ACT考试常见题型及答案解析

ACT考试常见题型及答案解析

一、单选题

1.If\fx=2x^2-4x+3\,whatisthevalueof\f2\（1分）A.3B.5C.9D.11【答案】C【解析】Substitute\x=2\intothefunction:\f2=22^2-42+3=8-8+3=3\.

2.Whatistheslopeofthelinepassingthroughthepoints1,2and3,6（1分）A.1B.2C.3D.4【答案】B【解析】Theslope\m\iscalculatedas\m=\frac{y_2-y_1}{x_2-x_1}=\frac{6-2}{3-1}=2\.

3.If\\sin\theta=\frac{3}{5}\,whatis\\cos\theta\for\\theta\inthefirstquadrant（2分）A.\\frac{4}{5}\B.\\frac{3}{4}\C.\\frac{5}{3}\D.\\frac{4}{3}\【答案】A【解析】UsingthePythagoreanidentity\\sin^2\theta+\cos^2\theta=1\,wefind\\cos\theta=\sqrt{1-\sin^2\theta}=\sqrt{1-\left\frac{3}{5}\right^2}=\frac{4}{5}\.

4.Whatisthevalueof\\log_28\（1分）A.2B.3C.4D.5【答案】B【解析】Since\8=2^3\,\\log_28=3\.

5.Inatrianglewithsides5cm,12cm,and13cm,whatisthemeasureoftheangleoppositethe13cmside（2分）A.30°B.45°C.60°D.90°【答案】D【解析】ThisisaPythagoreantriple,sothetriangleisarighttrianglewiththe90°angleoppositethe13cmside.

6.If\x^2-5x+6=0\,whatarethesolutionsfor\x\（1分）A.2,3B.-2,-3C.1,6D.-1,-6【答案】A【解析】Factorthequadratic:\x-2x-3=0\,so\x=2\or\x=3\.

7.Whatistheareaofacirclewithradius7cm（2分）A.\22\pi\cm²B.\49\pi\cm²C.\154\cm²D.\28\pi\cm²【答案】B【解析】Thearea\A\isgivenby\A=\pir^2=\pi7^2=49\pi\cm².

8.If\\frac{a}{b}=\frac{3}{4}\and\b=8\,whatisthevalueof\a\（1分）A.5B.6C.7D.8【答案】B【解析】Substitute\b=8\intotheequation:\a=\frac{3}{4}\times8=6\.

9.Whatisthesimplifiedformof\\frac{2x^3}{4x^2}\（2分）A.\\frac{x}{2}\B.\\frac{x^2}{2}\C.\x\D.\2x\【答案】C【解析】Simplifythefraction:\\frac{2x^3}{4x^2}=\frac{x}{2}\.

10.If\fx=x+1\and\gx=2x\,whatis\f\circg3\（1分）A.7B.8C.9D.10【答案】C【解析】Firstfind\g3=2\times3=6\,then\fg3=f6=6+1=7\.

二、多选题（每题4分，共20分）

1.Whichofthefollowingarefunctions（）A.\y=3x-2\B.\x^2+y^2=1\C.\y=\sqrt{x}\D.\y=\frac{1}{x}\E.\y=|x|\【答案】A、C、E【解析】A,C,andEdefinefunctionsbecauseforeach\x\,thereisexactlyone\y\.BandDdonotdefinefunctions.

2.Whatarethepropertiesofaparallelogram（）A.OppositesidesareparallelB.OppositesidesareequalC.OppositeanglesareequalD.DiagonalsbisecteachotherE.Allanglesare90°【答案】A、B、C、D【解析】Propertiesofaparallelogramincludeparalleloppositesides,equaloppositesides,equaloppositeangles,anddiagonalsthatbisecteachother.Eisapropertyofarectangle.

3.Whatarethesolutionstotheinequality\2x-35\（）A.\x4\B.\x4\C.\x8\D.\x8\E.\x1\【答案】A、E【解析】Solvetheinequality:\2x-35\→\2x8\→\x4\.

4.Whatarethecharacteristicsofarighttriangle（）A.Oneangleis90°B.ThePythagoreantheoremappliesC.Thesumofanglesis180°D.AllsidesareequalE.Oneangleisacute【答案】A、B、C【解析】A,B,andCarecharacteristicsofarighttriangle.Disacharacteristicofanequilateraltriangle.Eisnotnecessarilytrue.

