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SMK TUN ABDUL RAZAK
MATHEMATICS FORM 4
Teaching and Learning Module
I promise that I will study very hard to ensure that I will do very well for my Mathematics in
SPM
Signature : ___________________
Name : ___________________
School : ___________________
Class : ___________________
Year : ___________________
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Chapter 1 : Significant Figures
Significant figures --> relevant digits in a numbers, shows the level of accuracy.
1. State the number of significant in the following numbers
(a) 401 (b) 740102 (c) 20100
(d) 1.031 (e) 0.0109 (c) 0.0170
2. Round off the following numbers to 2 significant figures
(a) 4783 (b) 1541 (c) 1950
(d) 0.0147 (e) 0.1325 (d) 10.58
3. Round off the following numbers to 3 significant figures
(a) 1784 (b) 14780 (c) 3998
(d) 1796 (e) 1.0378 (d) 15.731
4. Round off the following numbers to 1 significant figures
(a) 4399 (b) 100.9 (c) 970
5. Calculate the following and round off the answer to number of significant figures as given in bracket.
(a)12
5882.0 (2 significant figures) (b) 82.0
4
895.4 (3 significant figures)
Five Rules
Non Zero( 1, 2, 3, 4, 5, 6, 7, 8, 9 )
Zero ( 0 )
--> Significant
Front
Integer Decimal
In between Back
--> Significant
--> Significant
--> Not Significant
--> Depend onlevel of accuracy
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(c) 15.9 ( 0.73 + 1.4 ) (3 significant figures) (e) 7.5 + 0.5 4.18 (2 significant figures)
6. Given the perimeter of the rectangle ABCD is 73.8 cm. Find the area of the rectangle ABCD and give your
answer in three significant figures
Chapter 1 : Standard Form
Standard Form --> write numbers in the form of A 10 n with 1 A < 10 and n is integer
1. Express the following in standard form.
(a) 5121 (b) 80500
(c) 74000000 (d) 0.0134
(d) 0.0000074 (e) 0.000108
2. Write the following as single number
(a) 7.54 10 5 (b) 3.36 10 - 4
3. Calculate the following and give your answer in standard form.
(a) 4300 + 89000 (b) 0.589 0.0027
(c) 4.5 70.5 (d) 45 0.06
4. Calculate the following and give your answer in standard form.
(a) 4.2 10 8 + 4.5 10 7 (b) 3.4 10 - 5 + 4.9 10 - 6
A D
CB
14.5 cm
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(c) 5.4 10 - 5 + 9.3 10 - 6 (d) 3.7 10 - 3 3 10 - 4
(e) 8.7 10 6 4.5 10 4 (f) 1.7 10 - 3 9 10 - 4
5. Calculate the following and give your answer in standard form.
(a) 7.4 10 5 2.7 10 - 3 (b) 4.4 10 2 2.5 10 4
(c)3
3
106
108.4
(d) ( 4.55 10 - 4 ) ( 5 10 - 7 )
6. Find the area of the right angled triangle ABC and give your answer in standard form.
7. Find the volume of a cone with the radius of the base is 10.5 cm and the height is 20 cm.
hrV,7
22 2cone
A85 cm
C
B
190 cm
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Chapter 2 : Quadratic Expressions
Quadratic Expressions one unknown, highest power is 2 and some are product of two linear expressions.
I. Determine whether the following are Quadratic Expressions or not.
(a) 2x - 6 (b) x 2 + 4x (c) x 2 + 8
(d) x 2 + xy + 3 (e) x 2 + y 2 (e) 4x ( 2x + 1)
2. Expand the following
(a) 4x ( 5 x) (b) ( 2 x )( x + 3 )
(c) )4y2(5y2
1
(d) (2 3x) 2
3. Factorise the following completely
(a) 18x 2 54 (b) 3y 2 9x
(c) x 2 81 (d) 3x 2 12
(e) 8x6x 2 (f) 15x8x 2
(g) 54x3x 2 (h) 24x5x 2
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(i)1x7x122 (j)3x7x42
()6x13x52 (l)8x2x62
4.AlisalaryisRM200xpermonthfor(x2)monthsandRM210xpermonth(x6)months.Writehistotalincomeforthatperiodoftime.
