EAS 2.6 Sample

12
7/14/2019 EAS 2.6 Sample http://slidepdf.com/reader/full/eas-26-sample 1/12 Contents uLake Ltd Innovative Publisher of Mathematics Texts Achievement Standard . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Expanding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Factorising . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Exponents . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Changing the Subject . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 Logarithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 Rational Expressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 Linear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 Linear Inequations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 Quadratic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 Quadratic Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 The Nature of the Roots of a Quadratic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 Excellence Questions for Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 Practice External Assessment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 Order Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

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Transcript of EAS 2.6 Sample

  • Year 12Mathematics

    Contents

    uLake Ltdu a e tduLake Ltd

    Innovative Publisher of Mathematics Texts

    EAS 2.6Robert Lakeland & Carl Nugent

    Algebraic Methods

    x2 + 5x + 6 = 3

    5 log 3

    h = V r

    2

    ? AchievementStandard .................................................. 2

    Expanding............................................................. 3

    Factorising............................................................. 6

    Exponents .............................................................. 12

    ChangingtheSubject ................................................... 16

    Logarithms............................................................. 21

    RationalExpressions.................................................... 29

    LinearEquations....................................................... 36

    LinearInequations...................................................... 42

    QuadraticEquations..................................................... 45

    QuadraticFormula...................................................... 50

    TheNatureoftheRootsofaQuadratic..................................... 54

    ExcellenceQuestionsforAlgebra.......................................... 57

    PracticeExternalAssessment............................................. 60

    Answers............................................................... 72

    OrderForm............................................................ 79

  • Year 12 Mathematics EAS 2.6 Published by NuLake Ltd New Zealand Robert Lakeland & Carl Nugent

    2 EAS 2.6 Algebraic Methods

    Thisachievementstandardinvolvesapplyingalgebraicmethodsinsolvingproblems.

    ThisachievementstandardisderivedfromLevel7ofTheNewZealandCurriculumandisrelatedto theachievementobjectives

    manipulaterational,exponential,andlogarithmicalgebraicexpressions formanduselinearandquadraticequations

    Applyalgebraicmethodsinsolvingproblemsinvolves: selectingandusingmethods demonstratingknowledgeofalgebraicconceptsandterms communicatingusingappropriaterepresentations.

    Relationalthinkinginvolvesoneormoreof: selectingandcarryingoutalogicalsequenceofsteps

    connectingdifferentconceptsorrepresentations demonstratingunderstandingofconcepts formingandusingamodel

    andrelatingfindingstoacontext,orcommunicatingthinkingusingappropriatemathematical statements.

    Extendedabstractthinkinginvolvesoneormoreof: devisingastrategytoinvestigateorsolveaproblem

    identifyingrelevantconceptsincontext developingachainoflogicalreasoning,orproof

    formingageneralisation

    andusingcorrectmathematicalstatements,orcommunicatingmathematicalinsight.

    Problemsaresituationsthatprovideopportunitiestoapplyknowledgeorunderstandingof mathematicalconceptsandmethods.Situationswillbesetinreal-lifeormathematicalcontexts.

    Methodsincludeaselectionfromthoserelatedto: manipulatingalgebraicexpressionsincludingrationalexpressions manipulatingexpressionswithexponents,includingfractionalandnegativeexponents determiningthenatureoftherootsofaquadraticequation solvingexponentialequations(whichmayincludemanipulatinglogarithms) formingandsolvinglinearandquadraticequations.

    Achievement Achievement with Merit Achievement with Excellence Applyalgebraicmethodsin

    solvingproblems. Applyalgebraicmethods,

    usingrelationalthinking,insolvingproblems.

    Applyalgebraicmethods,usingextendedabstractthinking,insolvingproblems.

    Algebraic Methods 2.6

  • Year 12 Mathematics EAS 2.6 Published by NuLake Ltd New Zealand Robert Lakeland & Carl Nugent

    6 EAS 2.6 Algebraic Methods

    Factorising

    There are four methods of factorising an expression. The flow chart below may be helpful in identifying the correct approach for a particular expression.

    No Yes

    ax2 + bx + c

    No Yes

    Does each termhave a common

    factor?

    Follow thecommon factor

    approach.Are there fourterms (2 pairs)which could be

    grouped?

    No YesAre there amaximum of

    three terms whichform a quadratic?

    Either yourexpression cannotbe factorised oryou have made awrong decision.

