Phương pháp giải các chủ đề căn bản hình học 12 lê hoành phò

516.0076 NGlTTJnS LE HOANH PHO PH561P I I I II I "T^t •^va ren luyer» -g, kl nang W T i l a m bai_' , NHA XUAT BAN DAI HOC QUOC GIA HA NOI NGlTr.TbS LE HOANH PHO CAC CHUDE CAN BAN T BOI D I / O N G H O C S I N H GIOI THIJVIBJTiWHBINHTHUAN N H A XUAT B A N D A I HOC QUOC GIA HA NQ\ 16 Hang Chuo'i - Hai Ba Trang - Ha Npi Dien thoai: Bien tap-Che" ban: (04) 39714896: Hanh chinh: (04) 39714899: Tong bien tap: (04) 39715011 Fax: (04)39714899 rjc Chiu trdch nhiem xuat ^ ^ ban: Gidm doc - Tong bien tap: TS.PHAM T H I TRAM NGUYEN KHOI Che' ban: N H AS A C H H O N G A N Siia bdi: NGOC L A M Bien tap: Trinh bay bia: MINH V O T H I THITA Ddi tdc lien ket xuat ban: N h a sach H O N G A N SACH L I E N KET CAC CHU DE CAN BAN HINH HOC 12 Ma so: 1L- 155OH2014 • In 2.000 cudn, khd 17 x 24cm tai Cong ty Co phan Van hoa Van Lang Giay phep xuat ban so: 463-2014/CXB/09-99 OHQGHN, 14/03/2014 Quyet dinh xuat ban so: 153LK-TN/QO-NXB OHQGHN In xong va nop luu chieu quy II nam 2014 Nham muc dfch giup cac ban hoc sinh I6p 10, Idp 11, Idp 12 nam vQng kien thufc can ban ve mon Toan tCr luc vao THPT cho den chuan bj thi Tot nghiep, tuyen sinh Cao dang, Dai hoc, tac gia da bien scan bo sach P H U O N G P H A P G I A I gom cuon: - CAC C H U D E CAN B A N DAI SO - CAC C H U DE CAN BAN HINH HQC - CAC C H U D E C A N B A N DAI S O - GIAI TICH 1 - CAC CHU DE CAN BAN HINH HOC 11 - C A C C H U D E C A N B A N GIAITl'CH - CAC C H U DE CAN BAN HINH HQC 12 TCr nen Toan can ban nay, cac ban c6 the nang cao dan dan, bo sung va md rpng kien thufc va phifdng phap giai Toan, ren luyen ky nang lam bai va tC/ng bade giai dung, giai gpn cac bai tap, cac bai toan kiem tra, thi cCf Cuon C A C C H U D E C A N B A N H J N H H Q C cd 15 chu de vdi npi dung la phan dang Toan, tdm tat kien thufc va phadng phap giai, cac chu y; phan tiep theo la cac bai toan chpn Ipc can ban minh hpa vdi nhieu dang loai va mifc dp; phan cuoi la bai tap cd hudng dan hay dap so Du da CO gang kiem tra qua trlnh bien scan song khdng tranh khoi nhufng sai sot ma tac gia chua thay het, mong don nhan cac gdp y cua quy ban doc, hpc sinh de Ian in sau hoan thien hon Tac gia L E HOANH PHb O CHU D E I KHOI Dfi DIEN Vfl PHEP D0I HINH DANG TOAN • KHOI DA DIEN Hinh da dien va khoi da dien - Hinh da dien gom mot so hint hgn da gidc phang thod man hai dieu kien: (1) Hai da gidc hai ki hodc khong co diem chung hodc cd moi dinh chung, hoac CO moi cgnh chung (2) Moi cgnh cua moi da gidc la cgnh chung cua dung hai da gidc - Hinh da dien chia khong gian lam hai phdn: phdn ben vd phdn ben ngodi Hinh da dien ciing vai phdn ben cua no goi la khoi da dien - Moi khoi da dien co the phdn chia dugc thdnh nhitng khoi tic dien Moi da gidc cua hinh H duac goi la mot mat cua khoi da dien Cdc dinh, cdc cgnh cua moi mat goi la dinh, cgnh ciia khoi da dien Cdc diem nam hinh H goi la diem cua khoi da dien Khoi chop va khSi long tru - Khoi da dien duac goi la khoi chop, khoi chop cut neu no duac giai hgn bai mot hinh chop, hinh chop cut Tuang tu cho khoi chop n-gidc, khoi chop cut ngidc, khoi chop deu khoi tit dien, - Khoi da dien duac goi la khoi Idng Iru neu no duac giai hgn bai mot hinh Idng tru, tuang tu cho khoi hop, khoi hop chu nhdt, khoi lapphuang - Phdn chia va lap ghep cdc khoi da dien: Moi khoi chop vd khoi Idng tru ludn CO the phdn chia dugc thdnh nhitng khoi tit dien bang nhieu cdch khdc Chiiy: 1) Dgc so O-le ciia khoi da dien loi: D6i v&i moi khoi da dien loi H, ta ki hieu D la so dinh C Id so cgnh, M la so mat cua H ihi co dgc so Z(H) = D-C + M = 2) Hinh lang tru deu: hinh lang tru dung (c6 canh ben vuong gdc vai mat day) va CO day la da giac deu 3) Hinh chop deu: day la da giac deu va cac canh ben bang Bai toan 1: Chung minh r^ng nSu khoi da dien c6 cac mat la tarn giac thi so mat phai la so chin Hay chi nhung khoi da dien nhu the voi so mat bang 4,6,8,10 Gidi Goi s6 canh cua kh6i da dien la C, s6 mat la M V i moi mat c6 ba canh va moi canh lai chung cho hai mat nen 3M = 2C Suy M la so chan Sau day la mot so khoi da dien so cac mat tam giac la 4,6, 8,10 Bai toan 2: Chung minh rang moi dinh ciia mot hinh da dien la dinh chung cua it nhat ba canh va la dinh chung ciia it nhat ba mat Gidi Ta dung phan chung Neu xuat phat tu mot dinh nao chi c6 hai canh, thi moi canh nhu thd la canh cua chi mot da giac, trai voi dieu kien dinh nghia cua hinh da dien Vay moi dinh phai la dinh chung cua it nhat la ba canh, va vi vay no ciing phai la dinh chung cua ba mat Bai toan 3: Chung minh rSng ndu khoi da dien c6 moi dinh la dinh chung cua ba canh thi s6 dinh phai la so chSn Gidi Gia su khoi da dien c6 C canh va c6 D dinh V i moi dinh la dinh chung cua ba canh va moi canh c6 hai dinh nen 3D 2C Vay D phai la so chan Bai toan 4: Chung minh rSng nSu kh6i da dien c6 cac mat la tam giac va m6i dinh la dinh chung cua ba canh thi la khoi tu dien Gidi Goi A la mot dinh cua kh6i da dien Theo gia thiet, dinh A la dinh chung cho ba canh, ta goi ba canh la AB, AC, AD Canh AB phai la canh chung cua hai mat tam giac, la hai mat ABC va ADB (vi qua dinh A chi c6 canh) c Tuomg tir, ta c6 cac mat tam giac ACD va BCD Vay Ichoi da dien