5.Whatarethetransformationsofthefunction\fx=x^2\（）A.\fx+2\shiftsthegraphleft2unitsB.\f2x\stretchesthegraphhorizontallyC.\fx-3\shiftsthegraphdown3unitsD.\-fx\reflectsthegraphoverthex-axisE.\fx+4\shiftsthegraphup4units【答案】B、C、D、E【解析】Aisincorrect;\fx+2\shiftsthegraphleft2units.B,C,D,andEdescribecorrecttransformations.

三、填空题

1.Thesumoftheinterioranglesofahexagonis______degrees.（4分）【答案】720【解析】Thesumoftheinterioranglesofan\n\-sidedpolygonis\n-2\times180\.Forahexagon\n=6\:\6-2\times180=720\degrees.

2.If\\tan\theta=\frac{3}{4}\,whatis\\sec\theta\（4分）【答案】\\frac{5}{4}\【解析】Usingtheidentity\\sec^2\theta=1+\tan^2\theta\,wefind\\sec^2\theta=1+\left\frac{3}{4}\right^2=1+\frac{9}{16}=\frac{25}{16}\,so\\sec\theta=\frac{5}{4}\.

3.Theprobabilityofrollinga6onafairsix-sideddieis______.（4分）【答案】\\frac{1}{6}\【解析】Thereis1favorableoutcomerollinga6outof6possibleoutcomes.

4.If\\frac{a}{b}=\frac{2}{3}\and\b=9\,whatisthevalueof\a\（4分）【答案】6【解析】Substitute\b=9\intotheequation:\a=\frac{2}{3}\times9=6\.

5.Thevolumeofacubewithsidelength3cmis______cm³.（4分）【答案】27【解析】Thevolume\V\isgivenby\V=s^3=3^3=27\cm³.

四、判断题

1.Theequation\x^2+y^2=4\representsacirclewithradius

4.（2分）【答案】（×）【解析】Theequationrepresentsacirclewithradius2,not

4.

2.If\\sin\theta=\frac{1}{2}\,then\\theta\couldbe30°or150°.（2分）【答案】（√）【解析】Thesinefunctionispositiveinthefirstandsecondquadrants,so\\theta\couldbe30°or150°.

3.Thesumoftheexterioranglesofanypolygonis360°.（2分）【答案】（√）【解析】Thisisapropertyofallpolygons,regardlessofthenumberofsides.

4.If\fx=x^2\and\gx=x+1\,then\f\circgx=x^2+2x+1\.（2分）【答案】（√）【解析】\f\circgx=fgx=fx+1=x+1^2=x^2+2x+1\.

5.Theslopeofahorizontallineisundefined.（2分）【答案】（×）【解析】Theslopeofahorizontallineis

0.

五、简答题

1.Explainthedifferencebetweenafunctionandarelation.（5分）【答案】Afunctionisaspecialtypeofrelationwhereeachinputx-valuehasexactlyoneoutputy-value.Arelationcanhavemultipleoutputsforasingleinput.Forexample,\y=x^2\isafunctionbecauseeachxhasoney,while\x^2+y^2=1\isarelationbutnotafunctionbecausesomex-valueshavetwoy-values.

2.Describethepropertiesofarhombus.（5分）【答案】Arhombusisaquadrilateralwithallfoursidesequalinlength.Oppositesidesareparallel,oppositeanglesareequal,andthediagonalsbisecteachotheratrightangles.Thesumoftheinterioranglesis360°.

3.Howdoyousolveaquadraticequationbyfactoring（5分）【答案】Tosolveaquadraticequationbyfactoring,rewritetheequationinstandardform\ax^2+bx+c=0\.Factorthequadraticexpressionintotwobinomials,seteachfactorequaltozero,andsolvefor\x\.Forexample,for\x^2-5x+6=0\,factorto\x-2x-3=0\,givingsolutions\x=2\and\x=3\.

六、分析题

1.Analyzethetransformationsofthefunction\fx=x^2\toobtain\gx=-2x-3^2+4\.Describethetransformationsstepbystep.（10分）【答案】Thefunction\gx=-2x-3^2+4\involvesseveraltransformationsof\fx=x^2\:-Horizontalshift:\x-3^2\shiftsthegraphright3units.-Verticalstretch:Thecoefficient-2stretchesthegraphverticallybyafactorof2andreflectsitoverthex-axis.-Verticalshift:+4shiftsthegraphup4units.Combiningthese,thegraphof\fx\istransformedto\gx\througharightshiftof3units,averticalstretchbyafactorof2withreflection,andanupwardshiftof4units.

