C6c em dga vdo tinh ch6t ndy thi ldm dugc nhfing bdi to6n nhu th€.. Suy ra tliiju phf,i chfne minh.
Trang 1SO GD-DT IIA TINH
TRU,ONG THPT NGUYEX TRUNG TIIITX
DE THr rntlDAr Hec LAN 2 (Kh6i a,n;
Nim hgc : 2010-2011
MOn : Tofn Thdi gian : 180 phut
c6u 1 (2 tli6m) Cho hdm s6 ! = x3 - 3x', + 2
1) Kh6o s6t sr,r biiSn thi6n vd vE AO tfri ( C ) cfa hdm s6.
2) Tim tr0n dudng thing (d): y : -2 nhffng di€m md tir d6 ke dugc Z tiep tuy{in t6i 1C;, eOng thoi 2 ti6p tuy6n do vu6ng g6c v6i nhau.
Ciu 2 ( 2 tli6m ) Giii chc phuong trinh vd hQ phuong trinh sau:
+ logr*, (*' - 2x + 1) = 6
2) 2cos' "r + cos 2x + sinx - 0
C6u3(2tti6m'l
1) Tinh tich phdn
+4x+4
2) Hoi co bao nhi€u s6 tU nhi6n gO- 7 chii s6, tuo cho chfi s6 dring sau nho h<vn chir
sO dring liOn tru6c n6 i
-aau!-( z rX!€aq)
Cho hinh ch6p tri g d6y ABCD ld hinh binh hdnh Goi B', D' lAn luo,t
ld trung di6m cira SB, SD 'D') cit SC tai C'
1) Chring minh SC:3 SC'
2)Tinhttr0 tich cria S.AB'C'D' theo th0 tictr V cria S.ABCD
Cffu5 (2tli6m)
1) Trong mat phdng tea dQ oxy cho : diemiu r(2;2), \)/).'5,5,,-.5,5, s, f?;1), H, (1,?l tu 3 chan dunng cao cfia tam gi6c ABC lAn luqt ha tri A, B, C Tim tqa dQ c6c dinh A, B, C
2) Cho hai s6 thgc avd b th6a a-b + 1 : 0.
Chring minh r6ng:
::::::::::::::::::::::::::-: ii€r ::::::::
4
I
?
x
)
2
dx
trongxuanht@yahoo.com sen to www.laisac.page.tl
Trang 2nUoNc oAN cnAvr roAN rnr rHrl on r,Ax rntl2 - KHor A-B
I
l
6 (1)
(2)
log ,*r(l - x)' : 6
+ 2) + log ,*r(l - x
,,
(x-2)'+
, (Y
y+
lop
_"(
<>
)
2)-+4
-1
)Br-,
6€
:*
og
y+
,r(*
y+
+k
:)=
o
ov+
-).2),
c)'
-x
-L
a
+
(1
v-
5)-,r!
,(1 ) ) v+2
I
X
;dr
v
4
Lg
(
)n.
.r0t
o
bl-r9t u
OT
ki, T
llc
3ul (1)
+2
let (r l+
-1, Di
+, pt
-) .?
v
: !=xt -3x'+2
+, TQp xd, Tim dugc y'; Tim c6c di6m tdi han; Tim y" 0,25
+, Tim c6c di6m d4c biQt: CUc dai (0;2); Cyc ti€u (272):; Di m u6n (1;0)
Oiem cdt tryc ox: (1-".6;0); (l;0); 1f *",6;0); Di6m c6 tsa dQ nguy€n (3;2); (-17
2)
0,25
+, Ve dO thi dgp, tron, c6 tinh d6i xring, di qua c6c di€m <l{c biet 0,25
+, Gid sri c6c Oiem cAn tim ld X(m;-2); Dudng ttrang (A ) di qua X v6i hp s6 g6c
k co pt ld: y : k(x-m)-2; khi d6 (l ) tiCp xirc ( C ) niSu vd chi n6u hQ :
ll*2 -6x-k
)
)3-r2
[x'-3x'+2 k(x-m)-2
(1),.
