GMAT模拟试题深度剖析及详细答案

GMAT模拟试题深度剖析及详细答案

一、单选题（每题2分，共20分）

1.If\x\isapositiveinteger,whatisthevalueof\x\when\3x+4\isdivisibleby7A.1B.2C.3D.4E.5【答案】C【解析】For\3x+4\tobedivisibleby7,\3x+4\equiv0\pmod{7}\.Thus,\3x\equiv-4\equiv3\pmod{7}\.Multiplyingbothsidesbythemodularinverseof3modulo7whichis5,weget\x\equiv5\times3\equiv15\equiv1\pmod{7}\.Thesmallestpositiveintegersatisfyingthisis\x=3\.

2.Whatistheareaofanequilateraltrianglewithaperimeterof18A.9√3B.12√3C.15√3D.18√3E.21√3【答案】B【解析】Thesidelengthoftheequilateraltriangleis\\frac{18}{3}=6\.Thearea\A\isgivenby\A=\frac{\sqrt{3}}{4}s^2\,where\s\isthesidelength.Thus,\A=\frac{\sqrt{3}}{4}\times6^2=9\sqrt{3}\.

3.If\fx=2x^2-3x+4\,whatis\f2\A.4B.6C.8D.10E.12【答案】C【解析】Substituting\x=2\intothefunction,\f2=22^2-32+4=8-6+4=6\.

4.Inabagcontaining10redballsand6blueballs,whatistheprobabilityofdrawingaredballfollowedbyablueballwithoutreplacementA.\\frac{1}{3}\B.\\frac{1}{6}\C.\\frac{1}{15}\D.\\frac{2}{9}\E.\\frac{5}{18}\【答案】E【解析】Theprobabilityofdrawingaredballfirstis\\frac{10}{16}=\frac{5}{8}\.Afterdrawingaredball,thereare15ballsleft,6ofwhichareblue.Thus,theprobabilityofdrawingablueballnextis\\frac{6}{15}=\frac{2}{5}\.Thecombinedprobabilityis\\frac{5}{8}\times\frac{2}{5}=\frac{1}{4}\.

5.If\\frac{a}{b}=\frac{3}{4}\,whatisthevalueof\\frac{2a+3b}{2a-3b}\A.\\frac{1}{2}\B.1C.2D.3E.4【答案】C【解析】From\\frac{a}{b}=\frac{3}{4}\,wehave\4a=3b\or\a=\frac{3}{4}b\.Substitutingthisintotheexpression,\\frac{2a+3b}{2a-3b}=\frac{2\times\frac{3}{4}b+3b}{2\times\frac{3}{4}b-3b}=\frac{\frac{3}{2}b+3b}{\frac{3}{2}b-3b}=\frac{\frac{9}{2}b}{-\frac{3}{2}b}=-3\.

6.Whatisthevalueof\\sin30^\circ\cos60^\circ+\cos30^\circ\sin60^\circ\A.\\frac{1}{2}\B.\\frac{\sqrt{3}}{2}\C.1D.\\sqrt{3}\E.2【答案】C【解析】Usingthesineadditionformula,\\sinA+B=\sinA\cosB+\cosA\sinB\,weget\\sin30^\circ+60^\circ=\sin90^\circ=1\.

7.If\x^2-5x+6=0\,whatisthesumoftherootsA.5B.6C.7D.8E.9【答案】A【解析】UsingVietasformulas,thesumoftherootsof\ax^2+bx+c=0\is\-\frac{b}{a}\.Here,\a=1\,\b=-5\,sothesumis\-\frac{-5}{1}=5\.

8.Whatisthesimplifiedformof\\frac{3x^2-12}{x^2-4}\A.\\frac{3}{1}\B.\\frac{3x-12}{x-4}\C.\\frac{3x-4}{x-2}\D.\\frac{3x+2}{x-2}\E.\\frac{3x-2}{x+2}\【答案】D【解析】Factoringthenumeratoranddenominator,\\frac{3x^2-4}{x^2-4}=\frac{3x+2x-2}{x+2x-2}=3\.

9.If\2^{2x}=16\,whatisthevalueof\x\A.2B.3C.4D.5E.6【答案】A【解析】Rewriting16as\2^4\,weget\2^{2x}=2^4\.Thus,\2x=4\or\x=2\.

10.Inacirclewithradius5,whatisthelengthofanarcsubtendedbyacentralangleof\72^\circ\A.\4\pi\B.\5\pi\C.\6\pi\D.\7\pi\E.\8\pi\【答案】A【解析】Thecircumferenceofthecircleis\2\pi\times5=10\pi\.Thelengthofthearcis\\frac{72}{360}\times10\pi=\frac{1}{5}\times10\pi=2\pi\.

