luyện thi đại học môn toán

Tuye'n chQtt & Giai thifu dethi Todn UQC - Nguyen Phu Khanh, Nguyen Tai nen A H = D H =^ => ( A B C ) Ke D K va Thu Cty TNHH MTV DWH £)e t u M CO the ke dugc hai tiep tuyen den ( C ) thi I M > R 2t^ + 4t +1 > BC±(AHD) (AHD) A H => D K „ , > ^ h o , c , < - : ^ ( ) (ABC) Phuong trinh di qua hai tiep diem A , B c6 dang: =>DAK = 45°, D A H = 45° ( t - l ) ( x - l ) + (t + 3)(y + ) - = => ADAK vuong can tai K : =>K = H : ^ D H l ( A B C ) Di^n tich tarn giac ABC la: SABC = •^ABACsin60° = — ^ a^Vs a^ AB) = 2V2t^+4t + 10 3t + l Xet f ( t ) = thoa dieu ki^n (•) 2V2t2+4t + 10 2t + 14 Voi t > — thi f (t) > thi ham so f (t) dong bien tren niia khoang Ta hai duang thSng DE va HE va bang DEH s/ Gpi CF la duong cao xuat phat tir C cua tarn giac deu ABC canh a ™ aVs , HE = - C F = 2 DH = =>DEH = arctan2 nen tan DEH = HE , ^(2t2+4t + 10)^ T " =IT Ke HE AB => DE A B Vay goc giiia mp (ABD) va (ABC) la goc giua ro /g om c ok Vay, goc giua hai mp ( D A B ) va ( A B C ) la DEH = arctan2 ce bo Cau 6: Xet ham so: f ( t ) = ln(t + ) - Vt+T voi t > - w fa va f ( t ) = o t = ^' ww f ' ( t ) doi dau t u duong sang am qua nen f (3) la gia tri Ion nhat Suy f (t) < f (3) < 0, do f(x) + f ( y ) + f ( y ) < hay In(x +1) + In(y +1) + ln(z +1) < Vx + + ^ y + + Vz + I I PHAN RIENG Thi sinh chi dupe chpn lam mpt hai phan (phan A hole B) Voi t < - | thi f ' ( t ) = o t = - L^p bang bien thien, suy f (t) < ^ hay d ( N , AB) < ^ DMng thiic xay t = -7 tuc M ( - ; - ) Vay, M ( - ; - ) thi gia trj Ion nhat bSng ^ la diem can tim Cau8.a:GQi M e A j = > M ( l + t i ; + t j ; l - t i ) , N e A j =>N(2-t2;3-t2;-2) => M N = (l - - tpl - - t i ; - + tj) Aj CO vecto chi phucng U j = ( l ; ; - l ) , v i la mat phSng ( P ) vuong goc Aj nen (p) c6 vecto phap tuyen n = U j =(l;2;-l) Gia thie't dau bai ( P ) cat Aj va A j tai M va N , the nen M N nam ( P ) , suy n M N = l ( l - t - t i ) + ( l - t j - t i ) - l ( - + t i ) = A Theo chi/ang trinh chuan = > t = - t i = > M N = ( t i - l ; - l ; - + t i ) M N = ^ ( t i - ) ^ + Cau 7.a: Duong tron c6 tam l ( l ; - ) , b a n kinh R = Giai thie't Vi M e ( d ) nentpa dp M ( t ; t + l ) up aS Taco: f ( t ) = ^ " , ^ ^' 2(t + l ) Ta c6: d ( N ; Taco: f ( t ) = 1 aJs The tich khoi t u dien ABCD la V = - D H S A B C = , ^„ 3t + l iL ie uO nT hi Da iH oc 01 / ADAH vuong can tai H nen co CF = Khang Vift M N = VlT o - ) ^ + = N / T T o tj =0 hoac tj =4 Do CO diem M(1;2;1) hoac M(5;10;-3) 123 TwygH chgn & Giai thifu dethi Toan hpc - Nguyen Phu Khanh, Nguyht Tat Thu ^ I t t S b : T i m d u g c tpa d p d i e m A ( ; 16; ) M $ t p h i n g (P) d i q u a M(l; 2; 1) c6 vecto phap t u y e n n = U j = ( l ; ; - l ) G(?i U j , i ^ , r i p Ian l u g t la cac vecto chi p h u o n g ciia d , A va vecto p h a p p h u o n g t r i n h la: l ( x - l ) + ( y - ) - l ( z - l ) = hay x + y - z - = t^yeh ciia ( P ) Gia s u u j = (a; b; c) a^ + b^ + c^ > M^i p h a n g (P) d i qua M(5; 10; - ) c6 vecto phap t u y e n n = U j = ( l ; ; - l ) Vi d c ( P ) p h u o n g t r i n h la: l ( x - ) + ( y - ) - l ( z + 3) = hay x + y - z - = (d':^)=45° hoac x + y - z - = = > a - b + c = b = a + c ( l ) ^ 2(a + 2b + c ) ' = ( a + b + c ) i^^^^^^^i (2) sVa^+b^+C iL ie uO nT hi Da iH oc 01 / Vay, C O m a t p h a n g ( P ) : x + y - z - = : n e n xx^Ln^ Tir (1) v a (2) suy ra: 14c^ + 30ac = z = a - b i , z - l - i = : a - l + ( b - ) i r : > | z - l - i | = ^ ( a - l f + ( b - ) ^ Thay vao (l) ta c6: ( a - i f + ( b - f + ( a + b i ) i + a - b i = l l + i Cau 9.b: D i e u k i f n: x > s/ a = b+2 a-b =2 up a =4 ro b =2 /g \c b = -1 | ( a - l ) ' + ( b - f =9 om Suy c6 so p h u c can t i m la z = - i va z = + 2i 2V5 c K h i d o |z| = V i va |z| = ok B Theo chUorng trinh nang cao bo C a u 7.b: D u o n g t r o n (C) c6 t a r n ! ( - ; 2), R = va d i e m I thuoc d u o n g thang A ce D u o n g t r o n ( C ) c6 t a m J ban k i n h R ' = l va tiep xiic ngoai v o i d u o n g tron w fa ( C ) suy q u y tich cua d i e m I la d u o n g tron ( K ) C t a m I ban k i n h R + R ' = =25 ww h a y ( K ) : (x + l f + ( y - f V-2 Ta = ll x = + 7t V o i 15a + 7c = 0, chpn a = 7, c = - , b = - , ta tim d u g c d : y = - t < : > ( a - l ) ^ + ( b - f + a - b + i ( a - b ) = l l + 2i (a-lf+(b-2)'+a-b z = 14 K h o a n g each ciia I t o i A la I o n nhat k h i I la giao d i e m cua d u o n g t h i n g d d i z = 14-15t va x > 0, y > Phuong trinh dau tuong duong: - y"* = | l + x^y^ j l o g i x - l o g j y V 5 Neu X > y t h i p h u o n g t r i n h cho v n g h i ^ m N e u x < y t h i p h u o n g t r i n h cho v n g h i ^ m Neu X = y t h i p h u o n g t r i n h t h u t r o thanh: x + V2x + = + Vx + O x - l =V ^ - V x + l = o(x-l) , + , ~^^~]) Vx + + V2x + l Vx + + 72x + l , = O o x= l v i l+- >/x + + V2x + l >0 V^y, h? p h u o n g t r i n h da cho c6 n g h i f m (x; y ) = ( l ; l ) qua J va v u o n g goc v o i A v o i d u o n g t r o n (K) d C O p h u o n g t r i n h : 4x - 3y +10 = Tpa dQ d i e m I thoa m a n h$: 4x - 3y +10 = | ( x - f l ) + ( y - f =25' x = 2,y = x = -4,y = -2 Voi l ( ; ) = > ( x - f + ( y - f = , v o i I ( - ; - ) = > ( x + ) ^ + ( y + ^ = Vay, k h o a n g each t u I t o i A I o n nhat bang 125 ^'^>A o£THiTHiirsdi9 + 51 *Su 9.a: Tim so phuc z thoa man z + —^ " z Theo chi/crng trinh nang cac I PHAN C H U N G C H O T A T CA CAC THI SINK Cau 1: Cho ham so y = 5i = Cau 7.b: Trong mat p h i n g tpa dp Oxy, cho hinh thang vuong ABCD, vuong tai - 3x^ + (3m - 3)x + c6 thi la ( C „ ) va D Phuong trinh AD: x - yyjl = Trung diem M eiia BC eo tpa dp M ( l ; 0) giet BC = CD = 2AB Tim tpa dp eiia diem A b) Tim m de ham so c6 eye d^i, eye tieu eiing voi diem l ( - l ; - l ) t^o Cau 8.b: Trong khong gian toa dp Oxyz, cho mat phang (?):2x + y + z - = iL ie uO nT hi Da iH oc 01 / a) Khao sat su bien thien va ve thi (C) cua ham so m = tam giae vuong tai I va m | t cau (S): x^ + y^ + z^ + 4x + 6y - 2z J-11 = T u diem M tren (P) dyng Cau 2: Giai phuong trinh: tiep tuyen M N den mat cau ( N la tiep diem) Tim M de M N ngan nhat, tinh sin^ X - Vscos^x - i s i n 2x Idioang each ngan nhat (sin x - cos x) - — s i n 2x Cau 9.b: Giai phuong trinh sau: ^ l o g ^ (x^ + 2x j - log j (x + 3) = logg - — ^ x + ( - y ) x + ( - y ) x - ( y +l) = Cau 3: Giai h§ phuong trinh: H(/dNGDiiNGlAl , Cau ,, ,^ r fln^x-31nx + I PHAN C H U N G C H O T A T C A C A C THf SINH -, r—dx J x { l n x ) Cau 5: Cho tam di?n Oxyz c6 xOy = yOz = zOx = a Tren Ox, Oy, Oz lay cac Caul: om trift tieu va doi dau qua moi nghi^m, nghla la phai eo: A'>0 ok c a^+b^+c^ II PHAN RIENG Thi sinh chi dugrc chpn lam mpt hai phan (phan A bo V i l y ' ^ ' ^ ^ = ° nen: jyi'^l) = ( m " ) x i + m + [y'(x2) = y(x2) = ( m - ) x + m + l fa w Cau 7.a: Trong mat phSng tpa dp Oxy, cho duong tron ( K ) : x^ + y^ = va hai Khi do: I A = ( x i + ; ( m - ) x i + m + 2), iB = ( x + ; ( m - ) x + m + 2) A, B) la hai diem thupe ( K ) va doi xung ww diem A (0; 2), B ( ; - ) Gpi C, D (A voi qua true tung Biet r i n g giao diem E eua hai duong t h i n g AC, BD n l m tren duong tron ( K J ) : x^ + y^ + 3x - = 0, hay tim tpa dp eua E Cau 8.a: Trong khong gian tpa dp Oxyz, cho hinh thoi ABCD eo dinh B thupc trye Ox, dinh D thupe mat p h l n g (Oyz) va duong eheo AC nam tren duong l-(m-l)>0om0 thoaman: (a + b - e ) ( b + c - a ) ( e + a - b ) = l Chung minh rang: s/ ro OABC CO the tich Ion nha't ' a +b +c a) Danh cho ban dpc up diem A, B, C cho OA = OB = OC = k > Tim dieu ki?n eua a de t u di^n Ta : Tmh tich phan: I = Tam giae lAB vuong tgi I nen c6: lA.IB = I o ( x i +l)(x2 + l ) + [ ( m - ) x j + m + 2][(2m-4)x2 + m + 2] = o ( m ^ - m + 17)xiX2 +(2m^ - ) ( x i +X2) + m^ + m + = (•) Theo dinh ly V i - et: xj + X2 = 2, Xj Xj = m - thing d : =^ = j Tim tpa dp cac dinh A, B, C, D ciia hinh thoi ABCP Khi (*) tro thanh: biet di?n tich hinh thoi ABCD b i n g ISyJl 126 (dvdt) 127 Taco: S ^ g c = f ^ ^ - ^ C s i n e o " 4m^ - 15m^ + 37m - = 0m = l Do'i chieu dieu ki$n ta c6 m = la gia trj can tim Gpi N la trung diem BC, AN la duofng cao AABC deu sinx - cosx = tanx = o x = — + krt, k e Z iL ie uO nT hi Da iH oc 01 / Taco: A N = — B C = k V S s i n 2 It sin X + -v/Scosx = o sin x + — = l o x = - + k i , k € Z 3j Cau 3: Dieu kien y > -1 ^ A G = ^AN = ^ s i n P 3 Xet AAOG vuong t^i O , ta c6: Phuong trinh thu nhat tuong duong vai: x^ + 2x^ + 2x - = y ^x^ + 3x + sj OG2=A02-AG2=k2 l - - s m ^ l 2) o ( x - l ) ( x ^ +3x + 5) = y ( x + x + 