Tim t9a de di~m Mtren C sao cho tam giac MAB co tr9ng Him ding thuQc C.. Tinh chitu cao eua hinh eh6p S.ABC va eosin cua goc 2 gifra hai dUOng thdng AC, SB... I 'fnrang Chuyen Le Quy
Trang 1Tru'(l'ng Chuyen U Quy Don llR - VT DE THI nIli' D';'I HOC - CAO DANG LAN 3 (2010 - 2011)
Mon: Toan - KhBi: D DE CHiNH THUC "-~ TIlt'; ginll/lull biii: 180 pltut )
I Phin chung eho tAt ca de thi sinh (7.0 tlMm):
f{;!W!j
Cau 1(2.0 tI;im): Cho ham s6 y = x-I (*)
x+l
1) Khao sat S\I biSn thien va ve d6 thi (C) cua ham s6 (*)
2) Cho hai di~m A(4;3) va B(-I;-2) Tim t9a de) di~m Mtren (C) sao cho tam giac MAB co tr9ng Him
ding thuQc (C)
Cau II (2.0 tliim):
· h !, sin4 x+sin2x cos4 x ' 2 ·
1) Gtal P uang tnnJ,.l: =
COS-Xl + i -xy(x +y) ==
2) Giai h~ phuang trinh:
{ l+xy=~x+y-l
Cau III (1.0 tliim): Tinh tich phful J-2-e-dx
o e X -9
Cau IV (1.0 tliim): Cho hinh chOp S.ABC co day la tam giac ABC vuong t~i C, BC a va BAC 30° Th~
:1
tich kh6i ch6p S.ABC bAng ~ va SA = SB = SC Tinh chitu cao eua hinh eh6p S.ABC va eosin cua goc
2
gifra hai dUOng thdng AC, SB
Cau V (1.0 tliim):
Chocacsot ' h\IC uang a, b ,c t oa a+ b +c;;:: Clmngml rang: - - + - - + - - ; ; : : +1 +1 10
II Ph~n rieng (3.0 tliim) : Thi sinh chi aU(fc lam m<J1 trang hai phdn (phdn A ha(ic phdn B)
A Theo chuong trinh Chu§n:
Cau VI.a (2.0 tliim):
1) T rong m~t phing v6i M tQa de) Oxy, cho tam giac ABC co C thue)c dUOng thing d: x - y - 3 :::: 0 ;
A( 1 ;2), B(2;7) va chiSu cao AH cua tam giac ABC bAng 1 Tim to~ de) C
2) Trong khong gian v6i M tQa dQ Oxyz, cho cac di~m A(0;2;-l), B(3;0;2) va C(2;-1 ;0), ViSt phtrang trinh m~t du (S) di qua A, B, eva (S) ti€p xuc v6i m~t phdng (P): x +y +2z +2 = 0
Cau VII a (1.0 tlMm): Trong m~t phing phuc cho s6 phuc z thoa man Iz -il== 2, tim t~p hgp cac di6m bi~u di~n cua 36 ph~c w =·iz +i-I
B Theo chU'ong trinh Nang cao:
Cau VI.b(2.0 tlMm):
1) Trong m~t phing v6i MtQa dQ Oxy, cho elip (E): ~5 + ~ =1 co hai tieu di6m la FI va F2 Di~m M
tren (E) tho a man ~ == 60° , Tinh di~n tfch tam giac MFIF2
2) Trong khOng gian v6i MtQa de) Oxyz, vi~t phuang trinh m~t ph~ng (P) di qua di€m A(3; 1;2), (P)
" d ' h! d x-2 y-3 z+l dA h"kh' ' h ' - d '(P)b~ 3
song song ven uang tang : - -= - -== - - ong t en oang cac glUa va ang
2 1 2
Cau VII.b(l.O tliim): Tim mQi gia trj Clla tham s6 m d~ d6 thj ham s6 y::::: (x + 2)(x 2 - mx +m 2 3) ti6p XUC v6i tr\lc hoanh
-IIIt1r -Thi s~n~ khO,n~ dU'9'c sir dl}ng tili li~u Can bi} coi thi khOnrg~ai thich gi them
11(,) va ten thl Sinh: , So bao danh:
Trang 2I
'fnrang Chuyen Le Quy Don BR VT DAp AN VAHtrONG DAN CHAM DE THI THU D~I HQC - CAO DANG
LAN 3 (2010 - 2011) Mon: Tolin - KhAi: D
CAu ,
(fi~m
1 Kh • , ao sat va ve , - d" b' 0 t I: y =- X -1 0.25
x+I
*T~p X3.C djnh: D = IR \ {-I}
0.25
y' = 2 2; ham sil d6ng bien trong timg khoang (-«>; -1), (-1;
x-+±«> <-+-1"
*Bang bien thien:
x -00 1 +00
Y
1 /
~
-00
u'
:; " ,
u
L
" -.-!
