luyện thi đại học môn toán

Cau 9b: Dat t = |z|, t > 0 thi ta c6 \zf (a.z + b) = -c.z
Nen ta c6 |c|.t < t^ (|a|.t + |b|) <=>t^ + t - l>0<=>t>
Va | c | . t > t ^ ( | a | . t - | b | ) o t 2 - t - l < 0 o t <
V^y

< |z| <

Cau 8a: Trong khong gian voi h# tpa dp Oxyz cho hai duong thang:
- 1 + 75

dj:2^ =X =^ ;

Cau 9a: Cho Z j , Z2 la cae nghi#m phuc ciia phuong trinh 2z^ - 4z + l l = 0 .
Z,

+|Z2

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DETHITHUfSdlO

(zi+z2r

B. Theo chUorng trinh nang cao

Cau 7b: Trong mat p h i n g voi h? tpa dp Oxy cho ba diem
vuong, AB d i qua E va CD d i qua F.

- 3mx^ + 4m^ c6 do thj (Cm)

Cau 8b: Trong khong gian Oxyz, t i m tren Ox diem A each deu duong
,
x—1 V z+ 2
'
thang d : — ^ = =
va mat phang (a): 2x - y - 2z = 0 .

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 t h i n g y = x .

s/
up
/g

ro

6 ( y - 2 ) 2 + ^ = x3y[(2y + 3)2-6'

.c

om

dx

ok

- i l + x + V l + x^

Cau 9b: (1 diem) Giai h ^ phuong trinh sau:

Ta

n
Cau 2: Giai phuong trinh: (sin^ x +1 j +1 = 73 sin 2x + 4 sin x + —
6J
-xy = 1

bo

Cau 5: Cho hinh chop S.ABCD c6 day ABCD la hinh chu nhat eo tam O va

2logi_x(-xy - 2x + y + 2) + log2+y (x^ - 2x +1) = 6
, (x,y6R).
logi-x (y + 5) - log2+y (x + 4)
=1

HirdNGDANGlAl
I. PHAN CHUNG CHO TAT CA CAC THI SINH
Caul:
a) B?n dpc t y lam.
b) Ta c6: y' = 3x^ - 6 m x , y' = 0<:>x = 0 v x = 2m .

Ham so' c6 hai diem eye trj o m ?t 0.

p h i n g (SAC) tao voi day mpt goc 60° . Tinh the tich khoi chop S.ABCD .

Cac diem eye tri ciia (C^,) la M ( 0 ; 4m^)va N ( 2 m ; 0 ) .

w.

ww

(a+b

fa

ce

AB = a, A D = a73 ; SO = SD . Mat phSng (SBD) vuong goc voi mat day, mat
Cau6: Chung minh rang ne'u a,b,c>0 t h i :
a+b
b+c
ic + a ^

Vb+c

Trung diem cua doan M N la l ( m ; 2m^ j v a M N = (2m;-4m3)
ya+c

I I . PHAN RIENG T h i sinh chi dugc chpn lam mpt trong hai phan (phan A
ho^c B)

Duong t h i n g d : y = x c6 vecto chi phuong la u = ( l ; l )
M , N doi xung nhau qua duong t h i n g (d) <=> M N 1 (d) va I G (d)

A. Theo chUorng trinh chuan

MN.u = 0

2m + (-4m^) = 0

m =0

Cau 7a: Trong mp voi h? tpa dp Oxy cho duong tron (C) c6 phuong trinh

2m'' = m

m(2m2-l) = 0

m =±

x^ + y^ - 2x + 6y - 1 5 = 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

I(l;l),E(-2;2),

F(2;-2). Tim tpa dp cac dinh cua hinh vuong ABCD, bie't I la tam cua hinh

I. PHAN CHUNG CHO TAT CA CAC THI SINH

Cau 4: Tinh tich phan: I = [ -

va hai diem A ( l ; - 1 ; 2 ) , B(3;-4;-2).

cho l A + IB dgt gia trj nho nhat.

.

Cau 3: Giai hf phuong trinh

=^

Xet vi tri tuong doi ciia d] va d 2 . Tim tpa dp diem I tren duong t h i n g d j sao

1 + 75

Tinh

Cau 1: Cho ham so y =

d 2 : ^ =^

Ke't hpp dieu kif n ta dupe m = ±

72

72
63


" ^

Cau 2: P h u o n g t r i n h t u o n g d u o n g v o i :

Vay nghi^m cua h§ da cho la:

sin"* X + 2sin^ x + 2 - 2\/3 sin xcosx - 2 V 3 s i n x - 2 c o s x = 0

sin x = 0
<=>-^ ^
<=> cosx = 1 <=> X = k 2 n .
[ V 3 s i n x + c o s x - l =0

u^-l
=>

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Ta
s/
up

.c

i

bo

ok
2
+

fa

ce

s = -5-M

w.

< 4P

ww

^
v 6 n g h i ^ m do
P = 13 — v l 3
4

P =

/g

P =1

om

"S = 2

ro

+ y , P = xy ta c6:

S= 2

2

I

u2

du

1
•+ —

u^u+lj

du

J

du = !n(u + l ) - ^ l n u

=1

SH 1 OD

SH 1 (ABCD)

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°.
Ta c6: A C = B D = 2a

O C - a => A O C D deu

G Q I E la trung diem cua O C ,
suy ra
^

H K =

1DE
2

=

- ^ .

4

Trong tarn giac vuong S H K ta c6:

^

S H = HKtan60" . ^ . 7 3 : . ^
4
4
Vay the tich cua khol chop la:
Vs.ABCD

= |SH.SABCD

B

=1^'^'^^^^^

Cau 6 : Ap dyng bat dang thuc y/x + ^Jy < ^2(x + y ) , ta c6 :
" g h i ? m ciia p h u o n g t r i n h :

a+b
c

4t2 + 2 ( 5 + 7 l 3 ) t +13 + 713 = 0 o t =
64

V2+1

nen S H la duong cao cua hinh chop.

[x + y + 2xy = 4

- 5 + VT3

1

x = l=:>u^72 + l

M|t khac (SBD) 1 ( A B C D )

T ^e tn uhgip voi•u'^^
, <[x^ + y ^ - x y - l
K
h^ ta co:

4-S

4

du

Cau 5: Goi H la trung diem cua C D

o x ^ ( 2 y + l f = - ( y - 4 ) ^ <=>x(2y + l ) = - y + 4 o x + y + 2xy = 4

2S^+3S2-llS-6-0

du

u ^
1+u

du

(8y^ + 24y^ + 6y +1) = - ( y ^ - Uy^ + 48y - 64)

S + 2P = 4

I

l + u^2

T r u hai p h u o n g t r i n h cvia h? ta c6 duoc:

S^-3SP-P = 1

2

1

12y^ - 48y + 63 = 8x^y^ + 24x^y^ + 6x^y

X

1 ^

dx = — 1 +

2u

2

x-' + y"* - xy = 1

S =

X = •

Doi can x = - l = > u = 7 2 - l ;

- xy = 1

12(y - if +15 = 2xVr(2y + 3 ) ^ - 6

Dat S =

-5-713?: 76713-14^
'

Cau 4: D§t u = x + 7x^+1 => ( u - x)^ = 1 + x^

(

d a cho t u o n g d u o n g v o i :
x^ +

4

=0

< » s i n ' ' X H-^N/Ssinx + c o s x - l )

Cau 3 :

-5-713+76713-14

(x;y) = ( l ; i ) .

I^^-^^l^^^Eli

7^r

+.

1

72U

b+c

/c + a
1 7^_^7bl
+ -/
> 72
t7?
7?J

r^^^7b^

7b_^7?1
721,7^

1 1 7b 1
n 7? 1 1 ^
7 b j 72l7^ 7 ? j 72l7^ 7 b J

7^j

72

7a

7a ^


'

-_1

1 1

Ap dune bat dang thuc — + — >
x y x

N/ZIVC

VbJ"^ V2l>/a

4

+y

3N/2 .

, ta c6 :

^ycJ

x/b.
2^y2a

2>/2b
=

Vb + N/c

272^

2V2b

^ 272^

Vb + Vc

Va + Vc

^

2N/2C

x/a+Vc

^ya^-^/b

Do do

^ 2V2b

Va + N/b ~ ^/2(b + c)
= 2

(a + b

pia

^

^2{a

+ c)

Va + c

2>^

"Vb + c

+

b)

2

+

11

Z2

Duong thang CD c6 phuong trinh dang:

(dpcm).

a ( x - 2 ) + b(y + 2) = 0 <=> ax + by - 2a + 2b = 0 .

V i d ( I , A B ) = d(IXD)=;>

A. Theo chUtfng trinh chuan

|3a-b| _ |a-3b|
<=> a = - b.
,
Va^ + b^ Va^ + b^

Suy ra phuong trinh A B : x - y + 4 = 0, C D : x - y - 4 = 0.

Ta

Cau 7a: Duong tron (C) c6 tam 1(1;-3), ban kinh R = 5. Gpi H la trung diem

Phuong trinh BC va D A c6 dang x + y + c = 0
d(I,BC) = d ( I , AB) = 2V2 =i.

s/

AB thi A H = 3 va I H 1 A B => I H = 4 .

up

Matkhac I H - d ( l , A )

/g

ro

Vi A l d : 4x - 3y + 2 = 0 => A: 3x + 4y + c = 0

om

= 4<::>C = 29,C = -11

fa

il^ = - u ^ .

Gpi A j la diem do'i xung cua A qua d j .

Suy ra A ( l ; 5 ) , B ( - 3 ; l ) , C ( l ; - 3 ) , D ( 5 ; l )
BC:x + y - 6 - 0 , D A : x + y + 2 = 0

•

Suy ra A ( - 3 ; l ) , B ( l ; 5 ) , C ( 5 ; l ) , D ( l ; - 3 ) .
Cau 8b: Gpi A ( a ; 0 ; 0 ) £ O x .

(a) qua

ww

Taco AB = ( 2 ; - 3 ; - 4 ) = > A B / / d i .

w.

Ma M ( 2 ; 0 ; - l ) 6 d i nhung M g d j =>d,//d2

c = 2,c = - 6 .

.
. . . . . .
|2at
Khoang each t u A deh m|it phang (a) : d ( A ; a ) = - j = = = = =

ce

Cau 8a: Vec to chi phuong cua hai duong thang Ian lugt la:

bo

ok

3x + 4y + 29 = 0 va 3x + 4 y - l l = 0.

= 272

BC:x + y + 2 = 0, D A : x + y - 6 = 0

•

.c

Vay CO 2 duang thang thoa man bai toan:

u7 = ( 2 ; - 3 ; - 4 ) , u^ - (-2;3;4)

=

3v'2.

<=> ax + by + 2a - 2b = 0 vai a^ + b^ > 0 .

hoac B)

lc-9|

; z, + Z 2

=1+

B. Theo chUtfng trinh nang cao
Cau 7b: Duong thang AB c6 phuong trinh dang: a(x + 2) + b(y - 2) = 0

I I . PHAN RIENG T h i sinh chi dirgic chpn lam mpt trong hai phan (phan A

d(I,A)-IH<::>

x/22

Z2

2

2

?=• + •

+ —!=

< ^2(x + y ) , ta c6 :
2N/2^

•3^/2'

Suy ra

>^=

A p dung bat d i n g thuc Vx + ^

i,

Cau 9a: Phuong trinh da c6 cac nghiem: zi - ^ -

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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 =

Taco: l A + IB = l A j + I B > A i B
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 AjB va d .
J 65 -21
Do A B / / d i => I la trung diem ciia AjB suy ra I
129 58
66

Theo gia thiet: d ( A ; ( a ) ) - C I ( A ; A )
-43
29

J

o

|2a|
!—=
3

Vsa^ - 24a + 36 <::> 4a
, 2 = oSa"^2 - -.^
A 2
24a + 36 <=> 4a
- r..
24a + -.^
36 = n0
3


CtyTNHHMTV

<=>4(a-3)^ =0<=>a = 3.

<=>

day, A D = ay/3 . G p i E, F Ian l u p t t r u n g d i e m ciia cac d o a n BC, D E . T i n h the

- x y - 2x + y + 2 > 0,

- 2x + 1 > 0, y + 5 > 0, X + 4 > 0

jfch h i n h chop F.ABC. C h u n g m i n h A F v u o n g goc v o l C D .

0 < l - X 5 t l , 0 < 2 + y;itl

Cau 6: C h o so t h u c d u o n g a, b thoa m a n : 6|a^ + b^ j + 20ab = 5(a + b ) ( a b + 3 ) .

