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

Nối tiếp nội dung phần 1 tài liệu Tuyển chọn 39 đề thử sức học kì môn Toán 12 nâng cao, phần 2 giới thiệu tới người đọc phần đáp án và hướng dẫn giải chi tiết các đề thi thử. Mời các bạn cùng tham khảo nội dung hi tiết.

Tuygn chpn 39 dg thii siic hqc ki mOn ToAn I6p 12 NSng cap - Ph?m Trpng Tha

1 "inh 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

HU6NG D A N - D A P S O

C a u l

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

Vay phu'dng trinh dii cho c6 nghiem x = -3, - ~ 2

2 Gidi phUifns^ trinh

3 Gidi he pluOfnii trinh

vag'(x) = 3x'^ - 6 x + 4 + = 3(x-l)- + l+ J >() Vxe(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 = jS^^^^.SH = l^^L^ (dvll)

183

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

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

The tich ci'ia khoi can ngoai ticp ciia hinh chop S A H E K

Cty TIMHH MTV D W H Khang Vijt

D E S O 3 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 T i m gia tri Idn nhal, gia tri nho nhat ciia ham soy = ( x +

CSu I I I (1,0 diem) Giiii he phifting trinh:

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

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

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

canh day

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

2 Goi M la trung diem AB 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 " Goi D la trung diem canh CC'.Biet rang hinh chieu vuong goc ciia A ' IC-n mat phang (ABC) trung vdi Irong tam tam giac A B C Tinh the tich khoi lifdien ABCD

185

T u y i n chpn 39 6i thCt siic hpc ki mOn Toan I6p 12 Nang cao - Phgm Trgng Thi/

2 PhiTdng irinh lie'p luycn vdi do ihj (C) lai diem M (dap so y x - 2)

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

Nen max y = y(l) = 3\/3, min y = y(±2) = ()

x e | - 2 ; 2 | X 6 | - 2 ; 2 |

Cfiu I I I Gidi he phifcfiif- trinh

T a c o - y ^ + 12x^ +3xy^ =7x-^ +3x^y + 7x + y - 2

» ( x - y ) ' ' + ( x - y ) = ( 2 x - l / + ( 2 x - l )

X c l ham so f"(i) = I-'+1 iren R.,phi/dng irinh tren c6 dang l ( x - y ) = f ( 2 x - l )

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

Cty TNHH MTV DVVH Khang Vigt

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

Gpi N la irung diem ci'ia CD ^ A N ^ = 5, SN^ = 11

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

^5 + l2-n 4l5_

" 2.S.2S " 10

Suy ra (SA, C M ) = arccos

A N - + S A ^ - S N ^ 2AN.SA

10

V

CSu V Tinh the tivh khoi td dien ABCD

Goi H, K Ian liTdl la hinh chieu A', D len mp(ABC)

Suy ra H la trong tarn AABC va D K = - A ' H •

TuyS'n chpn 39 thCf sufc hpc ki m6n Toan I6p 12 Nang cao - Phgm Trpng Thg

1 Khiio sal sif bicn Ihicn va vc do ihi (C) cua ham so'

2 V i o l phu'dng Irinh licp luycn ciia do ihj (C), bic'l r^ng lie'p luyen do vuon

goc vdi diTcIng lhang c6 phu'dng trinh y = x

C S u HI (3,0 diem) Cho hinh chop S.ABC c6 day A B C la lam giac deu canh ;

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

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

1 Tinh Ihc lich ciia khoi chop S.ABC iheo a

2 Xac dinh lam I , ban kinh va linh dien lich val cau ngoai licp hinh cho

S.ABC Iheo a

3 Tinh the tich cua khoi chop A B C N M iheo a

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

Thi sink cliidiMc lam mot trong haiphdii (phdn A Itogc plidn I})

A Theo chir<yn« trinh C h u a n

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

x+ l

1 Giai phiTdng trinh l o g 2 ( 2 ' ' - l ) l o g 2 ( 2 ' ' " " - 2 ) = 6

2 Giiii bat phu-dng Irinh 2" + 2"""^' - 3 < 0

Cfiu y'A. (1,0 diem)

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

B Theo chif(/ng trinh Nang cao

cau IVb (2,0 diem)

1 Cho X la so'thifc am Chtfug minh

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

^^oi lam giiic vuong, irong do c - b 1 va c + b 1 mtif :

Churng minh rang log^.^^ a + log^._^ a = 2 \og^^^ a.log<._h a (*)

Cfiu Vb (1,0 diem) T i m gia trj \6n nhal va gia tri nho nhal cua ham so'

1 Khdo sat subien thien va ve do thi (C) ciia ham so {6oc gia liT giai)

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

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

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

• The X = 0 viio {1) ta difdc: b = 1 • The x = 2 vao (1) ta diTdc: b = 5

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

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

+ Goi E la trung diem ciia BC, luc do: A E 1 BC, /

A E la hinh chieu ciia SE Icn mp(ABC) 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 (ABC), A E 1 BC

189

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

N c n goc giiJa 2 mat phang (SBC) va ( A B C )

chinh la goc hOp b("<i hai difitng thfing A E va S E

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

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

N c n I lii liim mill ciiii ngoai ticp 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

R = IA = N/OA-+ AP- - J - A E - S A

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

3 Tinh the tick ciia khoi chop A.BCNM theo a

+ T'l CO- ^ s A M N _ SA S M SN _ SM SN

^^.PMC " SA • SB • SC ~ SB • SC

+ Trong tam giac SAB c6 SA^ = SM.SB

TiTcJng tiT trong tam giac SAC ta c6:

PhiTdng trinh ( I ) v i c l lai t ( l + t) = 6 o t " + t - 6 = 0 o t = 2 hoSc t := - 3 + Khi t = 2 t h i l o g ( 2 ^ - l ) - 2 < ^ 2 ' = 5 < : > x = log.,5

Khi 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 " + Vay phiTdng trinh da cho c6 2 nghicm la x = log2 5, x = 2 logj 3 - 3

