39 tuyển chọn để thử sức học kỳ môn tóan nâng cao 10 bắc trung nam

Niic do ma ciic em c6 phifdng phap nhan IhiJc mot so mat cua the' g i d i xung quanh va biet each boat dong c6 hieu qua trong dfli song.. M o n Toan gop phan rat quan trpng trong viee rc

510.76

T527C Chuyen Nguyen Quang Dieu - Dong Thap) PHAM TRONG THlT

uyen chon

di thvc svcc hoc ky

I

FHAM TRONG T H U (GV T H P T Chuyen Nguyen Quang Dieu - Dong lhap)

Danh chohocsinh Idp 10 chuffng trinh ndng cao

On tap vd nang cao ki nojig lam bai

• HiJ V;EN 'um\

I ffiG N H A X U A T B A N fiAI H O r O l i n C G I A H A NOI

N H A xufifT B A N Dfli H O C oud'c Gin Ndl

16 Hdng Chuoi - Hai Bd TrUng - Hd Npi Dien thoai : Bien t a p - Che b a n : (04) 39714896;

Hdnh chinh: (04) 39714899; Tong bien t a p : (04) 39714897

Fox: (04) 39714899

Chiu trdch nhi^m xuat ban

Gidm doc - Tong bien tap : TS P H A M T H ! T R A M

Biin tdp T H A N H HOA

Che ban : C O N G TY K H A N G V I E T

Trinh bay bia C O N G TY K H A N G V I E T

Tong phdt hdnh vd doi tdc lien ket xuat ban:

1 D!CH Vg VAN HOA KHANG V I E T

f Dja Chi: 71 Oinh Tien Hoang - P.Da Kao - Q.1 - TP.HCM ^

Quyet dinh xuat b^n so: 337LK-TN/QD-NXBDHQGHN, cap ngay 31/07/2013

In xong va nop JOu chieu quy IV nam 201 3

Md^i iidi itdn

B6 sach rfVE/V CHON 39 DE THLf SL/C HOC KI MON TOAN I6p 1 0,

11 va 12 nang cao d i r a c bien soan va tuycn chon diTa trcn noi dung chiTcIng

trinh T H P T hicMi hiinh; bo sach toan niiy giup cac c m c6 dieu kicn lam qucn \6'\

cac dang dc thi hoc k i cf muTc do cao Ricng cuon 12 co them phan phu luc giup

cac cm tif k i e m tra, danh gia, bd sung kie'n thtfc ve toan T H P T cho minh nham

tao ncn toan can ban vffng chitc cho cac cm \i\S6c khi c h i n h thtjTc bU'dc vao ki thi Dai hoc, Cao dang

Hi vong bo sach sc gop phiin giiip cac em dat kct qua cao t r o n g cac ki t h i ,

dong thcJi l a mot cong cu hd trd cho cac bac phu huynh giiip cho con em hoc tap tot h(<n

Trong qua trinh bien soan, d\x tac gia dfi c d g i i n g nhiTng cuon sach van c o the

c o n nhiJng khiem khuyct ngoai y muo'n Chung tdi ni't mong nhan di/dc sif gop y

chan thanh ciia cac thiiy, co giao, cac em hoc sinh dc trong hln tai ban sau sach

dUcJc hoan chinh hdn

Tac gia rii't cam d n Nh;i xuii't ban Dai hoc Qudc gia Ha N o i , Cong ty T N H H

M T V D V V H Khang V i c t da dong vicn, khuycn khich vii tao moi dieu kien de

cud^n sach nay st'lm den tay ban doc

Website: phamtronglhu.com.vn

Tac gia PHAM TRQNG THLf

Kf HIEU DIJNG T R O N G B p S A C H

Vectd phap luyen V T P T Vcctd chi ptiiTdng V T C P Dieu phai chuTng minh dpcm

Y e u cau bai loan Y C B T

[Jat dang thu-f i J D T fhUifng trinh P T HC' phiftfng Irinh H P T Hal phiMng trinh U P T

7 loi khuySn cho thi sink phuung phdp gidi mQt bcU thi

Nhif chung ta da b i c i mon Toan la mon hoc chic'm mot vj tri ra't quan trong va then

chot, rat can thiet dc hoc cac mon khae tif ticu hoe cho den cac Idp tren M o n Toan

gitjp cac em nhan biet cac moi quan he ve so lifdng va hiiih khong gian cua the gidi

hicn Ihifc Niic) do ma ciic em c6 phifdng phap nhan IhiJc mot so mat cua the' g i d i xung

quanh va biet each boat dong c6 hieu qua trong dfli song M o n Toan gop phan rat quan

trpng trong viee rcn luyc'ii phifinig pliiip suy nghl, phifclng phap suy luan, phiTcIng phap

giai quyct van dc N o gop piian phat irien tri thong minh, each suy nghl doc lap, linh

boat, sang tao va vice iilnh thanh cac pham chat can thict cho ngiTrti lao dong nhiT can

cu, can than, eo y c h i vifdt kho khan, lam vice c6 kc hoach, eo ne nep vii liic phong

khoa hoc

Xua't phat tCf v i tri quan trpng ciia mon Toan, qua tlufe te giang day nhieu nam d

cap T H P T T o i nhan lhay rang de hpc sinh hoe lot mon Toan thi ngoai vice cac c m nam

vffng k i c n ihifc trong sach giiio khoa, ky niing tinh toan that tol ma con phai bie't phiTtJng

phap giai mot hai thi nhif liie nao trong lue dang thi dc eo d i e m cao M u o n lam diWc

dieu nay tin' sinh can phiii tufin thii Iheo cac biTdc sau day:

1) That binh tinh trong luc lam bai thi

2) Can doe that cham rai loan bp de, danh gia sc) bp dc) de, kho cija cua cac eau,

xem nhfrng can nao quen thupc, la vdi minh

3) Giai ngay lap tiJc cac eau ma ban thay de

4) M o t vai eau can thiel den sir suy nghl sau h(<n, thi sinh can phai doe ky eau hoi,

gach du'di eiic gia thiet va y e u eau ciia bai toan Dinh hiTcfng each giai, hinh dung do

phi'rc tap ciia eiieh giai dc c6 sir lira chpn diing dan

5) T r h i h bay bai giai thi sinh khong nen lam tat, m o i biTdc ncn vic't mot dong de de

kiem tra, v l giam khao chain bai thi theo ba rem nen eo mot birdc nao do sai thi van con

diem d nhfrng biftk- bien doi diing trifdc do Cach hay nha't la lam xong biTdc nao k i e m

tra bifiKc ay de phat hicn ngay cho sai

6) Trong qua trinh giai mot bai toan ne'u thi sinh gap kho khan giila ehiTng, c6 the

chijfa khoang trong trC-n giay thi de bo sung sau va nhanh chong ehuycn sang l a m cau

khae

7) K h i da hoan ta't bai t h i , neu con thCii gian thi sinh ncn doc l a i bai giai va ra scat

lai cac chi liet da trnih bay (thong thirilng cac loi thi sinh hay bo sot bift'lc l a m la tap xac

dinh, dieu k i c n c6 nghla ciia can bac ciian, ham so logarit, doi can khi dung phiTcfng

phap doi bien de tinh tich phan, loai bo ngiiiem ngoai lai trt)ng phi/dng trinh ) nham

hoan thicn bai thi tot h()n cho den het gid

Nhieu hpc tro toi day ap dung 7 l(1i khuyen tren da tn'f thanh thii khoa dai hpc cua

nhieu trirclng, nhiTng thanh cong nha't la loi ed hoc tro thii khoa " k e p " k h o i A va B ciia

trirdng D a i hpc Khoa hpc Tif nhien TP H C M va D a i hpc Y DiTdc TP H C M nam 2011

