Tuyển chọn và hướng dẫn giải 39 đề thử sức học kì môn Toán 12 nâng cao: Phần 2

Tuygn chpn 39

dg

Cty TNHH MTV D W H Khang Vi$l

t h i i siic hqc ki mOn ToAn I6p 12 NSng cap - Ph?m Trpng Tha

1. " i n h the tich ciia hinh chop S.ABCD va the tich chop S.BHKC.
2. Ohii'ng minh 5 diem S, A, H, E vii K ciing nam tren mot mat cau. Tinh the
tich c a khoi cau ngoai tiep cua hinh chop S.AHEK.
3. Go! M la hinh chieu cua H trcn canh SA. Tinh the tich cua hinh chop
M.AHEK.

X"

02

HU6NG D A N - D A P S O
Caul.
1. Klu'io scU si( hien thien vu ve do thj (C) ciia ham .vr^fdoc gia tiT giiii).
2. Viet phUcfnii trinh tiep tityen ciia (C), hiet tiep tuyen...(6dp so y = 24x - 43).
3. Xcic (linh ni de phU(/nf^ trinh...
PhifcJng trinh da cho c6 ba nghiem phan biel
YCBT
X + 1 > 0
<=> < 2 - x >()

x +3>0
-X--x

+ 6 = (x + 3)-

<:> <

x>-3
2x- + 7x + 3 = ()

<=>

Vay phu'dng trinh dii cho c6 nghiem x = -3, - ~ 2
2. Gidi phUifns^ trinh...
Dieu kien: 1 < x < 3 va x 2 .

182

5
<z> X = - •

X = -

3

2 < X <3

3

x=3

Vay phifctng Irlnh da cho c6 nghicMii =
3. Gidi he pluOfnii trinh...

Ta C O e ' - e ' = ! n ( x - l ) - l n ( y - l )
N/X -1 + y"* = 3x^ - 4 y + 5
Dieu kien: x, y > 1.
Xel ham so 1(1) = e' + ln(t - 1), t > 1.

e ' + ln(x - 1) = e-^ + ln{y - 1) (1)
V 5 ^ + y^ = 3 x ^ - 4 y + 5 (2)

Do do (I) <=> X =^ y.
Thay vao (2) la difdc:
5,) =
x / x ^ - - x ^ + 3x- - 4 x + 5 <=> g(x) = x-^ - 3 x - + 4x - 5 + V J T ^ = 0 (3)
T a c o : g(2) = ()

V - x ' - x + fi = x + 3
x>-3
X - -3
1
X= —
2

5

<=>

Ta CO I'd) = e' + —^-j- > 0, I > 1. Suy ra ham so 1(1) dong bien IrcMi (1; + <»).

-i-l

x"* - 3 x ^ + x - m = 2 + x - x [ m - l = x'* - 2 x - - 3 (*)
Y C B T o ( * ) c 6 ba n g h i e m x G ( - l ; 2).
Difa vao do Ihi (C) la c6 - 4 < m - 1 < - 3 <::> - 3 < m < - 2 .
C a u II.
I. GidiphUdny, trinh...
^

x-1

-5x+ 6

( 3 - x ) - ( x - l ) ( 3 - x ) - ( ) c : > 2 x - 2 - x + l = ()
ll
2 2 x - 4 - x + l = ()

<=> i

Ta C O 6 4 . 2 ^ ^ = 4^' ^""^"'^ o 4 ^ * - ' = ^^7^^

X"

X - .

4-2X-X+1=0

logjCx"* - 3x^ + x - m ) - l o g 2 ( x + l) = l o g 2 ( 2 - x )
l

log-,(x"* - 3 x " + x - m ) = log,{2 + x - x ^ )

,
= '^'g.,-y- + l o g 3 ( 3 - x )
(x-l)(3-x)
^
^ (x-2)(x-3) (x-l)(3-x)
- 5 x + 6 = log.,

phiAing Irlnh da cho ti/dng du"(
X--3
X =

1•
2

—

vag'(x) = 3x'^ - 6 x + 4 +

2^f^l

= 3 ( x - l ) - +' *l +'

J

2 V ^

>(). V x e ( l ;

nen (3) c6 nghiC'in duy nhal la x = 2.
Viiy he da cho c6 nghiem duy nhat lii (2; 2).

cau

III.

Tinh tite tich ciia hinh chop S.ABCD vd the tich chop S.BHKC.
Tam giac SUB vuong tai H co SBH = 30" nen SH = BH tan30" = aV3.
Dien lich hinh vuong ABCD la 5^^^,^ = AB" = 16a^dvdl).

The lich hinh chop S.ABCD la Vs^^^.y = j S ^ ^ ^ ^ . S H = l^^L^

(dvll).
183


TuyS'n chqn 3 9 dg \hil sifc hpc ki mOn ToAn Idp 12 Nang cao - Phgm Trgng T h g

Cty TIMHH MTV D W H Khang V i j t

Ta c6: A B E H

oo A B A K

B E B H

B E _ B H . B A _ 3a.4a ^ 12

^ B A " B K ^ B K "

^ S j ^ ^ B H
SBAK

BK^ ~25a-~25

B E ^ 3 12^_9_^S^^16
B K

4 25

16 ^
=
AHEK

25

16 I
=

25

"^AK 25

S,^,

25

,

96a^
, ,
3a.4a =
(dvdt).

2

t

''J'

25

Do do the tich ciia hinh chop M . A H E K la:
V„„„-x 4s„„,,d(M,,ABCD,, = i

DE SO 3

.

j,f f
^

.

^

=^

,dvu).

D E THI KHAO S A T CHAT Ll/dNG MON TOAN
THPT KHOI C H U Y E N DAI HOC VINH

Cau I . (3,0 diem)

Cho ham so y = - — - c6 do ihi la (C).
2-x
1. T m i d i e m M thuoc (C) biet hoanh do ciia no thoa m a n phiTcfng trinh

2. Cln'niii niinli 5 cJieiii S, A, H, E vd K ciin^i nam tren mot mat edit....

y"(x)-2.
2. Vie'l phiTcJng trinh tiep tuycn voti do thi (C) tai diem M t i m difdc c( cau 1.

Ta c6:

Cau \\. (2,0 diem)

• A D 1 A B va A D 1 S H iicn A D 1 S A ^ S A K = 9 0 " .
• S H I H K

ncMi

•CHIBK

vaBKlSH

*

1. Cho hiim so y = ^x"* + (3m - 2 ) x ^ + (1 - 2 m ) x + 3, m la lham so. T i m m de

SHK-9()".

ncMi B K l ( S K E )

SEK = 9 0 " .

ham so dat cifc ticu tai d i e m x = 1 .

V a y nam d i e m S, A , H , E va K ciing nam Iren mot mat can c6 diTctng kinh
la SK.

2. T i m gia tri Idn nhal, gia tri nho nhat ciia ham soy = ( x +
CSu I I I . (1,0 diem) G i i i i he phifting trinh:

Ta CO SK- = S H - + H K " = 3a- + lOa" - 13a- => SK = a V l 3 .

- y ^ + 12x- + 3xy- = Ix' + 3x^y + 7x + y - 2

The tich ci'ia khoi can ngoai ticp ciia hinh chop S . A H E K .
V=y'^

=y(aV13) =

3. Tinli the tich ci'ta lunh chop

^

2x- - xy + 3x - 5 = 0

(dvll).

Gfiu I V . (3,0 diem) Cho hinh chop ti'f giac deu S.ABCD c6 canh ben gap S l a "
canh day.

M.AHEK.

1. Cho A B = a^/2.Tinh khoang each gii7a hai duTlng thang A D va SC.

Ta co:
d ( M , (ABCD)) ^ A M ^ A M . A S _ A H ' _ 1
d(S, ( A B C D ) )

A S " A S - ~ AS"

4

d ( M . (ABCD))
SH

1
~4

' '

2. G o i M la trung d i e m A B . Tinh goc giffa hai du'i:(ng thang SA v a C M .
Cau V . (1,0 diem) Cho hinh lang trii tam giac A B C . A ' B ' C c6 day la lam giac
(leu, canh bcMi bang a va t;u) \(h diiy mot g(')c 6 0 " . G o i D la trung d i e m canh

^ d ( M . (ABCD)) = - S H = —
4
4
1X4

(dvcd).

CC'.Biet rang hinh chieu vuong goc ciia A ' IC-n mat phang ( A B C ) trung v d i
Irong tam tam giac A B C . T i n h the tich khoi l i f d i e n A B C D .

185


T u y i n chpn 39 6i

thCt siic hpc ki mOn Toan I6p 12 Nang cao - Phgm Trgng Thi/

Cty TNHH MTV DVVH Khang Vigt

H U d N G D A N - D A P SO

TCr do he phirPng irinh da cho c6 hai nghiem ( x ; y ) lii (1;()) va

3'

Cflu I .
cau IV.

1. Tini diem M tlnioc {C) hiet hoanhdo ...

I

Tap xac djnh D = :x\|2 .
T a c o y' = —
y
(2-xr

"

=

Tinh khodii,i; each iiiifa hai dudn}^ thdiifi AD vd SC.
AC = 2a , SA = iiS,

1
i
1
1
Suy ra — = — — +
+
h"
OS^
OBOC^

(2 - x)"* = I
Suy ra d i e m M can l i m c6 loa dp la (1; - 1 ) .

h =

2. PhiTdng irinh lie'p l u y c n vdi do ihj (C) lai d i e m M... (dap so y

x - 2).

CauII.

H a m so dal ci/c lieu l a i d i e m x = 1 khi •

y ' ( l ) = ()
y''(l)>0

[4m-2 = 0

<=> <

[6m-2>()

o

1
m = 2

D a i A B = 2 =^ SA = 2>/3
Gpi N la irung d i e m ci'ia C D ^ A N ^ = 5, SN^ = 11.

xe|-2;2|

C f i u I I I . Gidi he phifcfiif-

min

Suy ra (SA, C M ) = arccos

y = y(±2) = ().

10

CSu V . Tinh the tivh khoi td dien

trinh...

ABCD.

Goi H , K Ian liTdl la hinh chieu A ' , D len m p ( A B C ) .

iren R.,phi/dng irinh tren c6 dang l ( x - y ) = f ( 2 x - l )

Suy ra H la trong tarn A A B C va D K = - A ' H •
2
A
Taco A'H = ^ ,
2

V l f''(l) = 3 r + 1 > 0, V l e R nC-n ham so 1(1) ddng bien tren R.

AH = ^.
2

Do do l ( x - y ) = r(2x - l ) o x - y = 2 x - l o y ^ i - x .
'AHAB

The vao phifdng irinh con lai ta diTPc: 3x^ + 2x - 5 = 0 <=>

16

^ABC

1

f

•

s

3a

i>

16

5X =

186

2AN.SA

I,
V

X6|-2;2|

T a c o - y ^ + 12x^ + 3 x y ^ =7x-^ + 3 x ^ y + 7x + y - 2
»(x-y)''+(x-y) = (2x-l/+(2x-l)
X c l ham so f"(i) = I-'+1

AN-+SA^-SN^

^5 + l2-n__4l5_
" 2.S.2S " 10

l e l - 2 ; 2].

V i y ( - 2 ) = y(2) = (), y ( l ) = 3V3.
y = y ( l ) = 3\/3,

11

Ta CO cos(SA, C M ) = cos(SA, A N ) = cos S A N =

2\.

T a c o y' = ' * ~ ^ ^ ' ^ ^ " . y ' = ( ) o x
\/4-x^

max

(SBC)

2. Ti'nh f;dc /^iifa hai dUitii}^ thdufi SA vd CM.

2. Tim i^ici trj hhi illicit, f^id trj nlio nluit i im ham .so...

Nen

5a^

T a c o d ( A D , SC) = d ( A , (SBC)) = 2 d ( 0 ,
= 2h =

Ta c6: y ' = x ' + 2(3m - 2)x + 1 - 2 m , y " = 2x + 6 m - 4.

11

11

1. Tim m lie ham .so dut ci/c tieu tai...

Tap xac djnh D = |-2;

SO =

D a l h = d ( 0 , (SBC).

(2-x)^

Thco gia ihict — ^ — - = 2 o
(2-x)'

3

—

3

187


TuyS'n chpn 3 9

D E

Cty TNHH MTV D W H Khang Vi^t

thCf sufc hpc ki m6n Toan I6p 12 Nang cao - Phgm Trpng Thg

D E

S O 4

Sd

T H I H O CK i I M O N

GIAO DUG V AD A P

T O A N

L 6 P1 2

TAP THOA THIEN

I. P H A N C H U N G C H O T A T C A T H I S I N H (7,0

HUE

1 A-''.75

r

16

CAu n. (3,0 diem)

-25

0,5

^^oi lam giiic vuong, irong do c - b

V = — s i n " ' X - 2 s i n x trcn doan [0; 7i].
y 3

H U 6 N G D A N - D A P so
C S u I. Tinh i^ici tri cua cac hieu

c6 do thi (C).
X

thiic...

- 1

1. Khiio sal sif b i c n Ihicn va vc do ihi (C) cua ham so'.

1. T a c o A = (3-^)-^+(2""^) ^ _ ( 5 - ) 2 = 9 + 8 - 5 = 12.

2. V i o l phu'dng Irinh l i c p l u y c n ciia do ihj (C), bic'l r^ng lie'p luyen do vuon
goc v d i diTcIng lhang c6 phu'dng trinh y = x.

