Tuyển chọn và hướng dẫn giải 39 đề thử sức học kì môn Toán 12 nâng cao: Phần 1

'TTV ^

PHAM

TRONG

THU

^ G V T H P T Chuyen Nguyen Quang Dieu - Dong Thap)

uyen chon

^ thit site hoc ky

MON

TOAN

m NANG CAO i
Danh cho hoc sink tdp 12 chuang truth nangcap

,

9


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B A N

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H O C

Q U O C

Gin

Hii

N O I

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C O N G TY KHANG Vlf T

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C O N G T Y TNHH MTV
DJCH V g V A N H O A K H A N G V I | T

\

M^i nj&i itdn

B6 sach JU\EN CHON 39 DE THLf SLfC HOC KI MOM TOAN \dp 10,
11 va 12 nang cao diTdc bien soan va luyen chon diTa tren noi dung chtfcfng
trinh THPT hien hanh; bp sach toan nay giup cac em c6 dieu kien lam quen \di
cac dang de thi hoc ki d miJc dp cao. Rieng cuon 12 c6 them phan phu luc giup
cac em l\X kiem tra, danh gid, bo sung kien thtfc ve todn THPT cho minh nhkm
tao nen toan can ban viJng chac cho cac em triTdc khi chinh thuTc bxidc vao ki thi
Dai hoc, Cao dang.
Hi vpng bp sach se gop phan giiip cac em dat ket qua cao trong cdc ki thi,
dong thdi la mot cong cu ho trd cho cac bac phu huynh giup cho con em hoc tap
tot hPn.
Trong qua trinh bien soan, dij tac gia da c6' g^ng nhiftig cuon s^ch van c6 the
con nhffng khie'm khuyet ngoai y muon. Chung toi rat mong nhan di^Pc s\i gop y
chan thanh ciia cac thay, c6 giao, cac em hoc sinh de trong Ian tai ban sau sach
difdc hoan chinh hPn.
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M a so: 1 L - 1 1 5 D H 2 0 1 3
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so: 3 3 9 L K - T N / Q D - N X B D H Q G H N , c a p ngay 31/07/2013

In xong va n o p liAi c h i e u q u y IV n a m 2 0 1 3

Vectd phdp tuyen
Vectd chi phi/dng
Dieu phai chiJng minh

Yeu cau biii todn
Mat phdng
Ba't ddng thuTc
Phifcfng trinh
H$ phifdng trinh
Ba't phifdng trinh
Ve trdi
Ve'phai

VTPT
VTCP
dpcm
YCBT
mp
BDT
PT
HPT
BPT
VT
VP


7 loi khuy§n cho thi sink vi phicang phdp gidi mOt bdi thi

NhU chiing ta da bie't mon Toan la mon hoc chiem mot vi tri ra't quan trpng va then
cho't, ra't can thic't do hoc cac mon khac tiT tieu hoc cho de'n cac Idp tren. Mon Toan
giiip cac em nhan bie't cac moi quan he ve so' lUdng va hinh khong gian cua the gidi
hipn thifc. Nhd do ma cac em c6 phiTdng phap nhan thiJc mpt so mat ciia the' gidi xung
quanh va bie't each hoat dong c6 hieu qua trong ddi song. Mon Toan gop phan ra't quan
trong trong vice rcn luycn phifdng phap suy nghl, phUdng phap suy luan, phUdng phap

gidi quyet van de. No gop phan phat tricn tri thong minh, each suy nghl doc lap, linh
hoat, sang tao va vipc hinh thanh cac pham cha't can thie't cho ngifdi lao dpng nhiTcan
cii, can than, c6 y chi vu'pt kho khan, lam viec cd ke' hoach, cd ne nep va tac phong
khoa hoc.
Xua't phat tij' vi tri quan trong cua mon Toan, qua thifc le' giang day nhieu nam d
ca'p THPT. Toi nhan tha'y rang dc hoc sinh hoc to't mon Toan thi ngoai vice cac em n^m
vilng kie'n thilc trong sach giao khoa, ky nang tinh toan that tot ma con phai bie't phUdng
phap giai mot bai thi nhif the nao trong luc dang thi dc cd diem cao. Muo'n lam dU'dc
dieu nay thi sinh can phai tuan thii theo cac b\idc sau day:
1) That binh tinh trong luc lam bai thi.
2) Can doc that cham rai toan bp dc, danh gia sP bp dp de, khd cua cua cac cau,
xem nhifng cau nao quen thupc, la vdi minh.
3) Giai ngay lap ti'fc cac cau ma ban tha'y de.
4) Mot vai cau can thie't de'n sif suy nghl sau hdn, thi sinh can phai dpc ky cau hoi,
gach dirdi cac gia thie't vii ycu cau cua bai todn. Dinh hiTdng each giai, hinh dung dp
phiJc tap cua each giai de cd si/ li/a chpn diing din.
5) Trinh bay biii giai thi sinh khong nen lam tat, moi bifdc nen vie't mot dong de de
kiem tra, vi giam khao cha'm bai thi theo ba rem nen cd mot bifdc nao dd sai thi van con
diem d nhu'ng biTdc bie'n ddi dung trUde dd. Cach hay nha't la lam xong btfdc nao kiem
tra birdc ay de phat hicn ngay cho sai.
6) Trong qua trinh giai mot bai toan ne'u thi sinh gap khd khan giiJa chiifng, cd the
chiifa khodng trong tren gia'y thi de bo sung sau va nhanh ehdng chuyen sang lam cau
khac.
7) Khi da hoan tat bai thi, ne'u con thdi gian thi sinh nen dpc lai bai giai va ra soat
lai cac chi tic't da trinh bay (thong thiTdng cdc loi thi sinh hay bd sdt biTdc lam la tap xac
dinh, dieu kipn cd nghla ciia can bac chan, ham .so' logarit, doi can khi dCing phiTdng
phap doi bie'n dc tinh tich phan, loai bo nghicm ngoai lai trong phifdng trinh...) nham
hoan thien bai thi td't hcJn cho de'n he't gid.
Nhieu hpc ird tdi day dp dung 7 Idi khuyen tren da trd thanh thii khoa dai hpc cua
nhieu triTdng, nhi/ng thanh cong nha't la toi cd hpc tro thu khoa "kep" khoi A va B cua

trirdng Dai hpc Khoa hpc TiT nhicn TP. HCM va Dai hpc Y DiTdc TP. HCM nam 2011.
Chuc cac thi sinh dat ke't qua cao trong cac ki thi
PHAMTRpNG

THa

Ph^n I. B p BE THUr Sljrc HQC K I MON TOAN L6P

12

A. BQ DE THUf SlJC HQC K I I M d N TOAN LCJP 1 2
D E SO 1

DE

THLT SLTC

HOC K i I M O N TOAN L 6 P 1 2

Thdi gian lam bai: 120

pliut

Cau I. (3,0 diem) Cho ham .so y = -x + 1
2x-l

1. Khao sat siT bie'n thien va ve do thi (C) cija ham .so da cho.
2. ChtJ-ng minh rSng vdi moi m diTdng lhang y = x + m luon ciCt do thi (C) tai
hai diem phan biet A va B. Goi k ^ k j hin liTdt la he so goc cua cac tiep tuyen
vdi (C) tai hai diem A va B. Tim m de tdng k, + k2 dat gia trj Idn nha't.

C&u 11.(2,0 diem)
1. Tim gia Irj idn nha't, gia tri nho nha't cua y = -2sin-'x + 3cos2x -6sinx.
2. Tinh gia tri bieu iMc T = ^' '
3. Giai phu-dng trinh 6.9" -13.6^ +6.4" =0.
Cau III, (2,0 diem) Cho hinh chop ti? giac deu S.ABCD c6 canh day la a, canh
ben la aS. Tinh the tich khoi chop S.ABCD.
Cau l\. (2,0 diem)
1. Cho ham so' y = In

vdi X > -— . Chiang minh rang xy' +1 = e^
3
2 + 3x
2. Giai ba't phu-dng trinh log4(2x^ +3x +1) > logT(2x + 1).
CSu V. (1,0 diem) Cho ham so y - X- - 2 x + 2 m - l cd do thi (Cni).Tim m de
x-1

hiim so cd ciTc dai, ciTc tieu va khoang each giffa hai diem ciTc dai, ciTc lieu
bang 6.
;
DAP A N THAM KHAO

-4

Cau
Diem
Dap an
I
1. (2,0 diem) Khao .sat siT bie'n thien va ve dd thi (C) ciia ham...
(10
0,25

diem) a) lap xac dinh D = R \ [2


TuySn chon 39

Cty TNHH IVITV DWH Khang Vigt

th& sijfc hpc kl mOn Toan I6p 12 NSng cao - Phgm Trgng Thu

V i A' = (m +1)^ +1 > 0, Vm e K. Suy ra d luon luon cii (C) tai

b) Su-bien thien:

hai diem phan biet vdi moi m.

1

•<0, V x ^ - .
- Chieu bie'n thien: y' = - (2x-l)'
- Ham so nghich bien tren cac khoang

Goi X , va X j la cac nghicm ciia phUdng trinh (*), ta c6

1
va

-

- Gidi han va tiem can:
©

© hm y

x->-i»

=-00,

=+oo

=> tiem can diJng: x =—

Theo dinh li Vi-et

Xj +

X2

= -m,

XjX2

= - — ^ — , suy ra
0,25

kj + k 2 = - 4 m - - 8 m - 6 = -4(m + l)^ - 2 < - 2 .
Suy ra gia trj Idn nhaft kj + k2 la - 2 khi m = - 1 .

Bang bien thien:
—00

1. (1,0 diem) Tim gia trj \6n nhS't, gia trj nho nhS't...
II
(2,0
Ta C O y = -2sin' x + 3 ( l - 2 s i n x)-6sinx
diem)
= - 2 s i n ' ' x - 6 s i n ^ x - 6 s i n x + 3 (1)

+00

2
y'
y

(4x,X2 - 2 ( X j + X 2 ) + l f

2

v2.

x

0,25

4(x, + X 2 ) ^ - 8 x , X 2 - 4 ( x , + X 2 ) + 2
2

iim y

(2x2-ir

(2xi-ir

Hm y = lim y = -—=> tiem can ngang: y = - —•
\-*+'x

0,25

—

1

+00

1

2
-00

2

c) Do th\) qua diem A ( l ; 0 ) , B(0; - 1 ) .

0,25

Dat u = sinx, - 1 < u < 1
Ta CO (1) vie't lai y = -2u'' - 6 u ^ - 6 u + 3.
0,5

y' = _6u^ - 1 2 u - 6 = -6(u2 +2u + l ) = -6(u + I)2 <0,
y' = Oc>u = - l .

Xct y ( - l ) = 5, y ( l ) = - l l .
Suy ra maxy = 5 khi sinx = - 1 0 x = - — + k27r, k e Z .
xeK
2

-I

o
-1

0,25

miny = -11 khi sinx = 1 0 x = —+ k27t, k e Z .
xeR
2
2. (0,5 diem) Tinh...
•log,8-llog^l25 = log,8-ilog,125
0,25
= log,8-log,125 =

log,:J^.