Chapter2:QuadraticEquations
QuadraticEquationsoneunknown,highestpoweris2andcontains"="sign.
I.DeterminewhetherthefollowingareQuadraticEquationsornot.(a)2x26(b)2x+8=0(c)x2+4x=0
(d)x2+y+3=0(e)0yy
2 (e)x3x
3
2.Writethefollowingingeneralform 0cxbxa2(a)x=5-4x2 (b)x2=3(x3)+5
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(c) ( x + 2 )( x 3 ) = 6x (d) p2p
1
3. Ali pay RM 17.50 for x blue pens which cost RM ( x 3 ) each and ( x 2 ) black pens which cost RM2
xeach.
Write a quadratic equation in terms of x.
4. Determine whether the given value of x are roots of the quadratic equation.
(a)5
2,2,1x;2x3x5 2 (b) 3,5,3x;15xx2 2
5. Solve the following quadratic equations
(a) 1)3x( 2 (b) x4
1x5 2
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(c) xx
23 (e) 2)4x3(8x6
(c) 52
xx3 2
(e) )4x)(1x(8x2
6. Diagram shows the rectangle ABCD with the measurement AD = ( x + 6 ) cm and DC = x cm. Given E and F arethe midpoint of AD and DC respectively. The area of the triangle BEF is 60 cm 2. Find the value of x.
A D
x cm
CB
( x + 6 ) cm
F
E
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Sets : Basic Concept
A . Defining Sets
Descriptions A is the set of prime numbers that are less than 20
Set Notations A = { 2, 3, 5, 7, 11, 13, 17, 19 }
B. Venn diagram
Representing sets by geometrical diagram such as circles, triangles, rectangles or ovals.
C. Elements
Using , . Examples : 3 A, 11 A, 9 A.
Listing : A = { 2, 3, 5, 7, 11, 13, 17, 19 }
Number of elements : n(A) = 8
D. Empty sets
No elements, using { }, .
Example : Set B is odd numbers divisible by 2. B = { } or B = .
E. Equal sets
Two sets with same elements.
Given Set A is even numbers that are less than 10 and Set B is the first 4 multiples of 2.Since A = { 2, 4, 6, 8 } and B = { 2, 4, 6, 8 }Therefore A and B are equal sets, A = B.
Sets : Subset
Given M = { p, q, r, s, t }, N = { q, r, s }, P = { s, t, u }.
Since all the elements of N are the elements of M. N is subset of M, N M
Not all the elements of P are elements of M, P is not subset of M, P M
A A
8 2
7
5
11
17
13
19 3
M
t
r p
s
N q
u
PM
t
p
s
q r
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The subset of { a } are { a } and { }. A set with 1 element has 2 subset
The subset of { a, b } are { a, b }, { a } , ( b } and { }. A set with 2 element has 4 subset
The subset of { a, b, c } are { a, b, c }, { a, b }, { b, c }, ( a, c}, { a }, ( b }, { c } and { }. A set with 3 element has 8 subset
Therefore, the number of subset is 2 n, n = number of elements.
Sets : Universal Set
Universal set consist all element under discussion. Using .
Universal set, = { x : x is integer, 1 x 10 }, Set A = { x : x is multiples of 3 }, Set B = { x : x is multiple of 4 }
Sets : Complement of a Set
Complement of a set consist all elements in universal set that are not elements of that set.
