4. Transformation Dan Statistik

8/10/2019 4. Transformation Dan Statistik

1/81

Module PMR

1. Example of transformation arei. translation

ii. reflectioniii. rotationiv. enlargement

A. TRANSLATION

1. In a translation, all the pointsin a plane are moved in the samedirectionthrough the same distance.

2. Under a translation, the shapes, sizes and orientationsof the object

and the image are the same.

3. A translation is described in the form

b

a

here!

i. arepresents the horizontalmovement " +arefers to the rightmovements # -arefers to the leftmovements $

ii. brepresents the verticalmovement " +brefers p!ardmovements # -brefers do!n!ardmovements$

%. Example of translations!

"a$ &ranslation

24 means "b$ &ranslation

35 means

moved %units to the left moved ' units to the rightfolloed b( 2 units upards. folloed b( 3 units

donards.

". R#$L#%TION

109

object

image

' unitsto the right

3units

donards

object

image

% unitsto the left

2units

upards

CHAPTER 14 : TRANSFORMATIONS


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Module PMR

1. In a reflection, all the pointsin a plane are flipped overin the sameplane at a straight line )non as the a&is of reflection.

2. Under a reflection !"a$ the shape and size of the object and the image are the same.

"b$ the object and image formed the mirror imageof each other.

3. &he a&is of reflectionis the perpendiclar 'isectorof the line joiningan object point to its corresponding image.

%. Example of reflections!

"a$ *eflection inx+axis. "b$ *eflection in the line A

%. ROTATION

110

0- 2- 4

- 2

2

2 4

- 4

6

4

object

image

x

y

Axis of reflection

object

imageA

Axis of reflection


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Module PMR

1. In a rotation, all the pointsin a plane are rotatedabout a fixed point")non as centre of rotation$ in the same directionthrough the sameangle.

2. &he direction of rotation can be !

"a$ cloc)ise "b$ anticloc)ise

3. Under a rotation !

"a$ the shape and size of the object and the image are the same."b$ the centre of rotation is the onl( point that does not change

position."a$ the distances of the object and its image from the centre ofrotation

are the same

%. Example of rotations!

"a$ *otation through -ocloc)ise "b$ *otation through -oabout the point / anticloc)ise about "+2,

2$

"c$ *otation through 10o

about the centre A

111

/

centre ofrotation

object

image

cloc)iserotation

0- 2- 4

- 2

2

2 4

- 4

4

x

y

object

image

centre ofrotation

anticloc)iserotation

/centre of

rotation

image


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Module PMR

'. &he image of a rotation through -ocloc)ise is exactl( the image of arotation through 2oanticloc)ise.

. &he image of rotation through 10ocloc)ise and anticloc)ise isexactl( the same.

. &o find the centre of rotation hen object and its image are given,construct to perpendicular bisectors of to points on the object andtheir corresponding points on the image.&he centre of rotation is the intersection point of the to perpendicularbisectors.or example!

(. #NLAR)#*#NT

1. In an enlargement, all the pointsin a plane are movedfrom a fixedpoint " )non as centre of enlargement$ folloing a constant ratio.

112

4

54A4

5

A

Intersection point"centre of rotation$

6erpendicularbisector of AA4

6erpendicularbisector of 554


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Module PMR

2. &he constant ratio is )non as the scale factor.

3. 7cale factor, ) 8objecttheofsideingcorrespondofLength

imagetheofsideofLength

%. Under an enlargement!

"a$ object and its image are similar"b$ corresponding sides are parallel

'. Example of enlargement!

"a$ Enlargement at the centre / and "b$ Enlargement at thecentre scale factor 2 "+3,2$ and

scale factor 2

1

"c$ Enlargement at the centre & and scalefactor 3

113

0- 2- 4

- 2

2

2 4

- 4

4

x

y

object

imagecentre ofenlargement

/centre ofenlargement

object

imageA

A4

4

54

5

64

94

*4

6

9

*


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Module PMR

(#S%RI"IN) A TRAN$OR*ATION

"a$ &ranslation

b

a

Example !

