luyện thi đại học môn toán

Cau 9b: Dat t = |z|, t > thi ta c6 \zf (a.z + b) = -c.z Nen ta c6 |c|.t < t^ (|a|.t + |b|) t^ + t - l>0t> Va | c | t > t ^ ( | a | t - | b | ) o t - t - l < o t < V^y < |z| < Cau 8a: Trong khong gian voi h# tpa dp Oxyz cho hai duong thang: - + 75 dj:2^ =X =^ ; Cau 9a: Cho Z j , Z2 la cae nghi#m phuc ciia phuong trinh 2z^ - 4z + l l = Z, +|Z2 iL ie uO nT hi Da iH oc 01 / DETHITHUfSdlO (zi+z2r B Theo chUorng trinh nang cao Cau 7b: Trong mat p h i n g voi h? tpa dp Oxy cho ba diem vuong, AB d i qua E va CD d i qua F - 3mx^ + 4m^ c6 thj (Cm) Cau 8b: Trong khong gian Oxyz, t i m tren Ox diem A each deu duong , x—1 V z+ ' thang d : — ^ = = va mat phang (a): 2x - y - 2z = a) Khao sat sy bien thien va ve thi ham so' m = , b) Xac djnh m de hai diem eye trj cua thi ham so' doi xiing qua duong t h i n g y = x s/ up /g ro ( y - ) + ^ = x3y[(2y + 3)2-6' c om dx ok - i l + x + V l + x^ Cau 9b: (1 diem) Giai h ^ phuong trinh sau: Ta n Cau 2: Giai phuong trinh: (sin^ x +1 j +1 = 73 sin 2x + sin x + — 6J -xy = bo Cau 5: Cho hinh chop S.ABCD c6 day ABCD la hinh chu nhat eo tam O va 2logi_x(-xy - 2x + y + 2) + log2+y (x^ - 2x +1) = , (x,y6R) logi-x (y + 5) - log2+y (x + 4) =1 HirdNGDANGlAl I PHAN CHUNG CHO TAT CA CAC THI SINH Caul: a) B?n dpc t y lam b) Ta c6: y' = 3x^ - m x , y' = 0x = v x = 2m Ham so' c6 hai diem eye trj o m ?t p h i n g (SAC) tao voi day mpt goc 60° Tinh the tich khoi chop S.ABCD Cac diem eye tri ciia (C^,) la M ( ; 4m^)va N ( m ; ) w ww (a+b fa ce AB = a, A D = a73 ; SO = SD Mat phSng (SBD) vuong goc voi mat day, mat Cau6: Chung minh rang ne'u a,b,c>0 t h i : a+b b+c ic + a ^ Vb+c Trung diem cua doan M N la l ( m ; 2m^ j v a M N = (2m;-4m3) ya+c I I PHAN RIENG T h i sinh chi dugc chpn lam mpt hai phan (phan A ho^c B) Duong t h i n g d : y = x c6 vecto chi phuong la u = ( l ; l ) M , N doi xung qua duong t h i n g (d) M N (d) va I G (d) A Theo chUorng trinh chuan MN.u = 2m + (-4m^) = m =0 Cau 7a: Trong mp voi h? tpa dp Oxy cho duong tron (C) c6 phuong trinh 2m'' = m m(2m2-l) = m =± x^ + y^ - 2x + 6y - = Viet PT duong thing A vuong goc voi duong thing d : 4x - 3y + = va cat duong tron (C) t^i A, B cho AB = 62 I(l;l),E(-2;2), F(2;-2) Tim tpa dp cac dinh cua hinh vuong ABCD, bie't I la tam cua hinh I PHAN CHUNG CHO TAT CA CAC THI SINH Cau 4: Tinh tich phan: I = [ - va hai diem A ( l ; - ; ) , B(3;-4;-2) cho l A + IB dgt gia trj nho nhat Cau 3: Giai hf phuong trinh =^ Xet vi tri tuong doi ciia d] va d Tim tpa dp diem I tren duong t h i n g d j + 75 Tinh Cau 1: Cho ham so y = d : ^ =^ Ke't hpp dieu