5) R = Cau 2: Giai phuong trinh: tanx + cot2x = Nen (?) va (S) khong c6 diem chung Cau 3: Giai h? phuong trinh: Ta cua I len mat phang (?) s/ Cau 9.b: Dieu ki#n: x > ro up ?huong trinh cho tuong duong vol /g om c ok + 3)(x + 2)(x + l ) ] = « (x2+3x)(x2+3x + 2) =3 (*) bo o[x(x + 2)(x + 3)(x +1)] = log3 o [ x ( x + 2)(x + 3)(x +1)] = w fa ce D|t t = x^ + 3x, phuong trinh (•) tro thanh: t^ + 2t - = o t = -3, t = V6i t = l tuc x^ + 3x = x^ + 3x - = o x = ~^ thoa dieu ki?n ww Vol t = -3 tuc x^ + 3x = -3 o x^ + 3x + = phuong trinh v6 nghifm Vly, phuong trinh cho c6 nghi?m x = -3 + M sin4x x2-y2+l = ( ^ - V ^ - x ) n (67rx - 371^ jcosx.sin^ x + 47i(l + sin^ x) Cau 4: Tinh tich phan: I = '- = _ — ^ ^x Ma IN khong doi nen MN ngan nhat IM ngan nhat, tiic la M la hinh chieu o log3 (x^ + 2x) + logg (x + 3) = log3 - logg (x +1) 2(cosx-l) Vx+l+7y-3+x-y=2 GQi I la tam mat cau, tam giac IMN vuong tai N, ta c6: IN^ + MN^ = IM^ ]log3[x(x rKhangVi$t DETHITH(jfSd20 DUt AB = x =>BC = CD = 2x =>MH = ^ = - ^ Vay, AD = - Va AD = - s u y t = ^ 3 I )VVI Vl + sin-'x Cau 5: Cho hinh chop S.ABCD, day la hinh chu nhat c6 AB = 3, BC = 6, mSt phJing (SAB) vuong goc voi mat phMng day, cac mat phSng (SBC) va (SCD) cimg tao voi m | t phSng (ABCD) cae goc bang Biet khoang each giua hai duong thSng SA va BD bSng S Tinh the tich khoi chop S.ABCD va cosin goc giiia hai duong thing SA va BD Cau 6: Cho a, b, c la cac so thyc duong thoa man a + b + c = Chung minh rang : 8\/abc < 11 PHAN RIENG Thi sinh chi duQC chpn lam mpt hai phan (phan A hoac B) A Theo chiTorng trinh chuan Cau 7.a: Trong mat phang tpa dp Oxy, cho tam giac ABC vuong can tgi A, phuong trinh BA: 2x - y - = 0, duong thang AC di qua diem M ( - l ; 1) diem A nSm tren duong th3ng A: x - 4y + = Tim tpa dp cae dinh cua tam giac ABC biet rang dinh A eo hoanh dp duong Cau 8.a: Trong khong gian voi hq tpa dp Oxyz, cho hai duong thang x _ y + 1_ z 132 x - _ y +1_ z-4 133 Cty TNHHMTVjyWH Tuyen chqn & Gi&i thifu de thi Todn hqc - Nguyen Phu Khdnh , Nguyen Tat Thu Viet p h u o n g t r i n h d u o n g t h i n g A cat ca hai d u o n g t h i n g d ^ d j d o n g t h o i TU u - ' o AB.ACBC BC Theo bai toan, ta co: R = —— = —r- 4S \ A B C v u o n g goc v i m a t phang ( P ) : x + y - z + = Cau 2: D i e u k i ^ n : x 5^ m - , m e Z Cau 7.b: T r o n g m a t p h i n g toa d o O x y , cho d u o n g t h i n g Ian l u g t c6 p h u o n g iL ie uO nT hi Da iH oc 01 / „, u u A sinx cos2x 2{cosx-l) P h u o n g t r i n h cho t u o n g d u o n g : + =— '— cosx sin2x 2sin2xcos2x cos X - cos2x = cos X - o cos^ x = cos x sin2x sin2xcos2x t r i n h la ( d j ) : 2x - 3y - = v a ( d j ) : 5x + 2y - = Viet p h u o n g t r i n h d u o n g t h i n g d i qua giao d i e m cua ( d j ) , ( d j ) Ian l u g t cat cac tia Ox, O y tai A va B dat gia trj nho nha't 'AOAB J Cau 8.b: T r o n g k h o n g gian v o i h§ tga Oxyz, cho d i e m A(3; - ; - ) va mat p h i n g ( P ) : x - y - z + l = Viet p h u o n g t r i n h mat p h i n g (Q) d i qua A, v u o n g goc v o i mat p h i n g ( P ) biet rang mat p h i n g ( Q ) cat hai t r y c O y , Oz Ian l u g t ro up s/ C a u b : G i a i b a t p h u o n g t r i n h : l o g j Vx^ - x + + log^ V x - > ^ l o g j (x + 3) /g HI/OFNGDANGIAI om I PHAN C H U N G C H O TAT CA C A C T H I S I N H c Caul: ok a) D a n h cho ban doc X = cos X = — Giasu: A(0;-l), B ( - V m ; - m ^ - l ) , c(N/m;-m^-l) Ta CO d i ^ n tich t a m giac S ^ ^ B C = ^ B C d ( A , B C ) = m^%/m ( d v d t ) 134 « x =± - + k2n,keZ X e t h a m so: f { t ) = t^ + 2\/t v o i t > , ta c6: f ' ( t ) = 2t + - ^ > i : i f ( t ) vt dong bien t > , k h i p h u o n g t r i n h : f ( x + l) = f ( y ) o y ^ x + Thay vao p h u o n g t r i n h t h u hai ta dugc: Vx + + x - 2 = - x o =3 fx y = (thoa d i e u ki^n) x - x - = (5-x)' Vay, h? CO n g h i f m la { x ; y ) = (3;4) Cau «(67tx-371^1 COS x.sin^x 4: I = J^^ Tarn giac ABC can tai A nen canh day la BC v o l : BC = V m , AB = A C = V m + m * — + krt P h u o n g t r i n h t h u nha't t u o n g d u o n g : (x +1)^ + 2\/x + = y^ + ^ fa w m o i n g h i e m nen ham so da cho c6 cue t r i ww N e u m > t h i y ' = c6 n g h i ? m x = 0, x = - V m , x = \/m va d o i da'u qua X = D o i chieu dieu