2 l o g i _ x [ ( 1 - x ) ( y + 2 ) ] + 2log2+y (1 - x ) = 6

T i m gia t r j n h o nhat ciia bieu thuc

l o g i - x ( y + 5 ) - l o g 2 + 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 ) - I o g 2 ^ y ( x + 4)

(2)'

=1

P =

II.

y = -X-1

s/
up
ro

/g
ok

.c

om

DETHITHllfSdll

bo

I. 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

fa

ce

Cau 1 : C h o h a m so y = x^ - Sx^ + 1 , c6 d o t h j la ( C ) .

w.

a) K h a o sat s u bien t h i e n va ve d o thj ( C ) ciia h a m so.

ww

b) T i m cac d i e m A , B thupc d o t h i ( C ) sao cho tiep t u y e n ciia ( C ) tai A , B

dx
Cau 4: T i n h t i c h p h a n : I = fix^+2x

b^^

PHAN R I E N G T h i s i n h c h i d u p e ehpn l a m m p t t r o n g h a i p h a n ( p h a n A

Cau 8.a: T r o n g m a t phSng tpa d p O x y z , cho hai mat p h a n g ( P ) : x + ^ - 2 z + 5 = 0

V a y h? C O n g h i p m d a y nhat x = - 2 , y = 1 .

Cau 3: Giai p h u o n g t r i n h : 2(x^ + 2) = 5Vx^ + l

'

l ^ i ciia h i n h chu nhat bie't D n a m tren d u o n g thang c6 p h u o n g t r i n h : x - y - 2 = 0

K i e m tra d i e u k i ^ n ta thay chi c6 x = - 2 , y = 1 thoa m a n d i e u k i | n tren.

rilTt

'

f
4^
dinh A ( l ; l ) . G p i G 2 ; la trpng tarn tam giac A B D . T i m tpa d p cac d i n h con

= 1

Cau 2: Giai p h u o n g t r i n h : 5 cos 2x + - - 4 sin
V
6;
U2

+ 25

Cau 7.a: T r o n g mat phJing tpa d p Oxy, cho h i n h c h i i nhat A B C D , v o i toa dp cac

x+ 4
-x + 4 ^
2 «
rx = 0=>y = - l
= l - x < = > x ^ + 2 x = 0 <=>
^
x+ 4
[_x = - 2 = > y = l

song song v o i n h a u va A B = 4V2 .

a ^

b

Ta

o

b

-16

A. Theo chi/orng trinh chuan

(3).The vao (2) ta c6:

l o g i - x ( - X + 4) - l o g i _ , (x + 4) = 1 <:> l o g i _ ,

9

hoac B)

D a t l o g 2 + y ( l - x ) = t t h i (1) t r o thanh: t + ^ - 2 = 0 < » ( t - l ) ^ = O o t = l .
V o l t = 1 ta c6: 1 - X = y + 2 o

a

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H $ da cho

Khang Vjft

Cau 5: C h o t u d i ^ n A B C D c6 A B C la tarn giac deu canh b i n g 2a, A D v u o n g g6c

V a y A (3; 0; 0 ) .
C a u 9b: D i e u k i ^ n :

DWH

>

x + 9

j

va ( Q ) : X + 2y - 2z - 1 3 = 0 . Viet p h u o n g t r i n h mat cau (S) d i qua goc tpa dp

O, qua d i e m A (5; 2; 1 ) d o n g t h o i tiep xiic v o i ca hai mat phSng ( P ) va ( Q ) .
Cau 9.a: T i n h m o d u n ciia so p h u c z , bie't z^ + 12i = z va z c6 phan thuc d u o n g
B. Theo chUorng trinh nang cao
X^

y2

Cau 7.b T r o n g m a t phang tpa d p Oxyz, cho elip ( E ) : — + ^

= 1 va d u o n g

thSng d : X + y + 2013 = 0 . Lap p h u o n g t r i n h d u o n g thang A v u o n g goc v o i d va

c^t ( E ) tai hai d i e m M , N sao cho M N =

— .
3

Cau 8.b T r o n g mat phSng tpa d p O x y z , cho mat phang ( P ) : x - 2y + 2z + 2 = 0
va d u o n g th5ng ( d ) :

=

=

. M a t cau (S) c6 tarn I n a m tren d u o n g

thang ( d ) va giao v o i m a t phang ( P ) theo m p t d u o n g t r o n , d u o n g t r o n nay
Voi t a m I tao t h a n h m p t h i n h non c6 the tich Ion nhat. Viet p h u o n g t r i n h mat
cau (S), bie't ban k i n h m g t cau bSng 3N/3 .

Cau 9.b: Giai h$ p h u o n g t r i n h sau:

2x'^ + 2 x y - 3 x - y + l = 0
2=0

63
iHiiii'

69


Tuyen

ch(>n fy Gim

thieti

dethi

Todn

H Q C - Nguyen

Phu

Khdnh

, Nguyen

Tat

Cty

Thu.

WSQm DANGlAl

Cau 3: Voi ^

I . P H A N C H U N G CHO TAT CA CAC T H I SINH
Cau 1:
b) Goi A^a;a'' -3a^ + l j va B|b;b^ -3b^ +1

va a ?t b la cac diem thoa man

Do 2 tiep tuyen tren song song voi nhau nen ta phai c6 y'(a) = y'(b)

Cau 4: I = j -

L a i c 6 : A B = 4V2 o ^(a - b f + (a^ - b ^ + Sa^ + 3b2)

1

Ta
s/

=32

(2)

up

l + (2 + ab)^

ro
/g

.c

- 2X - 3 = 0 => X = - 1 hoac X = 3

4[Jl-2

!t

12

+ 2 sin

ww

X +-

= 1 o X = — + k27t
sm x + 12
12

Cau 5: Ta c6 E la trung diem B C
A E = -!-(AB + A C J .

F la trung diem D E => A F = ^ ( A D + A E

X + •

12

= - A D + - A B + - A C

2

-7 = 0

(*)

4

4

C D = A D - A C => A F . C D

1
1
-AD+-AB+-AC
2
4
4

A D - AC

= i A D 2 - i A D . A C + -!-AB.AD--AB.AC + - A C A D - - A C 2
2
2
4
4
4
4
AD^ =(aV3) =33^ , AC^ =4a2.
A D 1 AC , A D 1 AB => AD.AC = AD.AB = 0,

- X

t-2

ce
w.

fa

f X + —O
I
12j

Khi do phuong trinh (*) tro thanh: 5t^ + 2t - 7 = 0 => t = 1 tuc phai c6:

ll7t

t(l-2)

In

AD + i ( A B + Ac)

voi -1 < t < 1

Chu y\O the dat t =
12

i>

dt

hoac A ( 3 ; 1 ) , B ( - 1 ; - 3 ) thoa man

yeu cau bai toan .

12j

,

dt

Jt(t-2)

bo

Do do ton tai hai d i e m A ( - 1 ; - 3 ) , B ( 3 ; 1 )

X +•

xdx

ok

Vay a, b la nghiem phuong trinh:

Dat t = sin

+ 1 - X = 0 =>x = 3

om

(4 - 4t)(5 + 4t +1^) - 32 = 0 ; o t^ + 3t^ +1 + 3 = 0 o (t^ + l ) ( t + 3) = 0 ^ t = -3

Phuong trinh cho viet lai: 5sin^

3VlO-3x +3

Khi do:

D l t t = ab va thay a + b = 2 (do (l)) vao (2) ta duoc :

[n

+ X - 5 ) ( u - x + 2) = 0=> X = 3

Dat t = x^ + 2 => dt = 2xdx hay xdx = - d t

=\4i

o ( b - a ) ^ l + (2 + a b f = 32

o ^(b - a)^ + (b - af (2 + a b f = A^l

27

dx

]X''+2x

<x>3(a-b)(a + b ) - 6 ( a - b ) = 0 « ( a - b ) ( a + b - 2 ) = 0 =>a + b = 2 ( l )

= sin

Khang

, dat u = V l 0 - 3 x , dua phuong trinh ve h^:

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Huang 2: <=>(x-3)

Phuong trinh tiep tuyen tai B c6 h? so goc y'(b) = 3b^ - 6 b

- X

DWH

Huang 1: 9(l0-3x) = x"* +16x2 -8x^ c ^ ( x - 3 ) ( x + 2)(x2 - 7 x + 15) = 0

Phuong trinh tiep tuyen tai A c6 he so goc y'(a) = 3a^ - 6a

Cau 2: Ta c6: sin

MTV

Cdch khdc: Binh phuong 2 ve, ta duoc: 3 7 l 0 - 3 x - 3 = -x^ + 4x - 3

bai toan. Ta c6: y ' = 3 x ^ - 6 x

(a + b f - 4 a b

10

X
4x + 3u ^ fj.^ yg'fj^gQ yg'ja ^u,g,c
u^ + 3 X - 1 0 = 0

a) Danh cho ban dpc.

Hay


TNHH

AB.AC = AB.ACcosBAC = 2a.2a.cos60'' = 2a^.

4

2


Tuyen chgn & Gi&i thi$u dethi Todn hqc - Ngui/en Phti Khdnh , Nguyen Tat Thu.

Cty TNHH MTV DWH Khang Vift

Do do DC:

Suy ra AF.CD = 1 Sa^ -l.la^ ~^Aa^ = 0 =^ AF 1 CD.
2

V,F A B C

4

4

- y - 2 = 0.

Biet phuong trinh DC se viet dupe phuong trinh AB ma ABCD la hinh chii
jih$t

(dvtt)

-

X

^^^^ 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.
Cau 6: Tir gia thiet, chia ca hai ve cho ah, ta du(?c:

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:.

b^a

.20 = 5 ( a . b ) . 1 5 ^ > 2 J ^ i i ^ ( . ) .

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a

.

' ' G - X A = 2 ( X I - X C )

AG = 2GI <^

D a t t = -^ + - ^- , t h i (*) 6t + 20>10^/3(tT2)<=>t>H
b^a
a

b'

fa

h^'

b"^a

-2,

-3

b3
a^

=

fa

b] (a

b^

^b

a) [b

aj

.yG-yA=2(yi-yG)

2

Tir d o :

phuong trinh: (x - a)^ + (y - b)^ + (z - c)^ = R^
Mat cau (S) di qua diem O, A nen c6:

ro

/g

+b^

.c

14156
27

+{b-2f

fa

a + 2b-2c + 5

|a + 2 b - 2 c - 1 3

V+2^+2'
-1
o a + 2b-2c = 4 (3).

A. Theo chUcrng trinh chuan

T a g i a i h a ( l ) , (2) v a (3)

w.

(2)

M3t cau Hep xuc vol (P) va ( Q ) O d(l;(P)) = d ( l ; ( Q ) )

I I . P H A N R I E N G T h i sinh chi dugfc chpn lam mpt trong hai phan (phan A
hoac B)

ww

o 5 a + 2b + c = 15 ( l )

+{c-lf

d[{P),(Q)] <^a2+b2+c2=9

ce

Vay, (a;b) = ( l ; 3 ) , ( 3 ; l ) thi minP = 14156
27

={a-5f

9i C O , (P) va ( Q ) song song nen duong kinh ciia mat cau ( S ) la

om

, suy ra minP =

DA.DC = 0

Ta
up

s/

t>y

ok

27

C(4;2)

Cau S.a: Gia su mat cau ta dang d i tim c6 tam I(a; b; c) va ban kinh la R, nen c6

bo

V 3 y

X C = 2 X M - X A = 4

A D 1 DC

f'(t) = 3 6 t 3 - 4 8 t 2 - 2 2 t + 48 va f ' ( t ) > 0 vai V t > — , suy ra f ( t ) luon dong

3
10
bien tren nua khoang
-;+oo
14156

yi=-

D e d : x - y - 2 = 0=> D(x; x - 2)

-2i

Xethamso f ( t ) = 9 t ' ' - 1 6 t 3 - l l t ^ + 4 8 t - 3 2 v o i

^10^

2

,'^c=2yM-yA=2

P = 9t''-16t3-llt2+48t-32

P>f

3
'

Do ABCD la hinh chii nhat nen ta c61 la trung diem ciia AC.

n2

^a b^^
, —+ — -2
^b a

^

<=>

a + 2b-2c + 5 = a + 2 b - 2 c - 1 3
a + 2b-2c + 5 = -(a + 2 b - 2 c - 1 3 )

Cau 7.a:
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).
Vi D thuoc duang thSng x - y - 2 = 0 m a C thoa man phuong trinh nay.

Cau9.a: Gia su z = x + y i , ( x , y e

O

X'

.

+12i = z <=>(x + y i ) ^ + 12i = x - y i

- 3 x y 2 + ( 3 x 2 y - y ^ + 1 2 i = x-yi<=>

Do X > 0 r:i> ( l )

x3-3xy2=x

(1)

3x2y-y%12 = -y

(2)

x^ = 3y^ + 1 . The vao (2) ta dupe

3(3y^ + l j y - y ^ + 1 2 = - y <»2y3 + y + 3 = 0

(3)


Cty TNHH MTV DWH Khang Vi?t

Tuyen 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

Voi, t = — =^ I

Vay z = 2 - i => |zj = N/S

Cau 7.b: (A) C6 phuong trinhy = x + b,he M . Phuong trinh giao diem cua (A)

10

5

Cau 9.b:

3x^ + bx + 2b^ -10 = 0 (l)

^ ( ' ' M - X N ) ' = y hay (x,, +x^)^ -4x^.x^ = ^ (2)
4b

A^^-i^

_

-2=0

- 4 2b2-10 = 16
— <»b^ =9=>b = -3 hoac b = 3.

x =0

y=l

hoac

.c

om

Cau 8.b The tich khoi non la V = - S . h , trong do h = d(l;(P)), day la duong
•
3

ok
bo
ce

= 27-h^ ^ V = - f 2 7 - h 2 ) h

fa

A p dyng bat dang thuc trung binh CQng, trung binh nhan:

ww

w.

