& (2001 2012) & ... (2003 2012) - 20012011 2001-2012 1 14 2001 : : (6) 1.1. : () = () (). 8,5 .2. ,(=,,) :. () . . .1() 2 . () ... (). 2 .1. . 1 2 . . () + () < 1.. () = () 2() = (). 4 .2. . ,()= 41()= 125() :. 41 . 125 . 32. 61. 2,5 .3. , . ()= 31,()=41() =51. 2 3 . () 1. 201 . ( ) ) A B ( 2. 152 . ( ) ) B A ( 3. 54 4. 121 5. 2019 6 2 f(x) = x+x. A. f(x) +f (x) = 0. 8 . f (0,1). 8 . IR :2f 22f = 2. 9 3 4 3 80 . 4 . [ ) Fi45-550,2 55-650,5 65-75 75-85 . , . 8 . . 9 . 80 .. 65 . 4 . 55 75 . 4 4 5 4, , 50% 12,16% 10 . . . . 6 . , . 6 .4.000, 14 16 . 6 ., ,5. (CV). 7 5 1 28 2002 : : (4) 1. Ax1,x2,,xk , ,k, k . . i , xi , i = 1,2,,k; 3 . fi xi,i = 1,2,,k; 3 . : i) 0 fi 1 i = 1,2,,k ii)f1 + f2 + + fk = 1. 4 .1. , : ( ) = () + (). 8 1 2 .2. . . 5 . :i) P()ii) (). 2 2 f(x) = 1 xx 2+. . f. 4 . ) x ( f lim3 x. 4 . f. 7 . f y = 2x + 5. 10 3 10 , :8, 10, 13, 13, 15, 16, 18, 14, 14, 9. 2 3 ., . 6 ., . 6 . 10%, . 13 4, () + () 2( ). : f(x) = (x - P(AB))3 - (x - P(AB))3 ,x R. . P(AB) P(AB). 5 .f(x) 2) B ( P ) A ( Px+= . 13 .,, f(P(A)) = f(P(B)). 7 3 1 27 2003 : : (4) 1. f(x)=x f(x) = 1. 8 . f ; 6 . () . 6 ., . . . . . . (f(g(x))) = f (g(x)) . g (x) f , g . . , = . . . 5 1 2 2 55% ,40% 30%. . : . 5 . 5 . 7 . . 8 3 f(x) = 12 xx . . :.R.(-1,1) .R- {-1,1}. (1, + ) 5 .f(x) =.. () = 1 + ().. . 12 2 2 2 f xe2 x) x ( f+= .. . 9. xe1(x) f ) x ( f = +. 8. f (0, f(0)). 8 3 25 14, 10 11.. 15 . 12. 25 5000, (CV). 13 4 = {1, 2, 3, 4, 5, 6} f : IR IR , 2 4x kx x31f(x)2 3+ + = k.P(1) = P(3) = P(5) = 2P(2) = 4P(4) = 2P(6), : 3 3 . P(1),P(2),P(3),P(4),P(5),P(6). 8. , : : . 8. ,: f IR . 9 1.(,).. .2.,. . , .3. .4. .5.:(3).6. : 10:00.K 1 1 28 2005 : : (4) 1. : P(AB) = P(A) + P(B) P(AB). 10 . . ; 3 . ; 4. , . .f f(x)>0 ,f . 2 . ( ) 2g(x)(x) g f(x) g(x) (x) f ) x ( gf(x) + = , f, g . 2 2 2 . . 2 . P(A) > P(B). 2 2 BiB: 01052015254 8 12 16 20 i . : / [) xBiB BiB fBiB BiB. . FBiB[4, 8)[8, 12)[12, 16)[16, 20) 11 . . 8 . 10; 6 3 3 3 ,, :(i) , 87 . (ii) P(B),P(AB) 45,21k, =, k = 5 x 6 2x15 3x

5 xlim+ . . k. 5 .P(B),P(AB) . 8 . :(1) . 6 (2) . 6 4 f ) (0, x ,x1) x ( f + =. . f (1,1). 7 . (x, y) fxx yy,x,Oy . 4 4 , . 10 . ()x=5 sBxB = 2. ysByB . 8 U 1.(, ).UU . . 2. ,.U U. , .3. . 4. . 5.:(3) . 6. : 10.30 . K 1 1 8 2005 : : (4) 1 A.1.F(x),f(x)g(x)F(x) = f(x)+g(x). f,g, :F(x) = f(x) + g(x). 9 A.2. x, >0 , < 0 ;xx 4 ., , . . . 2 . x>0, (lnx)=x1. 2 2 2 ., fi vi, Fi, Ni. 2 . . 2 ., ()=(), ()=(). 2 . . 2 2 f(x) =lnx - x2 , R . . f. 3 .fx, . 5 ., (1,1) f y=3x-2 . 10 . (f(x) x lim2 x3). 7 3 3 3o 50% 20.81,5% (16,22) . x , s x s, 2 x , s 3 x_. .2 s 20 x = = 10 .*, ( 95% .s) x s, x_ + 5 .R, f(x)=9s x ) 4 x ( x2R2+ + . 10 4 ={1,2,3,4,5,6,7,8,9,10} .,, { } { } { } 2,6 B - A, 1,3,4 B A, 6 , 5 , 4 , 3 , 2 , 1 B A = = = I U+ = 21 - x1 x/ x . . (), (), (). 9 ., . 3 4 4 ., . 3 .s2,3,5,, ={ / s2 > 24}. 10 1. (,). . 2. . . . 3. . 4. . 5.: (3) . 6. : 10.30 . K 1 25 2006 : : (4) 1o A.fR. c . (cf(x))=cf(x), x R. 10 B.., ; 3 .f ; 4 ., ..f, x0A, f(x) f(x0) x x0. 2 .A, ,, . 2 1 2 . x0 :2x1x1= . 2 . . 2 2 50 . , : xi i0+4 15+8 24 3-1 42 50 . . 3 :. . 7 . . 7 . 3 . 8 2 3 3o x(x+4)2 . ., . x . 7 . 191 100, , . 8 . , , ; 10 4 f(x) = -2x2+kx +x 4+ 10, x0. .A (1,f(1)) xx,k=2 . 5 . =f(1) x 3 4 s=13(4) f 2., , 8. (i) (10, 16). 10 (ii) , . >0, , . 10 ( ) 1. (,). . .2. , . . . 3. . 4. .5.: (3) .6. : 10. 30 . K 4 1 1 3 2006 : : (4) 1 A. , :P() =1(). 9 .1f x0A; 3 .2 f ; 3 ., , . ., . 2 .:(x) =x, x IR. 2 2 2 .(CV) . 2 . . 2 .,. :P() P(AB) P(). 2 2 f(x)=ex(x2+x+9),IR . f (2,e2) y = e2x+3e2, : . =1 =6. 12 . f. 13 3o . 0,7 0,1. . . . 15 3 3 . 0,6 : i. .ii. . 10 4 2006 : xi fi[ ... ... )...0,1 [ ... 7 )...... [ ... ... )...0,3 [ ... ... )10... /////////////////////// 1 4 f42f2, : .c 2. 10 . , . 5 4 4 . i. x. 4 ii. s. 6 :===k1 ik1 i2i ii2i2 x x1 ) (s. ( ) 1.(,). . .2. , . . . 3. .4. .5.: (3) .6. : 10. 30 . 1 22 2007 : : (4) 1o A. () = () (). 8 B..f x0 ; 4 .() , . 3 1., , ,, .., Fi xi. 2 1 2 .f,g, :( ) ( ) ( ) . (x) g (g(x)) f x g f = 2 .ff(x0)=0 x0(,),f(x)>0(,x0)f(x)0 f3(x)= x, x>0 f4(x)=x, x . 4 2 f(x)=xex+3, x .. f(x)=f(x)+ex3 10 . x xe (x) flim2x0 x. 15 2 3 3o { } 5 , 4 , 3 , 2 , 1 , 0 , 1 = (1)=(0)=(1)=(2)=2(3)=2(4)=2(5). :{ } { } 1 2x 2, x 2x 1, x , 2 B , 3 x x 3, 1, A2 2+ + + = = x . . , (1), (0), (1), (2), (3), (4), (5). 7 .x ={1,3}.B A 8 . x=1 :113B) P(A ,117P(B) ,115P(A) = = =()(). 10 4 : :12,18,t3,t4,. . .,t25 B:16,14,t3,t4,. . .,t25. t3+t4+ . . .+t25=345. . Ax Bx 15 x xB A= =. 7 3 4 . , 2As2Bs2516s s2B2A=. 8 . 151CVA = , .BCV 10 ( ) 1. (,,). . 2. , . . . 3. . 4. . , .5. .6.: (3) .7. : 10: 30 . K 4 1 1 29 2007 : : (4) 1 A. f(x)=x f(x)=1. 8 . . . 4 . :i) ()ii) (). 3 1. , , . .fi , i=1,2,,, xi.i , :i=360fi,i=1,2,,. 2 2 2 .f,gg(x)0, ( )2) x ( g) x ( g ) x ( f ) x ( g ) x ( f) x ( g) x ( f = . 2 .Af f(x)>0 , f . 2 2. :f1(x)=ex x . f2(x)=x1 x0. f3(x)=x x . f4(x)=c x c . 4 2 f(x)=.1 x xx2+ . f(x). 5 . ) x ( f lim1 x . 8 . f(x) . 12 3 3 3o ={1,2,3,4,5}. , : = {x/ 0 n(x1) < n3},B = {x/ (x25x)(x1)= 6(x1)}. . () (). 8 . () = 41, (). 7 .()= 41()= 81, (X), =. 10 4 x1, x2, ..., x11 :7, 5, , 2, 5, , 8, 6, , 5, 3, ,,0, x 21) x ( = . 2 . x oox xx x lim =. 2 . , . 2 2 f(x)=xe1 x, x . . 1 x f(x) elim2x1 x. 7 . exf(x)=2x. 9 . f(x). 9 2 5 3 3o 5. ( ) : 2026 2632 2419 2220 1823 . . 5 .38 40 , ;( ). 5 .SASB . 7 . . 3 , 3 11 . 8 3 5 4 4 50% ,30% . ., , ; 7 . : , . .107) (51 9 . f(x)=x3 21 x2 + P(B) x x . f(x) . 9 ( ) 1. (,). . 2. , . . 4 5 5 . 3. . 4. . , .5. .6.: (3) .7. : 10: 30 . K 5 5 1 1 2008 : : (5) 1o A.f,g . (f(x) + g(x)) = f(x) + g(x). 9 B.. t1, t2, , t X. 3 . ; 3 ., .. , 10%. 2 . . 2 1 5 2 .fx0 A1, = f(x) limox xA1 , ( ) x x1of(x) lim A =( ). 2 .f f(x)0 ( )xx 1=) x=f(t),t0 (t0)=f(t0) ) f , x1, x2x1 f. y y Ox ) 0 , x ( x x Oy ( ). ) x ( f , 0 O , , . 7 5 5 5 3. , 10 , x y : + = f ( ). ) 1 ( f , 1 (xi,yi), i=1,2,,10 , ix 10 x_= . 2 sx = iy . 8 4. , , ( ) ( ) ( ) ) ( f 2 ) ( f ) ( f + 5 ( ) 1. (,). .2. . . .3. .4. . ,.5. .6. .7. : (3) .8. : 10. 30. .K .... 2003 2011 2003 - 2012 .... 2003 1. , , : P(AB)=P(A)+P(B)-P(AB). (7 ). n . 1, ,kX X K () ,

1 2, ,...,kn n n 1 2, ,...,kf f f X , 1ki iiX X f==(4). ( - ) .1. n , ix , in () %if ,%i in f = i=1,2,3,n ;(2 )2. 7 , 3, -2, -1, 0,1, 2, 3 . ;(2 )3. 1, ,nx x K ,x s . b,' x ' s 1 2, ,...,nax b ax b ax b + + + (2 )4. n in if ix . ia , ia . (2 ). , :1. : 162.1 : 153.2 : 17 .( 8, 1 1,3 2 0,7). (2 ). , . , t, 3 21 1( )3 2x t t t t = . . ;(4 ) 2,,2002, .[ - ) %if0 2 12 4 24 66 - 8 168 10 1810 12 1612 14 1414 - 16 1316 1818 - 20 51. 16 18, 46, . (5 )2. , 55872 , 10. . (4 )3. 11 13. (4 )4. . 0,34 , .,,,().(12 ) 3 : .1. . (8 )2. : iy in22 ( ) xPA323 ( ') x P A2() xP3 P(A) 1 10,2x R . . ( x ) yy (4 ). yx f1) ( = (13 ) 4 2 3( ) ( ( )) [7 ( ) 3] ln ( ) f x P A x P A x x x P B = + , 0 x >( ), ( ) PA P B , , , .1. '( ) f x (4 )2. P(A), f , =1, . (6 )3. 1( )3PA = 55(1)36f = , 3( )4PB = , . (7 )4. 1 1( )12 3PA B (8 ) .... 2004 1 . ( ) ( ) ( ) F x = f x+ gx . f , g , : ( ) ( ) ( ) F x = f x+ g x .8 . 4. ={1, 3, 5} ={2, 4, 6} ( ) ( ) 1 4:1. .2. .3. .4. .4 . t1, t2, t3, , t x=4, R=10 s =2. , 2t1, 2t2, 2t3, , 2t( *) 6 . , :( ) cf x c ,( ) ( )f x g x ,( )( )f xg x ( ) g x 0 , f ( g( x) ) .7 2 1, 23. , .. . 9 . , :: : 7 . , . 4 . , , . 5 3 . ( ). f ( x) =2x22x+1, x IR. 10 B. : P( A) , P( A ) , P( ) , P( ) . . . 6 . , :2 21 s=2P (A) -2 P(A)+14 5 . , : 2 CV 2 , P( A) =P( A ) .4 4 , ( C) , , . [ , ) xi i fi % [4, 2) [2, 0 ) [0,2 ) [2,4 ) : , . . , o x1 C =6 B. , .. fi % . 9 . . 5 , 0, 5C. 5 2005 1 1 .i) f . 2 ii)f ; 3 . fi xi, i=1,2,, , . 10 .()() . 1. . 2 2. i, i=1,2,3,4,5 x1, x2,x3,x4,x5, 2iN 4i 2i = + , =110. 2 3., . 2 4.:x = ,x . 2 5. P(A), P(B) , P(A) P(B) A B . 2 2005 2005 2 2

