SAT 2 数学2 college board官网全部原题加答案讲解 SAT Subject Test Math Level 2.docx
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SAT 2 数学2 college board官网全部原题加答案讲解 SAT Subject Test Math Level 2.docx
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SAT2数学2collegeboard官网全部原题加答案讲解SATSubjectTestMathLevel2
SATSubjectTestPractice-ResultsSummary
MathematicsLevel2
1YouranswerOmitted!
Whatisthedistanceinspacebetweenthepointswithcoordinatesand?
(A)
(B)
(C)
(D)
(E)
Explanation
Difficulty:
Easy
ThecorrectanswerisD.
Thedistancebetweenthepointswithcoordinatesandisgivenbythe
distanceformula:
.
Therefore,thedistancebetweenthepointswithcoordinatesandis:
whichsimplifiesto.
2YouranswerOmitted!
If,whatvaluedoesapproachasgetsinfinitelylarger?
(A)
(B)
(C)
(D)
(E)
Explanation
Difficulty:
Easy
ThecorrectanswerisE.
Onewaytodeterminethevaluethatapproachesasgetsinfinitelylargeristorewritethe
definitionofthefunctiontouseonlynegativepowersofandthenreasonaboutthebehaviorof
negativepowersofasgetsinfinitelylarger.Sincethequestionisonlyconcernedwithwhat
happenstoasgetsinfinitelylarger,onecanassumethatispositive.For,the
expressionisequivalenttotheexpression.Asgetsinfinitely
larger,theexpressionapproachesthevalue,soasgetsinfinitelylarger,theexpression
approachesthevalue.Thus,asgetsinfinitelylarger,approaches.
Alternatively,onecanuseagraphingcalculatortoestimatetheheightofthehorizontalasymptote
forthefunction.Graphthefunctiononanintervalwith“large”
say,fromto.
Byexaminingthegraph,theallseemverycloseto.Graphthefunctionagain,
from,say,to.
Thevaryevenlessfrom.Infact,tothescaleofthecoordinateplaneshown,the
graphofthefunctionisnearlyindistinguishablefromtheasymptoticline.
Thissuggeststhatasgetsinfinitelylarger,approaches,thatis,.
Note:
Thealgebraicmethodispreferable,asitprovidesaproofthatguaranteesthatthevalue
approachesis.Althoughthegraphicalmethodworkedinthiscase,itdoesnotprovidea
completejustification;forexample,thegraphicalmethoddoesnotensurethatthegraphresembles
ahorizontallinefor“verylarge”suchas.
3YouranswerOmitted!
Ifisafactorof,then
(A)
(B)
(C)
(D)
(E)
Explanation
Difficulty:
Easy
ThecorrectanswerisA.
BytheFactorTheorem,isafactorofonlywhenisarootof
thatis,,whichsimplifiesto.
Therefore,.
Alternatively,onecanperformthedivisionofbyandthenfindavalue
forsothattheremainderofthedivisionis.
Sincetheremainderis,thevalueofmustsatisfy.Therefore,.
4YouranswerOmitted!
Alisondepositsintoanewsavingsaccountthatearnspercentinterestcompounded
annually.IfAlisonmakesnoadditionaldepositsorwithdrawals,howmanyyearswillittakeforthe
amountintheaccounttodouble?
(A)
(B)
(C)
(D)
(E)
Explanation
Difficulty:
Medium
Afteryear,theamountintheaccountisequalto.Afteryears,theamountis
equalto,andsoon.Afteryears,theamountisequalto.Youneed
tofindthevalueofforwhich.Thereareseveralwaystosolvethis
equation.Youcanuselogarithmstosolvetheequationasfollows.
Since,itwilltakemorethanyearsfortheamountintheaccounttodouble.Thus,
youneedtoroundupto.
Anotherwaytofindistouseyourgraphingcalculatortographand
.Fromtheanswerchoices,youknowyouneedtosettheviewingwindowwithvalues
fromtoaboutandvaluesextendingjustbeyond.Theofthe
pointofintersectionisapproximately.Thusyouneedtoroundupto.
5YouranswerOmitted!
Inthefigureabove,whenissubtractedfrom,whatisthelengthoftheresultantvector?
(A)
(B)
(C)
(D)
(E)
Explanation
Difficulty:
Medium
Theresultantofcanbedeterminedby.
Thelengthoftheresultantis:
6YouranswerOmitted!
Inthe-plane,whatistheareaofatrianglewhoseverticesare,,and?
(A)
(B)
(C)
(D)
(E)
Explanation
Difficulty:
Medium
Itishelpfultodrawasketchofthetriangle:
Thelengthofthebaseofthetriangleisandtheheightofthetriangleis.Therefore,
theareaofthetriangleis.ThecorrectanswerisB.
7YouranswerOmitted!
Arightcircularcylinderhasradiusandheight.Ifandaretwopointsonitssurface,what
isthemaximumpossiblestraight-linedistancebetweenand?
(A)
(B)
(C)
(D)
(E)
Explanation
Difficulty:
Medium
Themaximumpossibledistanceoccurswhenandareonthecircumferenceofoppositebases:
YoucanusethePythagoreanTheorem:
Thecorrectansweris(B).
8YouranswerOmitted!
Note:
Figurenotdrawntoscale.
Inthefigureabove,andthemeasureofis.Whatisthevalueof?
(A)
(B)
(C)
(D)
(E)
Explanation
Difficulty:
Medium
Thereareseveralwaystosolvethisproblem.Onewayistousethelawofsines.Since,
themeasureofisandthemeasureofis.Thus,and
.(Makesureyourcalculatorisindegreemode.)
