luyện thi đại học môn toán

Cty TNHH MTVDWHTOiaitg V^

DETHITHUfSOl
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

(5^c em H Q C sinh than men!.

Cau 1: Cho ham so y =

"Luyen gidi de truoc ky thi dai hgc - Tuyen chon vd giai thieu de thi
Todn hgc" la mpt trong nhOng cuon thupc bp sach "On luy$n thi Dai hgc", do
nhom tac gia chuyen toan THPT bien soan.

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b) Gpi I la giao diem eua hai duong ti^m can. Tim diem A thupc do thj ( C ) ,

biet tam giac OIA co di?n tich bang i , voi O la goc tpa dp.
.
<,
1
sm2x

= 2sin
Cau 2: Giai phuong trmh: -7=cotx +
V2
smx + eosx
Cau 3: Giai bat phuong trinh: 8^ ^ ^

/g

ro

up

s/

Ta

Cau 4: Tinh tich phan: I =

3>6sl2x-3+-j^.
Vx+1

x+1

^^LLI^

2

xdx.

Tam giae ASC vuong tai S va nkm trong mat phSng vuong goc voi day, SA = a..

Tinh theo a the tich khoi chop S.ABC va khoang each tix C den mat phiing (SAB).
Cau 6: Cho cac so thye khong am a,b,e thoa a + b + e = l va khong co hai so
nao dong thoi bang 0. Tim gia trj nho nha't ciia bieu thuc:
P =1
.+
^
r + (e + l ) ( 3 + a + b ) .
(a + b)(b + e) (e + a)(a + b) ^
'

ok

bo

ce

fa

w.

ww

Mac du tac gia da danh nhieu tam huyet cho cuon sach, xong sy sai sot la
dieu kho tranh khoi. Chung toi rat mong nhan dupe sy phan bi^n va gop y quy
bau eua quy dpc gia de nhirng Ian tai ban sau cuon sach dupe hoan thi^n hon.

J

+

7t
X + —

Cau 5: Cho hinh chop S.ABC eo day ABC la tam giae vuong can tai B, AC = 2a.

.c

om

Mon Toan la mpt mon rat ua phong each tai tu, nhung phai la tai tit mpt
each sang tao va thong minh. Khi giai mpt bai toan, thay v i dung thoi gian de
luc Ipi tri nho, thi ta can phai suy nghT phan tich de tim ra phuong phap giai
quyet bai toan do. Do'i voi Toan hpc, khong eo trang sach nao la thua. Tung
trang, tung dong deu phai hieu. Mon Toan doi hoi phai kien nhan va ben bi
ngay t u nhirng bai tap don gian nhat, nhiing kien thiic co ban nhat. V i chinh
nhiing kien thue co ban moi giiip ban dpc hieu dupe nhij'ng kien thuc nang cao
sau nay.

1

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

Voi each viet khoa hpc va sinh dpng giiip ban dpc tiep can voi mon toan
mpt each t y nhien, khong ap luc, ban dpc tro nen t y tin va nang dpng hon;
hieu ro ban chat, biet each phan tich de tim ra trong tarn ciia van de va biet giai
thich, lap luan cho tirng bai toan. Sy da dang ciia h^ thong bai tap va tinh
huong giiip ban dpc luon hung thii khi giai toan.
Tac gia chii trpng bien soan nhung cau hoi mo, npi dung co ban bam sat
sach giao khoa va cau true de thi Dai hpc, dong thai phan bai tap thanh eac
dang toan co lai giai chi tiet. Hi^n nay de thi Dai hpc khong kho, to hop eua

nhieu van de dan gian, nhung chua nhieu cau hoi mo neu khong nam chae ly
thuye't se lung tiing trong vifc tim 16i giai bai toan. Voi mpt bai toan, khong
nen thoa man ngay voi mpt lai giai minh vira tim dupe ma phai co' gang tim
nhieu each giai nhat cho bai toan do, moi mpt each giai se eo them phan kien
thue mai on tap.

^ (C)
X

II. P H A N R I E N G

Thi sinh chi dxxtfc chpn lam mpt trong hai phan (phan A

hoac B)
A . Theo chUorng trinh chuan

Cau 7a: Trong mat phang Oxy cho tam giae ABC npi tiep duong tron (C) ec
phuong trinh: (x + 4)^ + y^ =25, H ( - 6 ; - 1 ) la trye tam tam giac ABC; M ( - 3 ; -2
la trung diem canh BC. Xae djnh tpa dp cac dinh A , B , C .
Cau 8a: Viet phuong trinh m|it cau (S) co tam nam tren duong than^

Thay rnat nhom bien soan
Tac gid: Nguyen Phu Khanh.

d:2iz2 = yzi = £zi
~3
2
2
( Q ) : x + 2 y - 2 z + 4 = 0.

va tiep xuc voi hai m^t phSng (P):x + 2 y - 2 z - 2 = 0 v


Tuyen chgn & Giai thifu dethi Todu hqc - Nguyen Phii Khdnh , Nguyen Tat Thu.

C t y TNHH MTV DWH Khang Viet

Cau 9a: Chung minh dang thuc sau:
1

Vay CO 4 diem thoa yeu cau bai toan: A,(2;3), A 2 ( 0 ; l ) , A 3 - ; 0

u ,

„2n-l_22"-l

2n

^"

2n + l
Cau 2: Dieu ki|n: •

(n la so nguyen duong, CJ^ la so to hop chap k ciia n phan tu).
B. Theo chUorng trinh nang cao

sm

Cau 8b: Trong khong gian Oxyz cho duong thang A: — ^ ~ ^ — ] ~
phang ( a ) : 2 x + y - 2 z + 3 = 0. Chung minh rang A va (a) cat nhau tai A. Lap

X

,^

n

X = - + k2K, X = -

Ta
up

s/

+ z^ = 0

ro

W^Z-5=:1

/g

H\i(}m DAN GIAI

om

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

.c

Cau 1:

.
_
2cosx = 0 o

cosx

2cos^x

sinx

smx + cosx

= 0

X = — + k7C

+ cos x = 2\/2 sinx cosx

2
n

.

Cau 9b: Tim cac so phuc z, w thoa

+ kjt

= sin2x

sm

X = — + k7l

phuong trinh mat cau (S) c6 tarn nMm tren A, di qua A va (S) cat m p ( a ) theo
mgt duong tron c6 ban kinh bang

4

cosx = 0

z +1

V

X* —

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diem M tren elip, hay tinh bieu thuc: P = F^M^ + FjM^ - 30M^ - F1M.F2M .

1

sinx + cosx ^ 0

cosx
sin2x
o-^r-— +
7 2 s i n x sinx + cosx

va di qua diem A sfS;- . Lap phuong trinh chinh t5c cua ( E ) va voi moi
V
^/

X—

X ?t k n

Phuong trinh

Cau 7b: Trong mat phang Oxy cho elip (E) C6 hai tieu diem I^(W3;0); I^(V3;0)

,

sinx ^ 0

4

4

kin

+

—

3

Ket hop dieu kien ta c6 nghiem cua phuong trinh la:
7t
X =

llTt

— + nn,

X=

„
+

1771

2nn, x =

„ ^

+ Inn, n&Z .

Cau 3: Dieu kiC^n: x > Ba't phuong trinh c=> 8 V 2 x - 3 + 3x^7+1 = 6 j ( 2 x - 3)(x + l ) + 4

ok

I

bo

a) Dpc gia t u lam

h = d(A,IO) = - ^ .2a-

Dodo

Nen

S^,OA4''°^4

2a-l
a-1

fa

,a^l.

a-1

1

2a2-4a + l
a-1

3
13
Ket hg-p dieu kien ta c6 nghiem bat phuong trinh la: - < x < —

a-1

a-1

«(2V2X-3-I)(4-3N/X+T)>0
7
13
>0<=>-( 2 V 2 X - 3 + I)(4 + 3 V X + T )
9

2a^-4a + l

S^oiA = 2 ^ 2a^-4a + l

o 4 ( 2 V 2 x - 3 - l ) + 3^/^(^-2^/2x-3)>0

(8x-13)(7-9x)

w.

A e ( C ) = ^ A a;

ww

GQi

2a-l

ce

b) Ta CO l ( l ; 2 ) = > O I = (l;2)=r>IO = \/5 va phuong trinh O I : 2 x - y = 0

Cau 4: Ta c6: I - V'^^'"^dx - 'fxe^^dx = A - B
23-^ - 5a + 2 = 0
2a^-3a = 0

a = 2,a = 2
a = 0,a = 2

, A4

Dat t = V l + 3 1 n x ^ l n x = l ( t 2 - l ) : ^ — = ^ t d t .
3^
/
X
3
Doi can x = l = > t = l , x = e=:>t = 2

-;4

U J


TuySii chgn & Giai thifu dethi Toan hqc - Nguyen Phu Khdnh , Nguyen Tat Thu.

2 »
.
Suy ra A = Jt-tdt = - t ^

c=0

14
9

du = dx

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

1 2x

dx = e2''dx

Suy ra 5 = ^ X 6 ^ "

ho?c B)
A. Theo chUerng trinh chuan
Cau 7a: Duong tron (C) c6 tarn I ( - 4 ; O ) , ban kinh R = 5

.2x

,

2J

2

2

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D|t

4

^

14 e2«-e2
Vgy I = — ^
9
4

= ^1;_2) laVTPTcua BC nen phuong trinh BC la: x - 2 y - l = 0

Do do tpa dp B, C la nghi^m ciia h?:

Cau 5: Ta c6 A B = B C = ^

-

= i ^

2

AC^

_

(c + a)(a + b)

Ta
s/
up

1
1
1
•+a + bl^b + c c + a,

V a y A { - 4 ; - 5 ) hoac A ( - 8 ; 3 ) .
Cau 8a: Vi mat cau (S) c6 tarn I e d

d(l,(P)) = d ( l , ( Q ) ) . R
6-3t

-t

2 - t | « > t = l = > l ( - l ; 3 ; 3 ) va R = l .

Cau 9a : Ta c6: ( l + x f " = C°„ + xC^„ +.... + x^^C^jJ
{l-xf".C^„-xC^„......x^"Ci^

( l - c ) ( l + c)

i-c^

D o d o : P > — i - + (c + l ) ( 4 - c ) = — ^ + 4 + 3c-c2
l-c^
^
'
l-c^
= - J _ + 4 ( l _ c 2 ) + 3c2+3c>2,

6

l { 2 - 3 t ; l + 2t;l + 2t).

Mat cau (S) tiep xuc voi hai mat phang (P) va ( Q ) nen

4
(a + b)(a + b + 2c)

(x + 4)^ + y 2 = 2 5

Vay p h u o n g t r i n h mat cau ( S ) : (x + i f + ( y - 3^ + (z - 3^ = 1 .

ww

1
(a + b)(b + c)

vol x, y > 0
x+ y

w.

y

4

^

27213

X

—

.c

^ S , . , „ = isE.AB =

1 1 4
Cau 6: A p dung bat dang thuc - + — >
Ta c6:

4

1BC =
4

ok

3V.S . A B C

AC

=

2x + y + 13 = 0

Giai h$ nay ta tim dupe ( x ; y ) = (-4;-5),(-8;3).

bo

Vay d ( C { S A B ) ) =

-7-

=M= ^ A ^ ^ l ^ ^ E H

BC

Suy ra SE = VsH^ + H E ^ = ^

a^>/3

ro

va ™

=

ce

Ve H E I A B ^ S E I A B

2

/g

Do do Vs.ABC = 3 S H . S , i A B C = 3- ^ - ^

om

1 aVs

Uijc

phuong trinh A H : 2x + y +13 = 0

Tpa dp A la nghi^m cua he:

fa

n^^A v

(x + 4)^ + y 2 = 2 5

Do do B ( l ; 0 ) , C ( - 7 ; - 4 )
A H // I M

AC

x-2y-l =0

Giai h?nay ta dupe cac cap nghifm (x;y) = { l ; 0 ) , ( - 7 ; - 4 ) .

a72, suy ra S ^ g c = ^ ( ^ ^ j =

Gpi H la chan duang cao ha t u S ciia tarn giac S A C ri> S H 1 ( A B C )
AC = a7i::.SH = ^

I . Vay minP = 8.
a=b = -

DSng thuc xay ra

xf" - _ , ) 2 n . 2 ( x C L - ^ C L .
1
^^(Uxf"-(l-xrdx

-i_.4fl-c2]=8

!

2

~

x^-^C^ir')

(1)

( l + x p ' - ( l - x )2n+l

,2n+l

2n + l

2n + l
0

(2)


Cty TNHH MTV DWH Khang Viet

Tuyen chqn & Gi&i thifu dethi Todn hgc - Nguyen Phu Khdnh, Nsuuen Tat Thu.

Ma:

j(xC2„ +

Tu ( l ) suy ra w'' =-z^

x3cL+... + x2"-i

w

3

-

Suy ra (2) » w^.|z|^° = z^ <=>
4

v2n
-2n

2n

Tu (1), (2) va (3) suy ra:

fz = 0

-2n
W =

2n-l
-2n

(3).

s

+

0:

z= l

• w= -l:

Z

5

,

= !=>

w

1

=

z =1

= 0 <=> w = 0, w = - 1
z5

v6 nghiem

2

=l

(z) = 1

Thu lai ta thay cap (w,z) = (-1,1) thoa yeu cau bai toan.

^2n-l
ic^„ . i c ^ -'-Cl +- + ^1 C
^^

22"-l

OETHITHllfSOZ

B. Theo chUorng trinh nang cao
/

= z^ =>

w

^

=ic^„+ici,+...+^c
2n
2n

. Tu {2) suy ra

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^2

Z

V^

Cau7b:Giasu (E): —
+ ^ = 1 voi a,b>0.
a^

I. PHAN CHUNG CHO TAT CA CAC THI SINH
Cau 1: Cho ham so y = x^ - 3x2 - 3m (m +1) x - 1

a2=b2 + 3

3

^c:>a2=4,b2=l.

