0
Cõu1(2,0im).Choh ms
3 2
3 2y x x mx m = + + + -
(
m
lthams)cúthl
( )
m
C
.
a)Khosỏtsbinthiờnvvth cahm skhi
0m =
b)Xỏcnh
m
( )
m
C
cúcỏcimccivcctiunmvhaiphớatrchonh
Cõu2(1,0im).
Giiphn gtrỡnh:
2cos6 2cos4 3 cos2 sin 2 3x x x x + - = +
Cõu3(1,0im).
Tớnh:
( )
2
1
0
x
x
x x e
I dx
x e
-
+
=
+
ũ
Cõu4(1,0im).
a) Giiphn gtr ỡnh:
2 3 6 36
log log log logx x x x + + =
b) Tỡmshngkhụngphthucvo x trongkhaitrinnhthcNiutn
2
3
2
n
x
x
ổ ử
+
ỗ ữ
ố ứ
(vi
0x ạ
), bitrng
*
nẻ Ơ
v
( )
2 1
5 4
9 4
n n
n n
C C n
+ +
+ +
- = +
Cõu5(1,0im).
Chohỡnh chúp .S ABCD cúỏy ABCD lhỡnh chnhtvi 3 2AB a AD a = = . Hỡnh chiu
vuụnggúcca
S
lờnmtphng
( )
ABCD
lim H thucc nh AB saocho 2AH HB = .Gúc
gia mtphng
( )
SCD
v mt phng
( )
ABCD
bng
0
60
.Tớnh theo
a
th tớch khi chúp
.S ABCD
vtớnhkhongcỏchgiahaingthng
SC
v AD.
Cõu6(1,0im).
Trong mt phng ta Oxy cho tam giỏc cõn ABC cú ỏy BC nm trờn ng thng
:2 5 1 0d x y - + =
, cnh AB nm trờn ng thng
:12 23 0d x y
Â
- - =
. Vit phng trỡnh
ng thng AC b itnúiquaim
( )
31M .
Cõu7(1,0im).
Trong khụng gian Oxyz , cho
( ) ( ) ( )
100 , 020 , 003A B C .Vi t phng trỡnh mt phng
( )
P iqua ,O C saochokhong cỏcht An
( )
P bngkhong cỏcht B n
( )
P .
Cõu8(1,0im).
Gii h phngtrỡnh:
( )
2 2 2 2
3
5 2 2 2 2 5 3
2 1 2 12 7 8 2 5
x xy y x xy y x y
x y x y xy x
ỡ
+ + + + + = +
ù
ớ
+ + + + + = + +
ù
ợ
.
Cõu9(1,0im).
Chobasthcdng , ,a b c thamón
2 2 2
3a b c + + = .
Tỡmgiỏtrinh nhtcabiuth c
( )
1 1 1
8 5S a b c
a b c
ổ ử
= + + + + +
ỗ ữ
ố ứ
Ht
Thớsinhkhụngcsdngtiliu.Cỏnbcoithikhụnggiithớchgỡthờm.
Hvtờnthớsinh: Sbỏodanh:
SGD&T
TRNGTHPT
CHUYấNVNHPHC
KHOSTCHTLNG
CCMễNTHITHPTQUCGIALN3NMHC2014 -2015
MễN:TONKHI12A+B
Thigian180phỳt(Khụngkthigiangiao)
thigm01trang
23
1
TRNGTHPTCHUYấNVNHPHC.
(Hngdnchmcú 5trang)
HNGDNCHMKSCL LN3 NM2015
Mụn:TON 12AB
I.LUíCHUNG:
1)Nuthớsinhlmbikhụngtheocỏchnờutrongỏpỏnnhng vnỳngthỡchosim
tngphnnhthangimquynh.
2)Vicchitithoỏthangim(nucú)tronghngdnchmphimbokhụnglmsai
lchhngdnchmvphicthngnhtthc hintrongcỏcgiỏoviờnchmthihhosỏt.
3)imtonbitớnhn0,25im.Saukhicngimtonbi,ginguyờnktqu.