Complement of A is written as A'
Given Universal set, = { x : x is integer, 1 x 10 } and Set A = { x : x is square numbers }
Therefore, A = { 1, 4, 9 } and A' = { 2, 3, 5, 6, 7, 8, 9, 10 }
Sets : Relations between Set, subset, universal set and complement of a set
1. Given set = { x : x is an integer, 1 x 12 }, set A = { x : x is prime numbers }, set B = x : x is even numbers }.(I) List the element of
(a) Set A (b) Set B
(c) Set A' (d) Set B'
(II) Find(a) n(A) (b) n(A' )
(b) number of subset of set B
A B
2
3 1
5
4 9 8
7
6 10
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2. The venn diagram shows the element of set , set A, set B and set C. Find
(a) Set B' (b) B' (c) n (C' )
3. The venn diagram show the number of elements of set , set A, set B and set C. Find
(a) n (A) (b) n(B) (c) n(C)
(d) n(A' ) (e) n(B' ) (f) n(C' )
4. Diagram shows the number of elements of set , set A and set B. Find(a) Value of x (b) n(A' ) (c) n(B' )
5. Shaded the region of P'
(a) (b)
A B
2 3
1 5
4 9
8
7
6
10
A
4 3
CB
52
61
BA
x + 1x 68
QP
QP
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Sets : Intersection
1. Given = { 1, 2, 3, 4, 5, 6, 7, 8, 9 }, A = { 2, 4, 6, 8 }, B = { 3, 6, 9 } and C = { 1, 2, 3, 6 }. Find
(a) A B (b) A C
(c) A B C (d) ( A C )'
(c) ( B C )' (d) ( A B C )'
2. Shaded the region of the following venn diagramsa) P Q b) P Q
c) P Q' d) ( P Q )'
3. Diagram shows the number of elements of set , set A, set B and set C. Given n( A B) = n(C), find the value ofx.
4. There are 34 students in class 5A, 19 of them are badminton club members, 18 of them are computer clubmembers and 5 of them are not members of any of this two clubs. Find the number of students that are membersof both of these two clubs.
5. There are 36 students in class 5B, 21 of them are joining football club, 18 of them are joining reading club and 8of are joining both of the clubs. Find the number of students not joining any of these two clubs.
Q
P
QP
Q
P
QP
BA
x + 16 78
C
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Sets : Union
1. Given = { 11, 12, 13, 14, 15, 16, 17, 18, 19, 20 }, A = { 12, 14, 16, 18, 20 }, B = { 11, 20 } and C = { 12, 15,
18 }. Find
(a) A B (b) A C
(c) A B C (d) ( A C )'
(c) ( B C )' (d) ( A B C )'
(c) B C' (d) (A' C
2. Shaded the region of the following venn diagramsa) P Q b) P Q
c) P Q' d) ( P Q )'
3. Diagram shows the number of elements of set , set A, set B and set C. Given n( A ) = n( B C), find the value ofx.
3. There are n students in class 5C, 20 of them are joining football club, 18 of them are joining reading club, 8 of arejoining both of the clubs and 7 of them are not joining any of these two clubs. Find the value of n.
P
QP
QP
QP
BA
x + 26 510
C
Q
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Sets : Combined operations
1. Given = { 6, 7, 8, 9, 10, 11, 12 }, A = { 6, 8, 10, 12 }, B = { 6, 9, 12 } and C = { 8, 12 }. Find
(a) (A B) C (b) (A B) C'
(c) (B C)' A (d) ( A C )' B
2. Shaded the region of the following venn diagrams
a) P Q R b) (P Q)' R
c) (P R) Q d) P Q'
3. Diagram shows the number of elements of set , set A, set B and set C. Given n( ) = 32, find(a) the value of x.(b) n [ A (B C)](c) n [ C (A C)]
PQP
QP
RP
BA
x + 2
6 510
C
Q
x
3
R R
Q
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Statistic:Mode,medianandMean
Foreachofthefollowingdata,findthemode,themedianandthemean.
(a)Score123456
Frequency254753
(b)Mars234567
Numberofstudents365372
Statistic:ModelclassandMean
Foreachofthefollowingdata,findthemodalclassandthemean.
(a)Age61011-151520212526303135
Frequency259851
(b)Mars11011202130314041505160
Numberofstudents368742
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Statistic:FrequencytablesandHistograms
Thefollowingshowtheheight,incm,of44students.(a)Basedonthedatashown,constructagroupedfrequencytableusingtheclasses136140,141145,146150andsoon.(b)Usingthescaleof2cmtorepresent5cmonthex-axisand2cmtorepresent1studentonthey-axis,drawaHistogramforthedata.
