SEMINAR TEKNIK SEMINAR TEKNIK MENJAWAB MENJAWAB

SOALAN PMRSOALAN PMR

MATEMATIKMATEMATIK(KERTAS 1 &2)(KERTAS 1 &2)

PEMBAHAGIAN MARKAH PEMBAHAGIAN MARKAH

KERTAS 1 : (OBJEKTIF – Aneka Pilihan)KERTAS 1 : (OBJEKTIF – Aneka Pilihan)Bil soalan Bil soalan : 40 : 40 MarkahMarkah : 40 : 40 TopikTopik : Semua: SemuaMasaMasa : 1 Jam 15 Minit: 1 Jam 15 Minit* Boleh Guna Kalkulator* Boleh Guna Kalkulator

KERTAS 2 : (SUBJEKTIF)KERTAS 2 : (SUBJEKTIF)Bil soalan Bil soalan : 20 : 20 MarkahMarkah : 60 : 60 TopikTopik : Pilihan: PilihanMasaMasa : 1 Jam 45 Minit: 1 Jam 45 Minit

* Tidak Boleh Guna Kalkulator* Tidak Boleh Guna Kalkulator

JUMLAH MARKAH: K1 + K2 = 100 MARKAHJUMLAH MARKAH: K1 + K2 = 100 MARKAH

TOPIK² PENTING MATEMATIK TOPIK² PENTING MATEMATIK (Kertas II PMR)(Kertas II PMR)

BILBIL TOPIKTOPIKPMRPMR

0505 0606 0707 0808 0909

1 INTEGERS / DIRECTED NUMBERSINTEGERS / DIRECTED NUMBERS

2 WHOLE NUMBERS / FRACTIONS / WHOLE NUMBERS / FRACTIONS / DECIMALSDECIMALS

3 SQUARES, SQUARE ROOTS, CUBES SQUARES, SQUARE ROOTS, CUBES & CUBE ROOTS& CUBE ROOTS

4 LINEAR EQUATIONS I & IILINEAR EQUATIONS I & II

5 ALGEBRAIC EXPRESSIONS I,II,IIIALGEBRAIC EXPRESSIONS I,II,III

6 TRANSFORMATIONS I & IITRANSFORMATIONS I & II

7 STATISTICS I & IISTATISTICS I & II

8 SOLID GEOMETRY I SOLID GEOMETRY I

9 LOCI IN TWO DIMENSIONSLOCI IN TWO DIMENSIONS

10 GEOMETRICAL CONSTRUCTIONSGEOMETRICAL CONSTRUCTIONS

TOPIK² PENTING MATEMATIK TOPIK² PENTING MATEMATIK (Kertas II PMR)(Kertas II PMR)

BILBIL TOPIKTOPIKPMRPMR

0505 0606 0707 0808 0909

11 SCALE DRAWINGSCALE DRAWING

12 ALGEBRAIC FORMULAEALGEBRAIC FORMULAE

13 INDICESINDICES

14 LINEAR INEQUALITIESLINEAR INEQUALITIES

15 GRAPHS OF FUNCTIONSGRAPHS OF FUNCTIONS

16 TRIGONOMETRYTRIGONOMETRY

SOALAN DAN SKIMA JAWAPAN SOALAN DAN SKIMA JAWAPAN PEPERIKSAAN PEPERIKSAAN

PERCUBAAN PMR 2010PERCUBAAN PMR 2010

INTEGERS / DIRECTED INTEGERS / DIRECTED NUMBERSNUMBERS

1

YearYear TopicTopic

20052005 Directed Numbers Directed Numbers

20062006 Directed NumbersDirected Numbers

20072007 IntegersIntegers

20082008 --

20092009 Directed NumbersDirected Numbers

[ 2 marks][ 2 marks]

Q.1: Directed NumbersQ.1: Directed Numbers

47

428

416

Calculate the value of )4(4

3128

Answer :Answer :

20

(1)

(1)