(27-h^j + (27-h^j + 2h2 >3^(27-h^)^2h2 hay ( 2 7 - h 2 ) h < 5 4
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-2f.(z.4f
74

=27

<=>

' 2 ^ = 2
x + y -1 = 0

1
x=—
2
41
y=±

2N/X^V =2

y=0

OETHITHijfs6l2

up

ro

/g

Vay, CO hai duong thang thoa man yeu cau de bai la y = x + 3, y = x - 3

tron giao tuyen ciia mat phang voi mat cau.

2x-l = 0

x =l

Ta

2b2-10

hoac

(3)

s/

Ap dung dinh ly Vi-et cho phuong trinh ( l ) : •

= r^ - h^ =^

2x2 + 2 x y - 3 x - y + l=0

(2x-l)(x-l) + y(2x-l) = 0

MN = l ^ c . M N ^ = f o ( y , - y , f . ( x , - x , f = f

Taco:

10 \
= 27
11

2x^ + 2 x y - 3 x - y + l=0

y =x+ b

Tu (2) va (3) suy ra:

Z--

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vaa (E) la :

14
y+ —

29
X-11

B. Theo chiforng trinh nang cao
y=X+b

29 _14 .10 phuong trinh mat cau ( S ) :
11' 11 '11.

I. PHAN CHUNG CHO TAT CA CAC THI SINH

Cau 1: Cho ham so y = x"* - x^ +1, c6 do thj la ( C ) .
a) Khao sat su bien thien va ve do thj (C) cua ham so .
b) Tim tren do thj (C) nhCrng diem A sao cho tiep tuyen tai A cat (C) tai
hai diem B, C khac A va B, C nSm ve 2 phia doi voi A.
7
3(cOtX + l)
r(
771^
,.
Cau 2: Giai phuong trinh: 3cot x + -^^
^-4V2cos x + —
=1
(1)
4 j
smx
Cau 3: Giai phuong trinh:
(13 - 4x) 7 2 ^

+ (4x - 3) VS - 2x = 2 + 8N/-4X2 + 16X - 1 5 1

ich phan: I =
Cau 4: Tinh tich

V

(2),

dx

p

7===

O ^ + N/X + N/X + I

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 lula hinh chieu vuong goc cua A len SB va SC. Tinh the tich cua hinh chop
A.BCNMva cosin goc giira'MN va AB.
75


>^iy 1 r v n n ivit v uv

Tuye'n chtfn b Giai thi?u dethi Todtt hgc-Nguyen

Phu Khdnh , Nguyen Tat Thu.

PHAN CHUNG CHO TAT CA CAC THI SINH

a^b^
12
Tim gia tri nho nhat ciia bieu thuc: P = —
+ — + 12ab
a^ + b^ ab
I I . PHAN RIENG Thi sinh chi dugc chpn lam mpt trong hai phan (phan A
hoac B)

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:

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A. Theo chUtfng trinh chuan
Cau 7,a: Trong mat phSng Oxy, cho tarn giac ABC vuong tgi C, biet A ( 3 ; 0 ) ,

y = (4a3-2a)(x-a) + a''-a2+l

dinh C thuoc true tung, diem B nam tren duong thSng A: 4x + 3y - 1 2 = 0. Tim

Phuong trinh hoanh do giao diem cua (d) va (C) la:

tpa do trpng tam tam giac ABC, biet dien tich tam giac ABC bang 6.
Cau 8.a Trong i
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

B(2;3;-2), C(0;4;-3) va mat cau (S): ( x - l f + ( y - 2 f + ( z - 3 f =25. Gpi mat

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

phang (P) di qua A va cat (S) theo giao tuyen la mpt duong tron c6 ban kinh

Theo bai toan thi g ( x ) = 0 c6 2 nghiem phan bift x ^ X j sao cho: X j
nho nha't. Lap phuong trinh duong thang (d) d i qua B nam trong mat phSng

A' = - 2 a ^ + a > 0

s/

Cau 2: Dieu kien sinx

ro

/g

Z 2 , biet rSng: Z j - izj + Z j =2

6

0<=> x k 7 t , k € Z .

om

cosx + 1

o 3 cos'><+3jinx__4^cos
smx
sin^ x

.c

B. Theo chi/tfng trinh nang cao

„
1
<=>0
„
3(cotx + l )
r
771
=1
3cot2x + - ^ ^ :
^ - 4 V 2 cos x + smx

up

(lW3i)
y . Tim tap hp-p diem bieu dien cho so phuc
16(1+ i)

(xi-a)(x2-a)<0

Ta

2
3

Cau 9.a: Cho so phuc Z j =

rT^i

Hl/OfNGOANGlAl

Cau 6: Cho a, b > 0 thoa man a + b - ab < 0 .

(P) va each C mpt khoang bang

v i i i\mmg

ok

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 di qua A cat (C)

bo

o3

ce

tai M , N sao cho khoang each t u M den A I bang mpt nua khoang each t u N
den A I .

4

=1

cos X „ e o s x + sinx - 4 ( s i n x + eosx) = l
+3
sin^ X
sin^ X

w.

fa

> 3cos^ X + 3(sin X + cosx) - 4(sin x + cosx)sin^ x = sin^ x

Cau 8.b Trong khong gian voi h? true tpa dp Oxyz, cho cho mat phSng ( p ) :

ww

> (sinX + c o s x ) | 3 4 s i n ^ x j + 3eos^ x - s i n ^ x = 0

x - 2 y + 2z + 6 = 0 va cac d i e m A ( - 1 ; 2 ; 3 ) , B(3;0;-1), C(1;4;7). Tim diem M

| > (sin X + cosx)(3 - 4sin^ x) + 3(1 - sin^ x] - sin^ x = 0

thupc (P) sao cho M A ^ + M B ^ + M C ^ nho nhat.
> (sinX + cosx)(3 - 4sin^ x) + 3 - 4sin^ x = 0
( l + 73iV^(2-i)
Cau 9.b: Tim tat ca cac so thuc b, c sao cho so'phuc ^
la nghiem
(i-75i)'(,.i)'

cua phuong trinh: z^ + 8bz + 64c = 0.

> ^3 - 4 sin^ x j(sin x + cos x +1) = 0
I

3-4sin^x = 0
sinx +cosx+ 1 = 0

T H I : 3-4sin2x = 0 0 3 - 2 ( 1 - c o s 2 x ) = 0c>2cos2x = - l « c o s 2 x = - ^
77

7/;


3

2x =

3

(keZ)
+ k2n

X =

N/2+1

= -!<=> sin

^

Khi do: I = 2

71^
X + —

4

V

x = — + k27t
2
X = 7t + k27t

, -

Sai lam :

1 + N/X+N/X + 1

2v;^

Cach i.-Dat Vi

w

x = - — + k7t, x = —+ k7r, X = - — + k27t,k G .

2

5

C a u 3: Dieu kien: - < x < -

s/

V5-2X

up

|(2x-3).|(5-2x)

ro

|(2x-3)4(5-2x) V2x-3

Ta

2
2
( 2 ) 0 (13 - 4x)V2x - 3 + (4x - 3) VS - 2x = 2 + 8J(2x - 3)(5 - 2x)

/g

= 2 + 8 ^ ( 2 x - 3 ) ( 5 - 2 x ) (*)

j u + (7u^ + 3v^) V

=

2(2 + Suv)

ok

(Su^ +

.c

om

Dat u = N / 2 X - 3 > 0, V = \/5-2x >0. Khi do phuong trinh (*) bien doi thanh

4

Vs

75

SB

5

ce

Luc do phuang trinh (•*)

thanh: t^ - 4t^ +1 + 6 = 0
(t + l)(t - 2)(t - 3) = 0 <=> t = 2 thoa dieu ki^n.
Vait.= 2 hay ^(2x - 3)(5 - 2x) = 1 <^ [2(x - 2)]^ = 0 <» x = 2
Vay, phuang trinh cho c6 nghiem duy nhat x = 2

tro

I

BC1

(SAB)

1 -

t^-t + i '

dt

1
1
3-V2-ln(V2+l'
2L

1
2t^J

2

^1

SM SN SM 1 ..^

V " " S B ' S C " SB'2 ' '

3

"

5

=> B C l A

M ^

A M

1

(SBC)

=> A M 1 ( B C N M ) => A M

la duong cao hinh chop
1
= -^AM.Sp
BC
SC

SsBc - S s N M = ^ | § ( S B S C - S M S N )

Ta c6: SB = aVs,

AC = 2a => SC = 2aV2, A M =

AC = AS = 2a, ASAC vuong can tai A
N la hinh chieu cua A len SC => N la trung diem SC

C h i i y : 13-4x = 3 + 2 ( 5 - 2 x ) , 4 x - 3 = 3 + 2(2x-3)
Khi do phuang trinh (*) bien doi thanh |3 + 2u^jv + (3 + 2v^ j u = 2 + 8uv

V2+1

Cdch 2; Ta CO

SBCNM =

t^ - 2

1

4 = > V , = | V (2)

V-2^AABC .SA = -

(**)

ww

3(u + v)(2 - uv) + 7uv(u + v) = 16uv + 4

V

5

sin BSC = sin MSN =

fa

- uvj + 7uv(u + v) = 16uv + 4

Lai dat t = u + v, V2 < t < 2 => uv = -Y—.

SM

w.

<=> 3(u + v)^u^ +

4a

V

h-

w
w
w
,r
= V s ^ N ' V 2 = VA.BCMN'V = VS^BC'

2

bo

o 3 ^ u ^ + v^j + 7uv(u + v) = 16uv + 4

78

1+ V x - V ^

Vay, nghiem phuang trinh cho la:

3

t

1

Cau 5:

dt

t^(t + l)
1

t-ln

2

/,
(keZ)

Ket hgp dieu kien =>x = - — + k27t, k e Z

3

t3U.,\
^(t.l)

t^-l

t^+1

1-

^ 1 1 1

4

3

.dx =

2t ,

. Thoa man dieu kien.

4

X + — = 71 + — 4- k27t

4

t-1

x=

+ kTT

x + — = — + k27i
7t

(k e

71
X + —

TH2: sinx + cosx + l = 0<=>N/2sin

4: Dat t = >/x + Vx + 1

O

3

3

7t

k7l

X = — +

k27t

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2x = — +

1 BC
^BCNM

2SC

SBSC-SM

SC

= iBC SB--SM
2

2

SN = aN/2,SM =

4aS

Sa^N/is
10

79


^ A . B C N M - 3''^^•SBCNM -

r nho nhat khi I H Ian nhat, ma I H < l A => I H Ion nhat khi va chi khi H = A ,

—

j^hi do l A 1 (P) => (P) CO vecta phap tuyen n = l A = ( l ; l ; - 2)

M P / / A B , M P n S A = P => ( M N , A B ) = ( M N , M P ) = N M P
. AB a
SA
va MP =
= —, SP =
=a
2
2
2
ASMN cp:
M N ^ = SM^ + SN^ -2SM.SN.COSMSN = SM^ + SN^ - 2 S M . S N . — = —
SC
5

Gpi u = (a;b;c) la vecto chi phuong ciia d, theo de bai ta c6 h?:
"•" = 0

a^b^

s/

Ta
/g

ro

up

Xet ham so f (t) = —^
+ — + 12t vol t > 4
t2-2t
t
I I . PHAN RIENG Thi sinh chi dupe chpn lam mpt trong hai phan (phan A
hoac B)

om

A. Theo chUOng trinh chuan

.c
ok
bo
ce

w.

= r A B . d ( C ; A B ) , trong do: AB = 5|b-l|, d(C;AB) =

Theo bai toan, ta c6: - . 5 b - 1
2

ww

SAABC

3c-12

a = -b=>a = l , b - - l , c - 0
a = — b = > a = 43,b = 5,c = 24
5

Gia su Z 2 =

y i , (x,y e

bieu dien boi diem M ( x ; y ) . K h i do ta c6:

x + y i - i ( l - i ) + l + i = 2<=> x + yi = 2 < » x ^ + y ^ = 4
Vay tap hp-p diem bieu dien cho so'phuc

la duong tron tam O, ban kinh 2.

