Teknik Menjawab Soalan Matematik PMR

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E B C A D E F A G E O A B D C F E BENGKEL MATEMATIK ( PMR ) “TEKNIK MENJAWAB SOALAN DALAM KERTAS 2” Oleh LAI JUN SIEW SMK SUNGAI MAONG 2008

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Transcript of Teknik Menjawab Soalan Matematik PMR

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BENGKEL

MATEMATIK

( PMR )

“TEKNIK MENJAWAB

SOALAN DALAM KERTAS 2”

Oleh LAI JUN SIEW SMK SUNGAI MAONG 2008

Teknik Menjawab Soalan Kertas 2 matematik PMR

Lai J S , SMK Sg Maong, Kuching. 2009 1

Format Pentaksiran Matematik Baru PMR (2004)

Bil Perkara Kertas 1 ( 50/1 )

Kertas 2 ( 50/2 )

1 Jenis Instrumen Ujian Objektif

Ujian Subjektif

2 Jenis Item Aneka Pilihan dan

Gabungan

Respons Terhad

( TunjukkanLangkah Kerja dan jawapan )

3 Bilangan Soalan 40 soalan ( Jawab semua)

20 soalan ( Jawab semua)

4 Jumlah Markah 40

60

5 Tempoh Ujian 1 jam 15 minit

1 jam 45 minat

6 Wajaran Konstruk

Pengetahuan - 40% Kemahiran - 60%

Pengetahuan - 30% Kemahiran - 65% Nilai - 05%

7 Cakupan Konteks

Semua bidang

pembelajaran dari Tingkatan 1 hingga

Tingkatan 3

Semua bidang pembelajaran dari Tingkatan 1 hingga Tingkatan 3

R : S : T = 5 : 4 : 1

R : S : T = 5 : 2 : 3

8

Aras Kesukuran Rendah - R Sederhana - S Tinggi - T

Keseluruhan

R : S : T = 5 : 3 : 2

9 Alatan

Tambahan

a. Kalkulator Saintifik b. Buku Sifir Matematik c. Alatan Geometri

a. Buku Sifir Matematik b. Alatan Geometri

Teknik Menjawab Soalan Kertas 2 matematik PMR

Lai J S , SMK Sg Maong, Kuching. 2009 2

ANALYSIS OF PMR MATHEMATICS PAPERS ( 2004 – 2008 )

Number of Questions 2004 2005 2006 2007 2008

Topics

P1 P2 P1 P2 P1 P2 P1 P2 P1 P2

1 Whole Numbers 0 0 1 1 1 0 1 0 1 2 Number Patterns and

Sequences 3 0 3 0 2 0 2 0 1

3 Fractions 1 1 0 0 2 1 1 0 2 1

4 Decimals 1 0 0 0 0 0 1 1 1

5 Percentages 1 0 1 0 1 0 1 0 6 Integers and Directed

Numbers 1 1 1 1 0 1 1 1 1

7 Algebraic Expressions 0 3 0 3 0 3 0 3 3

8 Basic Measurements 3 0 0 0 1 0 1 0 3 9 Lines and Angles 1 0 1 0 2 0 0 0 1

10 Polygons 2 0 5 0 3 0 4 0 4

11 Perimeter and Area 1 0 0 0 2 0 3 0 2 12 Solid Geometry 2 1 3 0 4 0 4 1 3

13 Squares, Square Roots, Cubes and Cube Roots

0 1 0 1 0 1 0 1 1

14 Linear Equations 0 1 1 1 1 1 1 1 1 1

15 Ratios, Rates and Proportions 3 0 4 0 3 0 3 0 2 16 Pythagoras’ Theorem 3 0 3 0 1 0 1 0 1 17 Geometrical Constructions 0 1 0 1 0 1 0 1 1