2.Atrianglehasanglesmeasuring45°,45°,and90°.Ifthehypotenuseis10cm,findthelengthsoftheothertwosides.（15分）【答案】Ina45°-45°-90°triangle,thelegsarecongruent,andthehypotenuseis\\sqrt{2}\timesthelengthofeachleg.Lettheleglengthbe\x\.Then,\x\sqrt{2}=10\,so\x=\frac{10}{\sqrt{2}}=5\sqrt{2}\cm.Therefore,thelengthsoftheothertwosidesare\5\sqrt{2}\cmeach.

七、综合应用题

1.Arectangulargardenhasaperimeterof60metersandalengththatistwicethewidth.Findthedimensionsofthegardenandthearea.（25分）【答案】Letthewidthbe\w\metersandthelengthbe\l\meters.Giventhat\l=2w\andtheperimeter\P=2l+2w=60\:-Substitute\l=2w\:\22w+2w=60\→\4w+2w=60\→\6w=60\→\w=10\meters.-Find\l\:\l=2w=2\times10=20\meters.-Calculatethearea\A\:\A=l\timesw=20\times10=200\squaremeters.Therefore,thedimensionsare10metersby20meters,andtheareais200squaremeters.---标准答案

一、单选题

1.C

2.B

3.A

4.B

5.D

6.A

7.B

8.B

9.C

10.C

二、多选题

1.A、C、E

2.A、B、C、D

3.A、E

4.A、B、C

5.B、C、D、E

三、填空题

1.

7202.\\frac{5}{4}\

3.\\frac{1}{6}\

4.

65.27

四、判断题

1.（×）

2.（√）

3.（√）

4.（√）

5.（×）

五、简答题

1.Afunctionisaspecialtypeofrelationwhereeachinputx-valuehasexactlyoneoutputy-value.Arelationcanhavemultipleoutputsforasingleinput.Forexample,\y=x^2\isafunctionbecauseeachxhasoney,while\x^2+y^2=1\isarelationbutnotafunctionbecausesomex-valueshavetwoy-values.

2.Arhombusisaquadrilateralwithallfoursidesequalinlength.Oppositesidesareparallel,oppositeanglesareequal,andthediagonalsbisecteachotheratrightangles.Thesumoftheinterioranglesis360°.

3.Tosolveaquadraticequationbyfactoring,rewritetheequationinstandardform\ax^2+bx+c=0\.Factorthequadraticexpressionintotwobinomials,seteachfactorequaltozero,andsolvefor\x\.Forexample,for\x^2-5x+6=0\,factorto\x-2x-3=0\,givingsolutions\x=2\and\x=3\.

六、分析题

1.Thefunction\gx=-2x-3^2+4\involvesseveraltransformationsof\fx=x^2\:-Horizontalshift:\x-3^2\shiftsthegraphright3units.-Verticalstretch:Thecoefficient-2stretchesthegraphverticallybyafactorof2andreflectsitoverthex-axis.-Verticalshift:+4shiftsthegraphup4units.Combiningthese,thegraphof\fx\istransformedto\gx\througharightshiftof3units,averticalstretchbyafactorof2withreflection,andanupwardshiftof4units.

2.Ina45°-45°-90°triangle,thelegsarecongruent,andthehypotenuseis\\sqrt{2}\timesthelengthofeachleg.Lettheleglengthbe\x\.Then,\x\sqrt{2}=10\,so\x=\frac{10}{\sqrt{2}}=5\sqrt{2}\cm.Therefore,thelengthsoftheothertwosidesare\5\sqrt{2}\cmeach.

七、综合应用题

1.Letthewidthbe\w\metersandthelengthbe\l\meters.Giventhat\l=2w\andtheperimeter\P=2l+2w=60\:-Substitute\l=2w\:\22w+2w=60\→\4w+2w=60\→\6w=60\→\w=10\meters.-Find\l\:\l=2w=2\times10=20\meters.-Calculatethearea\A\:\A=l\timesw=20\times10=200\squaremeters.Therefore,thedimensionsare10metersby20meters,andtheareais200squaremeters.。