Doi vdi nhung phuang trinh mu c6 3 ca so {hoac nhiiu han, chdng han chtia a^, , f thi ta chia cd hai ve cua phuang trinh cho hoac 6^ hoac r^, dt gidrii xuong con hai ca so.. Doi vdi nhUn[r]
(1)X = (thoa man dieu kitni) •t=>3x = t o a n (Dvi' b i D H - 0 B ) Gidi hat phucing trinh : Chi:fcfng ^ y S x ^ - G x + T - 4x + < (1) iiai Ta c6 Cac bai toan phi:fdng trinh, he phL^dng trinh, bat phifcfng trinh de thi dai hoc (1) o , I 8x^ - Gx + > ; < X VSx^ - 6x + < x - <^ ^ 4x - > <" (4x - 1) 8x2 _ 6^._^ < _ 1)2 \ n U X G ( -oc; X G [2 X I <^ X 5.1 e -1 ; + o c -;+oo X G (-oo;0] U 8x22x > G ^;+oc)u{i Phifdrig trinh, bat phtfdng trinh chiJa can p nghiem la T = K h i klui: (Ian can ta tliu'tJng dung inpt so phep bien d5i tUdng dUdng sau day: (1) v ^ = < = > { 22^2 (2) va<i?^||iS^ ; ^ > B t o a n ( D e D H - 0 A ) I lis, ^ (7) VA=VB^A = B- (9) <yA> B <^ A> B'^; ; (6) '/A^^^lill (8) (10) \/A = B A = B^; VA < B <^ A < B ' \ Giai bat phMng trinh : \/5x - - s/a; - > v/2x - f 5x - > ^ -X ^^^^Q - >"T iai Dieu kien \ 2x-4^ VA>D^ (3) (5) ; X > K h i (*) tUdng ditdng v6i v/5x - > v/x - + \/2x - ^ x + > v/(x - l ) ( x - 4) ^ 5x - > 3x - + 2sy{x - l ) ( x ~ 4) \/(x - l ) ( x - 4) < x + <^(x - l ) ( x - 4) < (x + 2)2 <^ 2x2 _ 6x + < x2 + 4x + <=>x2 - lOx < <^ X G (0; 10) i v ' ^ ' "et hdp vc5i dieu kien ta dudc x G [2; 10) Vay tap nghiem ciia (*) la [2; 10) Bai toan ( D H - 0 D ) Gidi phudng trinh : t o a n ( D e D H - 0 A ) Gidi bat phUdng trinh : " \/x + + V x T T - v^xTT = Vx - G i a i Dieu kien x > - K h i phudrig t r i i i h da cho tiWug ditdng vdi 2\Jx + + 2^Jx + i = \/x + + 4=>4 (x- + + 2v^x + 1) = X + + 8v/x + + 16 328 vx - • ^ ,^,,s iai Dieu kien / x "- 116 >> 00 I X - > ^^ f x G ( - o o ; - ] U [ ; + o o ) X G (3; +00) I 329 ^ ^ ^ [4.+^) ^ ., (2) K h i V)at plutdiig t n n h da cho tUdiig difdng vdi \/2(x2 - 16) + x - > - x < ^ y2(x'^ lU x2 10 , > i -! 2(x2 Ta 16) > 2x < 16 > 2x > - 16) Vay (*) <^ / e 10 - 2x 0 > (10 - ^ 2x)2 (1) /?+4= vx U (3 + oc) < = > ( > - Do > ^^2(v7J)'^-5v/^ + v^>2 2>0^ X > X < < Ket hdp vcJi dieu kien va nghiein x = 0, ta difdc tap nghiem cua (1) la CO (1) ^ { X G f - o o ; 4] U [4; +oo) ^ ^ +°^) '^^'^"^ U[4;+oo) ^*^") B a i t o a n ( D H - B - P h a n c h u n g ) Gidi phuong trinh Taco ^ I 2x^ - 32 > 100 - 40x + 4x2 ^ ^ oc' 5] - 20x + 66 < ^ ^ <^ X e (-oo; X X G (10 (10 - \/34; G i a i Dieu kien - - lO+s/U) <;=> X G ( l O - VM; (thoa dieu kieii) X V6i dieu kien T a c o ( ) ^ I Je <^ < > Ta c6 + 1 + \/0^ i: ] + (3x + l ) = < x < ta c6 - + - ; • + (3X+ 1) > l + v/6^ s/3FTl+4 •\ • ( i } Vay (*) <^ 7: = Plntdng t r i n h da cho c:6 nghieni uhat x — X f_oo;-v/6]u[v/6;+oo) /rf(x-5)(3x + l) = + > 0, k h i <2 = ^ + + T l i a y vao (2), t a d\fdc t+ Vt'^ - > <=> \/t'^ - > - i «• = o , v/x+4= + \ / - t ' + - - > Vx V X - i < ^2 - G > - f, > « - ' > - t + «2 + +4 + v/3x + + + \Jx^ - 4x + > 3\/S G i a i Dieu kien; x > + \/3 hoac < x < - v/3 • Do thay x = la nghieni ciia (1) • X c t X > Chia (1) cho ^ x , dUdc bat phitdng t n n h tUdng dUdng \/X < x < Phitdng t r i n h titdng ditdng V3x B a i t o a n ( D H - 2 B ) Gidi hat phUcJng trinh : Dat t = ^/x -f 4= '' (v'3x + - 4) + ' ( - V ' e - x ) + 3x^ - 14x - = > Vay tap nghieni ci'ia bat phitdng t r i n h la (lO - \/34; +oo) X + _ = (x G R ) + 3x2 _ - VG-x V'Sx+l + 132 < _ (3) /.^ (4) B a i t o a n ( D e D H - 0 A - P h a n c h u n g ) Gidi phiMng tnnh ; v / ^ + v / ^ ^ - = (x G G i a i Dieu kien - 5x > u-* = 3x - x < - Dat u = <yix - K h i 5u'^ + 10 _ + ^ X = — ^ =^ - 5x = - ~ Thay vao (1) t a ditdc + ^ / ^ - = < , / ^ 331 = 8-2u1!• fnf^ • - (3) ^ ^ ^ ( - bv'^) = (8 - 2n)2 G i a i T a c6 24 - 15u^ = 64 - 32w + 4u^ ( xij + x-2 = \3 _ ^2y ^ J.2 ^y2 { 155^^+47/^ - 32M + 40 = ^ " = V a y \/3x - x = - ( t h o a m a n dieu k i e i i ) = - <^ i - = - B a i t o a n ( D H - A - p h a n c h u n g ) Gidi phuong , , , - ! trinh : > 1 - \/2(x2 - X + 1) • ^ ( xrj + x-2 = = Q ^\^ - y) {2x - y + 1) = _2xy-^j - +v/5 i^Zl • •'^ = hay I - - v/5 hay y = v/5 y = -v/5 G i a i D i e u k i e n x- > T a c v^2(x'-^ - X + 1) > 2x^ - x + > 1 - yj2(x'^ - x + 1) < VTiy l ) a t plutong t r i n h d a c h o t i r d n g diWng v d i X e CO b a nghieni ( ; ) , a i t o a n ( D e D H - 0 B - P h a n c h u n g ) Gidi f v/2(x2 _ ,7 + 1) <^ v^2(x2 - x + 1) < + v/^ - X ^/^•<l r l + ( x - x + l ) < ( l + ^^;-x)''^ XW + X + = 7?/ { + + = 131/2 he phuang tnnh ^, ^ / ^ m\ (x-,yeR) v ^ - x > G i a i H o d f i d i o t i t r t n g d u r i n g vcfi (v^)'-v/:^-i<o 2(x2 - X + 1) < + X + x^ + 2vx - 2x - 2xv^ (do y = k h o n g thoa m a n he) ^1 + V - - ^ x^ - x + - 2v/5 + x v / J < / + ^5 ( x + v ^ - 1)'' < 1= - D a t a = x + - , b= y - X v/x = - X = t ± \/5 > ^ ( l - x p = <^ X = v/5 a = -5 = 12 g = taco l a ngliieiu d u y n h a t c i i a I f = X = 3?/ 32/ + - = - - 2/ ^ y ^ S l:/>(i ,ifi,l.^,> phirniig t i i n l i d a c h o ^{ V 5.2 H e phifcfng t r i n h d a i so Bai toan (DH-2012D) Gidi he phuang Khi{ |J = i ^ \1=0 trinh , 2xy -y 332 = ^ ^ ^ { ^ = taco r + l r nj + X - = x ^ - xhj + x' + i / - ,J, — - — ^ / a + = ^ I o2 + a - 20 = x X + - K c t h p j ) vc'Ji d i r u k i f u i x > t a diWc x = V a y t a co a = Khi{ X 2/ y / a ++ 66 == 77 I a'-^ - = 13 ^ + ^ j: < x^ - x + = X X - S u y r a a2 = x^ + - + + v/5 + v/5 4=> X + v/J - 75^' = -^5 j x = 12^ 12y + - = ^ = 12 -5 i 333 (4) 12?X-f 5j/+ { = Vay he c6 hai nghiem (x; v) = {3;1) vk {x;y) B a i t o a n 11 ( D H - 0 B ) Gidi ,, he '^1 G i a i Dieu kien x^Qvky r vo nghiem) ' xl' = (l^-Y '^y=—;Tx'^ + 3x=—— ^ ^ 25^ -t; 65^ + 455 + 82 = ' , D o t va y la nghiem ciia phitdng t r i n h ^•2 + 2A; + ^ = ^ K h i he titdng diTdng v d i [ix'v = y' + (1) \.T?y2 = +2 (2) Vay (t, y) = = y'^ - (x;y) = = {y - x){y-\-x) : > ( r - y ) [ x y + (x + ? y ) ] = ^ [ V2' B' K h i y = X, thay vao (1) t a ditdc T i t iy 'ixy + (x + y) = khong th6 xay T o m lai { y = } nghiem day nhat cua he da clio Giai/le/;/ii/(/n(7 Mn/t f x'^ - 3x^ - 9x + 22 = f G i a i Dat L = -x r Dat = y + <, P = 2J chung) Gidi he phuang trinh ^,^iiv i x/ | /3 + I f'' + y ' + t + y = ^• _ + ) + ^ • i / /, + x = i /;^ + x2 = i;x)= / x ( x + y) = l X = / x ( x + y) = L I X = ^ 22 • ( S'-^ - SPS + 3(5^ - 2P) - = 22 ^ ^ / f+ x = ^ \t + x ) - t e = ,i;x) = a;2 Ta dudc he 334 r + 3y2 - 9y I + 3,2 ^ x(x + y + l ) = = x ( x + y) He (2) ird t h a n h : Bat Ho trcl t h a n h ^* nen nghiem ciia he: G i a i De thay x = khong thoa m a n (1) K h i x 7^ 0, he (1) titdng ditdng v6i; va 3x = ^ - ^ ^ suy x > va y > Vay t r i t d n g hdp BcM t o a n 12 ( D H - 2 A ) 2}- x-^ la mot nghiem ciia he = ^-4^ 1' , f :c(x + y + ) - = 3x''* - x'^ - = ^ (x - l)(3x^ + 2x + 2) = ^ X = Vay I y Z I ^^ = -2 ^ a x = -t B a i t o a n 13 ( D e D H - 0 D - P h a n J^-5_Y^\^^)^0 { 2' - j , ( i ; y) Lay (1) t i l t (2) theo ve, ta dxtdc 'ixy{x-y) A; G , H^J; f p _ ^i X •^ • 'i* = • = \ 22 53 - P + ( ^ 2P) - = 2;1 pith ft) \ : ( y I X=:2' f t luan: Tat ca cac nghiem cua he phudng t r i n h (1) l a (-;rf=(2;^^).(x;») = |i,i.t 335 s, ] (5) B a i t o a n 14 ( D H - 2011A) Gidi he phuang 5.3 trinh y - x / + 3?y^ - ( x + J/) = (1) 5x2 xy{ (^^.^u) P h u ' d n g t r i n h lifdng g i a c N h i i n g b a i t o a n l i e n q u a n d e n lUOng giac, n e u t a v i e t A: (/, m , n ) 5.3.1 ^ (x'^ + - 1) _ 2(.T?/ - 1) = tat TCm If t h u y l t : y = t h a y vao (1) t a d i t d c oc - + _ V a y ( x ; y) = ( ; ) ( x ; y) = ( - ; - ) T r i r d n g h d p : x^ + • - 2.T - - = <^ 3.T - - + ^ = <^3.r'' - 6.T2 + = <^ ( T ^ - l ) ^ = <^ x'^ = x = ±1 ri0«l*fftAAr?»f ^'^^ cos u = cos V <^ u = V + k2n u = -v + k2-n s i n u = sint) (3) t a n u = t a n v <^ I t = i ; +/cTT ; (4) c o t u = cot i ; <:4' u = V + kn r^n'-yi'^'l ' ' , ,o j j / ^ n {k € sin u = I ^ u = ^ + (3) sin u = - (5) s i n u = <=> u = k2n; • l i v f ',1,1 • : ,*, V , (2) cos u = u = A:27r; + y) = <!^4x''^y - x y + 2y^ - x ^ = <=> = + ^-2^; (4) cosii = - /CTT; (6) cos u = <^ u = ^ + k n <^ u = TT + A-27r; (3) D i e u k i e n c n g h i e m N e u y ^ t i n tCt (3) s u y r a x = 0, m a u t h u a n vdi x'^ + y'^ = 2, v a y t i e p t h e o t a c h i x e t y ^ K h i d o c h i a c a l i a i v c c u a (3) c l i o y^ t a d i W c x = x = (3)^ • P h U d n g t r i n h cos u = m c6 n g h i e m k h i va c h i - < 771 < 2y y Chu y X = y t a dUdc X = <^ { y2=Ji - 2y t a ditrtc ^ ( x ; y) = ( ± ; - ±1) Vdi moi m Vdi -\<m<l u = a r c s i n m + k2-K u — -K - a r c s i n m 'lk2 cos u = m u = arccos 77z + k2-n u = - arccosTTi + A;27r eR ^ (^^2) l'^ ^ 33G | ,5.-, - =5 TOu = arctan m +/TTT 77^ <^ u = a r c c o t 771 + kn G Z) {k G Z) (/C C h u y e n d o i gixJa s i n v a c o s i n , t a n g v a c o t a n g C a c n g h i o n i c i i a he l a ta cd c o t 7i x + y2 = -t to-a ' ta c6 s u i u = m <^ tan u \ f x = 2y { , • P h i t O n g t r i n h s i n ?i = 771 CO n g h i e m k h i v a c h i - < m < L y Vdi f• = T h a y v a c (1) t a d u d c < ^ x y - x y + y ' - x ^ - y^ - x'^y - x y ^ = { JVy2 , Z) ' (1) ban ' u = v+k27r u = IT - V + k n (2) ^-m.^ r T r u ' d n g h d p d a c b i e t l a n g h i e m c u a h§ x y - x y + / - (x^ + y'^)ix Vdi m a khong C o n g thufc n g h i e m c u a c a c p h u ' d n g t r i n h I t f d n g g i a c c d (1) 5.x - CO c h u t h i c h g i t h e m t h i t a h i d u k (/ m, n ) l a nhiTng so n g u y e n G i a i T a c6 Trvrdng hdp K/ , , .,, ,, (1) s i n x = cos ^ — - x j ; (2) cosx = sin - xj (3) t a n x = cot ; (4) cotx = tan - xj - x^ 337 ,., ; , (6) D S i d a u h a m so lifdng giac C h u y Sau day la mot so cong thiic rat hay gap cd lien quan den so : I (1) - sinx = sin(-x) ; (2) - cos x = cos (TT - x ) ; (3) - tanx = tau(-x) ; (4) - cotx = c o t { - x ) C a c bulctc g i a i m o t p h u ' d n g t r i n h lufdng g i a c Bu:dc D a t dieu kien dg phuong t r i n h xac d i n h Bvtdc Giai phvtdng t r i n h Bu'dc K e t hdp vdi dieu kien de ket luan nghiem , j ^-.