de-cuong-on-tap-hoc-ki-2-toan-9-de-cuong-on-tap-hoc-ki-2-toan-9

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Ä &lj1* Ð1 7°3 +Ð& .Ỵ - 10 +Ð& 12 -2013
MƠN TỐN
I./Ë 7+8$
3+ê1 Ô, 6ễ:
1. 1aL GXQJ
2
SQK QJKD 3KmkQJ WUuQK EF KDL PÝW Q Oj SKmkQJ WUuQK
Fy G¥QJ D[ζ +bx
0), +c = 0(a
WURQJ ÿy [ Oj QDEF Oj FiF VÕ FKR WUmßFKD\ FzQ JĐL Oj KË VÕ
2
&KR SKmkQJ WUuQK E±F KDL
D[ E[ F D 
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&Ð1* 7+ì& 1*+,Ê0 7+8 *Ð1

∋

∋ ' b' 2 ac

b2 4ac

∋ ! 0 SKmkQJ WUuQK Fy QJKLËP SKkQ ELËW
∋ '! 0SKmkQJ WUuQK Fy QJKLËP SKkQ ELËW
∋b
∋b

∋b'
∋b'
'
'
x1
; x2
x1
; x2
2a
2a
a
a
∋ 0 SKmkQJ WUuQK Fy QJKLËP NpS
∋ ' 0 SKmkQJ WUuQK Fy QJKLËP NpS
b
b'
x1 x 2
x1 x 2
2a
a
∋ 0 SKmkQJ WUuQK Y{ QJKLËP
∋ ' 0 SKmkQJ WUuQK Y{ QJKLËP
2. 1aL GXQJ
D 3KmkQJ WUuQK WUQJ SKmkQJ Fy 4G¥QJ D[ E[ F D  )
&iFK JL§L »W W [2 YòL W ã WD Fy SKmkQJ WUuQK EF KDL 2WKHR Q W DW + bt + c = 0
-! JL§L SKmkQJ WUuQK WuP W • ! [
E 3KmkQJ WUuQK FKíD Q ó PX
- %mòF 7uP .;
- %mòF 4X\ ìQJ Yj NKủ PX
- %mòF *LĐL 37 YùD WuP mỗF

- %mòF .W OX±Q&K~ ê ÿÕL FKLÃX YßL .;
F 3KmkQJ WUuQK WtFK Fy G¥QJ $
%& &iFK JL§L $
%&
$
KR»F % KR»F &
3. 1aL GXQJ
2
SQK Ot 9L ±ét: 1ÃX SKmkQJ WUuQK
D[ E[ F D  Fy KDL QJKLËP
1 [ 2, x thì:
−
S
°
°
→
°P
°
↓

x1
x1x 2

b
a

x2
c
a

&K~ ê Ç NLÇP WUD SKmkQJ WUuQK E±F KDL Fy QJKLËP WD NLÇP WUD PÝW WURQJ KDL FiFK
DF WKu 37 Fy KDL QJKLËP SKkQ ELËW
VDX
2) ∋τ KR»F ∋ ¶ τ WKu 37 FR KDL QJKLËP
0ÝW VÕ EjL WRiQ iS GéQJ ÿÏQK Ot 9LpW
a) x1+ x 2=
c) x12 + x22 = (x1+ x )2 ±2 2x .x1 , 2

b
,
a

b) x1.x 2=

c
,
a

d) x13 + x23 = (x1+ x )2 ±3 3x .x1 (x2 +1x )

2

−u v S 2
S τ 4P WKu X Y Oj KDL QJKLËP
↓uv P

SQK Ot 9L ±pW ÿ+R 1ÃX Fy KDL VÕ X Yj Y VDR
→ FKR

2

FëD SKmkQJ WUuQK
[ ± Sx + P = 0.
&iFK WtQK QK1P QJKLOP FoD SKñïQJ WUuQK E5F2KDL D[ E[ DF 

c
a

- 1ÃX D E F WKu SKmkQJ WUuQK Fy QJKLËP
Oj [2 = 1; x. =
1

1


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c
a

- 1ÃX D ± E F WKu SKmkQJ WUuQK Fy QJKLËP
Oj 2[ = -1; x. =
1
4. 1aL GXQJ
−∋ τ 0
°
e) Có 2 nghiêm duong khi →
P! 0
°S ! 0