2 Giiii hat phiOrni; trinh 2^ + 2"""^' - 3 < 0 {*)

Ta c6: {*) « 2" + 2 2 " " - 3 < 0 ( I )

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

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

Khi 0 < l < 2 thi 0 < 2" < I c> X < 0

Vay lap nghicm bat phi/cmg lrinhS = (-00; ( ) ) u ( l ; + o o )

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

Ti\p xac dinh I ) = x

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

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

2 Clnoxi^ minh

A p dung dinh l i Pilago la c 6 : a2 + b2 = c2 a2 = (c - b)(c + b) {**)

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

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

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 ihicn va vc do 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 Ihiing d song song V('<i di/^tng l h a n g y = - 3 x

va licp x i i c \(Vi d o I h i (C)

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

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

C a u H I (3,0 diem) Cho h i n h chop S.ABC c6 lam giiic A B C v i i o i i g l;ii 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

1 Chiifng minh rang lam giac SBC vuong lai B ; o:

2 Tinh I h c lich ciia khoi chop S.ABC;

3 Tinh 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;

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

CdulVii (2,0 diem)

1 Giai phi/dng i r i n h l o g 2 ( x 2 - 3 ) +1 = 21og4(6x - 10) ' '

2 Giai bal phiTOng I r i n h 9 " < 3""" + 4

^&u\d (1,0 diem) '

T i m gia Irj Idn nha't va gia tri 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

IJ Theo chi/<yng trinh NSnK cao

cau IVb (2,0 diem)

1 Giiii he phifoliig Irinh

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 = - 3 x - 1 )

cau I I

2 (x^ + 1)' 2x

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

2 Churnii minh ran^

Vay tarn giac SBC vuong tai B

2 Tinh the tich ciia khdi chop S.ABC;

• Tam giac ABC vuong tai B ncn

Cty TNHH MTV DVVHKhang Vi?t

II Dicn tich tarn giiic ABC: S^^^ = ^AB.BC = ^.l.>/3 = ^ (cm^)

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

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

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

Ban kinh mat cau R = - S C = - \ / s A - + AC" = — 1(1

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

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

Vay phu'(<ng trinh da cho c6 mol nghiemx = 2

2 Giai iuit plnfOni; trinli

Dat I = 3 \ >()

BPT da cho trtf lhanh l " - 3l - 4 < 0 o -1 < l < 4 => 3" < 4 <=> x < log3 4

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

Cau Va llni i^id trj l('fn nhat vd }>id tri nhd nlid't ciia hdni so ':

Tapxacdjnh !) = ((); 3|

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

Ta C O y' = 0 o x = - l g l ( ) ; 3|

x = 2

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

Vily max y = y(0) = 10, min y = y(2) = - 1 0

X f l l l ; 1| x £ | ( ) ; 1|

Cau I V b

Gidi he phUifnii trinh

Dieu kiC'n x 0 vii y ^ 0

195

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

V i j y he phufdng Irinh da cho c6 hai n g h i e n i (x; y) la (2; 2 ) , ( - 4 ; 4)

2 Gidi pliUdiiii tniih

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

Plnrcfng tnnh da cho ud lhaiih r - 3l - 4 = 0 <=>

t = - l (loai)

1 = 4

V('<i 1 = 4 Ihi 2 " " " " = 2- o x - - X = 2 o x = - 1 hoilc x = 2

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

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

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

1 - x Dao ham y' =

V2x - X "

Ta C O y' = 0 o X = 1

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

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

D E SO 6 D E THI HOC KI I MON TOAN LdP 12 S d G I A O D U C V A D A O T A P B E N T R E

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 (7,0diem)

X

3x-Cilii I (3,0 diem) C"ho ham so y = — + —— - x (1)

3 4

1 Khao sat si/ h i c n IhiC-n va ve do thi (C) ciia ham so (1)

2 V i e t phifitng trinh tiep l u y c n vc'Ji do ihj (C) b i c t r i i n g l i c p l i i y e n song soiv^

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

C a n \\ (2,0 diem)

\ T\m g i a - l r i 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 d a l cifc dai l a i x = 1

1 %

Cty TNHH MTV DWH Khang Vi?t

QS^u I I I (2,0 diem) Cho k h o i chop S A B C D c6 day A B C D Mx hinh thoi, canh

j,:-ing a, B A D = 6 0 " , S A = S B = S D = a

1 T i n h the lich ciia k h o i chop da cho

2 Xac dinh l a m 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) ' fUi siiih clii diMc lam mot troiifi hai pliaii (pbdii A hoac phdii li) * '

\ Theo cliiJ'(/n{» trinh C h u a n ,,; i j

CaulVa (2,0 diem)

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

1 Ciiai pliiAtng tnnh ( i ) khi m = 3

2 T m i gia tri m de phiTOng trinh (1) c6 hai n g h i e m phan b i c l x , , x-,lh6a man X | X - , = 8

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

Ham so da cho l i e n liic Iren doan | 1 ; e|

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 = -AC.BD = -.aV3.a = (dvdl)

Ta lai CO SH = VsA" - A H " - a - - (dvcd)

Vay V = - (dvtl)

2 Xcic <Jinh tarn va tinh ban liinh ciia mat can

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

.IS = SK.AS SH a •a

1 Gidi phi /cfniLi tnnh (1) khi m = 3

Khi ni = 3 la c6 logjX - 3k)g2X - 4 = 0 <=> log^x = -1

logoX = 4

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^ = log^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, IT va t, + h = 3 <=> 2m - 3 = 3 <=> m = 3

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

Trong ASOA vuong lai O, la c6: SO = VsA" - O A " = 7(2a)- - a" =

-AS-The 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

Dal I = 2^, I > 0 Phu-dng Irlnh Iren ltd ihiinh r - 4l + 3 0 o 1 = 1 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)

Dall = 2 \) Vdi xe[0; 21 ihi le|l;4|

YCBT o t- - 4l + 2m - 3 = 0 (='=) c6 nghiem trcn doan [1;4]

