500 bài toán Ôn thi vào lớp 10
DESCRIPTION
gTRANSCRIPT
-
500 bi ton n thi vo lp 10
1
Rt gn biu thc
Bi 1 A=
++
+
121:
11
112
xx
x
xxx
x a) Rt gn A b) Tnh A bit
x=2
32 c)Tm xZ AZ d) Tm GTNN ca A e)Tm x
A=1/3
g) So snh A vi 1 h) Tm x A > 1/2
Bi 2 B= x
xx
+
1)1( 2
:
+
+
+
xx
xxx
x
xx
11
11
a)Rt gn B b)Tm x
B=2/5 c)Tnh B bit x= 12-6 3 d) Tm GTNN v GTLN caB e) So snh B vi 1/2 g) Tm x
B > 3x
Bi 3 C=
+
+ xxxx
x
123:
325
3522
a)Rt gn C=x23
1
b)Tm GTNN ca C vi C=1
1.
1+xC
c)Tnh C vi x=32
2
d)Tm x
C>0
e)Tm x Z C Z g)Tm x C= 5 x
Bi 4 E=
+
+
+
+
xx
x
xx
x
xx
xx 21
11:
12 a)Rt gn E=
1xx
b)Tm x
E > 1
c)Tm GTNN ca E vi x > 1 d)Tm x Z E Z e)Tnh E ti
512 =+x
g)Tm x E = 9/2
Bi 5 G=
+
+
+
++
+
+
11
11
:111
1x
x
x
x
x
x
x
x
x
x a)Rt gn G =
x
x
412 +
b)Tm GTNN ca G vi x>0 c)Tnh G ti x = 17- 4 13 d)Tm x
G = 9/8 Bi 6 K=x
x
x
x
xx
x
+
+
+
312
23
6592
a)Rt gn K=31
+
x
x
b)Tm x K
-
500 bi ton n thi vo lp 10
2
g) Tnh K bit x-3 2x + =0 h) So Snh K vi 1
Bi 7 M=
+
+
+
+
12
111
:11
11
xx
x
xx
x
x
x a)Rt gn
M=12
4++ xx
x
b)Tm x M= 8/9 c)Tnh M ti x= 17+12 2 d)Chng minh M 0 e)So snh M vi 1 g) Tm GTNN, GTLN ca M
Bi 8 N=
+
+
32
23
69
:19
3x
x
x
x
xx
x
x
xx a)Rt gn N=
23x
b)Tm x N4 e) Tnh R ti x=11-
6 2
Bi 11 S=
+
++
12
11
:1
1aaaa
a
aa
a a)Rt gn S=
11
++
a
aa
b)Tm a S=2a c)Tm GTNN ca S vi a>1 d)Tnh S ti a=1/2
e)Tm a Z S Z
Bi 12 Y=
++
+
+
11
1.
221
2333
xx
x
x
x
xx
xx a)Rt gn
Y=22
+
x
x
b)Tm x Y=x c)Tm xZ Y Z d)Tm
GTLN ca Y
-
500 bi ton n thi vo lp 10
3
Bi 13 P = 3 6 411 1
x x
xx x
+
+ a) Rt gn P=
11
+
x
x
c)Tm x Z P Z d)Tm GTNN ca P e) Tnh P ti x=6-
2 5
Bi 14 P = xx
xx
xx
xx
x
x
+
+
++ 1122
a) Rt gn P=x
xx 222 ++
b) Tm GTNN ca P c) Tnh P ti x = 12+ 6 3
Bi 15 P = 2
221
11
11
+
+
x
xx
x
x
x a) Rt gn P=
x
x1 b) tm GTLN ,
GTNN ca P c) Tm x P =2 d) Tnh P ti x= 3-2 2 e ) Tm x P > 0
g) So snh P vi -2 x
Bi 16 P = 1
11
211
++
+
+
+
xx
x
xx
x
x
x a) Rt gn P =
1++
xx
x b) tm
GTLN ca P
c) Tm x P = -4 d) Tnh P ti x=6-2 5 e ) Tm x P < -3
g) So snh P vi 1 h) Tm x Z P Z
Bi 17 P = 1)1(22
1
2
++
++
x
x
x
xx
xx
xx a) Rt gn P = 1+ xx b) Tm
GTNN ca P
c) Tm x P = 3 d) Tnh P ti x=7+2 3 e ) Tm x P > 3 g) So snh P vi 1/2
Bi 18 P =
++
+
++
11
11
:223
aaaa
a
aa
aa a) Rt gn P =
a
a
21+
b Tm x
P = 3
d) Tnh P ti x= 15-6 6 e ) Tm x P>3 g) So snh P vi 1/2
Bi 19 P = 11
21
1:
11
+
++
xxxx
x
xx
x a) Rt gn P =
12
+
x
x c) Tm x
P =5
-
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b) Tm GTLN , GTNN ca P=1P
e ) Tm x P>0 d) Tnh P ti x=5-
2 6
Bi 20 P = 1212
111
2
++
+
+
x
x
xx
x
x
xx
xx
xxxx a) Rt gn P =
1+++
xx
xx b)
tm GTLN , GTNN ca P c) Tm x P = 2 d) Tnh P ti x= 8+2 10 e ) Tm x P>1
Bi 21 P= 1
11
11
2
++
++
+
xxx
x
xx
x a) Rt gn P=
1++ xxx
b) Tm GTLN , GTNN ca P c) Tm x P =1/3 d) Tnh ti x= 22- 4 10
Bi 22 P=
++
++
+ 22
11
12333
xxxx
xx a) Rt gn P=
1
1
x
x
+
b) Tm GTLN
ca P
c) Tm x P = 4 d) Tnh P ti x=17+12 2 e ) Tm x P< 2 g) So snh P vi 3
Bi 22 P =
+
+
+
xx
x
xx
x
x
x
x
x
324
35
:9
433
33
a) Rt gn
P=2
4x
x
b) Tm GTNN ca P vi x>4 c) Tm x P = 3 d)Tm x P > 4 x
Bi 23 P =
+
+
+
52
25
10325
:125
5a
a
a
a
aa
a
a
aa a) Rt gn P
=2
5+a
b) Tm GTLN ca P c) Tm a P = 2 d) Tnh P ti a= 4 - 2 3 e ) Tm a P > 2
Bi 24 P = 23
:2
42
+
+ x
x
xx
x
x
x a) Rt gn P=
34
+
x
x b) Tm GTNN ca P
c) Tm x P = -1 d) Tnh P ti x=11-4 6 e ) Tm x P>-1 g) So snh P vi 1
-
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5
Bi 25 P = ( )( )( )
12
1126
131
2
2
+
+
aaa
a
aa
a a) Rt gn P=
115++
+
aa
a
b) Tm GTLN , GTNN ca P c) Tm x P = 1 ) Tnh P ti x= 7-2 6
Bi 26 P =
+
+
18
11
11
:1
11
3x
x
x
x
x
x
xx
xx a) Rt gn P =
x
x
44+
b) Tm GTLN , GTNN ca P c) Tm x P = 8 h) Tm x Z P Z
d) Tnh P ti x= 10-2 21 e ) Tm x P >5 g) So snh P vi 4
Bi 27 P = 1+ 121
21
12
+
+
x
xx
xx
xxxx
x
xx a) Rt gn P b
Tm GTLN , GTNN ca P c) Tm x P = 3 d) Tnh P ti x= 13- 4 10
Bi 28 P =
++
++
+
+ 1
21
1:
223
22 xxx
xx
x
x
x
x
x a) Rt gn P= ( )1.2 3++xx
b) Tm GTLN , GTNN ca P c) Tm x P = 3 d) Tnh P ti x= 15+6 6
e ) Tm x P >4 g) So snh P vi 2
Bi 29 P = 4 1 3: 12 3 3 2
x x x x
x x x x
+ +
a) Rt gn P =
12
+
x
x
b) Tm GTNN ca P c) Tm x P =1/2 d) Tnh P ti x= 5+2 6
e ) Tm x P > -1 g) So snh P vi 1
Bi 30 P =
+
+ 12
11
:1
221
1xxxxxx
x
x a) Rt gn P =
11
x
x
+
b)Tm x P =x3
1 c) Tm GTNN ca P d) Tnh P ti x=7-2
Bi 31 P =
+
+
+
+
+
12:
32
23
652
x
x
x
x
x
x
xx
x Rt gn P =
14
x
x
+
-
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6
b) Tm x P = 3 c) Tm x Z P Z d) Tnh P ti x= 5 2 6
e ) Tm x P>2 g) So snh P vi 2 h) Tm GTLN , GTNN ca
P=1P
Bi 32) P = x :
++
+++
+
12
11
11
xx
x
xxx
x Rt gn P = 1++ xx
b) Tm x P = 6 e ) Tm x P >3 g) So snh P vi 3 x h) Tm
GTNN ca P
Bi 33) P = ( )
12
23
233
+
++
+
+
x
x
x
x
xx
xx Rt gn P =
3 82
x
x
+
+ b) Tm x
P = 7/2 c) Tm x Z P Z d) Tnh P ti x= 13 4 10 e ) Tm x P> 10/3
g) So snh P vi 3 h) Tm GTLN , GTNN ca P
Bi 34 P=
+
+
+
4
7221
2 xx
x
x
x
x:
+
12
3x
x a) Rt gn P =
25
+
x
x
b) Tnh P bit x= 9-4 5 c) Tm GTNN ca P d) Tm xZ PZ
Bi 35 P =
+
+
+
xx
x
xx
x
x
x
x
x
23
22
:4
422
22
a) Rt gn P =3
4x
x
b) Tm x P = -1 c) Tm x Z P Z d) Tnh P ti x= 15 4 14
e ) Tm x P > 4 g) So snh P vi 4 x h) Tm GTLN , GTNN ca P
vi x>9
Bi 36 P =
++
+
+
141:
11
112
xx
x
xxx
x a) Rt gn P =
3xx
b) Tm x P = - 2 c) Tm x Z P Z d) Tnh P ti x= 23 4 15
e ) Tm x P >1 h) Tm GTLN , GTNN ca P=31
x
x
+. P
Bi 37 P = 33
12
321926
+
+
+
+
x
x
x
x
xx
xxx a) Rt gn P =
316+
+
x
x
-
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7
b) Tnh P ti x= 7- 4 3 c) Tm GTNN ca P b) Tm x P = 7 c) Tm
x Z P Z d) Tnh P ti x= 17 12 2 e ) Tm x P < x h) Tm GTNN ca P
Bi 38 P = x
x
x
x
xx
x
+
+
+
+
312
43
12712
a) Rt gn P = 42
x
x
b) Tnh P ti x= 2 347 c) Tm x 2AA < d) Tm x P = 2
c) Tm x Z P Z e ) Tm x P > 1 h) Tm GTLN , GTNN ca P= P .
42
x
x
+
Bi 39 P = x
x
xx
xx
xx
xx 111 ++
+
+
a) Rt gn P = x
xx 12 ++ b) Tm x
P= 9/2
c) Tm x Z P Z d) Tnh P ti x= 25 6 14 g) So snh P vi 4
h) Tm GTLN , GTNN ca P
Bi 40 P = 1
461
31
++
x
x
xx
x a) Rt gn P =
11
+
x
x b) Tm x
P = -1 c) Tm x Z P Z d) Tnh P ti x= 11 4 6 e ) Tm x P > 2
g) So snh P vi 1 h) Tm GTNN ca P
i) Tnh P ti x = 347347 ++ k) Tm x P < 1/2
Bi 41 P = xx
x
x
x
x +
++ :
11
a) Rt gn P=x
xx 1++ b) Tm x
P = -1 c) Tm x Z P Z e ) Tm x P > 2x + g) So snh P vi 1
h) Tm GTLN , GTNN ca P b) Tnh P ti x = 15
815
8+
Bi 42 P =
+
++
1322
:933
332
x
x
x
x
x
x
x
x a) Rt gn P =
33x
+
b) Tm x P = c) Tm x Z P Z b) Tm x khi x= 16 c) Tm GTNN ca N
-
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8
Bi 43 P = 1 1 1 2 1
:12 2 2 2
+ + + +
+ +
x x x x x
xx x x x Rt gn P =
1x
x b) Tm x
P =2 c) Tm x Z P Z
Bi 44 P = 2 1
: 111 1
x x
xx x x x x
+ +
+ a) Rt gn P =
11
x
x x
+ +
b) Tm x P = -1/7 c) Tm x Z P Z d) Tnh P ti x= 9
g) So snh P vi 1 h) Tm GTLN , GTNN ca P
Bi 45 P = 2 9
93 3x x
xx x
+ +
+ a) Rt gn P =
53x
b) Tm x P = 5
c) Tm x Z P Z d) Tnh P ti x= 11 6 2 e ) Tm x P >0
Bi 46 P = 3 2 22 3 5 6
x x x
x x x x
+ + ++ +
+ a) Rt gn P =
12x
b) Tm x
P = -1 c) Tm x Z P Z d) Tnh P ti x= 6 4 2 e ) Tm x P > 1
Bi 47: Cho biu thc: P=
+
++
++
+
+
652
32
23
:1
1xx
x
x
x
x
x
x
x
a) Rt gn P b)Tm gi tr ca a P
-
MATHVN.COM - www.mathvn.com 500 bi ton n thi vo lp 10
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9
Bi 52: Cho biu thc: P=
++
+ 11:
11
12
x
x
xxxxx
x
a) Rt gn P b)Tm x P 0
Bi 53: Cho biu thc: P=
+
+
++
+a
a
a
aa
a
a
a
11
.