二、多选题（每题4分，共20分）

1.WhichofthefollowingarefunctionsA.\y=3x-2\B.\y=x^2\C.\x=y^2\D.\y=\frac{1}{x}\E.\y=2\【答案】A、B、D、E【解析】Functionsmusthaveauniqueoutputforeachinput.\y=3x-2\,\y=x^2\,\y=\frac{1}{x}\,and\y=2\allsatisfythiscondition.\x=y^2\doesnot,asoneinput\x\cancorrespondtotwooutputs.

2.WhatarethepropertiesofaparallelogramA.OppositesidesareequalB.OppositeanglesareequalC.DiagonalsbisecteachotherD.Allanglesare\90^\circ\E.Adjacentanglesaresupplementary【答案】A、B、C、E【解析】Aparallelogramhasoppositesidesequal,oppositeanglesequal,diagonalsthatbisecteachother,andadjacentanglesthataresupplementary.Itdoesnotnecessarilyhaveallanglesas\90^\circ\whichwouldmakeitarectangle.

3.WhichofthefollowingareirrationalnumbersA.\\sqrt{4}\B.\\sqrt{2}\C.\\frac{1}{3}\D.\\pi\E.\\frac{22}{7}\【答案】B、D【解析】\\sqrt{4}=2\isrational.\\sqrt{2}\and\\pi\areirrational.\\frac{1}{3}\and\\frac{22}{7}\arerational.

4.WhatarethepropertiesofacircleA.AllpointsonthecircleareequidistantfromthecenterB.AcirclehasinfinitelinesofsymmetryC.AcirclehasnoverticesD.AcirclecanbedividedintosectorsE.Acirclehasacircumference【答案】A、B、C、D、E【解析】Allthegivenpropertiesaretrueforacircle.

5.WhichofthefollowingaretrueabouttrianglesA.Thesumoftheinterioranglesofatriangleis\180^\circ\B.AnequilateraltrianglehasallsidesequalC.Arighttrianglehasoneangleof\90^\circ\D.ThePythagoreantheoremappliestoalltrianglesE.Atrianglecanhavetwoobtuseangles【答案】A、B、C【解析】Thesumoftheinterioranglesofatriangleisalways\180^\circ\.Anequilateraltrianglehasallsidesequal.Arighttrianglehasoneangleof\90^\circ\.ThePythagoreantheoremappliesonlytorighttriangles.Atrianglecannothavetwoobtuseangles.

三、填空题（每题4分，共20分）

1.If\fx=x^2-4x+5\,whatis\f3\__________（4分）【答案】8【解析】Substituting\x=3\intothefunction,\f3=3^2-4\times3+5=9-12+5=2\.

2.Theareaofarectanglewithlength10cmandwidth6cmis__________squarecm.__________（4分）【答案】60【解析】Theareaofarectangleislength\\times\width,so\10\times6=60\squarecm.

3.If\\sin\theta=\frac{3}{5}\,whatis\\cos\theta\for\\theta\inthefirstquadrant__________（4分）【答案】\\frac{4}{5}\【解析】UsingthePythagoreanidentity\\sin^2\theta+\cos^2\theta=1\,weget\\left\frac{3}{5}\right^2+\cos^2\theta=1\.Thus,\\cos^2\theta=1-\frac{9}{25}=\frac{16}{25}\,so\\cos\theta=\frac{4}{5}\.

4.Thesumofthefirst10naturalnumbersis__________.__________（4分）【答案】55【解析】Thesumofthefirst\n\naturalnumbersisgivenby\\frac{nn+1}{2}\.Thus,\\frac{10\times11}{2}=55\.

5.If\3x-7=11\,whatisthevalueof\x\__________（4分）【答案】6【解析】Adding7tobothsides,\3x=18\.Dividingby3,\x=6\.

四、判断题（每题2分，共10分）

1.Thesquarerootofanegativenumberisarealnumber.（）（2分）【答案】（×）【解析】Thesquarerootofanegativenumberisnotarealnumber;itisanimaginarynumber.

2.Themedianofadatasetisalwaysequaltothemean.（）（2分）【答案】（×）【解析】Themedianisthemiddlevaluewhenthedataisordered,whilethemeanistheaverage.Theyareequalonlyincertaincases.

3.Theequation\x^2+1=0\hasrealsolutions.（）（2分）【答案】（×）【解析】Thesolutionsto\x^2+1=0\are\x=\pmi\,whichareimaginarynumbers,notreal.

4.Theareaofacircleisgivenby\\pir^2\.（）（2分）【答案】（√）【解析】Thisisthecorrectformulafortheareaofacirclewithradius\r\.

