Cambridge International Examinations CAMBRIDGE INTERNATIONAL … · 2019-07-26 · Cambridge International Examinations Cambridge International General Certificate of Secondary Education

This document consists of 8 printed pages.

IB16 06_0607_11/4RP © UCLES 2016 [Turn over

*7320481266*

Cambridge International Examinations Cambridge International General Certificate of Secondary Education

CAMBRIDGE INTERNATIONAL MATHEMATICS 0607/11

Paper 1 (Core) May/June 2016

45 minutes

Candidates answer on the Question Paper.

Additional Materials: 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. Do not use staples, paper clips, glue or correction fluid. You may use an HB pencil for any diagrams or graphs. DO NOT WRITE IN ANY BARCODES. Answer all the questions. CALCULATORS MUST NOT BE USED IN THIS PAPER. All answers should be given in their simplest form. You must show all the relevant working to gain full marks and you will be given marks for correct methods even if your answer is incorrect. The number of marks is given in brackets [ ] at the end of each question or part question. The total number of marks for this paper is 40.

2

© UCLES 2016 0607/11/M/J/16

Formula List

Area, A, of triangle, base b, height h. A = 12

bh

Area, A, of circle, radius r. A = πr2

Circumference, C, of circle, radius r. C = 2πr

Curved surface area, A, of cylinder of radius r, height h. A = 2πrh

Curved surface area, A, of cone of radius r, sloping edge l. A = πrl

Curved surface area, A, of sphere of radius r. A = 4πr2

Volume, V, of prism, cross-sectional area A, length l. V =Al

Volume, V, of pyramid, base area A, height h. V= 13

Ah

Volume, V, of cylinder of radius r, height h. V = πr2h

Volume, V, of cone of radius r, height h. V = 13

πr2h

Volume, V, of sphere of radius r. V = 43

πr3

3

© UCLES 2016 0607/11/M/J/16 [Turn over

Answer all the questions. 1

Shade 32

of this shape.

[1]

2

Draw a sector inside this circle. Draw a chord inside this circle. [2] 3 Write down all the factors of 21. [2]

4

© UCLES 2016 0607/11/M/J/16

4 Work out.

(a) 16 + 8 × 4

[1] (b) 16 – 8 ÷ 4

[1] 5 Complete the mapping diagram.

16

11

9

5

2

19

15

7

1

[1]

6 Jenny shares $40 between her two sons in the ratio 3:1. Work out how much each son receives.

$ and $ [2]

5


7 Tick the shapes that have both line symmetry and rotational symmetry.

Rectangle Kite Parallelogram

Rhombus Isosceles Triangle

[2]

8 The diagram shows a child’s solid building block in the shape of a cuboid 2 cm by 5 cm by 10 cm.

5 cm

10 cm

2 cm

Find the total surface area of the cuboid. cm2 [3]

6

© UCLES 2016 0607/11/M/J/16

9 Write down the next two terms in the sequence.

18, 18, 16, 12, 6, …

, [2]

10 The Venn diagram shows two sets A and B. U = {1, 2, 3, 4, 5, 6, 7, 8, 9}

A

U

B

4

26

8

1

93

7

5

(a) Complete the following.

(i) A = { } [1]

(ii) B′ = { } [1]

(iii) A ∩ B = { } [1] (b) What is the mathematical name given to the numbers in set A? [1]

(c) Circle the statements which are correct for this Venn diagram.

A ∪ B = U 7 ∉ A n(B) = 4 A ∩ B′ = {4} [2]

7


11 110° NOT TO

SCALE

140°

r°

80°

Find the value of r. r = [3]

12 A car travels 100 metres in 8 seconds. Find its speed in kilometres per hour. km/h [2]

13 Describe the single transformation that maps y = f(x) onto y = f(x) + 3.

[2]

14 An archer hits the target with probability 107

.

He takes 50 shots at the target. How many times does he expect to hit the target? [1]

15 Write down all the integers that satisfy the following inequality.

–3 x < 2 [2]

Questions 16 and 17 are printed on the next page.

8

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. To avoid the issue of disclosure of answer-related information to candidates, all copyright acknowledgements are reproduced online in the Cambridge International Examinations Copyright Acknowledgements Booklet. This is produced for each series of examinations and is freely available to download at www.cie.org.uk after the live examination series. 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 2016 0607/11/M/J/16

16 (a) Factorise. (i) 3x + 6 [1]

(ii) p2 + pq [1]

(b) Expand the brackets and simplify. x – 3(2x – 7) [2]

17 Solve the following simultaneous equations. 2x + y = 8 3x + 2y = 12

x =

y = [3]

This document consists of 11 printed pages and 1 blank page.

DC (LK/AR) 115879/3© UCLES 2016 [Turn over

Cambridge International ExaminationsCambridge International General Certificate of Secondary Education

*9989739186*

CAMBRIDGE INTERNATIONAL MATHEMATICS 0607/21Paper 2 (Extended) May/June 2016 45 minutesCandidates answer on the Question Paper.

Additional Materials: 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.Do not use staples, paper clips, glue or correction fluid.You may use an HB pencil for any diagrams or graphs.DO NOT WRITE IN ANY BARCODES.

Answer all the questions.CALCULATORS MUST NOT BE USED IN THIS PAPER.All answers should be given in their simplest form.You must show all the relevant working to gain full marks and you will be given marks for correct methods even if your answer is incorrect.The number of marks is given in brackets [ ] at the end of each question or part question.The total number of marks for this paper is 40.

2

0607/21/M/J/16© UCLES 2016

Formula List

For the equation ax bx c 02 + + = x a

b b ac2

42!

=- -

Curved surface area, A, of cylinder of radius r, height h. rA rh2=

Curved surface area, A, of cone of radius r, sloping edge l. rA rl=

Curved surface area, A, of sphere of radius r. rA r4 2=

Volume, V, of pyramid, base area A, height h. V Ah3

1=

Volume, V, of cylinder of radius r, height h. rV r h2=

Volume, V, of cone of radius r, height h. rV r h3

1 2=

Volume, V, of sphere of radius r. rV r3

4 3=

sin sin sinAa

Bb

Cc

= =

cosa b c bc A22 2 2= + -

sinbc A2

1Area =

A

CB

c b

a

3

0607/21/M/J/16© UCLES 2016 [Turn over

Answer all the questions.

1 Work out.

(a) .0 04

8

................................................. [1]

(b) 5

4

4

1-

�� [2]

2 (a) Shade two more squares so that this shape has exactly one line of symmetry.

[1]

(b) Shade two more triangles so that this shape has rotational symmetry of order 3.

[1]

4

0607/21/M/J/16© UCLES 2016

3 By rounding each number to 1 significant figure, estimate the value of this calculation. Show all your working.

. .

.

52 3 99 6

11 37 289#

+

................................................. [2]

4 a 2 3 75 2 3# #= b 2 3 5

3 4# #=

Leaving your answer as the product of prime factors, find

(a)� b2,

................................................. [1]

(b) the highest common factor (HCF) of�a and b,.

................................................. [1]

(c) the lowest common multiple (LCM) of a�and b.

�� [2]

5


5 Luis has a large jar containing red, yellow, green and blue beads. He takes a bead at random from the jar, notes its colour and replaces it. He repeats this 200 times.

The table shows his results.

Colour Red Yellow Green Blue

Number of beads 26 72 64 38

Relative frequency

(a) Complete the table to show the relative frequencies. [2]

(b) (i) There are 5000 beads in the jar altogether.

Estimate the number of green beads in the jar.

................................................. [1]

(ii) Explain why this is a good estimate.

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

...................................................................................................................................................... [1]

6 Solve.

x x2 3

12-

+=

................................................. [3]

6

0607/21/M/J/16© UCLES 2016

7 U = {Integers from 1 to 18} F = {Factors of 12} M = {Multiples of 3} E = {Even numbers}

(a) Complete the Venn diagram by putting the numbers 2, 3, 4, 8, 12, 15 and 18 in the correct subsets.

