148292108 Teknik Menjawab Soalan Matematik PMR

E

BC

A

D

E

F

A

G

E

O

A

B

D

C

FE

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



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



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 Enlargeme nts 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



Guidelines For Answering Questions in Mathematics paper 2 ( PMR )

1. FRACTIONS.

1. )21

41

1(121

- 2. )65

21

(31

3 + 7)51

321

4( .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 22 = 4 32 = 9 42 = 16

52 = 25

62 = 36 72 = 49 82 = 64

92 = 81 102 = 100 112=121 122 = 144 132 = 169 142 = 196 152 = 225 16 = 256 Squares

172 = 289 18 = 324 19 = 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= 729 103=1000

2)43

(45.0 1 +--- )7(4.021

2 2 ---

4 0 .1 82 .

0 .9

3.2 50.0816.0

=0.081600

=8

=200

=

=

-=

-=

-

=

=

=

=

+=

+

=

+=

++-=

+---

3 .2 51 .

0 .0 8

=

3. 24 28 14 =



3 1 =1 38 =2 3 27 =3 3 64 =4 3125 =5 3 216 =6 3343 =7 3512 =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. 3272964

6

=

+=

+-

4. ALGEBRAIC EXPRESSIONS

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

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

2. Factorize each of the following expressions. (i) 4p 8pq (ii) y(2x y) + 5xy (iii) (m3) ( 6 2 m)

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

832 Express (i)

2

xyx

yx -

-

+=

+-=

--=

--

=-

-

2

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

5 1 0n

n n-

-



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

20

o

o

o

o

120

10872

60

AprilMarch

February January

10360 60

6020

360 1206012

360 726018

360 10860

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.



6. Linear Equations

1. Solve the equation 5k = 3k 8 2. Solve the equation 6x 3 = 3(4+x) 3. Solve the equation k + 52

(8 3k) = - 19

5k = 3k 8 5k 3k = 8 2k = 8 k = 4 4. Solve the equation 8y 2 = 3y + 8

5. Solve the equation 7 234x

x- = +

7. Indices 1. Simplify (rs 5- ) 3 s 15 2 Simplify (3pq3 ) 2 (q 3 ) 4 p6 q8

- + =

=

=

+ - =

=

=

3. Simplify 2m4 9m 5- 4. Find the value of

232 4

5

3 819

-

5. Simplify ( )22 1 3 44 2m n m n- 6. Given that 43 81x- = , calculate the value of x. 8. Algebraic Formulae

1. Given that p 2r

= pr

, express p in terms of r. 2. Given that 2 4k

t-

= , express k in terms of t.

=

=

= +

=

=

=

2 22

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 ) 5mn

-= , express m in terms of n

5. Given that A = 1 ( )2

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

9. Trigonometry

adjacent side

hypotenuse opposite side

A B

C tan =

sideadjacent side opposite

sin =

hypotenuseside opposite

cos =

hypotenusesideadjacent

5

=

=

=

=

=

-=

--=

-=+

-=-+

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.



1. In the diagram below, given that 43

sin =x , calculate the value of cos y.

=

=

=

=

=

=

=

+=

+=

Hence =

!

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

(i) Given that ,54

cos =x calculate the length of QR .

(ii) Hence, calculate the value of tan y

"#

"#

"#

"#

=

=

=

=

$#

$#

=

=

=

=

-=

Hence

= =

=

=

=

=

#% $#

$%

!

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 ABC 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



Guidelines for questions in Paper 2 (subjective) in PMR 1. In the Cartesian plane as shown, triangle ABC is the image of triangle ABC under a reflection in the line M. (i) Draw the image of triangle ABC. (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 = d istance o f a po in t on the im age from cen tre o f en largemen td istance o f the correspond ing po in t on the ob ject from the cen tre o f en largemen t

x

2. The diagram below is drawn on a tessellation of congruent triangles . The shape ABCD 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, PQRS 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



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 PQR 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

- < -

< -

x

(ii) 4x 20 (iii) 24

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

-

))))

->

-)))

->-

>- >- *+, -



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



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 = x. 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 = x2

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:




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.

%*/062*/(!.*+/!(//(&2 %*/0"4!**+/!2*!*+/7&+&/%*/08&.&,5/*!9!2*+/!2*

%*/0&:/4*+/7&+

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



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



(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



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



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 ingT h e len g th o f the ac tua l ob 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 ) 1n

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 : 12

. 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

25 251 1

75 3 50 225 25

= =

= =



18 . ANGLES (a) Angles In Circle Student must be able to knoww the concept


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 = 180o

140o 115o

xo

P Q

R

T S

U

60O

40O

xO

A



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,

22

(

)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

148292108 Teknik Menjawab Soalan Matematik PMR

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Transcript of 148292108 Teknik Menjawab Soalan Matematik PMR