2. (1,0 diem) ChiJng minh rang v(Ji mpi m...
Phifcfng trinh hoanh do giao diem cua do thi ham so da cho va
—x +1
diTdng thang d: y = x + m la
=x+m
2x - 1
c : > 2 x 2 + 2 m x - m - l = 0(*)
A

UJ

V 8

0,25

<

f

5


Cty TNHH MTV DWH Khang Vijt

Tuyin ch(?n 39 dg tht> sijfc hpc kl mOn Toan I6p 12 Nang cao - Phgm Trpng Thu
3. (0,5 diem) G i a i phifcTng trinh...

e

v2x
Ta c6: 6.9" - 1 3 . 6 " + 6.4" = 0 <=> 6.

Datt =

.2,

-13.

v2.

v2.

+ 6 = 0(*)

2. (1,0 diem) G i a i ba't phrf(/ng trinh...
"x<-l

, t > 0. PhiTcfng trinh (*) trd thanh

6t^ - 1 3 t + 6 =

v2y

v2y

X>
Taco

v2y

Ill

T i n h the tich khoi chop S . A B C D

(2,0

<=>2x^ + 3x + l > 4 x ^ + 4x + l

G o i O la tarn c u a hinh vuong A B C D . V i S . A B C D la hinh chop
A S A O v u o n g tai O c6

T =
5a^

a
V
(1,0
diem)

2

aVlO

Dod6V,^3C^=-.SO.S^3C^

[

2

.

T i m m de ham S(Y c6 ciic dai, ci/c t i e a . .
Tap xacdinh D = 1R\{1).

2

a

=•

o -

A ' = - 2 + 2m > 0

AB = 6 o

0,5

<=> m > 1.

A B ^ = 36 <=> ( X y - x ^ ) ' + ( y B - y A ) ' = 3 6

c : > ( X p - x ^ r + ( 2 x , 5 - 2 - 2 x ^ + 2 ) 2 =36

2 + 3x

5(x,j - x ^ r = 36 o

2 + 3x

(Xjj + x ^ )^ - 4xyX^ =

0,5
y

2 + 3x

(2 + 3x)^

2 + 3x

(x-1)^

Goi A ( x ^ ; y ^ ) , BCxjj; y n ) la hai diem cifc t r i

2 + 3x

Ta CO x y ' + 1 = x

(x-l)2

g(l) = l - 2 + 3 - 2 m ^ O

^

6

g(x)

n g h i c m phan bict khac 1
(dvtt).

1. (1,0 diem) ChuTng minh...
2

x^ - 2 x + 3 - 2 m

H a m .so c6 ciTc dai va ciTc t i c u k h i va chi k h i g ( x ) = 0 c6 2

1 a^/io 2 a-'Vio

Dao h a m y ' =

0,25

D a o ham y'

'

0,5

K e t hdp (*) ta du'rtc tap nghicm cua bat phu'cfng trinh la

so^ = S A 2 - O A 2

IV
(2,0
diem)

1
—
2

<ii> 2x'^ + X < 0 <=>-—< X < 0.
2

tiJ g i a c d c u n c n S O 1 ( A B C D ) => S O 1 O A .

3

0,25

log4(2x^ + 3x + l ) > l o g 4 (2x + l ) ^

o

•SO =

(*)

log4(2x^+3x + l)>log2(2x + l )

V a y nghicm cua phifdng trinh da cho la x = ±1.

= (aV3)2

« x > - i

2x + l > 0

C^X=±1.

_2 _
~ 3 "

6

2 x - + 3x + l > 0 ^ _

Dieu kien •

" 2

0 »

0,25

Ttr (1) v a (2) suy ra dpcm.

_3

diem)

(2)

2 + 3x

+ 1=

2 + 3x

(1)

- 4(3 - 2m) = — o
5

m = — ( t h o a man).
10

7


Tuyin chpn 39 6i

D±

S6

IM

Cty TNHH MTV DWH Khang Vi$t

sOc hoc ki mOn ToAn Iflp 12 Nang cao - Pham Trpng Thu
D E

2

THlIr SOC

H O C

Kl

I M O N

T O A N

L6P

12

Thdi gian lam bai: 120 phut

Cau I. (3,0 diem) Cho ham so y = x"^ - 2x^ + 4 (I).
1. Khiio sal siT bicn Ihicn va ve do Ihj (C) cua ham s6' (1).
2. Tim cac giii Irj cua m de phUtlng trinhx'* - 2 x " - l o g 2 m = 0 c6 4 nghiem
phan bicl.

C&u U. (2,0 diem)

1. Tim gia iri Idn nhiil ciia ham so' f(x) = x - ' + 3 x ^ - 7 2 x + 90 t r e n [ - 5 ; 5].
2. Tinh gia trj bieu ihufc A =

49

3. Giai bal phi/dng irinh 2-""" -21.2"^-"^^'+ 2 >().
Cfiu HI. (2,0 diem) Cho hinh chop S.ABC c6 day ABC la lam giac vuong tai B,
SA 1 (ABC). SA = iiyfl, BAC = 30", BC = a, M la trung diem ciia SB. Tinh the
tich cua khoi tiJ dien MABC.
Cau IV. (2,0 diem)
1. Cho ham so y = c''' sin5x.ChiJng minh ring: y''-4y' + 29y = 0 (*).
3 \ 2 y =972
2. Giai he phi/dng Innh l o g ^ ( x - y ) = 3
Cau V. (1,0 diem) Cho ham so y = -

-1
0

0 - 0

-1
0 +

0
0

'

1 31^

+00

Dodo
- Ham so dong bien Ircn moi khoang ( - 1 ; 0) va (1; + oo).
- Ham so nghich bien tren moi khoang ( - oo; -1) va (0; 1).
• Cyc trj:
- Ham so dat cifc lieu x = ±1 va y^.^ = y( ± 1) = 3.
- Ham so dat ciTc dai x = 0 va y^^ = y(0) = 4.
• Gioi han: lim y = + o o ; lim y =+CXD.
Bang bien thien
X

- 0 0

+00

c) Do thi (C): Qua cac diem

+00

-

0

+

+00

,(±2; 12).

^ ^ c 6 do thi (C). Viel phiTdng Irinh
x-1

tie'p tuyen ciia (C) c6 he so' goc la -1 .
D A P

A N

T H A M

K H A O

Cau
Dap dn
Diem
I
1. (2,0 diem) Khao sat si/ bien thien va ve do thj (C) cua ham...
(3,0 a) Tap xdc diiih D = R.

0,25
diem) b) Sir bien thien:
• Sir bien thien:
0,5
•?
X ~ 0
• Chieu bien thien y'= 4x(x" -1), y'= 0 o
8
Lx = ±i

2. (1,0 diem) Tim cac gia trj cua m de phtf(yng trinh...
Ta CO x"* - 2x- - log2 m = 0 <=> x** - 2x- + 4 = logj m + 4 = k.
De phmJng trinh da cho c6 4 nghiem phan biel<=> diTdng thing
y = k cat do thj (C) tai 4 diem phan biet . DiTa vao do thj (C)
cua ham so , la c6 3 < k < 4
<=> 3 < log, m + 4<4<=>-l< log2 m < 0 <=> 1- < m < 1.
9


Tuy^n chpn 39 dg thif silc h(?c kl m6n Join I6p 12 Nang cao - Phgm Trpng Thu

II

(2,0
diem)

1.(1,0 diem) Tim gia tri lXet ham so g(x) = x"^ + 3x^ - 72x + 90 I r e n D - [ - 5 ; 5]
x= 4eD
Ta CO g'(x) = 3x^ + 6x - 72, g'(x) = 0 <=>

_x = - 6 g D
Ta xet g(5) = -70, g( - 5 ) = 400, g( 4) = --86
=> max f(x) = max | g(-5) g(4) g(5) - 400 khi x = - 5 .
1-5;51
1-5;5]^
2. (0,5 diem) Tinh...
--sin logy4

'+

4 2

logy 4 ^^l-21ogy4 ^

Cty TNHH MTV DWH Khang Vi$t

Dien tich tam giac ABC la S^^^ = ^ AB.BC =
0,5
0,5

Goi H la trung diem cua AB thi
MH//SA z:^ MH 1 (ABC) va MH =

SA

(dvdt).

aV2

3

21og,,4

3

0,25

.logy 4 ~ 4 '
25-^'°g:252 ^ 25'°^5-'

=

52l«g52

^ 5log54

^ 4_

4^l(.g7 2 ^.^2 logy 2 ^.710^74

—+ 4 •4 = 19.
4

3. (0,5

0,25

Giai bat phUcyng trinh...

Ta CO 2^"^' - 21.2-*2''+3) + 2 > 0 (*)
^22x+l_2i2-((2x+lH2l^2>0

Dal u = 2'''"^^ u > 0. Bat phiTdng trinh (*) trd thanh
u

21

<=> u 4u
<

0,25

+2 > 0 o 4 u ^ + 8u-21>0.
2

(loai) hoac u > — (nhan)
•
•
2
'

^ 2 ^ " + ' >|<=>2x + l > l o g 2 ^ o x > ^ l o g 2 3 - l .
Vay tap nghicm cua bat phufdng Irinh da cho la

III

0,25

10

Tinh the tich ciia kho'i tu" di^n MABC.

Trong tam giac ABC vuong tai B co
BAC = 30", BC = a =:> AB = BCtan30° = aVJ.

a^V6
(dvtt).
12

IV 1. (1,0 diem) ChiJng minh...
(2,0
diem ) T a c o y' = (e^'') sin5x + e^''(sin5x)'
= 2G^^ sin 5x 4- Se^'^cosSx = e^^ (2 sin 5x 4- 5cos5x)
y" = ^e^") (2sin5x 4- 5cos5x) 4- e^''(2sin5x 4- 5cos5x)'

T = — logo 3 - 1 ; 4-00
(2,0
diem)

The tich cua khoi chop M.ABC la
1 ayfl
V A B C = 3MH.S^ef,-- —

0,5

= 2e^''(2sin5x 4- 5cos5x) + e^''(10cos5x - 25sin5x)
= -216^" sin5x + 20e^'' cos5x.

Dodo
VT(*) = - 2 le^^ sin5x + 20e2'' cos5x - 40^" (2sin 5x 4- 5cos5x)
4-290^" s i n 5 x = 0 .

2. (1,0 diem) Giai h$ phxidng trinh..
Dieu kien: x > y.


luyen cliyii M

de l l i i j ;.ut ligc ki iiion l u a i i lup U' f^jiig

Ta

CO

<

3X

2y

=972
^ <=>

logy:^(x-y) = 3

['hjni liuiig

llii/

Cty TNHH MTV

<=>

= y + 3

[ 3 y + - \2
<=> <^
x-y=3
[x = y + 3

Tinh: A =

(1,0

C f i u I I I . (2,0 diem)
0,25

V i e t phifc/nK t r i n h t i e p t u y e n c i i a ( C ) c 6 h $ s o ' g o c l a - 1
PhiTdng i r i n h tie'p luyc'ri c 6 d a n g d : y = - x + b

x-1

0,25

3. G o i M la t r u n g d i e m c a n h S A . T i n h t h e t i c h ti? d i e n

MABC.

