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高考完全解读·理科数学·课标本·16版(答案)_图文

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2 1 D  【 N={ x | x - 3 x + 2 }={ x | 1  】 ≤0 ≤x ≤

 

x } , A = { x | 0< x } , . ≤2 ∩瓓 ≤2 $; C RB 7 A  【 x | x< 2 m } , { x | x  】 > B={ 1瓓 ≥ RB= 2 m } . ∵A , ∴2 m , ∴m , . ?瓓 ≤2 ≤1 $; A RB 8 D  【 1 , 1 ) , ( 1 , 2 ) , ( 1 ,  】 &' BT78L ( 4 ) , ( 2 , 1 ) , ( 2 , 2 ) , ( 2 , 4 ) , ( 4 , 1 ) , ( 4 , 2 ) , ( 4 , 4 ) , } 9W. 9 A  【 ∵ B= { x | ( x + 2 ) ( x - 1 )< 0 } , ∴ B= 】 { x | - 2< x < 1 } . ∵A= { x | x } , ∴A ≥1 ∩B= ?. x < 1 0C  【 P= x 0< 】 ~5②, ? M =
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2 } , { 0 , 1 , 2 } , { 1 , 2 } . ! M= "# M ∩N= 2 C  【 | x - 1 | < 2 2<x - 1< 2 , 1< 】 ?- $- x < 3 , (- 1 , 3 ) . %&' A= ()*+,+-./, [ 1 , 4 ] . [ 1 , 3 ) . 01&' B= "# A ∩B= 3 B  【 M∪ N2345 M 645 N-789 】 {- 1 , 0 , 1 , 2 } , . :-&', $ M∪N= ;B 4 C 5 C  【 { x | - 3< x < 3 } , { x | x 】 <= A= 瓓 ≤ RB= - 16 x>5 } , )={ x | -3<x< "# A ∩( 瓓 RB 3 } x | x 16 x > 5 }= { x | - 3< x 1 } . ∩{ ≤- ≤- 6 C  【 4 } i = 4 ,  】 > M∩ N={ ?4 ∈ M, "# z z =- 4 i , . ;C 7 C  【 ={ x | x 2 } , x | - 4 】 瓓 ≤- ! T={ ≤ RS x } , ) { x | x } , . ≤1 $( 瓓 ∪T= ≤1 ;C RS 8 B  【 x | x>26 x<0 } ,  】 @A A={ B B= { x | - 5< x < 5 } , { x | - 5< x < 06 "# A ∩B= 槡 槡 槡 2< x < 5 } , ACDE; A , BCFG; AH B ∪ B=R 槡 , C D . . IJKLMNOP CH CQDE $; B 9 B  【 < y < z , y < z < x , z < x < y 】 RST x UL , y , z VW:XYZ x [\]^_-`WFa+, 0 = 1 , y = 2 , z = 3 , w= 4k ]ij x bcdefgh, 2 , 3 , 4 ) , ( 1 , 2 , 4 ) , y , z , lRm, n( ∈S ∈S oB ( w ) , ( x , y , w ) ∈S ∈S :X. 1 0A  【 x | x 0 ) , } ,  】 j T={ ∈ (-∞, nx ∈Z V= { x | x 0 ,+∞ ) , } 0 } , ∈( nx ∈Z ∪{ 01 TO VO5pfqr, { , 5pf]qr, !j T= s+} V= { } , T , V , B , t+ 01 O5pfQqr $uv C , D , . ;A
2 0 1 6 Ⅱ.

     

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1 A  【 】 o??R-???, >5

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x 2 D  【 ∵ ?x , e > 0 , ∴AD; ∵ ,+ y =  】 ∈R x 2 x 2 = x 2 , 2 ) , ∴B 2 Hy L??, ?? ( ?? 2 =x,

∵?a = b = 0?, a + b = 0 , D; B 0????m?, ∴CD; a > 1 , b > 1 , b > 1 , >]_?-./0? a ∴DFG. 3 D  【 =- b , a | =| b | ” 】 ?R“ ?a || -?? a | =| b | , =- b ” , . R=“ ?| |a $; D 4 A  【  】 ? R ? ? ? ? R - ? ? R - ? f.
2 2 “ a + b + c = 3 ” a+ b + c ” , “ a +b + -??[“ ≠3 2 2 2 2 c ” a +b +c < 3 ” . ≥3 -??[ “ ()??R[ “ , ” , , ” , ?p |q |???R[ “ ? ?p | ?q $? 2 2 2 + b + c = 3 , b +c ” R“ ?a | a+ ≥3 -??R[ 2 2 2 “ + b + c , b + c < 3 ” . ?a ≠3 |a +

1 B  【 x | , 1 , x | 】 >| ≤1 1- ≤x ≤1 % B={ - 1 } , { x | 0< x } . ≤x ≤1 "# A ∩B= ≤1 2 B  【 A= { x | - 3< x < 3 } , B= { x | 】 >w?1, x } , { x | x < 3 } . ≤2 xb+y0? A ∪B= 3 B  【 A 3 , 4 } , ( A ) hz 】 {Rm1, ∩ B={ ∩B ∪ C= { 3 , 4 , 5 } , . ;B 4 B  【 ∵ M ={ 1 , 2 , 3 } , ∴瓓 { 4 , 5 , 6 } ,  】 UM = ∴( ) { 4 , 5 } . 瓓 ∩N= UM 5 C  【 = { 3 , 4 , 5 } , ) 】 {Rm1瓓 |( 瓓 ∪B= UA UA { 2 , 3 , 4 , 5 } , . ;C 6 C  【 = { x | x <- 46 x > 0 } , A= { x | 0 】 瓓 ≤ RB

5 ①④ 【  】 ~ ①, ]i? P=??¤?V?, P A | +| P B | A B | =| A C | +| C B | , , <= | ≥| $ C[ A B , C-T??. t B C-§¨??, ~②, ? C[ R △A B-T?= D . D A | +| D B| +| D C | = ?? A 5[ | 3 | D C | . C A | +| C B| ?| ≤



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、 B 、 C 、 D[??-?W?, P[??¤ ③, ]i? A B?-??, ∴| P A | + ? O= P°§± A ?V?, P B | +| P C| +| P D| O A | +| O B| +| O C | + | ≥| | O D| | B C | +| C D| +| A B| . ≥2 > P-?m.?, C?%0j_5?, ?? O?·°±? B $ ③ D. B C D- ~ ¨ ± A C , B D^ ? 5 O ~④, ??? A P[??¤?V?, P A | +| P C| A C | , ?, >5 | ≥| | P B | +| P D| B D| , ∴| P A | +| P C| +| P B | + ≥| | P D| A C| +| B D| =| A O| +| O B| +| O C | + ≥| O D| , | %O=????W??-?V-T??. a b 6 A  【 = , $%】 >F???, 1 $a ≤b ? s i nA s i nB s i nA i nB , . ≤s ;A 7 B  【 l n ( x + 1 )< 0 x + 1< 1 1< x < $%】 ?0< ?- 0 , 1 , 0 ) 0 ) x < 0 ” B(- [ (-∞, -??&, "# “ l n ( x + 1 )< 0 ” [“ -??]????. 8 A  【 = 1 , : y = x + 1H?^?5 $%】 ?k |§± l ( 0 , 1 ) , (- 1 , 0 ) A B-?? S ??, "# △O A B= △O 1 1 × 1× 1= , k = 1 ” A B-??= "# “ ?“ △O 2 2 1 1 ” ; A B- ? ? = , , ? △O | k= ±1 "# 2 2 1 “ A B-??= ” k= 1 ” , k= 1 ” △O ? "# “ [ /“ 2 1 “ A B-? ? = ” △O - ? ? B ] ? ? ? ?, $ 2 . ;A 9 A  【 B C D=?? ” , $% 】 ?“ ??? A |~¨± “ A C D ” A C D ” ⊥B :X; B?~¨± “ ⊥B :X, | “ A B C D ” , “ ??? ?L0?=??_ "# ??? A B C D=??” A C D ” [“ ⊥B -??]????. a b 1 0C  【 , b???-?? $%】 , ??[H a | a || b | a b = 1 aH b-??^?, > Ba ∥b ??, | a | | b | a . ?, Hb -???0?^?. $; C
2 0 1 6 Ⅱ.

4 B  【 , $%】 <= “ ?[ ” -??[ “ ]?[ ” "# “ , b ?a ?[t+, | a+b[t+ ” -? ?R[ “ , b + b . . ?a ]?[t+, |a ][t+” $; B 5 D  【 , $%】 ~5 A ??5?V??-??§±^?6???, <?[? ??OP0?[??、 , R; ~5 B ???5?W^\?§-??¤-? , <?[?R; ~5 C § ?§±0?[??-, , ±b 0??5?? α¤, <?[?R; ~5 D , §± a H?? βKL?}?, <? a ∥β [?? . R. ??"?, ;D
2 6 B  【 1 , 2 ) , x -a ” $%】 ??1“ ~?m x ∈[ ≤0

, ∴ a> 4[?R=?-V =??R, ?à? a ≥4 . W??]???? 7 A  【 p : | 4 x-3| x-3 , $ %】 ≤1 ? -1 ≤4 ≤1 ∴ 1 2 ; q : x -( 2 a+1 ) x+a ( a+1 ) ≤x ≤1 ≤0 ? 2

( x - a ) [ x - ( a + 1 ) ] , ∴a + 1 . ≤0 ≤x ≤a >Rm? p [q -??]????, 1 a ≤ , 1 2 |0 $L ≤a ≤ . 2 a + 1 , ≥1

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8 A  【 “ , ” “ $%】 {Rm?, ??p |q [?R, ? q , ” , ” |?p [??R, "# “ ? ?q |p [?R, “ , ” ?p |?q [??R, $p [ ?q -??]?? ??. 9 A  【 “ p , q $%】 ∨q [??R ” ?p Tá?LVW p ?p[? "# “ ∨q [??R ” ? [??R, /“ , “ ?p p , ?p [?R ” ?“ [??R ” "# “ R” p , p [?R” ?“ ∨q [??R” "#“ ∨q [?? ?p[  ? R ” R” [“ - ? ? ] ? ? ? ?, $ . ;A 1 0D  【 A 、 B 、 CQ]?1? α . $%】 {Rm, ⊥β ~ , , m , , >l ∥m ⊥ β1 l ⊥β !l ?α <?L α ⊥ 5D . . β ??"?, ;D    &'()* 
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1 D  【 “ , y\ = ? +, $ %】 ① - ? ? R: ?x | x y = 1 ” “ [??R; ② -??R: ??]^_-` ¨?][?_`¨? ” [??R; ③ -???R: 2 2 x + m= 0KL?+h, 1 ” “ ?x - | m> [?? R; ?R④[?R, "#?-???R?[? . R, $; D 2 A  【 d = b c , b , c , d{?:_ $%】 ①a ]V?? a = d =- 2 , b = c = 2 , ?+?, ?j a $ ① DE; ②? a b a = 2 , b =- 4 1 1 , , b ?, < 1]× > $a ??? b a ( | x | )= l o g x | , f ( | - x | )= ]FG, $②DE; ③∵f 2| l o g - x | = l o g x | , ∴f ( | x | )= f ( | -x | ) . $③F 2| 2| G. $]FG-L①②. 3 D  【 “ , ” , $ %】 ?p |q -???R[“ ? ?q ” . . |?p $; D

1 C  【 $% 】 ???R-??[????, ??? , | x | + x≥0 ” ?è?~éê?, $?R“ ?x ∈R 2 , | x + x 0 ” , . ?x ∈R $; C ??=“ 0 0| 0< 2 C  【 , $%】 ????è “ ?” ê=ì°?è “ ?” . í????#??, $; C 3 B  【 : $% 】 ???R-??=c??R, "# p > 0 , x + 1 ) e> 1-??[ ?p : 0 , ?x ?L( ?x 0> x 1 ) e≤1 . ?1( 0+ 4 D  【 $% 】 {Rm, ?R p [??R. > x>2 ? x > 1 , > 1 x > 2 , x > 1 ” x > 2 ” Bx ? <? “ [“ -? / ?]????, $?R q [?R, | ?q [?? p ?q ) . R, ∧( [??R, ;D
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2 2 5 D  【 4 a c x +b x +c , 】 > b- ≤0?]? a ≥0

?[<= a -ò?]G?, "# A]FG; ? b= 2 2 0?, > c b > c b , “ >a ?]? a "# B]FG; ~ 2 , ” , ∈R L x≥ 0 -??[“ ì° x ∈R L ?m x 2 x< 0 ” , . "# C]FG. ;D 6 B  【 】 >Rm0??R p =?R, ?R q = ∴( ?p ) ??R. ∧q =??R. 7 C  【 ∵p p 】 |?p 1 [??R , 1 =?R ; 2[ ∴q : p | ?p ∨p ?R, 2 =??R , 1 1 2 [??R , q : p p , q : ( ?p ) p , q : ∧ [?R ∨ =?R 2 1 2 3 1 2 4 p ( ?p ) . ∴ q , q , C . ∧ =??R ??R[ ; 1 2 1 4 8 C  【 , qQ =  ? R,  】 ?R p $p ∧ q=  ?R. 9 D  【  】 <???R-??[c??R, $? : , 2 x . . -??=?p ?x ∈A ?B $; D Rp 1 0 - 4< m<- 2  【 f ( x )= m ( x - 2 m ) ( x + m+ 】 3 ) =ó?,+, , f ( x )< 06 g ( x )< 0 , ?~?x ∈R ????±÷ 0 . ??ù, % m< ?ú? 1"3.



?[FG-; ~5③, $?[]FG-. ??"?, . FG?R-%?[②, ;B 2 4 B  【 A T?R-??é= “ , x +  】 ?x ∈R 2 x + 3 ” ; CTé=??]????; D T= ≥0 ?R.
2 5 B  【 , = 0?, x = 0 , . 】 ~5 B ?x $; B 3 4 6 B  【 x , < 06 x > 1 , ∴?R p 】 ? x< |x =

- i nx - c o s x = 2 s i nx ?s ?R; 槡

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7 B  【 : y = f ( x )= l n [ ( 1- x ) ( 1+ 】 ~5?R p x ) ] , 1 - x ) ( 1 + x )> 0 , 1 < x < 1 , ∴,+ f ( x ) &( 1- 1 , 1 ) , ∵f (-x )= -?? ' = (- O5??~?, l n [ ( 1+ x ) ( 1-x ) ]=f ( x ) , ∴,+ f ( x ) =t, ∴?R p= ? ? R; : y=f ( x )= ~5?R q +, x e - 1 , ( x ) , ,+ f -? ? ' = R O 5 ? ? ~ ?, x e + 1 1 1 - x x - x e - 1 e 1- e ∵f (- x )= -x ( x ) , = = x = -f e + 1 1+ e 1 1+ x e ∴,+ f ( x )= s , +, ∴ ? R q=  ? R. ∴( ?p ) . ∧q [?R, $; B 8 D  【 ( a+ 1 )- 】 ??§±??, |?kl a , =- 36 a= 2 , 2??§ 2× 3= 0 h1 a ?? a= 3 , ±(', "#?§±??? a=- "#?R p = ?; ?)? `? ]° ? ? β- ? *, |]??? , . α ∥β "#?R q =. $; D 9 B  【 0?, s i n( )= s i nα+ s i nβ , 】 j α= α+ β A [ ? ? R;j φ = x + s i n2 π (x )= ?,, + f 2

ú? 1 f ( x )= 0-?( x = 2 m , x m- 3 , x nx 1 2 =- 1- 2= 3 m+ 3 . x , 1?, 2 m< 1 , ①? x % m>- ??ü( x 1> 2 1= 1 % m< . 2 x , 1?, m- 3< 1 , ②? x % m<- ü( x 1< 2 2 =- 4 , 4< m<- 1 . % m>- $- x , 1?, x x 2< 1?k ③? x % m =- 1= 2 1= 2 =- l??. ∴kl??( 1 ) 4< m< 0 . - m-je??= - 2 ) - 4 ) ,f ( x ) g ( x )< 0 , kl??( ?x ∈(- ? ∞, 4%0. àkló?,+-?(?5 - 1?, m- 3<- 4n m< 0 , ①? m>- ?( x 2 =- 1?, 2 m <- 4n m < ?h. ②? m<- ?( x 1= , 2 . 1?, f ( x )= -( x + 0 h1 m<- ③ ? m= - 2 2 ) ∴]kl( 2 ) . ≤0!:X, ∴kl( 1 ) ( 2 ) 4< m<- 2 . - m-je??[ -
2 0 1 6 Ⅱ.

π = c o s 2 x B[?R; [t,+, ~5 2 3 2 ( x )=x +a x +b x +c , `?,+ f ?x → -∞ ?, y y ( x ) ?x !f ° R? ∞, →- → +∞ ?, → +∞, 3 2 , x a x b x $ ?x ∈R -?+[,-].-, 0 0+ 0+ 0+

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1 A  【 “ 】 ???R-??[c??R, ?” ê , “ ” , . =“ ?” ≥” ê=“< $; A 2 D  【  】 ? R [ " # ? è - ? ? ? R. $? . ;D 3 B  【 p ” , q 】 ~5①, >“ nq =?R1 p T á?LVW[?R, <? ① ]FG; ~5 ②, $

2 c = 0 , C[??R; ( x )= 0?, l n x + l nx - a = ?f 2 1 1 1 2 + nx 0 , = l n x + l nx = l - ≥- , |L a 2 4 4 2 a > 0 , f ( x ) = l n x + l n x - a D "#? ,+ L / ?, . [??R. ??0?; B 1 1 0t >-  【 ?p 】 =?R, |p =??R. ] 2 5 2 , - h, x + 2 x-2> 0L 45 1 > _? t %t 2 5 2 2 5 2 , ¤-h. x < ?, < L° 1 ? 1< 2 - 2 x 2 5 x 2 1 1 2 1 2 1 - <1 , =2 - "# 2 - ∈ x 2 x x 2 x 1 1 - , 0. >- . $t 2 2

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1 1 x 2 = , =- 1 ; > 0 , l o g x | = , = %x ?x || %x 2 2 2
1 2 2 = 2 6x

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3 ①③ 【 x , y ) , x , y ) $%】 ~5?m( ∈M ì°( ∈ 1 1 2 2 M , x y y 0 , ?1 x ? 8 [~5? + ??mV 1 2+ 1 2< x , y ) , x , y ) , ?( ?9 °? + ?ì° : V? ( ? 1 1 2 2 0 ° . =l nx ??W?H???:- ; ¨ü5 9 °y 1 , 0 ) , -?+?j?( |]ì°:V????W? 0 ° , H???:- ; ¨ ü5 9 "# ② ]k l./ P ; , <?①③-?+=> 0?, ①③ ?kl./ P $;①③. 4 D  【 x $% 】 ??,+-hz?Lm?, àkl , ≠0 4n x . h1 x ≥- ≠0 {x x + 4 , ≥0 ( x + 3 ) l n 5 A  【 ∵f ( x )= , ∴ ??,+ f ( x ) $% 】 x 1- 2 槡 x + 3> 0 , 3< x < 0 . Lm?, à? %- x 1- 2 > 0 ,

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2 5 C  【 , 16 $% 】 > Rm01 x -x>0 h 1 x>

x < 0 , 0 ) 1 , "#"g,+-?? ' = (-∞, ∪( + . ∞) 1 12 61  【 + =- 4 × - $%】 >w?$1 f - 2 2 3 2=1 , , = !>,+-23= 2 01 f 2 1 = 1 f- . 2

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Ⅰ. 
 



