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

Tài liệu Tuyển chọn 39 đề thử sức học kì môn Toán 12 nâng cao được biên soạn nhằm giúp người đọc làm quen với các dạng đề thi học kỳ ở mức độ cao. Phần 1 giới thiệu các đề thi, mời các bạn cùng tham khảo nội dung chi tiết.

NHi^ X U A ' T B A N Dni H O C Q U O C G i n Hii N O I

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Chiu trdch nhi^m xuat ban

Gidm ddc - Tong biin tap : IS. P H A M T H j T R A M

SACH LifiN KET

TUYEN CHQN 39 THI THLf SL/C HQC KJ MON TOAN 12 NANG C A P

M a so: 1 L - 1 1 5 D H 2 0 1 3

In 2 0 0 0 c u o n , k h d 1 6 x 2 4 c m

Tail Cty T N H H M T V IN A N MAI T H j N H D L / C

Dja c h l : 7 1 , Kha V a n Can, P H i e p Binh C h a n h , Q Thu Dure, TP H o C h i M i n h

S6' xuat b 5 n : 4 2 0 - 201 3/CXB/05 - 5 8 / D H Q G H N ngay 0 3 / 0 4 / 2 0 1 3

Quyet d j n h xuat hkn so: 339LK-TN/QD-NXBDHQGHN, cap ngay 31/07/2013

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

M^i nj&i itdn

B6 sach JU\EN CHON 39 DE THLf SLfC HOC KI MOM TOAN \dp 10,

11 va 12 nang cao diTdc bien soan va luyen chon diTa tren noi dung chtfcfng

trinh THPT hien hanh; bp sach toan nay giup cac em c6 dieu kien lam quen \di

cac dang de thi hoc ki d miJc dp cao Rieng cuon 12 c6 them phan phu luc giup

cac em l\X kiem tra, danh gid, bo sung kien thtfc ve todn THPT cho minh nhkm tao nen toan can ban viJng chac cho cac em triTdc khi chinh thuTc bxidc vao ki thi

Dai hoc, Cao dang

Hi vpng bp sach se gop phan giiip cac em dat ket qua cao trong cdc ki thi, dong thdi la mot cong cu ho trd cho cac bac phu huynh giup cho con em hoc tap tot hPn

Trong qua trinh bien soan, dij tac gia da c6' g^ng nhiftig cuon s^ch van c6 the

con nhffng khie'm khuyet ngoai y muon Chung toi rat mong nhan di^Pc s\i gop y

chan thanh ciia cac thay, c6 giao, cac em hoc sinh de trong Ian tai ban sau sach difdc hoan chinh hPn

Tac gia ra't cam Pn Nha xuat ban Dai hpc Quoc gia Ha Npi, Cong ty TNHH MTV DVVH Khang Viet da dpng vien, khuyen khich va tao mpi dieu kien de cuon sach nay sdm den tay ban doc

Dieu phai chiJng minh dpcm

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

NhU chiing ta da bie't mon Toan la mon hoc chiem mot vi tri ra't quan trpng va then

cho't, ra't can thic't do hoc cac mon khac tiT tieu hoc cho de'n cac Idp tren Mon Toan

giiip cac em nhan bie't cac moi quan he ve so' lUdng va hinh khong gian cua the gidi

hipn thifc Nhd do ma cac em c6 phiTdng phap nhan thiJc mpt so mat ciia the' gidi xung

quanh va bie't each hoat dong c6 hieu qua trong ddi song Mon Toan gop phan ra't quan

trong trong vice rcn luycn phifdng phap suy nghl, phUdng phap suy luan, phUdng phap

gidi quyet van de No gop phan phat tricn tri thong minh, each suy nghl doc lap, linh

hoat, sang tao va vipc hinh thanh cac pham cha't can thie't cho ngifdi lao dpng nhiTcan

cii, can than, c6 y chi vu'pt kho khan, lam viec cd ke' hoach, cd ne nep va tac phong

khoa hoc

Xua't phat tij' vi tri quan trong cua mon Toan, qua thifc le' giang day nhieu nam d

ca'p THPT Toi nhan tha'y rang dc hoc sinh hoc to't mon Toan thi ngoai vice cac em n^m

vilng kie'n thilc trong sach giao khoa, ky nang tinh toan that tot ma con phai bie't phUdng

phap giai mot bai thi nhif the nao trong luc dang thi dc cd diem cao Muo'n lam dU'dc

dieu nay thi sinh can phai tuan thii theo cac b\idc sau day:

1) That binh tinh trong luc lam bai thi

2) Can doc that cham rai toan bp dc, danh gia sP bp dp de, khd cua cua cac cau,

xem nhifng cau nao quen thupc, la vdi minh

3) Giai ngay lap ti'fc cac cau ma ban tha'y de

4) Mot vai cau can thie't de'n sif suy nghl sau hdn, thi sinh can phai dpc ky cau hoi,

gach dirdi cac gia thie't vii ycu cau cua bai todn Dinh hiTdng each giai, hinh dung dp

phiJc tap cua each giai de cd si/ li/a chpn diing din

5) Trinh bay biii giai thi sinh khong nen lam tat, moi bifdc nen vie't mot dong de de

kiem tra, vi giam khao cha'm bai thi theo ba rem nen cd mot bifdc nao dd sai thi van con

diem d nhu'ng biTdc bie'n ddi dung trUde dd Cach hay nha't la lam xong btfdc nao kiem

tra birdc ay de phat hicn ngay cho sai

6) Trong qua trinh giai mot bai toan ne'u thi sinh gap khd khan giiJa chiifng, cd the

chiifa khodng trong tren gia'y thi de bo sung sau va nhanh ehdng chuyen sang lam cau

khac

7) Khi da hoan tat bai thi, ne'u con thdi gian thi sinh nen dpc lai bai giai va ra soat

lai cac chi tic't da trinh bay (thong thiTdng cdc loi thi sinh hay bd sdt biTdc lam la tap xac

dinh, dieu kipn cd nghla ciia can bac chan, ham so' logarit, doi can khi dCing phiTdng

phap doi bie'n dc tinh tich phan, loai bo nghicm ngoai lai trong phifdng trinh ) nham

hoan thien bai thi td't hcJn cho de'n he't gid

Nhieu hpc ird tdi day dp dung 7 Idi khuyen tren da trd thanh thii khoa dai hpc cua

nhieu triTdng, nhi/ng thanh cong nha't la toi cd hpc tro thu khoa "kep" khoi A va B cua

trirdng Dai hpc Khoa hpc TiT nhicn TP HCM va Dai hpc Y DiTdc TP HCM nam 2011

Chuc cac thi sinh dat ke't qua cao trong cac ki thi

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

1 Khao sat siT bie'n thien va ve do thi (C) cija ham so da cho

2 ChtJ-ng minh rSng vdi moi m diTdng lhang y = x + m luon ciCt do thi (C) tai hai diem phan biet A va B Goi k^kj hin liTdt la he so goc cua cac tiep tuyen vdi (C) tai hai diem A va B Tim m de tdng k, + k2 dat gia trj Idn nha't

C&u 11.(2,0 diem)

1 Tim gia Irj idn nha't, gia tri nho nha't cua y = -2sin-'x + 3cos2x -6sinx

2 Tinh gia tri bieu iMc T = ^' '

3 Giai phu-dng trinh 6.9" -13.6^ +6.4" =0

Cau III, (2,0 diem) Cho hinh chop ti? giac deu S.ABCD c6 canh day la a, canh

ben la aS Tinh the tich khoi chop S.ABCD

Cau l\ (2,0 diem)

1 Cho ham so' y = In

2 + 3x vdi X > -— 3 Chiang minh rang xy' +1 = e^

2 Giai ba't phu-dng trinh log4(2x^ +3x +1) > logT(2x + 1)

CSu V (1,0 diem) Cho ham so y - X- -2x + x-1 2 m - l cd do thi (Cni).Tim m de hiim so cd ciTc dai, ciTc tieu va khoang each giffa hai diem ciTc dai, ciTc lieu bang 6 ;