F1

6

9

1014 16

E

5 13

7 11 17

M

U

[2]

(b) List the members of

(i) ( )E F M, , l,

................................................. [1]

(ii) E M F+ +l l.

................................................. [1]

8 Solve. ( )x x2 3 2 3 12+ -

................................................. [3]

7


9

NOT TO SCALE

130°O

C

D

A

B

A, B, C and D are points on the circle centre O. Angle BOD = 130°.

(a) Find angle DCB.

Angle DCB�=� ................................................ [1]

(b) Find angle BAD.

Angle BAD�=� ................................................ [1]

8

0607/21/M/J/16© UCLES 2016

10 Factorise completely.

(a) 12x2 – 27xy

................................................. [2]

(b) 4a2 + 8ab – ac – 2bc

�� [2]

11 Rationalise the denominator.

7

1

�� [1]

9


12

q

r

b

a

p

Write the vectors p, q and r in terms of a and b .

p = ��

q = ................................................

r = �� [3]

10

0607/21/M/J/16© UCLES 2016

13

30 60–30–60 90 120 150 180 210 240 270 300 330

2

–2

y

x°0

The graph of y�=�asin (x + b)° is shown in the diagram. Find the value of a and the value of b.

a = ................................................

b = ................................................ [2]

11

0607/21/M/J/16© UCLES 2016

14

O P

Q

y

x

NOT TO SCALE

The diagram shows a sketch of the graph of y�=�ax2�+�bx� O is the point (0, 0), P�is the point (4, 0) and�Q is the point (8, 96).

Find the value of a and the value of b.

a = ................................................

� b�=�� [3]

12

0607/21/M/J/16© UCLES 2016

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.

To avoid the issue of disclosure of answer-related information to candidates, all copyright acknowledgements are reproduced online in the Cambridge International Examinations Copyright Acknowledgements Booklet. This is produced for each series of examinations and is freely available to download at www.cie.org.uk after the live examination series.

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.

bLank page


DC (LK/CGW) 115323/3© UCLES 2016 [Turn over


*0011989208*



1 hour 45 minutes


Additional Materials: Geometrical Instruments Graphics Calculator



Answer all the questions.Unless instructed otherwise, give your answers exactly or correct to three significant figures as appropriate.Answers in degrees should be given to one decimal place.For r, use your calculator value.You must show all the relevant working to gain full marks and you will be given marks for correct methods, including sketches, even if your answer is incorrect.The number of marks is given in brackets [ ] at the end of each question or part question.The total number of marks for this paper is 96.

2

0607/31/M/J/16© UCLES 2016

Formula List

Area, A, of triangle, base b, height h. A = bh2

1

Area, A, of circle, radius r. A = rr2

Circumference, C, of circle, radius r. C = 2rr

Curved surface area, A, of cylinder of radius r, height h. A = 2rrh

Curved surface area, A, of cone of radius r, sloping edge l. A = rrl

Curved surface area, A, of sphere of radius r. A = 4rr2

Volume, V, of prism, cross-sectional area A, length l. V = Al

Volume, V, of pyramid, base area A, height h. V= Ah3

1

Volume, V, of cylinder of radius r, height h. V = rr2h

Volume, V, of cone of radius r, height h. V = r h3

1 2r

Volume, V, of sphere of radius r. V = r3

4 3r

3



1 (a) Write 356.31

(i) correct to 1 decimal place,

.................................................................. [1]

(ii) correct to 2 significant figures,

.................................................................. [1]

(iii) correct to the nearest 100,

.................................................................. [1]

(iv) in standard form.

.................................................................. [1]

(b) (i) Calculate . .16 8 9 612 - .

Write down all the figures shown on your calculator, giving your answer as a decimal.

.................................................................. [1]

(ii) Myrto estimates that the answer to part (b)(i) is 300.

(a) Find the difference between Myrto’s estimate and your answer to part (b)(i).

.................................................................. [1]

(b) Write this difference as a percentage of your answer to part (b)(i).

............................................................ % [1]

4

0607/31/M/J/16© UCLES 2016

2 (a) Write 4 × 4 × 4 × 4 × 4 × 4

(i) as a power of 4,

.................................................................. [1]

(ii) as an integer.

.................................................................. [1]

(b) Find the value of

(i) 4 44 2+ ,

.................................................................. [1]

(ii) 4 44 0- .

.................................................................. [1]

(c) Write 4

42

10

as a power of 4.

.................................................................. [1]

5


3 Tingwei buys 2 kg of cheese. The cheese costs $13.50 for one kilogram.

(a) Work out how much Tingwei pays for the 2 kg of cheese.

$ ................................................................. [1]

(b) He uses all the cheese to make 200 cheese balls.

Find the mass, in grams, of one cheese ball.

............................................................. g [1]

(c) (i) He sells all these cheese balls at a school fair for $0.25 each.

Work out how much money he received.

$ ................................................................. [1]

(ii) The profit goes to the school charity.

Work out how much money goes to the school charity.

$ ................................................................. [1] (d) The school fair makes a total profit of $460.

Write the profit that Tingwei made as a fraction of $460. Give your answer in its simplest form.

.................................................................. [2]

6

0607/31/M/J/16© UCLES 2016

4 The number of strawberries in each of 20 boxes is listed below.

32 28 27 32 33 28 34 28 29 29

28 28 33 31 33 33 30 29 29 26

(a) Complete the frequency table.

Number of strawberries 26 27 28 29 30 31 32 33 34

Frequency 1 1 1

[2]

(b) Find

(i) the range,

.................................................................. [1]

(ii) the mode,

.................................................................. [1]

(iii) the median,

.................................................................. [1]

(iv) the mean.

.................................................................. [1]

(c) One of these boxes of strawberries is chosen at random.

Find the probability that it contains

(i) exactly 33 strawberries,

.................................................................. [1]

(ii) fewer than 30 strawberries.

.................................................................. [1]

7


5 (a) A B C D5 22

1= - -

(i) Find the value of A when B = 2, C = 3 and D = 6.

.................................................................. [2]

(ii) Find the value of B when A = 12, C = 1 and D = 4.

.................................................................. [3]

(b) Find the value of 7p − 4q when p = −3 and q = –2.

.................................................................. [2]

(c) Rearrange 2y = 3x − 9 to make x the subject.

x = ................................................................. [2]

(d) The mass of 1 pomegranate and 2 kiwi fruit is 480 g. The mass of 1 pomegranate and 6 kiwi fruit is 840 g.

Find the mass of 1 pomegranate and the mass of 1 kiwi fruit. Show all your working.

1 pomegranate = ............................. g

1 kiwi fruit = ............................. g [4]

8

0607/31/M/J/16© UCLES 2016

6 30 people were asked where they were going on holiday. The results are to be shown in a pie chart.

Country India Spain South Africa United States Australia

Number of people 5 12 3 6 4

Sector angle 60° 48°

(a) Calculate the sector angle for Spain.

.................................................................. [2]

(b) Complete the pie chart. Label each sector.

AustraliaIndia

[3]

9


7 (a)

F

E

A

C

B105° NOT TO

SCALE

DG

H

AFB and CGD are parallel lines. EFGH is a straight line and angle AFE = 105°.

Find

(i) angle EFB,

Angle EFB = .................................. [1]

(ii) angle CGF.

Angle CGF = .................................. [1]

(b)

NOT TOSCALE

A

O

D

C B

r°

s°

p°

70°

q°

AOB and COD are diameters of a circle, centre O. The lines AD and CB are parallel and angle CAB = 70°.

Find the values of p, q, r and s.

p = ................................

q = ................................

r = ................................

s = ................................ [4]

10

0607/31/M/J/16© UCLES 2016

8

NOT TOSCALE

2.5 km

40°

H

C SB

The diagram shows four straight cycle tracks HB, HC, BC and CS. BC = CS and HC = 2.5 km. Angle HBC = 90° and angle BHC = 40°.

(a) Abimela cycles from home, H, to school, S, each day along cycle track HC and CS.

(i) Use trigonometry to find the distance BC.

.......................................................... km [2]

(ii) Find the distance Abimela cycles to school.

.......................................................... km [1]

(b) One day track HC is blocked and she has to cycle along tracks HB, BC and CS.