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1 B  【 f ( x ) g ( x ) ( x ) g ( x ) $%】 =s,+, =t,+, $f f ( x ) | g ( x ) | | f ( x ) | g ( x ) =s,+, =s,+, =t | f ( x ) g ( x ) | . ,+, =t,+, $; B 2 B  【 =l nx $%】 <=~+,+ y -?? ' ][
- x R , ; $OPuv;C C <= *+, + y=e , % x 1 y = , ; °??' ¤? Q ??, $uv;C A e

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2 5 D  【 ( x )= x - 1? f ( x )= x + x 】 ,+ f R]

; uv;C A?;C B ; [t,+?][s,+, x - x - x x C f ( x ) =2 -2 , f ( - x ) =2 -2 = C T | x - x x - x ( 2- 2 )=-f ( x ) , ( x )= 2- 2 =s - "# f x - x ; ( x )= 2 + 2 , ,+, uv;C C ;C DT f | - x x x - x f (- x )= 2 + 2= f ( x ) , ( x )= 2+ 2 =t "# f . ;D ,+, 6 3  【 ( x ) = 2 】 <= f -?+O5§± x ~?, " ( x )= f ( 4 - x ) , f (- x )= f ( 4+ x ) , (- x )= !f #f f ( x ) , ( x )= f ( 4+ x ) , (- 1 )= f ( 4- 1 )= "# f |f f ( 3 )= 3 . 7 (- 1 , 3 )  【 2<x < 2?,  】 >Rm0?, ?- f ( x )> 0 . f ( x - 1 ) ( x ) -? + [> f -? + ? S ? ( x - 1 )> 0 , 1< T 1W??UV 1×-, ?f |- x < 3 . 5 8 ( x )  【 】 >5,+ f [ 23 = 4-s,+, 1 6 2 9 4 1 3 4- + + f = f 2× " # f 4 4 6 7 3 7 3 4- f2× f- = +f - = -f - 6 4 6 4 7 3 π 5 f s i n = . =- + 6 1 6 6 1 6 3 3 x 9 -  【 ( x )= l n ( e + 1 )+ a x 】 ,+ f =t, 2 - 3 x 3 x (- x )= f ( x ) , n ( e + 1 )- a x = l n ( e + $f %l +, 3 x 1+ e 2 a x 1 ) +a x , n 3 a x=l ne , @A1 l % x 6 x =2 e + e 3 x 1+ e 2 a x 3 x 2 a x + 3 x 3 x e , 1=e ( e + 1 ) , a?1 e + " 3 x 6 x= e+ e 3 a x + 3 x = 0 , =- . #2 h1 a 2

1 2 1 x+ x . 2 2 1 2 1 , 0 ]?,f ( x )= x + x= ?x ∈ [-1 2 2 1 2 1 1 1 x + - , =- ?, f ( x ) "#? x j1 2 2 8 2 1 J?e - . 8 2 ,  【 2 4 - ( x ) 】 >Rm0? f =s,+, n 3 ∴f ( m x - 2 )+ f ( x )< 00? °??'¤=A,+, ( m x - 2 )< f (- x ) , ∴m x - 2<- x . ?= f ( m )= x ·m- 2+ x , m 90O5 m-V?,+ g ∈ [- 2 , 2 ] , , 2 ] g ( m)<0! 0 1 ? m∈ [-2 ?, 2 (- 2 )< 0 , g ( 2 )< 0 , 2< x < . :X, %g h1 - 3 2 x + x + 1 =1+ 5 C  【 f ( x )= 2  】 ( ) R m, x+ 1 x x , ( x )= 2 [s,+, (-a )= 1+ Bh $f 2 x + 1 x+ 1 h (- a )= 1- h ( a )= 2- [ 1+h ( a ) ]= 2-f ( a )= 2 4 . 2- = , $; C 3 3 6 B  【 , f (- x )=-| - x | =-| x | = 】 ~5;C A
2 f ( x + 1 )= 2 f ( x )= x + x , ( x )= "# f

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f ( x ) , ( x ) , ,+ f =t,+, "# AD; ~5;C B 1+ x > 0 , 1<x < 1 , > 1- ,+?? ' O5?? 1- x > 0 (- x )= l g ( 1- x )- l g ( 1+x )=-f ( x ) , ~?, !f ( x ) , ,+ f = s , +, " # BF G ; ~5;C C - x x f (- x )= 2 + 2 =f ( x ) , ( x ) ,+ f =t,+, " 3 3 , f (- x )= (- x ) - 1=- x - # CD; ~5;C D 1 , f ( x ) f ( - x ) f ( x ) - f ( - x ) , f ( x ) $ ≠ n ≠ %,+ . R][s,+, ?][t,+, "# DD. $; B 7 C  【 = f ( x ) 】 {Rm1, ,+ y -? + O5§

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( x ) < ? , + y=f [ t , +, n ± x=0~ ?, f (- 2+ 4 )=f (-2 )+f ( 2 ) , ( 2 )=f ( 2 )+ %f f ( 2 ) , ( 2 )= 0 , ( x + 4 )=f ( x ) , "# f "# f %,+ y = f ( x ) f ( 20 1 4 )= [ # 4= 2 3 - 2 3 , +, f ( 4× 5 0 3+ 2 )= f ( 2 )= 0 , . ;C 8 D  【 ( x )= f ( x ) ( x ) 】 >f π- 1,+ f -? + π x ( x )=e +s i nx1 , + ° ~ ?. >f 2 π π - , ??Q ? A, ( x )=f ( ) >f π -x 1, 2 2 f ( 2 )= f ( 2 ) , f ( 3 )= f ( 3 ) , , 2 , 3 π- π- !1 π- π- π π ∴f ( )>f ( 1 )>f ( Q45 - , , π -2 π- 2 2 3 ) , ∴f ( 2 )> f ( 1 )> f ( 3 ) . 9 A  【 ( x - 2 )=f ( x + 2 ) , ( x + 4 )=  】 >f 1f f ( x ) , ∴f ( x ) T = 4 , f ( - x ) =- f ( x ) , - 23 ?' ( l o g 0 )=f ( 1+l o g 0 )=f ( l o g 0- 3 )= Lf 22 21 21 - f ( 3- l o g 0 ) , ∵ 3 -l o g 0∈ ( -1 ,0 ) , 21 2 1 1 4 1 3 - l o g 1 0 ∴f ( l o g 0 )=- 2 - =- - =- 1 . 22 5 5 5 = O5 x

2 0 1 6 Ⅱ.
2 1 C  【 ( x )=x +x][s,+;  】 ,+ f ,+

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1 0 C  【 f (- x ) = -f ( x ) , $ % 】{ R m, | 2- 2 - k ·2 k ·2 - x x , 2 -k ·2 ) · %( - x x= - x - x 2 + k ·2 2+ k ·2 x - x - x x x - x ( 2 + k ·2 )=( 2 +k ·2 ) (- 2 +k ·2 ) , 2 ∴k = 1 , k =± 1 , . ;C
- x x x - x

1 2 a= 9 . 4 7 C  【 < 0?, ( x )= $%】 ?a ?'?+01,+ f ( a x - 1 ) x | 0 , +∞ ) 0?, | °( ¤? Q ? A; ? a= f ( x )=| x | , ( x ) 0 , +∞ ) ?'?+01 f °ZJ( ¤ > 0?, ( x )= ?a ?'? + 01,+ f ?Q?A; 1 1 , ? , + | ( a x - 1 ) x | ∞ ¤= A ,+, ° 0 2 a a 1 1 , a ”[ “ ° ¤ = ? , +, $“ ≤0 ,+ 2 a a f ( x )=| ( a x - 1 ) x | 0 ,+∞ ) °ZJ( ¤? Q ? A ” -??????.

& ' ( # $ )   * + , -

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1 A  【 ∵,+ y = f ( x ) $%】 hfV: -? + O5§ = 1~?-????[ f ( x )= f ( 2- x ) , ∴x + ±x 2 m x + 1 = ( 2 - x )+ m ( 2 - x )+ 1 , m+ 2 ) x = @A1( m+ 2 , ∴m+ 2= 0 , 2 . % m=- 2 ∵f ( x )= x + m x + 1-~?y? X = x = hfó: m m - , ∴- = , 2 , . 1 % m=- $; A 2 2
x 2 A  【 ( x )= a $%】 ,+ f ° R?[?,+, _W a < 1 ( > 0n a ) ; ( x )= ( 2- 50< ò' a ≠1 ,+ g 3 a ) x 0 , 2 , ° R?[A,+, _ W 5 2-a> % a< a < 1 < 2 , a < 2 0< a < 1 , A . <= 0< ?a ? $; / 0 , , α> α ≤0 3 B  【 $ %】 > 6 2 1 α= -46 - 4 α = 4 , α= 2 , . α= $; B 2

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8 A  【 ∵f ( x ) ∴f ( 1 )=- f (- 1 )= $%】 [s,+, - 3 . 9 ( 1 ) ′ ( x )= 2 x +b . , $? f >R?, ~?m- x ∈R 2 2 2 x + b b x + c , ( b - 2 ) x + c - b ≤x + % x+ ≥0! : 2 b 2 ( b - 2 ) - 4 ( c - b ) , 1 . X, "# Δ= ≤0 oB c ≥ + 4 , 5[ c ≥1 nc ≥2

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4 ( 0 , 1 )  【 = f ( x ) $%】 ??,+ y -?+?ú? 2 k < 1?, ( x )= k "3. |? 0< O5 x -? X f L ?W]?-?(.

c + ( c - b )> 0 . 2 x+c ) -f ( x )=( 2 c -b ) x+ $? x ≥ 0?, L( c ( c - 1 ) . ≥0 2 f ( x ) x + c ) . %? x ≥0?, ≤( ( 2 ) 1 ) c b | , >| b | >( ?, ≥| |? c ?, L M≥ 2 2 2 f ( c )- f ( b ) c - b + b c - b c + 2 b . = = 2 2 2 2 b + c c - b c - b 1 b c + 2 b = , 1<t < 1 , = 2- , &t |- B,+ 1+ c b + c t 3 1 g ( t )= 2- (- 1< t < 1 ) ∞, . -e'= - 2 t 1+



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2 69  【 ∵f ( x )= x + a x + b 0 , + , $%】 -e'=[ ∞) 2 2 a a 0 - = , b = . ∴b 4 4 1 2 2 ∴f ( x )= x+ a x + a = 4

, + ∞ ). [3 2

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1 C  【 f ( x )= f ( x ) ( x ) , $%】 ≠x ()ó?,+ 1 2 1 2 x x 1+ 2 x = 0 , c ) -?+~?., [?~?y, B? ( 2 x x 1+ 2 = x x , c ) O5 x -~??( ?°ó?,+ 1+ 2 2 ∴f ( x x )= c , . -?+?, $; C 1+ 2 2 B  【 $%】 °?V§¨ [\ =| f ( x ) | = Pù?? y ?y a x-1- ? + ? ú ? 4" =a x - 1 >?+0?? y 3, 2 = x - 4 x Hy ^ ] ?ò'R 2 x=a x-1? L > x -4 m, 6 , = VWh1 a=- §± y

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x 6 A  【 ( x )= 2 + 2 x , ( x ) 0 ,+∞ ) $%】 ?f |f °( a b a 2 a= 2 + 3 b > 0 , ?=A,+, >2+ db 12+ b 2 a > 2 + 2 b , ( a )> f ( b ) , b , %f $L a> % ACF

g ( x ) 2 ) ??Q??, ° (-∞,- ?? Q ??, " = f ( x ) - 2 ) . °(- ??Q?A. ;D #,+ y ∞, 1 a 2 槡 1 0  【 2 o g , $%】 >w? 4 = ? a=l ! 4 2= 2
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( 1 , 1 ) ; +[A,+, uv A xb ( ??0#uv ) : 1 , ;C B o~+,+? +q 0<a< H p ,+? : 1 , +rs; ;C C o~+,+? +q a> Hp, +?+rs. 5 C  【 = l o g 】 xbTJ??tü?. <= a ∈ 2π
1 ( 1 , 2 ) , b =l o g 0 , c =π-2∈ ( 0 , 1 ) , π< "# a> 2 c > b .

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a c a c 6 B  【 , 1 0 =b , ∴5 = 1 0 ,  】 >w?1 5 =b d d c c d c a ∵5 = 1 0 , ∴5 = 1 0 , 5 , ∴d c = a , . |5 = $; B

, ∴c < a < b . - 2< 0 71  【 l g5 l g槡 2 0 = l g ( 5 ×槡 2 0 )= l g 1 0 = 1 . 】 槡+ 槡 8 C  【 ( x ) ( x + 2 )=f ( x ) 】 >5 f [kl f -t 0 , 1 ] f ( x )=x , ( x ) ,+, n? x ∈[ ?, $f [ 23 0"3, = 2- 23 ,+, ?? + ?ú? 1 (), = l o g x | ?[t,+, ?? + O5 y y~?, +y 4| . v$?w?u-??}L 6W, $; C

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0 ú? 1 1 1 1 1  【 =l o g =- 1 9 f , ∴ ff 】 2 2 2 3 2

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Ⅰ. 
 


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1 A  【 1"3, 】 ?ú? 1 °?V§¨ [\ Pù e, y x + 2 , y l nx , y x+ 3<? y 1= 2 =- 3= 4 =- y A , y y B , ? +, $1 y H -?? d H -?? 1 2 3 4 ú? 9
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x 2

1 A  【 a< 1 , a+ 1< 2 , 】 >w?1 0< "# 1< ( x ) ( x ) 0 , (),+ f - t , +, 0#7. f °( + ( a + 1 )> f ( 2 ) . ??Q??, "# f ∞) 2 22  【 ( a b )= 1 , b = 1 0 , ( a )+ 】 >f 1a 5[ f
2 2 2 f ( b )= l ga +l gb =2 ( l ga+l gb )=2 l ga b= 2 l g 1 0= 2 .

= x b = x . ef a A< B ∵f ( x ) ∴0= f ( a )< f ( b ) . ° R?=?QA,+, ( x ) 0 , + !∵g °( ?=?QA,+, ∞) ∴g ( a )< g ( b )= 0 . ∴g ( a )< 0< f ( b ) .

3 C  【 = l o g 】 <=,+ y °?? ' ¤[ A , 4x o g l o g , o g l o g , +, "# l !l "# a= 4 9> 46 4 9= 23 c > b , . $; C
2 4 D  【 ( x )=4-x  】 <=,+ f = t , +,

g ( x ) ( x ) ·g ( x ) [s,+, "#,+ f =s,+, , B . ??+ O5??~?, uv A !? x>0?, g ( x )= l o g , > 1 g ( x )> 0 , x < 1?, ?x ?, ? 0< 2x
2 g ( x )< 0 ; f ( x )=4-x , f ( x )<0 , ? x>2?, ?

1 ú? 1 2 D  【 ( x )= x c o s x + s i nx , 】 hfV: &f ∵f (- x )=- x ·c o s x - s i nx =- f ( x ) ,



∴,+ y = x c o s x + s i nx . =s,+, 0uv B c o s x + s i nx = 0 , a nx =-x , &x 1t °?V§¨ [ = t a nx =- x \PT<?,+ y ?y -?+?ú? 1 2"3, = x c o s x + s i nx >?0?,+ y - / ?L VWx5 π 、 C , . × πIJ, 0uv A $; D 2    3 ú? 1 4 ú? 1 8 B  【 f ( x )=g ( x ) $%】 L?W]^_-?(, % ( x )=| x - 2 | + 1H g ( x )= k x -?+L 2W ,+ f ??, <??ù:

& ' ( # $ )   * + , 2 ú? 1 ( x )=x c o sx+s i nx , (-x )= hf ó: & f | f - x c o s x - s i nx =- f ( x ) , ∴f ( x ) =s,+. ∵s,+-? + O5??~?, B BT? + ]O ∴ uv B ; = 5??~?, ?x π y = 1 , ?, B> C 2

5 ú? 1 = k x : y = >?0??§± y x5 l 1 IJ?, ò'Rm, % 1 x : y = x Hl 2 2

π = ?, y , ∴uv C ; = T?+?? x ≠1 ?x π?, 2 y =- , =π?, y > 0?, ∴ uv A , π B AT, ?x . $; D
x 3 C  【 , ; $% 】 >w ? 3 -1 ≠0 ?x ≠0 uv A ! 3 x x 3 > 0 ∵x < 0?, 3 - 1< 0 , x < 0 , ∴y =x , $u 3- 1 2 x x[ 3( 3- x l n 3 )- 3 ] ; ′ = , x l n 3< 0 !y ? 3- vB x 2 3 - 1 ) ( 3 x > , y ′ < 0 , . > 0 ?, "# D]ò'. $; C l n 3 f ( x ) f ( x ) f ( x ) 1 2 n 4 B  【 = = …= , = f ( x ) $%】 %y x x x 1 2 n =k x -?+H y -??- [\ kl??_?. ! , 3 , 4 . ??á??L?W, áyL?W, $n 0j 2 1 - 5 B  【 ( x )= l n ( x + 1 )- x , g ′ ( x )= $%】 &g x + 1 - x 1= , 1<x < 0?, g ′ ( x )> 0 , > 0?, ?- ?x + 1 x g ′ ( x )< 0 , ∴g ( x ) g ( 0 )= 0 , f ( x )< 0 , 、 uv A m a x= C , , . !>??'0uv D $; B 1 x 6 D  【 ( x )=a - , 1?, f ( 0 )= $%】 &f ? a> a 1 1- ∈( 0 , 1 ) , a< 1?, "# AH BQD; ? 0< a 1 f ( 0 )= 1- < 0 , . "# CD D~, $; D a

1 < k < 1 , . ;B 2

9 B  【 f ( x )=| c o s x | ·s i nx , $%】 >Rm?, ?x ∈ 1 π , ] ?, f ( x )=c o s x ·s i nx = s i n2 x ; ?x ∈ [0 2 2 1 , i n2 f ( x )=- c o s x ·s i nx =- s x , π ?, $ (π 2 ] 2 . ;B
2 0 1 6 Ⅱ.

1 B  【 6"3, $%】 ?ú? 1 °?V [\ PT < ? ( x )= ,+f

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4 x - 4 , x , ≤1

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+, 0?L 3W??.

7 B  【 $%】 >Rm?kl??-?,+? + ?L 3Hú? 1 4?z{|, ú? 1 3T, ′ , ú? 1 ? BO5??-~?? B )?0?: < 0?, x x 0 , y y 0 , ?a $ BFG. 1+ 2> 1+ 2< 4T, ′ , ú? 1 ? AO5??-~?? A )?0?: > 0?, x + x < 0 , y + y > 0 , C , D . ?a QD 1 2 1 2 6 ú? 1 2 (- 2 , 0 )  【 ( x ) 7" $%】 <? f -? +, ?ú? 1 = mH y = f ( x ) 3, ?'?+0?, ?§± y -?+ màkl - 2< m< 0 . L`W??,

1 0

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> 1?= A , +; ?=?,+, °x B ? ,+ y= l gt ( x ) < 1?=?,+, =A,+, "# f ° 0<x x > 1?=A,+; ( x ) < !,+ f =t,+, "# x - 1?, f ( x ) - 1<x < 0?, f ( x ) =?,+, =A, f ( x ) ( 1 )=f (- 1 )=l g 2 , +. -J ?e= f ?Jü e. ②⑤DE, ①③④FG. 9 B  【 ( x ) 0 , + 】 <=t,+ f °ZJ[ ?[ ∞) (- 1 )= 0 , ( 2 x - 1 )< 00@= A,+n f "# f f ( | 2 x - 1 | )< f ( 1 ) , 2 x - 1 | < 1 , |L | h1 x -j 0 , 1 ) , . e??[( $; B
2 1 0B  【 x - 2 )-( x - 1 ) , 1 】 &( ≤1 1- ≤x ≤

7 ú? 1 3 D  【 y  】 ?x 6DA ü?, ? 6DA ü, $y . ?x -AüBAü, $uv B = ù????z: ?x π ?, }U "~-? ~ ¨= 2

     

π O G= . 1× ∠F 0g1 l ?? T - hi = 1- 2 2 槡 1- 2 2 2 3- 6 π E= = 槡 槡. 1-槡 , c o s = $?? B 4 2 s i n6 0 ° 3 1 23 C= = 槡, = B E+ B C+ C D= !$? B $y s i n6 0 ° 3 2 3- 6 2 3 6 3- 2 6 槡 2 B E+ B C= 2× 槡 槡 + 槡 = 槡 . 3 3 3 2 3 槡 + 2 3 槡 6 3- 2 6 3 43 槡 槡 < = 槡, <= 3 2 3 ( x ) . "#,+ f -?+[ù??. $; D 4 D  【 , 】 >w?, ,+-?? ' = R "# BD ; , C ; x = 0 , f ( x )= E !,+=t,+ "# DE & l n2 , ≠0 "# ADE.
x 5 D  【 > 1 < c < 1 , f ( x )= l o g ( c + 】 ?? c ?[ 0 c

2 , ∴f ( x )=

- 2 , - 1 , ≤x ≤2 = f ( x )- c ?y {x x - 1 , x <- 16 x > 2 .



( x ) Hx yUL?W?}?, < ?,+ f -? + ? 8"3, 2 , - 1 ] ú? 1 ??+ c -je??[(- ∪ ( 1 , 2 ] .

8 ú? 1    &'()* !*
 

a f ( a )= , 2 t ) ?[?QA ,+, $éL | ? R0 b f ( b )= , 2 x x x x ( x )= , o g ( c +t )= , a@=g f %l % c+ c 2 2

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Ⅰ. 
x 1 B  【 9 ( x )= 2 | l o g x | - 】 ?ú? 1 "3, ,+ f 0 . 5

1 =| l o g | = -/?W+_W5,+ y H,+ y 0 . 5x
- x 2 ?+-??W+. >? + ?, L?W??, "#

t = c L?W]^_-?+(? t -je???R, m ( m> 0 ) , =c 0@= t = m- m, &c = | c +t 2 = t = m- m ( m> ?R?V?0a@=g,+ y Hy 0 ) -?+ L?W??? t -je?? ? R, ?' 1 , . ??01 t ∈ 0 4 l nx 6 D  【 1?, y=e -( x- 1 )= 1 ;  】 ? x> ? 1 - l nx 0< x < 1 y = e - ( 1- x )= + x - 1 . ?, ?'? x . z0?, ;D
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1 2 2 B  【 x 】 °?V??§¨[\ P¤?? y 1=

2 7 B  【 ( 4 ) g (- 4 )= a × l o g 4< 】 ()Rm> f a x - 2 0 , a < 1 , = a ( 0< a < 1 ) 10< <?*+,+ y , A , C , y =l o g | x | ( 0< a < B ?+%0G? uv a 1 ) . -?+?',+-?Q.0?, $; B 2 1 x + 1 x | + 8 ①③④  【 f ( x )= l g =l g| ,  】 | x | | x |

Hy 2=

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2 ( x )=x - +? + ? L V W ? ?. <?,+f

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?L 1W/?.