t1, t2, ,t100t , s 0 F ( )( ) ( ) = = t 2s x 4, x 0 x 4F x x 224 s, x 4, = [ 0, +). ) x 4 F ( ) ( ) ( )Fx t 2s x 2 = + . 7 ) t1, t2, , t100 . 10 ) () g ( )( ) F xg xt 2s= 1 1A ,g4 4 . 8 3 ,(..),4.000 , . 100 1.400.. 3.360 2.000.. ) x , s R . 12 ) . 1.200.. 2.000.. 7 ),,6%, , . 6 2005 3 4 ,P(K),() , () 0. ) ( ) ( ) ( )2 1f x x P xP K2 = + , xR xoR ( ) 2 5 P K2. :i)xo = ()+() 6 ii)() = 2() 4 ), , : P(), P(K), P(), (), (), (), (), (), (), () 14=( ) ( ) = 2 P K 3. i) : , , , . 12 ii) . 3 2006 1 1. x1 , x2 , ,x X , i) fi xi, i = 1,2,,ii) 0 fi 1 i = 1,2,,iii) f1 + f2 ++ f =1(9 ).,. ( ) ( ) ( ) B P A P B A P + = (8 ). 1 ()=0.3 () 0.3 0.8 0.7 0.12 ()=0.3, ()=0.4, ()=0.1 () 0.6 0.7 0.8 0.9( - ) 0.6 0.7 0.8 0.9 3 ()=0.8 , ()=0.2 ( ) 0.6 0.7 0.8 0.9() = 0.5 0.6 0.7 0.8()= 0.1 0.2 0.3 0.4 4 ()=0.3 ()=0.6 ( - )= 0.1 0.2 0.3 0.4(8 ) 2 + = x x x f 6 ) (3. 2111 3lim221 = +xx xx f 9 :i) = 2(7 )ii) = 5(6 )iii) f () xx(6 )iv)xf .(6 ) 2006 2 3... xi ififi% NiFiFi%12 110 1603 704 0.0755 400.i).(4 )ii) (3 )iii) (4 ). . :: (3 ): (3 ). . :: (4 ): (4 ) 4 x1 , x2 , , x .. x1 ,x2 , ,x 20% 25 ,i) ii) 2 2221,..., ,vx x xiii)10,,.(9 ). 2 , . 30% 60%50% , :i) ii), .(8 ) 2006 3. . 1 . 0,8% .(8 ) 2007 11 11. ; 32. ;( ) 4B. f , :( ) cf (x) cf (x),c IR = ' 8.() (), () () .1. f 0x A 0ox xlim f (x) f (x ) f . 22. . 23. . 24.O (CV) . 25. . 2 2007 22 2 f (x) = x2 + lnx. . 5. . 5. . 7. : 1 x3 ) x ( f x1 xlim 8 3 80 ,530,5.30,,10% 15 40.. (, , ). 8. . 8. 23 . 4 2007 33., 60 ; 5 4 , , , () () = 15() R :(),(),(),(). : .0 < R 1 4 .R = P(AB) + P( ) 7. f (x) = 5P(A)x 5P(B) 1,x 1x 1 5P(A B)+3,x 1 = IR :. () = () +52 7. R = 1 4. () = 1 () = 0 3 2008 11' & 11. R ; 32.={1,2,}. {i}. 43. f(x)=x2 , f(x) = 2x. 8. .. f x0 f0 000 hf(x h) f(x )(x ) limh+ += . 2. O f (x0, f (x0)) f (x0) f (x) xx = x0. 2. . 2. . 2. . 2 2008 22 2 f(x)=242 2xx+ .. f. 6. (x, f(x)) f xx. 5. 2lim ( )xf x=16 6. xi,i=1,2,3,4x,=40. =2lim ( )xf x :Xi vi fi Ni Fi1 42 34 0,2 1 8 3100.60,40 10 . 40 .. . :: : 7. ,4. 39,6, . 6 2008 33.100 2,5% 26 , 33 . 6.100 2 21 1( ) 250000i i i ii iv x v x = = = ,.22 2 11( )1[ ]i iii iiv xS v xv v= . 6 4 = {1,2,3,4,5,6,7} m x, 5ex, x+4, -7x, 1 ( x ). g(x)=mx2-2x+3 ( x ). m=2. 9. ={ / g (1, g(1)) xx}. N E. 8. , , , 3 23 2 ( ) ( )( ) 2008,12 2PA B PAh x x x x = + + + x . 8 2009 1 1 ' . & 1

.,, ( ) ( ) ( ) = I . 9 .. f ; 3 . ; 3 ., , , , . . x>0 ( )xx2x= . 2 . >x . 2 . . 2 .f 1 2x x , 1 2x x . 2 .,.A( ) ( ) PA PB A B . 2 2009 2 2 2

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( )( )( ) ( )( )( ) ( )2PA B PB Af x x xPA PB 2PA PB= + + + ,UR ( ) ( )K 1 f 1 , y=x+2. ) . 7 )( )3PA B4= U Cf f 103 , () (). 6 )( )1PA2= ( )1PB4= ( ) ( ) g x 6f x 12x 2019 = + . 6 )()Cf( ) ( )K1 f 1 , ( )1 1 1M x y , , ( ) ( )2 2 2 10 10 10M x y M x y , , ..., , 10() 1 2 10x x x , , ..., 596 Sx=2 . 6 2010 1 1 ' . & 1 A.( ) x f ( ) x g , ( ) ( ) ( ) x g x f x F + =( ) ( ) ( ) x g x f x F + = . 9 B.. . 3 .ixi ; 3 ., , , , . . f , A x1( ) ( )1x f x f x 1x . 2 . i i i360 =

. 2 . , ( )B A . 2 .:( )( )( ) ( ) ( ) ( )( ) ( )2x gx g x f x g x fx gx f + = 2 . . 2 2010 2 2 2 ( ) ( ) 15 x 1 x ln x f2+ + + + = , 15 . . ( ) x f 3 . : .( ) x f . 5 .( ) + 2 x x1 xx f lim221 x. 5 . () ( ) x f x y=15 x y + + = . 6 . () 0 , 1 , 9, 1050. 6 3 , . . 4E4 E1=2E18 E3=8 E5=iv 0 481216 20 2 46 N = 50, : . . 3 2010 3 3 ., . 10 . . .10 17. 6 . 10 16. 6 4 { } , , , 6 , 3 , 2 = { } , , = ,:( )21A P = ( )( )( ) ( )( )2PP 6 P33 P2 P 2 = = = = . . . 5 .,,( ) 2010 x 20 x 12 x3x f2 3+ + = ( ) ( ) 1 f , 1 A 48 . 4 . ( )5 3 x 232 x 2x g2 = . g(x) :{ } x = x B . 4 .160420R 43 x . {c = c }. 4 . , , , A . 8 2011 () 1 1

' . & A 1. A :( ) ( ) PA 1 PA= . 8 2.x1, x2, ,x , 1,2,, ii=1,2,, : i. . ii. i ; 4 3. f 0x . 3 4., . . f ( )0f x 0=( )0x , ,( ) f x 0>( )0, x ( ) f x 0 10%, .. :12 = x14 = +xs x16 2 = +xs x 14 16 :% 5 , 13 %268 95= :540 5 , 13 401005 , 1340000 = = . 5 3 99 . : :17 5 12 5 = + = + =x :2 = =xS S:1722= =sCV 11,7%. 5%. : Keystone3 1) , i, xi .) fi i xi . : fii= i = 1, 2, , .)i) 0 i i = 1, 2, , 10i . 0 fi 1 i = 1, 2, , .ii) f1 + f2 + + f =1 ... ... 2 1 2 1= =+ + += + + +. 1. 1. . 150 . ..2 ..()() P(A) = =.2..(i)P() = 1.(ii)P() = 0 2() x+1 0, x -1 Af=-{-1}()234612lim ) ( lim3 3= =+= xxx fx x() =++ += +=2') 1 ()' 1 ( 2 ) 1 ( )' 2 (12) ( 'xx x x xxxx f2002 : Keystone4=+ +=+ +2 2) 1 (2 2 2) 1 (2 ) 1 ( 2xx xxx x2) 1 (2+ x() } 1 { ox 2 ) ( ' =ox f:2) 1 (2) ( '+=xx f: + = =+22) 1 ( 2 2 2) 1 (2ooxx = + = + 0 1 ) 1 ( 0 2 ) 1 ( 22 2o ox x = + = + + + 0 ) 2 ( 0 ) 1 1 )( 1 1 (o o o ox x x x ( 0 =ox2 =ox ) (0,f(0)) = (0,0) B(-2,f(-2)) = (-2,4). : (0,0)) 0 )( 0 ( ' ) 0 ( = x f f yx y 2 0 = x y 2 = (-2,4)) 2 )( 2 ( ' ) 2 ( + = x f f y) 2 ( 2 4 + = x y4 2 4 + = x y8 2 + = x y: ( ) : y = x+ f (0,0).: = f '(0) = 2 0 = 20 - = 0 y = 2x y = 'x+' f B(2,4).: ' = f '(-2) = 2 4 = 2(-2)+ = 8 y = 2x+8 : Keystone5 3xiiixi8 1 89 1 910 1 1013 2 2614 2 2815 1 1516 1 1618 1 1810 130)1. 131013010 x x81 ii i= = ==2. :8 9 10 13 13 14 14 15 16 18: 13,5214 132t t6 5=+=+=3. = 13, 14.) R = 18 - 8 = 10. s2 :( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) [ ]= + + + + + + + =2 2 2 2 2 2 2 2 213 18 13 16 13 15 13 14 2 13 13 2 13 10 13 9 13 8101s[ ] 9109025 9 4 2 9 16 25101= = + + + + + + =3 s s2= = 133xsCV1= = 23%.).yi,i= 1,2,,1010%0,9.x y 9 , 0 = , sy = 0,9 sx1x x2CVxsx 0,9s 0,9CV = == . : Keystone6 4) : P(A)+P(B) 2P(A B).P(A)+P(B) - P(A B) P(A B) P(A B) P(A B)) : ( ) ( ) = x B) P(A x 3 B) P(A x 3 (x) '2 2f: f '(x)=0 ( ) ( )2 2B) P(A x 3 B) P(A x 3 = 0 B) P(A x B) P(A xB) P(A x B) P(A x + = = P(B) P(A) B) P(A B) P(A 2x B) P(A B) P(A+ = + = = 2P(B) P(A)x+=: f '(x)>0 ( ) ( ) 0 B) P(A x 3 B) P(A x 32 2> ( ) ( ) 0 B) P(A x B) P(A x B) P(A x - B) P(A x > + + ( ) ( ) [ ] 0 B) P(A B) P(A 2 B) P(A - B) P(A > + x( ) ( ) [ ] ) 1 ( 0 P(B) P(A) 2 B) P(A - B) P(A > + x: A B A B P(A B) P(A B) : P(A B) P(A B): P(A B) < P(A B): P(A B) - P(A B) < 0: (1) 2x < P(A)+P(B) 2P(B) P(A)x+ xx 10e > x 1 > 0 x < 1. : 1 +f(x) + f(x) f ( , 1] , f [1, ) + , f x = 1 1 11 2 1f ( 1) ee e + = = = . . x x x xx 2 x 1 x 2 x 1 1f (x) f '(x)e e e e+ + + = + = = . . : 00 2f (0) 2e+= = 00 1f '(0) 1e = = . 1 fCA(0, f (0)):y f (0) f '(0)(x 0) = , , y 2 1(x 0) = y = x + 2. 2 H fCA(0, f (0)) y f '(0)x = + , :y x = + . , f (0) 0 = + = f(0) = 2. fCA(0, f (0)):y x 2 = + . 3 __x 14 = 25 25i __i 1xx25== , 25ii 1x1425== 25ii 1x 14 25 350== =. ___10x 11 = 10 10i ____i 110xx10== , 10ii 1x1110== 10ii 1x 11 10 110== =. . ___15x 25 25 10i i i ____i 11 i 1 i 115x x x350 110 240x 1615 15 15 15= = == = = = = . . : 225i25i 1 2 2ii 1x1s x25 25== | | | \ = ` ), { }221 350 1 122.500 1 100s 5.000 5.000 5.000 4.900 425 25 25 25 25 25 = = = = = ` ` ) ) s = 2. : __s 2 1CV14 7x= = = . 4. : (1) + (2) + (3) + (4) + (5) + (6) = 1, 2(2) + (2) + 2(2) + (2) + 2(2) + (2) = 1 9(2) = 1 1P(2)9= . : 1P(2) P(4) P(6)9= = = , 2P(1) P(3) P(5) 2P(2)9= = = = . . = {2, 4, 6}, 1 1 1 3 1( ) (2) (4) (6)9 9 9 9 3 = + + = + + = = = {1, 3, 5}, 2 2 2 6 2( ) (1) (3) (5)9 9 9 9 3 = + + = + + = = . . f R : 2f '(x) x 2kx 4 = + , 2 2 2( 2k) 4 4 4k 16 4(k 4) 4(k 2)(k 2) = = = = + . k = 1, = 12 < 0 2f '(x) x 2x 4 0 = + > x R , f R, . k = 2, = 0 2f '(x) x 4x 4 = + , f '(x) 0 > x ( , 0) (0, ) + ,fR, . k = 3, 4, 5, 6 > 0, f R, . : = {2, 3} 2 1 3 1( ) (1) (2)9 9 9 3 = + = + = = . : Keystone 4 1 . 3 : : P(A B) = P(A) + P(B) P(A B) ( ) + () + () ( ), (1) () + () A B . (1) (): ) () () () () () () () ( + = P(A B) = P(A) + P(B) P(A B). (additive law). . . . . . (, ). . 2 . / [ ) xi vi fi Ni .. Fi [4, 8)650,150,1 [8, 12)10100,2150,3 [12, 16)14250,5400,8 [16, 20)18100,2501 501 .13,2.506605018 10 14 25 10 10 6 5x = = + + + = 2005 : Keystone 5. 10 5 + 5 = 10 . 3 . 431 531 x3lim5) 1)(x (x5) 3(xlim5 6x x15 3xlim 5 x 5 x25 x=== =+ = . . 43= , =45,21,43X . 145> , 45 P(A B), P (B). = 21,43P(B)} B), {P(A . B B A P (A B) P (B) P (A B) P (B) P(A B) < P (B). 21B) P(A = , 43P(B)= . . (1) P(A B) = P (A) + P (B) P (A B). 2143P(A)87 + = , 85P(A)= . (2) : 812185B) P(A P(A) B) P(A = = = . 4 .1 : x1f(x)= (0, +), 2x1f(x) = . f (1, 1) = f(1) = -1. y = -x + . (1, 1) , 1 = -1 + = 2. y = -x + 2. 2 : f (1, 1) : y f(1) = f(1) (x 1). 1 f(1)= 11112 = = ) ( f . y 1 = 1(x 1) y = x + 1 + 1 y = x + 2. : Keystone 6 . M(x,y) BA O x'xy'y12 (x,y) x1f(x)= 1, 2 xxyy. = 2x + 2y = 2(x + y) (1) x1y= , (1) : + =x1x 2 . + =x1x 2 (x) x (0, +). (x) (0, +) ( )( )2 222x1 x 1 x2x1 x2x11 2 (x)+ == = . (x) = 0 : ( )( )1 x 1 x 0x1 x 1 x22 = + = =+ . x = -1 x (0, +). : Keystone 7 : ' (x) (x) 01 - + x+ . (1) = 4 x = 1, (x) . (1, 1). . : 3 2 x y = + = 66[ \= = . 2005 1 A1. h 0 :h) x ( g ) h x ( gh) h ( f ) h x ( fh) x ( g ) x ( f ) h x ( g ) h x ( fh) x ( F ) h x ( F ++ += + + += + :) x ( ' g ) x ( ' fh) x ( g ) h x ( glimh) h ( f ) h x ( flimh) x ( F ) h x ( Flim0 h 0 h 0 h+ = ++ += + , F(x) = f(x) + g(x). 2. __x__x 0 s, , __sCVx= . : __x 0 > : __sCVx= , __x 0 12(x 2) > 0 x 2 > 0 x > 2. :2 +f(x) +f(x) f [2, ) + , f ( , 2] , f x = 2