B. Theo chUtfng trinh nang cao
Cau 7.b:

Cdch 1: ^i^i(c) = A M . A N - A I ^ -

=

466 > 0,

Hon nira
PA/(C) =

3c-12

= 6=> ( b - l ) ( c - 4 ) = 4 (2)

T u ( l ) va (2) suy ra:
Cau 8.a Gpi I la tam mat cau (S), l A = ( l ; l ; - 2 ) = : > I A = N / 5 < R = > A nam tron

2 A M ^ = 2 M N 2 = 466

=>MN =

V233

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

m|t cau. Vi the (P) cat mat cau (S) theo duong tron c6 ban kinh r, trong d

Vi M la trung diem ciia A N nen ta c6: N ( 2 X + l;2y -14)

r=

Diem N thuoc duong tron nen ta c6: (2x)^ + (2y - 9)^ = 169.

80

- I H ^ va H la hinh chieu ciia I len mat phSng ( P ) .

=l-i

suy ra A nam ngoai duong tron.

fa

AC.BC = 0 <r>9b + 4bc + c^-4c = 0 ( l )

~2

( U i f . ( l * i f ( l + i) = ( 2 i f ( U i ) = - 4 ( l . i ) ^ z , = - j ^ = i < ^

12
12
+ — + 12ab hay P > —
+ — + 12t
a^b^-2ab ab
t^-2t
t

Gia thiet tam giac ABC vuong tai C ta c6:

2

_3

Cau 9.a: Ta c6 ( l + Vsi)^ = 1 + Sx/Si + 3.3i^ + 3>/3i^ = -8

Hon nua a^b^-2ab >a^+b^ >2ab=> ab> 4 . Dat t = a b , t a c 6 t > 4

Taco: AC = {-3;c),BC = (-3b;4b + c - 4 )

a^+b^+c^

a^+b^+c^

Cau 6: T u dieu kien bai toan ta c6 (a + b)^ < a^b^ o a^ + b^ < a^b^ - 2ab

Cau 7.a: Gia su rSng: B (3b; -4b + 4), C (O; c).

41

a+b
c =2
(b + c f + ( a - 2 c ) ^ + ( a + 2b)^ ^ 3

ASPN c6: NP2 = SP^ + SN^ - 2SP.SN.cos ASC = SP^ + SN^ - 2SP.SN.— = a^
SC
2V3O
AMPN c6: NP^ = N M ' ' + MP^ - 2MN.MP.cosPMN
cosPMN =

40

Ta c6: P >

[a + b - 2 c = 0
(b + c) + ( a - 2 c ) +(a + 2b)

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d(C;(d)) =

CB,u

81


C t y TNHH

Ta

CO

h#:

+ 5f =169

{x-lf+{y

, ^ ,

. ,

Tau 2: Giai h? phuong trmn:
^

( 2 x ) ^ + ( 2 y - 9 f =169

I ^2x + y + 5 x - y

GA^ + GB^ + GC^ C O dinh, v i the MA^ +MB^ +MC^ nho nhat khi M G c6 do

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Gpi (d) la duong thang di qua G va vuong goc voi (P)

Cau 5: Cho lang try dung ABC.A'B'C c6 day ABC la tam giac vuong tai A,
BC = 3a, A A ' = a va goc giira A'B voi mat phSng trung true doan BC bSng 30".

Nen C O phuong trinh : - — - = ^ — - = - — - .
^
^
1 - 2
2

Tinh theo a the tich khoi lang try ABC.A'B'C va khoang each giua hai duong

Toa do diem M la nghiem ciia h? phuong trinh:

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

x=0
y = 4=>M(0;4;1).

1
-2
2
x - 2 y + 2z + 6 = 0

•

c=5

.c
bo

ww

w.

fa

b+c-3=0

ce

<:*(l + 2i)^ + b ( l + 2i) + c = 0 o ( 2 b + 4)i + b + c - 3 = 0

ok

Theo gia thiet ta c6 (8 + 16i)^ + 8b(8 + 16i) + 64c = 0

b = -2

z

Ta
s/
up
ro

-

,
,
- 8 ( l + 2i) = 8 + 16i

om

3

8(2-i)

/g

(-8)'(2-i)

2b + 4 = 0

V

+ ——+
.
x+2y y+z x+z
I I . PHAN RIENG T h i sinh chi dugc chpn lam mpt trong hai phan (phan A
ho$c B)

+ 3.3i2 -3^31^ = -8, ( l + if = 2i

(l + V 3 i ) ' ' ( 2 - i )
Do do ^
6

X

nhat va gia trj nho nhat ciia bieu thuc P =

z=l

Cau 9.b: Ta c6 ( l + \/3i)^ = I + SN/SI + 3.3i^ + 3^31^ = - 8

( l - S i f =l-3Si

=8

i o i + (x^ - x ) ^ x - 2
Cau 4: Tinh tich phan: I = f
^
dx
3
x^-3x + 2

dai nho nhat, nghia la MG 1 (P) hay M la hinh chie'u cua G len (P).

z-3

„

=0

Cau 3: Giai he phuong trmh: <

tim dugc G(1;2;3) va MA^ + MB^ + MC^ = SMG^ + GA^ + GB^ + GC^. Ta thay

y-2

1 +cotx

Vict

f 7 2 x + y - J5x + 3y = - 2

Cau 8.b: Gpi G la trpng tam cua tam giac ABC, ta c6: GA + GB + GC = 0, tir day

x-1

2sm X + cos2x - smx

MTV DWH Khang

OETHITHiJSOlS

A. Theo chUorng trinh chuan

Cau 7.a: Trong mat phang Oxy, cho cac duong tron ( C j ) : (x - 1 ) ^ + y^ = ^ va
(Cj):

( x - 2 ) ^ + ( y - 2 ) ^ = 2 . Viet phuong trinh duong thang d tiep xiic voi

duong tron ( C j ) 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

.

I. PHAN CHUNG CHO TAT CA CAC THI SINH

B. Theo chUorng trinh nang cao

Cau 1: Cho ham so y =

c6 do thi la (C)

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 d i qua M ( - 6 ; 3 ) va cat duong tron

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).
82

(C) tai hai diem phan bi^t A, B sao cho tam giac lAB c6 di$n rich bSng

va

AB>2.
83


Tuye'n

chgn

& Giai

thifu

dithi

To&n

hQC-Nguyen

PUu Khunh

, N\;iii/rn

I'at Thu.

C a u 8.b: T r o n g k h o n g gian tpa dp O x y z , cho hai d i e m A ( 3 ; - 2 ; 3 ) , B ( - 5 ; 1 0 ; - 1 )
va m a t p h a n g ( 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 =

TH3:

biet d :

c-2 = -

_ y+ 1_ z-1

2 ~ " ^ ~

^

TH4:

2

b-2 =

— = .^-iZi.

C a u 9.b: Cho so phuc z thoa m a n : z

1 + 31

5

HUdfNGDANGlAl

o l .

Ta
up

s/

K(b;l).

ro

Xet h a i t a m giac A H C va B K A ta c6:

/g

B K A = A H C = 90°

fa

THI: I
b-2 =
c-2

o

b=c

b-2= -

+ mn

yj2x + y- 75x + 3y = - 2

+ y + 5x + 3y - 4y = 8

Dat u = yJ2x + y > 0, v = ^ ^ x + Sy > 0=> y = 2 v ^ - 5 u ^
i do, h ^ cho t r o t h a n h :
u +2=v

u - v = -2
2C..2'

" u +v2-4(2v^-5u2j

=8

u + (u + 2 ) ^ - 4

2(u + 2 f - 5 u 2

(*)<»13u2 - 2 7 u - 3 6 = 0 « u = 3 = > v = 5

{c-lY

Ta

CO

he: <

( k h o n g thoa)

J2x + y = 3

f2x + y = 9

fx = 2

px

[5x + 3y - 2D

[y = 5

+ 3y = 5

2b
b-1
o
2c
c-1

10

iu4:I=

c-1

= -

4

Vay, h? cho c6 n g h i ^ m (x; y ) = (2; 5)

2c

TH2:

84

ww

2b
b-1

-71
—

2

(b-2f=

w.

c-2 =

^

^J2x

ok

ce

(b-ir
o
X

( k h o n g thoa)
b = -2,c = |

c-1

C a u 3: H # p h u o n g t r i n h cho viet lai:

.c
BK
~

c-1

2c

bo

/

n\ 4 b
{c-2) =
Suy ra A A H C = A B K A => j

b-2 =

b = - , c = -2

Do'i chieu d i e u k i e n ta dupe: x = — + k n hoac x = — + k27t.
4
'
2

om

CAH

AB = AC

IAH

<

b-1

„
.
,
7t kjt , „
1 ^

hoac s m x = 1 <=>x = — + — hoac x = — + k2Tt.
4
2
2

G p i H , K la h i n h chieu cua C va B tren d u a n g thang y = 1.

c u n g p h u goc

<=>

2b

b = 3,c = - l

,

a) D a n h cho ban dpc.
(u 3 b - l '
,
, b < l v a C c;
b) G p i B o;
I b - 1 ,
I c - l j

KAB = HCA

c-2 = -

b =-l,c = 3=>B(-l;2),C(3;4)

| 2 s i n ^ x - l s i n x + c o s 2 x = 0 <=> c o s 2 x = 0

C a u 1:

•

b-1

2b

b-1
<=> i
2c
b-2 = c-1

P h u o n g t r i n h cho t u o n g d u o n g v o i

I . 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

Suy ra H ( c ; l ) v a

c-2 =

2b
2c

Cau 2: D i e u k i ^ n :

T i m p h a n t h y c cua so'phuc z^°^^ .

•

b-1
2c
b-2 = c-1

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X

2b

c-2 =

10

b = c ( k h o n g thoa)

f——!
dx+f

dx = A + B
7
jJ x.,2
2 - 3 1x. . +. 12
i•' xY -- 2
,

10^

^

J ^

dx = ( l n | x - 2 | - l n | x - l | )
A = ] ^
dx= f
j x ^ - 3 x +2
jvx-2 x - 1 .
, 8
, „ , 1 6
= l n - + ln2 = ln —
9
9

10

= 8 (*)


Tuyen chpn & Giai thifu de thi Todn hqc - Nguyen Phii Khdnh , Nguyin TA't Thu.

B =

x-2


3t^

3^+6

69

+ 6t

Cty TNHHMIV

I I . P H A N R I E N G T h i s i n h c h i dugic c h p n l a m m g t t r o n g h a i p h a n ( p h a n A
hoac B)

f^. Theo chi/orng trinh chuan

true tpa d o n h u h i n h ve

C a u 7.a: ( C j ) - c6 tarn I | ( l ; 0 ) va ban k i n h R j =

A(O;O;O);A'(O;O;a);B(m;0;O);C(0;n;O)

iL
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Taco

Trong do m = AB, n = A C .

( C 2 ) CO tarn 12(2; 2) va ban k i n h

Tarn giac A B C v u o n g tai A , BC = 3a =>
BC = ( - m ; n ; 0 )

+ n ^ = 9a^

la VTPT ciia m p

d qua I2 (2; 2 ) , c6 vecta phap tuyen n (a; b) ^ 0 c6 p h u o n g t r i n h :
a(x-2) + b(y-2) = 0

Ta lap d u g c p h u a n g t r i n h sau

/

s/

Ta

1
4 9a
= - <=> m =

up

^

ro

f(x,y,z) = — ^ + ^ ^ +

/g
om
.c

ok

^ v ^ _ Z _ ^ > 0

(y + z)(z + x) Vx + ^y

ce

bo

ethi#u: f(x,y,z)-f x,y,7xy

Ta xet

fa

x

-

Vy

ww

y

w.

y

K h i d o P = f(x,y,z)>

V i X = m a x { x ; y ; z | nen dat t =

Vy

voi l < t < 3

Xet f ( t ) = - ^ — + —
voi t6ri;3'
^'
t2+2

t+1
L J
5:
CO

X

/

X

dtiepxuc (Cj) khi d(li;d) -

f(x,y,^) =

Ta

= >/2

vatiepxiic ( C i ) .

A ^ = ( m ; 0 ; - a ) la V T C P ciia A ' B

-m

v2

Gia s u d la d u o n g thang can t i m va d cat ( C 2 ) tai A , B nen d q u a l 2 ( 2 ; 2 )

t r u n g true doan BC

C a u 6 : Dat

Vift

7
4
Vay, m i n P = —, m a x P = —
6
3

. D o i b i e h v o i t = \/x-2

, , 16 69
Vay, I = l n — + —
^
9
4
C a u 5: C h o n

I)VVHKhang

-2ft3-2)(t-2)
f'(t) = -^^
va f ( t ) = 0 < ^ t = 2 thoa bai toan
(t^.2f(t.lf

1
o

-

a + 2b
= =

1

=-=

~

,

<=> a^ + Sab + 7b^ = 0

<=> (a + b)(a + 7 b ) = 0 <=> a = - b hoac a = - 7 b

V o i a = - b , suy ra d : x - y = 0

V o i a = - 7 b , suy ra d : 7x - y - 1 2 = 0
Vay, CO 2 d u o n g thang can t i m : x - y = 0, 7 x - y - 1 2 = 0
C a u 8.a: G o i M ' la giao d i e m cua d va A, suy ra M ' ( 2 - t ; l - 2 t ; l + 1 )
=i.MM' = (3-t;2-2t;3 + t).
Gpi I la t a m cua m a t cau. Ta suy ra tpa d o d i e m I ( - 1 ; 2; l ) , p h u o n g t r i n h 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 d i e m ciia A v o i mat cau, neu goi J la t r u n g d i e m cvia A B t h i ta
CO OJ v u o n g goc v o i A B va OJ = 3, h o n nua I M = Syjl
Hay d ( M ; ( P ) ) = 3 ^

=> J M = 3

. ^^'^^
^ 3 o | t - 5 | = \ / 6 t ^ - 8 t + 22
V 6 t ^ - 8 t + 22

<=> t = - 1 hoac t = —.
5
P h u o n g t r i n h d u o n g thang can t i m d i qua 2 d i e m M va M '
^au9.a:

z^-2>/2z + 8 = 0 c6 A' = - 6 = (>/6i)^

P h u o n g t r i n h cho c6 2 n g h i e m phuc la: Z j 2 = V2 ± \f6i
87


Tuyen chgn b Gioi thifu Ai thi Toan htfc - Nguyen PM Khdnh , Nguyen Tai

Cty TNHH MTV DWH

Thu.