kien, n g h i f m p h u o n g t r i n h da cho la: x = ± - + k27:,k e Z x>-l Cau 3: D i e u kien: y>3 N e u m < t h i y ' = c6 n g h i e m x = va d o i da'u t u ( - ) sang ( + ) nen c6 cue t r j ( k h o n g thoa bai toan ) ce bo cos Ta tai d i e m phan biet M va N cho O M = O N b) T a c o : y' = x|x^ - m j = 4m Vay, m = thoa m a n yeu cau bai toan B Theo chi/tfng trinh nang cao AB 4m^%/rn m ' ^ - m N / m + = m = (thoa man) z - = va 17 z + z = z z V > cho + Khang Vie , ' \/l + sin^ x u = 67IX - 37r^ Dat dv = , dx + j47tVl + sin^ xdx d u = 6ndx cosx.sin^ X sm X t sin^x 135 Cty TNHH MTV P W H khang ruye'n chgn & Giai thifu dethi Todn hQC - Nguyen Phu Khdnh, Nguyen Tat Thu Khi I = (67rx - 3n^ j^Vl + sin' = 471^ Taco: f ( t ) = | Zau 5: Ha SH A B => S H ( A B C D ) (do (SAB) ( A B C D ) = AB) Nh$n thay, ( - t ) ^ - t ^ = ( l - t ) ( - t ) > Ke H K CD => t u giac H B C K la hinh chu nhat Ta thay B C ( S A B ) Dang thiic xay a = b = c = iL ie uO nT hi Da iH oc 01 / CD ( S H K ) => SKH = ( ( S C D ) , ( A B C D ) ) JI PHAN R I E N G Thi sinh chi du-grc chpn lam mpt hai phan (phan A SBH = SKH = > A S H B = ASHK (g - c - g) => HB = HK = BC = holcB) Do A la trung diem HB Ta thay Z7 A B D K la hinh binh hanh A Theo chi/orng trinh chuan => BD//AK Cau 7.a: Ggi diem A e A => A(4yo - 6; yg) BD//(SAK) ma SA e ( S A K ) => d(BD,SA) = d ( B D , ( S A K ) ) = d ( D , ( S A K ) ) = d ( H , ( S A K ) ) = ^6 = h h^ HS^ •+ HA^ Ta up s/ V V 5 V D|t t = abc, ta w ww ro om + c^ = (a + b + c)^ - 3{a + b)(b + c)(c + a) < 27 -24abc CO V 0 AB va n = (l;4;-2) ciing phuong AB = tn I l +k-m =t - k - m = 4t + k - m = -2t k =0 o t = -l=^A(2;3;2),B(l;-l;4) m =2 =>0 n h o nhat OH' P h u o n g t r i n h da cho t u o n g d u o n g : K h i d o A B nhan O M l a m vec t o phap tuyen Ta vie't d u o c p h u o n g t r i n h A B ' ^ + s/ ' a^' 1+ 3- /g a ^ D a t t=J , t>0 ok V^AOAB/ (x - 2) > ^ l o g ^ i (x + 3) « ilog3 (x^ - x + ) - i l o g ( x - ) > - i l o g ( x + 3) « log3 [(x - 2)(x - ) ] > log3 (x - 2) - log3 (x + 3) Olog3[(x-2)(x-3)]>log3 'x-2^ x+3 «('^x2-9>l« x/lO X>^[W Ket h(?p v o i dieu ki?n, ta dupe ng hi^m cua bat p h u o n g t r i n h da cho la: x>VlO w fa ce bo a t^ +1 X e t h a m s o : f ( t ) = 4.—^ — voi t > (3t4.l) ro ^ om •AB = c ^ AB ;0 , B up Theo bai toan, ta t i m du^c: (3a + h Ta Cach 2: P h u o n g t r i n h d u o n g thang d c6 dang: a ( x - 3) + b ( y - l ) = , ( a , b > O) ilog3 (x2 - 5x + 6) + \\og^-, ww Gia t r j n h o nhat ciia f ( t ) la - dat dug-c k h i t = hay a = 3b P h u o n g t r i n h d u o n g thang can t i m la: 3x + y - = C a u 8.b: Gia s u n g la m p t vecto phap t u y e n ciia ( Q ) Khi n Q l n p ( l ; - l ; - l ) M a t p h ^ n g ( Q ) cat hai tryc O y va Oz tai M ( ; a ; ) , N ( ; ; b ) phan bi?t cho O M = O N nen a = b o a 138 = b^O hoac a = - b * 139 Tuyen chgn & Giai thifu dethi Todn hgc - Nguyen Phu Khdnh, Nxuyht Tai Thu Cau 9.^: Tinh modun cua so'phiic z, biet: z = (2 - i)"^ + ( l + i)'* - ^ — i - OETHITHUfSdzi Theo chUomg trinh nang cao Cau 7.b: Trong mSt phSng tpa dp Oxy, cho hinh vuong ABCD eo phuong trinh duong thing AB: 2x + y - = 0, va C, D Ian lupt thupc dupng thing d j : 3x - y - = 0, d j : x + y - = Tinh di|n tich hinh vuong I P H A N C H U N G C H O T A T C A C A C T H I S I N H Cau 1: Cho ham so : y = x'^ - 3x^ + mx +1 c6 thi la (C^^) a) Khao sat sy bien thien va ve thi (C) cua ham so m = x = -t Cau 8.b: Trong khong gian Oxyz, cho duong thang (d): y = + 2t va mSt cau [z = - - t iL ie uO nT hi Da iH oc 01 / b) Tim m de ham so c6 cue dai, cxfc tieu Gpi ( A ) la duong thang di qua hai diem eye dai, cue tieu Tim gia trj ion nhat khoang each tir diem I - ; — den U 4j duong thSng ( A ) cosx + yfz sin f I X 7t — (S): x^ + y^ + z^ - 2x - 6y + 4z -11 = Viet phuong trinh mat phing (p) vuong goc duong thang (d), cat mat cau (S) theo giao tuyen la mpt duong tron c6 ban kinh r = \ Cau 9.b: Tim so phue z thoa man ( l - 3i) z la so thuc va z - + 5i = 4j Cau 3: Giai phuong trinh: sVZx + l + 2x = loVx-3 +13 HMGDANGIAI ixe" (e^+lj + l Cau 4: Tinh tich phan sau: I = f ^ —dx P H A N C H U N G C H O T A T C A C A C T H I S I N H Caul: a) Danh cho ban dpc s/ up b) Taco y' = 3x^-6x + m Ham so c6 eye dai, eye