Tuyen chgn & Gi&i thi$u dethi Todn hqc - Ngui/en Phti Khdnh , Nguyen Tat Thu.. Cdch i:Go\ la giao diem 2 duong cheo hinh chCr nhat ABCD.. Biet phuong trinh DC se viet dupe phuong trinh A
Trang 1Cau 9b: Dat t = |z|, t > 0 thi ta c6 \zf (a.z + b) = -c.z
I PHAN CHUNG CHO TAT CA CAC THI SINH
Cau 1: Cho ham so y = - 3mx^ + 4m^ c6 do thj (Cm)
a) Khao sat sy bien thien va ve do thi ham so' khi m = 1,
b) Xac djnh m de hai diem eye trj cua do thi ham so' doi xiing nhau qua
duong thing y = x
Cau 2: Giai phuong trinh: (sin^ x +1 j +1 = 73 sin 2x + 4 sin
Cau 3: Giai hf phuong trinh
n
x + —
6J -xy = 1
6 ( y - 2 ) 2 + ^ = x3y[(2y + 3)2-6'
dx Cau 4: Tinh tich phan: I = [ -
- i l + x + Vl + x^
Cau 5: Cho hinh chop S.ABCD c6 day ABCD la hinh chu nhat eo tam O va
AB = a, A D = a73 ; SO = SD Mat phSng (SBD) vuong goc voi mat day, mat
phing (SAC) tao voi day mpt goc 60° Tinh the tich khoi chop S.ABCD
Cau6: Chung minh rang ne'u a,b,c>0 t h i :
a + b b + c ic + a ^
(a+b Vb+c ya+c
I I PHAN RIENG Thi sinh chi dugc chpn lam mpt trong hai phan (phan A
ho^c B)
A Theo chUorng trinh chuan
Cau 7a: Trong mp voi h? tpa dp Oxy cho duong tron (C) c6 phuong trinh
x^ + y^ - 2x + 6y -15 = 0 Viet PT duong thing A vuong goc voi duong thing
d : 4x - 3y + 2 = 0 va cat duong tron (C) t^i A, B sao cho AB = 6
62
Cau 8a: Trong khong gian voi h# tpa dp Oxyz cho hai duong thang:
d j : 2 ^ = X = ^ ; d 2 : ^ = ^ = ^ va hai diem A(l;-1;2), B(3;-4;-2) Xet vi tri tuong doi ciia d] va d2 Tim tpa dp diem I tren duong thing d j sao
cho lA + IB dgt gia trj nho nhat
Cau 9a: Cho Z j , Z2 la cae nghi#m phuc ciia phuong trinh 2z^ - 4z + l l = 0
Cau 8b: Trong khong gian Oxyz, tim tren Ox diem A each deu duong , x — 1 V z + 2 '
thang d : — ^ = = va mat phang (a): 2x - y - 2z = 0 Cau 9b: (1 diem) Giai h^ phuong trinh sau:
2logi_x(-xy - 2x + y + 2) + log2+y (x^ - 2x +1) = 6 logi-x (y + 5) - log2+y (x + 4) = 1
Cac diem eye tri ciia (C^,) la M(0; 4m^)va N(2m; 0 ) Trung diem cua doan M N la l ( m ; 2m^ jva M N = (2m;-4m3) Duong thing d : y = x c6 vecto chi phuong la u = ( l ; l )
M , N doi xung nhau qua duong thing (d) <=> M N 1 (d) va I G (d)
M N u = 0 2m'' = m
Ke't hpp dieu kif n ta dupe m = ±
Trang 2" ^
Cau 2: Phuong trinh t u o n g d u o n g voi :
sin"* X + 2sin^ x + 2 - 2\/3 sin xcosx - 2 V 3 s i n x - 2 c o s x = 0
12y^ - 48y + 63 = 8x^y^ + 24x^y^ + 6x^y
T r u hai p h u o n g t r i n h cvia h? ta c6 duoc:
(8y^ + 24y^ + 6y +1) = - ( y ^ - Uy^ + 48y - 64)
o x ^ ( 2 y + l f = - ( y - 4 ) ^ <=>x(2y + l ) = - y + 4 o x + y + 2xy = 4
T ^ n u •u'^^ , [x^ + y ^ - x y - l
Ket hgip voi h^ ta co: <