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

b) Tim tat ca cac gia tri cua tham so m de ham so (l) c6 hai cue tri ciing dau.

4b2

s/

la^

1

Ta

Theo gia thiet bai toan ta c6 h^

Cau 2: Giai phuang trinh :

(l + tanx)(2cos2x-l)
rr
'- = 2V2 cos3x .
'

up

Suyra(E):^ + I - = l .

ro

sm x +

'(x2+l)y4+l = 2xy2(y3-l)

om

/g

XetM(xo;yo)e(E)^^ + y ^ = l = ^ y 2 = i - i .

Cau 3: Giai h$ phuong trinh:

xy2 |3xy'* - 2j = xy"* (x + 2y) +1

(voi x,y e

ok

.c

Suy ra P = (a + exg )^ + (a - exp )^ - 2(x^ + y2 j _ (a^ - e^x^)

71 '

fa

ce

bo

x= -l
x-l_y_z+l

Cau 8b: Xet h^ phuong trinh : < 2 ~ T ~ ~ ^ <=> y = - l ^ A ( - l ; - l ; 0 )
2x+y-2z+3=0
z=0

ww

w.

Goi I la tarn cua mat cau, suy ra I (l + 2t; t; -1 -1). Theo gia thiet bai toan ta c6

• t = l r : > l ( 3 ; l ; - 2 ) , R = IA = 2 7 6 = > ( S ) : ( x - 3 f + ( y - l f + ( z + 2 f =24
• t = -3t:>l(-5;-3;2),R = IA = 2V6=>(S):(x + 5 f + (y + 3 f + ( z - 2 f =24
Cau 9b: Tim cac so phiic w,z thoa:

8

w''' + z^=0

(1)

w^z-5=l

(2)'

Cau 4: Tinh tich phan: I =

'^l (x-l)sin(lnx) + xcos(lnx)
^
^ ^dx.

Cau 5: Cho hinh chop S.ABCD c6 day ABCD la hinh thoi canh a, BAD = 60°
va SA = SB = SD. Mat cau ngoai tiep hinh chop S.ABCD c6 ban kinh bang
va SA > a . Tinh the tich khoi chop S.ABCD .
5
Cau 6: Cho cac sothuc duong a,b,c thoa man a + b + c = 1.
3bc
2ca ^ 5
;— +
> —.
c + ab a + bc b + ca 3
I I . PHAN RIENG Thi sinh chi du(?c chpn lam mpt trong hai phan (phan A
hoac B)
A. Theo chUorng trinh chuan
^,

,

.

,

I

Chung mmh rang:

2ab

+

Tuyen chiftt b Giai thifu dethi

Todn HQC

- Nguyen

Phu Kh,\nh , Nguyen

Cau 7a: Trong mat phSng Oxy cho tam giac ABC npi tie'p duong tron (C):
(x-1)^ +{y-lf

Cau 8a: Trong khong gian Oxyz cho hai duong th^ng:

= m ( m + l ) x j - m ( m + l ) - 2 x j - 3 m ( m + l ) x j - 1 = |m^ + m + l j ( - 2 x j - l )

fx = l + t

Tuongty y2=(m^ + m + l j ( - 2 x 2 - l )

3 v —4 z - l
•
/\
==
vam|itph5ng ( a ) : x + y + z - l l = 0.
2
1 3

X —

Do do yiy2 > 0 <=> (2xj + l)(2x2 +1) > 0 o 4x,X2 + 2{xj + X2) +1 > 0

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z=l

Viet phuong trinh duong thang A c3t hai duong thang A,, A j va mat phang

o -4m (m + l ) + 5 > 0 < = > 4 m ^ + 4 m - 5 < 0 o — — — < m <
V
;
2

(a) lanluqrttai A , B , M thoa man A M = 2MB dong thoi A l A j .

Cau 2: Dieu kien: sin

Cau 9a: Gpi zi la nghi^m phuc c6 phan ao am cua phuong trinh z^ - 2z + 5 = 0.
2z-z^+l
= 1.
Tim tap hp-p cac diem Mcbieu dien so phuc z thoa:
z + zf+2

s/

up

lap phuong trinh cac canh ciia hinh vuong.
Cau 8b: Trong khong gian Oxyz cho diem A{3; 2; 3) va hai duong th3ng

Ta

Cau 7b: Trong m^t phSng voi h^ toa dp Oxy cho hinh vuong ABCD biet
M ( 2 ; 1 ) ; N { 4 ; - 2 ) ; P(2;0); Q ( l ; 2 ) Ian lupt thupc c^inh AB, BC CD, A D . Hay

cos2x = 2

X =

ok
bo

w.

HMGDANGIAI
Cau 1:

ww

I. PHAN CHUNG CHO TAT CA CAC THI SINH
a) Ban dpc ty lam
b) Tap xac djnh D = M.

Taco: y ' = 3 x ^ - 6 x - 3 m ( m + l ) => y' = O o x ^ - 2 x - m ( m + l)=:0
Ham so CO hai eye trj khi va chi khi ( l ) c6 hai nghifm phan bi^t x,,X2
o A'= 1+ m ( m + l ) = m^ + m + 1 > 0 dung voi Vm .
10

^

x)

x=±- +
6

k7t

Cau 3: H?

o

- + nT[,

4

X

= ± - + nn, n
6

G

Z .

x V + 2xy2+l + y*-2xy-'=0
3xV-2xy2-xV-2xy-'-l =0

fa

ce

— tie'p xiic voi Parabol y = x + m .

x?.tJ + k K . l

Ket hpp voi dieu ki^n ta c6 nghi^m cua phuong trinh da cho la:

va C ciia tam giac ABC biet di chua duong cao BH va d2 chua duong trung

Cau 9b: Tim m de do thj ham so' y =

^0; c o s x 5 t O o x = - 7 + k7i;
4

.

kn

7t

cos2x = 0

diem A ciing n^m trong mpt mat phang. Xac dinh toa dp cac dinh B

tuyen C M ciia tam giac ABC.

4j

2

o 2cos2x - 1 = 2cos3xcosx = cos4x + cos2x o 2cos2 2x -cos2x = 0

.c

d2cva

om

/g

ro

, x-2
y-3
z-3
« . x-1 y-4
z - 3 ^, ,
• . ^ - ^.i
dj : — — = — = — — va d2 : — ^ = ^ = — . Chung minh duong thang

7t 1
X + —

COS X (sin X + COS

B. Theo chiToTng trinh nang cao

di,

Vift

yj = x ^ - 3 x j - 3 m ( m + l ) x i - l = x , ( x ^ - 2 ) » i ) - ( x j - 2 x j j - 2 x j - 3 m ( m + l ) x i - l

giac ABC bang 12. Tim tpa dp cac dinh cua tam giac ABC.

:

Khang

V i X , langhiemciia ( l ) nen X j - 2 x j = m ( m + l ) . Suy ra:

=10. Diem M(0;2) la trung diem canh BC va di^n tich tam

y = -2 + t ,

DWH

CtyTNHHMTV

Tat Thu.

x2+24 + ^-2xy

y

x+

=- l

y

<=>

3 x V - ^ - x 2 - 2 x y - ^ =0
y'
y

3x2y^-2xy-

(do y = 0 khong la nghi^m ciia h?)
D l t a = x + ^ , b = xy,tac6he:

-2xy = - l

a2-2b = - l
a2-3b2+2b = 0

f
I

0 ' =0

x + —•

y J

a2=2b-l
W-4b

+l =0

b=l
a = ±l

1


Tuyen chgn & Giai thieu dethi Tomi h^c - Nguyen huu Khdnh , Nguyen

a=l
b=l

-7-

1

X =•

2

X = —

y

xy = 1

hoac

y=

Cty TNIIU Af IV DWH Khang Viet

TatThu^
X =

1+
2

Mat khac: S^BCD = ^S^^BD =

Vay the tich khoi chop S.ABCD la: V = |SH.SABCD = ^ ^ ' ^ =
Cau 6: Bat d3ng thuc can chung minh tuong duong voi
2ab
3bc
2ca
^5
(c + a)(c + b)^(a + b)(a + c)^(b + c)(b + c ) ~ 3 '

e'2

o 2ab(l - c ) + 3bc(l - a) + 2ca(l - b) > | ( l - a)(l - b)(l - c).

c2 „:

Cau4:Tac6I= j sin(lnx) + cos(lnx) d x - | ^'"^'"'^)(jx
1
71

o2

e2

"

.c

om

/g

ro

= (xsin(lnx) + cos(lnx)) ^ = e 2 - l .
Cau 5: Tu gia thiet, suy ra ABD la tarn giac deu nen SABD la hinh chop deu.
Goi H, O Ian luot la tarn ciia tarn giac ABD va hinh thoi ABCD.

up

s/

Ta

I x'sin(lnx) + x.(sin(lnx)) dx - [sin(lnx)d(lnx)
1
-I
1
o2

Suy ra S H I ( A B C D )

ww

w.

fa

ce

bo

ok

Mat phSng trung true canh SA cat SH tai I, ta c6 I la tarn mat cau ngoai
tie'p hinh chop S.ABD.
Vi ASFI - ASHA, suy ra — = — =^ SA^ = 2SI.SH
SH SA
Ma A H = - A O = ^ ^ S H 2 = S A 2 - ^ .
3
3
3
Nen ta c6 phuang trinh
2\

2^
12a' S A ^ - ^
SA^=4Sl2 SA^-^
<::>SA^2 = 2a' (loai)
SA^ = 2a2 => SA = aV2
12

SH =

.

~

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1 = -1 y 'X- =y - l = 0
<=>
he v6 nghiem.
b=l
y
1

y^ + y + l = 0
xy = 1
Vay nghifm ciia he da cho la: (x;y) = -1±V5 i + Vs^
71

,,2

4 11
<=> ab + 4bc + ca > 16abc <=>- + — + ->16.
a b c
11
4
Ap dung bat dang thuc - + — >
ta c6:
X
y x+y
4 1 1 4
4 ^ 16
,
- +—+- > - +
>
= 16 (dpcm).
a b c a b+c a + b + c
Dang thuc xay ra khi a = i , b = c =

II. PHAN RIENG T h i sinh chi dirg-c chpn lam mgt trong hai phan (phan A
hoac B)

{x-l)%(y-lf =10^ y=x+ 2 ^

A. Thee chUorng trinh chuan

x2=4

Cau 7a: Duong tron (C) c6 tam l(l;l)/ suy ra MI = (l;-l).
ViBCdiquaM va vuonggoc voi MI n e n B C : x - y + 2 = 0.
Toa dp B, C la nghiem ciia he:
"x = 2,y==24| a - b + 2|
Taco: d(A,BC) = l ^ — B C = 4V2 =>SAABC
Nen[ xta- CyO+!T2-=b0 + 2| = 6 <=> a =[x'=4
b + 4,a = b -Lx8.= -2,y = 0
Suyra
• a = bB(2;4),C(-2;0)
+ 4 thay vao (l)hoac
ta c6:B(-2;0),C(2;4) .
Gpi(bA(a;b),
a - l f + ( b - l f= 0<:^b
=10 (l)
+ 3 f + ( suyra
b - l f (=10c^b2+2b
= 0,b = -2
13


Tuyen chon t-^ Giai thifu aJthi Todn hpc - Nguyen Phu Khanh, Nxiiuen Tat Thu.

Cty TNHH MTV DWH Khang Vift

Do ABCD la hinh vuong nen d(P;AB) = d(Q; BC)

• a = b - 8 thay vao ( l ) ta c6: (b - 9)^ + (b -1)^ = 10 v6 nghi^m.
V$y A(0;4) hoac A ( 2 ; - 2 ) .

Hay

Cau8a: Vi A e A j , B e A 2 nen A ( l + a;-2 + a ; l ) , B(3 + 2b;4 + b ; l + 3b)

±

^

lO'lO'S

,B

23

'21

Zj

5_
= 1 - 21

/g

GQI M ( X ; y) diem bieu dien so' phuc z, suy ra z = x + yi

.c

z + zf + 2 (voi ( x ; y ) ^ ( l ; 4 ) )

ok

+1

o 4x2 + 4 (y +1)^ = (x -

^

ce

bo

2z-Zi

_

fa

z + Zj + 2

= l o

om

Taco: 2 z - Z i + 1 = 2x + 2 ( y + l ) i ; z + z^+ 2 = ( x - l ) + ( y - 4 ) i
2z-Zj+l

ww

V^y tap hp-p M la duong tron c6 phuong trinh :

3x2 + 3y2 + 2x + 16y-13 = 0.
B. Theo chi/orng trinh nang cao

Cau 7b: Gia su duong t h i n g AB qua M va c6 vec to phap tuyen la n(a;b)
^a^ + b^ 7t 0 ) suy ra vec to phap tuyen ciia BC la: fij ( - b ; a ) .
Phuong trinh AB c6 dang: ax + by - 2a - b = 0
BC CO dang: - bx + ay + 4b + 2a = 0
14

AD:2x+y-4=0.

;CD:-x+y+2=0.

0, M Q M J = ( - 1 ; 1; O) :

De thay t
a,b M o M i = 0

A, d i , d2 nam trong mpt

mat phang.
Taco B(2 + t;3 + t ; 3 - 2 t ) = : > M

t + 5 t+ 5
-;3-t
. 2 ' 2

Do M 6 d 2 = > t = - l = > B ( l ; 2 ; 5 ) , M ( 2 ; 2 ; 4 ) .
C ( l + c;4-2c;3 + c ) . D o AC 1 B H : ^ A C . i ^ = 0 c

= 0=>C(l;4;3)

Cau 9b: Hai duong cong da cho tiep xiic nhau <=> h^ phuang trinh sau c6 nghi^m:
x - x +1
x-1

w.

o 3 x 2 + 3 y 2 + 2 x + 16y-13 = 0 .