"i$&ranslation

5

3

"b$ *eflection in the line " a&is of reflection $Example!"i$ *eflection in the line A

"c$ *otation through "angle of rotation$ "direction of rotation$ about

"centre of rotation$Example!"i$ *otation through -ocloc)ise about "1,3$

"d$ Enlargement at centre "centre of enlargement$ ith scale factor:::::::Example!"i$ Enlargement at centre / ith scale factor 2.

%ommon #rrors

estion #rrors %orrect Steps

In the diagram belo, A4454;4is the image of A5; in

transformation


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Module PMR

"b$

24

"c$ &ranslation

4

2

(RA T# I*A)# O$ TRANSLATION

115

A4

4

5

;

A

54

;4

2 4 6 8

2

4

6

8

A

2 4 6 8

2

4

6

8

A

2 6

4

6

8

A

2 4 6 8

2

4

6

8

A

2 4 6 8

2

4

6

8

A

2 4 6

2

4

6

8

A

4

6

8A

4

6

8

A4

6

8

A/ TRANSLATION

1) 2) 3)


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Module PMR

&ranslation

2

5 &ranslation

4

4 &ranslation

5

0

&ranslation

4

3

&ranslation

1

6

&ranslation

2

3

&ranslation

0

5

&ranslation

4

1

&ranslation

3

4

STATIN) T# TRANSLATION

116

1) 2) 3)

7) 8) 9)

4) 5) 6)

A

A

A0A0


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Module PMR

"/ ROTATION

(RAIN) T# I*A)# O$ ROTATION

5loc)ise rotation of -o Anticloc)ise rotation of -o

about the point "','$ about the point ",3$

117

A

AA0 A0

A0

A

A

A

A0

A0 A0

4) 5) 6)

7) 8) 9)

i$. ii$.

&ranslation

1

5


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Module PMR

3$ 3$.

118

2 4 6 8

2 4 6 8

2

4

6

8

A

2 4 6 8

2

4

6

8

A

2

4

6

8A

2

4

6

8

A

2

4

6

8

A

2 4 6 8

2

4

6

8

A

1)

2)

1)

2)


11/81

Module PMR

(RA T# I*A)# O$ ROTATION

5loc)ise rotation of -o *otation of 10o

about the point "%,3$ about the point "','$

1$.

119

1).

2 4 6 8

2

4

6

8

A

2).

2

4

6

8

A

2

4

6

8

A

2 4 6 8

2

4

6

8A

iii$. iv$..

a)

2).


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Module PMR

(#T#R*IN# T# %#NTR# O$ ROTATION

i$. 5loc)ise rotation of -o

120

3).

2 4 6 8

2

4

6

8

A

2

4

6

8

2 4 6 8

2 4 6

8

A

3).

2

4

6

8A

"

%

(

A0

(0

"0

1).

2

4

6

8

A

%"

A0

"0

%0

2).


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Module PMR

"a$5entre of rotation 8 ==== "b$5entre of rotation 8==.....

"c$ 5entre of rotation 8 ==== "d$ 5entre of rotation 8==

121

2 4 6 8

2

4

6

8

A

"

"0

(0

A%0

( %

2 4 6 8

2

4

6

8

A

A0

"

"0

%0

(0

(%

3). 4).

6).

2

4

6

8A

%

"

(

A0

%1

(02

4

6

8

A

"

(

#

%

A0

#0(0

%0 "0

5).


14/81

Module PMR

TO $IN( T# %#NTR# O$ ROTATION

ii$.. Anticloc)ise rotation of -o

1$. 2$.

"a$ 5entre of rotation 8 ==.. "b$ 5entre of rotation 8 ==..

122

2

4

6

8

A0

%0"0

A

"

%

2 4 6 8

2

4

6

8

A

(

"

A0

"0

%0

(0

2 4 6 8%

"0

2 4 6 8

2

4

6

8

A0

"

(

A%

(0 %02

4

6

8

A0

%0

A

"

(%

(0

"082 4 62

4

6

8

2

4

6

8

A0"0

%"

(

A

a)

3).

b)

4).


