2020年全国大学高等数学考试及答案解析

经典word整理文档，仅参考，双击此处可删除页眉页脚。本资料属于网络整理，如有侵权，请联系删除，谢谢！2020年全国大学高等数学考试试题41cosx,x0f(x)在x0）axb,x01212()abBabDab2C)ab0'f(x)f(x)f(x)0）(A)f(1)f(1)(C)f(1)f(1)Bf(1)f(1)Df(1)f(1)3ax与x3na(x1)的()nnnn1n114xy1，yxy3x，设D2xyfx,y在Dfx,y()D1ydfrcos,rsinrdrsin23142sin2frcos,rsinrdr1sin2d312sin41dfrcos,rsindrosin231x42sind1frcos,rsindrsin2132sin24（5）设是nE为n）BET()ETDE2TC)E2T200210100021,B020,C020A）001001002()C,CBC,CC)C,CDC,C若()PAPBPABPAPBPABPAPBPAPBPP22X,YEX2,EY1,DX3EXXY2()335541f(x)f(3)(0)1x22y3y0yy'''(x,y)|xy1D22xy122Laxyz12y3z)(x(x,y)N{XYY0}________.dy2f(e,cosx)f(u,v)2yxdx2x0，x0kknln1limn2nnk1y3x3y20y(x)x3y(x)3u(x),v(x)(x(x)]u(x(x)(xv(x)u(x),u(x),,u(x)f(x)u(xu(x)u(x)12n12nfx()2xy,z222,0LA0,2,0B0,x,zyzxzxyy(xyzI.2222Lβαα，=+k.αα,αβ=2α+2kα，β=2α内R3,12311322313为R3123αα,αξk0,123123ξ.2xx2xx8xx2xxf(x,x,x)22222131312132312QYQXy21ya22（8本题满分分）设随机变量X,Y相互独立，且X的概率分布为20y11P(XP(X，Yf(y)其他20,()求PYEY)ZXY()求n次XnX,X12nN(,)ZXin)n2iiZ,ZZ。12n()求Zi()利用一阶矩求2020年全国大学高等数学考试及解析41cosx,x0f(x)在x0）axb,x01212()abBabDab2C)ab01x1cosx12112lim,f(x)lim在x0b.选A.axax2ax02ax0'f(x)f(x)f(x)0）()ff(C)ff(Bff(Dff(f(x)0f(x)0f(x)f'(x)或(2)C(1)(1)0f'(x)0f'(x)(2)3ax与x3na(x1)的()nnnn1n1ax2a(x1)nnnn1n1a(x1)(0,2)nnn1na(x1)(0,2)x与x3na(x1)的3nnnnn1n114xy1设D2xy，yx，y3xfx,y在Dfx,y()D1dfrcos,rsinrdrsin23142sin2frcos,rsinrdr1sin2d312sin41dfrcos,rsindrsin23142sind1frcos,rsindrsin2132sin24yDd1f(rcos,rsinrdr，3f(x,y)sin21D2sin4ox（5）设是nE为n）BET()ETDE2TC)E2T(E)0(E得)x0ET0。TTE的1得个即Er()TTTT个200210100021,B020,C020A）001001002()C,CBC,CC)C,CDC,CE)0A1003r(2E)1A~020002B0BE由3r(2EB)2C∴A~CBC若()PABPAPBPABPAPBPAPBPAPBPP22AB,ABBP(AB)P()且P()P(B)P(AB)P(B)P(AB)P()P(B).2X,YEX2,EY1,DX3EXXY2()3355Y2)]E(XXY2X)E(X)E(XY)2E(X)E[X(X22D(X)E(X)E(X)EY)2E(X)232212252.41f(x)f(0)(3)1x26f(0)(x)(x11f(x)2nn2n1x1(x)22n0n0f(x)(2n(2nn2)xf0n2n3n22y3y0yy'''e(ccos2xcsin2x)c,cyx12122301i1,22csin2x)eccos2xx12D(x,y)|xy122xy122Laa1P2Q2,,由积分与路径无关知【解析】y(xyx(xy222222PQa1yxxyz12y3z)(x14(x2y3z)dxdydz6zdxdydz