BC:2x+y-6=0;

M|it phiing (P) di qua d j , d2 c6 phuang trinh: x + y + z - 8 = 0

Ta

Cau 9a: Ta c6: z^ - 2z + 5 = 0 o z - 1 ± 2i

CD:x-2y-2=0

Taco "^b = (-3; -3; -3)

U's'sJ

z

9

Nen

14'

s/

y

5.-1
5__
-17
17

AB:x-2y=0;

Cau8b:di qua Mo(2;3;3) covectochi phuang a = ( l ; l ; - 2 )

up

X

b = -a

di qua M j (l;4;3) c6 vecto chi phuong b = ( l ; - 2 ; l )

17 17 9^

ro

Va phuong trinh A :

>

AD:-x-y+3=0

Mlitkhac A l A j =>AB.Uj =0<=>2a-3b-8 = 0 (2)
Tir ( 1 ) va ( 2 ) suyra a = — , b = - =>AB =
10
5
21
23
14

"b = -2a

• b = - a . Khi do
A B : - x + y + l = 0 ;BC: - x - y + 2 = 0

+ l i 2 b + 6 ^ 2 b + l - l l = 0 o 2 a + 12b-17 = 0 ( l )

f

<=>

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Gpi M ( x ; y ; z ) , t u A M = 2MB, ta c6: y - a + 2 = 2(4 + b - y ) <=>'{y = a+2b+6
3
z - l = 2(l + 3b-z)
z = 2b + l
Vi M € ( a ) nen i

2 + b2

-

• b = -2a suy ra phuong trinh cac canh can tim la:

a + 4b + 7
x=3

x - a - l = 2(3 + 2 b - x )

3b + 4a

-b

(.-if

2

=x

+m

(I)

= 2x

Taco: (2)<=>x(2x2 - 5 x + 4) = 0 o x = 0 thay vao ( l ) ta dupe m = - l .
V$y m = - 1 la gia tri can tim.


Tuyeh chgn & Giai thifu dethi Toan hgc - Nguyen Phu Khdtth , Nguyin

ct;/

T^tThu^

in M I \ nvH

Kttang

vtet

B. Theo chUofng trinh nang cao
Cau 7b: Trong mat

OETHITHUfs63

phcing

voi he toa do Oxy cho hai d i e m A ( l ; - l ) va B ( 4 ; 3 ) .

Tim toa dp cac diem C va D sao cho ABCD la hinh vuong.
I . PHAN CHUNG CHO TAT CA CAC THI SINH

Cau 8b: Trong khong gian voi h^ toa dp Oxyz cho duong thiing

Cau 1: Cho ham so y = x"* - (3m + 2) x^ + 4m c6 do thi la ( C ^ ) , voi m la tham so

A : ^ = ^ = ~

va mat phSng ( a ) : x + 2y - 2z - 1 = 0 . Viet phuong trinh mat phang (p) chua A

b) Tim tat ca cac gia trj cua tham so m de do thi (C^) cat Ox tai bo'n diem

va tao voi (a) mot goc nho nha't.

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a) Khao sat su bien thien va ve do thi ham so da cho khi m = 0 .
phan b i f t A, B, C, D (x^ < Xg < x^ < x^) thoa BC = 2AB .

Cau 9b: Cho cac so phiic p, q (q * O). Chung minh r i n g neu cac nghi^m cua

3x
x
3
Cau 2: Giai phuong trinh: cosx + 2\/3cos—sin—= cos3x + —.
I—

phuong trinh x^ + px + q^ = 0 c6 modun bang nhau thi ^ la so thuc.

Cau 3: Giai bat phuong trinh sau: ^Vx'^ + x + 2 < x^ + 3.
e X |lnx

Cau 4: Tinh tich phan sau: I = ,
J

Hl/dfNG DANGIAI

I . PHAN CHUNG CHO TAT CA CAC THI SINH

+ ln^ xj dx

Cau 1:
1 + V l + X In

a) Ban dpc t u lam

X

Cau 5: Cho lang try A B C A ' B ' C c6 day A B C la tam giac can A B = A C - a ,

Ta

up

diem cac canh C C va A ' B ' , mat phang ( A A ' C ) tao voi mat phang ( A B C )

b) Phuong trinh hoanh dp giao diem ciia (C^,) va Ox:

s/

B A C = 120° va A B ' vuong goc voi day ( A ' B ' C ) . Gpi M , N Ian lupt la trung

S'^'Us''"^ + S'^"^

Dat t = x^, t > 0 . T a c 6 p h u o n g t r i n h : t ^ - ( 3 m + 2)t + 4 m = 0

= ^ . Tim gia trj nho

A=9m^-4m+4>0
phanbi^t t j , t2 (t, < t 2 ) » • S = 3 m + 2 > 0

ok

nha't ciia bieu thuc: P = a^ + b^ + c^ + 3(a.2^ + b.2'' + c^'^ .

om>0

(l)

bo

P = 4m>0

fa

ce

I I . PHAN R I E N G Thi sinh chi duqc chpn lam mpt trong hai phan (phan A
hoac B)
A. Theo chUorng trinh chuan

w.

Cau 7a: Trong mat phSng Oxy cho tam giac ABC c6 M ( 1 ; 0 ) , N ( 4 ; - 3 ) Ian lupt

ww

la trung diem cua AB, AC; D(2;6) la chan duong cao ha Kr A len BC. Tim tpa
do cac dinh ciia tam giac ABC.
Cau 8a: Trong khong gian Oxyz cho ba duong thMng:
, x-1
y+1 z-1
, x+1 y - 1
z
, ,

X
y+1
2 •"
•
== — va d o : — = ^
2
3 - 1
- 2 - 4

(*)

(C^) ck Ox tai bon diem phan biet khi va chi khi (*) c6 hai nghiem duong

om

thoa

.c

Cau 6: Cho cac so thuc a , b , C € [ 0 ; l ]

/g

ro

mot goc 30" . Tinh the tich khoi lang try A B C A ' B ' C va c6 sin ciia goc giua
hai duong thSng A M va C N .

x ' * - ( 3 m + 2)x^ + 4 m = 0

Suy ra BC = 2 A B < = > 2 ^ = 2
Mat khac, ta c6:

(^-7t^)ot2

.

2

Viet phuong trinh mat phSng (a) di qua d2 va cMt d ^ d j Ian lupt tai A , B

=4ti

t, + t 2 = 3 m + 2

5tj = 3 m + 2

tjt2=4m

t^ = m

o 9 m ^ - 1 3 m + 4 = 0<=>m = l , m = -

z+1
=

Khi do: A(-^;0),B(-7t^;0),c(7t;";0),D(7tr;0)

^3m + 2^'

=m

(thoa ( l ) ) .

4
Vay m = 1, m = - la nhCrng gia trj can tim.

sao choAB=: Vl3 .
Cau 9a: Tim tat ca cac so phuc z thoa dieu kien: z^ = 4z .
16

Cau 2: Phuong trinh o 2(cos x - cos3\^+ 4Vs c o s : ^ i n - - 3 = 0
' ~

7

/

«

17


iux/ni

thou

Ir Cioi

ihicii

dc thi Toiin

lioc

- Ngtll/eii

Kluitih

,

'!'nt

Ihu.

<=> 4sin 2x.sin x + 2^3 (sin 2x - sin x) - 3 = 0

Nen B'K = A ' B ' s i n 6 0 ° = ^

o 2 s i n 2 x ( 2 s i n x + N/3)-V3(2sinx + >/3J = 0<=>(2sinx + V3)(2sin2x-V3)

sin2x =

+

X = — +

kn,

3

V3

Kzn,

X = —

6

X = — +

3

k2n
, keZ .

The tich khoi lang try: V = A B ' . S ^ A B C =

X = — + k7i

3

Nen ( C ' N , A M ) = ( E M , A M )

Cau 3: Dieu ki?n: x^ + x + 2 > 0 <=> (x + l)|x^ - x + 2 j > 0 <=> x > - 1 .

Vi A B ' 1 C ' N = > A E 1 E M = : > ( C ' N , A M ) = A M E

Bat phuong trinh o 5^(x + l)|x^ - x + 2J < 2(x + 1 ) + 2|x^ - x + 2 J

Taco:

x+1

„ x+1
^ ^„
/ x+1
, „
<2—
+ 2.Dat t =
-, t>0
x^-x + 2
x^-x + 2
x^ - X + 2

« 5

Ta c6: 5t <2t^ + 2 o t > 2 v

AE = i A B ' = - ; E M 2 = C ' N 2 =
2
4

t<-.
2

2

4

2

om

.

^ ox (in x + In^ x)dx ^ e in x (l + In x)dx
/
—— J
1
Cau 4: Ta c60 I — J
J 1 + Vl + x l n x
J l + Vl + x l n x

.c

va x >

—

1

=2 j ( t ^ - t ) d t =
1

ww

| L _ J _

w.

Suy ra

3

^XA2

A T : 2 T:X.2

AM

= AE + EM =

29a^

2(C'B'2+C'A'2)-A'B'2

•EM =

. . . a%/29
=> A M =
4

16
ME
Vliy cosAME = — = 2,,
^
M A V 29

Cau 6: Xet ham so f (x) = 2" - x - 1 , c6

f (x) = 2 ' ' l n 2 - l = * f ' ( x ) = 0 o x = l o g 2 ^ = xo
Lap bang bien thien va ket hop voi f (O) = f ( l ) = 0 ta suy ra dupe
f ( x ) < 0 , V x e [ 0 ; l ] hay 2 ' ' - x - l < a V x e [ 0 ; l "
Mat khac \/x,y,zeM,
x^ +

fa

Dat t = Vl + x l n x =>xlnx = t ^ - ! = > ( ! + l n x ) d x = 2tdt

1=

ok

^ 5 + V33

bo

-l
.

ce

^5-733

/g

Ket hgp dieu kien ta c6 nghiem cua bat phuong trinh da cho la:

.

s/

x>

ro

x2-x + 2

5 + >/33
V

up

/ " " " ^ — >4<=>4x^ - 5 x + 7 < 0 (v6nghiem)
x^ - X + 2
1
x +1
1
2 c
o n
5-N/33
• t<-<:>^5
•<-<=>x^-5x-2>0<»x<

Ta

• t>2<=>

2

C

Gpi E la trung diem cua A B ' , suy ra M E / / C ' N

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sinx = - - 2

Suy ra A B ' = B'K.tan30° = | ,

ta c6:

+ z'^ - 3xyz = ^ ( x + y + z) (x - y)^ + (y - zf + (z - x)^

Do do neu x + y + z<0=>x"' + y'' + z^<3xyz
1+e

2

(2N/i7T-3)-

Cau 5: Ta c6: BC^ - A B ^ + AC^ - 2AB.ACcos A = Sa^ => BC = aVs
Gpi K la hinh chieu ciia B' len A ' C , suy ra A ' C ' 1 { A B ' K )

T u d o d a n den: 8" - x^ - 1 < 3 x . 2 ' ' o 8" - 1 < x^ + 3x.2^ V x 6 [ 0 ; l "
Suy ra P>8^ +8'' + 8^-3 = 7
D i n g thuc xay ra khi va chi khi a = 0,b = 0,c = 1 va cac hoan v j
V?y m in P = 7 .

Do do A K B ' = ( ( A ' B ' C ' ) , ( A A ' C ' ) ) = 30°.
Trongtamgiac A ' K B ' c6 K A ^ ' = 6 0 ° , A ' B ' = a
18

1<


Tuye'tt chqn & Giai thifu ttethi Todn hoc - Nguyen Phii Khdnh , Nxm/en Tat Thu.
II. PHAN R I E N G

Cty TNHH MTV DWH

T h i sinh chi dirge chpn lam mpt trong hai phan (phan A

Suy ra trung diem cua AC tuong ving la

hoac B)
A. Theo chUcrng trinh chuan

1 1

2'2

Cau 7a: Ta c6 M N = (3; -3) va M N // BC nen phuong trinh BC:x + y - 8 = 0
Suy ra B ( b ; 8 - b ) . Do M la trung diem AB nen

(p) chua A nen

Cau8a:Tac6 A e d , ^ A ( l + a ; - l + 2 a ; l - a ) , Bedg ^ B ( - 2 b ; - l - 4 b ; - l + 2b)

COS(t) =

Tu AB = V]3=>(x + l f + 4 x 2 + ( x - 2 ) ^ = 1 3 c ^ x = -1,x = -

6d2 => A e ( a )

Xet T =

Ta

S u y r a n = AB,u =(7;-6;-4) la VTPT cua ( a ) .

up

s/

Phuong trinh ( a ) : 7 x - 6 y - 4 z + 13 = 0.

ro

_2

/g

3

om

Suy ra n = -3AB,u = (-14; 11; 5) la VTPT cua ( a ) .

ok

.c

Phuong trinh ( a ) : 14x - l l y - 5z + 25 = 0 .
Ta xet:

=2

= 4z => z^ = 4z.z = 4|z|^ = 16 o | z ^ -4j^z^ + 4J = 0 <=> z = ±2,z = ±2i

ww

Do do:

=4

Thu 1^1 ta thay bon nghi^m nay thoa phuong trinh .
V^y phuong trinh c6 5 nghif m: z = 0, z = ±2, z = +2i.
B. Theo chi/tfng trinh nang cao
fx = 4 - 4t

Cau 7b: Phuong trinh B

C : .
Gpi C ( 4 - 4 t ; 3 + 3t)
[y = 3 + 3t
^
'
Ta CO BC^ = AB^ = 25 o
+9t2 = 25 <» t = ±1
Do do CO hai diem C(0;6) va C(8;0) thoa man yeu cau bai toan.

