ÔN TẬP TOÁN 10 ĐỢT 3 (KHTN HN)

NGAN
HANc
cAu rrol D4r
s6
to-oqT
g
Tfrrdng THPT
Chuy€n
Khoa
hgc
Tq
nhi€n
I.NQi
dung
1.
Phuong trinh
(15
cdu).
i
2. Phudng
trinh
vO
tf
(15
cau).
3.
HQ
phuong
trlnh
(15
cau).

4. Lrrong
gi5,c (25
cau).
II.Phin ngdn hi,ng cAu h6i
'
Phrrdng trinh(cdc
bei
tfi 1-15)
1.
Giei
phrrong
tr)nh
2l"l-lr+t|:2
2.
Giei
vd
bi€n lu$n theo
m
m(mr-1)
+2:(m*2)r
3. Giei
vd bi€n lu6n theo
m
mn
_
m
_
B
:fn
r*I

4.
Giai vd biQn
ludn
theo m
(m
-
I)r2
-
(2m
+
1)r
*
m
*
4
:
0
5.4ra-7r2
-5r-1:0
6. 3(z
+
5)(r
+
6)(r
*7)
:s,
7. ra- 1or3+2612-
1or*1:o
8. ra
+

3r3
-
r4r2
-
6r
I
4:0
9.
(z
+
5)(r
+
6)(r
+
8)(z
+
9)
:40
10.
.(z
+
1)(r
+
2)(r
+
3)(r
+
4)
:72012
1,I.

(r
+
2)4
+
(r
+
8)a
:
242
12.
(r
-
Z)u
+
(x
-
4)6
:64
13. (r2
*
3r
-
4)3
+
(2r2
-
5r
*3)3
:
(3r2

-
2r
-
I)3
14.
(r2
-
6r
-
9)'
:
r(r'
-
4r
-
9)
15.
ra
:24r
*32
Phddng
trinh v6
ti(cdc bai td 16-30)
16. 2J4r+1*Ji+2:J;r+6
l
17.
\/i
+
\ttn +9:
\6+

L
+
\/n +Z
ls.
2\ET3
+
JifiiT!
:2{2r
+
\/{FTtor
+
s
ts.
J67
-Trl4
-
,fip r
1r:
I
20.
Jr
-
2
+
JTTZ
+
r/7
4
:
6

-
2r
21.
\E+
1+
JT4
+
1/@
+\@
-fi
:
5
22.
ft-l+JT-:tr2-6r*11
n.
\Lgar+
6r
+
T
+
rlbirTm
+
u
:
{
-
2r
-
12
-

2a.
{r
+
2
-
4t/n
-
2
+
1/r
+
7
-
6t/r
-
2
:
3
25.
+Er+L+W-r-{r+A.
.:
26. (\E+r-r)(\fr4+1)-z
27.
2(r2
+
2)
:
5\F
+
t

2s.
{m-r*{i:s
29'
12*t/r+5:5
30. z3*1:2VrF1
HQ
phrrong
trinh(cdc
bli tfi 31-45)
31.
Giei
vd biQn ludn theo
a
32. Tim m dd tre c6 nghi€m (r,y)
th6a mdn ry
< 0
lr-mY:1
I
lmr+u:3
JJ.
l"r-(a+L)y:1
Itr-
2)r+2a:a-2
lr'*ry+y2:19
I"
*, *
ry:11
!{*+r)(E+1)
:6
l"'*a2+r:2(r*u)

I**i**:s
'lr,l
r
1
r
l
_
[;+i+7+P:8
I"
*u2
:7
Itt*a5:13+a3
34
35.
36.
37.
38.
39.
I*'*1:3x-u
,
Ir'f1:3a-r
/-
)t/t}-r*1/Y-2:4
I/TO u
+
tlFZ:
4
13a2
44.
Tim

m
dd
tre
sau
c6
nghigm
("a+r*a:m*r
\r'o*ra2:m
45.
Tim
a
dd
he
c6
nghiQm
duY
nh6t
Io@n+i)
:E+t-ltl
l"*a2:r
Ldgng
giSc(c6c
bai
tfi
46-70)
25r,
46. Cho
sina:
t,n
.

