# Bengkel Peningkatan Maths Spm

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JOM SKOR A

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a + 90% = 126/140

K1 = 35/40 dan k2=91/100

A 80% = 112/140

K1 = 33/40 dan k2 = 79/100

a- 75% = 105/140

K1 = 30/40 dan k2 =75/140

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SEMUA MANUSIA DILAHIRKAN

DENGAN POTENSI YANG

TINGGI..

KITA YANG MENCORAKKAN

HIDUP KITA.

BERUBAHLAH..

TENTUKAN MASA

DEPAN HIDUP

KITA

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jawab soalan mudah dahulu

1. Plans and elevations 12 m

2. graph inequalities 3 m

4. Area & perimeters

6m5. Volumes - 4m

6. MATHEMATICAL REASONING -5M

7. Matrices

6m8. Graph of functions 12 m

9. Statistics 12m

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PLANS AND ELEVATIONS

1. Drawing only.

2. Not necessary to label the drawing.

3. Rotation is accepted.

4. Extra lines (added) / not drawn

(missed) [dashed/dotted/solid]==> no marks given !

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4 cm

K

B

D

C

H

G

F

E

A

L

M

J

5 cm

7 cm

2 cm

2 cm

Diagram 15.1

Rajah 15.1

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plan

5CM

5CM 2CM

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4 cm

K

B

D

C

H

G

F

E

AL

M

J

5 cm

7 cm

2 cm

2 cm

3 cm

4 cm

1 cm

P

Q

R

S

T

U

V

W

X

Y

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The elevation of the solid as viewed from X.

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Elevation as viewed from y

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REGION REPRESENTING INEQUALITIESIN TWO VARIABLES

1. Draw a specific straight line 1 M

2. The region correctly shaded 2 M

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INEQUALITIES

> - I>ESAR

< - I

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INEQUALITIES IN GRAPH OF FUNCTION

PAPER 2 - 2009, 2007, 2005, 2003

Y - KE ATAS GARISAN,

Y - KE BAWAH GARISAN

X -

KE KANAN,X - KE KIRI

GARIS PUTUS-PUTUS UNTUK > ATAU

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y -x + 5

Y

2x - 2

X > 1

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1. Write in the general form,

ax2 + bx+ c= 0.

[ a, b, c~ constants ( Integer )]

2. Factorise completely

3. Write down the values of unknown

1M

2M

1M

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(2y

5)(y+ 7) = 4 72 2y

2y

2

+ 14y -5y

35 = 8y -14

2y2 + y - 21 = 0

(2y + 7 ) ( y 3 ) = 0

Y = , y = 32

7

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Circles ; Sector - Perimeter and

Use the correct formulae and substitutethe related values to calculate

i) the perimeter of the diagram,

ii) the area of the shaded region.

6M

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Circumference & Area of a circle

~ Circumference : 2r= d~ Area : r2

R

Q

P

O

jLength of arcPQR

Area of asectorOPQR

2

360 r 2

360 r

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Given POQ = 60, ROT= 45 and POT= 90. Using =, calculateDiberi POQ = 60, ROT = 45 dan POT = 90. Dengan menggunakan = , hitungkan

(a) the perimeter, in cm, of the whole diagram.

perimeter, dalam cm, seluruh rajah itu.

PERIMETER = PO + OT + TS + SQ + QP

cm3.71@3

214@3171

223

1742

217

222

360

60714

7

222

360

301421

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b) the area, in cm2, of the shaded region.

luas, dalam cm2, kawasan yang berlorek.

AREA 0F THE SHADED REGION :

3.205@3

616@

3

1205

3

225231

14

7

22

360

1521

7

22

360

60

cm2

2 2

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Solid Geometry ;

~ Volume

Use the correct formulae and substitutethe related values to calculate the volumeof the composite solid.

4M

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VOLUMES OF THE SOLIDS :

cm1400

770630

107722

21103)2814(

21

2

3

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Mathematical Reasoning.

1. A Statement. 1M

2. A Quantifier. 1M

3. Operations involving Not or

No, And and Or,.

4. Implication: The converse5. Argument.

6. Deduction and Induction.

1M

1M1M

2M

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MATHEMATICAL REASONING PAPER 2 ONLY.

1. STATEMENT / PERNYATAAN AYAT YANG MAKSUDNYA

JELAS BENAR ATAU PALSU.

2. PENGKUANTITI

ALL / SEMUA --- UNTUK SEMUA KES

SOME --- BENAR BAGI BEBERAPA KES SAHAJA.

