Luyện giải đề trước kỳ thi đại học tuyển chọn và giới thiệu đề thi toán học phần 1

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!.
"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.
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.
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.

Cau 1: Cho ham so y =

^ (C)
X

1

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

+

J

^^LLI^

2

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

x+1

Cau 4: Tinh tich phan: I =

7t
X + —

xdx.

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

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) ^
'
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

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.

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.
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

^"

C t y TNHH MTV DWH Khang Viet

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
Cau 7b: Trong mat phang Oxy cho elip (E) C6 hai tieu diem I^(W3;0); I^(V3;0)
va di qua diem A sfS;- . Lap phuong trinh chinh t5c cua ( E ) va voi moi
V
^/
diem M tren elip, hay tinh bieu thuc: P = F^M^ + FjM^ - 30M^ - F1M.F2M .
,

X—

1

z +1

V

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

sinx + cosx ^ 0

X* —

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

4

.
_
2cosx = 0 o

+ kjt

sm

X

cosx

2cos^x

sinx

smx + cosx

= 0

X = — + k7C

cosx = 0
+ cos x = 2\/2 sinx cosx

= sin2x

sm

X = — + k7l

2
n

,^

n

X = - + k2K, X = -

.

4

4

kin

+

—

3

Ket hop dieu kien ta c6 nghiem cua phuong trinh la:

+ z^ = 0

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

X ?t k n

Phuong trinh

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

sinx ^ 0

7t

W^Z-5=:1

X =

H\i(}m DAN GIAI

llTt

— + nn,

X=

„
+

1771

2nn, x =

„ ^

+ Inn, n&Z .

Cau 3: Dieu kiC^n: x > -

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

Cau 1:
I

a) Dpc gia t u lam

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

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

A e ( C ) = ^ A a;

2a-l

,a^l.

a-1

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

Dodo

Nen

S^,OA4''°^4

2a-l
a-1

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

2a2-4a + l
a-1

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

2a^-4a + l
a-1

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

a-1

«(2V2X-3-I)(4-3N/X+T)>0
(8x-13)(7-9x)

1

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
D|t

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

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 = ^

-

n^^A v

2

Tpa dp A la nghi^m cua he:
1 aVs

Uijc

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

va ™

Suy ra SE = VsH^ + H E ^ = ^
3V.S . A B C

2

=

AC

_

4

=

1BC =
4

—

4

^

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

1

-t

vol x, y > 0
x+ y
1
1
1
•+a + bl^b + c c + a,

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

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

Vay p h u o n g t r i n h mat cau ( S ) : (x + i f + ( y - 3^ + (z - 3^ = 1 .
Cau 9a : Ta c6: ( l + x f " = C°„ + xC^„ +.... + x^^C^jJ
{l-xf".C^„-xC^„......x^"Ci^

4
( 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).

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

(c + a)(a + b)

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

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

27213

y

2x + y + 13 = 0

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

^ S , . , „ = isE.AB =

X

(a + b)(b + c)

-7-

AC^

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

a^>/3

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

BC

Vay d ( C { S A B ) ) =

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

= i ^

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

(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

Suy ra (2) » w^.|z|^° = z^ <=>
^2

4

v2n
-2n

2n

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

fz = 0

-2n

s

Z

. Tu {2) suy ra

= z^ =>

+

w

W =

2n-l
-2n

(3).

^2n-l
ic^„ . i c ^ -'-Cl +- + ^1 C
^^

0:

z= l

• w= -l:

v6 nghiem

2

Z

5

,

= !=>

w

1

=

z =1

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

=l

(z) = 1

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

OETHITHllfSOZ

B. Theo chUorng trinh nang cao
/

-

^

=ic^„+ici,+...+^c
2n
2n

3

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

Theo gia thiet bai toan ta c6 h^

3
la^

1

^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

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

Cau 2: Giai phuang trinh :

(l + tanx)(2cos2x-l)

rr
'- = 2V2 cos3x .
'

sm x +

71 '

