Gio vin: Nguyn nh Thng cng n tp hc k I

Nm hc 2012 - 2013
CNG TON 10 HK I
NM HC 2011 2012
PHN I: I S
CHNG I: TP HP MNH
Bi 1. Lit k cc phn t ca cc tp hp sau:
1/
{ } 10 n 4 N n A
2/
{ } 6 n N n B <
*
3/
{ } 0 3 4n n N n C
2
+
4/
( )( ) { } 0 3 2x x 3x 2x N x D
2 2
+
5/
{ N n E
n l c ca } 12 6/
{ N n F
n l bi s ca 3 v nh hn } 14
7/
{ N n G
n l c s chung ca 16 v } 24 8/
{ N n H
n l bi ca 2 v 3 vi n nh hn } 16
9/
{ N n K
n l s nguyn t v nh hn } 20 10/
{ N n M
n l s chn v nh hn } 10
11/
{ N n N
n l s chia ht cho 3 v nh hn } 19 12/
{ N 1 n P
2
+
n l s t nhin v nh hn } 4
13/

'

+
+
N
1 n
3 n
Q
n l s t nhin v nh hn } 6 14/
{ N n R
n l s chia 3 d 1 v n nh hn } 30
Bi 2. Lit k cc phn t ca cc tp hp sau:
1/
{ } 3 k 5 Z, k 1 3k A
2/
{ } 0 9 x Z x B
2

3/
{ } 3 x Z x C
4/
{ 2k x x D
vi Z k v } 13 x 3 < <
5/
{ } 6 x 3 2x Z x E + < +
6/
{ } 4 2x 5 x Z x F + +
7/ { ( )( ) } 0 x 3 x 2 3x x Z x G
2 2
+ 8/
Z k
k
2 k
H
2

'
+

vi } 4 k 1 <
Bi 3. Lit k cc phn t ca cc tp hp sau:
1/
{ } 5 x 3 R x A <
2/
{ } 1 x R x B >
3/
{ } 3 x R x C
4/
{ } 3 x R x D
5/
{ } 2 1 x R x E
6/
{ } 0 3 2x R x F > +
7/
{ ( ) } 1 x 2 x R x F
2 2
+ < 8/
{ ( ) 0 5 3x 2x x R x G
2
+
Bi 4.
1/ Tm tt c cc tp con ca tp hp sau: { } d c, 2,3,
2/ Tm tt c cc tp con ca tp
} { 4 x N x C
c 3 phn t
3/ Cho 2 tp hp { } 1;2;3;4;5 A v { } 1;2 B . Tm tt c cc tp hp X tha mn iu kin: A X B .
1
Gio vin: Nguyn nh Thng cng n tp hc k I
Nm hc 2012 - 2013
Bi 5. Tm
A \ B B; \ A C; A B; A
1/ A l tp hp cc s t nhin l khng ln hn 10;
{ } 6 x Z x B
*

2/ ( ) [ ] 10;2011 B , 8;15 A 3/ ( ) [ ] 1;3 B , 2; A +
4/ ( ] ( ) + 1; B , ;4 A 5/
} { } { 8 x 2 R x B ; 5 x 1 R x A <
CHNG II: HM S BC NHT V BC HAI
Bi 6. Tm tp xc nh ca cc hm s
1/
2 x
3x
y
+

2/ 3 2x y 3/
4 x
x 3
y

4/
( ) x 5 x 3
5 2x
y

5/ 3x 4 1 2x y + + 6/
10 3x x
x 5
y
2

7/
3 x
5 2x
y

8/
5 6x x
5x
2 x
x
y
2
2
+
+

9/
1 x
3x
1 x
2x
y
2
+
+
+

10/
x
3 x
1 2x y

+ + 11/
5 4x x
3 5 2x
y
2

+
12/
1 x 2 x x
5 x
y
2
+ +

13/
x x
4 x
y
2

+
14/ 1 x 2 x y
2 3
+ + 15/
1 x
x 2 x 2
y
+
+ +

16/
1 x
2x 3 1 x
y

17/
x x
x 1
y
2

+
18/
2x 3
1
2 x y
3

+
19/
( ) 2 x x 3
2x 5 4 x
y
2
+

20/
2 x x
3 2x
y
2
+ +
+

Bi 7. Xt tnh chn l ca hm s:
1/ 3x 4x y
3
+ 2/ 1 3x x y
2 4
3/
5 x 2 x y
4
+
4/
1 x
1 2x 3x 2x
y
2 4

