maths module f4 2011a

8/3/2019 Maths Module F4 2011a

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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 : ___________________

Prepared by Tan Sze Haun Page : 1


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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


(a) 1784 (b) 14780 (c) 3998

(d) 1796 (e) 1.0378 (d) 15.731


(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


(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


(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)



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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 )

maths module f4 2011a

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