^.^ t j i o T Ren luyen k y nang b i i n doi tich Bai tap 15 va chii y sau day t u y ddn gian n h i m g no thirrJng x u a t hion klii t a bien doi m o t phirdng t r i n h nao phu'dng t r i n h t i c h Bai • o.) h) c) d) e) h) t o a n 15 ChUng minh rang sin^.x = (1 - c o s x ) ( l + cosx) ; cos^.x = (1 - s i n T ) ( l + s i n x ) ; cos2x = (cosx - sinx )(c osx + s i n x ) ; + sin2x = (sinx + cosx)^ - sin2x = (sinx - cosx)^ ,• sin.x + cosx + tanx = ; cosx sin X + cos X I + cot X = ; ^ sinx \/2sin(x+ - ) = s i n x + cosx ; i) j) + cos2x + sin2x = c o s x ( s i n x + cosx) ; - cos2x + sin2x = s i n x ( s i n x + cosx) / n) Giai a) sin^x = - cos x = (1 - c o s x ) ( l + cosx) b) cos^x = - sin^x = (1 - s i n x ) ( l + s i n x ) [ c) cos 2x = cos^x - sin^x = (cos x - sin x)(cos x + sin x ) d) + sin2x = sin^x + s i n x c o s x + cos'^x = (sinx + cosx)^ e) - sin2x = sin^x - s i n x c o s x + cos'^x = (sinx - cosx)'^ • , , sin X sin x + cos x f) + t a n x = + = cos X COS X , COS X sin X + cos x g) + c o t x = + - : = : smx smx h) v ^ s i n ( x + ^ ) = \/2 ^sinxcos ^ + cosx sin = sin x + c o s x i) + cos2x + sin2x = 2cos^x + sinx cosx = cosx(siux + cosx) j ) - cos2x + sin2x = 2sin^x + s i n x c o s x = s i n x ( s i n x + cosx) • (2) l+tan''a= (3) + cot^ a = (5) + cos a = cos — ; (7) + s i n a = (^sin — + cos — j sin' Q ( Trong cac de t h i tuySn sinii vao Dai hoc, Cao dftng n l i i m g nam gan dhy, da so cac bai toan ve giai phUdng t r i n h ludng giac deu rdi vao m o t hai dang: Phildng t r f n h dua vo dang tich hoac phudng t r i n h chrra an d man « 5.3.2 „ , cos^ a (1) ^ ' ; cos 2a (4) (6) - cos a = sin'^ — ; (8) - s i n a = |^sin — - cos — j Oi\'^ CK = cos^ a ' cos 2a l-cot''a= : ( ; > 5.3.3 - tan^ Q a\ (X -I P h u ' d n g t r i n h b a c h a i , b a c b a d o i vofi m o t h a m so h f d n g giac PhUdng t r i n h bac hai, bac ba doi vdi mot liam so lifdng giiic la nhijfng phudng t r i n h dang ai^ + f»t + = 0, ai^ + t)t^ + + c/ = 0, vdi i la m o t liain so ktdng giac nao ' B a i t o a n 16 (Duf b i D H - 0 B ) tan ,;^K,I Giai phmng ( i - ) -—=3 tan ,X - trinh cos 2x - (*) COS'^ X G i a i Dieu kien cosx ^ va s i n x ^ K h i tau"' (*) <^ tan <^ - cot .1 X - tan x = ^ + tan^ X tan X = COS X = 2x - cos'^ h X —2 sin X ^ 4=> cos^ X tanx y ^ '2 h tan^ x = t a n x <^ tan^ x + = <^ tan'^ x = - O t a n x = -1 4^ t a n x = tan X = - ^ + A,-Tr {k e Z) (thoa man dieu kien) ' B a i t o a n 17 ( D H - 0 B ) G'ldi phuang trinh s i n x — = 3(1 — s i n x ) taii^x G i a i Dieu kien cosx ^ 4^ x ^ - + nn {n e Z ) K h i (1) tildng dudng sin X ~ 3(1 - s i n x ) siu'"^ x = -^^ TT^ cos'-^ X <^ „ sm x - 3(1 - s i n x ) sin^ x (1 - s i n x ) ( l + s i n x ) (7) sin X — „ siii'^ X , sin i + sin x — — sin x = sin x = ——: + sinx Neu 4- 6^ > a sin ^ — s i n x = - ( v nghieni) •»2sin^ X + s i n x - = <^ <^ cos s,,,,.,, ^ ^ Vay (1) tLfdng duong vdi X = — + A;27r (1 + sin X 4- cos 2x) sin 4- tan (1)^ ? ^ „ h + sin :^sin a { sm ^ Bai = CO (*) = 4=cosx ^ trinh , 4^^f.^,,.,.,„.,^^,,j,| ^ X \/3 4- — cos ^ '^'^^ X v^*^'"*^ X = i sinx = - - V3COS.T = ^ 0 ~ X = - + A;27r 2^ X = - - + yl-27r (fceZ) B a i t o a n 20 ( D H - B - P h a n c h u n g ) Gidi phMng X = + 7^6 k2n x = — + k2n ^ TT x - J = J + /.•27r - sin^ x + sin x + = : TT TT = - <^ cos x cos - 4- sin x sm — = - ^ <^ cos ^x ~ ^ ) ~ 6/ va s i n x + cosx 7^ 0) X s.n:,' 4-\/3cos,7; = siii'^ - + cos^ o - sm , s i n x = ^oai, dieu kien cosx 7^ 0) Gidi phuong 4- sinx 4- vScos.T = <^ sin.r 4- V2 dang hdn t o a n 19 ( D H - 0 D ) Giai-Ta , ^ cosx (1 + s i n x + cos2x) (sinx + c o s x ) c o s x <^ : — = cos s i n x + cosx <^ s i n x = sin f —— I V 6/ sniQ ^siii ^ 4-cos 1" X •O^ + s i n x + cos2x = (do dk cosx 1' = (pliUdng t r i n h xlii biett^'ach giai) I thiic htcfng gidc se nhanh V2 (1 + s i n x + cos2x) ( s i n x cos — + cosx sin —) <^ sin X 4- cos 2x = trinh l if ihih = —^ cosx + ^) c X = va n) — - : cos C h u y Neu \a\ |cj hoac \b\ \c\ ta grdt phiCOng trinh nhd hang Giai phucfng X « + A) X COS a :os(.7: B a i t o a n 18 ( D H - A - p h i a n c h u n g ) , : sin.r 4- (vdi cos a = s i n x = sm — G i a i DiSu k i « { ™ - $ th'i ta lam i i h u sau : Bien doi phitdng t r i n h trinh (sill 2x + cos 2x) cos X + cos 2x - sin x = (/,eZ) G i a i Tap xac cliiih K T a c6 (1) ^ (!)<!=> cos 2x cos X 4- cos 2x + sin 2x cos x - sin x = Vay nghiem ciia phuong t r i n h la x = - ^ + fc27r, x = ^ + 6 5.3.4 k2TT (k eZ) cos2x(cosx 4- 2) + sinx(2cos^x - 1) = •!•.>.< <^ cos 2x(cos X 4- 2) + sin x cos 2x = cos 2x(cos X + sin x 4- = 0) Phu'dng t r i n h bac nhat doi vdi s i n x v a c o s x • cos 2x = X e t phiTdng t r i n h n s i n x + 6cosx = c{vd\ + C a c h giai • Neu 4- 6^ < t h i ket luan phitdng t r i n h v6 nghiem 340 s i n x 4- ^ 0) ^ COST 4-2 = ^vo nghieni 1^ 4- 1^ < (-2)^ COS 2x = <!=> 2x = ^ + A-TT 341 X = ^ 4- A-^ (keZ) i<\ (8) B a i t o a n 21 ( D H - D - P h a n c h u n g ) Grdi phUdng di dua phuang V B a i t o a n 2 Gidi cdc phuong trinh sau : a) siir' X - \/3 cos^ x = sin x cos^ x - v/3 siii^ x cos x {De DH-2008B) b) cos^ X + s i n x - 3sin- x c o s x = {DH Hue J'J'JS) s i n x — cos2x + s i n x — cosx - = **) r -H G i a i Plur(5iig t r i n h da cho titdng diWng sin ; X cos r - cos x - (1 - sin^ x) + sin x - = trinh < ^ c o s x ( s i n x - 1) + (2sin^x + s i n x - 2) = Giai a) Tap xac d i n h R <t=>cosx(2sinx • X e t cosx = - 1) + ( s i n x - l ) ( s i n x + 2) = <^(2sinx - l ) ( c o s x + s i n x - f 2) = cosX + s i n X + = f v nghieni l " + l''^ < (-2)"'^j <^sinx = X = J + /.•27I TT G X = — dd cho ve phitong ^, X— ^ + ; , K h i s i n x = ± , thay vao phuong t r i n h da cho t a difdc ± = , dieu vo If cluing t o cosx = khong thoa man phitong t r i n h TT - = - 2smx trinh bac nhat theo s i n x vd cos2x trinh i • X e t cos X 7^ Chia cii hai ve ciia phirung t r i n h cho cos'' x t a diroc ' • tan'' X - 4- k'ln = tanx - V^tan^ x ^ tan^ X + \/3 tan^ x - ^ t a n x + \/3J = 4=> ( t a n x + v/3) (tan^x - 1) = 5.3.5 P h u - d n g t r i n h d n g c a p b a c h a i d o i v d i since v a coscc Xet pluWng t r i n h h a n g so v a + b sin x cos x + r: co.s''' x = d , vdi u, b, c, d l a nluTng </ sin'^ + (:^ t a n x = — \/3 tanx = ±1 K h i d = 0, plnfrfng t i i n h t r c n dUrtc goi la ])lntflnK t i i n h t l n i a n nhat (dang c a p ) bac h a i doi vdi s i n x va cosx C a c h giai • K i n i t r a xeni cosx = (•^'= ^ -Vki^^ c6 thoii m a n p h i W n g t r i n h hay kliong? X ^ ^ X Nghiem ciia phirdng t r i n h la - - + ^-7r, x = ± - + /c7r(A: G Z ) b) Neu cosx = t h i thay vao pinning t i i n h da cho t a dime ±1 = , diou vo If chiiTig to cosx = khong thoa man phuang t r i n h K h i cosx 7^ Chia ca hai ve plntdiig t r i n h cho cos''x t a ditdc • K h i cosx ^ 0, chia ca hai vd ciia p h u o n g t r i n h cho co.s^x, d u a vc phiWnu, + t a n x (1 + t a n ^ x ) - tan'"^ x = t r i n h Hfic hai thco t a n x tan^ X ^ tan^ x + t a n x + = C h u > Phitang trinh ddng cap bac la phiMng trinh c6 dang nhiC aau: <!=^ ( t a n x - 1) (tan^ x - t a n x - l ) = • D a n g c h i n h t a c : a sin'' X + sin^^ x cos x + c sin x cos'^ x + d cos'' x = tanx = tan.r = \/2 <=i> t a n x = + \/2 • D a n g m d r o n g ( h a y c o n goi l a phu'cfng t r i n h b a c - ) : a sin'' X + sin'^ x cos x + c sin x cos'^ x + d cos"' x 4- ( m sin x + n cos x) = N h a n x e t TUdiig tU ban doc hay diia each gidi pliUOng bac ba doi xi6i s i n x va cosx Con doi vdi- pliiCOug trinh thay ni = ni(sin'- x + cos" x) va n = n(sin'^ dang cap bac {xcrn bdi loan N h a n x e t 7a cdn c.6 thigidi vd cos X bang c/uii sU dung trinh ddng cap bac - , bhig + cos'^ x) la dita vc phitdng each X = — + A'TT X = a r c t a n ( l - ^ ) + nTT X = arctan(l + ( ^ ^ Z ) - + mix LiTu y B a i t a p 22rt) dUdc giai bang each dUa ve phiWng t r i n h tfch B a i t o a n 23 (Du" b i D H - 0 A ) Gidi phuang trinh ••" tr'inh 2^2cos^ ( x - J ) - c o s x - s i n x = 22) phiiOng cdc cong = - - + kn Q = ± — + kn trhih ddng cap bac hai doi vdi sin-'' thiic •) - c o s 2a l + cos2n; r, sui o = , cos a = , snir\cosa = - s n i za 2 312 G i a i Ta c6 (1) (1) ^ [V2 (cosx + s i n x ) — cos X — sin x = (9) £ + siu'' X + cos^ X sin x + cos x siii^ x - cos x - sin x = <=> (; • K h i cosx- = (hay x = ^ + kn) t a c6 s i n x = ± T h a y vao (2) t a dUdc ( ± l \:5 ) ' ' - ( ± l ) = 0(ckmg).- X = - - (loai) /f27r B a i t o a n 25 ( D H - 2 B ) /• Vay X = ^ + A-TT la nghiem ciia (1) ty fj-n Gidi phuong iwnk trmh |; (cosx + \/3sinxj cosx = cosx - \/3sinx + (1) • K h i COST ^ 0, chia cii hai ve cho cos^x (a (htuc + tan'* T + tan r + tan'^ x - ( + tan'^ x) - t a n x ( l + tan^ x) = TT TT <^2 t a n x - = <t=> tan x = <^ t a n x = tan - <^ x = - + WTT 4 Htf«3ng d S n (1) <=> 2cos'^x + \/3sin2x = cosx - \/3sinx + 2cos'^ X - + \/3sin2x = cosx - \/3sinx Vay nghiem cua phu'dng t r i n h da cho la ^ 5.3.6 X = ^ + kiT cos2x + \/3sin2x = cosx - \/3sinx J + niTT (/c, m e X = - cos 2x H sm 2x = - cos x 2 <=>,cos(^ - 2x) = cos(^ + x ) Z) P h L f d n g t r i n h d a n g a s i n cc + b c o s a; = c s i n u + d c o s u, vori a ^ + b^ = c^ + d^ B a i t o a n 26 ( D H - 0 B - P h a n C a c h giai Chia ca hai ve cho Vo^ + b'^ dc (hra vc phiWng t r i n h cd ban, B a i t o a n 24 ( D e D H - 0 A - P h a n chung) (1 - s i n x ) cosx (1 + s i n x ) ( l - sinx) Gidi phMng trmh <^ cosx , , =:v/3 (*) !