Phuong trinh duong t h i n g AB la:

= 2N/2 + 3.2.76.i + 3.V2.(V6i)^ + (76i)^ = -I6V2,

(V2 +

Khang Vift

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 ( 2 t ; - l - t ; l + 2t)

,1671

A = MN=J>u^ = M N = (2t + 3 ; - t - 8 ; 2 t + l )

iL
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, 2013 , ,2013
Zj
+Z2

cos(p = -

Cdch l.-Gpi H la hinh chieu ciia I tren AB, suy ra Gpi H la trung diem cua AB

hay AB = 2 A H . Dat A H = x ,(0 < x < 3).

^2^

SAIAR = T I A . A B

'AIAB

2

= -h-x^2x
2

o x" -

Ta

a . b i - i ^ = ^

/g

= 8 => I H = 2^2
4

2

3
AIB
2

IH
"lA

4N/T4

, ket hgp gia thie't suy ra M A + MB = A B .

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).
88

l + 3i

5

10

5

o 10a + lObi - a + 3b + (b + 3a)i = 12 + 14i o 9a + 3b + (lib + 3a)i = 12 + 14i
-1IOO6
9a + 3b = 12
' ^ U z = l + i^z20i2 (1+i)'
= ( 2.aoo6
i ) ^ " * = - 2,1006
b= l
l i b + 3a = 14
Vay, phan thvc cua z^°^^ la -2^°''^

ok

2

Ket h(?p gia thuyet suy ra: c o s — = - = > d ( l ; AB) = l A . c o s — = 1
2
3
2
7a-4b
hay , ,
= 1 « . 483^ - 56ab + ISb^ =0 o (4a - 3b)(l2a - 5b) - 0
Va^ +

Cau S.b: De tha'y, AB =

^

a.bi-fe::^lKli^=^

1
= 3

w.

De tha'y, cos

9

bo

2^/2

AIB

fa

AIB

=> cos

ww

hoac cos

4%/2

ce

=^IA.IB.sinAIB = -5-R^sinAIB, sinAIB =

.c

om

Gpi H la hinh chieu cua I tren AB thi I H ^ = lA^ SAIB

up

ro

a^+h^>0.

s/

nhan. Suy ra I H = 1

Cach 2: (C) c6 tarn I ( l ; - l ) , ban kinh R = 3. Duong th^ng A d i qua M c6 dgng:
a(x + 6) + b ( y - 3 ) = 0,

(2t + 3)^+(t + 8)'+(2t + l ) ^ 3

6 + 7i
Cau 9,b: Cho so phuc z thoa man : z - (1)
l + 3i
5
Gpi so phuc z = a + bi (a, b € i?) => z = a - bi thay vao ( l ) , ta dupe:

+8=0

Voi X = 1 => AB = 2 khong thoa.
A B = 4>/2

_2VlO

o t = -4 ho|ic t = —

<=>x = l hogc x = 272
Voi X = 2N/2

4t + 6 +1 + 8 + 4t + 2

2V10

B. Theo chi/crng trinh nang cao
Cau 7.b:

DETHITHUfSOH
I. PHAN CHUNG CHO TAT CA CAC THI SINH
Cau 1: Cho ham so y = x^ -3x2 + 2, c6
jj^j ^Q'^
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 p h i n g gioi han boi do thj (C) va

duong thSng NP. Tinh ti so

S
Sj
89


Tuyen

Todn HQC - Nguyen

chgn & Gidi thifu dethi

Phu Khdnh

, Nguyen

Cty TNHH

Tat Thu.

Cau 2: Giai phuong trinh: sinx + Vscosx + 2 = v2 + cos2x + V3sin2x

Vift

Cau 9.b: Giai phuong trinh sau tren t^p so phuc C: z * - z ^ + ^ + z + l = 0

Cau 3: Giai phuong trinh: (2x - 4 ) V 3 x - 2 + Vx + S -yjsx^ + 7 x - 6 = 5x - 7

HMG

Tax-' +lnfx +Vx^ +1
Cau 4: Tinh tich phan: I = J
^
^dx
0

MTV DWH Khang

DANGIAI

I. P H A N 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

Vx^+1

fx =1+X
b) Thvrc hi?n phep bien doi phep tinh tien • ^ . Trong h f true mdi duong

SAB = SBC = 30".

cong (C) CO phuong trinh Y =

Cau 6: Cho 0 < a < b < c < l . Tim gia trj Ion nha't cua bieu thuc :

Xj^ = m 5" 0, khong mat tinh tong quat c6 the coi m > 0

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Cau 5: Tinh the tich khoi chop SABC biet AB = AC = a, BC = ^ a , SA = a^Is va

P = (a2-b2)(b-c) + c2(l-c)

II. PHAN R I E N G T h i sinh chi dupe chpn lam mpt trong hai phan (phan A
hoacB)

Tiep tuye'n tai M c6 phuong trinh:
y = (Sm^ - 3 j ( X - m ) + (m^ - 3m) = (3m^ - 3)x - 2m^

A. Theo chUorng trinh chuan

Hoanh dp giao diem N ciia tiep tuye'n tai M va (C) la nghiem cua phuong

Cau 7.a: Trong mat phang tpa dp Oxy, cho hinh vuong ABCD, c6 canh AB di

trinh: X^ - 3X = (sm^ -3^X-2m^

s/

khi duong tron (C) :(x - 2^ + (y - 3)^ = 10 npi tiep ABCD.

Tiep tuye'n tai N c6 hoanh dp X = - 2 m c6 phuong trinh:
y = (l2m2 - 3 ) ( X + 2m) + (-8m^ + 6m) = (l2m^ -3)x + 16m^

up
ro

om

/g

(Q): x + 2 y - 2 z + 2 = 0 va duong thang (d) c6 phuong trinh ^^=-^=-^Y~-

.c

Lap phuong trinh mat phang (P) chua duong thang (d) va tao voi mat

Hoanh dp giao diem P cua tiep tuyen tai N va (C) la nghiem cua phuong
trinh: X^-3X=(l2m^-3)x + 16m^ o ( X + 2m)^(X-4m) = 0<=^X-4m v i X ^ - 2 m
Di^n tich hinh phSng giai han boi do thj (C) va duong thSng M N la:
x 3 - 3 X - ( 3 m 2 - 3 ) x + 2m3^X=

bo

ok

phang (Q) mpt goc a voi sin a = J - .

ce

V6

fa

w.

ww

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

J ( X - m ) ' ( X + 2m)|dX

-2n«

Cau 9.a: Tim so phuc z bie't 2^z + l j + z - l = ( l - i ) z ^ .
B. Theo chUtfng trinh nang cao

(X + 2m) = 0 « X = - 2 m vi X ^ m .

c^{X-mf

Ta

qua diem M ( - 3 ; - 2 ) , va x^ > 0 . Tim tpa dp cac dinh cua hinh vuong ABCD

Cau 8.a: Trong khong gian voi he tpa dp Oxyz, cho mat phang

- 3X. M la diem thupc (C) c6 hoanh dp

[ ( X - m ) ^ + 3 m ( X - m ) ^ dX =

I

(X-m)^

27m^

+ m(X-m)^

|-2m"

-2m

I

i#n tich hinh phSng gioi han boi do thj (C) va duong th5ng NP la

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.
90

^2

x3-3X-(l2m2-3)x-16m3dX=

-2 m

j (X + 2m)

(X-4m)|dX

-2 m'
i4m

J (X + 2m) - 6 m ( X + 2m)

{X + 2m)''
,

^ .3
^ + 2m(X + 2m)
dX = •i^

= 108m*
Jl-2tii

91


Tuye'tt chQtt b Giai thifu de thi Todn hgc - Nguyen

PUu

I id

Cty TNHH MTV DWH Khang Vift

I'hu.

o2cos X

I/

<=>2cos X

I

7t

>

+2

1

+2

4 cos

+ COS

6,
7t

>

6,

X

v2

6

^

Ta
s/
up

ro
/g

ce

<»3(x-l)(>/)rH3 + V3x-2)(V3x-2-N/x + 3 - l ) = 0 o x = l hoac x = 6

^

x3

^

fa

16

.HM =

7i

4
-

—

32

V,S A B C = 2V....

. . . . J,
j ( b - c) + c^ ( l - c),

Taco f ( 0 , b , c ) = - b 2 ( b - c ) + c 2 ( l - c )

73 In x +Vx^ +•

7x ^ + 1

33^ 3a2

- - B C . H M - ^ a ^ => YsBCH = -SH.SAn(-H
'ABCH
16

Cau 6: Dat f (a,b,c) = (a^ -

w.
dx +

x2

-dx

Xethi?u: f ( a , b , c ) - f ( 0 , b , c ) = a 2 ( b - c ) < 0 v i b < c = > f ( a , b , c ) < f ( 0 , b , c )
Khi do P < f ( 0 , b, c) = - b ^ (b - c) + c^ ( 1 - c).

Xet I] = 0J V^x ^ + 1 dx = 0J Vx^
, +1.xdx

Xet f ( b ) = - b 2 ( b - c ) + c 2 ( l - c )

D^t t = Vx^ +1

Ta c6: f ' ( b ) = -7,\p- + 2bc va f'(b) = 0 <::^ b = 0 hoac b = —

x^ = t^ - 1 va xdx = tdt

Suy ra 1= J - ^ ^ t d t = J ( t 2 - l ] d t =
92

ww

u = 73x-2,u>0 va v = Vx + 3,v>0

SA 1 (BCH)

SM2=SC2-CM2=-^^^
16

^ABCH

0

4 ^ . . . . . . ..

H S = H A => VsBCH = VABCH => VgABC = 2VsBCH

2 ISa^
HM2 = S M 2 - S H 2 16

=0

ok

.c

V3X-2+1

om

2x-5-v^r+3-V3x-2

< » 3 ( x - l ) ( 3 X - 2 ) - ( X + 3)-(NA
,

In^ (2 + N/S)
2

=> S C = a = A C => C H 1 S A

bo

-1 = 0O3{X-1)

dx=

))Q

V

S C ^ = S A ^ + A C ^ - 2SA. A C cos 30° = a^

<»(V3x-2-l)(2x-4-Vx + 3)-(3x-3) = 0

>/3x-' + ln x +Vx^ +;
Cau 4: 1= J —
x^+l
0

In2(2 + V3)

=> S B = S C o S M 1 B C
A B = A C => A M 1 B C

7C^

+ 1 = cos X —
o cos X —
6
6

o(2x-4)(V3x-2-l)-Vx + 3(73x-2-l)-(3x-3) = 0

Chuy-.Dat

d In x +Vx^ +1

S A chung

(2x-4)V3x-2 + > / ^ - ^ ( x + 3)(3x-2) = 5 x - 7

N/3X-2 + 1

x+ 7x^+1

Cau 5: S A B = S A C => A S A B = A S A C

Cau 3: Phuong trinh cho tuong duong voi:

.3(x-l)

7i

dx^

AB =A C

= - - = c o s — < » x = - —+ k27t hoac x = —+ k2n
ocos X —
6
2
3
6
2

2 x - £ - V x + 3_

V:x ^ + l

Vly, I = I i + l 2 = 3+

2x —
3
71

0

j-^r

= lin2rx+>/?7Tn ^

s

1
1 + —cos2x + — s i n 2 x
2
2

Sin—Sin X + cos—cos x + 2 = . 2 1 + COS—cos2x + sin—sin2x
6
6
3
3
/

75 In X + Vx^ + 1

iL

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0 2

,

Xet l2=

S2 lOSm" 16
Cau 2: Phuong trinh cho tuong duong v o i :
1 .
^/3
+ 2=
—sinx + — c o s x
2
2

Khunii

4
3

23
Suyra maxf(b) = f ^ ] = -||c3 + c2 va P < f ( c ) voi f(c) = - ^ c 3 + c 2
V 3 y
27
2/


Suy ra maxf (c) = f

2(z + l ) + z - l = ( l - i ) | z | ^ c;>2(a-bi + l ) + a + b i - l = ( l - i ) ( a 2 +b^'

+ 2c va f (c) = 0 » c = 0 hoSc c = i18|

Ta c6: f'(c) = - y

'18^
.23.

