ACTM 2019 Precalculus Regional Contest · d. 460.9 YPZ[\ e. Unable to determine 17. Determine the...
Transcript of ACTM 2019 Precalculus Regional Contest · d. 460.9 YPZ[\ e. Unable to determine 17. Determine the...
ArkansasCouncilofTeachersofMathematics2019PrecalculusRegionalExam
Page1
Workthemultiplechoicequestionsfirst,choosingthesinglebestresponsefromthechoicesavailable.Indicateyouranswerhereandonyouranswersheet.Then,attemptthetiebreakerquestionsattheendstartingwithTieBreaker#1,then#2,andfinally#3.Turninyouranswersheetandthetiebreakerpageswhenyouarefinished.Youmaykeepthepageswiththemultiple-choicequestions.Figuresaren’tnecessarilydrawntoscale.Anglesaregiveninradiansunlessotherwisestated.
1. Given𝑓 𝑥 = 𝑥$ − 6𝑥,expand𝑓 𝑥 − '$.
a. 𝑥$ − 6𝑥 − '
$
b. 𝑥$ − 5𝑥 − '')
c. 𝑥$ − 6𝑥 + '')
d. 𝑥$ − 7𝑥 + ',)
2. Considerthegraphsandtheirequationsbelow.
Whichcoefficient(A,B,C,orD)hastheleastvalue?
a. Ab. Bc. Cd. De. Unabletodetermine
3. Identifytheverticalasymptote(s)ofthefunction𝑓 𝑥 = -../01)
a. 𝑥 = 0b. 𝑥 = 8c. 𝑥 = ±8d. 𝑥 = 64e. Therearenoverticalasymptotes.
ArkansasCouncilofTeachersofMathematics2019PrecalculusRegionalExam
Page2
4. Given𝑓 𝑥 = 𝑥$ − 4and𝑔 𝑥 = '.,whatisthedomainofthefunction𝑓 𝑔 𝑥 .
a. −'
$, '$
b. −2,2 c. −'
$, 0 ∪ 0, '
$
d. −∞, 0 ∪ 0,∞ e. −∞,−2 ∪ 2,∞
5. AhomeownermowsthelawneveryWednesdayafternoon.Letℎbeafunctionthatdeterminestheheightofthegrass(ininches)intermsoftheamountoftimethathaspassedsinceWednesdayafternoon(indays).Whichofthefollowingisthecorrectinterpretationoftheequationℎ 2 = 5?
a. OnMondayafternoon,thegrassis2inchestall.b. OnFridayafternoon,thegrassis5inchestall.c. OnWednesdayafternoon,thegrasshasjustbeencut.d. Twotimestheheightofthegrassis5inches.
6. Determinethevalueofsin 𝜃 iftan 𝜃 = − '$-andcos 𝜃 > 0.
a. sin 𝜃 = − -'$
b. sin 𝜃 = − ','$
c. sin 𝜃 = − -',
d. sin 𝜃 = − '$',
e. Cannotbedetermined
7. Thegraphof𝑓(𝑥)isshowntotheright.Statetheinterval(s)of𝑥onwhich𝑓isincreasing.
a. −2.12,0.786 b. −∞, 1.015 ∪ −2.052,∞ c. 1.015,−2.052 d. −∞,−2.12 ∪ 0.786,∞ e. Unabletodetermine
ArkansasCouncilofTeachersofMathematics2019PrecalculusRegionalExam
Page3
8. Findallvalidvaluesfor𝜃inthefunctionbelow.Assumethat𝑘isaninteger.
𝑊 𝜃 =sin 𝜃
cos 𝜃 + 2
a. 𝜃 ≠ −2b. 𝜃 = −2c. 𝜃 ≠ 𝑘𝜋d. 𝜃 = 𝑘𝜋e. Noneoftheabove.
9. Forthefunction𝑓 𝑥 = 2 − $,𝑥,findtheaveragerateofchangeof𝑓(𝑥)as𝑥increases
from−3to1.
a. 2b. −$
,
c. 4d. )
,
e. Unabletodetermine
10. Markworksasacookandspendsallthemoneyheearnsongraphictees.Let𝑤beafunctionthatdeterminestheamountofmoneyMarkhasearned(𝑚)intermsofthenumberofhourshehasworked(ℎ).Let𝑔beafunctionthatdeterminesthenumberofgraphicteesMarkcanbuy(𝑛)intermsoftheamountofmoneyMarkhasearned(𝑚).Whichofthefollowingcompositionsoffunctionsmakessense?
a. 𝑤 𝑚 𝑛 b. 𝑤 𝑔 𝑚 c. 𝑔 𝑤 ℎ d. ℎ 𝑚 𝑛 e. Unabletodeterminewiththeinformationgiven.