5/
( ) x x x
3 2x x
y
3
2 4
+
+

6/
x
2 x 2 x
y
+

7/
2 x
x 2x
y
3

8/
1 x
x 2 x 2
y
+
+ +

9/
2 x
2 5x 2 5x
y
2
+
+

10/
4x
2x 1 2x 1
y
+ +

2
Gio vin: Nguyn nh Thng cng n tp hc k I
Nm hc 2012 - 2013
Bi 8. Kho st s bin thin v v th cc hm s:
1/
2 3x y
2/
5 2x y +
3/
3
5 2x
y

4/
2
3x 4
y

Bi 9. Xc nh
b a,
th hm s
b ax y +
sau:
1/ i qua hai im ( ) 0;1 A v ( ) 3 2; B
2/ i qua ( ) 3 4; C v song song vi ng thng 1 x
3
2
y +
3/ i qua ( ) 1;2 D v c h s gc bng 2
4/ i qua ( ) 4;2 E v vung gc vi ng thng 5 x
2
1
y +
5/ Ct trc honh ti im c honh 3 x v i qua ( ) 2;4 M
6/ Ct trc tung ti im c tung l 2 v i qua
1) N(3;
Bi 10.
1/ Vit phng trnh ng thng i qua ( ) 4;3 A v song song vi ng thng
1 2x y : +
2/ Vit phng trnh ng thng i qua ( ) 2;1 B v vung gc vi ng thng 1 x
3
1
y : d +
Bi 11. Xt s bin thin v v th cc hm s sau:
1/ 3 4x x y
2
+ 2/ 2 x x y
2
+ 3/ 3 2x x y
2
+ 4/ 2x x y
2
+
Bi 12. Tm ta giao im ca cc th hm s sau:
1/
1 x y
v 1 2x x y
2
2/
3 x y +
v 1 4x x y
2
+
3/
5 2x y
v 4 4x x y
2
+ 4/
1 2x y
v 3 2x x y
2
+ +
Bi 13. Xc nh parabol 1 bx ax y
2
+ + bit parabol :
1/ i qua hai im ( ) 1;2 A v ( ) 2;11 B 2/ C nh ( ) 1;0 I
3/ Qua ( ) 1;6 M v c trc i xng c phng trnh l 2 x 4/ Qua ( ) 1;4 N c tung
nh l 0
Bi 14. Tm parabol c 4x ax y
2
+ , bit rng parabol :
1/ i qua hai im ( ) 2 1; A v ( ) 2;3 B 2/ C nh ( ) 2 2; I
3/ C honh nh l 3 v i qua im ( ) 2;1 P
4/ C trc i xng l ng thng 2 x v ct trc honh ti im ( ) 3;0
Bi 15. Xc nh parabol c bx ax y
2
+ + , bit rng parabol :
1/ C trc i xng
6
5
x , ct trc tung ti im
A(0;2)
v i qua im ( ) 2;4 B
2/ C nh
4) 1; I(
v i qua
3;0) A(
3/ i qua
4) A(1;
v tip xc vi trc honh ti 3 x
4/ C nh ( ) 1 2; S v ct trc honh ti im c honh l 1
3
Gio vin: Nguyn nh Thng cng n tp hc k I
Nm hc 2012 - 2013
5/ i qua ba im
C(3;2) 1;6), B( A(1;0),
Bi 16.
1/ Cho parabol ( ) ( ) 0 a bx ax y : P
2
+ , bit ( ) P c trc i xng l ng thng 1 x v ( ) P qua ( ) 1;3 M .
Tm cc h s
b a,
2/ Cho hm s c bx 2x y
2
+ + c th l mt parabol ( ) P . Xc nh
c b,
bit ( ) P nhn ng thng 1 x
lm trc i xng v i qua ( ) 2;5 A
3/ Cho hm s c 4x ax y
2
+ c th ( ) P . Tm a v c ( ) P c trc i xng l ng thng 2 x v nh
ca ( ) P nm trn ng thng
1 y
CHNG III: PHNG TRNH V H PHNG TRNH
Bi 17. Gii cc phng trnh sau:
1/ 3 x 1 x 3 x + + 2/ 1 x 2 2 x +
3/ 1 x 2 1 x x 4/ 14 3x 7 5x 3x
2
+ +
5/ 2 4 x + 6/ ( ) 0 6 x x 1 x
2

