Phần 1 tài liệu Tuyển chọn 500 bài tập Toán 10 giới thiệu các bài tập theo chuyên đề thuộc phần Đại số bao gồm: Tập hợp và hàm số, phương trình và bất phương trình bậc nhất một ẩn, hệ phương trình bậc nhất hai ẩn, bất đẳng thức,... Mời các bạn cùng tham khảo nội dung chi tiết.
P=^7 MAU THONG - LEMAU THAO n^uyinchgn 500 bai t$p TOAlV 10 THLf VIEW TJNH biNH THUAN NHA XUAT BAN HA NQI LCfl N O I MlfC i f f C DAII Phan I 500 Bam b a i t o a n loTp 10 dUcrc s o a n t h e o c a c y e u cau : s d t s a c h giao k h o a h i e n h ^ n h SO Chuyen de Tap hop va h a m so 05 Chuyen d l Phuong t r i n h v^ bat phucfng t r i n h bac n h a t mQt an 30 Cdc k i e n thufc ca hAn difcfc t a p t r u n g v a o tifng c h u y e n de Chuyen d§ He phuong t r i n h bac n h a t h a i a n 37 T r o n g m o i chuyen de, chiing toi t o m li/cfc p h a n h' t h u y e t , chi r a c&ch Chuyen de B a t d i n g thufc 43 van d u n g l i t h u y e t v a o cAc b a i t a p tis d& d e n k h o , n a n g cao d a n theo Chuyen de N h i thiic bac n h a t : f(x) = ax + b 63 Chuyen de Phifong t r i n h b§c h a i mot an 66 Chuyen de He phuong t r i n h bac h a i doi xufng doi v d i x va y 78 Chuyen de He phuong t r i n h d i n g cap bac hai 84 Chuyen de Dau ciia t a m thiic bac h a i 89 Chuyen de 10 B a t phiTOng t r i n h bac h a i 98 htfdng tiep can miJc c a c de t h i t u y e n sinh D a i hoc, t r i c h d a n v a g i a i m o t so de t h i t u y e n sinh D a i hoc c a c n a m 2002, 2003, 2004 D a c b i ^ t , trUdc m o i b a i g i a i , c h i i n g toi d u a p h a n hiTdng dSn, phan n a y h e t siJc q u a n t r o n g , no giiip b a n doc t i m duoc h i i d n g g i a i quyet b ^ i t o a n , torn t ^ t qua t r i n h g i a i , de k h i g i a i xong b a n c h i c a n n^m difoc c a c y c h i n h la dii Sdch dugc chia lam hai phan : Chuyen de 11 Xac d i n h gia t r i ciia t h a m so m de t a m thilc bac h a i CO dau k h o n g t h a y d6i P h a n : D a i so (300 bai) P h a n : H i n h h o c (200 bki) Sau m o i p h a n , c h i i n g toi c6 p h a n t o a n t d n g h o p n h ^ m giiip 'hoc s i n h on l a i c a c k i e n thiJc ccf b a n , t a p t r a I d i c a c c a u h o i tr^c n g h i e m 109 Chuyen de 12 So sanh so a v d i h a i n g h i e m ciia phuong t r i n h bac h a i 113 Chuyen de 13 Phirorng t r i n h va bat phuong t r i n h c6 chiJa gia t r i tuyet doi 129 Chuyen de 14 Phifcfng t r i n h v^ bat phucrng t r i n h c6 chufa cSn thufc 137 gid v a k i e m t r a l a i c a c k i e n thiJc d a h o c t r o n g m o i h o c k i Chuyen de 15 T d n g hop cudi n a m 153 M6i de kiem ^h^n D&t bi^t : Cuoi s a c h c6 10 de k i e m t r a n h k m giiip hoc s i n h t\i d a n h tra c6 : Cau h o i l i t h u y e t Cau h o i trftc n g h i e m Toan t u l u a n M a c dii d a CO g S n g n h i e u k h i b i e n s o a n , n h i t o g c h ^ c k h o n g t r a n h k h o i t h i e u sot, c h i i n g toi m o n g n h a n dufoc sU d o n g gop y k i e n x a y diTng ciia quy d o n g n g h i e p , c a c e m hoc s i n h de I a n t a i b a n s a u , s a c h se h o a n c h i n h h o n HiNMHOC Chuyen de Vecto • Chuyen de True - Toa dp t r e n true 188 Chuyen de He true toa dp Descartes vudng goc 194 Chuyen de T i so luong giac 209 Chuyen de Tich v6 hudng ciia h a i vectP 222 Chuyen de He thiic lupng t r o n g t a m giac 235 Chuyen de G i a i t a m giac - f n g dung thifc te 244 Chuyen de He thiJc lUcJng t r o n g di/dng t r o n 250 Chuyen de Tdng hop cudi nSm 256 DQ De k i e m t r a Hoc k i I kiem tra 172 290 290 De k i e m t r a Hoc k i I I va cuoi n a m T6P HOP V6 HflM ci.H.y6n so K i e n thurc ca ban I) Tap xac djnh c u a ham s o y = f(x) • Tap xac d i n h ciia h a m so' y = f(x) la : • Cach t i m tap xac d i n h ciia h a m so'y = f(x) BU hay y i > y : H a m so nghich bien t r e n k h o a n g (a; b) ni) T i n h c h i n 1^ c u a mQt h a m so Gia sii h a m so y = f(x) c6 tap xac d i n h D thoa m a n d i l u k i e n Vx, x e D • D$c bi^t a) Neu b = t h i y = ax va t h i 1^ dudng t h i n g d i qua goo tpa =>-xeD y (Ta bao D la tap doi x i i n g qua O) H a m so y = f i x ) chSn t r e n D Dinh nghIa Dinh nghia f(- X) = f(x), V X e D O f ( - X) = - f i x ) , T i n h chat Do \ H a m so' y = f(x) l e ' t r e n D V X e D a < T i n h chat t h i ham so chSn n h a n t u n g l a m true doi xufng true [b = D t h i h a m so' le n h a n go'e to a l a m t a m doi xiifng Neu b) a = t h i y = b la h a m so' h k n g v a t h i la dudng t h i n g eung phLfOng v d i true h o a n h Ox (Ta bao h a m so' h i n g c6 t h i l a dudng y t h i n g n i m ngang) y - X ->x O b y = b -y s IV) Ham s o b a c n h a t : y = ax + b (1) V) Dinh nghia H a m so' bae n h a t la h a m so c6 dang : y = ax + b (a ?