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> 0?, f ( x )= 2 x - 6+l nx , ′ ( x )= 2+ ?x <= f 1 > 0 , ( x )= 2 x - 6+l nx 0 ,+∞ ) "#,+ f °( x ( 1 )= 2- 6+l n1= - 4< 0 , ?? Q ? A, <= f f ( 3 )= l n3> 0 , ( x )=2 x- 6+l nx° " #, + f ( 0 , + ( x ) ?Ln?LVW / ?. ??, ,+ f ∞) . -/?W+= 2 1 1 1 1 4 4 3=e =e 4 C  【 + 4× - - $% 】 <= f 4 4 1 1 1 1 2 2 2< 0 , f = e +4× -3=e - 1> 0 , "# 2 2 1 1 x , . f ( x )= e + 4 x - 3-/?"°ZJ= 4 2

,+L 2W/?. 2 B  【 s i nπ x - x + 1= 0 , s i nπ x =x - $%】 &2 |2 1 , ( x )=2 s i nπ x , g ( x )=x-1 , ( x )= &h | f 2 s i nπ x - x + 1- / ?W+ ? R a @ =? W,+ h ( x ) ( x ) h ( x )= 2 s i nπ x Hg ?+-??W+?R. -J ?F 23 = T= 2 π = 2 , <??W,+ π 2" -?+, ?ú? 2 ∵h ( 1 )=g ( 1 ) , 3, 5 5 h > g , 2 2 g ( 4 )= 3 > 2 , g (- 1 )= 2 ú? 2 , ∴?W,+?+-??V}L 5W, ∴f ( x )= - 2 2 s i nπ x - x + 1-/?W+= 5 .

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5 B  【 $% 】 °?V [\ PT ? ? < ? , + f ( x ) , g ( x ) -?+?ú 2 1 , ( x )= ? "3 ?X f g ( x ) L?W]^_?(_W5?W,+?+ L ? W ] ? - ? ?'? + 0?, ?§ ?, 1 ú? 2 = k x 2 , 1 ) ±y -? ? ü5 [\ ??H? ( , ±=x - 1-? ? ?ò'Rm, $ ?? n?5§± y 1 k < 1 . < 2 6 C  【 ( 1 )= 6-l o g 6> 0 , f ( 2 )= $% 】 <= f 2 1= 2> 0 , f ( 4 )= 3- l o g 2 2= 3 1 - l o g < 0 , " 2 4=- 2 2 ( x ) 2 , 4 ) , . #,+ f -/?"°ZJ=( $; C

3 D  【 3" $ %】 ?ú? 2 [ f ( x ) ]+1=01 &f 3, f [ f ( x ) ]=- 1 . > 0?, ?k °??§¨ [\ Pù < ? ( x ) -ü n ? +d § ,+ f =- 1 , = ±y c mק± y - 1H,+ f ( x ) -? + L

3 ú? 2

2W??, 、 t , 0 , 0<t ???[\??[ t |t 1 2 1< 2 < 1 ; = t = t , k<?§± y ?'? + 0?, § 1H y 2 = t ( x ) ±y -?+L 2W]?-??, § 1 H,+ f = t ( x ) ±y -?+L 2W]?-??, < 2 H,+ f =f [ f ( x ) ]+ 1L 4W / ?. ???,+ y ??, ? k < 0?, = f [ f ( x ) ]+ 1L 1W / ?, ,+y ?' m . ;C?, ;D
2 4 D  【 ( x ) =x -a x+1 ° Z J $ % 】< = f

7 A 8 ( 0 , 1 ) 9 , + ∪( ∞)
2 9 B  【 ( x )=( x - 2 ) + 1 , $%】 >w? g "#??

2 , 1 ) , ( 2 )= 2 l n 2 1 , 2 ) , 2 , 1 ) ?=( !f ∈( 0??( ( x )= 2 l nx ( x )= ?5,+ f ?+-ù?, $,+ f 2 2 l nx ( x )= x - 4 x + 5-? + L 2 -?+H,+ g W??.
3 2 1 0A  【 ( x )=x +a x +b x +c $%】 <=,+ f L?

, 3 ? L / ?," # x - a x +1 =0 ° (1 2 ) 1 , 3 ?Lh. a x + 1= 0 , = x + , > x- 1a ? (1 2 ) x
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1 g ( x )= x + , ( x ) 1 ,+∞ ) , (-∞,- 1 ) |g °( x , 1 ) ??Q? A, ° (-1 ? ? Q ? ?, <= x < 3 , ( x ) "# g ° 1 < 2

, x , ′ ( x )= W?e? x 0?O5 ? ,+-? X f 1 2
2 3 x + 2 a x + b = 0L?W]_-?( x , x . |? X 1 2

3 ( f ( x ) )+ 2 a f ( x )+b = 0L?W]_-?(, % f ( x )= x ( x )=x , ?? X (-W+ 8 [?? 16 f 2 ( x )= x ( x )=x W?X f 1? f 2 -]_?(-W+ = f ( x ) I?, k?'?+0q?,+ y -? + H§ = x = x W]?-??, $" ±y 1 ?§± y 2 }L 3 g?XL 3W]?-?(.
2 0 1 6 Ⅱ.



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??Q?A, "#? 1 0 , . "# a ∈ 2 3

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5 B  【 ( x )= l o g + x - 2-? $%】 hfV: ,+ f 3x 0 ,+∞ ) , 0 ,+∞ ) ?'=( ín° ( ?? A, ?+ ( 1 )= -1<0 , f ( 2 )=l o g 2>0 , ,-] ., !f 3 ∴,+f ( x )=l o g +x - 2L?V- / ?n / ? 3x 1 , 2 ) °ZJ( ¤. =l o g =-x + 2-? + hfó: ??,+ y Hy 3x ( , 1 , 2 ) ?#) ] ?q ????- ?[\ ° Z J ( . ¤, $; B

1 C  【 =| l o g | - $%】 &y 2x


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= 0 , l o g | = %| 2x


, l o gx | °?V [ \ P ù ? ? y=| ? y= (1 2) , -?+( ?#) $???+L 2W??, % (1 2)


1 2

6 C  【  】 ?x ≥1 f ( x )= f ( x - 1 ) , ?, ( x + 1 )=f ( x ) , 1f f ( x ) "# - 23 = 4 ú? 2 1 , ( x )= k ( x + <= g ) , ( x ) 1 , 0 ) , ( x )- 1 "# g -?+I??(- <= f g ( x )= 0L?W]?-?+(, = f ( x ) "# y -? = g ( x ) = f ( x ) +H y -?+L?W??, <? y 4"3, ( 5 , 1 ) 1 , 0 ) ?+, ?ú? 2 0?, H(- "° 1 ( 4 , 1 ) 1 , 0 ) §±-??= , H (- "°§±-? 6 1 ?= , ???+L?W??, |k -je??= 5 1 1 , . 6 5

13 1 0 , ∴y =l nxH y= x- ? + ° ( 1 , e ) , e< 4 4 1 3 e , e ) ∴ §± y =a x = x ( ?mL 1W??, °y 4 1 = x ( x ) ?y IJ?, H,+ f -? + L 2W? e 1 1 , , ∴a . ∈ $; B ?, 4 e

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2 7 C  【 ( x )= 0 , 3 x-| x - 4 | ,  】 >f 1 m= ? 2 g ( x )= 3 x -| x - 4 | , 2?, g ( x )= ?x ≥26 x ≤- 3 2 2 5 2 2 - x -| x- 4 | = 3 x-x + 4=- x + , 3 ? 2 4 2 2 - 2< x < 2?, g ( x )= 3 x -| x - 4 | = 3 x + x - 4= 2 3 2 5 2 x + - , = g ( x )= 3 x -| x- 4 | ?? y ?+ 2 4 2 5"3, ( x )=| x- 4 | - 3 x + m ?ú? 2 ??,+ f 2 5 6< m< 6 , UL?W]?- / ?, | m<- 6 - 4 2 5 - ∪(- 6 , 6 ) . . ∞, %m ∈ - $; C 4

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6 ú? 2 9 B  【 ( x )= 0 , x 0 , 2 ) = 06 x = 】 >f ∈[ 01 x 1 , %°VW 23 ¤, ,+-? + H x yL?W? 0 , 6 ) = 6?, ?, °ZJ [ ?}L 6W??. ?x ? $}L 7W]?-??. [ò'?g-??, x 1 0A  【 =a  】 ? X e +x x eH -h x 1 %=,+ y 1= y a-x? + - ? ? ( ? 0= )- ? [ \, =? A ?X l nx + x =a-h x 2 %=, nx a-x +y Hy ? 2 =l 0= 7 ú? 2 ) +-?? ( ?=? B -? 、 BO5 y = x x x [\, |A ~?. ?| -eJ? 1- 2| | A B | 、 B?]±-??Q= ?, -UVJ?, ?? A 1 x 1 , ′ = e = 1h1 x 0 , y ′ = = 1h1 x &y 1 1= 2 2= x 1 , x x , . "# | -J?e= 1 "#; A 1- 2|    &'()* 
Ⅰ. 
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5 ú? 2 1 8 B  【 ∵y = l nx ( x > 1 ) , ∴y ′ = , 】 ? ] ?= x 1 ( x , y ) , ∴]±?X= y - y ( x - x ) , ∴y - 0 0 0= 0 x 0 1 1 l nx ( x - x ) , = a x ??H y ^?, | a= , 0= 0 x x 0 0 1 l nx 1= 0 , ∴x , ∴ a= . =a x ?§± y H 0- 0 =e e 1 1 y = x + 1???, , §± = y= x ? x=1?, 4 4 1 1 1 l nx - x = l n 1- < , = e l nx - x = 0 ?x ?, 4 4 4 1 1 3 3 l ne - e>0 , l nx- x=l ne - ? x=e ?, 4 4

8 ú? 2

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3 20 0 0  【 $%】 ? ? ° Jl*? ?, ?X ?= 2× ( 0+ 1 0+ 2 0+ …+ 1 9 0 ) ( ; ?) ? ? ° l*? 2W 1 0+ 0+ 1 0+ 2 0+… + 1 8 0 ) ??, ?X ?= 2×( ) ( 3 6 0 ) ; 3 , ( ? ??? ? ??°l*? W?? ? ( 2 0+ 1 0+ 0+ 1 0+ 2 0+ …+ 1 7 0 ) ( X?= 2× ?) ( 2 0? ) ; …, ??? 3 {? ? ?. ef ? ? °TJ 0 、 1 1W ? ?, 9 0+ ? X ? J ?, = 2×( -? 1 8 0+ …+ 0+ 1 0+ 2 0+ …+ 1 0 0 )= 20 0 0 ( . ?) 4 D  【 , , $%】 ???QAU?= x ????e= a
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15 0 0 4 0 0 ( x ) ?f j 1 J ? e, h 1 x= . >5 2 0 0- 3 x 9 4 0 0 2 5 0 4 4< < 4 5 , ( 4 4 )=T 4 4 )= , f ( 4 5 )= Bf 1( 9 1 1 3 0 0 T ( 4 5 )= , f ( 4 4 )<f ( 4 5 ) , = 4 4? ? : $? x 3 1 3 2 5 0 ( 4 4 )= . ????-?JJ?, nJ??J= f 1 1 > 2?, T ( x )> T ( x ) , ②? k >5 k =Fa+, $ 1 2 k , ≥3

15 0 0 15 0 0 3 7 5 = . ?? ≥ ( 1+ k ) x 2 ( 1+ 3 ) x 5 x 2 0 0- 0 0- 0- 3 7 5 ( x )= , ( x )= m a x { T ( x ) , T ( x ) } , KT φ $? 1 5 0- x T ( x ) ( x )=m a x { T x ) , T x ) } [A,+, |f ≥ 1( 3( 10 0 0 3 7 5 , m a x { T ( x ) , T ( x ) }= ( x )= m a x . φ 1 x 5 x 0- 7 5 10 0 0 3 ( x ) , T ( x ) = >,+ T -? Q .?, ? 1 0- x 5 x 4 0 0 4 0 0 < ( x ) = . 6< ?φ j1J?e, h1 x >5 3 1 1 1 1 2 5 0 2 5 0 3 7 , ( 3 6 ) =T 3 6 )= > , ( 3 7 )= Bφ φ 1( 9 1 1 3 7 5 2 5 0 > , T ( 3 7 )= $???:???? - J? ? 1 3 1 1 2 5 0 . Jü5 1 1 < 2?, T ( x )< T ( x ) , ③? k >5 k =Fa+, $ 1 2 k=1 , ( x ) =m a x { T x ) , T x ) }= ?? f 2( 3( 20 0 0 7 5 0 , . m a x x 1 0 0- x 20 0 0 x ) , T x ) = >,+ T - ? Q . ?, ? 2( 3( x 8 0 0 7 5 0 ( x ) = , ?f j1J?e, h1 x ?¤①1 0 0- x 1 1 2 5 0 , ? ?, ?? ? : ? ?? ? - J? ?J= ü5 9 2 5 0 . 1 1 = 2?, ??"?, ?k ? : ? ?? ? -?J J?, , B , C` z C ? - ? + ? ? = 4 4 , ??, ?? A 8 8 , 6 8 . 1 2 2 7 ( 1 ) = 0 , x - ( 1+ k ) x = 0 , &y 1k >? § m? 2 0 2 0 2 0 k > 0 , k > 0 , = $x ≤ ?R???? x 2= 1 1+ k k + k 2 0 = 1 0 , = 1?j_?. ?n ? ? k "# ¨ - J ü 2 0??. ?X= 1 ( 2 ) > 0 , <= a "#¨?0?TS\ 1 2 2 > 0 , . 2= k a - ( 1+ k ) a ?ì° k ?3 :X 2 0 2 2 2 2 0 a k + a + 6 4= 0LF( ?O5 k -?X ak - 2 2 2 (- 2 0 a )- 4 a( a + 6 4 ) ?7?? Δ= ≥0 . ( ?a ≤6 " # ? a] H I 6 ? ?) ?, 0?T S\. 7 8 ( 1 ) t = 0 . 5?, P- ?[\ x 7 t = , ? ??? P= 2 1 22 = x , 3 . A P| = 1 P- ?[\ y >| ±?X y P= 4 9

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x ∴ a槡 2 0- x 0- , ≥1 2 1 2 0- x 0!:X. %a ≥ 槡 ~0 ≤x ≤2 2 ( x )= Bf 1 2 0- x 5 , -Jüe=槡 2槡

1 2 , = 1?_?:X, <= t+ 2 ≥2 ?n ? ? t "# t
2 2 v 4 4× 2+ 3 3 7= 2 5 , 5 . ≥1 %v ≥2 5? ? ? ? ? ? ? <?, ??° - ? ± á ? [ 2

     

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2 0 1 6 Ⅱ.

∴a 5 . m i n= 槡 7 B  【 8 0 0+ 0 . 6 x < 1 . 1 x 】 >Rm?: ?, ?? > 16 0 0 . hI1 x ?????????'à, 8 ( 1 ) ACS-x ? = f ( x ) ???á= x ?7?, ? BCS-x ? = g ( x ) ( x )= 7, ? 7, >R?1 f 1 1 k x , g ( x )= k x , ( 1 )= , . >R?? f $k 1 2槡 1= 4 4 5 5 ( 4 )= , . !g "# k 2= 2 4 1 5 ( x )= x ( x ) ; g ( x )= 槡 x ( x ) . oB f ≥0 ≥0 4 4 ( 2 ) h ( x )=f ( x )+g ( 1 0-x )= ( 0 0 ) . ≤x ≤1 =槡 1 0- x , = 1 0- t , = ?t |x $y


1 C  【 %, , 】 ??W x ???F-W?= a |?
2 ( 1+ 1 0 %) = 1 . 2 1 a , ??W??? - W? = a | 1 . 2 1 a ( 1-x %)=a , %= , % 1-x >Rm? 1 1 . 2 1

2 1 % = ≈1 7 . 3 5 5 %, 7 . 4 %. $x $0j 1 1 2 1 2 A  【 ( 0<x < 6 ) , 】 ? ?w?? - U = x ?? 2 4- 4 x , = x × = 2 x ( 6-x )=- 2 ( x - |y ??= y 2
2 3 ) + 1 8 , ∴? x = 3?, y Jü. 3 3 A  【 = k x ( k ) 】 >RS0?? ??? y ≠0 [

1 5 1 0- x + 槡 x 4 4 1 2 ( 1 0-t )+ 4

= 4?, y = 2 , VWp ,+ ??, >w?01, ?x 2 1 1 3 3 k × 4 , = 3= , = x . = % 2= h1 k $y &y 2 3 2 4 3 1 1 13 1 1 3 , = x, = . | % x= , h1 x 2 5 6 2 5 6 3 2 8 2 4 ( 1 ) 1 2 0  ( 2 ) 8 0  【  】 <= ?@ ?J- A ?, = 6 0? t = 1 8 0 z?:?P?? ?A ?, Bn? t , ?z?:?^_ k ?'RT ? ?-?W,+O z? :?H? ? ?J-?@OPé?b P0?, a t+ b t + c F??. ,+ Q= 2 a ( t - 1 2 0 ) + m , ()Rm1 Q= ? 2T+) ? ? 01


5 5 12 5 5 1 t - t =- t + t + =- 2 4 4 4 2 4

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5 6 5 = ?, h ( x ) , = 3 . 7 5 . ?t ?? x m a x= 2 1 6 . 7 5?7, BCS?? 6 . 2 5? "#? ACS?? 3 6 5 7?, 1×Jüx? ?7. 1 6 9 ( 1 ) ( x ) >Rm?, ? ? ~ ?? ? ? -,+ ?? f 1 0 , 10 0 0 ] -???g[: ?x ∈[ ?, ( x ) ( x ) ( x ) ①f [A,+, ②f ≥1! :X; ③f ≤ !:X. x ( 2 ) ( x )= + 2 : 1 0 , ①~5,+ ?? f ?x ∈[ 1 5 0 10 0 0 ] f ( x ) ( x ) ?, [A,+, |f ≥1ef!:X; x x ( x )= 1 0 , 10 0 0 ] + 2 B??,+ f ≤ °[ ?! 1 5 0 5 9 x 0 0! :X, 2 9 x ) 2 9 0 , :X, a?% 2 ≥3 B( m i n= x ∴f ( x ) ≤ ]!:X. 5 $?,+??]ò'???g. ( x )=4 l gx-2 : 1 0 , ②~5 , + ? ? f ? x ∈[ 10 0 0 ] f ( x ) ( x ) = f ( 1 0 ) = ?, [ A , +, | f m i n . ∴f ( x ) 4 l g 1 0- 2= 2> 1 ≥1!:X; x 4 l g e 1 ( x )= 4 l gx - 2- , ′ ( x )= - . ?g |g 5 x 5 x 5

{

2 a ( 6 0- 1 2 0 ) + m= 1 1 6 , a = 0 . 0 1 , | "# 2 8 0 , a ( 1 0 0- 1 2 0 ) + m= 8 4 , m=

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2 Q= 0 . 0 1 ( t - 1 2 0 ) + 8 0 , 2 0?, $?? ?? += 1

07 / 1 0 0k g . z?:?j×J?e 8 5 A  【 9"3- [\ P, 】 ?X?ú? 2 5[>
2 = a x ( a< 0 ) , R??????±-? X = y ?? A-[\ = ( 4 ,-h ) , ( 3 , 3-h ) , |C ? ???-

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1 5

l g e - 1 4 l g e 1 2 0? , g ′ ( x )= - ≤ = ?x ≥1 x 5 5 2 1 l g e- , ( x ) 1 0 , 10 0 0 ] < 0 "# g °[ ?[?,+, 5 ( x ) ( 1 0 )= 4 l g 1 0- 2- 2= 0 . oB g ≤g

1 = 1 y ′ | , ( 0 , 0 ) : y = °? P ? - ] ±= l x = 0= 2 0 c o s x , : y =t a nx ( 0 , 0 ) <?0? j ± C °? P í? ? 1 y ′ = , y ′ | 5§± l -?*, ④FG; ~5⑤, x = 1= x 1 , ( 1 , 0 ) : y =x - 1 , ( x )= °? P ?-]±= l &h 1 x - 1 x - 1- l nx ( x > 0 ) , ′ ( x )= 1- = , 01 h ? x x ( x ) h ( 1 )= 0 , - 1 nx , V?01 h $x ≥l 0? m i n= : y = l nx ( 1 , 0 ) j± C °? P í? ?5§± l -ù *, ⑤DE. 4 ( e , e )  【 ′ =l nx +x · $%】 >Rm1 y 1 = 1+ x

& ' ( # $ )   * + , -

x x x ∴4 l gx - 2- ≤0 , l g x - 2 ∴f ( x ) %4 ≤ , ≤ 5 5 5 !:X. $?,+??ò'???g. 6 4 - 4 ( 0 ) , ≤x ≤4 x 1 0( 1 ) = 4 , = 8- <= a "# y 2 0- 2 x ( 4< x 0 ) . ≤1 6 4 - 4 , , |? 0 ≤x ≤4?, > ≥4 h1 x ≥0 "# 8- x ; ?? 0 ≤x ≤4

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x 0?, 0- 2 x , , ? 4< ≤1 >2 ≥4 h1 x ≤8 "#? x . ? 4< ≤8 , ??, 01 0 ≤x ≤8 %V??? 4W??- ?è, Léê??J0ì 8?. 1 ( 2 )? 6≤ x≤ 1 0 ?,y =2 × 5- x + 2

, x - y + 1= 0-? ? = 2 . ( m , n ) , l nx §± 2 ?P l nm= 2 , e , = e l ne = e , | 1+ h1 m= "# n %? P-[\=( e , e ) . 2 x 5 - 4  【 y = , y ′ = x , ∴y ′ | 4 , y ′ | $%】 x = 4= x =- 2= 2 - 2 , 4 , 8 ) , , ? P- [\ = ( ? Q - [\ = (-2 2 ) , ∴°? P?-]±? X = y - 8= 4 ( x - 4 ) , % y = 4 x - 8 . - 2=- 2 ( x + 2 ) , = °? Q?-]±?X= y %y y = 4 x - 8 , - 2 x - 2 , ( 1 ,- 4 ) , h 1A | A?y =- 2 x - 2

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1 6 1 6 a - 1 =1 0-x+ -a=( 1 4- [ 8-( x - 6 ) ] 1 4- x

1 6 a x )+ - a - 4 , 4- x 4 , 8 ] , <= 1 ∈[ B1 ≤a ≤ 1 4- x 4 , a 4 , 8 ] , 4- x = 4 a "# 4 ∈[ $?n?? 1 ?, 槡 槡 y a - a - 4 . LJ?e= 8 槡 a - a - 4 , 4- 1 6 2 , &8 ≥4 h1 2 ≤a ≤4 槡 槡 4- 1 6 2 . 6 . "# a -J?e= 2 ≈1 槡    &'()* 
Ⅰ. 
x 15 x + y +2=0  【 e +31, y ′ = $%】 > y= -5 x - 5 e , = y ′ | 5 , "#]±-?? k "# ] ± x = 0 =-

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4 . ?[\= -
3 6 D  【 y ′ = 4 x + 2 a x , 1 , a + 2 ) $%】 j±°?(- ?