[x + y + 2xy = 4 Dat S = X + y , P = xy ta c6:
M|t khac (SBD) 1 ( A B C D ) S H 1 ( A B C D ) nen S H la duong cao cua hinh chop
Gpi K la hinh chieu cua H len canh A C , suy ra goc H K S chinh la goc giira hai mat phang ( S A C ) va mat day nen H K S - 60°
7 2 U 7 b j 72l7^ 7 ? j 72l7^ 7 b J 1
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Trang 3A Theo chUtfng trinh chuan
Cau 7a: Duong tron (C) c6 tam 1(1;-3), ban kinh R = 5 Gpi H la trung diem
Suyra l A + IB dat gia trj nho nha't bang A j B , dat dup-c khi A i , I , B thang hang
I la giao diem ciia A j B va d
B Theo chUtfng trinh nang cao
Cau 7b: Duong thang AB c6 phuong trinh dang: a(x + 2) + b ( y - 2) = 0
<=> ax + by + 2a - 2b = 0 vai a^ + b^ > 0 Duong thang CD c6 phuong trinh dang:
Suy ra phuong trinh A B : x - y + 4 = 0, C D : x - y - 4 = 0
Phuong trinh BC va D A c6 dang x + y + c = 0
Khoang each t u A deh m|it phang (a) : d ( A ; a ) = - j = = = = =
( a ) qua M o ( l ; 0 ; - 2 ) va c6 vecto chi phuong u = ( l ; 2; 2 ) D3t M Q M I = u
Do do: d ( A ; ( a ) ) la ihionj'; cao ve tu A trong tam giac A M Q M I
j/ A ^ 2.S,^MoMi ['•'^^o;"] V8a2-24a + 36
= > d ( A ; A ) = 1- = -^;— i =
Theo gia thiet: d ( A ; ( a ) ) - C I ( A ; A )
|2a| Vsa^ - 24a + 36 , 2 o 2 -.^ A 2 r -.^ n
o ! — = <::> 4a = Sa"^ - 24a + 36 <=> 4a - 24a + 36 = 0
3 3
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Trang 4l o g i - x ( y + 5)-log2+y(x + 4) = l
j l o g i - x ( y + 2) + log2+y (1 - x) - 2 = 0 (1)
l l o g i - x ( y + 5)-Iog2^y(x + 4) = 1 ( 2 ) ' Dat l o g 2 + y ( l - x ) = t thi (1) tro thanh: t + ^ - 2 = 0 < » ( t - l ) ^ = O o t = l
Vol t = 1 ta c6: 1 - X = y + 2 o y = - X - 1 (3).The vao (2) ta c6:
K i e m tra dieu k i ^ n ta thay chi c6 x = - 2 , y = 1 thoa man dieu k i | n tren
Vay h? CO nghipm day nhat x = -2, y = 1
DETHITHllfSdll
I PHAN C H U N G C H O T A T CA C A C T H I S I N H
Cau 1: Cho ham so y = x^ - Sx^ + 1 , c6 do thj la ( C )
a) Khao sat su bien thien va ve do thj (C) ciia ham so
b) T i m cac diem A, B thupc do thi (C) sao cho tiep tuyen ciia (C) tai A , B
song song v o i nhau va A B = 4V2
Cau 2: Giai p h u o n g trinh: 5 cos 2x + - - 4 sin rilTt >
x + 9
Cau 3: Giai p h u o n g trinh: 2(x^ + 2) = 5Vx^ + l
dx Cau 4: T i n h tich phan: I = f-
i x ^ + 2 x
63
CtyTNHHMTV DWH Khang Vjft
Cau 5: Cho t u d i ^ n ABCD c6 ABC la tarn giac deu canh b i n g 2a, A D vuong g6c
day, A D = ay/3 Gpi E, F Ian lupt trung diem ciia cac doan BC, DE Tinh the
jfch hinh chop F.ABC Chung m i n h AF vuong goc vol CD
Cau 6: Cho so thuc d u o n g a, b thoa man: 6|a^ + b^ j + 20ab = 5(a + b)(ab + 3)