WHOLE NUMBERS / WHOLE NUMBERS / FRACTIONS / DECIMALSFRACTIONS / DECIMALS

2

YearYear TopicTopic

20052005 Whole Numbers Whole Numbers

20062006 FractionsFractions

20072007 DecimalsDecimals

20082008 FractionsFractions

20092009 --

[ 2 marks][ 2 marks]

2

5242.11

Q.2: DecimalsQ.2: Decimals

express the answer as a decimal5

2242.11 Calculate the value of

Answer :Answer :

542.11

42.6

(1)

(1)

SQUARES, SQUARE ROOTS, SQUARES, SQUARE ROOTS, CUBES, CUBE ROOTSCUBES, CUBE ROOTS

3

[ 1 + 2 =3 marks][ 1 + 2 =3 marks]

2005 2006 2007 2008 2009

Q.3: Squares,Square Q.3: Squares,Square Roots, Cubes Roots, Cubes & Cube Roots & Cube Roots

3 027.0(a)

Answer :Answer :

)3.0()3.0()3.0(3

3.0 (1)

Q.3: Squares,Square Q.3: Squares,Square Roots, Cubes Roots, Cubes & Cube Roots & Cube Roots

23

(b)

Answer :Answer :

9

(1)

(1)

23 647

2)4(7

247

ENLARGEMENT (SIMILARITY)ENLARGEMENT (SIMILARITY)

4

[ 2 marks][ 2 marks]

2005 2006 2007 2008 2009

- -

Q.4: Enlargement Q.4: Enlargement (Similarity)(Similarity)

In Diagram 4, triangle XYZ and triangle FGH are similar.

GFH

HFG

State:(a) The angle in triangle FGH which corresponds to

YXZ

Answer :Answer :

70°70° 70°70°

or GFH

YXZ

Q.4: Enlargement Q.4: Enlargement (Similarity)(Similarity)

In Diagram 4, triangle XYZ and triangle FGH are similar.

State:(b) The side of triangle FGH which corresponds to the side XY of triangle XYZ.

Answer :Answer :

70°70° 70°70°

FG or GF

TRIGONOMETRYTRIGONOMETRY

5

[ 3 marks][ 3 marks]

2005 2006 2007 2008 2009

Q.5: TrigonometryQ.5: Trigonometry

In Diagram 5, JKL and LMN are right angled triangles.

(a) Calculate the length, in cm, of LM.

Answer :Answer :

Given that:

5

3sin x

(3)(3)

(5)(5)

(4)(4)

6 cm

8 cm

KL = 6cm LM = KM – KLLM = 19cm – 6cmLM = 13cm

(1)

(1)

13 cm

5 cm5 cm

Q.5: TrigonometryQ.5: Trigonometry

5

3sin x

In Diagram 5, JKL and LMN are right angled triangles.

Given that:

(3)(3)

(5)(5)

(4)(4)

6 cm

8 cm

(b) Find the value of tan y°.

Answer :Answer :

13 cm

12

5tan y (1)

TRANSFORMATIONS TRANSFORMATIONS

6 & 7

YearYear TopicsTopics MarksMarks

20052005 ReflectionsReflections Translations &Translations &

RotationsRotations2+4=62+4=6

20062006 ReflectionsReflections TranslationsTranslations 2+2=42+2=4

20072007 ReflectionsReflections -- 22

20082008 ReflectionsReflections RotationsRotations 44

20092009 ReflectionsReflections -- 22

Q.6: Transformation I Q.6: Transformation I (Translation)(Translation)

Diagram 6 in the answer space shows polygon Q drawn on the square

grid of 1 unit. Q‘ is the image of Q under a translation . Draw the image of Q.

6

7

Q’

Q.7: Transformation II Q.7: Transformation II (Enlargement)(Enlargement)

Diagram 7 shows two triangles, ABC and A’B’C’, drawn on a CartesianPlane. A’B’C’ is the image of ABC under transformation G. Describe in full transformation G.

