GIẢI ĐỀ THI HSG LỚP 12 NĂM 2010 - NAM ĐỊNH

GIẢI Đ THI HC SINH GII TỈNH NAM ĐỊNH NĂM 2010 A. Trc nghim: B. T" lu%n: Câu I:                   x x x x x π π π − + + = ⇔ − + + =        x x x⇔ − + + =       x x x⇔ + + =       x x x⇔ + + =         x x x =  ⇔  + + =          x k x x π =   ⇔  + = −          x k x x π π π =   ⇔  + = −          x k x k x x m π π π π π π = =     ⇔ ⇔   + = − + = − +          x k k Z x m m Z π π π = ∈   ⇔  = − + ∈    !"#$%& '( )# *+ x k k Z π = ∈      x m m Z π π = − + ∈          ,  ,         x y x y x x y y x y x y x y   − = + − = +   ⇔   − = − =     -./$   u x v y  = ≥   = ≥   $0$ 1'!2 *               ,                  u v u v u v u v u v u v u v u v u v u v    − = + − = + − = − +    ⇔ ⇔    − = − = − =                u u v uv u v  + − =  ⇔  − =   341 v =  *$%5$ 6      u u  =   =   78# *+-  v ≠  *             u u u vv v u v  + − =  ⇔   − =  9./$    u u t do v v ≥  = ≥  >  $0'!2$            t t t t t t t t t+ − = ⇔ + − = ⇔ = = = − < :; 9< t = $ & u = $$%5$ 6    v − = 78# *+< t = $ & u v= $$%5$ 6    v =     v v do v u⇔ = ⇒ = > ⇒ = -   =    x u x v y y  = = =    ⇔ ⇔    = = =      *# *+>1 ?$ =  x y =   =   C'ch kh'c: +)Co ,   ,              x x y y x y x x y y y y y y x y x y x y    − = + − = + + − = +    ⇔ ⇔    − = = + = +              y y y y x y  + − = +  ⇔  = +                @   A      y y y y y y y y y y x y x y   ≥ ≥       ⇔ + − = + ⇔ + − − + =     = + = +             =  ,          y y y x y y y y x y x y  ≥   ≥   =    =     ⇔ − + = ⇔ ⇔     = =     = +     = +     -  *# *+>1 ?$ =  x y =   =  . x > -B0  :# :# :#         :# :# :# :# :#        x x x x x x x x x x x x x+ ≤ ⇔ + ≤ ⇔ + ≤ ⇔ ≤ :#    :#   :#   x x⇔ ≤ =     :#   :#    x x x⇔ ≤ ⇔ − ≤ ≤ ⇔ ≤ ≤ C$'( # *+    x≤ ≤  C'ch kh'c: +) . x > -./$  :#  u u x x= ⇔ = -B 1'!2C$         -     u u u u u u+ ≤ ⇔ ≤ ⇔ ≤ ⇔ ≤      :#    u x x⇔ − ≤ ≤ ⇔ − ≤ ≤ ⇔ − ≤ ≤ -C$'( # *+    x≤ ≤ Câu II: 9B0    D    y x m m x x= − − − E)'F 76:G$H$%GI76   DD     y x m m x= − − − E)'F 76:G$H$%GI 0J6+K';$L$M1$; x = N 76 ON    D         DD        y m m m y m m  = − − − =   ⇔ ⇔ =   > − − − >     m = $ & 6+K';$L$M1$; x = -  C'ch kh'c: 9J6+K';$L$M1$; x = $ &  D        y m m m= ⇒ − − − = ⇒ = 9  m =  6+K$%5$ 6          x y x x C= + − + -B0   D  y x x x= + − E)'F 76:G$H$%GI  D    y x x x+ = ⇔ = − = = - B BPB$0$ ? m = $Q+C$- m = $ & 6+K';$L$M1$; x = - C  m =  6+K$%5$ 6        x y x C= − + 9.!R#$ S#>'T107< *K#N !"