* <=> - sin 3x + — 4x = cos + ^) = cos Q 3x Bai t o a n 27 344 „ cos 4x H "'^ 3sinx-sin3x cos 3x = cos 4x - - + X = - ^ + A;27r k2iT (Du" bi ^ = 42 + DH-2005A) Tim ufihiem ' y tren khodng (0; TT) nia trmh - 2x^ X + ^ = - 2x + k27r x + - = - - + x + A;27r „ 4x = - - 3x + k2ir phUdng ^2x + ^) /r: v cos 3x = (*) sm - sm 3x + cos - cos 3x = cos 4x cos4x = cos ^3x — — j - \/3(l - cos2x) \/3 • o , o sm X = - sin Zx + —- cos 2x 2 rr ^ , o • COS X COS — - sni X sm - = sin zx cos — + cos 2,x sin — ^ - Hill ox + ^ 1 <^ - cos X ^ trmh <^ sin 3.r + \/3 cos 3.7- = cos4.r cosx - \/3sinx = sin2x + \/3cos2x f & w ff: „ <^ - s i n ^' + sin x cos x = y ( + sin x - sin^ x) sin2x = Vs + V^sinx Gidi pliUdng G i a i Tap xac d i n h E Ta ccj sin3x = s i n x - sin"* x (*) <^ (1 - s i n x ) cosx = >/3(l + s i n x ) ( l - s i n x ) cos X chung) s i n x + cosx sin2x + \/3cos3x = 2(cos4x + sin'* x ) va s i n x ^ i K h i G i a i Dio\ kien s i n x 7^ (4 sm x sin^ - - N/S cos 2x = + cos^ 345 X - 37r (*) (10) Giai T a c6 Bai (*) <^ 2(1 - C O S T ) - <F^2-2cos,i; \/3cos2.T = >/3cos2z = - - <=>-2cosx = N/3COS2X - — COS X v/3 = - y - X = COS COS sin2x 2x COS - O c o s f2x - - sin 2x sni - \ ^ cosx = x) 2x + - = TT - x + k2n 2x + - = X - TT + II2TT k2-K 57r x = -^^h2Ti Gidi pliMng , <^ - COS " ^ <^2cos^ ( x - ^ ) = c o s ( x - ^ ) ^cos(x ^ ) [ c o s ( x - ^ ) - \/3 sin 2x + cos 2x = c o s x ^ c o s x ( y S s i n x + cosx - = \/3cosx) Su* d u n g 7r = TT - + , /.-TT < ^ X = 27r 310 , h ,, A-TT (A- G cac x = — - + A;27r, x = A-27r c o n c6 t h e d i r a « • 2\/3 s i n x c o s x + c o s ^ x = cos x l) = ^ cong [ ^ thiJc b i c n dfii d6 giai phiTdng Viec sic dung cdc cong thiic biin ddi nhdm dUa phxldng trinh dd cho vi tr'mli tick, hoac cdc ph.iMiig trinh tiink phuong dd hict each gidi a ) C o n g thiJc b i e n doi t o n g t h a n h ti'ch a+b a-b cos — — - cos — — a + b a b s m a + s m = s m — - — cos — - — cos a + cos = can Khi nhom dd y den nhUng hicn nhdn B a i t o a n 30 = - (v6 n g h i e m ) + A-2 lifdng g i a c xuat ^ • c o s ( x - | ) = TT (1) C h i i y j: + A-TT giai t i t d n g t a n h i t c a c b a i t a p d d a n g D o i vdi bai t a p 29 trinh « • + cos 2x cos - + s m 2x s m - = sui x s m — + cos x cos ^ 3 V 6/ «)s ( x - - X = i ve p l n f d n g t r i n h t k l i n h i t s a u : — — - lo o ^/3 I1 ^3 2x + — s i n 2x = - s m x + — c o s x ^2 ^ ) <;=> + cos ^2x ~ ^ ) ~ — ) = cosx Lifti y H a i b a i tap 28 v a 29 k h o n g rdi vao d a n g t a d a n g x o t , n l u t n g ( o cacli 1; ho n g l i i e m [h) c h i + cos2x + \ / s i n x = 3(sinx + v^Scosx) + \ V a y n g h i e m ( l i a p h U d n g t r i n h l a x = ^ + kix, .3.7 (1) <^ + c o s x + \ / s i n x 4- = ( s i n x + ^ ) = cos x - cos ^ = c o s x -t^ c o s x c o s ( x - , G i a i T a co + cos - cos (2x - x ^ - ± - (6) (Du" bj D H - 0 A ) - ) ^ ft 2cos^x + 2\/3sinxcosx + = 3(sinx + \/3cosx) <^2 • >> ' ( G i a i P h u d n g t r i n h d a c h o titdng ditdng aj;<>i s i n x + ^ c o s 2x = c o s x - 18 < sin2x c h o n /; = D o t a co nghioni t l m o c k h o a n g ( ; r ) l a ] irinh \/3 sin 2x + c o s 2x = c o s x - JtOl'-Jia) D o X G (0; TT) nen ho n g h i e m {a) c h i c h o n A- = 0,^^ = B a i t o a n 28 Gidi phiMnff (DH-2012A) y ^ „ C O S x — - s i n 2x <F> C O S ^2x + ^ ) — cos(7r - + + cos ^ T - t o a n 29 cdc goc so hang cho , ^ a+b cos a - cos o = - s m — : — s m a+b a sin a sin = cos • sm ; chiia tong sin {hoac hoac hieu cosin) cdc tu chung (DH-2012D) cua goc , Gidi phmng 347 , goc vdi nhau, de Idm Irmh s i n 3x + c o s 3x - s i n x + c o s x = ^ cdc dd bang a-b b \/2 c o s 2x (1) (11) 44> - s i n x s i n x - 2sin x = - s i n x ( s i n x + sinx) = X = kn sinx = ^ 2x = -x + k2n I sin2x = s i n ( - x ) 2x = TT + X + k2n ' X = kn X = kn k2n x= — _k2n {k£ L ^~ x = n + k2n Giai (1) <^ (e()s3.T: + cosx) + (sin 3.7; - s i i i x ) = \/2cos2x <^ cos 2x cos X + cos 2J: sin x = \/2 cos 2x • ^ c o s x ( s i n x + 2cosx - V5) = Tnrdng lu.Jp : cos2x = ()<=> 2x = - 4- ^:7r <=> x = - + s i n x + 2cosx - \/2 = <=> s i u x + cosx = V2 cos a cos h = - [cos (a + b) + cos (a - b)] ; :0?) + = ^ cos a sin Vay nghicui ciia i)lurang t r i n h la n kn 77r >• • " sin a cos ^ = ^ [sin (a + 6) + sin (a - 6)] ; ^ " 12 ^ '^•27r v1,' f sin o sin = - - [cos (a + 6) - cos (a - 6)] ; J sin U S H i l t £ it^ioS ) C o n g t h i i c b i e n doi t i c h t h a n h t n g TnnJng lit.lp : s i n x + 2cosx - \/2 = K l i i = ^ [sin (« + fc) - sin (« - ) ] , ^ B a i t o a n 33 ( D H - 0 A ) Gidi phMng trinh •^• = i + y ' " ^ T + ' - ' ' ^ ' ^ ' = T + ' ' ' " - cos^ 3x cos 2x - cos^ x = B a i tcniii 31 ( D H 0 B ) Gidi phuong trmh : sin^ 2x + sin 7x - = sin x Hu'Qtng d S n C a c h T a CO 'A, G i a i Ta co (*) (1) <^ ^ cos 3x(cos 5x + cos x ) - cos^ x = sin7x - sinx - (1 - 2sin^ 2x) ==0 c o s x s i n x - COH4X = o (•os4x ( s i n r - 1) = 0 n 4x = - + kn 3.i- = ^ -f k2n :ir cos 4a: = sin3:r = - ^ cos 4x = sni 3x = sni — G kn " T o TT k2n x= — + 1 - ( c o s x + cos2x + cos4x + cos2x) - - ( + cos2x) = i cos 8x + cos 4x - = <^ cos^ 4x + cos 4x - = p a c h T a CO ,, k2n 18 I {keZ) (1) <^ (1 + cos6x)cos2x - (1 + cos2x) = cos6xcos2x - = - (cos 8x + cos 4x) - I = cos 8x + cos 4x - = t o a n 34 ( D H - 0 D - P h a n c h u n g ) Gidi phuang trinh B a i toan 32 ( D I I 0 G D ) Gidi phrtdng innh •' i cos3x + cos2x - cosx - = G i a i Ta vu VS cos 5x - sin 3x cos 2x - sin x = iai PhUfIng t r i n h t i M n g difdng: \/3cos5x - (sin5x + s i n x ) - sinx = <^ VScosbx - s i n x = s i n x ( * ) <?4> (cos3x - cosx) + cos2x - = <> sin 2x sin J - - (1 - cos2x) = 318 — r: • r • • cos5x - - sui5x = suix <^ sni ( 349 ^ \ - 5x I = s n i x (12) c ) C o n g thuTc h a b a c , n a n g c u n g - TT Gx = bx I - fc27r 4tx = - I A;27r sni a = - icos 2a i — ; u o i.u, + ^ds ; cos a = 2a TT T ~ X = L i f t i y Sau k h i d i i n g cong thiic bac, t a thu6ng dung cong t h i i c bien doi t i n g t h a n h t i c h nhiT d muc a) ^ B a i t o a n 35 ( D H - 0 D ) Giai phUdng cos^ X + sin'* a; + cos trinh B a i t o a n 37 ( D H - 0 B sin ^3x - - ) Gidi phuong trinh : sin^ 3x - cos'^ 4x = sin'^ 5x - cos^ 6x (1) ••\ G i a i Tap xac d i n h R Phitdng t i i n h tudng dudng G i a i Ta C O ( ) < ^ - 2sin^xcos^x + ^ sin ^4.T - 1 - cos 6x + sin2x - ^ + cos 8x _ - cos lOx 1 - - ( - cos4x) + - (siii2x - cos4x) ~ ~ ^ X = COS X = c o s l l x = cos7x - + cos4x + s i n x - 2cos4x - = ^ - cos4x + s i n x - = 2sin^2x + s i n x - = ^ sin2.T = sin2x = - (vo nghiem) = ^ + fc27r X = ^ e Gidi phuong ^2 <!^2 (cos^ X \ sin22x - - s i n x = 0<»3sin^2x + s i n x - = V ) (nhan) (loai) 7^ 2x = - + fc27r 44> TT X = - +/cTT K c t hflp vcii dioii kien t a dildc nghiom cua phUdng t r i n h la x = 350 2x Gidi phuang trinh : + sin x = sin x cos — (fc 2x = ^ + X + k2TV 2x - - = - X + G (sin^ Z) X x = - \ k2-K Jai t o a n 39 ( D u h i D H - 0 D ) (1) + + xj k2n (^-eZ) Gidi phuong trinh + cos'' x) + cos 4x + sin 2x = (1) P ~t~ cos 4x nen i G i a i V i sin'* x + cos"* x = - sin^ x cos^ x = - - sin^ 2x = ^ + fcTr, vdi (1) k la so nguyen le = s i n x + cos2x = <^ cos2x = s i u ( - x ) <i=^ cos2x = cos / sin2x = - o X sin x + cos 2x + sin 2x = sin x + sin 2x \/2 s i n x 7^ — K h i ( ) titdng dildng + sin^ x)^ - cos^ x sin'^ x (cos^ x + sin^ x) - - s i n x = sin2x = = ( ) <=> s i n x + cos2x + sin2x = s i n x ( l + cosx) + sin^ x ) - sin x cos x = X = X [ G i a i Ta co ^ \/2-2siux (cos^ X /CTT trinh (cos^ X + sin^x) - s i n x c o s x G i a i Dieu kien \/2 - sin x 7^ = Z) sin X + cos B a i t o a n 36 ( D H - 0 A ) X i B a i t o a n 38 ( D y h i D H - 0 B ) + fcTT {k — + ly, + cos 12x 2 ~ <^ cos 8x 4- cos 6x = cos 12x + cos lOx 4^2 cos 7x cos X = cos I x cos x 1 - - sin^ 2x + - (sin2x - cos4x) - - = 2x •Jrti: 4- cos4x + cos4x + sin2x = 351 2cos4x + sin2x + = (13) 2(1 - sin^ 2x) + sin 2x + = <^ sin^ 2x - sin 2x - = sin 2x- = - sin2x = - (v6 nghiem) ! B a i toan 42 (Dvi" b i D H - 0 B ) Gidi phuang trinh cos2x- + (1 + 2cosx)(sinx ~ cosx) = Giai Ta c6 (1) ttfdng duong v6i \~ '•''•^ (cosx - sinx)(siux + cosx) + (1 + 2cosx)(sinx - cosx) — <^(cosx - sinx)(sinx + cosx - - 2cosx) = « ' -;iJ.iu, 5.3.8 Phu'dng t r i n h difa ve dang tich De dua phUdng trinh dd cho ve phucfng trinh tich dieu quan nhdt vdn Id Idm dc phdt hicn rci nhun ti'C chung nlianh nhat Dan doc ncri xem Im chu y2d trang 339 vd bdi tap 15 d trang 338 de^ cd dinh hudng tot hdn qua trinh gidi bdi tap (1) Giai Ta c6 (1) sinx cos X , = tan X l)(2sinx + cosx) = sinx(2cosx - 1) (2cosx ~ l)(sinx + cos.r) = <^ cos ^ — sinx = — cosx TT X = ± - + k2n cos X = cos ^ ^ {k€ Z) ^ X = - - + /CTT tanx —1 Bai toan 41 ( D H - 0 B ) Gidi phuang trinh ^ cos ^ = ft tan - - X = TT + 2m7r TT X = — + 2n7r X = — + /cTT, X = TT + 2m7r, X = — + 2n7r (/c, m , n G B a i t o a n 43 (Di^ b i D H - 0 D ) Gidi phuang trinh 4sin^x + 4sin^x + 3sin2x + 6cosx = Giai Ta c6 (1) tiMng ditdng vdi sin'* X + sin'^ x + sin 2x + cos x = sinx + cosx + sin2x + (1 + cos2x) = <^4sin^x(siux + 1) +6cosx(sinx + 1) = sin X + cos X + sin x cos x + cos^ x = sinx + cosx + 2cosx(sinx + cosx) = • ^ ( s i n X + 1)(4 - 4cos^ x + 6cosx) = sin X + = ^ sin X = - cos^ X - cos X cos^ X - cos X - = sinx + cosx = l + 2cosx = (sinx + cosx)(l + 2cosx) = sinx = - cosx it) ~ f^y nghiem ciia (1) la (1) Giai Ta c6 (1) Phuong trinh sin ^ cos | - cos^ ^ = tUdng diMng vdi •«-(2cosx + sin X + cos X 4- sin 2x + cos 2x = cos^ TT TT , = tan - <^ x = — + kn A A X I X x\ cos - sm - - cos - = <^ V 2) 1) (2 sinX + cos x) — sin x cos x - sin x (2COS.T PhUdng trinh sinx = cosx tUdng duong vdi Bai toan 40 (DH-2004D) Gidi phuang trinh : (2cosx - l)(2sina; + cosx) = sin2x - sinx sui X = cos X sin - cos - - cos X — sin X = sinx — cosx — = cosx = tan X = X = + A,-27r tanx cos X cos X tan(-J) X X cosx = cos — 352 n - - + kn i ±— +k2n [ke Z) = ^1 - - = COS 27r — o ^ X = ± — + k2TT Bai toan 44 (Du- bi thi D H - 0 B ) Gidi phuang trinh (2 sin^ X - 1) tan^ 2x + 3(2 cos^ x - 1) = - = (14) B a i t o a n 48 (Du* bi D H - 0 D ) Gidi phuang G i a i Dieu kien: cos2x ^ K h i (1) tUdng dudng vdi sin2x + cos2x + s i n x - cosx - = , - cos 2x tan^ 2x + cos 2x = <J=> cos 2x(3 - taii^ 2x) = X t a n x = \/3 t a i i x = —\/3 - taii^ 2x = <t=> = - L Ht^dtng d a n Ta CO + ^ I \ (vdifceZ) <^ cos x-:(>•)!