2000年考研数学一真题解析.pdf

1、2000 (1) 1 2 0 2x xdx = . 2 222 232 1 xyz += ( ) 1, 2 , 2 . 3 30 xy y += . 4 1 2 3 12 1 1 23 2 3 120 x ax ax += a = . (5) A B 1 , 9 A B B A PA = . 1 () () , f xgx ( ) ( ) ( ) ( ) 0 fxgxfxgx B ( ) ()( ) ( ) f xga fagx C ()() ()() f xgx fbgb D ( )() ()() f xgx faga 2 () 2222 1 :0 , Sx y z az S += S A

2、1 4 SS xdS xdS = B 1 4 SS ydS xdS = C 1 4 SS zdS xdS = D 1 4 SS xyzdS xyzdS = 3 1 n n u = A () 1 1. n n n u n = B 2 1 n n u = C () 21 2 1 . nn n uu = ( D ) () 1 1 . nn n uu + = + 4 n () 1 , m mn . 0 x , S () () 2 0, x S xf x dydz xyf x dzdx e zdxdy = () f x () 0, + ( ) 0 lim 1, x fx + = ( ) f x . ()

3、 1 1 32 n n n n x n = + . R 0 P 0 P 0 k . () f x 0, () () 00 0, cos 0, fxd x fx x d x = = () 0, 12 , ( ) ( ) 12 0 ff = = . 6 A * 1000 0100 , 1010 0308 A = 11 3, ABA BA E =+ E 4 . B 1 6 2 5 . n n x n y n n x y . 1 1 1 n n x y + + n n x y 11 11 ; nn nn xx A yy + + = 2 12 41 , 11 = A 3 1 1 1 2 1 2 x y

4、= 1 1 n n x y + + . ( ) 01 pp = 0 12 , n xxx X . 2000 (1) 1 2 0 2x xdx = . . 4 () 11 2 22 2 00 0 21 11 s i n c o s 4 x x dx x dxx t tdt = = = 2 222 232 1 xyz += ( ) 1, 2 , 2 . 122 146 xyz + = . () 222 , 2 3 2 1 Fxyz x y z =+ () () () () () () 1, 2,2 1, 2,2 1, 2,2 1, 2, 2 2 2 , 1, 2 , 2 4 8, 1, 2,2 6

5、 12. | | | x y z Fx Fy Fz = = = = = = 122 146 xyz + = 3 30 xy y += . 2 1 2 C yC x =+ . p y = 3 0, pp x + = 3 . p Cx = 32 2 112 2 , 22 C CC yC xd xCxC C x = = += 4 1 2 3 12 1 1 23 2 3 120 x ax ax += a = . - 1 . () () 12 1 1 1 2 1 1 12 1 1 23 2 3 0 1 1 01 1 12002 310 0313 aaa aaa a a + + 1 a = 2 3 . 3

6、 a = 2 . (5) A B 1 , 9 A B B A () PA = . 2 . 3 () ()() 1 , 9 PA B PA B PA B = A B A B A B ( ) () PA B PA B = () () () () PAPB PAPB = () () () ( ) 11, PA PB PAPB = () PA = () PB ()()() () 2 1 1, 9 PA B PAPB PA = = () PA = 2 . 3 1 () () , f xgx ( ) ( ) ( ) ( ) 0 fxgxfxgx B ( ) ()( ) ( ) f xga fagx C (

7、)() ()() f xgx fbgb D ( )() ()() f xgx faga A . () () ()() () () () 2 0, fx fxgx fxgx gx gx = ()() ()() f xgb fbgx A . 2 () 2222 1 :0 , Sx y z az S += S A 1 4 SS xdS xdS = B 1 4 SS ydS xdS = C 1 4 SS zdS xdS = D 1 4 SS xyzdS xyzdS = C . S 0 x = 0 y = A B D C . , 11 44 SSS zdS zdS xdS = 3 1 n n u = A

8、 () 1 1. n n n u n = B 2 1 n n u = C () 21 2 1 . nn n uu = ( D ) () 1 1 . nn n uu + = + D . D A B C . () 2 1 1 ln n n n = () 22 1 1 ln n n nn u nnn = = A () 1 1 1 n n n = 2 11 1 n nn u n = = B () 1 1 1 1 n n n = () 21 2 111 111 212 nn nnn uu nnn = = + c . 4 n () 1 , m mn n 1 , m A 1 , m 1 , m . B 1

9、, m 1 , m . C 1 , m 1 , m . D () 1 , m A = ( ) 1 , m B = . D . . A 1 , m 1 , m 1 , m 1 , m ( ) 1 , , m rm 1 , m . 1 m = () () 11 1, 0 , 0, 1 TT = 1 1 . B . 1 m = () () 11 1, 0 , 0, 1 TT = 1 1 () () 11 1, 0 , 0 , 0 TT = 1 1 1 . C 1 , m 1 , m 1 , m () () 11 , , , mm rrm = 1 , m 1 m = () () 11 1, 0 , 0

10、, 1 TT = 1 1 . E D . ( ) 1 , m A = () 1 , m B = () () rA rB = () ( ) 11 , , , mm rrm = = 1 , m . 5 () , X Y X Y = + X Y = (A) () () . EXE Y = (B) () () () () 22 22 . EX EX EY EY = (C) ()() 22 . EXE Y = (D) () () () () 22 22 . EX EX EY EY +=+ B . () ( ) ()()()() ()() () () , , , , ,Cov Cov X Y X Y Co

11、v X X Cov X Y Cov Y X Cov Y Y Cov X X Cov Y Y DX DY =+ =+ = = () () ( ) () () () () 22 22 ,0 0Cov D X D Y EX EX EY EY = = = B . 1 0 2s i n lim . 4 1 x x ex x e x + + + 11 00 11 00 2s i n 2s i n2 lim lim 1 1, 44 1 11 2s i n 2s i n lim lim 0 1 1, 44 11 xx xx xx xx ex ex xx ee xx ex ex xx ee xx + + +=