(2,0 diem)

;

1

1. C h o h a m so y = I n ( c o s x ) . C h i y n g m i n h y ' t a n x - y " - 1 = 0.
2x + y = 4
2. G i a i he phu-tJng t r i n h
l o g ( x + y ) - l o g x = 21og3

c6 n g h i e m
x-1

;

+ b

d tie'p x i i c ( C ) <=>
x^-3x + 4

C h o h i n h c h o p S . A B C D c 6 t a t ca c a c c a n h d e u b k n g a.

1. C h i J u g m i n h r a n g S . A B C D la h i n h c h o p d e u .

C&ulV.
= -X

t h e o a va b.

2. T i n h t h e t i c h h i n h c h o p d o .

diem)
x^ - 3 x + 4

Iog5l20

3. G i a i phu-dng t r i n h l o g 3 ( 9 ' ' + 8) = x + 2.

_ (nhan).
x = 5

V a y n g h i e m c u a p h i f d n g I r i n h da c h o la ( x ; y ) = ( 5 ; 2 ) .
V

Khang Vl^t

2. C h o l o g o S = a, log3 5 = b.

3 \2

0,5

X

DWH

= (-x + b)'

|. •

0,25
x^ - 3 x + 4
<=>

x-1

= -x + b

C&u V . (1,0 diem)

(1)

T i m m d e h a m so y =

c o cU^c I n .

x +1

CO n g h i e m

x^-2x-l

= -1

DAP

(2)

AN

THAM

KHAO

(x-ir
T i f (2) o

2x'- - 4 x = 0 o

Cdu

X = 0 hoSc x = 2.

T h e X = 0 v a o (1) ta diTdc: b = - 4 , the x = 2 v a o (1) ta diTdc:

0,25

I
(3,0

b = 4.
V a y C O h a i t i e p t u y e n y = - x - 4 , y = - x + 4.

0,25

diem)

Dap
1. (2,0 diem)

Diem

K h a o s a t si/ b i e n t h i e n v a v e d d t h i ( C ) c u a h a m . .

a) Tap xdc diiih
b) Sir bien

an

0,25

D = M.

thien:

• C h i e u b i e n t h i e n : T a c6 y ' = 3 x - - 6 x ; y ' = 0

D E

SO

D E

3

THLT S Q C

HOC

Ki I M O N

T O A N L6P

12

Thdi gian lam bai: 1 2 0 phut

X

y'
C a u I . (3,0

diem)

C h o h a m so

+

0

2
-

y = x ' ^ - 3 x - + 4 (1)

- H i i m s o ' d d n g b i e n t r e n m o i k h o a n g (-oo;
- H a m so n g h i c h b i e n t r e n k h o j i n g (0; 2).

2. G o i d la du'cfng t h a n g d i q u a d i e m A ( 3 ; 4 ) va c 6 he so g o c la m . T i m m d e

• Circ t r i :

c u a ( C ) t a i M va N v u o n g g 6 c v d i n h a u .

t r i Idn

x = 2'
+00

0,5

0

I n n h o nha't c u a

+oo).

- H a m so d a t ciTc d a i t a i x = 0, y^^.^ = y ( 0 ) = 4.

• C a c g i d i b a n t a i v o ci/c:
nha't, g i a

0) va (2;

0,5

- H a m s o ' d a t cifc t i e u t a i x = 2, y ^ - j = y ( 2 ) = 0.

diem)

1. T i m g i a

x = 0

D o do

du'cing t h a n g d ci(t d o t h i ( C ) t i i i 3 d i e m p h a n b i $ t A , M , N sao c h o h a i tie'p t u y e n

l i m y = -oo ,

lim y =

+oo.

y = 2x^+3x^-12x + ltren

[-1;3].
12

0

-00

1. K h a o sat s i T b i e n t h i e n va v e d o t h i ( C ) c u a h a m so ( 1 ) .

C S u I I . (2,0

o

,

•

'1^

13


Cty

Tuyln chgn 39 66 this sufc hpc kl mOn Toan Idp 12 Nang cao - Phgm Trgng ThU

0

-00

y'

DWH Khang Vi^t

2. (0,5 diem) Tinh...

• Bang hien t h i c n :
X

TNHH MTV

+

2

0

-

0

.

+00

+

A — ^ ^ ^ ^ ^ _ ^ ^

+
120 = logs 2^-53 = 31og5 2 + l o g , 5 + l o g j 3
"

0,25

=

^
+1+
'
log2 5
log3 5

0,25

.y
f3

c) Do tlti [Q) qua d i e m iM 3 ; 4 ) , B ( l ; 2 ), C ( - l ; 0).

y

>

4

/^
-u

o

• A=

\\l
1

2

,

-+1

0
+ -

1
•-ri=

3b + ab + a
=

la
bj ill
ab^
3. (0,5 diem) G i i i i phifc/ng trinh...

i

Ta CO l o g ^ ( 9 ' ' + 8 ) = x + 2<::>9''+8 = 3"""-

(*)

Dat t = 3^^, t > 0. PhiTdng Irinh (*) t r d thanh
<=>
0,5

=>
[t = 8

X

- 9t + 8 = 0

0,25

<=>
[3^=8

[x = log3 8

V a y n g h i c m ciia phifring Irinh da cho la

3

0,25

77=

rx=o

0,25

[x = log3 8

III

1. (1,0 diem) Chrfng minh rang S . A B C D la hinh chop deu.

(2,0

Tur giac A B C D la hinh Ihoi v i c6 cac canh deu b ^ n g a. G p i 0

diem)

la giao d i e m cua A C va B D . T a m giac SAC can ( v i c6

0,25

SA = SC = a ) va 0 la trung d i e m A C nen SO 1 A C .
S

2. (1,0 diem) Tmv m de dififng thang d cat do thj ( C ) tai...
Phu'dng trinh hoanh do giao d i e m cua d va (C) la
x-^ - 3 x ^ + 4 = m ( x - 3 ) + 4 o ( x - 3 ) ( x ^ - m ) = 0.
T h c o de bai ta c6 m > 0, m ?t 9 va y'cVm ).y'( - V m ) = - 1

0,5
0,25

=> '3m -6N/m)(3m + 6Vm) - - 1 <=>9m^ - 3 6 m + 1 = 0
m =
<=>

6 + >y35
^

0,25

(thoa man).

———Y

6-V35
m =
II
(2,0
diem)

14

T a m giac SBD can ( v i c6 SB = SD = a ) va O la trung d i e m

3
1. (1,0 diem) T i m gia trj \(in nha't, gia tri nho nhat...

BDnen S O I AC.

.Tapxacdinh:D = [-l;3]
• y ' = 6x^ + 6x - 1 2 , y ' = 0 o

Suy ra SO .L m p ( A B C D ) .
X

1 hoac x = - 2 g D.

0,5

• Ta CO y(3) = 46, y ( l ) = - 6 , y ( - 1 ) = H .

0,25

•Vay max y = 4 6 k h i x = 3 , min y = - 6 k h i x = l .
xe[-l;3i
X6[-1;3]

0,25

V i SA = SB - SC = SD nen O A = OB = OC = O D va do do
A B C D la hinh vuong.

0,25
0,25
0,25

V a y S . A B C D la hinh chop deu v d i difcfng cao la SO.
15


'uy6'n chpn 39

6

dg

\hCl

SLfc

hpc ki man To^n I6p 12 Mang cao - Phgm Trqng ThJ

Cty TNHH MTV DWH Khang Vi^t

2. (0,5 diem) Tinh th^ tich hmh chop do.

V

V i ABCD la hinh vuong canh a nen OA= ^

• Trong tarn

giac vuong SAO ta c6 SO = VsA^ - O A ^ = Ja^ - — =

(1,0
diem)

Tim m de ham so'co c\ic dai va c\ic tieu...
Tapxacdinh: D = K \ { - I }
x^ +2x + 2m + l
Dao ham: y' = (x + i r

—

g(x)

0,5

(x + 1)^

Ham so da cho c6 cifc dai, cifc tieu o g(x) = 0 c6 hai nghiem
a^f2

Dod6V3^3C^=-.SO.S^3CO

a2=^
3 2
3. (0,5 diem) Tmh the tich ti? dien M A B C .

(dvtt).

phan biet khac - 1 o

'A' = - 2 m > 0
g ( - l ) = l - 2 + 2m + 1^0

« m <0.

0,5

Ke M H J_ mp(ABCD) thi M H la du-cJng cao cua hinh tiJ dien
MABC va M H =

so

aV2

OE

Dod6V^^3^=-.MH.S^3C-^

IV

1 a^/2

a^

a^V2
24

SO

4

DE

T H C T SCTC H O C Ki I M O N T O A I S I L 6 P 1 2
Thdi gian lam bar. 120 phut

(dvlt).
Cflu I . (3,0 diem) Cho ham so y = 2x3 - 3^2 ^ ^ ^

1. (1,0 diem) ChiJng minh...

1. Khiio sat su" bicn thicn va vc do thi (C) cua ham so (1).

^
, , (cosx)'
Ta C O y =
—- =
cosx

2. Tim phu'dng trinh cac du'dng thdng qua diem A

sinx

= -tanx

cosx

19^
12

; 4 va tie'p xuc vdi do

thi (C) cua ham so'.

y =•

C&ull. (2,0 diem)

COS^ X

1. Tim gist iri Idn nhat, giti trj nho nha't cua y = f(x) = x^ + e"''^' tren [-1; 1].

=> y' tan X - y" - 1 = - tan^ x + — \
1
cos x
= -tan^ x +1 + tan^ x - 1 = 0 (dpcm).
2. (1,0 diem) Giai h$ phrfcfng trinh...

2. Tinh gia tri bieu thiJc: A = logVlO + InVe - In —

log. 9 '
3. Gicii bat phu-cfng trinh 4x^ + x.2'*

Dieu kicn: x > 0 va x ?t - y .

21og2 3

+ 3.2^" >x^.2''" +8X + 12.

CSu I I I . (2,0 diem) Cho hinh chop S.ABC c6 day la tam giac vuong can tai A c6
2x + y = 4
Ta

AB = AC = 2a. Mat ben qua canh huyen vuong goc vdi mat day, hai mSt con lai

CO

<=>

log(x + y)^ - logx = 21og3

tao vdi day mot goc 30". Tinh the tich khoi chop.

y = 4-2x

Chuiy.

<

1. Cho ham so y = x'^e^"'-^". Chu-ng minh rang: xy' - y(12 + 2013x) = 0.

log(4 - x)^ = logx + log9 = log9x

x^ -17x + 16 = 0

<=>

X

=1

X=:16

o

ly-2
'x = 16
y = -28

Vay he phu'dng trinh da cho c6 nghiem (x; y) la:
^' ? - - r ; ^

:(1;2);/(1,6; - 2 8 ) .

log^x + log^y = 2

x =I

y = 4-2x

y = 4-2x

(2,0 diem)

2. Giai he phqdng trinh
x^y-2y + 9 = 0

(nhan).
Cau V. (1,0 diem) Cho ham so y =

x +2

CO do thi (C). Tim cac diem P, Q

x+1

thupc hai nhanh cua (C) sao cho do dai PQ nho nhat.


Tuye'n cligii 39 de tl)U LUC lioc ki 111611 loan iup '. Nang cao - Phgm Trgng ThU

Cty TNHH MTV DWH Khang Vi$t

D A P A N THAM KHAO

3. (1,0 diem) Giai ha't phif(/ng trinh...