ArkansasCouncilofTeachersofMathematics2019PrecalculusRegionalExam
Page4
11. Thegraphof𝑓isshowntotheright.Choosethegraphclosestto𝑓0'.
a. b.
c. d.
e. Noneoftheabove.
12. Determineifthefunction𝑓 𝑥 = $.Q,-.Q'
isone-to-one.Ifitisone-to-one,find𝑓0' 𝑥 .
a. 𝑓0' 𝑥 = − $.Q,-.Q'
b. 𝑓0' 𝑥 = ,.Q$.Q-
c. 𝑓0' 𝑥 = 0,.Q'-.Q$
d. 𝑓0' 𝑥 = 0.Q,-.0$
e. Thisfunctionisnotone-to-one
ArkansasCouncilofTeachersofMathematics2019PrecalculusRegionalExam
Page5
Forproblems13and14,considerthefollowingrationalfunction.
𝑅 𝑥 =3𝑥$
𝑥 + 𝑥$+ 50
13. Finishthestatement:As𝑥increaseswithoutbound,𝑅 𝑥 approaches…
a. −1, 0b. 3c. 50d. 53e. +∞
14. Finishthestatement:As𝑥approaches−1fromtheleft,𝑅(𝑥)approaches…
a. +∞b. −∞c. 50d. 0e. Unabletodetermine.
15. Maryboughtausedcarfor$5000andthesalespersontoldherthatthevalueofthecarisexpectedtodepreciateatarateof8%peryear.WhichofthefollowingmodelsthevalueofMary’scarwithrespecttothenumberofyearsthathavepassedsincesheboughtit?
a. 𝑀 𝑛 = 5000 − 0.08𝑛b. 𝑀 𝑛 = 5000 0.92 Wc. 𝑀 𝑛 = 5000 0.08 Wd. 𝑀 𝑛 = 5000 − 0.92𝑛e. Unabletodetermine
ArkansasCouncilofTeachersofMathematics2019PrecalculusRegionalExam
Page6
Usethefollowingtrianglefigureinproblems16and17.
16. Thelawofcosinesandthelawofsinescanbeusedtofindthelengthofthesidesofanarbitrary(non-right)triangle.Applyoneoftheselawstofindthelengthofside𝑎intheabovefigure.
a. 17.5𝑢𝑛𝑖𝑡𝑠b. 21.5𝑢𝑛𝑖𝑡𝑠c. 306.0𝑢𝑛𝑖𝑡𝑠d. 460.9𝑢𝑛𝑖𝑡𝑠e. Unabletodetermine
17. Determinetheareaoftheabovetriangle.
a. 55.3𝑢𝑛𝑖𝑡𝑠$b. 67.5𝑢𝑛𝑖𝑡𝑠$c. 96.6𝑢𝑛𝑖𝑡𝑠$d. 161.0𝑢𝑛𝑖𝑡𝑠$e. Theareacannotbefound.
18. Definelog_ 𝐴 = 10andlog_(𝐵) = 2.Usethesetoevaluatelog_b/
c.
a. 𝑥 = 5b. 𝑥 = 8c. 𝑥 = 18d. 𝑥 = 50e. 𝑥 = 98
ArkansasCouncilofTeachersofMathematics2019PrecalculusRegionalExam
Page7
19. Anangleof2.4radiansinterceptsacircleofradius3.2,asshowninthefiguretotheright.Findthelengthofthearc𝑆.
a. 1.33b. 2.00c. 7.49d. 7.68
20. Whichofthefollowingisapointontheunitcircleandtheterminalsideoftheangle−)e
,?
a. −'$, − 𝟑
𝟐
b. −'$, ,$
c. − 𝟑𝟐, − '
$
d. − 𝟑𝟐, '$
e. ,$, − '
$
21. Usethepropertiesoflogarithmstocombinethefollowingexpressionintoasinglelogarithm.2 log 𝑧 − 3 log 𝑥 + 1
a. log 𝑧$ 𝑥 + 1 ,
b. log i/
.Q' j
c. log $i,.Q,
d. log 2𝑧 − 3𝑥 − 3 e. Noneoftheabove.