1 ^ 1H
3
2. Ta CO B = logy - ~ - = log^. 3^ = - •

C S u H I . (3,0 diem)

cau I I .

Cho hinh chop S.ABC c6 day A B C la lam giac deu canh ;

m a l ben SBC help v d i m a l day m o l g()c bang 60°, SA l ( A B C ) . G o i M vji N la

1. Khdo sat subien

liTcn la hinh chieu vuong goc cua A Iren cac canh ben SB va SC.

2. Viet phmni^ trinh tiep tuyen ciia do thi (C), biet rani^ tiep

1. T i n h Ihc lich ciia khoi chop S.ABC iheo a.

thien va ve do thi (C) ciia ham so {6oc gia liT giai).

2x-l

S . A B C Iheo a.
d tiep xuc vdi (C) o

3. T i n h the tich cua k h o i chop A . B C N M iheo a.
I I . P H A N R I E N G (3,0 diem)
Thi sink cliidiMc
lam mot trong haiphdii

x-j

• = -x + b

(phdn A Itogc plidn I})

2x-l
1

diem)
x+l

• The X = 0 viio {1) ta difdc: b = 1.

diem)

• The x = 2 vao (1) ta diTdc: b = 5.

Vay CO hai tiep tuyen y = - x + 5,y = - x + l .

1- Tinh the tich ciia khoi chop S.ABC theo a.

diem)

+ Goi E la trung d i e m ciia BC, luc do: A E 1 BC,
1
-1 +J1+ • (2''-2-'')4'
l + ,|l + ^ ( 2 ' ' - 2 - ' ' r

188

(2)

Cfiu H I .

B. Theo chif(/ng trinh Nang cao

1. Cho X la so'thifc am. Chtfug minh

CO nghicm
• = -1

TCr (2) <=> (X - 1)- = 1 <=> X = 0 hoac x = 2.

T i m giii trj Idn nhal va gia tri nho nha't cua ham so y = sin^ x - ^ 3 sin x + 1 .

cau IVb. (2,0

(1)

(x-ir

2. Giiii bat phu-dng Irinh 2 " + 2"""^' - 3 < 0.
(1,0

= (-x + b)'

= -x + b

x-1

1. G i a i phiTdng trinh l o g 2 ( 2 ' ' - l ) . l o g 2 ( 2 ' ' " " - 2 ) = 6.

y'A.

CO nghi^m

'2x-l
x-1

A. Theo chir
Cfiu

tuyen...

Phu'dng trinh licp luycn c6 dang d: y = - x + b.

2. Xac dinh l a m I , ban kinh va linh dien lich v a l cau ngoai l i c p hinh cho

C a u I V a . (2,0

mtif :

1.

T i m gia trj \6n nhal va gia tri nho nhal cua ham so'

2. B = l o g 9 l 5 + l o g 9 l 8 - l o g 9 l 0

Cho ham so y =

1 va c + b

Churng minh rang log^.^^ a + log^._^ a = 2 \og^^^ a.log<._h a (*).
Cfiu V b . (1,0 diem)

diem)

C S u I. (1,0 diem) T i n h gia trj cua cac bicu thuTc sau:
1. A = 273 +

2. Cho a, b la do diii hai canh goc vuong, c lii canh do dai canh huyen cua

1-2^
1 + 2'

A E la hinh chieu ciia SE Icn m p ( A B C )

/
J

ncn SE 1 BC (theo dinh ly ba du-dng vuong goc).
( S B C ) n ( A B C ) = BC; SE c (SBC), SE 1 BC; A E c ( A B C ) , A E 1 BC.
189


Tuyfi'n chpn 39 (3i thil sOc hpc ki mOn Toan lop 12 Nang cap - Ph^m Trgng Thu

Cty TNHH MTV DVVIi Ktuny Vi

c a u IVa.

N c n goc giiJa 2 mat phang ( S B C ) va ( A B C )
chinh la goc hOp b("
1. Giaiphuoiifi

hay S E A = 60°

+

Dicu kicn: x > 0

+

( * ) c : > l o g , ( 2 ' ' - l ) . ( l + l o g 2 ( 2 ' ' - 1 ) ) = 6 (1)

+

Iog2(2'' - l ) . l o g 2 ( 2 ' ' ' ^ ' - 2 ) = 6 (*).

trinh

Trong ASAH viiong tai A c6
SA - AE.tan6()° = — . t a n 6 0 " - — •
Dat t = log2(2'' - 1 )

+

S^,^=iAE.BC =

1 73a
2

a=

2

S-d-

PhiTdng trinh ( I ) v i c l lai t ( l + t) = 6 o

4

1 3a V3a3

3

2

+

(dvlt).

4

8

tdni /, IHIH kiiilt va tiiili dien tick vat can iiiioai

2. Xcic Jjiili

ticp lunh chop

+

+

Goi O la trong tarn ciia tam giac A B C ihi O la tam difdng Iron ngoai t i c p

K h i t = 2 t h i l o g . ( 2 ^ - l ) - 2 < ^ 2 ' = 5 < : > x = log.,5.
K h i t = - 5 thi l o g , ( 2 " - 1 ) = - 3 o 2 " - 1 + 1 - - <=> X = l o g , - = 2 l o g , 3 - 3
8
8
8
"

S.Ai

theo a.

t " + t - 6 = 0 o t = 2 hoSc t := - 3 .

V a y phiTdng trinh da cho c6 2 nghicm la x = log2 5, x = 2 l o g j 3 - 3 .

lain

giac A B C .
Goi d la true ciia difring tron ngoai ticp tam giac A B C t h i d l ( A B C ) \ ; i

2. Giiii hat

trinh 2^ + 2"""^' - 3 < 0 {*).

phiOrni;

Ta c6: {*) «

2" + 2.2"" - 3 < 0 ( I )

d//SA.Trong m p ( d ; S A ) k c difiJng trung t r i f c A c i i a SA (P lii trung d i c m ciia SA).

D i l l t - 2"^ > 0, ( I ) v i c l lai r - 31 ^ 2 < 0 o 1 > 2 ho3c 0 < t < 1.

dirring thang A cat true d tai I , ta c6: I e d => l A = IS; I e A => l A = IB = IC.

Khi 1 > 2 Ihi 2"^ > 2<i> x > 1.

N c n I lii l i i m m i l l ciiii ngoai t i c p hnih chop S.ABC c6 ban kinh R = l A .
+

Tu" giac A O I P lii hnih chiJ nhiit c6 l A lii difdng chco

K h i 0 < l < 2 thi 0 < 2 " < I c> X

<

0.

Vay lap n g h i c m bat phi/cmg lrinhS = (-00; ( ) ) u ( l ; + o o ) .
R = IA =

N/OA-

+ AP- -

J

-SA
2

-AE

Cau V a . Tim i^ici tri hhi nhat va :^ia tri nho nini'l ciia ham .so...

12

Ti\p xac dinh I ) = x .
+

3. Tinh the tick ciia khoi chop A.BCNM
+

Hiim so' dii ' h.; irc't th;iiih g(t) = r - Vit + 1

theo a.

T ' l CO- ^ s . A M N _ SA S M SN _ S M SN
^^.PMC

+

D a o h a m g\'.i

Tinh gia i r i g

Trong tam giac S A B c6 SA^ = SM.SB

TiTcJng tiT trong tam giac SAC ta c6:

Ncn

V.S.AMN

Vs.ABC

^

V

190

^A.BCNM

.. S M

SB

=V

i i - ^ 3 , g'(t) = 0 <=> t = ^

e [-1; ]].

" SA • SB • SC ~ SB • SC

SB

+

D a l I = s i n x , I e ( - l ; 1|.

D i c n lich mat can: S^.^^ = 4 7 t R ' : = 4 7 i - ^ ^ ^ = Y^7ta'^(dvdt).

^S.ABC

SN_

SC

- V

SN

9

SC

13

81

81

169

169

^S.AMN

I-

SA2 + A B 2

'S.ABC

13

V

D o d o max y =
xeK

=

g ( l ) = 2 - N / 3 , g ( - l ) = 2 + V3.

/

max g(t) = 2 + ^3 o x = - - + k 2 7 t , k e Z .
tel-1; 1]

2
x.= - +

miny=

88 >/3a^

81
169

SB^

2

169

8

llS'd^
169

xeK

(dvtl)

m i n g(i) = — < z >
le|-l; Ij

>:\

k2Tt

3

, keZ.

4

x= —

«j ill

+

k27t

191


Tuygn chon 39 di \hil sCfc hpc ki mfln Toan Iflp 12 Nang cao - Pham Trpng Thu

Cty TIMHH MTV DWH Khang Viet

Dodo

Cfiu I V b .
Chiinii

1.

ininh...

( 2 - ' ' + 2 + 2--") =

-1 +

1 (2" -1)2

_l +i(2"+2-'') =
2

1

1+

mm

y = mm g(l) =

X6|0;7i|

I6|(); l | "

1-2'

2"

DE s o

l + - ( 2 " - 2 - " ) 2 = | + ^ l + i ( 2 2 " - 2 + 2-2")
4

1 +, , i ( 2 2 " + 2 + 2--") =
V4

1 ( 2 " + 1 ) 2 ^ 1 + 2^

I
l +i.(2"+2-") =
2

2"

2"

3

n

x =—
4
<=>
37t
x = —

,x+l

(do x < ( ) n c n 2 " <1).
TiTdng lir:

le|(): l |

24i

- 1 +,11 + ^ ( 2 " - 2 - ' ^ ) - =

Ta c 6 :

1

m a x y = m a x g ( l ) = ()<=> x = 0 h o a c x = jt.
xe|(); 7r|

5

DE

I. P H A N C H U N G CHO
(2)

T H I H O C K iI M O N T O A N L 6 P1 2
T R U d N G T H P T M O C H O A - L O N G AIM

T A T C A T H I S I N H (7,0 diem)

C a u I . (3,0 diem) Cho ham so y = \ - 3x - 1 c6 d o Ihj (C).
1. Khao sal S I / b i c n i h i c n va vc d o i h j (C) ciia h a m so'.
2. V i c l p h i f d n g I r i n h c u a di/clng I h i i n g d song song V('
1+'(2"-2-^)2
4

T i r ( l ) v a (2) t a c o :

va l i c p x i i c \(Vi d o I h i (C).

1-2'

(dpcm).

l+'(2"-2-")2
4
2.

Clnoxi^

1. Tinh dao h a m y ' ciia ham so da cho;
2. Chu'ng minh r a n g y ' < I , V x e K.

minh...

A p dung dinh l i Pilago la c 6 : a2 + b2 = c2

a2 = (c - b)(c + b) {**)

+

K h i a = 1 , dang ihiJc (*) luon diing vdi moi c - b ?t 1 va c + b ; t 1

+

Khi a

1, lay logaril c6 so a hai vc ciia dang ihiifc ("=*) la c 6 :

l o g , ( c - b ) + l o g , ( c + b) = 2 o - ; — i
C a u V b . llni

iiici

C a u H . (/,«J/ti/H) Cho hiim so y = h i ( x 2 + 1 ) .

tri W«

'that

D a l l = sinx; x e ();n

+ 1—^-

vd f^id tri nlid nhdt
•IG

= 2 : ^ dpcm.

ciia

ham so...

0;1

C a u H I . (3,0 diem) Cho h i n h chop S.ABC c6 lam g i i i c A B C v i i o i i g l ; i i B, c a n h
AC = 2cm, A C B = 3 0 " , di/iJng cao SA, gc')c giuTa diTctng lhang SB \ m a l phiing
(ABC)

•^ ^xm:

la 6 0 " .

1. Chiifng minh rang lam giac SBC vuong l a i B ;

o:

2. T i n h I h c lich c i i a k h o i chop S.ABC;
3. T i n h lam D v i i ban kinh ciia m i l l cilu ngoai l i c p hinh chop S.ABC. .

n.PHANRIKNG(J/>rf/^'//i;

4.3
H a m so da cho trcl lhanh g(t) = -^"^ " 2 l .

Thi sink chi didic Idm mot troitg hai phdii (phdn A hoac phdn

li)

A . T h e o chi/(/n}i t r i n h C h u a n

D a o h a m g'(l) = 4l2 - 2 , g'(i) = 0 o i 2

^}_^
1=

Tinh gia I r i g

192

2

CdulVii.
(2,0 diem)
1. G i a i phi/dng i r i n h l o g 2 ( x 2 - 3 ) +1 = 21og4(6x - 10).

2

^ , g ( i ) - - | . g(0)-o.

2

€ 0;1

'

'

2. Giai bal phiTOng I r i n h 9 " < 3 " " " + 4 .
^&u\d.

(1,0 diem)

'

T i m gia I r j Idn nha't va gia t r i nho nhii't ciia ham so y = 2x"' - 3 x 2 _ i 2 x +10
trendoan [0; 31.


Tuyg'n

chon 39 de iliii '.uc hoc ki mbn

ToAn k3p 12 Niang cao - Ph^m Trpng ThU

I

IJ. Theo chi/cau IVb. (2,0 diem)
1. Giiii he phifoliig Irinh

x + 3y = 8

Cty TNHH MTV DVVHKhang Vi?t
I

Dicn tich tarn giiic ABC: S^^^ = ^ A B . B C = ^.l.>/3 = ^

,

The tich khoi chop S.ABC: V = ^S^,j(_..SA = ^ . - y - . N / i = ^ ( c m ^ .

(cm^).