II.PN:
Cõu í Nidungtrỡnhby im
1
a
Chohms
3 2
3 2y x x mx m = + + + -
(
m
lthams)cúth l
( )
m
C
.
a)Khosỏtsb inthiờnvvthcahmskhi
0m =
1,0
ồ
Khi
0m =
hmstrthnh
3 2
3 2y x x = + -
ã TX: D R =
ã Sbinthiờn:
+)Chiubinthiờn:
2
0
3 6 , ' 0
2
x
y x x y
x
=
ộ
= + =
ờ
= -
ở
Hmsngbintrờncỏckhong
( ) ( )
2 , 0 -Ơ - +Ơ ,nghchbintrờn
( )
20 -
0.25
+)Cctr :Hmstcciti 2 ( 2) 2
CD CD
x y y = - = - =
Hmstcctiuti 0 (0) 2
CT CT
x y y = = = -
+)Giihn:
lim lim
x x
y y
đ-Ơ đ+Ơ
= -Ơ = +Ơ
0.25
Bngbinthiờn:
x
-Ơ 2 0 +Ơ
'y +0 0+
y
2 +Ơ
-Ơ
2
0.25
ã th:ct Ox ti
( )
( ) ( )
10 , 1 30 , 1 30 - - + - -
thnhnimunU(
10 -
)ltõmixng.
( Giỏmkhotv)
0.25
b
b)Xỏcnhm
( )
m
C cúcỏc imccivcctiunmvhaiphớatrchonh
1,0
ồ
Phngtrỡnh honh giaoim ca
( )
m
C vtrchonh l
( )
( )
( )
3 2 2
3 2 0 1 2 2 0 1x x mx m x x x m + + + - = - + + - =
0.25
( )
( ) ( )
2
1
1
2 2 0 2
x
g x x x m
= -
ộ
ờ
= + + - =
ở
0.25
2
( )
m
C
cúhaiimcctrnmvhaiphớaivitrc
Ox
( )
1PT
cúba nghim
phõnbit
( )
2
cú hainghimphõnbit khỏc 1 -
( )
3 0
3
1 3 0
m
m
g m
Â
D = - >
ỡ
ù
<
ớ
- = - ạ
ù
ợ
0.25
Vykhi
3m <
thỡ
( )
m
C
cúcỏcimccivcctiunmvhaiphớatrchonh
0.25
Chỳý hcsinhcú thgiitheocỏchphng trỡnh 0y
Â
= cúhainghimphõnbit
1 2
,x x
v
( ) ( )
1 2
0
Cé CT
y y y x y x ì = ì <
2
Giiphngtrỡnh : 2cos 6 2c os 4 3 cos 2 sin 2 3x x x x + - = +
1,0
ồ
PT
( ) ( )
2 cos6 cos 4 3 1 cos 2 2sin cosx x x x x + = + +
0.25
( )
cos 0
4cos5 cos 2 cos 3 cos sin
2cos 5 3 cos sin
x
x x x x x
x x x
=
ộ
= +
ờ
= +
ở
0.25
ã
( )
cos 0 ,
2
x x k k
p
= = + p ẻZ
ã
3 1
2cos5 3 cos sin cos5 cos sin cos5 cos
2 2 6
x x x x x x x x
p
ổ ử
= + = + = -
ỗ ữ
ố ứ
0.25
( )
5 2
6 24 2
5 2
36 30
6
x x k
x k
k
x kx x k
p p p
ộ
ộ
= - + p
= - +
ờ
ờ
ẻ
ờ
ờ
p p p
ờ ờ
= + = - + p
ờ ờ
ở
ở
Z
Vyptcú ba hnghim
( )
2 24 2 36 3
x k x k x k k
p p p p p
= + p = - + = + ẻZ
0.25
3
Tớnh
( )
2
1
0
x
x
x x e
I dx
x e
-
+
=
+
ũ
1,0
ồ
( )
( )
2
1 1
0 0
1
1
x
x x
x x
x x e
xe x e
I dx dx
x e xe
-
+
+
= ì = ì
+ +
ũ ũ
0.25
t
( )
. 1 1
x x
t x e dt x e dx = + ị = +
icn+ 0 1x t = ị =
+
1 1x t e = ị = +
0.25
Suyra
( )
1 1 1
0 1 1
1
1 1
1
1
x x
e e
x
xe x e
t
I dx dt dt
xe t t
+ +
+
-
ổ ử
= ì = ì = - ì
ỗ ữ
+
ố ứ
ũ ũ ũ
0.25
Vy
( )
( )
1
1
ln ln 1
e
I t t e e
+
= - = - +
0.25
4
a
Giiphngtrỡnh:
2 3 6 36
log log log logx x x x + + =
0,5
ồ
Phngtrỡnhxỏc nhvimi x R ẻ
p dngcụngthc
( )
log log log , 0 , , 1 1
a a b
c b c a b c a b = ì < ạ ạ
0.25
Phngtrỡnh
2 3 2 6 2 36 2
log log 2 log log 2 log log 2 logx x x x + ì + ì = ì
( )
2 3 6 36
log log 2 log 2 1 log 2 0x + + - =
( )
*
3
Do
3 6 36
log 2 log 2 1 log 2 0 + + - >
PT
( )
2
* log 0 1x x = =
Vynghimphngtrỡnhl.