18 Coordinates 2 0 2 0 3 0 2 0 2 19 Loci in Two Dimensions 1 1 1 1 1 1 1 1 1 1

20 Circles ( Area and Angles) 6 0 6 0 5 0 5 0 5

21 Transformations 1 2 1 2 1 2 1 3 1 4

22 Statistics 5 2 4 1 4 2 4 2 4 2

23 Indices 0 2 0 2 0 2 0 1 1

24 Algebraic Formulae 0 1 0 1 0 1 0 1 1

25 Scale Drawings 1 0 0 1 0 1 0 0 1

26 Linear Inequalities 1 1 1 1 1 1 1 1 1 1

27 Graphs of Functions 1 1 2 1 2 1 2 1 1

28 Trigonometry 0 1 0 2 0 1 1 2 1

40 20 40 20 40 20 40 20 40 20 Total

Teknik Menjawab Soalan Kertas 2 matematik PMR

Lai J S , SMK Sg Maong, Kuching. 2009 3

THE IMPORTANT TOPICS FOR PAPER 2

1. Fractions 2. Directed Numbers 3. Squares, Square Roots, Cubes and Cube Roots 4. Algebraic Expressions 5. Statistics 6. Linear Equations 7. Indices 8. Algebraic Formulae 9. Trigonometry 10. Transformations ( Reflections, Translations,

Roations and Enlargements 11. Inequalities 12. Solid Geometry ( Net of Solids) 13. Congruency ( Transformations) 14. Graphs of Functions 15. Geometerical Constructions 16. Loci In Two Dimensions 17. Scale Drawing 18. Angles in Circles and Angles Between Parallel Lines 19. Coordinates

THE IMPORTANT TOPICS FOR PAPER 1

1. Whole Numbers 2. Number Patterns and Sequences 3. Fractions 4. Decimals 5. Percenages 6. Integers and Directed Numbers 7. Algebraic Expressions 8. Basic Measurements 9. Lines and Angles 10. Polygons 11. Perimeter and Area 12. Solid Geometry – Volumes of Shapes 13. Squares, squares Roots, Cubes and Cubes Roots 14. Linear Equations 15. Ratio, Rates and Proportions 16. Pythagoras’ Theorem 17. Scale Drawings 18. Coordinates 19. Loci in Two Dimensions 20. Circles – Angles , Area & Circumference 21. Transformations 22. Statistics 23. Indices 24. Algebraic Formulae 25. Linear Inequalities 26. Graphs of Functions 27. Trigonometry

Teknik Menjawab Soalan Kertas 2 matematik PMR

Lai J S , SMK Sg Maong, Kuching. 2009 4

Guidelines For Answering Questions in Mathematics paper 2 ( PMR )

1. FRACTIONS.

1. )2

1

4

11(

12

1−× 2. )

6

5

2

1(

3

13 +÷ 7)

5

13

2

14( .3 ÷+

2. DIRECTED NUMBERS.

A. Calculate the following. Give your answers as decimals.

3. DECIMALS

3. Squares and square roots, cubes and cube roots.

Examples (Calculation can be carried out without using calculators):

12 = 1 2

2 = 4 3

2 = 9 4

2 = 16

52

= 25

6

2 = 36 7

2 = 49 8

2 = 64

92 = 81 10

2 = 100 112=121 12

2 = 144 13

2 = 169 14

2 = 196 15

2 = 225 16² = 256 Squares

172

= 289 18² = 324 19² = 361 202

= 400 252

= 625

1 = 1 4 = 2 9 = 3 16 = 4 25 = 5 36 = 6 49 = 7 64 = 8

81 = 9 100 = 10 121 =11 144 =12 169 =13 196 =14 225 =15 256 =16 Square

Roots

289 =17 324 =18 361 =19 400 =20 625 =25

13 = 1 23 = 8 33 = 27 43 = 64 53 = 125 63 = 216 73 = 343 83= 512 Cubes

93= 729 10

3=1000

2)4

3(45.0 1 +−−− )7(4.0

2

12 2 −−÷−

4 0 .1 82 .

0 .9

×

3.2 5

0.08

16.0 =

0.08

1600 =

8

=200

×1.