Find the distance HB.

.......................................................... km [2]

(c) Find the extra distance that Abimela now has to cycle to school.

.......................................................... km [1]

11


9

1

0

2

1 2 3 4 5 6 7 8

3

4

5

6

7

8

9

y

x

(a) On the grid, plot the points A(2, 3) and B(5, 7). Draw the line AB. [2]

(b) Write down the co-ordinates of the midpoint of AB.

( .................... , .................... ) [1]

(c) Find the gradient of AB.

.................................................................. [2]

(d) Find the equation of the line parallel to AB that passes through the point (0, 4).

.................................................................. [2]

12

0607/31/M/J/16© UCLES 2016

10

15 cm

12 cm

4 cm

NOT TOSCALE

The diagram shows 12 solid cylinders packed into a box. Each cylinder has radius 1 cm and length 15 cm.

(a) (i) Find the volume of one cylinder.

......................................................... cm3 [1]

(ii) Work out the volume of 12 cylinders.

......................................................... cm3 [1]

(b) The box measures 15 cm by 12 cm by 4 cm.

Find the volume of the box.

......................................................... cm3 [1]

(c) Find the volume of the box not taken up by the cylinders.

......................................................... cm3 [1]

(d) Write your answer to part (c) as a percentage of the total volume of the box.

............................................................ % [1]

13


11

2

2

−2

−4

−6

4

6

8

10

y

4 6−6 −4 −2 0 8 10x

P

The diagram shows a pentagon, P.

(a) Draw the image of P after a reflection in the y-axis. Label this image Q. [1]

(b) Draw the image of P after a translation by the vector 2

6-

J

LKKN

POO .

Label this image R. [2]

(c) Draw the image of P after an enlargement, scale factor 3, centre (0, 0). Label this image S. [2]

(d) Find the ratio

length of horizontal side of S : length of horizontal side of P.

.................... : .................... [1]

(e) Congruent Regular Similar

Choose a word from the list to complete the statement.

P and S are ……………..……… shapes. [1]

14

0607/31/M/J/16© UCLES 2016

12 The masses of 200 meerkats are recorded in the frequency table.

Mass (x grams) Frequency

x200 3001 G 5

x00 003 41 G 10

x00 004 51 G 26

x00 005 61 G 34

x00 006 71 G 40

x00 007 81 G 62

x00 008 91 G 18

x00 009 101 G 5

Total 200

(a) Write down the modal group.

.................... x1 G .................... [1]

(b) (i) Show that the midpoint of the first group is 250.

[1]

(ii) Find an estimate of the mean mass of these 200 meerkats.

............................................................. g [2]

(c) Complete the cumulative frequency table.

Mass (x grams) Cumulative frequency

x 300G 5

x 400G

x 500G

x 006G

x 007G

x 008G

x 900G 195

x 1000G 200 [2]

15


(d) Complete the cumulative frequency curve.

20

100 200 300 400 500Mass (grams)

600 700 800 900 1000

40

60

80

0

100

120

Cumulativefrequency

140

160

180

200

x

[3]

(e) Use your graph to find

(i) the median,

............................................................. g [1]

(ii) the inter-quartile range,

............................................................. g [2]

(iii) the number of meerkats with a mass of more than 850 g.

.................................................................. [2]

Question 13 is printed on the next page.

16

0607/31/M/J/16© UCLES 2016




13

0−3 3

10

y

x

−10

( )f x x x22= +

(a) On the diagram, sketch the graph of y = f(x) from x = −3 to x = 3. [4]

(b) Write down the equation of the vertical asymptote for this graph.

.................................................................. [1]

(c) Find the co-ordinates of the local minimum point.

( .................... , .................... ) [1]

(d) Write down the number of solutions of y = f(x) when y = 6.

.................................................................. [1]


DC (KN/SG) 115864/3© UCLES 2016 [Turn over


*8398422716*


Paper 4 (Extended) May/June 2016

2 hours 15 minutes


Additional Materials: Geometrical Instruments Graphics Calculator



Answer all the questions.Unless instructed otherwise, give your answers exactly or correct to three significant figures as appropriate.Answers in degrees should be given to one decimal place.For p, use your calculator value.You must show all the relevant working to gain full marks and you will be given marks for correct methods, including sketches, even if your answer is incorrect.The number of marks is given in brackets [ ] at the end of each question or part question.The total number of marks for this paper is 120.

2

0607/41/M/J/16© UCLES 2016

Formula List

For the equation ax bx c 02 + + = x a

b b ac2

42!

=- -

Curved surface area, A, of cylinder of radius r, height h. rA rh2=

Curved surface area, A, of cone of radius r, sloping edge l. rA rl=

Curved surface area, A, of sphere of radius r. rA r4 2=

Volume, V, of pyramid, base area A, height h. V Ah3

1=

Volume, V, of cylinder of radius r, height h. rV r h2=

Volume, V, of cone of radius r, height h. rV r h3

1 2=

Volume, V, of sphere of radius r. rV r3

4 3=

sin sin sinAa

Bb

Cc

= =

cosa b c bc A22 2 2= + -

sinbc A2

1Area =

A

CB

c b

a

3



1 (a) Annelise buys a car that is one year old for $13 600. The value of this car has reduced by 15% of the value when it was new.

(i) Calculate the value of the car when it was new.

$ ................................................................ [3]

(ii) After the first year the car reduces in value by 11% each year for the next 3 years.

Calculate the value of the car after these 3 years.

$ ................................................................ [3]

(b) Boris buys a car for $23 000. The value of this car reduces by 8% each year.

Find the number of complete years it takes for the value of the car to fall below $11 500.

.................................................................. [3]

4

0607/41/M/J/16© UCLES 2016

2 The frequency of a radio wave, f , is inversely proportional to the wavelength, L metres. A radio station broadcasts on a frequency of 93.7 and a wavelength of 3.2 m.

(a) Find a formula for f, in terms of L, writing any constants correct to 3 significant figures.

f = ................................................................ [3]

(b) Chat Radio broadcasts with a wavelength of 2.8 m.

Find the frequency of Chat Radio.

................................................................. [1]

(c) Allsports Radio broadcasts with a frequency of 0.35 .

Find the wavelength of Allsports Radio.

.............................................................. m [2]

5


3

–1

1

0

2

3

4

5

–1 1 2 3 4 5 6 7 8 9 10–2–3–4–5

–2

–3

–4

–5

A

y

x

(a) (i) Draw the image of quadrilateral A after it has been reflected in the y-axis and then rotated through 90° anti-clockwise about the origin. [3]

(ii) Describe fully the single transformation equivalent to reflection in the y-axis followed by rotation 90° anti-clockwise about the origin.

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

..................................................................................................................................................... [2]

(b) (i) Draw the image of quadrilateral A after a stretch, factor 3 with the y-axis invariant. Label the image B. [2]

(ii) Describe fully the single transformation that maps the quadrilateral B back onto quadrilateral A.

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

..................................................................................................................................................... [2]

6

0607/41/M/J/16© UCLES 2016

4

12 cm

40 cm

NOT TOSCALE

The diagram shows a solid trophy for a football tournament. The sphere on the top has a radius of 15 cm. The sphere rests on a cylinder with the same radius as the sphere and height 40 cm. The base is a cylinder with radius 25 cm and height 12 cm.

(a) Calculate the volume of the trophy.

...........................................................cm3 [4]

7


(b) The mass of the trophy is 15 kg. Each member of the winning team receives a model of the trophy made from the same material. The model is similar to the real trophy and one-fifth of the height.

(i) Calculate the total height of each model trophy.

............................................................ cm [1]

(ii) Calculate the mass, in grams, of each model trophy.

............................................................... g [3]

8

0607/41/M/J/16© UCLES 2016

5 In Kim’s game a player looks at a fixed number of objects on a tray for a length of time, t seconds. The player is then tested to find how many objects they remember.

The table shows the results for 10 players.