Chuc cac thi sinh dat k e l qua cao trong cac k i thi

PHAM TRQNG THI/

B p BE THfir Sljrc HQC K I MON TOAN LdfP 1 0

A BO BE THUf sure HQC K I I MON TOAN LtfP 10

D E SO 1 D E THlIr SllTC HOC KI I MON TOAN L 6 P 1 0

Thdi gian lam bai: 90 phut

C a u I (1,0 diem)

Cho hai tap hdp P = |x e K I |x - 2| < 3|, Q = |x e I I x + 2 > 4

1 V i e t cac tap hdp P va Q diTdti dang khoang, doan, ntj'a khoang hay hctp cija

cac khoang, doan, nihi khoang B i c u dicn cac tap hdp nay tren trijc so

2 Chiang minh rang C j ^ C P n Q ) = C ^ P u C j ^ Q

X| + X j la Ciic so' nguycn

2. Giai va bien hian phiTdng trinh -—— + ^ = 2 •

difclng lhang A M l a i E T i m loa do diem E V

1 M u - 1 » r , ^ ~ cosA cosB cosC a ^

2. Nhan dang tam giac A B C thoa man + + = — ( )•

a b c be

Cau\ (2,0 diem)

1 Giai phirong irinh X/4-3N/10-3T = X - 2 * i ^ , j

2 T i m gia tri nho nha't ciia y = x + —^— (voi x > 3 ) i •'.•-> J,

Cfiu V I (1,0 diem)

Cho tam giac ABC vuong tai A c o A B = 3, AC = 4 va Irung tuycn A D Tim

diem E e AC sao cho BE 1 AD

1 (1,0 diem) Ve do thj y = - x ^ + 4x - 3 (doc giii tif giai)

2 (1,0 diem) Tim m de phii'ofng trinh

1

/ - 2 - l \

/ • ' -3

y = - x ^ + 4 x - 3 va diTdng thiing y = m - 3

+ Hiim so y = - x ^ + 4 x - 3 la hiim so' chSn ncn do thi do'i

xifng qua true tung Khi x > 0 thi ham so' trd thanh

y = - x ^ + 4x - 3 Do do do thj ciia ham so y = - x ^ + 4 x - 3

bao gom philn do thj ham so y = - x ~ + 4x - 3 d ben phai true

tung vii phan do'i xiJng ciia no qua true tung. -^^M.'- ; V \

Thco do thi, phiTdng trinh dii cho c6 it nhat ba nghiem khi v&

chi khi -3 < m - 3 < 1 <=> 0 < i T i < 4

1 (1,0 diem) Txm cac gia trj nguyen cua m

Phu'dng trinh da cho co hai nghiem phan biet khi va chi khi

m ^ ! ] ( * ) •

m - 1 ;t 0 A' = 5 m - 1 > 0 m > -

5

Ta CO X| + X , = 2(m + l)

m - 1 = 2 + -m - l

De tong X| + X2 la so nguyen thi dieu kien can vii dii la m - 1

lii iTc'k- ciia 4 Co cac Iri/cJng hdp:

VcTi dieu kien do PT da cho tiTdng diTdng (m +1 )x = 6 (*)

»Vdi m = - 1 thi (*) VP nghiem nen PT da cho v6 nghiem

6

m + 1 iWvri'-V'J' ' Vdi m ^ - 1 thi CO nghiem x =

PhUdng irinh da cho c6 nghiem t r e n o

m ^ - 1 va m 2: PhMng Irinh da cho c6 nghicm x =

m +1

I V

(2,0

diem)

1 (1,0 diem) 'Vim to a do diem E

G lii Irong lam cua lam giac ABC ncn

2 (7,0 diem) Nhan dan^ tarn giac

Ta CO ( * ) « bccosA + accosB + abcosC = (**)

Tir a- - b- + c^ - 2bccosA ^ bccosA = ^(b^ + c^ - ) (1)

TiTdng liT:

accosB - ^(a- + c^ - b") (2); abcosC = ^(a^ + b" - c") (3)

The (1), (2) va (3) vao (**) va nil gon la diTdc:

b" + c" = a" c::> AABC vuong lai A

6

V 1 (1,0 diem) Giai phtfc/nj? trinh

(2,0 diem) PhiTclng irinh da cho liTOng diTdng

Vc'Ji — <x74 10 < — va

27 3 X - - 7 x + 15- ( 7^ X — + — > 0 , V x e R 4 0,25

Nc-n (*)<=> X = 3 Vay phirong trinh c6 mol nghicm x = 3

2 (1,0 diem) Tim jjia tr j nho nha't ciia

0,25

1 T i m lap hop con A , B ciia S sao cho A u B | l ; 2; 3; 4 | , A n B = | l ; 2 •

2 T i m cac lap C sao cho C u ( A n B ) = A u B

1 G i i i i phifiJng Irinh yjl-x' +x\J\ 5 = ^3 - 2x - x

2 G i i i i va b i c n luan phifdng Irinh m^x - m~ - 4 = 4m(x - 1)

C a u I V (2,0 diem) T r o n g m i l l phdng Oxy, Um d i e m M biel:

1 M N P Q la hinh binh hanh vdi N(2; 3), P ( - 6 ; - 3 ) , Q ( l ; 8)

Cho lam giiic A B C c6 A = 120", A B A C = - 6 va A M B C = - 1 6 ( v d i M la

irung d i e m ciia BC) T i n h do dai cac canh A B va A C

1 (0,5 diem) T i n i tap hop con A , B ciia S sao cho

Tir A u B = | i ; 2; 3; 4 | suy ra hai ph;1n liV 3 vii 4 phai Ihuoc

m p l vii chi mol irong hai lap A va B Do do c6 bon kel qua sau:

1 (0,5 diem) Ve do thj y = 3x + 4 (doc gia liT giiii)

2 (0,5 diem) X a c dinh a, c de do thj ham so'

V i do Ihi da cho di qua hai diem A ( l : - 3 ) , B(2; 5) ncn

- 3 = a - 4 + c a + c = l [a = 4

5 = 4 a - 8 + c 4a + c = i3 [c = -3

3 (1,0 diem) X a c djnh jjiao diem ciia hai do thj t r e n

Toa do giao d i e m ciia hai do ihj IrC-n la nghiem ciia he

V a y giao d i e m ciia hai do I h j ircn lii

1 (1,0 diem) Giai he phiTc/ns trinh

He phifting l i i n h da cho tiftJng dufcJng

Thdi gian lam bai: 90 phut

C S u I (1,0 diem) Cho lap hc.tp A = { x e /J |x| < s}, B = j x e Z 1 9 < < 26} Xac

dinh cac lap help A n B , A w B, A \, B \

C S u I I (2,0 diem)

