Luyện giải đề trước kỳ thi đại học tuyển chọn và giới thiệu đề thi toán học phần 2

Tuyen chgn & Giai thifu dethi Todn hgc - Nguyen Phu Khdnh, Nxuyht Tai Thu.. Viet phuong trinh mat phing p vuong goc duong thang d, cat mat cau S theo giao tuyen la mpt duong tron c6 ban

Tuyen chgn & Giai thifu dethi Todn hgc - Nguyen Phu Khdnh, Nxuyht Tai Thu

OETHITHUfSdzi

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: Cho ham so : y = x'^ - 3x^ + mx +1 c6 do thi la (C^^)

a) Khao sat sy bien thien va ve do thi (C) cua ham so khi m = 0

b) Tim m de ham so c6 cue dai, cxfc tieu Gpi (A) la duong thang di qua hai

diem eye dai, cue tieu Tim gia trj ion nhat khoang each tir diem I - ; — den

Cau 5: Cho hinh hpp dung ABCD.A'B'C'D' eo day la hinh thoi e^nh a, BAD=a

voi cosa=-, canh ben AA' = 2a Gpi M la diem thoa man DM = k.DA va N la

4

trung diem cua canh A'B' Tinh the tich khoi tu dien C'MD'N theo a va tim

kde C ' M I D ' N

Cau 6: Cho 3 so thuc khong am a, b, c thoa man a + b + c = 3 Tim gia tri nho

nhai cua bieu thuc: P = a + b^ + c^

II PHAN R I E N G Thi sinh chi dupe chpn lam mpt trong hai phan (phan A

hoac B)

A Theo chUorng trinh chuan

Cau 7.a: Trong mat phang tpa dp Oxy, cho tam giac ABC vuong tai A va diem

B(1;1) Phuong trinh duong thSng AC: 4x + 3y - 32 = 0 Tia BC lay M sao cho

5>/2 BM.BC = 75 Tim C biet ban kinh duong tron ngoai tiep tam giac AMC la — ^

Cau 8.a: Trong khong gian Oxyz, cho hai duong thSng (dj): = =

(d;): ^ 2 ~^ \ \g (P): x + y-2z + 5 = 0 Lap phuong

trinh duong thSng (d) song song voi m^t phSng (P) va cat (d^), ( d j ) Ian lupt

tai A, B sao cho dp dai doan AB nho nhat

140

Cau 9.^: Tinh modun cua so'phiic z, biet: z = (2 - i)"^ + ( l + i)'* - ^ — i -

0 Theo chUomg trinh nang cao Cau 7.b: Trong mSt phSng tpa dp Oxy, cho hinh vuong ABCD eo phuong trinh duong thing AB: 2x + y - 1 = 0, va C, D Ian lupt thupc 2 dupng thing

d j : 3x - y - 4 = 0, dj : x + y - 6 = 0 Tinh di|n tich hinh vuong

x = - t Cau 8.b: Trong khong gian Oxyz, cho duong thang (d): y = 1 + 2t va mSt cau

[z = -3-2t (S): x^ + y^ + z^ - 2x - 6y + 4z -11 = 0 Viet phuong trinh mat phing (p) vuong goc duong thang (d), cat mat cau (S) theo giao tuyen la mpt duong tron c6 ban kinh r = 4

Cau 9.b: Tim so phue z thoa man (l - 3i) z la so thuc va z - 2 + 5i = 1

HMGDANGIAI

1 PHAN C H U N G C H O T A T C A C A C T H I S I N H Caul:

a) Danh cho ban dpc

b) Taco y' = 3x^-6x + m Ham so c6 eye dai, eye tieu khi phuong trinh y' = 0 c6 hai nghi^m phan bi?t.Tuclaeanc6: A ' - 9 - 3 m > 0 o m < 3

x _ l _

3 3 2m -2 x + — + 1 m , Chia da thiic y cho y', ta dupe: y = y'

Gia sir ham so c6 eye d^i, eye tieu t^ii cae diem (xi;yi),(x2;y2) •

Vi y'(xj) = 0,y'(x2) = 0 nen phuong trinh duong thing (A)qua hai diem

eye dgi, eye tieu la: y = r2m_2^ + ^ + 1 hay y = —(2x + l)-2x + l

( \

Ta thay, duong thang (A) luon di qua diem co'djnh A —;2 so'goc

|*a duong thing lA la k = | Ke IH 1 ( A ) ta thay d(l;A) = I H ^ I A = |

I Ding thuc xay ra khi I A ± ( A ) o ^ - 2 = - i = - - < » m = l V^y, max d ( l ; A ) = | k h i m = 1

141

Tuye'n chgn b Giai thi?u dethi Todu hgc - Nguyett Phu Khanh , Nguyen Tat Thu

Cau 2: Phuang trinh cho tuong duong voi phuang trinh:

sin3x + 3sinx = 4sin^ x.cosx + 2cosx + sinx-cosx

<=> sin 3x + s inx = 2 sin x.sin 2x + cos x - s inx

o 2 sin 2x.cos X - 2 sin 2x.s inx = cos x - s inx

<=>(cosx-sinx)(2sin2x-l) = 0 o c o s x = sinx hoac 2 s i n 2 x - l = 0

Voi cosx = sinx <=> x = — +

Cau 3: Dieu ki#n: x > 3

Phuong trinh cho tuong duong: 5(2Vx-3 - N/2X + I ) = 2X -13

Nhan 2 ve voi bieu thiic lien hg-p va dat thua so chung:

( 2 X- 1 3 ) ( 5 - 2 N / X ^ - N / 2 X + I ) = 0 c^x = y hoac 2N/)r^ + N/2X +1 = 5

, f x < 6 2^/x^ + ^ / 2 ^ = 5<=>27(x-3)(2x + l ) = 18-3x<^• X^ - 8 8 X + 336 = 0

z:>X = 4

13 Vgy, phuong trinh cho c6 nghiem la: x = - y , x = 4

D^t P(b) = 3 - b - c + b2+c^ voi be[U,3'

Taco: P'(b) = 2 b - 1 va P'(b) = O o b = i

Tirdo, ta duoc P > m i n P = c''-c + —

4 Xet P(c) = c 3 - c + J voi ce[0;3"

Theo chi/orng trinh chuan

7.a: Gpi I la tarn duong tron ngoai tiep tarn giac AMC

Vi B n i m ngoai duong tron ( l )

nen ta c6: BM.BC = BM.BC ( l )

V^y, phuong trinh duong t h i n g (d) la: = =

Phuong trinh duong trung tryc I N cua A C => A C n I N = N C ( 8 ; 0 ) hoac Ijifc^^/ ^ 4^ Dfb 6

C(2;8) ^ ^' ^ ' ^ ' ^ ^

Cdch2.Tu M d\mg M K 1 B C , ( K € A B )

G<?i I la trung diem K C => I la tarn duong tron ngogi tiep tam giac A M C

(Do t u giac A K C M npi tiep)

Cau 9.a: Ta c6: (2 - i) = 3 - 4i va ( l + i)'' = (2i)^ = -4

Ta C O A A B C dong d^ng A M B K nen: — = — A B B K = M B B C = 75

M B B K

Phuong trinh duong t h i n g A B qua diem B(1;1) va c6 VTPT (3; - 4 ) :

3 x - 4 y + l = 0

V i A la giao diem cua A B va A C nen A ( 5 ; 4 )

V i ABCD la hinh vuong nen j C D l n

Cau8.a:Dat A ( - l + a;-2 + 2a;a), B ( 2 + 2 b ; l + b ; l + b)

Mat p h i n g (P) vuong goc duong thang (d) nen phuong trinh mat p h i n g (P) bdang: - x + 2 y - 2 z + D = 0

I

=>AB = (-a + 2b + 3;-2a + b + 3;-a + b + l )

Do AB song song v o i (P) nen AB 1 np = ( l ; l ; - 2 ) <=> b = a - 4

d(l,(P),) = V R = - r 2 D + 9

4 ^ o D + 9 = 9<=> D = 0

D = -18

Vay, CO hai mat phang can tim la: - x + 2y - 2z = 0, - x + 2y - 2z - 1 8 = 0

Cau 9.b: Gia su z = x + y i , khi do ( l - 3i)z = ( l - 3i)(a + hi) = a + 3b + (b - 3a)i

I PHAN C H U N G C H O TAT CA CAC THI SINH

Cau 1: Cho ham so: y = x'' - 3x + 2 c6 do thj la (C)

a) Khao sat su bien thien va ve do thj (C) cua ham so

b) T i m toa do diem M thuoc duong thang (d) c6 phuong trinh y = -3x + 2

sao cho t u M ke duoc hai tiep tuyen toi do thi (C) va hai tiep tuyen do vuong

goc voi nhau

Cau 2: Giai phuong trinh: sinxcos2x + cos^ x|tan^ x - 1 j + 2sin"' x = 0

Cau 3: Giai h# phuong trinh:

A B C = 60" Hai mat p h i n g ( S A D ) va ( S B C ) la hai tam giac vuong Ian lugt tai

A va C Dong thai cac mat phMng nay ciing hop voi m^t day mpt goc a Tinh

the tich khoi chop S A B C D theo a va a

T-ty MTV DWH Khung Viei

Cau 6: Cho a, b, c l a 3 sothuc duong thoa man: (a + c)(b + c) = 4c^

Tim gia tri Ion nhat cua bieu thiic: Q = — - — + —^— + ———

b + 3c a + 3c bc + ca

II PHAN RIENG Thi sinh chi du<?c chpn lam mgt trong hai phan (phan A hole B)

A Theo chUcrng trinh chuan

Cau 7.a: Trong mat phang tpa do Oxy, cho duong tron ( C ) : ( x - l ) ^ +(y+2)^ =4