Statistic:FrequencytablesandFrequencypolygons
Thefollowingshowthemass,inkg,of36students.(a)Basedonthedatashown,constructagroupedfrequencytableusingtheclasses2630,3135,3640andsoon.(b)Usingthescaleof2cmtorepresent5kgonthex-axisand2cmtorepresent1studentonthey-axis,drawafrequencypolygonforthedistribution.
168 173 156 175 144 163 138 142 156 154 152172 169 162 154 158 156 152 139 142 149 151
145 148 157 153 163 165 168 157 163 154 163
159 160 170 149 145 163 169 167 158 164 171
42 52 63 54 33 56 58 46 54 53 49 51
47 56 64 48 53 49 56 48 47 59 44 57
32 43 59 38 64 52 36 53 39 54 38 61
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Statistic:Cumulativefrequency,ogive,medianandinterquartilerange
Thetableshowsthefrequencydistributionoftheheightsof120plantsinanursery.(a)constructacumulativefrequencytableforthedistribution.(b)usingascaleof2cmtorepresent5cmonthex-axisand2cmtorepresent10plantsonthey-axis,drawanogiveforthedistribution.(c)fromtheogive,findi)themedian,
Ii)theinterquartilerange.
Height(cm)Frequency
10146
151910
202419
252923
303430
353922
404410
Statistic:Cumulativefrequency,ogive,medianandproblems
1.Thetableshowsthefrequencydistributionofthemarksof100students.
(a)constructacumulativefrequencytableforthedistribution.(b)usingascaleof2cmtorepresent10marksonthex-axisand2cmtorepresent10plantsonthey-axis,drawanogiveforthedistribution.(c)fromtheogive,findi)thenumberofstudentspassingthetestifthepassingmarkis46.ii)thepassingmarkif50studentspassedinthetest,iii)thepassingmarkif30studentfailedinthetest.
mars(cm)Frequency
10193
20296
303910
404917505925
606922
707912
80895
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7. The heights, in cm, of some plants are shown in the above frequency polygon.(a) State the modal class.(b) Calculate the mean height of the plants.(c) Based on the frequency polygon, constructs a cumulative frequency table for the distribution.(d) Using a scale of 2 cm to represent 5 cm on the x-axis and 2 cm to represent 5 plants on the y-axis, draw an
ogive for the distribution.(e) From the ogive, find
i) the median,ii) the interquartile range
Height (cm)
Numberofplants
2
12
4
10
8
6
3 383323 28181380
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MathematicalReasoning:Statement
1.Determinewhethereachofthefollowingsentencesisastatementornot.
(a)Lionscanfly.
(b)Itmightraintoday.
(c)13isaprimenumber.
(d)113isanimproperfraction.
(e)20+10
(b)64,8,,3,=
(c){1,2,3},{3},=,
(d)2,8,()3,=,
MathematicalReasoning:QuantifiersAllandSome
1.Determineifeachstatementsistrueorfalse.
(a)Alltriangleshavethreesides.
(b)Allmultipleof3ismultipleof6
(c)Allparallelogramshavetwopairsofparallelsides.
(d)Allprimenumbersareoddnumbers.
(e)Allcuberootofanumberarepositive.
(f)Alldecimalnumbersarelessthenone.
2.Determinewhetherthestatementcanbegeneralizedtocoverallcasesbyusingthequantifierallwithoutaffectingthetruthofthestatement.
(a)Thenegativenumber5islessthan0.(c)Theoddnumber5isaprimenumber.
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(b)ParallelogramPQRShasnoaxisofsymmetry.(d)Theevennumber6isdivisibleby3.
3.Constructatruestatementusingthequantifierallorsomebasedonthegivenobjectsandproperty.(a)Cuboids,8verticesand12sides.
(b)Regularpolygons,sidesofequallength.
(c)Pyramids,triangularbase.
(d)Commonfactorsof24and36,divisibleby3.