XetM(xo;yo)e(E)^^ + y ^ = l = ^ y 2 = i - i .
'(x2+l)y4+l = 2xy2(y3-l)
Suy ra P = (a + exg )^ + (a - exp )^ - 2(x^ + y2 j _ (a^ - e^x^)
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
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 3: Giai h$ phuong trinh:

Cau 4: Tinh tich phan: I =

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

(voi x,y e

'^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

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

giac ABC bang 12. Tim tpa dp cac dinh cua tam giac ABC.
Cau 8a: Trong khong gian Oxyz cho hai duong th^ng:
fx = l + t

y = -2 + t ,

:

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

X —

z=l

Viet phuong trinh duong thang A c3t hai duong thang A,, A j va mat phang
(a) lanluqrttai A , B , M thoa man A M = 2MB dong thoi A l A j .
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

M ( 2 ; 1 ) ; N { 4 ; - 2 ) ; P(2;0); Q ( l ; 2 ) Ian lupt thupc c^inh AB, BC CD, A D . Hay
lap phuong trinh cac canh ciia hinh vuong.
Cau 8b: Trong khong gian Oxyz cho diem A{3; 2; 3) va hai duong th3ng

, x-2
y-3

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

di,

d2cva

= 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 )
Tuongty y2=(m^ + m + l j ( - 2 x 2 - l )
Do do yiy2 > 0 <=> (2xj + l)(2x2 +1) > 0 o 4x,X2 + 2{xj + X2) +1 > 0
o -4m (m + l ) + 5 > 0 < = > 4 m ^ + 4 m - 5 < 0 o — — — < m <
V
;
2
Cau 2: Dieu kien: sin

7t 1
X + —

4j

I. PHAN CHUNG CHO TAT CA CAC THI SINH
Cau 1:
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

2

.

x?.tJ + k K . l
^

x)

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

7t

cos2x = 0
cos2x = 2

x=±- +
6

k7t

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

tuyen C M ciia tam giac ABC.

HMGDANGIAI

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

COS X (sin X + COS

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

— tie'p xiic voi Parabol y = x + m .

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

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

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

Khang

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

B. Theo chiToTng trinh nang cao
Cau 7b: Trong m^t phSng voi h^ toa dp Oxy cho hinh vuong ABCD biet

DWH

CtyTNHHMTV

Tat Thu.

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

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

y=

hoac

Cty TNIIU Af IV DWH Khang Viet

TatThu^
X =

1+
2

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

e'2

c2 „:

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

o2

e2

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

"

= (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.
Suy ra S H I ( A B C D )

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 =

.

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

,,2

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 '

~

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

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

a + 4b + 7
x=3

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

±

^

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

Mlitkhac A l A j =>AB.Uj =0<=>2a-3b-8 = 0 (2)
f

Tir ( 1 ) va ( 2 ) suyra a = — , b = - =>AB =
10
5
21
23
14
Va phuong trinh A :

X

y

17 17 9^
lO'lO'S

9

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

Zj

,B

23

14'

U's'sJ

5_
= 1 - 21

GQI M ( X ; y) diem bieu dien so' phuc z, suy ra z = x + yi
Taco: 2 z - Z i + 1 = 2x + 2 ( y + l ) i ; z + z^+ 2 = ( x - l ) + ( y - 4 ) i
Nen

2z-Zj+l

z + Zj + 2

= l o

2z-Zi

+1

2 + b2

-

<=>

"b = -2a

>

b = -a

• b = -2a suy ra phuong trinh cac canh can tim la:
AB:x-2y=0;

CD:x-2y-2=0

BC:2x+y-6=0;

AD:2x+y-4=0.

• b = - a . Khi do
A B : - x + y + l = 0 ;BC: - x - y + 2 = 0
AD:-x-y+3=0

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

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

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

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

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

a,b M o M i = 0

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

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

^

_

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

3b + 4a

-b

di qua M j (l;4;3) c6 vecto chi phuong b = ( l ; - 2 ; l )
'21

z

5.-1
5__
-17
17

Cty TNHH MTV DWH Khang Vift

x - x +1
x-1

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

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

A : ^ = ^ = ~

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

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.