Time in seconds (t) 30 40 50 60 70 80 90 100 110 120

Number of objects (n) 8 10 15 12 16 20 18 23 19 25

(a) Complete the scatter diagram. The first six points have been plotted for you.

n

26

24

22

20

18

16

14

12

10

8

6

4

2

0 10 20 30 40 50 60 70 80 90 100 110 120t

Numberof objects

Time in seconds [2]

(b) What type of correlation is shown by the scatter diagram?

................................................................. [1]

9


(c) (i) Calculate the mean time.

................................................................s [1]

(ii) Calculate the mean number of objects.

................................................................. [1]

(d) (i) Find the equation of the regression line. Give your answer in the form n mt c= + .

n = ................................................................ [2]

(ii) Errol looks at the tray for 85 seconds.

Use your equation to estimate the number of objects he remembers.

................................................................. [1]

10

0607/41/M/J/16© UCLES 2016

6 (a) These are the first four terms of a sequence.

5 8 11 14

Write down an expression in terms of n for the nth term, sn, of the sequence.

sn = ................................................................ [2]

(b) The nth term, tn, of another sequence is n n2 62 -+ .

Write down the first four terms of this sequence.

.............. , .............. , .............. , .............. [2]

(c) The nth term of a third sequence, un, is given by

u nt

2nn=+

.

Find an expression for un, in terms of n, giving your answer in its simplest form.

un = ................................................................ [3]

(d) The nth term of a fourth sequence is given by s un n+ .

Is 501 a term of this fourth sequence? Give your reasons.

.................................... because ................................................................................................................

............................................................................................................................................................. [2]

11


7

A

30 m 36 m

NOT TOSCALE

B

C D

68°

AB is a vertical tower of height 30 m. BC and BD are straight wires attached to B. A, C and D are on horizontal ground withC due west of D. Angle BCA = 68° and BD = 36 m.

(a) Calculate AD.

AD = ............................................................ m [3]

(b) Calculate AC and show that it rounds to 12.1 m, correct to 3 significant figures.

[3]

(c) Calculate the bearing of A from D.

................................................................. [3]

12

0607/41/M/J/16© UCLES 2016

8 (a)

2

0

–4

–3 3x

y

f ( ) logx x x1 2 2= + +^ h

(i) On the diagram, sketch the graph of f ( )y x= for values of x between –3 and 3. [2]

(ii) Solve f ( )x 0= .

x = .......................... or x = ........................... [2]

(iii) Write down the equation of the asymptote to the graph of f ( )y x= .

................................................................. [1]

13


(b) (i) On this diagram, sketch the graph of logy x2 1= +^ h for values of x between –3 and 3.

2

0

–4

–3 3x

y

[2]

(ii) Describe a similarity between the graphs in part (a)(i) and part (b)(i).

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

..................................................................................................................................................... [1]

(iii) Explain the differences between the graphs in part (a)(i) and part (b)(i).

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

..................................................................................................................................................... [2]

14

0607/41/M/J/16© UCLES 2016

9 Hamish travels from Perth to London by train. During the journey, the train stops in Edinburgh.

(a) The distance from Perth to Edinburgh is 65 km. The train travels at an average speed of 48.75 km/h for this part of the journey.

Find the time taken to travel from Perth to Edinburgh. Give your answer in hours and minutes.

............................... h ....................... min [3]

(b) The average speed for the whole journey from Perth to London is 119.5 km/h. The distance from Edinburgh to London is 632 km.

Find the average speed for the journey from Edinburgh to London.

........................................................ km/h [5]

(c) During the journey, the train travels through a tunnel of length 800 m. The train travels through this tunnel at 120 km/h. The train is 130 m long.

Calculate the time taken for the train to pass completely through the tunnel. Give your answer in seconds.

............................................................... s [3]

15


10 A is the point (–2, –1) and B is the point (6, 3).

(a) Calculate AB .

................................................................. [3]

(b) The point P has co-ordinates (x, y) and PA = PB.

Show that x y2 5+ = .

[5]

(c) If P is also on the line y x= , find the co-ordinates of P.

( ................................, ..............................) [2]

16

0607/41/M/J/16© UCLES 2016

11

60°

(3x – 1) cm

2x cm 9 cm

A

B C

NOT TOSCALE

(a) Use the cosine rule to show that x x7 4 80 02 - - = .

[4]

17


(b) (i) Solve the equation x x7 4 80 02 - - = .

Show all your working.

x = .......................... or x = ........................... [3]

(ii) Find the length of AB and the length of BC.

AB = .............................. cm

BC = .............................. cm [2]

(c) Find the area of triangle ABC.

.......................................................... cm2 [2]

18

0607/41/M/J/16© UCLES 2016

12 The table shows the masses in grams of 200 eggs.

Mass (m grams) 45 1 m G 50 50 1 m G 55 55 1 m G 60 60 1 m G 65 65 1 m G 70 70 1 m G 75 75 1 m G 80

Frequency 5 19 34 58 46 29 9

(a) Calculate an estimate of the mean mass.

.............................................................. g [2]

(b) On the grid, complete the cumulative frequency curve for the information in the table.

200

180

160

140

120

100

80

60

40

20

0

Cumulativefrequency

Mass (grams)40 45 50 55 60 65 70 75 80

m

[5]

19


(c) Use your graph to find

(i) the median mass,

.............................................................. g [1]

(ii) the interquartile range.

.............................................................. g [2]

(d) This table shows how the eggs are graded according to their mass.

Size Small Medium Large Very Large

Mass (m grams) m G 53 53 1 m G 63 63 1 m G 75 m 2 75

(i) An egg is chosen at random from the 200 eggs. Estimate the probability that the egg is Small.

................................................................. [1]

(ii) Two eggs are chosen from the 200 eggs.

Find the probability that both are Very Large.

................................................................. [2]


20

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13 (a) f ( )x x5 2= -

(i) Solve f ( )x1 = 2 .

x = ................................................................ [2]

(ii) Find and simplify f f ( )x^ h.

................................................................. [2]

(iii) Find f –1(x).

f –1(x) = ................................................................ [2]

(b) g(x) is a function with an inverse function g–1(x).

Write down the value of g(g–1(3)).

................................................................. [1]


DC (NF/SG) 117229/2© UCLES 2016 [Turn over

*7235961311*



1 hour


Additional Materials: Graphics Calculator



Answer all the questions.You must show all relevant working to gain full marks for correct methods, including sketches.In this paper you will also be assessed on your ability to provide full reasons and communicate your mathematics clearly and precisely.At the end of the examination, fasten all your work securely together.The total number of marks for this paper is 24.


2

0607/51/M/J/16© UCLES 2016


INVESTIGATION DIVIDINGRECTANGLES

This investigation looks at the connections between the rectangles made by dividing one rectangle into two smaller rectangles.

In this investigation

•• the length of a rectangle is always longer than its width

length

lengthwidth

width

•• the length and width of a rectangle are always a whole number of units•• the scale factor of any enlargement is greater than 1.

1 In the diagrams below, rectangle B is an enlargement of rectangle A.

(a)

B

A

Write down the scale factor of this enlargement.

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

3

0607/51/M/J/16© UCLES 2016 [Turnover

(b)

B

A

Write down the scale factor of this enlargement.

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

(c)

AB

4

NOT TOSCALE

For this pair of rectangles, the scale factor is 10.

Work out the length of rectangle B.

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

(d)

AB

3

1 5

NOT TOSCALE

Work out the length of rectangle B.

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

4

0607/51/M/J/16© UCLES 2016

2 A rectangle is cut into two smaller rectangles, A and B.

A B

When B is an enlargement of A, the original rectangle is called a scale-rectangle.

A B

width

widthlength

length

AB

widthwidth

lengthlength

Example

A 4 by 10 rectangle is cut as shown.

A B4

10 2 8

4

A B

2

44

8

AB

24

48

NOT TOSCALE

48

2length of A

length of B= = and

2

42

width of A

width of B= =

So B is an enlargement of A with scale factor 2.

Thismeansthattherectanglewithdimensions4by10isascale-rectanglewithafactorof2.

5


(a)

A B

The diagram shows a 3 by 10 rectangle. This is a scale-rectangle.

Show that it has a factor of 3.