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

0.

五、简答题（每题4分，共20分）

1.Explainthedifferencebetweenanequilateraltriangleandanisoscelestriangle.__________（4分）【答案】Anequilateraltrianglehasallthreesidesequalandallthreeanglesequal,eachmeasuring\60^\circ\.Anisoscelestrianglehasatleasttwosidesequalandatleasttwoanglesequal,buttheanglesoppositetheequalsidesarenotnecessarily\60^\circ\.

2.WhatisthePythagoreantheorem,andwhendoesitapply__________（4分）【答案】ThePythagoreantheoremstatesthatinarighttriangle,thesquareofthelengthofthehypotenusethesideoppositetherightangleisequaltothesumofthesquaresofthelengthsoftheothertwosides.Itappliesonlytorighttriangles.

3.Defineafunctioninmathematics.__________（4分）【答案】Afunctionisarelationbetweenasetofinputsandasetofpermissibleoutputs,suchthateachinputisrelatedtoexactlyoneoutput.

4.Whatisthedifferencebetweenanacuteangleandanobtuseangle__________（4分）【答案】Anacuteangleisananglethatmeasureslessthan\90^\circ\.Anobtuseangleisananglethatmeasuresmorethan\90^\circ\butlessthan\180^\circ\.

5.Explaintheconceptofacirclescircumference.__________（4分）【答案】Thecircumferenceofacircleisthedistancearoundthecircle.Itiscalculatedusingtheformula\C=2\pir\,where\r\istheradiusofthecircle.

六、分析题（每题12分，共24分）

1.Arectanglehasaperimeterof30cmandalengthof10cm.Whatisthewidthoftherectangle__________（12分）【答案】Width=5cm【解析】Theperimeterofarectangleisgivenby\P=2l+w\,where\l\isthelengthand\w\isthewidth.Given\P=30\cmand\l=10\cm,wehave\30=210+w\.Solvingfor\w\,\30=20+2w\,so\2w=10\and\w=5\cm.

2.Acirclehasaradiusof7cm.Whatistheareaofthecircle__________（12分）【答案】Area=\49\pi\squarecm【解析】Theareaofacircleisgivenby\A=\pir^2\.Given\r=7\cm,wehave\A=\pi\times7^2=49\pi\squarecm.

七、综合应用题（每题25分，共50分）

1.Atraintravelsataconstantspeedof60km/h.Howfarwillittravelin3hours__________（25分）【答案】Distance=180km【解析】Thedistancetraveledisgivenby\\text{Distance}=\text{Speed}\times\text{Time}\.Givenspeed=60km/handtime=3hours,wehave\\text{Distance}=60\times3=180\km.

2.Thecostofabookis$

20.Ifthepriceisincreasedby25%,whatisthenewprice__________（25分）【答案】Newprice=$25【解析】Theincreaseinpriceis\25\%\of$20,whichis\

0.25\times20=$5\.Thenewpriceis\20+5=$25\.---标准答案

一、单选题

1.C

2.B

3.C

4.E

5.C

6.C

7.A

8.D

9.A

10.A

二、多选题

1.A、B、D、E

2.A、B、C、E

3.B、D

4.A、B、C、D、E

5.A、B、C

三、填空题

1.

82.

603.\\frac{4}{5}\

4.

555.6

四、判断题

1.（×）

2.（×）

3.（×）

4.（√）

5.（×）

五、简答题

1.Anequilateraltrianglehasallthreesidesequalandallthreeanglesequal,eachmeasuring\60^\circ\.Anisoscelestrianglehasatleasttwosidesequalandatleasttwoanglesequal,buttheanglesoppositetheequalsidesarenotnecessarily\60^\circ\.

2.ThePythagoreantheoremstatesthatinarighttriangle,thesquareofthelengthofthehypotenusethesideoppositetherightangleisequaltothesumofthesquaresofthelengthsoftheothertwosides.Itappliesonlytorighttriangles.

3.Afunctionisarelationbetweenasetofinputsandasetofpermissibleoutputs,suchthateachinputisrelatedtoexactlyoneoutput.

4.Anacuteangleisananglethatmeasureslessthan\90^\circ\.Anobtuseangleisananglethatmeasuresmorethan\90^\circ\butlessthan\180^\circ\.

5.Thecircumferenceofacircleisthedistancearoundthecircle.Itiscalculatedusingtheformula\C=2\pir\,where\r\istheradiusofthecircle.

六、分析题

1.Width=5cm

2.Area=\49\pi\squarecm

七、综合应用题

1.Distance=180km

2.Newprice=$25。