    Is the coefficientin front of the x

    term 1?2

    No YesFollow thecomplex

    quadratic (a 1)approach.

    Follow the simplequadraticapproach.

    Follow thegrouping of pairs

    to termsapproach.

    FactorisingRewritinganexpressionasaproductofitsfactorsiscalledfactorising.

    Theexpression4x+36couldbewrittenas4(x+9)becausewhenwemultiplyitoutweget4x+36.

    Factorisingisthereverseprocedureofexpanding.

    Weareonlyrequiredtofactoriseanexpressionifitconsistsofasum(ordifference)oftwoormoreterms.Wemustthenfactorisealloftheexpressionnotjustpartofit.

    Weadoptdifferentfactorisingtechniquesfordifferenttypesofexpressions.

    Common FactorsWhenwestudytheexpressionbelowweseethatallthetermsthatmakeituphaveafactorincommon,sotheappropriateapproachistoextractthehighestcommonfactor.Forexample,eachtermintheexpression

    4x3y2+6xy520x2y7zhas2,x,andy2ascommonfactors.Weidentifythesebyfirststudyingthenumbersatthefront(coefficients)andseeingthat2isthehighestcommonfactor,thenwestudythexcomponent(x3,xandx2)andnotexisthehighestcommonfactorandfinallytheycomponent(y2,y5andy7)andnotey2isthehighestcommonfactor.

    Wewritetheseontheoutsideofthebracketsanddividethemintothecomponentofeachterm.

    4x3y2+6xy520x2y7z =2xy2(2x2+3y310xy5z)

    2dividesinto4,6and20togive2,3and10.

    xdividesintox3,xandx2togivex2,1andx.

    y2dividesintoy2,y5andy7togive1,y3andy5.

    Ourexpressionbecomes

    4x3y2+6xy520x2y7z=2xy2(2x2.1+3.1.y310xy5z)

    =2xy2(2x2+3y310xy5z)

    (x+1)(x4)=x23x4

    Factorising

    Expanding uLake LtduLake Ltd Innovative Publisher of Mathematics Texts

  • 13EAS 2.6 Algebraic Methods

    Year 12 Mathematics EAS 2.6 Published by NuLake Ltd New Zealand Robert Lakeland & Carl Nugent

    ExampleEvaluate641/2

    Itispossibletodothisonacalculator,butbyhanditbecomes

    641/2 =1

    641 2/

    = 164

    =18

    ExampleSimplify(4x2)3x(5x4)2

    Weplace(5x4)2onthebottomandsimplify

    (4x2)3x(5x4)2 = ( )( )45

    2 3

    4 2xx

    =453 6

    2 8xx

    = 6425

    6

    8xx

    =6425 2x

    Example

    Simplify4 2 5

    75xx

    /

    Simplifyingeachlineindependently

    =4 2 5

    7 5xx

    /

    /

    =4x1

    =4x

    4 2 575

    xx

    /

    Example

    Simplifyandwritewithapositiveexponent ( )( )

    23

    1

    2mmn

    Writingeachlinewithapositiveexponent

    =

    12

    9 2 2m

    m n

    =12m

    x1

    9m2n2Simplifying

    =1

    18 3 2m n

    Investigate your calculator now. Make sure you are capable of calculating negative powers and fractional indices.

    On the TI-84 Plus to find 641/2 you enter:

    To convert your answer to a fraction enter:

    () 1

    (4 ^6

    ENTER2 )

    On the Casio 9750 to find 641/2 you enter:

    To convert your answer to a fraction enter:

    6 ^ () 1

    a b/c 2 ) EXE

    (4

    F D

    MATH 1 ENTERFrac

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  • 23EAS 2.6 Algebraic Methods

    Year 12 Mathematics EAS 2.6 Published by NuLake Ltd New Zealand Robert Lakeland & Carl Nugent

    AchievementUsetheequivalentlogstatement(y=bxisequivalenttologby=x)tosolvethe following.