kif n ta dupe m = ± 72 72 63 " ^ Cau 2: P h u o n g t r i n h t u o n g d u o n g v o i : Vay nghi^m cua h§ da cho la: sin"* X + 2sin^ x + - 2\/3 sin xcosx - V s i n x - c o s x = sin x = -^ ^ cosx = X = k n [ V s i n x + c o s x - l =0 u^-l => iL ie uO nT hi Da iH oc 01 / Ta s/ up c i bo ok + fa ce s = -5-M w < 4P ww ^ v n g h i ^ m P = 13 — v l P = /g P =1 om "S = ro + y , P = xy ta c6: S= 2 I u2 du •+ — u^u+lj du J du = !n(u + l ) - ^ l n u =1 SH OD SH (ABCD) Gpi K la hinh chieu cua H len canh A C , suy goc H K S chinh la goc giira hai mat phang ( S A C ) va mat day nen H K S - 60° Ta c6: A C = B D = 2a O C - a => A O C D deu G Q I E la trung diem cua O C , suy ^ H K = 1DE = - ^ Trong tarn giac vuong S H K ta c6: ^ S H = HKtan60" ^ : ^ 4 Vay the tich cua khol chop la: Vs.ABCD = |SH.SABCD B =1^'^'^^^^^ Cau : Ap dyng bat dang thuc y/x + ^Jy < ^2(x + y ) , ta c6 : " g h i ? m ciia p h u o n g t r i n h : a+b c 4t2 + ( + l ) t +13 + 713 = o t = 64 V2+1 nen S H la duong cao cua hinh chop [x + y + 2xy = - + VT3 x = l=:>u^72 + l M|t khac (SBD) ( A B C D ) T ^e tn uhgip voi•u'^^ , ( u - x)^ = + x^ ( d a cho t u o n g d u o n g v o i : x^ + =0 < » s i n ' ' X H-^N/Ssinx + c o s x - l ) Cau : -5-713+76713-14 (x;y) = ( l ; i ) I^^-^^l^^^Eli 7^r + 72U b+c /c + a 7^_^7bl + -/ > 72 t7? 7?J r^^^7b^ 7b_^7?1 721,7^ 1 7b n 7? 1 ^ b j 72l7^ ? j 72l7^ b J 7^j 72 7a 7a ^ ' -_1 1 Ap dune bat dang thuc — + — > x y x N/ZIVC VbJ"^ V2l>/a +y 3N/2 , ta c6 : ^ycJ x/b 2^y2a 2>/2b = Vb + N/c 272^ 2V2b ^ 272^ Vb + Vc Va + Vc ^ 2N/2C x/a+Vc ^ya^-^/b Do ^ 2V2b Va + N/b ~ ^/2(b + c) = (a + b pia ^ ^2{a + c) Va + c 2>^ "Vb + c + b) + 11 Z2 Duong thang CD c6 phuong trinh dang: (dpcm) a ( x - ) + b(y + 2) = ax + by - 2a + 2b = V i d ( I , A B ) = d(IXD)=;> A Theo chUtfng trinh chuan |3a-b| _ |a-3b| a = - b , Va^ + b^ Va^ + b^ Suy phuong trinh A B : x - y + = 0, C D : x - y - = Ta Cau 7a: Duong tron (C) c6 tam 1(1;-3), ban kinh R = Gpi H la trung diem Phuong trinh BC va D A c6 dang x + y + c = d(I,BC) = d ( I , AB) = 2V2 =i s/ AB thi A H = va I H A B => I H = up Matkhac I H - d ( l , A ) /g ro Vi A l d : 4x - 3y + = => A: 3x + 4y + c = om = 4C = 29,C = -11 fa il^ = - u ^ Gpi A j la diem do'i xung cua A qua d j Suy A ( l ; ) , B ( - ; l ) , C ( l ; - ) , D ( ; l ) BC:x + y - - , D A : x + y + = • Suy A ( - ; l ) , B ( l ; ) , C ( ; l ) , D ( l ; - ) Cau 8b: Gpi A ( a ; ; ) £ O x (a) qua ww Taco AB = ( ; - ; - ) = > A B / / d i w Ma M ( ; ; - l ) d i nhung M g d j =>d,//d2 c = 2,c = - |2at Khoang each t u A deh m|it phang (a) : d ( A ; a ) = - j = = = = = ce Cau 8a: Vec to chi phuong cua hai duong thang Ian lugt la: bo ok 3x + 4y + 29 = va 3x + y - l l = = 272 BC:x + y + = 0, D A : x + y - = • c Vay CO duang thang thoa man bai toan: u7 = ( ; - ; - ) , u^ - (-2;3;4) = 3v'2 ax + by + 2a - 2b = vai a^ + b^ > hoac B) lc-9| ; z, + Z =1+ B Theo chUtfng trinh nang cao Cau 7b: Duong thang AB c6 phuong trinh dang: a(x + 2) + b(y - 2) = I I PHAN RIENG T h i sinh chi dirgic chpn lam mpt hai phan (phan A d(I,A)-IH x/22 Z2 2 ?=• + • + —!= < ^2(x + y ) , ta c6 : 2N/2^ •3^/2' Suy >^= A p dung bat d i n g thuc Vx + ^ i, Cau 9a: Phuong trinh da c6 cac nghiem: zi - ^ - iL ie uO nT hi Da iH oc 01 / ^ M o ( l ; ; - ) va c6 vecto chi phuong u = ( l ; 2; ) D3t M Q M I = u Do do: d ( A ; ( a ) ) la ihionj'; cao ve tu A tam giac A M Q M I j / A ^ 2.S,^MoMi ['•'^^o;"] V8a2-24a + 36 =>d(A;A) = 1- = -^;— i = Taco: l A + IB = l A j + I B > A i B Suyra l A + IB dat gia trj nho nha't bang A j B , dat dup-c A i , I , B thang hang I la giao diem ciia AjB va d J 65 -21 Do A B / / d i => I la trung diem ciia AjB suy I 129 58 66 Theo gia thiet: d ( A ; ( a ) ) - C I ( A ; A ) -43 29 J o |2a| !—= Vsa^ - 24a + 36 4a , = oSa"^2 - -.^ A 24a + 36 4a - r 24a + -.^ 36 = n0 CtyTNHHMTV 4(a-3)^ =0a = day, A D = ay/3 G p i E, F Ian l u p t t r u n g d i e m ciia cac d o a n BC, D E T i n h the - x y - 2x + y + > 0, - 2x + > 0, y + > 0, X + > jfch h i n h chop F.ABC C h u n g m i n h A F v u o n g goc v o l C D < l - X t l , < + y;itl Cau 6: C h o so t h u c d u o n g a, b thoa m a n : 6|a^ + b^ j + 20ab = 5(a + b ) ( a b + ) l o g i _ x [ ( - x ) ( y + ) ] + 2log2+y (1 - x ) = T i m gia t r j n h o nhat ciia bieu thuc l o g i - x ( y + ) - l o g + y ( x + 4) = l j l o g i - x ( y + 2) + log2+y (1 - x ) - = (1) l l o g i - x ( y + ) - I o g ^ y ( x + 4) (2)' =1 P = II y = -X-1 s/ up ro /g ok c om DETHITHllfSdll bo I P H A N C H U N G C H O T A T CA C A C T H I S I N H fa ce Cau : C h o h a m so y = x^ - Sx^ + , c6 d o t h j la ( C ) w a) K h a o sat s u bien t h i e n va ve d o thj ( C ) ciia h a m so ww b) T i m cac d i e m A , B thupc d o t h i ( C ) cho tiep t u y e n ciia ( C ) tai A , B dx Cau 4: T i n h t i c h p h a n : I = fix^+2x b^^ PHAN R I E N G T h i s i n h c h i d u p e ehpn l a m m p t t r o n g h a i p h a n ( p h a n A Cau 8.a: T r o n g m a t phSng tpa d p O x y z , cho hai mat p h a n g ( P ) : x + ^ - z + = V a y h? C O n g h i p m d a y nhat x = - , y = Cau 3: Giai p h u o n g t r i n h : 2(x^ + 2) = 5Vx^ + l ' l ^ i ciia h i n h chu nhat bie't D n a m tren d u o n g thang c6 p h u o n g t r i n h : x - y - = K i e m tra d i e u k i ^ n ta thay chi c6 x = - , y = thoa m a n d i e u k i | n tren rilTt ' f 4^ dinh A ( l ; l ) G p i G ; la trpng tarn tam giac A B D T i m tpa d p cac d i n h = Cau 2: Giai p h u o n g t r i n h : cos 2x + - - sin V 6; U2 + 25 Cau 7.a: T r o n g mat phJing tpa d p Oxy, cho h i n h c h i i nhat A B C D , v o i toa dp cac x+ -x + ^ « rx = 0=>y = - l = l - x < = > x ^ + x = ^ x+ [_x = - = > y = l song song v o i n h a u va A B = 4V2 a ^ b Ta o b -16 A Theo chi/orng trinh chuan (3).The vao (2) ta c6: l o g i - x ( - X + 4) - l o g i _ , (x + 4) = l o g i _ , hoac B) D a t l o g + y ( l - x ) = t t h i (1) t r o thanh: t + ^ - = < » ( t - l ) ^ = O o t = l V o l t = ta c6: - X = y + o a iL ie uO nT hi Da iH oc 01 / H $ da cho Khang Vjft Cau 5: C h o t u d i ^ n A B C D c6 A B C la tarn giac deu canh b i n g 2a, A D v u o n g g6c V a y A (3; 0; ) C a u 9b: D i e u k i ^ n : DWH > x + j va ( Q ) : X + 2y - 2z - = Viet p h u o n g t r i n h mat cau (S) d i qua goc tpa dp O, qua d i e m A (5; 2; ) d o n g t h o i tiep xiic v o i ca hai mat phSng ( P ) va ( Q ) Cau 9.a: T i n h m o d u n ciia so p h u c z , bie't z^ + 12i = z va z c6 phan thuc d u o n g B Theo chUorng trinh nang cao X^ y2 Cau 7.b T r o n g m a t phang tpa d p Oxyz, cho elip ( E ) : — + ^ = va d u o n g thSng d : X + y + 2013 = Lap p h u o n g t r i n h d u o n g thang A v u o n g goc v o i d va c^t ( E ) tai hai d i e m M , N cho M N = — Cau 8.b T r o n g mat phSng tpa d p O x y z , cho mat phang ( P ) : x - 2y + 2z + = va d u o n g th5ng ( d ) : = = M a t cau (S) c6 tarn I n a m tren d u o n g thang ( d ) va giao v o i m a t phang ( P ) theo m p t d u o n g t r o n , d u o n g t r o n Voi t a m I tao t h a n h m p t h i n h non c6 the tich Ion nhat Viet p h u o n g t r i n h mat cau (S), bie't ban k i n h m g t cau bSng 3N/3 Cau 9.b: Giai h$ p h u o n g t r i n h sau: 2x'^ + x y - x - y + l = 2=0 63 iHiiii' 69 Tuyen ch(>n fy Gim thieti dethi Todn H Q C - Nguyen Phu Khdnh , Nguyen Tat Cty Thu WSQm DANGlAl Cau 3: Voi ^ I P H A N C H U N G CHO TAT CA CAC T H I SINH Cau 1: b) Goi A^a;a'' -3a^ + l j va B|b;b^ -3b^ +1 va a ?t b la cac diem thoa man Do tiep tuyen tren song song voi nen ta phai c6 y'(a) = y'(b) Cau 4: I = j - L a i c : A B = 4V2 o ^(a - b f + (a^ - b ^ + Sa^ + 3b2) Ta s/ =32 (2) up l + (2 + ab)^ ro /g c - 2X - = => X = - hoac X = 4[Jl-2 !t 12 + sin ww X +- = o X = — + k27t sm x + 12 12 Cau 5: Ta c6 E la trung diem B C A E = -!-(AB + A C J F la trung diem D E => A F = ^ ( A D + A E X + • 12 = - A D + - A B + - A C -7 = (*) 4 C D = A D - A C => A F C D 1 -AD+-AB+-AC 4 A D - AC = i A D - i A D A C + -!