3. Tinh tarn D vd ban kinh ciia mat cau ntfoai tiep hinh chop S.ABC.

2. Giaiphir(1ngtrinh2^^ ' ' - 2 ^ ^ ^ ^ ^ = 3 .

.

C&uVh. (1,0 diem)

Hai goc SAC va SBC lii hai g()c vuong nen tam D cua mat cau ngoai ijcp

hinh chop S.ABC la trung diem canh SC.

Tim gia tri Idn nhal va giti tri nho nhat cua ham so y = V2x -

.

.

Ban kinh mat cau R = - S C = - \ / s A - + AC" = —

2 -

2

-

2'"'"-

cauiva.

H U d N G D A N - D A P SO

1. Giai pludfnii

cau I.

trinh...

1. Khdo sat sU hien thien va ve do thi (Q ^uu ham .w (doc gia lif giai).
2. Viet phU(fng trinh ciia ^/wvV/i.i; thani^ d son}; son;; iY>(...(dap so y = -3x -1).

,
f x - - 3>()
r
Dicu kien <^
o x > V3.
6x~l()>()

cau I I .

V('
2
. (x^ + 1)'
1. Tinh dqo ham y' cua ham so da clw y = In(x + l ) = > y =
x^ + l

log2(2x- - 6 ) = : l o g 2 ( 6 x - l ( ) ) o 2 x - - 6 = 6 x - 1 0 o 2 x ' - 6 x + 4 = ()

2x
\ + '

<=> x - 2 (thoa man dieu kien).

2. Churnii minh ran^...
Ta C O

y' < 1

1(1

Vay phu'(< 1 o (x -1)^ > 0 (dung moi x e K;

x" +1

2. Giai iuit plnfOni;

trinli...

Dat I = 3 \ >().

cau III

BPT da cho trtf lhanh l " - 3l - 4 < 0 o

-1 < l < 4 => 3" < 4 <=> x < log3 4.

1. Chifn}- minh rein); tarn aide SBC vuon^ tai B;
Ta

CO

Vay lap nghiem ciia bat phu'ttng trinh da cho lii S = (-oo; log^ 4).

S

Cau

BC 1 A B (do AABC vuong lai B)

Va. llni

i^id trj l('fn nhat vd }>id tri nhd nlid't ciia hdni so...

':

Tapxacdjnh !) = ((); 3|.

B C I S A (do S A l ( A B C ) )

Dao hiim y' = 6x" - 6x -12

=>BC J . ( S A B ) = > B C 1 S B .
Vay tarn giac SBC vuong tai B.

Ta

2. Tinh the tich ciia khdi chop S.ABC;

CO

y' = 0 o

x = - l g l ( ) ; 3|
x =2

• Tam giac ABC vuong tai B ncn

Ta tinh gia trj y(()) = 10. y(3) = 1, y(2) = -10.

AB = AC.sinACB = l (cm).

Vily max y = y(0) = 10, min y = y(2) = - 1 0 .
X f . l l l ; .1|

• AB la hinh chicu vuong g6c cua SB tren

x £ | ( ) ; .1|

Cau IVb.

mp(ABC)nenSBA=60".
SA
Tarn giac SAB vuong tai A nen tan 60" = —
AD

r

Gidi he phUifnii
Dieu kiC'n x

trinh...

0 vii y ^ 0.

SA = V3 (cm).
195


TuyS'n chgn 39 ai thCf SLIC hpc ki m6n Toan I6p 12 Nang cao - Phgm Trgng ThJ

Ta

log,x^=21ogy ^jx-^=y-*

^

X

=y

CO
X +

<=> i

x + 3y = 8

3y = 8

x = 2

hoac

y =2

X +

Cty TNHH MTV DWH

hoac •

3y = 8

x = -y
X +

3y = 8

QS^u I I I . (2,0 diem) Cho khoi chop S . A B C D c6 day A B C D Mx hinh thoi, canh
j,:-ing a, B A D = 60", S A = SB = SD = a.

x = -4

1. Tinh the lich ciia khoi chop da cho.

y=4 •

2. Xac dinh lam va linh ban kinh ciia mat can ngoai l i c p l u ' d i c n S A B D .
II. T H A N R I K N ( ; (3,0 diem)

V i j y he phufdng Irinh da cho c6 hai nghieni (x; y) la (2; 2 ) , ( - 4 ; 4).

'

fUi siiih clii diMc lam mot troiifi hai pliaii (pbdii A hoac phdii li)
2. Gidi pliUdiiii

tniih...

CaulVa.

Plnrcfng tnnh da cho ud lhaiih

= 2- o x -

t = - l (loai)
r - 3l - 4 = 0 <=>

- X= 2 o x

Ta

CO

man
so...

X =

X|X-,

=8.

Cau V a . (1,0 diem)
Mot khoi non c6 difcJng sinh b;lng 2a va dien lich xung qnanh cua mat non

1-x

y' = 0 o

(2,0 diem)

2. T m i gia tri m de phiTOng trinh (1) c6 hai nghiem phan b i c l x , , x-,lh6a

Tap xiic dinh I) - |(); 2].

V2x

,,; i j

1. Ciiai pliiAtng tnnh ( i ) khi m = 3.

= - 1 hoilc x = 2.

C a n Vb. 77/;; i;ici tii h'ln nhd'l va i;id trj nlw nluit cua luiin

'

Cho phirdng Irinh l o g ; x - ( 2 m - 3 ) I o g 2 X - 4 = 0 ( m la tham so) ( I ) .

1= 4

V a y phiMng t n n h da cho c6 hai n g h i c m x = - 1 , x = 2.

Dao ham y' =

*

\ Theo cliiJ'(/n{» trinh C h u a n

D a l 1 = 2 ' ' ' ~ \ >().

V('
Khang Vi?t

bang 2na'' (dvdl). T i i i i i the lich ciia khoi non da cho.

- X "

B. Theo chir(/ng trinh Nanj» cao

1.

Cau IVb. (2,0 diem)

* :

Ta linh gia tri y(()) = 0, y(2) = 0, y( 1) = 1.

Cho phif(
Vay

1. Giiii phiriJng trinh (1) khi m = 3.

m a x y = y ( l ) = l , min y = y{()) = y(2) = 0.

'

(1).
'

'

2. T i m gia tri m de phifctng trinh (1) c6 nghiem tren doan |0; 2 ] .
Cau Vb.f//y^/^'/;/)

D E SO 6

D E THI HOC KI I MON TOAN L d P

12

S d GIAO DUC V A D A O T A P B E N TRE

I. P I I A N C H U N t ; C H O T A i C A I H I S I N H
X
3xCilii I. (3,0 diem) C"ho ham so y = — + —— - x
3
4

Mot khoi non cd gilc ('J dinh bhng 60" va diC-n lich day b;lng 9;i (dvdl). Tinh
ihO lich ciia klioi non da cho.

H U d N G D A N - D A P SO

(7,0diem)
cfiui.

•

(1).
'• KhiU) Silt si( hie'}} thicn vd vP do thi (C) ci'ui ltd/)} so ... (doc giii l i f giiii).

1. Khao sat si/ hicn IhiC-n va ve do thi (C) ciia ham so (1).
2. V i e t phifitng trinh tiep luycn vc'Ji do ihj (C) bict riing l i c p l i i y e n song soiv^

9^
Vict pli}((t}}i; ti'inh tiep tiiycn rdi do tl}i (C) biet... y = - x ; y = - x + —
16

vdti diA-^ng lhang y = - x + 8.

Cau I I .

C a n \\. (2,0 diem)

'• T'lii} i^id tii nhd t}hd't vd ^id trj Idn nhd't ciia lidm sd...

\. T\m gia-lri nho nhat va gia Hi k'Jn nhat ciia ham so y = r(x) = x - - 8 1 n x
tren doan | 1 ; e).
2. T m i m dc ham so y = x"^ - 3 m x - + 3 ( m - - l)x - nr
1%

Ham so da cho lien liic Iren doan | 1 ; e|.
Ta c6: r'(x) = 2 x - - ,

+ 1 dal cifc dai l a i x = 1

l"(x) = () o
x = - 2 g | l ; c\

107

Tuy6'n

chpn

3 9 6i

thCi s i f c

h p c kl m 6 n T o i n

lOp 1 2

N i n g

c a o -

Phjim

Trpng

Ta linh ("(1) = 1, f(2) = 4 - In 256, f(c) = c" - 8.
Vay max f(x) = f( 1) = 1; min l(x) = 1(2) = 4 - In 256.
X6|I;L'|

x e | l : c |

2. Tint in de ham so...
T a p x a c d j n h D = R.
T a c o : y' = 3x^ - 6 m x + 3(m- -1), y" = 6 x - 6 m .

Ham so da cho dat ciTc dai tai x = 1 <=>[y'(i) = ()
y''(i)3 - 6 m + 3(m- - l ) - ( )
6-6m<0

m = ()
m = 2 o m = 2.
m >1

Cau III
1. Tinh the tich ciia khoi chop ch'i cho.
Goi V la the tich ciia khoi chop S.ABCD. Vi S.ABCD la hinh chop dcu do do
S H l ( A B D ) (vc'ti H la lam ciia lam giac ABD), suy raV = is^i3CD-^^ ™
'ABCD

= - A C . B D = -.aV3.a =

Ta lai

CO

(dvdl).

SH = VsA" - A H " - a - -

(dvcd).

(dvtl).
Vay V = 2. Xcic nfioai tii'p ti( tlicn SABD.

S.ABD la liir dicn dcu do do lam 1 ciia mat
cau ngoai licp luf dicn S.ABD lii giao diem
cua Iruc SH va dueling Irung IriTc Kl ciia doan
thang SA nam trong mat phang (SAO).

IS SK
ASKI dung dang ASHA nen AS SH
a
aV6
SK.AS
•Vay R = 1 S - .IS =
•a
aN/6
SH
23 ^
Cau IVa.
1. Gidi phi/cfniLi tnnh (1) khi m = 3.

Cfy

T h U

• (dvcd).

log^x = -1

T N H H

M T V

D V V H

Khang

Vi$t

X=—

2•
x = l6
2 Tim }iid tri ni de phucfnif trinh (I) cd hai n^iiiem phdn hiet...
Dill I = lognX (x > ()).Ta CO i, + t^ = l o g ^ x , + log^Xo = log^XjX^ = i o g j X = 3.
Phu-cJng tnnh da cho Iri'l lhanh l^ - (2m - 3)1 - 4 = 0 (*)
< '
YCBT o (*) CO hai nghiem I,. I T va t, + h = 3 <=> 2m - 3 = 3 <=> m = 3.
Q§i}i\'A. Tinh the ticii ciia khoi ndn da cho.
.j,
Goi SA va SO Ian liTcU la du'dng sinh va chieu cao ciia hlnh non.
Ta CO S^^ = ji.OA.SA «> 27t.a- = 7t.OA.2a => OA = a.
Khi ni = 3 la c6 l o g j X - 3k)g2X - 4 = 0 <=>

logoX = 4

Trong ASOA vuong lai O, la c6: SO = VsA" - O A " = 7(2a)- - a" = -ASThe lich ciia hlnh non can lim la V = ^Tt.OA'.SO = Z L I L ^ (jvii).

cau ivb.

1. GidiphU(fnii trinh (I) khi m = 3.
Khi m = 3, la c6 4^ - 4.2'' + 3 - 0
1=1

Dal I = 2^, I > 0. Phu-dng Irlnh Iren ltd ihiinh r - 4l + 3 0 o 1 = 3"
• Vdi I = 1 ihi 2" = 1 « X = 0. • Vdi t = 3 Ihl 2'' = 3 <^ X = logj 3.
Vay phifitng trinh dil cho c6 hai nghiem la x = 0, x = logT 3.
2. Tim i^id tri m de phidmi^ trinh (I)...
D a l l = 2 \)
Vdi x e [ 0 ; 21 ihi l e | l ; 4 | .
YCBT o t- - 4l + 2m - 3 = 0 (='=) c6 nghiem trcn doan [1;4].
Dill 1(1) = r - 4 l + 2 m - 3 , 16[1;4J.
Ta CO I'd) = 2t - 4 , f'(I) = 0 0 1 = 2 .
Bang bien thiC-n ciia ham so' f(t) Iren doan [1;4].
4
2
t
1
+
0
I'd)
2m-6
2m - 3
I'd)
2m-7
199


Cty TNHH MTV DVVH Khang Vi$t

Tuyln chpn 39 6i thif silc hqc ki man Toan I6p 12 Nang cao - Phjm Trpng Thu

Difa vao bang bicn thicn ta lhay (*) c6 nghiOm ircn

J,, I ' H A N R I K N ( ; (2,0 diem)

1;4

fjiisiiili
1
7
<=> 2m - 7 < 0 < 2m - 3 <=> - < m < - •
2

2

li)

^ Theo clurf/nK trinh C h u a n

.
'

vlii diMc lam mot troiig hai phdn (pitdii A liogc p/idii

^fiu V a . (2,0 diem)

•

1, T i n h dao hiim ciia cac ham so' sau:

CAu Vb. 77/;// the tick ciia khni non da cho.

b) y = e''.ln(2 + sinx).

a ) y = x^e^",

Goi SA, SB la difilng sinh ciia hlnh non va SO lii chicu cao ciia hinh non.
Ta lhay thic'l d i c n qua liuc la lam giac SAB dcu

2. Vie'l phiAJng liinh lie'p luye'n ciia do ihi h a m so y =

x-4
X

. S^,.^ - 7iR- « 9Tt = 7iR- =?• R = 3.

iU ,

,bie'l liep luye'n

1

^(,ng song vt'li du'Cing ihiing d: 3x - 4y = 0.
\\ Theo chifofnj; trinh Nan}? cao

2

2

CSu V b . (2,0 diem)

. V = ^ 7 t R - h - ^ n . O A - . S O = 97tN/3 (dvu).