Tim gia trj nho nhat ciia bieu thuc
A Theo chi/orng trinh chuan
Cau 7.a: Trong mat phJing tpa dp Oxy, cho hinh chii nhat ABCD, voi toa dp cac
f 4^
dinh A ( l ; l ) G p i G 2 ; - la trpng tarn tam giac ABD T i m tpa dp cac dinh con
l^i ciia hinh chu nhat bie't D nam tren duong thang c6 phuong trinh: x - y - 2 = 0
Cau 8.a: Trong mat phSng tpa dp Oxyz, cho hai mat phang ( P ) : x + ^ - 2 z + 5 = 0
va ( Q ) : X + 2y - 2z - 1 3 = 0 Viet phuong trinh mat cau (S) d i qua goc tpa dp
O, qua diem A (5; 2; 1 ) dong thoi tiep xiic voi ca hai mat phSng ( P ) va ( Q ) Cau 9.a: Tinh m o d u n ciia so phuc z , bie't z^ + 12i = z va z c6 phan thuc duong
B Theo chUorng trinh nang cao
X^ y2
Cau 7.b Trong mat phang tpa dp Oxyz, cho elip (E): — + ^ = 1 va duong
thSng d : X + y + 2013 = 0 Lap phuong trinh d u o n g thang A vuong goc voi d va
c^t ( E ) tai hai diem M , N sao cho M N = —
3 Cau 8.b Trong mat phSng tpa dp O x y z , cho mat phang ( P ) : x - 2y + 2z + 2 = 0
va duong th5ng ( d ) : = = Mat cau (S) c6 tarn I nam tren duong thang (d) va giao voi mat phang ( P ) theo mpt d u o n g tron, d u o n g tron nay Voi tam I tao thanh mpt hinh non c6 the tich Ion nhat Viet p h u o n g trinh mat
cau (S), bie't ban kinh mgt cau bSng 3N/3
Cau 9.b: Giai h$ p h u o n g trinh sau: 2x'^ + 2 x y - 3 x - y + l = 0
2 = 0
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Trang 5Tuyen ch(>n fy Gim thieti dethi Todn H Q C - Nguyen Phu Khdnh , Nguyen Tat Thu
WSQm DANGlAl
I PHAN CHUNG CHO TAT CA CAC T H I SINH
Cau 1:
a) Danh cho ban dpc
b) Goi A^a;a'' -3a^ + l j va B|b;b^ -3b^ +1 va a ?t b la cac diem thoa man
bai toan Ta c6: y' = 3x^-6x
Phuong trinh tiep tuyen tai A c6 he so goc y'(a) = 3a^ - 6a
Phuong trinh tiep tuyen tai B c6 h? so goc y'(b) = 3b^ - 6 b
Do 2 tiep tuyen tren song song voi nhau nen ta phai c6 y'(a) = y'(b)
Vay a, b la nghiem phuong trinh: - 2X - 3 = 0 => X = - 1 hoac X = 3
Do do ton tai hai diemA(-1;-3),B(3;1) hoac A(3;1),B(-1;-3) thoa man
yeu cau bai toan
Cau 2: Ta c6: sin - X = sin f O X + —
Phuong trinh cho viet lai: 5sin^ X +
-12 + 2 sin X + •
12 - 7 = 0 (*) Dat t = sin X + •
Cty TNHH MTV DWH Khang Vie
Cau 3: Voi ^ < x < , dat u = V l 0 - 3 x , dua phuong trinh ve h^:
X 4x + 3u ^ fj.^ yg'fj^gQ yg'ja ^u,g,c + X - 5 ) ( u - x + 2) = 0=> X = 3 u^ + 3 X - 1 0 = 0
Cdch khdc: Binh phuong 2 ve, ta duoc: 37l0-3x - 3 = -x^ + 4x - 3
+ 1 - X = 0 =>x = 3
Dat t = x^ + 2 => dt = 2xdx hay xdx = - d t Khi do:
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Trang 6Tuyen chgn & Gi&i thi$u dethi Todn hqc - Ngui/en Phti Khdnh , Nguyen Tat Thu
Suy ra AF.CD = 1 Sa^ -l.la^ ~^Aa^ = 0 =^ AF 1 CD
