Tam giae SAB vuong t~i S, goe gifra SB va m~t phfutg ABCD bing 30° .Tinh theo a th8 tich cua kh6i chop S.ABCD va khoang each gifra hai dUOng thfutg AB, Sc.. Vi~t phuang trinh dUOng thing
Trang 1Tnrong Cbuyen Le Quy Don BR VT DE THI TmJ D~I HQC - CAO DANG LAN 3 (2010 - 2011)
Mon: Toan - Kb8i: B
I Ph§n chung eho tit eli cae thf sinh (7.0 ([iim):
Cau 1(2.0 ([iim): Cho ham s6 y:::; ~ (*)
x l
O
1) Khao sat S\I' bi~n thien va ve d6 thi (C) eua ham s6 (*)
2) Vi~t phuang trinh tiep tuy~n d eua (C) bi~t d e~t hai tIVe to~ dQ t~i cae di8m A va B thoa man tr9ng tam tam giae OAB thuQe dUOng thing: x 4y = 0, (0 la g6e to~ dQ)
Cau II (2.0 ([iim):
q · h 'nh 2sinxeos2x+sin2x 1
1) tal P uang trl : = +eosx
2) OiM b~t phuang trinh: x log~ x + 6 < 3xlog2 X
Cau III (1.0 ([iim): Tinh tieh ph§n 1= j£x+idx
o 8x+3
Cau IV'(1.0 ([iim): Cho hinh chop S.ABCD co ABCD la hinh vuong e~nh a va m~t ben (SAB) vuong goe v6i
m~t day Tam giae SAB vuong t~i S, goe gifra SB va m~t phfutg (ABCD) bing 30° Tinh theo a th8 tich cua
kh6i chop S.ABCD va khoang each gifra hai dUOng thfutg AB, Sc
Cau v (1.0 ([iim):
T' 1m +f t ' La ca gta r! eua am so m sao COy P uang n " t' , th J- h h~ h t 'nh {2X2Y +9X = my2 2 co ung a ng lym 'd' b h'~
xy2 +2y = x
II Ph§n rieng (3.0 ([iim) : Thf sinh chi aU(lc lam m6t trong hai phdn (phdn A hoi;ic phdn B)
A Theo ehU'ong trinh Chuin:
Cau VI.a (2.0 ([iim):
1) Trong m~t phing v6i h~ t9a dQ Oxy, eho tam giae d€u ABC n9l ti~p dUOng trim
(C): x 2 + y2 -4y-4 0 va c~nh AB co trung di€m M thuQc dUOng th~g d: 2x y-l =O Vi~t
phuang trinh dUOng thing AB va tim to~ dQ di8m C
2) Trong khong gian v6i h~ t9a dQ Oxyz, cho eac di€m A(3;-I;I), B(-1;0;-2), C(4;1; 1) va D(3;2;- 6)
Vi~t phuang trinh m~t cAu tiep xuc v6i hai dUOng thing AC va BD l§n luqt ~i A va B
Cau VII.a (1.0 ([iim): Trong eac s6 phuc z thoa man I; 2-3il 5, tim s6 phue z sao cholz+4-5il d~t gia tri nh6 nh~t
B Theo chU'ong trinh Nang cao:
Cau VI.b(2.0 ([iim):
1) Trong m~t phfutg v6i h¢ t9a dQ Oxy, cho tam giac ABC co dUOng thfutg AB di quadiSm 0; phan giac trong goc A thuQe duemg th~g d l x y + 1 = 0 va dUOng eao ke tir B thuQC dUOng th~g
d 2 : 4x+ 3y 48:::; O Tim to~ dQ cae di€m A va B
2) Trong khong gian v6i h~ t9a dQ Oxyz, cho hinh thoi ABCD co tam I thuQe duemg thing
d: x-I = y - 3 z - 2
