INST SPECIMEN PA A U D P S RU END R I 1A O VEL NS TO N DID...L, 2 L, 3 L, and 4 L ZLWK UHVSHFWLYH HTXDWLRQV 1 L: 13 22 31 x yt z · ¸ ¸ ¸ ¹ 2 L:: 13 22 31 x yp z · ¸ =+ ¸ ¸

SPEC/5/MATM

E/SP1/ENG

/TZ0/XX

MATH

EMATICS

STAN

DA

RD LEV

ELPA

PER 1

SPECIMEN

INSTRU

CTION

S TO CAN

DID

ATES

y�W

rite your session number in the boxes above.

y�D

o not open this examination paper until instructed to do so.

y�You are not perm

itted access to any calculator for this paper.y

Section A: answ

er all questions in the boxes provided.y

Section B: answ

er all questions on the answer sheets provided. W

rite your session number

on each answer sheet, and attach them

to this examination paper and your cover

sheet using the tag provided.y

At the end of the examination, indicate the num

ber of sheets used in the appropriate box on your cover sheet.

y�U

nless otherwise stated in the question, all num

erical answers should be given exactly or

correct to three significant figures.y�

A clean copy of the Mathem

atics SL formula booklet is required for this paper.

y�The m

aximum

mark for this exam

ination paper is [90 marks].

11 pages

1 hour 30 minutes

© International Baccalaureate O

rganization 2012

Examination code

XX

XX

–X

XX

X

Candidate session number

00

0112

SPEC/5/M

ATME/SP1/EN

G/TZ0/X

X– 2 –

0212

Full marks are not necessarily aw

arded for a correct answer w

ith no working. Answ

ers must be supported

by working and/or explanations. W

here an answer is incorrect, som

e marks m

ay be given for a correct m

ethod, provided this is shown by w

ritten working. You are therefore advised to show

all working.

SEC

TIO

N A

(44 Marks)

Answer all questions in the boxes provided. W

orking may be continued below

the lines if necessary.

1. [M

aximum

mark: 7]

In the following diagram

, A

B o

=u

and BD o

=v

.

A

B

C

D

E

u

v

The midpoint of A

D o

is E and BD1

DC

3=

.

Express each of the following vectors in term

s of u and Y࣠.

(a)

AE o

[3 marks]

(b)

EC o

[4 marks]

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SPEC/5/M

ATME/SP1/EN

G/TZ0/X

X– 3 –

Turn over 0312

2. [M

aximum

mark: 5]

There are nine books on a shelf. For each book, x is the num

ber of pages, and y is the selling price in pounds (£). Let r be the correlation�FRHI¿FLHQW�

(a)

Write dow

n the possible minim

um and m

aximum

values of r�[2 m

arks]

(b)

Given that

0.95r=

, which of the follow

ing diagrams best represents the data.

[1 mark]

��F��

)RU�WKH�GDWD�LQ�GLDJUDP��'

��ZKLFK�tw

o of the following expressions describe

the correlation between x and \࣠?

perfect, zero, linear, strong positive, strong negative,w

eak positive, weak negative

[2 marks]

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

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

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

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

xx

AB

CD

yy

x

y

x

y

SPEC/5/M

ATME/SP1/EN

G/TZ0/X

X– 4 –

0412

3. [M

aximum

mark: 6]

A

toy car travels with velocity

1m

sv

� for six seconds. This is show

n in the graph below

.

v (metres per second)

y

x

v

0 1 2 3 4

12

34

56

t (seconds)

(a)

Write dow

n the car’s velocity at 3

t=.

[1 mark]

(b)

Find the car’s acceleration at 1.5

t=.

[2 marks]

(c)

Find the total distance travelled.[3 m

arks]

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SPEC/5/M

ATME/SP1/EN

G/TZ0/X

X– 5 –

Turn over 0512

4. [M

aximum

mark: 5]

A

data set has a mean of 20 and a standard deviation of 6.

(a)

Each value in the data set has 10 added to it. Write dow

n the value of

(i) the new

mean;

(ii) the new

standard deviation.[2 m

arks]

(b)

Each value in the original data set is multiplied by 10.

(i) W

rite down the value of the new

mean.