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Module PMR

"c$ 5entre of rotation 8 ==== "d$ 5entre of rotation 8 ===

%/ R#$L#%TION(RAIN) T# I*A)# O$ R#$L#%TION

123

A

A A

1)

.2) 3)


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Module PMR

(#T#R*IN# T# A2IS O$ R#$L#%TION

124

A

A

A

A

A

A

5) 6)

7) 8) 9)

J K

K

J

L

M

M

L

J

K

K

J

L

M

M

L

J K K J

L

MM

L

1) 2) 3)

P

4)

P

P

5)

4)


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Module PMR

(#T#R*IN# T# S%AL# $A%TOR AN( %#NTR# O$ #NLAR)#*#NT

7cale factor 8 ===.. 7cale factor 8 ===..

125

2 4 6 8

2

4

6

8

A

A0

2 4 6 8

2

4

6

8A

A0

7)

2 4 6

2

4

6

8A0

A

8)

2 4 6 8

A0

A

2 4 6 8

(/ #NLAR)#*#NT

2

4

6

8

2

4

6

8

A

A0


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Module PMR

5entre of enlargement 8===. 5entre of enlargement 8 ==

7cale factor 8 ===.. 7cale factor 8 ===.. 5entre of enlargement 8 ===. 5entre of enlargement 8===

126

2 4 6 8

2

4

6

8

A0

A

2 4 6 8

2

4

6

8

A

A0

2 4 6 8

2

4

6

8A

A0

2 4 6 8

2

4

6

8

AA0

7cale factor= 5entre of enlargement=

7cale factor= 5entre of enlargement=


19/81

Module PMR

(RA T# I*A)# O$ #NLAR)#*#NT

i$. 7cale factor 2, about "2,3$ ii$. 7cale factor 3, about "0,%$

127

4

6

8

"4

6

8

"

2 4 6 8

2

4

6

8

A

2 4 6 8

2

4

6

8

A

1) 1)

2) 2)


20/81

Module PMR

(#S%RI"# A TRANS$OR*ATION

128

A A0

2 4 6 8

2

A

A0

2 4 6

2

4

6

8

A

A0

2

4

6

8

A0

A

2

4

6

8

A0 A

5) 6)

2) 3)

2 4 6 8

2

4

6

8

A0

A

2 4 6 8

2 4 6 8

2

4

6

8

%

2 4 6 8

2

4

6

8

%

3) 3)

&ranslation

23


21/81

Module PMR

(#S%RI"# A TRANS$OR*ATION

129

2 4 6

2

4

6

8 A

A0

2 4 6 8

A

A0

2 4 6 8

2

4

6

8

A

A0

2 4 6 8

2

A

A0

2 4 6 8

2

4

6

8

A0

A

2 4 6

2

4

6

8

A0

A

2

A

A0

11) 12)

2

4

6

8

A0

A

14)

2

4

6

8

A0

A

15)

8) 9)


22/81

Module PMR

(RAIN) %ON)R3#NT 4OL5)ON7tarting from line >?, dra pol(gon @>?B6 hich is congruent to pol(gonA5;E

130

2 4 6 8

2

4

6

8

A

A0

2 4 6 8

2

4

6

8A

A0

17)

2 4 6

2

4

6

8

A0 A

18)

A

B

D

C

E

"1$F

J

K

"2$ A

J

K

B

CD

E

F

K

J

A"3$

E

F

D

CB

"%$

K

A

F

E

B

CD


23/81

Module PMR

4*R past 6ear 7estions

899:

1$. In ;iagram 2, 64 is the image of 6 under transformation ;.

;escribe in full transformation ;. C 2 marks DAnswer :

2$. ;iagram 3 in the anser space shos pol(gon ? and straight line 69dran on a grid of eual suares. 7tarting from line 69, dra pol(gon

7&UFG hich is congruent to pol(gon ?.

131

y

1242 8 10

4

2

6

0

10

8

12

6x

P

P

"'$

J

K

A

B C

DE

F

"$

K

A

B

J

CDE

F


24/81

Module PMR

C 2 marks D

8994

?44

@>

?

6


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Module PMR

899?

-$. &ransformation < is a reflection in thex+axis

1$. 11$.