n <
f
.Tinh
tan(a
-
i)
47. Cho
cosr
*
cos!:
1,sina *
siny:
]'Tinh
cos(z
-
y)'
48.
Tinh cos
18".
4e'
rinh
o:#r-*h,
50'
chrlng
minh
rhng
cos(a
-
b)cos(a
+

b)
:
cos2
a
-
sin2
b
40.
41.
42.
43.
(r
,
y
-26
)ui;-
5
I"
-
a2
:24
I"'*2ry*r:u2
\r(r
+
a)
:2
(r'+g2+22:L
{rt+ as+23:I
["n*ya+za:1
IrO*r+L:7a

\*'a'*ry+r:
(r@+u*1)
:3
tt"*
v)2+r:1
51.
Chrlng
minh
rHng
cos2 e
*
sin2
E
*
sin(r
*
g)
sin(r
-
a)

I
52.
Gie
st
cos
r cosa
cos
z
f

0.Chr1ng
minh
rbng
sin(z
-
g),
sin(y
-
z),
sin(z
-
r)
-
n
*ar"*g
-
"*g"oa"
-
"oar"or"
-'
53.
Cho
sinr
f
O.Chrlng
minh
rB,ng
sin
5r
*"

:
2cos
4r
*2cos2r
*
1
54.
Chrlng
minh
rb,ng
;,."-
.
t*a
'U*z
zlr
sinz *sin3r*sinz
-
sin(r*
a
*
z):
4sin
Tsinf
sin,
55.
Chrlng
minh
rXng
cot r
-

tanr
-
2tan2r
-
4tan4r
:
8cot
8r
bb.
I rnn
sin
350
sin 65'
sin 80"
A_
sin
200
*
sin 50'
*
sin
70'
57.
Tinh
58.
Cho
tano:2.Tinh
A
:
4sin2

10o
*
4 sin2
50o
cos
20"
+
cos 80o
sin 2a *
sin
4a
A
1
*
cos
2a
*
cos4a
59.
Chrlng
minh
rli,ng
sin2(a
-
0
:sin2
a
-
sin2
P

+
zsinasinBcos(a
-
B)
60'
ch(rng
minh
rang
3
+
4cos
2a
I
cos4a
,4
ffi:col'-0
61.
Chrlng
minh
rb,ng
"
,inartcos'r:]*f;"or+,
62.
Chirng
minh
ri.ng
.,
sin3
acos3a *
sin3acos3

a
:
9
sin4a
4
63.
Cho
cos
r
*
cos
E
*
cos
z
:O'Chrlng
minh
rbng
cos 3z
*
cos
39
*
cos
3z
:
12 cos
r
cos1
cos

z
64.
Chrlng
minh
rXng
t'a,
'tr
2n
24n
1T
sin,, +sin,,
+."+sin,,
:cott
4
r
65.
Tinh
21
41
6n
8n
lOn iZtr
L4tr
,9
:cosfr
+cos
r,
+co,s15
*cos
r,

*cos
1b
+cos
r,
*cos-
66.
Tinh
P
:
cc
21
4tr
8zr
rsrcosg"otg
67.
Tinh
^7t
^2r
.8tr
A*cos';*cos";*cos"-
.,,,
5'-
15
15
rii
68.
Cho sin(2a +
B)
:7sinp.C$dng.
minh

rbng
Stan(o +
0):4tan61
69.
Tinh
A:
tango
-
tan
27o
*
tan63o
+
tan810
?0.
Chrlng
minh
rXng
1r
Zn
Bn
1
cos;-cos;*cos;:;
l(lz
NGAN nANc
cAu
Hor niivn Hoc
ro
odr
g

(Di,ng
cho hqc si,nh
Kh6i 10', Tradng
THPT
chuy€n KHTN)
I. NOi dung
1. Vector,
riitjm vd rlrrdrrg ttr6'ng
(10
bbi).
2. Drrdng trdn
(10
bb,i).
3.
Ellipse
(10
bbi).
II. Ngdn
hhng cdu
h6i
1. Vector, di6m
vb
dttdng thlng
Cdu
1l Trong h€ tqa dO
Descartes cho
vecto , d
1*o,
A"),1
@r,ga).