D A P A N T H A M K H A O -4

I

(10 diem)

1 (2,0 diem) Khao sat siT bie'n thien va ve dd thi (C) ciia ham

I

(10 diem) a) lap xac dinh D = R \ -

TuySn chon 39 th& sijfc hpc kl mOn Toan I6p 12 NSng cao - Phgm Trgng Thu

b) Su-bien thien:

- Chieu bie'n thien: y' = - - 1

( 2 x - l ) '

•<0, V x ^ -

- Ham so nghich bien tren cac khoang 1 va

- Gidi han va tiem can:

© Hm y = lim y = -—=> tiem can ngang: y = - —•

2 (1,0 diem) ChiJng minh rang v(Ji mpi m

Phifcfng trinh hoanh do giao diem cua do thi ham so da cho va

—x +1 diTdng thang d: y = x + m la = x + m

2x - 1

c : > 2 x 2 + 2 m x - m - l = 0 ( * )

Cty TNHH IVITV DWH Khang Vigt

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

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

1 (1,0 diem) Tim gia trj \6n nhS't, gia trj nho nhS't

II

(2,0 diem) Ta C O y = -2sin' x + 3 ( l - 2 s i n x ) - 6 s i n x

= - 2 s i n ' ' x - 6 s i n ^ x - 6 s i n x + 3 (1)

0,25

II

(2,0 diem)

Dat u = sinx, - 1 < u < 1

Ta CO (1) vie't lai y = -2u'' - 6 u ^ - 6 u + 3

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

0,5

II

(2,0 diem)

Xct y ( - l ) = 5, y ( l ) = - l l

Suy ra maxy = 5 khi sinx = - 1 0 x = - — + k27r, k e Z

xeK 2 miny = -11 khi sinx = 1 0 x = —+ k27t, k e Z

xeR 2

0,25

II

(2,0 diem)

2 (0,5 diem) Tinh

II

(2,0 diem)

<

f

5

Tuyin ch(?n 39 dg tht> sijfc hpc kl mOn Toan I6p 12 Nang cao - Phgm Trpng Thu

3 (0,5 diem) G i a i phifcTng trinh

~ 3 "

v2y

C ^ X= ± 1

v2y v2y Vay nghicm cua phifdng trinh da cho la x = ±1

I l l

(2,0

diem)

Tinh the tich khoi chop S A B C D

G o i O la tarn cua hinh vuong A B C D V i S A B C D la hinh chop

( x - l ) 2 ( x - 1 ) ^

Ham so c6 ciTc dai va ciTc ticu khi va chi k h i g ( x ) = 0 c6 2 0,5

nghicm phan bict khac 1

o - A' = - 2 + 2m > 0 <=> m > 1

g ( l ) = l - 2 + 3 - 2 m ^ O Goi A ( x ^ ; y ^ ) , BCxjj; y n ) la hai diem cifc tri

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

c : > ( X p - x ^ r + ( 2 x , 5 - 2 - 2 x ^ + 2 ) 2 = 3 6 5(x,j - x ^ r = 36 o (Xjj + x ^ )^ - 4 x y X ^ = y

0,5

- 4(3 - 2m) = — o m = — ( t h o a man)

5 10

7

Tuyin chpn 39 6i IM sOc hoc ki mOn ToAn Iflp 12 Nang cao - Pham Trpng Thu

D E THlIr SOC H O C Kl I M O N T O A N L 6 P 1 2

Thdi gian lam bai: 120 phut

D± S6 2

Cau I (3,0 diem) Cho ham so y = x"^ - 2x^ + 4 (I)

1 Khiio sal siT bicn Ihicn va ve do Ihj (C) cua ham s6' (1)

2 Tim cac giii Irj cua m de phUtlng trinhx'* -2x" -log2m = 0 c6 4 nghiem

phan bicl

C&u U (2,0 diem)

1 Tim gia iri Idn nhiil ciia ham so' f(x) =

2 Tinh gia trj bieu ihufc A =

x-'+3x^-72x + 90 tren[-5; 5]

49

3 Giai bal phi/dng irinh 2-""" -21.2"^-"^^'+ 2 >()

Cfiu HI (2,0 diem) Cho hinh chop S.ABC c6 day ABC la lam giac vuong tai B,

SA 1 (ABC) SA = iiyfl, BAC = 30", BC = a, M la trung diem ciia SB Tinh the

tich cua khoi tiJ dien MABC

Cau IV (2,0 diem)

1 Cho ham so y = c''' sin5x.ChiJng minh ring: y''-4y' + 29y = 0 (*)

3\2y =972 log^(x-y) = 3

2 Giai he phi/dng Innh

Cau V (1,0 diem) Cho ham so y = - ^ ^ c 6 do thi (C) Viel phiTdng Irinh

x-1 tie'p tuyen ciia (C) c6 he so' goc la -1

diem) b) Sir bien thien:

• Sir bien thien:

- Ham so dong bien Ircn moi khoang (-1; 0) va (1; + oo)

- Ham so nghich bien tren moi khoang ( - oo; -1) va (0; 1)

• Cyc trj:

- Ham so dat cifc lieu x = ±1 va y^.^ = y( ± 1) = 3

- Ham so dat ciTc dai x = 0 va y^^ = y(0) = 4

• Gioi han: lim y = + o o ; lim y =+CXD

Bang bien thien

0 + 0 - 0 +

c) Do thi (C): Qua cac diem ' 1 31^ ,(±2; 12)

2 (1,0 diem) Tim cac gia trj cua m de phtf(yng trinh

Ta CO x"* - 2x- - log2 m = 0 <=> x** - 2x- + 4 = logj m + 4 = k

De phmJng trinh da cho c6 4 nghiem phan biel<=> diTdng thing

y = k cat do thj (C) tai 4 diem phan biet DiTa vao do thj (C) cua ham so , la c6 3 < k < 4

1

<=> 3 < log, m + 4<4<=>-l< log2 m < 0 <=> - < m < 1

9

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

=> max f(x) = max | 1-5;51 1-5;5]^ -5) = 4 g(-5) 00, g( g(4) 4) = - g(5) -86 - 400 khi x = - 5 0,5

1.(1,0 diem) Tim gia tri l<?n nh&t

Tinh the tich ciia khói tu" di^n MABC

Trong tam giac ABC vuong tai B co

BAC = 30", BC = a =:> AB = BCtan30° = aVJ

Cty TNHH MTV DWH Khang Vi$t

0,25

0,25

0,25 IV

(2,0 diem )

0,25

0,5

Dien tich tam giac ABC la S^^^ = ^ AB.BC = (dvdt) Goi H la trung diem cua AB thi

MH//SA z:^ MH 1 (ABC) va MH = SA aV2

The tich cua khoi chop M.ABC la

V A B C = 3MH.S^ef, — 1 ayfl âV6 12 (dvtt)

1 (1,0 diem) ChiJng minh

Taco ý = (ế') sin5x + ế'(sin5x)'

= 2G^^ sin 5x 4- Sế^cosSx = ê^ (2 sin 5x 4- 5cos5x)

y" = ^ê") (2sin5x 4- 5cos5x) 4- ế'(2sin5x 4- 5cos5x)'

= 2ế'(2sin5x 4- 5cos5x) + ế'(10cos5x - 25sin5x)

= -216^" sin5x + 20ế' cos5x

luyen cliyii M de lliij ;.ut ligc ki iiion luaii lup U' f^jiig [ ' h j n i l i u i i g llii/

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

0,5

2 (1,0 diem) Tmv m de dififng thang d cat do thj (C) tai

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

<=> => <=>

[ t = 8 [ 3 ^ = 8 [x = log3 8

rx=o Vay nghicm ciia phifring Irinh da cho la

[x = log3 8

0,25

III

(2,0 diem)