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 ,
B A C = 120° va A B ' vuong goc voi day ( A ' B ' C ) . Gpi M , N Ian lupt la trung
diem cac canh C C va A ' B ' , mat phang ( A A ' C ) tao voi mat phang ( A B C )
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 .
Cau 6: Cho cac so thuc a , b , C € [ 0 ; l ]

thoa

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

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

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

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

= ^ . Tim gia trj nho

om>0

(l)

P = 4m>0

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

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

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

(*)

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)
sinx = - - 2
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 )
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

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

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

t<-.

2

• t>2<=>

/ " " " ^ — >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<
2

x2-x + 2

5 + >/33
V

x>

2

4

^5-733

-l

.

^ 5 + V33

va x >

^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
2

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

.

C

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

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

« 5

Suy ra A B ' = B'K.tan30° = | ,

.

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

^ 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

f ( x ) < 0 , V x e [ 0 ; l ] hay 2 ' ' - x - l < a V x e [ 0 ; l "

—

Mat khac \/x,y,zeM,

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

x^ +

Suy ra

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=

| L _ J _
1

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

1+e
3

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

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

hoac B)
A. Theo chUcrng trinh chuan

Suy ra B ( b ; 8 - b ) . Do M la trung diem AB nen

A(2-b;b-8)

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

2'2

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

-a + 2b + c = 0
a- c+d= 0

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

I a + 2b - 2c I

2 + b2+c2

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

6d2 => A e ( a )

(a + 2b-2c)^

=>T =

(4b-cf

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

Xet T =

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

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

• Voi x = - = > A B =
3

7 _8
3'

3'

_2
3

Phuong trinh ( a ) : 14x - l l y - 5z + 25 = 0 .
Cau 9a: Ta thay z = 0 thoa phuong trinh
Ta xet:

a^+b^+c^

5b2+2c2+4bc'

2

c ^ ( 2 T - l ) t 2 + 2t(2T + 4) + 5T-16 = 0.

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

1
27
• Voi T = - phuong trinh c6 nghiem * =
•
• Voi T 5-^ ^ de phuong trinh c6 nghiem t khi va chi khi

z^Q

= 4z => z ^ = 4

Do do:

=4

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

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

(2T + 4 f - ( 2 T - l ) . ( 5 T - 1 6 ) > 0 o 0 < T <

=2

Thu 1^1 ta thay bon nghi^m nay thoa phuong trinh .

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

>D(5;-4).

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

Tu AB = V]3=>(x + l f + 4 x 2 + ( x - 2 ) ^ = 1 3 c ^ x = -1,x = • Vai x = l r ^ A B = (0;2;-3),tac6 u = (2;3;-l) laVTCPciia d j va A(-1;1;0)

^9 _ 1^
2' 2

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

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

Tu

. D ( - 3 ; 2 ) va I

(p) chua A nen

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

Khang Vi

Suy ra trung diem cua AC tuong ving la

1 1

Cau 7a: Ta c6 M N = (3; -3) va M N // BC nen phuong trinh BC:x + y - 8 = 0

Matkhac: A D 1 M N

Cty TNHH MTV DWH

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 ^ ) =

>-1

p2 2(ac + b d )

p
r= > ^ =- - ^
^ + 2 = m > 0 = > - i - = ±N/m l a m p t so thirc.

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
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)

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^

q_

q

z,

p

p.zi

k^

p

1 '

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

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

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

Zj

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 .

DETHITHUfs64
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 a u 1: C h o h a m so y = x"* - 3x + 1 ( l )
a) Khao sat s y bien thien va ve d o thj ( C ) cua h a m so ( l ) .
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 :
x|'' -3|x| = m-' - 3 m

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 ) .

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 ) .

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

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

- X =,

4x-3x^y-9xy^

V

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

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 .

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

HI/dNG DANGIAI

x2+2

3y

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

_ 6x-y
(x2-l)%3 =
C a u 3: Giai h ^ p h u o n g t r i n h :

= - y - " ^ 6 nhat.

d i ^ n deu.