•» B a i t o a n 45 (Du* bi t h i D H - 0 A ) Giai phuong trinh sin i n ( x - ^ ) + 4sinx + B a i t o a n 46 ( D H - 0 D ) X 1=0 (1) t o a n 49 {Dii ' + 2cosx)(sin2x - 1) = <^ 27r TT X = x = ^ + /c27r B a i toan 47 ( D H - 0 D ) X = (1) <^4cos^ x(cosx - 2) = cosx = <^ x = ^ + kit, vdik e Z -l<kn<U-l^-\<k<'-^-l'''A'\e{0,l,2,3} Do X = - , X = y , X 57r = y , = (1) ^ ^ " ' y r \T - j - ^ , o x \- TT COS + - - 3x _ , „ , ( _ _ _ J TT \ j COS [:r + / 3x TT V ^ = - j = V/2COS y ^^-^ = ,/2c0Sy v/2cos 3x — 2y 3x' ^ 3x cos ^ = \/2cos y V, :.Sk,irte '• cos y 3x <^ cos — 4- \ / c o s ( x + J) 1.):) = 7n X = (1) cos y = / cos r \T X ( X + - J = C O S y = L X = - + /c27r - r + A.-27r Vay ughioni ciia ( ) la B a i t o a n 50 (D\i + ^ + ^•27r, X = -TT + bi D H - 0 B ) Giui phamuj klr^ 355 {k e Z) trinh s m ( x + - j - M u ( x - - - j = - y 354 A;27r TT 3x 4cos'^x - Scos^x = e [0; 14] <^ < ^ + fcyr < 14, hay 37r 3x /- Tim x e [0; 14] nghiem dung phxiOng trinh (cos3x + c o s x ) - 4(1 + cos2x) = 0^ TT /5x Mn ^ y COS I ^ + KIT G i a i V i cos3x = 4cos'^x - 3cosx ncn ( ) tUdng ditdng vdi ^ TTN < = s m ( y - - ±—+k2TT — + ^ cos ^' ~ ~2 sin 2x = cos 3x — cos 2x + cos x — = Vay x = ^+k-K 7r\ / S X l i a i Ta cd (1) sin X cos^ X + sin X cos X - - cos X = <=!>2sinxcosx(l + 2cosx) - (1 + 2cosx) = cos \ bi D H - 0 B ) Gidi phuong trinh « " ' i y - J - ' " n - J G i a i Tap xac d i n h R Phildng t r i n h (1) tUdng dttdng vdi r = (2 sin x - 1) - (2 sin^ x - sin x + 1) = X trinh sin x ( l + cos 2.T) + sin 2x = + cos x cos tjuml ( s i n x - 1) (cosx - s i n x + 1) = |,, + sin x + 2) = Gidi phUdng (1) cos.r ( s i n x - 1) - ( s h i r - l ) ( H i n x - 1) = Htrdng d a n Tap xac d i n h E Ta c6 ( ) tiidng dudng vdi sin x( v/3 cos - ^ (1) <^ s i n x c o s x + - 2sin^x + s i n x - cosx - = (Thoa m a i l dicu kien) trinh (1) (15) G i a i T a c6 <^ s i n x c o s ^ x + (1 - s i n x ) (1 - cosx) = + c o s x sin X cos^ x + (1 - sin x) (1 - cos x) = cos^ x ( l ) ^ s i n ( x + ^ ) - [ s i n ( x - | ) + sin ^ ] = s i n (^x + ^) c o s ^ x ( l ~ s i n x ) = (1 - s i n x ) (1 - cosx) ~ sinxcos (^x - 1- - sin- X = 02cos2x = - cosx = < ^ COS s i n X cos COS Ta = ^ ^) CO (1) X = TT - + x - - cos sin X = X = — 27r ^ (fceZ) B a i t o a n 51 (Drf b i D H - 0 A ) - J) G i d i p/ii/cfn^ M n / i = sin (a; - ^ ) (sin2x - cos2x) = — ^ (sinx - cosx) + — S sin X - = cos tan X sin 2x ^ X B a i t o a n 52 ( D H - 1 B ) ' COS X COSX = COS (1 X + cos x ) (1 - sin X cos x + sin x - cos x ) = (2 (3 tan X = ^ - = tan / 7r\ -— « - x = - - + V 4/ fc7r Xet (3) D a t u = s i n x - cosx = \/2sin(x - ^ ) Dieu kien |u| < \/2 K h i X = ± - + A:2;r X = — + = - sin X cos x <^ sin x cos x = KTT Giai phudng trinh + sinx (sin sinx ^ — = -1 ix cos.- (2 cosx - l ) ( s i n x - cosx) = cosx = X = Ta CO phudng t r i n h (2) titdng dildng v d i <=> s i n x ( c o s x - 1) - cosx(2cosx - 1) = TT COS - + sin^ s i n x + cosx = - s i n x c o s x + s i n x - cosx = s i n x ( c o s x - 1) + cosx - 2cos^ X = X = X (sinx + c o s x ) ( l - s i n x c o s x ) - (cosx + s i n x ) ( c o s x - s i n x ) = O sin 2x - cos 2x - sin x + cos x - = s i n x cos x - sin x + cos x - (1 + cos 2x) = cos + cos^ (1) <^ (sinx + c o s x ) ( l - s i n x c o s x ) - cos2x = + ^ ^ X G i a i Tap xac d i n h R Ta c6 G i a i Ta c6 (1) <^ — Gidi /j/it/^n^? trinh^h ,ic$m !' B a i t o a n 53 ( D y b i D H - 0 D ) sin^ '1 x=±?+fc27r X = ^ + fc27r, X = TT + k2-n, x = ± ^ + k2'K {k G Z ) , + KTT X = - + fc27r sin (2x ^ Nghicni ciia plnrong t r i n h da cho la X = - + A;27r (2) (3) -1 X = cosx= - = - + /c^ sin X = [ 2cos^ X + cosx - = <^ va (3) tUrtng diMng k2TT cos cos (^x ~ ^ ) (-^ " s i n x ) = *' T h a y vao (3) t a difdc - + u = < ^ - l + ti^ + 2u = < ^ u ^ + 2u + l = 0<!->u = 2x + cosx + s i n x Vay G i a i V i (1 - s i n x ) (1 - cosx) = - (sinx + cosx) + s i n x c o s x nen (1) sin X cos^ x + [sin x cos x - (sin x + cos x)] = cos 2x 356 X = n27r •y2sin(x - ^ ) r= - sin(x - ^ ) = s i n ( - ^ ) 357 X = — +m2TT -l- (16) Nghiem cua phUdiig t i i n h da cho la ' X = - - 5.3.9 + fcTT, X = 7i27r, x = ^ ^ + m27r (fc, n , m G B a i t o a n 54 ( D H - 0 A ) Giai phMng trinh • ( - l > „ s i n ^ c o s x + ( l + c o s ^ x ) s i n x = l + siu2x (1) ^ (1) <^ cos X + siu^ X cos X + sin x + cos^ x sin x = (sin x + cos x)^ o (sin X + cos x ) + sin X cos x(sin x + cos x ) = (sin x + cos x)^ ( s i n x + cosx) [ + s i n x c o s x - ( s i n x + c,osx)] = s i n x + cosx = + sin X cos X - (sin x + cos x) = (2) (3) ' * B a i t o a n 55 ( D H - 1 D ) Giai phxtdng trinh sin 2x + cos x — sin x — = t a n x + \/3 y ; T^-i - r c o s x G i a i Dieu kien < R \n X ^ - V + — - f rwK X ( m , n e Z ) K h i ^ kTr{keZ) (1) ^ sin2x + c o s x - s i n x ^ X c t (3) D a t ?/ = s i n x + cosx = v/2cos (.r - J ) , |u| < ^2 K h i -1=0 cos X (sin x + 1) - (sin x + 1) = sin X u'-^ = siii'-^ X + sin x cos x + cos'^ x = + sin x cos x sin x cos x = ( s i n x + 1) (2cosx - 1) = u2 - • sinx = - COS r = cos — o T h a y vao (3) t a ditdc - u = 0<^u^-2u+l + = 0<^u=l Vay = <^ cos (2; - ^ = COS X = - - + A:27r X = ± - + I X = ' ''-vrs!'; G i a i Di6u kien { ^osx J C h u y Ta cd the giai plnMng trinh (3) nhanh han hang each bien doi (3) <^ (1 - s i n x ) ( l - cosx) = \ V IJ, 1" ' 'tiii'- ^ ^'"^^ 7^ x 7^ (1) m e Z K h i , cos^'x siir X , 9„ „ + 4cos2 2x = <4> + 4cos^ 2x = sin X cos X sm X cos X cos 2x , „ <^ , + cos-^ 2x = <^ cos 2x + cos 2x = sin 2x siii2x (1) <^ cosx smx cos2x = ( t h a dk) — ^ = -2cos2x L sin2x 358 - k2-n t a n X = cot X + cos^ 2x k2n x = - + fc27r, x = fc27r, x = - ^ + / c r ( f c G Z ) 1^ K e t hdp vdi dieu kicn ta d u d t ngliiOm ciia (1) la x = - + A-27r, /r G Z X = - + A-27r - Nghiem ciia phiTdng t i i n h da cho la lift; cosx = B a i t o a n 56 (Du" bj D H - 0 A ) Giai phKdng trinh v/2cos hdp V(3i loai phudng t r i n h k h i giai neu khong can than rat dg dan den lay t h i i a hoac thieu nghiem Dieu quan dau tien di giai dang la dat dieu kicn va kidm t r a dieu kicn xac d i n h Thong thitcJng t a hay d i m g dudng t r o n hidng giac hoac phUdng t r i n h nghiem uguyen de loai nghiem M o t phitdiig phap rat liieu qua la ket hdp dieu kien, loai nghiem t r o n g tfrng budc bien doi, ban dye hay theo doi phUdng phap t h o n g qua Idi giai ciia cac bai tap 56 57, 58, 60, ,,,, Ta CO (2) tiTdng ditdng vd\ s i n x = - c o s x < » t a n x = - = t a n ( ^ - ^ ) <=^x = d m S u v a phifdng p h a p ket •t i Ti'A Z) t i G i a i Ta c P h v f d n g t r i n h c h i J a an nghiem 2x = - + kn ^ 359 s i n x c o s x = - (thoa d k ) (17) n TT kn Dieu kien s i n x 7^ (vi k h i s i n z 7^ t h i cosx 7"^ - ) K h i kTT - =I + T sin4;r = — - - 4x= cos X sm X •£ — s i n- x + + cosx cos X + cos'^ X + sin^ x = sin x + sin x cos x <^ cosx + = s i n x ( l + cosx) — s i n x (do + cosx 7^ 0) (2) ^ + fc27r •I! '&Vay tat, ca cac nghiem ci'ia (1) la a' o SUIX = - = sin TT — X = - X = — L i m y Viec doi rhicii cos2x = vdi diou kion sin2x' ^ nhanh hdn va d6 hdn nlneu so vdi viec doi chieu x = - + — vdi dieu kien x ^ — (giong n h u Giiii phMm/ cotx + smx ( l + Gidi phuang trinh G i a i Dieu kien cos X sinx cos X SUl X COSX SUl X COS X >- SUl X + sinx I + t a n x t a n I) =4 (1) + sinx cos cos ^ C O S (x - X COS X X sin + SUIX = sm X (1) , ; G i a i Dieu kien s i n x ^ va cosx 9^ va t a n x 7^ - K h i ' „ i h S U l X - - sm 2x SUIX cos x + s i n x cos X - sin X cos 2x cos x , ' ^ : = : h s i n x ( s m x - cosx) sm X cos X + sin x <^ (cos^ X - sin^ x) = cos 2x cos x sin x - siii^ x(cos^ x - sin^ x) cosx , 1= cos2xcosx cos2x = c o s x s i n x - sin^ X - = = cos = 4<=>-^ r h sm X - - sin 2x cos2x ( c o s x s i n x - sin^ x - 1) = I) trinh cos 2x = cos 2x cos x sin x - sin^ x cos 2x = I/ COS X COS COS X COS sni X cos X cos ^ + sin x sin i + suix1 ^ gjj^ x 7^ K h i (1) titdng dudng vdi s i n x sin | \ -t- k2TT k2Ty + tan.T ^ sin X 7^ sin - 7^ cos2x cot X - = (1) COS X k2n »I B a i t o a n 59 ( D H - 0 A ) troug phUdng t r i n h dai so) B a i t o a n 57 ( D H - 0 B ) + <=> X COS f C O S X COS = 4<^ COS X sin 2x 1=4 — J2 x = - ^ (fc G Z) T i e p theo t a giai (3) Ta CO (3) titdng (hrdiig = X = <(=^sin2x = ^ ( t h d a dk) <^ sin2x = sin ^ <^ Ta C O (2) ^ 2x = J + A;7r <=^ X = J + ^ I (2) (3) + -sin2x fcTT + fc7r — cos2x Y ' ' = <t=>sin2x + c o s x - = (4) (fc e z ) V i 1^ + 1^ < 3^ nen (4) vo nghiem Vay sau k h i ket hdp vdi dieu ki?n t a c6 ket luan: Nghiem cua phUdng t r i n h da cho la B a i t o a n 58 (Du* bi D H - 0 D ) tan I ^ — x^ Gidi phUdng + G i a i Ta c6 (1) 4=>C0tx + sm + sm COS X X + cos 3G0 X trinh = = X ^ = (1) (2) C a c h k h a c Ta cd + ^ (fc = ' ± , ± , ) Z ;i;,.>: , , V,^JX —sinx (cos X — sin x) c o s X / \ cos (1) <^ = ^.^ i ^- s i n x ( s m x - cosx) cos X + sm X sm X (cosx + s i n x ) cosx + sinx = ^ (cosx — s i n x ) cosx + s i n x smx 361 (18) — sin X = (cos X + s i l l x) cos X COS X cos siiix sin X + X Bai + sin X t o a n 61 ( D H - 0 B ) - tan X X + sin x = — sin2x Khi = (vo = sin2x + sinx - „ 2sinx phMng = trinh cos x = 2cot2x sin2x + sinx „ =-A=cos^ \ = ^ - 1 cos x sinx sin2x sin2x sin^ x + s i n x s i n x - cos x - Bai toan 62 ( D H cos^ x - cos x - = kir ±Uk x = (^eZ) K e t h d p v d i d i e u k i e n t a dUdc n g h i e m c i i a (1) l a x = ± ^ G i a i D i e u k i e n s i n x 7^ K h i d o (1) cos2x (1) sni2x > cos x + 2sin22x - = cos x + - 2cos^2x - = o Gidi ^ ^ ^ j ^.^ ^ j , sinx <^2 cos x + sin^ x - = 0,±2,±4, ) d\i hi D H - 0 A ) I/: ; , „ + 4sin2x — — s i u 2x smx cosx 2cos2x , „ 44- — — 1- sin x = — sin2x ' ~" sin2x d a cho l a + '^ik ^ d o (1) t i t d n g d i r d n g v d i cosx nghiem) V a y sau k h i k e t h d p vcii d i e u k i e n t a c6 k e t l u a n : N g h i e m c u a p h i t d n g t r i n h t o a n 60 ( D e (1) " ( (fc e Z) sin2x + cos2x - Bai : - cos x \ ==0 — + /CTT x = j trinh G i a i Dieu kicu tanx = 1 ^ ~ -sm2x+ = phMng I; cot c o s x = s i n x ( t h o a m a n d i e u ki§n) - s i n X cos X + s i n ^ x = X Gidi = - 2011A, P h a n chung) Gidi - + k-K v i A; £ Z phuang trinh + s i n x + cos x /- „ - Jl.4 : = = v2smxsm2x." K\ ,t-,, l+cot^x , SV I'- = cos x <^ - cos^ x + s i n x s i n x - cos x = cos x <=> - cos^ x + sin'^ x cos x - cos x = cos x ll) G i a i D i e u k i e n s i n x 7^ K h i d o <^ - cos^ x + cos x ( sin^ x - 1) = cos x <^ - cos^ x - cos X cos x = cos x <=> cos x (cos x + cos X + 2) (2) (3 (4 + s i n x + cos x - 2\/2 cos X <^ S i n sin X = Ta CO (3) ^ (4) 2x = J + A-TT ^ X = ^ + ^ (yl- G Z) T a c6 <^ cos'^ <^ cos (A; e 1, c o s x = - v a o (2) deu k h o n g t h o a m a n V a y cac n g h i g m c i i a (1) c h i n h l a cac n g h i e m c i i a ( ) T h e o t r e n t a dUric cac n g h i e m c i i a (1) l a 362 X + s i n x - 2\/2 cos x = ^cos X X + s i n x - \/2 j = cosx = ( t h o a man dieu kien) cos X ^=7 + ^(^GZ) = 'nip ^khong thoa man dieu kien^ Z) C a c h k h a c T h a y s i n x = 0, n g h i a l a c p s x = x + c o s x + s i n x - 2\/2cosx = m.!i ,.,2 cos^ X + c o s X + = ( v o n g h i e m v i A = - = - < ) V a y n g h i e m c i i a (1) l a x = ^ + ^ \/^sinxsin2x sin^ X (1 + s i n x + cos x ) = 2\/2sin^ x c o s x =0 cos2x = (thoa dieu kien) cos x + cos X + = (1) ^ s i n ^ x ( l + s i n x + c o s x ) = X + sin = - + X = A:7r X - - = A27r \/2 X = - + k2ix 3G3 c osx = ^^'^^ ^ u ^ v/2cos ( ^ : - - ) = fthoa man dieu kien) ^ ' \/2 (19) B a i t o a n ( D H - 0 A ) Gidi phiCOng sin G i a i D o sin [x (1 + cosx) [1 - cosx - (1 + s i n x ) ] = + X = sin — V4 sin —— <^ - cos^x - (1 + c o s x ) ( l + s i n x ) = trinh 77r = cos.T va sin - x \ < = > ( ! + cosx) [cosx + sinx] = (1) X cosx = - cosx = - s i n x (thoa d k ) (thoa d k ) c o t x = —1 - — ( s i n T + cosx) nen = phvTdng t r i n h ( ) c dieu kien la 5.4 MSsxSo ^ s i n ^ ^ x ^ ! ^ , m e Z K h i (1) tilOiig duong v i K h i gap phudng t r i n h , bat phitdng t r i n h c6 chiia cac so n ! , P „ , thudng tien hanh n h u sau : Bvtdc D a t dieu kien • D o i vdi n ! = 1.2.3 n , digu }<ien la n G N • D o i v d i Pn = n\, dieu kien la n G N* + sin x cos X (sinx + cosx) s i n x + cosx <^—: — ^ ( s u i x + cosx) ' sin X cos X <*=>(sinx + cosx) I — sin X cos X + y/2 = , dieu kien la TT tan s i n x 4- cosx = sin2x + — ^/2 = ^ X = x = - - - sin 2x = - V2 + kn — o a6o.1 man / X trinh 7i"\ : 3; „ sill ( X - T I tan X - cos'' G i a i Dieu kiCn c o s t hat phudng trinh Tim cdc so nguyen + •cos2x - sin X sin^ x = • cos2 X — sin X — cos^ k! l.'f 4=> + sin X ^ + cos X 364 _ (1) (n-3)! <^{n ^ X = = I n thoa « - + 2' '(n-2)!.2! ) ( n - l ) n + ( n - l ) n < 9n < 9n ^ ""^ n (n^ - 3n + + n - - 9) < • ; '' , r i > , n e Z <^ - < 71 < " " " ^ " " " n G { , } L i f t ! y K h i giai phudng t r i n h , bat phudng t r i n h c6 chiia cac Q + cos x = X • - sin2 X 1 - cos^ COSX + cos duong + ^ ' " ^ < 9n (n^ - 2n - 8) < o''°<S ^n^ - 2n - < (1)^ tiU'ano' G i a i Dieu kien n G Z , n > T a c6 x 7^ ^ + m r rn G Z K h i k<nM) K e t hdp v d i dieu kien de ket luan nghiem B a i t o a n 65 (Du* b i t h i D H - 0 A ) Gidi phudng n,keZ 0< Bxidc Bien d o i va r u t gpn de t i m nghiem Biidc + fcTT (Thoa man dieu kien) B a i t o a n 64 ( D H - 0 D ) ta v, (keZ) x= +k7r X = PhLfdng t r i n h , b a t phtfdng t r i n h c chiJa c a c s nl Pn, Ai C^- i t h i can c k y nang r u t gon, gian itdc T a thudng d i m g cac k i t h u a t sau : I n! (n-1)! 1.2.3 (n- l ) n • 1.2.3 (n-1) n! 1.2.3 (n-2)(n-l).n (n-2)! 1.2.3 (n-2) 365 = (n - l ) n , V (20) 1.2.3 (n-3)(n-2)(n- l).n = (n-2)(n-l).n, 1.2.3 (n-3) n! {n-3)! • I) • • n! va tttdng t u cho (n-ky: Bai toan 68 ( D y b | D H - 0 A ) Cho tap A gom n phdn tit, n>7 Tim n biet rdng so tap gom phdn tit cua tap A b&ng hai Ian so tap gom phdn tii cila tap hap A Giai Vdi dien kien n € N va n > 7, yen can bai toan tndng dutdng v6i Q^ B a i toan 66 (Du* bi thi D H - 2003D) 71m so tie nhien n thoa man C7 - ^ 2Clcl + ClG^-^ = 100 Giai Dicu kiCu n G Z, n > Ta c6 c2c;j-2 + 2Clcl MCl + C^C^-^ = 100 o (C2)2 + 2C^C^ + (C^)2 = 100 ^Cl? = 100 n! Cl + C3 = 10 ^ 2,^^ _ n! +, 3,^^ _ = 10 ^ 3^, _ 3^ ^ ^3 _ 3^2 ^ ^ ( n _ l ) n ^ (n - 2)(n - l ) n ^ = 60 ^" 2C^ ^ - ^ =2 " 7!(n-7)! 3!(n-3)! 7!(n-7)! 3!(n-3)! 1 = 24.5.6.7" ( n - ) ( n - ) ( n - ) ( n - ) < ^ (n - 6) (n - 5) (n - 4) {n-3) = 2.4.5.6.7 ^ {n - 6) (n - 5) {n - 4) (n - 3) = 5.6.7.8 Dat t = ( n - ) ( n - ) = 7i2-9?i+18 Khi ( n - ) ( - ) = l - + 20 = t + vav Ta dndc t{t + 2) = 1680 <^ <2 ^ 2( - 1680 = 0^ t = 40 t = -42 T? - 977, 772 - 9T7, <i=>n'^ - n - = - ^ ( n - 4)(n2 + 4n + 15) = < ^ n - = 0<=>n = 7)2 - 971 r? - 971 + 18 = 40 + 18 = -42 71= -2 n = 11 - 22 = + 60 = Ket hdp vdi dien kien ta dUdc 71 = 11 Bai toan 67 (Du" bi D H - 0 B ) Tim x,2/ G N t/ida /le Bai toan 69 ( D H - 0 D ) Tinh gid tri hiiu thvcc M = >12 + C3 = 22 + C | = 66 14 Giai Dion kion x, y eN; x>2,y>3 X! ^ (.T-2)! 3!0y-3)! + (y-3)! 6x2 - 6x 2(y3 _ T i i (1) ta C O / y! rhng Cl^, Khi ho da cho trd = 22 (,_l),+ kz^fcli)^=22 (y-2)(y-l)y + ^ - l l ^ = 6 ^ = 66 2!(x-2)! :r7T + 27/ = 132 - 6x2 ^ 12x2 + i x + x2 - X = 132 ^ 11x2 (.) » j" + IJU 3i!i±411 + 2.!" + ^.'L + 2!(n++'I.2)! = 149 2!(n+l)! ! ( i - 1)! , 1) 71(71 + 2!.7i! + (n + )(n + 2y - 60 = <^ (y - ^j^^^y (2) _ n^; _ 132 = 5)(y2 + 2) + (71 + 2)(n + 3) + ^ 6n2 + 24n - 270 = o 7i2 + 47i - 45 = <!=> ^ X = X =-3 ( , , + 1) =—c! + ^) = 149 71 = (do n > 0) 4- Bai toan 70 (Du" bi D H - 0 D ) Tim so nguyen Idn hdn thoa man 2P„ + - P „ A = 12 + 2y + 12) = ^ y = Lifu y Co the giai iihauh hdii bang each dat n = ( * ) ^ (71! (x-2)!'^ (y-3)!' (*) Giai Dien kien n G N va 71 > V i (x - 6)(2 - y) = 2x + 6y - xy - 12 nen Ket luan: x = va y = 366 + ta duoc Ta loai x = - va thay x = vao (1) diTdc / - 3y2 (*) Giai Dien kien 71 G Z, ri > Ta c6 V^>y^^= 264 - + 2^2^, + 2C2+3 + Cl^^ = 149 (71 G Z, n > 0) n2 + n + 27t2 + 671 + + 2n2 + 10?i + 12 + n^ + 77i + 12 = 298 + y3 _ 3y2 + 22/= 132 ^ ) + 2y) + x2 - X = 132 (2) - 3y2 - biet (71+1)! - 6) - Tl! (n-2)! = 0<i=^ n! = n! = (21) n! = ! (n - l ) n= = n = -l(loai) 7i n = n - 2= (3) (a'")" = a"»" ; (4) L n = (6) K e t hdp v d i dieu kien t a dUdc n { , } Bai t o a n 71 (Duf b i D H - 0 B ) d2- Tren rang di c6 10 diem c6 2800 tarn gidc phdn biet, cd dinh tren Cho hai dudng d2 c6 n diem Id cdc diim th&ng phdn song dieu (gia di va ( n > 2) biet dd cho Tim n thoa song kien Biit (4) a'°g« ^ = 6, log„ hinh vuong ABCD Tim n biet so torn Idn luat gidc Tren cho 1,2,3 van cdc canh diem AB,BC,CD,DA phdn co ba dinh, lay tit n + diem biet B a i t o a n 73 ( D H - 0 D ) khdc cua A,D,C,D dd cho la 439 G i a i Ngu n < t h i so t a m giac co ba d i n h lay tir 7i + difim d a cho khong vUOt qua Cg = 84 < 439 (loai) Vay n > V i nioi t a m giac dUdc tao thaiih ling vdi m o t t hop chap ciia n + p h i n t i i , neu t r o n g n + diem khong CO ba diem nao tha,ng hang N h u i i g tren canh CD c6 b a d i e m , tren canh DA CO n diem nen so t a m giac tao la C^+g -C^ - C^ • T h e o gia t h i e t t a co C'+6 - Cl - C l - 439 ^ (n + ) ( n + 5)(n + ) _ ^ ; ^ - - n! 3!(n-3)! = 439 + 3/1^ - n = <^mi^ + 72jt - 2520 =-Q<^ n^ + An - 140 = n = -14(loai) C h u y Khi gidi phUOng trinh, bat phuong trinh mu thl cdc cong thiCc sau day thudng xuyen ducic sit dung a'^.a" = a"*+" ; (2) 368 ;v, ,.,U:a \>'r < ấ <^ x > j / _ phuang ,, = a'"-" ; tnnh , ,„ •\ " : - 2^^ + = (*) 2^^ + = (*) : ^ 2 ^ ^ ' - ^ - ^ ' ^ 22^(2^'-^ - 1) - 4(2^'-^ - ) = -« <t:^(2^'-^-l)(22^-4) = ^ ^ - ! _ 20 x2 - X = 2 ^ - 22 2x = ^ X = X = Tap nghiem ciia (*) l a = { , } B a i t o a n 74 ( D u b i D H - 0 B ) Gidi 2^-i+4x-16 X - bat phuang ^ - ^ + 4x - 16 trinh : , ;ifl; , > -'it' G i a i Dieu kien x 7^ T a co x - Phifcfng t r i n h , b a t phifdng t r i n h m u (1) nghid) j, '^ij n = 10 Ket luan n = 10 5.5 • Gidi < ^ u ^ + 15/1^ + 7 i + - - dd co G i a i Tap xac d i n h D = E K h i _ (n - 2)(n - l ) n ^ ^ tren log^a n = 20 + 8n - 560 = <^ I ;j ~ 1^28 (loai) bi D H - 0 A ) thtic = 6, log„ 6" = a log„ 6, log„ = 2x^+x t o a n 72 ( D y cdc cong (3) Vdi a la hang so va < a < t a co Vay n = 20 Bai co mat PhuTdng p h a p S i i dung cac cong thilc sau : (1) Vdi a la hang so va < a t a co a"^ = a'' x = y (2) Vdi a la hang so va a > t a co < a^^ a; < y - r r T = 2800 ! ( n - ) ! + n 2!8! <!=!