Cdu

Dap an

I

1. (2,0 diem) Khao sat sir bicn thien va ve do thi (C) cua ham...

Diem

Ba't phU'dng trinh da cho tu'dng du'dng
4 ( x 2 - 2 x - 3 ) > 2 ' ' ^ x 2 - 2 x - 3 ) <=>

(3,0 Doc gia tiT giai_cach giai tiTdng tiT cau I . l de so' 3.
diem) 2. (1,0 diem) Tim phrfcfng trinh cac dtft/ng thang qua diem...

Ta CO hai tracing hdp
-2x-3<0

Dirdng thing d qua A vdi he so goc k c6 phuTdng trinh
y =k x
d

TrUdn^ hap 1 :
0,25

+4.

2^ - 4 > 0

7

r

19^

• 2x^-3x^+5 = k X
V

+4

0,25

(1)

3

<=> V2 < x < 3 .

x^-2x-3>0

x < - l hoacx>3

2''^ - 4 < 0

x2<2

-72


Vay tap nghiem cua bat phu'dng trinh da cho la

(2)

T = (-V2; - l ) u ( V 2 ; 3).

The k tir (2) vao (1) ta dmc:
2 x - ' - 3 x ^ + 5 = ( 6 x ^ - 6 x ) x - — +4
0,25

c^8x^ - 25x^ + 1 9 x - 2 = () o ( x - l ) ( 8 x ^ -17x + 2) = 0

Tinh the tich Ithoi chop S.ABC
Ill
(2,0
Ke SO 1BC
SO l ( A B C )
diem)
( v i ( S B C ) n ( A B C ) = BC).

<=> X = 1 hoac x = 2 hoSc x = - •
8
• Vdi X = 1 => k = 0. Tiep tuyen dj : y = 4.

O tren AB va AC.

• Vdi x = 2=> k = 12. Tiep tuyen d j :y = 12x-15•

Ta

Goi H, K Ian lu'dt la hinh chie'u cua

CO

0,25

l
,
21
.

,
21
645
• V d i X = — => k =
. Tiep tuyen d^ : y =
x+
8
32
•
32
128
1. (0,5diem) Tim gia trj Icjfn nhat, gia trj nho nhat...

A C l SO

S K I AC

A C l OK

O K I AC

Do do ((ABC), (SAC)) = SKO = 30°

II
(2,0 Taco r(x) = 3x2+20^"+'
diem)
V i r ( x ) > 0 , V x e [-1;11.

Tu-dng tiT ta c6 SHO = 30°.
Ma OK =

0,25

Taco 0 K =
max l"(x) = 1(1) = 1 + e^

min f(x) = f ( - l ) = - 1 + - •
xe|-l;l|

va OH = 2?:^
BC

BC

Suy ra, f(x) dong bicn trcn [ - 1 ; 1 ].

xel-l;ll

<

12 ^

6x2-6x = k

Dodo

-1 < X
<=> i
x2>2

Trui'fnf' hop 2:

la tiep xuc vcfi (C) khi va chi he phu'rtng trinh sau c6 nghiem:

f ^

_ 2 x -3)(2''^ - 4 ) < 0

0,25

AR

= ^ 0 la trung diem BC.

a
= a; SO = OK tan 3 0 ° = - p .

S

2
The tich cua khoi chop S.ABC la:

Q

2. (0,5 diem) Tinh...
1
'
'
1 1
. log^/lO + l n ^ y i ^ - l ^ - = logl02 +lnc2 - l n e ~ ' = - + - + 1 = 2.

e
2 2
^ 21og23_ 21og23 _ 2 1 o g 2 3 _ ^
0,25
Vay A = 4.

V. ARC = - SO.S, HP = - SO. AB. AC = - •

0,25

S.ABC

IV
(2,0
diem)

3

ABC

• 2a.2a = — a ^ (dvtt)
9

^

1. (1,0 diem) Chitng minh...
y' = 12x" .e^"'^" + x'^2013.e2'"-''' = x".e2"'3''(12 + 2013x)
•xy'=x'^e^'"''''(12 + 2013x).
Vay x y ' - y ( 1 2 + 2013x) = 0.

18

^

'


TuyS'n chpn 39 6i thi( sufc hpc ki mOn To&n I6p 12 MSIng cao - Ph^m Trpng Thu

2. (1,0 diem) Giai h§ phtfcfng trinh...
Dieu kien x >0, y >0.
Bien doi he phiTdng trinh da cho thanh
log^(xy)::=2
xy=9
(1)
x2y-2y + 9 =0
[ x - y - 2 y + 9 = 0(2)
Tir(l) suy ra y = - (3). The (3) vao (2) ta diTdc x^ + x - 2 = 0.
X

Giai phi/dng trinh nay ta c6 x = 1 va x = - 2 (loai).
Vay he phifdng trinh da cho c6 nghiem (x; y) = (1; 9).
V Tim cac diem P, Q thuQc hai nhanh cua (C) sao cho do dai...
(1,0 T a c 6 y = ^ = l + '
diem)
X + 1
X+ 1
Cho P -1 + a; 1 + - , Q - l - b
la hai diem thuoc hai
l

a;
nhanh cua (C) vdi a, b la0cac so'di/dng. Ta c6:
1
(I p
- + — = (a + br 1 + — PQ^ = (a + b f +
[ (ab)- J

Ap dung BDT Co-si:
P Q 2 =(a + b)2
1+ - 1

(ab)2

SO

5

tren

''••2

2. Trnhgia iri bie'u ihtfc: A = 27'°''% Si""''+9"°'"'.
3.

Giai phu-dng trinh iogjyCx^ - 5 x + 6)^ = - l o g — — +Iog9(x-3)^.

C&u III. (2,0 diem) Cho lang tru ABC.A'B'C'co A'.ABC la hinh ch6p tam giac
deu canh AB = a, canh ben A'A = b . Goi a la goc giiJa hai mat phang (ABC)
va (A'BC). Tinh tan a va the tich khoi chop A.'BB'C'C .
CHulW. (2,0 diem)

l . C h o h a m s o y = e~'' sinx.GiaiphU'dngtrinh y''-i-4xy'-i-3y = 0.
x~-4x + y-t-2 = 0
2. Giai he phi/dng trinh 21og ( x - 2 ) - l o g y = 0 ( ^ ' y ^
2

CSu V. (1,0 diem) Cho ham so y = ^

V2

^ c6 do thi (C). Tim m de diTdng

x-1
thang d c6 phifdng trinh y = - x + m c^t (C) tai hai diem phan biet.
DAP A NTHAM KHAO

> (27^)2.2.—= 8.

ab

Dodo minPQ = >/8 <=>a = b = l.
Vay hai diem can tim la P(0; 2), Q(-2; 0).
DE

Cty TNHH MTV DWH Khang Vi?t

DE THCT SOC H O C Ki I M O N T O A N LdP 1 2
Thdigian

lam bai: 120 phut

Cau I. (3,0 diem) Cho ham so y = x^ - 3x + 2 co do thj la (C).
1. Khao sat sir bien thien va ve do thj (C)cua ham so tren.
2. Goi d la du'cJng th^ng di qua diem A(3; 20) va c6 he so' goc m. Tim m de
di/cJng thang d cat (C) tai ba diem phan biet c6 hoanh do Idn - 2.
Cau 11. (2,0 diem)
1. Tim gia tri Idn nha't, gia trj nho nhat cua y = f(x) = - ^ x ^ + x + ln(l - x)

Cdu
Dap an
I
1. (2,0 diem) Khao sat siT bien thien va ve do thi (C) cua ham...
(3,0 Doc giii tif giai_cach giiii tu'dng tif cau 1.1 de so 3.
diem) 2. (1,0 diem) Tim m de difc/ng thang d cit (C)...
Du'dng thang d co phi/dng trinh y = m(x - 3) + 20.
PhiTdng trinh hoanh do giao diem cua d va (C) la
x^ - 3 x + 2 = m ( x - 3 ) + 20 c ^ ( x - 3 ) ( x ^ +3x + 6 - m ) = : 0
rx-3=o
X- + 3 x + 6 - m - 0 ( l )
De d cat (C) tai ba diem phan biet co hoanh do Idn - 2 thi PT
(l)phai CO hai nghiem phan biet Idn hdn - 2 va khac 3.
Dat t = x + 2thi(l)trc( thanh f(t) = t^ - 1 + 4 - m = 0 (2)
YCBT thi PT (2) CO hai nghiem duTdng phan bi^t khac 5

Diem

0,25
0,25
0,25
0,25
21

Cty TNHH MTV D W H Khang Vigt

Tuy^n chqn 3 9 dg thil sutc hgc ki mfln Toan Idp 12 NSng cao - Phgm Trgng Thii

A = 4m -15 > 0

N e u l < x < 2 thi

S = l >0
P
= 44 -- m
p =
m>()
r(5) - 24 - m
II
diem)

J5
— < m < 4.
4

o
;t

V a y phiTdng trinh da cho co tap nghiem la ^ = -j - j

0

1. (0,5 diem) Tim gia tri l
I

f'(x) = O c : > - x + l

Tinh the tich khoi chop A . ' B B ' C ' C .

(2,0

Gpi H la t a m cua A A B C vii M la trung d i e m cua B C .
Do

A ' . A B C la chop tam giac deu n c n A ' H la diTdng cao cua

hinh chop A ' . A B C dong thdi cung la chieu cao cua hinh lang

=0 ox2-2x=0<^

1-x
T i n h 1(0) = 0; l ( - 2 ) = - 4 + In3; f

x =0
X = 2 (loai)

= --ln2.
8

BCIAM

iru A B C . A ' B ' C ' . T a c o

BCl(A'AM)

BCIA'H
=> B C l A'M
Suy ra goc giifa m p ( A B C ) va m p ( A ' B C ) la A ' M A = a.

max l"(x) = f(0) = 0;

min

III
diem)

T a CO r ' ( x ) = - X + 1 - 1-x

Dodo

3 x ^ - 1 4 x + 15 = 0 c i > x = - -

l"(x) = r ( - 2 ) = - 4 + ln3.

2.(0,5 diem) Tinh...
3

.27'"83^^=3-^'«g3f^

/3'°83f^

=6-^=216.

-5^=625.
2
2 aVJ a\/3
AH = - A M = - - ^ - = - ^
3
3
2
3
1
aN/^
HM = - A M = — ^
3
6

\
Vay

A = 2 1 6 + 6 2 5 + 16 = 857.