1 X a c dinh cac he so' ciia parabol y = a x - + bx - 3 b i c l rang parabol di qua

d i e m A ( 5 ; - S) va c6 Iruc do'i xiJng x = 2 V c parabol t i m dU'dc

2 Cho parabol (P): y = x " ^ - 4 x + 3 Xac dinh m dc (P) va diTdng th^ng

d: y = mx - m ^ + 12 cal nhaii tai 2 d i e m c6 hoiinh do trai dau

T r o n g m a l phring O x y cho tam giac A B C c6 A ( 2 ; 1), B ( l ; 3), C (

-1 Tim loa do Irong l a m G ciia lam giac A B C

2 T i m loa do Irirc l a m H ciia lam giac A B C

X a c djnh c a c tap h(/p

I

(1,0 diem)

1 (1,0 diem) X a c djnh c a c he so'ciia parabol

I I

(2,0 diem )

V i do i h i da cho d i qua d i e m A(5; - 8 ) nen

Tir (1) va (2) suy ra a = - 1 ; b = 4 0,25

I I

(2,0 diem )

Ve y = - x - + 4 x - 3 ( d o c g i a tif g i i i i ) 0,5

2 (1,0 diem) X a c dinh m de ( P ) va di/tfns thang

Phifdng i n n h hoiinh do giao d i e m cua (P) va 6\i(1ng thilng d la

X - - 4 x + 3 = m x - m ^ + 1 2 < = > x ^ - ( 4 + m ) x + m^ - 9 = 0 0,5

Parabol (P) v i i during thang d cat nhau t a i 2 diem c6 hoanh do

m i i da'u ichi v i i chi k h i - 9 < 0 <=> - 3 < m < 3 0,5

I l l 1 (1,0 diem) G i a i phU"(/njj trinh

2 (1,0 diem) Giai va bien luan phifc/ng trinh

Vc'Ji X ^ phifdng trinh da cho Ird lhanh 3mx = 3 m - 7 0,25

• Vc'Ji m ^ 0 ihi x = — ^ • N g h i c m thoa man dieu k i c n khi

1 (0,5 diem) T i n i toa do tnin;; tam G ciia tam giac A B C

T o a do Irong t a m G ciia tam g i a c A B C l i i

Dang Ihu'c xay ra khi vii chi khi x = y = z = ^ •

Thdi gian lam bai: 90 phut

Cfiu I (1,0 diem) Cho so ihiTc m

1 Giai phi/dng Irinh 3x^ - 2 x - l = X - X

2 Giiii phiTdng trinh 2x'* + - 16x^ + 3x + 2 = 0

Cfiu IV (2,0 diem) Trong mat phang Oxy cho lam giac A B C c6 A(2; 6),

B ( - 3 ; - 4 ) , C(3;()) .•

1 T i m toa do D la chan cua dutfng phan giac trong cua goc A t f y >

2 T i m Ipa do lam cua diTdng tron noi tiep lam giac ABC

1.(0,5 diem) Tim m

I

(1,0 diem)

Ta C O B c A <=> 3m - 1 < - 1 < 1 < 3m + 7 0,25

I

(1,0 diem) lm<0

o <^ o - 2 < m < 0

I

(1,0 diem)

2 (0,5 diem) Tim m

I

(1,0 diem)

De A n B = 0 thi la c6 hai tru'(1ng hdp:

• 1 < 3 m - I <=> m > - •

3

8 2 Vay A n B = 0 khi m < — hoac m > - •

3 - 3

0,25

I I

(2,0 diem)

I (1,0 diem) X a c dinh cac he S(Ya, b c cua parabol

I I

(2,0 diem)

V i do thi (P) di qua diem A(0; I) nen c = 1, do do phifdng trinh parabol (P) c6 dang y = ax^ + bx + 1

0,25

I I

(2,0 diem)

I'4a-2b + l = l

V i (P) qua B ( - 2 ; 1), C(3; 2) nen la co he <^

[9a + 3b + 1 = 2 0,5

17

T a l h a y phiTdng trinh da cho k h o n g c6 n g h i c m x = 0 n c n la c h i a

hai vc c i i a phifdng Irinh dii cho x " , la di/tfc

Tuyg'n chgn 39 de thif sifc hqc ki mfln ToanTflp 10 Nflng cao - Phjm Trgng Thu~

V I

(1,0

diem)

T i m dac diem cua tam jjiac A B C thoa

Ta CO S^y(, = bcsin BcosC

Thdi gian lam bar 90 phut

Cflu I (1,0 diem) T i m tap xac djnh ciia cac h i i m so':

V 4 X + 8

x + 1

-C&uU (2,0 diem)

Cho hai difcfng lhang d | : y = m x - 4 v a d-,:y = - m x - 4

1 Chtfng m i n h rang vdti m o i m, d, vii d ^ l u o n ciCl nhau l a i m o t d i e m co djnh tren true lung

2 T i m m dc lam giac tao thiinh bdi d , , d j va true hoanh c6 d i e n lich la 8

C S u I I I (2,0 diem) ,

1 Cho phiTdng l r i n h ( m - l)x*^ + 2(m - l ) x + m + 3 = 0 (1) T i m m de phiTdng

trinh ( I ) CO hai n g h i c m phan b i e l x , , x , thoa main x^ + x^ + 3 X | X 2 = 1

1 (0,5 diem) T i m tap xac djnh ciia ham so

(1,0 diem)

1 (1,0 diem) T i n h t o a d o c a c d i e m C , G

IV

(2,0 diem)

V i G e O x v a C e O y n c n t o a d o G { x ; 0 ) v a C ( 0 ; y ) 0,25

IV

(2,0 diem)

Do l i n h chii't I r o n g l a m c i i a l a m giac A B C la difdc I C = 3 I G

(*)

f 3 x - 9 = - 3 f x - 2 ( * ) « ( - 3 ; y + 2) = 3 ( x - 3 ; 2 ) « c^^

[ y + 2 = 6 [ y = 4

0,25

IV

(2,0 diem)

V a y G ( 2 ; ( ) ) v i i C ( ( ) ; 4 ) 0,25

IV

(2,0 diem)

2 (1,0 diem) T i n h c h u v i c i i a t a m g i a c A B C

IV

(2,0 diem)

A B = ( 4 ; - 2 ) ^ A B = N/16 + 4 =2V5 0,25

IV

(2,0 diem)

B C = ( - 5 ; 7 ) =^ B C = V25 + 4 9 = V 7 4 0,25

IV

(2,0 diem)

C A = (1; - 5 ) = > B C = N/1 + 2 5 = V 2 6 0,25

IV

(2,0 diem)

C h u v i t a m g i a c A B C la A B + B C + C A = 2Vs + ^ + V26 0,25

V

(2,0 diS'm)

1 (1.0 diem) G i i i i h e Dhif(/ni! t r i n h

V

(2,0 diS'm)