cliinh la khoi tu dien ABCD Bai toan 5: Chung minh rang, so goc cua tat ca cac mat gap doi so canh cua khoi da dien Suy so goc chan Gidi Goi so goc la G va so canh cua khoi da dien la C Trong moi mat la da giac thi so goc bang so canh, ma so canh dugc tinh Ian nen G = 2C, do G chEn Bai toan 6: Chung minh khong ton tai khoi da dien c6 mot so le mat va moi mat lai CO mot so le canh Giai Gia su ton tai khoi da dien c6 so mat la M le va moi mat chua so le canh Ci, i=l,2, ,M Ta CO so goc cua khoi da dien: G = C, + C + + C M ^ G le: v6 ly Vay khong ton tai kh6i da dien thoa de bai Bai toan 7: Hay phan chia mot khoi tu dien ba khoi tu dien bed hai mat phang Giai A Cho kh6i tu dien ABCD Lay diem M va N phan biet nam giiia C va D Bang hai mat phang (ABM) va (ABN), ta chia khoi tu dien da cho ba khoi tu dien: ABCN, ABNM va ABMD Bai toan 8: Hay phan chia mot khoi tu dien bon khoi tu dien bai hai mat phang Giai A Cho kh6i tu dien ABCD Lay diem M nam giiia A va B, diem N nam giua C va D Bang hai mat phang (MCD) va (NAB), ta M/_ chia khoi tu dien da cho bon khoi tu dien: B• AMCN, AMND, BMCN va, BMND Bai toan 9: Hay phan chia mot khoi hop nam khoi tu dien Giai B Co t h i phan chia khoi hop ABCD.A'B'C'D' nam khoi tu dien sau day: ABDA', CBDC, B'A'C'B, D'A'C'D va, BDA'C DANG TOAN PHEP Ddfl HINH Phep den hinh - Mot phep hien hinh F khdng gian duac goi la phep dai hinh neu no hdo toan khoang each giita hai diem hat ky: neu F bien hai diem bat ki M, N Idn luat hai diim M', N' thi M'N' = MN Phep dai hinh bien duang thdng duang thdng, matphdng matphdng Hap cua nhung phep dai hinh la phep dai hinh Cdc phep d&i hinh dqc biet - Phep dong nhdt: Phep dai hinh bien diem M bdt ki chinh no - Phep tinh tien: Phep tinh tien theo vecta v Id phep bien hinh bien moi diem M diem M' cho M M ' = v - Phep doi xung qua duang thdng (phep doi xung true): Cho duang thdng d, phep doi xicng qua duang thdng d la phep bien hinh bien moi diem thudc d chinh no vd bien moi diem M khdng thudc d diem M' cho mat phdng (M, d) d Id dmrng trung true cua dogn thdng MM' - Phep doi xirng qua mot diem (phep doi xung tdm): Cho diem O, phep doi xicng qua diem O la phep bien hinh bien moi diem M diem M' cho OM + OM ' = , hay O la trung diem cua MM' - Phep ddi xicng qua mat phdng (P) la phep bien hinh bien mdi diem thudc (P) chinh no vd bien mdi diem M khdng thudc (P) thdnh diem M' cho (P) Id mat phdng trung true cua dogn thdng MM' Hai hinh bang - Hai hinh da dien goi Id bdng neu eo mot phep dai hinh bien hinh thdnh hinh Ddi vai cdc khdi da dien Idi: Neu phep dai hinh F bien tap cdc dinh cua khdi da