20

2 + b2+c2

(a + 2b-2c)^
a^+b^+c^

=>T =

(4b-cf

5b2+2c2+4bc'
2

c ^ ( 2 T - l ) t 2 + 2t(2T + 4) + 5T-16 = 0.
1
27
• Voi T = - phuong trinh c6 nghiem * =

•
• Voi T 5-^ ^ de phuong trinh c6 nghiem t khi va chi khi
(2T + 4 f - ( 2 T - l ) . ( 5 T - 1 6 ) > 0 o 0 < T <

fa

= 4z => z ^ = 4

I a + 2b - 2c I

c ,
^
(4-t)
Datt = b^0),khid6T=
^
/
b ^
'
5 + 2t2+4t

ce

z^Q

w.

Tu

bo

Cau 9a: Ta thay z = 0 thoa phuong trinh

a- c+d= 0

Vi 0 < (t) < - nen (j) nho nhat khi cos^ Ion nhat.

• Vai x = l r ^ A B = (0;2;-3),tac6 u = (2;3;-l) laVTCPciia d j va A(-1;1;0)

3'

-a + 2b + c = 0

Coi (j) la goc giua hai mat phang (p) va ( a ) , suy ra

Suy ra A B - ( - a - 2 b - l ; - 2 ( a + 2b);a + 2 b - 2 ) , d5t x = a + 2b

3'

>D(5;-4).

Cau 8b:Goi PT (fi):ax + by + cz+ d = 0=> nj, = (a;b;c) va n,^^ = ( l ; 2 ; - 2 ) .

AD.MN = 0 o 3 b - 3 ( l 4 - b ) = 0 « b = 7

7 _8

^9 _ 1^
2' 2

Vay taco C(0;6) va D ( - 3 ; 2 ) hoac C(8;0) va D ( 5 ; - 4 ) .

A(2-b;b-8)

VayA(-5;-l),B(7;l),C(l3;-5).

• Voi x = - = > A B =
3

. D ( - 3 ; 2 ) va I

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Matkhac: A D 1 M N

Khang Vi

53

Do do d) nho nhat o t = - — o 13b = -10c .
^
10

Ket luan PT mat ph^ng (p) can tim la : 7x + 1 0 y - 1 3 z - 2 0 = 0.
Cau 9b: Goi z, = a + bi, Zj = c + di la hai nghiem ciia phuong trinh da cho
Ta c6:

Zj + Z j

= - p , ^1^2 =

•

•a p ' J z i + z ^ f _ z ? + z^ ^
Suy ra
ZiZ
z,z
1^2
1^2
Do

a2+b2=c2+d2 = k


Tuyi'n chqn Ct Giai tItiC'u dethi Totitt hqc - Nguyen Phu Khanh , Nnuyen Tat Thu.

C a u 6: (1 d i e m ) C h o cac s o ' d u o n g a, b, c thoa m a n (a + b + c)

Nen

z,z
1^2
A p d y n g bat d a n g thuc Bunhia, ta c6:

(ac + b d < (a2 +

)(c2 + d ^ ) =

fl

1

1^

Va

b

c)

= 16.

a2+2b2
T i m gia t r j Ion nhat va gia t r j nho nhat ciia bieu thuc P = ab

>-1

I I . P H A N R I E N G T h i sinh chi duQC chpn lam mpt trong h a i phan (phan A
hole B)

p2 2(ac + b d )
p
r= > ^ =- - ^
^ + 2 = m > 0 = > - i - = ±N/m l a m p t so thirc.

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A. Theo chUtfng trinh chuan

C a u 7a: (2 diem) T r o n g m a t p h ^ n g h ^ tpa dp O x y , cho h i n h t h o i A B C D c6 tarn
Cdch2:Ta
Do
„

Suy

c6 z , Z 2 = = q ^ ,

= k

q.q =

z2+pz,+q2=0^^^_p^z^^
•Z^--i-

[q

9t

0 =>

z , ;^

= k^

l(2;l)vaAC = 2BD.Diem M

q_

q

z,

p

p.zi

k^

p

d d i qua diem A ( - 1 ; 0 ; - 1 )

Zj

thupc d u a n g thSng A B ; d i e m N(0; 7) thupc

v a cat d u a n g thang d ' : ^ ^ - ^ ~ ^ i ^ ~

goc g i i i a d u o n g t h a n g d v a d u o n g thang d " : ^^—^ =

V | y z la so t h y c .

Ta
s/
up
ro

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

/g

C a u 1: C h o h a m so y = x"* - 3x + 1 ( l )

.c

ok

b) Xac d j n h m de p h u o n g t r i n h sau c6 4 n g h i e m t h y c p h a n b i ^ t :

om

a) Khao sat s y bien thien va ve d o thj ( C ) cua h a m so ( l ) .

bo

x|'' -3|x| = m-' - 3 m

3y

C a u 4: T i n h tich phan I =

- X =,

V

x''-3x + 2

x + 3y
-dx.

C a u 5: C h o h i n h chop S.ABCD c6 d a y A B C D la h i n h t h o i canh a, B D = a. Tren
canh A B lay M sac cho B M = 2 A M . Gpi I la giao d i e m cua A C va D M , SI v u o n g
goc v o i mat p h a n g day va mat ben ( S A B ) tao v o i day m p t goc 60".
T i n h the tich cua k h o i chop S.IMBC .

B. Theo chuomg trinh nang cao
C a u 7b: Viet p h u o n g trinh canh A B p h u o n g trinh d u a n g thang A B c6 h§ so goc
duong), A D cua h i n h v u o n g A B C D biet A ( 2 ; - 1 ) va d u o n g cheo B D c6 phuong
trinh: x + 2 y - 5 = 0 .
C a u 8b: C h o ba d i e m A ( 5 ; 3 ; - 1 ) , B ( 2 ; 3 ; - 4 ) , C ( 1 ; 2 ; 0 ) .

Chung m i n h r^ng tam

giac A B C la t a m giac d e u v a t i m tpa d p d i e m D sao cho t i i d i # n A B C D la t u

d i ^ n deu.
C a u 9b: T i m so p h u c z sao cho z^ va

la hai so p h u c lien h p p ciia n h a u .
z

fa
w.

4x-3x^y-9xy^

10 4 + ( x ^ - x ) . ^
3

ww

x2+2

C a u 3: Giai h ^ p h u o n g t r i n h :

ce

C a u 2: Giai p h u o n g t r i n h : 4cos^ 3xcos2x + cos8x = \/3sin4x + 2cos2x
_ 6x-y
(x2-l)%3 =

= - y - " ^ 6 nhat.

C a u 9a: T i m p h a n t h u c va p h a n ao cua so p h u c z biet rang z ^ - 1 2 = 2 i ( 3 - z ) .

DETHITHUfs64

22

1 '

C a u 8a: T r o n g k h o n g gian v o i h f tpa d p Oxyz, lap p h u o n g t r i n h d u a n g thSng

p
z,
k^
p.z,
z,
p
r a : z = = + -=i- = — = + ^—i= —L + J l - z
z,

'

d u o n g thSng C D . T i m tpa d p d i n h B biet B c6 hoanh d p d u o n g .

= -i- + .

q

0

Z j .Zj

HI/dNG 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
Caul:
a) Ban dpc t u l a m
b) P h u o n g t r i n h d a cho la p h u o n g t r i n h hoanh d p giao d i e m g i i i a d o t h i ( C )
cua h a m s o : y = x ^ - 3 | x | + l va d u o n g thSng ( d ) : y = m ^ - 3 m + l ( ( d ) cung
p h u o n g v o i true hoanh)
Xet h a m so: y = x - 3|x| + 1 , ta c6:
23


""f"

Tuyen chgn &

''^'t'" T"''" 'wc - Nguyen

Phi, Khauh , Nguyen

Tat

Cty TNHH

Thu.

+) H a m so la m o t h a m chan nen
V i X = 0 k h o n g la nghiem cua he, nen ta c6:

d o n g thoi Vx > 0 t h i

x^

x3

iL
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(a-b)(a2+ab + b2+2) = 0 ^ [ a - ' - 2 a + l = 0 ^ [ ( a - l ) ( a 2 + a - l ) = 0

- 1 < m-' - 3 m + 1 < 1

a =b

-2 < m < - V 3

a = l,a =

1

1'^'

• a = b = 1 , ta c6: •!

ro

up

2cos8x = \/3sin4x - c o s 4 x = 2sin 4 x - ^

da cho t u o n g d u o n g v o i :
( x ^ - 2 x + 4 ) ( x 2 + 2 ) = 6x5y
.

. 2 ^ 4 x - 3 x - y - 9 x -2.2
y

^ ^

'

x^ + 8 = 6x^y

.c

om

x + 3y

0

9y^ - 6xy + x^ =

x + 3y

(x^ - 3xy + 9y2 )(x + 3y) = 4x ^ jx^"* + 27y^ = 4x
24

C a u 4: Ta c6: I 3y

3xy
10

x^ + 8 = 6x^y

VVs-i

hoac

-VN/5-1

-2

-775 - 1

V7f^'

3
10

10

X +

X =

-2

-

{ x ; y ) = ±N/2;±-

x^ + 8 = 6x^y
O

3

D o i chieu dieu kien, ta c6 nghiem cua he da cho la:

w.

3y > x

ww

C a u 3: Tir p h u o n g t r i n h t h u hai, ta c6

X

y =

fa

da cho.

y =

2

X

la nghiem ctia p h u o n g t r i n h

, keZ

X -

hoac •

u
- 1 + V5
, x^^
2
• a=b=
ci> -i
3y _ 7 5 - 1

ok

x~ — + k 6
2

^

^75-1

2

bo

8x = - — + 4x + k27r
3

ce

7t
, n
x = — + k—
18
6

/g

.
71
2n
-4x
4x + — = cos
6

8x = — - 4 x + k2n
3

. X

s/

<=> 2cos2xcos6x + cos8x = \/3sin4x o cos8x + cos4x + cos8x = 7 3 s i n 4 x

x = 7^

1

'y-:

Ta

2 c o s 2 x ( l + cos6x) + cos8x = \/3sin4x + 2cos2x

n
<=> cos8x = cos —
2

+ Vs •

-1

0 < m < N/3

C a u 2 : P h u o n g t r i n h da cho t u o n g d u o n g v o i

o

fa = b

a=b

1 + a- = 2 b

da cho CO 4 nghiem phan bi^t la:

m

l + b-^=2a

^

dieu kien cua m de p h u o n g t r i n h

x2

1 + a-'' = 2b

2
3v
Dat a = - r - > 0, b =
, ta t h u d u g c h^:

+) D o thi ( C ) la:

- 3m + 2 > 0

X

, 27y^ _ 4

y = |xf-3|x| + l = x ^ - 3 x + l

+) D y a vao d o thj ( C ) ta suy ra

Khutig

1

( C ) nhan tri,tc O y l a m true d o i x u n g ,

m-' - 3 m < 0

MTV DWH

• A=4

j

^ x^-3x + 2
dx

dx + I =

10.

j

dx = A + B

^

• =4
[7
3. M' (x - ll ) ( - 2 ) "
i x-2

x-1

dx = 4 l n

• Dat t = 7 x - 2 = > X = t'^ + 2 => dx = 3 t ^ d t .

x-2
x-1

10

9

Vipt


Cty TNHH MTV P W H Khang Vift

Tuyen chgn C* Gi&i thifu dethi Todn hgc - Nguyen Phu Khanh , Nguyen Tat Thu.

Doican x = 3

Cau 6: Dat b = ay,c = ax; x, y > 0, t u gia thiet ta c6:

t = l , x = 10 => t = 2

2(t^ + 2)3t2
2
B= \^
^
dt = 3 j ( t - % 2 ) d t = 3 '-.2,
4

1 1
x2 + y + - - 1 3 x+y+i=o n
(1 + x + y) 1 + - + - = 16 <=>
x yj
j,b,c t o n t a i k h i (*) c6 ngiii^m x , y > 0 hay la:

69 . . . , 69 , , 1 6
= — . Vgy 1 = — + 41n — .
4
"^-^
4
9

Cau 5: G(?i H la hinh chieu ciia
1 len AB, suy ra AB 1 (SIH)

y>0

(SAB)

iL
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hi
Da
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01
/

=> S H I la goc giua mat ben

y+ i-13
y

va mat day

Do do S H I = 60" .
suy ra A B D = 60" va B D = y

2
o • y + - -30
l
y>

tam giac B D M ta c6:
n

up

BM.sin60''
fs
2^=r~-^ 4
^
= ^ - ^ c o s BDM = sin BDM = ^
OI = OD.tanBDM = - ^ ^ l

I

+ 161>0<=>

1
y + -<7
y .

K h i d o p = y + -^,khao sat f ( y ) = y + - voi y ^

.c

maxP = f

{

7-3S

21 + 375

AB

AB

ce

OB

a 73

fa

AI.OB

bo

Ta CO A A H I - AAOB
Al

w.

Suy ra SI = IHtan60° = —
^
8

, dat duQ-c khi

1 1

min P = f(72) = 272,

dat du(?c khi

b - ^/2a

1
V?y VsMlCB=iSI.SM,CB
26

a^Vs a^Vs lla^Va

1 3a Ila^x/S
3 8
48

48

3-S
^

nghi^m cua phuong

c = xa

trinh ( v / 2 + l ) x 2 - ( 1 3 V 2 - 3 ) X + N/2+1 = 0.