112 3
3
a) Rt gn P b)Xt du ca biu thc P. a1
Bi 54: Cho biu thc: P= .11
11
12
:1
+
++
++
+
x
x
xx
x
xx
x
a) Rt gn P b)So snh P vi 3
Bi 55: Cho biu thc : P=
+
+
+
aa
aaa
a
aa
11
.
11
a) Rt gn P b)Tm a P< 347
Bi 56: Cho biu thc: P=
+
++
1322
:933
332
x
x
x
x
x
x
x
x
a) Rt gn P b)Tm x P1
Bi 60: Cho biu thc : P= 121
2
++
+
+
a
aa
aa
aa Rt gn P
b)Bit a>1 Hy so snh P vi P c)Tm a P=2 d)Tm gi tr nh nht ca P
Bi 61: Cho biu thc P=
+
+
+
+
++
+
+ 111
1:1
111
abaab
aba
abaab
aba
a)Rt gn P b)Tnh gi tr ca P nu a= 32 v b=3113
+
-
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10
c)Tm gi tr nh nht ca P nu 4=+ ba
Bi 62: Cho biu thc : P=
+
+
+
+
+
+
11
11111
a
a
a
a
aa
aa
aa
aa
aa
a)Rt gn P b)Vi gi tr no ca a th P=7 c)Vi gi tr no ca a th P>6
Bi 63: Cho biu thc: P=
+
+
11
11
21
2
2
a
a
a
a
a
a a)Rt gn P
b)Tm cc gi tr ca a P0 x 1
Bi 66: Cho biu thc : P=
++
+
+
121:
11
12
xx
x
xxx
xx
Rt gn P b)Tnh P khi x= 325 +
Bi 67: Cho biu thc: P=1 3 2 1
:42 4 2 4 2
x
xx x x
+
+
a) Rt gn P b)Tm gi tr ca x P=20
Bi 68: Cho biu thc : P=( )
yxxyyx
xyyx
yxyx
+
+
+
233
:
a) Rt gn P b)Chng minh P 0 Bi 69: Cho biu thc :
P=
++
++
+ bababa
bbaaab
babbaaab
ba:
31.
31
a) Rt gn b)Tnh P khi a=16 v b=4
Bi 70: Cho biu thc: P=12
.
12
1121
+
++
a
aa
aa
aaaa
a
aa
a)Rt gn P
b)Cho P=61
6+
tm gi tr ca a b)Chng minh rng P>32
-
MATHVN.COM - www.mathvn.com 500 bi ton n thi vo lp 10
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11
Bi 71: Cho biu thc: P=
++
+
+
35
53
15225
:125
5x
x
x
x
xx
x
x
xx
a) Rt gn P b)Vi gi tr no ca x th P61
Bi 74 Cho biu thc: P=33
33
:112
.
11xyyx
yyxxyxyxyxyx +
+++
++
+
+
a) Rt gn P b)Cho x.y=16. Xc nh x,y P c gi tr nh nht
Bi 75: Cho biu thc : P= x
x
yxyxxx
yxyx
+
11
.
222
2
3
a) Rt gn P b)Tm tt c cc s nguyn dng x y=625 v P - C c) Tm gi tr ca C C2 =
40C.
Bi 77: Cho biu thc M = 25 25 5 21 :25 3 10 2 5
a a a a a
a a a a a
+
+ +
a) Rt gn M b) Tm gi tr ca a M < 1 c) Tm gi tr ln nht ca M.
Bi 78: Cho biu thc 4 3 2 4:2 2 2
x x x xP
x x x x x
+
= +
a) Rt gn P b) Tm cc gi tr ca x P > 0 c) Tnh gi tr nh nht ca P
Bi 79: Cho biu thc P = ( )
( )( )2 2
2
1 3 2 1 2
1 13 1
a a
a a aa a
+ +
a) Rt gn P. b) So snh P vi biu thc Q = 2 11
a
a
-
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12
80 Cho biu thc A = 3 1 1 1 8
:1 11 1 1
m m m m m
m mm m m
+
+
a) Rt gn A. b) So snh A vi 1
Bi81: Cho biu thc A = 2 1 2
11 1 2 1
x x x x x x x x
x x x x
+ + +
a) Rt gn A. b) Tm x A = 6 6
5
c) Chng t A 2
3 l bt ng thc sai
Bi 82: Cho biu thc P = 3 1 2
:2 22 2 1 1
x x x x
xx x x x x
+ +
+ +
+ + a)
Rt gn P
b) Chng minh rng P > 1 c) Tnh gi tr ca P, bit 2 3x x+ =
d) Tm cc gi tr ca x : ( ) ( ) ( )2 2 . 5 2 2 . 2 4x P x x+ + = +
Bi 84: Cho biu thc P = 2 1
.
11 2 1 2 1x x x x x x x x
xx x x x x
+ + +
+
a) Rt gn P b) Tm gi tr ln nht ca A = 5 3
.
xPx x
+
c) Tm cc gi tr ca m mi x > 2 ta c: ( ) ( ). 1 3 1P x x m x x+ + > + Bi 90: Cho biu thc:
1x
2x
2x
3x
2xx
3)x3(xP
+
++
+
+=
a/ Rt gn P b/ Tm x 4
15P <
Bi 91: Cho biu thc:
+
=
2x
x
x
2x:
x2
3
x2x
4xP
a/ Rt gn P ; b/ Tm x x3 - 3xP =
b/ Tm cc gi tr ca a c x tho mn : ax1)xP( +>+
Bi 93. Cho x3
1x2
2x
3x
6x5x
9x2P
+
+
+
=
a. Rt gn P. b. Tm cc gi tr ca x P
-
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13
Cu 94. Cho biu thc ( )( )a 3 a 2 a a 1 1P :
a 1 a 1 a 1a 2 a 1
+ + + = + + +
a) Rt gn P. b) Tm a 1 a 1 1P 8
+
Cu 95. Cho biu thc x 1 2 xP 1 : 1
x 1 x 1 x x x x 1
= + +
+
a) Tm iu kin P c ngha v rt gn P.
b) Tm cc gi tr nguyn ca x biu thc P x nhn gi tr nguyn. Cu 96 .Cho
a a a aP 1 1 ; a 0, a 1a 1 1 a
+ = +
+ +
a) Rt gn P. b) Tm a bit P > 2 c) Tm a bit P = a . Cu 97.
1.Cho biu thc
x 1 x 1 8 x x x 3 1B :x 1 x 1x 1 x 1 x 1
+ =
+
a) Rt gn B.
b) Tnh gi tr ca B khi x 3 2 2= + . c) Chng minh rng B 1 vi mi gi tr ca x tha mn x 0; x 1
.
Bi 98(2)
1) Cho biu thc:
P = a 3 a 1 4 a 4
4 aa 2 a 2
+ +
+
(a 0; a 4) a) Rt gn P. b) Tnh gi tr ca P vi a =
9.
3) Rt gn biu thc: P = x 1 x 1 2
2 x 2 2 x 2 x 1
+
+ (x 0; x 1).
Cu 99 (2)Cho biu thc:
A = x 2 x 1 x 1
:2x x 1 x x 1 1 x
+ + +
+ + , vi x > 0 v x 1.
1) Rt gn biu thc A. 2) Chng minh rng: 0 < A < 2.
-
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14
Cu 100 (2)Cho biu thc: A = ( )2 x 2 x 1x x 1 x x 1
:x 1x x x x
+ +
+ .
1) Rt gn A. 2) Tm x nguyn A c gi tr nguyn.
A = 2
2
x 1 x 1 x 4x 1 x 2003.
x 1 x 1 x 1 x
+ + +
+ .
101) Tm iu kin i vi x biu thc c ngha. 2) Rt gn A. 3) Vi x Z ? A Z ?
102) Rt gn biu thc : A = 1 1 3
1a 3 a 3 a
+
+ vi a > 0 v a 9.
103) Rt gn biu thc sau : A = ( )x x 1 x 1 x xx 1 x 1
+
+ vi x 0, x 1.
104) Cho biu thc : Q = x 2 x 2 x 1
.x 1x 2 x 1 x
+ +
+ + , vi x > 0 ; x
1.
a) Chng minh rng Q = 2
x 1 ; b) Tm s nguyn x ln nht Q c gi tr nguyn.
Cu 105 ( 3 im )
Cho biu thc :
++
+
+=
12
:)1
11
2(xx
x
xxx
xxA
a) Rt gn biu thc .
b) Tnh gi tr ca A khi 324 +=x Cu 106 : ( 2,5 im )
Cho biu thc : 1 1 1 1 1A= :
1- x 1 1 1 1x x x x
+ + + +
a) Rt gn biu thc A . b) Tnh gi tr ca A khi x = 7 4 3+ c) Vi gi tr no ca x th A t gi tr nh nht .
Cu 107 ( 2,5 im )
Cho biu thc : A = 1 1 2
:2
a a a a a
aa a a a
+ +
+
a) Vi nhng gi tr no ca a th A xc nh .
b) Rt gn biu thc A . c) Vi nhng gi tr nguyn no ca a th A c gi
tr nguyn .
cu 108: (2 im) Cho biu thc: 1,0;11
11
+
+
+= aa
a
aa
a
aaA .
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1. Rt gn biu thc A. 2. Tm a 0 v a1 tho mn ng thc: A= -a2
cu 109: Rt gn biu thc:
1,0;1
11
1
+
+
= aaa
aa
aaM .
cu 110: Cho biu thc: yxyxyx
xyxyx
yxyx
yS >>
++
= ,0,0;2
: .
1. Rt gn biu thc trn. 2. Tm gi tr ca x v y S=1.
cu 111: Cho biu thc 1,0;1
1>
++
= xxxx
x
xA .
1. Rt gn biu thc A. 2 Tnh gi tr ca A khi 2
1=x
bi 112: Cho biu thc: 4,1,0;21
12
:1
11>
+
+
= xxxx
x
x
x
xxA .
1. Rt gn A. 2. Tm x A = 0.
Bi 113: (2 im)
Cho biu thc:
+
+
+=
x
1x
1x
x:
1x
1
)1x(x
1xB
a) Tm iu kin i vi x B xc nh. Rt gn B. b)Tm gi tr ca B khi
223x = .
phng trnh bc hai cha tham s Bi 1 Tm m cc phng trnh sau v nghim , c mt nghim , c hai nghim phn bit , c hai
nghim tri du , c hai nghim m , c hai nghim dng ,
a) x2 -3x +m 2 = 0 b) x2 - 2(m-1)x + m2 -m+1=0 c) x2 2x + m 3
= 0
d) x2 2(m+2) x + m +1= 0 e) (m 1 )x2 + 2(m 1)x m = 0 g) x2 2(m+1) x +
m 4 = 0
Bi 2 Cho pt 2x2 - 7x + 1 = 0 .Khng gii pt hy tnh gi tr ca biu thc A = (x1-1)(x2-1) vi x1,x2 l
nghim ca pt
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Bi 3 Cho pt mx2- 2(m+1)x +m 5 = 0 a) Xc nh m pt c 1 nghim duy
nht
b) Xc nh m pt c hai nghim tho mn h thc (x1+1)(x2+1) = 3
Bi 4 Cho pt x2- 2mx+4m - 4 = 0 . Tm m pt c hai nghim tho mn 4
13111
2
2
1=
++
+
x
x
x
x
b) Vit h thc lin h gia x1 v x2 m khng ph thuc vo m
Bi5 Cho pt x2 5x +2m- 1=0
a) Vi gi tr no ca m th pt c hai nghim phn bit b) Tm m 3
191
2
2
1=+
x
x
x
x
Bi 6 Cho pt x2 2(m+1)x + 2m + 10 = 0
a) Tm m pt c 2 nghim phn bit b) Tm GTNN ca biu thc
A=10x1x2+x12+x2
2
c) Vit h thc lin h gia x1 v x2 m khng ph thuc vo m
Bi 7 Cho pt (m- 4)x2 2mx + m 2 = 0 a) Gii pt vi m=3
b) Tm m pt c nghim x=2 , tm nghim cn li c) Tm m pt c 2
nghim phn bit
d) Vit h thc lin h gia x1 v x2 m khng ph thuc vo m
Bi 8 Cho pt mx2- 2(m+3)x + m 2 = 0 a) Vi gi tr no ca m th pt c hai nghim
phn bit
b) Tm m tho mn h thc 3x1x2 2(x1+x2) + 7 = 0
c) Vit h thc lin h gia x1 v x2 m khng ph thuc vo m
Bi 9 Cho pt x2 4x + m 1 = 0 . Tm m pt c hai nghim tho mn x1 = 2x2
Bi 10 Cho phng trnh x2 (m 3)x m = 0 a) Chng t pt lun c hai nghim phn bit
b) Tm m pt c nghim bng -2 . Tm nghim cn li
c) Tm m pt c hai nghim x1 , x2 tho mn h thc : 3(x1+x2) x1.x2 5 d) Vit h thc lin h gia x1 v x2 m khng ph thuc vo m
Bi 11 Cho pt x2 2x + m 3 = 0 a) Tm m pt c hai
nghim
b) Vi gi tr no ca m th pt c hai nghim tho mn h thc x13 + x2
3 = - 20
Bi12 Cho pt x2 2(m+3)x + m2 + 8m + 6 = 0 a) Tm m th pt c 2 nghim x1, x2 tho mn x12
+ x22 = 34
b) Vi gi tr ca m tm c khng gii pt hy tnh biu thc A = 1
2
2
1
x
x
x
x+
Bi 13 Cho pt x2 2(m+1) x + m 4 = 0 a) Chng minh pt lun c hai nghim phn bit
vi mi m
b) Tm m pt c hai nghim x1 , x2 tho mn h thc x12 + x2
2 = 40
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c) Vit h thc lin h gia x1 v x2 m khng ph thuc vo m
Bi 14 Cho pt x2 2(m+2) x + m +1= 0 a) Chng minh pt lun c hai nghim phn bit
vi mi m
b) Tm m pt c hai nghim x1 , x2 tho mn h thc (2x1 -1)(2x2 - 1)+3=0
c) Vit h thc lin h gia x1 v x2 m khng ph thuc vo m
Bi15 Cho pt x2 (2m+3)x + m = 0 a) Gii pt vi m = 2
b) Chng minh pt lun c hai nghim phn bit vi mi m
c) Vit h thc lin h gia x1 v x2 m khng ph thuc vo m
Bi 16 Cho pt x2 2(m+1)x + m 4 = 0 a) Chng minh pt lun c hai
nghim phn bit
b) Tm m pt c hai nghim tri du d) Lp pt c cc nghim l 1/x1 v
1/x2
c) Chng minh biu thc M = x1 ( 1- x2) + x2(1- x1) khng ph thuc vo m
e) Vit h thc lin h gia x1 v x2 m khng ph thuc vo m
Bi 17 Cho pt (m 1 )x2 + 2(m 1)x m = 0 b) Tm m pt c
hai nghim m
a) Tm m pt c nghim kp , hai nghim tri du m tng c gi tr m
Bi 18 Cho pt x2 2(m 1)x 3 m = 0 a) Chng t pt lun c hai nghim vi mi m
b) Tm m pt c hai nghim tho mn x12 + x2
2 10 c)Vit h thc lin h gia x1 v x2 m khng ph thuc vo m
Bi 19 Cho pt x2 (2m+1)x + m2+ 2 = 0
a) Tm m pt c hai nghim x1,x2 sao cho x12 + x2
2 t gi tr nh nht
b) Tm m pt c hai nghim x1 , x2 sao cho x1+ 2x2 = 4
Bi 20 Cho pt (m 2)x2 2mx + m - 4 = 0 a) Vi m bng bao nhiu th pt trn l pt bc
hai ?