D a y la he chu'a d o i xu'ng, d e difdc he d o i xiiTng la d a l l = - x v i i

f '*

I" + v~ + I + V = '

diTcIc h e n h i T s a u : <^ ^ (*)

[ l y + t + y = 1 [S = t + y -^^^ '

Thdi gian lam bai: 9 0 phut

C S u I (1,0 diem) T i m t a p x a c d j n h c i i a cac h i i m so:

1 (0,5 diem) Tm\p x a c d i n h c i i a h i i m s6'

I (1.0 diem) H;im so y - • — ^ — du'dc x a c d i n h k h i <

x 2 - 3 x + 2

\-\>0

X - - 3 x + 2 ^ 0

0,25

lUyBII L'lHIII j y ge inu sue not kl mn loAn I6p 10 N5ng cao - Phjm Trgng-Thu

V i i y phiTdng I r i n h d i i c h o c 6 h a i n g h i e m la x = - 3 ; x = 6

I V

(2,0 diem)

1 (1,0 diem) G i a i h y phif(/n^ trinh

H e phifcfng I r i n h d a cho lifcing difitng

x'^y" + x - + y - + x y ( x + y ) + x y + x + y - 2 = 0

l - ( x + y ) + xy = 6

1 Cho cac tap hdp so A = (-4; 5], B = ^-V2; + o o j Hay xac djnh cac tap

hdp sau va bicu dien tren true so A u B; A n B; A \; B \

2 Hay tim tat ca cac lap con cua tap hdp X = |1; 2; 3|. ^, i

CSu II (2,0 diem) Cho parabol (P): y = -x^ + 2x + 3 * ':

2 DuTcJng thang d: y = 2x -1 citt (P) tai hai diem A va B Tmi toa do A, B va

tinh do dai doan AB s '"

CliU III (2,0 diem) *;

1 Cho phu^dng trinh (m - l)x^ + 2(m - 4)x + m - 5 = 0 Tim m de phiTdng Irinh

CO hai nghiC-m phan biet x,, X2 thoa man he ihufc - i - + = 3

2 Giai phiTdng trinh x^ - 5 x - 1 3 = X " + X + 1

C&ul\ (2,0 diem) Trong mat phang Oxy cho A(-2; 1), B(4; 3) va C(-l;y)

1 Xac dinh gia tri y de tam giac ABC vuong tai C

2 Xac dinh gia trj y de tam giac ABC co trong tam G

Cfiu \ ( 2 , 0 d i e m )

mx + y = -2 2x + y = -m Tim m de he phu'dng trinh c6 nghicm

1 Cho he phm^ng trinh duy nhat thoa man y^ = x

2 Chiang minh a) (sina + cosa)" + (sina - cosa)^ =2

b) CO.S20" + C O S 4 0 " + C O S 6 0 " + - + C O S 1 6 0 ' ' +cosl8()° =-1

Cfiu VI (1,0 diem)

Cho tam giac ABC Goi A', B', C Ian lifdt la trung diem cac canh BC, CA,

AB ChiJ-ng minh rang AA' + BB' + CC' = 0

ffil •••• <(it'' • • •

29

2 (0,5 diem) H a y tini t a t ca cac tap con ciia tap h(/p

0 , | i} , { 2 t , { 3 l , { ; 2 ! , { 2 ; 3 | { ; 3}, { l ; 2; 3 0,5

H

(2,0

diem)

1 (1,0 diem) L a p ban^ hien thien va vc paraboi (P)

Doc giii tif giai

2 (1,0 diem) T i m toa do A , B va tinh do dai doan A B

Toa do giao diem A va B eua (P) va d la nghiem ciia he

(2,0

diem)

1.(1,0 diem) Tim m de phifc/n^ t r i n h co hai nghiem phan bi^L

PT da eho eo hai nghiem phan biel x,, X2 va X|X-, ^ 0

1 (1,25 diem) Xac dinh gia t r i y de tam giac A B C vuong t a i C

2 (0,75 diem) Xac djnh gia t r i y de t a m giac A B C c6 t r u n g tfim

Do G la trong lam cua AABC ncn

1 (1,0 diem) Tim m de h ^ phiff/ng t r i n h c6 nghicm

luyen engn j y BB Iw '•Ale hoc. ki man luail \fi]} lU NJliy lJU ^ n i j i i i iiyiig

Vay CO hai gia l i i cua m la m = - 1 hoac m = -3

2 (1,0 diem) ChiynK niinh

a) Ta C O

VT ^ s i i r a + 2sinacosa + cos'cx + s u r a - Zsinacosa + cos'a

= 2(sin"a + cos~a) = 2

b) V i cos2()" = cos( 180^' - 160^') = -cosl60"

cos4()" -cosdSO" - 140") =-cosl40"

cos6()" = cos( 1KO" - 120") = -cosl 20"

cosSO" -cos(180" -100") = -cosl00"

DE S O S OE T H C T SOC HOC KI I MON TOAN L 6 P 10

Thdi gian lam bai: 90 phut

Cau I (1,0 diem) Cho hai lap hi.tp A = jn e Nl n < I I j , B = jx e x l x - i

1 Tim lap hdp A n B V i c l kcl qua difc'li dang licl kc

2 Tim lal ca cac lap hdp C sao cho C c A va C c B Bai loan c6 tat ca bao

nhieu nghicm?

Cfiu U (2,0 diem)

1 Tim parabol (P); y = a x - + bx + 2,bict parabol c6 dinh 1(2; - 2 )

2 Lap bang bicn Ihicn va \ do thj ciia ham so' paiabol (P) vi'Ji a, b vCra lim

Cau III (2,0 diem)

1 Cho phift^ng Irinh (m - I)x" - 2(ni + l)x + m - 2 = 0.Tim m dc phiMng Irinh

CO nghicm phaii bicl x,, x-, Ihoa man he Ihifc — + —

Cfiu IV (2,0 diem) Trong mat phang Oxy cho lam giac ABC c6 cac diem

M(l; 4), N(3; 0) P ( - 1 ; 1 ) liin lu-i.n la Iriing diem cac canh A B , AC, BC •

1 Tim Ipa do cac dinh cua lam giac ABC

2 Tinh do dai liung luyen AP ciia lam giac ABC

Cau\l (1,0 diem) Cho = 5N/2, b =6, Ci, b) = 135" Tinh (a - 2b)(b - 2a),

DAP AN THAM KHAO

I

(1,0 diem)

1 (0,5 diem) Tim tap h(/p A n B Viet ket qua dxiiii danj; liC't ke

I

(1,0 diem)

T a c o A = {0; 1; 2; 10}

Malkhcic x - l < 2 o - 2 < x - l < 2 o - l < x < 3 Siiy ra B = ( - I ; 3)

0,25

I

(1,0 diem)

I

(1,0 diem)

2 (0,5 diem) Tim ta't cii cac tap h(/p C sao cho

I

(1,0 diem)

Ta C O C c: A va C c B => C c A n B 0,25

I

(1,0 diem)

T i r d 6 C = 0 , jo}, [ i j , {2}, {O; 1}, {O; 2}, {l; 2}, {O; 1; 2}

Biii loan c6 lal ca 8 nghicm

0,25

II

(2,0 diem )