Cdch i:Go\ la giao diem 2 duong cheo hinh chCr nhat ABCD V i G la trpng
tam tam giac ABD nen A, G, I thang hang Theo tinh chat trpng tam tam giac ta
(5 3^
de dang tim ra toa dp diem I - ; - Vi I l a trung diem AC nen biet tpa dp A, I
ta se tim ra tpa dp C(4;2)
V i D thuoc duang thSng x - y - 2 = 0 m a C thoa man phuong trinh nay
Cty TNHH MTV DWH Khang Vift
Do do DC: X - y - 2 = 0
Biet phuong trinh DC se viet dupe phuong trinh AB ma ABCD la hinh chii jih$t ^^^^ phap tuyen AB ta se biet phap tuyen A D t u do viet dupe phuong trinh A D Tpa dp D la giao diem ciia A D va DC Ta tim dupe D
Vi I la trung diem BD nen ta tim dupe diem B
Cdch 2;Gpi I la trung diem ciia BD Theo tinh chat trpng tam ta c6:
Trang 7Tuyen chqn &• Giai thifu dethi Toan hqc - Ngtiyen Phii Khdnh , Nguyen Tai Thu
Giai phuong trinh (3) ta dugc y = -1 => = 4 Do x > 0 nen x = 2
Vay z = 2 - i => |zj = N/S
B Theo chiforng trinh nang cao
Cau 7.b: (A) C6 phuong trinhy = x + b,he M Phuong trinh giao diem cua (A)
Vay, CO hai duong thang thoa man yeu cau de bai la y = x + 3, y = x - 3
Cau 8.b The tich khoi non la V = - S h , trong do h = d(l;(P)), day la duong
DSng thuc xay ra khi 2h2 = 27 - h^ <=> h = 3
Vay, max V = 1871 khi h = 3
Hon niia: I € d =:> l(2 +1 ;-2t ; - l + 3t)
d(l;{P)) = h = 3 o | l l t + 2| = 9 » t = - l hoac \ = ^
Voi, t = -1 => 1(1 ;2 ;-4) phuong trinh mat cau (S):
( x - l ) ^ ( y - 2 f ( z 4 f =27
Cty TNHH MTV DWH Khang Vi?t
Voi, t = — =^ I 29 _14 10 11' 11 '11 phuong trinh mat cau (S):
Cau 5: Cho hinh chop S.ABC, day ABC la tarn giac vuong tai B c6 AB = a,
BC = a73, SA vuong goc voi mat ph5ng {ABC) va SA = 2a Gpi M, N iSn lu<?t
la hinh chieu vuong goc cua A len SB va SC Tinh the tich cua hinh chop A.BCNMva cosin goc giira'MN va AB
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Trang 8Tuye'n chtfn b Giai thi?u dethi Todtt hgc-Nguyen Phu Khdnh , Nguyen Tat Thu
Cau 6: Cho a, b > 0 thoa man a + b - ab < 0
a^b^ 12 Tim gia t r i nho nhat ciia bieu thuc: P = — + — + 12ab
a^ + b^ ab
I I PHAN R I E N G T h i sinh chi dugc chpn lam mpt trong hai phan (phan A
hoac B)
A Theo chUtfng trinh chuan
Cau 7,a: Trong mat phSng Oxy, cho tarn giac ABC vuong tgi C, biet A ( 3 ; 0 ) ,
dinh C thuoc true tung, diem B nam tren duong thSng A: 4x + 3y - 1 2 = 0 Tim
tpa do trpng tam tam giac ABC, biet dien tich tam giac ABC bang 6
Cau 8.a Trong i<h6ng gian voi h^ true tpa dp Oxyz, cho ba diem A(2; 3; 1),
B(2;3;-2), C(0;4;-3) va mat cau (S): ( x - l f + ( y - 2 f + ( z - 3 f =25 Gpi mat
phang (P) d i qua A va cat (S) theo giao tuyen la mpt duong tron c6 ban kinh