1 (1,0 diem) Chrfng minh rang S.ABCD la hinh chop deu

III

(2,0 diem) Tur giac A B C D la hinh Ihoi v i c6 cac canh deu b^ng a Gpi 0 la giao diem cua A C va BD Tam giac SAC can (vi c6

SA = SC = a ) va 0 la trung diem A C nen SO 1 A C

0,25

III

(2,0 diem)

S

———Y

III

(2,0 diem)

Tam giac SBD can (vi c6 SB = SD = a ) va O la trung diem

III

(2,0 diem)

III

(2,0 diem)

'uy6'n chpn 39 d g \hCl SLfc hpc ki man To^n I6p 12 Mang cao - Phgm Trqng ThJ

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

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

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

D o d 6 V 3 ^ 3 C ^ = - S O S ^ 3 C O a^f2

3 2 a 2 = ^ (dvtt)

3 (0,5 diem) Tmh the tich ti? dien M A B C

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

y = 4 - 2 x log(4 - x)^ = logx + log9 = log9x

y = 4 - 2 x x^ -17x + 16 = 0 <=>

Tim m de ham so'co c\ic dai va c\ic tieu

Tapxacdinh: D = K \ { - I }

x^ +2x + 2m + l Dao ham: y' = -

(x + i r

g(x) (x + 1)^

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

'A' = - 2 m > 0 phan biet khac - 1 o

g ( - l ) = l - 2 + 2m + 1^0 « m < 0

0,5

0,5

O E S O 4 D E T H C T SCT C H O C Ki I M O N T O A I S I L 6 P 1 2

Thdi gian lam bar 120 phut

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

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

2 Tim phu'dng trinh cac du'dng thdng qua diem A 19^

12 ; 4 va tie'p xuc vdi do thi (C) cua ham so'

C&ull (2,0 diem)

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

2 Tinh gia tri bieu thiJc: A = logVlO + I n V e - In — 21og2 3

log 9 '

3 Gicii bat phu-cfng trinh 4x^ + x.2'* + 3.2^" >x^.2''" +8X + 12

CSu I I I (2,0 diem) Cho hinh chop S.ABC c6 day la tam giac vuong can tai A c6

AB = AC = 2a Mat ben qua canh huyen vuong goc vdi mat day, hai mSt con lai tao vdi day mot goc 30" Tinh the tich khoi chop

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

Cty TNHH MTV DWH Khang Vi$t

3 (1,0 diem) Giai hát phif(/ng trinh

Bát phÚdng trinh da cho túdng dúdng

- 1 < X < 3

x 2 > 2 <=> V2 < x < 3

Truífnf' hop 2:

x ^ - 2 x - 3 > 0 2''^ - 4 < 0

x < - l h o a c x > 3

x 2 < 2 -72 < x < - l Vay tap nghiem cua bat phúdng trinh da cho la

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

I l l

(2,0 diem)

Tinh the tich Ithoi chop S.ABC

Ke SO 1BC SO l ( A B C ) ( v i ( S B C ) n ( A B C ) = B C ) Goi H, K Ian lúdt la hinh chiéu cua

Do do ((ABC), (SAC)) = SKO = 30°

Tu-dng tiT ta c6 SHO = 30°

The tich cua khoi chop S.ABC la:

V ARC = - SỌS, HP = - SỌ AB AC = - • • 2ạ2a = — a ^ (dvtt)

S.ABC 3 ABC ^ 9 ^ '

IV

(2,0 diem)

1 (1,0 diem) Chitng minh

ý = 12x" ê"'^" + x'^2013.e2'"-''' = x".e2"'3''(12 + 2013x)

•xý=x'^ế"''''(12 + 2013x)

Vay x y ' - y ( 1 2 + 2013x) = 0

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

2 (1,0 diem) Giai h§ phtfcfng trinh

Giai phi/dng trinh nay ta c6 x = 1 va x = -2 (loai)

Vay he phifdng trinh da cho c6 nghiem (x; y) = (1; 9)

Dodo minPQ = >/8 <=>a = b = l

Vay hai diem can tim la P(0; 2), Q(-2; 0)

DE S O 5 DE THCT SOC H O C Ki I M O N T O A N LdP 1 2

Thdigian lam bai: 120 phut

Cau I (3,0 diem) Cho ham so y = x^ - 3x + 2 co do thj la (C)

1 Khao sat sir bien thien va ve do thj (C)cua ham so tren

2 Goi d la du'cJng th^ng di qua diem A(3; 20) va c6 he so' goc m Tim m de

di/cJng thang d cat (C) tai ba diem phan biet c6 hoanh do Idn - 2

Cau 11 (2,0 diem)

1 Tim gia tri Idn nha't, gia trj nho nhat cua y = f(x) = - ^ x ^ + x + ln(l - x)

Cty TNHH MTV DWH Khang Vi?t

tren ' ' • • 2

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

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

C&u III (2,0 diem) Cho lang tru ABC.A'B'C'co A'.ABC la hinh ch6p tam giac

deu canh AB = a, canh ben A'A = b Goi a la goc giiJa hai mat phang (ABC)

va (A'BC) Tinh tan a va the tich khoi chop A.'BB'C'C

CHulW (2,0 diem)

l.Chohamsoy = e~'' sinx.GiaiphU'dngtrinh y''-i-4xy'-i-3y = 0

2 Giai he phi/dng trinh x~-4x + y-t-2 = 0 21og (x-2)-log y = 0 ( ^ ' y ^

1 (2,0 diem) Khao sat siT bien thien va ve do thi (C) cua ham

I

(3,0 diem) Doc giii tif giai_cach giiii tu'dng tif cau 1.1 de so 3

I

(3,0 diem) 2 (1,0 diem) Tim m de difc/ng thang d cit (C)

I

(3,0 diem)

Du'dng thang d co phi/dng trinh y = m(x - 3) + 20 0,25

I

(3,0 diem)

PhiTdng trinh hoanh do giao diem cua d va (C) la

De d cat (C) tai ba diem phan biet co hoanh do Idn -2 thi PT (l)phai CO hai nghiem phan biet Idn hdn - 2 va khac 3 0,25

I

(3,0 diem)

Dat t = x + 2thi(l)trc( thanh f(t) = t^ - 1 + 4 - m = 0 (2)

YCBT thi PT (2) CO hai nghiem duTdng phan bi^t khac 5 0,25

21

Tuy^n chqn 39 dg thil sutc hgc ki mfln Toan Idp 12 NSng cao - Phgm Trgng Thii

N e u l < x < 2 t h i 3 x ^ 1 4 x + 15 = 0 c i > x =

-V a y phiTdng trinh da cho co tap n g h i e m la ^ = -j - j

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

G p i H la t a m cua A A B C v i i M la trung d i e m cua B C

D o A ' A B C la chop tam giac deu n c n A ' H la diTdng cao cua hinh chop A ' A B C dong thdi cung la chieu cao cua hinh lang

Tuy§'n chpn 39 thif sifc hpc ki m6n Toan I6p 12 MSng cao - Pham Trpng Jhd

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

Giai phufc^ng trinh nay ta c6 x = 3 va x = 0 (loai)

Vay he phiTcJng trinh dii cho co nghiem (x; y) = (3; 1)

11m m de 6\iHn^ thanjj d c6 phiTc/ng t r i n h y = - x + m cat (C)

PhiTdng trinh hoanh do giao diem cua (C) vdi d la

x~ + x - I

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

d cat (C) tai hai diem phan biet

o (*) CO hai nghiem phan biet khac 1

Cty TIMHH MTV DVVH Khang Vi?t

o i S p 6 D E THCT S O C H O C K i I M O N T O A N L 6 P 1 2 Thdi glan lam bar 120 phut

Cflu I (3,0 diem) Cho ham so y = (1) ' ,

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

2 T i m M G (C),biet rSng tiep tuyen vc'Ji (C) tai M ciit Ox, Oy hin lUdt tai A,

B tao thanh tarn giac O A B c6 dien tich bilng — (vdi O la go'c toa dp)