=f ′ (- 1 )=- 4- 2 a= 8 , ]±-? ? k h1 a= - 6 .
1 α- 72  【 y ′ = x , k = y ′ | $%】 α ]±-??: α= x = 1=

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2- 0 . = 2 1- 0 8 1 1 y ′ =2 a x- , y ′ | a-1=0 ,  【 $ %】 x = 1 =2 x 2 1 ∴a = . 2 9 ( 1 ) ( x ) 0 , + , ,+ f -??'=( ∞) a x b x-1 b x-1 x f ′ ( x )= a e l nx + e - 2e + e . x x x ( 1 )= 2 , f ′ ( 1 )= e . >Rm01 f = 1 , b = 2 . $a 2 x-1 x ( 2 ) 1 ) f ( x )= e l nx + e , >( ?, x 2 - x ( x )> 1_W5 x l nx > x e - . oB f e ( x )= x l nx , ′ ( x )= 1+ l nx . ?,+ g |g 1 , ?, g ′ ( x )< 0 ; "#? x ∈ 0 e

+ 2=- 5 ( x - 0 ) , x + y + 2= 0 . ?X= y %5 b 2 2 - 3  【 =a x + I? P ( 2 ,- 5 ) $% 】 >j± y x b b 5= 4 a +  ( 1 ) . ′ = 2 a x - 2, 01 - !y "#° 2 x b 7 a - =-  ( 2 ) . ? P?-]±?? 4 2 4 1 ) ( 2 ) =- 1 , b =- 2 , + b =- 3 . >( h1 a "# a 2 3 ①③④ 【 y ′ = 3 x , y ′ | = 0 , $% 】 ~5 ①, "# x = 0
3 l : y = 0[j± C : y = x ( 0 , 0 ) °? P ? - ] ±, < 3 : y = x °? P ( 0 , 0 ) ?0?j± C í? ?5§± l ′ =2 ( x+1 ) , -? *, ① F G; ~ 5 ②, <= y y ′ | = 0 , l : x = - 1 C : y = ( x + "# ][ j ± x =- 1

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1 ) °? P (-1 , 0 ) ? - ] ±, ② D E; ~ 5 ③, y ′ = c o s x , y ′ | = 1 , P ( 0 , 0 ) : °? ? ] ±= l x = 0 y = x , C : y = s i n x P ( 0 , 0 ) <?0?j± °? í? 1 y ′ = 2 , ?5§± l - ? *, ③ F G; ~ 5 ④, c o sx



, + g ′ ( x )> 0 . ∞ ) ?, (1 e 1 1 , ) ?? Q ??, + ( x ) ∞ ) ?? $g °(0 °( e e ?x ∈ ( x ) 0 ,+∞ ) Q?A,o B g °( ?-J?e=

1 6



( )

1 1 =- . e e

2 - x - x ( x )= x e - , ′ ( x )= e ( 1- x ) . ?,+ h |h e 0 , 1 ) h ′ ( x )> 0 ; 1 ,+∞ ) "#? x ∈( ?, ?x ∈( h ′ ( x )< 0 . ?, ( x ) 0 , 1 ) 1 , +∞ ) $h °( ?? Q ? A, °( ?? Q ( x ) 0 , + ( 1 )= ??, oB h °( ?-Jüe=h ∞) 1 - . e > 0?, g ( x )> h ( x ) , ( x )> 1 . ??, ?x %f
2 0 1 6 Ⅱ.

x 0 0 e s i nx . ′ ( 0 )=e c o s 0-e s i n0= 1 , "# f %?? π a nα= 1 , 1 α= . ¨ αkl t 4 2 2 2 8 D  【 | P Q| = ( a - c ) +( b -d ) , Q 】 23 P

1 D  【 f ′ ( x )= 3 m+ c o s x , 】 <=ì°^\?§ 3 m+ c o s x ) ( 3 m+ c o s x )=- 1 , -]±, "#?( 1 2 2 : 9 m +3 ( c o s x +c o s x ) m +1+ a? 1 ? X 1 2 c o s x c o s x 0 , O 5 m - ? X L h, " # Δ= 1 2=
2 9 ( c o s x c o s x ) -3 6 ≥ 0! : X, "#?L° 1- 2 c o s x c o s x 1 , - 1-? ? H TVW= : VW= 1 2 2 m = 0 , 0 . ??:X, ???X1 9 "# m= f ( x ) x x A  【 ( x ) =a g ( x ) , ∴ =a .   】∵ f g ( x ) f ( x ) x ∵f ′ ( x ) g ( x )> f ( x ) g ′ ( x ) , ∴ ′ =( a ) ′ = g ( x ) f ′ ( x ) g ( x )- f ( x ) g ′ ( x ) x =al na>0 , na>0 , %l 2 g ( x ) f ( 1 ) f (- 1 ) 5 5 - 1 ∴a > 1 . ∵ + = , ∴ a+a = , g ( 1 ) g (- 1 ) 2 2 f ( n ) f ( x ) x f ( n ) n ∴a = 2 , ∴ = 2, ∴ = 2, ∴+? g ( n ) g ( x ) g ( n ) n ( 1- 2 ) 2 n + 1 ∴S =2 -2>6 2 , =_ ? + ?, n = 1- 2 ∴n + 1> 6 , > 5 , ∴n , . %n -J?e= 6 $; A 1 x a B  【 f ′ ( x )= 2l n 2+ , ′ ( a )= 2l n 2+ 】 >f x 1 1 a a = 0 , n2= - , ·2 l n2= -1 , 1 2l |a % a a a a 2 l n2 =- 1 . C  【 f ′ ( x )=-a s i nx , g ′ ( x )=  】 {Rm1, 2 x + b , ′ ( 0 )= g ′ ( 0 ) , a s i n0= 2× 0+ 5[L f %- b , = 0 , f ( 0 )=g ( 0 ) , 1 , |b ! m= % m =a= <? a + b = 1 , . ;C B  【 ∵f ( x )= s i nx , f x )=c o s x , f ( x )= 】 0 1( 2 - s i nx , f ( x )=- c o s x , f ( x )= s i n x , …, ∴f ( x )= 3 4 n f ( x ) , f ( x ) = f ( x ) =- s i n x . $ n + 4 20 1 4 2 1 1 C  【 ( x ) 】 >p,+ f -?+I ? A , , 4 2 1 x , ′ = . 01 p , + = y=槡 ??,+= y | 2 x 槡 1 = 1 y ′ = , "#? A? - ] ± ? X = x = 1 2 4 1 1 y - = x - , x - 4 y + 1= 0 . %4 2 4 x x B  【 ( x )=ec o s x , ′ ( x )=e c o s x - 】 >f 1f

: y=x+t ??- h i - ? ?, ? l = j ± y= 2 x + 3 l nx = x + t - -]±, ?Ra@=???± y 3 =x+2- h i - ? ?. ′ = -2 x+ = Hy &y x 1 ( x > 0 ) , = 1 , ( 1 ,- 1 ) . h1 x |]? A !∵]? : y = x +t 1= 1+t , ∴t =- 2 , ∴l : y = °l ?, 1- 2+ 2 4 x - 2 , ∴ ? ? ? ± J h i d= 2 , = =2槡 2 槡 2 槡 2 ∴d = 8 . 9 B  【 ( x ) 】 >,+-? +, 0?,+ f [? Q ?A-, "#,+? + ??mV? ? - ? ,+e = 2? ?ü5 /, ín>? + 0?, ,+? + ° x x = 3 k , - ]±?? k ü5° ? ] ±? ? "# 1 2 f ′ ( 2 )> f ′ ( 3 ) . ( 2 , f ( 2 ) ) , B ( 3 , f ( 3 ) ) , KA ?§± f ( 3 )- f ( 2 ) A B , B- ? ? k= =f ( 3 )- |§ ± A 3- 2 f ( 2 ) , k > k 0 , ′ ( 2 )> >,+?+, 0? k %f 1> 2> f ( 3 )- f ( 2 )> f ′ ( 3 )> 0 . . $; B
2 1 0( 1 ) f ′ ( x )= 3 a x + 6 x - 6 a , f ′ (- 1 )= 0 , a - 6- %3

     



[

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6 a = 0 , ∴a =- 2 . ( 2 ) 0 , 9 ) , = §± m!I??( ?§± m[ j ± y 2 g ( x ) x , 3 x 6 x 1 2 ) , -]±, ?]?=( 0+ 0 0+
2 ∵g ′ ( x )= 6 x 6 , ∴ ] ±? X = y - ( 3 x 6 x 0+ 0 0+ 0+ 1 2 )= ( 6 x 6 ) ( x - x ) , 0+ 0 0 , 9 ) 1 , ?( ??, 1x 0 =± 1?, = 9 ; ?x ]±?X= y 0 =- = 1 , y = 1 2 x + 9 . ?x ? ] ±? X = 0

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′ ( x )= 01 - 6 x+ 6 x + 1 2= 0 , =- 16 >f h1 x x = 2 , =- 1?, y = f ( x ) =- 1 8 ; °x -]±?X= y = 2?, y = f ( x ) = 9 . °x -]±?X= y ∴y = f ( x ) = g ( x ) = 9 . Hy -V??]±[ y 2 ′ ( x )= 1 21 - 6 x + 6 x + 1 2= 1 2 , ∴x = 06 !> f x = 1 . = 0?, y = f ( x ) = 1 2 x - 1 1 ; °x -]±?X= y = 1?, y = f ( x ) = 1 2 x - 1 0 ; °x -]±?X= y ∴y = f ( x ) = g ( x ) = 1 2 x + 9 . Hy -?]±][ y y =f ( x ) =g ( x ) = 9 , ??"?, Hy -? ] ±[ y = 0 . ??ì° k    &'()* 
Ⅰ. 
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1 4



1 D  【 ( x )= k x - l nx , ′ ( x )= k - 】 <= f "# f 1 . ( x ) 1 ,+∞ ) <= f °ZJ( ?? Q ? A, "# x 1 1 > 1?, f ′ ( x )= k - ≥0! :X, ?x %k ≥ ° x x



1 7

1 , + > 1 , ZJ( ?!:X. <= x "# 0< ∞) 1 , . . "# k ≥1 $; D

1 < x

x 2 C  【 ( x )=e -l nx , 0 , $%】 90,+ f <=° (

1 x x 1 ) ′ ( x )= e - ∈(- e - 1 ) , ( x )= e - ?f $f ∞, x

& ' ( # $ )   * + , -

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x e f ( x ) 、 BD; ( x )= , -ü?, $A 9 0 ,+ g | 2 x x x x x x e - e ( x - 1 ) e e g ′ ( x )= 2 = , ( x )= ° $,+ g 2 x x x x x ( 0 , 1 ) ( x )> g ( x ) , x e >x e , ??Q ??, $g 1 2 2 1
1 2

. $; C 3 A  【 f (-x )=l n ( 1-x )-l n ( 1+x )= $ %】 f ( x ) , ( x ) =l n ( 1+x )- - $ ① F G; <= f 1+ x 2 x l n ( 1- x )= l n , 1 , 1 ) !? x ∈ (- ?, 2∈ 1- x 1+ x 2 x (- 1 ,1 ) ," # f n 2 =l 1+ x

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l n

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1 ) | f ( x )| | x | ( x )- 2 x , ( x )= ?, ≥2 ?f ≥0 &g f ( x )- 2 x =l n( 1+x )-l n ( 1-x )-2 x ( x 0 , ∈[
2 1 1 2 x 1 ) ) , ′ ( x )= + - 2= , <= g " 2≥ 0 1+ x 1- x 1- x

( x ) 0 , 1 ) g ( x )= f ( x )- #g °Z J [ ?? Q ? A, 2 x ( 0 )= 0 , ( x ) x , ( x ) = 2 x ≥g %f ≥2 !f Hy ?= f ( x )| | x | s,+, "# | ≥2 : X, $ ③ F G, $ . ;A 4 D  【 ∵ ,+ y =f ( x ) $% 】 ° ,+, 1 ∴f ′ ( x )= 2 x + a - 2≥0° x ∴a ≥ 1 2 x ° 2 - x


0 ú? 3 >,+-?Q.1: ( 0 , x ) 1 f ′ ( x ) f ( x ) - ? x 1 0 J?e ( x , x ) x 1 2 2 + ? 0 Jüe ( x , + ∞) 2 - ?

, + ∞ ) ?[ A (1 2

, + ∞ ) ?!:X. (1 2

1 ∴f ( x )< 0 , f ( x )> f ( 1 )=- a >- . . $; D 1 2 2 8 ( 1 ) f ( x ) + , f ′ ( x )= 1+ a - -??'=(- ∞, ∞)
2 2 x - 3 x . ′ ( x )= 0 , &f 1x 1=

, + ∞ ) ?!:X, (1 2 1 1 + ( x )= - 2 x ∞ ) ??Q??, Bg °( , 2 x 1 = ?, g ( x )= 3 , ∴a . ?x ≥3 2 5 B  【 1 , 1 ] f ′ ( x )> 0 $%】 >?0?? x ∈ [- ?, ′ ( x ) ( x ) 1 , 1 ] nf PAü???, <? f ° Z J [- ??Q?An?A±VP??G.
x 6 C  【 = 1?, f ( x )=( e - 1 ) ( x- 1 ) , $% 】 ?k x f ′ ( x )= x e - 1 , ∵f ′ ( 1 )= e - 1 , ∴f ( x ) = 1 ≠0 °x

- 1-槡 4+ 3 a , x 2= 3

- 1+槡 4+ 3 a , x x , 1< 2 3 ′ ( x )=- 3 ( x - x ) ( x - x ) . "# f 1 2 < x > x f ′ ( x )< 0 ; ?x 16 x 2 ?, x < x f ′ ( x )> 0 . ?x 1< 2 ?, f ( x ) ( - , x ) ( x $ ° ¤? Q ??, ° ∞ 1 ? 2,+ ∞) ( x , x ) ¤?Q?A. 1 2 ( 2 ) > 0 , 0 , x 0 . <= a "# x 1< 2> x 1 , ( 1 ) , f ( x ) 0 , 1 ] ①? a ≥4?, ≥ > ? °[ ?? 2 , f ( x ) x = 0 x = 1 Q?A "# ° ?? ? ??j1 J?e?Jüe. 4?, x 1 . 1 ) f ( x ) 0 , x ] ②? 0<a< >( ?, °[ 2< 2 x , 1 ] , f ( x ) ??Q?A, °[ ?? Q ?? <? ° 2

?]?j×?e; x 2 = 2?, f ( x )= ( e - 1 ) ( x - 1 ) , f ′ ( x )=( x - ?k
x x x x 1 ) ( x e + e - 2 ) , ( x )= x e + e - 2 , &H

′ ( x )= x e+ 2 e> 0 , x 0 , + . |H ∈( ∞) ( x ) 0 , + YZ H °( ?=A,+, ∞)





1 8

- 1+槡 4+ 3 a ?j1Jüe. 3 ( 0 )= 1 , f ( 1 )=a , 1?, f ( x ) !f "#? 0<a< ° x = 1?j1J?e; 1?, f ( x ) 0? x= 1? ??j1 J ? a= ° x= ?e; a < 4?, f ( x ) = 0?j1J?e. ? 1< °x 3 x + 3 x - 3 a , x , ≥a 9 ( 1 ) ( x )= 3 <= f x- 3 x + 3 a , x < a , 2 x+ 3 , x , 3 ≥a ′ ( x )= 2 "# f 3 x- 3 , x < a . , , >5 -1 ≤x ≤1 ①? a ≤ -1?, L x ≥a $ 3 f ( x )= x + 3 x - 3 a , ( x ) 1 , 1 ) M( a )= ?? f ° (- ?[ A ,+, <?, f ( 1 )=4-3 a , m( a ) =f (-1 ) = -4-3 a , $ M ( a )- m ( a )= ( 4- 3 a )- (- 4- 3 a )= 8 . 1< a < 1?, ②? - 3 a , 1 ) , f ( x )=x + 3 x - 3 a° ( a , 1 ) ?x ∈( ?[ A 3 1 , a ) , f ( x )= x - 3 x + 3 a° (- 1 , ,+; ?x ∈(- a )? [ ? , +, M( a ) =m a x { f ( 1 ) , " #, 3 f (- 1 ) } , m ( a )= f ( a )= a . ( 1 )- f (- 1 )=- 6 a + 2 , >5 f 1 3 1<a M( a )-m( a )= -a - <?, ?- ≤ ?, 3 3 a + 4 ; 1 3 a < 1?, M( a )- m ( a )=- a + 3 a + 2 . ? < 3 3 , ( x )= x- 3 x + 3 a , ③? a ≥1?, Lx ≤a $f ( x ) 1 , 1 ) M( a )= ?? f ° (- ?[?,+, <?, f (- 1 )= 2+3 a , m( a ) =f ( 1 ) = -2+3 a , $ M ( a )- m ( a )= ( 2+ 3 a )- (- 2+ 3 a )= 4 . ??, , a 1 , ≤- ?8 1 3 ? -a - 3 a + 4 , - 1< a ≤ , ? 3 M ( a )- m ( a )=? 1 3 3 a + 2 , < a < 1 , ? -a + 3 ? ?4 , a . ≥1 ( 2 ) ( x )= f ( x )+ b , &h | 3 x + 3 x - 3 a + b , x , ≥a h ( x )= 3 x- 3 x + 3 a + b , x < a , 2 3 x + 3 , x , ≥a h ′ ( x )= 2 3 x- 3 , x < a . 2 f ( x )+b ] , 1 ] <=[ ≤ 4~ x ∈ [-1 ! : X, % - 2 ( x ) 1 , 1 ] ≤h ≤2~ x ∈[- !:X, 1 ) "#>( ?, h ( x ) , 1 ) ①? a ≤ -1?, ° (-1 ? [ A , +, h ( x ) 1 , 1 ] ( 1 )= 4- 3 a+b , °[- ?- J üe[ h (- 1 )=- 4- 3 a+ b , 4- 3 a+ b J?e[ h |- ≥ - 2n 4- 3 a + b , ≤2 rs; 1 1<a h ( x ) 1 , 1 ] ②? - ≤ ?, ° [- ?- J ?e 3 x = x 2=