•A'H = V A ' A ' - A H ^

3. (1,0 diem) Giai phtfc/ng trinh.,

-^V9b2-3a^

lDicu kicn
A A ' H M vuong tai H nen tana =

x>3

A'H

2V 3 b 2 - a 2

HM

PhiTcing trinh da cho tiTdng diTdng
log^^x^ - 5 x + 6J = log3

+ log3|x-3
V

2

<=> X

\

x -

-5x +6 =

I
Ncu

x-3

^

,

( X " - 5x + 6 j = log3 ^ x - P
->

SABC^BC- AM=

The tich k h o i chop A ' . A B C la
x-3

(*)

2

1
V.=-SABCA'H = -.
3
3

c^S 1

2
a •V3b^a2
9b^-3a^ =
12

The tich k h o i 13ng tru A B C . A ' B ' C la

x > 3 t h i (*) < i > x^ - 6x + 9 = 0 (v6 n g h i c m ) .
2:


Tuy§'n chpn 39

Cty TIMHH MTV DVVH Khang Vi?t

thif sifc hpc ki m6n Toan I6p 12 MSng cao - Pham Trpng Jhd

D E THCT S O C H O C K i I M O N T O A N L 6 P 1 2

oiSp 6

Thdi glan lam bar. 120 phut

V a y the lich khoi chop A.'BB'C'C la
V = V2-V,=
IV
(2,0
diem)

a^Vsb^-a^

Cflu I. (3,0 diem)
(dvtt) •

'

,

2. T i m M G (C),biet rSng tiep tuyen vc'Ji (C) tai M ciit Ox, Oy hin lUdt tai A,
B tao thanh tarn giac O A B c6 dien tich bilng — (vdi O la go'c toa dp).
4
•
,

T a c o y ' = e ^ (cosx - 2xsinx).
y" = e ^ ( - 4 x c o s x + 4x s i n x - 3 s i n x ) .

•

•

Cfiu I I . (2,0 diem)
^
., . , ,
,,
, ,
,
1 + sin^ X + cos^ X
1. T i m gia Iri kJn nhat, gia tri nho nhal cua y =
T
T —
1 + sin x + cos X

- 4 6 " " x^sinx = 0 ;

<=> X = 0 h o a c sin X = 0 <=> X = k ; : , k e Z .

^ logy 16+2 log I 5

2. (1,0 diem) G i a i he phiTi/n^ trinh...
2. T i n h gia t r i b i e u thiJc: M = 3

D i c u k i c n x > 2, y > 0.

log25 4-log | 3

^ +5

5

3. G i a i bat phU'cfng trinh log-,^ X + 31og2 X > ^ i o g ^ ^ 16.

B i c n doi he phiTdng trinh da cho thanh
x - 4 x + y + 2 =0

x " - 4 x + y + 2 = 0 (1)

Cflu I I I . (2,0 diem) Cho hinh chop tam giac deu S.ABC c6 canh ben bKng a va

2 1 o g ^ { x - 2 ) = 21og^y

x - 2 =y

tao vdi milt day A B C g o c a . G o i O la tam cua tam giac deu A B C .

(2)

1. T i n h theo a the tich cua khoi chop S.ABC.

The (2) vao (1) ta diTdc x " - 3 x = 0 .

2. G o i M, N Ian liTdt la trung d i e m cua A B va A C . Mat ph^ng (P) qua M N va

G i a i phufc^ng trinh nay ta c6 x = 3 va x = 0 (loai).
V a y he phiTcJng trinh dii cho co nghiem (x; y ) = (3; 1).
V
(1,0
diem)

(1)

1. Khao sat s i f b i e n thicn va ve do thj (C) cua ham s 6 ' ( l ) .

1. (1,0 diem) G i a i phxidn^ trinh...

Do do y " + 4 x y ' + 3y = 0

Cho ham so y =

song song \6\O cat SA tai d i e m E. ChiJng minh ^^"^^^ ^ — •
Vs.ABC

11m m de 6\iHn^ t h a n j j d c6 phiTc/ng t r i n h y = - x + m c a t (C).
PhiTdng trinh hoanh do giao d i e m cua (C) v d i d la
x~ + x - I
x-1

= - x + m o 2 x " - mx + m - 1 = 0 {*•) (x

CSiuW.

(2,0 diem)

1. T i m diio h a m ciia cac ham .so:
a) y = ( x ^ - 2 x + 2)e''.

1)

b) y = S"^" + 3''cos2x.

d cat (C) tai hai d i e m phan biet
o

16

(*) CO hai nghiem phan biet khac 1

log
2. G i a i he phi/dng trinh

(3y-l) = x
2

(x; y e R).

A = m=-8m + 8>0
2 - m +m-l?i0

<=>

(ihoa)

rii<4-2>/2
m > 4 + 2V?

C&u\.

(1,0 diem)

T i m m de du'cJng t h i n g y = m ch dU'cJng cong y = ^ ^ "^^—x -1

t a i hai d i e m

phan biet A , B sao cho O A vuong goc OB (\6i O la goc tpa do).

Vay m < 4 - ly/l hoac m > 4 + 2 V2 i h i d cat (C) tai hai d i e m
phan hiC't.
94

25


TuyS'n chpn 39

Cty TNHH MTV DWH Khang Vigt

thtJ sifc hpc ki mOn To^n \dp 12 Nang cao - Ph^m Trpng Jhu

miny= min f(t) = - tai t = - 1 <=> x = - + — , k e Z.

D A P A N THAM KHAO

Cdu

x e R - ' Tinh...
l e | - l ; II
2. (0.5 diem)

Dap an
Dieim
1. (2,0 diem) Khao sat stf bien thien va ve do thj (C) cua ham...

-U.!:ylfi+21og| 5

(3,0 Doc gia tiT giai_cach giai ttfc^ng tiT cau I.l de so' 1.
diem) 2. (1,0 diem) Tim M e (C), biet rang tiep tuyen vtfi (C) tai M ...
2x, . PhiTdng trinh tiep tuyen d cua (C) tai M
Goi M
la y-y„ = l " ( x , , ) ( x - x ^ , )

2x^
^ 2x^, 2x
+ l (x„+l)^ (x„ + l)2
d cat Ox tai A => A(-x,^ • 0).
o

2

y=

,

Tac6:M = 3

/

4

2

(X

L2\>=-\,-1

1

+1)2

4

2

4

0,25

~ 2 T2
1

[log T X < - 4
—
<=>
o X < 16'

4

X„+l

[log2X>l .

2

1. (0,5

Tim gia trj l(?n nha't, gia tri nho nhat.
l + l-"^sin^2x 2 - ' ^ ( l - c o s 4 x )
.
Ta CO y =
4
8
_ 13 + 3cos4x
l^l-Um'-lx
2 - ' ( l - c o s 4 x ) ~ ^ ^ + 2cos4x
2
4
Dat t = c o s 4 x , - l < t < l .
16 >0, V t e [ - 1 ; 1].
(14 + 2tr

14 + 2t
Suy ra ham so tang trcn [-1; 1]
Do do max y = m a x f(l) = 1 tai t = 1 o x = — , k € Z.
t e | - l ; l|

'

2

0,25
0,25
0,25

x>2

Kct h(Jp dicu kien (*) la diTdc

. Vay M,(1;1),M2 — ; - 2

Ham so r(t)=:i^^t^ c 6 V(i) = -

1/;

0,25

^

5
5
3. (1,0 diem) Giai bat phUc/ng trinh...

Dicu kien x > ( ) (*).
Vdi dicu kien trcn ba't phUdng trinh da cho tifdng diTdng

^2

x„ = - - = > y ( , = - 2

xeK

+5

V"^3^,^3,og5.^2^^^^32

0,25

2x^ + Xy + 1 = 0 (v6 nghiem)

1

(2,0
diem)

2

log25 4 - l o g | 3

,

2<-x„-l = 0

II

4

l o g ^ - x + 3 1 o g T X > - l o g , 2"^ c=> log^"x + 3 1 o g 2 X - 4 > 0

K

- ^ « ' x 2

'

_ ^I()gy4-21()gy5 ^^I(>g25 4+I(>g,3

d cat Oy tai B => B
S

6

0 < X<
x>2

—

16-

0,25

Vay tijip nghiem ciia BPT da cllola T = fo; — u l 2 ; + o o ) .

III

1. (1.0 diem) Tinh theo a the tich cua khoi chop S.ABC.
(2,0 Vi SO 1 (ABC) nen SAO = a => SO = asina, AO - acosa
diem) => AB = a\/3cosa.
The tich khoi chop deu la
V,S . A B C - 3' . 4 ^ .SO ''^ 4^ cos^asina (dvtt).
2. (1,0 diem) Chtfng minh...
Goi I la giao diem ci'ia MN va AO.
AE AT
Suy ra I la trung diem cua MN vii EI // SO ^ — = — ( D

0,5
0,5

0,5
27


Cty TNHH MTV DVVH Khang ViSt

Tuyg'n chpn 39 dg thCf siJc hpc ki mOn Toan I6p 12 Nflng cap - Phgm Trpng Thu

Vay h0 phu'dng trinh dii cho c6 n g h i e m (x; y) =

AI=-AF

\

Goi F la trung diem ciia BC. Ta c6 <

V
(1,0
diem)

AO=-AF
I
3
S.

0,25

7

Tmi m de diicVng thang y = m cat difcfng cong...
. Goi d: y = m va (C): y =

x" + mx - 1

. Phu'dng trinh hoanh do giao d i e m ciia d va (C) la
X

0,25

+mx-l

:=m<=>x" = l - m (X 1) C'^)
x-1
. d cat (C) tai hai diem phan biet A, B c6 hoanh d o x j , X2 khi
jm

(*) CO hai n g h i e m X , , X 2 k h a c !<=><;
\)
m;^0

0,25

m
iB

.0A10B<=>—• — = - l c ^ —
= -lc:>m2 + m - l = 0
-d-m)
Xi x->

T i r ( l ) va (2) => — = - .
AS 4
Do ^5

IV
(2,0
diem)

AN

AE

V<.^3C

AC

AS

AB

m=

0,5

"^AMNE _ A M

_

0,5

(thoa dieu kien (**)).

3

16

o i so 7

1. (1,0 diem) Tim dao ham ciia aic ham sO...

D E T H l T SCTC H O C K l I M O N T O A N L 6 P

a) y' = (x^ - 2x + 2)'e'' + (e'' / ( x ^ - 2x + 2)

12

Thdi gian lam bai: 120 phut

0,5
= ( 2 x - 2 ) e ' ' + e ' ' ( x ^ - 2 x + 2) = x^e''.

Cfiu I. (3,0 diem) Cho ham so y = f(x) - x^ - 3x^ + 3mx +1 - m, c6 do thi (C,^).

b) y' = 5'*\ln5.(4x)' + (3")'cos2x+(cos2x)'3''

1. Khao sat su" bien thien va ve do thi (C) cua ham so khi m
0,5

= 4.5'^''.ln5 + 3^1n3.cos2x - 2,sin2x).
2. (1,0 diem) Giai hC' phrfoTng trinh...

dai D(x,; y, );circ tieuT(x,; y , ) . ChiJng minh rang

Dieu kien y

0.

2. Tim m de ham so' c6 ci/c tri. Vdi dieu kien viTa tim, gia suf ham so' c6 cifc
^ — ^

= 2.

(X,-X2)(XjX2-1)

0,25
Cfiu II. (2,0 diem)

Bien ddi he phiTcfng trinh da cho thanh .