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

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

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

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

2 G o i M, N Ian liTdt la trung diem cua A B va AC Mat ph^ng (P) qua M N va

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

TuyS'n chpn 39 thtJ sifc hpc ki mOn To^n \dp 12 Nang cao - Ph^m Trpng Jhu

D A P A N T H A M K H A O

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

2 (1,0 diem) Tim M e (C), biet rang tiep tuyen vtfi (C) tai M

Goi M 2x, PhiTdng trinh tiep tuyen d cua (C) tai M

Cty TNHH MTV DWH Khang Vigt

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

Vdi dicu kien trcn ba't phUdng trinh da cho tifdng diTdng

log^-x + 31ogTX>-log , 2"^ c=> log^"x + 3 1 o g 2 X - 4 > 0

1 (1.0 diem) Tinh theo a the tich cua khoi chop S.ABC

III

(2,0 diem) Vi SO 1 (ABC) nen SAO = a => SO = asina, AO - acosa => AB = a\/3cosa 0,5

III

(2,0 diem)

The tich khoi chop deu la

V, - ' ^ SO ''^ ^ cos^asina (dvtt) S A B C 3 4 4 0,5

III

(2,0 diem)

2 (1,0 diem) Chtfng minh

III

(2,0 diem)

Goi I la giao diem ci'ia MN va AO

AE AT Suy ra I la trung diem cua MN vii EI // SO ^ — = — (D

0,5

27

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

Goi F la trung diem ciia BC Ta c6 <

The (1) v£lo (2) \h rut gon lai ta difdc 6y^ - 3 y = 0

Giai phufdng trinh nay ta c6 y = ^ va y = 0 (loai)

Vay h0 phu'dng trinh dii cho c6 n g h i e m (x; y) = 7

Tmi m de diicVng thang y = m cat difcfng cong

Goi d: y = m va (C): y = x" + mx - 1

Phu'dng trinh h o a n h do giao d i e m ciia d va (C) la

X + m x - l

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

Thdi gian lam bai: 120 phut

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

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

2 Tim m de ham so' c6 ci/c tri Vdi dieu kien viTa tim, gia suf ham so' c6 cifc dai D(x,; y, );circ t i e u T ( x , ; y , ) ChiJng minh rang ^ — ^ = 2

( X , - X 2 ) ( X j X 2 - 1 )

Cfiu II (2,0 diem)

I T i m gia tri gia tri nho nha't cua y = sin"^x-cos2x+ sinx+ 2 tren khoang

6i thtf sifc hgc kl m a n Toan I6p 12 NSng cao - Ptigm Trpng Tha

C S u I I I (2,0 diem) Cho hinh lang try drfng A ' B ' C ' A B C c6 day la tarn gi^c

vuong A B C tai B Gia suf A B = a, A A ' = 2a, A C ' = 3a G o i M la trung d i e m

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

C a u I V (2,0 diem)

1 Cho ham so y = 0" ln(2 + sinx) ChuTng minh (2 + s i n x ) ( y ' - y ) = e'^cosx

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

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

x + 1

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

DAP AN THAM KHAO

D e h a m so c6 ciTc t r i t h i phiTdng trinh y ' = 0 c6 hai n g h i e m

phan bietc:> Ay, = 9 - 9 m > 0 < = > m < l

H a m SO y CO the vie't lai y = sin-'x - (1 - 2 s i n ^ x ) + sinx + 2

hay y = sin'^x + 2sin"x + sinx + 1

T i n h the tich ti? di^n l A B C

T r o n g tarn giac vuong A ' A C ta c6:

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

/ 1 ^

/ ^ / ' ^ / 1 ^

The tich cua ti? dien lABC la:

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

Ne'u P, Q la hai diem tren (C) doi xiJng nhau qua difdng th^ng

d : y = x +1 thi phufcJng trinh diTdng PQ (vuong goc vdi d) c6

I e d ncn + 1.0iai ra ta co m = 1

4 4 Khi do hoanh do hai diem P, Q la nghiem cua phU"dng trinh

2 x - - I = 0 c ^ x = ± ' Vay hai diem can tim la P

3 3

1 Khao sat siTbicn thicn va ve do thi (C) cua ham so da cho

2 Tim tren do thi (C) hai diem phan biet M , N doi xiJng nhau qua true tung

CSu I I I (2,0 diem) Cho hinh chop S.ABCD c6 day ABCD la hinh vuong canh a

Canh ben SA vuong g6c vdi day, M la diem di dong tren canh CD, H la hinh

33

Tuygn chpn 39 gj thCf sOfc hpc k1 mOn To^n lap 12 NSng cao - Phgm TrpnQ Thu

chie'u cua dinh S Icn BM Tim vj tri cua diem M tren CD de the tich kho'i chop

S.ABH la Idn nha't Tinh the tich Idn nhat do, biet SA = h

Cfiu IV (2,0 diem)

1 Cho ham so y = sin( In x) + cos(ln x) ChiJng minh y + xy' + x^y' = 0

2 Giai bat phiTdng trinh (^^fl + T^sJ + |V7^^V48 ]

Cfiu V (1,0 diem) Cho ham so (C): y =

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

DAP AN THAM KHAO

I

(3,0

diem)

1 (2,0 diem) Khao sat stf bien thien va ve d6 thj (C) cua ham

Doc gia i\i giai_cach giai tifdng tif cau I.l de so' 3

2 (1,0 diem) Tim tren do thj (C) hai di§m phSn bi$t M, N.,

Goi M(xj; y,) va NCxj; yj) la hai diem thupc (C)

0,25

II

(2,0

diem)

1 (0,5 diem) Tim gia tri Idn nhfi't, gia trj nho nhS't

H^m so'y = lien tuc tren

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

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

V a y tap nghiem cua bat phUdng trinh da cho la T = ( - 2 ; 2) 0,25

Cty TNHH MTV DVVH Khang Vi$t

V

(1,0 diem)

Tim a de (P) tiep xiic ( C )

(P) tiep xiic (C)

X - X + 1 2

= x + a

x - 1 x"^ - x + 1

c(5 nghiem

- 2 x ( x - i r

(2)

(2)c> x^^l 2

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

« x - 0 (3) The (3) vao (1) ta dUdc a = - 1 Vay a = - 1 thi (P) tiep xiic vdi (C)

1 Khiio sat sif bicn thien va vc do thi (H) cua ham so (1)

2 Goi I la tam doi xtfng cua (H) T i m diem M thuoc ( H ) sao cho tiep tuyen

cua (H) tai M vuong goc vdi dU'dng thang I M

37

luyeii cliyii 39 cic Uui ;,u, liyu ki nion ToAn Iflp 12 NSng cao - Phgm Trpng Thu

sin a

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

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

6 cos(a + 30°) cos(a -30°)

Cfiu l \ ( 2 , 0 d i e m )

1 Cho ham so y = yjlx-x^.Chtfng minh y-'y" + 1 = 0

2 Giai he bat phiTdng trinh 49'' + 7" - 2 > 0 > —

Cfiu V (1,0 diem) Tim nhSng diem tren do thi (C): y = x ^ + x - 1 , x - i each deu hai

true toa do

1 (2,0 diem) K h a o sat s\i b i e n t h i e n va ve do t h j (C) ciia ham

Doc gia tU'giai_cach giai tifdng tyf cau I.l de so' 1

2 (1,0 diem) Tim diem M thuQC (H) sao cho tiep tuyen

M(x„ ; y,,) e (H) o y„ = (x„ ^ 1)

Tiep tuye'n A tai M c6 he so goc:

1 k,=y'(x.,) = -

I la giao diem cua dU'dng tiem can

x„-l = l x„-l = - l <=> x„=0^y„=l

Vay CO hai diem can tim M,(2 ; 3), MjCO; 1)