b) Gii pt vi m = 2 c) Tm m pt c hai nghim phn bit ?
d) Gi s pt c hai nghim x1 , x2 . Tnh x12 + x2
2
Bi 21 Cho pt x2 (m-2)x - m2+ 3m - 4 = 0
a) Chng minh rng pt lun c hai nghim phn bit vi mi m
b) Tm m t s gia hai nghim ca pt c tr tuyt i bng 2
Bi 22 Cho pt x2 2(m +2)x +m +1 = 0 a) Gii pt vi m = 2
b) Tm m pt c hai nghim tri du
c) Gi x1 v x2 l cc nghim ca pt . Tm m x1( 1- 2x2) + x2(1- 2x1) = m2
Bi 23 Cho pt x2 (m 1)x m2 +m 1 = 0 a) Gii pt vi m = - 1
b) Chng minh rng pt lun c hai nghim phn bit vi mi m c) Tm m 1x + 2x = 2
Bi24: Cho phng trnh : ( ) 0224 2 =+ mmxxm (x l n )
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a) Tm m phng trnh c nghim x=2 .Tm nghim cn li
b)Tm m phng trnh 2 c nghim phn bit c)Tnh A = 2221 xx + theo m
Bi25: Cho phng trnh : ( ) 04122 =++ mxmx (x l n ) a)Tm m phng trnh 2 c nghim tri du
b)Chng minh rng phng trnh lun c 2 nghim phn bit vi mi m
c) Chng minh biu thc M= ( ) ( )1221 11 xxxx + khng ph thuc vo m. Bi26: Tm m phng trnh : a) ( ) 0122 =+ mxx c hai nghim dng phn bit b) 0124 2 =++ mxx c hai nghim m phn bit
c) ( ) ( ) 012121 22 =+++ mxmxm c hai nghim tri du Bi 27: Cho phng trnh : ( ) 021 22 =+ aaxax a)CMR phng trnh trn c 2 nghim trI du vi mi a
b) Gi hai nghim ca phng trnh l x1 v x2 .Tm gi tr ca a 22
21 xx + t gi tr nh nht
Bi 28:Vi gi tr no ca m th hai phng trnh sau c t nht mt nghim s chung:
( )22 3 2 12 0x m x + + = (1) ( )24 9 2 36 0x m x + = (2) Bi 29: Cho phng trnh : 0222 22 =+ mmxx a)Tm m phng trnh c hai nghim dng phn bit
b) Gi s phng trnh c hai nghim khng m, tm nghim dng ln nht ca phng trnh
Bi 30 Cho phng trnh: 0142 =+++ mxx a)Tm iu kin ca m phng trnh c nghim
b)Tm m sao cho phng trnh c hai nghim x1v x2 tho mn iu kin 102221 =+ xx
Bi 31: Cho phng trnh ( ) 052122 =+ mxmx a) CMR phng trnh lun c hai nghim vi mi m
b) Tm m phng trnh c hai nghim cung du . Khi hai nghim mang du g ?
Bi 32: Cho phng trnh ( ) 0102122 =+++ mxmx (vi m l tham s ) a)Gii v bin lun v s nghim ca phng trnh
b)Trong trng hp phng trnh c hai nghim phn bit l 21; xx ; hy tm mt h thc lin h gia
21; xx m khng ph thuc vo m
c)Tm gi tr ca m 22212110 xxxx ++ t gi tr nh nht
Bi 33: Cho phng trnh ( ) 0121 2 =++ mmxxm vi m l tham s a) CMR phng trnh lun c hai nghim phn bit 1m b)Tm m d phng trnh c tch hai nghim bng 5, t hy tnh tng hai nghim ca phng trnh
c) Tm mt h thc lin h gia hai nghim khng ph thuc vo m
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19
d)Tm m phng trnh c nghim 21; xx tho mn h thc: 025
1
2
2
1=++
x
x
x
x
Bi 34: Cho phng trnh : 012 =+ mmxx (m l tham s)
a)CMR phnh trnh c nghim 21; xx vi mi m ;
b)t 2 21 2 1 2B 6x x x x= + Tm m B=8 ; Tm gi tr nh nht ca B v gi tr ca m tng
ng
c)Tm m sao cho phng trnh c nghim ny bng hai ln nghim kia
Bi 35: Cho f(x) = x2 - 2 (m+2).x + 6m+1 a)CMR phng trnh f(x) = 0 c nghim vi
mi m
b) t x=t+2 .Tnh f(x) theo t, t tm iu kin i vi m phng trnh f(x) = 0 c 2 nghim ln
hn 2
Bi 36 Cho phng trnh : ( ) 05412 22 =+++ mmxmx a)Tm m phng trnh c nghim
b)Tm m phng trnh c hai nghim phn bit u dng
c) Xc nh gi tr ca m phng trnh c hai nghim c gi tr tuyt i bng nhau v tri du
nhau
d)Gi 21; xx l hai nghim nu c ca phng trnh . Tnh 22
21 xx + theo m
Bi 37: Cho phng trnh ( ) 0122 =+++ mxmxx a)Gii phng trnh khi m=
21
b) Tm cc gi tr ca m phng trnh c hai nghim tri du
c)Gi 21; xx l hai nghim ca phng trnh . Tm gi tr ca m :
21221 )21()21( mxxxx =+
Bi 38: Cho phng trnh 032 =++ nmxx (1) (n , m l tham s) a) Cho n=0 . CMR phng trnh lun c nghim vi mi m
b) Tm m v n hai nghim 21; xx ca phng trnh (1) tho mn h :
=
=
71
22
21
21
xx
xx
Bi 39: Cho phng trnh: ( ) 052222 = kxkx ( k l tham s) a)CMR phng trnh c hai nghim phn bit vi mi gi tr ca k
b) Gi 21; xx l hai nghim ca phng trnh . Tm gi tr ca k sao cho 1822
21 =+ xx
Bi 40: Cho phng trnh ( ) 04412 2 =+ mxxm (1) a)Gii phng trnh (1) khi m=1
b)Gii phng trnh (1) khi m bt k c)Tm gi tr ca m phng trnh (1) c mt
nghim bng m
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Bi 41:Cho phng trnh : ( ) 0332 22 =+ mmxmx a)CMR phng trnh lun c hai nghim phn bit vi mi m
b) Xc nh m phng trnh c hai nghim 21, xx tho mn 61 21
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2) Gi hai nghim ca phng trnh l x1 v x2. Tm cc gi tr ca m tho mn 5x1 + x2 = 4.
Cu 51 Cho phng trnh: x2 + 4x + 1 = 0 (1)
1) Gii phng trnh (1). 2) Gi x1, x2 l hai nghim ca phng trnh (1). Tnh B = x13 + x2
3.
2) Cho phng trnh : x2 - (m + 4)x + 3m + 3 = 0 (m l tham s).
a) Xc nh m phng trnh c mt nghim l bng 2. Tm nghim cn li.
b) Xc nh m phng trnh c hai nghim x1, x2 tho mn x13 + x2
3 0. Cu 52 Cho phng trnh: (m 1)x2 + 2mx + m 2 = 0 (*)
1) Gii phng trnh khi m = 1. 2) Tm m phng trnh (*) c 2 nghim phn bit.
Cu 53 Cho phng trnh x2 2 (m + 1 )x + m2 - 2m + 3 = 0 (1).
a) Gii phng trnh vi m = 1 .
b) Xc nh gi tr ca m (1) c hai nghim tri du .
c) Tm m (1) c mt nghim bng 3 . Tm nghim kia .
Cu 54 Cho phng trnh x2 ( m+1)x + m2 2m + 2 = 0 (1)
a) Gii phng trnh vi m = 2 .
b) Xc nh gi tr ca m phng trnh c nghim kp . Tm nghim kp .
c) Vi gi tr no ca m th 2221 xx + t gi tr b nht , ln nht .
Cu 56 Cho phng trnh : x2 + 2x 4 = 0 . gi x1, x2, l nghim ca phng trnh .
Tnh gi tr ca biu thc : 2
21
221
2122
21 322
xxxx
xxxxA+
+=
Cu 57 Cho phng trnh x2 ( 2m + 1 )x + m2 + m 1 =0.
a) Chng minh rng phng trnh lun c nghim vi mi m .
b) Gi x1, x2, l hai nghim ca phng trnh . Tm m sao cho : ( 2x1 x2 )( 2x2 x1 ) t
gi tr nh nht v tnh gi tr nh nht y .
c) Hy tm mt h thc lin h gia x1 v x2 m khng ph thuc vo m .
Cu 58 Cho phng trnh : x2 mx + m 1 = 0 .
1) Gi hai nghim ca phng trnh l x1 , x2 . Tnh gi tr ca biu thc .
2212
21
22
21 1
xxxx
xxM
+
+= . T tm m M > 0 .
2) Tm gi tr ca m biu thc P = 12221 + xx t gi tr nh nht .
Cu 59 Cho phng trnh : 2x2 ( m+ 1 )x +m 1 = 0
a) Gii phng trnh khi m = 1 .
b) Tm cc gi tr ca m hiu hai nghim bng tch ca chng .
Cu 60 Cho phng trnh (m2 + m + 1 )x2 - ( m2 + 8m + 3 )x 1 = 0
a) Chng minh x1x2 < 0 .
b) Gi hai nghim ca phng trnh l x1, x2 . Tm gi tr ln nht , nh nht ca biu thc : S
= x1 + x2
Cu 61 Cho phng trnh : x2 ( m+2)x + m2 1 = 0 (1)
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a) Gi x1, x2 l hai nghim ca phng trnh .Tm m tho mn x1 x2 = 2 .
b) Tm gi tr nguyn nh nht ca m phng trnh c hai nghim khc nhau .
Cu 62 Gi s x1 v x2 l hai nghim ca phng trnh : x2 (m+1)x +m2 2m +2 = 0
(1)
a) Tm cc gi tr ca m phng trnh c nghim kp , hai nghim phn bit .
b) Tm m 2221 xx + t gi tr b nht , ln nht .
Cu 63 Cho phng trnh : 2x2 + ( 2m - 1)x + m - 1 = 0
1) Tm m phng trnh c hai nghim x1 , x2 tho mn 3x1 - 4x2 = 11 .
2) Tm ng thc lin h gia x1 v x2 khng ph thuc vo m .
3) Vi gi tr no ca m th x1 v x2 cng dng .
Parapol v ng thng Bi 1 Xc nh to giao im ca (P) : y=2/3x2 v (d) : y = x+3 bng phng php i s v th
Bi2 Cho (P) : y= -x2 v ng thng (d) : y= - x+3 a) Xc nh giao im ca
(P) v (d)
b) Vit pt ng thng (d) vung gc vi (d) v tip xc vi (P)
Bi 3 Cho (P) : y = ax2 (a#0) v (d) : y = mx+n
a) Tm m,n bit (d) i qua hai im A(0;-1) v B(3;2) b) Tnh a bit (d) tip xc
vi (P)
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Bi 4 Gii bng th pt x2- x 6 = 0
Cho hm s y= 1/3x2 : (P) v y= - x+6 : (d) . Hy v (P) v (d) trn cng h trc to ri kim tra li
bng php tnh
Bi 5Cho (P) : y= x2/4 v im A(-3/2;1) ` a) Vit pt ng thng (d) i qua A v
tip xc vi (P)
b) V trn h trc to th (P) v (d)
Bi 6 Chng minh : ng thng (d) : y = x+1/2 v (P) : y = -x2/2 tip xc nhau . Tm to tip im
?