16

1

4

3

12

1

)4

2

4

5(

12

1

)2

1

4

5(

12

1

)2

1

4

1(1

12

1 1.

=

×=

−×=

−×=

−×

2

12

2

5

8

6

3

10

6

8

3

10

)6

5

6

3(

3

10

)6

5

2

1(

3

13 2.

=

=

×=

÷=

+÷=

2.3

20.3

20.750.45

2)4

3(0.45

=

+=

++−=

+−−−

3 .2 51 .

0 .0 8

×=

3. −24 ÷ 28 − 14 =

Teknik Menjawab Soalan Kertas 2 matematik PMR

Lai J S , SMK Sg Maong, Kuching. 2009 5

3 1 =1 3 8 =2 3 27 =3 3 64 =4 3125 =5 3 216 =6 3 343 =7 3 512 =8 Cube

Roots 3 729 =9 31000 =10

A. Find the value of

3 343.0 .1 − 1.69 .2

3 729.0 .3

3 343 7=

3 0.343 0.7=

3 0.343 0.7− = −

B. Calculate the value of :

1. 169)7( 2+− 2.

232 5)4121( ÷− 3. 32

729

646 ×

62

1349

1697)( 2

=

+=

+−

4. ALGEBRAIC EXPRESSIONS

1. Simplify each of the expressions to its simplest form.

(i) 10x –3y + ( 2x – 3)² (ii) ( y– 5 )² – 25 + 4y (iii) 5(xy – 4 )– 8 ( xy – 2 )

10x – 3y + ( 2x – 3)² = 10x – 3y + (2x – 3) (2x – 3 )

= 10x – 3y + 4x² – 6x – 6x + 9 = 4x² – 2x – 3y + 9

2. Factorize each of the following expressions.

(i) 4p – 8pq (ii) – y(2x – y) + 5xy (iii) (m–3)² – ( 6 – 2m)

4p – 8pq = 4p(1–2q) – y(2x – y) + 5xy = –2xy + y² + 5xy

= 3xy + y² = y ( 3x + y )

3. m.lowest ter itsin fraction single a as 5

832 Express (i)

2

xy

x

y

x −−

5xy

87x

5xy

83x10x

5xy

8)(3x10x

5xy

8)(3x

5xy

5x2x

5xy

83x

y

2x

2

22

22

22

+=

+−=

−−=

−−

×

×=

−−

2

1 2 5( ii) E x p res s a s a s in g le frac tio n in i ts lo w es t te rm .

5 1 0

n

n n

−−

Teknik Menjawab Soalan Kertas 2 matematik PMR

Lai J S , SMK Sg Maong, Kuching. 2009 6

5. STATISTICS

a. Construct or complete the pictogram, bar chart, line graph and pie chart base on the information given.

1. The table below shows the number of computers sold in a computer retail shop in the period of the first four

months of 2004.

Month Number of computers

January 10

February 20

March 12

April 18

2.

Day Mon Tue Wed Thurs Fri

Number of watermelons 100 x 90 140 160

The table above shows the number of watermelons harvested in a farm from Monday to Friday. If the total number of

watermelons harvested for five days is 610,

(i) Find the value of x .

(ii) Complete the pictogram by drawing the correct numbers of for Tuesday and Thursday.

Monday

Tuesday

Wednesday

Thursday

Friday

3. The diagram below shows the scores obtained by 15 police cadets in a shooting competition.

(a) Using the data, complete the frequency table .

Score Frequency

1

2

3

4

Draw a bar chart / pie chart to represent all the information

in the table on a grid

Number of Computers

20

o

o

o

o

120

10872

60

AprilMarch

February January

10360 60

60

20360 120

60

12360 72

60

18360 108

60

o

o

o

o

× =

× =

× =

× =

1 , 2 , 4 , 3 , 1 , 2 , 4 , 1 , 3 , 2 , 1 , 3 , 1 , 1 , 3

(b) State the mode.

(c) State the median.