^ 0) • Tap xAc d i n h : D = R • T i n h chat : • y = ax + b -00 — " a < a > -00 +00 X - 00 + GO + y = ax + b y \ y y \ 00 ~0 a < y c - 00 - 00 + 00 - 00 H a m so (1) : H ^ m so (1) : • Dong bie'n t r e n k h o a n g (0; + » ) • D n g bie'n t r e n k h o a n g (- oc; 0) • Nghieh bie'n t r e n k h o a n g (- oo; 0) • N g h i e h bie'n t r e n k h o a n g (0; + • a > X 00 +00 ^ Do thi : D6 t h i h a m so' bae n h a t la mot dudng t h i n g o (1) B a n g bie'n t h i e n a < +00 / I H a m so (1) la h a m so^chin =>D6 t h i h a m so (1) eo true doi xiJng la Oy => H a m so' (1) nghieh bie'n t r e n R x (1) Tap xac d i n h D = R a < + 00 {a^O) • • y = a x % c (a ^ 0) + bx + c Dang => H a m so (1) d6ng bie'n t r e n R -00 y = ax 1) a > a > X (1) Hamsobachai: D t h i la mot parabol c6 : ->x D i n h S(0 ; c) - True doi xiJng 1^ Oy B a i a) Cho t^ip A c6 n phan tiif (n S: 1) Chtfng minh A c6 2" t^p b) Cho A = |1, 2, 3, 4, 5, 6| Gpi B la tgp ciia A cho B chiiTa ma khong chuTa Hoi c6 bao nhieu tap B ? * HUdng a < a > 2) Dang y = a x % bx + c B i e n doi ve dang y ~ a X = X (a ^ 0) X + x b) H i n h t h a n h tap B : • T i m tap A ' = A \ | l , 21 • T i m cac tap ciia A' r i t l i e m so' vao mSi tap ta c6 tap B 4ac '2a = aX^- 4a 4a a) Xet m e n h de P(n) "A c6 n phan tuf (n > 1) thi A c6 2" tĐp con" ã Ta CO P ( l ) diing v i neu A c6 m o t p h a n tilr t h i A c6 2^ = tap con, la tap va A 2a b (Ta t r d ve dang y = ax^ + c) + 2a + » X X - X + =c y 4a - Do t h i la mot parabol c6 : , J - h Dmh S b 2a A Gia sii m e n h de P(n) diing k h i n = k (k > , k e N ) ta chuTng m i n h P(n) diing k h i n = k + - T h a t vay, k h i A c6 k p h i n tiJf, ch^ng h a n : A = Au = |ai, a2, thi A + X ,, • X CO 2^ tap con, dp la | |, | a i l , |a2l, , |ai, 32, , aki , aiJ Do k h i A diioc t h e m mot p h a n tiir (au + i ch^ng han) t h i A c6 t h # m 2'' tap hop bfing each t h e m p h a n tuf ak + vao cac tap n6i t r e n , la |au + 1), |ai, ak + i l , ^ a ^ y - ' GIAI Bang bien t h i e n oc D i i n g phufong phap qui nap (2) A = b^ - 4ac - a) b + vdri dan - X , l a i , a2, => A ^ M e n h de P(n) diing k h i n = k + (dpcm) • • b) -A CO , ak, ak + i l 2'' + 2'' = 2''' ' tap • T i i tap A = I I , 2, 3, 4, 5, 6| ta c6 tap A' = A \ | l , 2| = {3, 4, 5, 61 ( A c A) • T i m t a t ca tap ciia A ' (cac tap k h o n g chufa va 2) roi t h e m p h a n tii: vao m i tap ciia A ' ta se dugc tap B theo yeu cau d l bai Vay so' tap B = so tap ciia A ' = 2^ = 16 Cho bleu thii-c T = J^J—- + a' - 1-2 B a i -a^ + a2 Hay tim tat ca gia tri ciia a de T d\X(fc xac dinh va rut gon T ? * HUdng • ddn Chu y : a^ = a ^ va (x - y)^ = x^ - 2xy + y^ T xac d i n h k h i mau ^ GlAl • • T xac dinh • - a + a^ a - Rut gon T = ?t l |al?il a a -1 - a a9^+l + (|a|-lf a a - 1- a " (N-1)(N + 1) a - a a -1 = a| + =>y= -(x-l)-(2x+l) = - x -i (|a| - ) % — P h u a n g t r i n h cua ( A C ) t a se CO m o t phucfng t r i n h bac h a i a n so x, y y = a x + b = a + b a = 3 = a(0) + b b = G i a i he Phirang t r i n h AC- = 4AB- (AC) (1) (2) (1) (2) D a p so : y = 3x + ( B C ) ( A M ) n e n ( B C ) c6 h e so goc =:> P h u c f n g t r i n h ( B C ) c6 d a n g y (he so goc = ) - - - x + m ( A B ) : y = x + 4, ( A C ) : y = - - x + AC- = 4ABG i a i he f(l) 1(2) c=> (X + x>0) 1)^ + (y - 2f ^ = 20 (1) (2) D a p s o ' : C ( ; 0) 131 B a i t a i C h o h a m s ' y = ^ + x + ^ - x 2x + a) V e t h i h a m so : y = b) B i ^ n l u ^ n t h e o m s o n g h i $ m phUc(ng t r i n h : + 2x + + * HU&ng + 2x + + - x + = m (1) a) Chufng m i n h t h i h a m s o (1) c t r i i c d o i xu'ng b) T r e n k h o a n g (0 ; 2) h a m so' (1) t S n g h a y g i