C a u 2: Giai p h u o n g t r i n h : 4cos^ 3xcos2x + cos8x = \/3sin4x + 2cos2x

22

'

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,

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

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

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

MTV DWH

1

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

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^

X

, 27y^ _ 4
x3

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

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

Khutig

l + b-^=2a

^

fa = b

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

1 + a- = 2 b

da cho CO 4 nghiem phan bi^t la:

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

a=b

- 1 < m-' - 3 m + 1 < 1
a =b

-2 < m < - V 3

m-' - 3 m < 0

m

+ Vs •

-1

0 < m < N/3

- 3m + 2 > 0

a = l,a =

1

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

n
<=> cos8x = cos —
2

7t
, n
x = — + k—
18
6

8x = - — + 4x + k27r
3

x~ — + k 6
2

X

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

, keZ

.

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

0

^ ^

'

x^ + 8 = 6x^y

9y^ - 6xy + x^ =

x + 3y

VVs-i

hoac

-VN/5-1

x^ + 8 = 6x^y

-2

-775 - 1

V7f^'

3
10

10

C a u 4: Ta c6: I X +

3y

3xy
10

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

-2

-

{ x ; y ) = ±N/2;±-

x^ + 8 = 6x^y
O

X =

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

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

3

2

y =

3y > x
x + 3y

y =

^75-1

X

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

X -

hoac •

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

.
71
2n
-4x
4x + — = cos
6

8x = — - 4 x + k2n
3

. X

2

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

^

1

'y-:

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

x = 7^

1'^'

• a = b = 1 , ta c6: •!

• 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

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

=> S H I la goc giua mat ben
(SAB)

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:

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:
IVl

n

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

sin60"

sin BDM

y)

>0

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)

MD =

^

I

BM.sin60''

fs
2^=r~-^ 4
^
= ^ - ^ c o s BDM = sin BDM = ^

tan BDM = . — . !
,-1=
\^ BDM

OI = OD.tanBDM = - ^ ^ l

+ 161>0<=>

1
y + -<7
y .

^y^-^T—•

y + —<13
y

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

la trung
maxP = f

{

7-3S

21 + 375

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

b=
, dat duQ-c khi

diem A O

^
2

a

3-S

c=•

Ta CO A A H I - AAOB
IH

Al

AI.OB

OB

AB

AB

a 73

min P = f(72) = 272,

_ A M AI _ 1

b - ^/2a
c = xa

hoac B)
A. Theo chUorng trinh chuan
Cau 7a: Goi N ' la diem dol xung

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

26

nghi^m cua phuong

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

1 1

SAABO ~ A B • A 0 " 3 ' 2 ~ 6

1
V?y VsMlCB=iSI.SM,CB

^

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

Suy ra SI = IHtan60° = —
^
8
SAMI

dat du(?c khi

a^Vs a^Vs lla^Va
4

1 3a Ila^x/S
3 8
48

48

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 ) =

va

1

IH^

1

1

l A ^ " IB^

5

416^ ^

~

»

DWH

Khang

Viet

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

= 2

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

Cau 8b: Ta c6 A B = BC = C A = 3N/2 nen tarn giac A B C deu.
Gpi G la trong tarn cua tarn giac ABC.

2
4b + 2

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

=:5ob

SuV ra G

'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

Vay B ( l ; - 1 ) .
Cau 8a: Goi B = d ' n A =:i> B(1 + 2t;2 + t ; - 2 - t) =:> A B = (2t + 2; t + 2 ; - t - ] )
cos ( d " , A ) =

MTV

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

„

Matkhac B€AB=>B b ; - ^

8 + 3-1

Cty TNHH

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 ) '

6t^ +14t + 9

3 V 6 t 2 + 14t + 9
5
f
- - t
v3
y

i o n nhat.

(2
- + 5t
3

Vay CO hai d i e m D ( 2 ; 6 ; - l ) va D
14t2+18t

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

\

-

(6t2 + 14t + 9)

L a p bang bien thien ta suy ra d u o c m a x f ( t ) = f
' t = — =^AB =
7

1 _

'

9^

9

I

7

5

z^

3

10._2__7^
3'

3'

1

_ cos2(p-isin2(p

r^(cos2(p +isin2(p)

r^

3,
-5

thoa yeu cau bai toan.
.5

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

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

4 5 2

. x+1_ y _ z +1
. Phuang trinh A :
7'7'7^

-4
Do do

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

2
2
= 18 c ^ t = - ; t = —
3
3

+ -^ +t

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

z=-^+t
3
V i D A = 3V2

Suy ra ( d " , A ) nho nhat

3

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

va

z

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

a =3

a=3

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

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 .

