M
O
9/5/ MATS
D
/ SP
2 / EN G
lT Z1 / XX
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22æ7404
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nternational Baccalaureate'
Baccalauréat lnternational
I
Bachillerato lnternacional
MATHEMATICAL STUDIES
STANDARD LEVEL
PAPER 2
Friday
I
May 2009 (morning)
t hour 30 minutes
I
NSTRUCTIONS TO CAN DI DATES
.
.
'
Do not open this examinatíon paper until instructed to do so.
Answer all the questions.
Unless otherwise stated in the question, all numerical answers must be given exactly or correct
to three significant figures.
I
2209-7404.
@
pages
International Baccalaureate Organization 2009
-2-
MO9/5/N4A
T
SD/SP2ÆNG I T Z I, IXX
Pleqse slart each question on q nsvÌ page. You are advised lo shou, all worlcing, where possible. Yl/here an
answer is wrong, some marlrs may be givenfor correcl melhod, provtded lhis is shown by wrìllen wurking.
Solutions þundfrom a graphic dßptay calculalor should be ntpporled by suilable working, e.g. if graph's
are ttsed lofind a solulion., you shottld skeÍch Íhese as parl of your answer.
1.
fMaximum mørk: I7J
The diagram shows the cumulative frequency gaph for the time I taken to perform
a certain
øsk by 2000 men.
2000
\\10
I
750
I
500
()
c)
ct
o
h
Ø9
t250
t¡{
q)
I
000
R'
=
O
150
500
250
0
0
5
l0
15
20
2s
30
Time I seconds
(a)
Use the diagram to estimate
S
(Ð
the median
(iÐ
the upper quartile and the lower quarlile;
(iii)
the interquartile
time; t3
,.
"'
lange. ['.o - lO =
lO 54C[4 rnarks]
l.¡l SoC-.
(7'h i s qu
2209-7404
\G 4e
e.s t i.on c
on, li,tnt e,s
on
th
e
J o ll owin g pa ge)
M09/5/IyIAISD/SP2ÆNG lT Zr
3
-)
(Question
(b)
I
continued)
forn the task.
Find tbe nurnber of men who take morc th¿¡n I I
Looo (c)
lxK
leSÒ
55 % of the men took less than p seconds to
7oooL.l5) :
3{o rALA p
> t
lloO è
tZ,56e.ç-,
[3 ntorhrJ
12 marksJ
The times taken for the 2000 rnen were groupecl as shown in the table below
(d)
Frequency
5</<10
500
l0</<15
tì50
15<t <20
CI
20<t <25
b
Vy'rite down the value
of
(Ð cr; lE
- laSD > So Ò
b.
- tglo > l5Ò
(iÐ
(e)
Time
2-ooo
l2 marksJ
Use your graphic display calculator to find an estimate
(Ð
the mean
(ii)
the standard deviation of the
of
time; \3 ã¡{time. S =.t,.1 I 5¡n-e.
1'3
marks
I
Everyóne who perfonns the task in less than one standard cleviation below the mean
will receive a bonus. Pedro takes 9.5 seconds to perform the task.
(Ð
Does Pedro receive the bonus'/ Justify yor¡Í answer.
\A -q,q\
2209-'7404
t 3,5q
[3 marhsJ
A/o bo-nr,rg
ftrn
over
-42.
[Maximum
(i)
MO 9/5 /N4ATSD/SP2ÆNG
NZl XX
nark: 2lJ
Sharon and Lisa share a flat. Sharon cooks dinner three nights out of ten.
If Sharon does not cook dinner, then Lisa does. If Sha¡on cooks dinner
the probability that they have pasta is 0.75. If Lisa cooks dinner the probability
that they have pasta is 0.12.
(a)
[3 marksJ
Copy and complete the tree diagram to reprcsent this ìnformation
Pasta
0.7s
,lf
Sharon
a,
a-t
Not Pasta
.12
Pasta
,1
Lisa
Not
cooks_diqner ancl they do not have
(b)
Find the probability that
(c)
Find the probability that they do not þave pas
(d)
Given
L_isa
GiI,ft\ =.últ
Pasta
pasta.
[2 marksJ
mar*s,
"?îäÏ:ääî î'E?X'sù
bqrr3
=,
that they do noi have pasta, find the probability that Lisa
[3 marlu/
cooked dinner.
,Îq \
(This Eteslion conlinues on thefollou,ing page)
2209-'7404
-5-
MO9/5/[4ATSD/SP2ÆNG ITZ
I
IW
(Queslion 2 continued)
(ii)
A survey was carried out in a year 12 class. The pupils were asked which
pop groLrps they like out of the Roclcers (R), the Salseros (S), and the Bluers (B).
The results are shown in the following diagram.