~' ~ ^
4 ' ' + 2 ' ' = 3 y 2 (2)

I . Tim gia tri gia tri nho nha't cua y = sin"^x-cos2x+ sinx+ 2 tren khoang
0,25
V 2' 2

The (1) v£lo (2) \h rut gon lai ta difdc 6y^ - 3 y = 0.
Giai phufdng trinh nay ta c6 y = ^ va y = 0 (loai).
1
Suy ra 2^^ = - < t ^ x = - l .
2
28

2. T i m gidi han lim
x->0
0,25

ln(2x + l ) - l n ( 3 x + l )

3. Giai phu'dng trinh logx+3 f 3 - V x 2 - 2 x + l

]_
2
29

Tuygn chpn 3 9 6i thtf sifc hgc kl m a n Toan I6p 12 NSng cao - Ptigm T r p n g T h a
Cty TNHH M T V D W H Khang Vi$t

CSu I I I . (2,0 diem) Cho hinh lang try drfng A ' B ' C ' . A B C c6 day la tarn gi^c
vuong A B C tai B . Gia suf A B = a, A A ' = 2a, A C ' = 3a. G o i M la trung d i e m

y ' = O o t = - U ( - l ; 1) hoac t = - - e ( - l ; ! ) •

cua A ' C va I la giao d i e m cua A M vii A'C. T i n h the tich tiJ dien l A B C .
^
Dodo

C a u I V . (2,0 diem)
1. Cho ham so y = 0" ln(2 + sinx). ChuTng minh (2 + s i n x ) ( y ' - y ) = e'^cosx.

23 ,
.
min y = — k h i s i n x =
27
xe . — -; —

2. (0,5 diem) T i m gidi han..
,.
ln(2x + l ) - l n ( 3 x + l ) ^
ln(2x + l )
hm
= 2hm
x^o
2x

C S u V . (1,0 diem) Cho h a m so y = x - l + — ! — c 6 do thi ( C ) . T i m tren (C) hai
x+1

log^^3

diem)

Diem

Dap an

f 3 - Vx2-2x + l l = i o

V

J

0
1. (2,0 diem) K h a o sat si/ bien thien va ve do thj (C)...

+

3 - x - l > 0

2. (1,0 diem) T i m m de ham so'co ci/c trj...

(l)c^3-|x-l| =

Dao ham y ' = f(x) = 3 x " - 6x + 3m.

. Vdi - 2

0,25

< X <

VxT3

<=>

0,25

X

< X 7t

|x - 1 | ) = 1 (1)
2

0,25

-2

-3
+ X

<=> - 2

< X <

4.

-1

- Vx + 3 o (x + 2)^ = x + 3

-3 + V5

0,25

=-

2

<=>x^ + 3 x + l = 0 o

(1)

-3

l o g , ^ 3 (3 -

(2)

1: (2) <=> 3

phan bietc:> Ay, = 9 - 9 m > 0 < = > m < l .
Ta CO y , = r ( x , ) = x ] - 3 x j + 3 m x , + 1 - m

Z

3;tl

D i e u liien

Doc gia lU'giai each giai tu'dng tif cau I . l dc so 3.

De ham so c6 ciTc t r i t h i phiTdng trinh y ' = 0 c6 hai nghiem

0,25

3. (1,0 diem) G i a i phtftfng trinh..

DAP AN THAM KHAO

(3,0

^ ,.
ln(3x + l )
3 lim
0
3x

= 2-3 = - l .

d i e m d o i xufng nhau qua dufcing thang d : y = x + !.

I

0,25

3

2 2

2. G i a i bat phiTdng Irinh 25''+^ + 9 " + ' > 3 4 . 1 5 \

Cdu

1

X

-3-7F

(loai)

y^ = f ( x 2 ) = x"2 - 3x 2 + 3mx2 + 1 - m (2)
L a y (1) trir (2) theo ve la diTdc:

V d i l < x < 4 : ( 2 ) o 3 - ( x - l ) = Vx + 3 c ^ ( 4 - x ) ^ = x + 3

0,25

x ] - X T j - 3 | x j - X 2 j + 3m(x, - X 2 ) = y , - y j
<::>(x, - X 2 ) [ ( x ,
Ma

X, + X 2

=2;

+ X 2 ) ^ - X j X j

x,X2

X

II
(2,0
diem)

=

9 + yf29

0,25

TxiO) la CO

0,25

2
X

y , - y 2 = ( x , - X 2 ) ( 4 - m - 6 + 3m) = 2(x, - X j K x i X j - 1 )

9-4^

< = > x ' ' - 9 x + 13 = 0 o

- 3 ( x , + X 2 ) + 3 m ] = y j - y 2 (3)

=m.

=

(loai)

V a y tap nghiem cua phu'dng trinh da cho la
0,25
T =

- 3 + ^/5

9-y[29

0,25

Suy ra dpcm.
1. (0,5 diem) T i m gia tri gia trj nho nhS't...
III

H a m SO y C O the vie't l a i y = sin-'x - (1 - 2 s i n ^ x ) + sinx + 2
hay y = sin'^x + 2sin"x + sinx + 1
Dat t = sinx, t e ( - l ; 1).
Ta c6: 1(1) = t'' + 2 l ^ + 1 + 1, r'(l) = 31" + 4t + 1

0,25

(2,0
diim)

T i n h the tich ti? di^n l A B C .
Trong tarn giac vuong A ' A C ta c6:
0,25
A C = \/9a2-4a2 =aV5.
Tir do trong tam giac vuong A B C , t h i
0,25
B C = V5a2 - a ^ - 2 a .

30

11


ruygn chpn 39 6i thiT site hpc kl mOn Toan Iflp 12 NSng cao - Ph^m Trpng Thi/

Do(AA'C'C)l(ABC)nen

A'

Cty TNHH MTV D W H Khang Vigt

,

M ,

C'

X-1 +

ke I H J. AC (H e AC) =^ IH 1 (ABC).
IH
Thco dinh li Ta-lct ta c6

CI

2a

/

=

CA'
V,CI^AC^2=.
=2
lA'
A'M
l A ' + CI
3

X =

Vn3'

AA'

/

/

/

/

/

^

1

^d _ _ D

K

^

'
1

^

3m+ 1
y =-X +m = •

0,75
^

^

\
H

CI

,
, , 3m+ 1
I e d ncn
4

C

(dvtt)

Vay hai diem can tim la P

0,25

2 + sinx
0,25

Vay (2 + s i n x ) ( y ' - y ) = c''cosx.
2. (1,0 diem) Giai ba't phU't/ng trinh...
\

UJ

<=>

f e^y>1

0,5

2x--I = 0c^x = ± '

0,75

0,5

-34

4

-

1

1

+1 , Q

+ 9>0

0,25

l3

[ X >0

0,5

DE S O 8

DE THCT S O C H O C Kl I M O N T O A N L 6 P 1 2
Thdi gian lam bai: 120 phut

C&ul. (3,0 diem)
11
Cho ham so y = - — + x ^ + 3 x - — .
3
3
1. Khao sat siTbicn thicn va ve do thi (C) cua ham so da cho.
2. Tim tren do thi (C) hai diem phan biet M , N doi xiJng nhau qua true tung.
Cfiu 11.(2,0 diem)
In^x
1. Tim gia trj Idn nha't va gia tri nho nhat cua ham so y =
tren dean
x

1;
2. T i n h A = 2008.

cos-^-log,9-log,6
49" 3

|^]yog^4
It

Vay tap nghiem cua bat phifdng trinh da cho la
T = (_«,; - 2 ] u [ ( ) ; + « 3 ) .
V
Tim t r e n (C) hai diem do'i xitng nhaa..
(1,0
Ne'u P, Q la hai diem tren (C) doi xiJng nhau qua difdng th^ng
diem)
d : y = x +1 thi phufcJng trinh diTdng PQ (vuong goc vdi d) c6

19

+1

(ngU'rtc lai).

X .
. , c"cosx
=
ln(2 + sinx) +
2 + sinx
e''cosx

-

,
+ 1.0iai ra ta co m = 1.

Khi do hoanh do hai diem P, Q la nghiem cua phU"dng trinh

1. (1,0 diem) Chtfiig minh...

IV
(2,0
diem) Taco y' = (^'') ln(2 + sinx) + c''(ln(2 + sinx))'

>34i5x

m-1

i

V , A R r = ' S . R p . I H - ' AB.BC.IH- ^ a.2a."^'^ - "^'^
lABC
3 ABC
6
3
9

2,;x+l^gx+l

m-I

Toa do trung diem I cua PQ la
^

'

2
,u
2^^,
4a

=>
=
= —=>IH = - A A =
AA'
CA' 3
3
3
The tich cua ti? dien lABC la:
IH

= - x + m<=>2x - ( m - l ) x - m = 0 ( x ? t _ i ) .
X +1

trong (AA'C'C)

dang d':y = - x + m.

0,25

2x-y
3. Giai he phu'cJng trinh

32''~y-26.3

2

=27.

^log^Cx-y)^^

0,5

CSu I I I . (2,0 diem) Cho hinh chop S.ABCD c6 day ABCD la hinh vuong canh a.
Canh ben SA vuong g6c vdi day, M la diem di dong tren canh CD, H la hinh

Phu'dng trinh hoanh do giao diem cua (C) va d'la
33


Cty TNHH MTV D W H Khang VH
Tuygn chpn 39 gj thCf sOfc hpc k1 mOn To^n lap 12 NSng cao - Phgm TrpnQ Thu

chie'u cua dinh S Icn BM. Tim vj tri cua diem M tren CD de the tich kho'i chop
S.ABH la Idn nha't. Tinh the tich Idn nhat do, biet SA = h.
Cfiu IV. (2,0 diem)
1. Cho ham so y = sin( In x) + cos(ln x). ChiJng minh y + xy' + x^y' = 0.

X

-X +

X-1

Tim a de (P) tiep xuc vdi (C).

xe|l;c'l

e

min y = y(l) = 0•

xe|l;L•'|
2. (0,5 diem) Tinh...
(I
cos-log79-log,6

2. Giai bat phiTdng trinh (^^fl + T^sJ + |V7^^V48<14.
]
Cfiu V. (1,0 diem) Cho ham so (C): y =

Dodo max y = y(e^) = — ;

49 3 7

1 vflva. nparabol
u w n ^
2
a r a n n i (PV
x + a.
(P): vy == x''

-7

D A P A N THAM KHAO

Diem
Cdu
Dap an
1. (2,0 diem) Khao sat stf bien thien va ve d6 thj (C) cua ham...
I
(3,0 Doc gia i\i giai_cach giai tifdng tif cau I.l de so' 3.

diem) 2. (1,0 diem) Tim tren do thj (C) hai di§m phSn bi$t M, N.,
Goi M(xj; y,) va NCxj; yj) la hai diem thupc (C)
. f x 2 = - x . ;tO
0,25
M va N doi xtfng nhau qua Oy <=> \
11

11

X2
= — ^ + X2+3x7- —

0,25

=

0,25

^X2=3

Vay hai diem phai tim la

\
J,
V

16^
3;

• ^^1

V

> —
3

J

II 1. (0,5 diem) Tim gia tri Idn nhfi't, gia trj nho nhS't.
(2,0
lien tuc tren l;ediem) H^m so'y =
Ta c6 y' =

4
16

v5y

5 1255
= 2008 — =
3. (7,0 diem) Giai hg phifrfng trinh...
16 2
Dieu kien x > y.
Ta c6:
Vay A = 2008 4 16

32x-y_26.3 2 ^27,
^log3(x-y)^^

3

-3

x,=-3

,

36

2. 2x-y

2x-y

3 2 -26.3
^log3(x-y)' ^ 2

2

- 2 7=0

2x-y

xj=3
Xj

log.