2f (2) 6(2 4 2 3) 6(4 8 3) 6 = + = + = . 4 x 1 : x 12x 1+ x 12 0x 1+ x 1 2(x 1)0x 1 x 1+ x 1 2(x 1)0x 1+ x 1 2x 20x 1+ + 3 x0x 1, 1 < x 3 x, x = 2 x = 3, = {2, 3}. A B {2, 6} = ,2, 6 A 2, 6 B . A B {1, 3, 4} = , 1, 3, 4 A 1, 3, 4 B . 1, 2, 3, 4, 6 A . 5, 5 (A B) , ,5 A , A B {1, 2, 3, 4, 5, 6} = 5. : = {1, 2, 3, 4, 6} = {1, 3, 4, 5}. . ( ) 5 1( )( ) 10 2 = = = , ( ) 4 2( )( ) 10 5 = = = , ( ) 2 1( )( ) 10 5 = = = . . , : = {1, 4, 5}, ( ) 3( )( ) 10 = = . . :{3} = , ( ) 1( )( ) 10 = = . ( ) ( ) . , , :(( ) ( )) ( ) ( ) ( ) ( ) ( ) ( ) = + = + = 4 2 1 4 2 2 4 2( ) ( ) 2 ( ) 210 10 10 10 10 5+ = + = + = = = . . , 3, 5 __3 5 9x 33 3 + + = = = 2__ vii 12x xsv= = :2__ 3i2 2 2i 1 2x x( 3 ) (3 3 ) (5 3 )s3 3= + + = = =2 2 2 2 2 2( 2 ) 0 (2 ) 4 4 83 3 3 + + + = = = . :2s 24 > 28243> 23 248 > 2 > 9 || > 3 < 3 > 3. , = 4, 5, 6, 7, 8, 9, 10, = {4, 5, 6, 7, 8, 9, 10} ( ) 7( )( ) 10 = = . : Keystone4 1. . 30.. . 142.. . 16.. - - - - 2.339 1350 2 1 4 8 5 4=== + + + + + +aaa a a a a.xivixiviNi0 7 0 71 23 23 302 12 24 423 2 6 444 6 24 50 50 7750775024 6 24 23 0=+ + + += x ..121 1226 25= == =+ ++ += == =+ ++ += == =t t. 3. 254508) ( = = A P .2006 : Keystone5 3. . () = x + (x + 4)2. () = x. 24) () () () (+ +==x xxN A NA P ,R x( ). x 0 x . 2) 4 ( 0 + ++ + + ++ + x x x1) 4 (02 + ++ + + ++ + x xx..) () () ( 8 2 0 16 10191419122= = = + =+ + = x x x xx xxA P x = 8 () = 8 + (8 + 4)2= 152 > 100. x = 8 . x = 2 () = 2 + (2 + 4)2= 38 < 100. x = 2 . , () = (2 + 4)2 = 36, 19183836= ==) () () (N K NK P. 04 2+ += xx xxx f ,) () ( . f :[ ]0416222+ + = xx xxx f ,) () ( . x4ff+-0117+ f x = 4 1714 = ) ( f . () f. : Keystone6 () = {f(1), f(2),f(3),f(4),f(5), ... }, f(A).) 4 ( ) ( f x f ) , 0 [ + + + + x f(A) 171) 4 ( = == = f . () 171 4. 4. 0 10 4 22 + + + = x x kx x x f , ) ( x > 0 xk x x f24 + + = ) ( Cf (1, f(1)) xx2 0 2 4 0 1 = = + + = k k f ) ( . k = 2 10 4 2 22+ + + = x x x x f ) ( f(1) = 14 (1, f(1)) (1,14). Cf , y= 14..14 ) 1 ( = == = = == = f x2 13 4 = = =132(-13)- s , ) ( f(i) 14 = x s = 2 : X18 2013,50% 2,35% 0,15%8 100,15% 2,35% 13,50% 34% 34%12 14 16 3 8 2000 310015 0= = , (10, 16) 81,5% = 2000 , 1630 20001005 81= , .(ii) 10 , 0 14 , 071142cv > >> > = == = = == = = == =xs : Keystone7 . > 0 , + 142.6 6 0 1 0 1 0 4 1 2 10 0142 + + , , , , ,. = 6. 2006 1 A. = :( ) = () + () () = () + () 1 = () + (). .1. f x0 , f(x) f(x0) x x0. x0 , f(x0) f. B.2. , . . ., , (). .( ) x ' x = x R,x>0, (). .(CV) , (). . , (). .P( ) 1 = P( ) 1 = , P( ) ( ) ( ) 0 ( ) 1 , (). : 2. f R x 2 x x 2f '(x) e ( x x 9) e (2 x ) e ( x x 9 2 x ) = + + + + = + ++ + 2A(2, e ) f, 2f (2) e = 2 2 2e ( 2 2 9) e ++ =4 2 9 1 + + =4 2 8 + = 2 4 + = 4 2 = () f 2 2y e x 3e = + , :2f '(2) e = 2 2 2e ( 2 2 9 2 2 ) e +++ + = 4 2 9 4 1 + ++ + = 8 3 10 + = () 2006 () () : 8 3( 4 2 ) 10 + = 8 12 6 10 = 8 6 12 10 = 2 2 =1 = () 4 2 1 6 = = . . = 1 = 6 : x 2 x 2f (x) e (x 6x 9) e (x 3) = + = x 2 x 2f '(x) e (x 6x 9 2x 6) e (x 4x 3) = ++ = + . : f '(x) 0 = x 2e (x 4x 3) 0 + = 2x 4x 3 0 + = x = 1 x = 3 f '(x) 0 > x 2e (x 4x 3) 0 + > 2x 4x 3 0 + > x < 1 x > 3. :1 +f(x) + +f(x)3 f ( ,1] , f [1, 3], f [3, ) + , f x = 1

1 2f (1) e (1 3) 4e = = . f x = 3

3 2f (3) e (3 3) 0 = = . 3 : = , = . : ( ) ( ) = ( ) ' = , (( ) ( )) 0, 7 =(( ) ') 0,1 = . (( ) ') 0,1 = 1 ( ) 0,1 =( ) 1 0,1 = ( ) 0, 9 = (). , :(( ) ( )) ( ) P( ) ( ) ( ) ( ) ( ) = + = + (( ) ( )) ( ) ( ) = () . () () :0, 7 0, 9 ( ) = ( ) 0, 9 0, 7 = ( ) 0, 2 = . ( ) 0, 2 0 = , , . . : = ( ) = 0,6. : ( ) ( ) ( ) = , 0,6 = () 0,2 () = 0,6 + 0,2 = 0,8. : ( ) ( ) ( ) ( ) = + , 0,9 = 0,8 + () 0,2 () = 0,9 0,8 + 0,2 = 0,3. i. , () = 0,3. ii. , ( ) ( ) ( ) 0, 3 0, 2 0,1 = = = . 4. c . a 1 , : [a, a + c), [a + c, a + 2c), [a + 2c, a + 3c), [a + 3c, a + 4c). To :a 3c a 4c 2a 7c2 2+ + + += : 2a 7c102+=(). 2 : a + 2c = 7 (II). (), () a, c. 2a 7c102a 2c 7+ =+ = 2a 7c 20a 7 2c+ = = 2(7 2c) 7c 20a 7 2c + = = 14 4c 7c 20a 7 2c + = = 4c 7c 20 14a 7 2c + = = 3c 6a 7 2c= = c 2a 7 2c= = c 2a 7 2 2= = c 2a 3= =. . a = 3, c = 2 : [3, 5), [5, 7), [7, 9), [9, 11) 3 542+= , 5 762+= , 7 982+= , 9 11102+= . 1f 0,1 = , 3f 0, 3 = , 4 2f 2f = . 1 2 3 4f f f f 1 + + + = , : 2 20,1 f 0, 3 2f 1 + + + = 2 2f 2f 1 0,1 0, 3 + = 23f 0, 6 = 2f 0, 3 = , 4 2f 2f 0, 4 = = : xi fi [3, 5)40,1 [5, 7)60,2 [7, 9)80,3 [9, 11)100,4 1 . i. : xi fi i ix f2i ix f[3, 5)40,10,41,6 [5, 7)60,21,27,2 [7, 9)80,32,419,2 [9, 11)100,4440 1868 : ki i __ k ki 1 ii i ii 1 i 1x vvx x x f 8v v== == = = = ii. : ki i __i 1x vxv== __ ki ii 1x v v x== 22__k2__i i 2k k ki 1 2 2 2 2 ii i i i i2 2i 1 i 1 i 1v x x vv 1 v xs x v x x fv v v v v== = = | || | | | \ \ = = = ` ) , 2__ k2 2 2i ii 1s x f x 68 8 68 64 4== = = =, s = 2. : Keystone3 1. ( . . 152).. ( . . 22). ( . . 87)1. , , 2.f1(x) = x 1, x f2(x) = 1/x, x > 0f3(x) =,21xx > 0f4(x) = x, x 2. f :f(x) = (xex + 3) = ex + xex = ex + f (x) 3 f (x) = xex + 3 xex = f (x) 3.= += x xe 3 (x) f ex xe (x) f2x x0 x2x0 xlim lim== +== 1) (x xe xx x3 3 e xx x3 (x) fx0 x2x0 x20 xlim lim lim11 0e1 xe0 x0 xlim ===. ( 1 xe(x) gx= {1} .) 3. = {1, 0, 1, 2, 3, 4, 5}, () = 1 (1) + (0) + (1) + (2) + (3) + (4) + (5) = 1. (1) = (0) = (1) = (2) = 2(3) =2 (4) = 2(5) = .(1) = (0) = (1) = (2) = , (3) = (4) = (5) = /2.2007 : Keystone4 + + + + (/2) + (/2) + (/2) = 1 112 2 11 2 3 8 1234 = = = + = + .(1) = (0) = (1) = (2) = 2/11 (3) = (4) = (5) = 1/11.. {1, 3} {1, 3, x2 x 3}1 {1, 3, x2 x 3}.x2 x 3 = 1 x2 x 2 = 0 x = 2x = 1. x = 2 : = {2, 3, 8, 3} = {3} {1, 3} x = 2 . x = 1 : = {2, 0, 1, 3} = {1, 3} x = 1 .. x = 1 = {1, 3, 1 } = {2, 0, 1, 3}.() = (1) + (3) + (1) 115112111112= + + = .() = (2) + (0) + (1) + (3) 117111112112112= + + + = .( ) = (1) + (3) 113111112= + = .( ) = () ( ) 112113115= = .( ) = () + () ( )== () + 1 () [() ( )] == 1 () + ( ) 1171131171 = + = 4.1525345 3025... 18 1225 4 3=+=+ + + + +=t t txA1525345 3025... 14 1625 4 3=+=+ + + + +=t t txB : Keystone5.[ ]225232 2 2) 15 ( ... ) 15 ( ) 15 18 ( ) 15 12 (251 + + + + = t t SA[ ]225232 2 2) 15 ( ... ) 15 ( ) 15 14 ( ) 15 16 (251 + + + + = t t SB( )25161 1 3 32512 2 2 2 2 2= + = B AS S ..( ) ( )= = 2 22222 215 25162516BBAAB AxSxSS S( ) ( ) ( ) = = 22B22B2A15 2516CV225115 2516CV CV( ) ( ) = = 251612251CV225 25162251CV2B2B( )251CV5 153CV25 2259CVB B2B= = = . 2007 1 A. f(x) = x, x R. x R h 0, :1hx h xh) x ( f ) h x ( f= += +.:1 1 limh) x ( f ) h x ( flim0 h 0 h= = + , 1 ) ( = x f ,x R. .. , : ( )( )( ) = () () . B.. .( ) 1 = . .( ) 0 = . 1. . i ix if oi i360 f = i i360 f = , (). . f,g g(x) 0, '2f (x) f '(x)g(x) f (x)g'(x)g(x) g (x) = , (). .f f(x) > 0 , f , (). : 2. 2x x 1 0 + , xR , = 3 < 0. : fA =R . . x 1 : 2 2x 1 x 1x 1 1 1lim f (x) lim1 1 1 3 x x 1 ( 1) ( 1) 1 = = = = + + + +. 2007 . f R 2 2 2 22 2 2x x 1 x(2x 1) x x 1 2x x 1 xf '(x)x x 1 x x 1 x x 1 + + + = = = + + +. : f '(x) 0 = 221 x0x x 1= + 21 x 0 = 2x 1 = x = 1 f '(x) 0 > 221 x0x x 1> + 21 x 0 > 2x 1 1 x, x = 2 x = 3, = {2, 3}. : (x2 5x)(x 1) = 6(x 1) (x2 5x)(x 1) + 6(x 1) = 0 (x2 5x + 6)(x 1) = 0 (x 1)(x 2)(x 3) = 0 x = 1 x = 2 x = 3, = {1, 2, 3}. . : = {1, 4, 5} = {4, 5}, : A B A B' = = P(A B) P( ) 0 = = ,B A' = (B A') ( ) 1 = = . . :( ) (2) (3) = + , 1(2) (3)4 + = . :(1) (2) (3) (4) (5) 1 + + + + = , 1(1) (4) (5) 14 + + + = 1(1) (4) (5) 14 + + = 3(1) (4) (5)4 + + =' ' {1, 4, 5} = , 3( ' ') (1) (4) (5)4 = + + = . . :' {1} = = , ( ) (1) = 1(1)8 = . = {2, 3} = {1, 2, 3}, = : ={1} ={1, 2} ={1, 3} ={1, 2, 3}. () = {1} 1(1)8 = . () = {1, 2, 3} 1 1 1 2 3(1) (2) (3) ( ) ( )8 4 8 8 8 + + = + = + = + = . : 1 3( )8 8 . 4 ( , , ) , : 2, 3, 5, 5, 5, 6, 7, 8. (11), . = 6, ( , , ), 6 < < . 2 R = 8 : xmax 2 = 8 xmax = 2 + 8 xmax = 10, , = 10. 6, : 2 3 5 5 5 6 7 8 10611+ ++++ + ++ ++= 51 + + = 66 + = 66 51 + = 15 = 15 () . 2 2 2217 + + = = 10, () : 2 2 2(15 ) 10 217 + + = 2 2225 30 100 217 0 + + + = 22 30 108 0 + = 215 54 0 + = , = 9 15 9 15 32 2 = = , 15 3 1892 2+ = = = 15 3 1262 2 = = = . = 9 () = 15 9 = 6, < . = 6 () = 15 6 = 9, . : (, , ) = (6, 9, 10). . 2__ k2x i ii 11s v x xv = = , : 2 2 2 2 2 2 2 22x(2 6) (3 6) 3 (5 6) 2 (6 6) (7 6) (8 6) (9 6) (10 6)s11 + + + + + + + = =2 2 2 2 2 2 2 2( 4) ( 3) 3 ( 1) 2 0 1 2 3 4 15 9 3 1 2 0 1 4 9 1611 11 + + + + + + + ++ ++ + ++= = =15 9 3 0 1 4 9 16 5811 11++ + + + ++= = , x58s11= . xx__58s 5811CV6 6 11x= = = . 58 110 6 11 > , 2258 110 6 11 > 58 136 11 10> 5810 > 3611 > 580 > 396, , x1CV10> . . __x xs , Y = X + c, Z = cX, : __ __y x c = + , __ __z c x = y xs s = , z xs | c | s = . 1 2Y c X c = + , c1 > 0. : __ __1 2y c x c = + , y 1 x 1 xs | c | s c s = = , c1 > 0. y 1 xs c s = y xs 2s = , c1 = 2. __ __x 6, y 9 = = , __ __1 2y c x c = + c1 = 2, : 29 2 6 c = + 29 12 c = + 2c 9 12 = 2c 3 = . : Keystone 3 1 .:(c x f = ) ( )..28. . .: . 96 . . . - - - - - . 2 . 11 11 1) (2 2+== x exxexx f exx x 2111lim1) (lim121=+= x xx fxex x. .= = = xx xxee x e xexx f2) )( 1 ( ) 1 ()1( ) (x xxxx xexex eex e e =+ = =2 ) 1 1 ( ) 1 (2 2 :xexe x f exx x = = 22) ( . . ex > 0, x R, : i)f (x) = 0 x = 2 ii)f (x) > 0 2 x > 0 x < 2 iii)f (x) < 0 2 x < 0 x > 2 : 2008 : Keystone 4xff -+1e22+- f (, 2] [2, +), 0 ) 2 ( = f . f x = 2, 21) 2 (ef = . 3 .225110518 22 24 26 20= =+ + + +=Ax 245120523 20 19 32 26= =+ + + +=Bx . /11192238= , /352440= 111935 24 11 1111124111122411 2> > > , 3 , 3 11 11 3,3 > 2436,3 > 24. . 4 . :5 , 0 ) ( = A P , 3 , 0 ) ( = B A P . . ) ( B A P . = + = ) ( ) ( ) ( ) ( B A P B P A P B A P = + ) ( ) ( ) ( 1 A B P B P A P= + = )] ( ) ( [ ) ( ) ( 1 B A P B P B P A P = + )] ( ) ( 1 B A P A P= = )] ( ) ( [ 1 B A P A P1077 , 0 3 , 0 1 ) ( 1 = = = B A P . . ) ( ) ( B A P B P B A B107) ( B P . 3 , 0 ) ( = B A P: = = 3 , 0 ) ( 5 , 0 3 , 0 ) ( ) ( B A P B A P A P512 , 0 ) ( = = B A P . ) (51) ( ) ( B P B P B A P B B A . .) ( 3 ) (2B P x x x f + = . f ). ( 12 1 B P = 51) ( B P 512) ( 12 B P 0575121 ) ( 12 1 < = B P < 0, 0 ) ( > x f R x , f . 2008 1 A1. F(x) f (x) g(x), x = + R. h 0 :h) x ( g ) h x ( gh) h ( f ) h x ( fh) x ( g ) x ( f ) h x ( g ) h x ( fh) x ( F ) h x ( F ++ += + + += + :) x ( ' g ) x ( ' fh) x ( g ) h x ( glimh) h ( f ) h x ( flimh) x ( F ) h x ( Flim0 h 0 h 0 h+ = ++ += + , F(x) = f(x) + g(x), (f(x) + g(x)) = f(x) + g(x). .. (s2) t1, t2, , tv __x , svt xivi2121= =( ) . B.. , , =. . . ,10%, (). . , (). .f 0x 1l , 01x xlim f (x)= l , ( )0vv1x xlim f (x)= l ,v, (). .f f(x) < 0 , f , (). . , (). : 2008 2. 4108 = , : o4f 360 108 = 4o108f360= 43f 0, 310= =() : f1 = f2 (I) 1 2 3 4f f f f 1 + + + = (), () : 1 1 3f f f 0, 3 1 + + + = 1 32f f 1 0, 3 + = 3 1f 0, 7 2f = () __x 70 = , 4i ii 1x f 70== 1 2 3 430f 50f 70f 90f 70 + + + = , (), (), () : 1 1 130f 50f 70(0, 7 2f ) 90 0, 3 70 + + + = 1 1 130f 50f 49 140f 27 70 + + + = 1 1 130f 50f 140f 70 27 49 + = 160f 6 = 16f60= 11f10= . () 21f10= , () 37 1 5f 210 10 10= = . .i. = 50, : 1 11 50v f v 50 510 10= = = = , 2 21 50v f v 50 510 10= = = = , 3 35 250v f v 50 2510 10= = = = , 4 43 150v f v 50 1510 10= = = = . : ix iv if[20, 40)3050,1 [40, 60)5050,1 [60, 80)70250,5 [80, 100)90150,3 501 ii. To 60 3 4v v 25 15 40 + = + = . iii. 1 . : 1 1F f 0,1 = = , 2 1 2F F f 0,1 0,1 0, 2 = + = + = , 3 2 3F F f 0, 2 0, 5 0, 7 = + = + = , 4 3 4F F f 0, 7 0, 3 1 = + = + = . 204050 6070 8010010,70,20,1Fi (40, 0,1) (60, 0,2) y = x + . : 0,1 400, 2 60= + = + 0,1 400,1 20= + = 0,1 400, 005 = = 0,1 40 0, 0050, 005 = = 0,10, 005 = =, : y = 0,005x 0,1. (50, ) , : = 0,00550 0,1 = 0,15. (80, 0,7) (60, 0,2) y = x + . : 0, 7 800, 2 60= + = + 0, 5 200, 2 60= = + 0, 0250, 2 60 = = 0, 0250, 2 60 0, 025 = = 0, 0251, 3 = = , : y = 0,025x 1,3. (70, ) , : = 0,02570 1,3 = 0,45. : = 0,45 0,15 = 0,3 = 30%. 2 50 70, [50, 60) [60, 70) [40, 60) [60, 80) , :2 31 1 1 1f f 0,1 0, 5 0, 05 0, 25 0, 3 30%2 2 2 2+ = + = + = = . 3 . 0 p 1 < < , 0 1 p 1 1 1 < < , p 1 < 0 0 1 p 1 1 1 + < + < + , p + 1 > 0. [0, 1] p 1, p + 1 ./ 0 p 1 < < , : 3 2p p p < , 0 p 1 < < . : P(B A) P(A B) > . 4 y > 0 ( ) . :x + 2y = 200 2y = 200 x 200 x 1y 100 x2 2= = . xyy y > 0 1100 x 02 > 1100 x2> x < 200 x > 0, : x(0, 200). . To 21 1xy x 100 x 100x x2 2 = = , 21f (x) 100x x , x (0, 20)2= . . f (0, 20) f '(x) 100 x = . : f '(x) 0 = 100 x = 0 x = 100. f '(x) 0 > 100 x > 0 x < 100, 0 < x < 100. : 0 100 200f(x) + f(x) f (0, 100), f (100, 200), f x = 100