0.25
b
Tỡmshngkhụngphthucvo
x
trongkhaitrinnhthcNiu tn
3 2
2
n
x
x
ổ ử
+
ỗ ữ
ố ứ
vi
0x ạ
,bit
*
nẻ Ơ
v
( )
2 1
5 4
9 4
n n
n n
C C n
+ +
+ +
- = +
0,5
ồ
T githit
( )
( )( )( ) ( )( )( )
( )
2 1
5 4
5 4 3 4 3 2
9 4 9 4
6 6
n n
n n
n n n n n n
C C n n
+ +
+ +
+ + + + + +
- = + - = +
15n ị = .Khiú
( )
15
30 5
15 15
15
3 32 2
3
15 15
0 0
2 2
2
k
k
k
k k k
k k
x C x C x
x x
-
-
= =
ổ ử ổ ử
+ = =
ỗ ữ ỗ ữ
ố ứ ố ứ
ồ ồ
0.25
Shngkhụng phthuc vo x tng ngvi
30 5
0 6
3
k
k
-
= =
Vyshngkhụngph thucvo x l
6 6
15
.2C
0.25
5
Cho hỡnh chúp .S ABCD cú ỏy ABCD l hỡnh ch nht vi 3 2AB a AD a = =
TớnhtheoathtớchkhichúpS.ABCDvtớnhkhongcỏchgiahaingthng
SC v AD .
1,0
ồ
(Tvhỡnh).K
( )
HK CD K CD ^ ẻ .Khiú:
( )
CD HK
CD SHK CD SK
CD SH
^
ỹ
ị ^ ị ^
ý
^
ỵ
.
Vygúcgia
( )
SCD
v
( )
ABCD
lgúc
ã
0
60SKH =
0.25
Trongtamgiỏ cvuụng
0
: tan 60 2 3SHK SH HK a = = .Thtớchkhụi chúp
.S ABC D
l
3
.
1 1
. .3 .2 .2 3 4 3
3 3
S ABCD ABCD
V S SH a a a a = = =
0.25
Vỡ
( ) ( ) ( )
( )
, , .SBC AD d AD SC d A SBC ị =
Trong
( )
SAB k AI SB ^ ,khiú
( )
BC AB
BC SAB BC AI
BC SH
^
ỹ
ị ^ ị ^
ý
^
ỵ
m
( )
SB AI AI SBC ^ ị ^
0.25
Vy
( ) ( )
( )
2 2
. 2 3.3 6 39
, ,
13
12
SH AB a a a
d AD SC d A SBC AI
SB
a a
= = = = =
+
0.25
6
Trongmt phngta Oxy cho tamgiỏccõn
ABC
cúỏy
BC
nmtrờnng
thng :2 5 1 0d x y - + = , cnh AB nmtrờnngthng :12 23 0d x y
Â
- - = .Vit
phngtrỡnh ngthng
AC
bitnúiquaim
( )
31M
.
1,0
ồ
VTPTca
( )
: 2 5
BC
BC n = -
r
,VTPT ca
( )
: 12 1
AB
AB n = -
r
,
VTPTca
( )
( )
2 2
: , 0
AC
AC n a b a b = + >
r
.Ta cú
ã
ã
0
90ABC ACB = <
ã
ã
( ) ( )
cos cos cos , cos ,
AB BC BC CA
ABC ACB n n n n ị = =
r r r r
0.25
4
2 2
2 2
. . 2 5
145
9 100 96 0
. . 5
AB BC CA BC
AB BC CA BC
n n n n a b
a ab b
n n n n
a b
-
= = - - =
+
r r r r
r r r r
12 0 9 8 0a b a b + = - =
0.25
Vi 12 0a b + = Chn 12, 1a b = = - thỡ
( )
12 1
CA
n AB AC = - ị
r
(loi)
0.25
Vi
9 8 0a b - =
Chn 8, 9a b = = nờn
( ) ( )
:8 3 9 1 0AC x y - + - =
: 8 9 33 0AC x y ị + - =
0.25
7
Trongkhụnggian Oxyz ,cho
( ) ( ) ( )
100 , 020 , 003A B C
.Vitphngtrỡnhmt
phng
( )
P iqua ,O C saochokhongcỏcht
A
n
( )
P bngkhongcỏcht
B
n
( )
P .