Cau 7a: Goi N ' la diem dol xung

4

a

c=•

hoac B)
A. Theo chUorng trinh chuan

SAABO ~ A B • A 0 " 3 ' 2 ~ 6

Do do S^^icB - S \ABC ~ ^ A A M I "

^
2

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

ww

_ A M AI _ 1

7-3V5 7 + 3N/5 ta tim dupe

b=

ok

diem A O
IH

^y^-^T—•

y + —<13

y

la trung

om

tan BDM = . — . !
,-1=
\^ BDM

SAMI

>0

ro

sin BDM

^

/g

sin60"

MD =

Ta

M D ^ = BMi±.RD.L-2JBMJ5D.cos60" = Z^l

s/

IVl

Mat khac:

y)

y >0

nen ap dung dinh l i c6 sin cho

BM

.

V + —<13
y

Do tam giac ABD deu nen

MD

-4(y + l)

48

lla^Vs
(dvtt).
384

cua N qua tam I thi ta c6 N ' ( 4 ; - 5 )
va N ' thuoc canh AB.
Suy ra M N ' =

V
nen phuong trinh AB: 4x + 3y - 1 = 0 .


Tuyen

ch
Phu Klidnh

, Nguyen

»

Tii't Thu.

V i A C = 2 B D nen A I = 2 B I .
Goi H la h i n h chieu cua I len AB, ta c6: I H = d ( l , A B ) =
1

IH^

1

1

l A ^ " IB^

5

416^ ^

Matkhac B€AB=>B b ; - ^

~

= 2

Viet

A B : x - 3 y - 5 = 0 va A D : 3 x + y - 5 = 0 .

Gpi G la trong tarn cua tarn giac ABC.

2
4b + 2

,b>0=^IB2=(b-2f +

SuV ra G

=:5ob

'8 8

5
3)

3'3'

= (-3; 15; 3) nen p h u o n g t r i n h true cua

va u = A B , A C
x =^ - t
3

Cau 8a: Goi B = d ' n A =:i> B(1 + 2t;2 + t ; - 2 - t) =:> A B = (2t + 2; t + 2 ; - t - ] )

3 V 6 t 2 + 14t + 9

5
f
- - t
v3
y

i o n nhat.

Vay CO hai d i e m D ( 2 ; 6 ; - l ) va D

14t2+18t

\
3

10._2__7^
3'

3'

9

I

7

5

4 5 2

1 _
z^

1

_ cos2(p-isin2(p

r^(cos2(p +isin2(p)

r^

ce

fa

T a c o : a ^ + b ^ - 1 2 = 2 i ( 3 - a - b i ) = 2 ( 3 - a ) i + 2b
a =3

Do do

• r'''[cos(-5(p) + i s i n ( - 5 ( p ) ] = -i-[cos(-2(p) + isin(-2(p)
r

ww

a 2 + b ^ - 1 2 = 2b " ^ j b ^ - 2 b - 3 = o ' ^ | b = - l , b = 3
r2

Suy ra phan t h u c va phan ao la 3 ; - 1 hoac 3 ; 3 .

C a u 7b: D o A B C D la h i n h v u o n g nen p h u o n g t r i n h A C : 2x - y - 5 = 0
G o i I la tarn cua h i n h v u o n g , suy ra I = B D n A C = > l ( 3 ; l )
r^C(4;3)^AC2=20.G
(kGZ)o^

5(p = 2(p + k2rt

B. Theo chi/orng trinh nang cao

B(5-2yo;y„)

D o 2 A B 2 = A C ^ O ( 3 - 2 y o ) ^ ( y o + 1 ) ' = 10 «

.5

= r'''(cos5(p + isin5(p)

z

w.

a=3

-5

thoa yeu cau bai toan.

3= 1
la hai s o p h u c lien h o p k h i va chi k h:i z^

va

bo

Cau 9a: G o i z = a + bi v 6 i a ; b e R

3,

1 r
= — cos(-2(p) + i s i n ( - 2 ( p )

.c

. x+1_ y _ z +1

. Phuang trinh A :
7'7'7^
-4

ok

' t = — =^AB =
7

9^

/g

L a p bang bien thien ta suy ra d u o c m a x f ( t ) = f

'

ro

(6t2 + 14t + 9)

om

-

up

C a u 9 b : D a t z = r(cos(p + i s i n c p ) , (pe [0;27:), t h i

6t2-.14t + 9

3

2
2
= 18 c ^ t = - ; t = —
3
3

+ -^ +t

s/

X e t h a m so: f ( t ) =

(2
- + 5t
3

Ta

6t^ +14t + 9

3

z=-^+t
3

V i D A = 3V2

Suy ra ( d " , A ) nho nhat

y = .^ + 5 t = ^ D
3
3

d u o n g tron tarn giac A B C la

2 I

- 2 t - 2 + 2t + 4 - 2 t - 2
3^(2t + 2 ) ' + ( t + 2 ) ' + ( t + l ) '

a =3

Khang

Cau 8b: Ta c6 A B = BC = C A = 3N/2 nen tarn giac A B C deu.

Vay B ( l ; - 1 ) .

cos ( d " , A ) =

DWH

i d u a n g thcing A B c6 he so goc d u o n g nen p h u o n g t r i n h

IHN/S

„

MTV

V a i yo = 2 = ^ B ( 1 ; 2 ) = > D ( 5 ; 0 ) .

iL
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Da
iH
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01
/

va

8 + 3-1

Cty TNHH

=>z = c o s - 5 ^ + i s i n ^ ^
3
3

r = l
k27t
(p = -

( v i (pe[0;27i) nen k = { 0 ; l ; 2 } . )

„
,
k27: . . k27t
,. ,
fr> 1 i
Vay so p h u c can t i m la z = cos — ^ +1 sin — j - v o i k = [u;V,2

yo = 0,yo = 2

• Voi y„=0=^B(5;0)=>D(l;2)
28

29


Tuye'n chqn & Giai thi^i dethi Todu hqc - Nguyen Phu Khanh, Nguyen Tai Thu.

Cau 9a; Cho so phuc z thoa man |z| = 1. Chung minh rang:

DETHITHUfSOS

l<|l + z^| + |l + z + z^|<5.
B. Theo chiTorng trinh nang cao

I. PHAN CHUNG CHO TAT CA CAC THI SINH

Cau 7b: Trong mat phSng Oxy cho duong tron (C): (x - 2)^ + (y -1)^ = 10 . Tim

Cau 1: Cho ham so y = x'' - 3(m + l)x^ + 3m(m + 2)x - 12m + 8 ( C ^ )

tpa dp cac dinh ciia hinh vuong M N P Q , biet M triing voi tam cua duong tron

a) Khao sat su bien thien va ve do thi ham so khi m = 0

^C); hai dinh N , Q thupc duong tron (C); duong thang P Q di qua E(-3;6) va

iL
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b) Tim m de do thj ( C ^ ) c6 hai diem cue trj A , B sao cho A M + B M nho

>0.

nhat voi M ( 3 ; 3 ) .

Cau 8b: Trong khong gian Oxyz cho hinh chop S.OABC c6 day OABC la hinh

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

thang vuong tai O va A ( 3 ; 0 ; 0 ) , AB = OA = i o C ,

y ^ + ( 4 x - l f = ^4x(8x + l )

S(0;3;4) va y c > 0 . Mpt

m l t ph^ng (a) di qua O va vuong goc voi SA cSt SB, SC tgi M va N . Tinh the

40x^ + x = y V l 4 x - l

tich khoi chop S O M N .
Cau 9b: Tim tap hpp cac diem M trong mat phang phuc bieu dien so'phuc z

Cau 4: Tinh di^n tich hinh phang gioi han boi cac duong
y = x ; y = x|2 + tan^xj va x = ^ .

/g

ro

up

om

P = 7 2 x 2 + 2 y 2 - 2 x + 2y + l + 7 2 x 2 + 2 y 2 + 2 x - 2 y + l+,y2x2+2y2 + 4x + 4y + 4 .

HlTdNG D A N G I A I

Ta

s/

Cau 5: Cho hinh chop S.ABCD c6 day ABCD la hinh thang vuong tai A va D,
tam giac SAD deu c6 canh bang 2a, BC = 3a . Cac mat ben tao voi day cac goc
bang nhau. Tinh the tich ciia khoi chop S.ABCD .
Cau 6: Tim gia trj nho nhat ciia bieu thuc

sao cho - ^ - i ^ la so'thuc duong.
z-i

I. PHAN CHUNG CHO TAT CA CAC THI SINH
Cau 1.
a) Ban dpc tu lam
b) TXD: D = ^
Taco: y' = 3 x ^ - 2 ( m + l ) x + m ( m + 2) i:>y' = O o

A. Theo chUcrng frinh chuan

Suy ra ( C ^ ) luon c6 hai diem cue trj A,B voi mpi m.

bo

ok

.c

II. PHAN RIENG Thi sinh chi duQc chpn lam mpt trong hai phan (phan A
hoac B)
5 3
8'8

Tim tpa dp tam duong tron npi tiep va tam duong tron bang tiep goc A cua
tam giac ABC.

ww

w.

fa

ce

Cau 7a: Trong mat phang Oxy cho tam giac A B C c6 A ( 1 ; 3 ) , B ( - 2 ; 0 ) , C

Cau 8a: Trong khong gian Oxyz cho ba duong th3ng
fx = -2t
. x - l y + 1 z-1
, x+1 y-1
z
, ,
di :
^^—^
, dj :
= ^ ^ = — va d , : y = - l - 4 t .
1
-1
-1
z = - l + 2t
Viet phuong trinh mat phSng (a) di qua d j va cat d^dg Ian lupt tai A, B
sao choAB = Vl3 .
30

Voi

Xj

Xj

=m

Xj

=m + 2

= m => y j = m'^ + 3m^ - 12m + 8 => A ^ m ; m^ + 3m^ - 12m

Voi X j = m + 2=>y2 = m'^+ 3 m ^ - 1 2 m + 4

+ sj

B^m+ 2;m^+ 3 m ^ - 1 2 m + 4J

Ta c6: AB = (2; -4) => AB = 2V5
D o d o : A M + B M > A B = 2>/5
D3ng thuc xay ra khi va chi khi AC = kAB, k > 0

(l)

Ma AC = | 3 - m ; - m - ' - 3 m ^ + 1 2 m - 6 J nen (l) tuong duong voi
3-m

m''+3m^-12m + 6

>0<:>

m <3
m^ + 3m2-10m = 0

o m = 0,m = 2,m = -5

31


Tuyen chpit b Gi&i thicu de thi Todit hoc - Nguyen Phii Khdnh , Nguyeu Tat Thu.

Vay m e {-5,0,2} la nhung gia tri can tim.

1
X = —

Cau 2: Dieu kien: cos2x ^0 .

Vay he da cho c6 nghi?m duy nhat

8

Phuang trinh da cho tuong duong voi
5 3
1
— +—cos4x = — sin 4x + tan 2x - 2 <=> 3cos4x = 4 sin 4x + 8 tan 2x - 21
8 8

2
2t
•, cos4x =
Dat t = tan2x, ta c6: sin4x ^
l + t^
1 + t^

"au 4: Phuong trinh hoanh do giao diem cua hai duong da cho

iL
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Da
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x = x 2 +tan^ X I <=> x = 0
va x|2 + t a n 2 x j - x : = x ( l + tan^ x j > 0 Vxe

Ta duQC phuang trinh :
-t')
1 +1^

8t

+ 8 t - 2 1 <=>3 l - t 2 U 8 t + 8 t ^ + 8 t - 2 1 t 2 - 2 1

1 +1^

nen dien tich can tinh la: S = j x | l + tan^ x dx
0

o 4t-^ - 9t^ + 8t -12 = 0 o (t - 2)(4t^ - 1 + 3) = 0

u =

c:>t = 2<=>tan2x = 2ci>x = -arctan2 + k—,

2

D|t

keZ

2

.c

80x^ + 2x = 2 y V l 4 x - l

bo

fa

(l).

w.

o ( y - V l 4 x - 1 9 6 x 2 _20x + 2 - ^ 4 x ( 8 x +1)

ce

y^ - 2 y V l 4 x - l + 14x - 1 + 96x^ -20x + 2 = ^4x{8\ l)

ok

Cpng hai phuong trinh ciia he voi nhau ta dupe:

ww

Ta c6: V T ( I ) > 96x^ - 20x + 2 = i 3(8x - i f + 8x +1 > | ( 8 x +1)
= -!-[l6x + 8x +1 + 2 ] >-^\6x{8x

+ 1)2 = 3/4x(8x +1) = V P ( l )

8

y =Vl4x-l =

Ta

X

dx

V

= tan X

4

S = X tan X f - f tan xdx = — + In I cos x 4I = ^ - i l n 2 ( d v t t ) .
0

J

A

Cau 5: Goi I la hinh chieu vuong goc cua S tren(ABCD), tuong tu nhu v i du
tren ta cijng c6 I la tam duong tron npi tiep hinh thang ABCD.
Vi tu giac ABCD ngoai tiep nen AB + DC = AD + BC = 5a
Di^n tich hinh thang ABCD la S = - ( A B + D C ) A D = -.5a.2a = Sa^
Goi p la nua chu vi va r la ban
kinh duong tron npi tiep ciia hinh
thang ABCD thi
AB + DC + A D + BC

10a

= 5a

S 5a^
va S = pr=>r = — =
= a=>IK = r = a.
p
5a

Tam giac SAD deu va c6 canh 2a nen

1

x =—
Suy ra (l) •

d v = 1 + tan

s/
up
om

/g

ro

Cau 3: Dk : x > — .
14
+16x^-8x + l = 3/4x(8x + l )

(du = dx

n

Ket hap dieu kien, ta c6 x = ^arctan2 + k ^ , k e Z la nghi^m cua phuang
trinh da cho.

X

. Thu lai h$ ta thay thoa man.

a\
S K ^ ^ = aV3::^SI = VsK2-IK2 ^Vsa^-a^ =a^/2
2

Vayv4si.s,3CD4a^-5^'~32

33


Tuye'n chgn ۥ* Gicri tltij-u dethi Toan hqc - Nguyen Phti Kliault, Nguyen Tii't Thu.