>5(n - l ) n + 45n = 2800 <^ cdc so hang i^V^MH aO = l ; a " = - ^ } D a n g D i / a v e c u n g m o t c d so 10! n! rang S JMTCO tren G i a i So t a i n giac co m o t dinh tliuoc d\a liai d i n h t h u o c ^2 la lOC^ So tarn giac c6 m o t d i n h thuoc d2 va hai d i n h thuoc d\a n C f o ' Theo gia thiet ta C O lOC^ + nC^o = 2800 ^ 10 thiet : (a6)" = a " " ; X > 2^-^ > + 4x - 16 > 4x - x < 2^'^+4x-16<4x-8 X > •>x-l > X < 2i~i < X > X - > < ? „ X - 1< Vay tap nghiem ciia bat phitong t r i n h la ( - 0 ; ) U ( ; + 0 ) 369 ; :fhoA'.i.o (22) D a n g D a t a n p h u n ^ X G ^ I f ^ n PhU'dng p h a p M u c dich cua viec dat an p h u la d u a ve phudng t r i n h m6i dcfu gian hdn K h i dat u = (vdi < a ^ 1) t i n phai c6 dicu k i c n w > C h u y Doi vdi nhung phuang trinh mu c6 ca so {hoac nhiiu han), chdng han chtia a^, , (f thi ta chia cd hai ve cua phuang trinh cho hoac 6^ hoac r^, dt gidrii xuong hai ca so Doi vdi nhUng phuang trinh viu c6 hai r.a so thi ta dun, tic cimq mot ca so hoac Idqari.t hai vc B a i t o a n 75 ( C a u a l a d e D H - 0 D ; c a u b l a d e D H - 0 B ) Gidi cdc phuang trinh sau : {0,-1,1,-2} Tap nghiein ciia (*) la = { , - , , - } , ., f ( i ) B a i t o a n 77 (DvT b i D H - 0 B ) Gidi bat phuang X + I _ 22X+1 _ 5^gx < trinh : G i a i Tap xac d i n h D = E T a c6 /gA'^ fey ( l ) ^ ^ - ^ - - ^ < 4=>3 ' - ] - - ^ <0 4; \4j 2x 2^'-^-2^+^-^'=3 ; a) b) (\/2 - l ) ^ + (v/2 + ^ - 2v/2 = ^3 - ^ s » « - n ; -'[2) 3Y Giai a) PhUdng t r i n h da cho c6 tap xac d i n l i la R T a c6 •A < loga 2.I3 amii Vay tap nghiem ciia (1) la = ( - 0 ; log a Dat u = 2^^-^, v i dieu G m r kien u > u - - = <^ T h a y vao tren t a dUdc B a i t o a n 78 (Du" b i D H - 0 D ) Gidi bat phuang - 3-u - = o u = (do u > ) Vay 2' x ' - x = <^ 2'^'-^ = 2^ - X= 2x-x2 x'-^ - X - = <^ X = x = G i a i Tap xac d j n h l a M T a c6 u^V2 gx •-2x + l ( N / - ) " = V/2 + [ (V/2-1)^^ = ( v / - ) Ket hdp v d i dieu kien u>0, B a i t o a n 76 (Du* b i D H - 0 B ) Gidi phuang trinh : g.^2+^_l - 10.3^'+^-2 + = 'ff > 3X^+x ^ 3x2+x^9 ' <u<S t a diTdc < u < D o < 3'='"2'' < ^ x^ - 2x < (*) <^x2 - 2x - < <^ - ^ < X < + \/2 Tap nghiem ciia bat phifdng t r i n h la = [1 - \/2; + \/2] G i a i Tap xac d i n h D = M T a c6 <3^9-^'-^^-2.3^'-2^-3<0 -2u-3<0^-1 Ket hian: Phifdng t r i n h c6 t a p nghieni la S = { - , } 10.3 2x-x2 Dat u = 3"^ - x > 0, t h a y vao t a diWc (^/2-1)^ = ^ - t: r, If b) Phitdng t r i n l i da cho c6 t a p xac d i n h la M D a t u = (\/2 - l)"^ > 0, thay - 2V2u + = -< ,UV ' < -1 Phifdng t r i n h da cho c6 t a p nghiein la = { - ; } vao t a dirdc u + - - 2y/2 = 0^u^ u trinh : ^Ut'i i < ^2 , ( - ) _ 10.3^'+^ + = " 3x2+x ^ 30 3x2+x ^ 32 ^ 370 B a i t o a n 79 ( D y b i D H - 0 D ) Gidz bat phuang -,2x-'-4x-2 -2 - 2 ' - ' - _ < 371 trinh : , d -/ (23) G i a i Tap xac d i n h D = R T a c6 (1) ^ i.22^'-4^ B a i t o a n 81 ( D H - Q D - P h a n ^-2<0 l^'^'^.f*^^^'' (2) G i a i Tap xac d i n h R Phitdng t r i n h tUdng ditdng T h a y vao (2) t a ditdc -u^ ^{u ^24^+2v^+2 _ 2X^+4x-4 -2<0^u^ - 8u - 32 < u - 4)(u2 + 4w + 8) < ^ tt < _ <t=» 2^'-''^^ < 2^, hay x-^ 2^'-4^ _ 24 24x _ - v/3; + v/3 ^2V5+2 _ , ,, 2x3-4\ Q , X = J _ v / T = x3 - (*) 24 D a n g P h i / d n g p h a p h a m so '* — Q D§ thay x = la nghiem ciia (*) De x la nghiem ciia (*) t h i Tap nghiem ciia (1) la A" = [ l - \/3; + \/3 ' = 2x^-4 22Vx+2 _ -2x-2<0^xe '[ _ 24+2v'x+2 _|_ ^ ^ 2 ^ - ^ ' ( - - ) Vay ^ ' < ;vi j trmhi,,, - 8.22^-^' - < ^ i ( ^ ' - ^ ) ^ - (2^'-2^) Dat u = ^ ' > C h u n g ) Gidi phuong >> x ^ - > ^ x > ^ Phifdng phap • N h a m nghiem x = a ciia phUdng t r i n h • C h i i n g n i i n h phitdng t r i n h eo nghiein d u y nhat b^ng each xet h a i triTdng Do x e t h a m so /(x) = 2\/x + - x^ + 4, Vx e \fl; + o o ) ' ' " hop X > a vh x < a • Co k h i t a phai khao sat hain so, lap bang bien t h i e n ni6i suy l a dildc k§t qua tren C h i i y dung cdc cong iJiHc sau: (1) Vdi a Id hhng so vdQ < a^l ta c6 (2) Vdi a la hang so vda> \ cd < a" ^ x < y x > y (3) Vdi a la hang so vd < a < I ta c6 a"" < a" B a i t o a n 80 ( D H - 0 A ) nghiem ciia phitdng t r i n h da cho la x = 1, x = <^ x — y = Gidi phiCdng trinh 5.6 C h i i y 10 Khi gidi G i a i PhUdng t r i n ^ d a cho c6 t a p xac d i n h la M T a eo x = thoa m a n /8\ \12/ /27^^ + = 12, Tren R, xet hai h a m so lien tuc /(.) = ( ^ ) % ; f?Z^^ 12 V i /(x) la h a m so nghicli bien tren E, g{x) la h a m so dong bien t r e u nen x = l a nghiem d u y nhat ciia pliUdng t r i n h da cho 372 phiMng cong thxic sau day thudng phUdng t r i n h da cho T i e p theo t a c6 Phu'dng trinh, b a t phifdng trinh logarit „ : 3.8^ + ^ - ^ - ^ = 3.8"^ + 4.12'^ - 18-' - 2.27^ = - 3x2 < 0, Vx > Suy r a h a m so /(x) nghich bien \/x + v ^ ; + o o ) , b d i vay x = la nghiem d u y nhat cua (*) V i the t a t ca cac Ta CO / ' ( x ) = trinh, xuyen bat phUdng (1) loga = ; (2) (3) log„a'' = ; (4) (5) log„(6c) = log„ + log„ c ; (6) (7) loga (8) (9) log„ = ^ log„ h ; , log„ c (11) = a log„ ; '°^^'=log.6 = trinh mu vd logarit dUdc siC dung : _ h ~ (10) (12) 373 logfctt ' thi cdc (24) (13) log„ logf, a = ; (14) log„« f = - log„ c (15) Ig a = log a = logio a ; (16) In a = lege a a Vay tap nghieni ciia (1) la = J,,.,,: B a i t o a n 83 ( D u " bi D H - 0 D ) Gidi phuang C h u y 1 Khi dnt dicu kicn dd car phudng trinh c.6 nghm ta can nhd rang (1) Doi vdi thl dieu kien cua cd so a /a < a 7^ 1, x y (2) Doi vdi log„ b thi dieu kien cua cd so a laO< a ^ I, dieu kien cua b 161og27x3a:-31og3^x2 am thi y = x" xdc dinh vdi moix ^0 Neu a khong nguyen < 27x3 ^ < 3x 7^ X > x2 > dinh vdi moi x > D a n g Du'a ve c u n g m o t c d so ^ nguyen thi y = x" xdc 16 log27:,3 ^ y U > logo V i " log3^ X - > / ^ U > V { { 11 < V { u > B a i to»m 82 (Duf bi D H - 0 A ) Giai phUdng ;vi , + ; log3 X ' f Q<x^\ G i a i Dieu kien < 3^, > f / =logi 9x\ log^ trinh - (1) 3x^ = i - X 4=> x2 = l x2 = - ^) ^ (loai) (loai) (nhan) (loai) 374 16 log(3^)3 <^ - - l o g ^ X = log3^ <^ X = + ^ log4 = if ^ ^ log2 [(x log2 log2 (4x) 1|] = (x + 3) (x - 1) = 4x X < (x + ) ( l - x) = 4x r ^ n ^ { x j l Khido (4x) ^ X > trinh log2 ( x ) ^ j a ^ ^ J 1| = {1} f x > - log2 (x + 3) + log2 |x - 4^ X Gidi phuang (x - x + 3>0 + 3) |x - - logg^ X (x + 3) |x - 1| = 4x x^ - 2x - = X < x^ + 6x - = X > XG{3,-1} - 3x2 + = X = X = -1 X = X = ->/2 + 3) ^ X > K h i + log^ = log^ [^x log3^ X = ^x-l|^>0 (*) X (1) ^ ^ < x , ^ i <^ l o g ^ X = ' \^ \ ( u > log„u>log„t;^| t;>0 ^ V l27 ;^ Phuang t r i n h da cho C O tap ngliiem la = (3) Neu < a < t h i (o" > a" O ?i < ?)) va = B a i t o a n 84 ( D y bi D H - 0 A ) f u > log„ = > u > v) va ^ v.i x' aO{ a- > X , log3^ X = (2) Neu a > t h i ( a " > a" < < x ^ - - Iog3^ X 16, log„ x = log„ y <^ ^ ^ K h i Phufdng p h a p Si'r dung cac cong thilc sau : (1) Neu < a 7^ t h i X X G i a i Dieu kien c6 tap xdc dinh thuoc vdo so mu a Neu a nguyen dudng thi y = x" xdc dinh vdi m p i x e E Neu a = hoac a = trinh •••11: ldb>0 (3) Ham so luy thiia y = x" {v^} log,(3.T3) ^ log^ ^9,^ _ ^ X < G {-3± x = ^-3 + v ^ (doO<x,^l) v/12} Vay phuong t r i n h (*) c6 tap nghieni la {3, \/l2 - 3} B a i t o a n 85 ( D e D H 0 - B ) Gidi bat phuang logo,7 ( T^+xN loge — — r < \ + / 375 g trinh (25) Dat i = ( +x ^ x'^ + X > x + > x + x"^ + x > loge x + G i a i Difiu k i c n < vTTx + Vl - Ta CO x + x2 + (6x + 24) X - x + B a i t o a n 86 (Du* bi D H 0 A ) Gidi bat phudng 2x + 3\ ^""Hr^^x + lJ X + y X xe G (-00; (-00; x + -1 -2)U(-l;+oo) X € B a i t o a n 87 ( D H - 1 D , ""."j'i phan Chung) X + x/l - X = \/l -x2 2\/l - x = <^2 + B a i t o a n 88 ( D e D H - 0 D ) = 1^ Gidi phuang = 2x + x+ < 4.2^ - > < = » ^ > ^ ^ log2 2^ > log2 ^ ^ > Taco 3^2 I (-00; X > log2 | x + < ( l ) ^ l o g ( ^ + 15.2^ + 27) >0 ^ -2) Gidi phuong (4^ + 15.2"^ + 27) / 1 f = = log2 V4.2^-3 V4.2^-3 \ = <^ 4"^ + 15.2'' + 27 = 16.4'^ - 24.2"^ + <^ 15.4'^ - 39.2'' - 18 = ^ 5.4"' - 13.2'' - trinh Dat u = 2^^ > 0, thay vao (2), t a ditdc (xeR) (1) r i + x > l-x>Q ^ { - 13u - = f x > - l x<\1 < x < K h i Ket hdp vdi dieu kien cua u t a dUdc u = Bdi vay (1) ^ log2(8 - x2) - log2(Vl + X + v/1 - x) - ^ - x^) = + log2(/rT^ + v / l - x ) - x^ = ( v T T x + v^l - x) 2^^ = ^ log2(8 376 = trinh i, log2(8-.x2) + l o g i ( v / r + ^ + v / r ^ ) - - G i a i Dieu kien I x ">l! G i a i Dieu kien x + x + I - 4/2 - 16/, + 32 = log2 (4- + 15.2 + 27) + log2 log: 2x + + = 4/, o >0 log ( 2x + ^ , „ I log2 - - - r ^ log2 2^ > log2 logi x+ + Ket hdp vdi dieu kien suy (1) c6 nghiem nhat x = trinh G i a i B a t phifdng t r i n h da cho tUdng ditdng vdi > logl ' •> t;'!if Vay V'l + log2 •»(/ - 2)2 (/2 + i + 8) = <^ f = > Tap nghiera ciia bat phvtdng t r i n h la (-4; - ) U (8; +oo) logl e - III -4t^ > < ^ x e ( - ; - ) U (8;+c3o) 2x + \ v/l - T2 = Thay vao (2) t a ditdc > (thoa dieu kien) x2 - 5x - 24 dieu kien ( > K h i f2 = + ^ - T ' x'^ +x x'^ + X logo,7 {logg - ^ q p j j < = logo,71 ^ loge > = loge x + X, X + (2) 2^^ = 2'°S2 4^ X = log2 3(tlioa dieu kicn) Do X = log2 la nghiem nhat ciia (1) 377 * (26) B a i t o a n 89 (Du* b i D H - 0 B ) Gidi phuang trinh Dat u = log2 X , thay vao (2) t a dUdc l o g ^ y / ^ i ^ - l o g i ( - x ) = iog8(.T-if G i a i Dieu kien < x < K h i : + :c (1) u - (firf y j i d i ••jWf|): (1) ^ log2 Vx + 1+ log2(3 -x) '-• + Vay log2 X = ^ = log2(x - 1) ^ log2(x + 1)(3 - x ) = log2(x - 1) <^ (x + 1)(3 - x ) = X - X — X = o - x + 2x + = x - l < ! = > x ^ - x - = < ^ x = 23 log2 X = log2 x = v^ B a i t o a n 92 (Du" bi D H - 0 D ) Gidi phuang l ]^ trinh 2(log2 X + 1) log, X + log2 ^ = - vTr ^ < » l + u + 2u = u < ^ u = l < ! ^ u = - 1+ u 1+ u j/:iQf<.:> ;:; ; 'A • G i a i Dieu kien x > K h i Ket hdp v6i dieu kien t a dudc x = ^ nghiem nhat ciia (1) B a i t o a n 90 ( D t f b i D H - 0 B ) Gidi phuang (!)<!=> (I0g2 X + 1) I0g2 X - = <^ (log2 X ) + log2 X - = trinh log2 X = l g X = l o g ( x - l ) + l o g ^ ( x - l ) = -2 log2X = l o g 0_ l()g2 X = log2 2 (1) B a i t o a n 93 ( D y b i D H - 0 D ) G i a i D i e u kien - < x 7^ K h i Gidi phuang trinh log3(3^-l).log3(3-+'-3)=G (1) <^ log; |x - 1| + log3(2x - 1) = < ^ l o g | x - l | ( x - 1) = ^ "=4X = ^ ^ | x - l | ( x - 1) = G i a i Dieu kien { '"^r^i i > Q <^ 3-'' > ^ x > K l i i ^ ^ x> (x - l)(2x - 1) = ( l ) ^ l o g ( - - l ) ( l + log3(3-^-l)]=6 - < x < ( x - l ) ( x - 1) = -3 Dat w = log3 (3^ - 1), thay vao (2) ta ditdo 2x2 - 3x - = •hi X > v.