0,25

0,25

0,25 0,25

Cty TNHH MTV DVVH Khang Vi$t

I [ 1 (0,5 diem) T i m gia t r j Idn nha't, gia t r j nho nhaft

» ^ log2, 2 + - = 0

l + log2 2x ^ 2 Dat t = log, 2x thi ta c6 : - - + - = 0 ^2 1 + t t 2

^ = -2

o 6t-2(t + l) + 5l(t + l) = 0 o 5 r + 9 t - 2 = 0 o

' ~ 5 logT 2x = -2

3. (0,75 diem) G i a i b a t phifdng trinh

Bat phu-dng trinh da cho lu-clng diTdng T -4.2" - 3 > 0

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

Tam giae ABC ean dinh A nen diTcing eao A D eung h-i Irung

tuyen: DB = DC , SB lao vdi mal day goc a nen SBA = a 0,25

liilc dpem

-cos a — s m a

U 4 ) [ a - 3 0 " ) (3) 0,25

40

IV

(2,0 diem)

1 (1,0 diem) Chufng minh

(2) 7'' < - 2 (loai) 7" >1

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

V

(1,0 diem)

Tim nhiynjj diem tren do thj

NhiTng diem tren do Ihi (C) each deu hai Iriie loa do ehinh lii giao diem cua (C) va hai du^cing phiin giae y = - x , y = xeiia goc loa dp

x ^ ±

M , v2 2y , M -

V2 N/2

• 2 ' 2

41

Tuygn chpn 39 ai thCf sijfc hpc ki mOn To^n Idp 12 h m j - Phgim Trgng Thi/

D E S O 1 0 D E THCT SLTC H O C Ki I M O N T O A N L 6 P 1 2

Thdi gian lam bai: 120 phut

Cfiu I (3,0 diem) Cho ham so y = Sx'' - Qx^ +1 (1)

1 Khao sat su" bie'n thicn va vc do thj (C) cua ham so (1)

2 DiTa vao do thi (C) hay bien luan theo m so nghiem ciaa phu'dng trinh:

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

3 Giai phi/dng trinh l o g ( x - 2 ) - - l o g ( 3 x - 6 ) = log2

Cau I I I (2,0 diem) Cho lang tru dii-ng A B C A ' B ' C c6 day la tam giac deu Mat

phang (A'BC) tao vc'^i day mot goc a va tam giac A ' B C c6 dien tich la S Tinh

the tich khoi lang tru A B C A ' B ' C

C'duW (2,0 diem)

1 Tinh gidi han:

^ , e ^ ' ' - e^ V 3 x 2 - 6 x + 4

A = hm x->i t a n ( x - l )

2 Giai he phu'dng trinh:

, 2 _

j X ' - x y + y " _

16

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

Cau V (1,0 diem) Cho ham so y = 2 x - 3

3 - x CO do thi (C) Vie't phi/dng trinh tiep tuyen cua (C) tai giao diem ciia (C) vdti true tung

Cty TNHH MTV DWH Khang Vigt

D A P A N T H A M K H A O

I

(3,0 diSm)

1 (2,0 diem) Khao sat s\i bien thien va ve do thi (C) cua ham

I

(3,0 diSm)

Doc gia tiT giai_cach giai tu'Ong tiT cau I l de so 2

I

(3,0 diSm) 2 (1,0 diem) Di/a vao do thj (C) hay bi$n luan

I

(3,0 diSm)

1 • y

\ = 1 - m

I

(3,0 diSm)

3 /

I

(3,0 diSm)

i / o

/ 49 / 32

X

I

(3,0 diSm)

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

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

0,25

I

(3,0 diSm)

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

giao diem cua do thj (C): y = Su"*-9u^ + l,u e [ - 1 ; 1] va dirdng thing d : y = 1 - m cung phiTdng vdi true hoanh, nen so nghiem cija phiTdng trinh da cho la so giao diem cua (C) v^ d

0,25

I

(3,0 diSm)

Dya vao do thi ta c6 ke't luan sau

l u y i M i ciioii '.'.:) ilil llui M I C !HU ki ;iioii lo.ui Nang cao - Phaiii li-jng Thl/

49 81

• - — < l - m < { ) o l < m < — : Phifdng trinh (1) c6 4 nghicm

• 0 < l - m < l o O < m < i : PhiTdng trinh (1) c6 2 nghicm

• 1 - m = 1 c=> m = 0: Phifdng trinh (1) c6 1 nghicm

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

Cty TNHH MTV DVVH Khang Vigt

Vay tap nghicm cua BPT da cho la T = (-oo; ())u — ; + CO

= i B C I 2

-2 cos a _ B C V 3

4 cos a

„ 1 S.cosa Suy ra BC = 2 •

V 73

A ' A = A E t a n A E A ' = ^ ^ ^ ^ tan A E A ' = V\/3Scosa tana

The tich lang tru A B C A ' B ' C la

t->0 sint + lim t^O

cost sint

: lim

t^O

e ^ ' - l t 2t sint •2 CO St + e lim

-3t.cost t->0 smt

t l + Vst^ +1

2 (1,0 diem) Giai h$ phrfdng trinh

Dieu kien xy > 0

He da cho tiTctng diTdng x^ - xy+ = 4

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

Viet phtfofng trinh tiep tuyen cua (C) t^i giao diem cua

Giao diem cua (C) vdi true tung la A(0; - 1 )

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

T a c o y' = L _ ^ ^ y ' ( x ) y ' ( 0 ) = l

( - x + 3)^ 3 Vay phu'dng trinh tiep tuye'n can tim la y = ^ x - 1

Thdi gian lam bai: 120 phut

Cfiu I (3,0 diem) Cho ham so y = x ' ^ (m + l)x^ - 3mx - 2 (I) (m la tham so)

1 Khao sat siT bien thien va ve do thi (C) cua ham so (1) khi m = -1

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

C&u 11.(2,0 diem) f

1 T i m gia tri Idn nha't va gia tri nho nhat cua ham so y = x + 3 + — tren

; - 4 ; - i :

2 Cho log3 5 = a Tinh log^75 3375 theo a

3 Giai bat phiTcfng trinh 2^^+' - 21.2"(2x+3) ^2 > 0

CSu I I I (2,0 diem) Cho hinh lang tru du-ng A B C A ' B ' C c6 canh ben A A ' = 2a,

c a n h d a y B C = a, B A C = 120°.Tinh di$n tich xung quanh va the tich cua hinh

tru ngoai tiep hinh lang tru I

C&u V (1,0 diem) Cho ham so y = x^ - x + 1

x - 1 CO do thi (C) T i m tat ca nhi?ng

diem M tren do thi (C) sao cho tong khoang each tiT M den hai diTcfng tiem can

1 (2,0 diem) Khao sat srf bid'n thien va ve d6 thj (C) cua ham

I

(3,0 diem) Khi m = - l : y = x^ + 3 x - 2

I

(3,0 diem)

b) Su-bien thien:

47

Tuygn chpn 39 i3i thCt sijfc hpc kl m6n Toan I6p 12 NSng cao - Pham Trqng Thu

- Ta C O y' = 3x^ + 3 > 0, Vx e R

- Ham so dong bien tren (- oo; + oo)

• Gioi ban: lim y = -co; lim y = + 0 0

Cly TNHH MTV DWH Khang Vi$t

Tatinh y(-4) = y(-l) =-2; y(-2) = -l

0,25

Ba't phu-dng trinh (*) Ird thilnh

21

u + 2 > 0 c : 4 u ' + 8 u - 2 I > 0 4u

Tinh dien ti'ch xung quanh va the ti'ch cua hinh try

III

(2,0 diem ) Gpi (0) va (0') Ian Urdt la tarn ciia cac du'dng tr6n ngoai tiep AABC va AA'B'C Ta c6 0 O' // AA' va OO' - AA' - 2a 0,5

III

(2,0 diem )