7/
1 x
4
1 x
1 3x
2

+
8/ 4 x
4 x
4 3x x
2
+
+
+ +
9/ 5 2x 7 4x 10/
1 x 1 2x x
2
+
11/ 4 16 2x x + 12/ 10 2 3x 9x +
13/ 1 2x 9 6x x
2
+ + 14/ 3x 2 3x x 4
2
+ + +
15/ 2 3 x 1 2x + 16/ 2 3x 2 x 10 3x + +
17/ 10 2 3x x 3x x
2 2
+ + 18/
2 2
x 5x 10 5x x 3 +
19/ ( )( ) 0 5 3 x x 3 4 x 4 x
2
+ + + + 20/ ( )( ) 0 10 4 x x 2 2 x 3 x
2
+ + +
Bi 18. Gii cc phng trnh sau:
1/
2 x
2 2x
2 x
2
1 x

+ 2/
3 x
2x 7
3 x
1
1

+
3/
( ) 2 x x
2
x
1
2 x
2 x

4/ 10
2 x
2 x x
2

+
+
5/
2 x
2 3x
x
2 x
4

6/ 4
3 2x
3x
2 2x
1 x

+
7/ 4
3 2x
3x
2 2x
1 x

+
8/ 0 3
2 x
1 2x
1 x
1 x
+

+
9/ 1
1 x
1 3x
1 x
5 2x

10/ 3
1 2x
3 x
1 x
4 2x

+
+
+

Bi 19. Gii cc phng trnh sau:

1/
5 3 2x +
2/
3 x 1 2x +
4
Gio vin: Nguyn nh Thng cng n tp hc k I
Nm hc 2012 - 2013
3/
2 3x 5 2x +
4/
1 2x 3 x + +
5/
1 x 4 2x
6/
6 5x x 2 2x
2
+
7/
2 x 3x 2 x
2

8/
5 6x x 5 5x 2x
2 2
+ + +
9/
0 4 2 x 2 x
2

10/
2 x 2 4x x
2
+
11/
11 4x 1 2x 4x
2
+ +
12/
1 4x 1 x
2
+
13/
1 2x 4 5x 2x
2
+
14/
0 8 2 x 4 x 3x
2
+ + +
Bi 20. Gii cc phng trnh sau:
1/ 0 4 3x x
2 4
+ 2/ 0 3 x 2x
2 4

3/ 0 6 3x
4
4/ 0 6x 2x
2 4
+
Bi 21. Cho phng trnh 0 3m m 1)x 2(m x
2 2
+ . nh m phng trnh:
1/ C 2 nghim phn bit 2/ C nghim (hay c 2 nghim)
3/ C nghim kp v tm nghim kp 4/ C mt nghim bng 1 v tnh nghim cn li
5/ C hai nghim tha ( )
2 1 2 1
x 4x x x 3 + 6/ C hai nghim tha
2 1
3x x
Bi 22. Cho phng trnh ( ) 0 2 m x 1 m x
2
+ + +
1/ Gii phng trnh vi 8 m
2/ Tm m phng trnh c nghim kp. Tm nghim kp
3/ Tm m phng trnh c hai nghim tri du
4/ Tm m phng trnh c hai nghim tha mn 9 x x
2
2
2
1
+
Bi 23.
1/ Chng minh rng vi mi 1 x > ta c 3
1 x
1
5 4x

+
2/ Chng minh rng:
3
1
x 7,
3x 1
4
3x 4 <

+
3/ Tm gi tr nh nht ca hm s:
x 2
3
3x 1 y

+ vi mi 2 x <
4/ Vi 4 x > hy tm gi tr nh nht ca biu thc:
4 x
1
x B

+
Bi 24.
1/ Chng minh rng: ( )( ) [ ] 1;5 x 4, x 5 1 x
2/ Tm gi tr ln nht ca hm s :
x) x)(2 (3 y +
vi mi 3 x 2
3/ Vi mi
1
]
1