O E SO 7

D E THr HOC Ki I MON TOAN L 6 P 12

1. a) Cho h a m so log-, 14 = a . T i n h log^y 32 theo a.
b) Cho hiim so y = e^'"''.Chii:ng minh nlng y'cosx - y s i n x - y " = 0.

S d GIAO DUC VA DAO TAO TIEIM GIANG

I. P H A N C H U N G C H O T A T C A T H I S I N H (8,0 diem)

liiycii song song vc'li difiJng l h a n g d : 3x + y - 2 = 0.

C a u I . (3,0 diem) Cho ham so y = x^^ - 3x- + 2 c6 do ihj (C).

Hl/dNG DAN - DAP SO

1. K h i i o sat sii" b i c n ihiC-n va vc do thi (C) ciia h a m so.
2. V i c l phifttng Irinh lic'p tuycn vi'Ji do ihi (C) tai d i e m c6 hoanh do bang I .
3. T i m d i e m M thuoc difclng lhang d: y = 3x - 2 s a o cho long khoang each lir
d i e m M U'Ji hai d i e m cifc l i i ciia (C) c6 gia iri nhi) nhal.
C i i u U. (2,0 diem)

-2
:
—
x + 2 , b i e l liep

X" - X

2. V i e t phifitng liinh lie'p luye'n ciia do thi hiim so y =

Cau I.
1. Klu'u) sal sif hien thicn va vc do thi (C) ciia ham .so ...{6C)c gia tif giiii).
2. Vict phififni^ trinh ticp titycn V('fi do ihi (C)

t//c''/?;...(dap so y = - 3 x + 3).

Tim dicni M thuoc dUtfnii ihcini^...

1. T i m m de phifdng Irinh x"* - 3x" + m = 0 c6 4 nghiem phan b i e l .

Toa do d i e m ci/c dai lii A((); 2) vii toa do diem cifc lieu lii B(2;

2. T i m gia Irj h'ln nhal vii gia Iri nho nhal ciia hiim so' r(x) = -x"* + 3 x " + I

PluriJng trinh di/itng thilng A B lii y = - 2 x + 2.

IrC-n doan |(); 2|.

-2).
;;: /

Ta C O M A + M B nho nhii't khi vii chi khi A , M , B lhang hang.

Cau l I I . r / , 5 ^ / ^ » ; ;
1. Giai phiriing Hinh 6.9'' - 13.6" + 6 . 4 " = 0 .

,

4
y = -2x + 2
X = —

Poa do d i e m M lii nghic'm ciia he

VayM

^4.
5'

y = 3x-2

2. G i i i i phifcJng Irinh l o g ; x - 9 1 o g j ^ x = 4.

2^
5

2
Cau

I V . (1,5 diem) Cho hinh chop tu- giac dcu S.ABCD eo day A B C D la hinh

vuong canli a, m i i l hen lao M'U day mol goc 60". G o i M, N, P, Q liln hfdl 1.^
irung diem ciia cac canh SA, SB, SC, SD.

iu I I .
Tini in dc phifdn}^

Innh...

1. T i n h 11 so ihe lich ciia hai khoi chop S.ABCD vii S.MNPQ.

E);lt u = x " , u > 0. Phifctng trinh da cho Iri'l thiinh u " - 3 u + m = 0 (*)

2. T i n h khoiing each lir A den mal ph;1ng (SCD).

Y C B T c ^ (^') C O 2 nghiem phan biel diriJng

200

J*'! '
201


Tuyin chpn 3 9 6i thCt s a c hgc kl m6n Toan I6p 12 NS; g cao - f h o r n i r g n g l l u

A = 9 - 4 m >()
S=3>0
P = m >()

Cty TNHH MTV DVVH Khang Vi§t

A = 9 - 4 m >()

Cfiu I V .

9
o 0< m< —•

<!> ! = 3 > {)
P =m>0

j

Jinh ti so the tick ciia hai khd'i chop S.ABCD

vd S.MNPQ

G o i O la tarn hlnh vuong A B C D vii

2. Tim i-id tri It'/n nhdt vd i^id tri nlid nhdt ciici hdni so...

H la irung d i e m ciia BC. Tarn giac
SOH vuong tai O va c6 SHO = 60".

H a m so' l'(x) - -x"^ + 3x" + 1 lien liic Ircn doan [0; 2J.

Suy r a S O = — , SH = a.
.

T a c o l ' ' ( x ) - - 4 x - ^ + 6 x = 2 x ( 3 - 2 x - ) , l ' ( x ) = ()<=> x = ()

The tich khoi chop S . A B C D la:
1

1
x =-J-g[();21
f

.

I — \

3

Ta linh 1(0) = 1, 1(2) = - 3 , f

|3

,3 i

Vs.MNPQ
Vay

:

max l'(x) = 1'
x.;|0;2|

2V,^^,

(dvlt).

6

SB

SC ^ ^ o 2 = 8

S M SN SP

2. Tinh khodns^ cdch tifA den mat phanii

mill fix) = 1 ( 2 ) - - 3 .

a'^\'3

2

T i so the tich ciia hai k h o i chop S.ABCD va S.MNPQ la
V.S.ABCD _ 2Vs.ABC _ SA

4

I

-> a\/3

YsAuro ~ ^ ^ A B C D - ^ ^ ~ 3

,

(SCD).

x.|():2|

4

Cau HI.

3a-V^.
/r
6 _'iv3

T a c o d ( A , (SCD)) =

s

1. Gidi phiOfnii trinh...
2x

Phifcfng Irlnh da cho li/cfng di/tJng 6

-13

^3^

+ 6 = 0.

cau V a .

^2,

.2;

1. Tinh dao ham ciia cdc ham so...
3
t= 2

, I > 0. PhiTcftig trinh da cho tri< lhanh 61" - 1 3 l + 6 = 0c:>

Dal I -

a) y ' = ( x - ) .c'^ + x ^ ( e ^ ^ ) = 2x.e^'' + x^,4x)'e-^'' = 2 x . e ^ M + 2 x ) .

2

v2y

t= 3

V d i t = - thi
2
v2.

= - <=> X = 1.
2

'3^
. Vi'Ji t = - Ihi
3
.2,

X

'3^
.2.

OX=:-l.

h) y ' ^ j e " ! .ln(2 + sinx) + e ' ' ( l n ( 2 + sinx))'
X cosx
\ » . .
X (2 + sinx)' X ,
= e'' ln(2 + sinx) + e''^-^
; — - = c^ ln(2 + sinx) + e
2 + sinx
2 + sinx

V a y phu'dng trinh dii cho c6 hai nghiCm x = ± 1 .
2. Gidi phidfnfi

2. Vietphidmi'

trinh...

D i c i i kiCMi X > 0.

3
13
Dap so y = — x ± —
' ^ 4 4

Cfiu V b .

V(1fi d i c i i k i c n trcn ta difctc phifcing trinh
logT X = - 1

l o g ; x - 31ogT X - 4 = 0 o
log-, X = 4

Tinh...
1
<=>

X = -

2 •
x = 16

1
V a y phiftJng trinh da cho c6 hai nghicm x = —, x = 16.

202

Irlnh tiep tuyen ciia do thi ham so...

_log2 2 _
5
• log^y32 =
log^721og2 7
Suy ra log^y 32 =

• logo 14 = a => log2 7 = a - 1 .

5

2(a-l)

203


Tuy6'n chpn 39

Cty TNHH MTV DVVH Khang Vi$t

thCf sCrc hpc ki mOn Toan Mp 12 IMang cao - Ph^m Trpng Thif

HUdNG D A N - D A P S O
Cfiul.

. y" = ( c ^ ' " ' ' ) ' fc.sx + c'"'"''(cosx)' = c^'"" (cos^x - sinx).

1. Kluio sat sir hien thien vd ve do thi (C) ciia hum .sY;'...(doc gia liT g i a i ) .
The y', y", y vao b i c u ihifc y'cosx - ysinx - y" la diTcJc dpcm.

2. Tim 111 de ham sd (I) c6 cUv dai tai x = 0.

2. PhiAliiii Irlnli l i c p l i n e n ciia do l l i i : y = - 3 x - 3, y = - 3 x - 19
. y ' = -4x" + 4 m x , y ' = 0 <=>
X

D E THI HOC Ki I MON T O A N L d P 12

D E SO 8

S d G i A O DUC V A D A O T A O A N GIAIMG

• N e u m > 0.

-co

m
I. P H A N C H U N ( ; C H O T A T C A T I U ' S I N H
C a u I . (3,0 diem) Clio h a m so y = -x"^ + IwvC

=m

diem)

+

y'

\fm

0
0

--

0

+

0

-

- 2 (1).

1. Khao saUsir b i c n i h i c n va \ do ihj (C) ciia liani so ( 1 ) khi ni = 2.

y

2. T\m m d c h a m so' (1) c6 ci/c d a i lai x = 0.
H a m so' d a i ci/c lieu lai x - 0.

C a u H . (1,0 diem) Tim gia Iri k'Jn nha'l va gia Iri nho nhal ciia h a m so' y = •
x-2
t r c n doan [ - 1 ; 11. T i f do siiy ra

cosa + 1

• N e u m < 0.

< 2 veil m o i a.

-co

m

0

y'
C a u III. (2,0

+00

0

cosa-2

+

diem)
y

1. R i i t g o n A = l o g , 25.1og^ 3 \ l o g ; ; 2.
2. G i i i i phMng

H a m so d a i cifc dai l a i x = 0.

Irinh 3.9"^"'' - 4 . 3 " ' + 9 = 0.

C a u I V . (2,0 diem)

V a y ni < 0 thcni man de.

Cho hlnh chop lam giiic dcu S.ABC c6 do dai canh day

C a u I I . Tim i^id tri h'fn nhcit vd };id tri nho nhdt ciia ham sd...

bang a. T a m giiic SAB vuong can lai S.
1. T i n h ihc licli ciia khoi chop S.ABC ihco a.

Tapxacdjnh

2. Ttr B kc diriJng cao B H ciia lam giac A B C . T i n h Ihco a ihc lich khoi n'f
d i c n H.SBC iCr do suy ra khoiing each lir H den m a l phang (SBC).

3
Ta c6: y ' = - •<0,VxeD.
(x-2)^

I I . P H A N U I K N C ; (2,0

Suyra

diem)

Tin sillh clii dii'
m a x y = y ( - l ) = 0, m i n y = y ( l ) = - 2 hay 0 > y > - 2 , V x e D.
xel)

li)

A. Theo chifcfnji trinh C h u a n

B . Theo chif(/nj» trinh Nang cao

xel)

D a l t = cos a => I e D theo Iren la di/dc

<2:

cosa + 1
cosa - 2

C a u V a . (1,0 diem) Giai phircJng Irinh log2(x + I ) - log^Cx - 1 ) = 2.
X — 1
C a u V i a . (1,0 diem) Tim d i e m ciTc Iri ciia hiim so y = In —
x^+3

D = | - l ; 1].

uIII.
.

Riiti^on...
A ---- l o g , 25.log., 3 - . l o g , 2 = l o g , 5-.log , 3\\og^ 2

C S u V b . (1,0 diem) Giai phiWng Irinh ^ l o g ^ x - 0 , 5 = Ktg^
= 2 • - • l o g , 5. l o g , 3. logs 2 = 3 l o g , 5. l o g , 2. l o g , 3 = 3.
C a u V I b . (1,0 diem) Tinh A = ( 0 , 5 ^ ) ^ log^ ~ ã
204

<2,Va.

ôi


Tuyfi'n chon 39
2.

thir sifc hpc ki mOn Toin I6p 12 Nang cao - Ph?m Trgng Thu

GuiipliUtfni^

x+1

^
, .
5
= 4 o x + 1 = 4 x - 4 <=> X = - •
x-1
3
So vdi dicu kien thi phiTdng irinh da cho c6 nghicm la " = ^

trinli...

<:>

Phifdiig tnnh da cho liTdng diTcJng 9"^ -12.3"^ + 27 = 0
Dal

1=3'", 1>1.

Phifdng irinh ircn Ird lhanh l ^ - 1 2 l + 27 = 0 <=>

.

V d i t = 3 l h i 3' = 3 c : . x ^ = l o x = ± l .

.

Vdi I = 9 ihi 3'^ = 3- o x^ = 2 <=> X = ±V2.

Cflu V i a . Tim diem ci/c tri ciia ham so...

1-3

Tap xiic djnh D = (1; +oo).

t =9

Ham so da cho vict lai y = ln(x -1) - ln(x^ + 3).

Vay phifdng Irinh da cho c6 4 nghicm Ui x = ±1, x = ±\/2.

X

1. Tinh the tich khoi chop S.ABC theo a.

lich khoi UV dicn da cho

AB = ancn SA = SB =

Vj,

AB

= ^-SA.SB.SC

X = - 1 (loai)
x=3

1

+00

3
+

0

^
Vay diem ci/c dai can tim la 3; I n 6

Suy ra V = 6

Cfiu Vb. Gidi phUifnfi trinh...

aV2

f 2 ~

aV2

2

Dieu kien

(dvll).