Goi R la ban kinh cua hinh try ngoai tie'p hinh lang tru ABC;A'B'C' Khi do R la ban kinh cua cac dir5ng tron (0) va (O')

Dien tich xung quanh cua hinh tru la S,, = 47taR

Ap dung dinh li sin vao tam giac ABC, ta c6:

1,0

49

Tim tdt ca nhifng diem M tren do thj sac cho tong khoang each

Goi M(x^,; y^) e (C) vdi y^ = x^ + 1

^ 0 - 1

Do thi (C) cua ham so da cho c6 tiem can dtfng dj : x - 1 = 0,

tiem can xien d^ : y - x = 0

Khoang each tiT M den tiem can diJng:

Khoang each tir M den tiem can xien:

4~2 ~4i ^ 0 - 1 Tong khoang each nay la (theo BDT Co-si)

Dfing thiJc xay r a o

4~2

x„ = 1 ± Vay CO 2 diem can tim co hoanh do la x^ = 1 ± ^

0,5

0,5

Ojlso 1 2 DE THCT SOC H O C Ki I M O N T O A N L 6 P 1 2

Thdi glan lam bai: 120 phut

CSu I (3,0 diem) Cho ham so y = x^ - (2m + 1 + (m^ - 3m + 2)x + 4 (1)

1 Khao sat siT bie'n thien va vc do thj (C) cua ham so (1) khi m = 1

2 Tim m de do thj ciia ham so' (1) c6 ciTc tri va hai diem ciTc tri nam ve hai Phia ciia true Ox

Tuygn chgn 39 (Jg thil sCfc hgc ki mfln ToAn lap 12 IViang cao - Phgrn Trgng Tha

Cau I I I (2,0 diem) Cho hinh chop S.ABC c6 day ABC la tam giac vuong tai B

va BAC = a.Mat ben SAB la tam giac deu canh a nam trong mSt phang vuong

goc v6i day Tinh the tich cua khoi chop S.ABC

C&uW (2,0 diem)

1 Tim tap xac dinh ciia cac ham so':

a)y = log(sinx) b) y =

T^^igCcosx)-2 Giai he phu'dng trinh

DAP AN THAM KHAO

I

(3,0

diem)

1 (2,0 diem) Khao sat s\i bien thien va ve do thj (C) cua ham

Doc gia tu" giai_cach giai tJdng tu* cau 1.1 de so' 3

2 (1,0 diem) Tm m de do thj cua ham so (1) c6 cxic trj

II

(2,0

diem)

I (0,5 diem) Tim gia trj Wn nha't, ^ia trj nho nhat

Tap xac dinh D^[S\S\

3 (1,0 diem) Giai phi/(/ng trinh

Dat t = 2'', t > 0 PhiTdng trinh da cho trd thanh

Tinh the tich liho'i chop S.ABC III

(2,0 diem)

Goi H la trung diem cOa AB; do SAB la tam giac deu nen

Trong tam giac vuong ABC ta c6 BC = ABtana

Dien tich lam giac ABC la S^^^ = - A B B C = - a ^ tana 0,5 The tich ciia khoi chop S.ABC la

V s A B C = j W S H = ^ t a n a ( d v t t )

0,5

TuySn chpn 39 <3i thii site hQC ki mOn Toin I6p 12 Nang cao - Phgm Trqng Thu

IV

(2,0

diem)

1 (1,0 diem) Tim tap xac djnh cua cac ham s6'

a) y = log(sinx) xac dinh o 0 < sin x < 1

Tim tat ca cac gia trj cua m de

Phi/cfng trinh hoanh do giao diem cua diTcJng th^ng d va do thi

(C) la x 2 - l = - X + m <=> 2x^ - mx - 1 = 0 (*)

V I A = m" + 8 > 0 , Vm nen (*) c6 hai nghiem phan b i e t x , , X2

Suy ra diTiJng lhang d cat do thi (C) tai hai diem phan biet A ,

Cty TNHH MTV DWH Khang Vi?t

D E THlIr SCTC HOC KJ I MON TOAN LdP 12

Thdi gian lam bai: 120 phut

SO 13

C&u I (3,0 diem) Cho ham so y = x'' + mx + 2 (1) (m la tham so)

1 Khao sat sir bien thien va ve do thi (C) cua ham so (1) khi m = 1

2 Tim m de do thi ham so (1) cat true hoanh tai mot diem duy nha't

3 Giai phi/ctng trinh log(10.5'' +15.20") = x + log25

C§u I I I (2,0 diem) Cho hinh chop S.ABC c6 day ABC la tam giac vuong tai A,

AB = a, ABC = 60",tam giac SAB deu Goi H la hinh chieu vuong goc cua A tren BC Hinh chieu vuong g()c cua dinh S tren mat phang (ABC) la diem nam tren di/c^ng thiing A H

1 Tinh the tich khoi chop S.ABC

2 Tinh goc giffa hai mat phang (SAC) va (ABC)

Cau IV (2,0 diem)

DAP AN THAM KHAO

I

(3,0 diem)

1 (2,0 diem) Khao sat s\i bien thien va ve do thj (C) cua h a m

-I

(3,0 diem)

Doc iziii tu" siiiii each giai Mdnz tiT cau I l de so 3

I

(3,0 diem) 2 (1.0 diem) Tmi m de do thi ham s o d ) c a t true hoanh tai

I

(3,0 diem)

Tuyg'n chpn 39 (SJ thCf sutc hpc ki mfln To^n Idp 12 Mang cao - P h j m Trpng Thi/

Dac) h a m y ' = 3x^ + m

D c do thi h a m so (1) cat true hoanh tai mot d i e m duy nhat k h i

va chi k h i h a m so da cho ddn dieu trcn x hoac dat hai cifc tri

3 (1,0 diem) G i a i phifc/n^ trinh

Phu-dng trinh da cho tu'cing difdng v6\

-^2Y

0; l o g ,

1 (1,0 diem) Tinh the tich khoi chop S A B C

• G o i O la hinh chicu vuong goe ciia S t r c n m p ( A B C ) ; O thuoc A H

Tuy^n chpn 39 <3i thCf silc hpc ki mOn Join I6p 12 Nang cao - Pham Trpng Thu

• T a m g i a c S A O c o SO = VsA^ O A ^ = ^ a ^ | = a ^

-The tich cua k h o i chop S.ABC la:

The tich cua k h o i chop S.ABC la:

VsABC =^SO.S^BC =^SO.AB.AC = ^ (dvtt)

0,25

2 (1,0 diem) T i n h g ( k uiffa hai m a t p h a n g ( S A C ) va ( A B C )

• Ha O M 1 A C = A M (1); do A C 1 SO, suy ra A C _L mp(SOM)

Til' ( 1) CO f ( x ) = r(y), suy ra x = y 0,25

Thay vao phu^dng trinh thiJ hai cua he ta diroc

(x'' + l) ( x 2 + X - 1 ) + x(x - 2) - 1 0 (x^ - l)(x'' + x-* + X + 2) = 0 0,25

Ta lha'y x"* + X - ' + x + 2 = (x + l ) - ( x ^ - x + l ) + l 0,25

V

(1,0 diem)

Khoang each tiT 1(1; 2) den tie'p tuye'n d bang

1 K h a o sal sir bien ihien va ve do thj (C) ciJa ham so

2 G o i M la mot diem di dpng tren (C) c6 hoanh dp > 1 T i c p luyen tai M

c^l hai liem can cua (C) tai A va B T i m M do dien tich lam giac O A B nho nhat (V(3i O la goc loa do)

C&ull (2,0 diem) *

1 T i m gia Iri Ic'Jn nha'l va gia Iri nho nhat cua ham so '

y = e'' (e^" - 9) + 36^" + 1 Iren doan [ 0 ; l n 2 ]