nho nha't Lap phuong trinh duong thang (d) d i qua B nam trong mat phSng
(P) va each C mpt khoang bang
( l W 3 i ) 16(1+ i )
2
3
Cau 9.a: Cho so phuc Z j = y Tim tap hp-p diem bieu dien cho so phuc
Z 2 , biet rSng: Z j - izj + Z j =2
B Theo chi/tfng trinh nang cao
Cau 7.b: Trong mat phSng Oxy, cho diem A ( - 1 ; 14) va duong tron (C) c6 tam
l ( l ; - 5 ) va ban kinh R = 13 Viet phuong trinh duong thSng d d i qua A cat (C)
tai M , N sao cho khoang each t u M den A I bang mpt nua khoang each t u N
cua phuong trinh: z^ + 8bz + 64c = 0
>^iy 1 r v n n ivit v uv v i i i\mmg rT^i
Hl/OfNGOANGlAl
PHAN CHUNG CHO TAT CA CAC THI SINH Cau 1:
3) Danh cho ban dpc
b) Gpi A(a;a'* -a^ + l ) la diem thoa man debai
Ta eo: y ' = 4x'' - 2x Phuong trinh tiep tuyen (d) cua (C) tai A la:
y = ( 4 a 3 - 2 a ) ( x - a ) + a ' ' - a 2 + l
Phuong trinh hoanh do giao diem cua (d) va (C) la:
x 4 _ x 2 + l = ( 4 a 3 - 2 a ) ( x - a ) + a ' * - a 2 + l » ( x - a f (x^+2ax + 3 a 2 - a ) = 0
c>x = a hoac g ( x ) = x^+2ax + 3a^-a = 0
Theo bai toan thi g ( x ) = 0 c6 2 nghiem phan bift x^ X j sao cho: X j <a < X2
- 4 ( s i n x + eosx) = l
> 3cos^ X + 3(sin X + cosx) - 4(sin x + cosx)sin^ x = sin^ x
> ( s i nX + c o s x ) | 3 4 s i n ^ xj + 3eos^ x -sin^ x = 0
| > (sin X + cosx)(3 - 4sin^ x) + 3(1 - sin^ x] - sin^ x = 0
> ( s i nX + cosx)(3 - 4sin^ x) + 3 - 4sin^ x = 0
> ^3 - 4 sin^ x j(sin x + cos x +1) = 0
Trang 9Ket hgp dieu kien =>x = - — + k27t, k e Z
Vay, nghiem phuang trinh cho la:
Dat u = N / 2 X - 3 > 0, V = \/5-2x >0 Khi do phuong trinh (*) bien doi thanh
(Su^ + ju + (7u^ + 3v^) V = 2(2 + Suv)
o3^u^ + v^j + 7uv(u + v) = 16uv + 4
3(u + v)^u^ + - uvj + 7uv(u + v) = 16uv + 4
3(u + v)(2 - uv) + 7uv(u + v) = 16uv + 4 (**)
<=>
t^ - 2 Lai dat t = u + v, V2 < t < 2 => uv = -Y— Luc do phuang trinh (•*) tro
thanh: t^ - 4t^ +1 + 6 = 0 (t + l)(t - 2)(t - 3) = 0 <=> t = 2 thoa dieu ki^n
t-1 2t ,
N/2+1
.dx = t ^ - l dt
1-t^+1 t^(t + l)
AC = AS = 2a, ASAC vuong can tai A
N la hinh chieu cua A len SC => N la trung diem SC SN = aN/2,SM = 4aS
Trang 10SC
2V3O
AMPN c6: NP^ = NM'' + MP^ - 2MN.MP.cosPMN cosPMN =
40
Cau 6: Tu dieu kien bai toan ta c6 (a + b)^ < a^b^ o a^ + b^ < a^b^ - 2ab
Hon nua a^b^-2ab >a^+b^ >2ab=> ab> 4 Dat t = ab,tac6 t > 4
a^b^-2ab ab + — + 12ab hay P > — + — + 12t t^-2t t
Xet ham so f (t) = —^ + — + 12t vol t > 4
t 2 - 2 t t
I I PHAN RIENG Thi sinh chi dupe chpn lam mpt trong hai phan (phan A
hoac B)
A Theo chUOng trinh chuan
Cau 7.a: Gia su rSng: B (3b; -4b + 4), C (O; c)
Taco: AC = {-3;c),BC = (-3b;4b + c - 4 )
Gia thiet tam giac ABC vuong tai C ta c6:
AC.BC = 0 <r>9b + 4bc + c^-4c = 0 ( l )