{l + u) = G - < x < 2x2 _ 3x + = (v6 nghiem) r r X > 1' I 2x2-3x - = B a i t o a n 91 (Du* b i D H - 0 A ) log2 X log., 2x Gidi phuang < ^ X = (thoa d k ) 1 l0g2 X 378 + 1+ log3 (3'- - 1) = log3 (r trinh (1) r X > I ^ ^ ^ ^ ^ l [ i , x ^ 2- logy v/2x + u - G = <^ ^ = w = -3 Vay x = X = -0,5 X > log^2 + log2^4 = l o g ^ f < X ^ G i a i Di6u kien <^ < r ^ ^ i < 72x7^ u2 - l) = 3^ - = -3 3-'' = 10 x = log3^ ^ - = log3 27 "27 S t u, (Thoa man dieu kien x > 0) K h i l0g2 X -• + l0g2 X B a i t o a n 94 ( D e D H - 0 A ) (2) Gidi phuang trinh l o g - i ( x + x - 1) ^j^g^^^ ( r - 1)2 = 379 , (27) G i a i Dieu kien B a i t o a n 96 ( D H - 0 A ) Gidi hat phuang < 2x - 7^ •1: _ 2x^+x-l>0 log3 (4x - 3) + l o g i (2x + 3) < S! / < 2x < X + 'ty Taco < 2.x - ^ < + ^\) > G i a i Dieu kien { 2x + > • ^ log2x-i <^log3 + log2^_i (2x - 1) + 21og^+i (2x - 1) = + <^ log2x-i (x + 1) + log^+i (2x - 1) = : (2) (4x - 3)^ 2x + < log3 3^ ^ Khid6 r r + = 0<=> u= u = • logg (4x - 3)^ - logg (2x + 3) < (4x-3)^ 2x + <^x>- < (4x - 3)^ < (2x + 3) <^16x'' - 24x + < 18x + 21^ 16x^ - 42x - 18 < •t^Sx^ - 21x - < <^ - - < X < "et hdp dieu k i f n dUdc tglp nghiem ciia bat phudng t r i n h la = Dat u = log2a;-i (a; + 1), thay vao (2) t a dUdc u + - = 3-!^u''-3u u 43 ^ logg (4x - 3) + log (2x + 3) < f (l)<:^log2,_i(x + l ) ( x - l ) + l o g , + i ( ; - l ) ' = -i; trinh B a i t o a n 97 ( D H - 0 B ) Gidi hat phuang -; trinh log,(log3(9 72))<l Do log2x-l ( ^ + 1) = log2x-i (a; + 1) = ^ ^ = 2x - x + l = (2x-l)2 X + ^ G i a i Dieu kien log2x-i + 1) = log2^-i (2:f - 1) l0g2x-l ( X + I ) = l g , _ i ( x - ) ^^2 >^ r x = f 9^-72>0 { I o g ( ^ - ) > = log3l < X 7^ I X = X = 5.4-1 f 9^ - 72 > r ^ 7"^ Vay phudng t r i n h da cho c6 tap nghiem la B a i t o a n 95 ( D H - 0 D ) Giai bat phuang trinh G i a i Dieu kien x 7^ va logi ( x2 - x + ^2 - 3x + - 3x + {I x2 - x + X logi x^ - x + X K h i >0 9^ - 72 < 3^ > K h i - u - 72 < <^ - < u < f x^ - 4x + > € (-oc;0)U [2- ^ ; G (0;l)U(2;+oo) 9^ - 3^ - 72 < at u - 3^, vdi dieu kien u > T h a y vao tren t a dildc > logi < ( l ) ^ l o g ( ^ - ) < x = log3 3^ x^ - 3x + +v/2 < 0 < 3* < > •i^x& > i.ij.ihi .et hdp v d i dieu kien u > 0, t a dUdc < u < Do 3"^ < 3^ X < K c t hdp vdi dieu kien t a dUdc tap nghiem cua (1) la = (logg 73; 2] - ^ ; 1) U (2; + v/2 Tap nghiem ciia bat phudng t r i n h la [2 - \/2; l ) U (2; + v/2 B a i t o a n 98 ( D e dv^ h i D H - 0 A ) Gidi bat phuang l o g i (4^ + 4) > l o g i (2^^+^ - 3.2^) 381 trinh : • r » (28) G i a i Ta CO { ^ 4- + < - 3.2- u v ; 4^ + > 22X+1 _ 2x > 4-+4<22-+i-3.2- +4< Do kot hdp vdi diou kion < x 7^ suy r a tap nghiem cua bat phUdng trinh / ' ( x ) < a = ( ; e j \ { l } - 3.2 ^ B a i t o a n 101 ( D e d\i b i D H - 0 A ) loga <»22^ + < 2.2^'^ - - ^ " ^ - - - > Dat ?i = 2- > T h a y vao t i c u t a diWc ' { 5^ j< Ket hdp v d i dieu kien u > 0, ta diTdc u > D o ' 2- > loga 2- > 2^ <^ X > B a i t o a n 99 ( D e du" b i D H - 0 B ) '' i ^ • ' ) Gidi bat phuang trinh log2 ( x + \/2.T2 - T ) 2x2-x>0 X + \/2.T2-X>0 log2 f x + \/2x2 - X j > < Ta CO " *' Gidi bat phuang log2 (^x + \/2.T2 - x^ < logi ^ log2 ( x + \/2x2 - x ) > = log2 Vay tap iighigiu ciia bat phirdiig t r i n h da cho la [2; + o o ) r G i a i D i c u kien ( - o o ; - ] U [4; + o o ) u^-3u-4>0^ue i.?.; ,i, <^ log2 X < log2 e 4* X < e « - x + \/2x2 - X > <f=J> \/2x2 - X > - X trinh X > \e ( - o o ; ] U [ i ; + o o ) f X < \G ( - o o ; - ) U ( l ; + o o ) 2-x <0 2x2 - X > - x > (T 2x2 - X > - 4x + 3,2 logix + 21og.(x-l) + log,6<0 / G i a i D i c u kiCii r > T a c6 (-r — 1) ^^ Jr x r f6i < n l o g i x + l o-.o-, g i (x + ilog2 <r , xe(-o^"-4)U(l;2] ^x€(-oc;-4)U(l;+oo) log X + log ( x - 1) - logi < Tap nghiem cua bat phUdng trinh da cho la (-00; - ) U ( ; +00) log [x (x - 1)] < l o g i B a i t o a n 102 ( D e t h i D H - 0 B ) ox^ - x - > < ^ x e X (x - 1) > ( - o o ; - ] U [3; + o o ) B a i t o a n 100 ( D y b i D H - 0 D ) Cho ham so G i a i B a t phudng trinh c6 tap xac djnh la E T a c6 logs (4^ + 144) - logs < + log^ ( - + l ) ^ logs (4" + 144) - logs 16 < logs + logs ( " " ' + l ) /(x) = x l o g ^ (x > 0, X 7^ ) ' - + 144 ^log5 ( 16 Hay tinh f'{x) vd gidi bat phuang trinh f'{x) < < logs [5 {2^-' + 1)] ^ 2- G i a i Vai < X 7^ , t a C O / ( x ) = r - ^ logax trinh log5 (4- + 144) - logs < + log5 ( - ^ + l ) K e t hdp v d i dieu kien suy r a tap nghiem ciia b a t phudng t r i n h da cho la S'=[3;+oo) ' Gidi bat phuang Do ^ - + 144 < 80 + 1 < a m-/ < ( - + 1) ' • • iBOH i>'- j - + 144 < 20.2^ + 80 ^ (2-)^ - 20.2- + 64 < (log,x)^ - (log,x)^ • Vay -Hi t Dat u = 2- > 0, thay vao t a dutdc - 40u + 64 < <(=!> < ?/< 16 mil i'^iz (oc Bdi vay /'(a;) < ^°g2^ ^°g2e < ^ log2X - log2e < (log2 X) < 2- < 16 22 < 2- < 2^ <^ Bat phUdng trinh c6 tap nghiem la khoang (2; 4) 383 < X < ' ,I (29) B a i t o a n 103 ( D e dvt b i D H - 0 A ) Gidi bat phuang trinh G i a i Dieu kien x > T a c6 ,f,»»;-| /j«,|fjft^ ' f f q j f * Ta CO ham so g{x) nghich bien tren khoang (Ipg^ ; + o o ) , h a m so / ( x ) dong bien t r e n (log5 4;+oo) v i \ f O | wivfij B6i vay x = la nghiom nhat ciia phitdng t r i n h da cho C a c h T a C O ^log2x'°^2xi > iog2 (^2-^2'°^2xiJ^ ^^^^ ^ logs (5" - 4) = - x ^ logs (5^ - 4) = logs 5^-^ ^ 4^5^ _ = 5^-^ ^ 5^ - = 5.5-^ log2 X log2 X > - + log2 X ' "'^ Dat u = 5"^ > 0, thay vao t a dUdc ^°g2 ^- log2 X > -1 + - log2 X - 4= - M Dat u = log2 X , - 4?/ - = <^ li = (do V > 0) tluiy vao t a diWc s^*^'^ y - y u <1 u :s + l > < ^ U ^ - u + 2>0<!=> Do (16 10g2X<l log2X>2 l0g2X<l0g2 ^ log2 X > log2 ^ X X < Kot hdp v i dion kiou siiy r a t a p nghiom ci'ia b a t phirong t r i n h d a cho la S' = ( ; ] U [ ; + o o ) Vay 5^ = < ^ X = D o X = la nghiem nhat cua phitdng t r i n h D a n g P h u ' d n g t r i n h , b a t phifdng t r i n h logarit chdra t h a m s6 PhiTdng p h a p Phitdng t r i n h /(x) = m c6 n g h i f m k h i va chi k h i m thuQc tap gia t r i ciia ham so / ( x ) B a i t o a n 105 ( D e D H - 0 A ) C/io p/iu:(?n5 irm/i : log|x + \J\oglx + - 2m - = ( m /d tham s6) iii ' D a n g Phu'dng p h a p h a m so Phtfdng p h a p (tifdng t\i d a n g d t r a n g ) • N h a m nghieni x = a cua phudng t r i n l i • C h i i n g m i n h phudng t r i n h c6 nghiom nhat bang each xet h a i trirdng hdp X > a vk X < a a.) Gidi phuang trinh m = b) Tim m de phuang tnnh dd cho c6 it nhat mot nghiem ^ ^ , G i a i Dieu kien x > a) K h i m = 2, phudng t r i n h da cho trcl t h a n h C h u y 12 Sii dung cdc cong thiic sau : Cho cdc so duong x vd y Khi • Neu a> I thi log^, x > log„ y -i^ x > y • Neu < a < thi log„ x > log„ y <^ x <y B a i t o a n 104 ( D y b i D H - 0 D ) ' ' ' - Gidi phuang log5 (5^ - 4) = - X It = u + Vu + l - = <^ Vu + = - u ( u<5 r 5-u>o 25 - lOu + 1/2 ^ \ - \\u I C a c h D6 thay x = l a nghiom ciia plntdng t r i n h d a cho Tron khoang /(x) = ftOl'Mtelic!' vdi dieu kien u > 0, thay vao t a diTdc log3 X , G i a i Dieu kien ^ - > < i ^ ^ > < » x > logj (log5 4; + o o ) , x e t hai ham so / YifB l o g i x + \ / b g i ^ + T - = Dat trinh : thuoc doan ; ' ^ 1= u <5 <=> u = n = u = M+ + 24 = Vay logl X = 3, hay log5(5^-4);5(x)=l-x log3X = log-^x^-VS 384 \/3 ^ l o g X = log3 ^ [ l o g X = 385 log3 -V3 x = 3^ .f x =3-A '(:• : (30) K h i m = 2, phirong t r i u h da cho c6 tap nghiem la = { ^ ^ ' 5.7 3^}' T a thvrdng d u n g c a c phifdng p h a p s a u : • PhUdng phap the: r u t x theo y hohc y theo x, thay vao phUdng t r i n h b ) D a t u = logi X, v6i d i c u kicn u > 0, thay vao t a dUdc r'.ti Ta TV:, u + y/u+1 - m - l = « > u + v/uTT - = m 1;3 <^ < a; < ^ <^ logg < loga x < logg lai i • Bion dm hoac dat an p h u dfi dita ve cac ho da bict each giai nhit he doi x i i n g loai 1, he doi x i l n g loai 2, CO X G B a i t o a n 107 ( D H - 0 D ) 4" + k h i va chi k h i u G [0; 3] Do plnldng t r i n h da cho c6 ft 1; ^ nhat mot nghiem thupc 1;3'*^ G i a i T a c6 (2) o = 2m CO I t nhat m o t nghiem u G [0; 3], tiJc la m thupc tap gia t r i cua ham so lien > , V « G ; J ; /(O) = 0, /(3) = Vay m a x / ( u ) = 4, m i n / ( u ) = 0, nghia la tap gid t r i cua h a m so f{u) [0;3] [0;3] tren Khi Khi Khi Ket y = t a y = L ta y = 4, t a luan: He = y^y^2\y y - 2- vao (1) t a ditoc "^^^^ + j / ^ - 52/2 + tuc f{u) = u + v/iTTT - 1, Vu G [0; 3] T a c6 2vAi+T 2-+' k h i va chi k h i phitong t r i n h u + Vu + -1 , ] j Gidi he phrxdng trinh l o g | X < < logsx <V3^0< Vay X e H e m u v a logarit i = ^ y{y^ -r^y + A)=Q<^ y = ] dudc 2^ = 0, phUdng t r i n h vo nghiem _^ dUdc 2^ = ^ 2^ = 2" <^ x = ' " dUdc 2^^ = ^ ^ = 2 < ^ x = : ^ ( x = phUdng t r i n h c6 hai nghi?m la | J ~ va | J ^ 0; 3] la [0; 4] Do <16 plurong t r i n h da cho co i t nhat mot iighicm thupc doan , k h i va chi k h i 2m B a i t o a n 108 ( D H - 0 A ) G [0; 4] < 2m, < < m < B a i t o a n 106 (Dxl hi D H - 0 B ) Tim m dephiidng thuoc khoang (0; 1) : trinh sau c6 nghiem ^ - log X + m = (1) G i a i Dieu kien x > D a t u = logax Thay vao phitOng t r i n h da cho t a dupe u^ + u + m ^ O (*) X e {0;1) ^ < X < 1^ log2 /(w) = u2+w,Vii, G (-00; +00 c6 nghieni - x) - log4 - = (1) (2) ^ " cl6 ^{y>0 ^ - l o g (y - x) - log4 ^ 1 ^ ( y - x ) - = -<^y-x = log4 ^ y = - ^ x log4 = = log4 -1 3y - Thay vao (2), t a dUdc f V / + y ^ 25 ^ 9y2 + 16y2 = 25.16 ^ 25y2 = 25.16 ^ f{u) tren (-CXD; 0) B a i t o a n 109 ( D H - 0 B ) I Do m < T y 386 [ "j = t^ K g t hdp vai dieu kien t a dudc y = 4, k h i x = 3, thoa m a n dieu kien Vay = (3 4) la nghiem nhat ciia he phUdng t r i n h 0) Bang each khao sat h a m so t a dUdc tap gia t r j ciia ham so ° l o g i (y x < loga <i=^ l o g x < nen phudng t r i n h da cho C O nghiem khoang (0; 1) k h i va chi k h i phUdng t r i n h (*) am, tulc la - m thupc tap gia t r i ciia ham so lien tuc la G i a i D i e u kien { ^ ^ ^ f \^ y [ x^ + y^ = 25 • ( l o g v/x) Yi , Gim/if Giai he 387 | {^l^t^f^'y^^^ (2) (31) Giai Dieu kien | o ^ y < ^° 3x logg — = loggS (2) <=> 31og3(3x) - Sloggy = y 3x — = 3>i> y = x y Thay vao (1) ta ditdc , x = 1, X = «>1 + 2yJ{x-l){2-x) = ^ v/(7^^T)(2-^ = <!=J> V^^^ + = <^ X - + ^ ( ^ " ^ ^ ( ^ ^ + - X = Khi X = 1, ta CO 2/ = Khi x = 2, ta c6 y = Vay h§ phirong trinh c6 hai nghieinla(x;?y) = ( l ; l ) , (x;y) = (2;2) Bai toan 110 ( D y bi DH-2002B) Giai he ( x-4M+3 = ?og, y > ^ { y 1 Khi ta c6 Khi y = 1, ta CO X = 1, thoa dieu kien Khi y = 3, ta ditdc x = 9, thoa dieu kien Vay he phitdng trinh c6 hai nghiem la (x; y) = (1; 1) va (x; y) = (9; 3) Bai toan 111 (DH-2010D-phan rieng Nang cao) Giai hephUdng trinh / x^ - 4x + y + = { o g ( x - ) - l o g ^ y = (-^.yeR) Giai Dieu kien x > va y > He da cho viet lai x2 - x + y + = ^ f x2 - x + y + = I o g ( x - ) = l o g y ^ \2 = y f x= { x ^ ~ ' ; ' o ^ { f - = 0(loai) 2/ = lx= { Bai toan 112 (DH-2010B-phan rieng Nang cao) Gidi he phKcfng trinh ( log2(3y - 1) = 388 ^' y e m 1) = + 2^ = y _ 2^" + ^ + 2^ = y 2^ + y y= 2^ + X ^|3y-l==2^ 4^ + 2^ = y 2^ + y" = [ ( ^ + 2^) = (2^ + 1)2 2^ + (2^ + l ) ( ^ - - ) =d'1/''''<! •• x=-l (thoa man dieu kien) if (2) -y/iogTx = v/log2 J/ <^ log4 X = log2 y <^ log4 X = log4 y"^ <^x = Thay vao (1) ta dildc \^ = 3y2 log2(3y 4^ (1) \- v/iog2y = o (2) Giai Dieu kien { f„g> | Giai Dieu kien y > - va x y He da cho titdng dttdug O Bai toan 113 (Dij' bi DH-2002D) Gidi h$ phiCdng trinh r log^ (x3 + 2x2 - 3x - 5y) = logy (y3 + 2y2 - 3y - 5x) = \ \; f < X ^ 1, < y ^ Giai Dieu kien { x^ + 2x2 - 3x - 5y > j^^i [ y3 + y - y - x > ^ I log,(y3 + y - y - x ) = l o g , y ' ^ Iy3xf+ +2y22xf- -3y3x- -5x5y= =y^*x^ I x - x - y - (1) ^ \x = (2) Lay phUdng trinh (1) trii phUdng trinh (2) theo ve, ta dUdc 2(x2 - y ) - 3x + 3y - y + 5x = <^ 2(x - y)(x + y) + 2(x - y) = 4=^2(x-y)(x + y + l ) = ^ { x = - - y ^, Khi X — y, thay vao he (*) ta dildc r 2x2 - 3x - 5x = 2_ Q ^ r X = I 2x2 - 3x - 5x = ^ ^ 4x u 1^ ^ ^ Khi X = —1 - y, thay vao (2) ta dudc 2y2 - 3y + (y + 1) - O 2y2 + 2y + = (v6 nghiem) Ket hdp vai dieu kien suy (x; y) = (4; 4) la nghifm nhat ciia he 389 (32) Bai toan 114 (Dtf bi D H - 0 A ) Gidi he phuang trinh i - r, Bai toan 115 (Uxi bi D H - 0 A ) Gidi h$ / X + v/^!Z2i±2 = 3!'-i + r (1) Giai Dieu kien { Q J J ^ Giai Dicu kiGn x G R va iy € R Dat j/ = r - 1, t; = ?y - Khi (16 (1) viet, lai { + s/u^ + - (i; + v^7;2 + l ) = 3" - 3" v/x^TT + x v/;?:rT v/;r2:n: + \/j'2 > |x|4-x \/X^TI > + (3) 3" - 3" < Khi u = -2, + i ) = logy X = -2 = - " ( u + v/u2 + 1) V logy X = logy 7^"^ O X = J / ~ ^ Thay vao (2) ta dUdc I" (5) u ^ Vu^ + / - I n < 0, Vu G 2!^ 2i^ +2^ = 3^ = (2^>2^ r2^2 I 21^ > 2° \^ ^2^^2V>3 > ys; Do y > khong thoa man (3) Neu < y < tin \ >1 • , ^^^.^ , «(^\^^°, -ir-^^', I > 2' 2^! > la u = => I' = Bdi \-ay he da cho titdng dudiig vdi / x - l = y - l = ^ ( x = l ^ { y ^ l 390 , : -2*+2'>3 Do < ?/ < khong thoa man (3) q., Bcii vay (3) vo nghiom Do (x; y) = ^log2 ^; logg Vay ham so Ii nghich bien tren R, do phUdng trinh (5) chi c6 mot nghieni (3) Neu y > thi De thay u = thoa (5> Xet ham h{u) = 3-"(u + Vu^ + l ) , V u G R Ta c6 h'{u) = - - " ln3(u + \/u2 + 1) + 3-" ( + x = y Thay vao (2) ta dvtdc ta c6 Vi (3) va (4) mau thnan v5i (2) nen khong the c6 u > t; • Tudng ti.r, cung khong the co u < • Vay t-lii kha nSng u = v Khi thay vao u - s/u^ + = 3^ ta ditdc w = u = -2 2^-'+ (4) + (*) 3 2^ + 2^ = <=J- 2^ = - <^ X = log2 - (thoa man dicu kiOn) > Trong g{u) > g{v) > 0, hay g{v) - g{u) < 0, nghia la u + \/u2 + ^ 3« ^ 3-« s: , Khi n = 1, ta c6 log^ x — ^ logy x = logy X + v/"2 + - [logy x + log„ y) = log^ y , (2) Vay ham so / dong bien tren R De thay ham so ^(x) = 3^ dong bien tren M • Xet u > V Khi f{u) > f{v), hay f{u) - / ( r ) > 0, nghia la 1/ « ,;, Dat u = logy X , thay vao (*), ta dirdc Xet ham so /(x) = x + v/x2 + l , V x G R Ta co /'(x) = l + , [2) i (logy X + 1) = log^ 1/ o logy X + = log^ J/ ^' Lay hai phirong trinh trir ta ditdc u ° ^ ^ ^ ^ ° ^ " ^ ^ ^ i (2) ^ i logj^(xy) = log^ y^^ = 3" j; + s/v^^ | la nghiem nhat cua he Bai toan 116 (Du" bi D H - 0 D ) Gidi he { x^ + y = y^ + x 2^+y_2^-i = x _ y (1) (2) : iiKL ,M>^<r:! (33) G i a i H f phudng trinh xac dinh v6i moi x va y Ta c6 trail ntiiQ, iim'' • < ^ ( x - y ) ( x + y - l ) = 0<t:»f^ = f _ Khi X = y, thay vao (2) ta dUdc 22x _ 2X-1 ^ 2^'= = 2"^-^ 2x = X - <^ X = -1 Khi X = - J/, thay vao (2) ta dUdc 2* - - ' ' - = - 1/ - y 2-2' = + 2y y = Ket luan: He c6 hai nghiem la ( - ; - ) va (1; 0) B a i t o a n 117 ( D e D H - 0 A - N a n g c a o ) Giai he phuang trinh f log2(x2 + y2) = + log2(a;y) \^^-xJ/+y' = • /(x) < m,\/x e D 4=i> (C) nfim hoan toan phia ditdi d • f{x} < m,yx e D (C) khong c6 diem phia tren d • fix) > in, Vx e D (C) nam hoan toan phia tren d • fix) > m,Wx e D (C) khong c6 diem phia dudi d • Bat phudng trinh fix) < m c6 nghiem x G D va chi 3x diem M (x; fix)) nhn phia diTdi d • Bat phUdng trinh fix) > m c6 nghiem x G va chi 3x di^m M (x; fix)) n&m phia tren d • Bat phUdng trinh / ( x ) < m c6 nghiem x G D va chi 3x diom M (x; fix)) nkm phia dudi d hoac nam tren d • Bat phUdng trinh fix) > m c6 nghiem x G Z? va chi 3x digm M (.T; fix)) nhn phia tren d hoac nam tren d -xy - xy + y = ^ \2 - x2 + x2 = + y2 = \ - xy + y' = ! ' nun/(x) > m (neu i i i i n / ( x ) ton tai) «• B a i t o a n 118 ( D H - 0 A ) Tim m diphUdng 3s/x - + my/xTl ih — - Vay nghiem ciia he la (x; y) = (2; 2), (x; y) = ( - ; - ) X Dat t = Phufdng phap dung dao h a m Ta thitfJng dung phUdng phap doi vdi cac bai toan plnMng trinh, he phifdng trinh chi'ra tham so va yeu cau bai toan la tim cac gia t r i cua tham so m de phUdng trinh (he phUdng trinh, bat phUdng trinh) da cho c6 nghiem thoa man dicu kion nao Cach lam thucing la lap bang bicn thion ciia ham so, de tijf nhin thi va tra Idi cac cau hoi cua de bai Sau day la mot so luu y them giai toan Goi (C) la thi ciia ham so y = /(x) tren tap xac dinh D va la dUcJng (doan) tliang y = m tren tap xac dinh D ciia ham so / Khi : trinh sau c6 nghiem thitc : = 2\/x2 - /\ nmi t^X <v ^ : G i a i Dicu kien x > Chia hai ve cho v/x+T > ta difdc phifdng trinh tUdng difdiig: X - 392 G D dg • PhUdng trinh fix) = m cd nghiem x G Z) va chi m thuoc tap gia tri ciia ham so / tren D • fix) < m,yx e D <^ max fix) < m (neu m^x/(x) ton tai) \y = (thoa dicu kien) 5.8 G D de (neu m i n / ( x ) t6n tai) • fix) > m , Vx G D X € D de • Bat phUdng trinh fix) < m c6 nghiem x G D va chi i n i n / ( x ) < m (neu m ^ / ( x ) ton tai) G i a i Dieu kien xy > He da cho dudc viet lai \ e Z? de • Bat phifdng trinh / ( x ) > m c6 nghiem x G Z) va chi nmx/(x) > m 81 log2(x2 + y2) = log2(2xy) , '' ' _ , x + = + ^ / X - l /^N + m = 2{/—— VX + — t a x+ (1) cd < t < Phifdng trinh (1) trd 3/,2 + m = 21 PliUdiig trinh da cho cd nghiem va chi phifdng trinh (2) cd nghiem fit) ' = -3(2 + 2t, vdi ( G [0; 1) Ta cd T + f(t) t thoa man < ( < Xet ham so fit) (2) m = -3(2 + 2t \ ^ = - ( + 2, ^lim /(() = - -1 393 ,, (34) Tfr bang bien thien t a c6 k i t qua - < m < B a i t o a n 119 m., phuang B a i t o a n 121 ( D H - 1 D ) - ( D H - 0 B ) Chring minh vdi moi gid tri duang ciia tham trinh sau c6 hai nghiem thUc phdn biet : x^ + 2x - = y/m.{x G i a i Dion kion x > K h i birih phitrtng hai v6 t a c6 : - r x - ( y + 2)x2 + xy = m X - y = - 2m so G i a i HO da cho vi6t lai { (x - 2f{x + 4)^ = m ( x - 2) x = <t*(x - 2)[(x - 2)(x + 4)2 - m] = ( x - ) ( x + 4)2 = m (*) (x2 + 2x- Xet ham so f{x) I ^j^J^" ^ = i _ 2m \ = m ( x - 2) 8f { u = x2 -x= '•| = (x - 2)(x + 4)^, vdi x > Ta c6 : He da cho t r d t,,<i>Af ( f(x) n o 'jrif - (J ( \%iit • r - x)r^ ^ ^ = (1 - 2^1) - " I uir= m /'(.V) = Chuing to vdi m > t h i (weK)-r + v = l-2m (u / ' ( x ) = 3x2 + x > , V x > (diSu kion w > (^x-^y i u=:2x-y / ' ( x ) = (x + 4)2 + 2(x + ) ( x - ) liin^ / ( x ) (^'^R) \ 2) ' *i Tim m de he phuang trinh sau c6 nghi$m / ~ ) ,j, ^ \ ^" ~ 27n) - u] =m I " ^ f t; = l - m - u (1) ' (*) luon C O diiiig nghieiu x > 2, tiic la ^•1 phuong t r i n h da cho luon c6 diing nghiem Tim cdc gid tri cua tham so m de phUdng B a i t o a n 120 ( D H - 0 A ) trinh sau c6 dung nghiem Dat phdn biet + / 1 V v/2x ^/6~~^ m • f'{x) = 0^ '2V2i + \/2x X - 2y/r^ ^ r(x) 2S + 2^ ' u = B a i t o a n 122 -1 + Dau ciia / ( x ) la dau ci'ia - — = — T i t bang bien thien ciia / ( x ) suy V2x x l a phitdng t r i n h c6 dung nghiem k h i va chi k h i 2\/6 + v ^ < m < 3v/2 + - ^ ^ ^ (loai) — ^nhan^ (2) c6 nghiem u G \ 2-v/3 ;+oo hay m < —-— Tim m de he sau cd nghiem : < ») Hi n ' V2 3x2 - m x v / i + 16 = 2^ G i a i Ta c6 (*)^ 394 • - - v/3 ^ / ^ 3V2 + 6-~_ - u - 2u + Vay he c6 nghiem 1 " f(u) / X 4- 2u + l ' u — 2^2x{G-x) fl[x) <=>2x = - x<=>x = \/6 X + \ / ~ ^ K h i ham 1 Taco = (2u+l)2 v^+\/2^ + 2^/6^ + 2\/6^=:m G i a i X e t ham so f{x) = ^ + \/2^ + ^ / ^ xac diiih va lien tuc tren [0; 6] Ta c6 /(7/.) x < - + 5x 3x2 _ + 16 = rxG[l;4] \3x2-mxv^+16 = 395 f xG[l;4 3x^ + 16 X\JX (35) Xet ham so fix) = 3.T^ + 16 , v6i a; e [1;4] Ta c6 : -' • = = '^tB S 0, Vx e [1; 4) Nhir vay ham so / nghich bien tren [1; 4], do h? c6 nghiem va chi /(4) < m < / ( I ) < m < 19 ; Bai toan 123 Xdc dinh m di he sau c6 nghiem phdn biet l o g ^ ( r + 1) - l o g ^ ( T - 1) > log34 log2(x2 - 2x + 5) + mlog(^2_2^+5)2 = (1) (2) Giai Dieu kien : x > Tft (1) ta c6 l ° g v / ^ > logv/32<^ ^ > 2<^ ^ < ^ x e (1;3) (3) 2x-2 > 0, Vx > 1, suy (x2 - x + 5)ln2 ham so t{x) dong bien tren (1; +oo), do vdi < x < thi < t < va (2) trd Dat / = log2(x2 - 2x + 5) Ta c6 t'{x) = « + y = => - 5( = - m (4) rlj.il Tu; each dat t ta c6: Vdi moi < € (2; 3) thi cho ta dung mot gia t r i x G (1; 3) Suy he c6 nghiem phan biet va chi (4) c6 nghiem phan biet t e (2; 3) Xet ham so hen tuc f{t) = f - 5t vdi t f(t) f(t) t-.3- bien thien ta c6, he phifdng trinh c6 nghifm 25 25 phan biet <^ - — < - m < - 6 < m < — 4 396 2 + ^2 - - - ^ (36)