Ôn luyện theo cấu trúc đề thi môn Toán Phần 2 Ôn thi tốt nghiệp THPT TS.Vũ Thế Hựu, Nguyễn Vĩnh Cận

CHI DAN Xet dau tarn thuTc h(x) = 2x^ - 3x - 5, c6 h(x) = Oc^x = - l , x = - 2 5 h(x) > vdi X < - hoac x > — ; h(x) < vdi - < x < — Xet tLfcmg tir vdi tam thufc g(x) = 4x^ - 19x + 12 ta lap diigfc bang sau : X + ^ _ h(x) g(x) + g(x) = 4x2 - 19x + 12 + h(x) = 2x' - 3x - -1 0 - J r - - + + - 0 ^ + - + - + + TCf bang xet dau ta ket luan : f(x) > vdi X e (-oo; -1) u ' v4' 5^ u (4; 2y +oo) rs ^ 3^ f(x) < vdi X e f(n) = vdi X = - , X = f(n) khong xac dinh vdi n = —, x = 4 5x +phifong > t r i n h va bieu d ib) 4x + < tren true so 18.a) 2x2 Giai- bat l n x^ tap+ nghiem HI DAN a) Xet dau tam thufc ve trai : fix) = 2x2 _ 5x + A = 52 - 16 > 0, fix) = Xj = - = , X2 fix) > (f(x) cung dau vdi 2) vdi x < - hoac x > Vay tap nghiem 1] 2; + c o ) Bieu dien tap cua bat phirong t r i n h la: T = { -00; — I 2_ nghiem tren true so' (phan true so khong bi danh cheo) ^ ] ằ ô f ^ ã b) Lam nhiT cau a) Tap nghiem T = (-3; -1) -3 -1 32 S TS Vu The Hi^u - Nguygn VTnh CSn 19 Giai he bat phifofng trinh va bieu dien tap nghiem tren true so : (I) x' + x - > (1) x'-5x +4 Tarn thufc theo bien m : A(m) = -3m^ - 46m - 15 c6 cac nghiem : m i = -15, A = (2m + - 4(m + 4)(3m + 1) > < m2 = - — CO gia t r i diiong (trai dau vdi - ) vdi -15 < m < - — 3 Ket luan : Phirong t r i n h (1) c6 hai nghiem neu -15 < m < - - b) Neu m = phirong trinh (2) c6 dang : 5x + = c6 mot nghiem fm ^ PhUOng trinh (2) v6 nghiem neu \ ^ ' ^ • [A = (m + l ) ' - ( m - ) ( m - l ) < A = -7m^ + 38m - 15 c6 gia t r i am (cung dau vdi - ) neu m lay gia t r i ngoai khoang hai nghiem la : m.^ ^ — va m2 = 21 DS : m < — hoac m > Tim cac gia t r i cua tham so m de bat phufong t r i n h x^ - (3m - i)x + 3m - > (1) dutJc nghiem dung vdi moi x thoa man I x I > CHI DAN Tarn thufc fix) = x^ - (3m - l)x + 3m - c6 biet thufc A = (3m - 1) - 4(3m - 2) = 9(m - 1)' • Neu m = 1, A = 0=:> f(x) > Vx e R, f(x) = o x = 1, do f(x) > Vx: > Hoc 6n luy?n theo CTDT mfln Todn THPT S 33 22 Neu m vi => f ( x ) = x i = 1, x = m - Vdfi m < t h i m - < tap nghiem cua (1) la: ( - o o ; 3m - 2) u ( ; +oo) De X cho x | > thuoc t a p n g h i e m t h i p h a i c6 - < m - c^O < m < Vdfi m > t h i 3m - > tap nghiem cua (1) la: ( - o o ; 1) u (3m - 2; +oo) De t h o a m a n yeu cau b a i t o a n t h i p h a i c6 3m-2 af (-1) = 4[4 + (3m + 1) - m - 2] > af (2) = 4[16 - 2(3m + 1) - m - 2] > , S 3m + l -1 < —= < 2 -3 < m < - < m + < 16 m > — 12 m < — - m + 12 > m ' + m + 33 > 2m + > 23 12 — < m < — T i m cac gia t r i cua t h a m so m de phirong t r i n h (m - 2)x^ - 2mx + m - = (1) CO h a i n g h i e m thoa m a n dieu k i e n - < X i < < X £ TS Vy Th6' H^fu - NguySn Vinh CJn 34 CHI DAN - < x i < < X2 o af (4) = (m - 2)(10m - 35) < af (-6) = (m - 2)(50m - 75) > c>2 o m < t a xet them biet thufc A = (m + 1)^ - 4m = (m - 1)^ S „ m+1 „ m-3 o\ 2= 2= < (vi m < 2) 2 Vdi m = ta CO nghiem kep x i = X2 = < Vdi m < va m ^ t h i X i < X < 25 Giai, bien luan theo tham so m he bat phiTOng t r i n h (I) x ' - (m + l)x + 2(m - 1) < (1) x' - (m + 2)x + 3(m - 1) > (2) CHI DAN Tam thu-c ve trai (1) la : fix) = x^ - (m + l ) x + 2(m - 1) = o X = m - hoac x = • Neu m - < o m < t h i (1) c6 tap nghiem • Neu m - > Tj = [m - 1; 2] m > t h i nghiem cua (1) la : Ti = [2; m - 1] • Neu m = t h i T i = 121 Xet tam thufc ve trai cua bat phtTcfng trinh (2) : g(x) = x^ - (m + 2)x + 3(m - 1) c6 cac nghiem x = m - l v a x = • Neu m < tap nghiem cua (2) la : T2 = (-00; m - 1] u [3; +co) • Neu m > tap nghiem cua (2) la : T2 = ( - c o ; 3] u [m - 1; +00) • Neu m = : T2 = {3} Tap nghiem T = T i n T2 cua he (I) nhif sau : + Neu m < t h i (I) c6 nghiem chung nhat x = m - + Neu < m < t h i T = [2; m - 1] + Neu m = t h i (I) c6 hai nghiem chung T = {2; 3} + Neu m > t h i T = [2; 3] u !m - 1} HQC va 6n luy?n theo CTDT m6n Toan THPT SS 35 §4 PHlIC(NG T R I N H , B A T P H U C I N G T R I N H CHlfA D A U GIA TRI T U Y E T DOI KIEN THLfC Phx:ifofng t r i n h chtfa d a u g i a t r i t u y ^ t d o i La phucfng t r i n h t r o n g bieu thufc cua no c6 chufa a n n a m t r o n g dau gia t r i tuyet do'i Vt : I 2x - 1 - 3x = Dudng l o i g i a i phirong t r i n h a neu l a diTa t r e n d i n h nghia a > -a neu a < ta phan chia tap xac d i n h t h a n h nhufng tap nho de thay the bieu thufc c6 gia t r i tuyet doi bkng bieu thufc khong c6 gia t r i tuyet doi ti/cfng diTOng Dac b i e t : | A(x) | = | B(x) i o (A(x)f = (B(x)f' B a t phu:cifng t r i n h chtfa d a u g i a t r i t u y $ t d o i Cac phtrong t r i n h don g i a n chufa dau gia t r i tuyet doi : M o t so t i n h chat cija gia t r i tuyet doi can n h d k h i g i a i bat phuong t r i n h chufa dau gia t r i tuyet doi a.b I fix) I < g(x) - g ( x ) < fix) < g(x) c) I f(x) I > g(x) b) I fix) I < I g(x) I « a) o f ' ( x ) < g'(x) fix) < - g ( x ) hoSc fix) > g(x) -a a a + b + BAI TAP 26 G i a i cac phuong t r i n h : - 2x = (1) b) X + a CHI DAN X + a) Theo d i n h nghia I x + (A) (1) « (B) 2x + vdi - ( x + 4) v d i X > X < 3x - = (2) -4 -4 Jlx + ) - x = X > -4 - ( x + 4) - 2x = -4 X < 63 TS Vu Thg Hi^u - Nguyln VFnh CJn 36 -x + = _ (B) o x = - ; r-3x-4 =7 = > v nghiem X < -4 -4 Vay phuong t r i n h (1) c6 nghiem nhat x = - b) De CO bieu thufc khong chufa gia t r i tuyet doi ttrcrng duong ta lap bang sau (A) < X > "2 X 2x + -2x - 2x + 2x + 3x- -3x + Bieu thufc (2) -5x - = - X Nghiem X = — (loai) X = (loai) -3x + 3x - +5=3 5x + = X = — (loai) Vay phiiong t r i n h (2) v6 nghiem (theo bang c6 nghia la vdi x < — thi (2) CO bieu thufc : - x - l = = > x = khong thuoc khoang nen loai) -2 27 Tuong tir nhir vay, ta loai x = va x = — Giaix cac phiTOng t r i n h : ^-x 2x-4 = ( ) b ) x ' + 6x + + x^ - a) =30(2) CHI DAN a) x ^ - x = « x = 0; x = l , x - = o x = x^ - X x^-x -x^ + x vdi X < vdi < 2x - vdi 2x- -2x + vdi X < X > X < hoac x >1 Ta lap bang de d i theo doi X x^-x 2x- -00 x^ - X -2x + -x^ + -2x + Bieu thufc (1) x^ - 3x + = -x^ Nghiem 3±V5 (loai) X X + x^ - x^-x X -2x + +00 2x - 4 = x^ - 3x + = x^ + - +Vs 3±V5 — - — (loai) Hoc X - 4=3 - + V29 6n luyOn theo CTDT mSn Toan THPT S 37 Phifdng t r i n h (1) c6 cac nghiem : X j = - i + Vs _ _ - + V29 X b) Ta lap bang sau : X -00 -4 x'^- Ix^-ll -2{x^ + 6x + 8) 2|x^ + ex + 8l 2(x2 + x + 8) B i e u thiifc (2) 3x^ + 12x + 15 = 30 -2 -1 2(x^ + Gx + x^ - -x2-12x-17=30 2(x^ + 6x + g -x^ + (c) +CO 2(x^ + 6x + 8) x^ - (d) (e) Nghiem (c) o _^ ^2x + 15 = 30 (d) x^ + 12x + 17 = 30 (e) 3x2 _^ -^2x + 15 = 30 TCr cac phirong t r i n h cac khoang ta t i m dtfoc cac nghiem cQa (2) la : X = - , X = Giai cac phirong t r i n h : 2x-l -3x = b) 2x - X + =6 28 a) c) x - x = x - CHI DAN b) X = va X = — c) X = 1, X = ±3 a) X = 11 29 Giai cac bat phuong t r i n h va bieu dien tap nghiem tren true so a) | x - l | < (1) b) i'2x+*l| > ' (2) CHI