Bi 7 Cho (P) : y= x2/2 v (d) : y = ax+b . Tm a,b bit (d) ct (P) ti hai im c honh l 4 v -2
Bi 8 Cho (P) : y = x2/2 v ng thng (d) : y = x m
a) Vi gi tr no ca m th (d) khng ct (P)
b) Cho m = - 3/2 . Tm to giao im ca (d) vi (P) . V (P) v (d) trn cng mt h trc to
Bi 9 Trn cng mt h trc to cho (P) : y = x2/2 v (d) : y = -1/2x +2 a) V (P) v (d)
b) Tm to giao im ca (P) v (d)
c) Vit pt ng thng (d) //(d) v tip xc vi (P) v tnh to tip im
Bi 10 Cho hm s y = x2/2 (P) a) V (P)
b) Vit pt ng thng i qua A(2;6) , B(-1;3) . Tm giao im (P) v (d)
c) T M(-3/2;-2) v ng thng (d) //AB v tm s giao im (P) v (d) bng php tnh v th
Bi 11 Trn h trc to Oxy v (P) : y = -x2/4 v (d) : y = x+1 a) Nu v tr tng i ca (P) v
(d)
b) Vit pt ng thng (d) //(d) v ct (P) ti im c tung l - 4
Bi 12 Cho (P) : y = -x2 a) V (P)
b) Gi A v B l 2 im thuc (P) c honh l -1 ; 2 . Lp pt ng thng AB
c) Vit pt ng thng (d) //AB v tip xc vi (P) t suy ra to tip im
Bi 13 Cho hm s (P) : y = ax2 v (d) : y = - x +m a) Tm a bit (P) i qua im A(-1;2)
, v (P)
b) Tm m (d) tip xc vi (P) ( cu a) . Tm to tip im
c) Gi B l giao im ca (d) tm c cu b vi trc tung , C l im i xng vi vi A qua trc
tung . Chng minh C nm trn (P) v tam gic ABC vung cn
Bi 14 Trong mt phng to Oxy cho ng thng (d) c dng 2x - y a2 = 0 v (P) : y = ax2 vi
a l tham s dng
a) Tm a (d) ct (P) ti hai im phn bit . Chng minh rng khi A v B nm bn phi trc tung
b) Gi xA v xB l honh ca A v B . Tm GTNN ca T = BaBA xxxx .
14+
+
Bi 15 Tm tt c cc gi tr ca m hai ng thng y = 2x + m + 2 v y = (1 - m)x+ 1 ct nhau ti
mt im trn (P) : y = 2x2
Bi 16 Trong mt phng to Oxy cho (P) : y = - x2 v ng thng (d) c h s gc l k
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a) Vit pt ng thng (d)
b)Chng minh rng vi mi gi tr ca k th (d) lun ct (P) ti hai im phn bit A v B
c) Gi honh ca A v B l xA v xB . Chng minh 21 xx 2
d) Chng minh OAB l tam gic vung
Bi 17: Cho hm s : 22xy = (P) a) V th (P) b) Tm trn th cc im cch
u hai trc to
c) Xt s giao im ca (P) vi ng thng (d) 1= mxy theo m
d) Vit phng trnh ng thng (d') i qua im M(0;-2) v tip xc vi (P)
Bi 18 : Cho (P) 2xy = v ng thng (d) mxy += 2 .Xc nh m hai ng :
a)Tip xc nhau . Tm to tip im
b)Ct nhau ti hai im phn bit A v B , mt im c honh x=-1. Tm honh im cn li .
Tm to A v B
Bi 19: Cho ng thng (d) 2)2()1(2 =+ ymxm a)Tm m ng thng (d) ct (P) 2xy = ti hai im phn bit A v B
b)Tm to trung im I ca on AB theo m c)Tm m (d) cch gc to mt
khong Max
d)Tm im c nh m (d) i qua khi m thay i
Bi 20: Cho (P) 2xy =
a)Tm tp hp cc im M sao cho t c th k c hai ng thng vung gc vi nhau v tip
xc vi (P)
b)Tm trn (P) cc im sao cho khong cch ti gc to bng 2
Bi21: Cho (P) 221
xy = v ng thng (d) y=a.x+b .
Xc nh a v b ng thng (d) I qua im A(-1;0) v tip xc vi (P).
Bi 22: Cho (P) 2xy = v ng thng (d) y=2x+m a) V (P) b)Tm m (P) tip
xc (d)
Bi 23: Cho (P) 4
2xy = v (d) y=x+m a)V (P)
a) Xc nh m (P) v (d) ct nhau ti hai im phn bit A v B
b) Xc nh phng trnh ng thng (d') song song vi ng thng (d) v ct (P) ti im c
tung bng -4
c) Xc nh phng trnh ng thng (d'') vung gc vi (d') v i qua giao im ca (d') v (P)
Bi 24: Cho hm s 2xy = (P) v hm s y=x+m (d)
a) Tm m sao cho (P) v (d) ct nhau ti hai im phn bit A v B
b) Xc nh phng trnh ng thng (d') vung gc vi (d) v tip xc vi (P)
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c)Thit lp cng thc tnh khong cch gia hai im bt k. p dng: Tm m sao cho khong cch
gia hai im A v B bng 23
Bi 25: Cho im A(-2;2) v ng thng ( 1d ) y=-2(x+1) a)im A c thuc ( 1d ) ?
V sao ?
b)Tm a hm s 2.xay = (P) i qua A
c)Xc nh phng trnh ng thng ( 2d ) i qua A v vung gc vi ( 1d )
d)Gi A v B l giao im ca (P) v ( 2d ) ; C l giao im ca ( 1d ) vi trc tung . Tm to ca B v C . Tnh din tch tam gic ABC
Bi 26: Cho (P) 241
xy = v ng thng (d) qua hai im A v B trn (P) c honh lm lt l -2
v 4
a)Kho st s bin thin v v th (P) ca hm s trn b)Vit phng trnh ng
thng (d)
Bi 27: Cho (P) 4
2xy = v im M (1;-2) a)Vit phng trnh ng thng (d) i qua
M v c h s gc l m
b)CMR (d) lun ct (P) ti hai im phn bit A v B khi m thay i
c)Gi BA xx ; ln lt l honh ca A v B .Xc nh m 22BABA xxxx + t gi tr nh nht v tnh
gi tr
Bi 28: Cho hm s 2xy = (P) a)V (P)
b)Gi A,B l hai im thuc (P) c honh ln lt l -1 v 2. Vit phng trnh ng thng AB
c)Vit phng trnh ng thng (d) song song vi AB v tip xc vi (P)
Bi 29: Trong h to xoy cho Parabol (P) 241
xy = v ng thng (d) 12 = mmxy
a)V (P) b)Tm m sao cho (P) v (d) tip xc nhau.Tm to tip im
c)Chng t rng (d) lun i qua mt im c nh
Bi 30: Cho (P) 241
xy = v im I(0;-2) .Gi (d) l ng thng qua I v c h s gc m.
a)V (P) . CMR (d) lun ct (P) ti hai im phn bit A v B Rm b)Tm gi tr ca m on AB ngn nht
Bi 31: Cho (P) 4
2xy = v ng thng (d) i qua im I( 1;
23
) c h s gc l m
a)V (P) v vit phng trnh (d) b)Tm m sao cho (d) tip xc (P)
c)Tm m sao cho (d) v (P) c hai im chung phn bit
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Bi 32: Cho (P) 4
2xy = v ng thng (d) 22
+=xy a) V (P) v (d)
b)Tm to giao im ca (P) v (d)
c)Tm to ca im thuc (P) sao cho ti ng tip tuyn ca (P) song song vi (d)
Bi 33: Cho (P) 2xy = a) V (P)
b)Gi A v B l hai im thuc (P) c honh ln lt l -1 v 2 . Vit phng trnh ng thng AB
c)Vit phng trnh ng thng (d) song song vi AB v tip xc vi (P)
Bi 34: Cho (P) 22xy = a) V (P)
b) Trn (P) ly im A c honh x=1 v im B c honh x=2 . Xc nh cc gi tr ca m v n
ng thng (d) y=mx+n tip xc vi (P) v song song vi AB
Bi 35: a.V th hm s y = x2 (P)
b. Tm h s gc ca ng thng ct trc tung ti im c tung bng 1 sao cho
ng thng y :
1.Ct (P) ti hai im 2. Tip xc vi (P) 3.Khng ct (P)
Bi 36: Cho ng thng (d) c phng trnh: y = mx - 2
m - 1 v parabol (P) c phng trnh y
=x2/2
a) Tm m (d) tip xc vi (P). B.Tnh to cc tip im
Bi 37: Cho parabol (P): y =
2
4
x v ng thng (d): y = 1
2 x + n
a)Tm gi tr ca n ng thng (d) tip xc vi (P)
b)Tm gi tr ca n ng thng (d) ct (P) ti hai im.
c)Xc nh to giao im ca ng thng (d) vi (P) nu n = 1
Bi 38 .Cho parabol y=2x2 v ng thng y=ax+2- a.
1. Chng minh rng parabol v ng thng trn lun xt nhau ti im A c nh. Tm im A .
2. Tm a parabol ct ng thng trn ch ti mt im.
Bi 39. Cho (P): y = -2x2 v (d) y = x -3 Tm giao im ca (P) v (d)
b) Gi giao im ca (P) v (d) cu a l A v B trong A l im c honh nh hn; C, D ln
lt l hnh chiu vung gc ca A v B trn Ox. Tnh din tch v chu vi t gic ABCD.
Bi 40. Trong mt phng ta Oxy cho (P) c phng trnh 2
xy 2= . Gi (d) l ng thng i
qua im I(0; - 2) v c h s gc k. a) Vit phng trnh dng thng (d). CMR (d) lun ct (P) ti hai im phn bit A v B khi k thay i.
b) Gi H, K theo th t l hnh chiu vung gc ca A, B ln trc honh. CMR tam gic IHK vung ti I.
Bi 41. Cho (P) y = -2 2x
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a) Tm k ng thng (d): y = kx + 2 ct (P) ti hai im phn bit. b) Chng minh im E(m; m2 + 1) khng thuc (P) vi mi gi tr ca m.
Bi 42 Cho hm s y = 21
x2
(P) 1) V th ca hm s.(P)
2) Gi A v B l hai im trn th ca hm s c honh ln lt l 1 v -2. Vit phng trnh
ng thng AB.
3) (d) y = x + m 2 ct (P) trn ti 2 im phn bit, gi x1 v x2 l honh 2 giao im y.
Tm m x12 + x2
2 + 20 = x12x2
2.
Bi 43 Cho ng thng (d) c phng trnh y = ax + b. Bit rng (d) ct trc honh ti im c
honh bng 1 v song song vi ng thng y = -2x + 2003.
1) Tm a v b. 2) Tm to cc im chung (nu c) ca (d) v Parabol y = 21
x2
.
Bi44 Cho Parabol (P) : y = 221
x v ng thng (D) : y = px + q .
Xc nh p v q ng thng (D) i qua im A ( - 1 ; 0 ) v tip xc vi (P) . Tm to tip im
.
Bi45 : Trong cng mt h trc to Oxy cho parabol (P) : 241
xy = v ng thng (D)
: 12 = mmxy
a) V (P) .
b) Tm m sao cho (D) tip xc vi (P) .
c) Chng t (D) lun i qua mt im c nh .
Bi 46. Cho hm s y = x2 c th l ng cong Parabol (P) .
a) CMR im A( - )2;2 nm trn ng cong (P) . b) Tm m th (d ) ca hm s y = ( m 1 )x + m ( m R , m 1 ) ct ng cong
(P) ti mt im .
c) Chng minh rng vi mi m khc 1 th (d ) ca hm s y = (m-1)x + m lun i qua
mt im c nh .
Bi 47 Cho hm s : 4
2xy = v y = - x 1
a) V th hai hm s trn cng mt h trc to .
b) Vit phng trnh cc ng thng song song vi ng thng y = - x 1 v ct th
hm s 4
2xy = ti im c tung l 4 .
Bi 48 Cho hm s )(21 2 Pxy = a. V th ca hm s (P)
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b. Vi gi tr no ca m th ng thng y=2x+m ct th (P) ti 2 im phn bit A v B.
Khi hy tm to hai im A v B.
Bi 49 : (3,5 im)Cho Parabol y=x2 v ng thng (d) c phng trnh y=2mx-m2+4.
a. Tm honh ca cc im thuc Parabol bit tung ca chng
b. Chng minh rng Parabol v ng thng (d) lun ct nhau ti 2 im phn bit. Tm to
giao im ca chng. Vi gi tr no ca m th tng cc tung ca chng t gi tr nh
nht?
Bi 49 : Cho ng thng d c phng trnh y=ax+b. Bit rng ng thng d ct trc honh ti im
c honh bng 1 v song song vi ng thng y=-2x+2003.
1. Tm a v b. 2. Tm to cc im chung (nu c) ca d v parabol 221
xy =
Bi 50: Cho parabol (P) v ng thng (d) c phng trnh: (P): y=x2/2 ; (d): y=mx-m+2 (m l
tham s).
1. Tm m ng thng (d) v (P) cng i qua im c honh bng x=4.
2. Chng minh rng vi mi gi tr ca m, ng thng (d) lun ct (P) ti 2 im phn bit.
Bi51: Trong mt phng to Oxy cho :(P): y=x2 (d): y=2(a-1)x+5-2a ; (a l tham s)
a. Vi a=2 tm to giao im ca ng thng (d) v (P).
b. Chng minh rng vi mi a ng thng (d) lun ct (P) ti 2 im phn bit.
c. Gi honh giao im ca ng thng (d) v (P) l x1, x2. Tm a x12+x2
2=6.
Bi 52 Cho parabol y=2x2.Khng v th, hy tm:
1. To giao im ca ng thng y=6x- 4,5 vi parabol.
2. Gi tr ca k, m sao cho ng thng y=kx+m tip xc vi parabol ti im A(1;2).
Bi 53 Cho phng trnh bc hai : x2 2(m 1) x + m 3 = 0. (1)
1/. Chng minh rng phng trnh (1) lun lun c hai nghim phn bit vi mi gi tr ca m.