tieu phuong trinh y' = c6 hai nghi^m phan bi?t.Tuclaeanc6: A ' - - m > o m < om /g ro e^+l ^ Cau 5: Cho hinh hpp dung ABCD.A'B'C'D' eo day la hinh thoi e^nh a, BAD=a voi cosa=-, canh ben AA' = 2a Gpi M la diem thoa man DM = k.DA va N la trung diem cua canh A'B' Tinh the tich khoi tu dien C'MD'N theo a va tim kde C ' M I D ' N Ta I c Chia da thiic y cho y ' , ta dupe: y = y' x _ l _ 3 ok Cau 6: Cho so thuc khong am a, b, c thoa man a + b + c = Tim gia tri nho nhai cua bieu thuc: P = a + b^ + c^ II P H A N R I E N G Thi sinh chi dupe chpn lam mpt hai phan (phan A hoac B) bo Vi y'(xj) = 0,y'(x2) = nen phuong trinh duong thing (A)qua hai diem eye dgi, eye tieu la: y = r2m_2^ ww Cau 7.a: Trong mat phang tpa dp Oxy, cho tam giac ABC vuong tai A va diem B(1;1) Phuong trinh duong thSng AC: 4x + 3y - 32 = Tia BC lay M cho (d;): ^ = = ~^ \ \g (P): x + y - z + = Lap phuong trinh duong thSng (d) song song voi m^t phSng (P) va cat (d^), ( d j ) Ian lupt tai A, B cho dp dai doan AB nho nhat 140 + ^ + hay y = —(2x + l ) - x + l ( 5>/2 BM.BC = 75 Tim C biet ban kinh duong tron ngoai tiep tam giac AMC la — ^ Cau 8.a: Trong khong gian Oxyz, cho hai duong thSng ( d j ) : m , -2 x + — + Gia sir ham so c6 eye d^i, eye tieu t^ii cae diem (xi;yi),(x2;y2) • ce fa w A Theo chUorng trinh chuan 2m \ Ta thay, duong thang (A) luon di qua diem co'djnh A — ; so'goc |*a duong thing lA la k = | Ke IH ( A ) ta thay d ( l ; A ) = I H ^ I A = | I Ding thuc xay I A ± ( A ) o ^ - = - i = - - < » m = l V^y, max d ( l ; A ) = | k h i m = 141 Twygti chpn & Gi6i thi^ dithi Todii hoc Nguyen Phu Khatth , hi^micn Tat Thu CtyTNHHMTV Cau 8.b: Mat cau (S) c6 tarn l(2;-2;l), ban kinh R = 1, M(0;0;m) e O z M | t phMng ( A B C ) c6 vecto phap tuyen n = IM = (-2;2;m - l ) ; m^t phing (ABC) di qua D(1;2;5) nen c6 phuang trinh: ( x - l ) - ( y - ) - ( m - l ) ( z - ) = hay x - y - ( m - l ) z + m - = X = 2-2t Duong thing IM: y = - + 2t z = l + (m-l)t Gpi H la giao diem ciia ( A B C ) voi IM thi to? dp cua H la nghi?m cua h?; X = 2-2t X = 2-2t y = - 2+ 2t y = - + 2t z = l + (m-l)t z = l + (m-l)t 4m + 2x-2y-(m-l)z +5m-3 =0 t =m^-2m +9 Do MA la tiep tuyen ciia (S) nen tam giac MAI vuong tai A va AHIIM, cho n e n t a c o I A = I H I M O I H I M = (do H n i m tren tia I M ) , IH=(-2t;2t;(m-l)t) 0;0;-^ bo ok c Cau 9,b: Phuong trinh thu nhat + Vx^ +4 ^y^ + - y = O X + Vx^ +4 = y + ^y^ +4 ologj t +t fa 7t^+4 + i f(t) = l ce Xet ham so: f (t) = t + Vt^+4 , ta c6: ww w •>0 f(t) dongbiehtren R nen f (x) = f (y) o x = y Phuong trinh thu hai tro thanh: x^ - 8x +10 = (x + 2) V2x-1 (*) Dat u = V2x-1 voi u > 0, thay vao phuong trinh (*), Igp bi?t so t2+4 A = 25(x + 2)^ => u = ^^-^ hoac u = -^^-^ (khong thoa) Voi u = ^ ^ ta du(?c x + = 3V2x-l c6 hai nghifm x = 1, x = 13, ta tim duoc (x;y) = (l;l),(l3;13) 152 I PHAN C H U N G C H O T A T CA C A C T H I S I N H 2x + iL ie uO nT hi Da iH oc 01 / Cau 1: Cho ham so y = — c6 thi la (C) a) Khao sat su bien thien va ve thi (C) cua ham so' b) Tim m de duong thang (d): y = 2x + m cat thj (C) tgi hai diem phan bi?t cho tiep tuyen ciia (C) t^ii hai diem song song voi Cau 2: Giai phuong trinh: sin^ X sin^ 3x = tan 2x (sin X + sin 3x) cosx cos3x Cau 3: Giai phuong trinh: 2^x^ + 2J = sVx^+l Ta e22 + lnx(2 + ln2x) : Tinh tich phan: 1= f -i— klx Cau X In X Cau 5: Cho hinh chop S.ABCD c6 day ABCD la hinh vuong, SA vuong goc voi day Gpi M , N Ian lugt la trung diem ciia SB va AD Tinh the tich ciia khoi chop M.NBCD biet duong thang M N tao voi mat day mpt goc 30° va MN = 2a>/3 / b c^ Cau 6: Cho a,b,c e ri;3] Chung minh rang: — + — + — c b a yb c a) II PHAN R I E N G Thi sinh chi dvtqic chpn lam mpt hai phan (phan A hole B) A Theo chUtfng trinh chuan Cau 7.a: Trong mat phSng voi h^ tpa Oxy, cho hai duong thang dj :x-y-2=0, P2 : 2x + y - = Viet phuong trinh duong thSng A di qua goc tpa dp O cat r d j , dj Ian lupt tai A, B cho OA.OB = 10 Cau 8.a: Trong mat phang tpa dp Oxy, cho hinh chu nhgt ABCD c6 M(4;6) la trung diem ciia AB Giao diem I ciia hai duong cheo nam tren duong thSng (d) CO phuang trinh 3x - 5y + = 0, diem N(6; 2) thupc canh CD Hay viet phuang trinh cgnh CD biet tung dp diem I Ion hon i W s i ' \ Cau 9.a: Tim modun ciia so phuc z biet: z = + i ( l i ) s/ ro =>M om ^m^ - 2m + 9Jt = l o m + = l o m = DETHITHUfSCf23 up = o (-2t).