SAABC = r A B d ( C ; A B ) , trong do: AB = 5|b-l|, d(C;AB) = 3c-12
Theo bai toan, ta c6: -.5 b - 1
2
3c-12
= 6=> ( b - l ) ( c - 4 ) = 4 (2)
Tu ( l ) va (2) suy ra:
Cau 8.a Gpi I la tam mat cau (S), lA = ( l ; l ; -2)=:>IA = N / 5 < R = > A nam tron
m|t cau Vi the (P) cat mat cau (S) theo duong tron c6 ban kinh r, trong d
r = - I H ^ va H la hinh chieu ciia I len mat phSng (P)
a + b
CB,u 41 (b + c) +(a-2c) +(a + 2b) _ 3
~ 2
c = - 2 (b + c f + ( a - 2 c ) ^ + ( a + 2b)^ ^ 3
2 a^+b^+c^
B Theo chUtfng trinh nang cao Cau 7.b:
Bai todtt tro thanh: "Viet phuong
trinh duong thang qua A cat duong
tron (C) theo day cung M N = V233 "
Cdch2:Gia sir M ( x ; y ) vi M ^huQc duong tron nen ta c6: ( x - l ) +(y+5) =1^9
Vi M la trung diem ciia A N nen ta c6: N(2X + l;2y -14) Diem N thuoc duong tron nen ta c6: (2x)^ + (2y - 9)^ = 169
81
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Trang 11Ta C O h#: {x-lf+{y + 5f =169
( 2 x ) ^ + ( 2 y - 9 f =169 Cau 8.b: Gpi G la trpng tam cua tam giac ABC, ta c6: GA + GB + GC = 0, tir day
tim dugc G(1;2;3) va MA^ + MB^ + MC^ = SMG^ + GA^ + GB^ + GC^ Ta thay
GA^ + GB^ + GC^ C O dinh, v i the MA^ +MB^ +MC^ nho nhat khi M G c6 do
dai nho nhat, nghia la MG 1 (P) hay M la hinh chie'u cua G len (P)
Gpi (d) la duong thang di qua G va vuong goc voi (P)
Nen C O phuong trinh : - — - = ^ — - = - — -
( l - S i f =l-3Si + 3.3i2 -3^31^ = -8, ( l + if = 2i
( l + V 3 i ) ' ' ( 2 - i ) ( - 8 ) ' ( 2 - i ) 8 ( 2 - i ) , ,
^ 6 3 - • - 8 ( l + 2i) = 8 + 16i
Do do
Theo gia thiet ta c6 (8 + 16i)^ + 8b(8 + 16i) + 64c = 0
<:*(l + 2i)^ + b ( l + 2i) + c = 0 o ( 2 b + 4)i + b + c - 3 = 0
2b + 4 = 0 b = -2
b + c - 3 = 0 c = 5
OETHITHiJSOlS
I PHAN CHUNG CHO TAT CA CAC THI SINH
Cau 1: Cho ham so y = c6 do thi la (C)
x - 1 '
a) Khao sat su bien thien va ve do thj (C) ciia ham so
b) Tim tpa do hai diem B, C thupc hai nhanh khac nhau cua do thj sao cho
tam giac ABC vuong can tai A(2;l)
3 x ^ - 3 x + 2
Cau 5: Cho lang try dung ABC.A'B'C c6 day ABC la tam giac vuong tai A,
BC = 3a, AA' = a va goc giira A'B voi mat phSng trung true doan BC bSng 30" Tinh theo a the tich khoi lang try ABC.A'B'C va khoang each giua hai duong thing A ' B voi AC
Cau 6: Cho cac so x, y, z thoa man x,y,z e [ l ; 9 ] va x > y,x > z Tim gia trj Idn
A Theo chUorng trinh chuan
Cau 7.a: Trong mat phang Oxy, cho cac duong tron (Cj): (x -1)^ + y^ = ^ va ( C j ) : ( x - 2 ) ^ + ( y - 2 ) ^ = 2 Viet phuong trinh duong thang d tiep xiic voi duong tron (Cj) va cat duong tron ( C j ) theo day cung eo dp dai 2>/2 Cau S.a: Trong mat phMng Oxyz, cho duong thSng ( d ) : - = ~ ~ vamatcau (S): (x + l ) + ( y - 2 ) + ( z - l ) = 25 Viet phuong trinh duong thSng