#$%&    y k x= − + 9.!R#$ S#>$4EU7<N 76 ON  *01# *+V           x x k x x x k  − + = − +    − =  9V                      x x x x x x x x x k x x k   − − + = − + + =   ⇔ ⇔     − = − =         x x x x k  ± = =  ⇔   − =  9<EW$ &NW>$W 9<EW    − $ &NW ,  = >$ ,   = y x= + <EW    $ &NW ,  = − >$ ,   = y x − = + - C0'!R#$ S#'T1076$4EU7<:6W ,   = y x= +  ,   = y x − = + -  C'ch kh'c: XY       T x y C ∈         x y x⇔ = − + -B0      D  D  y x x y x x x= − ⇒ = − 9BQ$Z0  C $;     T x y $    D  y y x x x y= − +  ∆ -.Q$  ∆ T10N 76 ON      D  y x x y= − +       x x =   ⇔ ⇔ ±  =   9<E  W$ &  ∆ $W W 9<EW    − $ &  ∆ $ ,   = y x= = +  9<EW    $ &  ∆ $ ,   = y x − = = + - C0'!R#$ S#'T1076$4EU7<:6W ,   = y x= +  ,   = y x − = + - 9./$               t x t t t x t dt dx t dt dx= + = = ⇒ = + ⇒ = ⇒ = 76        t x t x  = +   = +   3G                          @ A   t dt d t I t t dt t dt tdt dt t tt t + = = − + − = − + − + ++ ∫ ∫ ∫ ∫ ∫ ∫     @ : A   t t t t= − + − + ,      @  :     :  A :       = − + − + − − + − + = + -   :   I = + Câu III: 9              x y x y x y+ − + − = ⇔ − + + = G$[+V76C)N\ IW- 976:6$4'M+Z0$4$14]]Z0N^$P]-3G MA IA⊥ 76 MB IB⊥ -76_+$%G '!R#$%`a'!R#N\ V]76 b  c    DA B C C= I - 9.!R#$%`a'!R#N\ V]G$[+     J − 76C)N\   D  R IM= -          R⇒ = − − + − = -3Ga $               x y x y x+ + − = ⇔ + + − = - '$Y0'd'M+76$ e0+( *                       x y x y x y x y x y x x y x x y x   + − + − = + = − +   ⇔ ⇒ − + = − +   + + − = + = − +       x y ⇒ − − = 7'!R#$ f#$Q$%$g#T1)$:6   x y− − = - 9 !"#$%& $ h'; i:6     x y z + + = -]$ 1dG      M M M x y z + + = 9BP -  OM ON = ≠ 1%0   O M OM O N ON  ≠ ≠   ⇒   ≠ ≠    uuuur ur uuuur ur -]63$ 1d$0j]G -    - - -   -   OM t ON t OM ON OM ON OM ON OM ON  = >   = = =   uuuur uuuur uuuur uuuur uuuur uuuur 9BP$0 - - - M N M N M N x t x y t y z t z =   =   =  -3G    - - -      M N M N M N N N N x x y y z z t x y z⇔ + + = ⇔ + + = 9BP761%0                   N N N M M M N N N N N N x y z x y z t x y z t x y z t+ + = + + ⇔ + + = + +         N N N N N N x y z x y z do t⇔ + + = + + >          =              N N N N N N N N N x y z x y z x y z⇔ + + = + + ⇔ − + − + − = 9$Y0'd'M+3$ e0+($       =           x y z− + − + − = :6$Z0+d$+/$k1K'F $[+           K 76C)N\ l  R = - m#$e'M+3_+$%G+d$+/$k1K'F - Câu IV: 9BP  SA ABCD AB⊥ ⇒ :6 &  41718##Z0n$%G76 SA AB⊥ G SAB∆ 718#$;-##o0n76 :6#  SBA∠ = GnW-$0  $0  - SBA a∠ = = -   a SM⇒ =    SM SA⇒ = 76]5#o0n76 9B0        QQ QQ    QQ SAD BCM MN AD SAD MN BC CD SN SD BC BCM AD BC =   ⊂  ⇒ ⇒ =  ⊂    I  9B0 m#+ Cg'p  &  n-)'M+JV$!"