Teknik Menjawab Soalan Kertas 2 matematik PMR

Lai J S , SMK Sg Maong, Kuching. 2009 7

6. Linear Equations

1. Solve the equation 5k = 3k – 8 2. Solve the equation 6x – 3 = 3(4+x) 3. Solve the equation k + 5

2(8 – 3k) = - 19

5k = 3k – 8 6x – 3 = 3(4 + x) 5k – 3k = –8 6x – 3 = 12 + 3x 2k = – 8 6x – 3x = 12 + 3

k = –4 3x = 15 4. Solve the equation 8y – 2 = 3y + 8

5. Solve the equation 7 234

xx− = +

7. Indices

1. Simplify (rs5−

)3

× s15

2 Simplify (3pq3

)2

× (q3

)4

÷ p6

q8

− +

× =

=

=

-5 3 15 3 15 15

3 0

3

(rs ) s r s

r s

r

+ −

× ÷ = × ÷

=

=

3 2 3 4 6 8 2 2 6 12 6 8

2-6 6 12 8

-4 10

(3pq ) (q ) p q 3 p q q p q

9p q

9p q

3. Simplify 2m4

× 9m5−

4. Find the value of

23

2 4

5

3 81

9

×

5. Simplify ( )2

2 1 3 44 2m n m n−

÷ 6. Given that 43 81x−

= , calculate the value of x.

8. Algebraic Formulae

1. Given that p – 2

r =

p

r, express p in terms of r. 2. Given that 2

4k

t

−= , express k in terms of t.

=

=

= +

=

=

=

2 p p -

r r

rp rp - 2

r

pr 2 p

pr - p 2

p( r - 1) 2

2 p

( r - 1)

2 2

2

2

2 4

2 4

2 4

2 16

16 2

k -

t

k - t

( k - ) ( t)

k - t

k t

=

=

=

=

= +

3. Given that 3 ( 2 7 )5

m

n

−= , express m in terms of n

5. Given that A = 1( )

2a b h+ , express a in terms of A, b and h

9. Trigonometry

adjacent side

θ

hypotenuse opposite side

A B

C

tan θ = sideadjacent

side opposite

sin θ = hypotenuse

side opposite

cos θ = hypotenuse

sideadjacent

5

=

=

15x

3

x

6k

13-

78-k

-7813k-

392

15k-2k

20192

15k-k

192

15k-20k

193k)(82

5k 8.

=

=

=

−=

−−=

−=+

−=−+

4. Given that p = 25 – 3q2 , express q in terms of p

6. The acceleration of a moving object is given by a =

v u

t

−. Find the value of u if a = 21, v = 81 and t = 2.

Teknik Menjawab Soalan Kertas 2 matematik PMR

Lai J S , SMK Sg Maong, Kuching. 2009 8

1. In the diagram below, given that 4

3sin =x , calculate the value of cos y.

cm 12LM

4

48LM

3164LM

4

3

16

LM 1.

=

=

×=

=

cm 13LN

169LN

169

25144

512LN 222

=

=

=

+=

+=

Hence =5

, cos y13

2. In the diagram below, PURQ is a rectangle. QRS and PTS are straight lines. T is the midpoint of UR..

(i) Given that ,5

4cos =x calculate the length of QR .

(ii) Hence, calculate the value of tan y

cm 10PR

4

40PR

854PR

5

4

PR

8 2.

=

=

×=

=

cm 6

36QR

36

64-100

810QR 222

=

=

=

=

−=

Hence

= =

= ×

=

=

=

RS QR 6cm

QS 6 2

12 cm

8, cos y

12

2

3

10. Transformations There are four types of transformations – translation, reflection, rotation and enlargement.

A. Translation

Students must be able to:

a. describe a translation in the form

b

a, a is the movement parallel to the x-axis and b is the

movement parallel to the y-axis

b. write the coordinates of the image of a point under a translation

c. draw the image when an object is under a certain translation.