108
529

(3a +1) - bi = a^ + b^ - (a^ + b ^ ) i «

108

12
18
Vay, maxP =
xay ra khi a = 0, b = — , c = — .
529

^
23
23

b = 3a + l

A. Theo chUorng trinh chuan

Vay, CO hai so phuc :

Cau 7.a: Duong tron (C)c6 tam I(2;3), ban kmh R = VlO

^

= i ; z, =
^

la l ( 3 ; - l ) ban ki'nh bang ^ A B =

<=>

10
b = 3a +1
3
1
+ — i
10 10

a=0
lb=i

ok

b = -3a chpn a = 1, b = -3 => A B : x - 3y - 3 = 0, v i A e AB

bo

fa

ce

Hon nua: l A = R72 » l A ^ = 20 o (3a +1)^ + (a - 3)^ = 20 o a = 1 (thoa)

w.

ww

Khi do A (6; l ) , I la trung diem cua AC => C(-2;5)
Cau S.a: Gpi vecto phap tuyen np =(a;b;c)cua mat phling (P)

2

^NAB

S'^

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

x = 2=>y = 0=>N(2;0)

x+y-2=0

x = 4=>y = - 2 = > N ( 4 ; - 2 )

Cau S.b: Gpi ad = ( 2 ; l ; l ) va np = ( l ; 2 ; - l ) la vecto chi phuong ciia (d) va vecto
phap tuyeh ciia (P).
Gpi a la vecto chi phuong cvia (d') nen a = ^ad;np ^ = (-3; 3; 3)
Gpi ( Q ) la mat phSng chua ( d ' ) , song song voi (d)

jGpi M ( l ; - 2 ; 0 ) e ( d ) , t a c 6 : d [ d ; d ' ] = d [ d ; ( Q ) = d M ; ( Q )

Cau 9.a: Gpi z = a + bi

3^a2+(a + c)^

{a,h&R)

C ^

=^

M ( m ; 4 - 4m) va N O . N M = 0, den day ta tim dupe tpa dp M

=>np =(a;a + c;c)
= ; | « ( a - c ) ( 2 a + c) = 0

2
" ^

trung true doan AB voi (C), nen tpa dp N thoa h^:

,nen ng = a;nf

o

2

'o'" "h^'t bang 2 khi N A = NB. N la giao diem ciia duong

Mat phSng (?) chua duong thSng (d) nen c 6 a - b + c = 0<=>b = a + c

3

10
b =i 10

^ .

Ta
s/
/g

.c

om

hoac a = -2 (khong thoa a > 0).

ro

up

nen A(a;7 + 3a) va a >0
o l A ^ = 20 o ( a - 2 f + ( 3 a + 4 f =20 « a = 0

^VBC^+CA^ =

N la diem tuy y tren (C) nen S^AB = ^ N A . N B <

T H I ; a = -3b chpn a = 3, b = - 1 =:i> A B : 3 x - y + 7 = 0, vi A e AB

5
1
sma = — => cos a = —f=
6

a=

hoac

Phuong trinh duong tron (C) ngoai tiep AABC: (x - 3)^ + (y +1)^ = 2

o (a + 3b)(3a + b) = 0 <=> a = -3b hoac b = -3a

ho$c a = - 1 (khong thoa a > 0).

b = 3a + l

a=

Cau 7.b: Nhan thay, A A B C vuong tai C suy ra tam duong tron ngosii tiep A A B C

+ b^ > 0 . Duong tron npi tiep ABCD nen AB tiep xiic voi duong tron (C)
k h i v k c h i k h i d ( l ; A B ) = R o 5a + 5b = >/To « 3 a 2 + 1 0 a b + 3b^ =0

nen A ( 3 + 3a;a) va a > 0

lOa^ + 3a = 0

B. Theo chUorng trinh nang cao

Gpi duong t h i n g AB d i qua M , c6 phuong trinh: a(x + 3) + b ( y + 2) = 0,

OA

<=> \

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3a +1 = a^ + b^

ho|ic B)

TBI',

b = a2+b2
a =0

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

Honnua: l A =

3a + l = a2+b2

= (0;3;-3) =i> ( Q ) : y - z + m = 0

<=> |m + 2| ^ 2 o m = -4 hoac m = 0
^Voi m = 0 = > ( Q ) : y - z = 0
IVoi m = : - 4 = > ( Q ) : y - z - 4 = 0
95


Tuye'n chiftt b Giai thifu dethi Todu hyc - Nguuni

J'hu Kluitih

, Vxiii/. »

Tii't Thu.

r

fx = 3 + t
THI:

(Q):y-z =0 ^ A :

hay A :

y = -t

Cty TNHH MTV DWH Khang Vift

Cau 5: Cho hinh chop S.ABCD c6 day ABCD la hinh vuong canh a. Gpi K la
trung diem cua AB, H la giao diem cua BD voi KC. Hai mat phSng (SKC),

z = -t
TH2: ( Q ) : y - z - 4 = 0 r > A :

( S B D ) ciing vuong goc voi m|t phSng day. Biet goe giiia mat phSng ( S A B ) va

x =7 +t

x+2y-z-3=0
y-z-4=0

ni^t ph5ng ( A B C D ) bMng 60°. Tinh the tich khoi chop S . A B C D va tinh ban

y = -t

hay A :

kinh m^t cau ngoai tiep hinh chop S.ABC.

z = 4-t

Cau 6: Cho a,b,c > 0 thoa man a + b + c = 3 .

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Cau 9.b: Ta c6 z = 0 khong thoa man phtrong trinh

Chung minh rang:

Chia hai ve cua phuong trinh cho z^ (vai z ?t 0) ta c6 phuong trinh:

r,2,

z +

n

1
z —

4

2 2

+

b

+^ >
c

+ b^ + c^

11. PHAN R I E N G T h i sinh chi dvetjic ch^n lam mpt trong hai phan (phan A
ho?c B)

= 0

1
,
1 3
D|itt = z — , phuong trinh tro thanh: 2t^-2t + 5 = 0 o t = - - - i ,

a

1 3
t=- + -

A. Theo chUoTng trinh chul(n

2 2

Cka 7.a: Trong mat phSng tpa dp Oxy, cho duong tron ((^:(x-4) +(y-6) =5

V6it = i - | i ^ z - i =i - | i o z =l - i , z = - i - i i

Diem A(2; 5), B(6; 5) nam tren (C). Dinh C ciia tam giac ABC di dpng tren
duong tron ( C ) . Tim tpa dp tryc tam H cua tam giac ABC biet H nam tren

Vdit = i . | i = > z - i = l . | i o z = - i . i i , z =l . i

up

s/

Ta

duong thang ( d ) : x - y +1 = 0.

/g

ro

OETHITHUfSOlS

.c

-3mx^ + 3(m^ - l ) x - m^ + m ( l ) , m la tham so

ok

Cau 1: Cho ham so y =

om

I. PHAN C H U N G C H O T A T CA C A C THI SINH

a) Khao sat sy bien thien va ve do thi ( l ) ciia ham so khi m = 1 .

Cau 8.a: Trong m|it ph3ng tpa dp Oxyz, cho t u di^n ABCD c6 B ( - l ; 0; 2),

C ( - l ; 1; 0), D(2; 1; -2). Biet r5ng OA eung phuong voi u = (0;0;l) va the tich tu
di?n ABCD b i n g | . L^p phuong trinh m3t cau ngoai tiep t u di?n ABCD va
6
l|p phuong trinh m$t phSng (a) tiep di^n voi m3t cau ngoai tiep t u di?n ABCD
t^i A.

bo

Cau 9.a: Gpi Z j , Z j la nghi?m cua phuong trinh : z^ - 2z + 4 = 0.

ce

b) Tim m de ham so ( l ) c6 eye dai, cue tieu dong thoi thoi khoang each tii

fa

diem eye tieu cua do thj den goe tpa dp O b i n g 3 Ian khoang each t u diem cue

ww

w.

dai cua do thj den O.

2 ( l + sin2x) + cos4x-\/3sin4x
r•inh :
7=
= 2V2 sii
Cau 2: Giai phuong trinh
sin2x-\/3eos2x
(x + yf
Cau 3: Giai h$ phuong trinh:

Cau 4 : Tinh tich phan: I = J-

+ 8xy = 2(x + y)(8 + xy)

1

1

^Jx + y

x^ - y

x2-l

Tinh A=

Zj + 2Z2
2

2

+

+ Z,Z2

Z2

B. Theo chUorng trinh nang cao

x+—
4

Cau 7.b: Trong mSt phSng tpa dp Oxy, cho 2 duong tron ( C ) : x^ + y^ = 9 va
( C ) : x^ + y^ - ISx - 6y + 65 = 0. T u diem M thupc ( C ) ke 2 tiep tuyen voi ( C ) ,
gpi A , B la cac tiep diem. Tim tpa dp diem M biet A B = 4,8.
Cau S.b: Trong m|t phSng tpa dp Oxyz, cho A A B C vuong can tai C voi
(5; 3 ; - 5 ) , B ( 3 ; - 1 ; - 1 ) . L i p phuong trinh duong th^ng d, biet d d i qua dinh C

dx

( x 2 - x + l)(x2+3x + l)

A A B C , nMm trong m§t phSng ( a ) : 2x - 2y - z = 0 va tgo voi mSt phSng

(p):2x + y - 2 z + 5 = 0 goc 45''.
97


Tuyin chgn &• Giai thi^u dithi

Todn hpc - Nguyen Phu Khdnh,

Cty TNHH MTV DWH

Nguyin Tat Thu.
2

1^

theo thii t\ so' mu cua x
3
giam dan, tim so' hang dung giiia cua khai trien biet h^ so' cua so' h^ng thu ba
la 5.

•Jl sin

HMGDANGIAI

72 sin

Cau 9.b: V6i n e N * , khai trien nhj thiic

X

X +

7t
X +

7t

X +

71

x=m - l

^

4

X= m + l

ro

s/

up

Khoang each tu diem eye tieu cua do thi den goc tpa dp O bang 3 Ian khoang
each tu diem eye d?i den O o OB = 30A

/g

.c

om

• (m +1)^ + (-2 - 2m)^ = 9r(m -1)^ + (2 - 2m)^o 2 m ^ - 5 m + 2 = 0

ok
bo
ce

ww

w.

fa

Voidieuki^n: sin2x->/3eos2x7tOotan2x?tVsox?t- + k - ( k e Z ) , t a e 6
6
2^
'
71
2 (1 + sin 2x) + cos 4x ~
^'^ _ 2 ^ sin
x+—
sin2x->/3eos2x
4j

o2sin^

4

1 •o
sm4x = 2sf2sm x + — -sm2x
4 2
2

4x + — = -2N/2sin f
cos 2x + ^
I
6J
I
3;
Ix+ —
4;
/

.x

+

N/3cos2x
^

2

n + 2eos^ 2x + i ^ ' - l + 2^/2sin
—
4;
I
6;

l

X+

N
71

cos 2x-.^^ = 0
4;
I
6;
-

2x + - = -sin 2x + -

= ^/2sin 2x +
= >/2sin
TT

l

6J
.

71

71

6

4^

6

6)

4;

, _

x = --k27r

«

6

4

7t

6

7t

3

, _

+ — + k27t

4

(keZ)

X =

<=>

4

Cau 3: Dieu ki^n:

57t

18

+

271

x=

6 4 + k27r.
x + -4 = - 2 x
7C
7t
«
7t
7t , _
X + — = 7l + 2 x + — + — + k27t

Ta

Diem c\rc tieu cua do thj la B(m +1;-2 - 2 m ) .

Cau 2:

2x + — = sm 2x + 7t6)
6J

I
l

= sin^ 2x + ^
I
6J

6;

+ k27t

4

Diem eye dgi cua do thj la A(m -1;2 - 2m);

7t

71

X + —= 7t-2x

=2

+ COS

x + - = 2x +

Ham so'CO cxfc d^i, eye tieu Vm e M .