.1.1 <4 4.1 1 ·IJ ,
V' "
(C)Y'~
,+1
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I
l
2 Tim di~m M:
*M thuQc (C) nen M(m; m-l), ABM III tam giac <::::> AM,AB khang cung phuong 0.25
m+l
tlq.t hoq.c kii m
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* m=Ohoacm=9
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* Ki6m tra di~u ki~n (*), ket lu~n M(9;4/5)
f - = - - j - - - " - - ' - - - 1 - - - + - - -
IT
1 G · • b t ' h sin
4
• lal p U'O'Ug nn : = cos x
x:;t: Jr +k Jr
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*:I: ~ k· 8 2
Jr
4
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*V6i dk tren, (1) <=> cos 2x =cos2 2x
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*cos2x = 0 (101;ii)
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I *cos2x = 1 <::::> x =mJr,(m E Z) (thoa man dk)
II X2 +y2 -xy(x+ y)=4
2 Giai he phU'O'Ug trinh: ~
{ I+xy= x+y-l
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S2 2P - SP = 4 (1)
{1 + P = " S -1 (2)
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*(1) <::::> (S +2)(S -2-P) = 0 <::::> S:::: P+ 2 (lo'ili S = -2, do dk)
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*The vao (2): 1 + P :::: -.11 + P <=> P :::: 0 V P = -1 Suy ra S = 2 ho~c S = 1
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* N h' A
(0'2)' (2'0)' [I+.J5 .1-.J5).[l-.J5.1+.J5)
g l~m , , " 2 ' 2 ' 2 ' 2
Trang 3r:I~II; r ~ -: :: -r -·-IV
v
VI.a
Tioh ehiSu cao hloh chOp S.ABC va khoang each:
*GQi H lit chan dUOng cao eua hinh chop, SA == SB == SC nen H lit
tam day, v~y H III trung di~m AB
a2 Jj a3
,
*AC = aJj => SABC = - 2 ; v = 2 nen SH = afj
* D\l'Ilg hinh chii' nh~t ABCD, 5Uy ra g6c (SB;AC) = g6c
(BS;BD) = rp, vai SB == SD == 2a; BD = afj
*N' en cosrp=4' fj
s
.,
,
, , , , ,'0 :
""";:: ,,, :
/~ \ ~~
;! ,
,-' \.:
A ~ (j_"~"*~"~ _
H\
C6th€dtinhgiavitrai: VT'2a+b+c+ 9 ; Saua6xetham f(x) x+~;x~9
PHAN TU CHQN A.Theo chaO'll!! tnoh chuio:
1 mnh hoc toa dO pbing:
* Phuong trinh dth BC c6 d{lIlg: a(x - 2) + bey -7) = O;a 2 + b 2 > 0
*Chieu cao AH = 1 nen 1 = d(A;BC) = .J <:=> b = Ov 5a + 12b = 0
a 2 +b2
*V~y BC: x - 2:; 0 ho~c BC: 12x 5y + 11 = 0
* Suy ra C(2 ;-1) ho~c C(-2617;- 4717)
c
2 mnh hoc toa dO khong gian:
* M~t cAu (S) qua A, B, C nen co tam I thuQc hai m~t trung tnrc (M), (N) cua AB, AC :
(M) : 3x - 2y +3z 4 = 0 ; (N) : 2x 3y + z =O
* Suy ra I c6 to~ dQ d{mg tham 56: ](1 + 7t; 1+3/; 1 51)
B
*(S)ti€p xucvai (P) nen d(I;(P» = IA =R<=> R;:: J6 = J(l +7t)2 +(31-1)2 +(2-5t)2
I=Ovt = 12 vAR= J6
83
* V~y phuang trinh (S) : (x - V +(y _1)2 +(z _1)2 == 6
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e
Tioh tfeh phan: 1= S 2
o e
2
* 1 == eX => dl == eX dx I = S-2-dt
I I -9
~
* ] =(t+~lnl/-31)2
2 t+3
* ] =1+-ln 2
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Trang 4VILa Sa phuc:
* z x + iy (x;y E lR), ta co x2 +(y -Ii == 4 (*)
* w = iz + i -1 == -(y+1) +(x + l)i co di~m bi~u dien M : XM = -Y
* Thay y -XM 1 va x = YM - 1 vao (*) ta duqc (XM + 2)2 + (YM _1)2
*V~y ~p hQP M la dubng tron tam
C6 thJ d;Jt w = x + i Y ; BiJu diln z thea x; y Tfr Iz - il = 2 suy ra kq
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1, YM = x +1
== 4
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VII.b
B.Theo chU'O"llg trinh nang cao:
1 mnh hoc toa do phing:
*FJ(-4;0), F2(4;0) va FJF2 = 8
* Gia sir M(x;y) ta e6 MFJ 5 + 4/5.x; MF2 = 5 - 4/S.x Theo dli Cosin:
82 = (S + 4/S.X)2 + (5 -4/S.xi - 2(5 + 4/S.x)(S - 4/S.x).cos60o
3J3
*Gilii tim duqc x2 13 25/16 Suy ra Iy I= - -
4
* V~y S(MF1F2) = 3J3
2 mnh hoc toa do khong gian:
2
* Pt (P): a( x - 3) + b( y - 1 ) + e( z - 2) = 0, vcri a2+ b2 + c > 0
(P) song song d nen : 2a + b + 2e = O
l-a+2b- 3cl
*Khoang cach (d;(P» = d(B;(P» = I
2 2 ' vcri B(2;3;-I) thuQc d
2
va +b +c
* d(d;(P» = 3 <:::;> 2c 2 - ac -lOa 2 =0
*Chon a = l;e = -2 , b = 2 ; ho~c a 2; c = 5; b = -14
V~y phuong trinh m~t ph!ng (P): x + 2y - 2z - 1 = 0 ho~e 2x - 14y + 5z
Tim tham sa m:
co nghl~mx.
{ X2 -mx+m2 -3+(x+2)(2x-m)=0
* H~ pt tren <:::;>
2
{ (x+2)(2x m)=O
* (I) c6 nghi~m khi m = ,
* (II) co nghi~m khi m =-1; m == ±2
2 = O
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