Guidelines for questions in Paper 2 (subjective) in PMR

1. In the diagram below, the ∆ ABC is under a translation

−4

3.Draw the image of ∆A’B’C’ of the object ∆ABC.

B Reflection

Students must be able to :

a. determine and draw the image of an object, given the axis of reflection and vice versa

b. determine and draw the axis of reflection, given the image and the object.

c. write the coordinates of the image of a point under a reflection.

P

Q R S

T

U

y

x

8 cm

16 cm

5 cm

x

y

K L

M N

2. In the graph below, Q is the image of

the object P under a translation

k

h .

Describe the translation

k

h .

2

4

6

8

2

4 6 8

10

0 x

P

Q

y

2

2

4 6 -2 -4

4

6

-2

-4

x

y

0

A

B

C

Teknik Menjawab Soalan Kertas 2 matematik PMR

Lai J S , SMK Sg Maong, Kuching. 2009 9

Guidelines for questions in Paper 2 (subjective) in PMR

1. In the Cartesian plane as shown, triangle A’B’C’ is

the image of triangle ABC under a reflection in the

line M.

(i) Draw the image of triangle A’B’C’.

(ii) Write down the coordinate of the point B’

C Rotation

Students must be able to:

a. determine the image of a point or an object under a rotation

b. describe a rotation given the object and the image

- must have (i) centre of rotation

(ii) angle of rotation

(iii) direction of rotation - example: 900 clockwise / 900 anticlockwise

Guidelines for questions in Paper 2 (subjective) in PMR 1.

2. The diagram is drawn on square grids. Draw the

image of triangle P under a anticlockwise rotation

of 90° about a point C.

D. Enlargement

Students must be able to:

1. determine the centre of enlargement.

2. determine and draw the image of an enlargement

3. find the scale factor (k) of an enlargement

4. describe an enlargement, given the object and the image

- must have (i) centre of enlargement

(ii) scale factor (k)

k = object of side ingcorrespond theoflength

image of side a oflength

OR k = distance of a point on the im age from centre of enlargem ent

distance of the corresponding point on the object from the centre of enlargem ent

x

2. The diagram below is drawn on a tessellation of congruent

triangles . The shape A’B’C’D’ is the image of the shape ABCD

under a reflection. Draw the axis of the reflection.

-4

-4 -2 4 2

4

2

0 -2

y

x

-6

6

-6 6

B

C M

A

P

C

A B

D

A’

B’

D’

C

C

2. In the diagran below, P’Q’R’S’ is the image of

PQRS under a 90° clockwise rotation about a

point C.

(i) Mark the point C in the diagram below.

(ii) Write down the coordinate of the point C

2 4

2

x

y

-2

-2

0

Q p

S

P’

Q’

S’

R

-4 R’

Teknik Menjawab Soalan Kertas 2 matematik PMR

Lai J S , SMK Sg Maong, Kuching. 2009 10

Guidelines for questions in Paper 2 (subjective) in PMR

1. The diagram below is drawn on square grids. Triangle PQR is

under an enlargement at centre C with the scale factor 2.

Draw the image triangle P’Q’R’ of the triangle PQR in the

diagram.

11. Inequalities 1. To determine the possible values of an unknown :

(a) Given that x ≥ – 2 and x is an integer, state the possible values of x.

x = –2, –1, 0 , 1 …….

(b) Given that 5 < x ≤ 9 and x is an integer, state the values of x.

2. Solve the following linear inequalities .

(i) x – 7 < – 4 (ii) x + 4 > 9 (iii) 5x – 7 ≥ 4x + 3

7 4

7 4

3

x

x

x

− < −

< −

<

3. Solve the following linear inequalities .

(i) 32

>x

(ii) 4x ≤ 20 (iii) 24

<−x

(iv) −−−− 2x > 8

x > 3 × 2

x >6

4. Solve the following simultaneous linear inequalities.

List all the values of x which satisfy both the inequalities, given that x is an integer.