(b)

BA

The diagram shows a 2 by 4 rectangle.

Is this a scale-rectangle? Write Yes or No and give reasons for your answer.

(c)

A B

3

21

m

AB

3

21m

NOT TOSCALE

The diagram shows a scale-rectangle with a factor of 7.

(i) Find m.

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

(ii) Write down the dimensions of the scale-rectangle.

........................ by ........................

6

0607/51/M/J/16© UCLES 2016

(d)

A B

3

w

75

AB

3

w75

NOT TOSCALE

The diagram shows a scale-rectangle with a factor of 5.

(i) Find w.

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

(ii) Write down the dimensions of the scale-rectangle.

........................ by ........................

3

A By

x z

The diagram shows a scale-rectangle with a factor of n.

(a) When x = 2 and n = 6,

(i) work out y,

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

(ii) find z,

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

(iii) complete this statement with a number,

z = ............ × x

(iv) write down the connection between your answer to part(iii) and the factor, n.

7


(b) When x = 2 and z = 18,

(i) find n,

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

(ii) work out the dimensions of this scale-rectangle.

........................ by ........................

(c) Use your answers to part(a) and part(b) to complete the second and third rows of the table. Complete the remaining rows of the table.

n x y z Dimensions

2 2 4 8 4 by 10

6 2 ...... by ......

2 18 ...... by ......

5 7 ...... by ......

1 4 16 4 by 17

5 20 ...... by ......

Question4isprintedonthenextpage.

8

0607/51/M/J/16© UCLES 2016

4

A By

x z

The diagram shows a scale-rectangle with a factor of n.

(a) Work out the dimensions of this scale-rectangle in terms of n and x.

........................ by ........................

(b) Show that, for any scale-rectangle, its dimensions are in the ratio

width : length = n : n2 + 1.





DC (LK/FD) 115851/4© UCLES 2016 [Turn over


*5654958878*


Paper 6 (Extended) May/June 2016 1 hour 30 minutesCandidates answer on the Question Paper.

Additional Materials: Graphics Calculator



Answer both parts A and B.You must show all the relevant working to gain full marks for correct methods, including sketches.In this paper you will also be assessed on your ability to provide full reasons and communicate your mathematics clearly and precisely.At the end of the examination, fasten all your work securely together.The total number of marks for this paper is 40.

2

0607/61/M/J/16© UCLES 2016

Answer both parts A and B.

A INVESTIGATION MOVING TRIANGLES (20 marks)

You are advised to spend no more than 45 minutes on this part.

Q

PA

R

S

1

B

This investigation is about finding the connection between AP and BQ as P and Q move.All triangles are right-angled and RS is one unit.In questions 1, 2 and 3, P is at A.

1

R

Q

P S B

Triangle PBQ is an enlargement of triangle PSR, with P as the centre of the enlargement.

(a) Write down the scale factor of the enlargement in the diagram.

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

(b) Complete the table for enlarging triangle PSR.

Scale factor Length of PS Length of PB

3 4

6 30

7 14

(c) Use one word to complete this statement.

PSR and PBQ are ........................................ triangles because one is an enlargement of the other.

3


2

P S B

Q

NOT TOSCALExR

2

1

(a) Show that x = 10 when PB = 20.

(b) Find the value of x when PB = 16.

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

(c) Find an expression for x when PB = y.

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

3

P S

R x

1

Q

B

NOT TOSCALE

4y

For this triangle, find an expression for x when PB = y.

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

4

0607/61/M/J/16© UCLES 2016

4 These diagrams show P starting at A and then moving towards B.

(a)

P S B

Q

R

3

1

6NOT TOSCALE

y

In this diagram P is at A.

Find the value of y.

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

(b)

PA S B

Q

R

2

1

6NOT TOSCALE

z

In this diagram, P and S have moved towards B. PS is one unit less than in part (a).

Find the value of z.

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

(c) Using your answers to part (a) and part (b), work out the value of AP.

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

5


5

P S

R

Q

B

x

y5

1

A P S

R

Q

B

x

z4

1

NOT TOSCALE

These diagrams show P starting at A and moving towards B. In the second diagram, PS has decreased by one unit.

Using the method of question 4, show that AP = BQ.

6

0607/61/M/J/16© UCLES 2016

6

P S

R

Q

B

x

yn

1

A P S

R

Q

NOT TOSCALE

B

x

z

1

These diagrams show P starting at A and moving towards B. In the second diagram, PS has decreased by one unit.

Show that AP = BQ.

7


7 In this question, RS is no longer one unit. P starts at A and moves towards B. In the second diagram, PS has decreased by one unit.

P S

R

Q

B

x

yn

PA S

R

Q

NOT TOSCALE

B

x

z

(a) When RS = 2, find an expression for AP in terms of x.

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

(b) When RS = m, find an expression for AP in terms of x and m.

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

8

0607/61/M/J/16© UCLES 2016

B MODELLING MUSICAL NOTES (20 Marks)

You are advised to spend no more than 45 minutes on this part.

This task is about the connection between musical notes.A musical note is made by a sound wave which is modelled by a sine function.

Here is the sine wave for the note A1, where the time, t, is measured in seconds.

1

t0

–1

155

255

Each note has a different frequency, which is measured in Hertz (Hz).The frequency of the note A1 is 55 Hz because the sine wave repeats 55 times per second.

1 The frequency of the note A2 is two times the frequency of the note A1.

On the grid below, sketch the sine wave for the note A2 for t055

2G G .

1

t0

–1

155

255

9


2 The 12 notes in a musical scale are A, A#, B, C, C#, D, D#, E, F, F#, G, G#. The notes on a piano repeat this scale. Notes in the same scale have the same subscript. (For example, A2, C#2 and F2 are all in the same scale.)

The frequency, f Hz, of each note on a piano is modelled by the function

( ) .f n 27 5 2n12#= where n is an integer from 0 to 87.

n 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 87

Note A0 A#0 B0 C0 C#0 D0 D#0 E0 F0 F#0 G0 G#0 A1 A#1 B1

(a) When n = 0 the frequency of the note is 27.5 . This is the note A0.

(i) Work out the frequency of the note when n = 3.

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

(ii) Write down the note when n = 15.

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

(iii) Work out the frequency of the note E0.

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

(b) Write down all the values of n that give the note A on this piano.

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

(c) Which note has the highest frequency on this piano? Calculate this frequency.

Note .................................... Frequency .........................................

10

0607/61/M/J/16© UCLES 2016

3 k times the frequency of a note gives the frequency of the next note. This means that ( ) ( )f fk n n 1= + .

Find the exact value of k.

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

4 (a) On the axes below, sketch the graph of ( ) .g x 27 5 2x12#= for x0 87G G .

5000

87

g (x)

x0

(b) Find the note which has frequency closest to 1400 Hz.

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

11


5 A different musical scale has 10 notes.

Q R S T U V W X Y Z

The frequency of each note is modelled by the function

( )h n a 2bn#= where n is an integer from 0 to 29.

When n = 0 the note is Q0 and the frequency of this note is 600 Hz. The frequency of the note Q1 is 1200 Hz.

(a) Write down the value of a.

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

(b) Find the value of b.

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

(c) Show that ( ) ( )h hk n n 1= + where k is a constant to be found.


12

0607/61/M/J/16© UCLES 2016




6 In musical scales the frequency of the note P1 is two times the frequency of the note P0. (a) A musical scale has 23 notes. The frequency of the first note is 75 Hz.

Work out the frequency of the second note.

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

(b) The first note in another musical scale has a frequency of 100 Hz. The second note has a frequency of 108 Hz.

Find the number of notes in this scale.

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

® IGCSE is the registered trademark of Cambridge International Examinations.


© UCLES 2016 [Turn over




MARK SCHEME

Maximum Mark: 40

Published

This mark scheme is published as an aid to teachers and candidates, to indicate the requirements of the examination. It shows the basis on which Examiners were instructed to award marks. It does not indicate the details of the discussions that took place at an Examiners’ meeting before marking began, which would have considered the acceptability of alternative answers. Mark schemes should be read in conjunction with the question paper and the Principal Examiner Report for Teachers. Cambridge will not enter into discussions about these mark schemes. Cambridge is publishing the mark schemes for the May/June 2016 series for most Cambridge IGCSE

®,

Cambridge International A and AS Level components and some Cambridge O Level components.