X + — + COS

-

4y

• ^(m +1)^ + (-2 - 2m)^ ^ 3^(m -1)^ + (2 - 2m)^

+ COS

[

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V2sin

71

<=>2sin^

—

4;
4;

y' = 0 o 3 x ^ -6mx + 3 ( m ^ - l ) = 0 o x 2 - 2 m x + m^ - 1 = 0 o

+ COS

4

7t
X + —

b) Ta c6: y• = 3x^ - 6mx + 3^m^ -1

o i.(si
s m x + cosx)
f + —cos4x
2

—

7t

I. PHAN CHUNG CHO TAT CA CAC THI SINK
Cau 1:
a) Danh cho b^n dpc.

o m = 2 ho$c

Khang

, 271

(keZ)

+ k —

9

X=

6 4
[x + y >0

, 271
k —

3
k27t

6

[x''-y>0

Phucmg trinh thu nhat tuong duong
(x + y)^ -16(x + y)-2xy(x + y) + 8xy = 0
c:>(x + y) (x + y)^-16 -2xy (x + y ) - 4 = 0
o

(x + y ) - 4

(x + y)(x + y+ 4)-2xy = 0

o

(x + y ) - 4

x^+y^+4(x + y) = 0

>o
o (x + y ) - 4 = 0 o y = 4 - x. Thay vao phuong trinh thu hai, ta dug-c:
-(4-x)

<=>x'^+x-6 = 0 o

x = -3,y = 7
x = 2,y = 2

V^y, phuong trinh da cho c6 nghi^m la: (x;y) = (-3;7),(2;2)

Twyew chQit & Giai

thi$u dethi

Todn

hifc - Nguyen

Phii Khanh,

Nguyen

Tat

Cty TNHH

Thu.

Do'i can: x = l = > t = 2, x = 2=>t = 2
. 1 . t - 1 2 1 f, 3 . 1 ^
l^
dt = - l n
I n — - I n - = 4i l n 11
t + 3;
4 t + 3 2 ' 4 l 11
5j

Ta xet hi?u M = a +•
2 . ( b - 2) 1 ^ ( b ; l £ f _ ( ^ 2 ^ , 2 ^ , 2 ) , b c

Cau 5: SJi vuong goc ( A B C D )

x2

Ke H M vuong goc A B => A B 1 ( S H M )
ABMH

= a"^

( S A B ) , ( A B C D ) = S M H = 60°

vuong can tai H c6

BM = BH = - = > B H = — = > S H = MH.tan60°= ^
^ABCD - 3 SABCD-SH - -^-y^ -

-be + a

a.

8

-bc(b2+c2)

Ta

/g

ro

up

s/

I

om

Gpi N la trung diem ciia SB.

ok
ce
fa

AC = ( c - 2 ; d - 5 ) , AB = (4;0), BH = ( h - 6 ; h - 4 ) , CH = ( h - c ; h + l - d )
H la tryc tam tam giac ABC nen c6:
^

CH.AB = 0

(1)
( h - 2 ) ( h - 6 ) + ( d - 5 ) ( h - 4 ) = 0 (2)

55
=>IP = Z ^ . _ 5 ! . > O P = ^ R = IcWoi2+OC2=a.
V108
OP
3N/3
Cau 6: Bat dang thiic can chung minh tuong duong voi bat dSng thiic

BH.AC = 0

(h-4)'+(d-6f=:5

APNI dong dang voi APOB

a V + b V + c V > (a^ + b^ + c2 )a V c 2

(3-af
81
a.^
^<3<:>:^a5-3a^+9a3-27a2+^a<3
4
8
2
8

Trong do: (c - 4)^ + (d - 6)^ = 5

w.
NP = ^ B P = i

a.1^.3

Cau7.a:Giasu C(c;d) va H(h;h + l ) , c;^{2;6 .

ww

BS = V S H 2 + H B 2 = ^ = > B P = ^ B S = ^ ,

>0

A. Theo chiTorng trlnh chuan

bo

Gpi I la giao diem cua d va SO
IS = lA = IB = IC
=> I la tam m|t cau
Gpi P la giao diem cua A va BS.

(3-af

a(b-cr

Xethamso f(a) = - a 5 - 3 a ' ' + 9 a 3 - — a ^ + — a v6iaG(0;3'
8
2
8
I I . PHAN RIENG Thi sinh chi dxxgc chpn lam mpt trong hai phan (phan A
hoac B)

.c

Trong mp (SBD) , d la trung tryc ciia SB, li

hay a2+.

a^(b-cf

abc

4

2
(b + c f
Ta can chung minh: a +-

Ta CO tarn giac ABC vuong can tai B.
Gpi O la giao diem cua AC va BD
tarn I ciia mat cau thupc A la tryc
cua duang tron ngoai tiep tam giac ABC,
A vuong goc (ABCD) tai O

100

(b + c)^

2

— —

Do BH = -BO=>OP = - S H = ^
3
2
2

Vift

a + b + c > ^a^ + b^ + c^ jabc nghia la (a^ + b'^ + c^ jabc < 3

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it-1

Khang

Ta can chung minh: abc(a + b + c)>^a^+b^+c^ja^b^c^ hay

X

( \ >

DWH

Hon nua: vai a,b,c> 0 luon c6 a^b^ + b^c^ + c^a^ > abc(a + b + c)

dx.

D$t t = X + - => dt =

MTV

Lay ( l ) t r u (2), ta duc^c ( d - 5 ) ( d - h - 3 ) = 0 o d = 5 hoac d = h + 3
4

Voi d = 5 thay vao (l) ta dugc: h^ - 8h +12 = 0 <=> h = 2 ho|c h = 6
Voi d = h + 3 thay vao (l) ta dug-c: 2h^ -14h + 20 = 0 o h = 2 hoac h = 5
101


Tuycii Wn)>i iV Giiri Ihifu ilc thi loan hoc - Nguyen

Phii Khaitli , Ngmjcn

w-

Ihu.

Ttit

Tpa dp diem C la nghi#m h§:
xc +2yc - 2 z c -12 = 0
2xc-2yc-Zc=0

CauS.a: OA cung phuong voi u = (0;0;l) A(0;0;a) voi a 0
M|t phang (BCD) c6 cap vecto la: BC = (0;1; - 2), BD = (3;1; - 4)
nen c6 vecto phap tuyen n = BC;BD = (-2;-6;-3)

2

-

2z2 ^ = 12,

z^Zj =

4,

2

ww

w.

fa

ce

bo

ok

.c

om

/g

ro

B. Theo chi/tfng trinh nSng cao
Cau 7.b: Duong tron (C) c6 tam O{0;0) c6 ban kinh R = 3.
OA^
Tu AB = 4,8=>OH=l,8 va M 0 = OH = 5
Gia sir M c6 tpa dp M(a; b) ta c6: + b^ = 25 (l)
Hon nua M e (C) nen c6: a^ + b^ - 18a - 6b + 65 = 0 (2)
Giaih? (l) va (2) ta tim dupe: M(5;0), M(4;3)
Cau S.b: Gpi I la trung diem AB khi do l(4;l; - 3 ) . Do AABC vuong tai C nen
de thay C thupc m|it cau tam I ban kinh la< AB = 3

2

2

Phuong trinh m5t cau (S) CO dang: ( x - 4 ) + ( y - l ) +(z + 3) =9
Gpi C(xc;yc;zc) la tpa dp can tim. De thay, CI 1 AB CI.AB = 0
o 2 ( x c - 4 ) + 4(yc - l ) - 4 { z c +3) = 0 o x^ +2yc - 2 z c -12 = 0
Hon niia, d di qua C dong thoi d lai nSm trong mat phang (a) nen C cung
phai n3m trong m|t phang (a), nen c6: 2X(- - ly^ -ZQ=0
C e (S) =^ (xc - 4)'+(yc - i f + (zc + 3)'= 9
in2

v uv vn t^-nurig

viyt

=>C(2;3;-2)

\

n

2a-5b

o 37a^ - 32ab -5h^=0

o a = b ho|c 37a = -5b

Vol, a = b CO the lay a = b = 1 =>c = 0=> u = (l;l;0)
Voi, 37a = -5b c6 the lay b = -37, a = 5 => c = 84 => u = (5; -37; 84)
tku 9.b:

Ta

Zj +

s/

2z2 = 3 + -s/si =>

/

3V2

- 2z + 4 = 0 c6 hai nghi^m phuc la Zj = 1 - Tsi, Zj = 1 + Tsi

up

Zj +

mi

Gpi u = (a;b;c) la vecto chi phuong ciia duong thSng (d) va (d) nkm trong
mlitphSng (a) CO vecto phap tuyen n = ( 2 ; - 2 ; - l )
Vithen.u = 0 o 2 a - 2 b - c = 0=>c = 2a-2b (*)
n = 2a + b - 2 c
d t^o voi (Pj mptgoc 45 , nen sin45

4 ' * ^ V A B C D = 3d(A;(BCD)).SBCD

o -5 = 1—. 3 a - 4 .— <» a = 3 hoac a = — .
6 3 2 2
3
Vithe'co 2 diem A(0;0;3) ho|c A 0;0;-Cau 9.a:

.\

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BC;BD

/

ii\nn

[ ( x c - 4 f + ( y c - i r ^ ( z c + 3 f =9

Dodo (BCD)coding: 2(x + l) + 6 ( y - l ) + 3z = 0 <»2x + 6y + 3 z - 4 = 0
'BCD

L-ty

Khai trien nhj thuc
X

n

X

1 theo thu ty so mii cua x giam dan, ta dupe:
—
3

= C°x"-lc^x"-^+lc2x"-2-...+

3,
3
9
H^ so cua so h^ng thu ba trong khai trien la: ^ C ^
.

3j

n

V

2
I^ Theo 6gia thiet,' ta c6:. 1 ^-C„
, n! = 45 o n 2 - n - 9 0 = 0 o n = 10
g n = 5 o 2!(n-2)!
I Voi n = 10, ve phai cua (*) c6 11 so hang nen so hang dung giija la so hang
6 la:

' if

r5 v 5 - _ 2 8 5
3J MO" - 27

28
I V^y, so hgng dung giua cua khai trien la: - ^ " ^

103


jjiupc mSt phSng (P) biet duong thiing A M vuong goc vol A va khoang each

DETHITHUrsdie

^ A den duong thang A bang
I . PHAN CHUNG CHO TAT CA CAC THI SINH

Cau9-a= Cho 2 sophuc Z j va Z j thoa man: |zj| = 3,

Cau 1: Cho hkm so: y = ( 2 - m ) x ^ - 6 m x ^ + 9 ( 2 - m ) x - 2 , c o d o t h j l a ( C ^ )

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Cau 7.b: Trong m5t phSng tpa dp Oxy, cho tam giac ABC c6 tryc tam H thupc
duang thang 3x - y - 4 =0, biet duong tron ngoai tiep tam giac HBC c6

+ •y/3cos4x = 3-4sin^ x

phuong trinh: x^ +y^ - x - 5 y +4 =0, trung diem canh AB la M ( 2 ; 3 ) . Tim
tpa dp 3 dinh tam giac ?.

Cau 3: Giai phuong trinh:
2 log 1 f 5x - 9 + Vx^-2x + 25l + logj (x^ - 2x + 36) = 0

Cau 8.b: Trong khong gian voi h? tpa dp Oxyz, viet phuong trinh m|t phSng
(p) di qua duong thSng d: " ^ =

Ta

Cau 9.b: Tim so phuc z thoa man:

ro

va m^t day bang 60*^. Gpi I, J Ian lupt la trung diem cua AB va AC. Tinh the

up

Cau 5: Cho hinh chop tam giac deu S.ABC canh day bang a, goc giira mat ben

/g

tich khol chop S.ABC va khoang each t u diem A den mat phSng (SIj) theo a .

ok

bo

Tim gia trj Ian nhat va nho nhat ciia bie'u thuc: P = x^ + y^ + 3

.c

om

Cau 6: Cho hai so duong x, y thoa man dieu k i f n : x > 1, y > 1 va 3(x + y) = 4xy.

ce

I I . PHAN RIENG T h i sinh chi dwgc chpn lam mpt trong hai phan (phan A

fa

hoac B)

w.

A. Theo chuorng trinh chuSn

^^^"^7^

**^P

^"'^

(^)"

+ y^ + z^ + 2x - 4y - 4 = 0 .

s/

Cau 4: Tinh tich

ww

Cau 7.a: Trong m^t phSng tpa dp Oxy, cho duong tron ( C ) :
x ^ + y ^ - 2 x - 2 m y + m^-24=:0

Co tam I va duong th5ng A : mx + 4y = 0. Tim m biet duong th^ng A c^t
duong tron (C) tai 2 diem phan bi^t A, B thoa man di?n tich lAB = 12.
Cau 8.a: Trong khong gian tpa dp Oxyz, cho d i e m M ( l ; - l ; 0 ) va duong thang
= -^Y"

= 737 .

g. Theo chiTorng trinh nang cao

B va C sao cho di^n tich tam giac OBC bang \fl3 .

A : ^^-^ =

Z1-Z2

Z2

b) Tim m de duong thJing d: y = -2 ck ( C ^ ) t^ii ba diem phan bi?t A(0; -2),

4

=4,

Timsophuc z = ^

a) Khao sat sy bien thien va ve do thj (C) cua ham so khi m = 1.

Cau 2: Giai phuong trinh: 2sin^

33

^

va m^t phSng ( P ) : x + y + z - 2 = 0 . Tim tpa dp diem A

= 18 + 26i.