(ii) Find the value of the new

variance.[3 m

arks]

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SPEC/5/M

ATME/SP1/EN

G/TZ0/X

X– 6 –

0612

5. [M

aximum

mark: 7]

(a)

Find e

d1

e x

xx

+³

.[3 m

arks]

(b)

Find sin3

cos3d

xx

x³

.[4 m

arks]

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SPEC/5/M

ATME/SP1/EN

G/TZ0/X

X– 7 –

Turn over 0712

6. [M

aximum

mark: 7]

The expression 6sincos

xx can be expressed in the form

sin

abx

.

(a)

Find the value of a and of E࣠.[3 m

arks]

(b)

Hence or otherw

ise, solve the equation 3

6sincos

2x

x=

, for 42

xS

Sd

d.

[4 marks]

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SPEC/5/M

ATME/SP1/EN

G/TZ0/X

X– 8 –

0812

7. [M

aximum

mark: 7]

Given that

1(

)f

xx

=, answ

er the following.

��D��

)LQG�WKH�¿UVW�IRXU�GHULYDWLYHV�RI�(

)f

x.

[4 marks]

(b)

Write an expression for

()(

)n

fx

in terms of x and Q࣠.

[3 marks]

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SPEC/5/M

ATME/SP1/EN

G/TZ0/X

X– 9 –

Turn over 0912

Do N

OT w

rite solutions on this page.

SEC

TIO

N B

(46 Marks)

Answer all the questions on the answ

er sheets provided. Please start each question on a new page.

8. [M

aximum

mark: 15]

Let

2(

)3(

1)12

fx

x

��

.

(a)

Show that

2(

)3

69

fx

xx

�

�.

[2 marks]

(b)

For the graph of f

(i) w

rite down the coordinates of the vertex;

��

�LL��ZULWH�GRZ

Q�WKH�y-intercept;

��

�LLL��¿QG�ERWK�x-intercepts.

[7 marks]

(c)

Hence sketch the graph of f .

[3 marks]

(d)

Let 2

()

gx

x=

. The graph of f may be obtained from

the graph of g by the follow

ing two transform

ations

a stretch of scale factor t �LQ�WKH�y-direction,

followed by a translation of

pq §·

¨¸

©¹ .

Write dow

n pq §·

¨¸

©¹ and the value of t�

[3 marks]

SPEC/5/M

ATME/SP1/EN

G/TZ0/X

X– 10 –

1012

Do N

OT w


9. [M

aximum

mark: 14]

Tw

o standard six-sided dice are tossed. A diagram

representing the sample space is

shown below

.

score on second die

VFRUH�RQ�¿UVW�GLH

6 5 4 3 2 1

65

43

21

Let ;࣠� be the sum

of the scores on the two dice.

(a)

(i) Find P

(6)

X=

.

(ii) Find P

(6)

X>

.

(iii) Find P

(7

|6)

XX

=>

.[6 m

arks]

(b)

Elena plays a game w

here she tosses two dice.

If the sum is 6, she w

ins 3 points. If the sum

is greater than 6, she wins 1 point.

If the sum is less than 6, she loses k points.

Find the value of k for which the gam

e is fair.[8 m

arks]

SPEC/5/M

ATME/SP1/EN

G/TZ0/X

X– 11 –

1112

Do N

OT w


10. [M

aximum

mark: 17]

Let

()

cos3sin

fx

xx

=+

, 02

xd

dS

. The following diagram

shows the graph of f .

A

B

x

y

(0 , 1)

The y-intercept is at (0,1), there is a m

inimum

point at A (

),pq

and a maxim

um

point at B.

(a)

Find (

)f

x c.

[2 marks]

(b)

Hence

(i) show

that 2

q �

;

(ii) verify that A

is a minim

um point.

[10 marks]

(c)

Find the maxim

um value of

()

fx

.[3 m

arks]

The function

()

fx

can be written in the form

cos(

)r

xa

�.

(d)

Write dow

n the value of r and of D࣠.[2 m

arks]

Please do not write on this page.

Answ

ers written on this page

will not be m

arked.

1212

Please do not write on this page.

Answ

ers written on this page

will not be m

arked.