M la diem di dpng tren duong thang d: x - y + 1 = 0 Chung minh rSng t u M ke dup-c hai tiep tuyen M T j , M T j toi (C) ( T j , Tj la tiep diem) va tim toa do diem M , bie't duong thang TjTj di qua diem A ( 1 ; - 1 )

Cau S.a: Trong khong gian toa do Oxyz, cho diem M(0; - I ; 2), N ( - l ; I ; 3)

Viet phuong trinh mat phang (R) d i qua M , N va tao voi mat phang (P):

2 x - y - 2 z - 2 = 0 mot goc nho nhat

Cau9.a: Chosophuc z = 1 - i

1 + i

11 ,2010 _^j,2011 ^^2016 ^22021

- 9

Tinh mo dun ciia so'phuc w = z

B Theo chUtfng trinh nang cao

Cau 7,b: Trong mat phang toa do Oxy, cho hinh thoi ABCD c6 phuong trinh

hai canh AB va A D theo t h u t u la x + 2y - 2 = 0 va 2x + y +1 = 0 Canh BD chua diem M ( l ; 2) Tim toa dp cac dinh cua hinh thoi

Cau 8.b: Trong khong gian toa dp Oxyz, cho mat cau (S): (x - 2)^ + (y + 2)^ + (z - 1 f = 1 Tim toa dp diem M thupc tryc Oz sao cho t u Mice dupe ba tiep tuyen M A , MB,

MC toi mat cau (S) va diem D ( l ; 2; 5) thupc mat phSng (ABC)

a) Khao sat su bien thien va ve do thj (C) ciia ham so

b) Goi M ( a ; b ) la didm cSn t i m M e (d) => b =-3a + 2 Tiep tuyen cua do thj (C) t?i diem (xo;yo) la y=(3x^-3J(x-Xo)+x^-3xo+^ Cau 9.b: Giai he phuong trinh:

Tuyin chgn & Giai thifu dethi Toi'ui hoc Nguyen I'lui Khdnh , Nguyen Tat Thu

Tiep tuyen d i qua M ( a ; b )

V|y CO hai diem thoa m a n de bai la: M

Cau 2: Dieu ki?n c o s x * 0

sinxcos2x + cos^ x t a n ^ x - l j + 2sin^x = 0

s i n x ^ l - 2 s i n ^ xj + 2sin^ x - 1 + 2sin^ x = 0 o 2sin^ x + s i n x - 1 = 0

Xet i{t) = t^+/t voi t ^ O T a c o : f ( t ) = 2t + ^ , f ' ( t ) > 0 v o l V t > 0

ham so' f ( t ) d o n g bien tren nua khoang [0;+oo)

f ^ > / x - 2 J = £ ( 3 - y ) o x - 2 = 3- y hay x = 5 - y , thay vao p h u o n g t r i n h t h u

Cty TNIIII M T V m'X'H KhangVi?

Cau 4: Dat: t = cos x = > d t = - s i n x d x

d u = - d t

t _ _ 1

Tuyi'n chgn b Giai thifu dethi Todn UQC - Nguyen Phii Khdnh , Nguyen Tilt Thu

Do tinh doi xurigcua x, y ta dat S = x + y, P = xy =>S + P = 3

=i> P = 3 - S > 0 hay S < 3 De ton tai S, P ta c6:

S 2 - 4 P > 0 < o S 2 - 4 ( 3 - S ) > O o S > 2

S ^ - 2( 3 - S ) + 3S ^ 3 - S ^ S ^ 3 3 3S + (3-S) + 9 S 2 S 2

Tom lai, voi 2 < S < 3, luon c6 Q :

A Theo chi/crng trinh chuan

Cau 7.a: Duong tron (C) c6 tam l ( l ; - 2 ) ban kinh r = 2, M nam tren d nen

M ( m ; m + 1) =:>IM = ^ ( m - l f + ( m + 3 f = ^ 2 ( m + l f + 8

Vi I M > 2 nen M nam ngoai (C), do do qua M ke Avtqc 2 tiep tuyeh toi (C)

Gpi J la trung diem I M nen toa dp diem J

Twygti chpn & Gi6i thi^ dithi Todii hoc Nguyen Phu Khatth , hi^micn Tat Thu

Cau 8.b: Mat cau (S) c6 tarn l(2;-2;l), ban kinh R = 1, M(0;0;m) e Oz M|t

phMng (ABC) c6 vecto phap tuyen n = IM = (-2;2;m - l ) ; m^t phing (ABC) di

qua D(1;2;5) nen c6 phuang trinh:

2 ( x - l ) - 2 ( y - 2 ) - ( m - l ) ( z - 5 ) = 0 hay 2 x - 2 y - ( m - l ) z + 5 m - 3 = 0

X = 2-2t

y = - 2 + 2t

z = l + ( m - l ) t Gpi H la giao diem ciia (ABC) voi IM thi to? dp cua H la nghi?m cua h?;

X = 2-2t

CtyTNHHMTV DWIi Khang Vi^t

Duong thing IM:

t = -m^-2m + 9

Do MA la tiep tuyen ciia (S) nen tam giac MAI vuong tai A va AHIIM, cho

nentaco I A 2 = I H I M O I H I M = 1 (do H nim tren tia I M ) , IH=(-2t;2t;(m-l)t)

f(t) dongbiehtren R nen f (x) = f (y) o x = y

Phuong trinh thu hai tro thanh: x^ - 8x +10 = (x + 2) V2x-1 (*)

Dat u = V2x-1 voi u > 0, thay vao phuong trinh (*), Igp bi?t so

A = 25(x + 2)^ => u = ^^-^ hoac u = -^^-^ (khong thoa) 3 2

Voi u = ^ ^ ta du(?c x + 2 = 3V2x-l c6 hai nghifm x = 1, x = 13, ta tim

Cau 1: Cho ham so y = — c6 do thi la (C)

a) Khao sat su bien thien va ve do thi (C) cua ham so'

b) Tim m de duong thang (d): y = 2x + m cat do thj (C) tgi hai diem phan

bi?t sao cho tiep tuyen ciia (C) t^ii hai diem do song song voi nhau

sin^ X sin^ 3x

Cau 2: Giai phuong trinh: cosx cos3x = tan 2x (sin X + sin 3x)

Cau 3: Giai phuong trinh: 2^x^ + 2J = sVx^+l

A Theo chUtfng trinh chuan

Cau 7.a: Trong mat phSng voi h^ tpa do Oxy, cho hai duong thang dj :x-y-2=0,

P2 : 2x + y - 5 = 0 Viet phuong trinh duong thSng A di qua goc tpa dp O cat

rdj, dj Ian lupt tai A, B sao cho OA.OB = 10

Cau 8.a: Trong mat phang tpa dp Oxy, cho hinh chu nhgt ABCD c6 M(4;6) la trung diem ciia AB Giao diem I ciia hai duong cheo nam tren duong thSng (d)

CO phuang trinh 3x - 5y + 6 = 0, diem N(6; 2) thupc canh CD Hay viet phuang trinh cgnh CD biet tung dp diem I Ion hon 4

Cau 9.a: Tim modun ciia so phuc z biet: z = iWsi'

1 + i (l.2i) .\

153

Tuye'n chgn & Giai thifu dethi ToAn hgc - Nguyen Phu Khdnh , S ^ i n i c n TalThu

B Theo chi/orng trinh nang cao

Cau 7.b: Trong mat phSng v o i h ^ tpa dp Oxy, cho ba diem A ( - l ; -1), B(0; 2),

C(0; 1) Viet p h u o n g t r i n h d u o n g t h i n g A d i qua A sao cho tong khoang each

t u B va C toi A la Ion nha't

x - l _ y + 3 _ z - 3

va

Cau 8.b: Trong mat phang toa dp Oxyz, cho d u o n g thang d : ^ ^ ~ ^ ^ ~ Y "

mat ph5ng (P); 2x + y - 2z + 9 = 0 Gpi A la giao diem cua d v o i (P) Viet phuong

trinh d u o n g thSng A nam trong (P) biet A d i qua A va vuong goc v o l d

(d) cat (C) tai 2 d i e m phan bi|t k h i va chi k h i p h u o n g trinh (*) c6 hai nghi^m

phan bi§t va khac 2 <=> •

y ' ( x i ) = y'(x2)<=>Xj + X2 = 4 o m = - 2

Cau 2: Dieu ki?n: cos x*0, cos 3 x ^ 0

P h u o n g trinh cho t u o n g d u o n g v o i p h u o n g trinh:

tan X sin X + tan 3x sin 3x = tan 2x(sin x + sin 3x)

<=> (tan X - tan 2x)sin x + (tan 3x - tan 2x)sin 3x = 0

s i n ( - x )

cos X cos 2 X -smx + - s m x

cos3xcos2x -sin3x = 0

sinx = 0 sinSx sinx _ ^ sin x = 0

2 (x + l ) + ( x 2 - x + l ) =5^(x + l ) ( x 2 - x + l ) (*)

Qjch i C h i a ca 2 ve p h u o n g trinh (•) cho x^ - x + 1 , ta dupe:

x + 1 ,x - x + 1 - + 1 5 / - ^ ( * ) D a t t = j - i i ± i - 0

^ : 2(u^ +v^) = 5uv « > ( u - 2 v ) ( 2 u - v ) = 0c:>u = 2v hoac v = 2u

THI: u = 2v <=> Vx + l = 2Vx^ - x + 1, binh phuong 2 veroi riit gpn ta dupe:

I 4x^ - 5x + 3 = 0 , phuong trinh nay v 6 nghi^m voi mpi xeR

TH2: V = 2u o Vx^ - x + 1 = 2Vx + l , binh phuong2 ve roi riit gpn ta dupe:

Ttiyen choii i '-f Ci&i Ihicii ile thi Toiiii hoc - Ni;iii/<'" f " ' Khunh , Nf^iiyctt Tat IhU

CtyTNHHMTV DVVH Khattg Vift

Tu bang bien thien suy ra f (a) > f(Sc] = 10 / + 5 2,

-v / -v b b c -vc D^t t = thi t e

L i p bang bien thien suy ra f (t) > f (l) = 12

11 PHAN RIENG Thi sinh chi dvtgc chpn lam mpt trong hai phan (phan A

Do A = A n d j nen tpa dp cua A la nghi^m ciia h$ phuang trinh:

2

x - y - 2 = 0 [y = kx

x =

-y =

i - k 2k

y = kx

x =

y =

2 + k 5k

Luang trinh cua duong thMng A la y = 3x, y = - x , y = j x

^u8.a:Gpi P(xp;yp) doixungvai M(4;6) qua I nen 4 + Xp =2xj

6 + y p = 2 y i

11 thupc (d) nen _ f f c Z p ] + 6 = 0 o 3xp - 5yp - 6 = 0 ( l )

157

Tuye'tt chpn & Gi&i thiC'u dethi Toan hqc - Nguyen fnu A/i.iim vynyen x » > ^ r »

B Theo chuorng trinh nang cao

Cau 7.b: Gia su A di qua diem A va c6 vecto phap tuyen la n = (a;b) ;t 0, nen

CO phuong trinh: a(x +1) + b(y +1) = 0

a + 3b d(B,A) = - ^ = = , d ( C A ) = a + 2b

ab>0 2=i>a = 2,b = 5=>A: 2x + 5y + 7 = 0

b 5 Cau 8.b: A = d n (P), tpa dg cua A la nghi^m cua h$

Vi Ac(P)' ; nen VTCP cua A la: u ^ = Ud;np =(-5;0;-5) cung phuong

vol vecto u' = (l;0;l) => A.: y = - l

z = 4 + t Cau 9.b: Neu x = 0, tu phuong trmh thii nhat suy ra y = 0 Khi do khong thoa phuong trinh thu hai

Neu X 5* 0, chia ca 2 ve phuong trinh dau cho x^, ta dugc:

Datu = x - l , t a d u g c phuong trinh: 2012" V u ^ + 4 - u =2 (**)

Xet ham so g(u) = 2012" f V u ^ T I - u

I Suy ra ham so' g(u) dong bien tren R Mat khac g(0) = 2 nen u = 0 la

•Wghi^m duy nhat ciia (* *) Tu do x = 1 va y ^ • Vay, hf phuong trinh c6 nghi^m duy nhat (x; y) =

DCTHITHUfSd24

I PH/VN C H U N G C H O TAT CA C A C THI SINH

Cau 1: Cho ham so y = -x^ + 2x^ -1 c6 do thi la (C)

a) Khao sat sv bien thien va ve do thi (C) cua ham so

b) Tim diem M nSm tren tryc hoanh sao cho tu do c6 the ke dug^c ba tie'p

tuyen den do thi (C)

2 ^ 2 2cos2x + 4 Cau 2: Giai phuang trmh: tan^ x + 9cot^ x + = 14

Cau 3: Giai h^ phuang trinh:

, sin"^ X dx Cau 5: Cho khoi chop S.ABCD c6 day la hinh thang can, day Ion AB b^ng bon

Ian day nho CD, chieu cao cua day bang a Bon duong cao cua bon mat ben

ung voi dinh S c6 dp dai bSng nhau va bang b Tinh the tich cua khoi chop

A Theo chUorng trinh chuan

Cau 7.a: Trong mat phSng voi h? tpa dp Oxy, cho duong tron (C)

x2 + yz - 2x + 4y - 20 = 0 va duong thang (d): 3x + 4y - 20 = 0 Chung minh d

tie'p xiic voi (C) Tam giac ABC c6 dinh A thupc (C), cac dinh B va C thupc d

trung diem canh AB thupc (C) Tim tpa dp cac dinh A, B, C biet tryc tam cu.^

tam giac ABC trung voi tam cua duong tron (C) va diem B c6 hoanh dp duong

Cau 8.a: Trong mat phSng tpa dp Oxyz, cho mat phing (P): 2x - y + 2x + 6 = 0

va duong thSng (A) : = = Viet phuang trinh duong t h k g (d)

160

Cty TNHH MTV DWH Khattg Viet

di qua A(-3; 0; 2) va cit (A) t^i B sao cho m | t cau tam B tie'p xiic voi hai m a

phSng (Oxz) va (P)

Cau 9.a: Goi z , , Z 2 , Z 3 , Z 4 la bon nghi^m ciia phuang trinh z'*-2:'-2zi^+6z-4s0

1 1 1 1 tren tap so phuc tinh tong: S = — + — + — + —

Z j ^2 Zg Z4

B Theo chi/Ong trinh nang cao Cau 7.b: Trong mat phSng voi h? tpa dp Oxy, cho hinh thang can ABCD c6 dien

tich bang 18, day Ion CD nam tren duong thang c6 phuong trinh: x - y + 2 = 0

Biet hai duong cheo AC, BD vuong goc voi nhau va cat nhau tai diem l(3;l) Hay

Viet phuong trinh duong thJing BC biet diem C c6 hoanh d p am

Cau 8.b: Trong khong gian voi h? tpa dp Oxyz, cho mat phJing (P): x + 2y-z + 5 = 0

va duong thSng d: ^ = Z l l = £Z^ Goi d' la hinh chieu vuong goc ciia d len mat phang (P) v a E la giao diem cua d va (P) Tim tpa d p F thupc (P) sao cho EF vuong goc voi d ' va EF = 5N/3

Cau 9.b: Viet so phuc sau duoi dang luong giac: z = —^-

<=>(tanx + 3cotx)^ + tanx + 3cotx-20 = 0 161

Cau 3: Dieu ki#n: x > 0

Nh^n thay, (O; y) khong la nghi?m ciia phuong trinh

Xet X > 0, phuong trinh thu 2 tro thanh:

Thay vao phuong trinh (*): x^ + x + 2(x^ + l)>/x = 6

Ve trai cua phuong trinh la ham dong bien tren (0;+oo) nen c6 nghi^m duy

'4^

nhat X = 1 va he phuong trinh c6 nghi?m

2.(x + 2sinx-3)cosx , ^ xcosx_, 2 (2sinx-3)cosx

Cau 5: Goi H la chan duong cao cua chop thi H phai each deu cac c^nh cua day

va trong truong hg-p nay ta chung minh dug-c H nSm trong day

Suy ra hinh thang can ABCD c6 duong tron npi tiep tam H la trung diem

doan M N vai M , N Ian lugt la trung diem cac canh AB, CD va M N = a

Duong tron do tiep xiic voi BC tai E

thi H M = H N = HE = - la ban kinh

2 duong tron va SE = SM = SN = b

b > ^ .SH = | V 4 b 2 - a 2

D a t C N = x thi BM = 4x,CE = x, BE = 4x Tam giac HBC vuong 6 H

a + b + c + 1 (a + b + c + 3)^

1 27 Dat t = a + b + c + 1 nen c6 t > 1 Liic nay, P <

1 27 Xet ham so': f (t) = v o l t > 1

A Theo chi/orng trinh chuan

Cau 7.a: Duong tron ( C ) c6 tam l ( l ; - 2 ) va ban kinh R = 5

Tuye'n chgn &• Gi&i thi$u dethi Todn hgc - Nguyen Phu Khdnh , Nguyen Tat Thu

Do I la true tam AABC va IH 1 BC =i> A e IH Ket hop A e (C) => la diem

Y A = - 6 A(-2;-6)

.yA=2yi-yH

Goi M la trung diem canh AB Do HA la duong kinh nen HM 1 AM

Tam giac HAB c6 HM vua la trung tuye'n vua la duong cao nen AHAB can

20-3b'l tai H =^ HB = HA = 2R = 10, B e d B b;-

Thay vabieu thuc S = \ \ \ \ - - l + \ — + — ^ = 7 z? zl 4 zl 4 (^^i)2 4

B Theo chUtfng trlnh nang cao

Cau7.b: AICD can tai 1(3; l), C(t;t + 2)e(d) voi t<0, IC = \/2t^-4t + 10,

AIAB vuong can

(AB + CD).(IH + IK)

SAU^T^=- <»36 = (x-3|%/2+4N/2J '271+x-3^'

A B / / d : x - y - 4 = 0 DI: x = 3

o 3 6 = (|x-3| + 4)^ <^|x-3| = 2 o x = l (khong thoa ) ho|ic x = 5 =>A(5;l)

- ^ B(3;-1) = AB n DI BC: x + 2y - 1 = 0

CauB.b: Ee(d)=*E(-3 + 2t;-l + t;3 + t)

Ee(P):x + 2 y - z + 5 = 0=>t = l=>E(-l;0;4) LaydiemM(-3;-l;3)€(d), ta c6: EF = ME.nj^ = (-1;1;1)

[x = -1 -1

Phuang trinh tham so duong thing EF: • y = t => F (-1 -1; t; 4 +1)

z = 4 + t Theo bai toan, ta c6: EF = 5>/3 =:> EF^ = 45 <=> t^ = 25 <=> t = -5 hoac t = 5

Vay, CO hai diem F thoa man la : (-6;6;10) hoac (4;-5;-l) Cau 9.b:

"wy^w chgtt & Gim thifu dethi Todn h<fc-Nguyen PhU Khanh , Nguyen lai

I PHAN CHUNG CHO TAT CA CAC THI SINH

Cau 1: Cho ham so y = '

a) Khao sat su bien thien va ve do thj (C) ciia ham so

b) T i m m de d u o n g th5ng A : y = x - m ck (C) tai hai diem phan bi^t A , B

sao cho khoang each tCr A den true hoanh gap hai Ian khoang each t u B den

true tung

Cau 2: Giai p h u o n g trinh: sin 3x + +sin 3 x l =

-Cau 3: Giai bat p h u o n g trinh: 3 V x ^ - 1 < 2x^ + 3x + 1

Cau 4: Tinh ti'eh phan: I = f — — — d x

^ J 7 - 2 e o s 2 x

Cau 5: Cho hinh chop S.ABCD c6 day la hinh vuong eanh a S A 1 ( A B C D ) SC

hpp voi mat phSng ( A B C D ) goc 60" Gpi M , N , P Ian lu(?t la trung diem ciia

cac eanh B C C D , A D Tinh goc giOa SM va NP Tinh khoang each t u diem A

den mat ph5ng ( S M P ) , the tich hinh chop S.ABCD

Cau 6: Cho cac so thuc khong am a, b, c thoa man a + b + c = 3 T i m gia t r j Ion

nha't va gia trj nho nhat cua bieu thiie P = a + \/b + \/c

I I PHAN RIENG T h i sinh chi dugc chpn lam mgt trong hai phan (phan A

hoac B )

A Theo chUtfng trinh chuan

Cau 7.a: Trong mat phJng v o i h? tga d p vuong goc Oxy, cho hinh thang

ABCD (AB // CD) Bie't hai d i n h B ( 3 ; 3 ) va C ( 5 ; - 3 ) Giao diem I ciia hai duong

111 m i > i ^ r m i \ i i H i i g T T ^ I

cheo n a m tren d u o n g thiing A : 2x + y - 3 = 0 Xac djnh tpa d p cac d i n h con lai cua hinh thang A B C D de C I = 2BI, tam giac ACB c6 di$n tich bang 12, diem I

CO hoanh dp d u o n g va diem A c6 hoanh dp am

Cau 8.a: Trong m|t phSng tpa dp O x y z , cho A ( 1 ; 2 ; 2 ) , B ( - 3 ; - 2 ; 2 ) , C(-2;-2;1)

Viet p h u o n g trinh mat cau d i qua A , B, C tiep xuc voi mat phiing ( O x y ) Cau 9.a: Cho so phue z thoa man |z| - 2z = - 3 + 6 i

Tinh gia trj bieu thuc z + IzP

B Theo chiTtfng trinh nang cao

Cau 7.b: Trong mat phang voi h§ tpa dp vuong goc Oxy, cho d u o n g tron (C) x2 + y 2 = 4 va d u o n g thSng (d) :x + y + 4 = 0 T i m diem A thupc ( d ) sao cho tir A ve dupe 2 tiep tuyen tiep xiie (C) tai M , N thoa man di#n tich tam giac A M N bMng 3%/3

Cau 8.b: Trong khong gian Oxyz, cho ba diem A ( l ; 2; 3), B(3; 0; -1), C ( l ; - 2 ; 0)

va hai d u o n g thang A , : i i z 2 ^ y + 2 ^ z - 3 £zi=yzi = LLl

a) Danh cho ban dpc

b) Xet p h u o n g trinh hoanh dp giao diem:

x + 1 -2x + 2 = X - m <=> 2x^ - ( 2 m + l ) x + 2 m + l = 0

o l - - s m

2

= - » cos^ 6x = 1 sin^ 6x = 0 o x = ^

2 6 Cau 3: Dieu kien: x > 1

Vay, bat phuong trinh da cho c6 nghi?m x > 1

Cau 4: Tmh t.ch phan: ^ = = - ( 2 c o s x - 3 ) ( 2 c o s x 3 )

^2cosx-3 2cosx + 3y

1 l"r( 1 Khido: I = - 1 ^.t = — I n

6^121-3 2t + 3 / 12

.Dat t = cosx=>dt = -sinxdx

0

1 2 t - 3 2t + 3 Cau 5: Goi K la trung diem cua AB thi M K // NP ( S M , N P ) = ( S M , K M )

Xet tam giac S K M c6:

168

Nen cosSMK = = ^ (SM,NP) = SMK = arccos

Taco: S A ± M P

A P 1 M P • (SAP) ± MP => (SAP) 1 (SMP) = SP Trong ( S A P ) ke A H 1 SP => A H 1 ( S M P ) => d [ A , ( S M P ) ] = A H

A H ^ SA^ AP2 63^

Cau 6: v^b + < ^2(b + c) = ^ 2 ( 3 - a ) Khido P<a + ^ 2 ( 3 - a ) , f(a) = a + ^ 2 ( 3 - a ) voi a€[0;3"

maxP = — k h i a = —, b = c = —

Xet g(a) = a +Vb + v/c voi a 6 [ 0 ; 3 ] Ta c6: g'(a) = l > 0 , suy ra g(a) dong bien voi mpi a e [ 0 ; 3 ] va g(a)>g(o) = \/b+ %/c = N/b + V 3 - b

Xet g(b) = Vb + V 3 ^ voi be[0;3"

minP^Vs khi a = b = 0,c = 3 hoac a = c = 0; b = 3

11 PHAN RIENG Thi sinh chi dupe chpn lam mpt trong hai phan (phan A

^au 8.a: Gpi I(a;b;c), R Ian luot la tam va ban kinh mat cau can tim

I Tpa dp D la nghi^m cua hf:

169

Ta CO phuong trinh m|t phMng (Oxy) la z = 0 Mat cau Hep xiic vai mat

c phang (Oxy) c:>d(l,(Oxy)) = R hay R =

Theo bai toan, ta c6 h?: IA = IA = IC IB

IA = d(l,(Oxy)) Cau 9.a: Gia sir z = x + yi (x, y e ^ )

B Theo chUorng trinh nang cao

Cau7.b:Diem A e d ^ A(a;-4-a).Dat MAN = 2a, O A = x>0

VoiOA = 4 o a^ + (4 + a)^ = 4 o a = -4 hoac a = 0

Vay, toa dp diem A can tim A (-4; O) hoac A(0;-4)

x = l - f '

x = 2 + 2t Cau 8.b: Aj: ] y = -2 -1 va :

z = 3 + t y=l+2f z = -l + t' Gpi mat phSng (P) di qua A(l; 2; 3) va vuong goc vol duang thSng Aj

nen nhan ^ = (2;-l;l) lam VTPT, suy ra phuang trinh m|t phSng (P) la:

2 x - y + z - 3 = 0

GpiN la giao diem ciia mat phSng (P) va duang thiing Aj, suy ra N la

x = l - f y = l+2t' ,

;hi?mcuah^: => t'=-1 => N(2;-l;-2)

2 x - y + z - 3 = 0 Phuong trinh duang thiing A di qua A(1;2;3) va nh^n A N = (l;-3;-5) lam

x = 1 + m

y = 2 - 3 m z=3-5m

157

Cau 9.b: Dieu ki?n: x>l y >0

Phuong trinh dau tuong duong voi log^ ^ (x -1) = l o g y o y = x -1 I' Thay vao phuong trinh thu 2 tro thanh: 9^ - 36.3^' - 243 = 0

3y =9 3^=27

y = 2 = > X = 3

y = 3 = > X = 4

OETHITH0fs626

I PHAN C H U N G CHO TAT CA CAC THI SINH

Cau 1: Cho ham so y = - - x ^ + x - - c6 do thi la (C)

a) Khao sat su bien thien va ve do thj (C) ciia ham so

b) Goi M la diem thuQC do thj (C) c6 hoanh do x = 2 Tim cac gia trj ciia m

de tiep tuyen voi (C) tai M song song voi duong thSng d: y=(m^ -4^+^^—

Cau 2: Giai phuong trinh:

cos 1 Ox + 2 cos^ 4x + 6 cos 3x.cos X = cos X + 8 cos X cos'' X

Cau 3: Giai h$ phuong trinh: x(3x-7y + l ) = - 2 y ( y - l )

^x + 2y + 74x + y = 5

, , , - T '^r In^x + lnx ,

Cau 4: Tinh tich phan: I = J

i(lnx + x + l )

Cau 5: Cho hinh lang try dung A B C A ' B ' C c6 day A ' B ' C la tarn giac vuong

tai B' Gpi K la hinh chieu vuong goc ciia diem A ' len duong thing A C Biet

goc giixa duong thing A ' K voi mat phing ( C A B ' ) bing 30" va A ' B ' = a,

A ' C = aVs Tinh the tich khoi tu dien KA'BC

A Theo chi/ofng trinh chuan

Cau 7.a: Trong mat phing Oxy, cho hinh binh hanh ABCD c6 B(1;5) va duong

cao A H CO phuong trinh x + 2y - 2 = 0, voi H thuQC BC, duong phan giac trong

cua goc ACB c6 phuong trinh la x - y - 1 = 0 Tim toa do dinh A, C, D

Cau 8.a: Trong mat phing Oxyz, cho ba duong thang

Cty TNHH MTV DWH Khang Vift

Viet phuong trinh duong thing A vuong goc voi dj va cat d j , dg Ian lugt

tai cac diem A, B sao cho AB = 7 6

Cau 9.a: Cho so phiic z thoa man |iz - 3| = |z - 2 - i| va |z + 3i| = |2z + i|

Tim modun ciia so' phuc z

B Theo chUorng trinh nang cao

Cau 7.b: Trong mat phing Oxy, cho duong tron (C): x^ + y^ - 8 x - 2 y = 0 va diem A(9; 6) Viet phuong trinh duong thing qua A cat (C) theo mpt day cung