59

Tuygn chpn 39 dg thCr sire hgc ki mfln Toan I6p 12 Ming cao - Phgm Trpng Tha

2 C h o h i n h c h o p tuf g i a c d c u S A B C D c6 tat ca ciic c a n h cCing b a n g a G o i

A', B ' C D ' l a n lu^m la t r u n g d i e m c i i a cac c a n h S A , S B , S C , S D C h i J n g m i n h

t i c p l u y c n A c i i a ( C ) bie't A song song A ' : y = - — x •

DAP AN THAM KHAO

Cau Dap an Diem

I

(3,0

diem)

1 (2,0 diem) K h a o s a t s\i b i e n t h i e n v a ve do thi ( C ) c u a ham

D o c gia t i f g i a i _ c a c h g i a i tUdng tu" c a u I l de so^ 1

2 (1,0 diem) Tim M de d i ^ n tich t a m g i a c O A B nh o nha't

1 (0,5 diem) T i m gia trj Idn n h a t , gia trj nho nha't

I I

(2,0 diem) T a c o y = e ^ ( e - ^ - 9 ) + 3 e - % l = 6 ^ ' ' + 3 e ' ' ^ - 9 e ^ + 1 (*)

2 (0,75 diem) C h i J n g minh

I I

(2,0 diem)

T a c o y ' = - e ^ " ^ \ s i n x 0,25

I I

(2,0 diem)

y " = e " ' ' ^ \ s i n 2 x - e ^ " ^ \ c o s x 0,25

I I

(2,0 diem)

T h e y ' , y , y " v a o v e ' t r a i c i i a b i e u thu'c c a n c h i l n g m i n h va r u t

I I

(2,0 diem)

3 (0,75 diem) G i a i b a t phirdng trinh

I I

(2,0 diem)

o 2 x ^ + 4 x > ( 6 - x ) ^ o x ^ + 1 6 x - 3 6 > 0

<=> X < - 1 8 hoac X > 2

0,25

61

Tuygn chpn 39 thCr sufc hpc k1 mOn Toan I6p 12 Nang cap - Phgm Trgng Thu

K e t hcJp dieu k i e n ta c6 tap nghiem c u a bat phuTcfng trinh da

I I I

(2,0

diem)

1 (1,0 diem) T i n h t h e tich kho'i c a u n g o a i t i e p h i n h chop

Giii suf SH la du'dng cao ciia hlnh chop S.ABC

K h i do V I SA = SB = SC nen moi diem 5

N a m tren SH each d e u A , B va C

T r o n g m p ( S A H ) , diTcIng trung triTc

ciia SA cat SH t a i O thi O la t a m mat

cau ngoai tiep hinh chop va ban k i n h

mat cau lii R = SO

G o i 1 la trung d i e m cua S A thi tiJ

giiic A H O I noi tiep n e n :

G o i S H la dirdng cao cua hinh chop d e u S A B C D t h i H la t a m

hinh vuong A B C D va SH d i qua t a m H ' cua hinh vuong

A ' B ' C ' D ' M o i d i e m nSm tren S H deu each d e u b o n d i e m A ,

B , C, D va cung each deu b o n d i e m A ' , B', C , D ' T r e n diTcfng

thang S H ta xac djnh d i e m O sao cho OA = O A ' thi O each d e u

0,5

62

Cty TNHH MTV DVVH Khang Vift^

tam d i e m A , B , C, D , A ' , B', C , D'tiJc la t a m d i e m do n a m tren mSt cau t a m O, ban k i n h R = O A D i e m O la giao d i e i n cua du'dng thang S H va mat phang trung triTc cua doan thang

A A '

Ta chu y rang S A C la tam giiic vuong can G o i I la trung d i e n i

A A ' , thi SIO cijng la tam giac vuong can dinh I n e n

0,5

I V

(2,0 diem)

1 (1,0 diem) T i m gidi han

A = lim

-+ ( l - V c o s x ) -+ In(l -+ x^)

= l i m x-»0

/ 2 e'' - 1

Tuygn chpn 39 dg mil sire hpc kl m6n Toan Idp 12 Nang cao - Ph?m Trgng Thu

<=>

l o g ^ ( x + y) + l o g ^ ( x - y ) = l (log ^ 3)log ^ (X + y) - log^ (X - y) = log ^ 3

G i a i he ta diTdc : <

x + y = 2 log (x + y ) = l

V i e t phifi/ny trinh tiep tuyen A cua

G o i tiep d i e m la A ( x , , ; y „ ) V i l i e p tuyen A song song

A ' : y = - - ^ x nen he so goc cua tiep tuyen la y ' ( x „ ) =

Thdi glan lam bai: 120 phut

CSu I (3,0 diem) Cho ham so y = x'* - 2 m x ^ + m (1) ( m la tham so)

1 K h a o sat s i f b i e n thien va ve do thi (C) cua h ^ m so (1) k h i m = 1

2 Bie't A la d i e m thuoc do thi ham so (1) c6 hoanh do b^ng 1 T i m m de

khoang each ttr d i e m B - ; 1 den tiep tuyen cua do thi h a m so (1) tai A Ictn

C a u I I I (2,0 diem) Cho hinh ti? d i e n A B C D c6 A B = C D = a, bon canh con l a i

bang nhau va biing b G o i I va J Ian liTdt la trung d i e m cac canh A B va C D

1 Chi'rng minh rang IJ vuong goc v d i A B va C D T m h do dai doan thang IJ

2 T i n h ihco a va b the lich k h o i lu' dien A B C D

I (2,0 diem) K h a o sat si/ bie'n thien va ve do thj (C) ciia ham

Doc gia tu'giai_cach giai tUctng tif cau I I de so 2

2 (1,0 diem) T i m m de khoanj; each lit diem

' V i A la d i e m thuoc do thj ham so (1) nen A ( l : 1 - m )

• y ' = 4x"^ - 4 m x = > y ' ( I ) = 4 - 4 m PhU'dng trinh tiep tuyen A cua do thj ham so' ( I ) tai A c6 phiTdng trinh y - (1 - m ) = y ' ( I ) ( x - 1 )

iiygii clioii 313 do llni MIC lio; ki IIKHI Join \6p 12 Nang cao - Phgm Trpng ThU

V a y ham so da I ciTc tieu tai —, I C T =

ham so dat cifc dai tai —, (QY) - ^

B a l phiTcJng Irinh da cho lifdng during

(3 logj x - 2)^ - 5(4 + 2 iog3 X) > 2 log3 x - 7

Cty TNHH MTV DWH Khang Vi^l

K c l h(tp v d i d i e u k i c n bat phirdng Innh c6 n g h i c m la

0 < X < —1=7 hoac X > 27 0,25

I I I 1 (1,5 diem) C h i t n ^ niinli ranj; I J vuonK R O C v('/i A B va CD

(2,0 diem)

Hai lam giac can A C D va B C D bhng nhau va co chung day

C D ncn AJ = BJ V a y lam giiic A B J U\m giac can dinh J

J ^ \

7 - - \^

c

Hai lam giac can C A B va D A B bring nhau va c6 chung d a y

A B nC-n C I = D I V a y lam giac C D I la lam giac can di'nh I ncn

2 (0,5 diem) T i n l i theo a va b the tich kho'i tuT dien A B C D

Chi'i y rang vi A B vuong goc v6i CI vii D I ncn

A B 1 m p ( C D l ) The lich ciia khoi lu" dien A B C D la:

T 1 ^

6 4 ; xac dinh khi 4''''^"^''' - — > 0 64 0,25

Tuygn chpn 39 ai thil SLfc hpc ki mOn Join I6p 12 Nang cao - Ph^m Trpng Tha

V a y tap xac dinh ci'ia ham so h"i D = : ; + <»

2 (1,0 diem) G i a i h$ phi^dns trinh

Cty TNHH MTV DWH Khang Vi$t

• Vc'Ji X,, = - - , la CO phu'dng Irlnh l i e p luye'n can t i m la

Thdi gian lam bai: 120 phut

Cau I (3,0 diem) Cho ham so y = x"* - (3m + 2)x - + 3 m c6 do thj ( C , ^ )