1,0
ồ
Do
( )
P
cỏchu A v B nờnhoc
( )
P AB
hoc
( )
P
iquatrungim
.AB
0.25
Khi
( )
P AB
( )
( )
( ) ( )
( )
000
: 2 0
, 630 2 10
qua O
P P x y
vtpt n AB OC n
ỡ
ù
ị ị - =
ớ
ộ ự
= - ị = -
ù
ở ỷ
ợ
uuur uuur
r r
0.25
Khi
( )
P iquatrungim
1
10
2
I
ổ ử
ỗ ữ
ố ứ
ca .AB Ta cú :
( )
( )
( )
( )
0 0
000
: 2 0
3
, 3 0 210
2
qua O
P P x y
vtpt n IC OC n
ỡ
ù
ị ị + =
ớ
ổ ử
ộ ự
= ị =
ỗ ữ
ù
ở ỷ
ố ứ
ợ
uur uuur
r r
0.25
Vyphngtrỡnhmtphng
( ) ( )
: 2 0, : 2 0P x y P x y - = + =
0,25
8
Giih phngtrỡnh:
( ) ( )
( )
2 2 2 2
3
5 2 2 2 2 5 3 1
2 1 2 12 7 8 2 5 2
x xy y x xy y x y
x y x y xy x
ỡ
+ + + + + = +
ù
ớ
+ + + + + = + +
ù
ợ
.
1,0
ồ
iukin:
2 2
2 2
5 2 2 0
2 2 5 0 2 1 0
2 1 0
x xy y
x xy y x y
x y
ỡ
+ +
ù
+ + + +
ớ
ù
+ +
ợ
.
Khihcú nghim
( )
( )
1
0x y x y ắắđ +
0.25
Tathy
( )
2 2
5 2 2 2 *x xy y x y + + +
dubngkhi
x y =
thtvy
( ) ( ) ( )
2 2
2 2
* 5 2 2 2 0x xy y x y x y + + + - luụnỳngvimi ,x y ẻĂ
Tngt
( )
2 2
2 2 5 2 **x xy y x y + + + dubngkhi x y =
T
( ) ( )
( )
( )
( )
1 1
2 2 2 2
* & ** 5 2 2 2 2 5 3VT x xy y x xy y x y VP ị = + + + + + + =
Dungthcxy rakhi x y =
( )
3
0.25
5
Th
( )
3 vo
( )
2 tac:
2
3
3 1 2 19 8 2 5x x x x + + + = + +
( )
4 iukin
1
3
x -
( )
4
( )
( ) ( )
2
3
2 1 3 1 2 2 19 8 0x x x x x x - + + - + + + - + =
( )
( ) ( ) ( )
2 2
2
2 2
3
3
2 2 0
1 3 1
2 2 19 8 19 8
x x x x
x x
x x
x x x x
- -
- + + ì =
+ + +
+ + + + + +
0.25
( )
( ) ( ) ( )
2
2 2
3
3
0
1 1
2 2 0
1 3 1
2 2 19 8 19 8
x x
x x
x x x x
>
ộ ự
ờ ỳ
ờ ỳ
- + + ì =
ờ ỳ
+ + +
+ + + + + +
ờ ỳ
ờ ỳ
ở ỷ
144444444444424444444444443
( )
( )
3
2
3
0 0
0
1 1
x y
x x
x y
ộ
= ắắđ =
- =
ờ
= ắắđ =
ờ
ở
. Tha móniukin
Vyh phng trỡnh cúhainghim
( ) ( ) ( ) ( )
00 & 11x y x y = =
0,25
9
Chobasthcdng , ,a b c thamón
2 2 2
3a b c + + = .
Tỡmgiỏ trinhnhõtcabiuthc
( )
1 1 1
8 5S a b c
a b c
ổ ử
= + + + + +
ỗ ữ
ố ứ
1,0
ồ
Nhn xột:
( )
2
5 3 23
8 , 1
2
a
a
a
+
+ vimi 0 3a < < dubngkhi
1a =
thtvy
( ) ( )
2
2
3 2
5 3 23
8 3 16 23 10 0 1 3 10 0
2
a
a a a a a a
a
+
+ - + - Ê - - Ê luụnỳng
vimi0 3a < < dubngkhi
1a =
0.25
Tngt
( )
2
5 3 23
8 , 2
2
b
b
b
+
+ dubngkhi 1b =
( )
2
5 3 23
8 , 3
2
c
c
c
+
+ dubngkhi 1c =
0.25
T
( ) ( ) ( )
( )
( )
2 2 2
1 , 2 & 3
3 69
1 1 1
8 5 39
2
a b c
S a b c
a b c
+ + +
ổ ử
ắắắắđ = + + + + + =
ỗ ữ
ố ứ
Dubngxyrakhi
1a b c = = =
0.25
Vygiỏtrnhnht ca 39S = tckhivchkhi 1a b c = = =
0,25
Chỳý: tỡmravphica(1)ta sdngphng phỏp tiptuyn