II. PHAN RIENG

Cau 6: Ta vie't PT duoi dang:
P = >/2

X

2

2j

2j

0

X+—

hole B)

A. Theo chUofng trinh chuan

V(x.l)^(y-Hlf

Cau 7a: Gpi K(x;y) la tam duong tron npi tiep tam giac ABC.

Ta CO BDT sau: Voi moi so'thuc a|,a2,bi,b2
+ b? + + b^ > ^(a, + a2)' + (b, + b2)' D
Thatvay (*) tuang dirong voi (ajbj-a2b,)>0 (diing)
Dang thuc xay ra khi va chi khi a^bj = ajbj.
Dat s = X + y . Ap dung (*) ta duoc:

AK.AB
AK.AB

(2)

Tir (l) va (2) suy ra: P > 2^8^+1 + |s + 2

(3)

3V2

s/
up
ro
/g

+\>-{s-S'f

om

>0,dod6:

(4)

.c

Ma

ww

w.

fa

ce

bo

ok

(5)
Lai c6: S-s + s + 2 >1 + S
Ket hop (3), (4), (s) suy ra: P > 2 +
Diing thuc xay ra khi va chi khi dSng thuc d (3), (4) va (5) dong thoi xay ra.
Khi do:
x=v =—

^ 2
x = - -6
s = -(V3-s)(s + 2)>0 y = Vgy minP = 2 +
34

AK.AC
AK.AC

-3(x-l)-3(y-3)

Dang thuc xay ra khi va chi khi x = y —

4i
s+-

. D^t dupe khi X = y = 6

COS(AK,AB) = C O S ( A K , A C )

(BK,BA)=(BK,BC)

COS(BK,BA) = COS(BK,BC)

AK.AB

AK.AC

AC
AB
BK.BA BK.BC

BK.BA BK.BC
BC
AB
BK.AB BK.BC
Ma AK = ( x - l ; y - 3 ) , B K = (x + 2;y),AB = (-3;-3) nen (*) tuong duong voi

\2

1
X —
X+—
y-2
2j
M|t khac theo B D T Cauchy2 - Schwarz
^(x.lf.(y.lf .Jl(s.2f = s+2
1

(AK,AB)=(AK,ACJ

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1

KAB = KAC
KBC = KBA

Ta c6:

Ta

1

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

-|(x-^)-f(y-3)
15N/2

21/ + 2 ) 3+ ^
3(x + 2) + 3y^yO<
3N/2
1572

8

2x-y

=- \ \

x - 2 y = -2

=

[y = r

Q

V|iyK(0;l).
Gpi j(a;b) la tam duong tron bang tiep goc A ciia tam giac ABC. Ta c6:
(AJ,AB)=(AJ,AC
(BJ,BC) = (BJ,AB)

AJ.AB AJ.AC
2a-b = -l
AB
AC
BJ.BC BJ.AB ' |2a + b = - 4 '
I BC
AB

5

_3^
4'"2j
Cau 8a: Ta CO A e dj A(1 + a; -1 + 2a; 1 - a), B 6 dg B(-2b; -1 - 4b; -1 + 2b)
Suy ra AB = ( - a - 2 b - l ; - 2 { a + 2b);a + 2 b - 2 ) , dat x = a + 2b
V9y J

r 5

Tu AB = >/l3=>(x + l)^+4x2+(x-2f =13<»x = -l,x = |
35

Cty TNHH MTV DWH Khang Vi?t

• Voi x = l = > A B = { 0 ; 2 ; - 3 ) , taco u = ( 2 ; 3 ; - l ) la V T C P cua d2 va A ( - 1 ; 1 ; 0 )
T a c o P ( l 5 - 3 x ; x ) va Q P = M Q

e d2 => A e ( a )
Suy ra n = A B , u

= ( 7 ; - 6 ; - 4 ) la V T P T c u a ( a ) .

Phuang trinh ( a ) : 7x - 6y - 4z +13 = 0 .

I

Suy ra n = - 3 A B , u

= ( - 1 4 ; l l ; 5 ) la V T P T ciia ( a ) .

3'

X = 3, ta C O P ( 6 ; 3 ) , suy ra tarn cua hinh vuong l ( 4 ; 2 ) nen N ( 5 ; 0 )

•

x = 5 , t a c 6 P (O; 5 ) , suy ra tarn ciia hinh vuong I ( l ; 3) nen N ( - 1 ; 2 ) .

M ( 2 ; 1 ) , N ( 5 ; 0 ) , P ( 6 ; 3 ) , Q ( 3 ; 4 ) va M { 2 ; 1 ) , N ( - 1 ; 2 ) , P ( 0 ; 5 ) , Q ( 3 ; 4 ) .

3/

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3'

•

Vay C O hai bo diem thoa yeu cau bai toan:

• Vai x = - = > A B =

3

( l 2 - 3x)^ + (4 - x)^ = 10 <=> x = 3,x = 5

Cau 8b: Do A B C D la thang vuong tai A va O, dong thoi A e Ox,yQ > 0, O C = 6
Nen ta suy ra dugc C ( 0 ; 6 ; 0 ) . Tuong tu ta c6:

Phuang trinh ( a ) : 14x - l l y - 5 z - 25 := 0 .
C a u 9a: V i |z| = 1 nen |z"| = Izl" = l , n e

Ta c6: SB = ( 3 ; - 3 ; - 4 ) , suy ra phuong trinh mat phang ( a ) : 3 x - 3 y - 4 z = 0.

Ta c6:

Vi SB = ( 3 ; 0 ; - 4 ) , S C = ( 0 ; 3 ; - 4 ) nen ta c6 phuong trinh:

1 + z'' + 1 + z + z^ < 1 + z^ +1 + Izl + z^ =5

1-z

SB:

1 + z^

l + z^

l-z'"*

2

2

. 1

2

up
/g

ro

M ( 3 ; 3 ; 0 ) va N

N/10

0;

bo

OS

fa

(x-2)%(y-ir = 10^ x= -l
3x + y + 3 = 0

ly =

0 •

t

96

72

36

(-12; 12;

-9),(oS A

OMJON =

.

3nn

Ta C O A M , B M bieu dien cac so'phuc z + i ; z - i , nen

M
Q(3;4).

la so'thuc duong
z-i

Khi do tga do Q la nghiem ciia h^

[y = 4

O M =

•

x + 3 y - 1 5 = 0.

3x + y + 3 = 0

A

C a u 9 b : G Q i A ( 0 ; - 1 ) , B ( 0 ; 1 ) la cac diem bieu dien so phuc z^ = - i ; Z g = i .

Q

Truong hop nay ta loai vi X Q > 0.

(x-2)^(y-lf =10^jx = 3 ^

7

Vay V s o M f , = : ^ ( d v t t ) .

w.

cua he

a = 3b, ta C O phuong trinh E Q : 3x + y + 3 = 0. K h i do tpa dp Q la nghi^m

ww

•

ce

< : > ( 5 a - 5 b f =10(a^ + b ^ ) » 3 a 2 - 1 0 a b + 3b^ =0<=>a = 3b,b = 3 a .

7

Suy ra O S = (0;3;4), O M = (3;3;0),

ok

=

z = -4t'

cua ( a ) voi SB, S C la:

.c

om

Phuong trinh E Q c6 dang: a(x + 3) + b(y - 6) = 0 <=> ax + by + 3 a - 6 b = 0
5a-5b

va S C :
Tir do ta tim dugc cac giao diem

l + z 3 + l - z 3 = 1.

C a u 7b: Ta c6 M ( 2 ; 1 ) va E Q la tiep tuyen cua ( C ) .

nen taco:

y=3

z = 4-4t

B. Theo chUorng trinh nang cao

Vi d ( M , E Q ) = N/To

x=0

Ta

l-z^

x = 3t

s/

1 + z^V 1 + z + z^ = 1 + z^

B(3;3;0).

•

j

khi va chi khi —
= k <=> A M = k B M <r> M A = kMB (k > O ) . Do do diem M nam
z-i
^
'
tren duong thang A B va nam ngoai doan thang A B .

\

37


Tuyen

CHQU

b Gicri

thij-u

tie thi Todn

hqc - Nguyen

P/iii Klidnh

, Nguyen

Tat

Cty

Thu.

TNHH

MTV

DWH

Khang

Vift

B. Theo chUcrng trinh nang cao

DETHITHUrSOe

C a u 7b: T r o n g m a t p h l n g O x y cho cho h i n h t h o i A B C D c6 A ( l ; 2), phucmg
t r i n h B D la: x - y - 1 = 0 . T i m toa dp cac d i n h con lai ciia h i n h t h o i , biet rSng
BD = 2 A C va B c6 t u n g d p am.

I. PHAN C H U N G C H O T A T C A C A C THI SINH
C a u 1 : C h o ham so y =

C a u Sb: T r o n g k h o n g gian O x y z cho d i e m A(3; 2; 3) va hai d u o n g t h i n g :

- 3 m x ^ + ( m - l ) x - m ^ + m (1).

d , :•

a) Khao sat s ^ bien thien va ve do thj ham so (1) khi m = 1
y = -2m^ tai ba diem phan biet A , B, C (voi

1

1

z-3
-

2

, ,
x-1
va d , :
"
1

y-4
-

1

Xac d j n h toa d p cac d i n h B va C cua tam giac A B C biet d i chua d u o n g cao B H

< X g < x^ ) sao cho doan thang

va d2 chua d u o n g t r u n g t u y e n C M cua tam giac A B C .
8(sinS + cosS)
(
f
tan X +
tan X
I
6j
I

—

Cau 9b: Ti'nh gia t r i bieu thuc:

^
673
+ 2=tan2x + cot2x

c_r"
-^r^
4-q2p4
,
, / i\kp2k
,
, ol004 p2008 01005^-2010
i»-*-2010
' ^ * - 2 0 1 0 * - 2 0 1 0 + - + V~U * - 2 0 1 0 + - + '^
^2010"-^
*-2010

3)

Hi;(3rNGDiiNGiiii

C a u 3: Giai p h u a n g t r i n h : Vx^ + 3 x + 6 + V2x^ - 1 = 3x + 1

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

^.
^, , , , , ,

, V
xlnx
,
C a u 4: T m h tich p h a n : 1=
^dx .
^•^(lnx + x + 1)

Cau 1 :

s/

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

ro

d j n h tam va t i n h ban k i n h mat cau ngoai tiep h i n h chop S.BCE.

/g

+ 2xyz = 1 . C h u n g m i n h rang:

<»(x-m)(x^ -2mx-2m^ + m - l ) = 0

om

C a u 6: Cho x , y , z > 0 thoa x^ + y^ +

b) P h u a n g t r i n h hoanh d p giao d i e m cua hai d o t h j

up

C D = 2a. Canh ben SD 1 ( A B C D ) va SD = a. Goi E la t r u n g d i e m cua D C . Xac

Ta

a) Ban dpc t u l a m

C a u 5: Cho h i n h chop S.ABCD c6 day la h i n h thang v u o n g tai A , D , A B = A D = a,

X

.c

8 ( x + y + z)^ < lOlx-* + y-' + z-*) +11(1 + 4xyz)(x + y + z) - 12xyz .

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

ok

ce

D o thj h a m so (1) c i t d u o n g t h i n g y = - 2 m ^ tai hai d i e m p h a n bi^t k h i va

fa

chi k h i (*) CO hai n g h i ^ m phan bipt X i , X 2 khac m .

ww

w.

t r i n h d u o n g phan giac t r o n g cua B va C Ian lupt la: d j : x - 2 y + l = 0 va

Hay
x-1

C a u 8a: T r o n g k h o n g gian O x y z cho hai d u o n g thang Aj : —y— =

y+1
^

V i f ( m ) < 0 nen x j < m < X 2 . D o d 6 : A ( x i ; - 2 m ^ ) , c ( x 2 ; - 2 m ^ )

Suy ra AC^ = ( x j - X2)^ = 4{3m2 - m + 1 ) = 4 3 ( m
I

73-

T i m tap h p p d i e m bieu d i e n so p h u c vv = 2z - i + 3 .

<=> m 6 R
f ( m ) = -3m'^ + m - l ? t O

= ——

1

Cau 9a: Cho so p h u c z thoa dieu k i ^ n : |z - 2 + 3i| = 5 .

A' = 3m^ - m + 1 > 0

z-2

va A2 : ^ = ^^—^ = Y • Viet p h u a n g t r i n h mat phang ( P ) chua d u o n g thang
va tao v a i d u o n g thang A2 m p t goc cp thoa coscp =

(*)

Gpi f ( x ) la v e t r a i ciia (*).

C a u 7a: T r o n g m a t ph5ng ( O x y ) cho tam giac A B C v o i A ( 2 ; - 1 ) v a p h u a n g

d j : x + y + 3 = 0 . Viet p h u a n g t r i n h canh BC.

=m

_x^-2mx-2m^+ m - l =0

bo

hoac B)
A. Theo chuorng trinh chuan

38

2

z-3

C h u n g m i n h d u o n g thang d i , d2 va d i e m A c i i n g nam t r o n g m p t m a t phSng.

A C CO dp dai nho nha't.
C a u 2: Giai phuong trinh:

y - 3

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b) T i m tat ca cac gia trj ciia tham so m de do thj ham so (1) cit duong thing

x-2

t

11'

—
6J

111 11

+— > —
12
3

D i n g t h u c xay ra k h i m = - . V^y A C nho nha't k h i m = i .
6
6
39


Tuye'n chQtt & Gi&i thifu dethi Toan h,n

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

cos

I

tan

X

71
X + —

I

*0

6)

3J

(X

\'guycti Phu Khanh , Nguyen Tat Thu.

