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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
3. Quadratics equations 4m
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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QUADRATIC EXPRESSIONS AND EQUATIONS
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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Solve the quadratic equation:
(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
Area of shaded region.
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
bcad
A11
CERITA ADA DUIT BAYAR CUKAI
ada duit, bayar cukai,
Jadi : songsangannya1 per ada duit tolak bayar cukai,
Duit ada, tolak bayar tolak cukai
Invrse matrices
,
maka
Jika tiada songsangan , maka ad bc = 0
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
anticlockwiseabout point (6,2) 3M
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.ppt8/2/2019 Bengkel Peningkatan Maths Spm
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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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