2/. Tm m phng trnh (1) c mt nghim bng 3 v tnh nghim kia.
3/. Tm m phng trnh (1) c hai nghim i nhau.
Bi 54 Cho hm s: 2
2xy = a)V th (P) ca hm s trn.
b)Trn (P) ly hai im M v N theo th t c honh l -2 v 1. Vit phng trnh ng thng
MN.
c) Tm m (P) v ng thng (d): 2mxy += khng c im chung.
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H phng trnh cha tham s
Bi 1 Cho h pt
=+
=+
1522
yxmyx
a) GiI h pt vi m=1
b) Tm m h c nghim duy nht tho mn y= x
Bi 2 Cho h pt
=
=+
122
ymxmyx
a) Gii h pt vi m =2
b) Tm cc s nguyn m h c nghim duy nht vi x>0 v y2y
Bi 3 Cho h pt
+=+
=
122
myxmymx
a) Gii h pt vi m = 1
b) Tm m h c nghim duy nht , tm nghim duy nht
Bi 4 Cho h pt
=+
=+
11
ymxmyx
a) Gii h pt vi m=2
b) Tm m h c nghim duy nht tho mn x,y>0
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Bi 5 Cho h pt
=+
=
1312)1(
ayxyxa
a) Gii h pt vi a = 2
b) Chng minh vi mi a h pt c nghim duy nht c) Tm a x y c gi tr
ln nht
Bi 6 Cho h pt
=
=+
myxymx 42
a) Gii h pt vi m = 2
b) Vi gi tr no ca m th h c nghim duy nht ? tm nghim ?
c) Tm m h c v s nghim ?
Bi 7 : Tm gi tr ca m h phng trnh ; ( )
( )
=+
+=+
2111
ymxmyxm
C nghim duy nht tho mn iu kin x+y nh nht
Bi 8: Cho h phng trnh :
=
=+
542
aybxbyx
a)Gii h phng trnh khi ba =
b)Xc nh a v b h phng trnh trn c nghim :
* (1;-2)
* h c v s nghim
Bi 9 Gii v bin lun h phng trnh theo tham s m:
+=
=
mmyxmymx
642
Bi 10: Vi gi tr no ca a th h phng trnh :
=+
=+
21
yaxayx
a) C mt nghim duy nht b) V nghim
Bi 11:Cho h phng trnh :
=+
=+
ayxayxa
.
3)1( a) Gii h phng rnh khi a=- 2
b)Xc nh gi tr ca a h c nghim duy nht tho mn iu kin x+y>0
Bi 12: Cho h phng trnh 4 3 6
5 8
x y
x ay
=
+ = a.Gii phng trnh.
b.Tm gi tr ca a h c nghim duy nht m.
Bi 13: Cho h phng trnh 2
3 5
mx y
x my
=
+ = Tm gi tr ca m h c nghim x = 1
Bi 14 : Cho h phng trnh :( 1) 3
.
a x ya x y a
+ =
+ = a) Gii h vi 2a =
b.Xc nh gi tr ca a h c nghim duy nht tho mn x + y > 0
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Bi15 Cho h phng trnh: ( )
( )
=+
=+
241211213
yxmymx
1. Gii h phng trnh.vi m=2
2. Tm m h phng trnh c mt nghim sao cho x1.
Bi17 Cho h phng trnh : mx y 2
x my 1
=
+ = 1) Gii h phng trnh theo tham s
m.
2) Gi nghim ca h phng trnh l (x, y). Tm cc gi tr ca m x + y = -1.
3) Tm ng thc lin h gia x v y khng ph thuc vo m.
Bi 18 Cho h phng trnh: x 2y 3 m
2x y 3(m 2)
=
+ = + 1) Gii h phng trnh khi thay m =
-1.
2) Gi nghim ca h phng trnh l (x, y). Tm m x2 + y2 t gi tr nh nhtl.
Bi 19 Cho h phng trnh: x ay 1
(1)ax y 2
+ =
+ =
1) Gii h (1) khi a = 2. 2) Vi gi tr no ca a th h c nghim duy nht.
Bi 20 Cho h phng trnh ( )a 1 x y 4ax y 2a
+ + =
+ = (a l tham s). 1) Gii h khi a = 1.
2) Chng minh rng vi mi a h lun c nghim duy nht (x ; y) tho mn x + y 2.
Bi 21 Cho h phng trnh :
=+
=
22 2
yxmmyx
a) Gii h khi m = 1 .
b) Gii v bin lun h phng trnh .
Bi 22 Cho h phng trnh :
=+
=+
1352
ymxymx
a) Gii h phng trnh vi m = 1
b) Gii bin lun h phng trnh theo tham s m .
c) Tm m h phng trnh c nghim tho mn x2 + y2 = 1 .
Bi 23 Cho h phng trnh .
=+
=
533
myxymx
a) Gii h phng trnh khi m = 1 .
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b) Tm m h c nghim ng thi tho mn iu kin ; 13)1(7
2 =+
+m
myx
Bi 24 Cho h phng trnh
=+
=
1272
yxyxa
a) Gii h phng trnh khi a = 1
b) Gi nghim ca h phng trnh l ( x , y) . Tm cc gi tr ca a x + y = 2 .
Bi 25 Cho h phng trnh :
=+
=+
1352
ymxymx
a) Gii h phng trnh khi m = 1 .
b) Gii v bin lun h phng trnh theo tham s m .
c) Tm m x y = 2 .
Bi 26 Cho h phng trnh :
=+
=+
643
ymxmyx
a) Gii h khi m = 3
b) Tm m phng trnh c nghim x > 1 , y > 0 .
Bi 27 : Cho h phng trnh:( )
=+
=++
ayaxyxa
241
(a l tham s) 1. Gii h khi a=1.
hai ng thng Bi 1 Cho hai ng thng (d1) : y = 3x+4 v (d2) x - 2y = 0 , mt im A(-1;1)
a) Xt v tr tng i ca A vi hai ng thng b) Tm giao im
(d1) v (d2)
c) Tm M (d3) : (m-1)x+(m-2) y + m+1 = 0 ng quy vi (d1) v (d2)
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Bi 2 Cho hai ng thng (d1) : y = ( 123
m)x + 1 2n v (d2) : y = (m+2)x +n 3 . Tm m , n
(d1)//(d2) ; (d1) (d2) Bi 3 Cho hai ng thng (d1) : y = (k+1)x +3 v (d2) : y = (3- 2k)x + 1 .
Tm k (d1)//(d2) , (d1) ct (d2) , (d1) ct (d2)
Bi 4 Trong mt phng to Oxy cho ba im A(2;5) ; B(-1;-1) v C(4;9)
a) Vit pt ng thng BC ri suy ra ba im A,B,C thng hng
b) Chng minh ba ng thng BC ; 3x- y -1= 0 v x-2y +8 = 0 ng quy
Bi 5 Cho ng thng (d1) : y = mx 3 v (d2) : y = 2mx +1 m
a) V trn cng mt h trc to (d1) v (d2) vi m = 1 . Tm to giao im B ca chng ?
b) Vit pt ng thng i qua O v vi (d1) ti A . Xc nh to im A v tnh din tch tam
gic AOB
c) Chng t (d1) v (d2) u i qua mt im c nh . Tm im c nh
Bi 6 Cho hai ng thng (d) : mx y =2 v (d) : (2 m)x + y = m
a) Tm giao im ca (d) v (d) vi m = 2
b) Chng minh rng ng thng (d) lun i qua mt im c inh B v (d) lun i qua mt im c
nh C
c) Tm m giao im A ca hai ng thng trn tho mn iu kin l gc BAC vung
Bi 7 Cho hm s : y= (m-2)x+n (d) Tm gi tr ca m v n th (d) ca hm
s :
a) i qua hai im A(-1;2) v B(3;-4)
b) Ct trc tung ti im ctung bng 1- 2 v ct trc honh ti im c honh bng
2+ 2 . c) Ct ng thng -2y+x-3=0
d) Song song vi ng thng 3x+2y=1
Bi 8: Cho ng thng (d) 343
= xy a)V (d)
b)Tnh din tch tam gic c to thnh gia (d) v hai trc to
c) Tnh khong cch t gc O n (d)
Bi 9 Vi gi tr no ca m th hai ng thng : (d) 2)1( += xmy (d') 13 = xy a) Song song vi nhau c) Ct nhau c) Vung gc vi nhau
Bi 10 Tm gi tr ca a ba ng thng : 1( ) 2 5d y x= 2( ) 2d y x= +
3( ) . 12d y a x=
ng quy ti mt im trong mt phng to
Bi 11 Cho A(2;-1); B(-3;-2) 1. Tm phng trnh ng thng qua A v B.
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2. Tm phng trnh ng thng qua C(3;0) v song song vi AB.
Bi 12 Cho hm s y = (m 2)x + m + 3. 1) Tm iu kin ca m hm s lun
nghch bin.
2) Tm m th ca hm s ct trc honh ti im c honh bng 3.
3) Tm m th ca hm s trn v cc th ca cc hm s y = -x + 2 ; y = 2x 1 ng quy.
Bi 13 Cho hm s y = (m 1)x + m + 3.
1) Tm gi tr ca m th ca hm s song song vi th hm s y = -2x + 1.
2) Tm gi tr ca m th ca hm s i qua im (1 ; -4).
3) Tm im c nh m th ca hm s lun i qua vi mi m.
4) Tm gi tr ca m th ca hm s to vi trc tung v trc honh mt tam gic c din tch
bng 1 (vdt).
Bi 14 Cho hai im A(1 ; 1), B(2 ; -1). 1) Vit phng trnh ng thng AB.
2) Tm cc gi tr ca m t y = (m2 3m)x + m2 2m + 2 song song vi t AB ng thi i qua
im C(0 ; 2).
Bi 15 Cho hm s y = (2m 1)x + m 3 1) Tm m th ca hm s i qua im (2; 5)
2) Chng minh rng th ca hm s lun i qua mt im c nh vi mi m. Tm im c nh y.
3) Tm m th ca hm s ct trc honh ti im c honh x = 2 1 .
Bi 16 Cho hm s y = f(x) = 21
x2
. 1) Vi gi tr no ca x hm s trn nhn cc gi tr :
0 ; -8 ; -1
9; 2.
2) A v B l hai im trn th hm s c honh ln lt l -2 v 1. Vit pt ng thng i qua A
v B.
Bi 17 Cho hm s : y = x + m (D)Tm cc gi tr ca m ng thng (D) :
1) i qua im A(1; 2003). 2) Song song vi ng thng x y + 3 = 0.3)Tip xc vi parabol y
= - 21
x4
.
Bi 18 a)Tm cc gi tr ca a , b bit rng th ca hm s y = ax + b i qua hai im A( 2 ; - 1 )
v B ( )2;21
b)Vi gi tr no ca m th th ca cc hm s y = mx + 3 ; y = 3x 7 v th ca hm s xc
nh cu ( a ) ng quy .
Bi 19 Cho hm s y = ( m 2 ) x + m + 3 .
a) Tm iu kim ca m hm s lun nghch bin .
b) Tm m th hm s ct trc honh ti im c hnh l 3 .
c) Tm m th cc hm s y = - x + 2 ; y = 2x 1v y = (m 2 )x + m + 3 ng quy .
Bi 20 Cho hai ng thng y = 2x + m 1 v y = x + 2m .
a) Tm giao im ca hai ng thng ni trn .
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b) Tm tp hp cc giao im .
Bi 21 Cho hm s : y = ( 2m + 1 )x m + 3 (1)
a) Tm m bit th hm s (1) i qua im A ( -2 ; 3 ) .
b) Tm im c nh m th hm s lun i qua vi mi gi tr ca m .
Bi22 Trong mt phng to cho im A ( 3 ; 0) v ng thng x 2y = - 2 .
a) V th ca ng thng . Gi giao im ca ng thng vi trc tung v trc honh l
B v E .
b) Vit phng trnh ng thng qua A v vung gc vi ng thng x 2y = -2 .
c) Tm to giao im C ca hai ng thng . Chng minh rng EO. EA = EB . EC
v tnh din tch ca t gic OACB .
Bi 23 Trong h trc to Oxy cho hm s y = 3x + m (*)
1) Tnh gi tr ca m th hm s i qua : a) A( -1 ; 3 ) ; b) B( - 2 ; 5 )
2) Tm m th hm s ct trc honh ti im c honh l - 3 .
3) Tm m th hm s ct trc tung ti im c tung l - 5 .
Bi 24: Cho ng thng d c phng trnh y=ax+b. Bit rng ng thng d ct trc honh ti im
c honh bng 1 v song song vi ng thng y=-2x+2003.
1. Tm a v b. 2. Tm to cc im chung (nu c) ca d v parabol 221
xy =
Bi 25: Cho hm s y = (m - 1)x + m (d)
a) Xc nh gi tr ca m ng thng (d) ct trc tung ti im c tung bng 2004.
b) Vi gi tr no ca m th gc to bi ng thng (d) vi tia Ox l gc t?
Bi 26: Vi gi tr no ca k, ng thng y = kx + 1:
a) i qua im A(-1; 2) ?
b) Song song vi ng thng y = 5x?
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Gii bi ton bng cch lp phng trnh Bi 1 Mt cng nhn d nh lm 72 sp trong thi gian d nh . Thc t ngi phi lm 80sp, mc
d ngi lm mi gi thm 1 sp song thi gian hon thnh vn chm hn so vi d nh 12 .