#m#$ 1d); nnn$ & - - - - S HIK S ABC V SH SI SK V SA SB SC = - B 7XYqr$ 0$ m$L:6 &  41718##Z076J$%Gn$ &nqr$ S# 6#76qQQJr HF SH AE SA ⇒ = -  ' - -  - - - -  - -  - - - -  S HIK SIK S ABC SBC HF SI SK ISK V HF S SH SI SK V AE S SA SB SC AE SB SC BSC ∆ ∆ ∠ = = = ∠ '+ s>H#Cg'p$%G$0 - -  - -  S MBC S ABC V SM SB SC V SA SB SC = = 76 - -    - - -   = S MNC S ADC V SM SN SC V SA SD SC = = = - - - - - - - - -       = = S MBC S ABC S MBCN S MBC S MNC S ABC S ADC S MNC S ADC V V V V V V V V V  =   ⇒ ⇒ = + = +   =   9:6 &  o $G  - - -      - - - - -      ABC ADC S ABC S ADC S ABCD ABCD a S S V V V SA S a a a ∆ ∆ = ⇒ = = = = =  - - - - -      - -   =  = l S MBCN S MBC S MNC S ABCD S ABCD a V V V V V= + = = = n1%0$ M$\ N K'0>*3]:6  - - - l   - = l S ABCD S MBCN S ABCD a V V V V= − = = -  C'ch kh'c: 9BP  SA ABCD AB⊥ ⇒ :6 &  41718##Z0n$%G76 SA AB⊥ G SAB∆ 718#$;- ##o0n76:6#  SBA∠ = GnW-$0  $0  - SBA a∠ = = -]$ 1d; n76   a SM =   SM SA⇒ = 76]5#o0n76 9B0        QQ QQ    QQ SAD BCM MN AD SAD MN BC CD SN SD BC BCM AD BC =   ⊂  ⇒ ⇒ =  ⊂    I 7635#o0n76- 9 nnn'8+d$718##76N 8#'t# S#  Y $'Eu  '               -  a A B a D a S a C a a M BP   SN SD= 7635#o0n76   SN SD⇒ = uuur uuur         a a N⇒ 9  -     - - - -    S ABCD ABCD a V SA S a a a= = = 9    - - -   -  -  - -   -  -   = l l S MBCN S MBC S MNC a a a V V V SB SC SM SN SC SM     = + = + = = + =     uuur uuur uuur uuur uuur uuur 9'    - - -  - -  - -   l l S ABCD S MBCN a a a V V V= − = − = Câu V: 9./$       x t x R x π = ∈ + - 9B0           -  -       x x x R x x x x x π π ∀ ∈ + ≥ = ⇔ ≤ ⇔ ≤ + + >?1vWwEf%0N 76 ON       x x ± = ⇔ = -     x R t π π ∀ ∈ − ≤ ≤ 9 ' 6+K$%5$ 6             f t t t t t π π = − + + − ≤ ≤ - 9B0$W        -   -  -     t t t t t t t t t t t t+ = + = + − = −                  f t t t t t t t t t π π = − + + = − − + − + − ≤ ≤           f t t t t t π π ⇔ = − − + − ≤ ≤ 9./$         u t t u π π = − ≤ ≤ ⇒ − ≤ ≤ $0$ 1'!2 6+K            f u u u u u= − − + − ≤ ≤ 9  f u :G$H$%G        −     76    D        f u u u u= − − − ≤ ≤ 9B%G     D     u f u   ∈ − =     # *+      u u=− = 9B0    =                 f f f− = − = = XB3Z0 6+K'( :6         =     b      c     x R u Max y Maxf u Max f f f ∈   ∈ −     = = − − = 6XB33Z0 6+K'( :6          +  +   +b      c     x R u y f u f f f ∈   ∈ −     = = − − =  . = ∠ '+ s>H#Cg'p$%G$0 - -  - -  S MBC S ABC V SM SB SC V SA SB SC = = 76 - -    - - -   = S MNC S ADC V SM SN SC V SA SD SC = = = - - - - - - - - -       = = S MBC S ABC S. +   =   9:6 &  o $G  - - -      - - - - -      ABC ADC S ABC S ADC S ABCD ABCD a S S V V V SA S a a a ∆ ∆ = ⇒ = = = = =  - - - - -      - -   =  = l S MBCN S MBC. = uuur uuur         a a N⇒ 9  -     - - - -    S ABCD ABCD a V SA S a a a= = = 9    - - -   -  -  - -   -  -   = l l S MBCN S MBC S MNC a a a V V V