↓
−∋ τ 0
°
f) Có 2 nghiêm âm khi →
P! 0
°S 0

J &y QJKLậP WUiL GâX DF

ầ SKmkQJ WUuQK2 D[ E[ F D 
∋τ 0
D &y QJKLËP NKL
! 0
E &y QJKLËP SKkQ ELËW ∋NKL
F 9{ QJKLËP NKL û

−∋ τ 0
↓P ! 0

G &y QJKLậP FQJ GâX NKL


1aL GXQJ +ậ SKmkQJ WUuQK
*LĐL- Kậ SKmkQJ WUuQK Fk EĐQ Yj mD mỗF Y GƠQJ Fk E§Q 3KmkQJ SKiS WKÃ 3KmkQJ SKiS
3KmkQJ SKiS ÿ»W Q SKé
FÝQJ
- &KR KË SKmkQJ WUuQK−
→

ax by c

↓a ' x b' y c'

(I)

a
b
ζ! ;
D Ç KË SKmkQJ WUuQK , Fy QJKLËP GX\ QKâW
;
a
b
a b c
E ầ Kậ SKmkQJ WUuQK , Fy Y{ VÕ QJKLËP !;
a b; c;
a
b
c
ζ ;
F Ç KË SKmkQJ WUuQK , Y{ QJKLậP ! ;
;
a
b
c
% 3+ê1 +ẻ1+ +é&:
&iF JyF ÿYL YcL ÿñeQJ WUzQ
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ErQ WURQJ ÿmáQJ WUzQ JyF Fy ÿÍQK ã ErQ QJRjL ÿmáQJ WUzQ &iF HP {Q ã 6*.
&iF F{QJ WKqF WtQK
Σ| 5 Oj EiQ NtQK & Oj ÿÝ GjL ÿmáQJ WUzQ
- Ý GjL ÿmáQJ WUzQFKX YL &Σ5 WURQJ ÿy

Σ Rn
- Ý GjL FXQJ WUzQ O
WURQJ Σ|
ÿy 5 Oj EiQ NtQK O Oj ÿÝ GjL FXQJ WUzQ Q Oj VÕ ÿR FXQJ
180

- Diên tích hình trịn: S = ΣR2

Σ R2n
lR
S WUzQ =
- 'LËQ WtFK KuQK TX¥W
WURQJ ÿy O Oj ÿÝ GjL FXQJ WUzQ Q Oj VÕ ÿR FXQJ
360

2

3. 0aW VY ÿSQK Ot TXDQ WUUQJ YI ÿñeQJ NtQK Yj Gk\ FXQJ
D 7URQJ PÝW ÿmáQJ WUzQ KDL FXQJ Ẹ FK³Q JLóD KDL Gk\ VRQJ VRQJ WKu EµQJ QKDX
E 7URQJ PÝW ÿmáQJ WUzQ ÿmáQJ NtQK ÿL TXD ÿLÇP FKtQK JLóD FXQJ WKu L TXD WUXQJ LầP
FởD Gk\ F QJ FXQJ â\
c) Trong 1 ÿmáQJ WUzQ ÿmáQJ NtQK ÿL TXD WUXQJ ÿLÇP Gk\ FXQJ NK{QJ SKĐL Oj mỏQJ NtQK
WKu FKLD FXQJ â\ WKjQK FXQJ EµQJ QKDX
G 7URQJ PÝW ÿmáQJ WUzQ ÿmáQJ NtQK ÿL TXD LầP FKtQK JLúD FởD PíW FXQJ WKu YX{QJ JyF
YòL Gk\ F QJ FXQJ â\ Yj QJmỗF OƠL
'-X KLOX QK5Q ELGW PaW Wq JLiF QaL WLGS
a 7í JLiF Fy ÿÍQK FQJ FiFK X PíW LầP Fế ẽQK PíW NKRĐQJ FiFK NK{QJ ÿÙL
0
b 7í JLiF Fy WÙQJ KDL JyF ÿÕL QKDX EµQJ
ϖ khơng