12、= = + + += += + = + , x x zfx y g yy =+ f g 2 . z x y 12 2 1 zy yf f g xyx =+ 2 11 11 222 12 2 22 223 1 2 11 22 232 3 11 1 11 zxxy f yx f f f x f f g g xy y y y y x x xy ffx y ffgg yyx x =+ + = + 22 , 4 L xdy ydx I x y = + L ( ) 1, 0 R () 1 R . 22 22 , 44 yx PQ xyxy = + () ()() 22 2 22 4 , 0 , 0 4 P

13、yxQ xy xy xy = + cos : 2 sin x t C yt = = 0, 2 , tC 22 0. 4 LC xdy ydx xy + = + 2 2 22 22 2 0 1 2 44 LC xdy ydx xdy ydx IId t xy xy = = + 0 x , S () () 2 0, x S xf x dydz xyf x dzdx e zdxdy = () f x () 0, + ( ) 0 lim 1, x fx + = ( ) f x . () () () () () 2 2 0, x S x xf x dydz xyf x dzdx e zdxdy xfxf

14、xx fxed V = = + S S () ()( ) ( ) 2 0, 0 x xf x f x xf x e x += () () () 2 11 1, 0 x fx fx ex xx + = () () x x e fxeC x =+ () 2 00 lim lim 1, xx xx eC e fx x + + = () 2 0 lim 0, xx x eC e + += 10 C + = 1 C = () () 1. x x e fx e x = () 1 1 32 n n n n x n = + . () ()( ) () 1 11 1 2 1 32 3 1 lim lim lim

15、 3 321 2 31 1 3 n n n n nn n nn n n n n a a n n + + + + + = + + + + 3 R = ( ) 3, 3 3 x = () () 3 11 2 32 n n n nn + 1 1 n n = 3 x = 3 x = () () () () () 32 11 1 1 32 32 nn n nn nn nn n = + + () 1 1 n n n = () () 1 2 1 32 n n n n n = + . 3 x = . R 0 P 0 P 0 k . 0 P . 1 , O 0 OP x 0 P () ,0,0 R 2222 x

16、 yzR += () , xyz 0, 0, yz = () () 2 22 2 22 xkxRyzd V x kxRyzd V + = + () () 2 22 223 2 22 5 22 000 5 4 8s i n 3 32 15 R xR yzd V x yzd V R d V ddrr d r R R + =+ += + = () () 2 22 2 223 6 2 28 31 5 x x R y z dV R x dV R xyzd V R + = = + + = 4 R x = . ,0,0 . 4 R 2 O 0 P 0 P O z 222 2 x yzR z += () ,

17、xyz 0, 0, xy = () () 223 223 kz x y z dV z kx y zd V + = + () 2c o s 222 4 22 000 5 4s i n 3215 R x yzd V dd r d r R + = = () 2c o s 222 5 22 000 67 2 0 6 4s i n c o s 64cos sin 3 83 R zx y zd V d d r d r Rd R + = = = 5 . 4 zR = 5 0, 0, 4 R . () f x 0, () () 00 0, cos 0, fxd x fx x d x = = () 0, 12

18、, ( ) ( ) 12 0 ff = = . () () 0 , Fx ftd t = ( ) ( ) 00 , FF = = () () () () () 00 0 0 0 0c o sc o scos sinsin | fxx d x x d F x Fx x Fx x d x Fx x d x = =+ = () () 0 sin Gx Fx t d t = () ( ) 00 , GG = = () 0, () sin 0, F = ( ) 0, . () () () 00 FFF = () Fx 0, , ( ) 1 0, () 2 , () () 12 0 FF = () ()

19、12 0 ff = = 6 A * 1000 0100 , 1010 0308 A = 11 3, ABA BA E =+ E 4 . B A * A 1 . A 1 * , AA A A A E = 1 * n AA = 3 * 8 A A = 2 A = 11 3, ABA BA E =+ A * A * 23, B AB AA AB =+= () * 26 , E ABE = () 1 1 * 1000 6000 0100 0600 62 6 1010 6060 0306 0301 BE A = = 2 2 A = 1 * , AA A A A E = () () 11 * 1000 0

20、100 22 1010 31 00 88 2000 0200, 2020 31 00 44 AAA A = = A E . () 1 3, AEB A E = () 1 3 B AEA = () 1 1 1000 1000 0100 0100 , 2010 2010 4 33 010 00 3 44 AE = = 1000 2000 6000 0100 0200 0600 3 2010 2020 6060 4 31 0301 010 00 3 44 B = 1 6 2 5 . n n x n y n n x y . 1 1 1 n n x y + + n n x y 11 11 ; nn nn

21、 xx A yy + + = 2 12 41 , 11 = A 3 1 1 1 2 1 2 x y = 1 1 n n x y + + . 1 1 1 52 1 65 6 31 56 nnnn nn n x xx y yx y + + =+ + =+ 1 1 92 10 5 13 10 5 nnn nnn x xy yxy + + =+ =+ 1 1 92 10 5 13 10 5 nn nn x x yy + + = 92 10 5 . 13 10 5 A = 2 () 12 41 ,5 0 11 = 12 , . 11 4 , 1 A = 1 A 1 1 = . 22 1 1 2 , 1 2 2 A = A 2 1 2 = . 3 11 1 2 11 1 1 2 1 2 nnnnn nnn xxx x AA AA yyy y + + = = n A . 12 41 , 11 P = 1 1 2 , PAP = 1 1 2 , AP P = 1 1 1 2 1 41 41 1