Dit
d n-l
:
ragb
-
rbAa.
a)
C-hrrng
minh ring
r)
a A
(:U
ri)?n7':-Tnd
+
.
+
iii) kA A l.b
:
(kD@ n b.)
(k,l
lh c6c s6
thuc bat
ky)
iv) ? n
17
+
?1:-t n.T
+d
n.?
(vector.? uat r.y;.

b) Chrlng
minh rbns
1"
n
71
:
l?ll?llsin(?,711.
c)
Cho
tam
gi5.c
ABC.Chrrng
minh r},ng
aB nfr
:
nd n BA
:
CA n CA.
d) Cho
tam
giSc
ABC.Chrlng
minh
rli.ng
Sepc:irUB
nTd.
cAu
2i
cho
A(2,5),

B(1, 1),
C(3,3)
a)
Tim toa dO
D sao cho tfi
gi6c
ABC
D lb hinh binh
hd,nh.
b)
Tinh diQn tfch
hinh binh
hiuth ABCD.
cau
g./fifi
M'
d6ixrlng M
qua
d trong
c6,c trrrdng
hgp sau
a) d : r
-
2a
*2
:
0
vd, M(I,4).
b) d : 2r
*

y
-2
:
0
vh, M(6,5).
c) d :
4r
-
14A
-
29
:0
vd, M(1,
2).
Cau
a./Cho tam
gitrc
ABC c6 trung
didm mOt
canh
Ib
M(1,2). Bi6t hai trung tuy€n
xu6t
ph6t
tt hai dinh
c6
phuong
tr)nh lAn
ltrOt Ih d1
:

rly-3
:
0
vb d2 : 2r-y*4
:
0.
Vi6t
phrrong
trinh c5,c canh
cta tam
gi6,c
ABC.
Cdu 5.
Cho
hinh chrl
nh6,t ABCD
c6
1(6;2)
lb
giao
cii6m ctra
hai drrdrrg
ch6o AC
vit" BD.
Diiim
M(1;1)
thuQc drrdng thd.ng
AB.
Trung ditim
,O

cira cqnh CD
nXm tr6n
drtdng thd,ng
r
+
A
-
5
:0.
Vi6t
phrrong
trinh
cpnh
AB.
Cdu 6.
Cho
tam
gi6c
ABC
c6 C(-3,1). Phan
giac
AD
c6
phrrong
trinh
r*3y
*I2
:
0,
drrdng

aao
AH c6
phuong
trinh r
*
7A
+'32:
0.
Lap
phrrong
trinh c6c
cpnh tam
gi6c.
Cdu 7.
Cho
tam
giftc
ABC c5 A(2,0),8(4,I),C(1,2).
a) Vi6i
phrrong
trinh
ph6,n gi6c
trong
g5c
A.
b)
Tirn
tsa
dO didm D
thuQc BC

)d,
chdn drrdrrg
ph6,n
giSc
ngodi
g5c
A.
Cdu 8.
Cho
didm
A(a,b),
a>
0,b > 0.
Vi6t
phrrong
trinh
dudng thhng
d di
qua.4
khOng
di
qua g6c
O cft tia
Or,Oy
tai M,.ly' sao cho MO
+
Ol/ nh6
nh6t.
CAu 9. Cho hai
hinh vuOng

ABCD vd AB'C'D'ctng hu6ng.
Chrlng minh
rX,ng
cdc
dudng
thd,ng
BBt,CCt vb" DDt ddng
quy.
Cdu
10.
Cho
A(a,0) vd, B(0,
b), a,b > 0.
M di chuy6n
tr€n
d,oan
OA, l/ di chuy6n
tr€n do4n
OB
sao cho AM
:
ON.
Chrrng minh
rXng
trung
tr',tc MN luOn
di
qua
ditim
c6 dinh

vb,
tim
tqa dO di6m d6.
2. Dddng trbn
CAu 11. Ld,p
phrtdng
tr)nh dttdng trdn
(C) trong c6c tnldng
hop
sau
a) Dudng trbn
(C)
di
qua
A(1,
3), B(5,6), C(7,0).
b)
Dudng
trdn
(C)
t6m I(-4,2),
ti6p
xirc
v6i
dudng
thd,ng
d:
3r*4g
-
16:0.

c) Drrdng trdn
(C)
ti6p xric d: r
t2A
-5:0
tai,4(3,1)
vd
di
qua
8(6,4).
d) Drrdng trdn
(C)
di
qua
A(I,2),8(3,1) c5 tdm nb,m
tr€n
d,: 7r
*3y
-l1
:
0.
e) Drrirng
trbn
(C)
di
qua
A(4,2),8(5, 1) ti6p
xfic
dudng
thd,ng

d,
:
r
-
29
*
5
:
0.
f)
Drrirngtrdn
(C)
c6 b6nkinh R: \/t
ti6pxfc
vdid:
r-A
-
6:0
tai A(4,-2).
g)
Dudngtrbn
(C)
ddngt6,mvdi
(f)
,
12+y2
-4r-6y-
17:0vbti6pxircvdi
3r-49*7:0.
Cdu 12.