c 7í JLiF Fy ÿÍQK NÅ QKDX FQJ QKuQ RƠQ WKÃQJ QếL KDL QK FzQ OƠL
GmòL JyF
L.
d 7ớ JLiF Fy JyF QJRjL WƠL QK EàQJ JyF WURQJ FởD QK ÿÕL GLËQ
+uQK KUF NK{QJ JLDQ
2


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D +uQK W 4XD\ KuQK FKó QK±W YzQJ quanh F¥QK FÕ ÿÏQK KuQK VLQK UD Oj KuQK WUé
- 'LËQ WtFK [XQJ TXDQK
2Σ5 O WURQJ ÿy 5 Oj EiQ NtQK ÿi\ O Oj ÿÝ GjL ÿmáQJ VLQK
xq = 6
- 'LËQ WtFK WRjQ SK«Q xq6 +62SÿD\ = 2ΣRl + 2ΣR2
- 7KÇ WtFK 9 6K ΣR2K WURQJ ÿy 6 Oj GLËQ WtFK ÿi\ K Oj FKLÅX FDR 5 Oj EiQ NtQK ÿi\
b) Hình nón: Quay tam giác vng 1 vịng quanh F¥QK góc vng cÕ ÿÏQK, hình sinh ra là hình
nón.
Σ5O WURQJ ÿy 5 Oj EiQ NtQK ÿi\ O Oj ÿÝ GjL ÿmáQJ VLQK
- 'LËQ WtFK [XQJ TXDQK
xq = 6
- 'LËQ WtFK WRjQ SKôQ xq6 +6SD\ = Rl + R2
- 7Kầ WtFK 9

1
1
2
Sh = ΣR K

WURQJ ÿy 6 Oj GLËQ WtFK ÿi\ K Oj FKLÅX FDR 5 Oj EiQ NtQK ÿi\
3
3

F +uQK QyQ FéW
Σ(R 1+ R O2 WURQJ ÿy 51 , R2Oj EiQ NtQK ÿi\ O Oj ÿÝ GjL ÿmáQJ VLQK
- 'LËQ WtFK [XQJ TXDQK
xq = 6
- 7KÇ WtFK 9

1
Σ(R12 + R22 + R1R 2K WURQJ ÿy K Oj FKLÅX cao,
1
2R , R Oj EiQ NtQK ÿi\
3

G +uQK F«X 4XD\ QđD KuQK WUzQ WkP 2 EiQ NtQK 5 YzQJ TXDQK ÿmáQJ NtQK FÕ ÿÏQK, hình sinh
UD Oj KuQK F«X
2
ΣR2= 6dΣ WURQJ
- 'LËQ WtFK P»W F«XGLËQ WtFK [XQJ TXDQK
ÿy U Oj EiQ NtQK G Oj mỏQJ NtQK
4
3

- 7Kầ WtFK KuQK FôX 9R3
II.%ơ, 743
')QJ 5~W JUQ
2 x x
x x 1

1 ãữ
x 2 ãữ
1
:

x 1ữ
x 1ữ
x


Bi 1 &KR ELầX WKớF
3
a) 5~W JĐQ 3

b/Tính

P khi x= 5 2 3
♣

2a
Bài 2 &KR ELÇX WKớF3
1



a 1
1 a

a) 5~W JẹQ 3

a ãữ a
a
ữ. 2 a 1
≠

2a a
a
1 a a

c) Cho P=

6
1

WuP JLi WUÏ FëD D"

