1998-数一真题、标准答案及解析

1998120112limxxxx→++−−=.14−.1()()()220220220114lim112211lim4112lim.24xxxxxxxxxxxx→→→++−−=++−+−−=−==−22111~2xx−−−22000011112121limlim24111lim41112121lim.44xxxxxxxxxxxxxxxx→→→→−−−++−⎯⎯→=−−−+=−−−+⎯⎯→=−()2110xx−→→3()1uλ+.0u→()()()22111,2!uuuouλλλλ−+=+++0x→()()22221111,281111,28xxxoxxxxox⎛⎞+=++−+⎜⎟⎝⎠⎛⎞−=−+−+⎜⎟⎝⎠()()2222022011111122828lim1lim414xxxxxxoxxoxx→→+−+−−+−⎛⎞⎜⎟=−+⎜⎟⎝⎠=−2()()1,,zfxyyxyfxϕϕ=++2zxy∂=∂∂.()()()'''''yfxyxyyxyϕϕ++++.()()()()()()()()()()()''22''''''''''''1,11zyfxyfxyyxyxxxzfxyfxyyfxyxyyxyxyxxyfxyxyyxyϕϕϕϕϕ∂=−+++∂∂=−++++++∂∂=++++3l221,43xy+=,a()22234lxyxyds++=∫v.12.al221,43xy+=223412xy+=()()2223421221212,lllxyxydsxydsxydsaa++=+=+=∫∫∫vvvlxxyylxyds∫v0.4An*0,AA≠AEn.A,λ()2*AE+.21Aλ⎛⎞+⎜⎟⎝⎠.()0,Axxxλ=≠()111,0AAxxAAxxxλλ−−=⇒=≠*,AAxxλ=()22*,AAxxλ⎛⎞=⎜⎟⎝⎠()22*1,0,AAExxxλ⎡⎤⎛⎞⎡⎤⎢⎥+=+≠⎜⎟⎢⎥⎣⎦⎢⎥⎝⎠⎣⎦()2*AE+21Aλ⎛⎞+⎜⎟⎝⎠5D1yx=20,1,yxxe===(),XYD(),XYX2x=.14.D22111112.eexDSdxdydxx===∫∫∫(),XY()()1,,,20,xyDfxy⎧∈⎪=⎨⎪⎩x()()12011,1220,xXXdyxefxfxdyx+∞−∞⎧=≤≤⎪==⎨⎪⎩∫∫()124Xf.1()fx()220xdtfxtdtdx−∫A()2xfx(B)()2xfx−C()22xfxD()22xfx−A.22uxt=−()()()()()2202200221122122xxxdddtfxtdtfudufududxdxdxfxxxfx⎡⎤−=−=⎢⎥⎣⎦=⋅=∫∫∫2()()232fxxxxx=−−−A3.B2.C1.D0.B.()()()()()()2322111,fxxxxxxxxxx=−−−=−+−+()fx0,1x=1x=−()fx2.3()yyx=x2,1yxyxα=++++0x→+αx+()0yπ=()1yA2π.Bπ.C4eπ.D4eππD.2,1yxyxα=++++2.1yyxxxα=+++++0x→+'21yyx=+()0,yπ=arctanxyeπ=()arctan41.xyeeπππ==4111222333abcabcabc⎡⎤⎢⎥⎢⎥⎢⎥⎣⎦333121212xaybzcaabbcc−−−==−−−111232323xaybzcaabbcc−−−==−−−A.B.C.C.A.111222333abcabcabc⎡⎤⎢⎥⎢⎥⎢⎥⎣⎦121212232323333aabbccaabbccabc−−−⎡⎤⎢⎥−−−⎢⎥⎢⎥⎣⎦{}{}11212122232323,,,,SaabbccSaaaacc=−−−=−−−.()111,,abc()333,,abc:{}1313131,,Saabbcc=−−−,312SSS=+.123,,SSS12,SS.5AB()()()()01,0,||PAPBPBAPBA=,A()()||PABPAB=(B)()()||PABPAB≠(C)()()()PABPAPB=.