CO do dai 4^/3

Cau 8.b: Trong mat phing Oxyz, cho hai diem M ( l ; 2; 1), N ( - l ; 0; -1) Viet

phuong trinh mp(P) di qua M , N cat Ox, Oy theo thu tu tai A va B (khac O) sao cho

a) Danh cho ban dpc

b) Tiep tuyen tai M c6 phuong trinh: y = -3x + —

Tiep tuyen voi (C) tai M song song voi duong thing d:

m ^ - 4 = -3 9m+ 5 14

x ( 3 x - 7 y + l ) = : - 2 y ( y - l ) (1) 7x + 2y+74x + y =5 (2)

Jiu 3: Dieu Ki?n:

cho viet lai:

17.-^

Tuye'u chyn & Giin thiju tic thi Toan h(fc-Nguyen PMi Khanh , Nguyen Tat 77in

Vay, the tich khoi t u di?n B A ' C K la :

a^^/l5

Cau 6: Ta c6: Q

Da,t

12(b + c - a ) 12(c + a - b ) 25(a + b - c ) 6: 0 = 4 9 - P = —5^ - + — - +

Theo chUtfng trinh chuan

Cau 7.a: B C d i qua B(1;5) va vuong goc A H nen B C : - 2x + y - 3 = 0

-2x + y - 3 = 0

x - y - l = 0 ^ Toa dp C la nghi^m cua h | : > C ( - 4 ; - 5 )

Tuyen chgn b Giai thifu dethi Todn hgc - Nguyen Phu Khdnh , Nguyen Tat Thu

Ggi A' la diem doi xung B qua duong phan giac (d): x - y - 1 = 0, BA n (d) = K

Duong thSng KB di qua B va vuong goc (d) nen KB c6 phuong trinh x + y - 6 = 0

Toa dp diem K la nghi^m cua h?: x + y - 6 = 0 >K 7 5 2 ' 2 >A'{6;0)

[ x - y - l = 0 Phuang trinh AC: x - 2y - 6 = 0, A = CA' n A H => A(4; -1)

Trung diem l(0;-3) cua AC, dong thoi I la trung diem BD nen D(-1;-11)

Cau 8.a: dj c6 vecto chi phuong la u i = ( - l ; l ; l ) Voi A e d j / B e d j

=:> A ( l + a;a;a), B(b;2b;b - 2) AB = (b - 1 - a;2b - a;b - 2 - a )

1 5 5 1

V o i y = - ^ x = - - ^ z = - - - i = : > V26

B.Theo chuorng trinh nang cao

Cau 7.b: Toa dp tarn duong tron la l(4;l);ban kinh R = ^

Goi A la duong th5ng qua A va cat duong tron tai M, N phuang trinh cua A

~CfyTNlni MIV nvvil K/.,),/x Vift

Cau 8.b: Gia sir (P) cat Ox, Oy, Oz Ian lupt tai A(a;0;0), B{0;b;0), C(0;0;c)

Cau 9,b: Co A' = 4(2 - i f + 2 ( l + i)(5 + 3i) = 16

Vay phuang trinh c6 hai nghifm phuc:

+ Z 2 = 9

DETHITHUfSd27

I PHAN CHUNG C H O TAT CA CAC T H I S I N K

Cau 1: Cho ham so y = x"* - 2mx^ + m c6 do thj (C^)

•s

a) Khao sat su bien thien va ve do thj (Cj) cua ham so'

b) Tim tat ca cac gia trj thuc cua m de do thj ham so (C^ ) y = x'* - 2mx^ + m

C O ba diem eye trj t^o thanh mpt tam giac c6 ban kinh vong tron npi tiep Ian

Hon 1

Cau 2: Giai phuong trinh: sin^ x=cos^ x+cos^ 3x

•au 3: Giai h? phuong trinh:

J 9 x + y 2x

y - 9 = 18 >

W ] 1x2 )

Tuyen cht?n & GiaithJQU aethi Toan HQC - Nguyen Phu RhAnh , Ni;,nf,'„ Tnf IHU

, VxV''+3xe''+e''+K

Cau 4: Tinh tich phan: I = I dx

Cau 5: Cho tu di^n deu SABC Gpi (P) la mat phSng di qua duong cao SO ciia

tu di^n; mat ph3ng (P) cat cac mat phSng (SBC), (SCA) va (SAB) Ian lupt theo

cac giao tuyen SM, SN, SP Cac giao tuyen nay Ian lupt tao voi mat phing

(ABC) cac goc a, p, y Chung minh: tan^ a + tan^ (3 + tan^ y = 12

Cau 6: Cho cac so thuc duong a, b, c doi mot khac nhau thoa man 2a < c va

A Theo chUorng trinh chuan

Cau 7.a: Trong mat phang tpa do Oxy, cho parabol (P): y = x^ + 2x - 3 Xet

hinh binh hanh ABCD A { - 1 ; - 4 ) , B(2;5) thuoc (P) va tam I cua hinh binh

hanh thupc cung AB cua (P) sao cho tam giac lAB c6 dif n tich Ion nhat Hay

xac djnh toa do hai diem C, D

Cau 8.a: Trong mat phing tpa do Oxyz, cho 3 diem A(l; 2; -1), B(2; 1; I ) ; C(0; I ; 2)

va duong thSng (d) c6 phuong trinh la: (d): i = ^^-^ = ^-i^ Hay lap

phuong trinh duong thMng ( A ) di qua true tam cua tam giac ABC, nam trong

mat phang (ABC) va vuong goc voi duong thang (d)

Cau 9.a: Mpt hpp chiia 5 bi xanh, 7 bi do va 8 bi vang Lay ngau nhien 8 vien

bi Tinh xac suat de lay dupe 8 vien bi c6 du ca 3 mau

B Theo chUcrng trinh nang cao

Cau 7.b: Trong mat phSng tpa dp Oxy, tam giac ABC can tai A, c6 dinh B va C

thupc duong thing di: x + y + 1 = 0 Duong cao di qua dinh B la d2: x - 2y - 2 = 0,

diem M(2;1) thupc duong cao di qua dinh C

Viet phuong trinh cac canh ben cua tam giac ABC

Cau 8.b: Trong mat phang Oxyz, cho duong thang d : = = - va hai

diemA(l;l;0), B(2;1;1) Viet phuong trinh duong thSng A di qua A, A i d

sao cho khoang each tir B den duong thang A la Ion nha't

Cau 9.b: Tim so phuc z thoa man z^/2^+l=|z|(2 + 6iz)

HlTtifNGDANGlAl

I PHAN CHUNG CHO TAT CA CAC THI SINH

Cau 1:

a) Danh cho ban dpc

b) m > 0 thi do thj ham so da cho eo 3 cue trj A(0;m), B(-Vii^;m-m2),c(>A^;m-m2) =^S.^^^ = ".^V;^,

p = vm^ +m + V m

L^i'^o r = ^ > l « V m - % m 2 >m^+ m -m-l>Q

Cau2: Phuong trinh dupe bien doi duoi dang: Il£2i^=ll£2£l^ + cos2 3x

o 2 eos^ 3x + (cos 4x + cOS2x) o 2cos^ 3x + 2 cos 3x.cos X=0

o (cos 3x + cOS5x) cos 3x=0 o 2 cos2x.cos x.cos 3x=0 eos2x = 0

cos X = 0 o cos3x = 0

cos2x = 0 cos3x = 0

2x=r-+k7t

2 Vay, phuong trinh c6 2 hp nghifm

= 9xy + + Z _ + 2 Dat:u = ^ , v ^

Cau 4: I = Ij I^, trong do: I, = /(xe" + l)dx va = f ^ ^ ^ i l ^ x

2 3 V 9 Khi do A H = AO = SO = h -

Dieu phai chung minh: tan^ a + tan^ p + tan^ y = 12

^ SO^ SO^ SO^ , ,

kez

OM^ ON^ ' OP^ SO^ 6a^

Vay,thay vi chung minh ( l ) ,

OM^ ON^ Op2;

(3cosm + >/3sinm)^ (3cosm - V S s i n m f ^2sm^

m IS^sin^m + cos^mj

a^

Vay, (2) dung Do do c6 dieu phai chiing minh

Cau 6: Gia thiet: 2a < c nen c6 - < -

c 2 Hon nua, theogia thiet ta cungc6: - — + — = 2<=> — = — - 1

2a =c 8a = 3b = 4c, ch^ng han ta chpn (a,b,c) = (3,8,6)

27 gjj^ Vay, gia trj Ion nhat cua P = ~

• t l - PHAN RIENG Thi sinh chi duoc chpn lam mpt trong hai phan (phan A

H k o a c B) Theo chi/orng trlnh chuan

^ K a u 7.a: I thupc cung AB cua (p) sao cho di?n tich lAB ion nha't <=> I xa AB

H K h a t , tuc I la tiep diem cua tiep tuye'n (d)//AB cua (P)

Phuong trinh duong thang A B : y = 3x - 11=> ( d ) : y = 3x+ c

(d) tiep xuc (P) tai diem I I

2'~4 •C 1 ;

-2 ) , D

181

Zau 5: Ta c6 AC = BD = 2a Gpi SO la duong cao va SO = h

Trpn h$ true tga dp Oxyz sao cho:

O(0;0;0), A(a;0;0), B(0;a;0), C(-a;0;0), D(0;-a;0), S(0;0;h)

Xac djnh tarn va ban kinh m|it cau ngoai tiep hinh chop

Do hinh chop S.ABCD la tii giac deu

nen I € OS =:i> l(0;0;Zo)