    173. log525=A 174.log8512=B

    175. log264=C 176.log20.5=D

    177. log96561=E 178.log2128=F

    179. log3G=3 180.log10H=3

    181. log2I=0 182. log2J=7

    183. log8K=4 184. log10L=1

    185. logM32=5 186. logN10=1

    187. logP1=0 188.logQ243=5

    189. logR216=3 190.logS0.25=1

    191. logT0.01=2 192.logU729=3

    193. log93=V 194.log1632=W

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  • 29EAS 2.6 Algebraic Methods

    Year 12 Mathematics EAS 2.6 Published by NuLake Ltd New Zealand Robert Lakeland & Carl Nugent

    2430

    5

    2 4x yx y =

    45

    5

    2 4x yx y Dividethroughby6

    =45

    34xyy Simplifythexterms

    =45

    133x y Simplifytheyterms

    =45

    3

    3xy

    Rational Expressions

    Rational ExpressionsArationalexpressionisanalgebraicexpressionexpressedasafraction.Ourrulesforsimplifying,adding(orsubtracting)andmultiplyingarethesameastherulesweusefornumericalfractions.

    Simplifying Rational ExpressionsIfthereisacommonfactoronthetop(numerator)andbottom(denominator)wecandividethroughbyit.

    Ifweexaminetherationalexpression 42

    2 3

    5 3a cab c

    we

    canseethattherearefactorsincommonbetweenthetopandbottom.Ifwewritetheexpressionoutdifferentlythesebecomemoreobvious.

    42

    2 3

    5 3a cab c =

    42. aa2. 15b

    . cc3

    3

    =2.a.15b .1

    = 25ab

    Iftheexpressionsonthetopandbottomaresums,thentheyhavetobefactorisedpriortosimplification.

    Example

    Simplify2430

    5

    2 4x yx y

    Look to group the same factors on top of each other.

    3 a b/c 4 EXE20

    a b/cSHIFT

    Investigate your calculator now. Make sure you are capable of simplifying both proper and improper fractions.

    On the Casio 9750 the fraction part of a rational expression (e.g. 24

    30) can be quickly

    simplified by using the fraction key on your calculator. In this case you enter:

    If the fraction is top heavy (e.g. 3024

    ) then the same procedure is followed, but when you get a mixed

    numeral as an answer (e.g. 1 14) then you convert it

    to a top heavy fraction by:

    On the TI-84 Plus the fraction part of a rational expression (e.g. 24

    30) can be

    quickly simplified by using the FRAC function on the calculator. In this case you enter:

    If the fraction is top heavy (e.g. 3024

    ) then the same procedure is followed.

    2 a b/c 0 EXE34

    2 0 MATH34

    1 ENTERFrac

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  • Year 12 Mathematics EAS 2.6 Published by NuLake Ltd New Zealand Robert Lakeland & Carl Nugent

    36 EAS 2.6 Algebraic Methods

    288. ( ) ( )x

    xx

    x+

    +2

    54

    6 289.( )( )

    ( )( )

    xx

    xx

    ++

    ++

    23

    15

    290. ( ) ( )x xxx

    + 2 42 291.

    3 14

    2 532

    ( ) ( )xx

    xx++

    292. ( )( )

    ( )( )

    xx

    xx

    + +12 2

    34 3 293.

    32 1

    25 4

    2xx

    xx( ) ( )+ + +

    1.7 Linear Equations

    Linear Equations ExampleSolveforx 2x+5=3(6+4x)Alinearequationisanequationwithasingle

    powerofxthatcanbesimplifiedtotheform

    x =constant

    Tosolvealinearequationwesuccessivelysimplifytheequationusingthefollowingsteps.

    1.Expandanybrackets.

    4(x+3) =2x14

    4x+12 =2x14

    2.Collectallxtermsontheleftandconstantson theright.

    4x+12 =2x14

    4x2x =1412

    3.Ifinmovinganytermwecrosstheequalsign weusetheoppositeoperation.

    2x =26

    x = 262

    =13

    2x+5 =3(6+4x)

    Expandthebracket 2x+5 =18+12x

    Subtract5 2x =12x+185

    Subtract12x 2x12x =13

    10x =13

    Divideby10 x =1310

    =1.3

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  • Year 12 Mathematics EAS 2.6 Published by NuLake Ltd New Zealand Robert Lakeland & Carl Nugent

    44 EAS 2.6 Algebraic Methods

    354. 4 25 353

    ( )m m m+ > 355. 3

    2 334

    k k k <

    d) GivethemaximumamountofmoneyKizzimustinvestontermdeposit(andstillgetatotalof$1200ininterest)withthebankandalsogivetheinterestshewillreceivefromeachinvestmentperyear.

    Amountontermdeposit

    InterestfromUnitTrust

    Interestfromtermdeposit

    Merit Answer the following questions involving inequations.