;2
2
1
x
hy tm gi tr ln nht ca biu thc:
2x) x)(1 (2 B +
4/ Tm gi tr ln nht ca biu thc:
2
x 4 x y vi 2 x 2
5
Gio vin: Nguyn nh Thng cng n tp hc k I
Nm hc 2012 - 2013
PHN 2: HNH HC
CHNG I: VCT
Bi 1. Cho 6 im phn bit
F E, D, C, B, A,
chng minh:
1/ DB AC DC AB + + 2/
EB AD ED AB + +
3/
BD AC CD AB
4/
EB AB DC CE AD + +
5/ AB CB CE DC DE AC + + 6/ CD BF AE CF EB AD + + +
Bi 2. Cho tam gic ABC
1/ Xc nh I sao cho
0 IA IC IB +
2/ Tm im M tha
0 MC 2 MB MA +
3/ Vi M l im ty . Chng minh: CB CA MC 2 MB MA + +
4/ Hy xc nh im M tha mn iu kin:
BA MC MB MA +
Bi 3.
1/ Cho tam gic ABC u cnh a. Tnh
AC AB ; AC AB +
2/ Cho tam gic ABC u cnh bng 8, gi I l trung im BC. Tnh
BI BA
3/ Cho tam gic ABC u, cnh a, tm O. Tnh
OC AB AC
4/ Cho hnh ch nht ABCD, tm O, AB = 12a, AD = 5a. Tnh
AO AD
5/ Cho hnh ch nht ABCD, bit AB = 4, BC = 3, gi I l trung im BC. Tnh
IB IA ; DI IA +
6/ Cho hnh vung ABCD cnh a, tm O. Tnh di ca
AB BC
;
OB OA +