3 ( a )= a + b , ( 1 )= 4- 3 a + b , [h Jüe[ h 3 3 b 2n 4- 3 a+b , 2- a + "# a + ≥- ≤2 oB - 1 3 a a + b a - 2n 0 ≤3 ≤6 ≤a ≤ . 3 3 2 ( a )= - 2-a + 3 a , ′ ( a )= 3- 3 a >0 , &t |t

1 , ?[A,+, t ( a ) ( a ) ( 0 )=- 2 , ° 0 $t ≥t 3

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{

{

2 a + b ; <? - ≤3 ≤0 1 a < 1?, h ( x ) 1 , 1 ] ③? < ° [- ?- J ?e[ 3 3 h ( a )= a+ b , (- 1 )= 3 a +b + 2 , Jüe[ h "# 2 8 3 a + b a+b+2 , a+ ≥ -2n 3 ≤2 h 1 - <3 2 7 b ; ≤0 h ( x ) , 1 ] ④? a ≥ 1?, ° [-1 ?-Jüe[ h (- 1 )= 2 + 3 a + b , ( 1 )=- 2 + 3 a + b , J?e[ h a + b + 2 a+b - 2 2 , a+ "# 3 ≤2n 3 ≥- h1 3 b = 0 . a + b 2 a + b . ??, 13 -je??[ - ≤3 ≤0
- x - (- x ) 1 0( 1 ) , (- x )= e + e = <=~?m x ∈R ?L f - x x e + e = f ( x ) , ( x ) "# f [ R?-t,+. x - x - x ( 2 ) e +e -1 ) 0 , >? ? ? m( ≤ e -1° ( + ?!:X. ∞) t - 1 x =e ( x> 0 ) , > 1 , = &t |t "# m≤ - 2 t- t + 1 1 - >1: X. -1+ ~?m t <= t 1 t - 1+ + 1 t - 1

     

1 1 +1≥ 2 ( + 1= 3 t - 1 ) · ," # - t - 1 t - 1 1 1 = 2 , =l n2 ≥- , ?n ? ? t %x 3 1 t - 1+ + 1 t - 1 ?_?:X. - ∞, <??+ m-je??[ -



(

1 . 3

]

1 x 3 ( 3 ) ( x )=e + x -a (-x +3 x ) , &,+ g | e 1 x 2 g ′ ( x )= e - x+ 3 a ( x - 1 ) . e 1 2 , x -1 , , ≥0 ! a>0 $ x >0 e g ′ ( x )> 0 . ( x ) 1 ,+∞ ) "# g [[ ?- ? Q A , ( x ) 1 ,+∞ ) ( 1 )= +, <?g °[ ?- J ?e[ g - 1 e + e - 2 a . x - x 3 1 ,+∞ ) , (-x >5ì° x ? e +e -a 0∈ [ 0+ 3 x )< 0:X, ( 1 )< 0 . ?n??J?e g 0
x e - ?x ≥ 1?,
0 0

{ {

- 1 e + e - 1 + e - 2 a < 0 , > . $e %a 2 ( x )= x - ( e - 1 ) l nx - 1 , ′ ( x )= 1- &,+ h |h e - 1 . ′ ( x )= 0 , = e - 1 , &h 1x x 0 , e - 1 ) h ′ ( x )< 0 , ( x ) 0 , e - 1 ) ?x ∈( ?, $h [(

1 9

?-?Q?,+; e - 1 , + h ′ ( x )> 0 , ( x ) e - ?x ∈( ?, $h [( ∞) , + 1 ?-?QA,+. ∞) ( x ) 0 , + ( e - 1 ) . "# h °( ?-J?e[ h ∞) ( 1 )=h ( e )= 0 , 1 , e - 1 ) cm× h "#? x ∈( ? ( 0 , e - 1 ) h ( e - 1 ) ( x )< h ( 1 )= 0 . ?, ≤h e - 1 , e ) e - 1 , + h ( x )< h ( e )= ?x ∈( ?( ?, ∞) 0 . ( x )< 0~?m- x 1 , e ) "# h ∈( :X.
- 1 e + e , e?( 1 , e ) h ( a )< 0 , ?, % a- 2 a - 1 e - 1 1< ( e - 1 ) l na , a ; oB e < a - 1 e - 1 = e e = a ; ②? a ?, e ,+∞ ) e-1 ,+∞ ) h ( a )> ③? a ∈( ?( ?,

1 , ( x ) $,+ f °x ∈

, e¤][? Q -, "# (1 e )

A , BD; x < 1?, l nx < 0 , ( x )< 0 , C ? 0< ?? f , ( x ) x , + D; ?? x ≥e |f °( ¤[ A ,+, ∞) 0 0 DFG. 5 C  【 f ( x ) $ %】 ° x= -2? j 1 ? ? e, % f ′ (- 2 )= 0 , <- 2?, f ′ ( x )< 0 ; x >- 2?, nx f ′ ( x )> 0 , =x f ′ ( x ) 0 , 0 ) 2 , 0 ) . ?? y I?( d (- <- 2?, x < 0 , f ′ ( x )< 0 , > 0 ; 2< x < ?x |y ?- 0?, x < 0 , f ′ ( x )> 0 , y < 0 ; > 0?, f ′ ( x )> 0 , ?x y > 0 , C . $ FG
2 6 D  【 | M N| ( x )=x - $% 】 - J ?e, %,+ h 2 1 2 x - 1 = , = ef x x x

& ' ( # $ )   * + , -

①? a ∈

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h ( e )= 0 , - 1> ( e - 1 ) l na , a . %a $e > - 1 e + e a - 1 e - 1 , e ?, e <a ; ?? " ?, ? a ∈ ? 2

a - 1

e - 1

h ′ ( x )= 2 x - l nx -J?e,

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2 槡 ( x ) [,+ h °???' ¤?V- ? ?e?, ? 2 2 槡 = . [J?e?, $t 2 7 ① 【 1 ) $%】 o? + ?0# q ×, ?x ∈ (-∞, f ′ ( x )> 0 ; 1 , 2 ) f ′ ( x )<0 ; ?, ?x ∈( ?, ? x ∈ ( 2 , + f ′ ( x )> 0 , ( x ) ?, "# f L?W ? e? 1 ∞) , = 2?,+j1 ? ?e, = 1?, ?2 n? x ?x +j1?üe. $?L①]FG.
2 8 ( 1 ) ′ ( x )= 3 a x - 6 ( m +a ) x+ 1 2 m= >Rm1 f

a - 1 e - 1 a - 1 a = e e =a ; e ,+∞ ) e > ?, ? a ∈( ?, e - 1 a .

2 0 1 6 Ⅱ.

1 A  【 ( x )= x s i nx + c o s x , f ( 0 )= 1> 0 , $%】 &f < (- x )= f ( x ) , ( x ) ′ ( x )= =f "# f =t,+, !f π π , ? f , x c o sx , ′ ( x ) >0 , π? ° 0 ° 2 2

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f ′ ( x )< 0 , (x )° "# f

π , ) ? ? A,° (0 2

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π , π ???. ?'#?òó, 1 ACò'Rm. 2

)

2 D  【 ∵f ( x )< f ′ ( x ) ·t a nx , ′ ( x ) s i nx - $%】 %f f ( x ) ′ ( x ) s i nx - f ( x ) c o s x > [s ]′=f i nx s i nx f (π f ( x ) 6) π , ) ??Q?A, , ∴,+ 0 < °( 0 oB 2 s i nx π s i n f ( x ) c o s x > 0 , ∴






(π 3) π π , f < f . % 3 π 槡 ( 6) ( 3) s i n


( x - 2 ) ( a x - 2 m ) , ( x ) 0 , 3 ) 3 >5 f °( ?? ? e 2 m = 2 , a . ?, $ "#m= a ( 2 ) ′ ( x )= 3 ( x - 2 ) ( a x - 2 m ) , >5 f $ 2 m 2 m 3 , ①? ≤06 ≥3 %m ≤06 m ≥ a ?, 2 a a 2%klRm. jx 0= 3 . ?? m ≤06 m ≥ a 2 m 2 2 , m< a ②? 0< < % 0< ?, ?2?ù: a x f ′ ( x ) f ( x ) 1 2 m 2 m , 0 0 a a

(

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m , 2 (2 a ) -

2 0

( 2 , 3 ) +



3 C  【 ( x )= f ( 4-x ) ( x ) $%】 >f ?,+ f -? + = 2~?, ·f ′ ( x )> 2 f ′ ( x ) x - 2 ) · O5 x >x 1( f ′ ( x )> 0 , > 2?, f ′ ( x )> 0 , f ( x ) %? x ? Q ? A; < 2?, f ′ ( x )< 0 , f ( x ) ∵2<a< 4 , ?x ? Q ??. a a ∴2 >4>3 , 1<l o g a <2 , ∴ f ( 2 ) > f ( 3 ) > 2 f ( l o g a ) . . $; C 2 4 D  【 f ′ ( x )= $% 】 >w?1,
2 1 l n x - 1 · ( x > 0 2 x l n x





?Q ?ü ?Q ?? ?Q 9 m+ 1 ?A e ?? e ?A

) , ′ ( x )= 0 , nx =± 1 , = e = nx ≠1 &f 1l 1x 6x 1 1 1 , ?, , 1, . f ′ ( x )>0 ; ?x ∈ 0 ?x ∈ e e e

(

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x 1 , e )?, f ′ ( x ) <0 ; e ,+∞ )?, ∈( ? x ∈( 1 f ′ ( x )> 0 . = ?x =e ( x ) $x ??[,+ f -? e üe????e?, ?[>,+-??'0? x ≠

2 m ( 2 ) ( 0 ) ( 3 ) , $f ≤f 6f ≥f a 3 2 - 4 m+ 1 2 m a 4 a + 1 2 m+ 1 + 1 m+ 1 , %- ≤1 6 ≥9 2 a 2 - m ( 2 m- 3 a ) m , %3 ≤a 6 ≥0 2 a 3 a a %m ≤ 6m ≤06 m= . 3 2 a m ?? 0< ≤ . 3 2 m 3 a 3 , < m< ?, ③? 2< < %a ?2?ù: a 2

( )

2 0

x f ′ ( x )

0( 0 , 2 ) 2 + 0

2 m 2 m 2 m , ) , 3 (2 a a ( a ) - 0 +



   &'()* 

 !+)

Ⅰ. 

?Q ?ü ?Q ?? ?Q f ( x ) 1 ?A e ?? e ?A

9 m+ 1

3 1 D  【 x=x , 】> 4 h1 x=0 6 x=2 6 x=

2 m ( 0 ) ( 2 ) ( 3 ) , $f ≤f 6f ≥f a 3 2 - 4 m + 1 2 m a + 1 4 a + 1 2 m+ 1 m+ 1 , % ≤1 6- ≥9 2 a 2 - 4 m ( m- 3 a ) m a , % ≤06 3 ≥4 2 a 4 a 06 m a % m= ≥3 6m ≥ . 3 4 a 3 a ?? ≤m< . 3 2 a 4 a ??"?, ?+ m-je??[ m ≤ 6m ≥ . 3 3

( )

-2 ( , ÷ ?) () ? ? ? - ù ú m ? 0 ?, §± 3 y=4 x H j± y=x °?V+?¤?:-qr? . ( ) =4 ∫ 2 C  【 ( 2 x+e) d x=( x +e) =( 1+ 】 ∫
3 0 1 x 2 x 1 0 0 0 e )-( 0+e ) =e , . <?; C

14 2 4 x - x) d x= 2 ?-??= ( x - x 0 4





     

3 B  【 ( x ) d x ′ ( x )=2 x , 】 <= f [ü+, "# f



2 1 0



( x ) =x +c ( c , "#0? f =ü+) "# x +c=
2 3 x + 2 1x +c x 3



(

)

, h1 c=-
1 2

2 , f ( x ) d x= 3 0





9 ( 1 ) ( x ) 1 , + , ′ ( x )= ,+ f -??'=(- nf ∞) 1 1 m , ′ ( 1 )= 0 , + >Rm? f $ m =- , ?? 2 x + 1 1- x f ′ ( x )= , 2 ( x + 1 ) , 1 ) f ′ ( x )>0 , ( x ) ?x ∈(-1 ?, ,+ f °ZJ (- 1 , 1 ) ??Q?A; 1 ,+∞ ) f ′ ( x )< 0 , ( x ) ?x ∈( ?, ,+ f °ZJ ( 1 , + ??Q??. ∞) 1 ∴,+ f ( x ) = 1?j1?üe, °x $ m=- . 2 ( 2 ) : h ( x ) = f ( x ) - g ( x ) = f ( x )- ?Z & f ( x )- f ( x ) 1 2 ( x - x )- f ( x ) , 1 1 x x 1- 2 f ( x )- f ( x ) 1 2 ′ ( x )= f ′ ( x )- . |h x x 1- 2 ( x ) x , x ) <=,+ f °ZJ( ?0 ?, |()w? 1 2 ( x , x ) , f ′ ( x ?? 0 ?: ì° x ∈ ? 1 0 1 2 0) = f ( x )- f ( x ) 1 1 2 +m , ′ ( x )= , ∴h ′ ( x )= ! f x x x + 1 1- 2 x x 1 1 0- - = , f ′ ( x )- f ′ ( x )= 0 x + 1 x 1 ( x + 1 ) ( x 1 ) 0+ 0+ ∴? x x , x ) h ′ ( x )> 0 , ( x ) ∈( ?, oB h ?Q? 1 0 ∴h ( x )> h ( x ) = 0 ; A, 1 x , x ) h ′ ( x )< 0 , ( x ) ?x ∈( ?, oB h ?Q??, 0 2 ∴h ( x )> h ( x )= 0 ; 2 x , x ) , ( x )> g ( x ) . $~?m x ∈( ?L f 1 2 ( 3 ) ∵λ 1 , 0 , 0 , x x ?Z: λ nλ λ 1+ 2= 1> 2> 2> 1> - 1 , ∴λ x + x - x = x ( -1 ) + x λ λ λ 1 1 2 2 1 1 1 2 2 = ( x - x ) > 0 , ∴ x + x > x , , x λ λ λ 2 2 1 1 1 2 2 1 ?? λ 1 1+ x x , ∴λ x x x , x ) , λ λ ∈( 2 2< 2 1 1+ 2 2 1 2 2 ) x , x ) , ( x )> g ( x ) , !>( ?~?m x ∈( ?L f 1 2 f ( x ) - f ( x ) 1 2 ∴f ( x x )> ( x x x )+ λ λ λ λ 1 1+ 2 2 1 1+ 2 2- 1 x x 1- 2 f ( x )= f ( x )+ f ( x ) . λ λ 1 1 1 2 2

d ( x +c ) d x=∫ x= ( x-2 ∫ 3)
2 0 0



(

13 x - 3

2 x 3 4

)

1 0

=-

1 . 3

2 x nx 】 <=,+ y=eH,+ y=l \= 2 【 e ?,+, ??+O5§± y=x ~?, !,+ y=e 1 , e ) , H§± y=e -??[\ = ( "# B ? C ?
x x ( e × 1- e d x ) =2 e - 2 e -??= 2 0 x





1 0

=2 e -

( 2 e-2 ) =2 , >ùú4? - 4??à ??, 1" g-4? P = S 2 B? = 2. S e F??

2 5 B  【 ( x ) =-x + 1 , 】>RT?+$? f |"g 2 -x + 1 ) d x=2 - x +x ??= 2 ( 0 3






(



)

1 0



4 . 3

6 C  【 x x=4 , 】> y=槡 d y=x-2 01, " x , y=x - 2 dy y"?:-qr??#> y=槡 ?? = 2 x ( x-x+2 ) d x= ( 槡 ∫


2 3 12 2 x - x + 3 2

)

4 0



1 6 . 3

x x 2 7 C  【 e + 2 x ) d x=( e +x ) 】 ( 0 0 )-e =e , . 1 $; C





1 0

1 =( e +



4  【 】 >Rm? 9 2 3 2 x , ∴ x 槡 3
a 0

2 x d x=a. = ! ( x )′ 槡 ∫ 3

3 2





2 =a . %

2 3 4 2 2 a =a , ∴a= . 3 9



5  【 】>w?01 4 1 1 0 x ,      x ∈ 0 , , 2 f ( x )= 1 -1 0 x+1 0 , x ∈ , 1, 2

{

[ (

] ]

2 1

1 2 1 0 x ,      x ∈ 0 , , 2 f ( x )= | y=x 1 2 -1 0 x +1 0 x ,x ∈ , 1, 2 1 <?,+?+, ?ú? 3 "3. "g??

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[ (

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3 3 2 1 3 2 =2 2 2 x ) d x=2 4 = . x- x 4×2- 0 3 3 3 0 0 1 x x 0 - 1 6 1-  【 x=e =e -e =1- $%】 ed - 1 - 1 e 1 . e

(

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& ' ( # $ )   * + , 1 0

S=

( 1 0 x) d x+∫ ( -1 0 x+ ∫
2 2 0
1 2

1 2



73  【 $% 】( )? ?? - é b 0 ? " g ? ? =

1 0 x ) d x =
2 5 x




3 π 2

| c o s x | d x=
3 π 2 π 2




π 2

c o sx d x+

x= o s xd ∫c
π 2 π 2

3 π 2

1 03 x 3

1 2





(



1 03 x 3



1 ú? 3

)


1 2

∫ 0 π 3 π π s i n -s i n 0-( s . i n -s i n ) =3 2 2 2




π 2

c o s x d x-

c o s x d x=s i nx

-s i nx

3 π 2 π 2



5 1 0 1 0 1 1 + - +5 - - × +5× 1 2 3 3 8 4 5 = . 4 =
1 2 2  【 x +s i nx ) d x= $ %】 ( - 1 3

(

) (

)





(

13 x - 3

c o s x

)

1 - 1



2 . 3

2 0 1 6 Ⅱ.
1 1 π 2 x 1 +e- - 2  【 1-x + e - 1 ) d x= $%】 ( 槡 - 1 2 e



x+∫ ( e -1 ) d x . 1-xd x <= ∫ 槡 ∫槡1-xd 1-xd x= 23???-?gC?-??, %∫ 槡 π , e -1 ) d x=( e -x ) =( e -1 )- B ( 2 ∫ 1 -2 ( e +1 ) =e- , (槡 1-x +e - "# ∫ e
2 x 2 - 1 - 1 - 1 1 2 - 1 1 x x 1 1 - 1 - 1 - 1 1 2 x - 1







4 3 2 ∵( 1 , 2 )=j± f ( x ) =x -x +x +  【 $%】 3 1?-?, 1 , 2 )? - ] ±-? ? = k , ?°? ( | 2 k= f ′ ( 1 )=( 3 x - 2 x + 1 ) | 2 , ∴ °?( 1 , 2 ) x = 1 = ( x-1 ) , x . ?-]±?X= y-2 =2 % y=2 2 ∴ ,+ y=2 x ( x ) =x H,+ g ?:-???ú 3 2 . B?C?"3 ? y=2 x , ( 2 , 4 ) , O ( 0 , 0 ) . > 1?W j ±-?? A 2 y=x , 1 ( 2 , 0 ) ,! S ×2 ×4 = 4 ,j ± ? B O B = △A 2 2 g ( x )= x H§± x= 2 , x y?:-??-?? 3 2 2 x 8 2 S= x d x= = , ∴y = 2 xH , + 0 3 0 3 2 g ( x )= x ?:-??-??= S ′=S O B -S= △A 8 4 4- = . 3 3

{



1 ) d x=

1 π +e- -2 . 2 e

2 a<b<c  【 $ % 】( ) ? ? ? - ù ú m ? a= x d x= ( 1-x ) d x . 1-x< ? 0<x<1 ?,
0 0 2 1-x< 槡 1-x , 0 , 1 )?`W,+ "#°ZJ( 槡 2 y=1-x , y= 槡 1-x , y= 槡 1-x -?+o?× , x=1 ?, °? x=0 ?`W,+-?+('. () a<b<c . ???-ùúm?1 3 2 3 - 4  【 ( x )=x + x f ′ ( 1 ) , ′ ( x )= $%】 <= f "# f 2 3 x +2 x f ′ ( 1 ) , ′ ( 1 ) =3+2 f ′ ( 1 ) , f ′ ( 1 )= "# f 2 2 1 1 4 3 - 3 . ( x ) d x= =- 4 . "# f x + x f ′ ( 1 ) 0 0 4 3











[

]

4 B  【 $%】 ?j±?X y=xH§±?X y=x ? . X?X?, h1 x=0 6 x=1 ?'??0?;C BFG. 3 2  【 5 ( x )-?+0? f ( x ) =a ( x+ $%】() f 3 2 ) ( x- 2 ) ( a<0 ) . ( x ) 0 , 4 ) <= f -?+I( ?, " , . ( x ) =-( x+2 ) ( x- # -a=1 % a=-1 "# f
2 2 )= 4-x . "# S =



2 ú? 3 2 2 1 3 2 9 s = ( 3 t + 2 ) d t = 3t = m  【 $%】 + 2 t 1 1 2 2 3 7 1 3 ×4+4- 3 +2 =1 0- = ( m ) . 2 2 2 2

∫ ( )

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)

2 1 0 4m  【 ( t ) =t -4 t +3 =( t -1 ) · $%】<= v

( t -3 ) , 0 , 1 ]d[ 3 , 4 ]? v ( t )≥ 0 , "#°ZJ[ 1 , 3 ] ( t ) . =4s ° ZJ[ ?v ≤0 "# t ?-?X=
2 s = ( t - 4 t + 3 ) d t + 0





( 4-x) d x= 2 ( 4- ∫ ∫
2 - 2 0





4 t +

3 ) d t



( t- ( t- 4 t + 3 ) d t+ ∫ ∫ + t -2 t +3 t (1 ) 3
2 2 1 3 3 2 1 0





2 2

t -2 t +3 t (1 ) 3
3 2

3 1

3 2 + 1t -2 t +3 t 3

(

)






∴c o s α= 7

1 . 5

4 4 4 + + =4 ( m ) , %/?° 4s ¤-?X 3 3 3 . = 4m     &'()* 
Ⅰ. 
 !+(

4 2  【 i nα =- 槡 1-c o s  】{Rm1 s α= 3 4 s i nα 4 t a nα = = . - , c o s 5 α 3 4 3 - .| s i n2 s i nα c o sα = 2 × × α =2 5 5 4 4 = -2 . - 2 5 5

8 A  【 o s 1-s i n2α = 】>Rm0? c α =- 槡

1 B  【 ∵c o s ( - 8 0 ° )=c o s 8 0 °=k , s i n 8 0 °= 】 1-k 2 1-k , ∴t a n8 0 °= 槡 ,B t a n1 0 0 °= 槡 k
2 1-k a n8 0 °=- 槡 ,$; B . -t k π , 2 D  【 ∵θ ∴2 】 ∈ π, θ ∈ 4 2 2

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)

     

9 A  【 ∵s i nα - c o sα = 槡 2 , ∴( s i nα -  】
2 c o s ) = 1-2 s i nα c o s , ∴2 s i nα c o s α α =2 α =- 1 , ∴s i n 2 . α =-1

[

]

, π, [π 2 ]

2 0 1 6 Ⅱ.
2 1 B  【 ∵r= 槡 6 4 m + 9 , ∴c o s 】 α=

8 m -

2 o s2 , ∴c o s2 1-s i n 2 $ c θ≤ 0 θ =- 槡 θ=



槡( )
7 槡 1- 3 8



=-

1 . 8 1-c o s 2 θ = 2

2 6 4 m + 9 槡 2 4 4 m 1 1 - , ∴m >0 , = , ∴m = . n 2 5 2 5 6 4 m +9 2



2 2 o s2 s i n , ∴s i n !c θ = 1 -2 θ θ=

4 4 2 2 2 B  【 s i n o s i n o s  】 α -c α =s α -c α= 2 2 s i n 1= α-

1- - 1 8 9 3 = . ∴s i nθ= , . $; D 4 2 1 6 3 ( 2-s i n 2 , 1-c o s 2 )  【 3 】 ?ú? 3 "3, ?