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

Difa vao bang b i c n thicn ta lhay (*) c6 nghiOm i r c n 1;4

1 7

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

2 2 ' •

CAu V b 77/;// the tick ciia khni non da chọ

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 l i u c la lam giac SAB dcu

2 2

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

S d GIAO DUC VA DAO TAO TIEIM GIANG

Ị 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)

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

1 K h i i o sat sii" b i c n ihiC-n va vc do thi (C) ciia h a m sọ

2 V i c l phifttng Irinh lic'p tuycn víJi 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 ÚJi hai d i e m cifc l i i ciia (C) c6 gia i r i nhi) nhal

C i i u Ụ (2,0 diem)

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

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

IrC-n doan |(); 2 |

C a u l I I r / , 5 ^ / ^ » ; ;

1 G i a i phiriing Hinh 6.9'' - 13.6" + 6 4 " = 0

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

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

vuong canli a, m i i l hen lao M'U day m o l goc 60" G o i M , N, P, Q liln hfdl 1.^

irung d i e m ciia cac canh SA, SB, SC, SD

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

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

200

Cty TNHH MTV DVVH Khang Vi$t

J,, I ' H A N R I K N ( ; (2,0 diem) fjiisiiili vlii diMc lam mot troiig hai phdn (pitdii A liogc p/idii li)

^ Theo clurf/nK trinh C h u a n

^(,ng song vt'li dúCing ihiing d: 3x - 4y = 0

\\ Theo chifofnj; trinh Nan}? cao

CSu V b (2,0 diem)

1 a) Cho h a m so log-, 14 = a T i n h log^y 32 theo ạ

b) Cho hiim so y = ế"''.Chii:ng minh n l n g ýcosx - y s i n x - y " = 0

X " - X - 2

2 V i e t phifitng l i i n h liép luyén ciia do thi h i i m so y = : — , b i e l l i e p x + 2

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

Hl/dNG DAN - DAP SO

Cau Ị

1 Klúu) sal sif hien thicn va vc do thi (C) ciia ham so {6C)c gia tif g i i i i )

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^

Toa do d i e m ci/c dai lii Ă(); 2) vii toa do d i e m cifc l i e u lii B(2; - 2 ) PluriJng trinh di/itng thilng A B lii y = - 2 x + 2 ;;: /

Ta C O M A + M B nho nhiít k h i vii chi k h i A , M , B lhang hang

Poa do d i e m M l i i nghic'm ciia he y = - 2 x + 2

Tini in dc phifdn}^ Innh

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

Y C B T c ^ (^') C O 2 n g h i e m phan b i e l diriJng J*'! '

201

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

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

O X = : - l

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

2 Gidi phidfnfi trinh

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

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

Tuy6'n chpn 39 thCf sCrc hpc ki mOn Toan Mp 12 IMang cao - Ph^m Trpng Thif

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

The y ' , y", y vao b i c u ihifc y'cosx - ysinx - y" la diTcJc dpcm

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

D E S O 8 D E THI HOC Ki I MON T O A N L d P 12

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

I P H A N C H U N ( ; C H O T A T C A T I U ' S I N H diem)

C a u I (3,0 diem) Clio h a m so y = -x"^ + IwvC - 2 (1)

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

2 T\m m d c h a m so' (1) c6 c i / c d a i l a i x = 0

C a u H (1,0 diem) Tim gia Iri k'Jn nha'l v a gia Iri nho n h a l ciia h a m so' y = •

t r c n doan [ - 1 ; 11 T i f do siiy ra

x - 2 cosa + 1

c o s a - 2 < 2 veil m o i a

C a u III (2,0 diem)

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

C a u I V (2,0 diem) Cho hlnh chop lam giiic d c u S.ABC c6 do dai canh day

bang a T a m giiic S A B vuong can l a i S

1 T i n h ihc l i c l i ciia k h o i chop S.ABC ihco a

2 Ttr B kc diriJng cao B H ciia lam giac A B C T i n h Ihco a ihc l i c h k h o i n'f

d i c n H.SBC iCr do suy ra khoiing each lir H den m a l phang ( S B C )

B T h e o chif(/nj» trinh Nang cao

C S u V b (1,0 diem) G i a i phiWng Irinh ^ l o g ^ x - 0 , 5 = Ktg^

C a u V I b (1,0 diem) T i n h A = ( 0 , 5 ^ ) ^ log^ ~ •

204

HUdNG D A N - D A P S O

C f i u l

1 Kluio sat sir hien thien vd ve do thi (C) ciia hum sY;' (doc gia liT giai) «i

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

V a y ni < 0 thcni man de

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

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

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

C a u I V

1 Tinh the tich khoi chop S.ABC theo a

Do S.ABC hinh chop dcu ncn cac mai ben

cua hinh chop h~i nhiJng lam giac bang nhaii

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

lich khoi UV dicn da cho Vj, = ^-SA.SB.SC ^

Tarn giac SAB viiong can lai S, canh huycn

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

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

^SBC

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

C a u Va Gidi phUdn^ tnnh

So vdi dicu kien thi phiTdng irinh da cho c6 nghicm la " = ^

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

Tap xiic djnh D = (1; +oo)

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

Dao ham y' = 1 2x - x ' + 2 x + 3

x ^ + 3 ( x - l K x ' + 3 )

y ' - 0 » X = - 1 (loai)

x = 3 Bang bie'n thien

Phi/dng trinh da cho viet lai la yjlog^ - ^ =^ ^'"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 VIb Tinh

l i i y f ' ! ; ! i ; i 3') l i f '.m: lio( ki m6n To^n I6p 12 NSng cao - Phgm Trpng Thg

D E SO 9 D E T H i HOC K i I M O N TOAIM L 6 P 12

T R U d N G T H P T A N M I N H - K I E N G I A N G

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 (1)