a m HUcfng a) dan d&n D a t f(x) - T i m tap xac d i n h D cua h a m so T i n h f ( - x ) theo x r o i so sanh f ( - x ) v^ f(x) a) G i a i giong bai b) So n g h i e m ciia phijang t r i n h l a so giao d i e m ciia t h i h a m so da ve va dudng t h i n g y = m b) L a y x i , X2 e (0 ; 2), t i n h y i = f(xi) v ^ y2 = f(x2) r o i t i n h y i - y2 Cho x i < X2 r o i t i m dau cua y i - y2 GIAI a) - oc -2x o GIAI a) V e t h i h a m s o y = | x + l | + | x - l | y= 2x X + X - - neu - < X < X - - y X y Ta CO • (x > 1) y = m X e [- 2; 21 < X < >0 - X e [ - ; 21 72 - 72 + X + = fix) X y i = f ( x i ) = 72"+"^ + 72 " '^i b) Lay x i , X2 e (0; 2) ta c6 : y2 Ta CO : y i - y2 = (72 + 72 + i p Xj Xi Xi + x^] + So n g h i e m phifong t r i n h (1) m > 2 (p - Xj + 72 + 72 - X2 X2 p - Xn + V2 - ^1 + Xi + 72 + ^2 - x^) ^2 + 72 72 X2 + 72 - X i Phuong t r i n h (1) l a phuong t r i n h hoanh giao diem ciia t h i m = 72 + = f(x2) — Xn = ( X i - X2) -^2 - 727^) + ( ^ - 72T1^) h a m so da ve va dudng thSng (d,„) : y = m m < - Vay h a m so (1) l a h a m so chSn nen t h i c6 true doi xu'ng l a true B i $ n l u g n t h e o m so n g h i e m p h i f d n g t r i n h (1) m = X f ( - x ) = ^2 + ( - x ) + ^2 - ( - x ) = (- < x < 1) I ••f: o o tung (dpcm) /< S -2-1 x Vay t a p xac d i n h ciia h a m so 1^ D = [- 2; 2| (day l a tap doi xiifng) neu x > ^2 - X + - neu x < - 72 + M a m so xac d i n h - D a tfix)= fx + > 2x b) (1) = ( X i - X2) \ ; (72 < x i < X2 < V6 so n g h i e m x e [- ; 1| + Xi + ^ T X + Xj, 72 - w - Xi, xi < => )(72 = r - X, \~ ^ ' 72 - + X2 ) ,,^, + X2 , - X2 > 72 + x i x i - X2 < 0 r va 72 - X2 < 72 + "2 • y i - y2 > => H a m so' (1) nghich bien t r e n k h o a n g (0; 2) 15 3f(x + l ) - B a i T i m h a i h a m so f(x) v a g(x) b i e t 2g(x + l ) = 12X+1 = - 4x - f(x) + 2g(x) • Hiidng • GIAI T a t h a y d i , d , da c6 he so goc I a n liigt l a a i = - , aa = va as = (2) a i , , as k h a c n h a u d o i m o t ddn D o i v d i ( ) , d a t t = X + t a c6 f ( t ) - bdi • (1) X ta CO 3f(x) - 2g(x) = 2g(t) = (3) G i a i h e (3) v a (2) D o i v d i (1), d a t t = x + l o x taco: • C o n g (2) v a (3) t a diTOc f l x ) = x v a o (2) t a CO = - X + y = X X = - y = d i , da, ds t a o t h a n h t a m g i a c A B C • 12 h a y f ( x ) = x - : 2x - + 2g(x) = - 4x - o B a i 10 X a c d i n h h a m so f(x) v a g(x) b i e t y ^ ( - 1) + m 11 T h a y t b d i x t a c6 f ( x ) - 2g(x) = 12x - 1 (3) mot V a y d i n d^ = A ( - 1; 2) = t - l , f ( t ) - g ( t ) = ( t - 1) + = t - d i , d , ds cSt n h a u tiTng d o i T o a g i a o d i e m A ciia d] va da l a n g h i e m h e phucfng t r i n h : sau t h a y t GIAI • => 3, t h a y f(x) g ( x ) = - 3x + o ds k h o n g qua A m ^ T a CO ai.aa = - => d i da => T a m g i a c A B C v u o n g t a i A KB't l u a n Vdri m * t h i d i , da, ds t a o t h a n h t a m g i a c A B C v u o n g t a i A ( - ; 2) f(x) - 2g(x) = 11 f ( l - 2x) + g ( l ~ 2x) = - Gx + • HU&ng ddn G i a i g i o n g b a i D a p so : f(x) = 2x + va g(x) = x - B a i 11 T r o n g m ^ t p h d n g t p a d p , g p i d i , da, da I a n lu' C : ys = yc = => T i n h OA, OB, A B (P): y = => a = b) = (B, C e (P)) S(2; - 1); C(0; 3), A(3; 0) T a CO S C = 74 + 16 = 2V5 , SA = N/2 , A C = 3N/2 D u n g d i n h l i Pitago ta duge t a m giac S A C vuong t a i A va c6 chu vi GlAl 2p = 4V^ + 2^5 + ax + b, (a ^ 0) = 2(272 + b = Dinh S b -A^ 2a' 4a = S(0; 1) b = O v a c = l o = 4a V i b = va c = 1, t a c6 (P) : y = ax^ + M a (P) qua d i e m (- 4; 5) nen = 16a + Vay (P) : y = ^ B a i 15 Trong mftt ph^ng tpa dp Oxy, cho ba diem : ^ a) Tim diem M tren true hoanh cho MA + MB ng^n nha't T a m gi^c OBO vuong can t a i ( ; 0) • HU&ng A M A + M B > A B (hang so) Vay m i n ( M A + M B ) = A B = 2N/2 BO = OB A d ; 2), B ( - ; - 1) va C(3 ; i ) Toa dp B va C ? = yc y _b_ 2a • - b ' + 4ac' 4a b = 2a » c = = S(2; - 1) nen : va - b ' + 4ac 4a ^ = - i -3 O B' -1 ^x Dap s6': a = 1; b = - va c = => (P) : y = x ' - 4x + 18 • 19 GIAI x>2 |x-2l (1) c=> C l i i i y o n €lo 14 = x - x (i) v6 n g h i e m • m = => (ii) v6 n g h i e m • m • m ^ I => (ii) CO n g h i e m : - => (i) CO n g h i e m : { m + 1) x = x = m + 2(2m + l ) Xet