21f (100) 100 100 100 10.000 5.000 5.0002= = = . 5.000 m2. .f '(100) 100 100 0 = = ,f '(101) 100 101 1 = = ,f '(102) 100 102 2 = = , f '(103) 100 103 3 = = ,f '(104) 100 104 4 = = . ix , i 1, 2, 3, 4, 5 = , :__0 1 2 3 4 10x 25 5 = = = . . __x xs , Y = X + c, : __ __y x c = + y xs s = . Y X c = + , c1 > 0. : __ __y x c 2 c = + = + , y xs s = yxy__ssCV CV'2 cy= = = +, c 2. , : x xx__s sCV CV2x= = =. :CV' 2CV = , :x xs s22 c 2= + 112 c = + 2 + c = 1 c = 2 1 c = 1. : Keystone 4 1 . 150 . . . , 65 . . . , , , , 2 ) + + + + + + = + + += ==4 3 64 8 3 5 3 6 24 /22 4 4 3 3 2 2 1 141vvvx v x v x v x vx x v xiii7 3 59 4 52 3 59 ) 13 ( 41332 15 3 1242 2 2 2 222= + = + + = + ++ + += v v v v vvv. )= + + + ===vx x v x x v x x v x x vvx x vSii i 24 423 322 221 14122) ( ) ( ) ( ) () ( = + + + =+ + + + + + =2016 4 1 3 1 7 4 64 3 7 6) 4 8 ( 4 ) 4 5 ( 3 ) 4 3 ( 7 ) 4 2 ( 62 2 2 2 9 , 420982064 3 7 24= =+ + += . ) 42 , 249 , 4CV2 = = =xSxS% 10 % 5520114022> = = = , . 2009 : Keystone 5 3 ) f= R. f R . : f (x) = 3x2 12x + , f (x) = 6x 12. : 2f (x) + f (x) + 15 = 3x2 2(6x 12) + 3x2 12x + + 15 = 3x2 12x 24 12x + + 15 = 0 = 9. ) x 1: 19 12 31) (222+ = xx xxx f19 3) 1 )( 1 () 3 )( 1 ( 3) 1 )( 1 () 3 4 ( 32+=+ =+ + =xxx xx xx xx x. : 32619 3lim1) (lim1 12 ==+= xxxx fx x. ) (xo, f (xo)) . x y 3 =3 ) ( 3o = = x f = + 3 9 12 3o2ox x0 4 4 0 12 12 3o2o o2o= + = + x x x x . 2 0 ) 2 (o2o= = x x (2, f (2)). 5 7 2 9 2 6 2 ) 2 (2 3 = + = f (2, -5). : + = x f y ) 2 ( , . + = x y 3 . . 1 2 3 5 = + = : 1 3 + = x y . : Keystone 6 4 . . xxxxxx f222221 1) ( == = . x > 0 : 2 0 ) ( = = x x f2 0 0 ) ( < < > x x f. 2 0 ) ( > < x x f f (0,2] [2, +). . f : 1 6 2 ln ) 2 (2+ + = f . . . f [2, +) : 2 < 3 < 4 < 5 < 8) 8 ( ) 5 ( ) 4 ( ) 3 ( ) 2 ( f f f f f > > > > . : = + + + + = = ) 2 6 4 8 (ln ) 1 6 2 (ln ) 8 ( ) 2 (2 2 f f R. 341ln 3 8 ln 2 ln + = + =, : . 6 4 ln 2 6 2 4 ln ) 4 (2 2 + = + + = f . = < + + + = 2 6 4 ln41ln 32 { } 0 5 62< + = . ) 5 , 1 ( 0 5 62 < + , { } 4 , 3 , 2 = . 1003) () () ( = = A P . 2009 1 .=, , :( ) = () + () () = () + () 1 = () + () P(A) = 1 (). .. f x1 , f(x) f(x1) x x1. x1 , f(x1) f. B.. __x , t1, t2, , tv vi __1 2 v i 1tt t .. txv v=+ + += = . . .f 0x l , 0x xlim f (x)= l , ( )0vvx xlim f (x)= l v1, (). . xf (x) e , x = R, xf '(x) e , x = R, (). . 50% 50%, (). . sR,R 6s , (). . , (). : 2009 2 f R . x 1, : 3 3x 1 x 1limf (x) lim( x 8) 1 8 8 = = = x 1limf (x) 7= 8 = 7 = 8 7 = 1. . = 1, : 3f (x) x 8 = . i. x 2, : 3 22x 2 x 2 x 2 x 2f (x) x 8 (x 2)(x 2x 4)lim lim lim lim(x 2x 4) 12x 2 x 2 x 2 + += = = + + = . ii. 0x 2 = , 30f (x ) f (2) 2 8 8 8 0 = = = = . f R , 2f '(x) 3x = . : 20f '(x ) f '(2) 3 2 3 4 12 = = = = . 1 0 0(x , f (x )) 0 0 0y f (x ) f '(x )(x x ) = . :y 0 12(x 2) = y = 12x 24. 2 0y f '(x )x = +, y 12x = + . 0 0(x , f (x )) (2, 0) = , 0 = 122 + = 24. y = 12x 24. 3 . : 150 = , o1f 360 50 = 1o50 5f36 360= =11vfv= , 1v 536 72= 172 5v36= 1v 10 = . 230 = , o2f 360 30 = 2o30 1f12 360= =22vfv= , 2v 112 72= 272v12= 2v 6 = . : 1 2 3 4v v v v 72 + + + = 4 3v 3v =: 3 310 6 v 3v 72 + + + = 3 3v 3v 72 10 6 + = 34v 56 = 356v 144= = , 4 3v 3v 3 14 42 = = = . . :oo o 33v 14 5.040360 360 70v 72 72 = = = = oo o 44v 42 15.120360 360 210v 72 72 = = = = . . 1 2 372v v v 10 6 14 30 362+ + = + + = < = , x4, = x4. 4 1R x x = 4i i __1 2 3 4 i 1 1 4x v10 x 6 x 14 x 42 x 10 x 6 ( 7) 14 3 42 xx72 72 72= + + + + + + = = = =

1 4 1 410 x 42 42 42 x 10 x 42 x72 72 + + + = = . :__1 44 1 410 x 42 x10R 72 x 52 10(x x ) 72 52x72 + + = + =4 1 1 4 410x 10x 10 x 42 x 52x 0 = + + = , __10R 72 x 52 + = . 4.. f (0, 1) 338f '(x) vx= . : f '(x) 0 = 338v 0x = 338vx= 338xv= 2xv= . f '(x) 0 > 338v 0x > 338vx> 338xv> 2xv> . :0 1f(x) +f(x)v2 f 20,v| (

