2022考研數(shù)學(xué)一真題及答案

20214yxsinxyx2sinx11xyxy2xx1y1xlim1lim(yx)limsin0且yxxxxxxyxf(x)g(x)f(0)(1x)f)x[]〕0f'(x)f(x)g(x)f'(x)0f(x)g(x)00f(x)f(x)g(x)f(x)f(x)g(x)[a,b]01x,x12f1)xx()f(x)f(x)112120,x,xx12011)f(x)f(x)f()(x)f()xg(x)f()xxf(x)，111212f(x)f(011)x)f(x)f(x)x(1212是f(x)g(x)[a,b]F(x)f(x)g(x)f(x)f(0)(1x)f)xF(0)F)01F"(x)f"(x)f(x)0010F(x)F()F()F(x)f(x)g(x)0f(x)g(x)1f(x)dyf(x,ydy01dxx100010100dx0011d2cossin00021cossin1d2cossin000201dxf(x,ydy1000112cossincossin00022211axbx11222【詳解】注意x3，，2xx，022x，23(ab)b(xaxbsinx)23222b4ba,b2a220ab0a00b0cd0c00dbc)2(adbc)2(adadbcadbc222222220ab0a00b0cd0c00da0ba0ba0d0b0c0c0dc0dababadbccdcdad(adbc)bc(adbc)(adbc)2lk，,,k,l1231323,,123,,1231001kl(,,)(,,)K3〔，〕k,l132312312kll，k陣K13233100010lk，,,k,l1231323000,,123P(B).,P(AB)0.3P(BA)〔〕P(AB)03P(A)P()P(A)P(A)P(B)P(A)0.5P(A)0.5P(A)．.6P(BA)P(B)P(AB)0.50.5P(A)0.2，P(A)8X,XX,X12121f(x),f(x)，隨機(jī)變量Y的概率密度為f(y)(f(y)f(y))，隨機(jī)變量21211121Y(XX)2212,,12121212,,12121212112y(f(y)f(yEY)，211212211212y(f(y)f(y，222221121211111EY)EY)E(X)E(X)E(X)E(X)2222222244211112121211111D(X)D(X)EXXD(X)D(X)244444121212264x1siny)y1sinx),)z．224x1siny)y1sinx),)處的法向量為【詳解】曲面z在點22z,z,1|(2,,)(x)()(y0)()(z)0，xy,0)2xyz10即．(xx,2．設(shè)f(x)為周期為4的可導(dǎo)奇函數(shù)，且f'(x)，那么f(7)．0,2(x)dxx212200xCf()0C，xf(x)x2xf(x)f(7)f()f)1．；4即f(x)2xyy(lnxlny)0y)e3．yylnxxyuxyxey)eyxe132x1．y1yz0x2zL2zdxydz．LxyzLPQR．LDxyyz0:D(x,y)|xy1．22xy122xx2axx4xxf(x,x,x)2a2221311323．f(x,x,x)xxaxx4xx22212311323(xax)(x2x)(4a)x2222313234a20,2．a(chǎn)52x,x．設(shè)總體X的概率密度為f(x,)2，其中是未知參數(shù)，0,其它X,X,,XCX是n2212nii1=．C2x525CXnnECXCn22于E(X)x2以．22是i222ii1i1521，C225n1x(t2(e)tdttlimx1．1xln(21)x11x)tdt11(t(e2x)tdt(t(e2tt1lim1limlim(x(e)x)12x1xxxln(1)xx2x1111limx(()x22xx2x22xf(x)yxyxy60yf(x)322x(3y2xyx)y(y2xy)0，222即y223y2x2260yxyxy60及x,y2．令322x6yy4y2xy4x)y(3y2xyx)y2y0〔(224,y,y'()0"()0，把x9f(x)x1y2．在yzz22f(u)zf(ecosy)(4zeyexx2xxy22設(shè)f(0),f'(0)0f(u)ecosyzf(u)f(ecosy)，設(shè)uxxzz2f'(ue,f"(ueyf'(uey;xy2x2xxx2zyz2f'(uey,f"(ueyf'(uey；x2x2xy2zz22f"(uef"(eye2xx2xxy22zz22(4zeye2x，xxy22f"(u)4f(u)uf(u)CeCeC,C2u2u12121u*．41CeCeuf(u)．2u2u412711f(0),f'(0)0C,C．12111eeu．f(u)f(u)2u2u4:zxy(z)2233(x)dydz(y)dzdx(zdxdy1z:,設(shè)1y11x223322(x)dydz(y)dzdx(zdxdy((x)(y)dxdydz122(3x3y76x6ydxdydz22(3x3y7dxdydz2drdr(rdz2113700r21z:在取下側(cè)上，y11x2233(x)(y)(z10，11(x)dydz(y)dzdx(zdxdy33=(x)dydz(y)dzdx(z43310a,0b，cosaacosbba,b22nnnnnnnnn1lima0；nn8abnnn1acosb0a,0bcosa22nnnnn0aab0ab，22nnnnnbalima0．nnnnn1n10a,0b，22nnababbaba,sinnnnnnnnn2222abba2sinsinnnnnacosacosb22nnnnbbnbnabba2nnnnbab2bn22222nbnnbbnnnabbnnnn1n112340111設(shè)A12030AXEA1234123412341001A01110111011101020013131203043100，90AXxx1442xx2x3x34120AX的一個根底解系．311xyz111xyzB43B22xyz23433xyz44()12341001234100()0111010011101012030010431101100112341002610111010010213100131410013141B216111xyz111123212xyzccc2，2，2，134313x1y2z3333010101xyz444E2c6c1c1231c3c12cB1231c4c1c123c1cc23c,c,c31210111111001002n與11100n111111001002A，B．11100n111111EA(n)n1，11110n,A的n；23n0AA~；00102EB(n)n100n10n,B的n；23n100(EB)Bn1Bn0n1Bn0BB~0111111001002n與11100n1112)P(X2)XiY服XP(XU(,i),i,2．求YEYF(y)PYy)PYy,X)PYy,X2)PYy/X)P(X)PYy/X2)P(X2)1PYy/X)PYy/X2)2當(dāng)y0F(y)0；11y3y224當(dāng)0y1F(y)y；211y1F(y)y222412當(dāng)1y2；當(dāng)y2F(y)1．0,y03y0y1