(

)

2 3 -1 =- . 5 5

3 B  【 x -y=0 】>¨ θ -?°§± 2 ?, 0 a nθ=2 , 1t o s o s -c -2 θ-c θ = =2 . ?? = 1-t c o s i nθ a nθ θ-s 4 4 A  【 i n π +α =c o s 】<= s α= , α= 5 2

P C⊥ x A D∥ x C A 、 y5 C ?, y? P 5 D?, , P

)

A B . P =2 , ∵ ?g= 1 , ∴∠B A P =2 >Rm?B A P= }V,$ ∠D A P c o s 2-

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π 2- 2

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3 3 i n a n ? ?+?-¨, "# s α=- , |t α=- . 5 4 | s i nα s i nα| + 5 D  【 , 】 ?? = >Rm? | c o s o s α| c α s i nα -?°?ó、 ?+?-¨??±?, H ¨α c o s 、 , = 0 . α-~e^_ ò?^? "#?? 6 D  【 i n 2 i nα , s i nα c o s 】> s α =-s 12 α= -s i nα , ! α∈ o s $c α =- i nα≠ 0 , %s , π, (π 2 )

3 ú? 3 4 C  【 c o s 3 0 0 °=c o s ( 3 6 0 ° - 6 0 ° )=c o s 6 0 °= 】 1 , . $; C 2 5 π π  【 c o s =s i n 2× π +φ , 】>Rm, 6 3 3

1 , t a nα =-槡 3 . 2 s i nα+c o s 1 α 7 C  【 ∵ =-2 , ∴t a nα = . 】 s i nα-c o s 3 α 2 c o s i nα c o s α+s α 2 ∴c o s α +s i nα c o sα = = 2 2 s i n o s α+c α 1+t a nα 6 = . 2 t a n α+1 5 8 A  【 a nα=2 , 】 <= t α =?`+?¨,"# 25 5 s i nα =- 槡 , c o sα =-槡 , i n α+ π = |s 5 5 4

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12 π π π i n2 k k %s π+ 6 2 π+ +φ = , +φ =2 6 2 3 3 5 π ( k ) . , π ∈Z <= 0≤ φ <π "# φ = . 6 6 5 π π 6 C  【 ∵s i n i n o s , 】 α +α =s +α =c 2 2

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3 1 0 π π s i nα c o s + c o s s i n =- 槡 , . α $; A 4 4 1 0 9 C  【 2 t a n( )-3 c o s π +β +5 =0 】 π -α 2 t a nα+3 s i nβ+5 =0 , @A= -2 ①

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t a n( )+ 6 s i n( )- 1=0 a nα- π +α π+β @A= t 6 s i nβ-1 =0 . ② i nβ , a nα =3 . > ①② &? s h1 t !α ='¨, ( 3 1 0 2 2 i n o s , i nα = 槡 . α+c α =1 h1 s )s 1 0

6 π  【 ∵f ( x )° $ %】

π , ? ( L ? Q ., [π 6 2] π =- f π , π f = f2 i n ω +φ (π ( 6) ∴s (2 )= 2) ( 3) 3 -s i n i n ω +φ ω +φ (π ) =s (π ). 6 2 >) ? f < ` ¨, +? + 01 2 π ω +φ = π + 3

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2 π π , ∴ω =2 , ∴T= =π . ω +φ 6 ω
2 2 2 2 7 ( 1 )> | a| =( 3 s i nx ) +( s i nx ) =4 s i n x , 槡 2 2 2 | b | = ( c o s x ) +( s i nx ) =1 , a | = | b | , d| 1

26 π ∴s i n3 o s α= 槡. +α = -c 5 2

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2 π T= =π , . $; B 2 2 A  【 y=s i n2 x $%】 -? + ? l ? T i n2 x+ UV1× y=s 1 W?? 2

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π f ( x )=s,+. ?, 2

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c o s 6 x c o s 6 x 4 D  【 ( x )= x - , f ( -x )= - $%】 ,+ f x = 2- 2x 2x - 2 -f ( x ) , =s,+. f ( x )→ +∞; ?x n x>0 ?, ?x n x< →0 →0 0?, f ( x )→ -∞; x - x 2 -2 f ( x )→ 0 ; ?x → +∞ ?, → +∞,
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2 7 π π 3 π π +2 k x- +2 k , + π≤ 2 ≤ π h1 2 3 2 1 2 1 3 , k . π +k π ∈Z 1 2

7 1 π π π 1 π <2 4 , < t + < , 0< !0 ≤t <? %1 6 1 2 3 6 8 . 0?á 1 8??M*à??+. t<1 °1 9 ( 1 ) ∵ ,+ f ( x ) , ∴A+ 1=3 , . -Jüe[ 3 % A=2 ∵ ,+?+-^-??~?yIJ- hi = ∴ J?F23 T=π , ∴ω =2 . π + ( x )-hz?= f ( x ) =2 s i n2 1 . $,+ f x- 6 π , 2

k π≤x ≤

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π π π π , ∴- <α- < , 2 6 6 3 7

1 5 5 π s i n = , ( 1 )= , ③ ]:X; >]±?X? f 6 2 2 1 f ′ ( 1 )= , ( 1 )+ f ′ ( 1 )=3 . |f ??, ①④ :X. 2 π +2  【 i n( x+θ ) ( ,  】,+ y= s ω ω >0 2 0< ) θ<π =t,+, 01 θ= π , o s x . "# y=c ω 2 2 π x , !| - J?e= π "# T=π = , 1 1 -x 2| ω π , +2 . × ω =2 "# ω +θ= 2

π π π = , $ α= . 6 6 3

2 0 1 6 Ⅱ.

1 C  【 , 】.buvf, ~5;C A > J üe= 101 | a | =1 , a | =2 , >23= π01 | ??r ;C A0#uv; ??01;C B]FG; ? s, x= 0?, f ( x ) =a , ≠0 $;C CFG. 2 B  【 x 】>Rm0? 2 0+ x 0 = π =k , k , π ∈Z $ 3

     

2 2 8 ( 1 )<= f ( x ) =s i n x-c o s x+2 3 s i nω x · ω ω 槡

k π π π - , k , x ∈Z $ k=0?, ∈ 0 =- 2 6 6

. $; B , 0, [ -π 2 ] 3 B  【 , T=4 , 】 >R?? A=2 π | ω= ( x ) =2 s i n ?f 1 , ? 2

c o s x+ o s2 x+槡 3 s i n2 x + λ= ω λ =- c ω ω π + 2 s i n2 . λ x- ω 6 ( x ) >§± x=π[ y=f ?+-V?~?y, 01

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4 D  【 f ( x )=s i n ( x + )+ c o s ( x + )= 】 ω φ ω φ π , 2 s i nω , <= J ?F 23 = π "# x+φ+ 槡 4 2 π =π , . ( -x ) =f ( x ) , "# ω =2 !<= f "# ω fπ = f - π ,% 槡 2 s i n 2× π +φ+ π = 8 8 8 4

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( 2 )> y=f ( x ) , -?+I? π, 1 f π =0 0, 4 4

( ) ( ) ( ) π 2 s i n i nφ = c o sφ , +φ+ ] , %s 槡 [ 2×( - π 4 8)
"# φ = π π π , ( x )= 2 s i n 2 x + + "# f 槡 4 4 4

) ( ) π π π =-2 s i n s i n = % λ =-2 × - ( 2× 5 4 6 4 6) π -槡 2 , ( x ) =2 s i n 2 . $f x- ) -槡 (5 6 3
> 0≤ x ≤ "# - 3 5 π π π 5 π , 1 - ≤ x- ≤ , 3 5 6 6 6 1 i n 5x- π ≤ 1 , 2≤ ≤s 1 -1-槡 2 3 6 2 s i n 5x- π -槡 2≤ 2 -槡 2 , ( x )° $,+ f 3 6 3 π -1-槡 2 , 2-槡 2 ] . 0 , ?-je??=[ 5 3 2 9 ( 1 ) f ( x ) =a ·b+ | b | + 2 3 2 2 2 =5 3 s i nx c o s x+2 c o s x+4 c o s x+s i n x+ 槡 2 5 2 =5 3 s i nx c o s x+5 c o s x+ 槡 2

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2 c o s 2 x , . <?; D 槡 x+φ -x+φ 5 C  【 i n =s i n , 】 ?fV: >s 1 3 3 s i n x φ c o s =0! :X, ∵φ∈ [ 0 , 2 ] , ∴φ = π 3 3

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3 π . 2 x+φ x φ π f ( x ) =s i n =c o s + - ?fó: 3 3 3 2 [t,+, |

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φ π - =k ( k ) . 0 , π ∈Z ! φ∈ [ 3 2 3 2 ] , ∴φ = π . π 2 6 ①④ 【 ( x ) 0 , 】ef ① :X; >Rm, ?f °[ 1 ]?[?,+, ?θ ∈ π s i nθ>c o s , θ , ?, (π 4 2)

53 1+c o s 2 x 5 = 槡s i n 2 x+5× + 2 2 2 π +5 =5 s i n2 , x+ 6

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5 π π f ( s i nθ )< f ( c o s ) , > , θ $ ② ] : X; ? 6 6

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1 1 ( 1- s i n 2 )= . α 2 6 2 C  【 a nα> 0? α° ? Ⅰ 6 ? Ⅲ +?, $% 】 >t i nαH c o s i n 2 2 s i nα c o s ?? s α??, $s α= α> 0 , . $; C 2 2 2 s i n B- s i n A 3 D  【 = $ %】 >F???01 2 s i nA 2 i nB (s ) s i nA


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( ) π =5 f i n2 x+5 ( x-1 ) s 2 π π =5 s i n x+ - ) +5 (2 6 6 π π π o π =5 5 i n o s2 s -s i n c x+ ) c x+ ) ] + [s (2 6 6 ( 6 6
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3 1+c o s 2 x 1 0 ( 1 ) f ( x ) = 槡s i n2 x+ +a = 2 2 π π +a+ 1, s i n2 ∴T = π . +2 k > π≤ x+ 2 2 6 π 3 π π 2 x+ ≤ +2 k ( k ) , +k π ∈Z 1 π≤ x ≤ 6 2 6 2 π +k ( k ) . π ∈Z 3

( 1 π π 0 , + ° ? ` + ?, ∴s i n( θ+ = > $θ 2 4) 4 5 2 2 5 π = -槡 , i nθ o sθ= -槡 , +槡 c oB 槡s 4) 5 2 2 5
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25 2 o s 1- s i n "# c α=-槡 α=- 槡 . 5 i n $s 2 π π + o s i nα=槡 × i n c c o s s α α+ (π ) =s 4 4 4 2

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2 π +k , + π [π 6 3

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π π π π 5 π ∴ - ≤2 x+ ≤ , ≤x ≤ , 6 3 6 6 6 1 π ≤1 ∴ - ≤s i n2 . x+ 2 6 ( 2 )-

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=1 .

 

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2 9 ( 1 ) ( x )=( a+ 2 c o s x ) c o s ( 2 x + ) <= f θ [s,

2 2 s i n 4 c o s 4 s i nα c o s 0 α+ α+ α 1 = , ? V ? a?0 2 2 4 s i n α+ c o s α 2 t a n α- 8 t a nα- 3= 0 , a nα= 36t a nα= h1 t 13 1 3 2 t a nα - , a n 2 =- . 5[ t α= 2 3 4 1- t a n α - c o s 7 0 1 ° 2 s i n 3 5 ° - 2 2 3 C  【 = =  】 c o s 1 0 ° c o s 8 0 ° c o s 1 0 ° ·s i n1 0 ° 1 - c o s 7 0 ° 2 . =- 1 1 s i n 2 0 ° 2 c o s - s i nθ 1- t a nθ θ 4 3+ 2 2  【 = . 】 ?? = 槡 c o s + s i nθ 1+ t a nθ θ π , ∵2 , 2 ) , ∴θ π. θ ∈( π π ∈ 2 2 t a nθ a n 2 = =- 2 2 , Bt θ 槡 2 1- t a n θ

     

(

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2 ∴槡 2 t a n - t a nθ - 2= 0 , θ 槡 2 t a n + 1 ) ( t a n - 2 0 . %( θ θ 槡 槡 )=

2 槡 a nθ =- a nθ = 2 ( . $t 6t ÷?) 槡 2 2 槡 1+ 1- t a nθ 2 ∴ = = 3+ 2 2 . 槡 1+ t a nθ 2 槡 1- 2 2 2 2 2 1 3 2 s i n 1 2 s i n s i n c o s α+ α+ α+ α 5 = =  【 】 s i n 2 4 2 s i nα c o s α α 2 2 2 2 3 s i n c o s 1 3× + 1 1 t a n 2 3 α+ α 3 α+ = = = . 2 t a nα 2 s i nα c o s 2× 2 4 α 3 1 - t a nθ 1 π -  【 6 t a n = , a nθ = θ= 】 1t 4 1 0 1 + t a nθ 2 s i nθ t a nθ 1 c o s θ = , ∴s i nθ c o sθ= 2 = 2 2 3 s i nθ + c o s a nθ + 1 θ t 1 3 3 = . 1 1 0 + 1 9

a + 2 c o sx c o s ( 2 x + By =t,+, "# y +, 1= 2= π ) 0 , ) , = , ( x )= θ =s,+, >θ ∈( π 1θ "# f 2 2 x ·( a + 2 c o s x ) , - s i n 2 = 01 - ( a + 1 )= 0 , =- 1 . %a (π 4) 1 α ( 2 ) 1 ) ( x )= - s i n4 x , >( 1f <= f (4 )= 2 >f 1 2 4 - s i nα=- , i nα= . %s 2 5 5 3 , o s π, oB c α=- , (π 2 ) 5 π π π i nα o sα = i n c o s +c s i n "# s ( α+3) =s 3 3 !α ∈ 4- 3 3 槡 . 1 0
2 0 1 6 Ⅱ.



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3 槡 3 π π 槡 s i nα c o s + c o s s i n + c o s , i nα+ α α= % s 6 6 4 2 3 3 1 3 1 c o sα =槡 , s i n α +槡 c o sα = , $ $ 4 2 2 2 4 s i n α+

33 3 3 s i nx ( )= 3 s i nα= 槡 ?s i nα= . ?f α >α ∈ 槡 槡 5 5 4 π 2 k , 2 k k ) o s ( )= π π+ ( ∈Z ?c α= , ng α 2 5 1 2α 2 s i n = 1- c o s α= . 5 2

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4 1 π π =- s i n α+ =- . i n α+ . |s $; B 3 3 4

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2 1 0 , 2 C  【 s i nα+ 2 c o s ) = 槡  】 >( α 01 2

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π 2 c o s = . 1 槡 4 + (2) f 2 θ

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x x 1+ 2 = 2

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x x x π x π 3+ 4 2 n - 1+ 2 n = 2 = 2 , …, ( n- 1 ) π+ , π+ 2 2 2 2 π , x … +x x 5 9 "# x π+ π+ π+ 1+ 2+ 2 n - 1+ 2 n= 2
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1 A  【 E B+F C= $% 】 2

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1 → → → → ?B ?A C )= ( A B+ A C )= D , . $; A 24  【 ∵O A= (- 3 , 1 ) , O B= (- 2 , k ) , $%】 k - 1 ) .

?→ ?→ → ?→ ?→ ∴A B=O B -O A=(-2 , k ) -(-3 , 1 ) =( 1 ,
A , A B !∵O =??^-??"~é-??, ? → → ?→ → ∴O A B , A ·A B= - 3× 1+ 1×( k-1 )= ⊥A %O - 4+ k = 0 , = 4 . %k

A - B 2 c o s 9 ( 1 ) 2 c o s B- s i n ( A- B ) s i nB+ >Rm0?, 2 c o s ( A+ C ) = [ 1 + c o s ( A - B ) ] c o s B- s i n( A - B ) ·s i nB- c o s B = c o s ( A- B ) c o s B- s i n( A- B ) s i nB 3 = c o s ( A- B+ B )= c o s A=- , 5 3 o s A=- . "# c 5 4 ( 2 ) 1 ) i nA= . i nB= >( 0? s >F???, 1s 5 b s i nA 槡 2 = . a 2 2 槡 = 4 2> 5= b , , o s B= . !a |∠B< ∠A %c 槡 2
2 3 2+ c - 2 5 槡 2 o sB= = , !> / ? ? ? 1 c h1 2 2× 4 2 c 槡 c = 16 c = 7 ( . ÷?) → ?→ B A ·B C → ? → A° B C? ? ? - ? ? = ?→ = $? ? B | B C |

?→ →

3 D  【 i n { | a + b | , | a - b | } i n { | a | , $%】 ~5 m Hm | b | } , ^?5?????-~¨± UV - t ?? H?-?U-t??? t, ? u -ü?OP]?, A , B ; | a + b | , | a-b | <? QD B T- t ü?H | a | , | b | 0 9 : 0 ' ¨ ` ¨ ? - ` ?, <?L
2 2 2 2 m a x { | a + b | , | a - b | } a | +| b | , . ≥| <?; D

49 0 °  【 O= $%】 >A

?→

1 → → ( A B+A C ) , C 01 O= B 2

C=? O-§#, B C -T?, $B "#A HA -;¨ 0 ° . =9 5 B  【 1 , 2 ) , b=( 3 , 1 ) , $%】 >5 a=( 5[ b- a = ( 3 , 1 )- ( 1 , 2 )= ( 2 , - 1 ) , . ;B 6 A  【 2 , 4 ) , b=(-1 , 1 ) , $ %】 < = a=( "# 2 a - b = ( 2× 2- (- 1 ) , 2× 4- 1 )= ( 5 , 7 ) , . ;A 7 1 2  【 ∵a , ∴s i n 2 = c o s , ∴2 s i n c o s = $%】 ∥b θ θ θ θ 2 1 π 2 , , c o s . ∵θ ∴2 s i nθ = c o s , t a nθ = . θ ∈ 0 θ 2 2

→ →

B A | ·| B C | c o s B 2 | 槡 = c ·c o s B= . ?B → 2 | C | π + 1- c o sx 6 3 1 0( 1 ) ( x )= i n x+ +槡 s <= f 2 2



?→

(

)

(

)

(

4 8 D  【 t B C1 A B= 5 , A D= 槡 5 . $%】 hR △A % 槡 5 4→ 4 ?→ → 4 4 ?A → D= A B= ( C B- C A )= a - b , . $; D 5 5 5 5
2 9 C  【 ∵a , ∴1× 2- m = 0 , ∴m=± 2 . $%】 ∥b 槡

π 6

3 1 π + ) - c =槡 s o s i n( x = ) -1 ( x+π 6 6) 2 2 2 3 π π - = s i nx , ( B ) =槡 , !<= f $ 6 6 2

+ s i nx

(

)

1 0( 1 ) ∵P A+P B+P C= 0 , P A+P B+P C= hfV: ( 1- x , 1-y )+( 2-x , 3-y )+( 3-x , 2-y )= ( 6- 3 x , 6- 3 y ) , ∴ 6- 3 x = 0 , = 2 , y = 2 , h1 x {6- 3 y = 0 ,

→ ?→ ?→

→ ?→ ?→

3 π 槡 s i nB= . <=¨ B='¨, "# B= . 2 3 2 2 2 i nC=2 s i nA , = 2 a , a - >s 1c " # b =a +4 π 2 2 2 2 2 a ·2 a c o s = 3 a , a + b , B C= "# c = $ △A 3 2 π π π π C= , A= - = . §¨`¨?, 2 3 2 6

P= ( 2 , 2 ) , O P | = 2 2 . %O $| 槡 → → → ?P ?P ∵P A+ B+ C= 0 , hfó:

?→

?→

O A- O P )+ ( O B- O P )+ ( O C- O P )= 0 , |(

?→ ? →

?→ ?→

?→ ?→

2 8

1 ?→ ?→ ? → ?→ ∴O P= ( O A+ O B+ O C )= ( 2 , 2 ) , 3 ∴| O P | = 2 2 . 槡 ?→ → → ( 2 ) ∵O P= mA B+ n A C , ∴( x , y )= ( m+ 2 n , 2 m+ n ) , x = m+ 2 n , ∴ y = 2 m+ n , n = y - x , ??^?, 1 m- y - x = t , 3 4 & >ú? ?, =x +t ( 2 , ?§± y I? B 3 ) t , ?, j1 J üe 1 $m - n . -Jüe= 1

x = 3 , y = , y + . C D #x β α α+ β= >5? P° △B 3 , ¤( MN?  ) S\, += α+β=y+ x , ? 3

?→

{

6"3, ú? 3 ?? PH ( 1 , 1 ) ( '?, α+ ?B =y+ β 4 ú? 3 x j1 J ü e, 3 1 = 3 6 ú? 3

? J ü e = 1+ 4 . 3

     

2 0 1 6 Ⅱ.