Goi H \A chan dirCtng cao cOa lam giac dcu ABC ncn H la trung diem AC
HC

VASBC

1

=

AC

= —
2

logj X > 0 _ J X > 1
<=>•!
ox>l.
x>0
x>0

Phi/dng trinh da cho viet lai la yjlog^

24

2 ,

2. Tinh theo a the tich khoi tii dien H.SBC tir do suy ru khodni^ each tit H...

Ta CO

(x-lKx'+3)

. y'-0»

y

Tarn giac SAB viiong can lai S, canh huycn

\C

x^+3

y'

Vay SA, SB, SC doi mot vuong g()c ncn ihc

'

-x' +2x + 3

hinh chop dcu ncn cac mai ben

cua hinh chop h~i nhiJng lam giac bang nhaii.

rr

2x

Bang bie'n thien

Cau I V .

Do S.ABC

1

Dao ham y' =

a'^

• VH.SBC - - ^ ( ' ^ ^ " ) -

- ^ =^ ^'"62 ^

o log2 X - 2^1og2 x +1 = 0 o (^log2 X - 1 ) ^ = 0 o , / i o g ^ = 1
o log2 X = 1 O X = 2.
So vdi dieu kien phifdng trinh da cho c6 nghicm 1^ x = 2.
Cfiu V I b . Tinh...

1
a
Dicn tich lam giac SBC la S^^,. =-SB.SC =

A = ( 0 . 5 ^ ) ^ . l o g i = (0,5)^-^.log 2 2 - ' = ( 0 , 5 / | - | l o g 2 2
2
2
\y 2
^
_1_
0 . _i_

r

Khoang each tir H den mat phang (SAB) la d
^SBC

16'

'2, •

32

C a u V a . Gidi phUdn^ tnnh...
Dicu kien: x > 1
Vdi dicu kien Iren phiTttng irinb da cho trd thanh log
206

x+1
= 2

x-1
207


l i i y f ' ! ; ! i ; i 3')

lif

'.m:

lio(

ki m6n To^n I6p 12 NSng cao - Phgm Trpng Thg

Cty TNHH MTV DWH Khang Vi^t

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

D E SO 9

TRUdNG THPT A N MINH - KIEN GIANG

I. P H A N C H U N G C H O T A T C A T H I S I N H (7,0 diem)
C a u L (3,0 diem) Cho ham so y = f(x) = ^x"* - Sx^ +1x

HU6NG DAN - DAP

so

c a u 1.
J _ Khiio sat su hien thien va ve do thi (C) ciia ham so... (doc gia tif giai).
9
2. Phu'dng trinh tiep tuyen v d i do thj (C) y = - x - 1 6

(1).

'

(
C

cau II.

1. Khao sal si/ b i c n i h i c n va vc do ihj (C) cua ham so (1).
2. V i c l phu'dng Irinh l i c p Uiycn v d i do thi (C) lai d i e m c6 hoanh do x „ , biei
rang f ' ( x „ ) = 6.

I

Tinh the tich klioi chop S.ABCD

.

Tam giac SAC vuong lai A va SC = 4a n e n SA = A C =

.

Canh hinh vuong bang 2a

.

Th(5 tich kho'i chop S . A B C D la V^^g^^ = { s A - S ^ e ^ ^ = - ^ a ' ^ (dvtt).
3

tlieo a.

"^^^
Isfla.
{/.i'if

C a u I I . (3,0 diem) Cho hinh chop S . A B C D c6 SA vuong goc vdi m i l l phang
( A B C D ) , day A B C D la hinh vuong, lam giac SAC can dinh A vaSC = 4a. Gpi
M la Irung d i e m ci'ia doan SC.

2. Chi'm^i minh M la tdm mat cau di qua cac dinh cda hinh chop S.ABCD...

1. T i n h the tich k h o i chop S . A B C D iheo a.
2. Chifng minh M la l a m mat cau di qua cac dinh cua hinh chop S . A B C D .
Tinh dien tich mat ciiu nay theo a.

.

Vi B C l S A v a B C l A B n e n B C l ( S A B ) .

Cling

Till sink chi dii{fc Idm mot trong haiphdn

thuoc mat cau difdng kinh SC, tarn M

/

&il

/^
//
B^^^

(plidn A hoqc phdn li)

A. Theo chii'(/ng trinh ChuS'n

• D i e n tich mat cau la S = 4nr- = 167ta^ (dvdt)

C a u I V a . (2,0

3. Tinh khoani; each tif diem S den mat phang (AMD) theo a.

1. G i a i phiTdng Irinh 27.9" + 242.3" - 9 = 0.
2. G i a i bat phiTdng Irinh l o g j (2x) +
C S u V a . (1,0

V

^ = < 5.
log4Vx

KC

\SAMD - T ^ S A C D

1

• Ta c() M A = M D = 2a = A D nen S^^^P

diem) T i m gia trj Wn nhal va gia trj nho nha't cua ham so

f(x) = 4x^ - x'* + 1 i r e n doan [1; 4].

=

''^^•'^^^^P

V-^"^^^

\
^^^X

/
^
. ...^

'^T^

V3a^

• Khoang each liT d i e m S den mat phang ( A M D ) 1^:
d(S, ( A M D ) ) =

\/\

^^^X.

\

SC
ban kinh r = -— = 2a.

I I . P H A N R I E N G (3,0 diem)

'^^
.

,

= ^(dvcd).

B . Theo chifcfns trinh Nang cao
Cfiu I V h . (2,0 diem)

Cfiu I I I .

1. G i a i phu'dng trinh 21og5X +17 logj x - 9 = 0.

Tim cUc trj ciia liam

C S u V h . (1,0 diem) T i m cac tiem can cua do thj cua harh so f(x) =

208

so...

Tap xac dinh D = M.

2. G i a i phu'dng irinh e" - 1 - ln(l + x) = 0.
2x^-x-4
x-1

—x" + 2x + 3
- , y' = 0 o
c"
pB^ng bien thien
Dao ham y ' =

'

\

/[

Do SBC = SDC = SAC = 90"nCMi S, A, B, C,

D

*

* i hSt;)

/

TUdng lir C D 1 S D .

.

•

^

SuyraBClSB.
.

3. T i n h khoang each tif d i e m S den mat phang ( A M D ) theo a.
C a u I I I . (1,0 diem) Tim ciTc tri cua ham so y =

J i,- iV:)

x = - 1 hoac x = 3.

209,


CfyTNHH MTV DWH

TuySn chpn 39 at thir sCfc hpc k1 mOn ToAn I6p 12 Nang cao - Ph?m Trpng Thif

X

—00

+

0

y'

D i c u k i c n x > 0.

3

-1

-

0

Dat t = log^ x ta diMc 21" + 17l - 9 = 0

6

o

ta
2 •

t = -9

y
-2c

^

H a m so dat cifc lieu lai x = - I , y c j = - 2 c .

X =

VJ.

.

W6i t = - thi l o g , x = 2
2

.

V(-Ji t = - 9 (hi l o g , X = - 9 <=> X = 5""^.

Difa v^o bang bien i h i c n ta Ihay:
.

Khanfl Vi$t

<r>

T

Vay phi/dng trinh da cho c6 hai nghicm x = 1/5, x = 5~^.
.

H a m s o d a t c i f c d a i t a i x = 3 , y^e

=-4"

2. Gidi phifcfn^ trinh...

c

•

Dicu kicn x > - 1 .

Cflu I V a .
X c t ham so y = c" - 1 - l n ( l + x ) . Ta c6 y ' = c '

1. Gidi phucfnfi trinh...

1+ x
Dat 1 = 3", I > 0 .
1
PhiTdng trinh da cho lnl tha nh 27t^ + 2421 - 9 = 0

•

r(x) = c ' ' d 6 n g bicn va ham so g ( x ) = : - ^ n g h i c h
1+ x

( - 1 ; + CO) ncn phifting trinh

t = - 9 (loai)
V d i t = - i - thi 3" = 3"'' «
27

V i ham so

o

y ' = 0 <=> c" = — c 6
1+ x

b i c n ircn

nghicm duy nha't x = 0.

Bang bicn thicn

X = -3.

x

V a y phiTrtng trinh da cho c6 nghicm x = - 3 .

0
0

-1

y'

+C0

+

.f

/

2. Gidi hat phuimi^ trinh...
y

D i e u k i c n 0 < x ^ 1.
Ba't phiTdng trinh da cho diTdc v i c t lai 1 + l o g j x
4
D a t t = log2 X ta

di/dc

1 +1

i^-4i

+Y^ 5 o

<5.

1

• TiV bang bicn thicn cua ham soy = e" - 1 - l n ( l + x ) ta suy ra phi/cing trinh

log.x

+ 4
<0

o

KO

0
t = 2

x = 4

da cho CO n g h i c m duy nhat x = 0.

Cflu V b . Tim cdc ti^m ciin ciia Jo thi ^'da ham so ...
•

V i l i m y = +C0,

C f l u V a . rim aid tri Ic'm nhd't vd aid tri nhd nhd't ciia ham so ...

Tapxacdjnh D = [l; 4].
D a o h a m r(x) = 1 2 x ^ - 4 x 1

•

'x = 0(loai)
T a c 6 f'(x) = O o

0 ^ ^ ^ ^ ^

l i m y = - 0 0 ncn di/ilng lhang x = 1 la t i c m can drfng cua

x->r

V i lim ( r ( x ) - 2 x - l ) = \'*t'

'•"an x i c n cua do thi ham so.

lim

= 0 ncn dir^lng lhang y = 2x + l l a t i c m

x->±/-X-l

^

-

_x = 3
Ta tinh gia trj f( 1) = 4, f(3) = 28, f(4) = 1.
Vay

max f(x) = 1(3) = 28,
xe|l;41

;1 1 ''.

m i n f(x) = f(4) = l .
X6|(); 2|

Cflu I V b .

1. Gidi phu211


Tuy^n chpn 39 dj thCf sufc hgc k1 mOn ToAn Mp 12 N3ng cao - Phgm Trgng Thu
Cty TIMHH MTV DVVH Khang Viet

DE THI HOC Kl II MON TOAN LdP 12

DE SO 10

T a c() B e ( C ) =^ - y . , = -x;^, + 3x.^ - 4

T R U d N G THPT C H U Y E N HA ISIpl - A M S T E R D A M

T i r ( l ) v a ( 2 ) = > 6 x , ^ - 8 = 0c:^x
C a u I . (2,5 diem)

C h o h a m so: y =

+ 3x- - 4

V(<1 x

=±

=>y
3

2. T i m t o a d o c a c c a p d i e m n a m t r c n d o thj ( C ) c u a h i i m so b i c l r a n g c h u n g
d o i xiJng n h a u q u a g o c l o a d o .

I

1

V a y hai cap d i e m
ja do.

j\-\Jl-\-d\.

'2V3
3

•'•J ••:"(

8V3'

8^/3'
'

\l

9

2

2. T i n h diC-n l i c h h i n h p h a n g g\6i h a n bc'Ji h a i difcmg l h a n g

( d , ) : x + y = (),

V

J

= costdt.Vi

x e | ( ) ; 1) ncMi t e

diem)

t ^ 0, V('*i x = 1

T r o n g m a t p h a n g l o a d o Oxy/. c h o h a i du'dng t h a n g
(d,):
'

x

y - 2

/.
= - va
-2
1

1

2

-it:

-sin'tcosldt =

diem)

= - ("(! - c o s 4 l ) d t = - I 8 J
'
8

2. Tinh dicn tich hinh phanj^ i^itfi

1. G i i i i bii'l phu'itng t r i n h : l o g , ' - . 2 ' ' + i ' + l o g , ( 4 ' ' + 1 4 4 ) > 2 1 o g i 4 .
4
V
.i
5
2. T i m so phiJc /. b i c l r a n g so phiifc / i h o i i m a n d i c u k i c n 1/ - i - 21 = \ / H ) \
z.7 = 25.

Ta c 6 :
+ (d|) c a t ( d J t a i A ( - 8 ; 8).

+ ( d , ) citt ( C ) t a i B ( 2 ; 8).

cau I.

V a y d i c n t i c h h i n h p h i i n g la

Kluio .sat sif hicn thien va vc do thj (C) cua ham so... ( d o c g i i i

ur

giiii).

2. Tim toa do cac cap diciii iiaiit treii do tlii (C) ciia hdiii .\o hict rciiii; cliuni^ di>'
xiJtn^i nhau qua j^oc toa do.
y „ = x.', + 3x,; - 4 ( 1 ) .

G o i B d o i xiiTng vcJi A q u a O t h i B ( - x ^ , ; - y^,) v d i x^, ^ 0 .

010

han ii(fi hai JiA//;,!; tlidni^...

+ (d|) c a t ( C ) t a i 0 ( 0 ; 0 ) .

HI/6NG D A N - D A P SO

f

I

S = -.8.8-(-

1

8- xMdx - 3 2
0

= 31 + 1 2 - 4 4

(dvdt).

n

4o

n

ciia (d|) v i i ( d ^ ) .

();2

sin'^t.cos'ldt
0

3. V i c t p h i A l n g t r i n h m a t c i i u (S) c 6 d\i(1ng k i n h U\n v u o n g g o c chung

G o i A(x^,; y , , ) e ( C )

d o i xij-ng n h a u qua g o c

n

7t

2. V i c t phUctng I r i n h m i l l p h a n g ( P ) chii'a ( d , ) v i i song s o n g v6\.

1.