dien Idi H thdnh tap cdc dinh ciia khdi da dien Idi H' thi F bien H thdnh H' Dinh ly: Hai hinh tic dien ABCD vd A 'B'C'D' bdng niu chiing eo cdc egnh tuang icng bdng nhau, nghia Id AB -= A'B', BC = B'C, CD = CD', DA = D'A', AC = A'C, BD = B'D' Bai toan 1: Cho hai diem phan biet A, B va phep doi hinh f biln A A, biSn B B Chung minh rang f bien moi diem M nam tren ducmg thang AB diem M Gidi Ta CO f(A) = A, f(B) = B Gia su diem M thuoc duong thang A B va f ( M ) = M ' K h i M ' thuoc duomg M n g A B va A M = A M ' , B M = B M ' Suy M ' triing M , turc la f bi^n M chinh no Vay f bien mpi diem M nam tren duong thang A B chinh diem M Cho tam giac A B C va phep dai hinh f bien tarn giac A B C chinh no, tuc la f ( A ) = A , f(B) = B, f(C) = C Chung minh rang f bien moi diem M cua mp(ABC) chinh no, tuc la f ( M ) = M Bai toan 2: Gidi V i f(A) = A , f(B) = B va f(C) = C nen f bign mp(ABC) mp(ABC) Bai vay nSu M thuoc mp(ABC) va f ( M ) = M ' thi M ' thuoc mp(ABC) va A M = A M ' , B M = BM', C M = CM' Neu M ' va M phan biet thi ba diem A , B, C thuoc duong thang trung true cua doan thang M M ' tren mp(ABC), trai vai gia thiet A B C la tam giac Vay f ( M ) = M Bai toan 3: Cho t u dien A B C D Chung to rang phep dai hinh bien moi diem A , B, C, D chinh no phai la phep dong nhat Gidi Gia su phep dai hinh f bien cac diem A , B, C, D chinh cac diem do, tuc la f(A) = A , f(B) = B, f(C) = C, f(D) = D Ta chung minh rang f bien diem M bat k i M That vay gia su M ' = f ( M ) va M ' khac vai M K h i v i phep dai hinh khong lam thay doi khoang each giCra hai dikm nen A M = A M ' , B M = B M ' , C M = C M ' , D M = D M ' , suy b6n di^m A , B, C, D nam tren mat phang trung true cua doan M M ' , dieu trai vai gia thidt A B C D la hinh t u dien Vay M ' triing vai M va do f la phep dong nhk Bai toan 4: Cho hai t u dien A B C D va A'B'C'D' c6 cac canh tucmg ung b ^ g nhau: A B = A'B', BC = B'C, C D = C D ' , D A = D'A', D B = D'B', A C = A ' C Chung minh rang c6 khong qua mot phep dai hinh bien cac diem A , B, C, D Ian luat cac diSm A', B', C , D' Gidi Gia su CO hai phep dai hinh f i va f2 deu bien cac diem A , B, C, D Ian luat cac di^m A ' , B', C , D' Neu f| va f2 khac thi eo it nhat mot diem M cho neu M i = f i ( M ) va M2 = f2(M) thi M l va M2 la hai diem phan biet Khi v i fi va deu la phep dai hinh nen A ' M i = A M va A'M2 = A M , vay A ' M i = A'M2, tuang t u B ' M i = B'M2, C'Mi = C'M2, D ' M i = D'M2, do b6n dikm A\, C, D' cung nam tren mat phang trung true cua doan thang M1M2, trai vai gia thi^t A'B'C'D' la hinh t u dien Do vai mpi diem M ta deu c6 f i ( M ) = f2(M), tuc la hai phep dai hinh fi va T2 trung I Vay CO khong qua mot phep dai hinh biln cac di6m A, B, C, D \hn lugt cac dilm A', B', C, D' Bai toan 5: Cho hai tam giac bang ABC va A'B'C (AB = A'B', BC = B'C, AC = A'C) Chung minh rang c6 dung hai phep dai hinh, mSi phep bien tam giac ABC tam giac A'B'C Cho truac tam giac ABC Co nhirng phep dai hinh nao bien tam giac ABC chinh no? Gidi Tren duong thang a vuong goc vai mp(ABC) tai A lay diem D khac A, tren duong thang a' vuong goc vai mp(A'B'C') tai A' c6 hai diem phan biet Di va D2 cho A'Di = A'D2 = AD Ta CO cac hinh tu dien ABCD, A'B'C'Di va A'B'C'Dz CO cac canh tuang ung bang Neu f la phep dai hinh bien tam giac ABC tam giac A'B'C thi hoac f bien D thanh.Dj hoac f bien D D2 Vay CO diing hai phep dai hinh bien tam giac ABC tam giac A'B'C Do la phep dai hinh fi h'lkn tu dien ABCD tu dien A'B'C'D, va phep dai hinh fi bien tu dien ABCD tu dien A'B'C'D2 Day la truong hop rieng hai tam giac ABC va A'B'C trung Vay ta c6 hai phep doi hinh bien ABCD chinh no: la phep dong nhSt va phep d6i ,xung qua mp(ABC) Bai toan 6: Cho tu dien deu ABCD va phep doi hinh f bien ABCD chinh no, nghia la bien moi dinh cua tu dien mot dinh cua tu dien Tim tap hop cac diem M khong gian cho M = f(M) cac truong hop sau day: a) f(A) = B, f(B) = C, f(C) = A b) f(A) = B, f(B) = A, f(C) = D c) f(A) = B, f(B) = C, f(C) = D Gidi a) Theo gia thiet f(A) = B va f(B) = C, f(C) = A Do f(M) = M va chi M A = MB = MC Suy tap hop cac diem M la true cua duong tron ngoai ti^p tam giac ABC b) Theo gia thiSt f(A) = B, f(B) = A, f(C) = D Do f(M) = M va chi M A = MB va MC = MD, tuc la M d6ng thai •nam tren cac mat phang trung true cua AB va CD Suy tap hop cac diem M la duong thing di qua trung dikm cua AB va CD c) Theo gia thilt f(A) = B, f(B) = C, f(C) = D Do f(M) = M va chi MA = MB = MC = MD Suy tap hop cac diem M gom mot dikm nhk la tam tu dien ABCD 10 Cau Tir he phuong trinh da cho, ta c6: S= X 5' -4P>0 0 + 12m^ + 21m + = f(m) +y=m P = xy = m - Vai < m < 2, he CO nghiem x, y > khi: • Khi T = Khao sat ham so f(m) vai m e Ta duac minT = ' ^1 m = — va maxT = 100 m = 2 Cau 7a Goi A(a, b) thi B(4-a; 2-b) V i A, B e (E) nen: 10 (4-a)^ ^ (2-b)^ Vay A z wn 4ViT , + 15 10 a =2+ va B 15 ;b = i - 10 10 15 15 va phuang trinh (d): 8x + 9y - 25 = Cau 8a Goi 1(1+t; l+2t; -2t) e d la tam ciia mat cku (S) cSn tim Do (S) tiep xuc vai (a) va mat phang (Oxy) nen: d(I, (a)) = d(I, Oxy))c^ 2(l + t ) - ( l + 2t)-4t + l 2t t - l = 3t « t = - l h o a c t = Vai t = -1 thi (S) c6 tam 1(0; 1; 2) va ban kinh R = nen (S) c6 phuang trinh x' + ( y + l ) ^ + (z-2)^ = 1 ^rix Vai t - - thi (S) IT'• CO ,^6 2^ tam I - ; - ; — va ban kinh R = — nen (S) c6 phuang trinh V5 5 V ^ 2^^ + + z+ X — 5 Cau 9a Dat z = x + iy, ( x, y e R) 25 Ta c6: z z + 3(z - z) = x^ + y^ + 3.2iy = x^ + y^ + 6yi Do do: z z + 3(z - z) = - 3i 338 + y" = x=± Vl5 6y = -3 Vl5 i hoac z = Vay z = , 2 Vl5 i ' 2 Cau 7b Nhan xet A ^ d Goi I(x, y) la tarn hinh vuong A B C D V T C P cua d la: u=(-l;2) Ta CO I d va A I ± u nen t i m dugfc n> c Goi B(a; b) V i I B = l A va B e d nen tim dugc B 1 va D 5'5 B 19^ 5' 19 va ngugc lai "5'5 ;D I 5'5 Do phuomg trinh cac canh hinh vuong A B C D la: ( A B ) : 3x - y = 0; ( A D ) : x + 3y - 10 = 0; (BC): X + 3y - = 0; (CD): 3x - y + = hoac ( A B ) : X + 3y - 10 = 0; ( A D ) : 3x - y = 0; (BC): 3x - y + = 0; (DC): x + 3y - = Cau 8b d d i q u a A ( l ; ; - ) v a c VTCP u = ( - ; ; - l ) A d i q u a B ( - l ; 0; l ) v a c V T C P n = ( ; l ; - ) Ta CO [ u V ] A B ^ nen d va A cheo Duong vuong goc chung IJ c6 VTCP: a = [ u , v ] = (-5; -8; -14) Mat p h i n g (P) chua d va IJ c6 V T P T np = [ u , a ] = (-50; -23; 31) va di qua A ( l ; 0; -2) nen c6 phuomg trinh: -50(x- l ) - ( y - ) + 31(z + 2) = hay 50x + 23y - 31z - 112 = Mat phang (Q) chua A nen IJ c6 V T P T n^ = [ V , a ] = (-30; 66; -27) va di qua B ( - l ; 0; 1) nen c6 phuomg trinh: -10(x + 1) + 22(y - 0) - 9(z - 1) = hay lOx - 22y + 9z + = Cau 9b Dat: z = x + y i , x, y e R va M ( x ; y ) la d i l m biku dien s6 phuc z tren mat phang phiic 339 Taco « z-i =|(l+i)z| t = V3 71 Ket hgrp nghiem, nghiem phuong trinh la: x = y + kTi ( k e Z ) C a u D K : x > - - , BPT da cho tro thanh: (X +1)- - ( x + \yj2x + 5+ 2(V2x + ) ' > (1) f Y Khix = - - t h i ( l ) « > - - + >0:dung \ J Do X = - ^ la nghiem bat phuong trinh Khix ^ - - t h i ( l ) « ( , )'-3( , V ^ x +1 Dodo: < X+ K ) + 2>0 \ f ^ yj2x +5 « V2x + - - < x < - l « - U x < - t = e;x = e=>t = e'^ Suyra:I= dx = Jx(e'+lnx) Vay: I = ln(e*+ ) - X X J dt , —= = lnt lnt — J t ' = InCe*" + ) - Cau Ap dung dinh ly duong trung tuyen tarn giac, ta c6: _ AB- + AD- BD- _ a- " =^ AC = a => BC^ = AB^ + AC^ ^ AB AC Suy SA»CD = 2S \ABC = /,':' -X Ve GH // OA (H e AB), GK SH Ta CO AB ± GH, AB SG => AB GK a GK ( S A B ) ^ G K = , GH = B " " = ^ Vay VS.ABCD = y Cau Dat a = X + 4, b = y + 5, c = z + (x, y, z > 0) Do do: (x + 4)^ + (y + 5)^ + (z + 6)^ = 90 x^ + y^ + + 12(x + y + z) - 4x - 2y = 13 Gia su X + y + z < x, y, z e [0; 1) Suy x^ < x; y^ < y; z^ < z nen: x^ + y^ + z^ + 12(x + y + z) - 4x - 2y < 13(x + y + z) < 13 (v6 ly) Do do: X + y + z > Vay: a + b + c = x + y + z + > Cau 7,a Ha A H vuong goc vai d' la durong