1 Khao sal si/ bicn ihicn va vc do ihj (C) cua ham so (1)

2 V i c l phu'dng Irinh licp Uiycn vdi do thi (C) lai diem c6 hoanh do x „ , biei

rang f'(x„) = 6

C a u I I (3,0 diem) Cho hinh chop S.ABCD c6 SA vuong goc vdi mill phang

(ABCD), day A B C D la hinh vuong, lam giac SAC can dinh A vaSC = 4a Gpi

M la Irung diem ci'ia doan SC

1 Tinh the tich khoi chop S.ABCD iheo a

2 Chifng minh M la lam mat cau di qua cac dinh cua hinh chop S.ABCD

Tinh dien tich mat ciiu nay theo a

3 Tinh khoang each tif diem 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 = •

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

Till sink chi dii{fc Idm mot trong haiphdn (plidn A hoqc phdn li)

A Theo chii'(/ng trinh ChuS'n

C a u I V a (2,0

1 Giai phiTdng Irinh 27.9" + 242.3" - 9 = 0

2 Giai bat phiTdng Irinh logj (2x) + ^ = < 5

1 Giai phu'dng trinh 21og5X +17 logj x - 9 = 0

2 Giai phu'dng irinh e" - 1 - ln(l + x) = 0

C S u V h (1,0 diem) T i m cac tiem can cua do thj cua harh so f(x) = 2 x ^ - x - 4

I Tinh the tich klioi chop S.ABCD tlieo a "^^^

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

Canh hinh vuong bang 2a {/.i'if

Th(5 tich kho'i chop S.ABCD la V^^g^^ = {sA-S^e^^ = - ^ a ' ^ (dvtt) J i,- iV:)

• Dien tich mat cau la S = 4nr- = 167ta^ (dvdt) ^

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

Tim cUc trj ciia liam so

Tap xac dinh D = M

—x" + 2x + 3 Dao ham y' = - , y' = 0 o x = - 1 hoac x = 3

c"

pB^ng bien thien

209,

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

V a y phiTrtng trinh da cho c6 n g h i c m x = - 3

2 Gidi hat phuimi^ trinh

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

V a y max f ( x ) = 1(3) = 28, m i n f(x) = f(4) = l

x e | l ; 4 1 X6|(); 2|

C f l u I V b

1 Gidi phu<ma trinh

CfyTNHH MTV DWH Khanfl Vi$t

V a y phi/dng trinh da cho c6 hai nghicm x = 1/5, x = 5~^

2 Gidi phifcfn^ trinh

1 + x Bang b i c n t h i c n

• TiV bang b i c n t h i c n cua ham soy = e" - 1 - l n ( l + x ) ta suy ra phi/cing trinh

da cho CO nghicm 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, lim y = - 0 0 ncn di/ilng lhang x = 1 la t i c m can drfng cua

Tuy^n chpn 39 dj thCf sufc hgc k1 mOn ToAn Mp 12 N3ng cao - Phgm Trgng Thu

DE SO 10 TRUdNG THPT CHUYEN HA ISIpl - AMSTERDAM DE THI HOC Kl II MON TOAN LdP 12

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

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 song v6\

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

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 = 2 5

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

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

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

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

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

+ (d|) c a t ( C ) t a i 0 ( 0 ; 0 ) + ( d , ) citt ( C ) t a i B ( 2 ; 8)

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

Cflu ITI

1 Ch 'rn,<^ ininh

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; 1; 4)

Taco M N = (1; 0; 1) va u,; U j = (-9; - 6 ; - 3 ) 3 > M N u,;u2

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

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 )

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

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

1 Gicii hat plufifiiii tnnh

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

<:r>4'' - 2 0 2 " + 6 4 < 0 < = > 4 < 2 " < 1 6 < r > 2 < x < 4

Tim so phijtc Goi z = x + yi (x, y € R ) -! ;;,

z-Taco z - i - 2 = V l O o ( x - 2 ) ^ + ( y - l ) 2 =10 (1) J;^ I

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

Vay z = 3 + 4i hoac z = 5

DE s o 11 THI HOC Ki li MON T O A N L 6 P 12

TRUdNG THPT CHU VAN AN - HA ISI6I

I PHAN C H U N G C H O T A T C A T H I SINH (7,0 diem) Cflu I (3,0 diem) Cho ham so' y = x'' + 3x~ + mx + m - 2 c6 do ihj (C„^)

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

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

Cflu II (3,0 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"

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

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

C&u Va (1,0 diem)

T i m t a p hctp cac d i e m I r e n m a t p h d n g t o a d o b i e u d i e n cac so' phiJc z th

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

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

C a u Vb (1,0 diem) ^

T i m e;ic so phiJc z thoa man z" = -15 + 8i vii z c6 phiin ao la so difdng

H U d N G D A N - D A P S O

C a u I

1 Khcio sat sU hien thien va ve do tlij (C) ciia ham 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

PhiAing Irinh h o a n h d o giao diem ciia do thi ) v a l i i i c h o a n h l a

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

De d o thi (C^, ) c a t Iriic h o a n h lai 3 diem phan biet

o x~ + 2x + m - 2 = ()c6 hai nghicMii phan biet khac - 1

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 max Rx) = 1( - In 7 2 ) = - I n x / 2 - - ; min r(x) = r ( - l ) = - l - e " -

3 Gidi hd't phUcfni^ trinh

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

4 " + 4 < 2 - " " " - 3.2" o 2-" - 3.2" - 4 > ( ) = > 2 " > 4 = > x > 2

Cau III Tinh the tich khd'i chop

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

a - 7 3 ,,;v T i n h A B = AC = , SA = , S A B C

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

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

2 - 4 - 3 - 4

R = d ( l , ( P ) ) =

3 = 3 (dvcd)

Phifdng irinh mat cau (S) can lim la (x - 1)' + (y - 2)" + (z - 3)^ = 9

2 77;;; toa do tiep diem ciia mat can (S) va mat pluuiii (P)

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

x = l + 2t

d : j y = 2 - 2 l , t e x ,

z = 3 - t Toa do tiep diem A can tim la nghiem ciia he phu'dng trinh:

X - 1 + 2t

y - 2 - 2 t

z = 3 - 1

2 x - 2 y - z - 4 = () A(3; 0; 2)

x - l + 2t

y - 2 - 2 t

z = 3 - 1 2(l + 2 t ) - 2 ( 2 - 2 t ) - ( 3 - t ) - 4 = 0

x = l + 2t

y = 2 - 2 t

z = 3 - t

t = l

Tuy^n chpn 39 cie thu hyc ki nion Toari lop 1J MSng cao - Ph^m Trpng Thu

Cflu V a Tim tap lu/p ciic diem tren m^t phdnff toa do bieu dien cdc so phiic z

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

Du-cJng lhang d di qua diem M((); I ; - 3 ) v a c6 VTCPuj =(3; 4; 1)

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

Mat phang (?) di qua A ( l ; 2; 1) va nhan lip = i i j , A M = ( - 1 5 ; 11; 1) la

VTPT CO phir(fng Irinh 15(x - 1 ) - 1 l(y - 2) - l(z - 1 ) = 0 o 15x - 1 l y - z + 8 = 0

2 Ttm khocini^ each f;iifa diem A yd dif<ynfi thdnfi d

Cong thuTc khoang each

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

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

t a i A ( - 2 ; l )

Cau II (2,0 diem)

1 Tinh tich phan I = j x | ln(x +1) + X N / X ' ' + 8 dx

2 Tinh tich phan J =

c&u III (2,0 diem)

-dx

1 Goi Zj, Z T , lii hai nghiem phtfc ciia phiTctng trlnh z ^ - ( 4 - i ) z - l - 5 - 5 i = 0

Tinh gia tri cua bieu thuTc T = ^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

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

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

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^

c6 dien tich bkng l l \ / 2

HUdNG D A N - D A P S6

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 )

219

Tuygn chqn 39 6i thif sijfc hpc ki m6n Jo&n I6p 12 Nang cap - Phgm Trgng Thu

T u y « n chQn 3 9 i j thCf sijfc hpc ki mOn T o i n Iflp 12 Nang cao - P h j t n Trgnq Thg

Theo gia thict ta difdc

1 K h i i o 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 )

C f i u I I (1,5 diem)

1 T i n h ti'ch ph an A - J : 3x^ + 2 x

-I ^

dx

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

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 ) ,

C ( 2 ; - l ; l ) , D ( 0 ; 3 ; l )

1 V i c t phiTctng trinh mat phang ( B C D ) s

2 T m i toa do hinh c h i c u 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 diem)

^hisiiih chi diMc lam mot trong haiphdn (phdn A hodc phdn li)

T h e o chifc/ng t r i n h ChuS'n

•^Su I V a (1,5 diem) G i a i phifdng trinh sau tren C (an z): z"* - 1 = 0

'-^u V a (7,5 diem) T i n h the tich k h o i Iron xoay han b d i cac diTcJng:

y = x^ - 3x + 2 va y = 0, khi quay quanh true Ox

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

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

4y = y = X , khi quay quanh true Ox

H U d N G D A N - D A P S6 Cflu 1

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

2 Vii't phUcfna trinh tiep tuyen ciia (C) t(^i diem (dap so y = -9x +16)

2 Tim toa do hinh chieu ciia A tren mdt phdng (BCD)

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

Cty TNHH MTV DWH Khang Vii

3 77m toa do diem ddi xan}- ciia A qua mdt phdnfi (BCD)

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

diem cua AA'nen ta c6:

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

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

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

X -3x + 2 = 0<=> x = l

x = 2 Khi do the tich cua khoi tron xoay cam tim la:

2

TuySn chpn 3 9 thCf s a c hpc ki mSn Toan Idp 12 NSng cao - P h ? m Trpng Thii 1"

C&u V b Tinh the tich kiwi tron xouy luiii hdi ccic dutrnji

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

71 (dvtt)

DE SO 14 DE THI HOC KJ II MON TOAN LdP 12

T R U d N G T H P T T R A M N G U Y E N H A N - P H U Y E N

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)

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

2 V i e t phifdng i r i n h tiep tiiyen ciia do thi (C) biot tiep t u y e n song song v d i

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

2 Tim toa do d i e m A ' la d i e m do'i xi'fng ciia A qua (P)

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

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

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 ^

2 V i e t phifctng Irinh mat cau l a m O, l i e p xiic v d i d ,

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

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

i HLTdNG DAN - DAP SO |

C a u I

.» i Khcio Slit sir hicii thien va ve do thi (C) ciia ham so (doc gia tii" g i i i i )

|:2 PhifOu}' iiinh tiep tuyen cihi do tlii (C) hiet (dap so' y = -24x + 40)

C 5 u I I Tinh tich phan

C a u I I I 77/;; niodun ciia \o phi'fc z hiet

Ta cd: (1 + i)z - ( 4 + 7i) = 2 - 5 1 (1 + i)z = 6 + 2i

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 c a o - P h g m T r g n g T h g

• 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

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

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

X = - 2 +1

- y = - l - 2 t , t e K

z = 2 + 2t

2 77m toa do diem A' la diem doi xiin^ cua A qua (P)

Goi H la hinh chieu vuong goc cua A len (?) Toa do H la nghi?m ciia he

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

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

, 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

o 2 x + 6y + 5z = 0

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:

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

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

Cfiu Vb

!• Viet phU(m}i trinh mat phdnfi

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

• Dirdng thing d j di qua M2(0; 4; 5) va c6 mot VTC? v = (0; - 2 ; 3)

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

2- Viet phuoni- trinh mCit cau tam O, tiep xiic vdi

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

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

' Vay ( S ) : x 2 + y 2 + z 2 = i 8

O M , ii

=Vii

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

• H la hinh c h i c u vuong goc cua O len d[ ==> H e d | => H ( l +1 ; 0; - 5 + t)

V i O H 1 u => O H u = ( ) i z > l + l - 5 + t = 0 = > l = 2 = > H(3; 0; - 3)

4 Viet phUifniL' trinh dudn}^ thcinf'

• 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 m a l phang ( P ) : x + / + 1 = 0

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

• M a B e (P) 5 + 3 l ' + 1 = 0 l ' = - 2 =0 B((); 8; - 1)