Page 2 Mark Scheme Syllabus Paper

Cambridge IGCSE – May/June 2016 0607 11

© Cambridge International Examinations 2016

Abbreviations

awrt answers which round to

cao correct answer only

dep dependent

FT follow through after error

isw ignore subsequent working

oe or equivalent

SC Special Case

nfww not from wrong working

soi seen or implied

Question Answer Marks Part marks

1 correct shading 1

2

Sector only correctly drawn.

Chord only correctly drawn.

1

1

Do not allow diameter

Allow diameter

3 1, 3, 7, 21 cao 2 B1 for 3 and 7 and no others

or for 3 factors and no wrong numbers

or for all 4 factors and one incorrect

4 (a) 48 1

(b) 14 1

5 29 1

6 30, 10 2 M1 for 40 ÷ 4 or better

7 Rectangle, Rhombus 2 B1 for one correct and only one incorrect

or for both correct and only one incorrect

8 160 3 M2 for 2 (2 × 5 + 5 × 10 + 2 × 10) oe

or M1 for areas of any two faces

9 –2, –12 2 B1 for –2 as first term in answer

If zero scored, SC1 for reverse order

10 (a) (i) {1, 4, 9} 1

(ii) {2, 4, 6, 8} 1

(iii) {1, 9} 1

(b) Square [numbers] 1

(c) 7 ∉ A

A ∩ B ′ = {4}

1

1




Question Answer Marks Part marks

11 120 3 M2 for 180 × 3 – (90 + 110 + 140 + 80) oe

or B1 for 180 × 3 oe

and M1 – (90 + 110 + 140 + 80) oe

12

45

2 M1 for 100

1000

or 8

3600 oe

If zero scored, SC1 for100

8

13 Translation

0

3

B1

B1

Accept 3 up or 3 in positive y-direction

14 35 1

15 –3, –2, –1, 0, 1 2 B1 for any 3 or 4 correct in the range –3 to 1

If zero scored, SC1 for –3, –2, –1, 0, 1, 2

16 (a) (i) 3(x + 2) cao final answer 1

(ii) p(p + q) cao final answer 1

(b) 21 – 5x cao final answer 2 M1 for –6x + 21 seen

17 Correct method to eliminate one

variable

[x =] 4

[y =] 0

M1

A1

A1

Dependant on the coefficients being the same

for one of the variables

Correct consistent use of addition or

subtraction

If zero scored, SC1 for

correct substitution and evaluation to find

other variable,

or if no working shown, but 2 correct

answers







MARK SCHEME

Maximum Mark: 40

Published


®,





Abbreviations



dep dependent



oe or equivalent

SC Special Case


soi seen or implied

Question Answer Mark Part Marks

1 (a) 200 1

(b) 11

20 oe

2 M1 for 20

a

– 20

b with a = 16 or b = 5

2 (a)

1

(b)

1

3 10 300

50 100

×

+

20

M1

A1

Accept any 3 from 4

4 (a) 26 × 38 × 52 1

(b) 23 × 32 1

(c) 25 × 34 × 5[1] × 73 2 B1 for 3 of 4 factors correct

5 (a) 0.13, 0.36, 0.32, 0.19 oe 2 B1 for 2 or 3 correct

(b) (i) 1600 1

(ii) Sufficient trials oe 1

6 x = 14 3 M2 for 3x – 2x – 2 = 12

or M1 for 3 2( 1)

6

x x− + = 2 or better





7 (a)

2

B1 for 1 or 2 numbers omitted or misplaced

(b) (i) 5, 7, 11, 13, 17 1FT

(ii) 8, 10, 14, 16 1FT

8 x < 1.25 oe 3 With no wrong working seen

M1 for 2x + 3 > 6x – 2

M1FT for 3 + 2 > 6x – 2x oe

M1FT for b

xa

< from ax < b oe

9 (a) 65 1

(b) 115 1FT FT 180 – their (a)

10 (a) 3x(4x – 9y) final answer 2

B1 for ( )23 4 9x xy− or ( )12 27x x y−

(b) (a + 2b)(4a – c) final answer 2 B1 for ( ) ( )4 2 2a a b c a b+ − +

or ( ) ( )4 2 4a a c b a c− + −

11 7

7

1

12 p = a + b oe

q = 2a + b oe

r = –2a + b oe

3 B1 for each

13 a = 2

b = 30

1

1





14 [a =] 3

[b =] –12

3 M1 for kx(x – 4)

M1 for substituting (8, 96) or b = –4a soi

OR

M1 for 20 4 4a b= + or b = –4a soi

M1 for 296 8 8a b= +

OR

M1 for ( )( )2

[ ] 2 4y a x= − −

M1 for substituting (8, 96) or b = –4a soi

If zero scored, SC1 for a = 3, or b = –12







MARK SCHEME

Maximum Mark: 96

Published


®,





Abbreviations



dep dependent



oe or equivalent

SC Special Case


soi seen or implied


1 (a) (i) 356.3 1

(ii) 360 1

(iii) 400 1

(iv) 3.56[31] × 102 1

(b) (i) 279.14 1

(ii) (a) 20.86 1FT FT 300 – their (b)(i)

(b) 7.47 or 7.472 to 7.473 1FT FT their (b)(ii) ÷ their (b)(i) × 100

2 (a) (i) 46 1

(ii) 4096 1

(b) (i) 272 1

(ii) 255 1

(c) 48 1

3 (a) 27 1

(b) 10 1

(c) (i) 50 1

(ii) 23 1 FT FT their 50 – their 27

(d) 1

20

2 B1 FT for 23

460

their





4 (a)

26 27 28 29 30 31 32 33 34

1 1 5 4 1 1 2 4 1

2

B1 for 4 correct entries

(b) (i) 8 1

(ii) 28 1

(iii) 29 1

(iv) 30 1

(c) (i) 4

20 oe isw

1FT FT 4

20

their

(ii) 11

20 oe isw

1FT FT 2 5 4

20

+ +their their

5 (a) (i) 1

2 M1 for 1

25 2 2 3 6× − × − × or better

(ii) 3.2 3 M2 for 5B = 12 + 2 + 2 or better (Allow 1

sign error e.g. –5B)

or M1 for ( )1

12 5 2(1) 42

= − −B or better

(b) −13 2 M1 for 7 × −3 − 4 ×−2 or better

(c) 2 9

3

+y oe final answer

2 M1 for correct first step

(d) 6 kiwi – 2 kiwi = 840 – 480 oe

kiwi =90

pomegranate + 2 × their 90 = 480 oe

pomegranate = 300

M1

A1

M1

A1 FT

OR

M1 for setting up two equations

M1 for eliminating one variable

A1 kiwi = 90

A1 pomegranate = 300

second A1 is FT

If no working shown SC1 for both

answers correct

6 (a)

144

2 M1 for [ ]12

36030

× seen or 48 × 3 or

6012

5×

(b) Fully correct answer 3 B2 for correct sectors but no labels

or B1 for 1 correct sector

or B1for correct 3 labels according to size





7 (a) (i) 75 1

(ii) 105 1

(b) [p = ] 70 1

[q = ] 20 1

[r = ] 20 1FT FT their q or 90 − their p

[s = ] 140 1FT FT 70 + their p or 180 − 2 × their r

8 (a) (i)

1.61 or 1.606 to 1.607

2 M1 for sin 402.5

=

BC or better

(ii) 4.11 or 4.106 to 4.107 1FT FT 2.5 + their (a)(i)

(b) 1.92 or 1.915… 2 M1 for cos 402.5

HB= or better

or M1 for 2.52 – their 1.612

(c) 1.02 or 1.016 or 1.02 to 1.03 1FT FT 2 × their (a)(i) + their (b) −

their (a)(ii)

9 (a) Correct points plotted

(2, 3) and (5, 7)