U
7
(a)
Write down
(b)
Find n(R')
n
fi
(Rn.lnB)
[2 marksl
(c)
,SnB like.
(d)
pils who like the Rockers and
like
rhesarseros.
There are 33 pupils in the class.
2209-7404
[2 ntarksJ
4^d 3\¡¿rS
theBruersbut do not
c)
marhJ
RABAS,
[2 marks]
2\+5X
(Ð
Findx.
(ii)
Find the nunrbcr of
=33
Y=4
pupils
who like'the Rocker,:i
[3 mctrksl
Turn over
MO 9/5/NÍATSD/SP2ÆNG IT ZI
-63.
IXX
jJ
lMaximttm murk: I
The vcrtices of quadrilaleral ABCD as shown in the cliagrarn are
C(-2,1)
and
D(-1,
A(3, 1), B(0,2),
-l).
\,
o(line
CD. -J -1.. : -&
r
(a)
Calc-ulate the gradient
(b)
Sbow that line AD is oerocndicular to linc
(c)
Fincl the equation
ctr,r'iÞ(-l-\)
r
-fr
t
,{{R,r-) , bf-l,-i) +E:e
where
Lines
[2 marksJ
-r+z-
AI!
a,b,c
CD. ,
of line CD. Give your
eZ.
answer
- | ; -l (-e\ + l¡
-\ : 14l.3
and CD intersect at point
E.
=
in the forn ax+ó1,'=c
b=-3
The equation of liue AB
: f'"\
is .r'+ 3r, = ó
---P¡¿
2-x+
) x -G
- z"x
*_ >rt -l
=
{rã-
à
The dist¿urce belween D ancl E
(Ð
).209-1404
is J2o
Firid the arca of triarrslc ADE.
-
f
Òñ 1[€
*
marksl
T
-a-
[2 marks I
åil*\*
.
4-
lÐ uC-\ s
[2 marksJ
I + e{n).h
\=-3
-74.
MO9/5/I\4A I SD/SP2ÆNG IT Z1
IXX
[Maximum mark: l7J
A chocolate bar has the shape of a triangular right prisrr ABCDEF as shown in
the diagran The ends are equilateral triangÌes of sicle 6 cm and the length of the
chocolate bal is 23 cm.
E
líagrum not lo scnle
23 cm
6cm
6 crn
¿?
'/
-/
c/_
D
23 cm'
23 cm
B
(a)
6cm
A
(Ð
Write down the size of angle
(iÐ
Hence
BAF. 6Oo
the triangular encl of
or othe
chocolate bar.
:
bar. " \
I
lS, t?
the
grhÈ
-
t-L
(b)
Find the total surface area ofthe chocolate
(c)
ItisknownthatI cmrofthischocolateweighs I.5g. Calculatetheweightofthe
(r,ff)(e) + Z(z"yk)
fsBr
grì"t
= 'l +5
[4 marks]
[3 marlaJ
[3 marksJ
chocolate bar.
A different chocolate bar made with the same mixtru'e also has the shape of a
triangularprism. The ends are triangles with sides of length 4 cm, 6 cm and 7 cm.
(d)
is
Show that the size of the angle between
correct to 3 significant figures. CO
S
(e)
r( [+)LàtsirïïL
A
ntarltsJ
O
The weight of this chocolate bar is 500 g
.t
2209-7404
86.4"
marksJ
Þ
5ao
-- X1,T
Turn over
-85.
[Maximtmmark:
M09/5 iIV{ATSD/SP2ÆNG lT Z I D<X
22J
f(x)=3x+),x+0. 3 X + l}X
(a) Differentiate 7 (x) with respeci ï. ? - 2..[ X '3
Considerthetunction
7
[3 marlaj
i,o
(b)
Calculate
/'(x)
when
r
=
I
[2 marksJ
2
part (b) to decide wheth
Justig'your answer.
(c)
(d)
vrG, 'rt
Solve the equation .f'(x)=O
(e)
The graph of
of
(Ð
.)
f
or û"S,U€
sing
[2 marlæJ
$
ìS &S,rcq-6 t rr*þ , _ A*ír -_:3
marksl
'
g_ 21X-3 = O
i;a = l- l3
/'has a local minimum atpoint P. Let lbe
(Ð
Write down the coordinates of
(iÐ
Write down the gradient of
(iii)
Write down the equation of L
P.
(a,r)
L D
l=â
[5 markJ
Sketchthe graph of the ftinction ,[, for -3<x(6 and -7l"vll5.
clearly the point P and any intercepts of the curve with the axes.
(g) (Ð
On your graph draw and label the tangent f"
(ii)
Iintersects the graph of .f at a second point.
ofthis point of intersection.
Inclicate
@lopÉ5"o
'Writc
X= - I
2209-7404
the tangent to tlre graph
atP.
clown thc x-coordinatc
[3 marksJ