Youcanalsousethelawofcosines:
Sinceisisosceles,youcandrawthealtitudetothetriangle.
9YouranswerOmitted!
Thefunctionisdefinedbyfor.
Whatisthedifferencebetweenthemaximumandminimumvaluesof?
(A)
(B)
(C)
(D)
(E)
Explanation
Difficulty:
Medium
Itisnecessarytouseyourgraphingcalculatorforthisquestion.Firstgraphthefunction
.Itishelpfultoresizetheviewingwindowsothe-valuesgofrom
to.Onthisintervalthemaximumvalueofisandtheminimumvalueofis
.Thedifferencebetweenthesetwovaluesis,whichroundsto.
10YouranswerOmitted!
Supposethegraphofistranslatedunitsleftandunitup.Iftheresultinggraph
represents,whatisthevalueof?
(A)
(B)
(C)
(D)
(E)
Explanation
Difficulty:
Medium
Itmaybehelpfultodrawagraphofand.
Theequationforis.Therefore,
.ThecorrectanswerisB.
11YouranswerOmitted!
Asequenceisrecursivelydefinedby,for.Ifand,whatis
thevalueof?
(A)
(B)
(C)
(D)
(E)
Explanation
Difficulty:
Medium
Thevaluesforandaregiven.isequalto.isequalto
.isequalto.isequalto
.
Ifyourgraphingcalculatorhasasequencemode,youcandefinethesequencerecursivelyandfind
thevalueof.Let,sincethefirsttermis.Define.
Let,sincewehavetodefinethefirsttwotermsand.Thenexamininga
graphortable,youcanfind.
12YouranswerOmitted!
Thediameterandheightofarightcircularcylinderareequal.Ifthevolumeofthecylinderis,
whatistheheightofthecylinder?
(A)
(B)
(C)
(D)
(E)
Explanation
Difficulty:
Medium
ThecorrectanswerisA.
Todeterminetheheightofthecylinder,firstexpressthediameterofthecylinderintermsofthe
height,andthenexpresstheheightintermsofthevolumeofthecylinder.
Thevolumeofarightcircularcylinderisgivenby,whereistheradiusofthecircular
baseofthecylinderandistheheightofthecylinder.Sincethediameterandheightareequal,
.Thus.Substitutetheexpressionforinthevolumeformulatoeliminate:
.Solvingforgives.Sincethevolumeofthecylinderis,the
heightofthecylinderis.
13YouranswerOmitted!
If,then
(A)
(B)
(C)
(D)
(E)
Explanation
Difficulty:
Medium
ThecorrectanswerisE.
Onewaytodeterminethevalueofistoapplythesineofdifferenceoftwoangles
identity:
.Sinceand,the
identitygives.Therefore,.
Anotherwaytodeterminethevalueofistoapplythesupplementaryangle
trigonometricidentityforthesine:
.Therefore,.
14YouranswerOmitted!
Alinehasparametricequationsand,whereistheparameter.Theslopeof
thelineis
(A)
(B)
(C)
(D)
(E)
Explanation
Difficulty:
Medium
ThecorrectanswerisB.
Onewaytodeterminetheslopeofthelineistocomputetwopointsonthelineandthenusethe
slopeformula.Forexample,lettinggivesthepointontheline,andlettinggives
thepointontheline.Therefore,theslopeofthelineisequalto.
Alternatively,onecanexpressintermsof.Sinceand,
itfollowsthat.Therefore,theslopeofthelineis.
15YouranswerOmitted!
Whatistherangeofthefunctiondefinedby?
(A)Allrealnumbers
(B)Allrealnumbersexcept
(C)Allrealnumbersexcept
(D)Allrealnumbersexcept
(E)Allrealnumbersbetweenand
Explanation
Difficulty:
Medium
ThecorrectanswerisD.
Therangeofthefunctiondefinedbyisthesetofsuchthat
forsome.
Onewaytodeterminetherangeofthefunctiondefinedbyistosolvetheequation
forandthendeterminewhichcorrespondtoatleastone.To
solvefor,firstsubtractfrombothsidestogetandthentakethe
reciprocalofbothsidestoget.Theequationshowsthatforany
otherthan,thereisansuchthat,andthatthereisnosuchfor
.Therefore,therangeofthefunctiondefinedbyisallrealnumbersexcept.
Alternatively,onecanreasonaboutthepossiblevaluesoftheterm.Theexpressioncantake
onanyvalueexcept,sotheexpressioncantakeonanyvalueexcept.Therefore,the
rangeofthefunctiondefinedbyisallrealnumbersexcept.
16YouranswerOmitted!
Thetableaboveshowsthenumberofdigitalcamerasthatweresoldduringathree-daysale.The
pricesofmodels,,andwere,,and,respectively.Whichofthefollowing
matrixrepresentationsgivesthetotalincome,indollars,receivedfromthesaleofthecamerasfor
eachofthethreedays?
(A)
(B)
(C)
(D)
(E)
Explanation
Difficulty:
Medium
ThecorrectanswerisC.
Acorrectmatrixrepresentationmusthaveexactlythreeentries,eachofwhichrepresentsthetotal
income,indollars,foroneofthethreedays.ThetotalincomeforDayisgivenbythearithmetic
expression,whichisthesingleentryofthematrixproduct
;inthesameway,thetotalincomeforDayisgivenby
thesingleentryof;andthetotal
incomeforDayisgivenby,thesingleentryof
.Therefore,thematrixrepresentation
givesthetotalincome,indollars,receivedfromthe
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