6zdzdxdy，10Dz1z)2zzD2z(x1112y3z)dxdydz6zdxdydz6z(1z)dz3(z2zz)dz.12322400(x,y)N{XYY0}________.12X~N(1,1),Y~N(0,1)XY{XYY0}{(X1)Y0}{X10,Y0}{X10,Y0}11111{XY0}{XY22222.dy2f(e,cosx)f(u,v)2yxdx2x0，x0dydxdy2f(1,1),f(1,1),'''dx2111x0x0x0yf(e,cosx)y(0)f(1,1)xdydxfefsinxf(1,1)1f(1,1)0f(1,1)'x''''12121x0x0dy2fefe(sinx)fe(sinx)fsinxfefcosx''2x''x''x''2'x'dx212dy2f(1,1)f(1,1)f(1,1)''''dx212x0dyf(1,1)'dx1x0dy2f(1,1)f(1,1)f(1,1)''''dx21112x0kknln1limn2nnk114kk111x1114n2ln(1)1xx)x)x)x1dx)lim2210n2n221xn000k1y3x3y20y(x)x3y(x)3y(1)1y(1)03x3yy33y'022y'0令得x1x6x6yy'3yy3y''022x1x1or，1将x将xy1y01,y1y10将x1,y0y''(1)20y(1)1；x1y(1)01故xu(x),v(x)(x(x)]u(x(x)(xv(x)u(x),u(x),,u(x)f(x)u(xu(x)u(x)12n12nfx()u(xh)v(xh)u(x)v(x)u(x)v(xhh0u(xh)v(xh)u(xh)v(x)u(xh)v(x)u(x)v(x)hh0v(xh)v(x)u(xh)u(x)u(xh)v(x)hhh0h0u(x)v(x)u(x)v(x)f(x)[u(x)u(x)u(x12nu(xu(x)u(x)u(xu(x)u(x)u(xu(x)u(x)12n12n12n2xy,z222,0，LA0,2,0B0,x,zyzdxzxyy(xyzI.2222L2π2xcosππ2siny，:22zcosπ[(2sincos)sin2sincos(1sin)sin]d22π2π2sinsincossin)sind222π2π222sindπ2202βαα，=+k.αα,αβ=2α+2kα，β=2α内R3,12311322313为R3123αα,αξk0,123123ξ.2+2k,2,+k1,,12313213201,,0201232k0k12012102022k0k1402kk1β,β,βR故为3123k,0kkkkk112233112233k0,k0,i即kk111222333ik2k2+2kk+k011312223133+2kkkk113223130即+2k,,k13213101即0102k0k0kkk0112231k0,kk0213kk,k0111312xx2xx8xx2xxf(x,x,x)2222213131213232QYyy的值及一个正交矩阵QX222a11111326122;Q0,fxQy3y6ya222361111326214f(x,x,x)XA111T12341a1yyXf(x,x,x)1T222，23122142|A01110a241a故r()，21422r()a2将aA1114122141106，|EA11232411(3E)x0A1由由；111(6E)x0A60211由(0E)x0A02313,,,,令PPAP161231230131211,2,1T,，611,,,TT2311132631201，QT则Q612336011326xQyf3y6y2212（7本题满分分）设随机变量X,Y相互独立，且X的概率分布为20y11P(XP(X，Yf(y)其他20,()求PYEY)ZXY()求z,0z14EY};(II)f(z)(I)Y2,2z39Zz23()EY)y2ydy102492PYEY)PY)2ydy330()F(Z)P(Zz)P(XYz)zP(XYz,X0)P(XYz,X2)PYz,X0)PYz2,X2)11PYz)PYz2)22z0F(Z)00,z20当zz3F(Z)121,z1,z即当zz1z20z1F(Z)21z1z2F(Z)z211(z2)22z3F(Z)22z0z01z,0z1221F(Z),1z2z211(z2),2z322231,zz0z1F(Z)f(Z)z'z22z3zn次XnX,X12nN(,)ZXin)n2iiZ,ZZ