ArkansasCouncilofTeachersofMathematics2019PrecalculusRegionalExam
Page8
22. Solvethefollowingequation.Giveallsolutionsintheinterval0 ≤ 𝜃 < 2𝜋.
tan 𝜃 +𝜋3
= 1
a. 𝜃 = ''e
'$, $,e'$
b. 𝜃 = $e,, -e,
c. 𝜃 = e)
d. 𝜃 = − e'$
e. Noneoftheabove.
23. Afiremanhasa20-footladderthatshecanusetorescuesomeonefromawindow.Thesafetyofficerdeterminesthatthisladdercanbeleanedagainstaverticalwallwithaminimumangleofelevationof50°andamaximumangleofelevationof75°.Giventhesetwoconstraints,whatisthedifferenceinminimumandmaximumheightsthattheladdercansafelyreach?
a. 4.0feetb. 8.5feetc. 15.3feetd. 19.3feete. Unabletodetermine.
24. Let𝜃beanangleinquadrantIVsuchthatcos 𝜃 = pq.Findtheexactvalueofcsc 𝜃.
a. csc 𝜃 = − 'rq
b. csc 𝜃 = q'r
c. csc 𝜃 = 'rq
d. csc 𝜃 = − q'r
25. Usetrigonometricidentitiestosimplifythefollowingexpression.sin 𝑥 sec 𝑥tan$ 𝑥
a. cot 𝑥b. 1c. cos 𝑥d. tan$ 𝑥e. Noneoftheabove.
ArkansasCouncilofTeachersofMathematics2019PrecalculusRegionalExam
Page9
TieBreaker#1Name:____________________School:___________________Given𝑓 𝑥 = 3𝑥$ − 5𝑥 + 1,fullysimplifythedifferencequotient:t uQv 0t u
v
ArkansasCouncilofTeachersofMathematics2019PrecalculusRegionalExam
Page10
TieBreaker#2Name:____________________School:___________________Yourgrandmotherhasdecidedtopushlifetotheedgeandtrybungeejumping!Sheisstandinginthecenterofabridgeandjumpswithnoinhibitions.Inthefunofitall,younoticethattherelationshipofherheightwithrespecttotheamountoftimethathaspassedsinceshejumpedformsaperfectparabolicshape.Thebridgeis100feetintheair.ThreesecondsafterjumpingGrandma’sbungeecordiscompletelytaughtandsheisatherminimumheightof40feetfromtheground.WriteaquadraticfunctionthatwouldrelateGrandma’sheightaboveground(infeet)withrespecttothenumberofsecondsthathavepassedsinceshejumped.
ArkansasCouncilofTeachersofMathematics2019PrecalculusRegionalExam
Page11
TieBreaker#3Name:____________________School:___________________Verifythefollowingtrigonometricidentity:
1sin 𝑥
−sin 𝑥
1 + cos 𝑥= cot 𝑥
ArkansasCouncilofTeachersofMathematics2019PrecalculusRegionalExam
Page12
ACTMPrecalculusKey20191. D
2. B
3. C
4. C
5. B
6. D
7. D
8. E
9. B
10. C
11. D
12. D
13. D
14. A
15. B
16. B
17. A
18. C
19. D
20. B
21. B
22. A
23. A
24. D
25. A
ArkansasCouncilofTeachersofMathematics2019PrecalculusRegionalExam
Page13
TieBreakerNumber1SolutionGiven𝑓 𝑥 = 3𝑥$ − 5𝑥 + 1,simplifythedifferencequotient:t uQv 0t u
v
ArkansasCouncilofTeachersofMathematics2019PrecalculusRegionalExam
Page14
TieBreakerNumber2SolutionYourgrandmotherhasdecidedtopushlifetotheedgeandtrybungeejumping!Sheisstandinginthecenterofabridgeandjumpswithnoinhibitions.Inthefunofitall,younoticethattherelationshipofherheightwithrespecttotheamountoftimethathaspassedsinceshejumpedformsaperfectparabolicshape.Thebridgeis100feetintheair.ThreesecondsafterjumpingGrandma’sbungeecordiscompletelytaughtandsheisatherminimumheightof40feetfromtheground.WriteaquadraticfunctionthatwouldrelateGrandma’sheightaboveground(infeet)withrespecttothenumberofsecondsthathavepassedsinceshejumped.
ArkansasCouncilofTeachersofMathematics2019PrecalculusRegionalExam
Page15
TieBreakerNumber3SolutionVerifythefollowingtrigonometricidentity:
1sin 𝑥
−sin 𝑥
1 + cos 𝑥= cot 𝑥