3. OPERASI - a. Not / bukan mengubah maksud

pernyataan b. menggabung (compound) 2

pernyataan dgn :

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p q P and q P or q

Benar / true Benar / true Benar / true Benar / true

Palsu / false Benar / true Palsu / false Benar / true

Benar / true Palsu / false Palsu / false Benar / true

Palsu / false Palsu / false Palsu / false Palsu / false

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4.IMPLIKASI / IMPLICATION

Jika P, maka Q / If p, then q

a. Satu implikasi sahaja :

Antecedent : m > nConsequent : m n > 0

Implication : If m > n, then m - n > 0.Converse : If m n > 0, then m > n.

b. 2 implikasi -- p if and only if q

Implication 1 : If p, then qImplication 2 : If q, then p.

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5. HUJAH / ARGUMENTS TIGA BENTUK

P1 : ALL A IS B P1 : IF P, THEN Q P1 : IF P, THEN Q

P2 : C IS B P2 : P P2 : ~ P

CONCLUSION: C IS A CONCLUSION : Q CONCLUSION : ~ Q

6. MAKING DECISION / MEMBUAT KESIMPULAN

INDUCTION / ARUHANFrom several cases formulaDEDUKSI / DEDUCTION - SPM 2008Formula use in several cases

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TEOREM IBNU HAITHAM

ca

b

a, b, c

3, 4, 5

5, 12, 13

6, 8, 10

7, 24, 258, 15, 17

9, 12, 1512

11

109

8

7

6

5

4

3

2

1

x

144

121

10081

64

49

36

25

16

9

4

1

2x

1728

1331

1000

729

512

343

216

125

64

27

8

1

3x

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1

3

1

210. a. 83 = 512 and 8 =

XX

= 3 if and only if y = 273. yb

3,27:2

27,3:1

3

3

ythenIfyI

thenyyIfI

c. Premise 1: IfP Q =P, thenP QPremise 2: ..

conclusions :P Q P

P Q

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c

2. Solving the matrix equation :

If M=

=

a b

d

x

y

p

q

=x

y

p

qM 1

4M

MATRICES

1. Using the identity matrix - concept of inversematrix

AB = I = BA to find the constant.

2M/3M

dc

ba

1

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1A

dc

baA

ac

bd

A11

Duit ada, tolak bayar tolak cukai

Invrse matrices

,

maka

8 It is given that the inverse matrix of

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31

12

21

3 n

m

8 It is given that the inverse matrix of

(a) Find the values of m and of n.

A =

IS

31

121A 1)1()3(2 1

21

13

71

21

13

a

b

cd

m

, n = 17

1

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72 qp73 qp

Using matrix method , calculate the values of p and of q

31

12

7

7

q

p

q

p

7

1

21

13

7

7

7

1

)7(2)7(1

)7(1)7(3

7

1

21

14

3

2

p = 2, q = -3

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THE STRAIGHT LINE

(Parallelogram: A quadrilateral with opposite sides that

are parallel)

(Trapezium : A quadrilateral with one pair of parallel line)

(Two parallel lines)

* Find/Calculate

i) the gradient of a straight line,

y2 y1

x2 x1m= or m= yintercept

xintercept2M

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ii) the equation of a straight line, y= mx+ c.

[ Using the formulae : y1 = mx1 + cor yy1 = m(xx1)]

2M

iii) x-intercept or y-intercept.

[ y= 0 to obtain xintercept orvice-versa!] 1M

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Garis lurus QR ialah selari dengan paksi-y dan garis PQ ialah selari dengan garis RS.Persamaan garis lurus PQ ialah 2x + y = 10.

a. Nyatakan persamaan garis lurus RQ.QR // paksi-y, maka persamaannya ialah x= 5

b. Cari persamaan garis lurus RS dan seterusnya, nyatakan pintasan-y bagi garislurus itu.RS // PQ, maka kecerunan RS = kecerunan PQ

maka m = -2Guna m = -2 dan R( 5,14), ganti dalam y=mx + cMaka 14 = -2(5) + c14 + 10 = c

C = 24 , maka persamaan RS ialah y = - 2x + 24. dan pintasan-y = 24

102102

xy

yx

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LINES AND PLANES IN 3D

IDENTIFY THE ANGLE BETWEEN :

1. LINE AND PLANE

BY : BOLD THE LINE AND SHADE THE PLANE

IDENTIFY THE ANGLE USING SAME ALPHABETS

INTERSECTION POINT

2. TWO PLANES- IDENTIFY THE ANGLE USING SAME ALPHABETS

-INTERSECTION LINE

3. IDENTIFY THE NORMAL LINE TO FORM THE RIGHT ANGLE TRIANGLE4. DONT FORGET--- SOH CAH TOA

MARKING : - NAME THE ANGLE ---- 1M

USE TRIGONOMETRY TO CALCULATE THE ANGLE - 1M,

STATE THE VALUE OF THE ANGLE --- 1M

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Kenal pastikan dan hitungkan sudut di antara satah OJM dengan tapak JKLM.