(i) x ≤ 3 and 23

4−>

−x (ii) 5 – 2 x ≤ 3 and 643

<+x

Solutions :

(i) x ≤ 3 and 23

4x−>

x ≤ 3 ………….. (1)

23

4x−>

− ……….(2)

64x −>−

x > − 6 + 4

x > − 2

thus, − 2 < x ≤ 3

Answer : x = −1, 0, 1, 2, 3

P

R C

Q

C

This is not the final answer yet

This is the final answer where all the values of x are listed

2. In the diagram below, triangle N is the image of

triangle M under an enlargement.

(i) Mark and label the point P in the diagram.

(ii) Write down the coordinates of the point P.

2 0

4

6

6

-2

2

-2

M

N

8

y

Teknik Menjawab Soalan Kertas 2 matematik PMR

Lai J S , SMK Sg Maong, Kuching. 2009 11

12. Solid Geometry

Net (layout) of solids

Students must be able to: a. understand that the net (layout) of a solid is a two-dimensional plan which, when folded, becomes the solid.

b. draw the net of a given solid using the correct scale given in the question.

Guidelines for questions in Paper 2 (subjective) 1. The Diagram shows a right pyramid with a square base.

Draw a full scale the net of the pyramid on the grid in the answer space. The grid has equal squares with sides of 1

unit. (PMR 2004)

2. Prism – consists of four rectangles and two triangles

3. Other solids and nets

(a) cube – consists of six equal squares

(b) cuboid – consists of six rectangles ( usually three different pairs of congruent

rectangles, depending on the measurements)

(c) prism – consists of three rectangles and two triangles (for the prism shown below)

– triangles may be replaced by other shapes depending on the shape of the cross section

5 unit

6 unit

A B

C D

P

A B

C D

P K

4 units

4 units

3 units

3 cm 3 cm

3 cm

4 units 2 units

6 units

3 units

1 cm

2 cm

4 cm

Teknik Menjawab Soalan Kertas 2 matematik PMR

Lai J S , SMK Sg Maong, Kuching. 2009 12

13. Congruency

Students must be able to

a. read the question carefully before drawing the congruent figure. Make sure that the figure drawn is

congruent to the given figure.

b. draw figure neatly and draw lines with ruler.

Guidelines for questions in Paper 2 (subjective)

1. The Diagram below in the answer space shows

polygon ABCD and straight line PQ drawn on

a grid of equal squares. Starting from the line PQ,

draw polygon PQRS which is congruent to polygon

ABCD

14. GRAPHS OF FUNCTIONS

1. Students must be able to :

Draw the Function Graph when a frequency table is given.

(i) Linear function. .The graph is a straight line;

(ii) Quadratic function. The graph is a curve.

(iii) Cubic function. For example : y = x³. The graph is a curve.

Q P

A B

C

D

Q P

A B

C D

R

S

R

Q

K L

M

N

y = x

y = –x

y = x2 y = – x

2

y = x3

y = – x3

2. The Diagram below in the answer space shows

polygon KLMN and a straight line QR is drawn

a grid of equal squares. Starting from the line

QR, draw polygon PQRS which is on

congruent to polygon KLMN.

Solution:

Teknik Menjawab Soalan Kertas 2 matematik PMR

Lai J S , SMK Sg Maong, Kuching. 2009 13

Guidelines for questions in Paper 2 (subjective) in PMR

2. The table below shows the values of two variables, x and y of a function

x -3 -2 -1 0 1 2 3

y -25 -5 3 5 7 15 35

Draw the graph of the function using a scale 2 cm to 1 unit on the x-axis and 2 cm to 10 unit on the y- axis.

Step 1 : Write down the ordered pairs ( -3, 6 ), ( -2, 1 ), ( -1, -2 ), ( 0, -3 ), ( 1, -2 ), ( 2, 1 ), ( 3, 6 ) Step 2 : Plot the points on the graph paper. Step 3 : Draw a curve to join the points .