DAN a) (1) - < 3x - < - < 3x < « — < X < 2x + l > p3 2x + < - b) (2) •/W/MW/////////////, x < -2 -2 x>l Giai bat phuong t r i n h va bieu dien tap nghiem tren true so' _|x2 - x - f < x - (1) 30 CHI DAN (1) o-(3x-3)0, g(x)>0' B a t phu!ofng t r i n h chufa c a n thiJc Dang CO b a n cua b a t phiTOng t r i n h chura can bac h a i f (x) > a) g(x) < 4Ux)>gix) g(x) > • f(x) > [g(x)]^ f (x) > b) V f O O < g ( x ) ojg(x)>0 f (x) < [ g ( x ) ] ' De g i a i cac b a t phiTOng t r i n h chufa cSn thufc, t a dtfa r a cac dieu k i e n xac d i n h r o i luy thCra m o t each t h i c h hop cac ve cua b a t phiTOng t r i n h de g i a m d a n cac dau can thufc, d a n d a n dtTa t d i b a t phifong t r i n h , he bat phufong t r i n h k h o n g chuTa cSn thufc Cung c6 t h e dat cac a n p h u hoSc b i e n l u a n cac ve cua b a t phuong t r i n h de t i m n g h i e m BAITAP 31 a) G i a i cac phiTOng trinh : x - V2x + = (1) b) V3x + - V x - = CHI D A N a) (1) V2x + = X - x-4 >0 2x + - ( x - f Hgc vJ On luy§n theo CTBT m6n To^n THPT S 39 < X > x ' - lOx + = b) ( ) o V3x + = + V x - c ^ o 9(x - 3) = (x - If 32 X > -irx = C:>X = 3Vx-3 = x - 3x + = + x + V x - x>3 x-3>0 o x^ - l l x + 28 = Vx e R Dat t = V e x ' - x + vdi t > 0, ta c6 : x^ - 2x = 7-t' va phirang t r i n h (1) dan den t = -1 +t =0 (loai) t =7 o Vex' - 12x + = t > 6x' - 12x + = t ' ex^ - 12x + = 49 c:> xi,2 = ± V s b) Nhan xet : x^ - 3x + = Dat t = x^ - (2) (2)o 33 3x + 3, ta \ + - > Vx X CO : G R 00 Do : (1) 4-x> N/X-2 + V4-X 34 Giai cac phuong t r i n h : a) ^ / ^ - ^ / I ^ = l (1) CHI = ox x' - x + l l = =2 b) ^/^r75+^x + 6- = fe + l l (2) DAN a) Cdc/i i : Lap phUdng hai ve (1) ta difoc : ( D o (x + 34) - (x - 3) - 3^x + 34.^/^r^[^x + 34 - ^ x - ] = (Ap dung hang dang thuTc : (a - b)^ = a^ - b^ - 3ab(a - b)) ( D o ^(x + 34)(x - 3) = 12 o (x + 33)(x - 3) = 1728 "x = 30 o x ^ + 31x - 1830 = o x = -6l' Cdch : DSt an so phu u = ^x + 34, v = ^ x - ta c6 : (1) o u - V = 1, u^ = 37 = (u - v)^ = u^ - 3uv(u - v) o = 37 - 3uv => uv = 12 u + (-v) = Ta dilgfc he phirong t r i n h u ( - v ) = -12 u va - V la nghiem cua phifcfng t r i n h - X - 12 = o Xi = - , X2 = ^x + 34 = '^x + 34 = -3 x = 30 hoac o X 61' ^/^r^ = ^/^r^ = -4 b) Dat u = ^x + 5, V = ^x + ta thay u^ + v^ = 2x + 11 Suy (u + v f = + v^ + 3uv(u + v) => 2x + 11 = 2x + 11 + 3uv(u + v) o uv(u + v) = o ^x + 5.^x + 6.^2x + l l = =0 o ^x + = ^2x + l l = 35 x = -5 o x = -6 X 11 = Giai, bien luan theo tham so m cac phifOng t r i n h : a) Vx + m + V x - m = V2m (1) CHI b) Vx^ - 2mx + + = m (2) DAN a) DK : m > • Ne'u m = t h i (1) c6 nghiem x = X > m >0 • Neu m > 0(1) o X 2x + 2Vx^ - m ^ = 2m o HQC > m >0 4^ m =m- X 6n luygn theo C T D T mOn Toan T H P T 41 45 G i a i bat phuong t r i n h : ^^^^^^^ + V^T^ > 4=^ (D Vx - Vx ^ (Trich de thi tuyen sinh DH khoi A - 2004) CHI DAN D K X D : x ^ - 16 > 0, X - > 0x > (1) » V2(x' - ) + x - > - x X >4 (A) fV2(x' - ) > 10 - 2x < X > 10-2x4 10-2x>0 (B) 46 X >4 2(x'-16) >(10-2x)' He (A) CO n g h i e m : x < 5, he (B) c6 n g h i e m 10 - ^/34 < x < Tap n g h i e m cua (1) l a : x > 10 - V34 G i a i bat phifofng t r i n h : a) V5x - - Vx - > V2x - b) (1) (Trich de thi tuyen sinh DH khoi A - x + l + Vx' - x + l >3V^ 2005) (2) (Trich di thi tuyen sinh DH khoi B - 2012) CHI DAN a) D K X D : 5x - > 0, X - > 0, 2x - > c:> X > B i n h phiidng h a i ve (1), chuyen ve t h i c6 : x> X > (1) « X + >V2(x-l)(x-2) [(x + 2f > ( x ' - 3x + 2) X > x ' - lOx < o < X < 10 b) D K X D : x > , x ^ - x + l > c : > < x < 2-S Ta CO X = l a m o t n g h i e m cua (2) X e t X > 0, chia h a i ve cho \fx t h i diroc : V x + hoSc x > + Vs Vx +Jx+—- >3 V X Dat t = Vx + 4= t h i X + i - t^ - t h i (2) c6 dang : Vx X t +Vt^-6>3 Vt'-6>3-t t > G i a i bat phtfOng t r i n h : V x + > — t a duoc < V x < —, hoSc Vx > Vx 2 Suy r a t a p n g h i e m cua (2) l a : < x < — hoSc x > 4 HQC V§ 6n luy§n theo CTDT mfln Jo&n THPT El 47 §6 HE PHUCfNG TRINH NHIEU AN KIEN THLfC H $ phi^ofng t r i n h b a c n h a t h a i a n , b a a n a) H e phuong t r i n h bac n h a t h a i a n (x va y) c6 dang: (I) ajX + b i y ^ C j a X + b y = C2 K i hieu D = (1) (2) = aib2 - a2bi goi la d i n h thufc cila he ( I ) b2 - C2 a2 C: b2 C2 b, Ci = Cib2 - - aiC2 - C2bi a2Ci Quy tdc Crame G i a i he phifcfng t r i n h bac n h a t Neu D he (I) CO nghiem nhat (xo; yo) xac dinh bdi cong thijfc D.,y Xo = D ' _ yo = D - N e u D = 0, ^ hoac Dy ;^ h e ( I ) v6 n g h i e m - N e u D = Dx = Dy = he ( I ) c6 v6 so n g h i e m l a t a p n g h i e m cua phiiong t r i n h : aix + b i y = C i hoac ciia a X + b y = C2 b) Gidi he phiiang trinh bac nhat hai an bdng phuang phdp thi + b j y = c, (1) Cho he phiiong t r i n h : ( I ) - aiX [aaX + \y = C2 (2) T r e n cung m o t m a t phSng t o a Oxy, ve cac dirorng thSng (di) c6 phiiOng t r i n h (1) v a duTcfng t h a n g ( d ) c6 phtfcfng t r i n h (2) K h i t o a (xo; yo) ciaa giao d i e m (di) v a ( d ) l a n g h i e m cua he ( I ) N e u (di) v a ( d ) giao he (I) c6 n g h i e m n h a t N e u d i // d2 he (I) v6 n g h i e m Neu d i v a d trCing nhau, he (I) c6 v6 so n g h i e m Toa m o i d i e m cua (di) (hay ( d ) ) l a m o t n g h i e m BAI TAPi 47 G i a i cac he phtfofng t r i n h sau: a) ( I ) 2x - 3y = - x-2 b) ( I I ) 3x + y = I2-X + y=7 + 5y - 48 S TS Vu Thg' Hi;u - Nguyen Vinh CSn CHI D A N a) A p dung quy tac Crame cho he ( I ) -3 -4 D = = 11 ^ 0, = -4 Dx -3 = 11, = 22 = ^ 2 ^ —i D 11 11 N g h i e m n h a t cua he ( I ) l a (1; 2) Xo = D — b) Dieu k i e n xac d i n h ox ^ DSt X = t h a y vao ( I I ) t a X dxsgc he phuong t r i n h bac n h a t v d i X va y [ (ir) 3X + y - - X + 5y = X - A p dung quy tSc Crame cho he ( I D t a difcJc y = 48 G i a i , b i e n l u a n theo t h a m so m he phiTcfng t r i n h (I) 32 17 23 17 81 32 23' y = 17 X = 6mx + (2 - m)y = (m - l ) x - m y = CHIDAN Ta t i n h d i n h thufc cua he ( I ) D = 6m 2-m m -1 - m = -Gm^ - (m - 1X2 - m) = -5m^ - 3m + D =0» - m ^ - m + = « m = - l hoSc m = + Neu m = - , D = 0, Dx = - ;t he ( I ) v6 n g h i e m 22 + N e u m = - , D = 0, Dx = ^0 he ( I ) v6 n g h i e m 5 + N e u m ^ -1, m ^ —, D = - m ^ - m + ^ 0, he ( I ) c6 n g h i e m n h a t (xo; yo) v d i , 2-m m +4 D - m Xo = D D 5m^ + 3m - 6m m-1 9m+ D -5m^ - 3m + ã Hoc 6n luyĐn theo CTDT mfln Jo&n THPT 49 49 Giai cac he phiTcrng trinh: X 2y _ 29 x + y + 15 a) (I) 2x y x + y + 15 X + 2y + 3z = 10 b) (II) 2x + y - z = l l 3x - 2y + z = (1) (2) (3) CHI DAN a) DKXD: x ^ - , y ^ - DM X = x + Y = y + thi (I) trd thanh: X + 2Y = 29 15 (D _8_ X - Y = 15 Ap dung quy tSc ^Crame , cho ^he =(I')2 ta^ tinh difdc X = —, Y = — Giai y+1 ^ x +2 Vay (3; 2) la nghiem cua he (I) X + 2y + 3z = 10 (1) b) (II) J2x + y - z = l l (2) 3x - 2y + z = (3) Nhan phiicrng trinh (1) v6i - dem cong vao phiTcfng trinh (2) Lai nhan phifong trinh (1) vdi - cong vao phifOng trinh (3) thi dufdc x + 2y + 3z = 10 (1') (II)i - y - z = -9 (2') - y - z = -24 (3') Lai tiep tuc nhan phiTOng trinh (2') cua (II)i vdfi - , cong vao phifong trinh (3') thi diTcfc x + 2y + 3z = 10 x = (11)2 - y - z = -9 y = 48z - 48 z = l ta c6 the lap rieng bang cac he Ghi chu: De thuc hien phep giai tren so cua he (II) va thiTc hien nhtr sau: 'I 10^ ^1 10^ 10^ -1 11 —> -1 -7 -9 -1 -7 -9 6; 48 48j 10 -8 -8 -24; 13 -2 50 E3 TS Vu The' HiAi - Nguyen Vinh C5n Mpt so phxiofng t r i n h h a i a n d a n g d a c b i $ t a) He gom mot phiiang trinh bdc hai, mot phiiong trinh ax + by + c = (1) (I) [ A x ' + Bxy + C y ' + D x + Ey + F = (2) bdc nhdt G i a i he (I) b k n g phLrong phap the b) He phiiong trinh dot xiing loqi I L a he phufdng t r m h co dang < l g ( x , y ) = (2) Trong f(x, y) va g(x, y) la cac bleu thufc doi xufng doi v i cac bien x, y Cdch giai: D a t a n so phu S = x + y, P = xy Dieu k i e n can va du de he c6 n g h i e m la - P > rf(x,y) = (1) c) He phiiong trinh doi xiing loai H: ( I I ) fly,x) = (2) Cdch giai: Di/a viec g i a i he ( I I ) ve g i a i he: (F) d) Phuang trinh (III) 'f(x,y)-f(y,x) =0 f(y,x) = dang cap fi(x,y) = gi(x,y) (1) f2(x,y) = g , ( x , y ) (2) T r o n g m i phuong t r i n h cua he l a m o t dSng thiJc cua cac da thufc dang cap cung bac Cdch giai: G i a i he ( I I I ) v d i x = hoSc v d i y = V d i x ;t dat y = k x hoSc v6x y dat x = k y r o i khuf a n de doi ve g i a i phirong t r i n h m o t a n BAI TAP 50 G i a i , b i e n l u a n theo t h a m so m he phi/cfng t r i n h 3x + 5y = 13 (1) (I) x ' + y ' = m (2) CHI DAN TCf (1) X = l i z ^ y The vao (2) t h i diroc \ 13-5y + y ' - m = o 52y^ - 130y + 169 - m = (2') B i e t thufc cua (2'): A' = 65^ - 52(169 - 9m) = m - 4563 4563 39 • Neu m < 468 = — , A' < 0, (2') v6 n g h i e m => (I) v6 n g h i e m HQC va 6n luy§n theo CTBT mOn Tcrin THPT S 39 Neu m = — , A' = 0, (2') c6 n g h i e m kep y = — => x = — ' 4 He (I) CO n g h i e m [9 5^ 4' Neu m > — t h i (2') c6 h a i n g h i e m y i = • ' n g h i e m cua he ( I ) 51 G i a i he phiTOng t r i n h xy + X 52 tCr suy r a h a i + y = 11 x V + xy^ = 30 (Trich de thi vdo DHGTVT - 2000) CHI DAN Dat X + y = S, x y = P, t a c6: fS + P = 11 S.P = 30 S v a P l a n g h i e m cua phtrong t r i n h : X =5 X =6 x +y= [xy = hoac X - I I X + 30 = +y= xy = G i a i cac he t r e n t a difoc cac n g h i e m (2; 3), (3; 2), ( ; 5), (5; 1) 3y = 52 G i a i he phtfong t r i n h : (I) 3x = y^+2 x^ + (Trich de thi tuyen sinh DH khoi B - 2003) CHI DAN TCr cac phifdng t r i n h cua (I) suy r a x > 0, y > 3xV = y' + '3xV = y ' + (I)« 3xy(x - y) = y^ - x^ ^ y ' x = x^ + x V = y^ + x-y = 3xV = y ' + C5> (x - y)(3xy + x + y) = 3x'-x'-2 = (A) 3x'y = y ' + 3xy (A)c^ + X (B) +y=0 x = y = y =X TCr dieu k i e n x > 0, y > Suy r a : 3xy + x + y > 0, do he (B) v6 n g h i e m V a y he (I) c6 n g h i e m n h a t x = y = 52 a TS Vij Thg' HUu - Nguyen Vinh CSn X y - +- = a y X 53 G i a i v a b i e n l u a n theo t h a m so a h e phtfcfng t r i n h (I) x +y = (Trich CHI de thi tuyen DHQG Hd Nqi khoi B - 1997) DAN D K : x y ^ 0, t a c6: < +y X < = axy x +y =8 (x + y ) ' = (a + 2)xy x +y = fS = - P > t a c6: (I) c^l (a + ) P = 64 Dat S = X + y , P = x y , d i e u k i e n • sinh N e u a = - he v6 n g h i e m = • N e u a ^ - , t a c6: « a+2 ^—^ > a+2 a < - hoac a > K h i X , y l a h a i n g h i e m c u a p h i i O n g t r i n h : z 64 - 8z + X = a +2 4+4 tufc l a y =4 - = zi,2 = ± a-2 x =4 - a +2 hoSc a-2 y =4+4 'a + a-2 a +2 a-2 a +2 a-2 a +2 + Vdi - < a < he v6 nghiem 2x 54 G i a i h e p h i / o n g t r i n h : + l y (I) (1) ^ X 2y + - = X y (Trich CHI de thi tuyen sinh (2) vdo DHQG Hd Noi khoi B - 1999) DAN C o n g v e vdfi v e , r o i trCr v e v d i v e t a dUOc: ^ x +y 3(x + y ) 2(x + y ) + = — xy xy o/ Y 2(x - y ) + _ xy II- ^(V - xy f (x + y ) V J < Y^ (X + -y) V ^1 = J xy ^ = xyj T i f r u t r a diigc h e p h i i o n g t r i n h sau: HQC va 6n luyen theo CTBT m6n Toan THPT 53 x +y =0 x - y =0 X +y =0 (loai) xy 1- — = xy v6 n g h i e m 1- xy = x - y =0 xy x = ^/2; y = - V X = - V ; y = V2 x =l , y - x =-l, y = - l ' 55 G i a i he phi/dng t r i n h : a) ( I ) (2) x ' + x y + y ' = 17 (1) x ' + x y + y^ = 1 b) ( I I ) x^-y^=7 (1) x y ( x - y ) = (2) CHI D A N a) V e t r a i c u a ( ) v a ( ) h e ( I ) l a cac d a thufc d a n g c a p b a c d o i v d i x , y R o r a n g vdfi x = h o a c y = h e ( I ) v n g h i e m G i a sijf X x'(3 + 2k + k ' ) = l l 0, d a t y = k x t a dixac: (1') x ' ( l + k + k ' ) = 17 ( ' ) C h i a v e vdfi v e (1") v a (2') t h i difotc k' + 2k+ 11 4k2 - 3k - 10 = ^ k = - - h o a c k = 3k' + 2k + 17 5 T h a y k = - — t a dtfoc y = — x , p h i f o n g t r i n h ( ) t r d t h a n h • 3x2 _^ x — x 16 -X = 11 X = ±4^3 +5^/3 T h a y k = t a se t i m dtTcfc x = ± , y = ± (4V3 N h i r v a y h e ( I ) c6 n g h i e m l a : -5V3 -4V3 5V3 ' , ( ; 2), (-i;-2) b) N h a n x e t rSng x = k h o n g n g h i e m diing he ( I I ) V d i x ;t 0, d a t y = k x t h i h e ( I I ) t r d t h a n h x ' ( l - t = ' ) = (1') (ID _f3 n ^ = - =:> t ' - t + = t(l - t) x ' t ( l - t ) = (2') => t = h o a c t = L a m t u o n g tii cau a) t a dtroc h e ( I I ) c6 h a i n g h i e m l a ( - ; - ) v a ( ; 1) C h o h e phucfng t r i n h vdi t h a m so m (I) (2) x^ + y ' + x y = (1) x ' - y', + m ( x + y ) = X - y +m V(Ji g i a t r i n a o c u a m t h i h e ( I ) c6 d u n g n g h i e m 54 E J TS Vu The Huu - Nguyen Vinh C$n CHI DAN (Do (x - y)(x + y) + m(x + y) = x - y + m 2 o x ' +[xy+' + y xy - =-30 o(A) hoSc (B) r ( x - y + m)(x + y - 1) = '^^ x^ + y^ + xy = X- y +m= x^ + y^ + xy = I x ' + y ' + xy = He (A) CO hai nghiem (2; -1) va (-1; 2) De he (I) c6 dung nghiem thi he (B) phai c6 nghiem trung vdi cac nghiem cua he (A) hoSc v6 nghiem Ro rang vdi moi m he (B) khong the c6 nghiem trung vdi cac nghiem cua he (A) Vay can t i m gia t r i cua m de he (B) v6 nghiem (B)o = [3x' + 3ax + a ' - = 0(2') PhucJng trinh (2') v6 nghiem neu A = 9a^ - 12(a^ - 3) < o a < -2V3 hoac a > 2>/3 MOT SO HE PHl/dNG TRINH DAI SO KHAC 57 Giai he phirong t r i n h : (I) CHI DAN (Do x^ + y + x^y + xy^ + xy = — (1) x ' + y ' + x y ( l + 2x) = - (2) (Trich de thi tuyin sink DH kiwi A - 2008) x^ + y + xy(x^ + y) + xy = —(x^ +y)^ + xy = - - Dat x^ + y = u, xy = V ta diroc he u + uv + v = — u'^ + V = — fu = u^ - u - uv = i u + V= — u U^ + U + u + V= 1- =0 + V= Hpc (A) u u' + u + - = u^ + V= (B) — 6n luy§n theo CTDT mSn Toan THPT El 55 (A)o + y =0 xy = - X (B) J5 \ 25 Vl6 +y = xy = - - He ( I ) CO h a i n g h i e m 58 G i a i he phirong t r i n h : ( I ) o X = 1, y = — y _J25 16 va (2) x^ + 2xy = 6x + (1) x" + x V + xV^ = 2x + (Trich de thi tuyen sinh DH khoi B - 2008) CHI DAN \2 3x + (x^ + xy)^ = 2x + (Do xy = 3x + - - x ' = 2x + (1') xy = 3x + - (2') x =0 (1') - (x^ + 12x^ + 48x + 64) = o x(x + 4)^ = X = -4 V i x = k h o n g n g h i e m dung (2') n e n k h o n g la n g h i e m cua he ( I ) 17 V d i X = - t a dirge y = — V a y he ( I ) c6 n g h i e m n h a t 59 G i a i he phtfong t r i n h : xy + X + = 7y a) ( I ) xY + xy + = 13y2 (x + y)^-4 + l -4; ' 17 j (1) (2) (Trich de thi tuyen sinh DH khoi B (1) x(x + y + 1) - = b) ( I I ) =0 2009) (2) (Trich de thi tuyen sinh DH khoi D - 2009) CHI DAN a) V d i y = k h o n g n g h i e m dung he ( I ) , do chia cac phtfong t r i n h cua y y X X + — + — = he oho y ?t t h i dugc X x ^ + -— h+ ^- = 13 y 56 y Ea 15 Vu Thg' H\ju - Nguyin Vinh CSn Dat a n phu x + — = u, — = v t a diroc: • y y u +V=7 u + V =7 u ^ - v = 13 X + + u - 20 = — = X H — 4, V = u = -5, v - -5 = y Giai cac he u = t h i dtrcfc x = l , y = — v a x = 3, y = [y l a cac n g h i e m cua he ( I ) , b) Dieu k i e n xac d i n h x ;t 0, he ( I I ) c6 the v i e t t h a n h Dat u = x(x + y ) t a ducJc he u + u + X = u^ + x^ = X ^ o X = (u + x)^ - 2xu = X ^ x(x + y) + X x^(x + y)^ + = X ' = u +x=3 ux - u va X l a cac n g h i e m cua phiicfng t r i n h : - 3z + = x =2 x =l hoac u = x(x + y) = u = x(x + y) = o X = 1, y = hoSc x = 2, y = — 60 Giai he phirang t r i n h : ( I ) x V - x y ' + y ' - 2(x + y) = (1) xy(x^ + y ' ) + = (x + y)^ (2) (Trich de thi tuyen sink DH khoi A - 2011) CHI DAN TCr phi/0ng t r i n h (2) cua he (I) t a c6: (2)c^ xyix^ + y^) + = x^ + y^ + 2xy o (x^ + y^ - 2)(xy - 1) = TCr he (I) tiicfng dirong v d i hcfp cua h a i he phiTcfng t r i n h x V - x y ' + y ' - ( x + y) = (A) (I)c^ i xy =1 x V - x y ' + y ' - 2(x + y) = (B) (1) (1) y^-2 + Thay x = - vao (1) t h i diicfc 3y' - 6y + - = ^ y^ = y • y TCr 3uy he (A) c6 cac n g h i e m ( ; 1), ( - ; - ) + Thay = x^ + y^ vao (1) ta diTcfcSxV - 4xy^ + 3y^ - (x^ + y^)(x + y) = 2y^ - 5xy2 + x V - x^ = (*) H Q C vk 6n luy$n theo CTDT m6n Toan THPT 57 - V d i X = 0, y = la n g h i e m ciia {*) nhiTng k h o n g l a n g h i e m ciia he (B) V(5i X ;^ 0, d a t y = k x t h i d\iac x^(2k^ - 5k^ + k - 1) = => k = — hoSc k = V d i k = t a t i m difcfc cac n g h i e m cua he (B) t r u n g v d i cac n g h i e m cua he (A) XTA- - 1 ^ ^ 2V10 ^ Vio Vcfi k = - t a dtfoc X = ± —, y = ± V a y t a p 2n g h i e m •cua he ( I ) l a : - ^ a;lU-l;-lj2VlO Viol f 2VIO yflO x = ^ - x ' - x + 22 = y + y ' - y 61 G i a i h e phiTcfng t r i n h : (I) 2 x +y -x+y=— (1) (2) (Trich de thi tuyen sinh DH khoi A CHI DAN X e t phuong t r i n h (1) cua he t a c6: ( D o (x - If - 12(x - 1) = (y + if - 12(y + 1) (x - 1)^ - 12(x - 1) = (y + (DoCD { \ X 2012) - 12(y + 1) A2 y ^ - + — = (2-) TCf (2') suy r a : - < x - - < va - < y + - < 2 1 hay — < x - l < — va — < y + l < — •^ 2 ^ X e t h a m so fTt) = t^ - 12t t r e n doan f (t) = 3t^ - 12 < V t 2' 3 ta thay [-2; 2] e Vay h a m so f i t ) = t^ - 12t n g h i c h b i e n t r e n cua he (!') t a c6: ( x - If - ( x - l ) = (y + ) ' - 12{y + l ) o x x = y + (*) The (*) vao (2') cua he ( D t a difoc \ 2' 3 V i vay tLf (1') = y + A2 y +— + ^ ^"2 = y = -— ho&c y = - — • 3] (1 w TCr suy r a cac n g h i e m cua he ( I ) l a : 58 l ^ (2, 2j 2y ' I ' TS Vu Thg Huu - Nguyin Vinh C$n 62 Giai cac he phiTdng t r i n h : (x + y) + a) xy; = b) = 49 (x^ + y^) + - „ X xy(x + l)(y + 1) = 12 y j x + xy + y ^ - l d) \xy(x + y) = -2 CHI x + y + x^ + y^ = x^ + y^ + xy = x*+yU xY = 21 DAN a) Tap nghiem \ 7-3V5^ ' 1; T + aVs 7-3V5 + ; 3V5 ; b) Tap nghiem (-3; -2), (-3; 1), (2; -2), (2; 1), (-2; -3), (1; -3), (-2; 2), (1; 2) c) Tap nghiem ( - ; 2), (2; - ) , ( - ; - ) d) Tap nghiem (1; 2), (2; 1), ( - ; - ) , (-2; - ) 63 Giai cac he phtfcfng t r i n h : X a) + xy + y = b) y + yz + z = z + zx + X CHI X = + y = 2xy + yz = 2y + zx = 2x DAN a) Tap nghiem (1; 0; 4), (-3; - ; - ) b) Tap nghiem (1; 1; 1) X + my - m = 64 Cho he phiTOng t r i n h : (I) x' + y' - X = (1) (2) a) Tim cac gia t r i cua m de he (I) c6 hai nghiem phan biet b) Goi (xi; y j , (X2; y ) la hai nghiem cua he da cho, hay chiJng minh (X2 - xi)^ + (y2 - yi)^ < Dau = xay k h i nao CHI DAN a) Phifofng t r i n h (2) cua he bieu dien diicjng trbn tam —; ban kinh V2 / - phtrong t r i n h (1) bieu dien diTdng thang DS: < m < ^ b) m = - H Q C va 6n luygn theo CTDT mfln Toan THPT 59 CHUYENilll.PHMlRilllfilJGIIIC KIEN THlfC C a c gia tri Ivfofng giac ciia goc (cung) Ixfofng giac T r e n diidng t r o n lufcfng giac l a y d i e m M D a t a = (OA, O M ) So goc Itfong giac (OA, O M ) bang so cung A M K h i cosa = OP, s i n a = OQ tana = sma cot a = cos a cos a sma Cac gia t r i cosa, sina, t a n a , cota l a cac gia t r i Itfong giac cua goc a hay cung a Ta c6: sin(a + k27r) - sina, k e Z A(1;0) cos(a + k.2n) = cosa t a n ( a + krc) = t a n a cot(a + k n ) = cota C a c gia tri li^cfng giac c u a mpt so cung dac bi$t Goc a (30°) 7t n (45°) 71 2 V2 V3 2 V3 71 (60°) 73 sina cosa tana cota 1 (90°) 271 (120°) 371 571 72 V3 (150°) (135°) -73 KXD 73 2 2 V3 V3 3 2 73 72 -1 -1 C a c gia tri lu'oTng giac ciia cac goc c6 l i e n quan dac a) Goc do'i 60 S 73 -73 bi^t : •> cot(-a) = - c o t a tan(-a) = -tana cos(-a) - cosa sin(-a) = - s i n a TS Vu Thg' Hi;u - Nguygn Vinh Can b) Gdc bit sin(7r - a) = s i n a tan(7i - a) = - t a n a cos(7t - a ) = - c o s a cot(7t - a) = - c o t a c) Gdc han n sin(7i + a) = - s i n a tanCn + a) = t a n a d) Gdc phu cos(7i + a) = - c o s a cot(7t + a ) = cota sin — - a = cos a cos — a = s i n a ; n - a = cot a tan 71 cot a = tana 71 e) Gdc hon — sin ' 71 ^ - cosa cos ' 71 > " " n = - cota cot tan a + — C a c c o n g thufc Ixfofng g i a c C o n g thufc lifofng g i a c cof b a n a) sin^a + cos^a = 71 + T: 2j - - tan a Va b) tana.cota = a c) + tan^a = a ;t — + k i ^ cos^ a = -sma ?t k— 2 (a ;t k n , k e Z) sm a C o n g thuTc c p n g cos(a + b) = cosacosb - sinasinb cos(a - b) = cosacosb + sinasinb sin(a + b) = sinacosb + cosasinb sin(a - b) = sinacosb - cosasinb tan a + tan b tan a - tan b tan(a + b) = tan(a - b) = - t a n a t a n b + t a n a t a n b cot a cot b - cot a cot b + cot(a + b) = cot(a - b) = cot a - c o t b cot a + cot b d) + cot^a = C o n g thufc n h a n d o i cos2a = cos^a - sin^a = 2cos^a - = - 2sin^a sin2a = 2sinacosa tan a tan2a = - tan^ a cot2a = cot' a - cot a HQC fin luy§n theo CTDT mfin Toan THPT EH 61 ... Vs _ _ - + V29 X b) Ta lap bang sau : X -00 -4 x''^- Ix^-ll -2{ x^ + 6x + 8) 2| x^ + ex + 8l 2( x2 + x + 8) B i e u thiifc (2) 3x^ + 12x + 15 = 30 -2 -1 2( x^ + Gx + x^ - -x2-12x-17=30 2( x^ + 6x +... (I) ajX + b i y ^ C j a X + b y = C2 K i hieu D = (1) (2) = aib2 - a2bi goi la d i n h thufc cila he ( I ) b2 - C2 a2 C: b2 C2 b, Ci = Cib2 - - aiC2 - C2bi a2Ci Quy tdc Crame G i a i he phifcfng... zi ,2 = ± a -2 x =4 - a +2 hoSc a -2 y =4+4 ''a + a -2 a +2 a -2 a +2 a -2 a +2 + Vdi - < a < he v6 nghiem 2x 54 G i a i h e p h i / o n g t r i n h : + l y (I) (1) ^ X 2y + - = X y (Trich CHI de thi