(D)()()()PABPAPB≠.C.()()||PBAPBA=()()()()PABPABPAPA=()()()()()()()1PABPAPAPABPAPBPAB−==−⎡⎤⎡⎤⎣⎦⎣⎦()()()PABPAPB=C.11:111xyzl−−==−:210xyzπ−+−=0l0ly.1lπ1πl{}1,1,1s=−π{}1,1,2n=−111132.112ijknsnijk=×−=−−−()1,0,1l1π1π()()13210,xyz−−−−=3210xyz−−+=.0l0210:3210xyzlxyz−+−=⎧⎨−−+=⎩0ly()2112xyzy=⎧⎪⎨=−−⎪⎩y()()22221212xzyy⎡⎤+=+−−⎢⎥⎣⎦2224174210.xyzy−++−=211:111xyzl−−==−1010xyyz−−=⎧⎨+−=⎩l()110,xyyzλ−−++−=()110.xyzλλλ+−+−−=1ππ()1120,λλ−−=−=2,λ=−1π()()13210,xyz−−−−=3210xyz−−+=.,λ0x()()()42242,2Axyxyxyixxyjλλ=+−+(),uxy(),uxy.()()()()42242,2,,,PxyxyxyQxyxxyλλ=+=−+QPxy∂∂=∂∂()()42410.xxyλλ++=1λ=−0x()1,0()244210220,0arctan.xyxxuxydxCxxyyCx⋅=−+++=−+∫∫C.yv..,m,B,ρ()0kk.yv().yyv=,OOy22,dymmgBkvdtρ=−−dyvdt=.,dyvdt=22,dydvdydvvdtdydtdy=⋅=,dvmvmgBkvdyρ=−−,mvdydvmgBkvρ=−−()()2lnmmgBmyvmgBkvCkkρρ−=−−−−+00,|yv==()()2ln,mmgBCmgBkvkρρ−=−−()2ln.mmgBmmgBkvyvkkmgBρρρ−−−=−−−()()212222,axdydzzadxdyxyz∑++++∫∫∑222zaxy=−−−a.1()()()22122221.axdydzzadxdyIaxdydzzadxdyaxyz∑∑++==++++∫∫∫∫2221:0xyaz⎧+≤∑⎨=⎩z()()()()11222224420400311132122122.2DaarIaxdydzzadxdyaxdydzzadxdyaaazdVadxdyaazdVaaadrdrzdzaaππππθπ∑+∑∑ΩΩ−−=++−++⎛⎞=−++⎜⎟⎝⎠⎛⎞=−−+⎜⎟⎝⎠=−−=−∫∫∫∫∫∫∫∫∫∫∫∫∫∫∫w2()()()()221222221211.axdydzzadxdyIaxdydzzadxdyaxyzxdydzzadxdyIIa∑∑∑∑++==++++=++=+∫∫∫∫∫∫∫∫22212,DyzIxdydzaxydydz∑==−−−∫∫∫∫yzDyOz222,0.yzaz+≤≤()()22231022222222.311,xyaDIdarrdraIzadxdyaaaxydxdyaππθπ∑=−−=−=+=−−−∫∫∫∫∫∫xyDxOy22xya+≤()222223200122,6aIdaaarrrdraaππθ=−−−=∫∫312.2IIIaπ=+=−2sinsinsinlim.1112nnnnnnnπππ→∞⎡⎤⎢⎥+++⎢⎥+⎢⎥++⎣⎦sinsinsin,1,2,3,,.1iiinnnininnnnπππ=++111sin11sinsin.1nnniiiiniininnnnnnnπππ===⋅++∑∑∑10112limsin1sin.1nninixnnnπππ→∞=⋅=⋅=+∑∫10112limsinsin.nniixnnπππ→∞===∑∫2.