1 (1,0 diem) Tim parabol

II

(2,0 diem ) Ta C O - — = 2 <=> 4a + b = 0

2a

0,25

33

Thay toa do dinh 1(2; - 2) vao (P): y = ax^ + bx + 2 ta diMc:

- 2 = 4a + 2b + 2 <=> 4a + 2b = - 4

0,25

f4a + b = 0 [a = l Giai he < <=> •

Vay parabol can lim la y = x~ - 4x + 2 0,25

2 (IJ) diem) L a p bang bien thien va ve d<1 thj (doc gia ti/ giai)

(thoa man dicu kien (*))

Vay gia in m can lim la m = 4

1 (1,5 diem) T i m tga do cac dinh cua tam giac A B C

Ti? giac A M P N la hinh binh hanh ncn ta c6 M A = PN (*)

1 (1,0 diem) Giai h^ phifi/ng trinh

Xel 2x-^ + 3 = 5y (1)

2 y ' ' + 3 = 5x (2) Lay (1) IrCr (2) ve iheo ve" la diTdc:

2(x-' -y"') = 5(y - X) <i> 2(x - y)(x^ + xy + y^) + 5(x - y) = 0 c:>(x-y)(2x"- + 2 x y + 2y" +5) = 0

0 ( X - y ) ^ x - + y" + ( X + y)- + 5J = 0 <=> X = y (3)

1 V I X - + y - + ( x + y)'' + 5 >0, Vx, y j

1^

CSu I V (2,0 diem) T r o n g m a l p h a n g O x y cho d i e m A ( - 2; 1) ^ |

1 T m i toa d p c i i a d i e m B d o i xufng vdti d i e m A qua goc tpa d p O , t i m tpa dp cua d i e m B ' d o i xi^ng v d i d i e m A qua true h o a n h

2 G p i C la m o t d i e m c6 l u n g dp b a n g 2 v i i l a m g i a c A B C v u o n g cf C T i m tpa dp ciia d i e m C

1 (0,5 diem) L a p mOnh de phii djnh ciia cac n i ^ n h de

b) 3x G X , x " < 4 x - 4 0,25

I

(1,0 diem)

2 (O.S diem) Viet cac tap h(/p sau bhne each liet ke

I

(1,0 diem)

I

(1,0 diem)

4a + 2b + c = 9 wH*(V^

^ 16a + 4b + c = 5 16a + 4b + c = 5

1 (1,0 diem) Tvm to a do cua diem B doi xtfng \6\m A

Ta bict riing hai diem doi xi^ng vc'Ji nhau qua go'c tpa do thi cac tpa dp tU"(1ng u'ng ciia chung doi nhau

Do do ncu A ( - 2 ; 1) thi B(2; - 1 ) ' " > '' *

Ta bie't rllng hai did'm doi xii"ng vc'Ji nhau qua true hoanh c6

hoanh dp bllng nhau va tung dp doi nhau d/ J / - :

Do do ncu A ( - 2 ; l ) t h i B'(-2; - 1 )

2 (1,0 diem) T i m toa dp cua diem C

Gpi C(x; 2 ) e { O x y )

T a c o BC = ( x - 2 ; 3), A C = (x + 2; I ) Tarn giac A B C vuong tiii C khi va chi khi BC.AC = 0

<=>(x~2)(x + 2) + 3.1 = Oc:>x^ - 4 + 3 = = 0 o x = ± l Vay C ( l : 2) h o a c C ( - l ; 2)

V

(2,0 diem)

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

Neil v > - 2 : He da cho In* thanh y + 2 = x - 2

3 ' 5

2 (1,0 diem) Chrfny niinh

D a l a = s i n ' x va b = cos-x => a + b = 1 0,25

V T = a-* + b'* = (a^ + b^ )^ - laW = ((a + b)^ - 2ab)^ - lii^h^ 0,25

= (1 - 2ab)- - 2 a - b ' = 1 - 4 a b + 4a^b^ - 2a^b^ = 1 - 4 a b + 2a^b^ 0,25

= 1 - 4sin^xcos^x + 2sin^xcos^x (dpem) 0,25

39

Thai gian lam bar 90 phut

C S u I (1,0 diem) T i m tap xac dinh ciia cac h i i m so:

C a u I V (2,0 diem) Trong mat phiing Oxy cho d i e m A ( l ; 3), B((); 2), C(4; 5)

1 T i m toa do ciia d i e m E ihoa man CE = 3 A B - 4 A C

2 T i m toa do ciia d i e m A ' d o i xiVng vdi d i e m A qua B

CfiU V I (1,0 diem) Cho tarn giac A B C co A B = 24cm, A C = 32cm, BC = 40cm

1 Chijrng m i n h tarn giiic A B C viiong lai A

2 T i n h do dai dift-fng tiling t u y c n B M ciia tarn giac A B C

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

(1,0 diem)

1 (0,5 diem) T i m tap xac djnh ciia ham s(Y

H a m so y = 2013 1 3 - 2 x > 0 -t- — du'dc xac dinh khi <;

1 (1,0 diem) T i m ^ia trj Win nha't, nho nhtYt (neu co) ciia h a m so'

2 (1,0 diem) T i m y^ia tri Win nha't, nho nha't (neu co) ciia ham s(Y

y = - X " - 4 x + 3 C O a = - 1 < OnC-n be l o m hUdng xiiong

Hoanh do dinh = = ^ =-2i[0;5]

V a y miny = r(5) = - 4 2 , maxy = 1(0) = 3

0,25 0,25 0,5

41

l u ^ u i i unfit J g u g i i i u J U L iiaiiu L a u - I iigiii i i g i i g

2 (1,0 diem) Tim toa do ciia diem A ' d(Yi xvtnp^ d i e m A qua B

V i A" d o i xiJng vc'Hi A qua B , n c n B la irung d i c m ciia A A '

1 (1,0 diem) G i a i hC' phi/oTnK trinh

H e phiTcJng i r i n h da cho lu'iJng diTdng

That gian lam bar 90 phut

C a u I (1,0 diem) T i m lap xac dinh ciia cac ham so':

Cau III (2,0 diem)

1 T i m m dc phiWng Irinh m - ( x - 2 ) + 7m = ( 6 - m)(x + m ) - 2 v6 so'nghiem

2 G i i i i phu'dng irinh X - + 8 x - l = 2x + 6

3 G i i i i phiftlng Irinh \l5x~ + 2 x - 7 S = 2x + 3

c a u I V (2,0 diem) T r o n g m a l phang Oxy cho d i e m A( 1; 0), B(3; 0), C((); 4)

G o i M , N , P Ian kfcn la trung d i e m cac canh BC, C A , A B

44

1 T i n h giii i r i ciia bieu ihi'rc A M B C + B N C A + CP.AB

2 T i n h cos A, cosB, cosC

1 (0,5 diem) T i m tap xac djnh ciia ham s<Y

V a y lap Xiic dinh ciia ham so la D = ( l ; 3 ) u ( 3 ; +oo)