1 D  【  】 >Rm01 c H d}±, |ì°?+ k = , λ , d , 1 , λ ?1 c=λ % h 1 k= - n c= 1=- , λ - a + b =- ( a - b )=- d , . $c Hd ??, $; D

9 (- 4 , - 2 )  【 ∵ b=( 2 , 1 ) ,  】 n aH b-? ∴? a = ( 2 , ) ( 0 ) . ?^?, λ λ λ<
2 2 2 ∵| a | = 2 5 , ∴4 2 0 , 4 , 2 . λ+ λ= λ= λ=- 槡

{

2 2 A  【 B = A B ·A C+ B A ·B C+ C A ·C B , 】 >A

→ → → → ?→ → ? → → → → → ?→ → ?→ → 2 B -A B ·A C =B A ·B C -C A ·B C , B · ?A |A → → ? → → → → ? → ? → A B- A C )= B C ·( B A-C A ) A B · C B = B C · ( ? ?B → ? → → ? → ? → → ? → C B ·( A B+ B C )= 0 , B ·A C= 0 B ?C "# C ?C ⊥ → A C , B C=§¨`¨?. $△A

∴a = (- 4 , - 2 ) . 1 0 # O=[\ ??, O A "°§±= x y? X??§¨ [ 7 " 3, \ P, ?ú? 3 | 1 3 A ( 1 , 0 ) , B - ,槡 . 2 2

?→

(

)

3 B  【 a + b = ( 1 , 2 )+( , 0 )=( 1+λ , 2 ) , 】 λ λ c = ( 3 , 4 ) , ∵( a + b ) , ∴( 1+ )× 4- 2× 3= λ ∥c λ 1 0 , . h1 λ= . $; B 2 4 B  【 C-T?, 】 >Rm01? P=±? A $ ;C BFG. 5 A  【 5"3, 】 ?ú? 3 A+ O B+ C O= 0 , A+ >O 1O ?O → ?→ B=O C , A C B "#??? O =? ? ? ? ?. < = O= ?O → B C-?~, A | = △A "# | ? → ? → O B| =| O C| , | "#???

7 ú? 3

2 π , O C= , ( c o s , s i nα ) , ?∠A αα ∈ 0 |C α 3

( [

])

?→ ?O → ?O → C= x A+ y B , >O 1

{

1 c o s x - y , α= 2 3 槡 s i nα= y , 2

? → ?→ ? →

?→

3 23 槡 = c o s , y = 槡s , + y = s i nα i nα "# x α+ "# x 3 3 2 π π , , c o s i nα= 2 s i n α+ , α+3 !α ∈ 0 "# 槡s 6 3

(

)

[

]

5 ú? 3

π x + y . j1Jüe 2 ? α= ?, 3    &'()* 
 

6 B  【 ∵B C=a+b , C D =a-2 b , ∴B D=  】

?→ ?B ?C ?B → → → → C+ D= 2 a - b . , B , D`?}±, ∴A B , D !∵A ? → → B=λB D , ∴2 a+p b=λ ( 2 a-b ) , ∴2= }±. ?A

O A C B=??. A O= 6 0 ° , A B= 3 0 ° . "#∠C %∠C

?→

?→

Ⅰ. 
2 2 22 1 B  【 b+t a | =b + 2 a ·b t +a t , 】 >5 | & 22 2 f ( t )= a t+ 2 a ·b t +b , Bt [?m?+, "#0

, p =- , ∴λ= 1 , p =- 1 . 2 λ λ 7 D  【 m , 1 ) , b=( 1-n , 1 ) ( 】 w? a=( ?T m , n , , ( 1- n )= 0 , =F+) ?a ∥b | m- % m+ n = 1 . ∴ 3+ 2 n 2 m+ 2 n n 2 m 1 2 m+ + = + = 3+ + ≥ m n m n m n n 2 m n 2 m

)- J ? e = 1 f(t

2 2 2 4 a b- ( 2 a ·b ) = 2 4 a

2 2 2 2 2 2 2 4 a b- 4 a bc o s b s i n θ 4 θ 2 2 = = 1 , b | s i n = %| θ 2 4 4 a

· = 3+ 2 2 , = ?j ?n ? ? 槡 m n m n 槡 1 2 + -J?e[ 3+ 2 2 , . $; D 槡 m n

, b | 1 $?? θ G?, || ?VG?. 2 D  【 B"°-§±= x B-T 】 #A y, #A B= 4 , C ( a , b ) , y, ?X§¨[\P, ?A ?±= y P ( x , 0 ) , B P 1 , A (- 2 , 0 ) , B ( 2 , 0 ) , P ( 1 , 0 ) . 0= 0

_?, $

8 D  【 D"°§±= x 】 # O=??, #O y?

?→ ?→ ( x , y ) , ∵O P =αO C+ X§¨ [ \ P, ?? P ? → O D , x , y )=α ( 0 , 1 )+β ( 3 , 0 )=( 3 , ) , β |( β α "

?→ ?→ ? → ( b ) , P C= a - 1 , b ) .


B= ( 1 , 0 ) , P B=( 2-x , 0 ) , P C=( a-x , "#P 0

?→

2 9

B ·P C B ·P C , <=!LP ≥P 0 0 2- x ) ( a - x ) - 1!:X, "#( ≥a a?01 x - ( a + 2 ) x + a + 1 ≥0!:X, 2 2 ( a + 2 ) - 4 ( a + 1 ) , , "# Δ= ≤0 % a≤0 = 0 , B-?§??±?, C= "# a % C° A "# A B C . . $; D 3 5 a | ·  【 $% 】 ? ? a° b? ? ?- ? ?= | 2 a ·b c o s 〈 a , b 〉= , b | = 2 , a · b=( e 3 e ) · !| 1+ 2 | b | 2 e 2+ 6 e ·e 2+ 6× 1= 1 2= ???-??= 1 = 5 , "#?? a° b 2
2 2

?→ ?→ ?→ ?→











x+ y+ 3 x y 槡 2 x . 2 y 槡 3 +1 + 4 x 2 1



( ) 槡


y x



3 y 槡 + + 1 x

& ' ( # $ )   * + , -

( 槡
<=

)

(

y 槡 3 + x 2

)



1 1 | x | ≥ , "# -Jüe 4 4 | b |

. =2
2 0 1 6 Ⅱ.

1 B  【 , b , x - 4= 0 , 2 y =- 4 , $%】 >a ⊥c ∥c 12 = 2 , y =- 2 , 2 , 1 ) , b=( 1 ,- 2 ) , a+ |x $ a=( b = ( 3 , - 1 ) .

a ·b 5 = . | b | 2

4槡 2  【 B , A D"°§± $%】 # A=??, ??# A y A y , ( 0 , =x y、 y, ? X??§¨ [\ P x |A 0 ) , B ( 2 , 0 ) , E ( 2 , 1 ) . ( x , 2 ) , B=( 2 , ?F |A 槡 槡 槡 0 ) , A F= ( x , 2 ) . ∵A B ·A F= 2 , ∴x = 1 , ( 1 , %F 槡

2 2 B  【 A B ·B C+A B =A B ·( B C+A B )= $ %】

?→ → → ?→ → → → → → A B ·A C< 0 , A B| | A C| c o sA<0 , ∴c o sA<0 , %|
∴¨ A= ∴△A B C[ ¨, . ¨`¨?, $; B







→ →

3 C  【 O A | =| O B| =| O C| B C $%】 >| ? O= △A

?B → → → ?→ 2 ) , F= ( 1- 2 , 2 ) . E= (2 , 1 ) , ∴A E ·B F= !A
槡 槡

2 . 槡 7 ?→ ?→ ?→ 5 B C 2 0 ° , A B | =  【 $%】 ??A HA -;¨= 1 n| 1 2 3 , | A C | = 2 , B ·A C=| A B | ·| A C | ·c o s 1 2 0 ° = "#A 1 → ? → → ? → 3× 2=- 3 - × . P C , P ·B C= 0 , >A ⊥B 1A % 2

?→ ?→ ?→ → ?→ ?→ ?→ → ?→ ∵P A ·P B=P B ·P C , ∴( P A-P C ) · -?~. ?P ? ? ? → → → → → → → B= C A ·P B= 0 , B ·P C= 0 , B C ·P A=0 , ?? A ? → ? → ? → ∴? P[△A B C- ? ~, N A + N B + N C = 0 > ? ?N → ?N → ?N → A+ B=- C , ?'???f-?????f|
B C-(~, . ? N=△A $; C

?→

?→ ?→

?→

?→

4 B  【 m , n ) p , q ) $ %】 ? a=( H b=( } ±, | m q - n p = 0 , 0 , { à “ ⊙” -??? a ⊙b= $A = m q - n p , = n p - m q , FG. >5 a ⊙b !b ⊙a <? a , , a= ⊙b= -b ⊙a $ B] F G. ~5 C >5 λ ( m , n ) , a ) m q-λ n p , ( a λ λ <? ( λ ⊙b=λ !λ ⊙ b )= ( m q - n p )=λ m q -λ n p , , λ $ CFG. ~5 D 2 2 2 2 2 2 a ) + ( a ·b ) =m q- 2 m n p q + n p+ ( m p + ( ⊙b
2 2 2 2 2 2 2 2 2 2 n q ) = m ( p + q )+ n ( p + q )= ( m + n ) ( p + 2 2 2 q )=| a | | b | , $ DFG.

→ ?→ → → → → A P ·B C=( B +A C ) ·( A C -A B )=0 , λA "# → → → → 2 2 | A C |- | A B |+ ( 1 ) A B ·A C= 0 , 9 λ λ- % 4- λ-

7 ( 1 )= 0 , 3 λ- h1 λ= . 1 2 6 1 ?→ ?B →  【 D-T?, E= C+ $%】 <= E= C "# B 2 1?→ ?→ 1→ → ?→ → ?C → ?A → E= D- D C=A D- A B , A C= A D+A B . <= 2 2

?→ ?→ ?A → → → → A C· B E =1 , C· B E =( D +A B )· "#A
1→ 1 → 1→ → ?A → D- A B =| | D| - A B| + A B· ( ?A 2 ) 2 2
2 2

5 - 3  【 B CT, A B= $ %】 () Rm, ° _ ? △A 2 A D= A C= 4 A E= 4 , E= B A+ A E=- A B+ |0?B 1 → ?→ 1→ → ? → → A C , C D =C A +A D = -A C+ A B ." # 4 2

?→ → →



1 → 2 1 → ?A → D= 1 , A B |+ | A B | ·c o s 6 0 ° = 1 , % 1- | " 2 2 #- 7 ②④ 9 8 -  【 a | | b | · $%】 >??-+??? -| ≤a 8 b a | | b | a | | b | a ·b ( a , b 〉= ≤| ?| ≥- ?n ? ? 〈 2 . 2 a - b | | a | - 4 a · b+ π?_?:X) >| ≤3 ?4 2 2 2 | b | 4 a ·b | a | +| b | | a | | b | ≤9 ?9+ ≥4 ≥4 ≥ 9 - 4 a ·b ·b | a | =| b | , 〈 a , ?a ≥- ( ?n?? 2 8 9 b 〉= ·b π?_?:X) ?a -J?e= - . 8 | x | | x | | x | 9 2  【 = = = $% 】 | b | | x e y e 2 2 1+ 2| x + y + 3 x y 槡 槡 6 1 → 2 1 → 1 → | A B |+ | A B | = 0 , A B | = . h1 | 2 4 2

?B → ?→ E ·C D=

1→ 1→ → A B+ A C· ( - A C+ A B = ( -→ 4 ) 2 )

9→ → 1 → 2 1 9 → 2 A B ·A C- | A B| - · | A C| = ×4× 8 2 4 8 4 c o s A C- ∠B 1 2 1 2 9- 8- 4=- 3 . × 4- × 4= 2 4

π a+ 2 b ) ·( a-b )=  【 $%】 ()w??? ( 3
2 2 - 2 , a | +a · b- 2 | b | = 4+ 2× 2× ?!?1 |

1 π c o s - 2× 4=- 2 , o s = , = . θ h1 c θ θ 2 3 71 8  【 CH B D?5 O?, A C= , $%】 ?A ? ∠P θ P ·A C=A P · 2A O =2 | A P| ·| A O| c o sθ= |A → 2 2 A P | = 2× 3 = 1 8 . 2 | 33 ?→ 82 7+ 槡  【 8"3, B · $%】 ?ú? 3 >w? O 槡 2

→ →



?→



?→

3 0

?O → C=3=3×2×c o s∠B O C , 1
O C=6 0 ° , ∠B > / ?? ? 0 1 B C= 7 , B CT? B C? oB△O 槡 33 A= 4?? A° -?= 槡 . >O 7 槡 4=g#-??, A # O=?~,

   &'()* "*
Ⅰ. 

 

1 B  【 】 >Rm01 8 ú? 3

1 1 A B ·B C ·s i nB= , ! 2 2

33 C-hiJ üe= 4+ 槡, ∴ △A B C?? ק± B 7 槡 33 1 33 +槡 × 7 = 2 7 + 槡. -Jüe= × 4 槡 槡 2 2 7 槡 2 9 ( 1 ) f ( x )= 2 s i nx c o s x + 3 ( 2 c o sx - 1 )= s i n 2 x + 槡 2 π π x + , 3 c o s 2 x = 2 s i n2 ∵ω= 2 , ∴T= = . π 槡 3 2

2 A B= 1 , B C= 2 , i nB=槡 , 4 5 ° "# s "# B= 6 槡 2 B= 1 3 5 ° . 5 ° C= ? B =4 ?, >/???01 A
2 2 A B + B C - 2 A B ·B C ·c o s B =1 , C= ?? A 槡

(

)

A B= 1 , B C= 2 , 9 0 ° , $1 A= Hw??? “ 2 ¨` 槡 1 3 5 ° . rs, ÷?. "# B= > / ???01 ¨?”
2 2 A C=槡 A B + B C - 2 A B ·B C ·c o s B= 5 . 槡 2 2 2 2 2 2 C  【 ( a- b ) + 601 a + b - c = 】 > c=

     

(

)

A+ ( 2 ) ∵f ( A )= 2 s i n2 A+ ∴s i n2

(

π = 1 , 3

)

2 a b - 6  ①. > / ??? d C=

(

1 π = . 3 2

)

π 2 2 01 a +b - 3 2 c = a b  ②. a b - 6= a b , b = 6 . "#>①②1 2 %a 1 3 3 3 π 1 槡 6× = 槡. a b s i n = × "# S B C= △A 2 3 2 2 2

π π 5 π π ∵0< A< , ∴2 A+ = , % A= . 2 3 6 4 B ·A C=| A B | ·| A C | c o s A= 2 , ∴| A B | ·| A C | = BA 槡 1 ?→ 2 ?→ 槡 , | A B | ·| A C | s i nA = . 2 |S B C= △A 2 2 3 3 o s 1 0( 1 ) ∵a , ∴ c x + s i nx = 0 , ∴t a nx =- , ∥b 4 4 2 c o s x - 2 s i nx c o s x 1- 2 t a nx 2 ∴c o sx - s i n2 x = = = 2 2 2 s i n x + c o s x 1+ t a n x 8 . 5 1 i nx + c o s x , - ( 2 ) f ( x )= 2 ( a + b ) ·b= 2s · 4 1 ( = c o sx ,-1 ) =2 c o sx ( s i n x+c o sx )+ 2 1 3 2 2 s i nx i n2 c o s x + 2 c o s x + =s x +c o s 2 x + = 2 2 3 a b π x + + . 2 s i n2 = , >F???1 槡 4 2 s i nA s i nB

?→ ?→

?→

?→

?→

?→

3 A  【 B+ C= , i n 2 A+ s i n ( A- 】 <= A+ π >s B+ C )= s i n( C-A-B )+ 1 i n2 A+ s i n2 B+ 1s 2

(

)

1 s i n 2 C= , i n[ ( A + B )+ ( A- B ) ]+ s i n[ ( A+ %s 2 1 B )- ( A - B ) ]+ s i n2 C= , s i nC c o s ( A- a?1 2 2 B )+ 2 s i nC c o s C= 2 s i nC [ c o s ( A-B )-c o s ( A+ 1 1 B ) ]= , s i nA s i nB s i nC= , i nA · a?1 4 %s 2 2 1 1 1 s i nB s i nC= . b s i nC= b c s i nA= ! S= a 8 2 2 1 1 222 3 c a s i nB , s i nA s i nB s i nC = < ? S = abc 2 8 1 222 1 222 3 abc . >1 ≤S ≤21 1 ≤ abc≤2, %8 ≤ 6 4 6 4 a b c 6 2 , 、 D]V?:X. + c > ≤1 <?;C C !b 槡 a > 0 , c ( b + c )> b c ·a , c ( b + c )> 8 , <? b ≥8 %b >0 , b ( a+ ;C AV ? : X. ! a+b>c <? a b )> a b ·c , b ( a+b )>8 , ≥8 %a ef]?1? a b ( a + b )> 1 6 2 , ;C B]V?:X. ??"?, 槡 . ;A 4槡 3  【 2+b ) ( a-b )=( c - 】 >F???1 ( 2 2 2 b ) c , a + b ) ( a - b )= ( c - b ) c , c- a= %( % b+ 2 2 2 b + c - a 1 b c , o s A= = , 0 , ) , "# c !A ∈( π " 2 b c 2 π 2 2 2 = , c - a = b c b c - 4 , = ! b+ ≥2 ?n?? b #A 3 1 c = 2 c , b c s i nA ?, _?:X, %b ≤4 $S ≤ B C= △A 2 1 3 槡 × 4× =3 , B C??-Jüe=槡 3 . |△A 2 2 槡 4 6 56 0  【 A C =2×4 6=9 2 , A B= ,  】 ° s i n6 7 °

(

)

3 3 2 2 π π 槡 ∴s i nA= , ∴A= 6 A= . % 槡 = , s i nA 槡 2 4 4 6 3 π π A+ = ∵ b>a , ∴ A= , ∴f ( x ) +4 c o s2 6 4

(

)