2 B1 for each correct point

(b) (3.5, 5) 1

(c) 4

3

2 M1 for rise

run

or B1 for 1.3

(d) 4

43

y x= + oe final answer

2 FT

FT (c) 4y their x= + oe

B1 for 4

3y their x k= + or 4= +y kx

10 (a) (i) 47.1 or 47.12 to 47.13 1

(ii) 565 to 566 1 FT FT their (a)(i) × 12

(b) 720 1

(c) 154 to 155 1 FT FT their (b) − their (a)(ii)

(d) 21.39 to 21.53 1 FT FT their (c) ÷ their (b) × 100





11 (a) (0, 2), (–1, 1), (–2, 1), (–3, 2), (–2, 3) 1

(b)

(2, –4), (3, –5), (4, –5), (5, –4), (4, –3)

2 B1 for translation of 6

k −

or 2

k

or B1 for 6

2

−

(c) (0, 6), (3, 3), (6, 3), (9, 6), (6, 9)

2 B1 for any enlargement centre (0, 0)

or correct shape, wrong position

(d) 3 : 1 1

(e) similar 1

12 (a) [ ]700 800x <� 1

(b) (i) ( )

[ ]200 300

2502

+

= oe

1

(ii) 638.5 2 M1 for multiplying midpoints by

frequencies (and adding) – implied by

127700

(c)

x < 300 5

x < 400 15

x < 500 41

x < 600 75

x < 700 115

x < 800 177

x < 900 195

x < 1000 200

2 B1FT for 2 correct entries

(d) Fully correct curve or ruled polygon

3FT FT only if increasing

B2FT for their 4 or 5 points plotted

correctly

or B1FT for their 3 points plotted

correctly





(e) (i) 662 (660 to 680) 1FT FT as long as it is an increasing curve

(ii) 230 (230 to 260) 2FT B1 for one correct quartile seen

( )756 5 or 526 5± ±

FT as long as it is an increasing curve

(iii) 12 (8 to 16) 2FT B1 for 188 ± 4 seen

or M1 for clear method seen on graph

FT as long as it is an increasing curve

13 (a) Fully correct sketch 4

B1 for minimum in first quadrant

B1 for crossing x-axis approximately

between –1 and –2

B1 for not crossing y-axis

B1 for correct overall shape

(b) x = 0 1

(c) (1, 3) 1

(d) 3 1FT FT their graph







MARK SCHEME

Maximum Mark: 120

Published


®,





Abbreviations



dep dependent



oe or equivalent

SC Special Case


soi seen or implied


1 (a) (i) 16 000 3 M2 for 13600 ÷ 0.85 oe

or M1 for 13600 = 85%

(ii) 9590 or 9587 to 9588

3 M2 for 13600 × 0.893 oe

or M1 for 13600 × 0.89k , k > 1 oe

(b) 9 years nfww 3

M2 for

11500log

23000

log0.92

or 23 000 × 0.92n = 11 500 and appropriate

sketch or at least 2 valid trials

or M1 for 23 000 × 0.92n [= 11500]

If 0 scored SC2 for 8 nfww or 8.3(1295..) nfww

2 (a) 300

oeL

3 M1 for k

fL

= soi oe

M1 (Dep on 1st M1)for substituting f = 93.7 and L

= 3.2 soi by 299.8 or 299.84

(b) 107 or 107.0 to 107.1 …

1FT FT their k

L oe only

(c) 857 or 856.5 to 857.1 … 2FT FT their k

L oe only

M1 for 0.35 = their k

L

3 (a) (i) Quadrilateral drawn at

(–1, –1), (–1, –2), (–3, –1), (–3, –3)

3 M2 for 3 pts correct

or M1 for correct reflection of A in y-axis

(ii) Reflection

y = –x oe

1

1

(b) (i) Quadrilateral drawn at

(3, 1), (6, 1), (3, 3), (9, 3)

2 B1 for any stretch with y-axis invariant or with

stretch factor 3

(ii) Stretch, y-axis oe invariant

(stretch factor) 1

3

2 B1 for any 2 correct





4 (a) 66 000 or 65 970 to 65 982 4 M1 for 4

3 × π × 153

M1 for π × 152 × 40

M1 for π × 252 × 12

(b) (i) 16.4 1

(ii) 120 3 M2 for 15000 ÷ 53 oe

or M1 for 53 or ( 1

5)3 seen

5 (a) 4 points plotted correctly 2 B1 for 2 or 3 correct

(b) Positive 1 Ignore comments on strength

(c) (i) 75 1

(ii) 16.6 1

(d) (i) 0.168t + 3.96 2 or m = 0.1684 to 0.1685, c = 3.963 to 3.964

B1 for n = mt + c with either m or c correct

or SC1 for 0.17t + 4[.0]

(ii) 18 1FT FT from their equation with t = 85, answer

rounded or truncated to nearest whole number

6 (a) 3n + 2 oe final answer 2 B1 for 3n + k or kn + 2 oe

(b) –3, 4, 15, 30 2 B1 for 2 or 3 correct in correct place or –6, –3, 4,

15

(c) 2n – 3 oe final answer 3 M2 for (2n –3)( n + 2)

or SC1 for (2n + a)(n + b) where ab = –6 or a +

2b = 1

OR

B1 for –1, 1, 3, 5

B1 for answer 2n + k or kn – 3

(d) No and e.g. 502 not a multiple of 5 oe

nfww

2 Dep on 5n – 1

M1dep for their (3n + 2) + their (2n –3) = 501 oe

Dependent on (a) and (c) linear

7 (a) 19.9 or 19.89 to 19.90

3 M2 for 36² – 30² soi by 396

or M1 for AD² + 30² = 36² oe

(b) 30 ÷ tan 68 oe

12.12...

M2

A1

M1 for 30

tan68AC

= oe

(c) 301 or 301.3 to 301.4

or 239 or 238.6 to 238.7

3 B2 for 31.3 or 31.30 to 31.35

or M1 for tan = 12.1 ÷ their (a) oe





8 (a) (i) Correct sketch f(x)=log(1 + 2x + x^2)

-2 -1 1 2 3

-4

-2

2

x

y

2

B1 RH branch through (0, 0) ,with asymptote

x = a (–ve a)

B1 for LH branch symmetrical, with asymptote

x = a (–ve a)

(ii) –2

0

1

1

(iii) x = –1 1

(b) (i) Correct sketch f(x)=2log(1+x)

-3 3

-4

-2

2

x

y

2

B1 for correct shape

(ii) Same right hand branch 1

(iii) e.g.

log(1 + 2x + x2) = 2 log(1 + x)

No log of a negative number

1

1

Independent

9 (a) 1 hour 20 minutes cao 3 M1 for 65 ÷ 48.75

M1 for correctly converting their time in hours to

hours and mins

(b) 140 or 140.4 to 140.5 5

M1 for 632 + 65 [km] soi by 697

M1 for their 697 ÷ 119.5 soi by 5.83...