(\ , f 2,1v ||

, f 2xv=

23 2 2 2 2 2222 2 4 4 4vf v v 2 2v 2v v 3v4v v 42vv| | = + = + = + = + = |\ | | |\ . .. ( )3 224v P(A) 3vP(A)+ = ( )2f P(A) 3v = ( )2f P(A) fv| |= |\ . 2fv| | |\ , 2P(A)v=() 2N(A) v 9v 8( )( ) v = = , () : 2v 9v 8 2v v = v2 9v 8 = 2 v2 9v 8 2 = 0 v2 9v 10 = 0, = 121 9 11v2= 9 11v 102+= = 9 11v 12= = (). v 10 = () : 2 1P(A)10 5= = . . : | | P(A' B) P(A') P(B) P(A' B) 1 P(A) P(B) P(B) P(A B) = + = + = 1 11 P(A) P(B) P(B) P(A B) 1 P(A) P(A B) 15 6= + + = + = + =30 6 5 2930 30+= = . : Keystone 4 1. : ( ) ( ) ( ) ( ) 1 2 1 2... ( ... )vt x t x t x t t t vxv + + + + + = =1 2...0vt t tx x xv+ + = = . 2. 86, 87, . 3. . 140, . . . 4. 1.1 x : 2 2 2( ) 1 2 1 1 1 2 1 2 2( 1 1)1 1 1 1f x x x x x x xx x x x + + + = = = = =2 2 22 22( 1 1)( 1 1) 2( 1 1)( 1)( 1 1) ( 1)( 1 1)x x x x x xx x x x x x + + + + = + + + += 22 2 22( ) 2 ( 1) 2( 1)( 1 1) ( 1)( 1 1) 1 1x x xx xx x x x x x x x = = + + + + + + 2 1 1( ) 1 2lim lim 111 1x xf x xxx x = = + +. B2.2 22 2 21 1 2 1( ) (2 1 1) 2 ( 1) (2 1)2 1 1 1xf x x x x x xx x x x x x = + = + = = + + + f x0 = 0 : 22 0 1(0) 10 0 1f = = +. 3.xx : = f (0) = 1, 0 < , 34= . 2010 : Keystone 5 1. c, [0, c) [c, 2c). 2 6, 26 42c cc+= = . 2. i i [0, 4)220 [4, 8)640 [8, 12)1045 [12, 16)1430 [16, 20)1825 160 511 1 1600(220 640 1045 1430 1825) 10 160 160 160i ix x== = + + + + = =. 52 2 2 2 2 2 211 1( ) 20(2 10) 40(6 10) 45(10 10) 30(14 10) 25(18 10)160 160i iis x x = = = + + + + 14000 25160= = . s = 5 . 3. 5CV 50%10sx= = =>10%. . 4. 2 3 41 1 1 140 45 30( ) 70 74 2 4 2( )( ) 160 160 160 16NAPAN + + + + = = = = =. :s2 : 552 2 2 2 111 1( ) (20000) ( ) 125 100 25160 160 160i iii iixvs xv x=== = = =. s = 5 . : Keystone 6 1. ( )21 ( ) 1( ) ( ( )( ) ( )x P Af x x P Ax P A x P A = = = (1 ( )) (1 ( ))( )x PA x PAx PA + + . 1( ) 0 ( ) 1 f x x P A = = + 2( ) 1 x PA = . x1 > P(A) () + 1 > () 1 > 0 , x2 < P(A) () 1 < () 1 < 0, x2. ( ) f x : ) x > P(A)x P(A) > 0 ) x > P(A)x P(A) > 0x P(A) + 1 > 1 > 0. ) ( ) 0 1 ( ) 0 1 ( ) ( ) 0 f x x P A x P A f x > + > < + < 1 ( ) 0 1 ( ). x PA x PA + < > + f: P( ) xf (x) f(x)+- +1+P( ) f ((), 1 + ()] [1 + P(A), +). f x = 1 + () ( ) ( ) ( )2 11 ( ) ln 1 ( ) ( ) 1 ( ) ( ) ( )2f P A P A P A P A P A P B + = + + + =21 1ln1 1 ( ) ( )2 2PB PB = + = . 2. f x0 = 5/3, 1 : 1 + () = 5/3 () = 5/3 1 () = 2/3. f (x0) = 0 1 1 1(1 ( )) 0 ( ) 0 ( )2 2f P A P B P B + = = = . 3. :( ) 1 ( ) PA B PA B = . ( ) ( ) ( ) ( ) PA B PA P B PA B = + ( ) ( ) ( ) ( ) PA B PA P B PA B = + = 2 1 5 13 2 6 3+ = . 1 2( ) 13 3PA B = = . 4. :[ ] ( ) ( ) ( ) ( ) 2 ( ) P A B B A PA P B PA B = + . [ ]2 1 1 2 1 2 1( ) ( ) 23 2 3 3 2 3 2P A B B A = + = + = . 2010 A A1. F(x) = cf(x) h 0 :h) h ( f ) h x ( fch) x ( f c ) h x ( f ch) x ( F ) h x ( F + = + = +. :) x ( ' f ch) h ( f ) h x ( fc limh) x ( F ) h x ( Flim0 h 0 h = + = + , F(x)=cf(x), f h 0f (x h) f (h)f '(x) limch+ = . 2.f , x1, x2 x1 < x2 : f(x1) > f(x2). A3.() .A4. )f,g , fg{ } x A, g(x) 0 , (). ) 00x xlim ( x) x = , (). )c , c2 c2+ , c cc2 2 + = , (). ) 1 , (). ) 1 2A { , ,..., }= 1 2(A) ( ) ( ) ... ( ) = + + + , (). : 2010 : = 60, R = 20 10 = 10, = 5. R 10c 25= = =, [10, 12), [12, 14), [14, 16), [16, 18), [18, 20]. : 3144 = , o3f 360 144 = 3o144f360= 3f 0, 4 =3v0, 4v = 3v 0, 4v = , 3v 0, 4 60 24 = =, 1 2f f =(), 1 2v vv v= 1 2v v =() 1 2 3v v v 48 + + = ,(): 1 1v v 24 48 + + = 1 1v v 48 24 + = 12v 24 = 124v2= , 1 2v v 12 = = . 5v 6 = , 55v 6f 0,1v 60= = = . 1 2 3 4 5v v v v v 60 + + + + = , :448 v 6 60 + + = 4v 60 48 6 = 4v 6 = , 44v 6f 0,1v 60= = = . 1. : ix iv if if % [10, 12)11120,220 [12, 14)13120,220 [14, 16)15240,440 [16, 18)1760,110 [18, 20)1960,110 601100 2. 5i i __i 1x v11 12 13 12 15 24 17 6 19 6x60 60= + + + + = = = 132 156 360 102 114 86414, 460 60+ + + += = = . 3. 10 14 1 2v v 12 12 24 + = + =. 4. 1 . : 1 1F f 0, 2 = = , 2 1 2F F f 0, 2 0, 2 0, 4 = + = + = , 3 2 3F F f 0, 4 0, 4 0, 8 = + = + = , 4 3 4F F f 0, 8 0,1 0, 9 = + = + = , 5 4 5F F f 0, 9 0,1 1 = + = + = . 1012 14 16 17 18 20 Fi 10,90,80,40,2 (16, 0,8) (18, 0,9) y = x + . : 0, 8 160, 9 18= + = + 0, 8 160,1 2= + = 0, 8 160, 05 = = 0, 8 16 0, 050, 05 = = 00, 05 = =, : y = 0,05x. (17, ) , : = 0,0517 = 0,85. : 1 = 1 0,85 = 0,15 = 15%. 2 17, [17, 18) [18, 20] [16, 18), :4 51 1f f 0,1 0,1 0, 05 0,1 0,15 15%2 2+ = + = + = = . 1. 2 4A' { , } = , 1 3' { , } = , : 1 3' { , } = = = , ( ) = () 2 4' { , } = = = , ( ) = (). 2. : 1 2 3 4( ) ( ) ( ) ( ) 1 + + + = , ( ) ( ) 1 + =1 114 2 + + = + 2( + 1) + ( 1)( + 4) = 2( + 4) 22 + 2 + 2 + 4 4 = 22 + 8 22 + 2 + 2 + 4 4 22 8 = 0 2 3 4 = 0, 2 = 25 3 52 =3 5 842 2+ = = = 3 5 212 2 = = = (). 3. = 4, : 4 1 5( ) ( )4 4 8+ = = =+ 4 1 3( ) ( )2 4 8 = = =. 4. 1 2 3 4' ' { , , , } = = , ( ' ') ( ) 1 = = . 1. f R 2 2__ __2 21 1f '(t) 3 t x t x300s 100s = = . : f '(t) 0 = 2__21t x 0100s = __t x 0 = __t x = . f '(t) 0 > 2__21t x 0100s > __t x 0 __t x . : +f(t) ++f(t)__x f R ( ), f R. 2. f R __ __2 21 1f ''(t) 2 t x t x100s 50s = = . : f ''(t) 0 = __21t x 050s| | = |\ __t x 0 = __t x = . f ''(t) 0 > __21t x 050s| | > |\ __t x 0 > __t x > . : +f(t) +f(t)__x f __, x| ( (\ , f __x, |+| , f __t x =

2__ __ __21f ' x x x 0100s| | | |= = ||\ \ . f(f) __t x = __f ' x 0| | = |\ . 3. f '(0) 1 =( )2 2__ __2__2 21 x xf ' 0 0 x10s 100s 100s| || | |= = = | |\ |\ , : 2__x110s| | | = | |\ __x110s = __x110s = __x10s = __s 110x = , x1CV10= , . 4. : 2__ v2ii 11s t xv=| |= |\ , 2__ v2ii 1t x vs=| | = |\ () : ( )2__1 121f ' t t x100s| |= |\ ,( )2__2 221f ' t t x100s| |= |\ , , ( )2__v v21f ' t t x100s| |= |\ , __1 2 vf '(t ) f '(t ) ... f '(t )yv+ + += =2 2 2__ __ __1 2 32 2 21 1 1t x t x ... t x100s 100s 100sv| | | | | | + + + |||\ \ \ = = 2 2 2__ __ __ 2__ v1 2 v2i2i 111t x t x ... t xt x100s100sv v= | | | | | | | | + + + ` ||| |\ \ \ ) \ = = = 222(I)2 221 vsvsvs 1100s 100sv v 100 100s v= = = = . 2011 A A1. ( ) ( ) = , ( ) ( ) = , : () = ( ) + ( ) ( ) = () (). A2. , , =. 3.fixi,, i , ,fii = , i=1, 2, 3, , . A4. ) , (). ) , (). ) ( ) f (g(x)) ' f '(g(x)) g'(x) = , (). ) , , (). ) , 10%, (). : 2011 2011 [0, 1], : (): 20 4 1 2104 10 | |2 || 0 ( ) 1| |2 1 12 2 () (): 70 5 14 + 7 75 14 4 7 354 4 7 320 20 3 720 20 () (), () : 3 720 20 () 1. ( )( )( ) = 1( )4 = , 1 ( )4 ( ) = () = 4() (V) 64 < () < 72, : 64 < 4() < 7264 72( )4 4< < 16 ( ) 18 < < . (),()=17(V) : () = 417 = 68. 2. = ,,, : () + () + () = 1 : 27 14 5 14 4 + + = 284 5 14 + = 24 5 2 1 0 + = 24 5 1 0 + = , 2 = 9 5 38 = 5 3 818 8+ = = = 5 3 2 18 8 4 = = = . = 1 (), 14 = . 3. 14 = , :21 1 1( ) 4 44 16 4| | = = = |\ 1 7 5 7 2 1( ) 54 4 4 4 4 2 = + = + = = . () = 68, : ( ) 168 4 = 68( ) 174 = = , ( ) 168 4 = 68( ) 174 = = , ( ) 168 2 =68( ) 342 = = . 4.,, , : 1 1 2 1( ) ( ) ( )4 4 4 2 = + = + = = . , , , , , , if %, : if %, ,,,, ,, 1x 10 = , 2x 12 = , 3x 14 = , 4x 16 = , 5x 18 = , () , 1f % 10 = , 2f % 20 = , 3f % y= , 4 Ef % y = , 5f % 10 = . 1. xx, y = y. __x 14.200 = = 14,2 5i i __ 5i 1i ii 1x f %x x f100=== = , : E E10 10 12 20 14 y 16 y 18 1014, 2100 + + + + = E E100 240 14 y 16 y 18014, 2100+ + + += E520 30y14, 2100+= E520 30y 1.420 + = E30y 1.420 520 = E30y 900 = E900y 3030= = , y = y = 30. 2. if %, : fi% 3020108 10 12 14 16 18 20 3. c c = 18 16 = 16 14 = 14 12 = 12 10 = 2. c 212 2= = . [10 1, 10 + 1), [12 1, 12 + 1), [14 1, 14 + 1),[16 1, 16 + 1), [18 1, 18 + 1). if % : ixif % [9, 11)1010 [11, 13)1220 [13, 15)1430 [15, 17)1630 [17, 19)1810 100 4. 15.000 = 15 [15, 17) [17, 19) 30%, 10%, . : 30% + 10% = 40%. 5. , = 80 . 40% 80, 40%80 = 32 . 1. f R, : 21 11 2x x x3 10 5| | + |\ R xe R 21 11 2x x x3 10 5e| | + |\ R . : 3 223 21 11 21 11 2 10x 11x 4xx x xx x x3 10 53 30 15 30f (x) e e e| | + + +|\ = = = 3 2 3 21 11 2 1 11 2 'x x x x x x3 2 23 30 15 3 30 151 11 2 1 11 2f '(x) e x x x e 3x 2x3 30 15 3 30 15 + +| | | |= + = + = ||\ \ 3 2 3 21 11 2 1 11 22x x x x x x23 30 15 3 30 1511 2 15x 11x 2e x x e15 15 15 + + + | |= + = |\ . : f '(x) 0 =3 21 11 22x x x3 30 1515x 11x 2e 015 + + =215x 11x 2 0 + = , 2 = 1 11 1x30= 11 1 12 2x30 30 5+= = = 11 1 10 1x30 30 3= = = .f '(x) 0 > 3 21 11 22x x x3 30 1515x 11x 2e 015 + +> 215x 11x 2 0 + > 1x3 .: +f(x) + +f(x)3152 f 1,3| (

(\ , f 1 2,3 5 ( ( , f 2,5 |+|

, f 1x3= , 1f3| | |\ , f 2x5= , 2f5| | |\ . 2. A B , () () 1x3= ,2x5= : 1P(A)3= 2P(B)5= . A B , A B A = , 1P(A B) P(A)3 = = . : P(A B) P(A) P(A B) P(A) P(A) 0 = = = , 2P(A B) P(A) P(B) P(A B) P(A) P(B) P(A) P(B)5 = + = + = = , 2 1 6 5 1P(B A) P(B) P(A B) P(B) P(A)5 3 15 15 = = = = = . 3. h R, : 21 3x 1x x5 2 3| | |\ R xe R 21 3x 1x x5 2 3e R . : 23 3 221 3x 13x 1 1 9x 6x 2x x xx x5 2 310 5 15 30h(x) e e e = = =)f (x) h(x) = 3 2 3 210x 11x 4x 9x 6x 2x30 30e e + =