1 π x + - . 2 s i n2 槡 4 2 1 π π 1 π π , , , ∴2 x + ∈ , !∵x ∈ 0 3 4 1 2 4

) [ ] [ ] 6- 2 ∴槡 槡 ≤s i n , ≤1 ( 2x+π 4) 4 3 1 1 ∴槡 - 1 2 s i n - ≤ 2- , ≤槡 ( 2x+π 4) 2 槡 2 2 π , ])-je? ( x )+4 c o s x %f ∈[ 0 ( 2A+π 6) ( 3
3 1 ?[ 槡 - 1 , 2- . 槡 2 2

(

[

]

3 1

A B B C B CT, = , △A > F ? ? ? 0 ?: s i n3 0 ° s i n 3 7 ° A B s i n3 7 ° ∴B C= 0 . ≈6 s i n3 0 ° 62 3  【 B CT, $%】 hfV: °△A ()F???, 槡

3 6 槡 6 1 3 + 槡 槡 槡 c o s A s i nB= × - × = . 3 3 3 3 3 1 1 B C-?? S = a b s i nC= × 3× 3 2× <?△A 槡 2 2 1 3 2 = 槡. 2 3 1 0( 1 ) >Rm1 3 槡 B , s i n 2 2 3 1 3 1 槡 i n 2 A- c o s 2 A= s i n 2 B- c o s 2 B , %槡 s 2 2 2 2 A- s i n2 1+ c o s 2 A 1+ B 槡 3 c o s 2 - = s i n2 A- 2 2 2

(

)

& ' ( # $ )   * + , -

A C B C 4 23 = , = 槡 , i nB= 1 , 1 "# h1 s s i nB s i n 6 0 s i nB s i nA ° 0 ° , 1 2 0 ° ) , 9 0 ° , 3 0 ° , <= B ∈( "# B= "# C= 1 B C-?? S ·A C ·B C ·s i nC= "#△A B C= △A 2 2 3 . 槡 A C B C T, = hfó: ° △A ( ) F ? ? ?, 1 s i nB B C 4 23 , = 槡 , i nB= 1 , "# h1 s <= B ∈ s i nB s i n6 0 s i nA ° ( 0 ° ,1 2 0 ° ) ," # B = 9 0 ° ," # A B = 4- ( 2 3 ) =2 , B C- ? ? " # △A 槡 槡 1 ·A B ·B C= 2 3 . 槡 2 7 - 1  【 b=3 c , $% 】 >w? d F???1 2 < 4
2 2

(

π π B- . = s i n2 6 6

)

(

)

, , B 0 , ) , A- >a ≠b 1A ≠B ! A+ ∈( π 12 B- 2

π + 6

S B C = △A

1 b - c = a , = 3 k , c = 2 k , , ]i? b ?T k ≠0 "# 4
2 2 2 b + c - a 1 a = 4 k , o s A= =- . "#c 4 2 b c

2 π π π , B= , = π % A+ "# C= . 6 3 3 4 a c 8 ( 2 ) = 3 , s i nA= , = , = . >c 1a 槡 i nA s i nC 5 s 5 3 < c , C , o s A= , >a 1 A< oB c 5 s i n B = s i n ( A + C ) =s i n A c o s C+c o s A s i nC= $

4+ 3 3 槡 , 1 0 1 8 3+ 1 8 B C-??= S = a c s i nB= 槡 . "#△A 2 2 5
2 0 1 6 Ⅱ.
2 2 2 b + c - a 1 B  【 o sA= = $ %】 >/???1 c 2 b c 6 b c 5 3 = , , i n( B+C )=s i nA= ! 0<A<π |s 2 b c 5

8 ( 1 ) >R?d/???1
2 2 2 B D = B C + C D - 2 B C ·C D c o s C= 1 3- 1 2 c o s C ,

① B D= A B+ D A- 2 A B ·D A c o s A= 5+ 4 c o s C .② 1 o s C= , 6 0 ° , B D= 7 . >①②1 c $ C= 槡 2 ( 2 ) B C D-?? ??? A 1 1 S= A B ·D A s i nA+ B C ·C D s i nC 2 2 = 1 × 1× 2+ × 3× 2s ° n6 0 (1 )i 2 2
2 2 2

= 2 3 . 槡
2 9 ( 1 ) B CT, i nA= 槡 1- c o s A= ° △A >Rm? s

3 π π 槡 , i n B =s i n A+ = ! B =A+ , "# s 2 3 2

(

)

4 = , . $; B 5 b + c 1+ c o s A b + c 2A = = 2 B  【 ∵c o s , ∴ $% 】 ? 2 2 c 2 2 c 2 2 2 b + c - a c ( 1 + c o s A )= b + c ·c o s A= b · = ?c ?c 2 b c 2 2 2 2 2 2 2 b c -a = 2 b , B C[ ?b + ? a +b =c "# △A . §¨`¨?. $; B 1-
2 2 2 3 A  【 c - b = 3 a c , $%】 >F???01 a + " 槡 2 2 2 a + c - b 3 a c 槡 3 π 槡 o s B= = = , "# B= . #c 2 a c 2 a c 2 6 2 2 4 A  【 B DT, D= x , A = B D + $%】 °△A ?B |B 2 A D - 2 B D ·A D ·c o s D A , ∠B 2 2 2 4 = x + 1 0 - 2 ·1 0 x ·c o s 6 0 ° , %1 2 1 0 x - 9 6= 0 , a?1 x - 1 6 , x 6 ( . h1 x ÷?) 1= 2 =-

3 槡 ( 5)



6 槡 3× 6 a s i nB 3 槡 c o s A= . = = = >F???01 b 3 s i nA 3 槡 3 3 2 . 槡 ( 2 )> B =A + 3 槡 - s i nA=- . 3 B+ C= , ( A+ B ) . i nC= > A+ π 1 C= π- "# s s i n[ A+B ) ]=s i n( A+B )=s i nA c o s B+ π-( π π = , o sB =c o s A+ 1 c 2 2

(

)

C D T,> F ? ? ? 1 ° △B

B D = s i n∠B C D

B C 1 6 , ∴B C= ·s i n 3 0 ° = 8 2 . 槡 s i n∠B D C s i n 1 3 5 °

3 2

6 2 2 5 槡  【 C= 2 a , A C= b , M =槡 a + b , 】 ?B |A 3 A C 2 2 A B= 槡 4 a + b , s i n∠A B M =s i n∠A B C= = A B b B M . B MT, = °△A >F???, 1 2 2 s i n∠B A M 4 a + b

x π 1 x 1 1 + c o s + = s i n + . 2 6 2 2 2 2

(

)

( 1 ) ( )= >w? f α

3 1 α π + , i n + = 1s 2 6 2 2

(

)



2 3 π , 4 k k , 5[ α= π+ , ∈Z 3 2 ∴c o s 2 π π π k - - 4 o s . 1 α π- ) = (2 ) =c (2 3 3 3

A M a b 2 2 , , a =b , % = h1 2 5[ 2 2 s i n∠A B M 1 4 a + b 槡 3 B C 2 a 6 槡 s i n∠B A C= = = . 2 2 A B 3 4 a+ b 槡 6 2 i nC t a nA c s π  【 = , 1 + + 】 >F???01 3 b s i nB t a nB 2 c s i nA c o s B 2 s i nC = 1 + + = 0 , o s A s i nB+ "#L c b c o s A s i nB s i nB s i nA c o sB +2 s i nC c o sA=0 , i n( A+B )+ % s c o s A= 0 , B CT, s i n( A+ B )= s i nC 2 s i nC °△A ≠ 2 1 π 0 , 2 c o s A= 0 , c o s A=- , A= . 5[ 1+ 2 3 7 2 π  【 a+b -c ) ( a+b+c )=a b ,  】 >( 1× 3 2 2 2 a+ b- c =- a b . 2 2 2 a + b - c a b - o sC = = = () / ? ? ? 1 c 2 a b 2 a b 1 2 π - , $ C= . 2 3 8 ( 1 ) s i nx + s i ny = s i n + y x - y + y x - y + - + s i n ( x2 ( x2 2) 2)

a+ b 槡





( 2 ) ()F????: ( 2 a - c ) c o s B= b c o s C 2 s i nA- s i nC ) ·c o s B= ?( s i nB c o sC s i nA c o sB=s i n( B+C )=s i nA ?2 ? 1 π c o s B= ?B= . 2 3 A π 1+ 3 1 1+ 3 + ∵f ( A )= 槡 , ∴s i n + = 槡? 2 6 2 2 2

     

(

)

A π π 2 π π + = 6 ?A= 6 π . 2 6 3 3 3 2 π π < A < , ∴A = , B C=_?`¨?. B0 <?△A 3 3 1 0( 1 ) t B CT, A C= 6 0 ° , A B= 1 0 0?, °R △A ∠B | B C= 1 0 0 3 t B DT, A D= 4 5 ° , A B= ?, °R △A ∠B 槡 1 0 0?, D= 1 0 0?, |B C DT, B C= 7 5 ° + 1 5 ° = 9 0 ° , °△B ∠D
2 2 C=槡 B D + B C = 2 0 0?, |D C D 12 0 0? / = = 7 2?? / "#45 - ±V v ?= 1 0 6 0 ?, "#?45KLH±. ( 2 ) t C DT, C D= 3 0 ° , B E= °R △B ∠B !<= ∠D 1 5 ° , B E= 1 0 5 ° , E B= 4 5 ° . "#∠C "#∠C

x + y x - y = 2 s i n c o s . 2 2 ( 2 ) , b ,c = 2 b . >a :_ , +?, 1 a+c >F?? A + C i nA + s i nC= 2 s i nB , 1 ) s i n · !>( 0? 2 ?1 s 2 A- C A+ C =2 s i nB=2 c o s s i n( A+C )=4 s i n · 2 2 A+ C A- C A+ C c o s , ∴c o s = 2 c o s , 2 2 2 A C A C ∴c o s c o s = 3 s i n s i n , 2 2 2 2 A C 1 a n = . ∴t a n t 2 2 3
2 2 2 a + c - b o s B = = > / ? ? ? 1: c 2 a c a + c2 2 2 a + c - 2 2 2 3 a + 3 c - 2 a c 3 × 2 a c - 2 a c = = ≥ 2 a c 8 a c 8 a c 1 1 π 2B , ∴B ∴t a n ≤ , ≤ . 2 3 2 3

C ET, °△B >F???0?

E B B C = , s i n 3 0 ° s i n4 5 °

B C s i n 3 0 ° B= = 5 0 6 "# E ?, %??45h6? 槡 s i n 4 5 ° 5 0 6 ?. 槡    &'()* 
Ⅰ. 
 

1 A  【 a 1 , a 3 S 3 , a 3 S 1 2= 3× 】 1= 2= 1= 3= 2=
1 2 3 4 , a S 8=3×4 , a S a 4 =3 3 =4 5 =3 4 =3×4, 6= 4 3 S 3× 4 . . $; A 5= n - 1 2 (- 2 )  【 1?, a S  】 ? n= 1= 1=

(

)

2 a+ 3 1

1 , 1 . h1 a 1= 3 ? n≥ 2 ?,a n =S n -S n - 1 = a (2 3
n - 1

2 1 a + - 3 n 3

∴t a n

A C 2B t a n ≥t a n . 2 2 2 x x 3 x 2 x c o s +c o s =槡 s i n + 4 4 4 2 2



1 2 2 = a - a , 2 a . %a n =- n - 1 3 3 n 3 n-1

)

∴{ a } , 2-_?+?, [OC= 1 ??= - n
n - 1 ∴a (- 2 ) . n=

9 f ( x )=槡 3 s i n

3 3

& ' ( # $ )   * + , -

1 1 1- 1  【 3 ( 1 )-  ( 2 ) ( 1 ) 1 $% 】 ? n= 1 0 0 1 6 3 2 1 1 S (- 1 ) a , . ?, 1a 1= 1- 1 =- 2 4 1 n S (- 1 ) ( S S )- n. ?n ≥2?, n= n- n - 1 2 1 S ?n =t+?, n - 1 =- n, 2 1 1 S S - n+1, ?n =s+?, n= 2 n-1 2 1 1 1 1 , S =- , S- 4 = oB S !> S 1 =- 3= 4 3 2 2 2 1 6 1 1 - , 0 , S a a . 1S |S 2= 3= 2+ 3= 3 =- 1 6 1 6 1 1 ( 2 ) 1 ) S S …+ S >( 1S 1+ 3+ 5+ 9 9 =- 2 - 4 - 2 2 1 1 1 …-1 , S , 1 0 1 =- 1 6 - 0 2 2 20 20 1 1 S S … +S 2 S 2 S !S 2+ 4+ 6+ 1 0 0= 3+ 3 + 5+ 5 + 2 2 1 1 2 S … +2 S , S …+ $S 7+ 7 + 1 0 1+ 1 1+ 2+ 1 =0 2 20 1 1- S 1 0 0 1. 1 0 0= 3 2

(

)

a 1 , . $; A 1= n π 7 30 1 8  【 ∵a n c o s + 1 , $%】 n= 2 π ∴a c o s + , 1= 0+ 1 1= 2 2 π a 2 c o s + 1=- 2+ 1 , 2= 2 3 π 1= 0+ 1 a 3 c o s + , 3= 2 4 π a 4 c o s + 1= 4+ 1 , 4= 2 5 π a 5 c o s + , 1= 0+ 1 5= 2 6 π a 6 c o s + 1=- 6+ 1 , 6= 2 … S ( - 2 + 4 ) + ( - 6 + 8 ) + …+ ( - 2 0 1 0 + 2 0 1 2 )+ 2 0 1 2= 20 1 2= 5 0 3× 2+ 20 1 2= 30 1 8 . 5 k ( 5 k - 1 ) 8 ( 1 ) 50 3 0  ( 2 ) ( 1 )  【 $%】 >?0? 2 a a ( n + 1 ) ( n . ∈N+) n + 1= n+ a 2 , a a 3 , …, a a n . "# a 2- 1= 3- 2= n- n - 1= - a = 2+ 3+ … + n , 8?1 a n 1 n ( 1+ n ) 1+ 2+ 3+ …+ n = . %a n= 2 = 4 , 5 , 9 , 1 0 , 1 4 , 1 5 , 1 9 , 2 0 , 2 4 , 2 5 , … ?, a ?n n? 5 , b = a , b = a , b = a , b = a , 9 av % 2 5 4 1 0 6 1 5 8 2 0 …, a ( k . "# b ∈N+) 2 k= 5 k a a . "# b 20 1 2= 5 × 10 0 6= 50 3 0 1 ( 2 ) 1 ) a × >( T-? ?IX 0? b 2 k - 1= 5 k - 1= 2 5 k ( 5 k - 1 ) 5 k ( 5 k - 1 )= . 2 2 9 ( 1 ) ∵a 0 , = 1 , S a 4- 1 , a &n L4 %4 n> 1= 2- 1=
2 a 4- 1 , ∴a 4 a 5 . 2 =槡 1+ 2- 2 2 ( 2 ) 4 S a n-1 , 4 S a ?n ≥ 2?, n= n + 1 -4 n - 1= n- 2 2 ( n - 1 )- 1 , a , 4 ??^?1 4 L n =a n + 1 -a n -4 2 2 a ( a 2 ) , a 2 , %a n + 1= n+ n + 1= n+ ∴{ a } o? 2C:, [?,= 2-_,+?, n ∴a a 3× 2=a 6 , a a 1 2× 2=a 5= 2+ 2+ 1 4= 2+ 2+ 2 2 4 , , a , a a ·a , !a 9:_?+?, La | 2 5 1 4 5= 2 1 4 2 ( a 6 ) = a ( a 2 4 ) , 3 , h1 a 2+ 2 2+ 2= 1 ) = 1 , a = a + 2 ( n 2 ) . >( 1a ! ≥ 1 n + 1 n ∴{ a } 1 , 2 ?, = -_ , +?, %a n [ O C= n= 1+ ( n - 1 )× 2= 2 n - 1 . 1 1 1 ( 3 ) 2 ) + + …+ >( 1 a a a a a a 1 2 2 3 n n + 1 1 1 1 = + + …+ 1× 3 3× 5 ( 2 n - 1 ) ( 2 n + 1 )

(

)



1 1 3 5+ 3 槡 ∵a , a 1 , ∴a  【 $ %】 n + 2= 1= 3= 1 + 2 6 a n 1 1 1 1 2 3 , a= a= a= = , = , = 2 5 1 3 7 2 5 9 3 1 + 1 + 1 + 2 3 5 8 1 5 = , , a , ! a % a 1= 20 1 0 =a 20 1 2 20 1 0 = 5 1 3 8 1 1+ 8 5- 1 1 2 槡 , ∴a ?a 20 1 0 +a 20 1 0 -1=0 20 1 0 = a 2 1+ 20 1 0 ( . ÷7) 1 5- 1 2 槡 = , ∴1+a = !a 20 1 0= 20 0 8= 1+ a 2 20 0 8 5- 1 槡 5+ 1 5- 1 槡 槡 , , % a {???01 a 20 0 8 = 20 0 6 = 2 2 5- 1 5- 1 槡 槡 a … =a , + $ a 20 0 4= 2 0 = 2 0 +a 1 1 = 2 2 8 1 3 5+ 3 = 槡 . 1 3 2 6

54  【 a } , a ( n+ 1 ) · $%】 ?+?= { |a n n + 1- n= ( n + 5 )

(2 3)

n + 1

-n ( n+4 )


2 =( (2 3) 3)
n 2 n + 1





·

2 ( n+ 6 n + 5 )- n- 4 n ( 1 0- n) , [2 ] =3 3


a a ; "#? n ≤3?, n + 1> n n 4 , a < a . ? ≥ ? n+1 n a a a a , a a a …, <?, 1< 2< 3< 4 4> 5> 6> = 4 . $a "# k 4 Jü, 6 A  【 a S S ( S S S S $% 】 1 0= 1 0- 9= 1+ 9)- 9= 1=

1 1 1 - + +… + ( - [ ( 1-1 3) ( 3 5) 2 n - 1 1 2 n + 1) ] = 1 2

3 4

? ? ? ? ? ? ? ? ? ? ? ? ? ? ?
0 3 }5 C



1 1 1 1- < . n + 1 2 2 2

(

)

. 20 1 2 1 75  【 ( S S )- ( S S )= + 】 2 ( n + 1 )- n + 1 2 n- n n + 2 2 1 1 1 1 1 1 - > + - = - 2 n + 3 n + 2 2 n + 2 2 n + 4 n + 2 2 n + 2 1 m 1 > 0 , S S , < "# S ≥S "# 2 n- n 2- 1= 3 n + 4 1 6 2 1 1 6 m"?j1-Jüa+= 5 . ?m< , 3 3 ( n + 1 ) ( n + 4 ) 83 5    【 】 >w?-??, <u 2 0#1???-=?H?T > ?-W+IJ-O P= 1 n = 1?, a 2+ 3= × ( 2+ 3 )× 2 ; 1= 2 1 n = 2?, a 2+ 3+ 4= × ( 2+ 4 )× 3 ; 2= 2 … >?<u0#?.: 1 a 2+ 3+… +( n+ 2 )= ×[ 2+( n+ 2 ) ]× n= 2 ( n + 1 ) ( n + 4 ) ( n + 1 )= , a 3 5 . 6= 2 a 3 1 n 9 ( 1 ) , = >w ? 1 a 1 n≠ 0| > a n + 1 = a + 3 a n n + 1 a 3 1 1 1 1 n+ , - = , 2 , % B = 3 a a a 3 a n n + 1 n 1 1 1 ∴ [# 2=OC, # =?,-_,+?. a 3 n 1 n + 5 1 3 ∴ = 2+ ( n - 1 )= , ∴a . n= 3 a 3 n + 5 n n ( 3 - 4 a ) 1 n ( 2 ) ∵b · = 1 , 1 ) , |>( 1b n n= a n ( n + 1 ) n 1 1 1 - + + ∴S b b … +b n= 1+ 2+ n = 1- 2 2 3 1 1 1 1 1 - - + = 1- …+ , O5 n n n + 1 3 4 n + 1 1 1 b , ∴ ≤S < 1 . ?Q?A, !S 1= 1= 2 2 n 1 0( 1 ) a 6 , a 1 2 . 2= 3=

1 0( 1 ) a a S 1 , >R?: λ n n + 1= n- a S 1 . 1a λ n + 1 n + 2= n + 1- ( a a )= a . ??^?1 a λ n + 1 n + 2- n n + 1 , a . >5 a ≠0 "# a λ n + 1 n + 2- n= ( 2 ) a 1 , a a S 1 , 1 . >R?, λ 01 a λ- 1= 1 2= 1- 2= 1 ) a 1 . >( ?, λ+ 3= a a a , 4 . &2 h1 λ= 2= 1+ 3 a 4 . $a n + 2- n= a } , >?01 { [ O C= 1 ? , = 4-_ , + 2 n - 1 a = 4 n - 3 ; ?, 2 n - 1 a } , a 4 n - 1 . { [OC= 3 ?,= 4 -_,+?, 2 n 2 n= 2 n - 1 , a a 2 , "# a n= n + 1- n= 4 , a } <?ì° λ= ?1+?{ =_,+?. n
2 0 1 6 Ⅱ.
n x F ( n

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