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SEMINAR TEKNIK SEMINAR TEKNIK MENJAWAB MENJAWAB
SOALAN PMRSOALAN PMR
MATEMATIKMATEMATIK(KERTAS 1 &2)(KERTAS 1 &2)
Maahad Ar-Rahmah, Solok Gaung, MelakaMaahad Ar-Rahmah, Solok Gaung, Melaka22 Ogos 201022 Ogos 2010
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
p
3
23 (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