Step 4 : Label the graph

1. The table below shows the values of two variables, x and y of a function.

x -3 -2 -1 0 1 2 3

y 6 1 -2 -3 -2 1 6

Draw the graph of the function using a scale 2 cm to 1 unit on the x-axis and 2 cm to 1 unit on the y- axis.

−4 −3 −2 −1 4 3 2 1 0

6

5

4

3

2

1

−1

−2

−4

−3

Teknik Menjawab Soalan Kertas 2 matematik PMR

Lai J S , SMK Sg Maong, Kuching. 2009 14

15. GEOMETRICAL CONSTRUCTIONS Students must be able to (a) construct angles of 30°, 45°, 60°, 90° and 120°

Teknik Menjawab Soalan Kertas 2 matematik PMR

Lai J S , SMK Sg Maong, Kuching. 2009 15

(b) Bisect a given angle Bisect a given line

Guidelines for questions in Paper 2 (subjective) in PMR

1. Set squares and protractor are not allowed for this question.

Diagram 7 shows triangle DEF

(a) Using only a ruler and a pair of compasses, construct Diagram 7 using the measurements given, beginning from the

straight

line DE provided in the answer.

(b) Based on the diagram constructed in (a), measure the distance in cm between oint D anf the point F.

[ 5 marks

]

DF = 7.05 cm

o45

5 cm

ED

F

2 Set squares and protractors are not allowed for this question.

Diagram below shows a triangle DEF.

(a) Starting with the straight line DE , construct the triangle DEF.

(b) (i) Hence, construction a perpendicular line FG from the point F to the line DE which is extended to DEG. DEG is

a straight line.

(ii) Measure the length of FG, in cm

D E

P

R

(e) construct a perpendicular line to

a line through a point on the line (d) construct a perpendicular line

to a line through a point outside

the line.

45o

5 cm

F

D

E

ED

F

E 3 cm

D

5 cm

120°

Teknik Menjawab Soalan Kertas 2 matematik PMR

Lai J S , SMK Sg Maong, Kuching. 2009 16

16. LOCI IN TWO DIMENSIONS

Students must be able to (a) Construct the locus of a point that is moving at the same distance between 2 fixed points , A and B.

A B

(b) Construct the locus of a point that is moving at the same distance from the straight line AB.

S

Q

R

P

A B

(a) Construct the locus of a point that is moving the same distance from a fixed point A.

(d) Construct the locus of a point that is moving the same distance from 2 straight lines which intersect each other.

B

Q

P

A

Guidelines for questions in Paper 2 (subjective) in PMR

1. Diagram below in the answer space shows four squares, PKJN, KQLJ, NJMS and JLRM. W, X, and Y are three moving

points in the diagram.

(a) W moves such that it is equidistant from the straight lines PS and QR. By using the letters in the diagram, state the

locus of W.

(b) On the diagram , draw

(i) the locus of X such that XJ = JN.

(ii) The locus of Y such that its distance from point Q and point S are the same.

(c) Hence, mark with the symbol × all the intersections of the locus of X and the locus of Y.

Q P

S R M

J N

K

L

A B

locus

A B

A B

P Q

R S locus

locus

A

locus

Teknik Menjawab Soalan Kertas 2 matematik PMR

Lai J S , SMK Sg Maong, Kuching. 2009 17

2. Diagram below in the answer space shows PQRS drawn on a grid of equal squares with sides of 1 unit. M, X and Y are

three moving points in the diagram.

(a) M is the point which moves such thst its distance from point Q and point S are the same. By using the letter in the

diagram, state the locus of M.

(b) On the diagram draw

(i) the locus for the point X that is constantly 5 units from the line QR,

(ii) the locus for the point Y that is constantly 7 units from the point R.

(c) Hence, mark with the symbol × the points of intersection of the locus X and the locus Y.