%A4T#R ? STATISTI%S

1.1 (ataJ a collection of information or facts.;ata can be collected b( J 5ounting, easuring, /bservation, Intervie or9uestionnaires. &he data collected should be recorded in reuenc( &ableb( using a tall( chart. A tall( chart shon belo !

eg 1 ! Tall6 %hart

ar) &all( ar)s

%0

'3

'2

148

A

5

;

6

54

;4

A4

4A

5;

E


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Module PMR

1.2 $re7enc6J the number of times a certain number, measurement, scoreor item occurs. reuenc( can be easil( obtained b( using a tall( chartshon b( eg 1 and a freuenc( table is used to organiHe a set of data.&he freuenc( table can be constructed either verticall( or horiHontall( that

shon belo.

eg 2 ! $re7enc6 Ta'le

ar) reuenc(

%0 '

'3

%

' '

2 3

1.3 The !a6s to represent and interpret data

A. 4ictograms + represents data in the form of a picture diagram, use a picture or a

s(mbol that is easil( understood.

eg 3 !

roup A

roup

roup 5

represents 1 students

. "ar %harts + represents data using vertical or horiHontal bars. It is a freuenc(

diagram using rectangles of eual idth.

eg % !

149

ar) 48 53 60 65 72reuenc( 5 7 4 5 3


42/81

Module PMR

Nm'er of

%lassrooms

School

&7*96

1.

1%.

12.

1..

0.

.

%.

2.

.

+ A dual bar chart can be constructed to represent to sets of data in thesame chart b( using separate bars or single bar.

eg ' !

Nm'er of%lassrooms

School

R+4

@99

?9

=9

:9

89

9

5. Line )raphs + used to represent changes in data over a period of time. &he data is

first represented b( points. &hen line segments are dran to join thepoints.

eg !

;. 4ie chart+ a circle, divided into sectors of various siHes, used for illustration and

comparison of different categories of data.

or a uantit( 6, the angle of the sector representing 6 is

150

MDays

20

40

60

80

100

Sa

le

ofcocola!e

ca"es

0 # $ # % S S

1%.

Nm'er of

%lassrooms

School

*96

12.

1..

0.

.

%.

2.

.


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Module PMR

Angle of sector B%&e'ecy of *

360#o!al f&e'ecy

B %&e'ecy of * 360

eg ! &(pe of ood 7old in A ;a(

1.% *ode, *edian and *ean.

A *odeJ the value or item hich occurs most freuentl( or highestfre7enc6. &he mode can determine from a given set of data, afreuenc( table, a pictogram, a bar chart, a line graph or pie chart.

eg0 ! , %, 3, %, 0, , ', %, 3

mode 8 % " freuenc(83 $

*edianJ the middle value hen all the data are arranged in an increasing or

decreasing order. If the number of values is odd, then the median isthe middle value. If the number of values is even, then the medianis the mean of the to middle values.

eg -a ! 1g, 12g, 10g, 1g, 1g 1, 1, 12, 1, 10

edian 8 12g

eg -b ! 1, 12, 10, 1, 1,1% 1, 1, 12, 1%, 1, 10

edian 8 "12K1%$ L 2 8 13

eg -c !

7core 1 2 3 % '

reuenc

(

% 1 3

&otal freuenc( 8 %KKK1K3 8 21edian 8 Bumber of the 11thscore

8 3eg -d !

7core 1 2 3 % '

reuenc(

% 3

&otal freuenc( 8 %KKKK3 8 2edian 8 Bumber of the 1thand 11thscore

82

32+ 82

5 8 2.'

151

40o60

o

140o

%&ie+ ,oo+le

&&y

,oo+le

%&ie+ ice

,oo+le

So/


44/81

Module PMR

5 *eanJ is the arithmetic average of a set of data.

ean 8 sum of all the values of data total number of data

eg 1a ! ', 2, , 3, %, , ', 2, 3

mean 89

325643625 ++++++++

89

368 4

eg 1b !

6rice "*$ 1' 2 2' 3 3'

reuenc( 3 % 2 % 3

ean 834243

)335())430()225()420()315(

++++++++

816

105120508045 ++++

816

4008 M25

#&ample @

;a( on &ue Ged &huAbsentees 1 12 13

&he freuenc( table shos thenumber of absentees on fourparticular school da(s. ased on theinformation given, find

a$ the da( ith the highest numberof absentees.