3 2 3 210x 11x 4x 9x 6x 2x30 30 + = 3 2 3 210x 11x 4x 9x 6x 2x + = 3 2 3 210x 11x 4x 9x 6x 2x 0 + + + = 3 2x 5x 6x 0 + = 2x(x 5x 6) 0 + = x = 0 2x 5x 6 0 + =x = 0 x = 2 x = 3, 2x 5x 6 0 + = 2 = 1 5 1x2=5 1 6x 32 2+= = = 5 1 4x 22 2= = = . ) x1 = 0 x2 = 2 x3 = 3, 1 1v 2x 1 2 0 1 0 1 1 = + = + = + = ,2 2v 2x 1 2 2 1 4 1 5 = + = + = + = , 3 3v 2x 1 2 3 1 6 1 7 = + = + = + = . : __1 1 2 2 3 3x v x v x v 0 1 2 5 3 7 0 10 21 31x13 13 13 13+ + ++ + += = = = . 2011 A 1. : () = () + () (), () () () (). : ) () () () () () () () (+=B A B A B A () = () + () (). 2.={1,2,...,v} . {i} (i) , : () 0 (i) 1, 1 i v () (1) + (2) + ... + (v) = 1 {i} . () 1 2A { , ,..., }= 1 2( ) ( ) ... ( ) + + + , P( ) 0 = . A3.fx0 , hx f h x fh) ( ) (lim0 00 + .fx0 f(x0), , hx f h x fx fh) ( ) (lim ) (0 000 += . A4. )x>0: ( )'1x2 x= , (). ) f f(x)>0,f , (). ) iN ix , (). ) __x s, __ __x s, x s + 68% , (). 2011 )() , , , (). : 1. (0, 24] ( t + ,4 t ) 1 2'(t) 1 4 12 t t = = . : '(t) 0 = 21 0t = 21t=t 2 =t 4 = . '(t) 0 > 21 0t > 21t>t 2 >t 4 > . :0 424 (t) +(t) (0, 4], [4, +), t = 4 ( ) 4 4 4 4 4 4 2 4 8 4 = + = + = + = . t(0,4]t[4,24], t(4, 24]. 2. 1 ( ) 4 4 = , 4 = 1 = 4 1 = 3. 3. = 3 : ( ) t t 4 t 3 = + . t(0, 24] ( ) t 0 =t 4 t 3 0 + =() x t = , () : 2x 4x 3 0 + = , 2 = 4 4 2x2= 4 2 6x 32 2+= = = 4 2 2x 12 2= = = . : x = 3 t 3 = t = 9. x = 1 t 1 = t = 1. 4. t 4 : 2 2t 4 t 4 t 4 t 42 t 21'(t) t 2t tlim lim lim lim(t 4)(t 4) t 16 t 16 t (t 4)(t 4) = = = = + + ( ) ( )( ) ( )t 4 t 4t 2 t 2t 4lim limt (t 4)(t 4) t 2 t (t 4)(t 4) t 2 += = = + + + + ( ) ( )t 41 1 1 1lim2 8 4 64t (t 4) t 2 4(4 4) 4 2= = = =+ + + +. 1. if % 0 i = 1, 2, 3, 4 :2x 0x 20 02x 0x 6x 0 + x 0x 20x 0x 0 x 6 , x 6. x 6 1 2 2 2f % f % f % f % 100 + + + = , 2x x 20 2x x 6x 100 + + + + = 2x x 20 2x x 6x 100 0 + + + + = 2x 2x 80 0 = , 2 = 324 2 18x2= 2 18 20x 102 2+= = = 2 18 16x 82 2 = = = (). x = 10, 1f % 10 = , 2f % 10 20 30 = + = , 3f % 2 10 20 = = 24f % 10 6 10 100 60 40 = = = . 2. c > 0 , [25, 25 + c), [25 + c, 25 + 2c), [25 + 2c, 25 + 3c), [25 + 3c, 25 + 4c). , . : 1 1F % f % 10 = = , 2 1 2F % F % f % 10 30 40 = + = + = ,3 2 3F % F % f % 40 20 60 = + = + = , 4 3 4F % F % f % 60 40 100 = + = + = . 25 25+c 25+2c 25+3c 25+4cFi% 50100604010 (25 + 2c, 40) (25 + 3c, 60) y = x + . : 40 (25 2c)60 (25 3c)= + + = + + 40 (25 2c)20 c= + + = 40 (25 2c)20c = + = 2040 (25 2c)c20c = + = 50040 40c20c = = 500c20c = =, : 20 500y xc c= . (,50)50, (50, 50) :20 50050 50c c= 1.000 50050c c= 50050c= 500c50= c = 10. 3. c = 10 [25, 35), [35, 45), [45, 55), [55, 65). : 4 = 50, = 50. iivf % 100v= , iif % vv100= . :11f % v 10 50 500v 5100 100 100 = = = = , 22f % v 30 50 1.500v 15100 100 100 = = = = ,33f % v 20 50 1.000v 10100 100 100 = = = = , 44f % v 40 50 2.000v 20100 100 100 = = = = . O : () ixivif % iNiF % i ix v[25, 35)305101010150 [35, 45)4015302040600 [45, 55)5010203060500 [55, 65)602040501001.200 501002.450 : 4i i __i 1x v2.450x 4950 50== = =. 4. x > 0 , 5 + x. :30 (5 x) 40 15 50 10 60 204050 x + + + + =+ 150 + 30x + 600 + 500 + 1.200 = 2.000 + 40x 150 +600 + 500 + 1.200 2.000 = 40x 30x 450 = 10x x = 45. 45 40 . 1. 0, 2 : 1(( ) ( ))3 + = () 3 1(( ) ')3 = () 1(( ) ')2 = () : 10 133 10 1310 12 + (V) : ( )3 1 3 3 1 1( ) 1 (( ) ') 13 3 3 + = === = = , ( )1 2 1 3( ) 1 (( ) ') 12 2 2 = === = = , , , : (( ) ( )) ( ) ( ) ( ) ( ) ( ) ( ) = + = + =( ) ( ) ( ) ( ) ( ) ( ) = + = . : 1 3 13 2 3 + = ( + 1)( 2) = 3( 3) ( 2) 2 2 + 2 = 32 9 + 2 2 + 2 + 2 + 32 9 + 2 = 0 22 9 + 4 = 0, 2 = 49 9 74 = 9 7 1644 4+ = = = 9 7 2 14 4 2 = = = (, 3 (IV)). = 4. 2. = 4, : 4 1 5(( ) ( ))3 4 12+ = =, 1 1( )3 4 12 = =, 4 3 1( )4 2 2 = =. = 600 x > 50 , x 50 . x 50( )600 = x(B)600 = . :(( ) ( )) ( ) ( ) 2 ( ) = + , : 5 x 50 x 1212 600 600 12= + 2 5 x 50 x12 12 600 600+ = + 7 2x 5012 600= 2x 50750= 2x 50 = 350 2x = 350 + 50 2x = 400 x = 200. 200 50 150 1( )600 600 4 = = = 200 1(B)600 3 = = . : 1 1 3 1 2 1P(A B) P(A) ( )4 12 12 12 6 = = = = = , N(A B) 1N( ) 6= () = 600 :N(A B) 1600 6= ( ) = 100. : 1 1 4 1 3 1P( ) P( ) ( )3 12 12 12 4 = = = = = , N( ) 1N( ) 4 = () = 600 :N( ) 1600 4 = 600N( ) 1504 = = . N( ) 1( )N( ) 12 = = () = 600 : N( ) 1600 12 = 600N( ) 5012 = = . ) ( ) = 150, ( ) = 100 N( ) 50 = . ) N( ) 1( )N( ) 2 = = () = 600 :N( ) 1600 2 = 600N( ) 3002 = = , N( ) 300 = . 3. : = = . 2 N(( ) ') 600 300 300 = = . y,300y(0 y 300) . , () = 2(), : y 300 y2300 300= y = 2(300 y) y = 600 2y y + 2y = 300 3y = 300 y = 100. 100200 . . : 1 2012 A A1. F(x) f (x) g(x) = + .: F(x h) F(x) (f (x h) g(x h)) (f (x) g(x)) + = + + + + f (x h) g(x h) f (x) g(x) = + + + (f (x h) f (x)) (g(x h) g(x)) = + + + . h 0 : F(x h) F(x) f (x h) f (x) g(x h) g(x)h h h+ + + = + . f, g , h 0f (x h) f (x)f '(x) limh+ = h 0g(x h) g(x)g'(x) limh+ = . h 0 : h 0 h 0F(x h) F(x) f (x h) f (x) g(x h) g(x)lim limh h h + + + | |= + |\ h 0 h 0f (x h) f (x) g(x h) g(x)lim lim f (x) g (x)h h + + = + = + . : (f (x) g(x)) f (x) g (x) + = +: F(x) f (x) g(x) = + .: 0 0 0F(x) F(x ) f (x) g(x) (f (x ) g(x )) = + + 0 0f (x) g(x) f (x ) g(x ) = + 0 0f (x) f (x ) g(x) g(x ) = + . x x0 : 2012 : 2 0 0 00 0 0F(x) F(x ) f (x) f (x ) g(x) g(x )x x x x x x = + . f, g , 000x x0f (x) f (x )f '(x ) limx x= 000x x0g(x) g(x )g'(x ) limx x=. x x0 : 0 00 0 0x x x x0 0 0F(x) F(x ) f (x) f (x ) g(x) g(x )lim limx x x x x x | | = + | \ 0 00 00 0x x x x0 0f (x) f (x ) g(x) g(x )lim lim f (x ) g (x )x x x x = + = + . : (f (x) g(x)) f (x) g (x) + = + 2. . A, , : N(A)P(A) N( )= =. 3., CV, __x 0 , : __ sCV x= = , __x 0 . 4. ) , (). ) f x0 f x, x = x0, (). ), A B ,()(), (). ) , , (). ) 00x xlim x x= , (). : 3 : 1. , (,50). , = 25. A 2. 1 125, [5,15)[15,25), [25,35)[35,45), 1 2 3 4 + = + . : + 4 + 3 6 = 2 + 8 + 24 2 = 3 + 64 3 = 2 + 6 = 8. 2 1 25, [5, 15) [15, 25) 50%, 1 2f f 0, 5 + = . : 4 3 6 14 3 6 2 8 2 4 3 6 2 8 2 2+ + = + + + + + + + + + + 4 2 17 4 2=+2(4 2) = 7 + 48 4 = 7 + 4 8 7 = 4 + 4 = 8. 3 1 25, [5,15)[15,25) [25, 35), [35, 45), 1 2 3 4f f f f + = + . : 4 3 6 2 8 27 4 7 4 7 4 7 4+ + + = ++ + + + + 4 + 3 6 = 2 + 8 + 2 4 2 = 3 + 64 3 = 2 + 6 = 8. : : 4 [5, 15), [15, 25), [25, 35), [35, 45). ix , :15 15x 102+= = ,215 25x 202+= = ,325 35x 302+= = 435 45x 402+= = . :1 = + 4 =12,2 = 3 6 18 = ,3 = 2 +8 = 24 ,4 = 2 = 6 1 2 3 4 = + + + = 60 . :11 12 1.200f %= 100 = 100 = = 20 60 60 ,22 18 1.800f %= 100 = 100 = = 30 60 60 , 33 24 2.400f %= 100 = 100 = = 40 60 60 , 44 6 600f %= 100 = 100 = =10 60 60 3 1 2 4f %+f %+f %+f %=100 . :1 1N = =12,2 1 2N = N + =12+18 = 30 ,3 2 3N = N + = 30 +24 = 54 4 3 4N = N + = 60 . :1 1F %= f %= 20 ,2 1 2F %= F %+f %= 50,3 2 3F %= F %+f %= 50+ 40 = 90 4F %=100. : () xi i fi % Ni Fi % [5, 15) 1012201220 [15, 25) 2018303050 [25, 35) 3024405490 [35, 45) 4061060100 60100 : 5 3. H : 4i i__i=1x 1012 +2018+3024 +406 120+360+720 +240x = = = 60 60 1.440= 2460=. ( ) : ( )422 2 2 2 i i2 i=1x x (10 24) 12+(20 24) 18+(30 24) 24 +(40 24) 6s = = 60 2 2 2 2( 14) 12+( 4) 18+6 24 +16 6 19612+1618+3624+ 256660 60 = =2.352 288 864 1.536 5.0408460 60+ + += = =2, : 2s = s = 84 9,17 . 2 24i i4i=1 2 2i ii=1x s1= x | | | \ ` ). () xi i xii2i ix [5, 15) 10121201.200 [15, 25) 20183607.200 [25, 35) 302472021.600 [35, 45) 4062409.600 601.44039.600 : 221.440 2.073.600s = 39.600 = 39.60060 60 601 160 ` ` ) ) { }5.04039.600 34.560 = 8460160= =2, : 2s = s = 84 9,17 . 4. 1 37 [37,45).[37,45) [35,45) 45 37 8 445 35 10 5= =. , 44 4f 0,1= 0, 08 = 8%5 5 = . : 6 2 , (37, k).37 [37,45), . 37B2090100k : (35, 90) (45, 100). y = x + . , : 90 35100 45= + = +

90 35100 90 45 35= + = +

90 3510 10 = = 90 351 = =

90 351 = =

551 = =. y = x + 55 , (37, k) : k = 37 + 55 = 92. 100% 92% = 8%. : = , = , : , : . : ( )23P 1=+ ,( )2 2P I 1+=+ ,( )2 1P I 1+ =+. 1. f ( )222 x 3 2f (x)x x+ =+ fA { 1, 0} = R x 1, : : 7 ( ) ( )( )( )2 2 222x 1 x 12 x 3 2 2 x 3 2 x 3 2lim limx xx(x 1) x 3 2 + + + +++ + +=( )( ) ( )22 222 2 x 1 x 1x 3 2x 3 42 lim 2 limx(x 1) x 3 2 x(x 1) x 3 2 + + = =+ + + + + + ( ) ( )22 2 x 1 x 1x 1 (x 1)(x 1)2 lim 2 limx(x 1) x 3 2 x(x 1) x 3 2 + = =+ + + + + + ( ) ( )2 2x 1x 1 1 12 lim 2 14x x 3 241 1 3 2+ += = =+ + + += . ()=1 , I ,, I , ( I) () 1 < = , . 2. : ( ) ( ) ( ) ( ) P I P P I P I = + . 23 2 ( 1)1 1+ + +=+

2 1 3 2 1 += + +

2 1 3 1 += + 2 3 =

2 3 0 =

( 3) 0 = = 0 ( 3) 3 = 0 = 3 (). 3. = 3 : ( ) ( ) ( )9 5 1 4 2P , P I , P I10 10 2 10 5= = = = = . : ( ) ( ) I I .( ) ( ) I , I , : ( ) ( ) ( ) ( ) ( ) P I I P I P I = + ( ) ( ) ( ) () ( ) () ( ) P P I 2P I = + = + 9 5 4 9 5 8 6 3210 10 10 10 10 5+ = + = = = . 4. N( ) = 32 ( )2P I5 =: ()32 2N 5=2()= 160 ()160N 8020= = . : 8 1. f (0, +). lnx(0,+), (0,+)ln2x lnx, x2.(0,+)1+ln2x, ,(0,+) f1+ln2xx2( ). : ( )22 22''1+ ln x x (x) (1+ln x)f (x) =2x12lnx x 1 ln xxx= ( )22 22 2 2lnx 12lnx 1 ln x (ln x 2lnx +1)= = =x x x . : f (x) = 0 ( )22ln 10 =xxlnx 1 = 0lnx = 1x = e. f (x) < 0 ( )22ln 10 0, x2 > 0 1 + ln2x >0, f(x) > 0, x(0, +) 0.0 0.5 1.0 1.5 2.0 2.5 3.00.51.01.52.02.53.0y = f(x)M(x, f(x))K(x, 0)(0, f(x))O : (OK) | x | x = =( ) | f (x) | f (x) = = . : 221 ln x( ) (OK) (O ) x f (x) x 1 ln xx+ = = = = + . 2E(x) 1 ln x = + , x > 0. : 9 1 + ln2x (0, +) 1 2ln xE'(x) 2ln xx x= = . : E(x) = 02ln x0x= 2lnx = 0lnx = 0x = 1. E(x) > 02ln x0x> 2lnx > 0lnx = 0x > 1. f (0, 1], [1, +) x = 1 E(1) = 1. x = 1 () = 1, 2() = f(1) =1+ln 1=1, () = () = 1, . E(x) : ln2x 0ln2x + 1 1E(x) 1E(x) E(1), x = 1, . 3.() f (1, f(1)), : 22(ln1 1) = f (1) = = 11 . () y = x + . x, y, z c : y x c = + , __ __y x c = + y xs s = , z cx = , __ __z c x = y xs | c | s = . : __ ________ ___ __ __y x x 1 x x 20 = + = + = + = + = +, y x x x xs s s | 1| s s 2 + = = = = = yy__s2CV| 20 |y= = +. yi(i=1,2,,10) y1CV10 2 1| 10 | 10| 10| 20 10 20 10 20 10 20 10 + 20 10 30. 4. :A , 0 < () 1 A B , 0 < ( ) 1. : A A B , P(A) P(A B) () A B A B , P(A B) P(A B) () P(A), P(A B), P(A B) (0,1] (0, ) + f (0, +), () () : f(P(A)) f(P(A B)) (III) : 10 f(P(A B)) f(P(A B)) (IV). (III), (IV) : f(P(A)) +f(P(A B)) 2f(P(A B)) .... .... 2003 1. , 151. , 85. 1. 100% 100 1 100ii i inn f n nn n= = = =2. ( , 96) sCVx= . ( 3) ( 2) ( 1) 0 1 2 3 007 7x + + + + + += = = . .3. , 99, :s a s b x a x = + = ' , '.4. , 70: 0360360i i ia n fn= =. : 16 8 15 1, 3 170, 7 128 19, 5 11, 9 159, 415, 948 1, 3 0, 7 10 10 + + + += = =+ +. ( , 27, 28) , u , ( ) '( ) ut x t =( ) '( ) ''( ) a t u t x t = = . , , : 2( ) '( ) 1 ut x t t t = = ( ) ''( ) '( ) 2 1 a t x t u t t = = = t 0, ( ) 0 a t < . , . 21. x, y () [4,6) [16,18) . % 100if = 85. 100 85 15 x y + = = . , , 4 y x = . 3 ,12 x y = = . 3% 12% .2. , 10 (16+14+13+12+5)%=60% . , 55872 . , : 10055872 9312060=.3. , 11 13, 16 14( )% 15%2 2+ = . , 15% , 1593120 13968100=.4. : , : : . , , , P(). P(A)=0,34 P(T)=2P(B). , P(A)+P(B)+P(T)=1. :0, 660, 34 ( ) 2 ( ) 1 3( ) 0, 66 ( ) ( ) 0, 223PB PB P B PB PB + + = = = = ( ) 2 ( ) 0, 44 P T PB = = . , . , :( ) ( ) ( ) 0, 34 0, 44 0, 78 PA T PA P T = + = + = . 0, 22 93120 20486, 4 20486 = . 31.::.,:={,,,}.:={,,}.,,:.43) () (= NA N2.`,( iy )P(A)=3/4, P(A')=1-3/4=1/4 P()=1 : iy in232 x 3243x 2x 3. :

83 68342483 24332322 2 2x xx x x x xy+=+=+ + =. 2 21 1 8 1( ),0,2 6 3 6 38f x x Rx x x x y = = = + + f ( ) :2 212 3'( ) 8(6 3 )xf xx x+= + : 3 1'( ) 0 12 3 012 4f x x x x = + = = = 1'( ) 0 8 (12 3) 0 12 3 04f x x x x + + . :x -1/2 -1/4 0'( ) f x + + 0 - -( ) f x., f , 14x= , 21 8 8 8 8 64( )1 1 6 3 3 3 34 36 ( ) 3 ( )4 4 16 4 8 4 8f = = = = = + 41. 2 32 2'( ) [( ( )) (7 ( ) 3) ln ( )]'3 [ ( )] 7 ( ) 3 ln 1f x P A x P A x x x P BP A x P A x= + == + 2. '(1) 0 f = , :23 [ ( )] 7 ( ) 2 0 PA PA + = . =25 , ,7 5 122 ,. 0 ( ) 1 6 6( )7 5 2 1 , 6 6 3PAPA+ = = = = =., , 1( )3PA= .3. 1( )3PA= , 3 31 7 1 2( ) ( 3) ln ( ) ln ( )9 3 9 3f x x x x x P B x x x x P B = + = + + , 1 2(1) ( )9 3f P B = + +( ln1=0). , , 55(1)36f = , :1 2 55 4 24 55 27 3( ) ( ) ( )9 3 36 36 36 36 36 4P B P B PB + + = + + = = = . , . 1 3 13( ) ( ) ( ) 13 4 12PA B PA P B = + = + = > , ,0 ( ) 1 PA B . , .4. 1( ) ( ) ( ) , A B A31 1( ) 12 31 1( ) ( ) ( ) ( )12 1213 1( ) ( ) 1 , 12 12PA B PA B PAPA BPA B PA P B PA BPA B PA B + .... 2004 1 . 31. .. 1 2 3 4 .8 - x2 -' x= =R'= max' min'=2min+2max=2(maxmin)=2R=20 s' = |c|s = 2s = 4 . 33. ( ). 2. ={1, 2, 3, 1, 12, 13, 2, 21, 23, 3, 31, 32} .= } , , , , , {2 3 1 3 3 2 1 2 3 1 2 1 = } , , , , , {3 2 1 3 2 1 = { } . ( 150 ), , :()= 5 , 0 =126=) ( ) (