Answer :Answer :

(13,0)

Transformation G is an Enlargement

Scale factor

Centre

(1)

(1)

(1)2

1

(13,0)

ALGEBRAIC EXPRESSIONSALGEBRAIC EXPRESSIONS

8 & 9

2 marks / [ 1 + 2 =3 marks]2 marks / [ 1 + 2 =3 marks]

2005 2006 2007 2008 2009

Q.8: Algebraic Expressions Q.8: Algebraic Expressions II (Simplification)II (Simplification)

Simplify each of the following expressions:

(a)9 – 2(a + 3)

(b) 2(3p – q) – (4q – p)

Answer :Answer :

(a) 9 – 2a – 6

= 9 – 6 – 2a

= 3 – 2a

(b) 6p – 2q – 4q + p

= 6p + p – 2q – 4q

= 7p – 6q(1)

(1)

(1)

Q.9: Algebraic Expressions Q.9: Algebraic Expressions III (Factorisation)III (Factorisation)

Factorise completely:

81m² – 9

Answer :Answer :

9(9m² – 1)

= 9(3m – 1)(3m + 1)

= 9(3²m² – 1²)

(1)

(1)

LOCI IN TWO DIMENSIONSLOCI IN TWO DIMENSIONS

10

[5 marks][5 marks]

2005 2006 2007 2008 2009

Q.10: Loci in Two DimensionsQ.10: Loci in Two DimensionsDiagram 10 in the answer space shows two intersecting lines, PQ and QR.

(a) On the diagram, construct (i) the locus of point X which moves such that it is equidistant from the lines PQ and line QR.

(ii) the locus of point Y which moves such that it is equidistant from the point Q and point R.

(iii) Hence mark with the symbol the intersection of the locus X and the locus Y.

Q.10: Loci in Two DimensionsQ.10: Loci in Two Dimensions

Answer:

1 2

3Locus X

1

Q.10: Loci in Two DimensionsQ.10: Loci in Two Dimensions

Answer:

Locus X

4

4

Locus Y

5

Q.10: Loci in Two DimensionsQ.10: Loci in Two Dimensions

Answer:

Locus X

Locus Y

Point of intersection

GEOMETRICAL GEOMETRICAL CONSTRUCTIONSCONSTRUCTIONS

11

[5 marks][5 marks]

2005 2006 2007 2008 2009

Q.11: Geometrical Q.11: Geometrical ConstructionsConstructions

11 (a) Diagram 11.1 shows a quadrilateral ABCD.

Measure using a protractor.ABC

AnswerAnswer : :

113° ± 1°

Q.11: Geometrical Q.11: Geometrical ConstructionsConstructions

11 (b) Diagram 11.2 shows a triangle PQR.

By using only a ruler and a pair of compasses, construct triangle PQR on the straight line PQ provided in the answer space.

Q.11: Geometrical Q.11: Geometrical ConstructionsConstructions

Answer :Answer :11 (b)

21

3

4

ALGEBRAIC EXPRESSIONS ALGEBRAIC EXPRESSIONS IIIIII

(Algebraic Fractions)(Algebraic Fractions)

12

[3 marks][3 marks]

2005 2006 2007 2008 2009

Q.12: Algebraic Expressions Q.12: Algebraic Expressions III (Algebraic Fractions)III (Algebraic Fractions)

p

p

3

)36(6

p

p

3

69

p

p

3

366

Express as a single fraction in its simplest form.

Answer :Answer :

p

p

3

)23(3

p

p

p 3

362

p

p23 (1)

(1)

(1)

ALGEBRAIC FORMULAEALGEBRAIC FORMULAE

13

[3 marks][3 marks]

2005 2006 2007 2008 2009

Q.13: Algebraic Q.13: Algebraic FormulaeFormulae

52 mp

m

)52( mpm

pmpm 52

Given that , express m in terms of p

Answer :Answer :

mp

m25

pmpm 52

ppm 5)21(

p

pm

21

5

(1)

(1)

(1)

LINEAR EQUATIONSLINEAR EQUATIONS

14

[1 + 2 = 3 marks][1 + 2 = 3 marks]

2005 2006 2007 2008 2009

Solve each of the following equations:

(a)

(b)

Q.14: Linear EquationsQ.14: Linear Equations

x315

10)2(4 yy

315

x

Answer :Answer :