phan giac goc B, ta c6 H(h; 2-h) Suy ra: A H - (h + 4; -h) ^ = (-1; 1) la VTCP cua d', nen h = -2 Vay H(-2; 4) Goi A' doi xumg A qua H, nen A' e BC va H la trung dilm AA' Do A'(0; 6) Ta CO B e d' B(b; 2-b), C G d o C(10-2c; c) Ma BC = 2BA nen A' trung didm BC b + 10-2c = b - c = -10 - b + c = 12 { - b + c = 10 { b = -10 c=0 Vay B(-10; 12) C(10; 0) 343 Cau 8.a Goi I la tarn mat cau, ta c6 I e A « Do do: O I = d ( I , (P)) — 2 Cau 9.a Ta c6: |z + i | = - ^ x ' + ( y + 1)' (2) Val d(0, (A'AB)) = OH va OA'H = 30° Dat OH = X OA' = 2x va O'A' = OA = Tarn giac OO'A' vuong tai O' nen a" + Vay d(00' A'B) = f 2xV , , =4x" —j= S ) aV3 X = — T = 2V2 aV3 2^2 Cau Ta c6 - + - + - > + + (a + b)^ + (b + c)^ + (a + b)(b + e) b c a b+c a+b a(a + b)(b + c) b(a + b)(b + c) e(a + b)(b + c) ^ 1 b e a « (a + b)^ + (b + c)^ + (a + b)(b + c) ^ a'e , , b"(a+b) ,2 , cb(b + c) < — + a" + ab + ac + — -+h + ab + e^ + be—^ b e a V i : (a + b)^ + (b + e)^ + (a + b)(b + c) - a^ + ac + c^ + 3b^ + 3ab + 3be T•^ ^ i , , a'e b"(a+b) cb(b + e) ' , Do ta can chung minh: — + — +— ' ^ 2b + 2bc + ab b e a a-c b-(a+b) cb(b + e) Ihatvav: — + + 348 a c b A + uA a c c2 b b + + - — + 2 c b + c a^ —+ " a ab + ( V a ^ +^ j ^ ) + 2b^ ^ ab + 2bc Dau + b' Cy + 2b^ (dpcm) xay a = b = c Cau 7a (C) c6 lam 1(1; -1) va ban kinh R = Goi M la trung diem H K , tam giac IH IHK vuong can a I nen I M ± H K va d(I, A) = GoiC (A): ax5) + by +^ c =2a (a^ + b^ Ta O P(2; e A + 5b + c^=0) Va d(I; A) = ~ « 2(a - b + c ) ' = 25(a^ + b^) (2) T u (1) va (2): 2(a + 6b)^ = 25(a^ + b^) (3) Cho b = 1, va giai (3) ta dugc a = -1 hoac a = 47 23' V a i a = -1 ^ c = -3 =^ (A): x - y + - Vai a 209 47 c = - 23 (A):47x + y - = 23 Cau 8a Goi M(a; b; c) V i M each deu ba diem A B C nen MA = MB { M A = MC (a - ) - + b" +c- = a" +(b - ) " + c" [(a - ) - + b ' + c' = a' + (b - 3)' +(c 'a-b =0 if fb = a { a - 3b - 2c = - [c = - a Do M(a; a; - a) V i M each deu diem A va mp(a) nen M A = d ( M , ( a ) ) a= ^ ( a - l ) ^ + a^ + ( - a ) - = (^ + ^^ + ^ ) ' o 6a^ - 52a + 40 = o 23 a= Chpn diem M toa nguyen la M ( l ; 1; 2) Cau 9a P(x) = (x" + x - ) ' = ^ C ^ ( x ' ) ' ' ( x - ) ' ' k =0 = 'fzc:x'^"'(-i)'"] = xzc^c:(-i)'"x"'-'' k-0 Vm = 349 Chon k, m cho 10-k-m = < m < k < ,la dugrc cac cap (k, m) la (4; 3) va (5; 2) k,m e N Vay he s6 cua la: - C^C^ + C^C; = - Cau 7b Tarn giac ABC deu c6 dinh C thupc Ox nen A, B doi xung qua Ox Goi A(xo; yo) va B(xo; -yo) thuoc (E): — + y^ = thi AC = AB = yo I nen ta c6: ^0 =2;.yo = 4yf3 Voi Xo = 2; yo = C = A hoac C = B (loai) Vay hai diem A, B la: 4V3 ;B 4V3 7' hoac A (2 W37' O 4V3' Cau 8b Vi M