SxQ =4S^AB =4.isA.SB.sina = 2(a^ +h^)sina

=>STP =SABCD +SxQ =2(a^ +h^)sina + 2a^

Cty TNHH MTV DWH Khang Vift

Cau 6: Khong mat tinh tong quat ta c6 the gia su: x > y > z Xethamso: f(x) =

Taco: f'(x) = (x + y + z)

1 1 1

— H + — (x + y + z) xyz + ^2x^ - yzj(y + z)

(2 + z)'^ ,chu'ng minh dupe f (z) < 27

I I PHAN RIENG Thi sinh chi dugc chpn lam mpt trong hai phan (phan A hoac B )

A Theo chUorng trinh chuan

Cau 7.a: (C): (x -3)^ + (y +1)^ = 4 c6 tam l ( 3 ; - l ) va ban kinh R = 2 Gia su duong thSng qua P c6 vec to phap tuyen

IH EH IE 2

IF 2 EP 4 r- 2 8

= > I H - - ^ = -==,EH = - ^ = - L ^ P H = P I - I H = 2 7 5 - - ^ = - ^

fuyen ch<?n & Gi&i thi(u de thi Todn hgc - Nguyen Phu Khanh , Nguyen lat IHu

Cau 8.a: (S) c6 tarn l ( l ; 2; 3) va ban kinh R = 9

Giasu(P) coVTPT n = (A;B;C), (A^ + + > 0)

(P)//BC nen n l B C = (-!;!;4) =^n.BC = 0 o A = B + 4 C ^ n = (B + 4C;B;q

(P) diqua A(13;-l; 0)=^ phuong trinh (P):

(B + 4C)x + By + C z - 1 2 B - 5 2 C - 0

B + 4C + 2B + 3C-12B-52C (P) tiep xiic (S) <^ d[I,(P)] = R « , , , = ^

ta dugc phuong trinh (P): 8x + 4y + z-100 = 0

Cau 9.a: Ta c6 Z j Z j = Z2Z2 = Z3Z3 = 1 nen

Z I Z 2 Z 3

Z i Z 2 + Z 2 Z 3 + Z 3 Z i | _

I Z 1 Z 2 Z 3

B Theo chUorng trinh nang cao

Cau 7.b: B thu^c d suy ra B :

Z1Z2 + Z 2 Z 3 + Z 3 Z 1

x = t [x = 7 - 2 m

•! , C thuQC d' cho nen C: \ [y = - 5 - t i y = m ( t - 2 m + 9) ^ m - t - 2 ^

Theo tinh chat trgng tarn: => XQ = = 2,yG = ^ = ^

^au 9.b: Dieu ki^n:

x^ - 4x + 3>0 x^ -4x + 35^1

x- 3 > 0

x- 3 ^ 1

X >3

T H I : Neu x > 4 thi log4 Vx^ -4x + 3 > log41 = 0 va log4(x -3) > log41 = 0

Do do bat phuang trinh tuong duang: log4(x - 3) < log4 -/x^ -4x + 3

< » x - 3 < V x 2 - 4 x + 3<=>Vx-3< >/x^ (dung V x > 4 ) TH2: Neu 2 + N/2 < x <4 thi log4 Vx^ -4x + 3 > log4 1 = 0

va Iog4 (x - 3) < iog4 1=0 Suy ra bat phuang trinh v6 nghi^m

TH3: Neu 3< x <2 + x/2 thi log4 \lx^-4x + 3 < log^ 1 = 0

va l o g 4 ( x - 3 ) < l o g 4 l = 0

Do do bat phuang trinh tuong duong: log4(x - 3) < log4 Vx^ -4x + 3

O X- 3 < V X 2 - 4 X + 3 O N / X - 3 < V X ^ ( d i i n g Vx€(2; 2 + V 2 ) )

245

DETHITHl]fS638

I PHAN CHUNG CHO TAT CA CAC THI SINH

Cau 1: Cho ham so y = -x^ + 3x - 2, c6 do thj la (C)

a) Khao sat su bien thien va ve do thj (C) cua ham so

b) Tim tpa dp cac diem tren duong thing y = -4 ma tu do c6 the ke den do

thj (C) diing hai tiep tuyen

Cau 2: Giai phuong trinh:

s

sin^x + sin^ X + sin^ — + X = 273 sin X + — 7t cosx

Cau 3: Giai phuong trinh: l o g 2 x + logg (2x +1) = 2

Cau 4: Tinh tich phan : I = j - cot X - tan x

I sin 2 X cos

8

^ x

Cau 5: Cho hinh hop chu nhat ABCD.A'B'C'D' c6 AB = 2, AD = 4, A A ' = 6

Gpi I, J la trung diem AB, C' D'

Gpi M, N thoa A M = mAD, BN = mBB' (O < m < l )

Tinh khoang each tu A deh(BDA') Xac djnh ban kinh r cua duong tron

giao cua mat cau (S) ngoai tiep A B D A ' va ( B D A ' )

Cau 6: Cho 0 < c < b < a < l Tim gia tr| Ion nhat cua bieu thuc:

P = a 2 ( b - c ) + b2(c-b) + c 2 ( l - c )

I I PHAN RIENG Thi sinh chi dug^c chpn lam mpt trong hai phan (phan A

hole B)

A Theo chUcrng trinh chuan

Cau 7.a: Trong mat phang Oxy, cho hai duong tron: (Cj): x^+y^ =13 va

( C j ) : (x - 6)^ + y^ = 25 cat nhau tai A ( 2 ; 3) Viet phuong trinh duong thing di

qua A va cat (Cj),(C2) theo hai day cung eo dp dai bang nhau

Cau 8.a: Trong khong gian voi h? tpa dp Oxyz, cho A(3;5;4), B(3;l;4).Hay

tim tpa dp diem C thupc m5t phing ( P ) : x - y - z - l = 0 sao cho tam giac ABC

can tgi C va c6 di?n tich bing 2>/T7

246

t Cau 9.a: Tim so'phuc z c6 modun bang 1, dong thoi so'phuc w = + 2z - 1 ca

modun Ion nhat

B Theo chUorng trinh nang cao

Cau 7.b: Trong mat phing Oxy, cho ABC npi tiep trong duong tron tam I va

A(3; 3) Diem M(3;-l) nam tren duong tron I va thupc cung BC khong chua

diem A Gpi D, E Ian lupt la hinh chieu ciia diem M len cac duong thang BC, AC

Tim tpa dp cac dinh B, C biet ring tryc tam tam giac ABC la diem H(3;1),

duang thang DE c6 phuong trinh la x + 2y - 3 = 0 va hoanh dp ciia B nho hem 2

Cau 8.b: Trong khong gian voi h^ tpa dp Oxyz, cho diem A (3;-2;-2) va mat phing ( P ) : x - y - z + l = 0 Viet phuong trinh mat phing ( Q ) di qua A, vuong goc voi mat phing (?) biet rang mat phing ( Q ) cat hai tryc Oy, Oz Ian lupt

tai diem phan bi^t M va N sao cho OM = ON

Cau 9.b: Cho so'phuc z thoa man z = 1

Tim gia trj Ion nhat ciia M = z'' - z + 2

HMGDANGIAI

I PHAN CHUNG CHO TAT CA CAC THI SINH

Cau 1:

a) Danh cho ban dpc

b) Gpi A la diem n i m tren duong thing y = -4 nen A(a;-4)

Duong thang A qua A voi hf so'goc k c6 phuong trinh y = k(x - a) - 4 Duong thang A tiep xiic voi do thj (C) khi va chi khi h? phuong trinh sau

I Phuong trinh (1) tuong duong voi: ^g(,)^2x2 _ ( 3 , , 2 ) x ^ 3 a 2 = 0

1 Qua A ke dupe hai Hep tuyeh deh (C) khi va chi khi (2) c6 2 gia trj k

Wiac nhau, khi do ( l ) c6 diing 2 nghi^m phan bi^t X i , X 2 , dong thoi thoa

= -3xi +3, k j = -3x2 + 3 CO 2 gia trj k khac nhau

x = l

chou & Gi&i titieu ae pan noc - nguyen i-nu r<nann -,

Vay, cac diem can tim la A ( - 1 ; - 4 ) , A (2;-4) hoac A

Cau 2: Ta c6 sin^ x + sin — X

Cau 3: Dieu kien: x > 0

Nhan thay x = 2 la nghiem cua phuong trinh cho v i log2 2 + log5(2.2 + l ) = 2

Ta chung minh x = 2 la nghif m duy nhat

That vay, ham so y = logj x,y = logj (2x +1) deu c6 cac co so Ion hon 1 nen

cac ham so do dong bien

248

Voi x > 2 , t a c 6 : log2 x > log2 2 = 1, log^ (2x + ] ) > logg(2.2 +1) = 1

Voi 0 < X < 2, ta c6: logj x < logj 2 = 1, logg (2x +1) < log, (2.2 +1) = 1

Cau 4: I = J - cot x - tan X

2 Xac djnh tam K va ban kinh R cua

mat cau (S) ngoai tie'p A B D A ' (S): x^ + y^ + z^ - 2ax - 2by - 2cz = 0

(A6(S))

4 - 4 a = 0 [a = l

1 6 - 8 b = 0 = > i b = 2 36-12c = 0 c = 3

t + 1 dt

B, D, A ' e ( S ) :

Vay, maxP = - ^ ^ khi (a;b;c) = ( 1 1

I N/3 J

I I P H A N R I E N G T h i s i n h c h i d u Q c c h p n l a m m p t t r o n g h a i p h a n ( p h a n A

h o a c B )

A Theo chUtfng trinh chuan

C a u 7.a: T u gia thiet: ( C j ) c6 tarn I = (0;0),R - yjl3

d cat ( C j ) tai A, C thi tpa dg cua A, C la nghiem ciia he:

Tu" do ta CO phuong trinh:

Ta CO AB = 4, trung diem AB la I(3; 3;4)

SAABC = ^ C I A B = 2V17 => CI = VT7 ^ ( 3 - a)^ + (8 - a)^ = V l 7

<=> a = 4 hoac a = 7 Vay,c6hai diem C(4 ;3;0), C(7;3;3)

Vay, CO hai so phuc can tim la Zj = ' ^2 ~ ~2 ~"2~' '

B Theo chUofng trinh nang cao

Cau 7.b: Theo gia thiet ta c6 A, H , M nam tren duong thang x - 3 = 0 suy ra D

la trung diem cua H M => D(3;0)

Duong thang BC di qua D va c6 vecto phap tuyen HD = (0;l) => BC: y = 0

Goi F la hinh chieu cua M len duong thiing AB suy ra F thuQC duong th^ng

Do C nam tren duong thSng BC nen C(u;0) Mat khac H la true tam non

HB vuong goc voi AC, khi do: AC.HB = 0 c ^ 3 u - 9 - 3 = 0 » u = 4=>C(4;0)

Cau 8.b: Gia sir ng la mot vecto phap tuyen ciia ( Q )

Khi do n ^ l n j ^ ( l ; - l ; - l )

M|t phSng (Q) cat hai true Oy, Oz tai M(0;a;0),N(0;0;b) phan bi^t sa^

fa = b 9t 0 cho O M = ON nen a = b «•

cry iNHHMl V UVVH J^nang vifi

i Khi do mat phang (Q) :2x + y + z - 2 = 0 va ( Q ) catOy, Qztai M(0;2;0) va

>j(0;0;2) (thoa man) Neu a = - b t h i M N = (0;-a;-a)//u(0;l;l) va n ^ 1 u

= (0;1;-1)

nen n ^ = u , n p

Khi do mat phSng (Q): y - z = 0 va (Q) di Oy, Oz tai M(0; 0; 0) va N(0; 0; 0)

V?y, ( Q ) :2x + y + z - 2 = 0 la m$t phSng can tim

|;au 9.b: Cdch /.-Gpi z = x + yi (x,y € R )

I Theo gia thiet, ta c6 |z| = 1 ^ yjx^ + = 1 y^ = 1 - x^ x 6 [ - 1 ; l ] ( l )

Khi do: M - z ^ - z + 2 (x^ - 3xy2 - x + 2) + (3xV - - y ) i

= ^6 - 2(cos(pcos3(p + sincp + sin3(p) + 4(cos3(p - cos(p)

= V6 - 2 cos 2{p - 8 s in2(p sin (p = ^ 6 - 2 ( 2 c o s ^ ( p - l ) - 16(l -cos^ (p)cos(p

= 2 A / 4 C O S ^ P - COS^ (p - 4cos(p + 2

253

Tuyen chQti & Giai thifu dethi Toan hgc - TQguyen Phu Khanh , Nguyen Tat Thu

Dat t = cos(p(-l < t < l ) ,ta c6 M = 2^41^-1^-41 + 2

Xet ham so f (t) = 4t^ - - 4t + 2, t e [ - 1 ; f

1 2 Taco: f (t) = 12t^ - 2 t - 4 va f (t) = O o t = - - hoac t = -

Cau 1: Cho ham so y = - +1, c6 do thj la (C)

a) Khao sat sy bien thien va ve do thj (C) ciia ham so

b) Tim tren do thi (C) nhung diem A sao cho tiep tijyen tai A cat (C) tai

hai diem B, C khac A va B, C nam ve 2 phia doi voi A

sin3x-cos3x Cau 2: Giai phuong trinh: cos2x + •

2 s i n 2 x - l • = sinx(l + tanx)

Cau 3: Giai phuong trinh: Vsx^ +14x + 9 -\/x^ - x - 2 0 = sVx + l

Cau 4: Tinh tich phan: I = cos* x sin^ xdx

0

Cau 5: Cho hinh lap phuong ABCD.A'B'C'D' canh a Tren canh AB lay dier

M dat A M = m (0 < m < a) Mat phing ( A ' M C ) cat C D ' tai N Chimg min

A ' M C N la hinh binh hanh Tim vi tri M de di$n tich A ' M C N nho nhat kM

do tinh goc giCra ( C M N ) , ( M N D )

Cau 6: Cho x, y, z la cac so duong thoa man dieu ki?n x + y + z = xyz

Chung minh rSng: 4(x + y + z)^ - ( x y + yz + zx +1)^ > xy + yz + zx

I

CtyTNHHMTV DWH Khang Vift

I P H A N R I E N G Thi sinh chi durgc chpn lam mpt trong hai phan (phan A o|c B)

Theo chucrng trinh chuan Cau 7,a: Trong mat phiing tga dp Oxy, cho 2 duong thSng dj : 2x - y - 1 = 0,

d2 : 2x + y - 3 = 0 Gpi I la giao diem ciia dj va'dj, A la diem thupc dj va A

CO hoanh dp duong khac 1 (O < < l ) Lap phuong trinh duong thSng A di qua A, cat dj tai B sao cho di$n tich AIAB bang 6 va IB = 3IA

Cau 8.a: Trong khong gian voi h? tryc tpa dp vuong goc Oxyz, cho mat ph^ng (P): X + y + z - 3 = 0 va duong thing A : ^ ^ = ^ = — Lap phuong trinh duong thing d, nam trong mat phang (P), vuong goc voi duong thing A va each

8 duong thang A mpt khoang bang - =

•\i66

Cau9.a: Timsophuc z thoa man ( z - l ) ( z + 2ij lasothycva |z|nh6nhat

B Theo chUomg trinh nang cao

Cau 9.b: Trong cac so phuc thoa man dieu k i | n |z + 1 + 2i| = 1 ,so phuc z nao

CO modun nho nhat

" P H A N C H U N G C H O T A T CA C A C T H I S I N H CSu 1:

a) Danh cho ban dpc

Hi b) Gpi A (a; a"* - +1J la diem thoa man de bai

Ta c6: y' = 4x^ - 2x Phuong trinh tiep tiiyeh (d) ciia (C) tai A la:

y = ( 4 a 3 - 2 a ) ( x - a ) + a ^ - a 2 + l

Phuong trinh hoanh dp giao diem ciia (d) va (C) la:

x4 _ x2 +1 = (4a^ - 2a)(x - a) + a^ - a H l « (x - af (x^ + 2ax + Sa^ - a) = 0

o x = a hoac g(x) = x^+2ax + 3a^-a = 0

Theo bai toan thi g(x) = 0 c6 2 nghi^m phan biet x^,x^ sao cho: Xj < a <

A' = -2a^+a>0 ^ 1

( x i - a ) ( x 2 - a ) < 0 6

Cau 2: Dieu kien: sin 2x ^ ^, cos X ^ 0

Phuong trinh da cho tuong duong:

eos2x + 3sinx-4sin^x-4cos3x + 3cosx ^^.^^^^

2 s i n 2 x - l

, 2 (sinx + cosx)(2sin2x-l) sin x(sin x + cos x)

2 s i n 2 x - l cosx sin X + cos X = 0

^ sinx cosx-smx + 1 =

cosx

o cos X - s i n x +

THI: sinx + cosx = 0 <=> tanx = - 1 o x = - - - + k 7 t , k e Z

4 TH2: cosx-sinx + l = - ^ ^ o ( c o s x - s i n x ) ( l + cosx) = 0

cosx cosx-sinx = 0

1 + cos x = 0 <=>

tan x = 1 cos x = - 1

hoac X = 8 (thoa man de bai)

Vay, nghifm ciia phuong trinh la: x = llJ^^ x = 8,

Chon h§ trijc tpa dp Axyz sao cho:

||| A{0;0;0), B(a;0;0), C(a;a;0), D(0;a;0),

sin^2x

X 1 , sm4x +

16 64 24

n

32

A'(0;0;a), B'(a;0;a), C'(a;a;a), D'(0;a;a) => M(m;0;0)

1 Chung minh A'MCN la hinh binh hanh z

= -a(a;m-a;m) ^

Taco: MA',MC

=>n(A'MC) =(a;m-a;m)

=i> (A'MC): ax + (m - a)y + mz - am = 0

Phuong trinh tham so ciia CD:

V?y A • MCN la hinh binh hanh X

2 Tim vj tri M de difn rich A'MCN nho nhat, rinh goc giiia (CMN),(MND)

Ta c6: S^'MCN = [MA',MC]| = 2aVm2-am + a2 = 2a^

^(SAMCNLn=a2Vi«m = |

V$y M la trung diem ciia AB => ACMN can tai C

fCIlMN GQI I la trung diem cua MN

T = 4(x + y + zf-(xy + yz + zx)^-3(xy + yz + zx)

Taco: 3(xy+ yz + zx)<(x + y + zf nen T<^^^-^^^^ + 3(x + y + zf

Tir X + y + z = xyz ta CO xyz > 3V3 , vi the neu t = (x + y + z)^ thi t > 27

Xet ham so: f (t) = ^ + 3t voi t ^ 27

II PHAN RIENG Thi sinh chi dugfc chpn lam mpt trong hai phan (phan A

ho9C B)

A Theo chUarng trinh chuan

Cau 7.a: I = dj n => tao dp ciia I la nghi^m cua h?