Cho
d: 3z*
4y-3:0 vA,drrdngtrdn
(C)
t
12
+A2
-r-7y:0
a) Tim tqa dO
giao
didm cria
d ve,
(C).
b) Lap
phuong
trinh ti6p
tuy6n cria
(C)
tai chc
giao
didm
d5.
Cdu 13. LQp
phrrong
trinh ti6p tuy6n
d
cria
(C)
bi6t
'

a)
(C)
:
12
+A2
-
2r
-
Bg
*1
:
0
vd d song
song vtli
dttdng
th8,ng
5r
*L2y-6
:
0.
b)
(C),
12
+
A2
-
6r
*2E
:0
vil d vu6ng

g6c
vrti
drrdng th&ng
3r
-
y*
6
:
0.
Cdu
14.
Cho dudng
trdn
(C)
: x2
+A2
-
4r
-2A:0
vd
A(3,
-2).
ViCt
phrrong
trinh
ti6p tuy6n cria
(C)
ke td / vh tim toa dd c5.c
ti€p diiim.
Cdu

15.
Cho
c6.c dudng trbn
(C):
12*y2:r
(C^),
12
+
a2
-
2(*+ 1)r
)-
4my-
5
:
0
a) Tim tQ,p hqp tdm
cria
(C-)
b)
Chrrng
minh
c5
hai drrdng trbn cria hs
(C*)
ti6p xirc (C).
Vi6t
phuong
trinh ti6p
tuy6n chung cria

hai
drrdng trdn
nA,y.
C6u
16.
a) Cho
,4(3,3)
vd
E|(6,4)
tim
c6c
didm
M
trln
trqc holinh
sao
cho
TfrB
:
45".
="
t;
Cho
drrdng
thEng
d
: Ar
*
By
:

Az
I82.
Ditim
p
di
chuy6n
tr€n
d.
Tr6n
tia
Of
i6y
e
sao
"Uo
Op.Oe:
1.
Chfng
minh
rXng
Q
luon
nBm trcn
drrdng
trdn
c6 dinh.
Hay
vi6t
phrr<rng
tr)nh

drrdrrg
trbn
iI6.
cAulT.Chodudngtrbn(C):12+a2:R2vddidmM("'96)ongob'idrrdngtrdn'
K6
ii6p
tuy6n
MT1,"MT2
tdi
(C)'
vli
Tr,?2
ld
ti6p
di6m'
a)
Vi6t-
phttdng
trinh
rludng
th&ng
"1?2'
U) Cia
su
M di
cluydn
treri
clrrdng
thd,ng
d

c6
dintr.
Chrlng
rnirrh
rX,ng
7r?z
ludn di
qua
rn6t
ditim
c5
dinh.
Cdu
18.
Cho
A(a,0)
vb,
.B(-a,0).
Chrlng
minh
rb,ng di6m
M th6a
man
ffi
:
n
udi
k >
0,
k

+
I
th\
lu6n
nB.m
tr€n
mOt
dudng
trdn
c6
dinh.
vi6t
phrrong
trinh
drrdng
trdn
d6.
Cdu
19.
Tr6n
mat
ph&ng tqa
dO
c,ho
A(m,O)
vh,
B(0,
n) sao
cho
m,n >

0
vb'm*n:
a
ttrung
dtii.
Drrang
tror,
ft,
OB)
c6t
dudng
trdn
(B'
OA)
tai
P,Q'
Chring
minh
rbng
rrO
riu"
di
qua
ai6-
"6
dinh
khi
A,
B
thav

d6t"
Cdu
20.
1l6n
md,t
phfr,ng tqa
dO cho
'4(b,
h),
F(0,
h),
B(c,h),C(a,0)'
D(-a'
0)'
a) Chrlng
minh
rbng
trl
gi5,c ABCD
ld
hinh
thang
vit" FC
:
FD'
b)
AC
JAt
nn
@i