nam tren duong thang d nen M(2 - m; + m; + m) Duong thang A di qua M va song song voi duong thang d c6 phuong trinh ^ x-2 +m _ y - - m _ z-4-m • ~ i ~ ^2 Vi N nam tren duong thang A nen: N(2 - m + 2n; + m+ n; + m - 2n) VaN nam tren (a) nen: (2 - m + 2n) - (3 + m + n) + (4 + m - 2n) - = n = -3 - m Suy N(-3m - 4; 0; 3m + 10) Ta CO M N = « (6 + 2m)^ + (3 + m)^ + (2m + 6)^ = "m = -2 m^ + 6m +8 = m = -4 SuyraM(4; 1;2), M ( ; - l ; ) Cau 9b DK: x > log ^4" Chia hai ve cua phuong trinh cho 3" va dat t = Suy raO < t < - I Phuongtrinh: V t - t ' + V t ' - t ' = Vi < t < nen: - (2t -1^) = (t - 1)^ ^ => 2t -1^ < 1; - (4t^ - 3t^) = (t - l)^(3t^ + 2t + 1) ^ ^ 4t^ - 3t^ < Do ^2x-i^ + V4t^ -3t'' < Nen diubingxay Vay phuong trinh c6 nghiem nhat x = ra: t = x = 350 * * Chii de 1: K H O I D A D I E N V A P H E P DC>I H I N H Chu d^ 2: K H O I D A D I E N D E U V A P H E P V I T U ' 27 Chu dg 3: T H E T I C H K H O I L A N G T R U V A K H O I H O P 46 Chu dg 4: T H E T I C H K H O I D A D I E N vA 58 Chu da 5: M A T cAu, Chu da 6: M A T T R U - H I N H T R U - K H O I T R U KHOI CHOP K H O I CAu 94 115 Chii de : M A T N O N - H I N H N O N - K H O I N O N 124 Chii da 8: T O A N C U C T R I K H O N G G I A N 135 Chii de 9: T O A D O K H O N G G I A N 159 Chu da 10: P H U O N G T R I N H M A T cAU 181 Chii de 11: P H U O N G T R I N H M A T P H A N G 192 Chii de 12: P H U O N G T R I N H D U C N G T H A N G 204 Chii de 13: T U ' O N G G I A O C U A D U C J N G T H A N G , MAT PHANG VA MAT CAU Chu de 14: G O C vA K H O A N G C A C H T O A DO Chii da 15: C A C H I N H K H O I V A L T N G D U N G T O A D O PHU L U C : CAC DE O N T H I T O N G HOP 224 266 308 335 351 nhiasachhongan corn v n Email: n h a s a c h h o n g a n GUI ' TRAC NGHI$M \ ^ 1^ \ / ^ / X- \ - L e T h a i T o - VTnh L o n g - D T : 9 - Duang / - T a y N i n h * D T : - 19Truang Chinh - Budn - 129 Phan \ \ y y C h u Trinh - Oa Nang 3852971 - 518 Cach ^ y- y chlm bia va c h i l : , NS HONG AN" - NS N h a T r a n g - C a m R a n h * D T : 3854496 - NS D o n g Ho - Rach Gia - K i e n G i a n g Phong '• djnh sach chinh pham, • DT: 3821317 - 25 LS L9i - Tp Thanh H a * DT: 3857099 - N S Lien SUPng, 127 Tran Qudc Tuan - T r a Vinh - 278 LeHong y M aThupt *DT: 3953408 - 01 Hai B aTrtfng - B u d n MS T h u $ t *D T : (0500) \ ^ ISBN: 978-604-939-542-0 - Q u y NhOn * DT: 3823453 M a n g T h a n g Tann - T h u D a u M o t - B D - V T h j S a u , / T o n DuTc T h a n g - L o n g X u y e n - 6H a n T h u y e n - TP H u e - 78 B a c h D i n g - D a N i n g - D T : ( 1 ) 8 75bnb 935092 / / Gia: 72.000d / / ^ /