Tnh nng sut d kin , bit mi gi ngi lm khng qu 20 sp
Bi 2 Mt ngi d nh i xe p t im A n im B cch nhau 36km trong mt thi gian nht
nh . Sau khi i c na qung ng ngi dng li 18 . Do n B ng hn ngi
tng thm vn tc 2km trn qung ng cn li . Tnh vn tc ban u v thi gian xe ln bnh trn
ng ?
Bi 3 Mt tu thu chy trn mt khc sng di 80km , c i ln v ht 8h20 . Tnh vn tc ca tu
khi nc yn lng ? ,bit vn tc dng nc l 4km/h
Bi 4 Mt i cng nhn xy dng hon thnh mt cng trnh ht 420 ngy cng th . Tnh s ngi
ca i bit nu vng 5 ngi th s ngy hon thnh tng thm 7 ngy
Bi 5 Hai ca n cng khi hnh t hai bn A , B cch nhau 85km i ngc chiu nhau . Sau 1h40 th
gp nhau . Tnh vn tc ring ca mi ca n bit vn tc ca n i xui ln hn vn tc ca ca n i
ngc l 9km/h v vn tc dng nc l 3km/h
Bi 6 Trong mt bui lin hoan , mt lp hc sinh mi 15 khch ti d . V lp c 40 hs nn phi
k thm mt dy gh na v mi gh phi ngi thm mt ngi th mi ch . Bit mi dy gh u
c s ngi ngi nh nhau v ngi khng qu 5 ngi . Hi lp hc lc u c bao nhiu dy gh ?
Bi 7 Mt t d nh i t A n B cch nhau 120km trong mt thi gian quy nh . Sau khi i c
1h t b chn bi tu ho 10 . Do n B ng d nh xe phi tng tc thm 6km/h na . Tnh
vn tc t lc u ?
Bi 8 Mt chic thuyn khi hnh t bn sng A . Sau 5h20 mt chic ca n chy t A ui theo
v gp chic thuyn cch bn A 20km . Hi vn tc ca chic thuyn bit rng ca n chy nhanh hn
thuyn 12km/h
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Bi 9 Mt nhm th t k hoch sn xut 3000sp . Trong 8 ngy u h thc hin ng mc ra
, nhng ngy cn li h lm vt mc k hoch mi ngy 10sp nn hon thnh k hoch sm 2
ngy . Hi theo k hoch mi ngy nhm sn xut bao nhiu sp ?
Bi 10 Mt i xe cn chuyn ch 120 tn hng . Hm lm vic c 2 xe phi iu i ni khc nn mi
xe phi ch thm 16 tn . Hi i c bao nhiu xe ?
Bi 11 Hai t khi hnh cng mt lc t A n B . Mi gi t th nht i nhanh hn t th hai
12km nn n B trc t th hai l 100 . Tnh vn tc mi t ? bit SAB l 240km
Bi 12 Mt ca n xui dng 42km ri ngc dng tr li 20km mt tng cng 5h bit vn tc ca
dng chy l 2km/h . Tnh vn tc ca n lc nc yn lng?
Bi 13 Mt phng hp c 360 gh ngi c xp thnh tng dy v s gh mi dy bng nhau . Nu
s dy tng thm 1 v s gh mi dy cng tng thm 1 th trong phng c 400 gh . Hi trong phng
c bao nhiu dy gh , mi dy c bao nhiu gh?
Bi 14 Mt ngi i t t A n B cch nhau 100km vi vn tc xc nh . Khi v ngi i ng
khc di hn ng c 20km nhng vi vn tc ln hn vn tc ban u l 20km/h nn thi gian v t
hn thi gian i l 30 . Tnh vn tc t lc i ?
Bi 15 Mt ca n chy xui mt khc sng di 72km ri ngc dng khc sng 54km ht tt c 6h
tnh vn tc tht ca ca n bit vn tc dng nc l 3km/h
Bi 16 Mt i sn xut phi lm 1000sp trong mt thi gian quy nh . Nh ci tin k thut nn mi
ngy tng 10sp so vi k hoch v vy vt mc k hoch 80sp m cn hon thnh sm hn d
nh 2 ngy . Tnh s sp i phi lm mi ngy theo k hoch ?
Bi 17 Hai vi nc cng chy vo b khng c nc th sau 2h55 th y b . Nu chy ring th vi
1 chy y b nhanh hn vi 2 l 2h . Tnh thi gian mi vi chy ring y b ?
Bi 18 Mt cng nhn d kin hon thnh mt cng vic trong thi gian d nh vi nng sut 12sp/h
sau khi lm xong mt na cng vic ngi tng nng sut 15sp/h nh vy cng vic hon thnh
sm hn 1h so vi d nh . Tnh s sp m ngi cng nhn d nh lm ?
Bi 19 Mt tam gic vung c cnh huyn l 20cm , hai cnh gc vung hn km nhau 4cm . Tnh
mi cnh gc vung
Bi 20 Mt t d nh i t A n B di 60km vi vn tc d nh . Trn na qung ng u t
i vi vn tc km vn tc d nh l 6km/h , trn na qung ng sau t i vi vn tc nhanh hn
vn tc d nh l 10km/h . V vy t n B ng thi gian quy nh . Tnh vn tc d nh ca
t?
Bi 21 Hai my cy cng cy mt tha rung th sau 2h xong . Nu cy ring th my 1 hon thnh
sm hn my 2 l 3h . Hi thi gian cy ring ca mi my xong cng vic ?
Bi 22 Qung ng AB di 270km . Hai t cng khi hnh t A n B . t th nht i nhanh hn
t th 2 l 12km/h nn n B sm hn t 2 l 40 . Tnh vn tc mi t ?
Bi 23 Tm hai s bit s ln hn s b l 3 n v v tng cc bnh phng ca hai s l 369
Bi 24 Mt on xe cn ch 30 tn hng t im A n im B . Khi khi hnh th thm 2 xe na nn
mi xe ch t hn d nh l 0,5 tn . Tnh s xe ban u ?
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Bi 25 Mt ca n xui mt khc sng di 50km ri ngc 32km th ht 4h30 . Tnh vn tc dng
nc bit vn tc ca n l 18km/h
Bi 26 Mt tu thu xui dng mt khc sng di 48km ri ngc dng 48km ht 5h . Tnh vn tc
tu thu bit vn tc dng nc l 4km/h ?
Bi27 Hai cnh ca mt hnh ch nht hn km nhau 10m . Tnh chu vi bit din tch hnh ch nht l
1200m2 ?
Bi 28 Trong mt phng hp c 70 ngi d hp c sp xp ngi u trn cc dy gh . Nu bt i
2 dy th mi dy gh cn li phi xp thm 4 ngi mi ch . Hi lc u phng hp c bao nhiu
dy gh ?
Bi 29 Mt tha rung hnh tam gic c din tch 180m2 . Tnh cnh y bit nu tng cnh y 4m
v gim chiu cao tng ng 1m th din tch khng i ?
Bi 30 Lc 6h30 anh An i t A n B di 75km ri ngh ti B 20 ri quay v A . Khi v anh i vi
vn tc ln hn vn tc lc i 5km/h . Anh v An lc 12h20 . Tnh vn tc lc i ca anh An?
Bi 31 Trn mt cng trng xy dng mt i lao ng phi o 420m2 t . Tnh s ngi ca i
bit nu 5 ngi vng th s ngy hon thnh tng 5 ngy ?
Bi 32 Hai cng nhn cng lm xong cng vic th ht 4 ngy . Nu ngi th nht lm mt na cng
vic ri ngi th hai lm nt th ht tt c 9 ngy . Tnh thi gian hon thnh ring cng vic ca mi
ngi ?
Bi 33 Lc 7h30 mt t i t A n B ngh 30 ri i tip n C lc 10h15 . bit SAB = 30km v SBC
= 50km , vn tc trn on AB ln hn vn tc trn on BC l 10km/h . Tnh vn tc ca t trn
on AB , BC ?
Bi 34 Tm hai s t nhin bit hiu ca chng l 1275 v nu ly s ln chia s nh th c thng
l 3 v s d l 125
Bi 35 Hai a im cch nhau 56km . Lc 6h45mt ngi i xe p t A n B vi vn tc 10km/h .
Sau 2h mt ngi i xe p t B n A vi vn tc 14km/h . Hi n my gi hai ngi gp nhau
v im gp nhau cch A bao nhiu km?
Bi 36 Mt xe ti v mt xe con cng khi hnh t A n B . Xe ti i vi vn tc 30km/h , xe con i
vi vn tc 45km/h . Sau khi i c 0,75 qung ng xe con tng thm 5km/h na nn n B sm
hn xe ti 2h20 . Tnh SAB
Bi 37 Mt my bm mun bm y nc vo mt b cha vi cng sut 10m3 . Khi bm c 1/3
b ngi cng nhn vn hnh tng cng sut my l 15m3/h nn b cha c bm y trc 48 .
Tnh th tch b cha ?
Bi 38 Mt tp on nh c d nh trung bnh mi tun nh bt 20 tn c , nhng khi thc hin
vt mc 6 tn mt tun nn hon thnh k hoch sm hn so vi d nh 1 tun v vt mc k
hoch 10 tn . Tnh mc k hoch nh?
Bi 39 Mt ca n i xui t A n B vi vn tc 30km/h ri tr v A . Thi gian i xui t hn thi
gian i ngc l 40 . Tnh SAB ? bit vn tc dng nc l 3km/h
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39
Bi 40 Mt t d nh i t A n B vi vn tc 40 km/h . Lc u i vi vn tc , khi cn 60 km
na th c na qung ng th ngi li xe tng tc thm 10km/h nn n B sm hn d nh 1h
. Tnh SAB ?
Bi 41 Mt ca n xui dng 108km ri ngc dng 63 km ht 7h . Mt ln khc ca n xui dng
81km ri ngc dng 84km cng ht 7h .Tnh vn tc ring ca ca n v vn tc dng nc ?
Bi 42 Hai ngi th cng lm mt cng vic ht 16h . Nu ngi th nht lm 3h v ngi th hai
lm 6h th c 25% cng vic . Hi thi gian lm ring xong cng vic ca mi ngi ?
Bi 43 Hai vi nc cng chy vo mt b cha khng c nc th sau 1h30 th y b . Nu m vi
th nht 15 ri kho li v m vi th hai 20 th c 1/5 b . Hi mi vi chy ring th sau bao lu
th y b ?
Bi 44 Trong thng u hai t sn xut lm c 800sp . Sang thng th hai t mt tng nng sut
15% , t hai tng nng sut 20% nn lm c 945sp . Tnh s sp ca mi t trong thng u?
Bi 45 Hai ca n cng khi hnh t A n B . Ca n mt chy vi vn tc 20km/h , ca n hai chy vi
vn tc 24km/h . Trn ng i ca n hai dng 40 sau tip tc chy . Tnh chiu di AB bit hai ca
n n B cng mt lc ?
Bi 46 Hai t cng khi hnh t hai im A v B cch nhau 150 km i ngc chiu nhau v gp
nhau sau 2h . Tm vn tc mi t bit nu vn tc ca t A tng 5km/h v vn tc t B gim
5km/h th vn tc t A gp i vn tc t B
Bi47 Mt t chy trn qung ng AB . Lc i t chy vi vn tc 35km/h , lc v t chy vi
vn tc 42km/h nn thi gian v t hn thi gian i l na gi . tnh AB ?
Bi 49 An i t A n B . on ng AB gm on ng v on ng nha, on ng
bng 2/3 on ng nha . on ng nha An i vi vn tc 12km/h , on ng An i vi
vn tc 8 km/h . Bit An i c qung ng AB ht 6 gi . Tnh qung ng AB ?
Bi52 Sau khi nhn mc khon , mt cng nhn d nh lm trong 5 gi . Lc u mi gi ngi
lm c 12 sn phm . Khi lm c mt na s lng c giao , nh hp l ho nn mi gi
lm thm 3 sn phm na . Nh nn hon thnh sm hn d nh 1/2 gi .tnh s sn phm c
giao ?
Bi 53 Mt mnh vn hnh ch nht c chu vi 34m . Nu tng chiu di 3m , chiu rng 2m th din
tch tng 45m2 . Tnh chiu di, chiu rng ca mnh vn ?
Bi 54 Mt ca n chy xui mt khc sng di 63km sau chy ngc dng 30km ht tt c 5h .
Mt ln khc ca n chy xui dng 42km ri ngc dng 45 km cng mt 5h . Tm vn tc thc ca
ca n ?
Bi 55 Qung ng AB di 60km . Mt ngi i t A n B vi mt vn tc xc nh . Khi i t B v
A ngi y i vi vn tc ln hn vn tc lc i l 5km/h . V vy thi gian v t hn thi gian i l 1h
. Tnh vn tc lc i
Bi 56 Mt mnh t c chiu di ln hn chiu rng 5m . Nu gim chiu rng 4m v gim chiu di
5m th din tch mnh t gim 180m2 tnh kch thc mnh vn ?
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Bi 57 Hai lp 9A v 9B c tng s 80 bn quyn gp c tng s 198 cun v . Mt bn lp 9A gp
2 cun , mt bn lp 9B gp 3 cun . Tm s hc sinh mi lp ?
Bi 58 Hai ngi cng lm mt cng vic ht 3h . Nu h cng lm 2h ri ngi th hai lm tip 4h
th xong cng vic . Tnh thi gian mi ngi lm ring xong cng vic ?
Bi 59 Trn qung ng AB di 200km c xe con i t A , xe ti i t B . Nu cng khi hnh th hai
xe gp nhau ti im cch A 120km . Nu xe con i trc xe ti 1h th chng gp nhau ti im cch
A l 96km . Tnh vn tc mi xe ?
Bi 60 Mt mnh t hnh ch nht c chu vi bng 40m . Nu tng chiu rng 2m v gim chiu di
2m th din tch tng 4m2 tnh kch thc mnh t ban u ?