71
+ —

COS

A

X

(a)

2^2x2-1 = 2 x - l

(b)

n

^0

, ,

6j

0

sin4x

7x2-4x-8 =0

(a) C O nghnjm x =
1

t a n 2 x + cot2x9!:0

X

^

I

X + —

.tan

X

3;

\

6>

= - tan

7t^
X + —

^

6;

(b)<=>

.cot x + — = - 1
I
6;

cho la:

N e n p h u o n g t r i n h da cho t u o n g d u o n g v o i :

<=> cos 4x — = - l o 4 x
3

—

, = 7t + k27i

K

kTt

<=> x = — + — ,

3
2

3

Dat t =

s/
up
fa

ce

<» 7x^ - 4 x - 8 + (3x + l ) ( x + 2) - 2 ( 3 x + 1 ) V 2 x ^ = 0

bo

T a c o : (2)<=>2(3x + 1 ) ^ 2 x 2 - 1 = lOx^ + 3 x - 6

w.

(3)

ww

x<-2
2N/2X^ - 1 = - x - 2 <=>
,
v 6 nghiem,
7x2-4x-8 =0

^

0, Vx € K . D o do p h u a n g t r i n h (3) t u a n g d u o n g v o i

7x^ - 4 x - 8 + (3x + l )

4x2 + 4 x - 5 = 0

= 0

3x + l

2 + 2>/T5

;X=

--i + S

.

rdx.

Inx +1

+1

> d t = - - i ^ ^ d x . D o i can: x = l = > t = l , x = e = > t = 1

V

Suy ra I

dt

_

iCt + l f

1

1

3 ( t + l)2 2

1

r

e2

3 l ( e + 2)2

l ]
4j

Cau 5: V i A B = D E = A D = a va
D A B = l v nen A B E D la h i n h
vuong. Tarn giac B D C c6 EB =
ED = EC = a nen v u o n g tai B,
BE 1 C D nen t r u n g d i e m M

ciia BC la tam

duong

tron

ngoai tie'p tam giac EBC. D u n g
A la true d u o n g

tron

ngoai

tie'p tam giac EBC t h i A song
song v o i SD
Dyng

mat

phang

t r u n g t r y c canh SC, m a t

(x + 2 ) ^ - 4 { 2 x 2 - l )
X + 2 + 2N/2X2 - 1

(7x2-4x-8) 1-

<=> X =

ro
(2)

om

x^ + 3 x + 6 = l l x ^ + 6 x - 2 ( 3 x + l ) V 2 x ^ - 1

.c

(1)

ok

3x + l - V 2 x ^ - l > 0

/g

P h u o n g t r i n h <=> Vx^ + 3 x + 6 = 3x + 1 - \ l 2 x ^ - 1

< = > 7 x ^ - 4 x - 8 + (3x + l ) ( x + 2 - 2 V 2 x 2 - l ) = 0

x=

Hnx + l

Cau 3: D i e u k i ^ n : |x| > i

nen x + 2 + 2V2x^ - 1

4(2x2 - 1 ) = 4 x 2 - 4 x + l

Cau 4: Ta c6: I = J

^/3 sin 4x + cos4x = - 2

Ke't h o p dieu kien ta c6 p h u a n g t r i n h da cho v 6 n g h i ^ m .

V i : x + 2 + 2\/2x^ - 1 = 0 «

2

Ta

<=> - 6 + 3(1 - cos4x) = 3N/3 sin4x o
71

- l + ^/6

> -

Inx

- 8 ( s i n ' ^ x + c o s ^ x ] + 2 = 3>y3sin4x<=>-8 l - - s i n ^ 2 x + 2 = 3>/3sin4x
^
'
V
4
;

.

X

2

D o i chieu v a i (1) va dieu kien bai toan ta c6 n g h i ? m cua p h u a n g t r i n h da

t a n 2 x + cot2x = •
sin4x

,

1

> -

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Ta c6: tan

212VT5

phang do cSt A tai I
D i e m I la tam m a t cau ngoai tie'p h i n h chop S.BCE
Ke SN // D M Cc4t M I tai N ta c6 S D M N la h i n h chu nhat v a i SD = a va

= 0

x + 2+ 272x2-1
40

41


Cty TNHHMTV

= 8(x"' 4-^^ +z-') + 2(x-' H-y-'
2
Ta

CO

4

2

4

DWH

KHangVm

+6xyz) + n ( l + 4xyz)(x + y + z ) - 2 4 x y z

2'
2 8(x'' + y"* + z^) + 24 (xy + yz + zx)(x + y + z) - 24xyz = 8(x + y + z)^

SI^ = SN^ + N I ^ = SN^ + ( N M - IM)^ = ^a^ + (a - IM)^

Suy ra dieu phai chirng minh. Dang thiic xay ra khi va chi khi x = y = z = i .
Ma IC^ = I M ^ + MC^ = I M ^ + — va R = IC = IS
2

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

nen - a ^ + ( a - I M ) ^ = I M ^ + — o I M = - a
2
V
;
2
2

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hoac B)
Theo chUorng trinh chuan
Cau 7a: Gpi A ] , A2 Ian lupt la cac diem dol xirng voi A qua d^ va d2 , ta tim

Vay ban kinh mat cau ngoai tie'p hinh chop S.BCE la:

dupe A , ( 0 ; 3 ) va A 2 ( - 2 ; - 5 )

R =,IM +—=
a.
V
2
2
Cau 6: Voi x, y, z > 0 ta chung minh hai bat dling thuc sau

•

x"'+ y^ + z"'+ 6xyz > (x + y + z)(xy + yz +zx)

(1)

l + 4 x y z > 2 ( x y + yz + zx)

(2)

Theo rinh chat duong phan giac ta suy ra A , , A2 G BC
Phuong trinh BC: 4x - y + 3 = 0 .

\

Cau 8a: Duong thang Aj di qua A ( l ; - 1 ; 2 ) va c6 u7 = ( l ; 2 ; - l ) la VTCP
Duong thiing A2 c6 U j = ( 2 ; - 2 ; l ) la VTCP

Chung minh (1): Khong giam tinh tong quat la gia su x = min {x, y, z}

Gpi n = (a;b;c) lamptVTPTciia (a)

s/

Ta

x^ + y"' + z^ + 6xyz - (x + y + z)(xy + yz + zx)

up

= x ( x - y ) ( x - z ) + (y + z - x ) ( y - z ) ^ >0

om

/g

Chung minh (2): Nhgn thay trong ba so x , y , z luon ton tai hai so sao cho

.c

chung cimg Ion han hoac cung nho hon i . Khong giam tinh tong quat ta

(3)

bo

z(2x - l ) ( 2 y - 1 ) > 0 o 4xyz > 2zx + 2zy - z

ok

gia sir hai so do la x, y

MM khac cos(p =

w.

U2

34J7h^

^ J a 2 + b 2 + ( a + 2b)2

'

^

• a = 5b, ta chpn b = 1 =>a =5,c = 7. Phuong trinh (a) la: 5x+y+7z-18=0.
Cau 9a: Ta c6 z = ——^—- nen dieu ki^n bai toan dupe viet lai nhu sau:

(4)

Theo gia thie't x^ + y^ + z^ + 2xyz - 1 suy ra xy < ^ ^ ^

—
n

|2a - 2b + c

• a = - b , ta chpn a = 1 => b = - l , c = - 1 . Phuong trinh ( a ) la: x - y - z = 0 .

| w - 7 + 7i| = 10.

fa

hay z + 2xy < 1

n.u-

o 3 a 2 =2a2 +4ab + 5b^ » a ^ - 4 a b - 5 b ^ =0c:>a = -b,a = 5 b .

ce

Ta chi can chiing minh: 1 + 2xz + 2zy - z > 2(xy + yz + zx)

ww

•

ro

Suy ra dieu phai chiing minh. Dang thiic xay ra khi va chi khi x = y = z > 0.

Ta c6: r i . u = 0 <=> a + 2b - c = 0 => c = a + 2b

< 1 => 1 - xy > 0

Gpi M ( x ; y ) la diem bieu dien so phiic w, ta eo: (x - 7 ) ^ + ( y + 7)^ = 100
Do do tap hpp ciia diem M la duong tron tam I ( 7 ; - 7 ) , ban kinh R = 10.

Do do (4) tuong duong voi: (z + xy)^ ^ ( l - xy)^ <=> 1 - (z^ + 2xyzj - 2xy > 0

B. Theo chUorng trinh nang cao

o ( x - y ) ^ > 0 (dung)

Cau 7b:Ta c6 AC 1 BD nen phuong trinh A C : x + y - 3 = 0 .
Gpi I la giao ciia hai duong cheo AC va BD, suy ra tpa dp ciia I la nghipm

Do do (2) dupe chiing minh. Dang thuc xay ra khi va chi khi x = y = z = ^ .
Sir dung cac BDT (1) va (2) tn c6:
10(x^ +y^ + z^) + n ( l + 4xyz)(x + y + z)-12xyz

ciia h f
[x-y-l =0

x=2

x+y-3=0

[y=l

.I(2;1)=>C(3;0)
4]

CtyTNim

TuifS'ti chpn & Gi&i thifu dethi Todtt hqc - Nguyen Phti Kltdnh , Nguyen Tat Tttu.

G p i B ( b ; b - 1 ) , ta c6 BD = 2 A C

x-1
Cau 4: T i n h tich p h a n : I = f,-dx.
^
jx(x-lnx)

IB = 2 I A

» ( b - 2 ) ^ + ( b - 2 ) ^ = 8 c 5 . b = 4,b = 0. Do b - l < 0 = > b = 0 .
Vay B ( 0 ; - 1 ) ,

Cau 5: C h o lang t r y A B C . A ' B ' C c6 day A B C la tam giac v u o n g tai A , A B = a,

D(4;3).

/i^C = a 7 3 ; A ' A = A ' B = A ' C . M a t ph5ng ( A ' A B ) tao v o i m a t p h ^ n g (ABC)

Cau 8b: Ta c6: d j qua M Q (2;3;3) c6 vecto chi p h u a n g a = ( l ; l ; - 2 )

inpt goc 60" . T i n h the tich k h o i lang t r u va c6 sin ciia goc giira hai d u o n g

d2 qua M i ( l ; 4 ; 3 ) c6 vecta chi p h u a n g b = ( l ; - 2 ; l )

th^ng A C va A ' B .

Ta CO a,b ?i 0 va a, b M o M i ' = 0

iL
ie
uO
nT
hi
Da
iH
oc
01
/

Cau 6: Cho cac so t h u c d u o n g x, y, z thay d o i .

G Q I ( a ) la m a t phcing d i qua hai d u o n g thang d ] , d 2 .

T i m gia t r j I a n nhat cua bieu thuc P =

Ta CO p h u a n g t r i n h cua ( a ) la: x + y + z - 8 = 0 . Suy ra A € ( a ) .
Goi B ( 2 + t ; 3 + t ; 3 - 2 t ) , s u y ra i v l f — ;
V 2

Cau 7a: T r o n g m a t phang v o i h ^ toa d o O x y , cho t a m giac A B C biet A ( 5 ; 2 ) .
Phuong t r i n h d u o n g t r u n g true canh BC, d u a n g t r u n g t u y e n C C Ian l u p t la
x + y - 6 = 0 v a 2 x - y + 3 = 0 . T i m toa d o cac d i n h cua t a m giac A B C .

Ta

2S

Cau 8a: T r o n g k h o n g gian v o i he toa d o O x y z cho d u o n g thang

^x2010

s/

.

=

2 0 1 0 / _ „ ^ r 7 n _ \0

bo
fa

+ Sx^ + (3 - m ) x + 3 - m ( C ^ ) .

tai ba d i e m c6 hoanh d o k h o n g nho h o n - 9 .
C a u 2: Giai p h u a n g t r i n h : ( c o s 2 x - 5 c o s x . 3 ) ( 2 s i n x - l ) ^
2cosx-l
^
C a u 3: Giai p h u a n g t r m h : ^ x - 1 + 44

3

Cau 8b: T r o n g h ^ toa d p O x y z , cho d u o n g thang d :

b) T i m ta't ca cac gia t r j cua tham so so m de ( C ^ ) ck d u a n g thSng y = - 1 4

, , oy

'

2

voi n la so'nguyen duong thoa man C „ + 2 n = A ^ + i .

Bi ( 2 ; 2 ) ,

C, ( - l ; 2 ) . .

a) Khao sat s u bien thien va ve d o thj ham so k h i m = 3.

,

Cau 9a: Xac d j n h so hang k h o n g p h u thupe vao x k h i k h a i trien bieu thuc

dp chan cac duong cao ha tu cac dinh A, B, C Ian lupt la: A i ( - l ; - 2 ) ,

ww

C a u 1: Cho h a m so y =

vamatphSng (P):2x + y - 2 z + 9 = 0 .

Cau 7b: C h o tam giac A B C nhpn, viet phuang trinh d u a n g thing A C , biet toa

w.

I. PHAN CHUNG C H O T A T CA CAC T H I S I N H

=^

B. T h e o chUtfng t r i n h n a n g c a o

ce

OETHITHUfSO?

= l^

Viet p h u a n g t r i n h d u a n g t h ^ n g A nam t r o n g ( P ) cat va v u o n g goc v o i d .

f(x) =

.c

S = 22010

ok

Vay

om

= 2.2201''(cos6707r) = 2.2

OTI^

ro

-20107t
-201
_ 22010 ( cos 2010n + i.s i.n 201071 + 2 2010
cos• + isin33
3
V
J
y

:lzl

up

+(l-iV3)

/g

M a ( l + iV3)

,

>/3z^ + xy

A. T h e o chUtfng t r i n h c h u a n

D o AC.a = 0 = ^ t = 0 ^ C ( l ; 4 ; 3 ) .