va hai dinh A(3; -1; 1), B(-1; 1;3) Vi~t phuang trinh duang thfutg CD
Cau VII.b(1.0 ([iem): Tim cae gia tri nguyen duang eua n de so phue z = 1~i.Jj la so thuan ao
-IIItT -Thi sinh khong dUQ'c SIT d\lng tili li~u Can be} coi tbi kbong giiii thich gi them
HQ va ten thi sinh: sA bao danb: .••.• ••.••
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Trang 2II
[
X = 7l' +k27l' ~
* <=> 7l' 7l' {\ ;(k E Z)
x = +k7l" X = +k7l'
6 ' 3
gjl1-;(::- ~\1il Tij '''/s [x = +
>
Ciu
I
1 Kh • ao sa va ve ' t ' _;l-A uO thO I: Y =- X
x-I
Thang
• di~m Ghi eM
Y I = -1 2 ; ham s6 nghich bi8n trong timg khoang ( 00;1), (1; +00)
*Ti~m c~: lim Y = 1 ~ TCN : y = 1; lim y = ±oo ~ TCD: x = 1 0.25
*Bang bi8n thien:
0.25
* I'll> thi'
y'
+00
~I
-00
tJ 1 JJ ) lJ
2 Viet phU'01l2 trinh tiep tuyen:
2
*Giao di€m d va Ox: A(a 2 ;O) , giao di€m d va Oy: B(O; (a-I) a 2 J.
TrQng tam cua tam giac OAB : G (~; a
2
2 J ' di6u ki~n a *0; 1
3 3(a-I)
* G thuQc dt: x-4y 0 <=> a =O;a =-I;a =3 (lo~i a 0)
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Trang 32 Giai bit phU'O'ng trinh: x log; x +6 <3x log2 X +log,fl x (1) J '
I
*(1) <=> (log2 X - 3)(x log2 X - 2) < 0
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* ¢:::> {::':<7<O /
log2 x -;>0 /
0.25
* /(x)=log2 x - 2 d6ngbiantren (O;+«» vaf(2)=Onen /(x)<0<=>0<x<2;/(x»0
x
*V~y nghi~m cua (1) : S = (2;8)
0.25
III
h A
T ' h ' h lD tic p an: I = 4J.J2X+ Idx
o 8x+3
3 2
14t -1
* _ 3 n.!.+_1 _ 1 l l (1 1 ln12t-11J3 *1== .!.,lm~ 0.25
- tl4 8(2t-1) 8(2t+l)[t= "4t+16 2t+1 1 21; 7 0.25
IV Tinh the rich kh6i S.ABCD va khoang each:
va SBA 30·
*Tinh SA = a 2' 'SB = aJ3 2 ' 'SH = a.fj 4 VA V ~y S_ABCD a312 .fj 0.25
*AB, SC cheo nhau va AB II (SCD)::J SC nen d(AB;SC) == d(H;SCD 0.25 D\ffig va CM duqc d(H;SCD) = HK ( HM II AD va HK -L SM)
III - a - {,' ,\, I 1 ! l - ,r::
19 l\ \( ::: f~ • Lt, )~ \ ~ Til ., v} C 1
v
Tim gia tn tham s8 d~ he pt co 3 nghiem: {2X 2Y +9X ml "''9' \6 , - Lt
x 2
*N~u x == 0 suy fa y = 0 va nguqc ll;li (x;y) = (0;0) 1a m¢t nghi~m cua h~, veri mQi m 0.25
*N~u xy'f::O: H~ ¢:::> -y+ -l-m;d~t u=_x ,v=-y
x 2
1
veri m6i (u;v) veri uv 'f:: 0 cho ~¢t ngh~m (x;y) thoa xy 'f:: 0
2U+ 9 =m ,
Bai toan <=> tim m d~ h~ pt 2v co dUng hai nghi~m (u;v) thoa UV'f:: 0
Hay phuong trinh 2u + ~= m co dUng hai nghi~m u khac 0;2