Bi 61 Hai ngi lm chung mt cng vic ht 7h12 . Nu ngi th nht lm ring trong 5h v ngi
th hai lm trong 6h th c 3/4 cng vic . Tnh thi gian lm ring xong cng vic ca mi ngi ?
Bi 62 Hai trng A v B c 420 hs thi t t l 84% . Ring trng A vi t l 80% , trng B
vi t l 90% . Tnh s hs mi trng ?
Bi 63 Mt ngi d nh i t A n B trong mt thi gian quy nh vi vn tc 10km/h . Sau khi i
c na qung ng ngi ngh 30 nn n B ng d nh ngi tng vn tc ln
15km/h . Tnh SAB
Bi 64 Mt t chy trn mt qung ng di 120km trong mt thi gian nht nh . Khi c na
qung ng ngi dng 3 nn n B ng gi ngi tng vn tc thm 2km/h trn qung
ng cn li . Tnh vn tc d nh ca t?
Bi 65 Mt i xe cn ch 36 tn hng . Khi lm vic c thm 3 xe na nn mi xe ch t hn 1 tn so
vi d nh . Tnh s xe ban u ?
Bi 66 Hai vi nc nu cng chy vo mt b th sau 1h48 th y . Nu m ring th vi th nht
chy y b nhanh hn vi th hai 1h30 . Tnh thi gian mi vi chy ring y b ?
Bi 69 Mt x nghip dt thm c giao dt mt s thm trong 20 ngy . Khi thc hin x nghip
tng nng sut 20% nn sau 18 ngy dt xong v vt mc 24 tm . Tnh s thm thc t ?
Bi 70 Theo k hoch hai t phi lm 110sp . Khi thc hin t 1 tng nng sut 14% , t 2 tng 10%
nn lm c 123sp . Tnh s sp theo k hoch ca mi t ?
Bi 71 SAB di 120 km . Hai xe my cng xut pht t A n B . Xe th hai c vn tc ln hn xe th
nht l 10km/h nn n B sm hn 30 . Tnh vn tc mi xe ?
Bi 72 Mt mnh t hnh ch nht c din tch 240m2 . Nu tng chiu rng 3m ,gim chiu di 4m
th din tch khng i . Tnh kch thc ca mnh vn ?
Bi 73 Hai my cy cng cy xong mt m rung th ht 4 ngy . Nu cy ring th my 1 cy xong
trc my 2 l 6 ngy . Tnh thi gian cy ring xong m rung ca mi my ?
Bi 74 Mt t may d nh may 600 o trong mt thi gian nht nh nhng do ci tin k thut nn
tng nng sut mi ngy 4 o nn xong trc thi hn 5 ngy . Hi mi ngy t may c bao nhiu
o theo d nh
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Bi 75 Mt phng hp c 120 ch ngi nhng do c 165 ngi n hp nn ngi ta phI k thm ba
dy gh v mi dy thm mt gh . Hi ban u c bao nhiu dy gh , bit rng s dy gh khng
qu 20 dy?
Bi 76 Hai cng nhn cng lm mt cng vic th ht 12 ngy . Nu ngi 1 lm 1/2 cng vic ri
ngi kia lm nt th ht 25 ngy . Tnh thi gian lm ring xong cng vic ca mi ngi ?
Bi 77 Mt xe i t A n B cch nhau 120 km . i c na ng xe ngh 3 nn n B ng gi
xe phi tng vn tc thm 2km/h trn na qung ng cn li . Tnh thi gian xe chy t A n B
Bi 78 Mt phng hp c 360 ch ngi c chia thnh cc dy c s ch ngi bng nhau . Nu thm
mi dy 4 gh v bt 3 dy th s ch ngi khng i . Hi s dy lc u ?
Bi 79 Hai vi nc cng chy vo mt b cn sau 6h th y . Nu vi 1 chy 2h , vi 2 chy 3h th
c 2/5 b . Tnh thi gian vi mi vi chy ring y b ca mi vi ?
Bi 80 Mt mnh vn c chu vi l 34m . Nu tng chiu di 3m v gim chiu rng 2m th din tch
tng 45m2 . Hy tnh chiu di v chiu rng ca mnh vn ?
Bi 81 vn chuyn 18 tn hng ngi ta iu ng mt s xe tI c trng ti bng nhau . Nhng
thc t ngi ta li iu ng xe c trng ti ln hn xe c l 1tn/xe nn s xe t hn d nh l 3 xe
. Tnh trng ti mi xe ban u ?
Bi 82 Hai bn sng cch nhau 120km , mt ca n xui dng t A n B ngh 20ri ngc dng v A
ht tt c l 2h28 . Tm vn tc ca n khi nc yn lng bit vn tc dng nc l 3km/h
Bi 83 Mt ngi i xe my t A n B cch nhau 24 km . Khi t B tr v A ngi tng vn tc
thm 4km/h so vi lc i nn thi gian v t hn thi gian i 30 . Tnh vn tc lc i ca ngi ?
Bi 84 Mt tu thu chy trn khc sng di 120km , c I ln v ht 6h45 . Tnh vn tc tu bit vn
tc dng nc l 4km/h
Bi 85 Hai tnh A v B cch nhau 180 km . Cng mt lc , mt t i t A n B v mt xe my i t
B v A . Hai xe gp nhau ti th trn C . T C n B t i ht 2 gi , cn t C v A xe my i ht 4 gi
30 pht . Tnh vn tc ca mi xe bit rng trn ng AB hai xe u chy vi vn tc khng i
Bi 86: Mt ca n xui dng t bn A n bn B ri li ngc dng t bn B v bn A mt tt c 4 gi
. Tnh vn tc ca ca n khi nc yn lng ,bit rng qung sng AB di 30 km v vn tc dng nc
l 4 km/h.
Bi 87: Mt ca n xui t bn A n bn B vi vn tc 30 km/h , sau li ngc t B tr v A .Thi
gian xui t hn thi gian i ngc 1 gi 20 pht . Tnh khong cch gia hai bn A v B bit rng
vn tc dng nc l 5 km/h
Bi 88 Mt t i t A n B vi vn tc d nh . Nu vn tc tng thm 20km th n sm hn 1h ,
nu gim vn tc 10km th n mun hn 1h so vi d nh . Tnh vn tc d nh ca t
Bi 89: Mt ngi i xe p t A n B cch nhau 33 Km vi mt vn tc xc nh . Khi t B v A
ngi i bng con ng khc di hn trc 29 Km nhng vi vn tc ln hn vn tc lc i 3
Km/h . Tnh vn tc lc i , bit rng thi gian v nhiu hn thi gian i l 1 gi 30 pht.
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Bi 90 :Hai ca n cng khi hnh t hai bn A, B cch nhau 85 Km i ngc chiu nhau . Sau 1h40
th gp nhau . Tnh vn tc ring ca mi ca n , bit rng vn tc ca n i xui ln hn vn tc ca n
i ngc 9Km/h v vn tc dng nc l 3 Km/h.
Bi 91 : Mt ngi i xe p t A n B vi vn tc 15 Km/h . Sau mt thi gian, mt ngi i xe
my cng xut pht t A vi vn tc 30 Km/h v nu khng c g thay i th s ui kp ngi i xe
my ti B . Nhng sau khi i c na qung ng AB , ngi i xe p gim bt vn tc 3 Km/h
nn hai ngi gp nhau ti C cch B 10 Km . Tnh qung ng AB
Bi 92 : Mt ngi i xe p t A n B cch nhau 50 Km . Sau 1 gi 30 pht , mt ngi i xe
my cng i t A v n B sm hn 1 gi . Tnh vn tc ca mi xe , bit rng vn tc ca xe my gp
2,5 ln vn tc xe p.
Bi 93: Mt x nghip ng giy d nh hon thnh k hoch trong 26 ngy . Nhng do ci tin k
thut nn mi ngy vt mc 6000 i giy do chng nhng hon thnh k hoch nh
trong 24 ngy m cn vt mc 104 000 i giy . Tnh s i giy phi lm theo k hoch.
Bi 94: Hai t cng nhn lm chung trong 12 gi s hon thnh xong cng vic nh . H lm
chung vi nhau trong 4 gi th t th nht c iu i lm vic khc , t th hai lm nt cng vic
cn li trong 10 gi . Hi t th hai lm mt mnh th sau bao lu s hon thnh cng vic.
Bi 95: Mt on xe d nh ch 40 tn hng. Nhng thc t phi ch 14 tn na nn phi iu thm
hai xe v mi xe phi ch thm 0,5 tn. Tnh s xe ban u.
Bi 96: Hai ngi i xe p t A n B cch nhau 60km vi cng mt vn tc. i c 2/3 qung
ng ngi th nht b hng xe nn dng li 20 pht n t quay v A. Ngi th hai vn tip tc i
vi vn tc c v ti B chm hn ngi th nht lc v ti A l 40 pht. Hi vn tc ngi i xe p
bit t i nhanh hn xe p l 30km/h.
Bi 97 Din tch hnh thang bng 140 cm2, chiu cao bng 8cm. Xc nh chiu di cc cnh dy ca
n, nu cc cnh y hn km nhau 15cm
Bi 98: Mt rp ht c 300 ch ngi. Nu mi dy gh thm 2 ch ngi v bt i 3 dy gh
th rp ht s gim i 11 ch ngi. Hy tnh xem trc khi c d kin sp xp trong rp ht c
my dy gh.
Bi 99: Mt my bm theo k hoch bm y nc vo mt b cha 50 m3 trong mt thi
gian nht nh. Do ngi cng nhn cho my bm hot ng vi cng sut tng thm 5
m3/h, cho nn bm y b sm hn d kin l 1h 40. Hy tnh cng sut ca my bm theo
k hoch ban u.
Bi 100: C hai my bm bm nc vo b. Nu hai my cng bm th sau 22h55 pht
y b. Nu mi my bm ring th thi gian my mt bm y b t hn thi gian my hai
bm y b l 2 gi. Hi mi my bm ring th trong bao lu y b?
Bi101: Mt xe ti i t tnh A n tnh B vi vn tc 40 km/h. Sau 1 gi 30 pht, mt
chic xe con cng khi hnh t A n B vi vn tc 60 km/h. Hai xe gp nhau khi chng
i c na qung ng. Tnh qung ng AB
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Bi 102: Mt i cng nhn gm 20 ngi d inh s hon thnh cng vic c giao trong
thi gian nht nh. Do trc khi tin hnh cng vic 4 ngi trong i c phn cng i lm
vic khc, v vy hon thnh cng vic mi ngi phi lm thm 3 ngy. Hi thi gian d
kin ban u hon thnh cng vic l bao nhiu bit rng cng sut lm vic ca mi ngi
l nh nhau
Bi 103: Mt ca n i xui t bn A n bn B, cng lc mt ngi i b cng i t bn A
dc theo b sngv hng bn B. Sau khi chy c 24 km, ca n quay ch li gp ngi i b
ti mt a im D cch bn A mt khong 8 km. Tnh vn tc ca ca n khi nc yn lng,
bit vn tc ca ngi i b v vn tc ca dng nc u bng nhau v bng 4 km/h
Bi 104: Mt nhm th t k hoch sn xut 1200 sn phm. Trong 12 ngy u h lm theo ng k
hoch ra, nhng ngy cn li h lm vt mc mi ngy 20 sn phm, nn hon thnh k hoch
sm 2 ngy. Hi theo k hoch mi ngy cn sn xut bao nhiu sn phm.
Bi 105: Mt nhm th t k hoch lm 120 sn phm trong mt thi gian d nh. Khi lm c
mt na s sn phm nhm th ngh gii lao 10 pht. Do , hon thnh s sn phm cn li theo
ng thi gian d nh nhm th tng nng sut mi gi thm 6 sn phm. Tnh nng sut d kin.
Bi 106: Mt cng nhn d nh lm 120 sn phm trong mt thi gian d nh. Sau khi lm c 2
gi vi nng sut d kin, ngi ci tin cc thao tc hp l hn nn tng nng sut c 3
sn phm mi gi v v vy ngi hon thnh k hoch sm hn d nh 1gi 36 pht. Hy tnh
nng sut d kin.
Bi 107:Mt t c k hoch sn xut 350 sn phm theo nng sut d kin. Nu tng nng sut 10 sn
phm mt ngy th t hon thnh sn phm sm 2 ngy so vi gim nng sut 10 sn phm mi
ngy. Tnh nng sut d kin
Bi 108: Mt on xe vn ti d nh iu mt s xe cng loi vn chuyn 40 tn hng. Lc sp
khi hnh on xe c giao thm 14 tn hng na do phi iu thm 2 xe cng loi trn v mi xe
ch thm 0,5 tn hng. Tnh s xe ban u bit s xe ca i khng qu 12 xe.
Bi 109: Mt ngi i xe my t A n B cch nhau 60 km ri quay tr li A ngay vi vn tc c.
Nhng lc v, sau khi i c 1 gi th xe hng nn phi dng li sa 20 pht. Sau ngi y i vi
vn tc nhanh hn trc 4 km/h trn qung ng cn li. V th thi gian i v v bng nhau. Tnh
vn tc ban u ca xe.
Bi 110 : Mt ngi i xe my t A n B ng di 120 km. Khi t B tr v A, trong 1gi 40 pht
u ngi y i vi vn tc nh lc i, sau khi ngh 30 pht li tip tc i vi vn tc ln hn vn tc
lc trc 5km/h, khi v n A thy rng vn qu 10 pht so vi thi gian i t A n B. Tnh vn tc
lc i.
Bi 111. Hai ngi cng khi hnh i ngc chiu nhau, ngi th nht i t A n B. Ngi th hai
i t B n A. H go nhau sau 3h. Hi mi ngi i qung ng AB trong bao lu. Nu ngi th
nht n B mun hn ngi th hai n A l 2,5h.