9

t =-•

2

x - 1 y - 2 /. - 1
d. :
==——•
-2

1
4

1. C h i J n g m i n h r a n g ( d j ) , (d-,) c h c o n h a u v a v u o n g g o c vi'Ji n h a u .

(2,0

'

r. Tinh tich phdn...

( d o ) : y = X v a difc^Jng c o n g ( C ) : y = x'^.

C S u IV.

3

iull.

0

C a u III. (3,0

^+--!—.
~ 9

"

diem)

1. T i n h l i c h p h a n I =

= ± ^ ,

(C).

1. K h i i o s a l sir b i c n i h i c n v a v c d o t h j ( C ) c u a h a m s o .

C A u 11.(2,5

(2).

8 x - ^
4

;

iVi.n


Cty TNHH MTV DWH Khang Vi«t

Tuy^n chpn 39 6i thCl silc hpc ki mOn Tocin I6p 12 NSng cao - Phjim Trpnp Thu

Cflu ITI.
1. Ch'rn,<^ininh...

<:r>4'' - 2 0 . 2 " + 64<0<=>4<2" < 1 6 < r > 2 < x < 4 .
Tim so phijtc z-

Di ting lhang ( d , ) di qua M(0; 2; 0) va c6 VTCP la u,' = (1; - 2; 1).
Direing lhang (d^) di qua N ( l ; 2; 1) va c6 VTCP la u^=(-2;
Taco M N = (1; 0; 1) va

Goi z = x + yi (x, y € R ) .

1; 4).

u,; U j = (-9; - 6 ; - 3 ) 3 > M N . u,;u2 = - 1 2 ^ 0 .

-! ;;,

Taco z - i - 2 = V l O o ( x - 2 ) ^ + ( y - l ) 2 =10

(1)

Ta

(2)

CO

z.z

= 25 <=> x" + y" = 25

Dodo ( d | ) , ( d 2 ) chc3onhau.

Tif (1) va (2) suy ra (x; y) = (3; 4) hoac (x; y) = (5; 0).

V i u,.Uj = • (-2) + (-2). 1 + 1.4 = 0 =^(d,) 1 (d^).

Vay z = 3 + 4i hoac z = 5..

J;^ I

2. Viet plur(fini trinh mat plidnfi (P) chCcu...
= (-9; - 6 ; - 3 ) = -3(3; 2; 1) lam

Mat phang (P) di qua M va nhan

THI HOC Ki li MON T O A N L 6 P 12

DE s o 11

TRUdNG THPT CHU VAN AN - HA ISI6I

VTPT. Do do phi/dng Irinh milt phang (P) la:
3( X - 0) + 2(y - 2) + (z - 0) = 0 hay 3x + 2y + z - 4 = 0.

I. PHAN CHUNG C H O T A T CA THI SINH (7,0 diem)

3. Viet phidfiiii trlnh mat can (S)...
Goi A(a; - 2 a + 2; a ) e ( d , ) va B(-2b + l; b + 2; 4b + l ) 6 ( d 2 ) .

Cflu I. (3,0 diem) Cho ham so' y = x'' + 3x~ + mx + m - 2 c6 do ihj (C„^).

Tir do suy ra AB = (-a - 2b + 1; 2a + b; - a + 4b + 1).

1. Khiio sat sif bicn thicn va vc do thj (C) ciia ham so' khi m = 0.

AB nho nhat khi no la doan vuong goc chung cua ( d , ) , ( d 2 ) .

2. Tim cac gia tri cua m dc do ihi hiim so (*) ci(t true hoanh tai ha diem

Khi do

AB.ii, = 0

phan biet.

| - a - 2 b + l - 2 ( 2 a + b ) - a + 4b + l = ()

Cflu II. (3,0 diem)

-2(-a - 2b + 1) + (2a + b) + 4(-a + 4b f 1) = 0

AB.ii. = 0

1
a= -6a+ 2 = 0
3
<=>i
<=><;
21b + 2 = ()

1 T ' u . ' u
u '
I
'^flnx.Vl + Inx ,

1. Tinh tich phan I = J
dx.

ri
• Do do A

4

1

3' 3 ' 3

,B

2?

40

21

21 21

Tarn I va ban kinh R cua mat cau (S) la:
_16 34 H) ^ „

2r2r21)

AB

^9^+6^+3^

2

21

Cau IV.
1. Gicii hat plufifiiii

(X

I

14

3. Giaiba'tphiWng trinh: log, (4" + 4 ) > log, (2"""" - 3 . 2 " ) .
2

21
16'

21;

CHUUI.
2

f
I

f
I

34^

+ y

—

+ /

21,

2

2lJ

7

2

(1,0 diem)

Cho hinh chop S.ABC c6 mat ben SBC la tam giac deu canh a; SA l ( A B C ) .
Tinh the tich khoi chop S.ABC biet so do g()c BAC = 120".
n. PHAN R I E N G (3,0 diem)

tnnh...

Bat phi/iJng trinh da c h o o iog^

<=> log, 80

2. Tim gia trj k'tn nhat va gia tri nho nhat cua ham so f(x) = x - e^" tren doan

N/I26
2

Vay phiTdng Irinh m i l cau (S):

.

X

l - l ; 0|.

21

I

,

13

^hi sinh citi dii{ic lam m(>t trong hai phan (phan A hoac phan B)
+ logj 16 + logj 5 > logj ^4" +144|
4

>log5 (4' + 1 4 4 ) 0 8 0

^1

^

1.2"+1

> 4 ' +144

Phan A.
Cflu IVa. (2,0 diem) Trong khong gian Oxyz cho diem 1(1; 2; 3) va mat phang
( P ) : 2 x - 2 y - z - 4 = ().
1. Viet phi/tJng trinh mat cau (S) tarn I va tiep xuc vdi (P).
2. Tim toa do tiep diem ci'ia mat cau (S) vii mat phang (P).
215


Cly TNHH MTV DVVH Khang Vig!

Tuyfi'n chpn 39 dl thif sure hgc ki man ToAn I6p 12 Nang cao - Phgm Trpng Tha

C&u V a . (1,0
Tim

diem)
cac

t a p hctp

diem

cac

Iren m a t p h d n g toa do b i e u d i e n

so'

phiJc z

th

man dieu kien /. + 3 - 4 i = 2.
Phan B.

'

CAu [Vb. (2,0diem)

.

D a o h a m f'(x) = l - 2 e - \) = 0 c ^ x = -ln>/2 v a r " ( x ) = - 4 c ' ' '

.

VI

.

T a t i n h g i a t r i r( - 1) = - 1 - e"", 1(0) = - 1 , !"( - In x/2) = - I n ^ 2 •

.

Vay

1. Viet phiTcing irinh mat phang (P) di qua A va dufdng thang d.

^y

^

T i m e;ic so phiJc z thoa man z" = - 1 5 + 8i vii z c6 phiin ao la so difdng.

HUdNG D A N - D A P SO

1. Khcio sat sU hien thien va ve do tlij (C) ciia ham
giao diem ciia do thi

)

va liiic hoanh

la

thi (C^,

o

x~ + 2x + m - 2 = ()c6 hai

)cat

Iriic

hoanh

A' = 3 - m > ( )
<=> m <

+2(-i)fm-2^()

; ,

4 " + 4 < 2 - " " " - 3.2" o 2 - " - 3.2" - 4 > ( ) = > 2 " > 4 = > x > 2 .
Cau III. Tinh the tich khd'i chop...

T i n h A B = AC =

, SA =

/I'M

, SABC

• The tich khoi chop can tim la V =
'

a•^^/^

36

a-73

,,;v

12
(dvtl).

1. Viet phidtiiii trinh mat can (S) tarn I va tiep xuc vdi

lai 3 diem phan biet
nghicMii

d]

2

Cau IVa.

x'^ + 3x- + mx + m - 2 = 0 <=> (x + l)(x^ + 2 x + m - 2) = 0
do

.)

m i n r(x) = r ( - l ) = - l - e " - .

Rx) = 1( - In 7 2 ) = - I n x / 2 - - ;

d]

.sv;'...(doe g i a tif g i a i ) .

2. Tim ccic i^id trj ciia m de do thi ham so cat true hocinh...
hoanh do

max

V i 4 " + 4 > 0, Vx nen ba't phiTctng trinh da cho tifdng diTdng

.

(P).

V i mat can (S) tiep xiic vc'li (P) nen ban kinh R cua (S) la
2-4-3-4

phan biet khac - 1

R=d(l,(P)) =

3.

3

= 3 (dvcd).

Phifdng irinh mat cau (S) can l i m la (x - 1)' + (y - 2)" + (z - 3)^ = 9.
2. 77;;; toa do tiep diem ciia mat can (S) va mat pluuiii

C a u 11.

(P).

PhiTcmg irinh tham so ciia difttng thang d qua 1 vuong goe v d i (S) la

1. 77/;/; tich phan...

x = l + 2t

dx
Dat t = Inx ihi dt =
D o i can: x = I

t =

d:jy = 2-2l, tex.
0, x =

e => t =

,

z = 3- t

I.

Toa do tiep d i e m A can tim la nghiem ciia he phu'dng trinh:

Do do:
____

1

1

I = Jl%/l7ldt-|(t + l)^/^+Tdt-jVrMdl=
I)

^-liwfl.

• Chu'ng niinh tam giac A B C can tai A

C a u I.

I

< 0 nen ham so dii cho dat ciTc dai tai x

3. Gidi hd't phUcfni^ trinh...

C a u Vb. (1,0 diem)

(-1)-

Ins/i) = - 2

X6[-I;

2. T i m khoiing each giffa d i e m A vii difctng thang d.

PhiAing Irinh

-

,

Trong mat phang Oxyz cho diem A ( l ; 2; 1) va di/dng thai

j X _ y - l _ z + 3

De

. Tim
77 f^id tri Win nhdt va f^id trj nhd nhdt ciia ham so...
. T a p x a c d i n h D = | - l : ()|.

0

0

4 4 v / 2 - 16
15

x - l + 2t

x = l + 2t

y-2-2t

y-2-2t

y = 2-2t

z = 3-1

z = 3 -1

z = 3-t

2 x - 2 y - z - 4 = ()

2(l + 2 t ) - 2 ( 2 - 2 t ) - ( 3 - t ) - 4

X -

+

+-7(1
^

+ 1)

1 + 2t

A(3; 0; 2).

= 0

t = l


Tuy^n chpn 39 cie t h u

hyc ki nion Toari lop 1J MSng cao - Ph^m Trpng Thu

Cty TNHH MTV DWH Khang Vijt

Cflu Va. Tim tap lu/p ciic diem tren m^t phdnff toa do bieu dien cdc so phiic z ...
O i

Goi z = x + yi (x, y G R)
Ta

C O /. +

3 - 4i = 2 <=> (x + 3) + ( y - 4 ) i = 2 0 J(x + 3)^ +(y-4f

so 12

THI H O C K i II M d N T O A N L 6 P 1 2
TRUCtNG

THPT C H U Y ^ N

=2

Tap hctp cac diem bieu dicn so phiirc z la di/itng iron c6 phU'cJng irinh:

TRAN H U N G D A O - BINH

T H U A N

2x +1
C&u I. (3,0 diem) Cho ham so y =

•

^

1. Khao sat siT bicn thien va ve do thj (C) cua ham so tren.

(C): (x + 3)2 + ( y - 4 ) 2 = 4 .

2. Tinh dien tich hinh phang gidi han bcli (C), true ho^nh va tie'p tuyen (C)
CSu IVb.

taiA(-2;l).

1. Viet phU(/nfi trinh mat phdn}- (P) di qua A va dui'/ni^ thdnff d.

Cau II. (2,0 diem)

Du-cJng lhang d di qua diem M((); I ; - 3 ) v a c6 VTCPuj =(3; 4; 1).
1. Tinh tich phan I = j x | ln(x +1) +

Taco A M = ( - l ; - l ; - 4 ) v a li^,. A M =(-15; 11; 1).

XN/X''+8

dx.

Mat phang (?) di qua A ( l ; 2; 1) va nhan lip = i i j , A M =(-15; 11; 1) la
2. Tinh tich phan

-dx.

J=

VTPT CO phir(fng Irinh 15(x - 1 ) - 1 l(y - 2) - l(z - 1 ) = 0 o 15x - 1 l y - z + 8 = 0.
c&u III. (2,0 diem)

2. Ttm khocini^ each f;iifa diem A yd dif
1. Goi Zj, Z T , lii hai nghiem phtfc ciia phiTctng trlnh z ^ - ( 4 - i ) z - l - 5 - 5 i = 0.

Cong thuTc khoang each
, AM
d =

347
26

(dvcd).

Tinh gia tri cua bieu thuTc T =

C&u Vb. 77/?; cdc so phiic z thoa man...

^1

7.2

2

Z,,Z2

2. Tim so phiirc z c6 modun bang 1 va ( z - i - 2 i ) ( z - I ) l a so thifc.

Goi z = x + iy ( X , y e R)

Cfiu IV. (3,0 diem) Trong khong gian Oxyz cho hai diem A(l;2;-1), B(3; 0; 5)

Ta C O

v^mat phang (P): x - 2 y + 2 z - 1 0 = 0.

z^ = - 1 5 + 8 i c ^ x ^ - y ^ + 2xyi = -15 + 8iz:>r^"y^
2xy = 8
Giiii he (*)ladirc
<=> {

xy = 4

x =l

iy = 4

"~'-'^(*)

1. Tim giao diem I ciia diTcJng lhang AB vdi mp (P).
2. Viet phi/dng trinh mat phang (Q) .song song mp (P) va cich deu A, B.
3. Tim toa do diem C tren mat phang (P) sao cho lam giac ABC can tai C v^

hoa
lacy = -4

c6 dien tich bkng l l \ / 2 .