2 (0,5 diem) Tini tap xac djnh ciia ham so

H i i m so y = ^-^^—-—^difdc xiic dinh khi

luyeii Liiyii J3 ue uiu JUL iigti lu iiiJiiU LJU - r iijiiii iigiig

iini-Giiii he (*) la dUilc a = - 2 , h = - 4 , c = 6

V a y ( P ) : y = - 2 x - - 4 x + 6

0,25 0,25

2 (7,0 rfigw/) L a p han}; hien thien va ve do thj (dQc 0a t\i ^iai)

L (0,5 diem) Tim m de phif</n}; trinh c6 v6 s(Y nshiem

x ' + 8 x - I = - 2 x - 6

x > - 3 x^ + 6 x - 7 = 0 <=>

x^ +l()x + 5 = 0

X = 1

\ -5 + 2^f5

Vay pbiftlng trinh da cho co hai nghicm x = 1, x = -5 + 2^f5

3 (0,75 diem) Giai phrf(/ns trinh

L (1,0 diem) Tinh gia trj bieu thtfc

T a c o P(2;()), M ^3 ^ ' , N -; 2 va lap cac toa dp

S = 5

<=> S- + 5S - 50 = ()<=>

S = - l ( ) Vdi S = 5 => P = 6 Luc do ta c6 he phiTdng trinh

0,25

47

2 (1,0 diem) Rut j^on

1 Tim paraboi (P): y = a(x - m)^ bict rang parabol c6 dinh I ( - 3 ; 0)va cal

true lung lai diem M(0; - 5)

2 Lap bang bicn ihicn va vc do ihj cua ham so bac hai Ircn vdi a, m viTa lim

3 Giiii phirong irinh x + ^|x + - + ^x + — =9

CSu I V (2,0 diem) Trong mal phang Oxy cho lam giac ABC vuong can lai C

va diem A ( I ; - I), B({); 2) ' | % / ,

2 Tinh do dai difcJng cao CH cua lam giac ABC r-na\\m<i,K*,\) J H

- lan'x - cot X = 7

Cau V I (1,0diem) Cho AABC c6 BC = iS, AC = 2^2, AB = S-^

1 Chi'rng minh BC^ = AC" + AB" - 2AC.AB

2 Tinh cac goc A, B, C

-Ml

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

I I (0,5 diem) Chitng minh

(1,0 diem )

1 (1,0 diem) Tim parabol (P)

H i i m so' y = a(x - m ) " <=> y = ax" - 2amx + a m "

D o ihi ham so hi parabol c6 dinh l ( - 3 ; 0) ncn

- 2 a m

X =

-2a

= m = -3 (a ^ 0)

Thay m = -3 vao h a m so la c6 y - a(x + 3 ) "

Do ihi ham so' ca'l iriic lung tai M ( 0 ; - 5) ncn

- 5 - a ( ( ) + 3 ) - <:^a = - ^ V a y (P): y = - ^ ( x + 3 ) -

0,5

0,25

0,25

2 (1,0 diem) I Sip ban}; bien thien va ve do thj (doc ^ia tii ^iai)

1 (0,75 diem) Tim m de phiTcfng trinh co n^hiem

I l l

(2,0

diem)

T a c o ( m - l ) ( X - l ) + m - 2 = ( ) < = > ( m - l ) x = l ( * )

• m = 1: (*) tr('-f thiinh Ox = l.phiTdng Irinh da c h o v6 nghiC-m

• m ^ 1: ( * ) CO n g h i c m x > 3 khi va chi khi — ^ — > 3

x + 3

= x - l (1)

= - x + l (2)

x ^ - 3 2x + l = ( x - l ) ( x + 3) x " = 4

<=> X = ±2

0,25

x * - 3 ( 2 ) ' » < <=>

2x + I = ( - x + l)(x + 3) 7 = - 2 ± 76

X - + 4 x - 2 = 0

Vay phirong trinh da cho c6 nghicm x = ±2, x = - 2 ± yffi

3 (0,75 diem) G i a i phifc/ns trinh

V i l i I + - = 3 hay I = - , ta co x = 6

2 2

V a y phifdng trinh da cho co mot n g h i c m la x =

6-1 (1,0 diem) T i m toa do diem C

b) V T = sin^x + — ^ — + 2siiix • —5— + cos^x +

—^-j-sin X —^-j-sinx cos x

1 -, + 2cosx lan x - c o r x

2 Cho l a m giac A B C Chu'ng m i n h a l n g :

a) sin A = sin(B + C), cos A = - c o s ( B + C)

A + B C b) sin = cos—•

2 2 c) cos(A + B - 2C) = - c o s 3 C •

C a u V I (1,0 diem) Cho l a m giac A B C c6 a - BC = 5cm, c = A B = 8cm va

g 6 c B = 77" -v i ,

1 T i n h canh A C j

II

(2,0

diem)

1 (1,0 diem) Tim parahol (P)

Phifdng trinh hoanh do giao diem cua (?) va d la:

ax^ + b x - 2 = x - 4 (*)

( ? ) cat diTcing lhang d tai hai diem c6 hoanh dp x = ± - 2 n e n

X = ± - 2 la hai nghiem cua phiTdng trinh (*), la c6 HPT

' 4 a - 2 b - 2 = ( - 2 ) - 4

4a + 2 b - 2 = 2 - 4 <=> i

4 a - 2 b = - 4 4a + 2b = 0

(t > 0 ) , phiTdng irinh (*) ird lhanh

2 l - - 3 l - 5 = 0<r>i = - l (loai) hoacl = - (nhan)

V

(2,0 diem)

Chitng minh rang A B C D la ti? giac noi tiep difgfc

x - y - 1 = 0

luyen cnpn j » ae inu sue npc KI mon loan lop lu ixiang cao - r-n^mi u^ng i n u

C§iu\ (1,0 diem) Cho ham so ("(x) = <^ , • \J '

1 T i m tap xac dinh cua ham so ^ , ^ j

V d i gia trmao ciia a i h i d cat (P) l a i hai d i e m phan biet V | , ,

2 X e l tinh chan, Ic ciia hiim so y = 3x - 2x + 7 ; v » « Vf^

C&uWl (2,0 diem)

1 T i m m dc phifttng irinh ( n r + 1 )x" + (3m - l)x + 6m = 0 c6 hai nghiem X| = - 1 va \-, = 3

2 Giai phifdng trinh x x + 4 = - 2 x " + 3x + 2

1 Giiii he phu'dng trinh

3 G i i i i phmtng Irinh x " + 3x - 57(x - l)(x + 4) = 10

C S u I V ( 2 , « < f/^;H).Tiong mat phang Oxy cho A(3; 5), B ( - 5 ; 1), C(0; - 4 )

1 Tinh do dai trung tuyen A M cua tarn giac A B C

1 Tinh canh BC, dien tich S cua lam giac

2 T i n h difdng cao h., va ban kinh R ciia difctng Iron ngoai t i c p tam giac

ir' .; i ' 'r'; • ' i

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

I I (0,5 diem) T i m tap xac dinh cua ham s(Y

(P) cal difcing lhang d lai hai diem phan b i c l o (*) co hai

<=> <=> <

A = 49 + 8a>0

a;tO 49-

a >

8

0,5

2 (1,0 diem) Xet tinh chSn, le ciia ham s<Y

Vdi

n

-moi x e D => -x G D Ta co

X) = 3( - X)"* - 2( - X ) - + 7 = 3x'* - 2 x ^ + 7 = lU) 0,5

I I I 1 (0,5 diem) T\m m de phi^oTnjj t n n h co hai nghi^m

X > -4 3x- + x - 2 = 0

X =

Vay phifWng trinh da cho co hai nghiem x = - 1 , = ~ '