6

2
3

b) &KớQJ PLQK UàQJ 3 !
a2
a
a
a 1

2a

Bi 3 &KR ELầX WKíF 3
a) 5~W JĐQ 3
F 7uP D ÿÇ 3

a
a

1

P
E %LÃW D ! +m\ VR ViQK P L
G 7uP JLi WỤ QKể QKâW FởD 3


3 a
ab b
a


Bi 4 &KR ELầX WKớc:P=


3a
a a

1
b b

a

ã a 1. a
ữ:
bữ
2a 2 ab

b
2b

a) 5~W JẹQ 3
b) 7uP QKóQJ JLi WỤ QJX\rQ FëD D ÿÇ 3 Fy JLi WỤ QJX\rQ
♣ 1
♥ a 1

Bài 5 &KR ELÇX WKớF 3

1 ã
a 1
ữ:

a a 2

a 2 •÷
a 1÷
≠

a) 5~W JĐQ 3
1
6

b) 7uP JLi WỤ FëD D ÿÇ 3 !

♣x

x 7
♥ x 4

Bài 6: Cho A= ♦
♦

1 • ♣ x 2
÷:♦
♦
x 2÷
≠♥ x 2

x 2
x 2
3

2 x•
÷
x 4ữ


YòL [ ! [ 4.


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a) 5~W JĐQ $

1
A

b) 6R ViQK $
YòL

Bi 7 : &KR ELầX WKớF

x x 1 x x 1 ã 2 x 2 x 1
.
ữ:
x 1
x
x
x ữ
x


A= ♦
♦

D 5~W JÑQ $

E 7uP [ ÿÇ $

F 7uP [ QJX\rQ ÿÇ $
Fy JLi WUÏ QJX\rQ
')QJ &iF EjL WRiQ OLrQ TXDQ ÿGQ SKñïQJ WUuQK E5F KDL PaW 1Q Yj iS GmQJ KO WKqF 9L-et:

D3KmkQJ WUuQK E±F KDL Yj KË WKíF 9L-et:
1. *L§L FiF SKmkQJ WUuQK E±F KDL:
a. 2x2 ± 5x + 1 = 0
b. 4x2+ 4x + 1 = 0
c. -3x2 +2x + 8 = 0
d. 5x2± 6x ± 1 = 0
e. -3x2 + 14x ± 8 = 0
g. -7x2+ 4x ± 3 = 0
2. 1 P QJKLËP FëD FiF SKmkQJ WUuQK E±F KDL VDX
K 5x2+ 3x ± 2 = 0
a.
b. -18x2+ 7x + 11 = 0
c. x2+ 1001x + 1000 = 0
d. ± 7x2± 8x + 15 = 0
3. 7uP KDL VÕ ELÃW WÙQJ Yj WtFK FëD FK~QJ
a. u + v = 14, uv = 40
b. u + v = -7, uv = 12
c. u + v = -5, uv = -24
d. u + v = 4, uv = 19
E3KmkQJ WUuQK WUQJ SKmkQJ Yj SKmkQJ WUuQK FKíD Q ã P¯X
2
2
a. x4± 8x ±
9=0
b. x4± 1,16x +
0,16 = 0
4
2
4
2

c. x ± 7x ± 144 = 0
d. 36x ± 13x + 1 = 0
2
e. x4+ x ±2 20 = 0
g. x4± 11x +
18 = 0
12
8
1
x 1 x 1
x 2 3x 5
1
k.
x 3 x 2
x 3

h.

16
30
3
x 3 1 x
2x
x
8x 8
l.
x 2 x 4
x 2 x 4

i.