A di qua diem M ( - 1 ; - 1 ; - 2 ) cat duong thing (d) va cat mat cau (S) tai hai diem A, B sao cho AB = 8
Cau 9.a: Goi Z j , Z j la 2 nghiem phuc ciia phuong trinh z^ - 2-N^Z + 8 = 0
Tinh gia trj cua bieu thuc z f + z f
B Theo chUorng trinh nang cao
Cau 7.b: Trong mat phang Oxy, cho duong tron (C): ( x - l ) + ( y + l ) =9 c6 tam I Viet phuong trinh duong thing A di qua M ( - 6 ; 3 ) va cat duong tron (C) tai hai diem phan bi^t A, B sao cho tam giac lAB c6 di$n rich bSng va
A B > 2
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Trang 12Tuye'n chgn & Giai thifu dithi To&n hQC-Nguyen PUu Khunh , N\;iii/rn I'at Thu
C a u 8.b: Trong khong gian tpa dp Oxyz, cho hai diem A ( 3 ; - 2 ; 3 ) , B(-5;10;-1)
va mat phang (P) : 2x + y + 2z - 1 = 0 Viet p h u o n g t r i n h d u o n g thSng A d i qua
M € (P) cat d va tao v o i d m p t goc cp c6 gia trj coscp = biet d:
c - 1 2b
b - 1 2c
c - 1 2b
b - 1 2c
Trang 13Tuyen chpn & Giai thifu de thi Todn hqc - Nguyen Phii Khdnh , Nguyin TA't Thu
2 thoa bai toan
Cty TNHHMIV I)VVHKhang Vift
7 4
Vay, m i n P = —, maxP = — 6 3
I I PHAN R I E N G T h i s i n h c h i dugic chpn l a m mgt trong h a i p h a n ( p h a n A hoac B)
f^ Theo chi/orng trinh chuan
C a u 7.a: (Cj)- c6 tarn I | ( l ; 0 ) va ban kinh Rj =
Voi a = - 7 b , suy ra d : 7x - y - 1 2 = 0
Vay, CO 2 d u o n g thang can tim: x - y = 0, 7 x - y - 1 2 = 0
C a u 8.a: Goi M ' la giao diem cua d va A, suy ra M'(2 - t ; l - 2 t ; l +1)
= i M M ' = ( 3 - t ; 2 - 2 t ; 3 + t )
Gpi I la tam cua mat cau Ta suy ra tpa do diem I ( - 1 ; 2; l ) , phuong trinh mat
phang d i qua I c6 vecto phap tuyen M M ' = (3 - t; 2 - 2t; 3 + t ) :
( P ) : ( 3 - t ) ( x + l ) + ( 2 - 2 t ) ( y - 2 ) + (3 + t ) ( z - l ) = 0
A, B la 2 giao diem ciia A voi mat cau, neu goi J la trung diem cvia A B thi ta
CO OJ vuong goc v o i AB va OJ = 3, hon nua I M = Syjl => JM = 3
Hay d ( M ; ( P ) ) = 3 ^ ^ ^ ' ^ ^ ^ 3 o | t - 5 | = \/6t^-8t + 22
V 6 t ^ - 8 t + 22
<=> t = - 1 hoac t = —
5 Phuong trinh d u o n g thang can tim di qua 2 diem M va M '
^ a u 9 a : z^-2>/2z + 8 = 0 c6 A' = - 6 = (>/6i)^
Phuong trinh cho c6 2 nghiem phuc la: Zj 2 = V2 ± \f6i
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Trang 14Tuyen chgn b Gioi thifu Ai thi Toan htfc - Nguyen PM Khdnh , Nguyen Tai Thu
(V2 + = 2N/2 + 3.2.76 i + 3.V2.(V6i)^ + (76i)^ = -I6V2,
Voi X = 1 => AB = 2 khong thoa
Voi X = 2N/2 A B = 4>/2 nhan Suy ra I H = 1
Cach 2: (C) c6 tarn I ( l ; - l ) , ban kinh R = 3 Duong th^ng A di qua M c6 dgng:
2 3 2 hay 7 a - 4 b , , = 1 « 483^ - 56ab + ISb^ =0 o (4a - 3b)(l2a - 5b) - 0
Va^ +
Cau S.b: De tha'y, AB = 4N/T4 , ket hgp gia thie't suy ra MA + MB = AB
Hon nua A, B nkm 2 ve 2 phia so voi mat ph5ng (P) suy ra M la giao diem
cua duong th^ng AB va m|t phSng (P)
Cty TNHH MTV DWH Khang Vift
Phuong trinh duong thing AB la:
x = 3 + 2t
y = -2 - 3t
z = 3 + t
Suy ra tpa dp diem M (-3; 7; O)
GQi giao diem cua A va d la N , c6 tpa dp la: N (2t;-l - t ; l + 2t)
a) Khao sat sy bien thien va ve do thj (C) cua ham so
b) M la diem thupc (C) c6 hoanh dp x^^ ^ 1, Tie'p tuyen tai M cat do thj
(C) tai diem thu hai N (khac M) Tie'p tuyen tai N cat do thj (C) t^i diem thu hai P (khac N) Gpi Sj la di^n tich hinh phSng gioi han boi do th} (C) va duong thiing M N , Sj la di?n tich hinh phing gioi han boi do thj (C) va
Trang 15Tuyen chgn & Gidi thifu dethi Todn HQC - Nguyen Phu Khdnh , Nguyen Tat Thu
Cau 2: Giai phuong trinh: sinx + Vscosx + 2 = v2 + cos2x + V3sin2x
Cau 3: Giai phuong trinh: (2x - 4 ) V3x-2 + Vx + S -yjsx^ + 7 x - 6 = 5x - 7
Tax-' +lnfx +Vx^ +1 Cau 4: Tinh tich phan: I = J ^ ^dx
A Theo chUorng trinh chuan
Cau 7.a: Trong mat phang tpa dp Oxy, cho hinh vuong ABCD, c6 canh AB di
qua diem M ( - 3 ; - 2 ) , va x^ > 0 Tim tpa dp cac dinh cua hinh vuong ABCD
khi duong tron (C) :(x - 2^ + (y - 3)^ = 10 npi tiep ABCD
Cau 8.a: Trong khong gian voi he tpa dp Oxyz, cho mat phang
(Q ): x + 2y-2z + 2 = 0 va duong thang (d) c6 phuong trinh
^^=-^=-^Y~-Lap phuong trinh mat phang (P) chua duong thang (d) va tao voi mat
phang (Q) mpt goc a voi sin a = J -
V 6
Cau 9.a: Tim so phuc z bie't 2^z + l j + z - l = ( l - i ) z ^
B Theo chUtfng trinh nang cao
Cau 7.b: Trong mat ph3ng tpa dp Oxy, cho tam giac ABC, c6 A(2; -2), B(4; 0),
C 3;N^-1J va (C) la duong tron ngoai tiep tam giac Duong thang d c6
phuong trinh 4x + y - 4 = 0 Tim tren d diem M sao cho tiep tuye'n qua M tiep
xuc voi (C) t^i N thoa man S^^Q dat gia trj Ion nha't?
Cau 8.b: Trong khong gian voi h$ tpa dp Oxyz, cho duong thSng
(d): i i ^ = y i l = £ vamatphSng (P): x + 2 y - z - 3 = 0
Viet phuong trinh duong thang d' thupc (P), vuong goc voi d va c6 khoang
each giCra d va d' bSng yfl
Cty TNHH MTV DWH Khang Vift
Cau 9.b: Giai phuong trinh sau tren t^p so phuc C: z * - z ^ + ^ + z + l = 0
H M G DANGIAI
I PHAN C H U N G C H O T A T C A C A C T H I S I N H Cau 1:
a) Danh cho ban dpc
fx =1+X
b) Thvrc hi?n phep bien doi phep tinh tien • ^ Trong hf true mdi duong
cong (C) CO phuong trinh Y = - 3X M la diem thupc (C) c6 hoanh dp
Xj^ = m 5" 0, khong mat tinh tong quat c6 the coi m > 0 Tiep tuye'n tai M c6 phuong trinh: