PHAN DOAN THO~ (Chu bien) TR!NH THUY HANG - LAI THANH HLJ'ONG Biil DllfrNG HOC SINH V AD LirP 1 0 MONTOAN (Tai bdn ldn thu hai, c6 sua chua) NHA XUAT

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1 PHAN DOAN THO~ (Chu bien) TR!NH THUY HANG - LAI THANH HLJ'ONG Biil DllfrNG HOC SINH V AD LirP 1 0 MONTOAN (Tai bdn ldn thu hai, c6 sua chua) NHA XUAT BAN GIAO Dl)C VI~T NAM

2 {[[W+bz0FkV43GmRt7u4DpvuYxd]]} MQT s6 Kf Hl~u va v1 T TAT sd: s6 do THCS : Trung hqc CC1 sa dpcm : di~u phili cht!ng minh tmdk : thoa man di~u ki~n 2

3 "Bai dttong h9c sinh vao lc':lp 10 mon T oan" nam trong b9 sach ct1ng ten gom 3 mon T oan, Ngu van va Tie'ng Anh. Cu6n sach OliQ'C bien soc;m cong phu nham giup cac giao vien va cac em h<;>c sinh 16'p 9 c6 m9t tai li u huu fch a~ on t~p, dt! thi vao 16'p 10 trong ca ki thi 09-i tra va thi vao cac 16'p chuyen, 16'p ch<;>n. N >i dung cu6n sach gom 3 phim: Ph&n 1: Luy~n t~p theo chu d~ Phim nay gom 9 chu d~ tr<;>ng tam trong chtio'ng trinh T oan THCS, moi chu de giup h<;>c sinh on t~p!9-i nhung kie'n thlic va nhung ki nang co ban c1a OliQ'C h<;>c trong chtio'ng trinh va c6 m9t s6 phan nang cao giup cac em c6 them tti li u luy n thi vao cac 16'p chuyen, 16'p ch<;>n. Ph&n 2: Luy~n t~p theo d~ Phan nay bao gom 25 de luy n t~p t6ng hq'p v6'i cau true dt!a tren de thi cua cac dia phtto'ng. Trong m >t s6 de c6 nhlihg bai (J mlic a > nang cao danh cho h<;>c sinh on t~p thi vao cac 16'p chuyen, 16'p ch<;>n. Khi lam xong nhlihg de nay, h<;>c sinh c1a atiq'c ' trang bi t6t cac ki nang giai toan a~ tlj tin ho'n bti6'c vao ki thi. 3

4 Phcin 3: Hudng d~n va dap s6 Ph&n nay au'a ra ht16'ng dan, lm giai, aap so cho cac bai t~p cua ph&n 1 va aap an cho cac de tl,l' luy<?n & ph&n 2. B9 Sckh OU'Q'C bien soc,m tu'ang aoi hoan chlnh, tfch hq'p nhieu phll'ang phap va ki nang SU' ph9m. Tuy v~y, tranh khoi nhcmg sai s6t, mong quy b9n a9c g6p y. kh6ng thj Nh6m tac gia xin chan thanh cam o'n Phong Giao di,ic Trung h9c, Phong Khao thf va KiJm ajnh chat lugng cua S& Giao d1,1c va Dao t9o Bac Ninh aa a6ng g6p nhieu y kien dj nang cao chat IU'Q'ng cuon sach. cac rae GIA 4

5 {[[W+bz0FkV43GmRt7u4DpvuYxd]]} Pha'n mot ~...? 1ft.... LUYEN TAP THEO CHU DE I. KIEN THUC Cd BAN CHU DE 1. BIEN DOI cac BIEU THUC CHUA can 1. Can b~c hai cua s6 ala s6 X rna x 2 = a. Can b~c hai s6 h9c cua s6 a~o la s6 x ~ 0 sao cho x 2 =a. 2. Dieu ki n ton t<:ti JA la A ~ o ; Htmg dfmg thuc can: JA2 =!AI. 3. Oic quy de phep tmin: - Quy tac khai phuong m t tich (hay quy tac nhan cac can b~c hai) : b =~.Jb \fa, b~o; - Quy tac khai phuong m t thuong (hay quy tac chia cac can b~ c hai) ~~ ~ \la;,o;b>o; - Dua thua s6 ra ngoai dau can: -fa2h = Ia! Jb ;,, {a>o: ajb=-fa2h - Dua thua so vao trong dau can: r-:;-:-; a<o : ajb=-'l./a 2 b - Q uy tac - kh' u mau - cua b. 11~, u t h- uc d um - dau - can: ~b ra = ~ll f;j; = lbl; Jab - Quy tac tn.1c can thuc a mau s6: ho~c rut gqn phan thuc (neu c6 the). a ajb Jb - - b- a _ a(jb +*-) Jb±*- - b-e 5

6 Oic d~ng bai t~p co ban: a) Tinh gia trt m t bieu thuc so (rut g<;m bi~u thuc so). b) Rut g<;m m<?t bieu thuc chua bien. Su d1,mg ket qua rut g<;>n de: - Tin~ gia tri ciia bieu thuc khi biet gia tri ciia bien; - Giai phuong trlnh, bat phuong trlnh (so sarth bieu thuc vai m(>t so); - Tim gia tri nh6 nhat, lan nhat ciia m t bitu thuc; - Tim gia tri nguyen ciia bitiu thuc Ung vai cac gia tr~ nguyen ciia bien. II. VI Dl) VI dl_l 1. Tinh gia tri cac bitu thuc: a) ' : a) 4 4 _ 4( ) 4( 15-3) = (15-3)(J5 +3) (J5 +3)(J5-3) = 4(15 +3)- 4(15-3) = = b) ~2-.JJ +~2+.J3 2+J3 2-J3 = ~ (2- J3 )2 + ~ (2 + J3 )2 = 12 - JJI JJI = 2 -- J J3 = 4 0 c) 2(J2 +J6) 3~2+J3 2J2(J2 +J6) 2(J2) 2 (l+j3) 4(1+../3) 4 3~4+2J3 = 3~(l+J3)2 = 3(1+J3) =3. ' d 2 Ch b ~ h' A - ( 2fx..[;... 3x+3) (2fx - 2_1) V I l_j 0 leu t UC - I + I +. I. "X +3 "X X "X -3 a) Rut g<;>n A. b) Tim cac gia tri ciia X de A < -.!. 0 6

7 a) DK: X ~ 0 va X :f:. 9; A = 2FxcFx -3)+Fx<Fx +3)-3x-3. 2Fx -2-Fx +3 <Fx - 3)(J;. +3). Fx -3 = 2x-6Fx +x+3fx - 3x - 3 Fx -3 cfx-3)cfx+3) Fx+l -3(Fx + 1) 1-3 = =--=-- Fx+3 ' Fx+1 Fx b) Theoa)tac6A= I vaix ~ Ovax:f: IX Fx >0, 1 3 Fx Fx +3 3(-vx +3) A<--~ <- - ~ - -+ >0~ rna 3(J;. +3) > 0 Vx ~ O;x :;t 9, suy ra 6 - J;. > 0 ~ J;. < 6 ~ 0 ~ x < 36. Ket hqp vai DK, v~y vai cac gia tr! cua X rna 0 ~X< 36; X :f:. 9' ta c6 1 A<--. 3 V, d 3 Ch b' ' h, B X+ J;. 1 J;. - 1 I l,l. 0 H~U t UC = ( I I + --) : x-vx+x+-vx+l x+1 x+l a) Rut gqn B. b) T, h.,. ' B '. 2 m gta tn cua vat x = + J3. 2. c) Vai X> 1, tlrn gia tr! nho nhat cua sj;.. a) DK: x ~ 0 ; x :;t 1. B = ( j;. ( j;. + 1) + _1_). X+ 1 = j;. + 1 X+ 1 = j;. + 1 (X+ 1)( j;. + 1) X+ 1 Fx -1 X+ 1. Fx -1 j;. -1. b) Vmx=2 + J3 =4+2J3 = cj3+1) 2 ~J;.=J Khid6:B= J3+ 3 = J3cJ3+1) 2 =J3(2+J3)=2J3+3. J

8 C) B 1 _ X +,./x _ X - 1 +,./x _ I 2 _ fv 2 -vx vx vx Fx-1 Fx-1 Fx-1 Fx-1 Vi x > 1 => Fx -1 > 0. Ap dt;~ng ba't dfing thuc Co-si cho hai s6 duong la Fx-1; /'-,taco: BFxz.2J2+3. -vx -1 D, I 2 I 2 I ' au bang xay ra <=::> -vx -1 = 1 <=::> (-vx - 1) = 2 <=::> -vx - 1 = v'2 -vx -1 <=::>,Jx = 1 + J2 <=::> x = 3 + 2J2 > 1 (thoa man). V~y minbfx = 2J2 + 3 <=::> x = 2J Chu j: Khi chl ra gia trj nhd nhdt (16'n nhdt) ciia P, ca'n phdi chi ra t6n tc;li x rna f(li d6 p d(ll gia trf nhd nhcit (16'n nhdt) d6. Vi du 4. Ch o b teu. "' t h, uc C = ( 3 - Fx- 3) : ( --=- X+ 2,./x ) Fx-1 x-1 x+fx-2,./x+2 a) Rut gqn C. b) Tim x d~ C = Fx. c) Timxd~(Fx+1).C=x-2.f~+10+.Jx-4. (*) a) DK: X 2. 0 va X * 1. C = 3( Fx + 1)-Fx + 3. X+ 2-,Jx (,Jx - 1) ( Fx -1)( Fx + 1) 0 ( Fx + 2)( Fx -1) = 3,./x +3 - Fx +3 cfx +2)(,./x -1) _ 2Fx +6 (,.)x - 1)(,.)x + 1).,./x + 2 -,./x + 1. b) C = -vx lx...- _,. 2Fx +6 = -vx lx (1) 1 -vx + 1 Ta c6: (1) <=::> x + Fx = 2,./x +6 rna Fx + 2 > 0 (x'?.o;x-::f:-1). <=::> x-fx - 6=0 <=::>(,./x-3)(,./x+2)=0, V~yC=,./x <=::> x=9. => Fx- 3 = 0 <=::>X= 9 (thoa man dieu ki~n). c) (,./x+1).c=x-2fx+lo+.jx-4 <=::> 2Fx+6=x-2Fx+IO+.Jx-4 <=::> x - 4,./x Jx- 4 = 0 <=::> ( Fx - 2) 2 +.Jx- 4 = 0 8

9 1-ix-2=0 ' <=> i <=>X= 4 (thoa man). L.Jx - 4=0 V~y (*) xay ra khi va chi khi X= 4. Vi d1,1 5. Cho bi~u thuc D= (Fx+l+FY+Fx 1):(-ix+l FY+1.JxY-1 FY+1 a) Rut gc,m D. b) T, h. '. ' D '. 2 r;; m g~a tq cua v01 x = -..,; J ; y = r;;...,;3 + 1 c) Tim gia tri nh6 nhat cua D neu -fx +.JY = 4. a) DK: x 2 0; y 2 0; xy :t. 1. Ta c6: D = ( -fx + 1)( jry - 1) + ( jry + Fx )( Jry + 1)- xy + 1. <FY+1)(jry-l) : cfx +l)(jry -1)-(jry +-/x)(jry +l)+xy-1 = --FY. (jxy + l)(jry -1) b) y= 13-1 (13-1)2 =2-13=x ~ D = -FY = -~r--( =3-) 2 = = c) Ta c6: x 2 0; y 2 0. D = -Jry = -Fx.JY = -J;..(4-f;.) = x - 4Fx +4-4 = ( Fx -2) V~yminD=-4 <=> J;. = 2 {.JY= <=>x=y=4. 2 Ill. BAI TAP 1. Rut gqn cac bi~u thuc sau: 1-13 a) +--; c) \j. J6-2fi' 9

10 2. R ut ' g9n b' 1eu :! t h' uc. ' p- - X+ y-2 FY' l,. 0 ' 0 1 C. 1 C, vat x > va y >. -vx - -yy -vx+-yy Tfnh gia trt cua bieu thuc t<;ti X= 1000 va y = Cho bieu thuc : P= [Fx - x+2 J [ J;. J;. -4] a) Rut g9n P..Jx+l..Jx+l b) Tim cac gia trj CUa X thoa man p < 0. c) Tim gia trt nho nhat cua P. 1-x 4. Cho bieu thuc : -( 1 1 ) [.Jx JxJ A- -vx I. I + I. -vx -vx x+-vx a) Rut g9n A. b) Tfnh gia trt cua A t<;ti X = 9. c) Tim cac gia trt cua X thoa man: J;. A= 6f; s. Cho bieu thuc: Q - [ a) Rut g9n Q., h.,.., Q. 3- J5 b) T m g1a tn. cua ta1. x = sf;. - 4 ] [ 2 +.Jx - Fx-2 2Fx-x. Fx Fx-2 FxJ. c) Tim m de c6 x thoa man: Q = mxf;.- 2mx Xet bieu thuc: P-[ x+ 3 Fx x+fxj [ J - cfx+2)(fx- 1) x - 1. Fx+1 Fx - 1. a) Rut g9n P. ' 1 Fx+1 b) Tim x de -- ~ 1. p 8 7. Cho bieu thuc : 10 a) Rut g9n P. ' 1 b) Tim x de P < -. 2 P=.Jx Jx-4 Fx- 1 Fx+1 x-1

11 Cho bieu thuc : a) Rut g9n P. P - Fx+l x+2 + Fx+l - x xfx -1 x+fx +I b) Tim gi<i tri nho nhat cua bi~u thuc rx.. p 9.Chobieuthuc: Q= x 2 -JX a) Rut g9n Q. b) Tim gia tri nho nhat cua Q. 2 x+jx+ 2 (x - l) x+-.fx+l J;. fx -1 c) Tim cac so nguyen x d~ Jx nh~n gia tr! nguyen. I. KIEN TH(/C co BAN 1. D~nh nghia CHU DE 2. Hit PHUONG TRINH B!C NHAT HAl AN H phuang trinh b~c nhat hai ~n la h c6 d;:tng {a~+ bby.= c, (I) ax+ y=c v6i a, b, a', b' la cac so thl!c; a, b kh6ng dong thai b~ng 0; a', b' kh6ng dong thai b~ng o. 2. N g h.,., h' (I)l' -.,. ( ) h, - {axo+byo =c 1~m cua ~ a c;:tp gta tq x 0 ; Yo t oa man,,, a x 0 + b y 0 = c. Nghi m cua h phuang trlnh la t9a d giao di~m cua hai duang th~ng (dl); (d2) c6 phuang trlnh la hai phuang trlnh cua h da cho. 3. Dieu ki~n d~ h~ co nghi~m duy nhat, co vo so nghi~m, vo nghi~m. a) He (I) c6 nghiem duy nhat <=> - "* -- hay ab'- a'b :::1= 0. a' b' abc {ab'-a'b=o b) H (I) v6 nghi m <=> - =-- *- - hay,, a' b' c' ac -a c *- 0. a b 11

12 , a b c {ab'- a'b=o c) He (I) vo so nghiem <=>-=-=- hay ' '. a I b I C I ac I- a I c = 0 4. Cac d:;mg bai thu<mg g~p - Giai h phuang trlnh ; - Tlm gia tr~ cua tham s6 d~ h c6 nghi m cho tfu()'c ; - Tlm gia tr~ cua tham s6 d~ h vo nghi m ; VO s6 nghi m ho~c c6 m t nghi m thmt man dieu ki n cho tru6c ; - Giai va bi n 1u~n h phuang trlnh ; - Tlm h thuc lien h gifra X, y khong ph1:1 thu(>c vao tham SO. II. Vi Dl) Vi d1,1l. Giiti cac h phuang trlnh sau: a) 3x+5y=1 { 2x-3y=-5; xji - yj3 = 1 { X+ yj3 = Ji. a) Ta c6 th~ nit X tlr phuang trlnh thu nhat theo y roi the vao phuang trlnh thu hai. Tuy nhien, each sau day cling thuo'ng duqc ap d~:~ng. Nhan dong thai hai ve cua phuang trlnh thu nhat v6i 2 va cua phuang trinh thu hai v6i 3, ta duqc h tuang duang : 6x+i0y = 2 <=>{3x+5y=.l<=> x = _ 22, 17 { 6x-9y = y=17 19 y=19 D /,' N h'' I ap so. g tern cua h e 1' a (--; ) b) The X = Ji - y J3 vao phuang trinh thu nhat, ta duqc b) 12 :::>x= l. Ddp s,;: Nghi~m ella h? lil (I ; J6:.J3} Vi d1,1 2. Giai cac h phuang trlnh sau bang phuang phap dung ~n ph~:~: a) _8_ +~ = 1 x-1 y =-- x - 1 y+2 12 b) { 3.Jx JY+l = 7 s.j x - 2-3JY+l = - 1.

13 {[[W+bz0FkV43GmRt7u4DpvuYxd]]} a) Di~u ki~n X :f: 1, y :f: - 2. Nh~n xet dng cii hai phuong trinh cua h ~ deu h, h. h h' c ua m p an t uc --;--. V' 1 t h e ' ta d at --= u ; - - = v. x - I y+2 x-1 y+2 Ta duqc h~ phuong trinh b~c nhat v6i hai ~n u; v: { 8u+15v=1 {8u+15v=1 jv= ; 1 1 :::> 2 :::> u+v=- 8u+8v= U=28. I I X ,_ {X -1 = 28..._ {X= 29 Suy ra,. -,...,.- thoa man di~u ki~n. _ 1_=_I y+2 = 21 y=i9 y +2 2I b) DK: x ~2 ; y ~- l. D~t.Jx - 2 =u; Ji+l = v tac6h~: 3u+2v=7 <=> {9u+6v = 21 <=> {l9u=i9 { 5u-3v=-1 10u - 6v=-2 3u+2v=7 u = 1 {~ = 1 {x = 3 { :::> ~ ~ :::> (th6a man). v=2 \IY+l=2 y=3 3x-2y=7 Vi du 3. Cho he phucmg trlnh { J,(5n+1)x - (n - 2)y = n- -4n - 3 Tim n d~ h~ c6 nghi~m (x; y) = ( 1; - 2). Thay (x;y) = (1;-2)vao (I) ta c6: 3-2.( - 2) = 7 <=>3 + 4 = 7. (1) (2) Dodo, (x;y) =(1;-2) langhi~mcua(l). Thay ( x; y) = ( 1; -2) vao (2) ta c6: (Sn + 1) + 2.(n - 2) = n 2-4n - 3 <=> 7n - 3= n 2-4n - 3<=> n(n - 11)=0 <=>[n=o n = 11. V~y v6'i n = 0 ho~c n = 11 thl h~ da cho c6 nghi~m ( x; y) = ( 1; -2). 3x+2y=-8 (1) Vi du 4. Cho he phuong trlnh { 3mx+(m+5)y=(m-l)(m+l) (2) Tim m d~ h~ c6 nghi~m (x; y) thoa man: 4x - 2y = - 6. (3) (I) 13

14 Di~u ki~n d~ h~ c6 nghi~m duy nha1 : 3o(m + 5)- 6m =t 0 :::> m =t 50 Do (x; y) Ut nghi~m cua h~ p1 (I) va 1hoa man (3) ~ (x; y) la nghi~m cua (1), (2), (3) Ke1, hqp (1) va (3) 1a c6: {3x + 2y = -8 {x = - 2 :::> 4 X - 2 y = - 6 y = -1. Thay X = - 2, y = - 1 vao p1 (2) 1a duqc : - 6m - (m + 5) = m 2-1 :::> m 2 m=-7+j33 + 7m + 4 = 0 :::> ( 1/m)o [ m= -7-Jfi V ~ y v6i m = ffi ho~c m = - 7-.ffi 1hl h~ (I) c6 nghi~m thoa man 4x- 2y =- 60 Nhtj.n xet: Khi nghi~m cua h~ thoa man m 1 h~ 1huc, ta coi h~ thuc d6 la m 1 phuong trlnh 1hu bao Luc d6 c6 hai huang d~ giai: Huang m t la giai h~ thay vao h~ thuc; Huang hai la ke1 hqp m t phuong 1rlnh 1rong h~ v6i h~ 1huc de tlm X, y sau d6 thay vao phuong trlnh con l~io 14 Vi dt,j 5. Giai va bi n lu~n h~ phuong 1rlnh sau: x+3y = 1 { 2 (m +1)x+6y=2m The X = 1-3y vao phuc1ng 1rlnh 1hU hai, 1a duqc v6i m la m t so thvco (m 2 + 1)(1-3y) + 6y =2m =>3(m 2-1)y = (m - 1l - Neu m = 1 thl phuong trlnh tren nghi m dung v6i mqi yo Do d6, h~ c6 nghi m Ia (1-3y ; y), y la m 1 so thl!c tuy yo - Neu m = - 1 thl phuang trlnh tren c6 d~ng Oy = 4, vo nghi mo V~y h~ vo nghi~mo 2 - Neu A' m -/.. 1 h' h ' h A, ho A (m -1) m -1 -r ± t 1 p uong tnn tren co ng Iym y = = (m -1) 3(m+1) Vay A he A co, ng ho 1em A l' a (--; 2 m -1 J m+1 3(m+1) Nhtj.n xet: V6i phuong 1rlnh ax+ b = 0 (1) + Neu a = 0 thl phuong trlnh ( 1) c6 d;~mg Ox + b = 00 - Khi b = 0 thl phuong trlnh ( 1) c6 d~ng Ox = 0 => phuong trlnh c6 vo so nghi~m o

15 - Khi b :f. 0 phuong trlnh (1) vo nghi~m. b + Neu a :f. 0 thl phuang trlnh (1) c6 m('>t nghi~m duy nhat - -. a Ill. BAI TAP 1. Giai cac he; phuong trlnh : a) c) 2x - 3y=ll { 5x-4y=3; (J2 -l)x- y = J2 { X+ ( J2 + l)y = 1; 2. Giai cac h~ phuong trlnh: a) ~+_32_ = 3 x+3 y - 2 ~+~=5 x+3 y-2 ' 3. Cho h~ phuong trlnh 2x-ay = b { ax+ by= l. ' {X =J2 a) Tlm a, b de h~ c6 nghi~m ~ y='>/3. b) Tlm a, b M M c6 vo so nghi~m. 4. Cho h~ phuong trlnh: b) {; - ~ = 1 8x-5y=3; 2x+ay= -4 d) (a latham so) { ax- 3y = 5 b) x+ay =a+1 { ax+y = 3a =-1 x+2y x-2y 15 1 = 2. x+2y x-2y (ala tham so). a) Giai va bi~n lu~n h~ phuong trlnh. b) Tlm a d~ h~ c6 nghi~m duy nhat thoa man di~u ki~n xy nho nhat., {(m+1)x-y=3 (1) 5. Tlm m de h~ phuong trlnh: mx+y =m (2) c6 nghi~m (x;y) duy nhat thoa man x + y > Cho h~ phuong trlnh: (m-3)x+y=2 { mx+2y =8 Tlm tat d. cac gia trt nguyen cua m d~.h~ c6 nghi~m nguyen. (1) (2) 15