17. SCALE DRAWINGS

Student must be able to knoww the concept

(b) Scale of a drawing = T h e len g th o f d raw in g

T h e len g th o f th e ac tu a l o b jec t

(c) Do scale drawing with the scale 1 : n

( If n < 1 , then the drawing is larger then the actual object ( eg, 1 : 1

2

or 1 : 1

3

)

( If n > 1 , then the drawing is smaller then the actual object ( eg 1 : 2 or 1 : 4 )

(d) The length of drawing = ( the length of the actual object ) × 1

n

Guidelines for questions in Paper 2 (subjective) in PMR

1. Drawing diagram with scale 1 : n .

(a) Perform a scale drawing of the following shape on the square grid with sides of 1 cm by using the scale 1 : 25.

50 cm

75 cm

175 cm

100 cm

(b) Diagram below shows a trapezium. On the grid in the anwser space, draw the diagram using the scale 1 : 200. The

grid

has squares with sides of 1 cm.

(c) Diagram below shows a trapezium. On the grid in the anwser space, draw the diagram using the scale 1 : 1

2

. The

grid has squares with sides of 1 cm.

5 cm

4 cm 2 cm

3 cm

16 m

10 m

6 m

S

1 1175 7 100 4

25 25

1 175 3 50 2

25 25

× = × =

× = × =

Teknik Menjawab Soalan Kertas 2 matematik PMR

Lai J S , SMK Sg Maong, Kuching. 2009 18

18 . ANGLES

(a) Angles In Circle

Student must be able to knoww the concept

Guidelines for questions in Paper 2 (subjective) in PMR

2. In Diagram below, O is the centre of the circle. KLM and LON are straight lines. Find the value of x.

(b) Angles Between Parallel Lines (i) Corresponding angles (ii) Alternate angles (iii) Interior angles

Guidelines for questions in Paper 2 (subjective) in PMR

1. In the diagram below, PQR and QTU are straight lines. Find the value of x.

2. Find the value of x

x = y x = y a = b x + y = 180o

x = y p + q = 180o

1. In Diagram below, O is the centre of the circle ABC, calculate the value of x.

O B x

°

60°

C

105

°

x

40°

K

T

M

N

L

O

x

y

p

q

a

b

x = y p = q x + y = 180

o

140o 115

o

xo

P

Q R

T S

U

60O

40O

xO

A

Teknik Menjawab Soalan Kertas 2 matematik PMR

Lai J S , SMK Sg Maong, Kuching. 2009 19

19. Coordinates

Guidelines for questions in Paper 2 ( subjective) in PMR

(A) The coordinates of a point

1. In the Cartesian plane , points P, Q and R are three vertices

of a rectangle. Write down the coordinates of the fourth

vertex of the rectangle

.

(B) Midpoint

1. In the diagram below, mark and state the coordinates

of the mid point of the straight line joining point P

and point Q .

(C) The Distance Between Two points

1 . Given that point U(-5,-2) and point W(6, -2), find the distance of UW.

(i) By using graph

2. Given that point P(2,3) and point Q(2, -5), find the distance of PQ .

3 – ( – 5 )

= 3 + 5

= 8

----- END ----

The coordinate of the fouth vertex = ( 4 , -3 )

)5 , 1(

)2

10,

2

2(

)2

19,

2

)1()1((

−=

−=

+−+−

(ii) By calculation

6 – ( -5 )

=6 + 5

=11

2. In the figure below

(a) mark the point P (-2, 1),

(b) state the coordinates of point Q.

The mid point =( -2 , 1 )

2

4

6

8

2 0

-2

-4

4 6 x

8 -2 -4

y G

H

2. Given that point G(-1,9) and point H(-1, 1), State the

coordinates of the midpoint of the straight line GH.

(i) By using graph

(ii) By calculation

P

2

4

6

8

0 2 4 6 -2 -4

-2

-4

x

y

4

2

−6 −4 −2 0 2

−2

The coordinate of Q is ___________

Q

4

2

6

−4

−2

0 2

4

P

R

x

y

4

2

-2 -4 2 4

-2

-4 Q

x

y

0

Teknik Menjawab Soalan Kertas 2 matematik PMR

Lai J S , SMK Sg Maong, Kuching. 2009 20