J 2 x 2 x 4 -y-l=:0 fx = l - 3 = 0=^ jy = l-^(^^ ^)

Tu gia thuyet dj c6 VTPT n^ =(2;-l), dz c6 VTPT =(2;1) Gpi cp la goc

4-1 cuadj vadj =>cos(p = ^ — ^ = |i :>sin(p = | =>S.,AB = - I A 3 I A - i = ^ ^

Gia thuye't: S^j^g = 6 => lA^ = 5 => IB^ = 45

VA € dj => A(a,2a -1) voi a > a*\

IA2=5 <»(a-l)2+(2a-2)2=5o5(a-l)2=5»a = 0 (loai) hoac a = 2

/ a = 2=>A(2;3)

VBedj =>B(a,3-2b)

"b = 4=>B(4;-5)

b = -2=>B(-2;7)

"di A(2;3), B(4;5) phuong trinh can tim la = o 4x + y -11 = 0

IB^ =(b-l)2 +(2-2b)2 =5(b-l)^ IB^ =45o(b-l)2 =9<»

^ ,<x-2 y - 3 can tim la - - ^ i

Voi A (2; 3); B(-2;7) phuong trinh can tim la ^ ^ ,«x-2 y - 3 4-2 -5-3

•an rim la = ± - 2 - 2 7-3 <:>x + y - 5 = 0

'u8.a:Tac6 (P) c6 VTPT np =(l;l;l),

4 coVTCP i^ = (l;3;-l), M(1;0;0)6A

d l A u J 1 np r < J "1 = (-4;2;2) GQI (Q) la m|t phang chua d va song song voi A Khi do ta chpn

= -2(4;l;7) suy ra (Q) codang 4x + y + 7z + d = 0

4 + d vTaco d(A;d) = d(A;(p)) = d(M;(P)) = ^ Ket hpp voi gia thiet ta dupe: 4 + d >/66 V66 4 + d =8<:*d = 4 ho$c d = -12

259

THl: Neu d = 4 => ( Q ) : 4x + y + 7z + 4 = 0

Ch(?n d i e m N f - i ; ^ ; - l l e (P)n(Q) = d

1 13 suy ra phuong trmh: d : ^ = ^ = — ^

TH2; Neu d =-12 =^ (Q) : 4x + y + 7z -12 = 0

Chondiem N(l;l;l) e (P)n(Q) = d

suy ra phuong trmh: d : — ^ = —— = ——

Cau9.a: Goi z = a + bi (a,b€R)

Theo gia thiet, ta c6: (z - l)(z + 2i) = [(a -1) + bi][a - (b - 2)i

= a(a-l) + b(b-2) + [ab-(a-l)(b-2)]i (z-l)(z + 2i) lasothirc o a b - ( a - l ) ( b - 2 ) = 0»2a + b - 2 = 0 o b = 2-2a

Khi do z = a+(2-2a)i

Ta CO |z| = ^a^ +(2-2a)^ = Vsa^ -8a+ 4 = ^ 4

a — 5 5 5

mm 5 5 5 - ^ 5 5 , khi a = -=>b = - Vay z = - + —i 4 , 2 4 2

B Theo chUtfng trinh nang cao

Cau 7.b: Di?n tich hinh tron npi tiep hinh thoi ABA'B' bang 47i ban kinh

duong tron r = 2

O la tarn hinh tron, ke OK ± AB'=> r = OK = 2

1

Xet tarn giac vuong OAB', ta c6: 1 1 1 1 1 • + — ^ 0 - = -.- + - ^ (1)

OK^ OA^ OB^ 4 b^

Tir gia thuyet:

IA = IB

lA = IC <»

Ie(a) a + 2b + 2c-1 = 0

<=> b + 7c = 6 5a - 4b - 3c = 6 <=>

AB,AC = (-25;-35; 5)

AB = (0;-l;-7), AC = (5;-4;-3) =:> n = cos ((a), (ABC)) = |cos(.^,p)| =

Goi S' la dien tich hinh chieu cua tam giac ABC len mat phang (a) Taco S' = SABc-':os((a),(ABC)) = : ^ ^ = ^ (dvdt)

;au 9.b: Cach 1: Gpi z = x + yi (x,y e R )

Theo gia thiet, ta c6:

z +1 + 2i| = 1 |x +1 + (y + 2)i| = 1 <» (x +1)^ + (y + 2)^ = 1 Tap hop cac diem M(x; y) bieu dien so phiic z la duong tron (C) c6 tam l(-l;-2), ban kinh R = l

Diem M(x;y) bieu dien so phuc zco modun nho nliat la mot trong cac giao diem ciia duong thang OI voi duong tron (C) (sao cho dp dai doan OM nho nhat) Phuong trinh duong th^ng OI la y = 2x

Toa dp diem M thoa h^:

(x + l f ^ ( y + 2f =1

y = 2x (x + l f = i y = 2x y = x = -l + -2 + hay x = l

l-i _2-Al

261

Tuye'n chqn & Giai thifu dethi Todn hgc - Nguyen Phu Khanh, Nguyen Tat Thu

Ta CO OMi = sje-lsis < OM2 = V6 + 2V5

Do do min z| = OM^ ,dat dug-c khi M s

V§y, so phiiccan tim la z = +

>min|z| = 76-2>/5 , d^tdugckhi sin(t + (p) = l o ( p = - | - t + k27t,k(

Khi do cos(p = sint =

Taco: = x2+y2 ^(x + l-l)2+(y + 2 - 2 f

= (x + l)^+(y + 2)^-2(x + l)-4(y + 2) + 5 = 6-2[(x + l) + 2(y + 2)], (do(*))

CtyTNHH MIV DWHKhangVift

Theo bat dSng thuc Bunhiac6'pxki,ta c6:

'-10 + 2N/5'

x = - l + V5

DETHITHlJfSd40

I PHAN CHUNG CHO TAT CA CC T H I SINH

Cau 1: Cho ham so y = -x^ + (3m + l)x2 - 2(3m + l)x + 8 c6 do thj la (C„) a) Khao sat su bien thien va ve do thj (Cj) cua ham so

b) Tim m de (C^j^) cat tryc hoanh tai ba diem phan bi^t lap thanh mpt cap sonhan

Cau 2: Giai phuong trinh 2\/3 sinx.(l + cosx) - 4cosx.sin^ ^ ^

Cau 3: Giai phuong trinh: 1 + logjCx -1) = log^_j 4 Cau 4: Tinh tich phan : I = f^"0 + x)^^

0 1 + x

Cau 5: Cho hinh hpp dung ABCD.A'B'C'D' day la hinh thoi canh 2a,

BAD = 60° Duong cheo AC tao voi day goc (a) Gpi I la giao diem cac duong cheo cua hinh hpp, O la tarn cua ABCD Tinh the rich hinh hpp va d(BD, AC') theo a va a Cho diem M thoa man IM = d, h'nh tong T cac binh phuong khoang each tu M den 8 dinh cua hinh hpp theo d, a va a Tu do tim vj tri M de T^jr,

Cau 6: Cho cac so thyc a, b,c e [l;2 thoa man 4a + 2b + c = 11

!, 33 1 2 3 11 Chung mmh rang: — < - + — + - < —

^ ^ 10 a b c 2

I iii/c" cluni C.icn tliiihi ilc I'm loan I K X \ ' X ' ( I / < ' " I ' I ' I I Khiiiih , Nguyen Tat Thu

II PHAN RIENG Thi sinh chi dugc chpn lam mpt trong hai phan (phan A

hoac B)

A Theo chi/orng trinh chuan

Cau 7.a: Cho hinh chu nh^it ABCD c6 phucmg trinh duong thang AB: x-2y+l=Q,

phuong trinh duong thkng BD: x - 7y +14 = 0, duong thang AC di qua M(2; l )

Tim toa do cac dinh ciia hinh chir nhat

Cau 8.a: Trong khong gian tpa dp Oxyz cho mat phang (a):3x-2y + z-29=0

va hai diem A(4;4;6) ,B(2;9;3) Gpi E, F la hinh chieu ciia A va B tren (a)

Tinh dp dai doan EF Tim phuong trinh duong thing A nam trong mSt phang

(a) dong thoi A di qua giao diem ciia AB voi (a) va A vuong goc voi AB

Cau 9.a: Cho so phiic z 0 thoa man dieu ki^n ' ^ ^ z 3 + - < 2

Chung minh: 1

z + —

z

<2

B Theo chUorng trinh nang cao

Cau 7.b: Trong mat phMng Oxy, cho elip : x^ + 4y^ - 4 = 0 T i m nhung diem N

tren elip (E) sao cho : FjNFj =60'' (Fi, Fz la hai tieu diem ciia elip (E))

Cau 8.b: Trong khong gian tpa dp Oxyz, cho mat cau (S):

(x + 1)^ + ( y - 2 ) ^ +(z + 3)^ =17 va mat phang (P):2x + 2y + z + 7 =0

Viet phuong trinh duong thang A di qua A(8; 0; -23) nam trong (P) va tiep

xiic voi mat cau (S)

Cau 9.b: Cho so phuc z^O thoa man z S2

z + i Tim gia tri nho nhat va gia tri Ion nhat ciia bieu thiic P =

HirOfNG D A N G I A I

I PHAN CHUNG CHO TAT CA CAC THI SINH

Cau 1:

a) Danh cho ban doc

b) Cdch i ; H o a n h dp giao diem ciia tryc hoanh va [C^) la nghifm ciia

Phuong trinh (2) xay ra <=> i

au 2: Phuong trinh cho tuong duong voi

2 Vs sin X + 2%/3 sin x cos X - 2 cos X (1 - cos x) = 3

o 2^\/3 sin X - cos X j - |3sin^ x - 2V3 sin x.cos x + cos^ x j = 0

o IV3 sin X - cos x jIV3 sin X - cos X - 2 j = 0 <=> Vs sin X - cos x = 0

Cau 3: Dieu kif n: • x - l > 0 x > l

x-1^1 x ^ 2

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