2x-y

X2=-x,
2 + 3^x |

— ' - +, „X|

2. -log^Q-log^e
^2
_ ^log-^Q-log^Se

-7

lnx-0
,y' = 0 o lnx = 2 '

21nx-ln X ,

Tinh gia tri y d ) = 0, y{Q^) = Ar, y C e ^ ) - - ^
e^
e-^

x =l

0,25

0,25

X=e

0,25

2

= 2 7 =3^

(x-y)2=2
2x-y

3 2 = - 1 (loai)
(x-y)^=2
2x-y =6
.(x-y)2=2

x = 6-V2

2x-y =6
<=>

x - y = V2

<=>

y=6-2V2

(nhan)

x = 6 + V2
(loai)
y = 6 + 2A/2
x - y = -N/2
Vay nghiem cua he phiTdng trinh la (x; y) = (6 - N/2; 6 - 2^y2).
2x-y =6

Cty TNHH MTV DVVH Khang Vi$t

Tuygn chgn 39 66 M sijfc hgc kl mOn ToAn Idp 12 Nang cap - Phgm Trpng Tha

III
(2,0
diem)

V

T i n h the tich \dn nhat...

(1,0
diem)

Dat C M = x ( 0 < x < a ) Q

<:

—

9

•^ABM ~ "^ABCD

Tim a de (P) tiep xiic

"^ADM "^BCM

/'*W

j

X

(C).
- X+ 1
x-1

2

= x +a

(P) tiep xiic (C)

/ ' * \

CO n g h i e m

x"^ - x + 1

= (x2+a)'

x-1

0,5

1,0
^

4-V-\---^^

/--^ A /

vaBH-

/

2

2(a-+x^)

D

^

^

x-1

c

x^^l

(2)c>

2

X " - 2 x =2x(x'' - 2 x + l )

0,5

« x - 0

(2)

Ci>x(2x'^ - 5 x + 4 ) = 0
>0 VxeR

(3)

0,5

The (3) vao (1) ta dUdc a = - 1 .

2

L i luan V < — = > V ^ , , = — khi x - a, titc la M ^ D .
12
'"^^ 12
IV

c(5 n g h i e m

x^ - 2 x - 2 x
(x-ir

The tich k h o i chop S . A B H l a :

2

2

= x + a (1)

0,5

V a y a = - 1 thi (P) tiep xiic v d i (C).

1. (1,0 diem) ChiJng minh...

(2,0
diem)

,

cos(lnx)-sin(inx)

0,25

X
y'' = ( c o s ( i n x ) - s i n ( l n x ) )

'1
r 1Y
— + — (cos(lnx)-sin(inx))

U ;

X
-sin(inx)-cos(lnx)

cos(!nx)-sin(Inx)

2. (1,0 diem) G i a i M't phMng
N h a n t h a y : |V7 + V48
Datl=

Vv + V48

J.JV? -

Kl I M O N T O A N L 6 P 1 2

SQC HOC
Thdigian

lam bai: 120 phut

Cho h a m s o ' y = -^^^—^ (1)
x-1
1. K h i i o sat sif b i c n thien va vc do thi ( H ) cua ham so (1).

0,25

2. G o i I la t a m d o i xtfng cua ( H ) . T i m d i e m M thuoc ( H ) sao cho tiep tuyen
cua ( H ) tai M vuong goc v d i dU'dng thang I M .

trinh...

C&u 11.(2,0 diem)

N/48 ) = 1

1. T i m gia trj U'ltn nhii't va gia tri nho nhat ciia ham so f ( x ) = e"''''^^(4x^ - 5 x )

, t>0.

0,5

PhiTcJng trinh da cho trd thanh t + - < 14 < »

THlIr

C&ul. (3,0 diem)
0,5

2cos(lnx)

x2
x^
x2
The" y , y ' , y " vao y + x y ' + x ^ y " , rut gon l a i ta diTdc dpcm.

DE

DE S O S

trendoan - ;
• [2

2.

2. G i a i phirang t r i n h log^^ 8 - l o g 2 x 2 + logy243 = 0.

- 14t + 1 < 0

* j

i
o

7 - V48 < t < 7 + V48

o ( V 7 + V48

< yJl + ^f4^

3. G i a i b a t p h i f d n g t r i n h 2 " ^ " ' - 2 ^ ^ ' " " ^ > 3 .
< yll + yf4S

o - 2 < X < 2.

V a y tap n g h i e m cua bat phUdng trinh da cho la T = ( - 2 ; 2).

0,25
0,25

"

C S u I I I . (2,0 diem) Cho hinh chop S.ABC c6 hai mSt b e n S A B va S A C vuong
g6c v d i mat day. D a y A B C la tam giac can dinh A , co

diTdng

cao A D = a,mat

ben SBC la tam giac deu canh SB lao v d i mat day goc a .
37


luyeii

cliyii 3 9 cic Uui ;,u,. liyu ki nion

ToAn

Iflp

12

Cty TNHH MTV DVVH Khang Vi$t

NSng cao - Phgm Trpng T h u

1. Chtfng minh rang SB^ = SA^ + AD^ + BD^.

1. (0,5 diem) T i m gia t r j Idn nha't, gia t r j n h o nhaft.
(2,0 Taco f'(x) = 3 e - ^ ' ' + 2 ( 4 x 2 _ 5 ^ ) ^ ( g ^ _ 5 ) g 3 x + 2
diem)
^c^^^^(\2x^-lx-5).
f (x) = 0 o X = 1 hoac x =

(loai).
I[

sin a

2. ChiJng minh the tich cua khoi chop b^ng V

6 cos(a + 30°). cos(a -30°)
Cfiu l \ ( 2 , 0 d i e m )
1. Cho ham so y = yjlx-x^.Chtfng
minh y-'y" + 1 = 0.
49'' + 7" - 2 > 0
2. Giai he bat phiTdng trinh
>—

3 -,- .
Tinh gia trj 1(1) = - e ^ 1"r 1A = — e 2 , t
2
.2. 2
13
Do do max f(x) = f 1 = - c 2 ; min f(x) = f(l) = - e ^
y
2. (0,75 diem) G i a i p h i r o n g t r i n h . . . 2' 2
X >0
Dieu kicn
1
1•
3

, deu hai

Cfiu V. (1,0 diem) Tim nhSng diem tren do thi (C): y = x ^ + x - 1 each
x-i
true toa do.
DAP AN THAM

KHAO

Cdu
Dap an
Diem
I
1. (2,0 diem) K h a o sat s\i b i e n t h i e n va ve do t h j (C) c i i a ham..
(3,0 Doc gia tU'giai_cach giai tifdng tyf cau I.l de so' 1.
diem) 2. (1,0 diem) Tim diem M thuQC (H) sao cho tiep tuyen...
M(x„ ; y,,) e (H) o y„ =
(x„ ^ 1)
Tiep tuye'n A tai M c6 he so goc:
k,=y'(x.,) = - 1

0,25

,

a,

Du'dng thang IM c6 he so goc k, = — =
'

^1

(X

^
log2, 2 + - = 0
l + log2 2x
^ 2
Dat t = log,
2x
thi
ta
c6 : 1 + t - -t + -2 = 0
^2
o 6t-2(t + l) + 5l(t + l) = 0 o 5 r + 9 t - 2 = 0 o

logT 2x = J

1
-1)'

Yeu cau bai loan <=> kikj = -1
x„-l = l <=>
x„-l = - l x„=0^y„=l
Vay CO hai diem can tim M,(2 ; 3), MjCO; 1).
38

»

logT 2x = -2
1

I la giao diem cua dU'dng tiem can
•I(1;2)=>IM = x„-l; 1 = (a, ; a 2 )
,

X ?t Vdi dieu kienX 9tdo—,phUdng
trinh da cho trd thanh
4
2
31og,,2-log,,2 + ^ = O o ^ - ^ - l o g , , 2 + ^ = 0

0,25

0,25

^ = -2
' ~5

1
8

X= X=

1 %/2
Vay phifdng Irinh da cho c6 tap nghiem la S = •{ 8- ; 2
3. (0,75 diem) G i a i b a t phifdng t r i n h . . .
Bat phu-dng trinh da cho lu-clng diTdng T -4.2"
- 3>0
2
4
Dat 1 = 2'' "''>() thi la c6: t

3>0
I

0,25

<=>l^ -3t-4>0<;=> t > 4
t < -1 (loai)

39


Cty TNHH MTV DWH Khang Vi$t

Tuy^n chpn 39 06 ihii sufc hpc kl mOn Toan Iflp 12 Nang cao - Phgm Trpng T h j

=>2''^"' > 4 c i . x - - x - 2 > ( ) c : >

X

>2

x<-l'

Vay lap nghiem cua ba't phiTdng irinh da cho la
T = (-<»; - l ) u ( 2 ; + o o ) .
III
(2,0
diem)

0,25

1. (1,0 diem) Chufng minh...
IV
(2,0
^ = .
^
diem) Taeo y = .
2V2x-x'.
V2x-x1-x
-V2x-x^ -y—"
.(1-x)
V2x-x^
•y =•
2x - X

1. (0,75 diem) Chiyns minh...
Tam giae ABC ean dinh A nen diTcing eao AD eung h-i Irung
tuyen: DB = DC , SB lao vdi mal day goc a nen SBA = a.

0,25

-1

hayy" =

S

(2x - x - ) \ / 2 x - x -

y"*

Suy ra dpem.
2. (1,0 diem) Giai he bat phirc/ng trinh...
4 9 ' ' + 7 x _ 2 > 0 (1)
X c l he

A

(2)

>v7.

v7.

(1)<:^(7'')- + 7^ - 2 > 0 « >
Trong eae lam giae vuong SAB va ADB la eo:
fsB^=SA-+AB^
^
^
<
^
^
^ = ^ S B - ^ S A ^ + AD^ + BD^ (1)
A B - = A D - + BD^

0,5

0,5

2

4x- =4x^.sin~a + a ' + x - =>x^ =

,a<60".

<=>i
x^ + x - 2 > 0

The lieh cua khoi chop S.ABC:
2 •> .
2a''sina
= -BC.AD.SA = - a x ' s m a =
— (2)
^
^
3(3-4sin2a)
' r > / - A i - v c A

X" +

0,5

X-1

x-1
x' +X-1
x-1

2

1 . 2 )
Ma 3-4sin^a = 4 — sm a = 4 r 3
-cos
a
—
s
m a
4
)
U
4
)

= 4c()s(a + 30").cos [ a - 3 0 " ) (3)

Tim nhiynjj diem tren do thj...
V
(1,0 NhiTng diem tren do Ihi (C) each deu hai Iriie loa do ehinh lii
diem) giao diem cua (C) va hai du^cing phiin giae y = - x , y = xeiia
goc loa dp.
PhU'dng irinh cho hoanh do giao diem la

3-4sin^a

40

o 2-x>0

Vay lap nghiem ciia ba'l phUdng Irinh da cho la T = (1; 2].

Trong lam giae SAB la eo SA = 2xsina; luT (1) suy ra

The(3) v a o ( 2 ) l a d liilc dpem.

