A

B

C

X

NOT TOSCALE

52°28°

A, B, C and D lie on a circle. The chords AC and BD intersect at X. Angle BAC = 28° and angle AXD = 52°. Calculate angle XCD. Answer(a)Angle XCD = [3]

(b)

O

P

S

R

Q

25x° 22x°

NOT TOSCALE

PQRS is a cyclic quadrilateral in the circle, centre O. Angle QOS = 22x° and angle QRS = 25x°. Find the value of x. Answer(b) x = [3]

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(c)

NOT TOSCALE

44°O K M

L

8 cm

In the diagram OKL is a sector of a circle, centre O and radius 8 cm. OKM is a straight line and ML is a tangent to the circle at L. Angle LOK = 44°. Calculate the area shaded in the diagram. Answer(c) cm2 [5]

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9 200 students take a Mathematics examination. The cumulative frequency diagram shows information about the times taken, t minutes, to complete

the examination.

200

190

180

170

160

150

140

130

120

110

100

90

80

70

60

50

40

30

20

10

30 40 50

Time (minutes)

60 70 80 900

Cumulativefrequency

t

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(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]

(iv) the number of students who took more than 1 hour. Answer(a)(iv) [2]

(b) (i) Use the cumulative frequency diagram to complete the grouped frequency table.

Time, t minutes

30 I t Y 40 40 I t Y 50 50 I t Y 60 60 I t Y 70 70 I t Y 80 80 I t Y 90

Frequency 9 16 28 108 28

[1] (ii) Calculate an estimate of the mean time taken by the 200 students to complete the

examination. Show all your working. Answer(b)(ii) min [4]

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10 (a) Complete the table for the 6 th term and the n th term in each sequence.

Sequence 6 th term n th term

A 11, 9, 7, 5, 3

B 1, 4, 9, 16, 25

C 2, 6, 12, 20, 30

D 3, 9, 27, 81, 243

E 1, 3, 15, 61, 213

[12] (b) Find the value of the 100 th term in (i) Sequence A, Answer(b)(i) [1]

(ii) Sequence C. Answer(b)(ii) [1]

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(c) Find the value of n in Sequence D when the n th term is equal to 6561. Answer(c) n = [1]

(d) Find the value of the 10 th term in Sequence E. Answer(d) [1]

LIBS TASK OIGMATHS 11 0580 21 2012 - WELCOME IGCSE · 5 The Tiger Sky Tower in Singapore has a...

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