3 (0,5 diem) Giai phiTifng trinh

x > I

X < - 4 (*) Dieu kicn (x - i)(x + 4) > 0 <=>

I'T da cho urcJng dirdng (x" + 3x - 4) - 5\/x- + 3 x - 4 - 6 = 0 Dal t = \/x- + 3x - 4 (t > 0) ta co phiAlng liinh

r - 5l - 6 = 0c:> I = -1 (loai)

I = 6 (nhan)

V('<i I = 6 ihi N/X- +3X - 4 = 6 o X- + 3x - 40 = 0

X = 5 (Ihoa man (*))

x = -X Vay phifclng trinh da cho co hai nghiC-m x = -8, x = 5 •

<=>

IV

(2,0 diem)

1.(1,0 diem) Tinh do dai trung tuyen A M ciia tam giac A B C

Toa do irung diem M ciia BC:

L a y (1) Irif ( 2 ) thco vc la difdc

1 (0,5 diem) T i n h c a n h B C , dien tich S ciia tam giac

Theo dinh l i cosin la c6:

Cau M (2,0 diem)

1 Xiic dinli a dc ba diA-Int: lhang d|: y = 5(x + 1), d , ; y = - 3 x - ( a + 6) va

d^: y = ax + 2a + 3pliaii bicl va dung qui ^

2 Xac diiih ham so y = 2x~ + ax - b bicl do ihi ham so' nay la parabol c6 di'nh

I ( - h 4)

Cau m (2,0 diem) *

I 1 Cho a, b, c la ba so' thifc ihoa man 5a + 2h + 3c = () ChiJng minh rang

phiftfng I n n h ax" + bx + c = 0 c6 nghicm

3 Giai phifdng Irinh (x" - 3 x ) \ / 4 - x -3x~ =0

CAM \\ (2,0 diem)

TrongmatphangOxycho A ( - l ; -1), B((); I ) , C(4; - i), D(3; - 3 ) !

1 Chi'rng minh liV giac ABCD la hinh chiT nhiil

2 Tinh clui \ va diC'n lich ciia hlnh chu" nhat do

Cau VI (1,0 diem) Cho lam giac ABC c6 AB = 2, BC = 4, CA = 3 Tinh:

1 Tinh AB AC l o i siiy ra cosA?

2 Goi G la Irong lam ciia lam giiic ABC Tinh AG BC

DAP AN THAM KHAO

diem) Ham so xac dinh khi m" - 4x~ > 0 o

Vay tap xac dinh cua ham so lii D =

V a y lap xac djnh ciia ham so lii D = |1; 3 ) u ( 3 ; + co)

1 (1,0 diem) Xac djnh a de ba dtf(/ng thang

d| va d^cat nhau tai diem A ^ - a - 1 1 - 5 a - 1 5 ^

d|, d-, d-, ddng qui o d j d i qua A

+ 2a + 3 -5a -15

• Vdi a = 13: Ta thay d,, d j , <ij la ba diTdng lhang phan biet

va dong qui Vay a = 13 thoa miin de

0,25

0,25

2 (1,0 diem) Xac dinh ham s d

0,25

63

+ N c i i a = 0 tCr giii thict siiy ra 2b + 3c = 0(1)

• NcLi h = 0, iir (1) => c = 0 K h i do phiTdng i r i n h da cho c6

n g h i c n i m o i x

• Ncu b ^ 0, thi phifdng Irinh da cho c6 n g h i c m x = - — = - •

b 3

TrU(!ni> hap 2

T\i gia i h i c l siiy la h = - ^ • Liic do

A = b- - 4ac = 5a + 3c •4ac = - ( 2 5 a - + 9 c - +4ac)

TCr do siiy ra phifttng Irinh da cho c6 nghicm

2 (0,75 diem) Giiii phifcTn^ t r i n h

Vay phiTcJog trinh da cho c6 hai nghiC-m x = 3, x = 4 •

3 (0,75 diem) G i a i phifi/n^ trinh

n c n It? giac A B C D la hinh

2 (1,0 diem) I'lnh c h u vi, dien tich hinh chff nhat t r e n

N h a n hai vc ciia phifcJng i r i n h (1) vdi 3 roi cong vdi phffcfng

trinh (2) ihco vc, nhan hai ve'ciia phiMng Irinh (1) vdi 4 roi

cong v(^i phifttng trinh (3) theo v c t a di/tJc:

7 x - 8 y = 2 2 fx = - 6

- y = 2 y = - 8 (5) The (5) viio (1) ta di/dc / = - !

Vay he phiTiJng trinh da cho cd nghicm ( - 6; - 8; - 7)

2 (1,0 diem) Tim gia trj l<tn nhSt

A p diing B D T Co-si cho 2 so khong a m 9, x - 9 , ta c6:

VI

(J,0

diem)

1 (0,5 diem) Tinh ABAC rdi suy ra cosA?

Ta CO BC = AC - AB => BC" = AC" + AB^ - 2AC.AB

=^x5.AB=i^^ii^^i^=^i±ii^=-i (*)

2 2 2 TCr do cos A = cos I' AC, A B AC.AB -3 1

AC AB 2.2.3

0,25

0,25

2 (0,5 diem) Finh

Goi M la trung diem cua BC Ihi AG = - AM = - ( A B + Ac)

Do do AG.BC = ^ ( A B B C + AC.Bc) = ^ ( - B A B C + CA.CB)

Ma BA.BC =

CA.CB =

BA-+BC CA- 4+16-9 11 CA-+CB^-BA- 9 + 16-4 21 Suy ra >\G.BC = - _ U 2A

1 Cho ham so y = V 2 -x -V2x + 5a.rim a dc lap xac dinh cua ham so la

doan CO chicu diii bang 1

2 Tim lap xac dinh ciia ham so y -\[\ 9 + 2Vx + 8 •

Cfiu II (2,0 diem)

1 Xac dinh a va b dc do ihi ham so y = ax + b di qua hai diem A -1; - va

B(2; - 2 )

2 Xac dinh cac he so a, b, c biel rang parabol (P):y = ax^ +bx + cc^l true

tung tai diem A(0;-1) va di qua hai diem B(l; 2), C(-2; 5)

Cfiu III (2,0 diem)

1 Cho phiTOng trinh (m - l)x^ - 2(m + 2)x + m + 1 = 0 c6 hai nghiem Xj, Xj

Vdi g'^ f""! nguyen nao cua m thi phu"dng trinh tren c6 hai nghiem thoa man

+ X', -x,x-, la so nguyen

2 Giai phiTctng trinh

3 Giai phu'dng trinh

x - 2 2x

- 3 = 5

-x"^ + X - 8

x + 1 (x + l)(4-x)