( 2 ) o V 2 - X 0
1 _X
Dal BD = X , la t o SB = 2BD = 2x.

^ = TSABC

x>0

x>0

2. (1,25 diem) Chi?n« minh...

C A

ox>0.

7" >1

B

\r

7'' < - 2 (loai)

=x

X- +

o

X-1 =x'

x~ + X - 1 = - x

X =—

2

- X

o
+X

= - X

x^±

Viiy nhCTng diem Iren (C) each deu hai Iruc toa do la:
0,25
M,

v2 2y

, M-

V2

N/2

• 2 ' 2
41


Tuygn chpn 39 ai thCf sijfc hpc ki mOn To^n Idp 12 h m j

- Phgim Trgng Thi/

Cty TNHH MTV DWH Khang Vigt

D E THCT SLTC H O C K i I M O N T O A N L 6 P 1 2

DE SO 1 0

DAP A NTHAM

Thdi gian lam bai: 120 phut

Dap an

Ccdu
Cfiu I. (3,0 diem) Cho ham so y = Sx'' - Qx^ +1 (1).
1. Khao sat su" bie'n thicn va vc do thj (C) cua ham so (1).
2. DiTa vao do thi (C) hay bien luan theo m so nghiem ciaa phu'dng trinh:

KHAO

Diem

1. (2,0 diem) Khao sat s\i bien thien va ve do thi (C) cua ham...
I
(3,0 Doc gia tiT giai_cach giai tu'Ong tiT cau I . l de so 2.
diSm) 2. (1,0 diem) Di/a vao do thj (C) hay bi$n luan...

8cos''x - 9cos^x + m = 0 vdi X G [0; TT] •

1 .

•y

Cfiu I I . (2,0 diem)
1. Tim gia tri Idn nhat va gia tri nho nhaft ciia ham so y = 2sin^x - 2sinx +1
tren

\= 1- m

71 571

6

6

3
-1

2. Giai bat phu'dng trinh 3.

+ 2.

-5>0.

4

/

i

v3y

\

/

/

o

X

1
3. Giai phi/dng trinh l o g ( x - 2 ) - - l o g ( 3 x - 6 ) = log2.
Cau I I I . (2,0 diem) Cho lang tru dii-ng A B C . A ' B ' C c6 day la tam giac deu. Mat
phang (A'BC) tao vc'^i day mot goc a va tam giac A'BC c6 dien tich la S. Tinh
the tich khoi lang tru A B C . A ' B ' C .
/

C'duW. (2,0 diem)

/

1. Tinh gidi han:
^ ,. e ^ ' ' - e ^ V 3 x 2 - 6 x + 4
A = hm
x->i
tan(x-l)
2. Giai he phu'dng trinh:

32

Xet 8cos'*x - 9cos^x + m = 0 vdi x e [0; 7t] (1)
Dat u = cosx, phUdng trinh da cho trd thanh:

0,25

8 u ' ' - 9 u ^ + m = 0 ( 2 ) . V i x e [ 0 ; 7 i ] n e n u e [ - l ; 1].

,2_

jX'-xy+y" _

49

16

Ta CO ( 2 ) » Su'* - 9u^ +1 = 1 - m (3). PT (3) la FT hoanh do

l o g 4 ( x 2 + y 2 ) = i- + log4(xy)

giao diem cua do thj (C): y = Su"*-9u^ + l,u e [ - 1 ; 1] va

0,25

dirdng thing d : y = 1 - m cung phiTdng vdi true hoanh, nen so
Cau V. (1,0 diem) Cho ham so y =

2x-3
3-x

CO do thi (C). Vie't phi/dng trinh tiep

tuyen cua (C) tai giao diem ciia (C) vdti true tung.

nghiem cija phiTdng trinh da cho la so giao diem cua (C) v^ d.
Dya vao do thi ta c6 ke't luan sau
49

81
o m > — : PhiTdng trinh (1) v6 nghi?m.
32
32
49
81
• 1-m=
o m = — : PhiTdng trinh (1) c6 2 nghiem.
32
32
• l-m<

42

0,25

43


l u y i M i ciioii '.'.:) ilil llui M I C ! H U . ki ;iioii lo.ui

Nang cao - Phaiii l i - j n g Thl/

Cty TNHH MTV DVVH Khang Vigt

49
81
• - — < l - m < { ) o l < m < — : Phifdng trinh (1) c6 4 n g h i c m .
• 0
PhiTdng trinh (1) c6 2 n g h i c m .

Vay tap n g h i c m cua BPT da cho la T = ( - o o ; ( ) ) u
0,25

3. (0,5 diem)

• 1 - m = 1 c=> m = 0 : Phifdng trinh (1) c6 1 n g h i c m .
1. (1,0 diem)

(2,0

Dat t = sinx

2

+

CO

G i a i phxidna trinh..

D i c u k i c n x > 2 (*).

' 1 - m > 1 <=> m < 0 : Phi/dng trinh (1) v6 nghicm.
II

—;

PhiTdng trinh da cho li/ctng diTdng

T i m g i a trj l(?n nha^t, gia trj n h o nha't.

2 ( l o g ( x - 2 ) - 1 o g 2 ) = log(3x - 6 ) o

2 1 o g ^ = log(3x - 6 )

diem)

Ta

CO

X

G

6'

t

O

T

G

—; 1
2

o

log

'x-2'

= log(3x-6)

K h i do y = 2 t - - 2 l + 1 = l ( t ) vdi t < — ; 1
x-2
Ta

l ' ( t ) = 4t - 2 , r'(t) = 0 c> t = - e — ; 1
2
2

CO

T i n h gia tri 1'0—'

_}_

0=

J, f

2'

V

2J

-J(l) = l .
2

xe

ft

6

min
xe

-

:i

;

(l

Ill

T i n h the' t i c h khcYi c h o p S . A B C

G o i E la trung diem ciia BC, ta c6: A E 1 BC => A ' E 1 BC

Sit

le

Dat u =

2

2

2

; 1

=

57t

_1_

1
-

ft

2. (0,5 diem)

2

1
- 1

2

min

y =

(dinh l i 3 du^ctng vuong goc) => ((A'BC),(ABC)) = A E A ' = a.

5

jiax

-

—:

=iBC.I-2-

G i a i bfl't phifofng t r i n h . .

2

4 cos a

-5u + 2 > 0 o

Vdi u > 1

v3y
f4'
—

l9y
44

X

>

cos a

_ BCV3

Bat phiTdng trinh da cho trd lhanh: 3u + —
u

2x

i B C . ^
co.sa
BC73

, u > 0.

V d i u < —:
3

2

2

v9y

o3u"

x = 2 '

(2,0
diim)

y =

x = 14

So d i c u k i c n (*) => nghicm cua ph^dng trinh da cho la x = 14.

Siiy ra:
max

= 3 x - 6 o x " -16x + 28-0c:>

2
u <3u>l

[V

„
1
Suy ra BC = 2

V

S.cosa

•

73

A ' A = A E t a n A E A ' = ^ ^ ^ ^ tan A E A ' = V\/3Scosa t a n a .
The tich lang tru A B C . A ' B ' C la

2

^
,
1
< — = > 2 x > l = > x > —•
3
2
f 4— l

5> 0

0

x<0.

V ^ S .A„B,C. .• A ' A ^ ^ ^ " ^ - A ' A

s

IS.cosa

A2

• 7 7 3 S c o s a t a n a = Ssina. V 73Scosa (dvtt).

45


Tuy^n chgn 39

IV

thO silc hgc kl mOn Jo&n Idp 12 Nang cao - Phgm Trpng Thu

1. (1,0 diem) T i n h gi('/i han...

OE SO

D E T H QS Q C HOC Ki I MON
Thdi gian lam bai: 120

11

(2,0

3(x-ir+1

diem)

A = lim -

0,25

tan(x - 1 )

:e
:

Cfiu I. (3,0 diem) Cho ham so y = x ' ^ (m + l ) x ^ - 3mx - 2 (I) ( m la tham so).
1. Khao sat siT bien thien va ve do thi (C) cua ham so (1) k h i m = - 1 .
C&u 11.(2,0 diem)

lant

0,25

2,.
(e^'-l)cost
lim
+
t->0
sint
lim
t^O

12

2. T i m m de ham so (1) dong bien tren R.

,2t

: lim t-»0

TOAN LdP
phut

e^'-l

t

2t

sint

cost
lim
t^O

t

1. T i m gia tri Idn nha't va gia t r i nho nhat cua ham so y = x + 3 + — tren
;-4;-i:.

sint

•2 CO St + e l i m
t->0 smt

f

2. Cho log3 5 = a. T i n h log^75 3375 theo a.

-3t.cost
l + Vst^ + 1

0,5

3. G i a i bat phiTcfng trinh 2^^+' - 21.2"(2x+3) ^ 2 > 0.
C S u I I I . (2,0 diem) Cho hinh lang tru du-ng A B C . A ' B ' C c6 canh ben A A ' = 2a,
c a n h d a y B C = a, B A C = 1 2 0 ° . T i n h di$n tich xung quanh va the tich cua hinh

2. (1,0 diem) G i a i h$ phrfdng trinh...

tru ngoai tiep hinh lang t r u .

D i e u k i e n xy > 0 .
He da cho tiTctng diTdng

0,25
x^ - x y +

x^ - x y + y^ = 4

x^ + y ^ =2xy
x = y

x = y

x2.4

x = ±2

(x-yr

=0

x^ - x y + y^ = 4

0,25

-2).

-1)

Phu-dng trinh tiep tuyen cua (C) tai A : y = y'(x^ )(x - x ^ ) + y ^

( - x + 3)^

4

log(x^ + y ^ ) = l + 31og2
log(x + y ) - log(x - y ) = log3
x^ - x + 1
x-1

CO do thi (C). T i m tat ca nhi?ng

diem M tren do thi (C) sao cho tong khoang each tiT M d e n hai diTcfng t i e m can
la nho nhat.

Giao d i e m cua (C) v d i true tung la A ( 0 ;

L_^^y'(x

2

b)y = ln1 + e^

C&u V . (1,0 diem) Cho ham so y =

0,25

V i e t phtfofng trinh tiep tuyen cua ( C ) t^i giao diem cua...

T a c o y' =

iM

,2x.

2. G i a i he phu'cfng trinh

V a y HPT da cho c6 hai nghiem (x; y) la (2; 2), (-2;
V
(1,0
diem)

a)y =

0,25

log4(x2+y2) = log4(2xy)

<=> i

f
„x

<=> i

=4

(2,0 diem)

1. T i n h cac dao ham cua:

=4

log4(x2 + y^) = log4 2 + log^Cxy)

X - xy + y

C&ul\.

I

V a y phu'dng trinh tiep tuye'n can t i m la y = ^ x - 1 .

AN THAM

Cdu
0,5

I
(3,0
diem)

).y'(0) = l .
3

DAP

0,5

KHAO

Dap an

Diem

1. (2,0 diem) K h a o sat srf bid'n thien va ve d6 thj ( C ) cua ham...
K h i m = - l : y = x^ + 3 x - 2
a) Tap xdc dfnh D = M .
b) Su-bien

thien:

• Chieu b i e n thien

0,25
0,5
47