Solutiona$ &hursda(

b$ the da( ith the loest number ofabsentees.

Solutionb$ &uesda(

#&ercise @

rade A 5 ; Ereuenc( 3 0 1% -

&he table shos the distribution ofgrades obtained b( % students inthe 6*.a$


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Module PMR

2

21

22

23

2%

*epresents 2 tourist

&he pictogram above shos the number of tourist to an island resort from1--- to 2%.

a$ Ghich (ear had the highest number of touristM

Solution :a$ Near 2%

b$ ind the total number of tourist for the to (ears hich had the samenumber of tourist.

Solution :b$ &he to (ears ere 2 and 22.

&otal number of tourists for 2 and 228 2"1x2$8 %

c$ Ghat as the total number of tourists for the six (earsM

Solution :c$ &otal number of s(mbols

8 0K1K-K1K%K1%8 ''&otal number of tourists 8 '' x 2

8 11

d$ If each tourist spent an average of * during their sta(, calculate thetotal amount of mone( spent b( tourist on the island in 21.

Solution :d$ Bumber of tourists in 21 8 - x 2

8 10 Amount spent 8 10 x

8 * 1 0 #&ercise 8

onda(

&uesda(

153


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Module PMR

Gednesda(

&hursda(

rida(

*epresents bols

&he pictogram shos the number of the bols of noodles sold in a schoolcanteen from onda( to rida(.

a$ Ghich da( has the highest salesM

b$ If the profit from each bol of noodles sold is *.2, find the total profit

for the five da(s.

c$ ind the difference in the profits obtained from the sales on onda( andrida(.

#&ample C 7ales igures of Gatches for ive7hops

Shop

Nm'erof!atches

Sales $igres of atches for $ive Shops

#(%"A


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Module PMR

a$ ;ifference 8 %' + 1' 8 2' atches

b$ If 7hop E had an increase insales of 2O the folloing month,

hile 7hop A sold 3 feeratches, hat as the total salesfor the to shopsM

Solution :b$ 7hop E !

12O of % 8 %0 atches

7hop A !2' J 3 8 22 atches

&otal sales for the to shops8 %0 K 228 atches

c$ If the average value of a atchas *1', calculate the totalvalue of the atches sold b( allfive shops in ;ecember.

Solution :c$ &otal number of atches

8 2'K 1'K %'K 32'K %8 1 atches

&otal value 8 1 x 1' 8 *2%

&he horiHontal bar chart above

shos the sales of an ice creamvendor for six da(s.

a$ /n hich da( did he sell themost ice creamM

b$


48/81

Module PMR

Timesec/

*agesh1s "est :99 m Times

899:899C8998899@8999@DDD


49/81

Module PMR

mil)

chocolatecof fee

te a

All the orm 3 students too) part in asurve( on their favourite hot drin)s.&he results ere shon in the pie

chart above.

a$ Ghich drin) as least preferredb( the orm 3 studentsM

Solutions :a$ il) as least preferred b( orm

3 students.

b$ Ghich as the most populardrin) among the orm 3

studentsM

Solutions :&ea as the most popular drin)among the students.

c$


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Module PMR

r 7ong bought a bas)et ofatermelons ith *1'.


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Module PMR

1. /btaining information from a pictogram.

roup A

roup

*epresent ' students

#rrors&otal numbers of students8 %K8 1

%orrect Steps&otal numbers of students8 " %K$ x '8 1 x ' 8 '

2. Error in determining the mode in a freuenc( table.

ar)s 1 2 3 :reuenc(

3 % 3 1 '

#rrors&he mode is 3.&he mode is '.

%orrect Steps&he mode is %.

3. Error in determining the median of a set data.#rrors', 2, %, 0, 2, 3, 1 "data not rearrangein order$edian 8 0

%orrect Steps1, 2, 2, 3, %, ', 0edian 8 3

%.

Salar6 Less thanR*@999

R*@99@ to

R*@


52/81

Module PMR

#rrors&otal number of non+fiction boo)ssold8 1 K 13 K 1 K 118 %%

%orrect Steps&otal number of non+fiction boo)ssold8 K 0 K K -8 3

'. &he table shos the distribution of the scores obtained in tossing a dice 3times. ind the mean score of each toss.