E. C6,c
drrdng
trbn
ngopiti6p
tam
gi6' (FAD)
vd
(FBC)
c6t
nhau
tai G
kh5c
F' Chrlng
minh
rllng
E,
F,G
th&ng
hhng'
3.
Ellipse
Cdu
21.
Cho
drrdng
trdn
(O) chrla
drrdng
trbn
(O'). Cho

drrdng
trdn
(1)
ti6p
xirc
trong
,.Oi
(O) vd, ti6p
xric
"ngoir,i
(O'). Chfng
*intt
t6,m
I luOn
nb,m
tren
mOt
Ellipse
c6
dinh'
cdu22.
cho
hinh
thang
ABCD,d6,y
ldn
AB
c6 dinh,
d6y
nh6 CD.:

b
kh6ng'd6i'
C-l
ai.rr";j"
;;"
a,,,o
An
+
BC
:
k >
0
khdng
adl. Cnrtng
minh
rB'ng
giao didm
1
cria
hai
drrdng
c.heo
iC,rJfO
f"O"
n},m
tr€n
mq1
Eilipse
c6 ainn
khi

C,
D
thav
doi'
Cdu
23. Cho
A,
B, C
thl,ng
ha,ng
tren
drrdng
thfrng
d theo
thf
tU
d6'
Drrdng
trdn
(1)
;6"
;;
trruy
Arii
ludn
ti6p
ii.
a
tai
,4.

Ti6p
tuy6n
kh5'c
d cira
(1) k6 tt
B, C
cdt
nhau
Lai
K.
a)
chrlng
minh
rhng
K
lu6n
nhm
tr6n
mot
Ellipse
(E)
c6
clintr
khi
(1) thay
dtii'
Uj Cito
a(-S,0),
B(-3,0),
C(3,0)

vi6t
phuong
trinh
ctra
(E)'
cdu
24.
X6c
dinh
do
ddi
hai
truc,
ti6u
crl,
tdm
sai,
tqa
d0
ii€u
di6m,
toa do
c6c
dinh,
phrrong,tr)nh
dudng
chud,n
ctra
c6c
Ellipse

sau
,r' a'
a)
.
+7:r.
LlI
t
,,
.iz u'
1
b)
25+T:no
Cdu
25. Tim t6,m sai cria Ellipse
bi6i
a) M6i ti€u didm
nhin
truc
nh6
du6i
g6c
vudng.
b)
Kho6,ng c6c
girla
hai dinh tr6n hai
truc
bb,ng ti6u crJ.
*2
Cdu

26.
Cho
Ellipse
(E)
:
?
*
r':
1. Tim c6c diiim tren (E)
sao cho
a)
C6
b6n
kinh
qua
ti6u didm nAy bB,ng 3
Idn bdn kinh
qua
ti6u
di6m
kia.
b) Nhin hai tiOu didm drrdi
g6c
120'.
c) Nhin hai ti€u
di6m dudi
g6c
vu.6ng.
Cd:u
27.

Vi6t
phuong
trinh chfnh t6c cria
Ellipse
(E)
bi6t
a) Ellipse
(E)
qua
M(-2\/8,2), dO dir,i trsc
nh6 lb 6.
b) Ellipse
(E)
qua
M(4,-rt),N(2\/r,3).
c)
Ellipse
(E)
qua
B(2,-Z),c6 t6,m ,ui
":25.
d) Ellipse
(E)
qua
CGrt,2)
vb dudng chud,n
r: *4.
CAu
28.
Cho

Ellipse
(,8)
: 4 *t: 1td,m O.A,B
thuQc (E)
thay
d6i sao cho
,w
\u)
a2
t
bz
-
OA
L
OB. Chrrng
minh rXng .4,B
lu6n ti6p xric
dudng trdn
co dinh.
Cdu 29.
Cho
Ellipse
(,8)
:
t
*
t:
1 t6,m O,
mOt
ti6u di6m lb F

c6
hob,nh
dO
'v
\u )
.
a2
,
bZ
-
r vv uvo.A^r
sv
duong. M di chuydn
tr€n
(E)
lu6n c6
tung dO dudng.
D6.t
p:ifrF
a)
Tinh MF theo a,b
vb"
9.
b) MF c|,t
(E)
tai
di6m thrl hai Mt. Chtrngminh
ri,ng
h
.

afu
Unu,rg adi khl
M
di chuyiSn.
CAu 30. Cho
Ellipse
(E)
:
4**:1t6.m
O,
ti6u didm
Jr1, Fz.M
di chuyi5n tron
(B)
a)
Chrlng
minh rH,ng M
n.M
F2
+
O
M2
IuOn
kh6ng ddi.
b)
Chrlng
minh
rB,ng
(MFt
-

1v7Fr)'
-
4MO2ludn
kh6ng
d6i.