F;iF ÿÏQK JLi WỤ FëD P ÿÇ SKmkQJ WUuQK Fy QJKLËP Fy KDL QJKLËP SKkQ ELËW Fy QJKLËP
NpSY{
QJKLËP
ÕL L PÛL SKmkQJ WUuQK VDX Km\ WuP JLi WỤ FëD P ÿÇ SKmkQJ WUuQK Fy QJKLËP NpS
a. mx2± 2(m ± 1)x + m + 2 = 0
b. 3x2+ (m +1)x + 4 = 0
2
c. 5x + 2mx ± 2m + 15 = 0
d. mx2± 4(m ± 1)x ± 8 = 0
ÕL L PÛL SKmkQJ WUuQK VDX Km\ WuP JLi WỤ FëD P ÿÇ SKmkQJ WUuQK Fy QJKLËP WtQK
QJKLËP FëD SKmkQJ WUuQK WKHR P
a. mx2+ (2m ± 1)x + m + 2 = 0
b. 2x2- (4m +3)x + 2m -2 1 = 0
2
2
c. x ± 2(m + 3)x + m + 3 = 0
d. (m + 1)x2+ 4mx + 4m +1 = 0
')QJ &iF EjL W5S YI KO SKñïQJ WUuQK E5F QK-W 1Q
D*L+L KO SKñïQJ WUuQK Fï E+Q Yj ÿñD ÿñkF YI G)QJ Fï E+Q
Bài 1: *L§L FiF KË SKmkQJ WUuQK
−3x 2y 4
−4x 2y 3
−2x 3y 5
1) →
;
2) →
;
3) →
↓2x y 5

↓6x 3y 5
↓4x 6y 10
−3x 4y 2 0
−2x 5y 3
4) →
; 5) →
;
↓5x 2y 14
↓3x 2y 14
Bài 2: *L§L FiF KË SKmkQJ WUuQK VDX

−4x 6y 9
6) →
↓10x 15y 18

4


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− 3x 2 2y 3 6xy
1) →
;
↓ 4x 5 y 5 4xy

− 2x - 3 2y 4 4x y
2) →
↓ x 1 3y 3 3y x

−7x 5y - 2
y 27
−2y - 5x
8
5
2x
° x 3y
°° 3
°
4
3) →
;
4) →
6y
5x
x
1
°
° 6x - 3y 10 5
y
°↓ 3
°↓ 5x 6y
7
E *L+L KO E9QJ SKđïQJ SKiS ÿ?W 1Q SKm
*L§L FiF KË SKmkQJ WUuQK VDX
1
2
− 2
− 3x
−x 1

° x 2y y 2x 3
°x 1 y 4 4
°x 1
°
°
°
1) →
;
2) →
;
3) →
3
5
° 4
° 2x
° 2
1
9
°↓x 2y y 2x
°↓x 1 y 4
°↓x 1

3
1

54
;
12

3y

y 2
5
y 2

7
;
4

−2 x 2 2x
−
y 1 0
°
°5 x 1 3 y 2 7
4) →
; 5) →
2
2
2
°↓3 x 2x 2 y 1 7 0
°↓2 4x 8x 4 5 y 4y 4 13.
F ;iF ÿSQK JLi WUS FoD WKDP VY ÿK KO Fy QJKLOP WKR+ PmQ ÿLIX NLOQ FKR WcF
Bài 1:
D ÏQK P Yj Q ÿÇ KË SKmkQJ WUuQK VDX Fy QJKLËP Oj - 1).
−2mx n 1 y m n
→
↓ m 2 x 3ny 2m 3
2
E ÏQK D Yj E ELÃW SKmkQJ WUuQK D[ - E[Fy KDL QJKLËP Oj [ Yj [ -2.
Bài 2: ÏQK P ầ mỏQJ WKÃQJ VDX ìQJ TX\
a) 2x y = m ;

x = y = 2m ;
mx ± (m ± 1)y = 2m ± 1
2
b) mx + y = m + 1 ; (m + 2)x ± (3m + 5)y = m ± 5 ; (2 - m)x ± 2y = - m2 + 2m ± 2.
Bài 3: &KR KË SKmkQJ WUuQK
−mx 4y 10 m
(m lµ tham sè)
→
↓x my 4
2. P
D *L§L KË SKmkQJ WUuQK NKL
E *L§L Yj ELËQ OX±Q KË WKHR P
F ;iF ÿÏQK FiF JLi WUL QJX\rQ FëD P ầ Kậ Fy QJKLậP GX\ QKâW [ \ VDR FKR [ ! \ !
−x my 2
↓mx 2y 1