(a) (b)

3

15x

5x

4y – 8 = y + 10

4y – y = 10 + 8

3y = 18

y = 18 ÷ 3

y = 6(1)

(1)

(1)

INDICESINDICES

15 & 16

[1 + 2 = 3 marks] / [3 marks][1 + 2 = 3 marks] / [3 marks]

2005 2006 2007 2008 2009

Solve each of the following equations:

(a)

(b)

Q.15: IndicesQ.15: Indices

34y 25234 baba

34yAnswer :Answer :

(a) (b)

12y

252324 baba

2568 baba

2658 ba83 ba

(1) (1)

(1)

Q.16: IndicesQ.16: Indices

4

18

4

12

4

14

342

41

824 342

22

11 342

Evaluate:

Answer :Answer :

922

1

9

(1)

(1)

(1)

LINEAR INEQUALITIESLINEAR INEQUALITIES

17

[3 marks][3 marks]

2005 2006 2007 2008 2009

List all the integer values of x which satisfy both the inequalities

and

Q.17: Linear InequalitiesQ.17: Linear Inequalities

54x 23

x

9x

Answer :Answer :

45x 32x

6x

96 x

x=7,8,9

(1) (1)

(1)

STATISTICS I (Charts)STATISTICS I (Charts)

18 & 19

2 marks / [ 1 + 2 =3 marks]2 marks / [ 1 + 2 =3 marks]

YearYear TopicTopic

20052005 Pie ChartPie Chart

20062006 Line GraphLine Graph

20072007 Vertical Bar ChartVertical Bar Chart

20082008 Pie ChartPie Chart

20092009 Horizontal Bar ChartHorizontal Bar Chart

Q.18: Statistics II (Pie Q.18: Statistics II (Pie Charts)Charts)

)10012590(360 x

45

315360

x

x)10012590(360 x

Diagram 18 is a pie chart which shows the distribution of a number of students who scored Grade A,B,C or D in their Mathematics Test.

(a) State the mode of the data

(b) Calculate the percentage of the students Scored grade D in the test.

Answer:

(a)

(b)

Grade C

5.12100360

45 %

(1)

(1)

(1)

Table 19 shows the favourite hobbies of a group of studentsTable 19 shows the favourite hobbies of a group of students

Q.19: Statistics I (Horizontal Q.19: Statistics I (Horizontal Bar Chart)Bar Chart)

On Diagram 19 in the answer space, the information for Reading is shown in the bar chart. On Diagram 19 in the answer space, the information for Reading is shown in the bar chart. Complete the bar chart to represent all the information in Table 19.Complete the bar chart to represent all the information in Table 19.

Q.19: Statistics I (Horizontal Q.19: Statistics I (Horizontal Bar Chart)Bar Chart)

Answer:Answer:

GRAPHS OF FUNCTIONSGRAPHS OF FUNCTIONS

20

[3 marks][3 marks]

2005 2006 2007 2008 2009

Table 20 shows the values of two variables, x and y, of a function.Table 20 shows the values of two variables, x and y, of a function.

Q.20: Graphs of Q.20: Graphs of FunctionsFunctions

Table 20Table 20Jadual 20Jadual 20

x -4-4 -3-3 -2-2 -1-1 00 11 22

y 66 22 00 22 88 1818 3232

The x-axis and the y-axis are provided on the graph paper on page 21

(a)(a) By using a scale of 2 cm to 5 units, complete and label the y-axis By using a scale of 2 cm to 5 units, complete and label the y-axis

(b) Based on Table 20, plot all the points on the graph paper (b) Based on Table 20, plot all the points on the graph paper

(c) Hence, draw the graph of the function.(c) Hence, draw the graph of the function.

Q.20: Graphs Q.20: Graphs of Functionsof Functions

Answer:Answer:

5

10

15

20

25

30

35

40

Q.20: Graphs Q.20: Graphs of Functionsof Functions

Answer:Answer:

SESI SOAL JAWABSESI SOAL JAWAB

Sekian, Terima KasihSekian, Terima Kasih