5(p = 2(p + k2rt

B. Theo chi/orng trinh nang cao
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^

B(5-2yo;y„)

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

=>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.

DETHITHUfSOS
I. PHAN CHUNG CHO TAT CA CAC THI SINH
Cau 1: Cho ham so y = x'' - 3(m + l)x^ + 3m(m + 2)x - 12m + 8 ( C ^ )

a) Khao sat su bien thien va ve do thi ham so khi m = 0
b) Tim m de do thj ( C ^ ) c6 hai diem cue trj A , B sao cho A M + B M nho
nhat voi M ( 3 ; 3 ) .
Cau 2: Giai phuong trinh: sin^ x + cos^ x = sin 2x cos 2x + tan 2x - 2
Cau 3: Giai h^ phuong trinh :

y ^ + ( 4 x - l f = ^4x(8x + l )
40x^ + x = y V l 4 x - l

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

Cau 9a; Cho so phuc z thoa man |z| = 1. Chung minh rang:
l<|l + z^| + |l + z + z^|<5.
B. Theo chiTorng trinh nang cao
Cau 7b: Trong mat phSng Oxy cho duong tron (C): (x - 2)^ + (y -1)^ = 10 . Tim
tpa dp cac dinh ciia hinh vuong M N P Q , biet M triing voi tam cua duong tron
^C); hai dinh N , Q thupc duong tron (C); duong thang P Q di qua E(-3;6) va
>0.
Cau 8b: Trong khong gian Oxyz cho hinh chop S.OABC c6 day OABC la hinh
thang vuong tai O va A ( 3 ; 0 ; 0 ) , AB = OA = i o C ,

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

tich khoi chop S O M N .
Cau 9b: Tim tap hpp cac diem M trong mat phang phuc bieu dien so'phuc z
sao cho - ^ - i ^ la so'thuc duong.
z-i

HlTdNG D A N G I A I
I. PHAN CHUNG CHO TAT CA CAC THI SINH
Cau 1.
a) Ban dpc tu lam
b) TXD: D = ^

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

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.

5 3
8'8
Tim tpa dp tam duong tron npi tiep va tam duong tron bang tiep goc A cua
tam giac ABC.

Voi

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

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^

1 +1^

8t

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')

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

+ 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

X

(du = dx

d v = 1 + tan
n

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

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 ) .
trinh da cho.

0

J

A

Cau 5: Goi I la hinh chieu vuong goc cua S tren(ABCD), tuong tu nhu v i du

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

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

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

Di^n tich hinh thang ABCD la S = - ( A B + D C ) A D = -.5a.2a = Sa^

Cpng hai phuong trinh ciia he voi nhau ta dupe:
y^ - 2 y V l 4 x - l + 14x - 1 + 96x^ -20x + 2 = ^4x{8\ l)

Goi p la nua chu vi va r la ban
kinh duong tron npi tiep ciia hinh

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

(l).

thang ABCD thi
AB + DC + A D + BC

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 =

= 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) •

10a

. 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.

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

X

X+—

2

2j

2j

0

hole B)

V(x.l)^(y-Hlf

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:
1

1

\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

(2)

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

(3)

1

Ma

>0,dod6:

+\>-{s-S'f

(4)

(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

. D^t dupe khi X = y = 6

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

A. Theo chUofng trinh chuan

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

Ta c6:

AK.AB
AK.AB

AK.AC
AK.AC

(AK,AB)=(AK,ACJ

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

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

4i
s+-

II. PHAN RIENG

3V2

-|(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'

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 ) .