(-2) + 2.(2t) + (m - l).(m - ) t = /g IH-IM DWIi Khang Vi^t 153 Tuye'n chgn & Giai thifu dethi ToAn hgc - Nguyen Phu Khdnh , S ^ i n i c n TalThu B Theo chi/orng trinh nang cao sinx = C a u 7.b: T r o n g mat phSng v o i h ^ tpa d p Oxy, cho ba d i e m A ( - l ; - ) , B(0; 2), sinSx C(0; 1) Viet p h u o n g t r i n h d u o n g t h i n g A d i qua A cho t o n g k h o a n g each _ cos 3x t u B va C t o i A la Ion nha't sin2x = sinx = cos X = X = k7t p h u o n g t r i n h da cho t u o n g d u o n g v o i : va (x + l ) + ( x - x + l ) = ^ ( x + l ) ( x - x + l ) (*) mat ph5ng (P); 2x + y - 2z + = Gpi A la giao d i e m cua d v o i (P) Viet p h u o n g t r i n h d u o n g thSng A n a m (P) biet A d i qua A va v u o n g goc v o l d iL ie uO nT hi Da iH oc 01 / Qjch i C h i a ca ve p h u o n g t r i n h (•) cho x^ - x + , ta d u p e : 2y(4y=•2+3x2) = x4(x2+3) x +1 C a u 9.b: Giai he p h u o n g t r i n h : ,x 2012'* ( ^ y - x + - x + ) = 4024 I PHAN CHUNG CHO TAT CA CAC THI SINH a) D a n h cho ban dpc x^-x + X Ta = 2x + m s/ < » x + { m - ) x - ( m +3)-0,x;^2 x-2 x+1 = o x ^ - x + = v n g h i ^ m v o i m p i xeM TH2: t = r tuc - J i l i _ ^ i « x - x - = (*) >0 (m-6) + ( m + ) > phan bi§t va khac • ^ , g(2)^0 y f ^ f ^ : ok m ^ + m + 60 > (luon diing) X = Cdch 2:Dat c om /g A _ bo ce fa C a u 2: D i e u ki?n: cos x*0, cos x ^ ww y ' ( x i ) = y'(x2)Xj + X2 = o m = - w Tai hai giao d i e m ke hai tiep tuye'n song song k h i va chi k h i 5-V37 „ hoac X = + ^y37 5-N/37 + N/37 2 v = V x ^ - x + l > — , k h i d o p h u o n g t r i n h (•) t r o u = 2v Vx + l = 2Vx^ - x + , binh phuong v e r o i riit gpn ta dupe: THI: I u=4 ^ > , 4x^ - 5x + = , phuong trinh v nghi^m voi mpi xeR TH2: V = 2u o Vx^ - x + = 2Vx + l , binh p h u o n g ve roi riit gpn ta dupe: X^-5X-3.0 P h u o n g t r i n h cho t u o n g d u o n g v o i p h u o n g t r i n h : tan X sin X + tan 3x sin 3x = tan 2x(sin x + sin 3x) 2(u^ + v ^ ) = u v « > ( u - v ) ( u - v ) = 0c:>u = 2v hoac v = 2u Voi V m €€#M tthhii du( d u o n g thSng cat d o thj h a m so tai hai d i e m c6 hoanh d p Vm 6-m - x+1 V^y, n g h i ^ m p h u o n g t r i n h cho la: x = ro ( d ) cat ( C ) tai d i e m p h a n bi|t k h i va chi k h i p h u o n g t r i n h (*) c6 hai n g h i ^ m up b) P h u o n g t r i n h hoanh d p giao diem: t = tuc THI: C a u 1: 2x + -+1 / - ^ ( * ) D a t t = j - i i ± i - - x+ lx^-x + (x^ - x + Khi d o p h u o n g t r i n h (• •) t r o thanh: t ^ - t + = t = hole t = ^ Hl/dNG DAN GIAI Xj ^ X j Ta CO Xj + X2 = cos X sin x = QiU 3: D i e u kien: x > - x-l_y+3_z-3 C a u 8.b: Trong mat phang toa dp Oxyz, cho d u o n g thang d : ^ ^ ~ ^ ^ ~ Y " o sinx _ ^ « x =^ h o S c x |Vay, nghi^m phuong trinh la: x = = ^ x-^ ' ^ ^ (tan X - tan x ) s i n x + (tan 3x - tan x ) s i n 3x = sin(-x) smx -sin3x = -smx + cos3xcos2x cos X cos X I : l = 21i+l2 v i l , = j i ^ ^ x , ' x'^.ln'^x e • l = ' f ^ x i x^ ^ J " ' ^ V d x D a t t = x l n x = ^ d t = (lnx + l ) d x „ X In X 154 155 choii i'-f Ci&i Ihicii ile thi Toiiii - Ni;iii/f(t) = 10t-t2+ — - t^ t ,tudaytac6: V3 Xethamso f ( t ) = t - t + A _ i vai t e X X Cau 5: Gpi I la trung diem cua A B ta c6 M I // SA =>MI1(ABCD)=^MI1(NBCD) IS va £'(t) = 2^^ 1^ t^ t L i p bang bien thien suy f (t) > f ( l ) = 12 11 P H A N RIENG Thi sinh chi dvtgc chpn lam mpt hai phan (phan A ho?c B) A Theo chUcrng trinh chuan e2 Khi do: I , = — I n x ^ Cau 7.a: Do A qua O, nen c6 phuang trinh dang: x = ho^c y = kx Neu phuang trinh A: x = 0, A = A n d i : x - y - = 0=> A ( ; - ) V^NBCD =|-MI-SNBCD A n d j :2x + y - = 0=>B(0;5)=>OA.OB = 10 (thoaman) Va goc giira M N voi mp day chinh Neu phuang trinh A: y = kx la goc M M = 30° Ta Do A = A n d j nen tpa dp cua A la nghi^m ciia h$ phuang trinh: = ^ s/ M N up Tam giac M N l c6 cos 30° =>SfjBCD = S A B C Q - S ^ j N =18a ~~^ = ~ ^ VN.MBCD = 3-MI-SNBCD = - ^ ^ ^ - " ^ = fa b c^ fa c b Cau 6: f(a) = —+ —+ - — - + —+ lb c a; [c b a) 2k 1-k'l-k y = kx x= 2+k 5k y= 2+k >B 5k ^ 2+k'2+k OOA'.OB'-lOOo +4k' 25 + 25k^ ( - k ) ' - (2 + k ) ' ww l./^63a2 2x + y - = Khi do: O A O B = 10 Mat khac: sin30° = o M I = MN.sin30° = a i = aVs ^ MN ^Ayfic Do B = A n d j nen tpa dp ciia B la nghifm cua h^ phuang trinh: om /g 9a' _ 63a^ fa 9a^ ^ ce x X = 18a' va S ^ N , = ^ A N A I • ^ ^ ^ ok c o x = 3aN^ x=i-k 2k y = 1-k w Khi do: SABCD = [ y = kx bo Vi NI^=AN^+AI^9a^= — + — 4 x-y-2=0 ro N I = MN.cos30° = 2aS.