    356. Kizziisplanningtoinvestatotalof$15000intwodifferentareas.Sheintendstoputsomeofthemoneyontermdepositfor12monthswithabankandwillreceive6.5%interestperannum.SheintendstoinvestthebalanceinaUnitTrustwhichoffers8.5%interestperannum.

    a) WriteanexpressionwhichwillgivethetotalamountofreturnfromboththetermdepositandtheUnitTrustinvestmentintermsofn,theamountofmoneyinvestedintheUnitTrust.

    b) IfKizziwishestoearnatleast$1200ininterestfortheyearfromherinvestments,formaninequalityusingyourexpressionfroma)abovetorepresentthis.DoNOTsolvetheinequalityyet.

    c) Solvetheinequalityfromb)toidentifytheminimumamountofmoneyKizzineedstoinvestintheUnitTrustinordertoreceiveatleast$1200interestfortheyear.

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  • Year 12 Mathematics EAS 2.6 Published by NuLake Ltd New Zealand Robert Lakeland & Carl Nugent

    54 EAS 2.6 Algebraic Methods

    The Nature of the Roots of a QuadraticThe Nature of the Roots of a Quadratic

    Inthissectionweexaminewhatdeterminesthenatureoftherootsofaquadraticequation.Arootisanothernameforthesolution(orxintercept).

    Sometimeswhenwesolveaquadraticequationweonlygetone(ortwoequal)solutions.Othertimeswegetnosolutionoradecimalanswerthatneedstobeapproximated.Thesolutionofaquadraticequationisgivenby

    x= b b ac

    a 2 4

    2

    Thepartofthequadraticformulathatdeterminesthenatureoftherootsisb24ac.

    Forexample,ifb24acisaperfectsquareandanexactsquarerootexiststhentherootsarealwaysabletobeexpressedasafraction.b24aciscalledthediscriminantandisrepresentedbythesymbolD.

    Insummaryifthediscriminantis

    positiveandaperfectsquare(e.g.1,4,9etc.),thentherootsarerational.Thismeansitispossibletowriteboththerootsasafractionandthequadraticequationfactorises.

    positiveandnotaperfectsquare(e.g.1.5,7,11etc.),thentwoirrationalrootsexist.Thismeansitisnotpossibletowritetherootsasfractions.

    zero,thenoneroot(ortwoequalrationalroots)exists.

    negative,thentherearenorealroots.

    ExampleDescribetherootsofthequadraticequationx210x5=0.

    D =b24ac

    =(10)24.1.(5)

    =120Thisispositive,butnotaperfectsquaresothequadratichastwoirrationalroots.

    On the Casio 9750 from the MENU select first EQUA then the Solver (F3).

    Press to delete any existing

    equation.Type in the discriminant.

    For the discriminant, D = B2 4AC you enter:

    Enter alongside each variable (ignoring D) the A, B and C values of the quadratic you are trying to find the discriminant for. For the example x2 10x 5 = 0 enter A = 1, B = 10 and C = 5.Place the cursor alongside the variable D and press to get the answer D = 120 (the discriminant).

    F2 F1

    ALPHA sin SHIFT . ALPHA log ^

    EXE

    On the TI-84 Plus select the MATH menu. Press 0 or scroll down until the cursor is on Solver and press ENTER. To delete any existing equation press the up arrow and then the CLEAR key.

    Type in the discriminant.

    For the discriminant, D = B2 4AC enter D B2 + 4AC, as it needs to be in the form, function = 0.

    Enter alongside each variable (ignoring D) the A, B and C values of the quadratic you are trying to find the discriminant for. For the example x2 10x 5 = 0 enter A = 1, B = 10 and C = 5.Place the cursor alongside the variable D and press to get the answer D = 120 (the discriminant)

    ALPHA x1 ALPHA APPS ^ 2

    ALPHA ENTER

    A graphics calculator can be used to calculate the discriminant by using the Solver.

    + 4 ALPHA MATH ALPHA PRGM ENTER

    2 4 ALPHA x, , T ALPHA ln

    F6

    D B

    A C

    D = B

    A C

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  • 57EAS 2.6 Algebraic Methods

    Year 12 Mathematics EAS 2.6 Published by NuLake Ltd New Zealand Robert Lakeland & Carl Nugent

    Excellence Questions for Algebra

    441. Aninventorhasdevelopedanewtoyandhas twomarketingoptions,self-promotion oftheproductorlicenseittoacompany.

    Ifhechoosesselfpromotionthereturnin thousandsofdollarsis

    R=9tt211 Ifhechoosestolicenseittoacompanythen thereturnis

    R=kt

    Forwhatvalueofkwillthereturnfromthe toycompanyalwaysbeinexcessofthereturn fromself-promotion?