HUdNG D A N - D A P S6

Ket hdp v(^i dieu kion phan ao du'dng la tim dU'cJc z = 1 + 4i.
Cfiu I.

1. Khdo sat su hien thien vd ve do thi (C) da ham so...(doc gia tif giai).
2. Tinh di?n tich hinh phdn^ ^i(?i han bi'ti (C). true hodnh vd tie'p tuyen, (C)...
Tac6 y = f(x) = ^ ^ = > f ' ( x ) = ^
x-1
(x-1)'
PhircJng trinh tie'p tuyen d cua (C) tai A ( - 2 ; l)lh y = f'(-2)(x + 2) + f ( - 2 )
2IX

219


Tuygn chqn 39 6i

thif sijfc

hpc ki m6n Jo&n I6p 12

Nang cap

Cty TNHH MTV DVVH Khang Vi$t

- Phgm Trgng Thu

2 3
2 f 2
a c6 B = - u du = - u
3 J
9
, 3, ,
287
ay I = - l n 3 +
•^
2
36

<=>y = - - ( x + 2 ) + l = - - x + - ^
3
3
3
• Goi B la hinh chicu cua A Icn Ox.
(C) cat Ox laiC

74
9

f/Aj/i ticli phdn...

. d c a t O x t a i D d ; 0).

In:

Dien lich hinh phitng can tim la:
1

1
"2/

2x + l

X-1

-2

; Ji
2+x-l

Ta

C O

c

-

c -dx
- » 1
0 c" +
+1

J=

dx
1
Diit t = c + — + 1 =>dl = c " -

- - - ( 2 x + 3 1 n | x - l | ) 2 = 3 - 3 I n 2 (dvdt).

Doi can: x = 0 thi t = 3; x = In 2 thi t = - .
2
7

CauII.
1. Tinh tick plicin...

Ta c6: J ~ f— = In
J t
3

Taco: I = | x l n ( x + l)dx + j x - V x ' ' + 8dx
1
i
2
+ Tinh A = xln(x + l)dx.

Dat

I. '///)// i^id tri cihi hieII thi'rc...
Bictso A = ( 4 - i ) -

• T i r z j = 3 + i suy la z, = \/l7).

V =•

Ta c6:

2

2J

X-1 +

x+1

4 ( 5 - 5 i ) = 1 2 i - 5 = (2 + 3i)-.

Phu'iJng triiih da cho c6 hai nghiOm la z, = 3 + i, Z T = 1 - 2i.

du = —!—dx
x+1

dv = xdx

A = — ln(x + l)
2
1

7
7
2 = l n — ln3 = l n - 3
2

6

Caulll.

1

u = ln(x + 1)

dx
c'^y

dx = 21n3 — ln2 —
2
2

— X + Inx + 1

• Tfr

Z-,

• Tif

Z|Z-,

=

Do do T =

1 - 2i suy ra z-, =

-

5 - 5 i suy ra z,z
1''2

10 + 5
5f2

77m so phiic z cd mddun...
Goi z = X + iy (x,y G !•?.) ^ z = x - iy

4

Ta c6: (z + 2i)(z - 1) = (x + (2 - y )i)(x - 1 + yi)
+ T i n h B = jx2\/x^+8dx.
- x ( x - l ) - y ( 2 - y ) + (xy + ( 2 - y ) ( x - l ) ) i

1

Dat u =

N/X^TS => U- = X^ + 8 => 2udu = 3x^dx

Doi can: x = 1 Ihi u = 3; x = 2 thi u = 4.
220

x^dx = y - d u .

li

= X" + y" - x - 2y + (2x + y - 2)i
221


T u y « n chQn 3 9 i j

thCf sijfc

Cty TNHH MTV DVVH Khang Vigt

hpc ki mOn T o i n Iflp 12 Nang cao - P h j t n Trgnq Thg

3
2

Theo gia thict ta difdc

2

1

+ y =1
2x + y - 2 = ()

X

X =

1

<=>(18t2 - 1 2 t + 2 4 ) . l l = ( l l V 2 ) - c o l 8 t ^ - 1 2 t + 2 = 0

X = —

5

hoac

<=> 9t^ - 6 l + 1 = 0 <=> (31 - 1)- = 0 o t = - •

l y = ()

3

10

Vay C

4.
V a y z = l , /. = - + - ! •
3

3 ' 3

cauIV.

D E T H I H O C K I I I M O N T O A N L 6 P 12

O i SO 13

1. rim f^iiii) diem I ciia difoiiii thunji AB vi'fi mp (P).

S 6 GIAO DUC VA D A P TAP QUANG BIIMH

x = l+t
Phmng irinh tham so ciia dufdng thang A B la y = 2 - t

,teE.

z = - l + 3t

x = l+t
y = 2-t

y = 2-t

H

-11

x - 2 y + 2z-10 = 0

1. Khiio sat sir bie'n thicn va ve do thi (C) cua ham so'.
2. V i c t phu-dng trinh tiep tuyen cua (C) tai d i e m M ( 2 ; - 2 ) .

X = —

3

z - - l + 3 t = ^ i y = - - Vay

z - - l + 3t

diem)

CSu I . (2,5 diem) Cho ham so y = -x"^ + 3x c6 do thj (C).

Toa do giao diem 1 ciia diTcfng lhang A B vdi mat phang (P) la nghiem cua he:

x =l+ t

I . P H A N C H U N G C H O T A T C A T H I S I N H (7,0

C f i u I I . (1,5
I

13'

3'

J

diem)

1. Tinh ti'ch ph an A -

Z=:4

J :3x^

+ 2 x - - dx

I ^

~ 9
2. T i n h tich phan B = J x^ix^ + Idx.

2. Viet phmrii'

trinli mat phdnfi (Q) som^ s
Vi (Q)//(P) nC-n (Q): x - 2 y + 2z + D = 0 ( D ^ - 1 0 ) .

CSu I I I . (3,0 diem) Trong khong gian Oxyz cho bon d i e m A ( l ; 2 ; l ) , B ( - 2 ; l ; 3 ) ,

D - 5

D + 13

Jl2+(_2)2+22

>/l2+(-2)-+22

Theo gia i h i e l d ( A ; Q) = d(B; Q) o

' D - 5

= D + 13

D-5 = -D-13

^

o

.

D = -4.

Vay ( Q ) : x - 2 y + 2 z - 4 = 0

C(2;-l;l), D(0;3;l).
1. V i c t phiTctng trinh mat phang ( B C D ) .
2. T m i toa do hinh chicu cua A tren mat phang ( B C D ) .
3. T\m toa do d i e m doi xu'ng cua A qua mat ph^ng ( B C D ) .
' I . P H A N R I E N G (3,0
^hisiiih

3. Tim toa do diem C tren mp (P) sao cho tam aide ABC can tai C...
Goi C ( 4 - 4 t ; - 3 - t ;

t)e(P)

V I A A B C can l a i C nen C H 1 A B
V d i H(2; 1; 2) la trimg diem cua A B
T a c o : C H = ( 4 t - 2 ; t + 4; - t + 2), A H = (1; - 1 ; - 3 ) .
Theo gia thict:
S^BC = 11V2 c > 2 S^ A^ Cp H„ = 1 1N^ o C H . A H = I IN/2

diem)

chi diMc lam mot trong haiphdn

(phdn A hodc phdn li)

Theo chifc/ng t r i n h ChuS'n
•^Su I V a . (1,5 diem) Giai phifdng trinh sau tren C (an z): z"* - 1 = 0.
'-^u V a . (7,5 diem) T i n h the tich khoi Iron xoay

han bdi cac diTcJng:

y = x^ - 3x + 2 va y = 0, khi quay quanh true Ox.
^ - T h e o ehil'(/n}» t r i n h N a n g cao
^'^u I V b . (1,5 diem)

G i i i i phi/dng trinh .sau tren C (an z):
4z

+ I

z-i

222

s

5

^
z-i

+ 6 = 0.

-,


Cty TNHH MTV DWH Khang Vi

Tuy^n chpn 39 at th& siitc hpc kl tnOn toin I6p 12 Nanp cao - Phgm Trpng Thi/

3. 77m toa do diem ddi xan}- ciia A qua mdt phdnfi

Cflu Vb. (1,5 diem) Tinh the tich khoi t r 6 n xoay gidi han bdi cdc diTdng:
4y =

(BCD).

Goi A' la diem doi xu-ng cua A qua mat phing (BCD). Khi d6, H la truni

y = X , khi quay quanh true Ox.

V = 2 x H - x ^ = f - l = |

H U d N G D A N - D A P S6

diem cua AA'nen ta c6:

Cflu 1.
1. Khdo sat su bie'n thien va ve do thi (C) ciia ham so... (doc gia tU" giai).

y A ' - 2 y H - y ^ = ^ - 2 = ll.

z^,=2z^^-z^ = 22

Vay:

A'

]1 ±
[r r 1

4

2. Vii't phUcfna trinh tiep tuyen ciia (C) t(^i diem ... (dap so y = -9x +16).
Cflu IVa. Gicii phu</nf> trinh ...

Cflu I I .
1. Tinh tich

phdn...

\^
/
1^
/
\
T a c 6 : A = j 3 x ^ + 2 x — dx = (x''+2x-ln|x J
X

2. Tinh tich

T a c d : z^ -1 = 0

Dod6: B = ( t ^ - l ) t ^ d t = J(t^-t^)dt =
I

X
58

3

5

-3x + 2 = 0<=>

x =l
x =2

Khi do the tich cua khoi tron xoay cam tim la:

15'

2

2

V = 71 J(x2 -3x + 2)2dx = 71 f(x^ -6x^ + 13x2 -12x +4)dx
(BCD).

T a c 6 : B C = (4; - 2 ; - 2 ) , BD = (2; 2; - 2 ) , [BC,BD] = 4(2; 1; 3).

Mat ph^ng (BCD) di qua B(-2; 1;

= 7t

nhan ti = (2; 1; 3) l^m mOt VTPT,

do d6 c6 phirong trinh m 2(x + 2) + ( y - l ) + 3(z-3) = O o 2 x + y + 3 z - 6 = 0.
2. Tim toa do hinh chieu ciia A tren mdt phdng

'x^
5

3x^ 13x^ , 2
'
—+ ——-6x^+4x
2
3

Cflu IVb. Cidi phUcfng trinh...

Dat t =

(BCD).

PT dirdng th^ng A di qua A ( l ; 2; 1) va vuong g6c vdi mSt phing (BCD) la:

4z + i
Z-1

Khi do phUWng t r i n h da cho trcl t h a n h t^ - 5t + 6 = 0 o

= 1 + 2t

' y = 2 + t , teR.
z = 1 + 3t

•

Toa do hinh chieu H cua A tren mat ph^ng (BCD) la nghi?m ciia h$:

z = l + 3t
2x + y + 3 z - 6 = 0

HO'

Phu-dng trinh hoanh do giao diem cua hai du-cJng y = x^ - 3x + 2 va y = 0 la:

Cflu I I I .

y=2+t

z = ±i

Cflu Va. Tinh the tich khoi tron xoay fiit'n han bdi cdc dudng...

Ddi c$n: x = 0 thi t = 1, x = >/3 thi t =2.

x = l + 2t

z = ±l

Vay phifdng trinh da cho c6 nghiem la z = ±1, z = ±i.

Dat t = Vx^ + 1 => t^ = x^ + 1 => xdx = tdt.

X

z2=l

= 1 0 - I n 2.

phdn...

1. Viet phUtmfi trinh mat phan^

( z ^ - l ) ( z 2 +1) = 0 o

^JtH.ll
^ 7 ' 14' 14^

.

Vdi t = 2 , t a c 6 l i t i = 2 o
z-i

4z + i = 2z - 2i
O Z

=

t=2
t = 3'

3.

1.

2

V(?it = 3 , t a c 6 ^ = 3 o ^ ^ ^ ^ = 3^-3' o z = ^ i .
Z;z-i
Vay phiTdng trinh da cho c6 nghiem la z = - - i z = -4i
2


TuySn chpn 3 9

thCf s a c hpc ki mSn Toan Idp 12 NSng cao - P h ? m Trpng Thii

C&u V b . Tinh the tich kiwi tron xouy

1"

C t y T N H H MTV D W H Khang Vi$t

luiii hdi ccic dutrnji...
2. Tim toa do d i e m A ' la d i e m do'i xi'fng ciia A qua (P).

Phi/dng trinh hoanh do giao d i e m ciia hai di/dng cong da cho 1^

3. V i e t phu'cing trinh mat phring (Q) chi'fa O A vii vuong goc (P).
"x = ()
= X

4. V i e t phu'ttng Irinh mat cau (S) cd ban kinh R =:3va tiep xiic vdi (P) u i i

O

x=4

?

M ( - 5 ; 1; 1).

K h i do the tich ciia khoi Iron xoay can tim l i i :
V =

7t

j

dx = 71

x ^ ^ 4 _ 128

x-^
3

16

«0

()~

B . T h e o chU'C a u V b . Trong khong gian vdi he loa do Oxyz cho 2 difdng thang chco nhau:

71 (dvtt).

15

x = 1 +I
d, :

DE SO 14

Iy

=0

z =

DE THI HOC KJ II MON TOAN LdP 12

x=0
(I e x )

va d-, :

-5 +1

y = 4-2t'

( t ' e iR).