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

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/

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

l-z^

1-z

x = 3t
SB:

1 + z^

l + z^

l-z'"*

2

2

. 1

2

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

=

7

7

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

OS

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

z = -4t'

cua ( a ) voi SB, S C la:

N/10

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

va S C :
M ( 3 ; 3 ; 0 ) va N

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

y=3

Tir do ta tim dugc cac giao diem

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

nen taco:

x=0

z = 4-4t

B. Theo chUorng trinh nang cao

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

B(3;3;0).

A

96

O M =

72

(-12; 12;

-9),(oS A

OMJON =

.

3nn
Vay V s o M f , = : ^ ( d v t t ) .

cua he

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

ly =

0 •

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
t

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

Truong hop nay ta loai vi X Q > 0.
x + 3 y - 1 5 = 0.
Khi do tga do Q la nghiem ciia h^
(x-2)^(y-lf =10^jx = 3 ^
3x + y + 3 = 0

36

[y = 4

M
Q(3;4).

la so'thuc duong
z-i

•

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
b) T i m tat ca cac gia trj ciia tham so m de do thj ham so (1) cit duong thing
y = -2m^ tai ba diem phan biet A , B, C (voi

< X g < x^ ) sao cho doan thang

y - 3

1

1

z-3
-

2

, ,
x-1

va d , :
"
1

y-4
-

8(sinS + cosS)

1

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.
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

(
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 :

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,

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

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
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.

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

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

<»(x-m)(x^ -2mx-2m^ + m - l ) = 0

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

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

X

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

(*)

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

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
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

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

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

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 =

=m

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

A. Theo chuorng trinh chuan

38

2

z-3

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 .

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

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
Ta c6: tan

X

^

I

X + —

.tan

X

3;

\

6>

= - tan

7t^
X + —

^

6;

.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 :
- 8 ( s i n ' ^ x + c o s ^ x ] + 2 = 3>y3sin4x<=>-8 l - - s i n ^ 2 x + 2 = 3>/3sin4x
^
'
V
4
;
<=> - 6 + 3(1 - cos4x) = 3N/3 sin4x o

.

<=> cos 4x — = - l o 4 x
3

71
—

(b)<=>

1

> -

X

2

- l + ^/6

> -

2

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

<=> X =

4x2 + 4 x - 5 = 0

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

,

212VT5

x=

2 + 2>/T5

K

.

Cau 4: Ta c6: I = J

rdx.
Hnx + l

+1

kTt

<=> x = — + — ,
3
2

3

--i + S

Inx

^/3 sin 4x + cos4x = - 2

, = 7t + k27i

;X=

Dat t =

Inx +1

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

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 .
1

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

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

P h u o n g t r i n h <=> Vx^ + 3 x + 6 = 3x + 1 - \ l 2 x ^ - 1
3x + l - V 2 x ^ - l > 0

(1)

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

(2)

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

ED = EC = a nen v u o n g tai B,

T a c o : (2)<=>2(3x + 1 ) ^ 2 x 2 - 1 = lOx^ + 3 x - 6
<» 7x^ - 4 x - 8 + (3x + l ) ( x + 2) - 2 ( 3 x + 1 ) V 2 x ^ = 0
< = > 7 x ^ - 4 x - 8 + (3x + l ) ( x + 2 - 2 V 2 x 2 - l ) = 0

vuong. Tarn giac B D C c6 EB =

(3)

BE 1 C D nen t r u n g d i e m M
ciia BC la tam

V i : x + 2 + 2\/2x^ - 1 = 0 «

nen x + 2 + 2V2x^ - 1

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 )

= 0

3x + l

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-

tron

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

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

duong

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

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

Gpi n = (a;b;c) lamptVTPTciia (a)
Ta c6: r i . u = 0 <=> a + 2b - c = 0 => c = a + 2b

= x ( x - y ) ( x - z ) + (y + z - x ) ( y - z ) ^ >0
Suy ra dieu phai chiing minh. Dang thiic xay ra khi va chi khi x = y = z > 0.

•

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

MM khac cos(p =

n.u-

—
n

U2

|2a - 2b + c

34J7h^

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

'

^

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

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

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

gia sir hai so do la x, y

• a = 5b, ta chpn b = 1 =>a =5,c = 7. Phuong trinh (a) la: 5x+y+7z-18=0.