1. ABJAD YANG SAMA IALAH JM2. SUDUT MESTI DARI O, O PULA SEGARIS / SESATAH DENGAN M3. O PULA TERGAK DI ATAS L4. MAKA SUDUTNYA IALAH OML

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1. Coefficient of the unknown to be

eliminated must be the same. 1M

2. Write in linear equation of one unknown

1M

3. Find the values of the unknowns 2M

SIMULTANEOUS LINEAR EQUATIONS

[ By elimination ]

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PROBABILITY

The probability of event A,

( )( )

( )

n AP A

n S

The probability of the complement of event A, (A)

)(1)'( APAP

List the sample space and the events.

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The probability of combined events~ or ( addition )

~ and ( multiplication ) 2M

A combined event, A or B,

~ P(A B) = P(A) + P(B)A combined event, A and B,

~ P(A

B) = P(A)

P(B)

PROBABILITY

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2 3 4 5 6A.LIST THE SAMPLE SPACE

(WITHOUT REPLACEMENT)

S = { (2,3), (2,4) , (2,5), (2,6), (3,2),

(3,4) , (3,5), (3,6), (4,2), (4,3) , (4,5),(4,6), (5,2), (5,3) , (5,4), (5,6), (6,2),(6,3) , (6,4), (6,5)}

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B. i. Probabilty of code begin with number 4 :

Code begin with 4 = {(4,2), (4,3) , (4,5), (4,6) }

P (code begin with 4)16

4

4

1

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GRAPHS OF FUNCTIONS

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GRAPHS OF FUNCTIONS

a. Completed the table / calculate the values of y.2M

b. (i) Both axes drawn in the correct direction and

uniform scale, 1M

(ii) All the points plotted correctly, 2M

(iii) Draw the graphs of function~ quadratic/cubic/reciprocal. 1M-2M

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c. Find the values of xor yfrom the graph drawn

1M/2M

d. (i) Write down/state the linear equation,

y= mx+ c 1M

(ii) Draw the straight line, y= mx+ c 1M

(iii) Find the intersection point and state the

value of xcoordinate 1M

x

y

TRANSFORMATIONS

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Describe, in full :

ROTATION

Rotation 1M

State the angle in degrees

& *direction of rotations 1M

(clockwiseoranticlockwise)

Centre of rotations 1M

(Coordinatesorthepointinthediagramgiven)

TRANSFORMATIONS

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ENLARGEMENT

Enlargement 1M

Scale factor k 1M

[ k: integer ! ]

Centre of enlargement 1M(Coordinatesorthepointinthediagramgiven)

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Area of Image = (Scale Factor) x2 area of object

=2

area of objectxk

Using the formula to find

3M

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TRANSLATIONS

h

k

Translation 2M

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Reflections 1M

state the axis of reflection

(In the line .) 1M

[Equationofthestraightlineortheline

thatjointwopoints]

REFLECTIONS

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Examples:

A rotation of 90o

An enlargement with scale factor 3centered at point H 3M

A reflection in the line y= 2 2M

2

3Translation 2M

STATISTICS

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STATISTICS

a) Complete the table given 4M

c) (i) Both axes drawn in the correct direction anduniform scale, 1M

(ii) Plot the points and draw

~ a histogram / a frequency polygon / an ogive 2M

b) Calculate the estimated mean 4M

http://new%20ceramah/Drawing%20Histogram.ppthttp://new%20ceramah/Drawing%20Frequency%20Polygons.ppthttp://new%20ceramah/Drawing%20Ogives.ppthttp://new%20ceramah/Drawing%20Ogives.ppthttp://new%20ceramah/Drawing%20Ogives.ppthttp://new%20ceramah/Drawing%20Ogives.ppthttp://new%20ceramah/Drawing%20Ogives.ppthttp://new%20ceramah/Drawing%20Frequency%20Polygons.ppthttp://new%20ceramah/Drawing%20Histogram.ppt
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d)(i) From the ogive, find

~3rd quartile, 1st quartile,

interquartile range , median 1M

(ii) State the information from the ogive /

histogram / frequency polygon 1M

EARTH AS A SPHERE

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EARTH AS A SPHERE

1 Finding the latitude orlongitude,

Location of a place2M

2 Find the distance

(a) Shortest distance (a great circle)

*difference in latitude 60*difference in longitude(equator) 60

(b) Along a parallel of latitude

difference in longitude 60 cos of latitude

2M / 3M

substended =

substended =2M

or

Distance

60

Distance

60 cos of latitud

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3 Calculate the

(a) average speed,

(b) time taken for the flight.

2M/3M

Speed Time

Distance

EARTH AS A SPHERE

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