1/

z = 5 + 3l'

1. V i e t phu'cing trinh m a t p h i l n g ( a ) chifa d, va song song v d i d ^ .

TRUdNG THPT TRAM N G U Y E N H A N - PHU Y E N

2. V i e t phifctng Irinh mat cau lam O, liep xiic vdi d , .
I . 1 ' H A N C H I J N ( ; C H O T A T C A T H I S I N H {6,0 diem)

3. T i m loa do diem H la hnih chieu vuong goc ci'ia diem O Icn du'dng thang d , .

C a u I . (3,0 diem) Cho ham so

4. V i e t phu'(tng Irinh dirdng lhang A di qua A ( - l ; 2; 0) vuong goc vdi d, va

y = -x'' +

.

1. K h i i o sal sir bien i h i c n va ve do thj (C) ciia ham so tren.

cat d ^ .

2. V i e t phifdng irinh tiep tiiyen ciia do thi (C) biot tiep tuyen song song vdi
difdng thang d: y = - 2 4 x + 2011.

i

HLTdNG DAN - DAP SO
Cau I .

C a u H . (1,0 diem)

.» i . Khcio
T i n h tich phan I = f .

,;Vx + 4

C a u m.

dx •

Slit

sir hicii thien va ve do thi (C) ciia ham .so ... (doc gia tii" giiii).

| : 2 . PhifOu}' iiinh tiep tuyen cihi do tlii (C) hiet... (dap so' y = - 2 4 x + 40).
C 5 u I I . Tinh tich

(1,0 diem)

phan...

Dat I = 7x + 4 r:> I - = X + 4 :;:> dx

Tim modiin ciia so phifc z biet: (1 + i)z - ( 4 + 7i) = 2 - 5 i .

D o i can: x ^ 0 => l = 2. x - 5

2ldl.

i = 3.

C a u I V . (1,0 diem)
Do do: 1 = ' ^ ' ' ' - ' ^ 2 ^ d i = 2 ' f ( r - 4 ) d t = 2 - - 4 t
i
3

Xac djnh m de ham so y = x'' - 3(m + I ) x - + 9x - m dat cifc tri tai x , , X j sao
cho x [ + x ; < 10. V i e t phiTctng Irinh difdng thang di qua hai d i e m ciTc t r i .

1

C a u I I I . 77/;; niodun ciia .\o phi'fc z hiet...
Ta cd: (1 + i)z - ( 4 + 7i) = 2 - 5 1

I I . P H A N K I K N G (4,0 diem)
Thi sink chi diMc lam mot troitfi hai plidn (phdii A hogc phdii B)

<r>

6 + 2i

z ^ 1+i

(1 + i)z = 6 + 2i

(6 + 2 i ) ( l - i )
•= 4-2i.
2

' A . T h c o chir(/n)i t r i n h C h u a n
Cfiu V a . Trong khong gian v d i he toa do Oxyz cho d i e m A ( - 2 ; - 1 ; 2) va mat
phiing ( P ) : x - 2 y + 2z + 5 = 0.
1. V i e t phifctng Irinh duTlng liiang d di qua A va vuong goc v d i mat p h i n g (P)
226

|

Suy ra

|z = y^A- + ( - 2 ) - = 2s[5.

Cau I V . Xdc dinli m de luini so...
•

Tap

•

Dao ham y ' ^ 3x" - 6(m + l)x + 9

Xiic

dinh: D ^

3 _ 14
2~

3


f
T u y g n c h p n 3 9 d g t h i f sijfc h q c k l m O n T o A n \6p 1 2 N&rtg

cao -

Phgm Trgng

Cty T N H H M T V

Thg

n = OA, n.

• Ham SO dat cure trj tai x,,X2<=>y' = 0 c6 hai nghi^m phan biel x, va X j
:=> 3x^ - 6(m + l)x + 9 = 0 c6 hai nghi^m phan biet x, va X 2
o A' = 9m^ + 1 8 m - 1 8 > 0 < » m ^

DWH

Khang

Vl»t

= (2; 6; 5).

, Phi/dng trinh mat phing (Q) can ilm la phiTcfng trinh mat phing di qua diem
^ va CO VTPT n, phiTcJng trinh (Q): 2(x + 2) + 6(y +1) + 5(z - 2) = 0

+2m-2>0

o 2 x + 6y + 5z = 0.

< = > m < - l - V 3 hoac m > - l + >/3 (1)
Theo dinh l i Vi-6t, ta c6: x, + X j = 2(m +1), x , X 2 = 3.

.

4, Viet phU(fn!i trinh mdt cau (S) co ban kinh...
, phi/dng trinh dtfcJng thing A qua diem M va vuong goc mat phing (?) la:

Khi d6:

( X j + X 2 ) ^ - 2 x , X 2 < 10 o

x f + x^ < 10 o

4(m + 1)^ - 6 < 10

X=

<i>(m + l)^ < 4 c : > - 2 < m + l < 2 c ^ - 3 < m < l (2)

y = l-2t,teK.
z = l + 2t

T i l f ( l ) v a (2) ta di/cfc:

-3.

hoac - l + > , ^ < m < l thi x f + X 2 < 1 0

T a c 6 : y = f'(x) - x - - ( m + l) + (-2m^ - 4 m + 4)x + 2m + 3

V i f ( X j ) = f ( X j ) = 0 nen gia tri ctfc trj cua ham so la

. Vi tam I cua mat cau (S) thuoc diTdng thing A ndn I(-5 +1; 1 - 2t; 1 + 2t)
(S) tiep xuc (?) o

• Do do, phtrcfng trinh di/dng thing di qua hai diem cifc tri la:

-5 + t - 2 ( l - 2 t ) + 2(l + 2t) + 5
:
.
= 3 o 9t = 9 o t - ± l .
Vl + 4 + 4

+ Vdi t = 1 thi tarn cua mat cau (S,)can tim la I , ( - 4 ; - I ; 3)

f ( X j ) = (-2m^ - 4 m + 4)x, + 2 m + 3, f ( x 2 ) - ( - 2 m ^ - 4 m + 4 ) X 2 + 2m + 3.

=>Phu-dng trinh mat cau (S,): (x + 4)^ + (y +1)^ + (z - 3)^ = 9.
+ Vdi t = - l thi tam cua mat cau (S2)cantimia l2(-6; 3; - 1 )

y = (-2m^ - 4m + 4)x + 2m + 3.

PhiTdng trinh mat cau (S2): (x + 6)^ + (y - 3)^ + (z +1)^ = 9.

Cfiu Va.

Cfiu Vb.

1. Viet phUcfn^ trinh dudn^ than}!; d di qua A vd vuonfi ^vk- vdi mat phdn^...

!• Viet phU(m}i trinh mat phdnfi...

PhiTdng trinh diTcJng thang d di qua A va vuong goc vdi (?) la:
X

= -2 + 1

• Dirdng thang d, di qua M , ( l ; 0; - 5 ) vS c6 mot VTCP u = (l; 0; 1).

• Mat phang (a) chdra dj va song song vdi d2 nen (a) di qua M j va c6 mot

z = 2 + 2t
2. 77m toa do diem A' la diem doi xiin^ cua A qua (P).

VTPT n = [ u , v ] = (2; - 3; - 2), phiTdng trinh (a) la:
2 ( x - l ) - 3 y - 2 ( z + 5) = 0 o 2 x - 3 y - 2 z - 1 2 = 0.

Goi H la hinh chieu vuong goc cua A len (?). Toa do H la nghi?m ciia he
phtfdng trinh:
y= -l-2t
z = 2 + 2t
x - 2 y + 2z + 5 = 0

• ,

• Dirdng thing d j di qua M 2 ( 0 ; 4; 5) va c6 mot VTC? v = (0; - 2 ; 3).

- y = - l - 2 t , teK.

x = -2-t-t

-5 + 1

2- Viet phuoni- trinh mCit cau tam O, tiep xiic vdi...
x = -3
y =l

' PhiTdng trinh mat cau (S) tam 0(0; 0; 0) bin kinh R c6 dang x^ +y^ +z
Vay H(-3; 1; 0)

z=0

A' doi xurng ciia A qua (?)<=> H ' la trung diem cua AA' => A'(-4; 3; - 2 ) .

O M , ii
Mat cau (S) tiep xuc vdi d j nen R = d ( 0 , d , ) o R =

' Vay

=Vii.

(S):x2+y2+z2=i8

3. Viet phUcmg trinh mat phdnf,' (Q) chiia OA vd vuon^ HOC (P).
.

Mat phing (Q) chura OA va vuong goc vdi (?) nen c6 1 VT?T
T 1 Q


Cty TNHH MTV DVVH Khang Vigt

Tuyfi'n chpn 39 66 thCf sutc hpc ki mfln Tocin I6p 12 Nang cap - Phgm Trpng Thu

3. Tim toa do ctiein H la liinh cliie'u vuonf^ f^ov ciia diem O ten dutlnfi thcwj^...
•

H la hinh chicu vuong goc cua O len d[ ==> H e d | => H ( l + 1 ; 0; - 5 + t ) .

.

Vi OH

p .
4. Viet
•

1

1 9
4
Cho cac so ihifc du'dng a, b, c thoa man — + — + — = I . D a l
a
b
c
la giii tri nho nhat ciia bieu ihiiTc P = 4a + b + 9e. T i m nghiem ciia phUttng

C 3 u V . (1,0 diem)

u => OH.u = ( ) i z > l + l - 5 + t = 0 = > l = 2=> H(3; 0; - 3).

phUifniL'

trinh dudn}^ thcinf'...

trinh

G p i (P) Iii mat phang di qua A va vuong g()c vc'Ji d, ncn (P) c6 V T P T
ri^=(l;

0; l ) . S u y r a phifdng Irinh mal phang ( P ) : x + / . + 1 = 0.

121

1 + tanx

72si

smx

1 + cot x

C&MW.

= Pmm .

(2,0 diem)

' '

1. Trong mat phiing vdi he toa dp Oxy cho d i c m A ( 2 ; - 1 ) vil dUctng lhang
.

G o i B = d . n ( P ) , B e d o ^ B ( ( ) ; 4 - 2 l ' ; 5 + 3t').

•

M a B e (P)

•

During lhang A can l l m la difOJng ihiing di qua hai d i c m A va B . DiTcIng

l ' = - 2 =0 B((); 8; - 1).

5 + 3l' + 1 = 0

X =

Ihiing A CO phiTdng Irinh:

-1 + s

d : 3x + 5y - 7 = 0. V i e l phiCdng trinh difcing thiing qua A va lao vc'iti d mot goc
bang 45".
2. Trong khong gian vc'Ji he toa dp Oxy/„ cho d i e m A ( l ; 1; 2) vii mat phdng
(P): X + y + z + 1 = 0 . M o t mill phiing song song \(U (?) vii ciit hai tia Ox, Oy Uin

y = 2 + 6s,

se

3
lu'dl lai B, C sao cht) lam giac A B C eo dien tich bang — (dvdt). V i e t phu'Png

/ = -s

Irinh mat phiing do.

D E S O 15

D E K I E M TRA MON TOAN L 6 P 12

, Cfiu V I I . (1,0 diem)

T R l / d N G THCS &THPT N G U Y E N K H U Y E N TP.HCM
T i m he so' kUn nhat trong cac he so'ciia khai Irien

C a u I . (3,0 diem) Cho ham so

y = x"' + n i \ 2.

X

7

7

HU6NG DAN - DAP so

1. Khao sal sif b i c n ihic-n va vc do ihi (C) ciia hiim so k h i m = 3.
2. T m i la't ca cac gia tri ciia tham so m dc do thi ham so cilt true hoilnh lai

2

—+—

Caul.
1. Khdo sat sU hien thicn va vc do thi (C) ciia ham .sY;',..(dpc gia l i f g i i i i ) .

mot d i c m duy nha'l.

2. Tim tat cd vac fiici tri ciia tham so m de do thi ham so cat true hoanh tai...

C a u I I . (2,0 diem)

•

Tap xiic dinh: D -

•

Dao hiim: y' = 3x" + m c6

•

Do Ihj ham so ciit triic hoanh tai mot d i e m duy nha'l khi va chi khi hiim so da

x.

1. G i a i phu-png trinh x" + 3(x - l ) \ / x - + x + 1 - 2x + 3 = ().
x ' ( x + 2y) + y ( 2 x - 5 y ) = 0
2. G i i i i he phi/dng Irinh

1

+ log

16 = 4 - l o g , y

A'---3m.

cho d(Jn dieu Iren :< hoiic dal hai ci/c trj y , , y , ciing phia v d i true hoiinh

log,2

'A'>()

<=> A ' < ( ) (1) hoiic
C a u I I I . (1,0 diem) T i n h tich phan

x-

(2)
y,y2 > o

+2x-2

ilx.

x-Vl

G i i i i (1): (1) o - 3 m

C a u I V . (1,0 diem) Cho tu-dien A B C D eo A B l ( B C D ) va A B = aV2. B i c l tam

< 0 <=> m > 0.

G i i i i (2):
^-3m > ( )

giac B C D c()BC = a, B D = a^/3

va (rung luyen

B M = - ^ ^ - Xae dinh tiim vii

tinh the lieh ciia khoi ciiu ngoai tiep ciia lU; d i c n A B C D .

(2)c^

m <()

l_m

1

.i

3
J

\

m

>()

~T

\

4

o

-3 < m < 0.

m ' + 27) > 0

27
J

231