M1 for subtracting their 1.33...(from (a))

M1 for 632 ÷ (their 4.4993)

(c) 27.9

3 M2 for 800 130

1000120

60 60

+

×

×

oe

or M1 for distance ÷ speed





10 (a) 8.94 or 8.944... or 4 5 3 M2 for 8² + 4²

M1 for 8 and 4 seen

(b) Gradient of AB = 1

2 oe

Gradient of perpendicular = –2 oe

y = (their–2)x + c

midpoint (2, 1)

Substitute (2, 1) to reach c = 5

OR

(x + 2)² + (y + 1)² oe

(x – 6)² + (y – 3)² oe

equating above two expressions

3 correct expansions

correct completion with no errors or

omissions

1

1FT

M1

B1

A1

B1

B1

M1

B1

A1

May be on diagram

(c) 5 5,

3 3

oe

2

M1 for x + 2x = 5 oe

11 (a) 9² = (3x – 1)² + (2x)²

–2(2x)(3x – 1) cos 60 oe

81 = 9x² – 6x + 1 + 4x² – 6x² + 2x oe

2

7 4 80 0x x− − =

M1

A2

A1

or B1 for 9x² – 3x – 3x + 1

Completion with no errors or omissions

(b) (i) ( ) ( ) ( )2

4 4 4 7 80oe

2 7

− − ± − − × × −

×

x = 3.68 or 3.678...

or –3.11 or – 3.107 to –3.106

M1

B2

or sketch of quadratic graph (any relevant one)

with 1 positive root and 1 negative root

B1 for either

(ii) [AB =] 7.36 or 7.356 to 7.357

[BC =] 10[.0] or 10.03 to 10.04

1FT

1FT

FT 2 × a positive root

FT 3 × a positive root – 1

(c) 31.9 or 32[.0] or 31.85 to 32[.00] 2FT M1 for 1

2 × their AB × their BC sin 60 oe





12 (a) 63.6 2 M1 for midpoints (47.5, 52.5, 57.5, 62.5, 67.5,

72.5, 77.5) soi

(b) Correct Curve

5 B4 for 5 points correct and joined or for 6 points

correct

or B3 for at least 3 correct points

or B2 for all correct cfs 5, 24, 58, 116, 162, 191,

200 seen

or B1 for at least 3 correct cfs or for increasing

curve with 6 points plotted at upper bounds

If 0 or 1 or 2 scored, SC3 for all points correct

but consistently translated to mid-interval or

lower bound.

(c) (i) 63 to 64 1 Dependent on increasing curve

(ii)

8.5 to 10.5 2 B1 for[l.qtile. =] 58.5 to 59.5 or [u.qtile. = ] 68

to 69

Dependent on increasing curve

(d) (i) 12 to 16

200oe

1FT

FT (their 'read off' at 53)/200 dep on increasing

cfs

(ii) 72

39800 oe 2 M1 for

200

k ×

-1

199

k where k = 8, 9 or 10

13 (a) (i) 2.25 oe 2 M1 for 1 = 2(5 – 2x) or 5 – 2x =1

2 oe

(ii) –5 + 4x final answer 2 B1 for 5 – 2(5 – 2x)

(iii) 5 – x oe final answer

2

2 M1 for 2x = 5 – y or x = 5 – 2y or 5

2 2

yx= −

3 1







MARK SCHEME

Maximum Mark: 24

Published


®,





Abbreviations



dep dependent



oe or equivalent

SC Special Case


soi seen or implied


1 (a) 3 1

(b) 2 1

(c) 40 1

(d) 15 1 C opportunity

2 (a) 9[ 3]

3= and

3[ 3]

1= oe seen

1

(b) 3

2 or 1.5 and

2

1 or 2 oe and No oe

1

(c) (i) 147 1 C opportunity

(ii) 21 by 150 or 150 by 21 1 FT their(i)

(d) (i) 15 1 C opportunity

(ii) 15 by 78 or 78 by 15 1 FT their(i)

3 (a) (i) 12 1 C opportunity

(ii) 72 1 C opportunity

(iii) 36 1 FT (ii)

2

their

(iv) n2 oe 1

(b) (i) 3 1 C opportunity

(ii) 6 by 20 or 20 by 6 1 C opportunity





(c)

n x y z Dimensions

2 2 4 8 4 by 10

6 2 their 12 their 72 12* by 74*

their 3 2 their 6 18 their y by 20

5 7 35 175 35* by 182*

4 1 4 16 4 by 17

2 5 10 20 10* by 25

3 3 for all 8 cells

*FT their y by (their z + 2)

*FT their y by (their z + 7)

*FT their y by 25

B2 for 6 or 7 cells correct

or

B1 for 4 or 5 cells correct

4 (a) nx [by] n2x + x oe 2 B1 for each

C opportunity

(b) 2: ( 1)nx n x+ oe seen

1

Communication seen in at least 3 of 1(d), 2(c)(i), 2(d)(i), 3(a)(i), 3(a)(ii),

3(b)(i), 3(b)(ii) or 4(a)

2 C1 if seen in 2 of these







MARK SCHEME

Maximum Mark: 40

Published


®,





Abbreviations


dep dependent



oe or equivalent

SC Special Case


soi seen or implied

A INVESTIGATION MOVING TRIANGLES


1 (a) 2 1

(b)

Scale

factor PS PB

3 4 12

5 6 30

7 2 14

3

B1 for each one correct

(c) Similar 1

2 (a) 2 1

20 10= oe

1

Allow, for example,

2 : 20 = 1 : 10 or

2 : 1 = 20 : 10 or

2 × 10 = 20 and 1 × 10 = 10 or

2: 20 and 1 : x so 2x = 20, x = 10 or

PS is double RS so PB is double QB

or equivalent

(b) 8 1 C opportunity

(c) 2

y oe 1 condone

2

y× 1

3 4

y oe

1 condone 4

y× 1

If 0 scored in 2(c) and 3, allow SC1

for answers of y = 2x and

y = 4x





4 (a) 18 1 C opportunity

(b) 12 1 C opportunity

(c) their 6 1FT strict FT their y − their z

5 [y =] 5x and [z =] 4x

[AP =] 5x – 4x = x

M1

A1

may be on diagram

Allow 2 marks for

y = 5x and z = 4x seen or clearly

indicated

[AP =] y – z = x

6 [AP =] nx – (n – 1)x = x 1 or nx – (nx – x) = x

or nx – nx + x = x

not from wrong working

or equating expressions for BQ

1

y z

n n=

−

and rearranging to show

that either y

y zn

− = with y

xn

= or

that 1

zy z

n− =

−

with 1

z

x

n

=

−

C opportunity

7 (a) 2

x

2 M1 for 1

2xn and

1( 1)

2x n − oe seen

or for x = 2AP

(b) x

m

1

C opportunity

Communication seen in 3 of 2(b), 4(a), 4(b), 6 or 7(b) 2 C1 if seen in two of them




B MODELLING MUSICAL NOTES


1 Correct curve over full domain.

2

B1 for at least one correct, complete

cycle e.g. over the domain

10

110t� �

or for a graph of incorrect shape but

that has 4 cycles over the full domain

or for a graph with more than 3

inaccurate t-intercepts with 4 cycles

over the full domain

or for a fully correct and accurate

sketch graph of the sine wave for the

note A0

2 (a) (i) 32.7[0] or 32.703 to 32.7032 isw 1

(ii) C1 1

(iii) 41.2[0] or 41.203 to 41.2035 isw 1 C opportunity

(b) [0, 12,] 24, 36, 48, 60, 72, 84 1

(c) C7 and 4190 or 4186 or 4186.0 or 4186.00

or 4186.009 to 4186.01

1

3 1

122 or exact equivalent

1

isw conversion to decimal, but decimal

answer only does not score

C opportunity

4 (a) Correct exponential shape 1 Intent of smooth curve;

must not cross x-axis;

condone graph not drawn on full

domain;

condone y-intercept at origin;

(b) F5 2 M1 for n = 68 soi e.g. f(68) or 68

1227.5 2×

C opportunity

-1

1

t

0

1

55

2

55





5 (a) 600 1

(b) 1

10 oe isw

1

(c) Uses an algebraic process to find either 1

10h( 1) 2 h( )their

n n+ = × oe

or k = 1

102their

or 1.07 or 1.071 to 1.072

1FT FT their value of b, provided b ≠1;

Allow 2b

k = isw

Condone k found by calculating the

ratio of at least 2 pairs of consecutive

values

e.g. h(2)

h(1) and

h(4)

h(3)

6 (a)

77.3 or 77.29 to 77.295

2 M1 for 232

k

where k may be a

constant or a variable seen

C opportunity

(b) 9 2 not from wrong working

M1 for 100 2 108n

× =

or 100 1.08 200n

× =

or 1.08 2n

=

or for 1.089 = 1.99… soi

or for two correct trials using a valid

relationship seen

C opportunity

Communication in 2 of 2(a)(iii), 3, 4(b), 6(a) or 6(b) 2 C1 if seen in 1 of them

Documents

Cambridge International Examinations CAMBRIDGE INTERNATIONAL … · 2019-07-26 · Cambridge International Examinations Cambridge International General Certificate of Secondary Education