Hir(]fNGDANGlAl
I . PHAN CHUNG CHO TAT CA CAC THI SINH
Cau 1:
a) Danh cho ban dpc.
b) Phuong trinh hoanh dp giao diem :
(2-m)x^-6mx2+9{2-m)x-2

= -2

o(2-m)x3-6mx2+9(2-m)x =0

o

X (2

- m) x^ - 6mx + 9 ( 2 - m ) ] = 0 ( l )

o x = 0 hoac ( 2 - m ) x 2 - 6 m x + 9 ( 2 - m ) = 0 ( 2 )
De' phuong trinh ( l ) c6 ba nghi?m phan bi^t A ( 0 ; - 2 ) , B va C thi phuong
trinh ( 2 ) CO 2 nghi^m phan bi?t khac 0

A = 9m2-9(2-m)So
2 - m ^0

Voi l < m ^ 2 thi ( d ) cat ( C „ ) tai 3 diem phan bi?t A(0; - 2 ) ,

m >1
m*2
B(XI;-2),

; C ( x 2 ; - 2 ) va X j ^ X j

Gpi h la khoang each t u goc O den duong th^ng d thi h = 2
104

105


Theo de bai ta c6:
S AOBC= ^h.BC

= VT3

Cich2: ( * ) « 5 ( x - l ) - 4 + ^ ( x - l ) ' + 2 4 = ^ ( x - l ) ' 4 - 3 5
=> BC = >/i3 o

Theo djnh ly viet ta c6:

Xi+X2

=

( x i + x j f - 4x5X2 = 13

D$t t = X - 1 , khi do (* *) tro thanh: 5t - 4 + Vt^ +24 = Vt^ +35

6m
2-m

« * 5 ( t - l ) + Vt^+24-5 + 6-Vt^+35=0

XiX2=9

6m
2-m

< - > ( t - l ) 5+
I
Vt2+24 + 5

14
- 36 = 13 <^ m = — hoac m = 14 thoa man dieu kien.
13
,

Cau 2: Phuong trinh cho <=> 1 - cos —
2

In 2

4x + \/3cos4x = 3 - 4 s i n ^ x

71

=

(keZ).

Vly, I = — +

36 + k —
3

s/

X

om

/g

= logj Vx^ - 2 x + 36

fa

w.

Neu 5 x - 9 < 0 < = > x < — phuong trinh v6 nghi^m
5

ce

' Cdch I ; (•) <=> (5x - 9)f\/x^-2x + 25 + V x ^ - 2 x + 3 6 l = 11

ok

.c

(•)

bo

o 5 x - 9 + V x ^ - 2 x + 25=>/x^-2x + 36

ro

Cau 3: Dieu ki?n: 5x - 9 + V x ^ - 2 x + 25 > 0
Phuong trinh cho viet lai: log2 5x - 9 + Vx^ - 2x + 25

ww

Neu 5 x - 9 > 0 o x > - , x e t f ( x ) = (5x-9)fVx^ - 2 x + 25 + Vx^ - 2 x + 36
5
V
y
Ta c6:
f { x ) = 5fVx2-2x + 36 + Vx2-2x+25l + ( 5 x - 9 )

Vx > —
5

f (x) dong bich tren

9

Vi the, phuong trinh cho c6 nghi^m x = 2
106

x-1
=• +>0,
, V x 2 - 2 x + 36
Vx2-2x+25;

va f(2) = l l

x-1

ln2

e^'^-e'^+l

+ kn
, 71

ln^2

In 2 2e2x _ g X

Ta

12

iln2

J x2dx =

up

4x + - = -2x + kin
6

7t

-dx
J
0 e^^-e"^!

In 2

>/3
1
<=> —cos4x — s i n 4 x = cos2x <=> cos 4x + — = cos2x
2
2
6;
x=

In 2 2e2''-e^
2e2x

c a u 4 : 1 = f x2dx+

o > / 3 c o s 4 x - s i n 4 x = 2^1-2sin^xj

4x + - = 2x + k27t
6

= i i i
=Oz:>t = l tuc x = 2
Vt^+35 + 6,

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Suy ra:

(**)

ln3

Cau 5: Gpi M la trung diem BC, ta c6:
(SBC)n(ABC) = BC'
A M 1 BC
SMIBC
=> goc giiia ( S B C )

va (ABC) la SMA - 60°

^

GQiO = A M n C I
=> O la trpng tam, true tam A A B C .
V i S . A B C la hinh chop deu nen SO 1 ( A B C )
AABC deu canh a => A M = ^
OM = - A M = — ,
3
6

,

BC = a ; S^ABC = - A M . B C = •
^'^^
2

ASOM vuong tai O => SO = OM.tan60° =
The tich hinh chop S.ABC la: V = - .5^ABC -^"^ " 3 '

= |
4

'2

24


luym

JH^H bf

Um

Nguyen PhuKIu'inh . Vynyew Tat

thtfu He Wi% loan hoc

Thu.

Cty TNHH MTV DWH

Ta c6: IJ ± A M (do IJ / /BC),IJ i . S O => IJ ± ( S A M )

D o d o f(S) lahamsodongbientrendo^n

Ggi E = IJ n A M . Trong mp ( S A M ) , ve A K 1 SE , ( K e S E ) , IJ 1 A K

Trong mp ( S A M ) , l^ii ve M H 1 SE , ( H e SE).

Gia tri Ion nhat cua bieu thiic P = ^

Vi SEO la goc nhpn nen SEA la goc tu => K nam ngoai doan ES.

2

M H =

=

SE

Taco: d ( l ; A ) = -

12

=

26

SiAB = ^ I H . A B = I H . B H = I H V R ^ - I H ^

3aVl3
26

Ta

T u do ta CO phuong trinh:

4

w.

= (x + y)^ - 3xy(x + y) + 3

Gia thiet 3(x + y) = 4xy suy ra i + — = ^
Til ( I ) , (2) va (3) suyra

4

S 3

Xethamso: f(s) = s 3 - - S - - + — voi 3 < S < 4
^ '
4
S
3
Taco: f'(S) = 3S S - 108

+-^>0

voi VSe(3;4),

ro

trinh nay c6 4 gia trj m thoa man: -3; 3; - — ;

^

Cau S.a: Phucmg trinh mat p h i n g ( Q ) chua A M va vuong goc voi A, nen c6
phuong trinh: 2 x - y + z - 3 = 0.
G
Gpi (d) la giao tuyen aia ( P ) va ( Q ) nen c6 ( d ) :

2'2

y=t
z = l-3t

xy

Gpi A e ( d ) zi> A ( l + 2 t ; t ; l - 3 t ) , ta c6 A N = ^
(3)

P-s'-^S-i+Ii

= 12 <=> 3m^ - 25 m + 48 = 0, phuong

fa

Jl_

ww

J_

m^+16

x = l + 2t

ce

o3= x^ + y^ +3

om

tuc

4.25 m

ok

(x-l)(y-l)>0

.c

S>2

x + y>2

bo

H a n nua: x > 1, y > 1 suy ra

/g

U i c o : s 2 - 4 P > 0 < » s 2 - 4 — > 0 o S < 0 hoac S>3

up

3S
Cau 6: Dat S = X + y, P = xy theo gia thiet ta c6: 3S = 4P hay P = — ( l )
4 • '

P

: 5 nen A cat duong tron (C) t^i hai diem phan bi^t

A , B. Gpi H la trung diem A B thi I H 1 A B va I H = d ( l ; A )

3aVl3

ayf39 "
12

5 m

s/

Vgy d(A,(SIj)) =

isO.AE

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SASEM = ^ S E - M H =

ho?c B )

/V. Theo chi/cmg trinh chuan
Cau7.a: Duong tron (C) c6 tam l ( l ; m ) v a b a n k i n h R = 5

aV39

a aVs

khi (x;y) = ( l ; 3 ) , ( 3 ; l )

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

AAKE = AMHE => A K = M H

EO = EM - O M = i A M - i A M = i A M = — .
2
3
6
12
a^ ^
ASOE vuong tai O ^ SE = V S O V E O ^ =
4
48

f(4) = ^ .
3

(3 3)
113
Gia trj nho nhat ciia bieu thiic P = — khi (x; y) =
2'2

do I J l ( S A M ) ^ A K l ( S I j ) , s u y r a d(A,(SIj)) = A K .

Ta CO M H // A K va E la trung diem A M

[3;4] va f{3) = ^ ,

Khang Vift

t u day rim dugc t = - l

ho$ct = ^
7

Ciu9.a:

\^+

yii,

= Xj + y j . i

(x,y6i?)

= x^y?=9
Tir gia thiet, ta c6:

'=x2

+ y2=16

(xi-x2f+(yi-y2f=37
109


—«

Tuyeh

chgn

& Gioi

thi?u

dethi

+ y2

+ y^.yj =

X1.X2

Todn

H Q C - NguySn

ziz^^ziz^

Z2

Z2Z2

Khdnh,

Nguyen

Tat

Thu.

CtyTNHH

+ x| + y 2 - 37
^
= -6

( y j . x j - x j . y j f = (x^ + yi).[xl
^_^zi

Phu

Xi.X2

/ X
l'Me(P)
Vi (d) chua trong (P) nen c6 < _ ^
u.n = 0

+ yi.y2+(yi-X2-Xi.y2).i

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~

8

4b-c

( H B C ) t^i 1 diem c6 tga dp (2; 2) nen H ( 2 ; 2 ) .
(

2

A
V

I

f

1'

1/

—
——

2,

(x + y . i f = 18 + 2 6 i o

2

Ta
s/
up

[ y j - y i

=

-Xfj

y B - y N

13 .
a; — - b
2

(5

•I
2

ce

bo

b_71(3-b) = 0 o a 2 + b 2 - ^ a - H b + 2 3 ^ 0
2 IV
/
2
2
2

Ia

1^ +
—2 ;
l
_5

'
2, " 2

5

2

ww

(

(
1'
+ b
^ 2 ;
I
a + b =5

13u 23 _
a^ + b^
a
b +— =0
2
2
2
• a = 2 => b = 3 khong thoa v i M s B

(2)
^'

—
2;

^ 5

(a-l)(a-2)^
a +b = 5

* a = l = > b = 43:>B(l;4),A(3;2),C(l;l).

Cau 8.b: Duong thSng (d): d i qua d i e m M ( 0 ; - l ; 2 ) va c6 vecto chi phuon.'!
u=(-2;2;l)
1 i n

/ 3 _ 3y^21 ^

/ 3 _ 3^y2 \

>

^

'

'

D|it y = tx ta dupe t = - = > x = 3,y = l =>z = 3 + i
3

fi^THITHUfSOl?

Cau 1: Cho ham so y = x^ + 3x2 - mx + 2 c6 ^ j - ^j^j

^

a) Khao sat sy bien thien va ve do thi (C) cua ham so khi m = 0.
b) Tim cac gia t r i cua m de ham so c6 eye dai va eye tieu sao eho khoang

w.

Tpa dg diem B la nghi?m cua h? ( l ) va (2)
2

y^ - 3yx^ =-26

^

I . PHAN CHUNG CHO TAT CA CAC THI SINH

fa

(2-a)-

ok

V i I M vuong goc voi BM nen IM.BM = 0
1

x3 - 3xy2 = 18

(x,y€

ro
- X g

/g

|Xj - X j

om

«

Ian lugt la tam duang tron ngogi tiep tam giac ABC, HBC

.c

the thi IJ = NB

Cau 9.b: Gpi so phuc can tim z = x + y.i

1
'-2)

= 3 <»56b2+76bc + 20c2=0

^2b + c f + 4 b 2 + 4 c 2

5l
= - va B(a;b)
2j

Gpi N la diem doi xung voi H qua M thi N ( 2 ; 4 ) .
/15^
G(?i I , J
2'2

a _2b + c

-2a + 2b + c = 0

Theo bai toan, suy ra d(l;(P)) = R

Cau 7.b: Truoc het, ta thay duang thang (d) da cho tiep xuc voi duang tron

la diem thuQC duang tron ( H B C ) nen c6:

o

d = b-2c

- b + 2c + d = 0

M$t cau (S) ta c6 tam l ( - l ; 2 ; 0 ) , ban kinh R = 3.

B. Theo chilomg trinh nang cao

Phuong trinh duang tron ( H B C ) viet l^i la:

Vift

Khi do (P) tro thanh: (2b + c) x + 2by + 2cz + 2b - 4c = 0

-6 ± 6V3.i _ -3 ± 3>/3.i
16

Khattg

tuyeh n = (a;b;c)viO.

16

~

DWH

Gpi phuong trinh mat phMng (P) c6 d^ng: ax + by + cz + d = 0 c6 vecto phap

+ y i ) - ( x i . X 2 + y j . y s f = 144 - 36 = 108

{zzf

MTV

each t u trung diem cua doan thSng noi 2 diem eye trj cua ( C ^ ) den tiep tuyeh
cua ( C ^ ) t?i diem c6 hoanh dp bang 1 la Ian nhat.
^ a u 2: Giai phuong trinh: 6(tan 2x - tan x) = sin 4x + sin 2x
Cau 3: Djnh m de phuong trinh: J — x^ + - i - + m - 4= = 2x c6 nghi^m duy nhat.
V3
%x2
^
C a u 4 :Tinh tich phan: I = f
^,dx
i(lnx + l ) '
111