co nghiQm ( 6n x )
(2) (/
0,25
ThC k tir (l) vdo (2) ta c6 pt: x' -3x' +2= (3r' - 6x)(x - m) - 2
<) x' -3x' +4=3x(x-2)(x-*) <> HoA x:2,hoirc
2x'-(3m-l)x+2=0 (3)
0.25
rO rdng kh6ng co ti0p tuyOn nao cdu d0 ra thi pt (3) phai co 2
(3*i - 6*,) (3*', - 6xr) - -1
Vdi x : 2 thl ta duqc
ryQt tiOp tuy0n co pt: y : -2,
vu6ng goc vdi ti6p tuyOn ndy NOn mu6n thoa yOu
nghi0m pb x,, x2 thoa: y' ( x, ) .V'(*r) - - 1 Tuc :
<+ gxixl - 18r, xr(x, * xr) + 36x,x, - -1
0,25
Do x, , xrld nghiQm cria (3), theo Viets ta c6: xt.xz =l;x, + x, =
55 NOn dugc 9 -9(3m-1) + 36: -1 Hay m=a; Gi6 tri m ndy thoa
27'
nghiQm k6t 1u0n eicm cdn tim tex1fi;-21
2\op (-xv -2x + v+ 2) + los^ (x' -2x +1\ =
3m-1
2
Trang 3i iDat t: log,-" (y + 2) ta co Pt sau:
-2 e t- 1 'e logr-r(Y'+Z)-I e Y:
-x-1
t+-t
0,25
2
D[t
t-x+-x
)
ta co dt - (1 - i>a* ; dOi cQn t
x
A9
:J+-2
-)
J
:
44
+, Thay gr;tri y vao pt (2) vd giai thi dugc x: 0 ; x - -2
+, Ket hqp diou kiEn ta c6 nghiQm |d : x: -2, y : 1
+, Thay 2cos' x = cosx.2cost t = cos'x(co sLx +1) thi pt vii5t lai ld:
Cosx Cos2x+ cosx* cos2x+ sinx:0 ecos2x(cos" i11+titt"*cosx:0
e (cosx*sinx) ftro, x -sin x)(cos x + 1) + 1l-0
<> (cosx+sinx) ft.o, " - sin x)(cos x + 1) + 1.1:0
; i;;-;;;ii- sin'x + cosx(l- sinx) + (1- sinx)l:0
(cosx*sinxx 1 -sinxx 1 *sinx*cosx+ 1 ) - 0
ept co 2hqnghiQm x- -+ + kn;x -! *2kn
( Ttry vho lflp lufln tI6 cho tti6m thirnh ph6n chfnh x6c tl6n 0,25 )
1,00
Trang 4I
0 \t )5
-'
(2di6m)
I,Xtrc dinh tdm O ctra hbh ABCD
+, Xd giao O' cfia B'D' vdi SO
+, Xd giao C' cira AO' v6i SC '
+,KeCPllSO(P€AS)
+, Xd giao K cira AC' vdi CP
+,Chung minh K lA trung diOm CP
+,Chung minh S ld trung diOm AP
+, Suy ra C' ld trgng tdm
HaySC-3SC'.
2 Chia th6 tich cria S.AB'C'D' thinh2
kh6i chop tam gt6c:
S.AB'C' vd S.AC'D', tdc6:
0,25
0,25
0,25
0,25
0,5 0
0,25
0,25 0,25 0,25
tU thi: v(s.AC'D')
11
f rfs .ABCD)
1, Tam gi6c ABC C6 AH1, BHz, CH3 ld c6c dudng cao Nguoc l4i tam giac
H'FzHr se c6 AHr, BFtrz , CH3 ld c6c dudng ghdn gi6c ( C6c em dga vdo tinh
ch6t ndy thi ldm dugc nhfing bdi to6n nhu th€ )
+, Ri€ng bdi to6n ndy cho d{c biQt hcrn:
< H ,H,H 3cAn tpi Hr.*, Dinh A nim tr€n trung tn;c(d) cria H2H3.
Dinh B vd C nim tr€n ducrng ttring (d') di qua H1 vd vu6ng g6c v6i (d)
+, Vi6t pt ducrng phdn gi6c cria goc do I{zHr ho p v6i HzH: ( c6 2 pt )
+, Giao ctra dt co pt thf nh6,tvdi (d) cho ta di6m A( 0; 0 )
j giao vdi (d') cho ta OiCm B( 1; 3 ),giao cria dt co pt thft 2 v6i (d') cho ta C(
2, Ta c6 :
^lo' +b' -za+6b+10 +^la' +b' -4a-4b+8:"/(a-1)'+ (b+3)' *,l@-2)' +(b-2)'
Trong mpt phEng tqa dO Oxy"ta lBp tli€m A(1;-3), eiem BQ;2) vd duong thang (d)
c6 pt: x - y +1 : 0 Khi d6 di6m M(a,b) ndm trOn (d) thi cap (a;b) thoa a-b+1:0
Ta co lfo- 1)' + (b +3)' + J @ -2)' + (b -2)' :AM + BM
L6y B' d6i xung vdi B qua (d) suy ra : B'(1;3)
0.25
0.25 0,25 0,25
Ta c6 ngay QJ AM + BM > AB' : 6 Suy ra tliiju phf,i chfne minh