u-2
*K~t qua: mE (-00;0) D (0; 1) u (25; +«»
0.25
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Trang 4C
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'h1 ::; }ir -::) M ( 11
' 1 / 'J I
VLa A.Then chllO'Dg tranh chuan:
1 moh hoc toa 46 phing:
*(C) co tam I(0;2), ban kinh R = 2.J2 ABC d~u nQi tiap (C) nen 1M = .J2
ie-1lr:
~r2 ;
'(::::
i)y _ *M thuQc d nen M(m;2m - 1), 1M = .J2 ~m =l;m = 2 .1: l~ - l\'\1' ~n\-3)
* AB qua M va co VTPT ta 1M',IC =-21M nen i:) N) -\ 4~1- \r
~- ,.::::)
'2-~ m = 1 ::::> AB : x - y 0 va C(-2 ;4) tv\ (1\.) 1\ ) i'"'N::: (1\.r11)1 \'il1'V\ t k tJ
* mp(p) qua A va vuong goc AC : x + 2Y - 2z+ 1= ° 1 ~ ~ (~'.)' 1 'J _ ~ )
* mp(Q) qua B va vuong goc BD : 2x + y - 2z - 2 0
*M~t cAu dn tim co tam I thu9c (P) va (Q) nen I( 3 + 2t ;2t ;2+3t)
*IA = IB ¢:> t = -1 V~y 1(1 ;- 2 ;-1) va R = 3; phuong trinh mcAu :
(x-l)2+(y+2)2+(z+1)2 =9
VIla
{
So Cach 1: *z x + iy, v&i x, Y E
~ 2-3i=5¢:>(x-2i+(y+3)2 25¢:> ;tE[O;27l')
* Iz+4-5il =)(X+4)2 +(y-5i =~125+20(3cost-4sint)
* 3cost -4sint ;::: -5,'v't E [O;27l') nen Iz +4 - 5il ;::: 5,'v'z
* Dkg thuc xay ra khi cost =-3/5;sint = 4/5 V~y z =
Cach 2:
* Tgp h(YJJ cac iii€m M(x ;y) biJu diln z trong mp Oxy la (C): (x - 2)2 + (y + 3)2 =25
* Iz +4 ,.5i! == )(X+4)2 + (y _5)2 == AM, vaiA(-4;5) nd:m ngocli (C) AMnhO nh&t khi M la giao iiiim (gdn A han) cila (C) va lA, vai 1(2;-3) la tam cua (C)
* lA: 4x + 3y + 1 = 0, giao iii€m cila (C) va lA fa B(-1;1) va C(5 ;-7)
*lB<lC V0-'Z -1 +ilast5phii:cthoayeucauMitoan
VLb B.Theo chllO'ng trinh nang cao:
1 moh hoc toa 40 phing:
*D d6i xUng 0 qua dj : D(-I;I).
*D thu9c AC (tc phdn giae) va AC vuong g6c d2 nen AC: 3x 4y + 7 =O
*A ta giao cua AC va d1 : A(3 ;4) *B ta giao eua OA vii d2 : JB(6 ;8)
*I thu9c d nen I( 1 +t;3+t;2+t)
* ABCD la hinh thoi nen IA.IB == 0 ¢:> t =-1 .V~y 1(0 ;2 ; 1)
*C, D d6i xung v&i A,B qua I nen C(-3;5; 1) va D(l ;3;-1)
x-I y-3 z+1
*Phuong trinh CD: - - = - - = -
VII.b sa
'-;;3 2( -7l' -7l') 1 ''-;;3 2( 7l'
*"j -z == cos-+zsm-; +lVj = cos-+zsm
* (J3 - iY =211 (cos -n7l'
6
(J3-it 211- 1 [ (-m'C 7l') , (-n7l' 7l')]
• * V~y z==' = cos - - - +zsm - - -
, -n7l' 7 l ' ~T'
* z thuan ao ¢:> cos( ) = O,n EN¢:> n = 6k- 5, k E H
6 3
I
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