π{}na()11nnna∞=−∑111nnna∞=⎛⎞⎜⎟+⎝⎠∑.1{}nalimnna→∞,anaa≥0a≥.()11nnna∞=−∑0a()11nnna∞=−∑.{}na11,11nnnaa⎛⎞⎛⎞≤⎜⎟⎜⎟++⎝⎠⎝⎠11,1a+111nna∞=⎛⎞⎜⎟+⎝⎠∑111nnna∞=⎛⎞⎜⎟+⎝⎠∑.2lim0.nnaa→∞=1,1nnnba⎛⎞=⎜⎟+⎝⎠11limlim1,11nnnnbaa→∞→∞==++111nnna∞=⎛⎞⎜⎟+⎝⎠∑.()yfx=[]0,1.1()00,1,x∈[]00,x()0fx[]0,1x()yfx=.2()fx()0,1()()'2fxfxx−10x.1()()1,xxxftdtϕ=−∫()xϕ[]0,1()0,1()()010.ϕϕ==()00,1,x∈()'00.xϕ=()()()01'0000.xxxfxftdtϕ=−=∫()()0100.xxfxfxdx=∫2()()()1,xFxxfxftdt=−∫()()()()()()'''20,Fxxfxfxfxfxxfx=++=−()Fx()0,1()0Fx=0xx=10x.2222224,xayzbxyxzyz+++++=xyPzξηξ⎡⎤⎡⎤⎢⎥⎢⎥=⎢⎥⎢⎥⎢⎥⎢⎥⎣⎦⎣⎦2244,ηξ+=,ab.P111111bAba⎡⎤⎢⎥=⎢⎥⎢⎥⎣⎦000010004B⎡⎤⎢⎥=⎢⎥⎢⎥⎣⎦EAEBλλ−=−11001010111004bbaλλλλλλ−−−⎡⎤⎡⎤⎢⎥⎢⎥−−−=−⎢⎥⎢⎥⎢⎥⎢⎥−−−−⎣⎦⎣⎦3,1ab==.111131111A⎡⎤⎢⎥=⎢⎥⎢⎥⎣⎦1230,1,4.λλλ===()00,EAx−=10λ=()11,0,1Tα=−.()0,EAx−=21λ=()21,1,1Tα=−.()40,EAx−=34λ=()11,2,1Tα=.123,,ααα31212312311111121,0,,,,,,,22333666TTTαααηηηααα⎛⎞⎛⎞⎛⎞==−==−==⎜⎟⎜⎟⎜⎟⎝⎠⎝⎠⎝⎠11123612036111236P⎡⎤⎢⎥⎢⎥⎢⎥=−⎢⎥⎢⎥⎢⎥−⎢⎥⎣⎦.An,k0kAα=,α10kAα−≠1,,,kAAααα−.011,,,,kλλλ−10110,kkAAλαλαλα−−+++=()110110,kkkAAAλαλαλα−−−+++=100.kAλα−=10kAx−≠00.λ=1210,kλλλ−====1,,,kAAααα−.I1111221,222112222,221122,22000nnnnnnnnnaxaxaxaxaxaxaxaxax+++=⎧⎪+++=⎪⎨⎪⎪+++=⎩()()()11121,221222,212,2,,,,,,,,,,,,,TTTnnnnnnbbbbbbbbbII1111221,222112222,221122,22000nnnnnnnnnbxbxbxbxbxbxbxbxbx+++=⎧⎪+++=⎪⎨⎪⎪+++=⎩.II()()()111121,2221222,212,2,,,,,,,,,,TTTnnnnnnnycaaacaaacaaa=+++12,,,nccc.III,AB,TABO=()TTTBAABO==,AnIIn.B,nII()22.nrBnnn−=−=A2nI,nAnIIII.,XY012XY−.ZXY=−,XYZ()()()()()()0,1,EZEXEYDZDXDY=−==+=()~0,1ZXYN=−()()()()22222221DXYDZEZEZEZEZDZEZEZEZ−==−⎡⎤=−⎡⎤⎣⎦⎣⎦=+−⎡⎤⎡⎤⎣⎦⎣⎦=−⎡⎤⎣⎦2222012222zzEZzedzzedzπππ+∞+∞−−−∞=⋅==∫∫21.DXYπ−=−()23,4,6Nn()1.4,5.40.95n()2212tzzedtπ−−∞Φ=∫z1.281.6451.962.33()zΦ0.9000.9500.9750.990X()3.4~0,16X