    Hint:Findthevaluesforkforwhichthe

    equation9tt211=kthasnosolutions.

    ExcellenceAnswerthefollowingequations.

    442. Findtherangeofvaluesofpforwhichthe quadraticequation

    px2+(p+3)x+p=0

    hasrealroots.

    443. Ifp(x)isapolynomialandisdividedby (xk)andaremainderisobtained,thenthat remainderisp(k). Ifthequadraticp(x)=x23x+5givesthe sameremainderwhendividedbyx+kasit doeswhendividedbyx3kfindthevalueof k,k0.

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  • Year 12 Mathematics EAS 2.6 Published by NuLake Ltd New Zealand Robert Lakeland & Carl Nugent

    60 EAS 2.6 Algebraic Methods

    Practice External Assessment Task Algebraic Methods 2.6

    Make sure you show ALL relevant working for each question.

    You are advised to spend 60 minutes answering this assessment.

    QUESTION ONE(a) Expandandsimplify(2x+5)(x3)(x2)

    (b) Writeasasinglelogstatement0.5log36+log2

    (c) i) Factorise3x213x+4

    ii) Solvetheequation3x24x+1=0

    iii) Forwhatvalueofkdoestheequation3x2+kx+4=0haveequalroots?

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  • 73EAS 2.6 Algebraic Methods

    Year 12 Mathematics EAS 2.6 Published by NuLake Ltd New Zealand Robert Lakeland & Carl Nugent

    Page 14 cont...

    108. 14109. 8110. 8111. 27x9

    112. 23x113. x2114. y115. m

    12

    116. qk

    32

    117. 9a4b6

    118. k2564

    2

    119. pq

    2

    3 2

    120. 944

    2 2p

    m n121. 12 2g

    122. yx9

    6

    123. 1613

    11pq

    124. x8

    125.

    1x4b

    126. 3ba127. x

    ya

    a

    4

    128. 12q x

    129. 2n1/4m1/2

    130. 2x4

    131. p5/4q1/2

    Page 15132. 4x3133. x3

    134. 14x

    135. x2

    9136. 110p137. 14x

    Page 15 cont...

    138. 6 2nm

    139. 4x9/2

    140. x2/3

    141. a)A= p910 5

    b)V=p

    2780 4

    c) p=1.5cm

    d)Width=4.13cm

    Depth=2.04cm

    Heightofbottle5.33cm

    Page 17142. u=vat

    143. I= VR144. c=ymx

    145. x= y cm

    146. pirA2=

    147. pirV433=

    148. a ts ut2 2

    2=-

    149. pipiR A r

    2=

    +

    150. h a bA2

    =+

    151. b hA a2= -

    152. M P t34 4

    =-

    153. R = 100IPT

    Page 18

    154. b = a2 c2

    155. T= 100IPR

    156. h= Emg157. F= 95 32

    C+

    158. x= k ab c( )+

    159. y= 20 103 x

    Page 19

    160. a = 23 1b bvv+

    161. a = 5 12d bss+

    162. x = cy ay+ 5

    163. b =ara r

    Page 20

    164. r =

    A

    noasris+ve

    165. r =

    3V

    3

    166. x= c ab

    2

    167. s av u2

    2 2=

    -

    168. r =

    3S4 noasris+ve

    169. x = ( )y a4252

    170. y = xk2

    171. b = 1215 2 2a x172. Halfsphere= pir3

    2 3

    TotalVol. = pi pir r h32 3 2+

    Makehthesubject

    r2h=V pir32 3

    h = pipi

    rV r3

    22

    3-

    h = pirV r3

    22 -

    Page 23

    173. 25 =5A A =2

    174. 512 =8B B =3

    175. 64 =2C C =6

    176. 0.5 =2D D =1

    177. 9E =6561 E =4

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    EAS 2.6 1EAS 2.6 2EAS 2.6 6EAS 2.6 13EAS 2.6 23EAS 2.6 29EAS 2.6 36EAS 2.6 44EAS 2.6 54EAS 2.6 57EAS 2.6 60EAS 2.6 73