dau x - = -^^ '- - m +1 2m m +1 m X X -1 - X + 2 X > 0 X < fx=^x Tuy m X va X < (x = 2^ cho vi^c gidi de gidi dUctc phUofng trinh, ta can chu y mqt so kie'n thiic +^ - X < phu'Ong t r i n h Giai tucfng duong phirctng trinh m = - Co m o t n g h i e m : x = ^/A m =1 Co m o t n g h i e m : x = - 1< m< m =0 Dieu kien |3x - > o 2x - > (dang N/A = B) GIAI x = > o Vay tap nghiem ciia phiTorng trinh la S = |4, 7| o x ^ Nhan xet 138 - 2)(2x - 3) = - 4x , phUofng trinh v6 nghiem vi : Tim phifgng trinh tiTcfng diTcfng : X > o = V3x - < t > t > t = -1 < (loai) 6t + = t^ t^ - 6t - = t = > (nhan) t = c:.x^-2x=7 ox^-2x-7 = 0o Bai 230 Giai phtfotng trinh : ^x^ - 3x - + = 2x ^3x - - 72x - > * Hudng ddn Tim phirorng t r i n h tircrng duang : IJap so : X = Tir dieu kien va nhan xet tren, ta c6 : Bai 231 Giai phvCdng trinh : 5x-l= * Hudng ddn - J ^ f = 3x - - 7(3x - 2)(2x - 3) + 2x - 7(3x - 2)(2x - 3) = - : v6 nghi§ m VI ve trdi = ( x - ) ( x - ) > ve phai= - < - 2>/2 x = + 2V2 _ Vay tap nghiem cua phuang trinh \k ?> ^ \l - Ta CO : 3x - > 2x - o x + > dung vi x > - (737^ X = , \ ^JQ \ Dang N/A = B + /2x + = D i l u kien : x > Binh phi/cfng hai ve roi dUa ve dang 7A , £*ap so : PhUcfng trinh v6 nghiem = B 139 Bai Vdri g i a tri nao ciia m t h i phi^cfng t r i n h 2x - = Jx - m c6 ^ HMng nghiem ^ HU&ng a i 3 X a c d i n h m d e phi^cTng t r i n h x + m = d&n D a n g N/A = B • B i e n ddi phucfng t r i n h v l phuorng t r i n h bac h a i v d i dieu k i e n x > • SiJf dung cac d i n h h ' d chuyen de 13 G i a i giong bai 232 ddn :m < p^p so A = B^ Bai G i a i b a t p h U o f n g t r i n h s a u : ^/2x + < x - 4c HUdng ddn B > < B V/A GIAI x - m = 2x - (dang V A ( x - m ) = (2x - 4)' { • [4x2 _ + m + 16 = < Xi < T r t f c f n g hdp • : X l < < X2 o ^ • o \\ X2 Bai X 2x + > X > < ni< > - Vl7 V X + N/17 + v/l7 A 7A 4 > 16 ddn (1) S Xl > B < >B fB > X2 A > B2 x A > < Xl < X2 o ( f - 16(9m + 16) > 4.f(2) > o < S m < 16 140 x-4 (hien n h i e n ) 41 m > Ket lugn 25 > 16 - 2x < 2x + /x - > 16 7x - > 16 - 2x f{2) > < + >( 3 G i a i b a t phifomg t r i n h : 2x + ^ • Hudng f(2) < o TriioTng h ^ p : ^> 4x2 _ Vay tap n g h i e m ciia bat phifcfng t r i n h l a / S = > o X 2x- - 7x + > ^ 4.f(2) < 16 - 50 + m + 16 < 2x - > X2 Xl o o x > PhiTOng t r i n h (*) c6 i t n h a t mot (xi, X2 la h a i n g h i e m ciia phu'cfng t r i n h (*)), D a t fix) = 4x2 _ < B^ 2x + < (2x - 3) Vay ta c6 m o t t r o n g h a i triJcfng hcfp sau : * A (;'^) n g h i e m x > X2 > 72x - < 2x - Phuang t r i n h da cho c6 n g h i e m o < < A = B) x > 2x - > Xi nghiem B > • * + c6 (I) (II) o o X > X > X < 4x2 _ O X > + 260 < > (16-2xf X < , i 65 - 765 < X < 65 - sf65 65 + N/65 « < X < - X > o 65 - N/65 Dap s6': Hap (I), (II), ta CO : x > < X < - 273 B a i 236 Giai phi^ofng trinh sau : ^x~- l)(x - 5) < - x • HiCdng d&n 59 3x^ - 12x + < < X < + 2V3 + 2S - 2x/3 3 Dang N/A < B Dap so : x < < x < i x > 4(x-3)(x-2) < (4-xf - 10 Vay tap nghiem ciia bat phu'cfng t r i n h la S = 3; + 273I B a i 237 Giai phtfofng trinh sau : x - j3'-2x < • HU&ng dan Dang ^fA Bai 239 Giai bat phUcfng trinh : 7x + • HUdng d&n Dap s6': x < Giai gio'ng bai 238 B a i 238 Giai bat phxidng trinh - 1- - > yjx - * HUdng dan Phirong t r i n h (*) c= Dieu kien de \/A xac dinh ( \[A • Chuyen ve dang -1 xac dinh o > yjx - + yJx-2 , sau dUa ve dang JA roi binh phuang hai ve, > Dieu kien i x - > X (1) x>3 -3 > yjx-l > Jx-3 * HUdng ';(( X > + 7x - ^x{x - 5) < + > ^ + + yJ2xV5 dan Dap so : - < x < -7+729 Bai 241 Giai bat phrfcfng trinh : ^x^ + 2x + + x'^ 2: - 2x dan A n so phu : f = x^ + 2x Di/a ve dang A > B GIAI + 2X + + x ^ > - x ^x^ + 2x + > - 2x - x Dat t = x^ + 2x ta c6 bat phucfng t r i n h : 142 - X x - l > x - + x - + 2^(x-3)(x-2) 2^(x-3){x-2) < - x Giai gio'ng bai 238 * Hudng Tim bat phucfng trinh tifong dirong (vdri dieu kien x > 3) X = Bai 240 Giai bat phi^o(ng trinh : < B 7x - - 7x - > x - X - Dap so': A > 0) GIAI » X > , • (1) o - < ^|5 (*) >B 143 t + > Jt + > - t o a- < ^(I) Vay o (II) x< ( - t f - Y + 2x>2 « ox" + 2x-2 x-3 -X >0 B a i G i a i b a t phifotng t r i n h : x + V 2 + x 2x V - 2x + X T r d t h a n h ^x^ V 2x + X -3 < X < _ + X t " V I " 2x X = ci.t"' 2t+l © 2x + X nghiem Si = ( - « ; - 3) > -2 < X < X > - < dung +4 S = © © 1-31 > x + ( h a i ve d e u diTcfng) o x^ + > (x + 3)^ o ^ x < Trd * 1 - Tap S3 - 131 Trd < : dung t # + X trinh + < ( x - ) ( x + 3) ix-3)^x^ Tror t h a n h - V l - > 2x + X - 2x + + x) T a CO b a t phiro-ng t r i n h : t + - > ( t > ) - + d u n g v i yjx-^ + > 0, x + < • Dieu kien t > ci- ( t - 1)^ > o + Trdr t h a n h ^ x ^ + > x + dan D a t a n so' p h u t = 0 00 x < - x = - - - + < t < phumig t r i n h l a : S = ( - x ; - l - V | u [ - l + V ; * HUdng < (x - 3)(x + 3) - + S V a y t a p n g h i e m ciia b a t - 00 Bat phuang ox'x>- + < t < t - - t + 14 < - S ci> (x - 3)yjx^ X + t < c:>t>2 - X x ft'< (II) (X - ) x / x ^ + 4 - t > t + > (I) GlAl (I) - t < + < x + ( h a i ve d e u dacfng) » x^ + x > S4 < (x + 3)^ = (3; + ») V a y t a p n g h i e m ciia b a t phifdng t r i n h l a : B a i G i a i b a t phu-omg t r i n h ( x - 3) • HU&ng S = Si u + < x^ - - u [ ; + 00) 00; - dan Bai 144 S2 u S3 u S4 = ( - • V e p h a i x^ - = (x - 3)(x + 3) • X e t d a u x - va x + t r o n g m o t b a n g x e t d a u c h u n g • T i m b a t phircfng t r i n h tifcfng du'cfng 244 a) T i m g i a t r i Idrn n h a t c u a h a m so y = b) Suf d u n g ke't q u a d a t i m difcfc de g i a i phitofng t r i n h : 7x - + 74 - X = - + ^4 - x - 6x + 11 145 • HUdng d&n a) dung bat Akng thiJc Cosi • Dieu k i e n de h a m s o xac d i n h • rr a + b Vab < — — - • Chufng m i n h y < h^ng s o ( k i e m t r a dau = xay ra) b) • Chon X nguyen • ThiJf l a i (phan dao) GIAI • • H a m so xac d i n h o X - > - X - + ^4 - x o >, < X < /7 r- (x-2) + i x - i I n r- (4-x) + l 5-x x - n 2' o dau fx Ket luan : b) - x yMax = x = - l , x = 0, x = l , phi/cfng t r i n h da cho • X = n g h i e m dung phiTcrng t r i n h da cho + J4 - X va = - X = (2) dong o x xay • Ta CO : a + b = ^a^ + b'^ , giai phuong t r i n h de t i m a, b tir t i n h duoc x GlAl a*^ = Dat b = ^x Vay Ve p h a i x^ - 6x + 11 = (x - 3)^ + > (dau " = " xay o kien B > phU0ng -2 b'^ = X - a^ + b^ = 2x - x-2 t r i n h da cho t r d t h a n h a + b = ^a^ + b^ (a + b)^ = a^ + b^ « a^ + 3ab(a + b) + b^ = a^ + b^ a = x 3ab(a + b) = o b = a = -b B i e n doi v l dang %/A = B dieu o o • HUdng d&n Nhd x = 3) c^-x = B a i 245 Tim nghi^m nguyen ciia phufcfng trinh : 146 thdi • [A > = ^2x - b = 3/x - Ve t r a i 7x - + ^4 - x < (dau " = " xay o ã ô Chon an so' phu „ = x ' - 6x + 11 (*) VP = [a = / x - l x = VT = ^ 5,520 : r • HUdng d&n (2) • ^ ' a y ( * ) c=> - ^ " ^ + 36 x = 2, x = 4, x = deu k h o n g n g h i e m d u n g B a i 246 G i a i phi^omg trinh 3/x - + 3/x - (1) iTng dung k e t qua t r e n d§ giai phumig t r i n h J^T^ - i , x < • = + — d [1) "-" - c + 36 > Thiirlai • + VToc < — yLN = X X Vay tap n g h i e m nguyen cua phifcfng t r i n h la S = |3| I I > 12 7x + = - x^ - • Theo bat dang thufc Cosi, t a c6 : nen X + o V i X nguyen nen ta c h o n : x e |- , 0, 1, 2, 3, 4, 5,) VP > Tim gia tri Idn nhat ciia ham so y = 12 7x + = -x^ - a) y < X + Ta p h a i c6 GIAI Vay y = x"^ + VT < G i a i phifcfng t r i n h x e (a ; b) ta se CO x e (a; b) + x + 12 yJx + • a = x = l • b = x = • a = - b o x - l = -(x-2) o x = Vay tap n g h i e m ciia phuong t r i n h la S = |1, 2, — I 2i 147 B a i G i a i p h i f o T n g t r i n h (4x - 1) • HMng + = 2x^ + 2x + (*) x = < — (loai) dan • D a t t = x/x^ + i X = — > — (nhan) ^ • ' • B i e n d6'i phUcfng t r i n h ve phucfng t r i n h bac h a i , a n so't 4x^ + > : v6 n g h i e m I GIAI p a p s o ' : T a p n g h i e m ciia phirong t r i n h la S = — Phirang t r i n h (*) (4x - 1) jx^ + = 2(x^ + 1) + (2x - 1) Dat t = Ta CO x/x^n (t > 1) ^ • B a i T i m n h i y n g so' n g u y e n a v a b s a o c h o phirfofng t r i n h x'^ + a x + b = (4x - l ) t = 2t'^ + 2x - 2t^ - (4x - l ) t + (2x - 1) = CO h a i n g h i e m xj v a X t h o a m a n d i e u k i ^ n - < X j < - l v a 41 Hi^ofng dan A = (4x - 1)- - 8(2x - 1) = 16x- - 24x + = (4x - 3)^ D a t f(x) = x^ + ax + b 4x - + 4x - = 2x - t = Vay t = • t l.f(-l) < 4x • 2x - (1) (ma t > n e n x - l > l (1) ^ x ^ + = 2x - (X > o x > l ) • 1) 3x^ - 4x = x^ + = (2x - ly X > fx = x = x > 1 X C a c h k h a c : (4x - 1) v/x" + = 2x''^ + 2x + > — O X = +2x + l > , t r i n h (*) l a x - l > • , , (hai ve ciia phiicfng t r i n h • cung dau) B i e n doi tUcfng ducrng ( k e m dieii k i e n x > - ) GIAI Theo gia t h i e t : Phudng t r i n h c6 h a i n g h i e m x j , X2 thoa m a n < X < n e n ta c6 : l.f(-l) < -a + b < - (1) l.f(l) < a + b -2a + b > - (3) l.f(2) > I 2a + b > - (4) , (3) + (4) 2b > - => b > luo'i Vay -4 < b < -1 ma b nguyen b =: - thay vao (4x - l)^(x^ + 1) = (2x^ + 2x + 1)^ 148 ' •—> (1) + (2) == 2b < - => b < - V X G R ox>i ThiJrlai - < x i < - l v a , - 1 • • *—• xT^ X2 Suy a va b bang each chiJng m i n h a e (a ; (3) va a nguyen (ti/orng D a t f(x) = x'^ + ax + b (* j , [x^ + > 0, Vx R Ta t h a y \n dieu k i e n doi vofi phuonii • - Di§u k i e n 2x^ ^2 —• iU doi vdi b) Vay t a p n g h i e m ciia phufong t r i n h la S = — • A ' W < I h e o de bai ta co : < l.f(-2) > l.f(2) > rru 1 t h i phiTcfng t r i n h (1) c6 h a i n g h i e m : (*) Phirong t r i n h c6 n g h i e m 'a t h a n h ax + b = (] = - m - ^ m ^ - m + , * Ta D i i u k i e n de phi/omg t r i n h V") c6 n g h i e m l a : A = a ^ - 4y(y - b) = - 4y^ + 4by + a^ > (xi < X2)- , X g = - m + iy2m^ - m + a = b = Tri^drng hgrp : y ^ o + l f ( m ) = m^ - m ( l - m) + - m^ = 2m^ - m + > 0, V m e R => a, b T r t f d n g hofp : y = 0, liic phUcfng t r i n h {'•') trd A>0 - 2X A ' = (1 - m ) ^ - ( - m^) = 2m^ - m GIAI CO xi < X2 y i < y < y2 ( y i va y2 la hai n g h i e m ciia tarn thufc bac h a i : - 4y^ + b y + a^) 150 = B^ • hai n g h i e m thoa m a n gia thiet • • -4 B > A v a g i a t r i n h o n h a t b S n g - • = fa| -=« ^ Ht/oing ddn PhUcrng t r i n h V A = B B a i T i m a v a b s a c c h o h a m so y = o - W = 2 b = -3 • HUdng -• a p a i T i i y t h e o m, g i a i v a bi§n l u $ n phifoTng t r i n h ^ x ^ - 2x + = x - m a = D a p so': b = a = D a o l a i , a = v a b = - t h i phufOng t r i n h d a cho l a : x =3 + y2 < a < — nguyen "xi Yi R — < a < — a 2a - > - CO -1 R h a i n g h i e m nky t h o a m a n gia t h i e t • = 151 m A' l.f(m) S — - m P h i r o n g t r i n h (1) + + - Co h a i n g h i e m m < x i < X2 x i v a X2 v^xi = X2 = l - i n = l > m + V6 n g h i e m \'6 n g h i e m + + +x v ^ x i - X2 - + - m - < m C o h a i g h i e m x i , X2 X i < X2 < m A TONQ HOP CaOl NAM 15 K I E M T R A K I E N T H Q C V E HAM S O V6 nghiem ' ' p a i 251- ( C a u h o i t r S c nghiem) C h o h a m so y = f(x) x a c d i n h t r e n k h o a n g ( a ; b ) L a y x i , X2 e ( a ; b) Co n g h i e m x = + A U (iui.yoii da cho Co h a i n g h i e m x i , X2 : / Phufong t r i n h t v a d&t y i = f ( x i ) , y2 = f(x2) ( x i 9^ X2) B i e t h a m so y = f(x) d o n g bie'n t r e n k h o a n g (a ; b) C a u n a o s a u d a y s a i ? a) ^ — ^ b) (y2 - y i ) ( x - x i ) > > c) x , > X2 y , > y2 d) ( y i - y ) ( x i - xg) > • Hii&ng ddn T h e o d i n h n g h i a h a m so' d o n g b i e n t a c6 X j - X2 v a y i - y2 c u n g d a u V6 V a y cau d s a i nghiem B a i 252 ( C a u h o i t r S c n g h i e m ) m ''' 2x - C h o h a m so y = f(x) = HP" (x - 1) D a t y = g ( x ) = f(x + 1) C a u nao sauday dung ? ^ ^ ^ 2x + a) g ( x ) = • Hiidng — x+2 , , b) g(x) = 2x + x+1 ^ , ^ c) g ( x ) = 2x + x+2 ^ ^ 2x + d) g ( x ) = x+1 g ( x ) = f ( x + 1) = ( x + ) - ( x + 1) + ddn 2x + Dap so': g(x) = ( c a u c) x + Bai 253 (Cau hoi tr^c nghiem) C h o h a m so y = f(x) = - x^ - 2x + C a u n a o s a u d a y s a i ? a) D o t h i h a m so l a p a r a b o l c d i n h S ( - 1; 6) •i b) f(2) = - c) f ( - X ) = x^ + 2x + d) T r u e d o i x i f n g c i i a t h i h a m so l a dtfotng t h S n g x = - * Hudng ' dan C a u c s a i v i f ( - X ) = - ( - x ) ^ - ( - X ) + = - x^ + x + D a p so : C a u c Jai ( C a u h o i t r S e n g h i e m ) C h o h a m so y = f(x) c t a p x a c d i n h D j = (- ; 5] v a h a m so' y = g(x) CO t a p x a c d i n h D = (0 ; 6) T a p x a c d i n h c i i a h a m so y = f(x) + g(x) l a : a) ( - ; 6) 152 b) ( - ; 0) c ) 10 ; 5) d) (0 ; 5] 153 • HU&ng d&n Tim Di o B a i 258 ( C a u hoi trSc nghi^m) D2 T i m t a p g i a t r i c i i a h a m so' y = - x^ - 4x + D a p so : ( ; 51 ( c a u d ) a ) (HMng B a i 5 ( C a u h o i trfic n g h i ^ m ) H a m so n a o s a u d a y l a h a m so c h S n t r e n t $ p x a c d i n h c u a n o a) f(x) = b) f(x) = c ) f(x) = x^+1 • HU&ng ddn H a m so' f ( x ) c h S n t r e n t a p x a c d i n h D c i i a n o x e D = > - X D < f(-x) = f ( x ) , Vx e c ) (- «>; - 3] d ) [- 3; + •») d&n H a m so y = - ( x % x ) = x + = - (x + 2)^ < ^ ' B a i ( T o a n ti^ l u g n ) a) K h a o s a t suf b i e n t h i e n v a v e t h i h a m so y = - x^ + 4x - b) T i m m de phi^oTng t r i n h I x^ - 4x + I - m = c d i i n g b o n n g h i ^ m • HUdng D b) [5; + 00) p a p s o ' : T a p g i a t r i c i i a h a m so' l a D = ( - co; 5| d) f(x) = x - 5] d&n a) Dap so': f(x) = (cau b) x - - 00 X ^ y + 00 \ B a i 256 ( C a u hoi trdc nghi^m) 2v - H a m so y = —= c t a p x a c d i n h D = R\|2| T i n h a ? X - 4x + a - a)a = b)a=:-5 c)a = d)a = -4 • HUdng dan H a m so k h o n g x a c d i n h o x^-4x + a - x = l = x = o a = 2x - w c6 t a p xac d i n h D = R \ i | D a o l a i a = t a CO y = Tac6|-A| D a p s o ' : a = ( C a u a) B a i 257 ( C a u hoi trgc nghi^m) Bie't h a m so y = f(x) l a h a m so l e t r e n t $ p D = [- a ; a ] , a > f(x) X e t h a m so y = g(x) = C a u nao sau day diing ? X a) y = g(x) l a h a m so le t r e n d o a n [- a ; a] b) y = g(x) l a h a m so le t r e n t^lp [- a ; 0) u (0 ; a ] c) y = g(x) l a h a m so c h d n t r e n d o a n [- a ; a] d) y = g(x) l a h a m so c h S n t r e n t ^ p [- a ; 0) u (0 ; a ] • Hitdng Phiromg t r i n h d a cho o V e t h i h a m so' y2 =I I - x^ + x - I = m - x^ + x - I suy tiT t h i h a m so y i = - x^ + x - X e t d a u ( - x^ + x x - 3) X - - x^ + x - 3 + + X - dan • y = g ( x ) CO t a p xac d i n h D = [ - a ; a|\10) = [ - a ; ) >^ (0 ; a| • , g(- f(-x) X) = -X = -f(x) - X = f(x) X x.< V x > 155 • l < x < = > - x + x - > c:> I- x^ + x - i P a i ( C a u h o i tr&c nghi#m) C h o b a t phifofng t r i n h m^(x - 1) > x - m + Vdti g i a t r i n a o c u a m t h i t^ip n g h i e m c i i a b a t phUofng t r i n h l a S = R ? = - x^ + x - ya = y , V a y d t h i h a m so' y2 = t h i h a m so y i k h i < x < b) m = - a) m = • Hiiffng d) m = c)lml = ddn • _ , ' B i e n d6'i b a t phifOng t r i n h v e d a n g A x > B r o i cho A = v a B < D a p so : m = ( c a u a) s i B a i 263 (Cau hoi trSc nghiem) C h o phi^dng t r i n h (x - l ) ( x + 4)(x - 2)(x - 7) = m (1) P h U d n g t r i n h d a cho cd d u n g b o n n g h i e m cat d o t h i h a m so y t a i bo'n d i e m p h a n D i f d n g t h S n g y = n, biet D a p so : < m < ^ N e u d a t t = x^ - 3x + t h i phUc^ng t r i n h (1) t r d t h a n h : a) t - 26t - m = c) t^ - 30t - m = b) t^ + 26t - m = d) t + 30t - m = • HUdng V i e t l a i p h u o n g t r i n h ( ) : ( x - l ) ( x - 2).( dan )( )= m D a p so : t ^ - t - m = ( c a u c) B a i 260 (Toan t \ lugn) [f(x) + g(x) = - (1) T i m h a i h a m so f(x) v a g(x) b i e t f{x + l ) - g ( x + l ) = x (x ^ 0) (2) B a i ( C a u h o i tvic B i e u d i l n phi^otng t r i n h x^ - 2x^ + (2m - l ) x ^ + 2x + = t h a n h ( H a i h a m so f(x) v a g(x) c u n g c t a p x a c d i n h l a D ) • HUdng phi^cfng t r i n h a n so t = x - — ? (x ^ 0) ddn X Tii ( ) , d a t t = X + cx> x = t - , t a cd : f(t) - g{t) = + G i a i he • B nghiem) hay f(x) - g(x) = t - f(x) = (1) t a diSac k e t qua (2) • ^ KIEM TRA C A C KIEN THQC g(x) = (3) a) t - 2t + m - = c) t - 2t + 2(m + 1) = b) t'^ - 2t + m - = d) M p t phi^cTng t r i n h k h a c • Hudng ddn V i e t l a i p h a o n g t r i n h ( c h i a h a i ve ciia p h i i c f n g t r i n h c h o x 2x + 1 ^ x - - x ^ 0) : + ( m - 1) = -4x + D a p so : t ^ - t + m + = ( c a u d ) x - V E PHUONG TRINH VA B A T PHJONG TRINH B a i ( C a u h o i trfic n g h i e m ) Phi;fc(ng t r i n h ^ x ^ - = 2x - m tifcfng drfdng v d i h § n a o s a u d a y ? B a i 261 (Cau hoi trdc nghiem) > X a c d i n h m d e b a t phi^cfng t r i n h (m^ - 2)x < m ( x + 3) - v n g h i e m ? a)m =- l • Hudng b ) m =2 c ) - l < m < d) D a p so k h a c D i i p so : m = ( C a u b) b) > c) 3x^ - m x + dan B i e n d o i b a t phacfng t r i n h ve d a n g A x < B r o i cho A = v a B > 156 a) X > + = 3x^ - m x + - =0 m m — x ^ - m x + + m'^ d) =0 X > — 3x^ - m x + - =0 157 • Hitc/ng dan Phucrng t r i n h V A Cdn fB > = B A nhaf: Phuang = B^ bac hai : ax + bx + c = (a 0) c6 hai nghiem C ; dau (xi < < xo) c:>P = xi.xo < — < (liic hien nhien a > D a p so : trinh trai A > 0) (Cau b) 3x^ - m x + + = B a i G i a i phtfcfng t r i n h : ^10 + x - 2x^ + x'^ - 10 = 3x - yjx^ - x - B a i 266 ( C a u hoi t r i e nghiem) PhUc(ng t r i n h m ( x - 1) = - 3x c n g h i f m x e (2 ; + oo) k h i : a) - < m < b) c ) < m < * HU&ng ; 4c Hiicfng i, • x = d ) - < m < - • => • A => X = f > > tii d a y t a se c6 , X = Kiem tra x = D a p so : B i e n d o i p h u a n g t r i n h ve d a n g A x = B r o i x e t : A = D i e u k i e n de phucfng t r i n h x a c d! ii ni h - < m < dan • dan n g h i e m diing phUdng t r i n h X = B a i Tviy t h e o m , g i a i v a b i ^ n l u # n phrfcfng t r i n h : ?t > m + D a p so : - < m < - y/x^ - X = a - X (a la t h a m s6') (Cau d) ^^ HUc/ng dan PhUtfng t r i n h VA = B o B a i 267 (Cau hoi trSc nghiem) G i a sijf phtfcfng t r i n h b a c h a i x** + a x + b = (b 0) c h a i n g h i $ m X i , X g L $ p phUo^ng t r i n h b ^ c h a i c h a i n g h i e m l a — v a — ? a) bx^ - a x + = c) bx^ - a x - = b) bx^ + a x + = d) bx^ + a x - = • Hii&ng T i n h S = - ^ ^ = ^ ^ ^ = vaP= Xj.X2 D a p so : bx^ + a x + = ^ ^ Xj^ X2 - x = a - x • HUdng b)m X a < + X fa'='x^ a > ( c a u c) X ; a < hay - (*) a^-a 2a - - X = a^ K e t qua Phufcfng t r i n h c6 h a i n g h i e m t r a i d a u ( x i < < X ) 158 - (2a-l)x P h u o n g t r i n h da cho v6 n g h i e m a^ => Phi/cfng t r i n h ("'=) c6 n g h i e m x = ' 2a - — a va d ) - l < m < ddn D a p so : - < m < x^ - x = a^ - 2ax + x^ • a - c ) - < m < l a - x > a = - => P h i i o n g t r i n h (*) v n g h i e m a a)m ^x CO : ddn > GIAI Ta Xj B o < a < : Phuofng t r i n h d a c h o v n g h i e m P = X1.X2 < < a < - h a y a > : P h i f o n g t r i n h d a cho c6 n g h i e m x - „ j 159 C K I E M TRA C A C K I E N T H O C V i H E PHaONGTRiNH B a i ( C a u h o i trie Ne-u p a l 274 ( C a u hoi trdc nghiem) nghi#m) , (mx + y = m - C h o h ^ phi/ofng t r i n h { , c a u n a o s a u d a v diinff ' [x - y = - m ^ = t h i X v a y l a n g h i $ m phvfcTng t r i n h b^lc h a i n a o = 40 |x'^ a) sau day ? phUofng t r i n h l u o n c n g h i e m b) K h i m = t h i h ? phifofng t r i n h v n g h i e m a) t'^ - 8t + 40 = c) t - 8t + 12 = c) K h i m = - t h i h$ phifdng t r i n h c6 v6 so n g h i e m b) t^ - 8t + 20 = d) t^ - 8t + 18 = d) B a c a u a , b, c d e u s a i • HUdng :f HUcfng ddn dan + = (x + y)^ - phucfng t r i n h x y , tiT d a y t a t i n h d u o c x y = L a p d i n h thiJc D = - m - r o i dung t'^ - S t + P = K h i m = - t h i h e p h u a n g t r i n h t r d t h a n h \6 v so [ X - y = D a p so : t ^ - t + 12 = ( C a u c) n g h i e m (x ; y ) Dap s o ' : C a u c B a i 272, (Cau hoi trSc nghiem) Tim dieu ki$n a.b = 2m^ - 2m c u a m de t o n t a i h a i so a v a b thoa m a n B a i 275 ( C a u hoi trac nghiem) a + b = 2m a) m e [- ; 3] c ) m e ( - c o ; - 3] u [1 ; + D a p so': X > Phifofng t r i n h nao sau day c6 h a i nghi§m x j , X2 thoa m a n d i e u k i $ n < xa < - ? a) 2x'' - 5x + = c) x^ + 6x + = b) x % 5x + = d) x^ + 6x + = * HUcfng ddn Phirang t r i n h (a) v6 n g h i e m — (cau b) PhUcfng t r i n h (b) va (c) c6 n g h i e m x = - A' > BJii 283 (Cau hoi tr^c nghi$m) V i nhdng gia tri nao ciia m thi phu^ong t r i n h x^ - 4x + m = c6 h^' nghiC'm p h a n bi^t va dUong ? a) m (4; + Qo) lfi4 b) m e (0; + oo) c) m e (0; 4) d)me(-oo;l' PhUcfng t r i n h (d) c6 l.f(-l) > D a p so : cau d S = _ < - l 165 ... + — ^ 1+ d 1+ c ) = 6abc Chi?ng minh abed < 1^ 81 * Hudng ddn TiX gia thiet ta c6 : 1+ a b c 1+ b + 1+ c - 1+ b - +c - +d bed d — > 33| (1 + b ) ( l + c ) ( l + d) + 1+ d (1) 59 T i r a n g tix 1+ b •... K-x' = 11 X ' ^ ( K ^ + K + 3) = 11 ( D x^ + 2Kx^ + 3K^x^ = 17 x^[3K^ 3x' + 2Kx- X = y = BvCdc : D a t y = K x ( K ^ 0) Ta [K = K GIAI • 2x^ + x y + y2 = 15 X = -2 y = - X = -11 = y = - 11 Til... (m - l)x^ + 2(m + 2)x + m - = Tim m de : * Hudng > m -1 m -1 4P = + X ( m - 1) + 1 = -^^ = 1+ m -1 m -1 m -1 S = + ^ ^ ^ 6x - = m = M e n h de (1) — X, 75 * HU&ng d&n * a) A' : i (m - 2? - m ( m -