— = 3a GQI X la canh cua hinh vuong A B C D k'+lj =(k' + k-2) 63a3N/3 o k'+l = k2+k-2 k'+l =- k ' - k +2 = 100 k=3 k =-l,k = ^ ~[2~ Luang trinh cua duong thMng A la y = 3x, y = - x , y = j x •(a): a'bc ^u8.a:Gpi P(xp;yp) d o i x u n g v a i M ( ; ) qua I nen V i a , b , c e [ l ; ] nen c ^ > > b va f'(a) = O o x = Vbc 11 thupc (d) nen 156 DVVH Khattg Vift T u bang bien thien suy f (a) > f(Sc] = 10 /- - - + - - 2, v / v b b c vc du = —dx dv = — d x e2 , Nf^iiyctt e u = Inx I,/ji^x.Dat Khunh iL ie uO nT hi Da iH oc 01 / Ttiyen + Xp =2xj + yp=2yi _ f f c Z p ] + = o 3xp - 5yp - = (l) 157 Tuye'tt chpn & Gi&i thiC'u dethi Toan hqc - Nguyen fnu A / i i i m vynyen x » > ^ r » Lai CO P M I P N 'PMPN = 0(xp-4)(xp-6) + ( y p - ) ( y p - ) = , T u (l) va (2), suy ra: 34yp^ - 162yp +180 = « yp = ho|c + 3i + 3i^ + i ^ ^ = (2) vol vecto u ' = (l;0;l) => A.: y = - l z=4+t 30 Cau 9.b: Neu x = 0, t u phuong trmh thii nhat suy y = Khi khong thoa phuong trinh thu hai ' Neu X 5* 0, chia ca ve phuong trinh dau cho x^, ta dugc: = l i ± ^ Itill^llllil =-14 + 2i •2 1-i Xet ham so f (t) = t^ + 3t, t € B Theo chuorng trinh nang cao I + a + 2b) om ) = 29 ^ bo w Cau 8.b: A = d n (P), tpa dg cua A la nghi^m cua h$ z= 3+ t +4 -u (**) u +4 -1 In 2012- I va ^ 0 ce fa x= l-t =2 Xet ham so g ( u ) = 2012" f V u ^ T I - u = tren - 2012" 2=i>a = 2,b = 5=>A: 2x + 5y + = b Vu^+4-u Taco: g'(u) = 2012" l n 2 f V u ^ - u + 2012" c 5^ )(a^ + ab>0 DSng thuc xay \ ok d , - = l = ( | a | 5|b|) < - j ^ i l ' ^ «'+b^^ r— Ta V a ^ up ^/a2+b2 s/ a + 2b ro (T / D a t u = x - l , t a d u g c phuong trinh: 2012" /g , a + 2b a + 3b De thay f (t) la ham so dong bien tren Thay vao phuong trinh thu hai, ta dugc: 2012 x - l J ( x - l ) ^ - ( x - l ) = CO phuong trinh: a(x +1) + b(y +1) = d(CA) = M ^ ' Do t u (*) ta dugc — = x hay 2y = x*^ Cau 7.b: Gia su A di qua diem A va c6 vecto phap tuyen la n = (a;b) ;t 0, nen Gpi d = d(B,A) + d(C,A) = + ^ = x3+3x X = ^(-14)'+22==10V2 a + 3b d(B,A) = - ^ = = , Va^+b >3 iL ie uO nT hi Da iH oc 01 / 1-i 2y A{0;-1;4) ^• Vay, hf phuong trinh c6 nghi^m nhat (x; y) = 2x + y - z + = G Taco: V T C P c u a d l a : u ^ = ( - l ; ; l ) , V T P T cua (P) la: Hp = ( ; l ; - ) Vi 158 Aid Ac(P)' ; nen VTCP cua A la: u ^ = Ud;np =(-5;0;-5) cung phuong 159 Cty TNHH MTV DWH di qua A(-3; 0; 2) va cit (A) t^i B cho m | t cau tam B tie'p xiic voi hai m a phSng (Oxz) va (P) DCTHITHUfSd24 Cau 9.a: Goi bon nghi^m ciia phuang trinh z'*-2:'-2zi^+6z-4s0 1 1 tren tap so phuc tinh tong: S = — + — + — + — I PH/VN C H U N G C H O T A T CA C A C T H I S I N H Cau 1: Cho ham so y = -x^ + 2x^ -1 c6 thi la (C) z,,Z2,Z3,Z4 la Zj a) Khao sat sv bien thien va ve thi (C) cua ham so ^2 Zg Z4 B Theo chi/Ong trinh nang cao iL ie uO nT hi Da iH oc 01 / b) Tim diem M nSm tren tryc hoanh cho tu c6 the ke dug^c ba tie'p Cau 7.b: Trong mat phSng voi h? tpa d p Oxy, cho hinh thang can ABCD c6 dien tich bang 18, day Ion CD nam tren duong thang c6 phuong trinh: x - y + = Biet hai duong cheo AC, BD vuong goc voi va cat tai diem l(3;l) Hay V i e t phuong trinh duong thJing BC biet diem C c6 hoanh d p am Cau 8.b: Trong khong gian voi h? tpa d p Oxyz, cho mat phJing (P): x + 2y-z + = tuyen den thi (C) ^ 2cos2x + Cau 2: Giai phuang trmh: tan^ x + 9cot^ x + = 14 sin2x Cau 3: Giai h^ phuang trinh: Khattg Viet x3(4y2+l) + 2(x2+l)>/^ =6 x^+ll + 2^4y^ + l l = x + Vx^+l va duong thSng d : ^ = Z l l = £Z^ Goi d ' la hinh chieu vuong goc ciia d len mat phang (P) v a E la giao diem cua d va (P) Tim tpa d p F thupc (P) s a o 2.