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

(3)

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

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 < ^ ^ ^

| w - 7 + 7i| = 10.
< 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

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

M a ( l + iV3)

,

.

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 .

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

+(l-iV3)

:lzl

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

y

OTI^

= l^

=^

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

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 .
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

2 0 1 0 / _ „ ^ r 7 n _ \0

= 2.2201''(cos6707r) = 2.2
Vay

>/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

S = 22010

f(x) =

'

3

2

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

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

OETHITHUfSO?

Cau 7b: C h o tam giac A B C nhpn, viet phuang trinh d u a n g thing A C , biet toa
dp chan cac duong cao ha tu cac dinh A, B, C Ian lupt la: A i ( - l ; - 2 ) ,
C, ( - l ; 2 ) . .

I. PHAN CHUNG C H O T A T CA CAC T H I S I N H
C a u 1: Cho h a m so y =

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

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

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

,

, , oy

C a u 3: Giai p h u a n g t r m h : ^ x - 1 + 44

Bi ( 2 ; 2 ) ,

_

^_
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
b) Phuang trinh hoanh dp giao diem

x'' + 3x^ + ( 3 - m ) x + 3 - m = -14<»x-^ + 3x^ + ( 3 - m ) x + 1 7 - m =0

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

+1

Taco: f'(x) = 2(x + l )

+1

-1

f(x)
f(x)

1

-

62

^

0

+00

c-l

cit
Suy ra I = J - y

=1-e-1

+00

+

^
Ta

, „ ^ AC
CO H K =
=

, suy ra

A ' H = HKtan60" = y ,
+00

12 "

-00

Yeu cau bai toan o

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

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

(x + l)2
Bang bien
x thien -9

dx .

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 nghi^m ciia phuong trinh da cho.

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.

Xethamso f ( x ) = (x + l ) ^ + — voi x > - 9
X

x-1

Vift

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

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

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

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

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 =

1
1
1
•
+
—
=

=
+
V3 + a 73+ b 73+ c

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 p dving BDT Co si ta c6:

^

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

4

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 =

.

^2m-c + 5

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

ll-2m-2c^

s

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

2x - y + 3 = 0

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

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

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

^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^
z-3

eCC

^4

3x - 3y + 23 = 0

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

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

.

Z=:4 + t

Cau 9a:
neN, n>3
Ta c6: Cl + 2n = A^+j <=> n ( n - l ) ( n - 2 )

+ 2n = (n + l ) n

-i8

=c

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

-Cl^iUxf

s 41^

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

=

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

So' hang khong phu thupc vao x chi c6 trong hai bieu thiic:

Suy ra: B = (2m - c;9 - 2m - 2 c )
Vi C la trung diem cua AB nen: C =

UA

x=t

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

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

48

A c ( P ) va A i d = >

Theo nhj thuc Newton ta c6:

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

y+3

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

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

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

X - ]

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

vaC^(l + x)^

X

Trong do c6 hai so' hang khong phy thupc vao X la: —Cg.C3 va Cg.C4.
Do do so hang khong phu thupc vao x la: -Cg.C^ + Cg.C^ = -98.
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
Khidotaco:

_

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. .

^ ^ J _ V56t^ - 304t + 416

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

>/6t2-20t + 40

w.

hoac B)

3t^-10t*20

(31^-lot+ 20)

11

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

=

^ .
-3

A. Theo chUorng trinh chuan

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

57
~Y'

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

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

x= y

x = y = 16

Cau 9a: Mpt hpp dung 40 vien bi trong do c6 20 vien bi do, 10 vien bi xanh, 6

1 .
x=y= -

vien bi vang, 4 vien bi trang. Lay ngau nhien 2 bi, tinh xac suat de 2 vien bi lay

x= y

2

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

<=> i log2X = 4

o

log2 X = -2

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

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

OETHITHIJfSOa

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

I. PHAN CHUNG CHO TAT CA CAC THI SINH
Caul:Chohamso y = x''-2(2m + l ) x 2 + 5 m - l

bang nhau.

(1)

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

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