Bài 4: &KR KË SKmkQJ WUuQK
→

D *L§L KË SKmkQJ WUuQK WUên khi m = 2.
E 7uP FiF VÕ QJX\rQ P ÿÇ KË Fy QJKLËP GX\ QK©W [ \ Pj [ ! Yj \
F 7uP FiF VÕ QJX\rQ P ÿÇ KË Fy QJKLËP GX\ QK©W [ \ Pj [ \ Oj FiF VÕ QJX\rQ
')QJ &iF EjL W5S YI KjP VY E5F KDL Yj ÿ[ WKS KjP VY \ 2 D[ (ζa 0 )
Bài 1 Cho (P) y x2 Yj ÿmáQJ WK·QJ G \ = 2x+m
a) 9Á 3
b) 7uP P ÿÇ 3 WLÃS [~F G
Bài 2 9Á ÿ× WKÏ KjP VÕ \

1 2
x

2

a) 9LÃW SKmkQJ WUuQK ÿmáQJ WK·QJ ÿL TXD KDL ÿLÇP $
-2 ) và B 1 ; - 4 )
b) 7uP JLDR LầP FởD mỏQJ WKÃQJ YùD WuP mỗF YòL ÿ× WKÏ WUrQ
Bài 3: Cho (P) y

x2
và (d): y=x+ m
4

a) 9Á 3
5


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b) ;iF ÿÏQK P ÿÇ 3 Yj G FW QKDX WƠL KDL LầP SKkQ ELậW $
Yj %
;iF ÿÏQK
c) SKmkQJ WUuQK ÿmáQJ WK·QJ G VRQJ VRQJ YòL mỏQJ WKÃQJ G Yj FW 3 WƠL LP Fy
tung ÿÝ EµQJ - 4
Bài 4: Cho (P) y

1 2
x ÿmáQJ WK·QJ G TXD ÿLÇP $
Yj % WUrQ 3 Fy KRjQK í OôP OmỗW Oj -2
Yj

4

và 4
a) 9Á ÿ× WKÏ 3 FëD KjP VÕ WUrQ
b) 9LÃW SKmkQJ WUuQK ÿmáQJ WK·QJ G
> ;24≅ sao cho tam giác MAB có
c) 7uP ÿLÇP 0 WUrQ FXQJ $
% FởD 3 WmkQJ ớQJ KRjQKx
í
GLậQ WtFK OòQ QKâW
(*kL ê FXQJ $% FoD 3
WñïQJ qQJ KRjQK ÿa x > ;24≅ Fy QJKD Oj $
-2; y A) và B(4; y )B
tính
y A;; yB )
')QJ *L+L EjL WRiQ E9QJ FiFK O5S SKđïQJ trình:
Bài 1
+DL { W{ NKãL KjQK FQJ PÝW O~F ÿL Wï $
ÿÃQ % FiFK QKDX NP Ð W{ WKí QK©W PÛL JLá
FK¥\ QKDQK KkQ { W{ WKí KDL NP QrQ ÿÃQ % VßP KkQ { W{ WKí KDL JLá 7tQK YQ WếF PL
[H { W{.
Bi 2: 0íW QKyP WKỗ ằW N KRƠFK VĐQ [XâW VĐQ SKP 7URQJ QJj\ ôX Kẹ OjP WKHR ÿ~QJ
NÃ KR¥FK ÿÅ UD QKóQJ QJj\ FzQ O¥L Kẹ m OjP YmỗW PớF PL QJj\ VĐQ SKP QrQ KRjQ WKjQK N
KRƠFK VòP QJj\ +ểL WKHR N KRƠFK PL QJj\ FôQ VĐQ [XâW EDR QKLrX VĐQ SKP
Bi 3:
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