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Bi 112. Mt mnh vn hnh ch nht c din tch l 1200m2. Nay ngi ta tu b bng cch tng chiu rng ca vn thm 5m, ng thi rt bt chiu di 4m th mnh vn c din tch 1260m2. Tnh kch thc mnh vn sau khi tu b. Bi 113. Mt ca n xui dng t A n B di 80km, sau li ngc dng n C cch B 72km, thi gian ca n xui dng t hn thi gian ngc dng l 15 pht. Tnh vn tc ring ca ca n, bit vn tc ca dng nc l 4km/h. Cu 121 Trong mt bui lao ng trng cy, mt t gm 13 hc sinh (c nam v n) trng c tt
c 80 cy. Bit rng s cy cc bn nam trng c v s cy cc bn n trng c l bng nhau ;
mi bn nam trng c nhiu hn mi bn n 3 cy. Tnh s hc sinh nam v s hc sinh n ca t.
Bi 114 Khong cch gia hai thnh ph A v B l 180 km. Mt t i t A n B, ngh 90 pht B
ri tr li t B v A. Thi gian t lc i n lc tr v l 10 gi. Bit vn tc lc v km vn tc lc i
l 5 km/h. Tnh vn tc lc i ca t.
Bi 115 Mt hnh ch nht c din tch 300m2. Nu gim chiu rng 3m, tng chiu di thm 5m th ta
c hnh ch nht mi c din tch bng din tch hnh ch nht ban u. Tnh chu vi ca hnh ch
nht ban u.
Cu 116 Mt ca n xui dng t bn sng A n bn sng B cch nhau 24 km, cng lc cng t A
mt b na tri vi vn tc dng nc 4 km/h. Khi n B ca n quay li ngay v gp b na tri ti
mt a im C cch A l 8 km. Tnh vn tc thc ca ca n.
Bai 117 Mt hnh ch nht c ng cho bng 13m v chiu di ln hn chiu rng 7m. Tnh din
tch ca hnh ch nht .
Bi 118 Khong cch gia hai tnh A v B l 108 km. Hai t cng khi hnh mt lc i t A n B,
mi gi xe th nht chy nhanh hn xe th hai 6 km nn n B trc xe th hai 12 pht. Tnh vn tc
mi xe.
Bi 119 Theo k hoch, mt t cng nhn phi sn xut 360 sn phm. n khi lm vic, do phi iu
3 cng nhn i lm vic khc nn mi cng nhn cn li phi lm nhiu hn d nh 4 sn phm. Hi
lc u t c bao nhiu cng nhn? Bit rng nng sut lao ng ca mi cng nhn l nh nhau.
Bi 121 Mt t d nh i t A n B trong mt thi gian nht nh . Nu xe chy vi vn tc 35
km/h th n chm mt 2 gi . Nu xe chy vi vn tc 50 km/h th n sm hn 1 gi . Tnh qung
ng AB v thi gian d nh i lc u
Bi 122 Hai t khi hnh cng mt lc i t A n B cch nhau 300 km . t th nht mi gi
chy nhanh hn t th hai 10 km nn n B sm hn t th hai 1 gi . Tnh vn tc mi xe t .
Bi 123: Hai ngi cng lm chung mt cng vic s hon thnh trong 4h. Nu mi ngi lm ring
hon thnh cng vic th thi gian ngi th nht lm t hn ngi th 2 l 6h. Hi nu lm ring
th mi ngi phi lm trong bao lu s hon thnh cng vic?
Bi 124: Qung ng AB di 180 km. Cng mt lc hai t khi hnh t A n B. Do vn tc ca
t th nht hn vn tc ca t th hai l 15 km/h nn t th nht n sm hn t th hai 2h. Tnh
vn tc ca mi t?
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Bi 125 Mt tam gic vung c cnh huyn bng 15 cm v tng hai cnh gc vung bng 21 cm.
Tnh mi cnh gc vung.
Bi 126: Hai i cng nhn cng lm chung mt cng trnh ht 144 ngy th lm xong. Hi mi i
lm ring th trong bao lu s hon thnh cng trnh ; Bit rng mi ngy nng sut lm vic ca i
I bng 2/3 nng sut lm vic ca i II.
Bi 127: Mt khu vn hnh ch nht c din tch l 60m2 v chiu di ln hn chiu rng 7m. Tnh
kch thc ca vn.
Bi 128: Mt hi trng c 300 gh c xp thnh nhiu dy nh nhau. Ngi ta mun sp xp li
bng cch bt i 3 dy th phi xp thm 5 gh vo mi dy cn li. Hi lc u hi trng c bao
nhiu dy gh v mi dy c bao nhiu gh
Phn HNH Hc Bi 1 Cho hai ng trn (O;R) v (O;R) ct nhau ti A,B (Ov O thuc hai na mt phng b AB )
.Cc ng thng AO v AO ct (O) ti hai im C,D v ct ng trn (O) ti E,F .Chng minh :
a) Ba im C,B,F thng hng b) T gic CDEF ni tip
c) AB,CD,EF ng quy d)A l tm ng trn ni tip tam
gic BDE
e ) MN l tip tuyn chung ca (O) v (O) . Chng minh MN i qua trung im ca AB
Bi 2 Cho ng trn tm (O) v mt im A nm ngoi ng trn . Cc tip tuyn vi ng trn
k t A tip xc vi ng trn ti B,C . Gi M l im tu trn ng trn khc B v C .T M k
MH BC,MK CA,MI AB . CM: a) T gic ABOC ,MIBH,MKCH ni tip b) BAO BCO= , MIH= MHK c) MIH ~ MHK d) MI.MK=MH2 Bi 3 Cho ABC nhn ni tip (O) . Gi BB,CC l cc ng cao ca ABC ct nhau ti H.Gi E l im i xng ca H qua BC ,F l im i xng ca H qua trung im I ca BC , Gi G l giao
im ca AI v OH . CM: a) T gic BHCF l hnh bnh hnh b) E,F nm trn (O)
c) T gic BCFE l hnh thang cn d) G l trng tm ABC e) AO BC Bi 4 Cho ng trn (O) ng knh AB . Mt ct tuyn MN quay quanh trung im H ca OB
.Chng minh:
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a) Khi ct tuyn MN di ng , trung im I ca MN lun nm trn mt ng c nh
b) T A k tia Ax MN . Tia BI ct Ax ti C . Chng minh t gic BMCN l hnh bnh hnh c) Chng minh C l trc tm AMN d) Khi MN quay xung quanh H th C di ng trn ng no
e) Cho AB=2R ,AM.AN=3R2;AN=R 3 . Tnh din tch phn hnh trn nm ngoi tam gic AMN Bi 5 Cho 1/2(O) ng knh AB=2R ,k tuyp tuyn Bx vi (O).Gi C,D l cc im di ng trn (O)
.Cc tia AC,AD ct Bx ti E,F ( F nm gia B v E). Chng minh
a) ABF ~ BDF b) T gic CEFD ni tip c) Khi C,D di ng th tch AC.AE=AD.AF v khng i
Bi 6 Cho ABC ni tip (O) .Tia phn gic BAC ct BC ti I v ct (O) ti M a) Chng minh OM BC b) MC2=MI.MA
c) K ng knh MN . Cc tia phn gic ca B v C ct AN ti P v Q . Chng minh 4 im P,C,B,Q thuc mt ng trn
Bi7 Cho tam gic ABC cn ti A c BC=6cm ng cao AH=4cm ni tip ng trn (O;R) ng
knh AA .K ng knh CC, k AK CC a) Tnh R ? b)T gic CACA , AKHC l hnh g ? Ti sao?
c) Tnh din tch phn hnh trn (O) nm ngoi ABC ? Bi 8 T mt im A nm ngoi (O) k tip tuyn AM,AN vi (O) , (M,N(O))
a) T O k ng thng OM ct AN ti S . Chng minh : SO = SA b) Trn cung nh MN ly im P khc M v N . Tip tuyn ti P ct AM ti B , AN ti C .Gi s A c
nh ,P l im chuyn ng trn cung nh MN . Chng minh chu vi ABC khng i ? . Tnh gi tr khng i y?
c) V ct tuyn AEF khng i qua im O ,H l trung im EF . Chng minh cc im A,M,H,O,N
cng thuc mt ng trn
d) Chng minh AE.AF=AM2 e) Gi K l giao im ca MH vi (O) .Chng minh NK//AF
Bi 9 Cho (O) , hai ng knh AB,CD vung gc vi nhau . M l mt im trn cung nh AC . Tip
tuyn ca (O) ti M ct tia DC ti S . Gi I l giao im ca CD v BM . Chng minh:
a) T gic AMIO ni tip b) MIC MDB= ; MSD 2 MBA=
c) MD phn gicAMB d) IM.IB=IC.ID ; SM2=SC.SD
e) Tia phn gic COM ct BM ti N . Chng minh : NI tgMBONM
= v CN BM
g) Gi K l trung im MB . Khi M di chuyn trn cung nh AC th K di chuyn trn ng no ?
h) Xc nh v tr ca M trn cung nh AC sao cho AM=5/3MB
Bi 10 Cho 1/2(O) ng knh AB . V tip tuyn Ax,By . T C l mt im bt k trn na ng
trn (O) v tip tuyn vi ng trn ct Ax , By ti E,F
a) Chng minh FE=AE+BF
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b) Gi M l giao im OE vi AC , N l giao im OF vi BC . T gic MCNO l hnh g ? Ti sao ?
c) Gi D l giao im AF v BE Chng minh CD//AE d) Chng minh
EF.CD=EC.FB
e) Khi C di chuyn trn (O) th M,N di chuyn trn ng no ?
g) Xc nh v tr ca C din tch EOF b nht Bi 11 Cho hai ng trn (O;R) v (O;r) tip xc ngoi ti C . Gi AC, BC l hai ng knh ca (O)
v (O) . DE l dy cung vung gc ti trung im M ca AB . Gi giao im th hai ca ng thng
DC vi ng trn(O) ti F . BD ct (O) ti G . Chng minh :
a) T gic AEBF l hnh thoi b) Ba im B,E,F thng hng c) 4 im M,D,B,F thuc
mt ng trn d) DF,EG,AB ng quy e) MF=1/2DE g) MF l
tip tuyn ca (O)
Bi 12 Cho 1/2(O) ng knh AB , M l mt im trn na ng trn . H MH AB ,v hai na ng trn (I) ng knh AH,(K) ng knh BH nm pha trong na (O) , ct MA,MB ti P,Q .
Chng minh :
a) MH=PQ b) PQ l tip tuyn chung ca (I),(K)
c)PQ2=AH.BH;MP.MA=MQ.MBd) T gic APQB ni tip e) Xc nh v tr
ca M chu vi , din tch t gic IPQK ln nht
Bi 13 Cho tam gic vung ABC , vung ti A , ng cao AH ni tip (O) , d l tip tuyn ca (O) ti
A . Cc tip tuyn ca (O) ti B,C ct d ti D v E a) TnhDOE b) Chng minh : DE = BD+CE
c) Chng minh : BD.CE=R2 d) Chng minh BC l tip tuyn ca ng trn ng knh DE
Bi 14 Cho tam gic ABC cn ti A , cc ng cao AD, BE ct nhau ti H . Gi O l tm ng trn
ngoi tip tam gic AHE . Chng minh :
a) ED=1/2BC b) DE l tip tuyn ca (O) c) Tnh DE bit DH = 2cm , HA =
6cm
Bi 15 Cho 1/2(O) ng knh AB . V tip tuyn Ax,By . T M l mt im bt k trn na ng
trn (O) v tip tuyn vi ng trn ct Ax , By ti C,D . Cc ng thng AD,BC ct nhau ti N .
Chng minh :
a) CD=AB+BD b) MN//AC c) CD.MN=CM.DB
d) im M nm v tr no trn1/2(O) th AC+BD nh nht?
Bi 16 Cho ABC cn ti A ,I l tm ng trn ni tip , K l tm ng trn bng tip ca gc A , O l trung im ca IK . Chng minh :
a) Bn im B,I,C,K thuc ng trn tm O b) AC l tip tuyn ca (O)
c) Bit AB = AC = 20cm , BC = 24cm tnh bn knh (O) d) Tnh phn gii hn bi (O) v t
gic ABOC
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Bi 17 Cho ABC vung ti A . V (A;AH) . Gi HD l ng knh ca (A) . Tip tuyn ca ng trn ti D ct CA ti E . Gi I l hnh chiu ca A trn BE Chng minh : a) BEC cn b) AI = AH
c) BE l tip tuyn ca (A;AH) d) BE = BH+DE
Bi 18 Cho hnh vung ABCD , im E trn cnh BC . Qua B k ng thng vung gc vi DE ,
ng thng ny ct cc ng thng DE v DC ti K,H . Chng minh: a) T gic BHCD ni tip
b) TnhCHK c) KC.KD=KH.KB d) Khi E di chuyn trn BC th H di chuyn trn ng no ?
Bi 19 Cho (O;R) c hai ng knh AB v CD vung gc vi nhau . Trn on AB ly im M (khc
O). ng thng CM ct (O) ti im th hai N. ng thng vung gc vi AB ti M ct tip tuyn
ti N ca (O) im P .CM: a) T gic OMNP ni tip b) T gic CMPO l
hnh bnh hnh
c) Tch CM.CN khng ph thuc vo im M d) Khi M di chuyn trn AB th P chay trn mt on
thng c nh
Bi 20 Cho ABC vung ti A (vi AB > AC) , ng cao AH . Trn na mt phng b BC cha im A v na ng trn ng knh BH ct AB ti E ,