• /^\201()

yJ5y^ + zx

hole B)

V i C 6 d 2 nen C ( l + t ; 4 - 2 t ; 3 + t )

l^

X' + y z

I I . PHAN R I E N G T h i sinh chi du
—; 3 - t
2

Ma M e d 2 =>t = - l = > B(l;2;5)

Cau 9b: Ta c6: (l + i Vs)""" + (l -

MTV DWH Khaiig Vift

_

^_
1^

d sao cho khoang each t u B den A Ion nhat.
Cau 9b: Gia

'

x - 1 __ y + 2 _ z
= — va hai
-1

le p h u a n g t r i n h :

22x-y ^2'< =2'""^
log2 x(log4 y - 1 ) = 4

^3x + 3(^/5rn" + 2)
x-5

l

45


Tuye'n ch(,m b Giai

dethi Toiin

thiftt

liQC

-

Cty TNHH MTV DWH Khang

Nguyen Phu Khanh, Nguyen TA't Thu.

Hl/OfNG DANGlAl
I. PHAN CHUNG CHO TAT CA CAC THI SINH
Cau 1:

x-^ -15x^ +72x-128 = 0

x=8

x^ -21x^

x=9

+146X-342 = 0

Cau 4: Dat t = x - In x => dt =

a) Ban dpc t u lam

Cau 5: Ggi H la hinh chieu ciia A' len (ABC). Do A ' A = A ' B = A ' C nen H la
tam duong tron ngoai tie'p tam giac ABC, suy ra H la trung diem canh BC.

+1

Gpi K la trung diem AB, suy ra HK 1 AB . Suy ra AB ± ( A ' K H ) nen A I C H

=> f (x) = 0 « x = 1

1^

la goc giOa hai mat phang ( A ' A B ) voi (ABC) nen A ' K H = 60".

(x + l)2
-1
^

Yeu cau bai toan o

+00

12 "

(*) c6 ba nghi^m phan bi^t khong nho Hon - 9 .

/g

Dieu do xay ra khi va chi khi 12 < m < 62.

ce

bo

ok

.c

om

Cau2:Dieu ki^n: cosx?t — o x ^ t i — + k27t
2
3
(2cos^ X - 5cosx+ 2 ) ( 2 s i n x - l )
Phuang trinh o
^—
^
= (>/3 - l)(cosx - 2)

w.

fa

o 2 s i n x - l = V 3 - l c > s i n x = — o x = — + n27t, x = — + 2n7t.
2
3
3

ww

Ket UQP dieu ki?n, ta c6 x = — + 2n7t, n € Z la nghi?m ciia phuong trinh.
3
Cau 3: Dieu ki?n: x ;>t 5.
FT < » ( x - 5 ) ^ ^ + ( x - 5 ) ^ / 3 5 r r 3 - ( x - 5 ) ( x - 7 ) - ^ 3 ( x 2 - l ) + 2 ^ ^ ^

^ j r ^ ( x - 5 - ^/3xT3) + (x - 7)(^/3xT3 - X + 5) = 0

«(x-5-^/3^)(^/^-x

+ 7) = 0 o

" x - 5 = ^'3xT3

:-7 = ^ / ^

, suy ra

A ' H = HKtan60" = y ,

+00

-00

, „ ^ AC
CO H K =
=

Ta

62

s/

f(x)

+

0

-

-

^
Ta

+00

up

f(x)

1

ro

Bang bien
x thien -9

=1-e-1

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+1

Xethamso f ( x ) = (x + l ) ^ + — voi x > - 9
Taco: f'(x) = 2(x + l )

dx .

c-l

cit
Suy ra I = J - y

o ( x + l v V l 6 = m(x + l ) c * ( x + l)^+ — = m (do x ^ - 1 (*)

X

la nghi^m ciia phuong trinh da cho.

Doi cgn: x = l = > t = l , x = e=>t = e - l

b) Phuang trinh hoanh dp giao diem
x'' + 3x^ + ( 3 - m ) x + 3 - m = -14<»x-^ + 3x^ + ( 3 - m ) x + 1 7 - m =0
X

x-1

Vift

S.\ABC

=

2^^-^^"~T~

The tich khoi lang try:
,,
.,„„
3a 3^73
V = A H.S ^ A BC = — •
AABC

2

3a3^/3
=

2

.
4

GQ'I E la trung diem ciia A'C

A

suy ra HE // A ' B nen goc giiia hai
duong thcing A ' B va A C la goc giiia
hai duong thang HE va AE.
BC
Ta CO A H = — = a, ggi M la trung diem HC
Suyra E M / / A ' H va E M = 1 A ' H = —
'
2
4
13a^
EM 1 (ABC), M H = -!-BC = - => HE^ = HM^ + ME^ =
^
^
4
2
16
.,.2
AM

AH^+AC^

HC^

^ ,
.
.
37a^
=
=
, AE = A M + ME =

2
4
4
16
^
AE^ + EH^ — A H ^
Ap dune dinh l i Co sin ta c6: cos AEH =
^ • ^
2AE.HE

46

7a2

17

47
itmiliiriu.;.


Tuye'ti chgn b Giai thi?u dethi Toan hpc - Nguyen Phu Khanh , Nguyen Tat Thu.

Cau6:Dat a =

=

=^

=> abc = 1 . P =

Matph3ng (P) c6 VTPT n = ( 2 ; l ; - 2 )

1
1
1
•
+
—
=
=
+
V3 + a 73+ b 73+ c

Duong thiing d c6 VTCP u = (-1;2;1)

1
2
1 1
,
= ,
<
+ 737a
^4(3 +a)
3+a 4
1
1
1
3
27 + 6(a + b + c) + ab + bc + ca
3

Suy ra P <
+
+
+- =
- i —
+ 3 + a 3 + b 3 + c 4 28 + 9(a + b + c) + 3(ab + be + ca) 4
^ , ,
. ,
27 + 6(a + b + c) + ab + bc + ca
3
Ta chune minn
—^—
< —.

A c ( P ) va A i d = >

A p dving BDT Co si ta c6:

28 + 9(a + b + c) + 3(ab + bc + ca)

Phuong trinh tham so'ciia A : y = - l
Cau 9a:

neN, n>3
Ta c6: Cl + 2n = A^+j <=> n ( n - l ) ( n - 2 )

4

Theo nhj thuc Newton ta c6:

ro

Cau 7a; Goi C = (c;2c + 3) va I = ( m ; 6 - m) la trung diem cua BC.

X - ]

y+3

z-3

- 1 2
1
2x + y - 2 z + 9 = 0

fa

w.

x=0
<» y = - l r > A ( 0 ; - l ; 4 )
z-4

Vi A nam trong (P) va cSt d nen A di qua A.
48

+Ci 1.(1 + x f - C i
X

+ x)^ + C^ (1 + x)^ -... + Cix« (1 + xf
X

So' hang khong phu thupc vao x chi c6 trong hai bieu thiic:
-Cl^iUxf

vaC^(l + x)^

.c

X

Do do so hang khong phu thupc vao x la: -Cg.C^ + Cg.C^ = -98.

^6 ^ '6^;
37^

3

3 ;

/_19.42^
Tpa do B
'[ 3 ' 3 /
Cau 8a: Tga dp giao diem A cua d va (P) la nghi^m ciia h^

=c

Trong do c6 hai so' hang khong phy thupc vao X la: —Cg.C3 va Cg.C4.

^4

3x - 3y + 23 = 0

ww

TQa dp ciia C la nghi^m ciia h^:

2x - y + 3 = 0

-i8

s 41^

ce

bo

+ 3 = 0 ^ m = --r::>I=
6
Phuang trinh BC: 3x - 3y + 23 = 0.

eCC

ok

. r 2 m - c + 5'\c
Nen 2

om

/g

Suy ra: B = (2m - c;9 - 2m - 2 c )
ll-2m-2c^

Ta

up

A. Theo chi/cTng t r i n h chuan

s/

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

s

+ 2n = (n + l ) n

neN, n>3
.
<=> n = 8 .
n2-9n + 8 = 0

f(x) = - - x ( l + x)

^2m-c + 5

.

Z=:4 + t

.

Vi C la trung diem cua AB nen: C =

n,u = (5;0;5).
x=t

Do
< » 3 ( a + b + c) + 5(ab + bc + ca)>24 (*)
ab + be + ca > 3
a+b+c>3
Suy ra ^ - ^ • Dang thuc xay ra khi a = b = c = l<=>x = y = z .
Vay maxP =

=

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/

^

UA

B. Theo chUoTng t r i n h n a n g cao

Cau 7b: Ta c6 CB^x = A B ^ j

Tii giac ABjHCj npi tiep => A B ^ j = A H C j

AHCi=V5C
T u giac A i H B i C noi tiep => A j H C = AjBjC

(1)
(2)
(3)

(4).
Tu (1), (2), (3) va (4) suy ra: CBpc = A ^ B ^ => AC la phan giac ngoai goc Bj
cua tarn giac A j B ^ C i .
Taco: A j B j : 4 x - 3 y - 2 = 0;BiCi : y - 2 = 0
Phuang trinh duang phan giac ciia goc tgo boi hai dt A j B j , BjCi


fy--^

4x-3y-2

= |y-2|« "x-2y + 2 = 0
2x+y-6=0

Cau 5: Cho hinh lang try ABCD.A'B'C'D' c6 day ABCD la hinh thoi, canh
C
bSng a, ABC = 60". Hinh chie'u cua A ' len mat phang (ABCD) la giao diem cua

Vay phuong trinh canh A C : 2x + y - 6 = 0.

AC va BD. Mat phSng (A'B'BA) tao voi mat day (ABCD) mpt goc 60° . Tinh

Cau 8b: Gia su A la duong t h i n g di qua A va d tgi M ( l - 1 ; -2 + t;2t) € A
_
^ ^ J _ V56t^ - 304t + 416
AM

V28t^ - 152t + 208

Cau 6: Cho cac so thyc a,b,c>0 thoa a + b + c = 1 . Tim gia trj Ion nha't ciia
•
,
.
T,
a
b
c
bieu thuc: P = ,
. + /.
+ /
. •
^ya + bc v b + ca Vc + ab

N/3t2-10t + 20

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

L. .

>/6t2-20t + 40

w.

iL
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01
/

Khidotaco:

the tich cua khol chop va khoang each giiia hai duong t h i n g BD va A'C.

hoac B)

3t^-10t*20

(31^-lot+ 20)

A. Theo chUorng trinh chuan

11

Cau 7a: Trong mat ph3ng Oxy cho tam giac ABC, duong cao xuat phat t u A c6
4
phuong trinh x + 2 y - 3 = 0, trung diem BC thupc Ox va G(0;-) la trpng tam

Lap bang bien thien ta dugc max f(t) = £{-2) = 48 .
Vay M ( 3 ; - 4 ; - 4 ) = > A : ^ = ^
1
-4
Cau 9b: Ta c6:

=

^ .
-3

tam giac ABC. Tim tpa dp cac dinh ciia tam giac biet S^pc -

Cau 8a: Trong khong gian tpa dp Oxyz, lap phuong trinh mat phSng (a) di

[22''-y + 2" = 2^''y
f2^(''"yW2''"y-2 = 0
[2''"y=l
<
o"
log2x(log4y-l) = 4

[log2x(log2y-2) = 8
[log2x(log2y-2) = 8

o

log2 X = -2

Ta
up

<=> i log2X = 4

1 .
x=y= -

.c

om

(logjx) - 2 1 o g 2 X - 8 = 0

x = y = 16

ro

2

s/

x= y

qua hai diem A ( 0 ; - l ; 2 ) , B(l;0;3) va tiep xiic voi mat cau (S) c6 phuong trinh:

/g

x= y

bo

ok

OETHITHIJfSOa
(1)

fa

Caul:Chohamso y = x''-2(2m + l ) x 2 + 5 m - l

ce

I. PHAN CHUNG CHO TAT CA CAC THI SINH

w.

a) Khao sat sy bien thien va ve do thi ham so (1) khi m = 0

ww

b) Tim tat ca cac gia trj m de do thj ham so' (1) cat Ox t^i bon diem phan bi^t
CO hoanh dp Ion hon - 3 .

Cau 2: Giai phuong trinh: sin x (N/2 sin 2x + V2 +1) = sin 5x - >/2 sin x cos 2x .
Cau 3: Giai phuong trinh : 8x^ - 13x + 7 = (1 + -)^2x^
Cau 4: Tinh tich phan: 1 = f G ^ d x
n^Vx^+l
50

.

- 2.

57
~Y'

(x-l)2+(y-2)2+(z + l)2=2.

Cau 9a: Mpt hpp dung 40 vien bi trong do c6 20 vien bi do, 10 vien bi xanh, 6
vien bi vang, 4 vien bi trang. Lay ngau nhien 2 bi, tinh xac suat de 2 vien bi lay
ra CO cung mau.
B. Theo chUcrng trinh nang cao

Cau 7b: Trong mat phSng voi h$ tpa dp Oxy cho 2 duong tron (Cj): x^ + y^ = 13
va (C2): (x - 6)^ + y^ = 25. Gpi A la giao diem ciia (Ci) va (C2) vai yA < 0 . Viet
phuong trinh duong thSng di qua A va cat (Ci), (C2) theo 2 day cung c6 dp dai
bang nhau.
Cau 8b: Trong khong gian voi h^ tpa dp Oxyz cho hai diem A(1;1;0), B(2;l;-1)
va duong t h i n g d : ^^-^ = - ^ - ^ ^ ~ ^ ' ^ ™

^^^^ ^ thupc duong thang

d sao cho A ABC c6 dipn tich nho nha't.

Cau 9b: Cho z =
P =

. Tinh gia trj ciia bieu thuc:

r if
V

zj

(2

3

f

4

n

1^
3
1 '
f 4^
z + — +^ + 3
+ z + 4
1^
z ;
V
z >

^
z J
51