MathematicalReasoning:Operationsonstatements
1.Formanegationforeachstatementusingthewordnot.Statewhetherthenewstatementistrueorfalse(a)32+42isequalto52.
(b)Allcongruentshapeshavethesamearea.
(c)Onem3isequalonemillioncm3.
(d)Allprismshaveatriangularbase
2.Identifythetwostatementsineachofthegivencompoundstatements.(a)Apyramidhasapolygonalbaseandatleastthreetriangularsurfaces.(b)Thesurfaceareaandthevolumeofasphereare
24 rand243
rrespectively.
3.Formacompoundstatementbycombiningthetwogivenstatementsusingthewordand.
4.Identifythetwostatementsineachofthegivencompoundstatements.
(a)48%canbewrittenas48100or0.48.(b)Agraphcanbeastraightlineoracurve.
5.Formacompoundstatementbycombiningthetwogivenstatementsusingthewordor.
(a)Ascaledrawingcabbebiggerthantheobject.Ascaledrawingcabbesmallerthantheobject.(b)isaemptyset.
isthesubsetforanyset.
(c)15isacommonmultipleof3and5.45isacommonmultipleof3and5.(d)Aregularheptagonhas7sides.Aregularheptagonhas7axesofsymmetry.
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6.Determinewhetherthefollowingstatementsistrueorfalse.(a)2+4=6and24=8.(d)46=2or3(2)=6.
(b)22=4and(2)2=4.(e)28or243 .
(c)2g=200gand2m=2000cm.(f)6isafactorof18or27.
MathematicalReasoning:Implications
1.Statetheantecedentandconsequentofthefollowingimplications(a)Ifx5=10,thenx=15.
Antecedent:
Consequent:
(b)IfAB=BC=AC,thenABCisanequilateraltriangle.
Antecedent:
Consequent:
2.Writemathematicalstatementintheformof"Ifp,thenq"onthefollowinginformations.(a)Antecedent:y=2xConsequent:y2=4x2(b)Antecedent:Xisdivisibleby9.Consequent:Xisdivisibleby3.
3.Writetwoimplicationsfromthefollowingstatements.(a)2x+3=5ifandonlyifx=1.
Implication1:
Implication2:
(b)AB=AifandonlyifAB.
Implication1:
Implication2:
4.Writemathematicalstatementintheformof"pifandonlyifq"onthefollowinginformations.(a)Implication1:Ify=6,then2y=12.Implication2:If2y=12,theny=6.(b)Implication1:IfX>Y,thenYX180,thenABCisareflexangle.(b)IfMisdivisibleby3,thenMisdivisibleby6.
MathematicalReasoning:Arguments
Form I
Premise1:AllAareBPremise2:CisA
Conclusion:CisB
Form II
Premise1:Ifp,thenq.Premise2:pistrue.
Conclusion:qistrue.
Form III
Premise1:Ifp,thenq.Premise2:Notqistrue.
Conclusion:notpistrue.
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Completethefollowingargumentsbyfillintheconclusion,premise1orpremise2.
Premise1:AllSMKTARstudentsaresmart.
Premise2:AliisSMKTARstudent.
Conclusion:
Premise1:Allmultipleof6idmultipleof3
Premise2:
Conclusion:Xismultipleof3
Premise1:
Premise2:Xisdivisibleisanevennumber.
Conclusion:Xisdivisibleby2.
Premise1
:IfmP,then21
mP.
Premise2:9P.
Conclusion:
Premise1:IfM>N,theM5
Premise2:X+25
Conclusion:
Premise1:Ifn(A)=0,thenAisanemptyset..
Premise2:
Conclusion:n(A)0.
Premise1:
Premise2:Nisnotdivisibleby5
Conclusion:Nisnotthemultipleof5.
MathematicalReasoning:DeductionsandInductions
Maeconclusionsbydeductionforthespecificcasebasedonthegeneralatatementgiven.Makeconclusionsbyinductionbasedonthepatternofanumericalsequence.
(a)Thenumberofsubsetforasetwithnelementsis2n.SetMhas3elements.