7/ Cho hnh vung ABCD c tm O, cnh bng 6 cm. Tnh di cc vect sau: DB CA v ; AD AB u + +
Bi 4.
1/ Cho hnh bnh hnh ABCD. Gi I l trung im ca AB v M l mt im tha
IM 3 IC
. Chng minh rng:
BC BI 2 BM 3 +
. Suy ra B, M, D thng hng
2/ Cho hnh bnh hnh ABCD. Chng minh rng: DB BC AB ; 0 DC DB DA +
3/ Cho hnh bnh hnh ABCD, gi O l giao im ca hai ng cho. Chng minh rng
0 OA OB BC + +
4/ Cho hnh bnh hnh ABCD, gi I l trung im ca CD. Ly M trn on BI sao cho BM = 2MI. Chng minh
rng ba im A, M, C thng hng
5/ Cho hnh bnh hnh ABCD c tm O, gi M l trung im BC. Chng minh rng: AD
2
1
AB AM +
6/ Cho hnh bnh hnh ABCD c tm O. Vi im M ty hy chng minh rng: MD MB MC MA + +
7/ Cho tam gic ABC. Bn ngoi ca tam gic v cc hnh bnh hnh ABIJ, BCPQ, CARS. Chng minh rng:
0 PS IQ RJ + +
Bi 5.
1/ Gi G v G ln lt l trng tm ca tam gic ABC v tam gic ABC. Chng minh rng:
GG' 3 CC' BB' AA' + +
6
Gio vin: Nguyn nh Thng cng n tp hc k I
Nm hc 2012 - 2013
2/ Cho hai tam gic ABC v ABC. Gi G v G ln lt l trng tm ca hai tam gic trn. Gi I l trung im
ca GG. Chng minh rng:
0 I C' I B' I A' CI BI AI + + + + +
3/ Cho tam gic MNP c MQ l trung tuyn ca tam gic. Gi R l trung im ca MQ. Chng minh rng:
a/ 0 RP RN RM 2 + +
b/ 4OR OP 2OM ON + + , vi O bt k
c/ Dng im S sao cho t gic MNPS l hnh bnh hnh. Chng t rng:
MP 2 PM MN MS +
d/ Vi im O ty , hy chng minh rng:
OP OM OS ON + +
;
OI 4 OS OP OM ON + + +
4/ Cho tam gic MNP c
PI NS, MQ,
ln lt l trung tuyn ca tam gic. Chng minh rng:
a/ 0 PI NS MQ + +
b/ Chng minh rng hai tam gic MNP v tam gic SQI c cng trng tm
c/ Gi M l im i xng vi M qua N; N l im i xng vi N qua P; P l im i xng vi P qua
M. Chng minh rng vi mi im O bt k ta lun c:
OP' OM' ON' OP OM ON + + + +
5/ Cho t gic ABCD v
N M,
ln lt l trung im ca on thng
CD AB,
. Chng minh rng:
a/
MN 2 DA CB DB CA + +
b/ MN 4 BC AC BD AD + + +
c/ Gi I l trung im ca BC. Chng minh rng:
( ) DB 3 DA NA AI AB 2 + + +
6/ Cho lc gic u ABCDEF c tm O. Chng minh rng:
MO 6 MF ME MD MC MB MA + + + + + vi mi im M bt k
Bi 6. Cho 3 im
C(4;4) 2;6), B( A(1;2),
1/ Chng minh A, B, C khng thng hng
2/ Tm ta trung im I ca on AB
3/ Tm ta trng tm G ca tam gic ABC
4/ Tm ta im D sao cho t gic ABCD l hnh bnh hnh
5/ Tm ta im N sao cho B l trung im ca on AN
6/ Tm ta cc im H, Q, K sao cho C l trng tm ca tam gic ABH, B l trng tm ca tam gic ACQ, A l
trng tm ca tam gic BCK
7/ Tm ta im T sao cho hai im A v T i xng nhau qua B, qua C
8/ Tm ta im U sao cho BU 5 AC ;2 BU 3 AB
Bi 7. Cho tam gic ABC c
1;1) P( N(3;0), M(1;4),
ln lt l trung im ca cc cnh BC, CA, AB.
Tm ta A, B, C
Bi 8. Trong h trc ta cho hai im
1) B(6; A(2;1);
. Tm ta :
7
Gio vin: Nguyn nh Thng cng n tp hc k I
Nm hc 2012 - 2013
1/ im M thuc Ox sao cho A, B, M thng hng
2/ im N thuc Oy sao cho A, B, N thng hng
CHNG II: TCH V HNG CA HAI VECT V NG DNG
Bi 9. Tnh gi tr cc biu thc sau:
1/ asin0
0
+ bcos0
0
+ csin90
0
2/ acos90
0
+ b sin90
0
+ csin180
0
3/ a
2
sin90
0
+ b
2
cos90
0
+ c
2
cos180
0
4/ 3 sin
2
90
0
+ 2cos
2
60
0
3tan
2
45
0
5/ 4a
2
sin
2
45
0
3(atan45
0
)
2
+ (2acos45
0
)
2
6/ 3sin
2
45
0
(2tan45
0
)
3
8cos
2
30
0
+ 3cos
3
90
0
7/ 3 sin
2
90
0
+ 2cos
2
60
0
3tan
2
45
0
Bi 10. n gin cc biu thc sau:
1/ A = sin(90
0
x) + cos(180
0
x) + cot(180
0
x) + tan(90
0
x)
2/ B = cos(90
0
x) + sin(180
0
x) tan(90
0
x).cot(90
0
x)
Bi 11. Cho tam gic ABC vung ti A, AB = a, BC = 2a. Tnh cc tch v hng:
1/
AC . AB
2/
CB . AC
3/
BC . AB
Bi 12. Cho tam gic ABC u cnh bng a. Tnh cc tch v hng:
1/ AC . AB 2/ CB . AC 3/ BC . AB
Bi 13. Cho tam gic ABC u cnh a. Tnh ) AC 3 AB (2 AB
Bi 14. Cho tam gic ABC c AB = 6; AC = 8; BC = 11
1/ Tnh
AC . AB
v suy ra gi tr ca gc A
2/ Trn AB ly im M sao cho AM = 2. Trn AC ly im N sao cho AN = 4. Tnh
AN . AM
Bi 15. Cho hnh vung cnh a, I l trung im AI. Tnh AE . AB
Bi 16. Cho tam gic ABC bit AB = 2; AC = 3; gc A bng 120
0
. Tnh AC . AB v tnh di BC v tnh di trung
tuyn AM ca tam gic ABC
Bi 17. Cho tam gic ABC c
C(2;0) 3), B(5; 1), A(1;
1/ Tnh chu vi v nhn dng tam gic ABC
2/ Tm ta im M bit
AC 3 AB 2 CM
Bi 18. Cho tam gic ABC c
C(9;8) 2;6), B( A(1;2),
1/ Tnh AC . AB . Chng minh tam gic ABC vung ti A
2/ Tnh chu vi, din tch tam gic ABC
3/ Tm ta im M thuc trc tung ba im B, M, A thng hang
4/ Tm ta im N trn Ox tam gic ANC cn ti N
5/ Tm ta im D ABCD l hnh bnh hnh v tm tm I ca hnh bnh hnh
6/ Tm ta im M sao cho
0 MC MB 3 MA 2 +
8
Gio vin: Nguyn nh Thng cng n tp hc k I
Nm hc 2012 - 2013
---Chc cc em thi tt---
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