()= 5 , 0 =126=) ( ) ( ()=()=0 . 2 1 3 . : f '(x) = (2x22x+1)' = 4x2 f '(x)=0 4x2=0 x=1/2 f '(x)>0 4x2>0 x>1/2 f '(x) < + ( ) ( ) ( ) ( ) ( ) f x 0 P P K x 0 x P P K < + < > +, f ( , ()+()] [()+(), +), x=xo x=P()+().i) :( ) ( ) = 205f x 2( ) ( ) ( ) ( )2 5f P K P 2 + = ( ) ( ) ( ) ( ) ( ) ( ) ( )2 2 1 5P K P P 2 2 + + + = ( ) ( ) ( ) ( ) ( )2 2 2 1 5P K P K P KP 2 2 + + = ( ) ( ) ( )( )( ) ( )P K 02P KP 2P K P 2P K = = )i): () () ()() () (). () < ()( () (ii)) , , : (), (), (), (), (), (), (), (), (), (). 2005 5 10 5 6 : ( ) ( )( )P K P K P K2+= = ( )1P K 4= = ( ) ( )1 1P 2P K 24 2= = =, , :( ) ( )2P K 3 = ( ) ( ) + = 2P K P 3 ( ) ( ) ( ) ( ) + = 2P K P K P P 3 ( ) ( ) ( ) + = 2P K P 2P K 3 ( )1 1 22 4 2 3+ =( ) =1P K 24

: ( ) ( ) ( ) ( )1 1 1 17P K P K 4 2 24 24 = + = + = . ii) : xi P()P()()()()() vi 213121 ( ). () () ()() () () i 3 2 1 xi 20056 () () ()() () () i 3 2 1 xi 2005 7 1 : . 10. 4 1 6 2 . 2006 1 1 . . 85 . . 150 . 1. 2. 3., 4.,, 2 i) ( )( )( )( )=+ + + + + + += + 1 1 11 1 1 1lim11 3lim2 22 21221x x xx x x xxx xx x ( ) ( )( )( ) ( )( )( )( )( )( )( )( )( )41822 4 223 12lim1 3 1 11 2lim1 3 11 2 3lim1 3 11 3lim2 12 1 2 22 21 2 22221==+=+ + + +==+ + + + =+ + + + +=+ + + + += x x x xx x x xxx x xx x xx x xx xxx x x2 4 24121= = = ii) + = x x x f 6 ) (3. R Af= . =2: + = x x x f 6 2 ) (3 f R : 6 6 ) (2 = x x f . :( )( ) 1 1 0 1 1 6 0 6 6 0 ) (2 = = = + = = x x x x x x f :( )( ) 0 1 1 6 0 ) ( > + > x x x f -1 1 +-+ x < -1 x > 1 f : x -11 + f + - + f Hf x = -1, 4 6 2 ) 1 ( + = + + = f : + 4 = 9 = 5 iii) fC )) ( , (0 0x f x A ) (0x f ., 1 1 0 ) (0= = = x x x f 2006 2 :1 ) 1 ( 9 ) 1 ( = = f f : (-1,9) (1,1). iv) f x :6 6 ) (2 = x x f:0 0 12 0 ) ( = = = x x x f0 0 12 0 ) ( > > > x x x f f : x 0 + f -+f x = 0 f f x = 0 3 . i) xi i fi fi% Ni Fi Fi% i iv x ( )i iv x x21500,12512,5500,12512,550200 21100,27527,51600,440220110 31200,3302800,7703600 4300.0757,53100,77577,512030 5900,22522,54001100450360 v = 40011001200700 ii) :3400120040051= = == ii iv xx=400(), , . :323 32201 200=+=+=t t iii) : ( )474007004005122= === ii iv x xs : 2747= = s : 10166 , 067327= > = = =xsCV 6 , 0 7> . 101> CV. CV>10%, . . : v1 + v2 = 50+110=160 () = 160, : ( )( )( ) 52400160= ==NA NA P . 2006 3 : v4 + v5 = 30+90=120 . ()=120, : ( )( )( ) 103400120= ==NB NB P. , . == = 511200 ) (ii iv x N 4 ,. 4 4v x =120 . ()=120, : ( )( )( ) 1011200120= == NNP ,. 1 1v x =50 . ()=50, : ( )( )( ) 241120050= == NNP 4 . i)x s . , 25,, x +25. 25 , 10%. : 10125 =+ xs (1) , CV 20%, : 5110020= =xs(2) : 255525 5525 5 10525 105110125=====+ ==+ ===+xss xss xs ss xx sxsxs ii) 2 2221,..., ,vx x x : ==+ + +viivxv vx x x122 22211 ... : ((((((

.

' ===vxxvsviivii2112 21 2006 4625 252525122 1221 12 = = = === =vxxvxvxvxviiviiviivii 65012==vxvii iii)s R 6 . xmin10, xmax : s x 6 10max+ .40max x . . : : B: 30% , :()=0,3 60%, 4 , 0 ) ( 6 , 0 ) ( 1 6 , 0 ) ( = = = B P B P B P 50% , 5 , 0 ) ( = B A Pi)B A . : + = ) ( ) ( ) ( ) ( B A P B P A P B A P + = ) ( 4 , 0 3 , 0 5 , 0 B A P2 , 0 5 , 0 7 , 0 ) ( = = B A P 20% . ii)AB . :1 , 0 2 , 0 3 , 0 ) ( ) ( ) ( = = = B A P A P B A P 10% :2 , 0 2 , 0 4 , 0 ) ( ) ( ) ( = = = B A P B P A B P 20% . . . , 1,,. 2006 5., 21 + v 21 v. : : A: : vvvvP2121) () () (+=+= = : vvvvAA P2121) () () (== = () 0,8% (), : 125162000016 , 02016 , 0 2016 , 0 1 1008 , 021211008 , 0) ( ) (= = ==+ = ++=++ = vvv v vvvvvA P P 125 2007 11 1.1. 139.2. 87.. 30..1 ,2 ,3 ,4 ,5 2. x > 0, = ( 0, +). f (x) = (x2 + lnx) = (x2) + (lnx) = 2x + x1 x > 0. f = (0, +) f (x) = 2x + x1 > 0. .. x 1) 1 x ( 21 x) 1 x )( 1 x ( 21 x) 1 x ( 21 x2 x 21 x3x1x 2 x1 x3 ) x ( xf2 2+ =+ === += 4 ) 1 x ( 2 lim1 x3 ) x ( xflim1 x 1 x= + = 2007 22 3 R = 305 = 25 c 5525 R= =. :[ 5, 10),[10, 15),[15, 20),[20, 25) ,[25, 30) :7,5,12,5,17,5,22,5,27,5 :4 = 30(1),2 = 4 3(2), f1% = 10 (3)3 + 4 + 5 = 40 (4): 1 + 2 + 3 + 4 + 5 = 1 + 2 + 3 + 4 + 5 = 80(4)1 + 2 = 40 (5)11f % 100=

110 10080= 81 = , (5) 2 = 32. (2) 3 =8(4) 5 =2, i: 8,32,8,30,2 80. fi% fi% = i100, i = 1, 2, 3, 4, 5 : 10,40,10,37,5, 2,5 100. i :1 = 1, i = i 1 + i,i = 2, 3, 4, 5 : 8,40,48,78,80., :F1% = f1%, Fi %= Fi 1 %+ fi%, i = 2, 3, 4, 5 : 10, 50, 60, 97,5, 100. : [ - , - ) xii fi% Ni Fi%[5, 10 ) 7,5 8 10 8 10[10, 15) 12,5 32 40 40 50[15, 20) 17,5 8 10 48 60[20, 25) 22,5 30 37,5 78 97,5[25,30) 27,5 2 2,5 80 100 80 100 2007 33. :.1..(23,0)OxOx (0, 82,5). H 82,5 .2.[20,23) 35[20,25), [20, 23) 35 f4%, , 35 37,5 = 22,5 f1% + f2% + f3% + 22,5% = 82,5%. 1 . 60 = 1 + 2 + 3 +12, 60 12 125 230 = . 20+2=22.2.60 75% , (), 77 5 10 15 20 25 30xi10097,5605010OFi%2382,5 (Fi%) 2275 2007 44 4 () () = 51()51) () () () (= () () = 51() = () + 51(1), () < (). () ()() ()() () < () ()(2)R = () ()(3). (2) :() < ()0 < () ()0 < R() 1 () 0,() () 1 R 10 < R 1. :R = () () = [ () + () () ] ()= [ () () ] + [ () ( ]= ( ) + ( )= ( ) + ( ) [ : = ]= ( ) + ( )= ( ) + ( ())= ( ) + ( )2 . Venn = ( ) :( ) + ( ) = ( ) + () ( )= [() ()] + [1 ()] [ 1( )]= () () + 1 () 1 + ( )]= () ()= R= ( ) 2007 55 . f (x) IR , x 1lim f (x) f (1)= 3 ) B A ( P 5 ) x ( f lim1 x+ =(4)x 1 : (1)1 5 P(B) x 5P(B) 1 5P(A)x 5P(B) 1 5x 1 x 1 + = 1 x1 ) B ( P 5 x x ) B ( P 5 += 1 x1 x ) 1 x )( B ( P 5 + =1 x] 1 ) B ( P 5 )[ 1 x (+ =1 ) B ( P 5 + =,1 ) B ( P 5 ] 1 ) B ( P 5 [ lim1 x1 ) B ( P 5 x ) A ( P 5lim ) x ( f lim1 x 1 x 1 x+ = + = = (4) , , : 5() + 1 = 5 () + 3 () = () +52 (5). (1) (5) : () = () +53 (3) :R = () () = [ () + () ()] ()= () + () 2()= () + 53 + () +52 2()= 1. () < 1, R = () () < 1, , , () 1 :() = 1.R = () () =1() = 0. 2008 11' & 1.1. . . 92.2. . . 149.3. . . 28-29. 2) 0 2 + x0 2 2 + x2 x 0 2 2 = + x 2 2 = + x4 2 = + x2 = x0 2 2 + x2 x . f ) [ ) ( + , 2 2 , 2 .) f(x)=0. :0 42= x x ) [ ) ( + , 2 2 , 2 42= x x ) [ ) ( + , 2 2 , 2 2 = x x ) [ ) ( + , 2 2 , 2 2 = x . xx (-2,0).)( )( )( )( )( )( )( )( )( )2222 22 2 2 22 2 2 22 2 2 22 24) ( ++ + + =+ + ++ + + = +=xx x xx xx x xxxx f = = ( )( )( ) ( )( )( )( )( ) 2 2 222 2 2 24 22 2 2 2+ + + =+ + + = ++ + + x xxx x xxx x x 2008 22( )( ) ( )( ) 16 4 4 2 4 2 2 2 2 2 lim ) ( lim2 2= = + + = + + + = x x x fx x) :xivifiNiFi1 4 0,1 4 0,12 16 0,4 20 0,53 12 0,3 32 0,84 8 0,2 40 1 40 1 - -1 , 040411= = =vvf41 1= = v N1 , 01 1= = f F4 , 0401622= = =vvf ,20 16 42 1 2= + = + = v v N5 , 0 4 , 0 1 , 02 1 2= + = + = f f F8 40 2 , 04 444= = = = v f vvvf404= = v N 14= F12 28 40 40 8 16 4 403 3 4 3 2 1= = = + + + = + + + v v v v v v3 , 0401233= = =vvf , 32 12 203 2 3= + = + = v N N8 , 0 3 , 0 5 , 03 2 3= + = + = f F F 2008 33 3) . 10 50 60 30 10 40 40 60 100N(A) 50P(A) 0, 5N() 100= = =1: 2: .( )1 2 1 2 1 2 1 2B (B B ) P(B) P(B B ) P(B ) P B P(B B )60 40 10 900,9100 100 100 100= = = + = + = =) :100ii 1t 6040 4040 2400 1600 4000== + = + = : 100i i ii 1 i 1 i 11y t t y t 100 39, 6 3960= = == = = = c.c 4 , : 4000 3960 = 4c 4c= 40 c = 10) : 13,5%13,5%34%34% 2,35% 2,35% 0,15% 0,15% x 3s x 2s x s x x s + x 2s + x 3s + 2008 44 2,5% ( 2,35% + 0,15% = 2,5%) 26 , x 2s 26 =(1) : 6040 4040x 40100 + = = (1) : 2s = 14s = 7. : 33 13,5% + 2,35% + 0,15% = 16% . 16100100 =16 .) :s2 = 2 2 2i i i ii i i 1 i 1 2 2 i 1i i 2i 1v x v x v x1v x S100 100 100 100= = == ( ( ' . ' .( = ( ( ( 2 22i i i i i 1 2 2 2 2 2 2 2 i 1i i i i 2i 1 i 12 2v x v x100 s 100 100 100 s 100 v x v x100 10010000 s 250000 S 25 S 5= == = = = = = = R6s, R30. 13,5%13,5%34%34% 2,35% 2,35% 0,15% 0,15% 40475461332619 2008 55 4. : 5 4 7 1 5 5 515 5x xxx e x x e xx e x+ + + + += = = + f(x)= ex-x+1,x .H f f(x) = ex-1. ( ) 0 f x 01 0 1 0x x xe e e e x 0 - + +- f (x) f (x) x 2 min 0 0 x=0 f (0) 2 m f = =B. m=2 : g(x)=2x2-2x+3. g g(x)=4x-2. Cg (1, g(1)) xx g(1)=04-2=0 2=4 =2 =-2( ) =2.={1,3,4,5,6,7} ( ) 6( )( ) 7NPN= =. B A = ( ) ( ) = ( )( ) ( )R x xA A + ++ = , 2008 x2x122 3x h2 3. h R ( ) R x A PA P + + = , 1 x ) ( x4) ( 2 3x h2= = + + = + = 4 1 ) ( 2 ) ( ) ( 2 3 ) ( 14) ( 2 34 ) (2 2 2A P A P A P A PA PA P=( ) 0 4 1 ) (2< + A P: ( , =, ) () 1 ( ) 1 0 ( ) 1 0 < P(A)+1 . ) 3 C 7 C ,43 2 7 4 () . : 1 120% 30% 40% 60%2 2+ + = . 3

)={1,2,3, ... ,25} , , ()=25 { } { } A / / 3 3, 6,9,12,15,18, 21, 24 = = ()=8 { } / f = ( )2f x x x 9 = + ( )( )2 0 36 0 6 6 0 < < + < 6 6 < < . ={1,2,3,4,5} ()=5 2x x x / lim 16 x = 2009 3 3 ( )( )( ) ( )2 22 2x x x x x x x lim limx x += = ( ) ( )( )x x xx x lim lim x x 2 x += = + = 2 16 8 . ={1,2,3,4,5,6,7,8}. ()=8. ) ( )( )( ) 8 32% 25= = = ( )( )( ) 8 32% 25= = = ) ( )( )( ) 5 20% 25= = = { } ( )( )( ) 1 3 4% 25 = = = ) ( ) ( ) ( ) ( )8 5 1 12 48%25 25 25 25 = + = + = = ( ) ( ) ( ) ( ) = + = ( ) ( ) ( ) 1 = + = ( ) ( ) ( ) ( ) [ ] 1 = + = ( ) ( ) ( ) ( ) ( ) ( ) 1 1 = + + = + = 5 1 211 84%25 25 25= + = = ( ) ( ) ( ) ( ) ( ) = = = ( ) ( ) ( ) ( ) ( )1 4%25= = + = = : Venn 4