Cfiu IV (2,0 diem) Trong mat ph^ng Oxy cho A(2; 3), B(-l; 1), OC = -3i - 2j

1 Tim toa do vectd x = AB - 2AC - 2CB j

2 Bieu ihj vectd OC theo cac vectd OA, OB Tim toa do E sao cho CM

trong tarn tam giac ABE

Cfiu V (2,0 diem)

1 Giai he phiTdng trinh 2x2-y2=l

xy + x' = 2

2 Cho ba so thiTc a, b, c ChuTng minh r^ng (ab + be + ca)^ > 3abc(a + b + c)

Cfiu VI (1,0 diem) Cho tam giac ABC vuong tai A c6 di/cJng cao AH Ke HD

vuong vdi AB va HE vuong goc vdi AC

- - - • 2 , ' '

1 Chu'ng minh AD.AB = AH i

2 Chu'ng minh tuT giac BDEC noi tiep mot difcJng tron

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

Cdu

I

(1,0 diem)

1 (0,5 diem) Tim a de tap xac dinh ciia ham so" la

Ham so xac dinh khi 2 - x > 0 2x + 5a > 0 <=> \ x<2

Ham so' da cho xac dinh khi x > -8

Vay lap Xiic dinh ciia hiim so la D = |-8; + oo)

0,25 0,25

1 (1,0 diem) V(^i gia trj nguyen nao cua m thi phuTcTng trinh

D i e u kien de phi/dng Irinh da cho eo hai nghicm la:

K e i hdp dieu kien la diTcic m e 1-1; 0; 2; 3; 5

3 (0,5 diem) (iiai phtfdng trinh

1 (1,0 diem) Tim toa do vectd x = A B - 2 A C - 2CB

Ta c(') X - A B - 2 ( A C + C B ) = AB - 2AB = - A B = (3; 2)

2 (1,0 diem) Bieu thj vectd P C then cac vectd OA, OB

Gia sur OC = m O A + nOB (m,n e K) (*)

1 (1,0 diem) G i a i h^ phif(/ng trinh

X»3l he phU'dng Irinh 2x- - y - = 1 (1)

xy + x- = 2 (2) Nhan phUdng Irinh (1) vdi 2 roi IrCr vcHi phifdng trinh (2) theo

ve va rut gon lai ta duTdc: 3x" - xy - 2y~ = 0 (3)

Tuyfi'n chpn 39 dg thCf sure hgc ki mOn Toan I6p 10 Nana cap - Phjim Trgng Thu

Giai he phifcJng Irinh <

Vay he phifdng trinh d;

3

y = — X

2 » i ->

BDT da cho liTcJng diTdng

a-b- + b-f-^ + c ^ i - - a^bc - a b \ -abc^ > 0 0,25

uiy I iMHfT MIV uvvH Miaog V1?T

2 (0,5 diem) Chi?ng minh t i t giac B DEC noi tiep mpt dtfcfng t r o n

2 Liel ke ciic phan lir ciia P va tinh C^,M

-C&u I I (2,0 diem) Cho h a m so y = x^ + (2m + 1 )x - m - 1 co do ihi (P^^)

1 Lap bang bicn ihien va ve parabol (P) ciia ham so vc^i m= - 1

2 Tim m de (P,^^ )cat true Ox lai hai diem phan bicl co hoanh do x,, Xjlhoa

man x^ + X T = X | X T +1

Cfiu I I I (2,0 diem)

1. Giiii phi/dng Irinh

2 Giai phiTdng Irinh

x - 2 1

x + 2 X x(x + 2) 2x - 3 3x + 5

3 Giiii phiTdng irinh x^ = 2N/X + 2 + 4

Cfiu I V (2,0 diem) Trong mal phang Oxy cho cac diem A(l; 3), 3(4; 2)

1 Tim loa do diem D Iren Ox each dcii hai diem A va B

2 Tim loa do diem C de liV giac OABC la hinh binh hanh Chufng minh OABC la hinhchiTnhal

Cho lam giac ABC, lay cac diem 1, J sao cho l A = 2IB, 3JA + 2JC = 0

uhiyng minh rilng diffJng lhang IJ di qua Irong lam G ciia lam giac ABC

TUyen cfion tTuf r.i'K hoc Ri mfri Toan Irip 10 fj.ihi; i:,io Phjiii Tipng mil'

2 (1,0diem) I m i m de (P,„)cat true Ox tai hai diem phan bi^t

Phifdng tiinh hoanh do giao diem ci'ia (P^ )^^* ^•'V'-' '^^

x - + ( 2 m + l ) x - m - l = 0 (*)

0,25

(P|jj )c;tl true Ox tai hai diem phan biel c6 hoanh do x,, X2

0 phiWng Irinh (*) c6 hai nghiem phan bict x,, X j 0,25

Vay phirong Irinh da cho c6 hai nghiem x = - 1 , x = 4

2 (0,5 diem) Giai pbrf</ng trinb

Ta CO 2 x - 3 3x + 5 2x - 3 = 3x + 5

2 x - 3 = -(3x + 5)

<=> x = -8 5x = -2

X = - 8 2-

X = —

5 Vay phifdng trinh da cho c6 hai nghiem x = -8, x = - - ^ •

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

Vx + 2 + 2 {*) Giai ( * ) :

1 (1,0 diem) T i m toa do diem D tren Ox each deu hai diem.,

VI D e Ox nen D(x; ())cach deu A va B suy ra

2 (1,0 diem) T\m toa do diem C de ttf ^iae O A B C

Goi C(x; y) la dinh ihiir lu-hinh binh hanh O A B C ( 1 )

Ta CO OC = (x; y), A B = (3; - 1 )

1 (1,0 diem) G i a i h$ phuf(/ng trinh

X c l he phifdng Irinh 2x- - x y + 3y" + 1 = 7 x + 12y (1)

2 ( I A + I B + I C ) = 5IJ 6IG = 5IJ 0,25

Suy ra I , J, G thang hiing nen dicing thang IJ di qua trong tam

D E SO 1 8 D E THLT SLTC HOC K l I M O N T O A N L 6 P 1 0

Thdi gian lam bai: 90 phut

C a u I (1,0diem) Cho ham so r(x) = x / l - x ^ + 7 3 - x

1 Tim tap xac djnh A eiia ham so' ('(x)

2 Gia su- B = | x e !R I - 2 < X < 2| Hay xac djnh cac tap hcJp A n B, A \

C a u I I (2,0 diem) Cho ham so y = - x " - 4x + 6 c6 do thj la parabol (P)

1 T i m toa do dinh va phiTdng Irinh triie doi xufng cua (P)

2 V e do thi ciia (P)

3 Difa vao do t h i , hay cho bict tap hdp cac gia trj cua x sao cho y > 0

c a u I I I (2,0 diem)

1 T i m ciic gia trj ci'ia m de phiTdng trinh x " - 2 ( m - l ) x + m ^ - 3m = Oco hai

nghiem x , , x-, thoa man he ihiifc x^ + x^ = 8

1 Cho biet hinh dang eiia lam giac A B C '

2 T i m do dai dirdng cao cua lam giiic A B C va tam d\Xl1ng Iron noi liep lam

giac A B C

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