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

Ihiing A CO phiTdng Irinh:

X = - 1 + s

y = 2 + 6s, se / = -s

T R l / d N G T H C S & T H P T N G U Y E N K H U Y E N T P H C M

C a u I (3,0 diem) Cho ham so y = x"' + n i \ 2

1 K h a o sal sif b i c n ihic-n va vc do i h i (C) ciia h i i m 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

mot d i c m duy nha'l

l o g , 2

C a u I I I (1,0 diem) T i n h tich phan x - + 2 x - 2

x - V l

ilx

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

giac B C D c()BC = a, B D = a^/3 va (rung luyen B M = - ^ ^ - Xae dinh t i i m vii

tinh the lieh ciia k h o i ciiu ngoai tiep ciia lU; d i c n A B C D

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

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 O x y / „ cho d i e m A ( l ; 1; 2) vii mat phdng

(P): X + y + z + 1 = 0 M o t m i l l phiing song song \(U (?) vii ciit hai tia Ox, Oy Uin

3 lu'dl l a i B , C sao cht) lam giac A B C eo dien tich bang — (dvdt) V i e t phu'Png Irinh mat phiing do

, Cfiu V I I (1,0 diem)

T i m he so' kUn nhat trong cac he so'ciia khai Irien

HU6NG DAN - DAP so

— + —

7 7

C a u l

1 Khdo sat sU hien thicn va vc do thi (C) ciia ham sY;', (dpc gia lif giiii)

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

Tuygn chpn 39 di t\\!i site hpc ki m6n ToAn I6p 12 NSng cao - Phgm Trgng Thu

• Vay vdi m > -3 thi do ihi ham so da cho c^t Ox tai mpt diem duy nha't

c a u I I

1. Gidi phu<mf> trinh

Phi/dng irinh da cho viet lai x^ + x +1 + 3(x - l)Vx^ + x + l - 3x + 2 = 0 (*)

Vay phifcfng trinh da cho c6 3 nghiem: x = 0, x = - 1 , x =

2 Gidi phMrifi trinh

+ Khi y ^ 0 : T a c 6 : ( 2 ) o

3 /• \ 2 / \

X + 2 X + 2 X + 2 — + 2 — •5 = 0

Dat t = - , phi/dng trinh tren diTdc viet lai: t"' + 2t^ + 2t - 5 = 0

So vdi dieu kien ta suy ra: x = y = 2 Vay he phiTdng trinh da cho c6 nghiem: (x; y) = (2; 2)

CSu IV Xdc d 'mh turn rd tinh the tick ciia khoi cdu

Goi O la tam difdng tron ngoai licp ^ giac BCD, qua O diTng di/dng lhang

(•l) vuong goc vdi mat phang (BCD), khi do

la true ciia diCi^ng tron ngoai ticp tam giac

va (d) song song vdi AB

Trong mat phang ( A B ; d ) diCng diTdng

; trirc (A) cua d(X)n AB, (A) c^t (d) tai I B

Tuygn chQn 39 ij thil sdc hpc ki mOn ToAn Iflp 12 NSng cao - Phgm Trpng Thu

= > I B = IC = I D = I A

V a y I la l a m ciia mat can ngin.ii ticp ciia t i i ' d i c n A B C D

Trong tam giac B C D , ta ci):

G o i E la l i n n g d i e m ciia A B , khi do tif gi.ic O B E I l i i hinh chu" nhat, sny ra

ban kinh ciia mat can (S) la

sin X = 0 (loai) cosx

1 Viet pliU(fnfi trinh dudriji thcin^ qua A va tao vdi d mot f-oc

Phift^ng trinh di/clng thang A qua A ( 2 ; - 1 ) c6 V T P T n = ( a ; b) l a :

a ( x - 2 ) + b(y + l ) - ( ) o a x + b y - 2 a + b = 0 (a^ + b^ ^ 0 )

I T h c o gia Ihict: cos(A, d ) = cos45" = — o

3a + 5b

2 2

o (3a + 5b)^ = 17(a- + b~)^ Ha" - 3 ( ) a b - Sb^ - 0 «

Vc'li a = : 4 b chon <! ^ ' la diTdc: A, :4x + y - 7 = 0

a = 4

a = 4 b _ _ b -

Viet phUifni^ truth mat phan^ Jo

Phirong trinh mat phSng (Q) song song voi mat phang (P) c6 dang:

x + y + / + m = () ( m ? t l ) (Q) c5l hai tia O x , Oy tai hai diem B, C thi m < 0, k h i do:

B ( - m ; 0; 0 ) , C((); - m ; 0 )

Ta c6:

B A = ( l + m ; 1; 2 ) '

C A = ( 1 ; 1 + m ; 2) C)icn tich tam giiic A B C l i i :

TijySn iiion 39 cip thu si(c hoc ki mOn ToAfi lap 12 N&ng cao - Ph^m Trgng Thif

=>PhiTdng trinh m"^ + 3m^ + 9m -9-0 khong c6 nghiem tren (-co; 0)

Do 66, tren (-oo; 0) thi (*) c6 mot nghiem duy nha't la m = - 1

Vay mat phang (Q): x + y + z - l = 0

Cfiu V I I rim h$ sdU'm nhdt tron}' cdc h$ so ciia khai trien

{a,^} tang khi 0 < k < 2 6

I P H A N C H U N G CHO T A T CA T H I SINH (8,0 diem)

Cfiu I (3,0 diem) Cho ham so y = f(x) = x"^ - 2x2 + 1

1 Khao sat sir bien thien va ve do thi (C) cua h^m so tren

236

Cty TrjHH MTV DWH Khang Vigt

2 Bien luan theo tham so'm so' nghiem ciia 2m - 2 + 2x - x"^ = 0

3 Tinh the tich kho'i tron xoay khi quay hinh phing gidi han bdi do thi (C),

Ox quanh true Ox

Cfiu I I (1,0 diem) Giai bat phiTdng trinh log^

^ 1 Xet vi tri tiTdng doi cua hai dir6ng thing d va d'