-AB.AD AB.AC + - A C A D - - A C 2 4 4 AD^ =(aV3) =33^ , AC^ =4a2 A D AC , A D AB => AD.AC = AD.AB = 0, - X t-2 ce w fa f X + —O I 12j Khi phuong trinh (*) tro thanh: 5t^ + 2t - = => t = tuc phai c6: ll7t t(l-2) In AD + i ( A B + Ac) voi -1 < t < Chu y\O the dat t = 12 i> dt hoac A ( ; ) , B ( - ; - ) thoa man yeu cau bai toan 12j , dt Jt(t-2) bo Do ton tai hai d i e m A ( - ; - ) , B ( ; ) X +• xdx ok Vay a, b la nghiem phuong trinh: Dat t = sin + - X = =>x = om (4 - 4t)(5 + 4t +1^) - 32 = ; o t^ + 3t^ +1 + = o (t^ + l ) ( t + 3) = ^ t = -3 Phuong trinh cho viet lai: 5sin^ 3VlO-3x +3 Khi do: D l t t = ab va thay a + b = (do (l)) vao (2) ta duoc : [n + X - ) ( u - x + 2) = 0=> X = Dat t = x^ + => dt = 2xdx hay xdx = - d t =\4i o ( b - a ) ^ l + (2 + a b f = 32 o ^(b - a)^ + (b - af (2 + a b f = A^l 27 dx ]X''+2x 3(a-b)(a + b ) - ( a - b ) = « ( a - b ) ( a + b - ) = =>a + b = ( l ) = sin Khang , dat u = V l - x , dua phuong trinh ve h^: iL ie uO nT hi Da iH oc 01 / Huang 2: (x-3) Phuong trinh tiep tuyen tai B c6 h? so goc y'(b) = 3b^ - b - X DWH Huang 1: 9(l0-3x) = x"* +16x2 -8x^ c ^ ( x - ) ( x + 2)(x2 - x + 15) = Phuong trinh tiep tuyen tai A c6 he so goc y'(a) = 3a^ - 6a Cau 2: Ta c6: sin MTV Cdch khdc: Binh phuong ve, ta duoc: l - x - = -x^ + 4x - bai toan Ta c6: y ' = x ^ - x (a + b f - a b 10 X 4x + 3u ^ fj.^ yg'fj^gQ yg'ja ^u,g,c u^ + X - = a) Danh cho ban dpc Hay J ^ i i ^ ( ) iL ie uO nT hi Da iH oc 01 / a ' ' G - X A = ( X I - X C ) AG = 2GI 10^/3(tT2)t>H b^a a b' fa h^' b"^a -2, -3 b3 a^ = fa b] (a b^ ^b a) [b aj yG-yA=2(yi-yG) Tir d o : phuong trinh: (x - a)^ + (y - b)^ + (z - c)^ = R^ Mat cau (S) di qua diem O, A nen c6: ro /g +b^ c 14156 27 +{b-2f fa a + 2b-2c + |a + b - c - V+2^+2' -1 o a + 2b-2c = (3) A Theo chUcrng trinh chuan T a g i a i h a ( l ) , (2) v a (3) w (2) M3t cau Hep xuc vol (P) va ( Q ) O d(l;(P)) = d ( l ; ( Q ) ) I I P H A N R I E N G T h i sinh chi dugfc chpn lam mpt hai phan (phan A hoac B) ww o a + 2b + c = 15 ( l ) +{c-lf d[{P),(Q)] y ok 27 C(4;2) Cau S.a: Gia su mat cau ta dang d i tim c6 tam I(a; b; c) va ban kinh la R, nen c6 bo V y X C = X M - X A = A D DC f'(t) = t - t - 2 t + 48 va f ' ( t ) > vai V t > — , suy f ( t ) luon dong 10 bien tren nua khoang -;+oo 14156 yi=- D e d : x - y - = 0=> D(x; x - 2) -2i Xethamso f ( t ) = t ' ' - t - l l t ^ + t - v o i ^10^ ,'^c=2yM-yA=2 P = 9t''-16t3-llt2+48t-32 P>f ' Do ABCD la hinh chii nhat nen ta c61 la trung diem ciia AC n2 ^a b^^ , —+ — -2 ^b a ^ a + 2b-2c + = a + b - c - a + 2b-2c + = -(a + b - c - ) Cau 7.a: Cdch i:Go\ la giao diem duong cheo hinh chCr nhat ABCD V i G la trpng tam tam giac ABD nen A, G, I thang hang Theo tinh chat trpng tam tam giac ta (5 3^ de dang tim toa dp diem I - ; - Vi I l a trung diem AC nen biet tpa dp A, I ta se tim tpa dp C(4;2) Vi D thuoc duang thSng x - y - = m a C thoa man phuong trinh Cau9.a: Gia su z = x + y i , ( x , y e O X' +12i = z (x + y i ) ^ + 12i = x - y i - x y + ( x y - y ^ + i = x-yi Do X > r:i> ( l ) x3-3xy2=x (1) 3x2y-y%12 = -y (2) x^ = 3y^ + The vao (2) ta dupe 3(3y^ + l j y - y ^ + = - y = Do x > nen x = Voi, t = — =^ I Vay z = - i => |zj = N/S Cau 7.b: (A) C6 phuong trinhy = x + b,he M Phuong trinh giao diem cua (A) 10 Cau 9.b: 3x^ + bx + 2b^ -10 = (l) ^ ( ' ' M - X N ) ' = y hay (x,, +x^)^ -4x^.x^ = ^ (2) 4b A^^-i^ _ -2=0 - 2b2-10 = 16 — b = -3 hoac b = x =0 y=l hoac c om Cau 8.b The tich khoi non la V = - S h , h = d(l;(P)), day la duong • ok bo ce = 27-h^ ^ V = - f - h ) h fa A p dyng bat dang thuc trung binh CQng, trung binh nhan: ww w (27-h^j + (27-h^j + 2h2 >3^(27-h^)^2h2 hay ( - h ) h < DSng thuc xay 2h2 = 27 - h^ h = Vay, max V = 1871 h = Hon niia: I € d =:> l(2 +1 ;-2t ; - l + 3t) d(l;{P)) = h = o | l l t + 2| = » t = - l hoac \= ^ Voi, t = -1 => 1(1 ;2 ;-4) phuong trinh mat cau (S): (x-l)^(y-2f.(z.4f 74 =27 ' ^ = x + y -1 = x=— 41 y=± 2N/X^V =2 y=0 OETHITHijfs6l2 up ro /g Vay, CO hai duong thang thoa man yeu cau de bai la y = x + 3, y = x - tron giao tuyen ciia mat phang voi mat cau 2x-l = x =l Ta 2b2-10 hoac (3) s/ Ap dung dinh ly Vi-et cho phuong trinh ( l ) : • = r^ - h^ =^ 2x2 + x y - x - y + l=0 (2x-l)(x-l) + y(2x-l) = MN = l ^ c M N ^ = f o ( y , - y , f ( x , - x , f = f Taco: 10 \ = 27 11 2x^ + x y - x - y + l=0 y =x+ b Tu (2) va (3) suy ra: Z iL ie uO nT hi Da iH oc 01 / vaa (E) la : 14 y+ — 29 X-11 B Theo chiforng trinh nang cao y=X+b 29 _14 10 phuong trinh mat cau ( S ) : 11' 11 '11 I PHAN CHUNG CHO TAT CA CAC THI SINH Cau 1: Cho ham so y = x"* - x^ +1, c6 thj la ( C ) a) Khao sat su bien thien va ve thj (C) cua ham so b) Tim tren thj (C) nhCrng diem A cho tiep tuyen tai A cat (C) tai hai diem B, C khac A va B, C nSm ve phia doi voi A 3(cOtX + l) r( 771^ , Cau 2: Giai phuong trinh: 3cot x + -^^ ^-4V2cos x + — =1 (1) j smx Cau 3: Giai phuong trinh: (13 - 4x) ^ + (4x - 3) VS - 2x = + 8N/-4X2 + 16X - ich phan: I = Cau 4: Tinh tich V (2), dx p 7=== O ^ + N/X + N/X + I Cau 5: Cho hinh chop S.ABC, day ABC la tarn giac vuong tai B c6 AB = a, BC = a73, SA vuong goc voi mat ph5ng {ABC) va SA = 2a Gpi M, N iSn lu