1 Khao sal sir bien i h i e n va ve do i h i (C) cua ham so k h i m = 0

2 r m i m de diftlng lhang d: y = - 1 cat ( C „ , ) l a i bon d i e m phan b i e l deu c6

hoiinh do nho bdn 2

Can U (2,0 diem)

1 T m i giii tri k'tii nhal va gia Iri nho nhat ciia ham so y = cos^'x - 6cos^x+5

2 ChiJng minh rang ham so y = x - in(! + e") luon luon ddng b i e n

3 G i i i i bat phu'ctng irlnh 3 ^ - 4 ' ' 1 < — •

3"+! _ 4^+1 7

cau in (2,0 diem) Cho hinh lang irii ABC A'B'C'co day la lam giac deu canh

a, canh ben A A ' = b T a m giac B A C va B ' A C la nhifng l a m giiic vuong dinh A

1 Chu'ng m i n h neu H la trong l a m lam giac A ' B ' C thi A H 1 m p ( A ' B ' C )

2 T i n h theo a va b ihe lich hinh lang trii da cho

^•^u IV (2,0 diem)

Tuygn chpn 39 <Si thCf sire hpc kl mOn Toan Iflp 12 Nang cap - Ph?m Trgng Thu

Cau V (1,0 di^'m) Cho ham so' y = + 2(m + l)x + m^ + 4m ^ ^ ^ ^

X + 2 ham so' (1) c6 ciTc dai va cifc lieu, dong ihcli cac diem ci/c Iri cua do ihi cilng vdi

go'c loa do O tao lhanh mot tam giac vuong tai O

DAP AN THAM KHAO

diem) Hiim so da cho xiic dinh Vi'h moi gia Iri ciia x Dat l = cos-x liii()<l<l

Khi do, ham so da cho Ire) lhanh y = 1" - 6 t + 5 0,25

v 4 <()<=>-< 4 < — o3 -1 < X < 1

I I I

(2,0 diem)

I (1,0 diem) Chtfng minh neu H la tron^ tam tam siac

Tir gia Ihie't A C 1 AB la suy ra AC' 1 A'B', ngoai ra vi C'H 1 A'B' nen la suy ra AH 1 A'B'

Tu-dng lir, lir giii Ihiel AB' 1 AC, la suy ra AB' 1 A ' C , ngoai

ra V I B'H 1 A'C nen la suy ra AH 1 A'C

B' Vay AHlmp(A'B'C')

2 (1,0 diem) Tinh theo a va b the tich hinh lanu tru da cho

Tuy^n chpn 39 6i thCf sure hpc ki mOn Toin I6p 12 Nang cao - Phjm Trpng Thi/

The tieh ciia hinh lang irii lii

o - log.| X. log^ 11 = 1 - log,| 11

o log, u( 1 - log V ) = 1 o log u = !

Dat u = logx. V = logy, la c6 he phiTdng trinh

[ ulog3 = vlog4 f 11 = -log4

[log4(log4+ii) = log3(log3 + v) [v = -log3

2)-• V d i m 0 Ihi ham so (1) eo eire dai vii eiTe lieu, suy ra

diem eiTe dai A( - 2 - m; - 2) va ei/e lieu B( - 2 + m; 4m - 2)

Cty TNHH MTV DWH Khang Vi$t

Oi so 17 D E T H L T SCrCThdi gian lam bai: 120 phut H O C Ki I M O N T O A N L d P 12

mx + 4

(1), vdi m la lham so ihi/c

Cfiu I (3,0 diem) Cho ham so y = ^ ^

1 Khiio sal sir bien ihien vii ve do ihj (C) eiia ham so (1) vt'Ji m = 1

2 Tim m de ham so da eho nghieh bien lien khoiing (-co; 1)

Cfiu II (2,0 diem)

3 Gidi ba'l phiMng trinh 2 = ^ ^ - ^ - ^ + 1 5 2 ^ - 5 < 2\

Cfiu I I I (2,0 diem) Cho ti? dicMi ABCD Biet AB = CD = a, AC = BD = b,

AD = BC - e

1 Tinh the lieh khoi eau ngoai tiep U? dien

2 ChiJug minh riing c6 mot mat eau tie'p xue vdi 4 mat eua tiJ dien

Cfiu V (1,0 diem) Cho ham so y - —c6 do ihi (C) va diTcIng lhang

X - 2 A: y = a(x + l) + I T i m a de du'itng lhang A e a l d o ihj (C) tai hai diem eo hoanh

^9 trai dau nhau

D A P A N T H A M K H A O

I

(3,0 diem)

f

1 (2.0 diem) Khao sat s\i bien thien va ve do thj (C) cua

ham-I

(3,0 diem)

f

Doe iiiii tir iiiiii eaeh iiiai tU"dnt; lif eau 1.1 de so 1

I

(3,0 diem)

f

2 (1,0 diem) Tim m de ham sf/da cho nghjch bien tren khoang

I

(3,0 diem)

73

TuySn chpn 39 ai thcf site hgc kl mOn ToAn Iflp 12 Nflng cao - Ph^m Trpng ThLf

1 (0,5diem) Tim gia t r i \(tn nha't, gia t r j nho nha't

Dat t = sin X thi 0 < t < i (do x e |();

4 •>

Khi do, ham so da cho Irc'l lhanh g ( l ) = 2t - - t

T a c o g ' ( l ) = 2 - 4 l - , g'(t) = ()c:>l = 4= ( v i t e [ ( ) ; 1])

V2 / 1 \

2 (0,5 diem) Chiang m i n h rang

X c l ham so i'(x) = lanx - x iron niJa khoang 0;

3 (1,0 diem) G i a i ha't phifc/ng trinh

BF^T da cho liTdng diMng

1 (7,5 diem) T i n h the tich kho'i c i u ngoai t i e p ttf d i g n Goi M , N , O Ian liTdt la Irung d i e m cua A B , C D , M N

= 0 A ^ = A M ^ +0M^ = —+ b 2 + c 2 - a 2 a ^ + b ^ + c ^

7 5

Tuye'n chqn 39 aj thif sure hgc kl mOn ToAn Idp 12 IMflng cao - Phgm Trpng Jha

Ta CO bon mat ciia tiJ dien la bon lam giac bang nhau Do do

bon du"(ing l i o n ngoai l i e p bon lam giac do c6 ban k i n h b;ing

nhau la r

0,2S

Suy ra khoang each 111" 0 den bon m i l l liJ dien b;lng nhau

d = 7R~ - 1"' Suy ra m a l cau l a m O ban kinh R' = V R " - r"

l i c p xi'ic vi'Ji 4 m a l liV dien

D a l t = - " ( t > 0 ) PhuTcJng trinh thiir nhii't cua he t r d

The' vao phu'dng trinh thi? hai cua he ta difdc

log(x + 2) + log(3x - 2) = 4 log 2 o (x + 2)(3x - 2) = 16 0,25

76

<=> 3x~ + 4 x - 2 0 = 0 c : > x = 2 hoac x = ~ y ' Vc'ti X = 2 => y = 2 (Ihoa dieu k i e n (*))

• V6i X = - y ^ y = - y (khong thoa dieu k i e n ( * ) )

V a y he phu'dng trinh da cho c6 nghiem (x; y ) = (2; 2) 0,25

V

(1,0 diem)

Tim a de dUc/ng thang A cat do thj (C) tai hai diem c6

Cho ham so' y = x"^ - 2mx^ + m - 1 (1), v d i m la tham so'

1 Khao sat sU" bien ihien va ve do thi (C) cua ham so' (1) k h i m = 1

2 Xac dinh m de ham so (1) c6 ba d i e m ci/c trj, ddng thdi cac d i e m ciTc tri

i J a do thi tao thanh m o t tam giac co ban k i n h du'tJng tron ngoai tic'p b^ng 1