SPEC/5/MATM

E/SP2/ENG

/TZ0/XX

MATH

EMATICS

STAN

DA

RD LEV

ELPA

PER 2

SPECIMEN

INSTRU

CTION

S TO CAN

DID

ATES

y�W

rite your session number in the boxes above.

y�D

o not open this examination paper until instructed to do so.

y�A graphic display calculator is required for this paper.

y

Section A: answ

er all questions in the boxes provided.y

Section B: answ

er all questions on the answer sheets provided. W

rite your session number

on each answer sheet, and attach them

to this examination paper and your cover

sheet using the tag provided.y

At the end of the examination, indicate the num

ber of sheets used in the appropriate box on your cover sheet.

y�U

nless otherwise stated in the question, all num

erical answers should be given exactly or

correct to three significant figures.y�

A clean copy of the Mathem

atics SL formula booklet is required for this paper.

y�The m

aximum

mark for this exam

ination paper is [90 marks].

11 pages

1 hour 30 minutes

© International Baccalaureate O

rganization 2012

Examination code

XX

XX

–X

XX

X

Candidate session number

00

0112

SP

EC

/5/M

AT

ME

/SP

2/E

NG

/TZ

0/X

X– 2

–

0212

Full marks are not necessarily aw

arded for a correct answer w

ith no working. Answ

ers must be supported

by working and/or explanations. In particular, solutions found from

a graphic display calculator should be VXSSRUWHG�E\�VXLWDEOH�Z

RUNLQJ��H�J��LI�JUDSKV�DUH�XVHG�WR�¿QG�D�VROXWLRQ��\RX�VKRXOG�VNHWFK�WKHVH�DV�SDUW�RI�your answ

er. Where an answ

er is incorrect, some m

arks may be given for a correct m

ethod, provided this is show

n by written w

orking. You are therefore advised to show all w

orking.

SEC

TIO

N A

(47 Marks)

Answer all questions in the boxes provided. W

orking may be continued below

the lines if necessary.

1. [M

aximum

mark: 5]

�,Q�DQ�DULWKP

HWLF�VHULHV��WKH�¿UVW�WHUP�LV�

7�

�DQG�WKH�VXP�RI�WKH�¿UVW��WHUP

V�LV��

��D��

)LQG�WKH�FRPPRQ�GLIIHUHQFH�

[3 marks]

(b

) F

ind th

e valu

e of th

e 78

th�WHUP�

[2 marks]

��

��

��

��

��

��

��

��

��

��

��

��

SP

EC

/5/M

AT

ME

/SP

2/E

NG

/TZ

0/X

X– 3

–

Turn over 0312

2. [M

aximum

mark: 7]

L

et 2

()

4e

3x

fx

x�

�

�, fo

r 05

xd

d�

(a)

Fin

d th

e x-intercep

ts of th

e grap

h o

f f��[3 m

arks]

(b

) O

n th

e grid

belo

w, sk

etch th

e grap

h o

f f��[3 m

arks]

1 2 3 4 50–1–2–3–4–5

12

34

5x

y6 7

(c)

Write d

ow

n th

e grad

ient o

f the g

raph o

f f at 3

x=

�[1 m

ark]

��

��

��

SP

EC

/5/M

AT

ME

/SP

2/E

NG

/TZ

0/X

X– 4

–

0412

3. [M

aximum

mark: 6]

A

ran

dom

variab

le X �LV�GLVWULEXWHG�QRUP

DOO\�ZLWK�P

HDQ��,W�LV�NQRZQ�WKDW�

3�

��

Xa

>=

�

��D��

5HSUHVHQW�DOO�WKLV�LQIRUP

DWLRQ�RQ�WKH�IROORZLQJ�GLDJUDP

�[3 m

arks]

��E��

*LYHQ�WKDW�WKH�VWDQGDUG�GHYLDWLRQ�LV��¿QG��D࣠��*

LYH�\RXU�DQVZHU�FRUUHFW�WR�WKH�

QHDUHVW�ZKROH�QXP

EHU�[3 m

arks]

��

��

��

��

��

��

��

��

��

��

��

��

SP

EC

/5/M

AT

ME

/SP

2/E

NG

/TZ

0/X

X– 5

–

Turn over 0512

4. [M

aximum

mark: 7]