) K(1,f(1)) (): y=x+1 ( )K f 1 1 = = ( )( )( ) ( ) [ ]( )( ) ( )2PA B PB Af x x 2 = ++ + ( )( )( ) ( )PA Bf x x = + 2009 4 4 ( )( )( ) ( )( ) ( ) ( )PA Bf 1 1 1 PA B = = = ++ ( ) 0 = .. = . ) ( )f1 10, C f 03 3 = ( )( )( ) ( )PB Af 0 =+ ( )( ) ( )( ) ( ) ( )1 PB A3PB A 3 = = + + ( ) ( ) [ ] ( ) ( ) ( ) ( )3 PB PA B 2 = + = ( ) ( ) ( ) ( ) ( )3 1 3 4 4 = + = = ( )1A2= . ) ( ) ( )g x 6f x 12x 2019 , x = + R ( ) ( )g x 6f x 12 = ( ) ( )( )( ) ( )P A Bg x 0 f x 2 x 2 x 2 = = = =+ ( ( ) ( ) ( )PA B = + ) ( ) ( )g x 0 f x 2 x 2 > > > g x=2 g(2)=6f(2)-24+2019 ( )( )( ) ( ) [ ]( )( ) ( )PA B PB A 1 1 7f 2 4 4 2 3 3 2 = + = + =+ + g(2)=2009. ) ( )21 1f x x2 3= + ( )f x x = y=x+ () f ( ) f 1 1 = = ():y=x+ ( ) ( ) ( )f1K 1, f1 C f1 1 6 = + = . K1 : y x6= ( )i K i i1M y x , i 1, 2,...106 = = 1 59 1y x 106 6 6= = = y xs s 2 = = yys2CV 20%y 10= = = 2010 1 1 ' . & 1 . (. .31) .. (. .149) .(. .66) .. . . . . 2 .0 1 x2 +, R x A R =..( ) ( ) ( )+ + + + = 15 x 1 x ln x f21 x1 x 2 x1 xx 2 1 x11 xx 222222++ +=++ += ++=.( ) = + 2 x x1 xx f lim221 x= ++ + + 2 x x1 x1 x1 x 2 xlim22221 x ( )( ) ( ) =+ + 1 x 2 x1 xlim21 x= + =+ 2 11 12 x1 xlim1 x030=. .( ) ( )0 0x f , x M fC . ( ) ( ) //:( ) = 1 x f0 =++ +11 x1 x 2 x20020 + = + + 1 x 1 x 2 x20 020 0 x 0 x 20 0= = . ( ) = + + = 15 1 ln 0 f 15 + , ( ) 15 , 0 + ( ) ( ) + = x 0 f y :( ) + = x y :() 15 0 15 + = + = + ( ) 15 x y : + + = . .0 x1= 15 y1+ = 1 x2= 15 1 y2+ + = 9 x3= 15 9 y3+ + = 10 x4=15 10 y4+ + = 2010 2 2 : 15 + ,1 15 + + , 9 15 + + ,10 15 + + =+ + + + +=29 15 1 15 =+ +210 15 2 5 15 + + = 50 50 5 15 = + + = + 45 15 = + 2025 15 2010 = 3 . , 50 = . . . [ - ) ixiif % fi iNiF % Fi 0 - 4240,08840,088 4 8670,1414110,2222 8 1210180,3636290,5858 12 1614130,2626420,8484 16 - 201880,1616501100 = 501100 = 50 = + 505 4 = + 50 8 62 = 42 62 72= = 424 = + + + 42 18 7 44 134= ..: 10 17 ( ) = + + =5 4 34121A N 24 2 13 9 = + +( )( )48 , 05024NA N) A ( P = = = 48% .: 10 16 ( ) = + + + =5 3 2 1v v21v v B N 28 8 9 7 4 = + + + . ( )( )56 , 05028NB N) ( P = = = 56% 4 . :( )( )( ) ( )32 2 63PP P Pk = = = =( )2PR = ( ) ( ) ( ) ( )12P A P P P = + + = ( )1 122 2P + + = ( )132P = (1) 2010 3 3 ( ) 1 P= ( ) ( ) ( ) 2 3 6 P P P + + ( ) ( ) ( )( ) 11 P P P + + + = 13 2 3 12 2 + + + + + = 142 2 + = 9 12 2= 19= :( )1218P = ,( )133P = ,( )169P = ,( )19P = ,( )29P = ,( )16P =.( ) 20 x 24 x x f2+ = () // ()( ) = 48 1 f = + 48 44 4 = ( ) 20 x 24 x 4 x f2+ = ( ) = 0 x f = + 0 20 x 24 x 421 x= ,5 x= x 1 5 +( ) x f +-+ ( ) x f 1 = 5 = { } 6 , 5 , 4 , 3 , 2 , 1 = . 0 5 3 x 20 3 x 2 5 3 x 22 / 3 x 5 3 x 22 / 3 x4 x2 / 3 x x:2 x= 3 x=5 x= 6 x={ } 6 , 5 , 3 , 2 B =. 4 % 5 , 24011604= = . :20 s 2 x = + (1) = x43R = x43s 6 x 3 s 24 =(2) (2) ( )( ) = s 2 20 3 s 241 = s 6 60 s 24 = 60 s 30 2 s = (1) = + 20 4 x 16 x= : 10181162xsCV > = = = , . c :2 s s = = c 16 c x x + = + = . : 101V C 101xs + 101c 162 + c 16 20 4 c c:4 c = 5 c =6 c ={ } 6 , 5 , 4 = 2010 4 4 .:{ } 5 , 4 = { } 3 , 2 = { } 6 , 5 , 4 , 1 = { } 6 , 5 , 3 , 2 = , { } 6 , 3 , 2 = ( ) ( ) ( ) = + = 5 P 4 P A P 1876192= +( ) ( ) ( ) = + = 3 P 2 P B P 18731181= +( ) ( ) ( ) ( ) ( ) = + + + = 6 P 5 P 4 P 1 P A P 181191619291= + + +( ) ( ) ( ) ( ) ( ) = + + + = 6 P 5 P 3 P 2 P A B P1812916131181= + + + 2011 () 1 1' . & A 1. .... 150-151 2. .... 70 3. .... 22 4.. . . . . 1. f ( )2f x 3x 2 x , x = R ( )( )f 1 3 2f 1 3 2 = + = ( ) ( ) ( ) f 1 3f 1 3 2 3 3 2 = + = 3 2 9 6 4 12 3 + = + = = 2. =3 ( ) ( )3 2 2f x x 3x 4 f x 3x 6x , x = + = R ( ) ( ) f x 0 3x x 2 0 x 0 x 2 = = = =( ) ( ) f x 0 3x x 2 0 x 0 x 2 > > < > 2011 () 2 2 ( ] x ,0 f . [ ] x 0, 2 f . [ ) x 2, + f . x0=0 f(0)=4 x0=2 f(2)=0. 3.: ( ) ( ) ( )3 2f 3 h 3 h 3 3 h 4 + = + + + ( ) ( ) ( )3 2h 0 h 0f 3 h 4 3 h 3 3 hlim limh h + + += = ( ) ( ) ( )( )2 22h 0 h 0 h 03 h 3 h 3 3 h hlim lim lim3 h 9h h + + += = = + = ( )3 2f 3 3 3 3 4 4 = + =()2f 3 33 63 9 = = . ( ) ( ) ( )( )h 0 h 0f 3 h 4 f 3 h f 3L lim lim f 3 9h h + + = = = = (3,4). (3,4) ( ) f 3 9 = y=9x+. 4 93 23 = += A y=9x23. 4.( ) f x 6x 6 = ( ) f x 0 6x 6 0 x 1 = = =( ) f x 0 6x 6 0 x 1 > > > y=f(x) x ( ) ( )1,f 1 ( ) 1, 2 . 2011 () 3 3 1. 35 55. c c35 c 55 2c 55 35 c 102 2+ + + = = = . 4 [20,30), [30,40), [40,50) [50,60). 2. 1 412 4 = =1 2 3 4 2 340 24 + + + = + = (1) 2 31225 35 45 455x40 + + + = 2 3 2 3300 35 45 220 1440 35 45 920 + + + = + = (2) (1) (2) ( )2 3 2 32 3 2 324 45 45 108035 45 920 35 45 920+ = = + + = + = 2 210 160 16 = = 38 = . 3. [ )xiifiFiFi% [20,30)25120,30,330 [30,40)35160,40,770 [40,50)4580,20,990 [50,60)5540,11100 401 (30,30), (,50), (40,70) 50 30 2030 30 = = 2011 () 4 470 30 40440 30 10 = = = 204 4 140 3530 = = = = . 4.( ) ( ) ( ) ( ) PA B PB A PA B PA B + = = ( ) ( ) ( ) PA PB 2PA B = + (1) ( ) ( ) ( )( ) ( ) ( )( ) ( )( )P A 0, 25 1 P A 0, 25 P A 0,75P A P B 1,1P B 0, 65 1 P B 0,65 P B 0,35+ + ( ) ( ) ( ) ( ) P A P B 2P A B 1,1 2P A B + ( ) ( ) ( )( ) (1)P A P B 2P A B0,55 P A B2+ ( ) ( )( )P A B P B A0,55 P A B2 + s sCV 0, 25x x= =1.( ) ( )2f x 12x 2x 2s x = + f x0=1 xx 0. ( ) ( ) f 1 0 12 2x 2s 0 x 2s 6 x 6 2s = + = + = = s s0, 25 0, 25 s 0, 256 2sx 6 2s= = = ( ) s 0, 25 6 2s s 1,5 0,5s s 1 = = = s=1 x 6 2 1 x 4 = =. f: ( ) ( )3 2 3 2503f x 4x x 2s x s 4x 6x 20130, 25= + + + = +. : ( ) ( )2f x 12x 12x 12xx 1 = = 2011 () 5 5 f f(0) = 2013 f(1) = 2011. ( ) s 0, 25 6 2s s 1,5 0,5s s 3 = = + = . 2.y x c = + . y x c 4 c = + = + ys s 1 = =ys1CV 0,1 0,1 0,1 4 c 10y 4 c + + 4 c 10 c 14 + c>0. 4 c 10 c 6 + c 6. 3.( )2 1PA4 2= =( )1PB2 5= x 4 = =( )1 1PB24 5 3= = . i. ( ) ( ) [ ] , 0,1 = = ( ) ( )1 19 9 = = (1) ( ) ( ) ( ) ( ) = + 1 1 52 3 6 = + + =(2) ( ) 22 25 1 5 1(1) 18 15 2 06 9 6 9 = = + = 1 22 13 6 = = ( ) ( )13 16 =( )16 = 5 1 2(2)6 6 3 = =( )23 = ( ) ( ) ( ) ( ) = + = 2011 () 6 6( ) ( ) ( ) ( )1 1 51 13 6 6= + + = + = ( )56 = iii. x 4 = , s 1 = x 2s 2 =: x 3s x 2s x s x x s + x 2s + x 3s + 12 3 45 6768% 95% 99,7% xi , xi 2 100 952,5%2= . 1005 2002,5 = = . (....) 2012_3.3() : 1 6 : : & / : 1 2012 1. . 151. 2. . 14. 3..84(T , . 4., , , , . B1. 250 =250 . c 5 R 2045 5= = . 10 x2=10 [,+c) [+c, +2c) :22 4 810 2 12 20 4.2 2c cx + + + + + += = + = = 2% f a = 1 3 4 53% 3 ,% ,% , %2 10 5a a af a f f f = = = = . 1 2 3 4 53% % % % % 100 3 100 20.2 10 5a a af f f f f a a a + + + + = + + + + = = : (....) 2012_3.3() : 2 6 xifi%fiiNiFi%Fixi i [4-8)6600,60150150600,60900 [8-12)10200,2050200800,80500 [12-16)14100,1025225900,90350 [16-20)1860,0615240960,96270 [20-24)2240,04102501001220 10012502240 i i i f = . 2. 5i122408, 96 250iixx== = = , 896 . Fi%. Fi% 100 - 96- 90- 80- 70- 60- 50- 40- 30- 20- 10- 48 12162024 x . i 4 50 0 4 5 4 4 3, 33 7, 338 4 60 0 6 = = + . x> . (....) 2012_3.3() : 3 6 3.130013 16 1316 123 4 5 , 310 6 4% 17, 5%4 + + = 17, 52850000 498750100 =. 4.8640 864072012=, 7,2 .i.4-7,2 7, 2 4 3, 20, 808 4 4= ==80% , 0,80 60 48% = . 48%. ii. 10025 200 15 400 10 2500 3000 4000 9500 + + = + + = 80150 120100 = 1 . 950079,16120= . 1. f(x) :( 0- 4 0). x x f=[0,4) (4,+). g(x):( 00) x x > x>0. g=(0, +). 2. ( )( ) ( )4 4 4 43 23 6 3 3lim ( ) lim lim lim P(A)4 4 2 2 2x x x xxxf xx x x x = = = = = + +. : 2P(B) 1 1 2P(B) 1( ) 216 8 2 2xg x xx x x x = + + = + + 2P(B) 1 4(4) .4 4 8g = + + 4= 2 ( ) 1 1 2 ( ) 1 1= 1 (4) 1 ( ) .4 4 4 2 4 4 2P B P Bg P B = = + + = = = (....) 2012_3.3() : 4 6 3.2 1( ) ( ) ( ) 3 2PA B P B = > = . 1( ) ( ) ( ) ( ) ( )6PA B PA B = = + = 3 1 1 9 6 2 1314 2 6 12 12 12 12= + = + = >. 2( )5PA B = . ( ) ( ) ( ) ( ) ( ) 1 ( ) ( ) PA B P P B PA B PA P B PA B = + = + = 1 2 1 2 9( ) 1 ( ) ( ) ( ) 12 5 2 5 10P P B PA PA B = + + = + = + = . .[ ] ( ) ( ) = P A B B A( ) ( ) ( ) ( ) ( ) ( ) = + = + = P B P B A PA PA B P B PA B3 2 1 2 3 1 4 15 10 16 94 5 2 5 4 2 5 20 20 20 20= + = + = + = . 1. 3 2( ) 4 4 4 ( 1) f x x x x x = + = x -10 1 + -4x+ + x2-1+ + f(x) + + f(x)

( , 1],[ 1, 0],[0,1],[1, ). f f f f +x1=-1( 1) 2 f = x3=1(1) 2 f = x2=0 f(0)=1. (....) 2012_3.3() : 5 6 2.i): 0 ( ) 1 f [0,1]. :( ) ( ) (0) ( ) 1 1 (A) 2 0 (A) 1 f f f ()=1 =. :( ) ( )4 2 2 2P(B) P(A) P (B) 2P (B) 1 1 ( ) 2 ( ) 0 f P B P B = + + = = ( ) 0 P(B)= 2 P B= . 0 ( ) 1 P B ()=0 B = . 2.ii.) = = : 1 1 10 ( ) 1 0 2 ( ) 2 0 2 1, < < < < < < = ()=12 , , :2 2 2( ) ( ) 1 4 ( ) 4 ( ) 4 2 4 3, < < < < < < = ()=34. :xii 11 23 35 41 =10 ) 5 63 3 32 2t t + += = =. (....) 2012_3.3() : 6 6 ) = 1( ) ( )2 = = ( )3 ( )4 = = =