2 Viet phi/dng trinh mat phing (P) song song vdi hai diTdng thing d va d', biet khoang each ttr d den (P) gap doi khoang each tilf d' den (P)

n P H A N R I E N G (2,0 diem)

Thi sink chi du^c lam mQt trong hai phdn (phdn A ho^c phin B)

A Theo chrfrfng t r i n h Chua'n

Cfiu Va (1,0 diem) Giai phiTdng trinh sau tren C (an z): 3z^ + 2z + 7 = 0

Cfiu V i a (1,0 diem) Trong khong gian Oxyz cho di/dng thing d c6 phi/dng trinh

x + 1 y + 1 z - 3

\k mat phing (P): 2x + y - z = 0 X6t vi tri d va (P)

I

2 - 1 3

B Theo chi/rfng t r i n h Nfing cao

Cfiu Vb (1,0 diem) Giai phiTdng trinh sau tren C (an z): Sz^ - 2z +13 = 0 Cfiu V I b (1,0 diem) Trong khong gian Oxyz cho di/cJng thing d c6 phiTdng trinh

Tuygn chpn 39 6i thCf sufc hpc ki m6n ToAn I6p 12 Nang cao - Ph?m Trgng Thu

HUdNG D A N - D A P SO

Caul

1 Kluio sell sir hien tliien va ve eld thi (C) ciici hum sY;' (doc gia ly giai)

2 Bien ludn theo thain so in so iif-hiem ciia

Taco 2 m - 2 + 2x x"* =()« x"* - 2x2 + 1 = 2m - 1(*)

(*) la phi/dng trinh hoanh do giao diem cua (C) va dU'dng thang d:y = 2m - 1

cung phiTdng true Ox, nen so giao diem cua (C) va du'cJng thang d la so nghiem

cua phu'dng trinh da cho

Nhin do thi:

+ Neu 2 m - l < ( ) o m < ^ : V6 nghiem

+ Neu 2 m - l = l<=>m = i: 3 nghiem phan biet

+ Neu ( ) < 2 m - l < l c : > ^ < m < l : 4 nghiem phan bict

+ Neu 2 m - l = 0 2m - 1 > 1 <=> 2'- 2 nghiem phan biet

m > 1

3 Tinh the tich khd'i tron xoay khi quay hinh phdnf; fii('/i han bdi do thj (C)

The tich kho'i tron xoay can tim la:

I I

V = 271 |"(x^ - 2 x - + ir dx = 271 f(x** -4x^ + 6x^ -4x^ + l)dx

=27t (x^ 4x'' 6x'^ 4x^ + + x

5 3 0 ~ 315 1 _ 25671 (dvtt)

Cau II Giai bat phuifni^ trinh

Bat phu'dng Irinh da cho tu'dng du'dng vdi

V - -cosx

dv = sinxdx

Ap dung cong thi'fc lich phan lifng phan, ta c6:

J = (-cosx.ln(cos\)) sinxdx -cos — - In 3 K

1 Xet 17 tri ti((fn^ doi ciia hai difdni^ thihm d va d'

DirtJng thang d qua A( - 1 ; 1; 2) va c6 VTCP a = (2; 3; 1)

Tuygn chpn 39 thO sure hpc kl mOn Toan Mp 12 Nang cao - Phjim Trgng Thtf

Difcfng t h i n g d ' qua B{2; - 2; 0) va c6 V T C P b = (1; 5; - 2)

T a c 6 [a, b] = ( - l l ; 5 ; 7 ) , A B - ( 3 ; - 3 ; - 2 )

[a, b].AB = - 3 3 - 1 5 - 1 4 = - 6 2 ?t 0 => d d ' ch^o nhau

62 Khoang each giflTa d va d ' la

Suy ra phiTcfng trinh mat p h i n g (?) c6 dang - I I x + 5y + 7z + D = 0

Theo gia thiet thi ta can c6: d(A, (?)) = 2d(B ( ? ) )

1 Khao sat su" bien thien va ve do thi (C) cua ham so tren

2 T i n h dien tich hinh p h i n g gidi han bdi do thi (C)cua ham so (1) va diTcfng thing y = 1

C&ulh (2,0 diem)

1 T i n h tich phan I = J x V 4 x + 5dx

- I

2 T i n h tich phan J = Jxcosxdx xcosxdx

C&u III (1,0 diem)

1 Vie't phu'dng trinh tham so", chinh t i c diTcfng t h i n g A d i qua hai d i e m A, B

2 V i e t phtfdng trinh mat cau (S) du-dng kinh A B 3 Chu-ng to rang dudng t h i n g d tiep xuc v d i mat cau (S)

l l P H A N R I E N G (2,0 diem) sink chi dUffc lam mgt trong hai phdn (phdn A ho^c phdn B) ' ^

TuyS'n chpn 39 dg thir sire hgc ki m6n Toan I6p 12 NSng cao - Ph^m Trpng Thu

A The«) chi/«/nf{ trinh C h u a n

Cfiu I V a (1,0 diem) G i i i i phiTdng trinh sau trcMi C (an z): / ( / + l)(z + 2) = 6

Cflu V a (1,0 diem) V i c t phiTdng trinh mat phang (P') d o i x i l n g v d i mat phang

( P ) : x + 2 y + / - l = { ) qua d i e m 1(2; 0; 1)

B T h e o chifrfng trinh Nang cao

C S u I V b (1,0 diem) G i i i i phiTdng trinh sau Ircn C (an / ) : /,•* - 1 6 = 0

C a u V b (1,0 diem) T i m mot d i e m M thuoc diTcJng thiing ' ^ ' Y ' ^ ^ ' J

cho M A CO do dai ngan nhal vc'Ji A ( l ; 2; 3)

H U d N G D A N - D A P S O

C a u 1

1 Khcio sat sU Men thien vu ve do thi (C) ciia ham so {&oc gia tif g i a i )

2 Tinh dien tich htnh phanfi f^i('fi han hdi do thi (C)cua ham so

Phifdng trinh hoanh do giao d i e m giffa do thj (C) va difdng thang y = 1 la:

Dat u = 7 4 x + 5 u^ = 4x + 5 2udu = 4dx => dx = udu

D o i can: u = - i thi u = 1, x = 1 thi u = 3

2 Vict phtr</ni'trinli iiujl cdii (Sjduoni; l<inli All

• V i mat cau (S)c6 difcVng kinh A B ncn tarn I ciia mat cau (S) la trung d i e m

A R

cua A B Ta c6 1(1; 0; I) va ban kinh R ciia (S) la R = — =r 3

2

*1

• V a y phiAfng irinh mat cau (S) cam l i m l a ( x - 1)' + y ^ = 9

2' Chifni^ to rdnii difoni^ thans^ d tiep M'IC vol mat cau (S)

During thang d d i qua A c6 V T C P li = (1; 1; 4) v i AB.u = 4.1+4.1 + (-2).4 = 0

Tuy^n chpn 39 6i thCf sijfc hpc kl mOn ^oin I6p 12 Nflng cao - Phgm Trgng Thu

GiSi (*)

A' = 4 - 6 - - 2 = ( N /2i)\

• can bac hai cQa A' la ±V2i

• PhiTdng trinh (*) c6 hai nghiem phan bi^t Ih z, = -2 + V2i, Zj = -2 - >/2i

Vay tap nghiem ciia phiTdng trinh da cho S = 1; -l + yfli; -2-\/2i|

Cfiu Va Viet phMn^ trinh m^t phdnf^iP') doi xiina vdi rtt^t phdng

Lay diem M(0; 0; 1) e (P) Goi M' la diem doi xtfng cua diem M qua I

Do d6 MA ng^n nhat khi t = 1 hay M(l; 3; 1)

DE SO 18 OE THI HOC Ki II MON TOAN L 6 P 12 sd GiAO pgc VA DAO TAP D 6 N G THAP

I PHAN CHUNG CHO TAT CA THI SINH (7,0 diim) i

Cfiu I (3,0 diem)

1 Tim mot nguyen ham F(x) cua ham so' f(x) = x^d - x) bie't r^ng F(l) = 0

2 Tinh cdc tich phan sau: , ,

a) A = xVl-xdx b) B = J -I (2x + l)e "dx

Cty TNHH MTV DWH Khang Vi $t|

Aull (1,0 diem)

Tim phan thi/c phan ao ciSa so' phtfc z = (2 - i)(l + 2i) - 3(1 - i)

Cfiu III (3,0 diem) Trong khong gian vdi he toa dp Oxyz cho bon diem

A(-l; 1; 2), B(l; 0; 1) C(-l; 1; 0) D(2; - 1 ; -2) ^

1 Viet phi/dng trinh mat phing (P) di qua ba diem B,C \k D

2 Viet phu"dng trinh mat cau (S) c6 tam A va tie'p xuc mat phing (P)

3 Tim toa do diem E nam tren dU'cJng thdng CD sao cho B la hinh chieu vuong goc ciia E U-en dU'dng thing AB

11 PHAN RIENG (5,0 if/em;

Thisinh chl dUttc lam mot trong hai phan (phdn A hog.c phan B)

A Theo chifring trinh Chufi'n

CHulVa (1,0 diem) *

Tinh dien tich hinh phang gidi han bdi do thj (C) cua ham so y = cos' x.sin2x

cacdifdngthang x = 0,x = ^ v a y = 0

Cfiu Va (1,0 diem) Giiii phiTdng trinh 4z^ - 2z +1 = 0 tren tap so phtfc

Cfiu Via (1,0 diem)

Trong khong gian vdi he toa do Oxyz cho diem P(l; 2; -3), Q(3; 3; 0), R(2; - 3 ; 1) va S(3; - 1 ; 4) Tim toa dp diem M nim tren diTdng thing PQ wk

diem N tren diTdng thing RS sao cho khoang each hai diem M va N dat gid tri nho nha'l

B Theo chif(/ng trinh Nfing cao BaiIVb.(7,0rf//m;

Cho hinh phang (H) gidi han bdi cac dU'cJng y = , x=0, x = l va y = 0

2x + l

Tinh the tich kho'i tron xoay tao ra khi quay hinh phIng (H) xung quanh true

hoanh

Cfiu Vb (1,0 diem) Tim mo dun eua so phtfc z, bietz^ = 1 -4>/3i

Cfiu VIb (1,0 diem) Trong khong gian vdi he toa dp Oxyz cho mat cau (S) tam 1(1; 2; -1), hin

kinh R = 1 va hai diemA(l; 0; -1), B(0; 1; -l).Tim toa dp diem M n^m tren

•ludng thing AB va N tren mat cau (S), sao cho khoang cdch gii?a hai diem M

Va N dat gia Irj nho nha't

245

Tuyfi'n chpn 39 6& thCf site hpc ki m6n Join Idp 12 l\iang cao - Phgm Trpng Thu

Cty TNHH MTV D W H Khang Vijt

Phi/dng t r i n h m a l p h a n g (P) c a n Tim lii

4 ( x - l ) 4 7 ( y - ( ) ) " ! ( / - ! ) = () o 4 x + 7y - / - 3 = 0

Vict pliif(fii\; tr'inh mut cuii (S) cd tdiii A vu ticp xiic mut phuni; (P)

Phi/dng I r i n h m i l l c a n (S) l a m A ( - i ; 1; 2), b a n k i n h R l i i : (x + 1 ) - + { y - l ) - + ( / - 2 ) - = R -

Tuy^n chpn 39 66 thd sijrc hoc M mOn Join Mp 12 Natuj I.TJ Phgrn Trpng Thtf

Nen phi/dng trinh da cho c6 hai nghi^m la

Cau I V b Tinh the tich khd'i trdn xoay tao ra khi quay hinh phdnfi (H) xunf*

The tich kho'i tron xoay can lim:

Cflu Vb 77w md dun ciia so phiic z, biet

Goi z = x + yi (x,y e IR)