C

onsid

er the lin

es 1

L,

2L

, 3

L, an

d

4L��Z

LWK�UHVSHFWLYH�HTXDWLRQV�

1L

:

13

22

31

xyt

z §·

§·

§·

¨¸

¨¸

¨¸

�

�¨

¸¨

¸¨

¸¨

¸¨

¸¨

¸©

¹©

¹©

¹

2

L:

13

22

31

xyp

z §·

§·

§·

¨¸

¨¸

¨¸

=+

¨¸

¨¸

¨¸

¨¸

¨¸

¨¸

©¹

©¹

©¹

3L

:

01

12

0

xys

za �

§·

§·

§·

¨¸

¨¸

¨¸

=+

¨¸

¨¸

¨¸

¨¸

¨¸

¨¸

�©

¹©

¹©

¹

4

L:

�42

xyq

z

�§

·§

·¨

¸¨

¸=

¨¸

¨¸

¨¸

¨¸

�©

¹©

¹

(a)

Write d

ow

n th

e line w

hich

is parallel to

4

L�

[1 mark]

(b

) W

rite dow

n th

e positio

n v

ector o

f the p

oin

t of in

tersection o

f 1

L an

d

2L�

[1 mark]

(c)

Giv

en th

at 1

L is p

erpen

dicu

lar to

3L��¿QG�WKH�YDOXH�RI��a�

[5 marks]

��

��

��

��

��

��

��

��

��

��

��

��

SP

EC

/5/M

AT

ME

/SP

2/E

NG

/TZ

0/X

X±��±

0612

5. [M

aximum

mark: 7]

�7KH�SUREDELOLW\�RI�REWDLQLQJ�KHDGV�RQ�D�ELDVHG�FRLQ�LV��7KH�FRLQ�LV�WRVVHG�� WLP

HV�

��D��

�L��:ULWH�GRZ

Q�WKH�PHDQ�QXP

EHU�RI�KHDGV�

��

�LL��)LQG�WKH�VWDQGDUG�GHYLDWLRQ�RI�WKH�QXP

EHU�RI�KHDGV�[4 m

arks]

(b

) F

ind th

e pro

bab

ility th

at the n

um

ber o

f head

s obtain

ed is less th

an o

ne stan

dard

GHYLDWLRQ�DZD\�IURP

�WKH�PHDQ�

[3 marks]

��

��

��

��

��

��

��

��

��

��

��

��

SP

EC

/5/M

AT

ME

/SP

2/E

NG

/TZ

0/X

X– 7

–

Turn over 0712

6. [M

aximum

mark: 7]

�7KH�IROORZ

LQJ�GLDJUDP�VKRZ

V�D�SROH�%7��P

�WDOO�RQ�WKH�URRI�RI�D�YHUWLFDO�EXLOGLQJ��T

he

angle

of

dep

ression fro

m T

to

a

poin

t A on th

e horizo

ntal

gro

und is

35!��

The an

gle o

f elevatio

n o

f the to

p o

f the b

uild

ing fro

m A

is 30!�

A

B Tdiagram

not to scale

�)LQG�WKH�KHLJKW�RI�WKH�EXLOGLQJ�

��

��

��

��

��

��

��

��

��

��

��

��

build

ing

gro

und

SP

EC

/5/M

AT

ME

/SP

2/E

NG

/TZ

0/X

X– 8

–

0812

7. [M

aximum

mark: 8]

A

circle centre O

and rad

ius r �LV�VKRZ

Q�EHORZ��7KH�FKRUG�>$

%@�GLYLGHV�WKH�DUHD�RI�WKH�

FLUFOH�LQWR�WZR�SDUWV��$

QJOH�$2%�LV��ș�

AB

O

r

ș

��D��

)LQG�DQ�H[SUHVVLRQ�IRU�WKH�DUHD�RI�WKH�VKDGHG�UHJLRQ�[3 m

arks]

��E��

7KH�FKRUG�>$%@�GLYLGHV�WKH�DUHD�RI�WKH�FLUFOH�LQ�WKH�UDWLR��)LQG�WKH�

valu

e of ș࣠�

[5 marks]

��

��

��

��

��

��

��

��

��

��

��

��

SP

EC

/5/M

AT

ME

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2/E

NG

/TZ

0/X

X– 9

–

Turn over 0912

Do N

OT w


SEC

TIO

N B

(43 Marks)

Answer all the questions on the answ

er sheets provided. Please start each question on a new page.