Conclusion:
(a)5=3+2(1)7=3+2(2)9=3+2(3)11=3+2(4)
Conclusion:Thenumericalsequencecanbewrittenas
(b)
Thevolumeofasphereis3r3
4,whereristheradius.
SphereNhastheradiusof7cm.
Conclusion:
(a)10=10(02)9=10(12)6=10(22)1=10(32)
Conclusion:
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Straight Line : Gradient12
12
xx
yym
1. Find the gradient of the following straight line :a) b)
2. Find the gradient the straight line passes through the points:a) (1, 4) dan (2, 6)b) (3, 3) dan (5, 1)
B Steepness and direction of inclination.
Straight Line : Interceptserceptintx
erceptintym
1. State the y-intercept, x-intercept and gradient of the following diagrams.a) b) c) d)
gradient g1 > gradient g2g1 = gradient zero
g2 = gradient undefined
(2, 9)
(4, 1)
x
x
g1
g2
(0, 4)
(3, 0)
y
x
x
g1
g2
6
4
y
x
(0, 6)
(3, 0)
y
x
4
2
y
x
g1 = positive gradient
g2 = negative gradient
(2, 1)
(4, 6)
x
g1
g2
x
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2. Find the x- intercept, if gradient = 2 3. Find the y-intercept, if gradient = 1
Straight Line : Equations of Straight Line
1. Write the equations of the following
a) m = 2, c = 3 b) m =2
1, c = 2
c) m = 3, c =3
2d) m = 2, c = 0
2. Find the equations for the following graphs.(a) (b)
3. Given the equation of each of the following straight line. Find the gradient and y-intercept of the straight line.(a) y = 5 3x (b) 2y = 3x + 16 c) 3y + 6x = 7
4. Write the equations of the following straight lines (g1, g2, g3, g4 and g5) :
3
x
2 x
( 4 , 0 ) x
2
4
(4 , 2 )
( 3 , 5 )
g1
g2
g4
g3
g5
4
x23
x(4,
General Equations : y = mx + c
m = gradient, c = y-intercept
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5 Find the equations for the following straight lines(a) passes throught (3, 4) and has a gradient of 2 (b) passes throught(2, 5) and has a gradient of 3
6. Find the equations for the following straight lines(a) Passes through points (1, 4) and (2, 6) (b) Passes through points (2, 3) and (4, 9)
7. Find the equations for the following straight lines.a) b)
8. Diagram shows the straight line. Find(a) The gradient of PQR,(b) Coordinates of point R
P(10, 8)
Q(2, 4)R
(1, 4)
x
(5, 2)
(2, 2)
x
(3, 3)
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9. Diagram shows the equations of straight line DEF. Find(a) Coordinates of point E,(b) The gradient of DEF.
10. Find the intersections point between the straight line y = 2x + 6 and 2y + x = 2.
Straight Lines : Parallel Lines
1. Find the equation of straight line AB, given AB parallel to CD and the equation of CD is y = 2x + 3.
1. Find the equation of straight line PQ, given PQ parallel to RS.
F
E
D 5y
1 0x
= 2 0
O
A
A (2, 9)
y
x
C
D
y = 2x + 3
P (3, 7)
y
O
Q
x
C (2, 5) D (4, 3)
Prepared by Tan Sze Haun Page : 31
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3. From the diagram, finda) Gradient of straight line QP,b) Y-intercept of straight line QP,c) Equation of straight line QR, given straight line QR is
parallel to straight line PO.
4. In the diagram, OABC is a parallelogram. Find
a) Gradient of straight line OA ,b) Equation of straight line BC,c) Coordinates of point B.
5. In the diagram, straight line PQ is parallel to straight line URV. Finda) Gradient of straight line PQ,b) Equation of straight line URV,c) X-intercept of straight line URV.
O
U
V
P ( 1 , 1 0 )
y
x
Q ( 2 , 1 ) R ( 4 , 2 )
OR
P ( 2 , 4 )
Q ( 0 , 8 )
y
x
O
B
A ( 4 , 1 )
y
x
C ( 1 , 2 )