7core 1 2 3 % '

reuenc( ' % 0 3 %

#rrors

ean 8438645

654321

++++++++++

8 .

%orrect Stepsean 8

438645

)4(6)3(5)8(4)6(3)4(2)5(1

+++++

+++++

8 3.%

#&tra #&ercise

@ *ode a. 2 g, ' g, 3 g, g, ' g,

% g, 3 g, g, ' g, g,' g, g

ode 8 ::::: g

b. 1' mm, 10 mm, 12 mm, 13 mm, 1'mm,1 mm, 12 mm, 1mm, 1' mm, 13 mm,1% mm, 1' mm

ode 8 :::::: mm

c. %, ', , 2, 1, %, , , ',

ode 8

d.

Bumber of matches pla(ed 1 2 3 % '

reuenc(1 % 3 2 3

ode 8 :::::::matches

e.


53/81

Module PMR

f.

Bumber /f goals 1 2 3 %

reuenc( 2 ' % %

ode 8 goals

8. *edian

a. %, 2, ', %, , 2, 1

edian 8

b. -, 2, , 1, 0, -,

edian 8

c. 2)g, %)g, 1)g, ')g, %)g, 2)g

edian 8

d. 3m, 1m, 3m, 1m, %m

edian 8

e.

Bumber /f goals 1 2 3 %

reuenc( 1 ' % % 3

&otal freuenc( 8 1 K ' K % K % K 3 8edian 8 Bumber of goals of the ::: th reuenc(

8 goalsf.

rade A 5 ; E

Bumbers of students 12 - 0 11

&otal freuenc( 8 K 12 K - K 0 K 11 8

edian 8

C. *ean Ha. 21, 3, 2', 2, 31, 3%

ean 8

b. ,% , ,0 ,' , , '

ean 8

c.

ass "g$ ' 0

reuenc( ' 2

ean 8

161


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Module PMR

%. &he pictogram in ;iagram belo shos the number of durians sold b(three fruits sellers. 5alculate the total number of durians sold b( three ofthem.

7amad5hong

uthu

represents 1 durians

'.

&he line graph shos the sale of chocolate ca)es in a ee). ind thedifference beteen the highest sales and the loest sales in the ee).

.

9 (a6

$riThredTes*o n

@9

89

C9

:9

9

&he line graph shos the number of cars par)ed at @eti ?umut over a

period of five da(s. Each car is charged * 0 per da(. Ghat is the totalcollection for the five da(sM

2 1 3 % 0 0 ' %' % 3 1 2 1 1 % 2 %

&he data shos the scores obtained b( 2 pupils in a game.

a$ ( using the data given, complete the folloing table.

7core 1 2 3 % ' 0

reuenc(

b$ 7tate the mode

162

MDays

20

4060

80

100

Sale

ofcoco

la!e

ca"es

0 # $ # % S S


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Module PMR

0. &he mean of ', (, 2(, 3(, 12,12, 1', 2 is 11. ind the value of (.

-. 5alculate the difference beteen mean and median of the numbers

', 11, 2, 23, 1', -, 3, 10.

1. ind the difference beteen mode and median of the numbers%, 3, ', -, -, -, 3, , -.

11. iven that the mean of 3, ', 1, 2, x, 1', ( is 0, find the value of xK(.

12. &he pie chart shos the number of coloured balls in a store.

5alculate the angle of the sector hich represents blac) balls.

13. ;iagram belo is a line graph hich shos the number of residents in aton from 1-- to 2%.

a$ ;uring hich (ear did the number of residents increase the mostM

b$


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Module PMR

estions 'ased on 4*R $ormat

@. &he pictogram belo shos the number of cars sold b( 7(ari)at aju@a(a from (ear 2% to 2.

899:

899

Represents ampung.

b$ ind the percentage of female to the hole populationM

166


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Module PMR

>.

%lass Raden *a6ang "ahagia Adil Fr IGlas

%ollectionR*/ 899 C99


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Module PMR

D.

899@

8998

899C

899:

899

Date post:	02-Jun-2018
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4. Transformation Dan Statistik

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