8. [M

aximum

mark: 13]

Each

day, a

factory

record

ed th

e num

ber ( x ) o

f boxes it p

roduces an

d th

e total

pro

ductio

n co

st ( y ��GROODUV��7KH�UHVXOWV�IRU�QLQH�GD\V�DUH�VKRZQ�LQ�WKH�IROORZ

LQJ�WDEOH�

x��

44

��43

50

31

��

57

y400

582

784

��

448

870

537

724

(a)

Write d

ow

n th

e equatio

n o

f the reg

ression lin

e of y o

n [࣠�

[2 marks]

�8VH�\RXU�UHJUHVVLRQ�OLQH�DV�D�P

RGHO�WR�DQVZHU�WKH�IROORZ

LQJ�

(b

) In

terpret th

e mean

ing o

f

(i) th

e grad

ient;

(ii) th

e y�LQWHUFHSW�[2 m

arks]

��F��

(VWLPDWH�WKH�FRVW�RI�SURGXFLQJ��ER[HV�

[2 marks]

��G��

7KH�IDFWRU\�VHOOV�WKH�ER[HV�IRU��HDFK��)LQG�WKH�OHDVW�QXPEHU�RI�ER[HV�WKDW�

WKH�IDFWRU\�VKRXOG�SURGXFH�LQ�RQH�GD\�LQ�RUGHU�WR�PDNH�D�SUR¿W�

[3 marks]

(e)

Com

men

t on th

e appro

priaten

ess of u

sing y

our m

odel to

(i) estim

ate the co

st of p

roducin

g 5

000 b

oxes;

(ii) estim

ate the n

um

ber o

f boxes p

roduced

when

the to

tal pro

ductio

n co

st

LV��[4 m

arks]

SP

EC

/5/M

AT

ME

/SP

2/E

NG

/TZ

0/X

X– 1

0 –

1012

Do N

OT w


9. [M

aximum

mark: 16]

L

et 2

1(

),

11

xh

xx

x�

z�

+�

(a)

Fin

d

1()

hx

��

[4 marks]

(b

) (i)

Sketch

th

e grap

h

of h

fo

r 4

4x

�d

d

and

58

y�

dd

, in

cludin

g

any

DV\PSWRWHV�

��

�LL��:ULWH�GRZ

Q�WKH�HTXDWLRQV�RI�WKH�DV\PSWRWHV�

(iii) W

rite dow

n th

e x-intercep

t of th

e grap

h o

f K࣠�[7 m

arks]

(c)

Let R �EH�WKH�UHJLRQ�LQ�WKH�¿UVW�TXDGUDQW�HQFORVHG�E\�WKH�JUDSK�RI��K࣠, th

e x-axis

and th

e line

3x=

�

(i) F

ind th

e area of R�

(ii) W

rite dow

n an

expressio

n fo

r the v

olu

me o

btain

ed w

hen

R is revolv

ed

thro

ugh ��

! about th

e x�D[LV�[5 m

arks]

SP

EC

/5/M

AT

ME

/SP

2/E

NG

/TZ

0/X

X– 1

1 –

1112

Do N

OT w


10. [M

aximum

mark: 14]

�$�URFN�IDOOV�RII�WKH�WRS�RI�D�FOLII��/HW��h b

e its heig

ht ab

ove g

round in

metres,

after t �VHFRQGV�

The tab

le belo

w g

ives v

alues o

f h and W࣠�

t (seconds)

12

34

5

h (m

etres)105

98

84

��

(a)

Jane th

inks th

at the fu

nctio

n

32

��

��

��

ft

tt

t �

��

� is a su

itable

PRGHO�IRU�WKH�GDWD��8

VH�-DQH¶V�PRGHO�WR

(i) w

rite dow

n th

e heig

ht o

f the cliff;

��

�LL��¿QG�WKH�KHLJKW�RI�WKH�URFN�DIWHU��VHFRQGV�

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