(x + 2sinx-3)cosx Cau 4: Tinh tich phan: 1^^ ^— dx , sin"^ X Cau 9.b: Viet s o phuc sau duoi dang luong giac: z = —^- '( n Snf sm — i.sm — I ) 2>/aVb^T?7l (a + l)(b + l)(c-Hl) II P H A N R I E N G ok bo Thi sinh chi dugc chpn lam mpt hai phan (phan A ce P= c om /g ro up Cau 5: Cho khoi chop S.ABCD c6 day la hinh thang can, day Ion AB b^ng bon Ian day nho CD, chieu cao cua day bang a Bon duong cao cua bon mat ben ung voi dinh S c6 dp dai bSng va bang b Tinh the tich cua khoi chop theo a, b Cau 6: Cho a,b,c la cac so thvrc duong Tim gia tri Ion nhat cua bieu thu-c: s/ Ta cho EF vuong goc voi d ' va EF = 5N/3 w fa hole B) ww A Theo chUorng trinh chuan Cau 7.a: Trong mat phSng voi h? tpa dp Oxy, cho duong tron (C) x2 + yz - 2x + 4y - 20 = va duong thang (d): 3x + 4y - 20 = Chung minh d tie'p xiic voi (C) Tam giac ABC c6 dinh A thupc (C), cac dinh B va C thupc d trung diem canh AB thupc (C) Tim tpa dp cac dinh A, B, C biet tryc tam cu.^ tam giac ABC trung voi tam cua duong tron (C) va diem B c6 hoanh dp duong Cau 8.a: Trong mat phSng tpa dp Oxyz, cho mat phing (P): 2x - y + 2x + = va duong thSng (A) : 160 = = Viet phuang trinh duong t h k g (d) Hl/dNG DAN GIAI 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 Cau 1: a) Danh cho b^n dpc b) M(m;0)6Oxvadu6ngth5ngdquaM: y = k(x-a) Gia su d tie'p xiic voi (C) tai diem c6 hoanh dp XQ h?: -x^+2x^-1 = k ( x „ - m ) , k =-4x3+4X0 (1) (2) CO nghiem x^ ° Thay ( ) vao ( l ) , tu ta c6: x^ - hoac 3x^ - 4mxo + = c6 nghi?m Xg Qua M ke dupe tie'p tuyen den d o thj (C) thi phai ton tai gia tri k phan bift De y: x = 0, x = ± thi k = nen c6 tie'p tuyen ^ , 2cos2x Cau 2: tan'^ x + 9cot x + + —-— = 14 sin2x sin2x tan^ x + 9cot'^ x + cotx - tan x + 2(tan x + cot x) = 14 (tanx + 3cotx)^ + tanx + 3cotx-20 = 161 Tuye'n chqn b Giai thifu dethi Todn HQC - Nguyen PM Khdnh , Nguyen Cau 3: Dieu ki#n: x > Nh^n thay, (O; y) khong la nghi?m ciia Tat Thu thi H M = H N = HE = - la ban kinh duong tron va SE = SM = SN = b phuong trinh Xet X > 0, phuong trinh thu tro thanh: SH = | V b - a b>^ 2) I 2y + y V y + l = ^ + ^ J - ^ + l (*) D a t C N = x thi BM = 4x,CE = x, BE = 4x •>0 V e vr do ham so dong bien nen f (2y) = f ^1^ Tam giac HBC vuong H iL ie uO nT hi Da iH oc 01 / Xet ham so f(t) = t + tN/t^Tl Ta c6: ('{t) = l + ^J^Tl+~ +1 nen H E = E B E C C ^ — = x < » x = - , 4 o2y = X ^ Thay vao phuong trinh (*): x^ + x + 2(x^ + l)>/x = Ve trai cua phuong trinh la ham dong bien tren (0;+oo) nen c6 nghi^m nhat X = va he phuong trinh c6 nghi?m V^y, '4^ U Ta -dx ^•' cSin' m' X sin^ X t Ll , —cotx 2; 2 (dvtt) '—^ + (a + b + c + l f >A L " 1 w ww Vay I = I i + l = N / - doan M N vai M , N Ian lugt la trung diem cac canh AB, CD va M N = a 27 27 Xet ham so': f (t) = • ( 2f Ta c6: f ' ( t ) = — - + 81 = ^ hay J d^t vol t > va f'(t) = 0, t > t = T u day, ta c6 f ( t ) < f | j C a u 5: Goi H la chan duong cao cua chop thi H phai each deu cac c^nh cua day Suy hinh thang can ABCD c6 duong tron npi tiep tam H la trung diem (a + b + c + 3)^ Dat t = a + b + c +1 nen c6 t > Liic nay, P < (2sinx-3)cosx 2f2sinx-3 / \ / r ^ dx= I d(smx) = V - sin^x ^ sm x va truong hg-p ta chung minh dug-c H nSm day 27 Luc do, bieu thuc da cho tro thanh: P < • a + b+c+ fa It n Duong tron tiep xiic voi BC tai E + up /g c l^r ok om f ' vsin^ X u2 a'' + b'^ + c^ + > bo 71 X N C 27 ce „ sin-' X D ys.ABCD=l-^-l^l^^ ro n ^rXCOSX • s/ ^ AB = 2a, suy S^BCD = ^ C a u 6: A p dyng bat d5ng thuc trung binh cpng - trung binh nhan 2.(x + s i n x - ) c o s x , ^ xcosx_, (2sinx-3)cosx C a u 4: 1= dx= f ^dx+ ^ dx sm X „ sm X 7t \^ CD = i " o P ^ Do gia trj Ion nhat cua P = 8 dugc a = b = c = 11 PHAN RIENG T h i siifh chi dugic chpn l a m mpt h a i phan (phan A hoac B) A Theo chi/orng trinh chuan C a u 7.a: Duong tron ( C ) c6 tam l ( l ; - ) va ban kinh R = 163 Tuye'n chgn &• Gi&i thi$u dethi Todn hgc - Nguyen Phu Khdnh , Nguyen Tat Thu 3-8-20| IH = d(l,CD) = 2V2 =^ CI = = V2t2-4t + 10 hoac t = - l = > C ( - l ; l ) = = = R Suy d tiep xiic voi (C) Gpi H la tiep diem ciia (C) va d Toa dp H la nghi?m ciia h? phuang trinh H ( a ; a + ) G ( d ) , IH = (a - 3;a +1), IH CD « a - + a +1 = ^ a = H(l;3) ^ D(3;5) =^ CD = 4^2 (IC): y = 1, A(x;l) IC (x > 3) SAU^T^=- A(5;l) AB//d:x-y-4 =0 - ^ B(3;-1) = AB n DI BC: x + 2y - = DI: x = CauB.b: E e ( d ) = * E ( - + 2t;-l + t;3 + t) E e ( P ) : x + y - z + = 0=>t = l=>E(-l;0;4) L a y d i e m M ( - ; - l ; ) € ( d ) , ta c6: EF = ME.nj^ = (-1;1;1) /g ro up s/ Ta = o ( b - f + r2o-3b -2 y = 100 om 4 - c , BI=(-11;2) AC = c + 2; ok AC.BI = o - l l ( c + 2) + ^ i j ^ = o c = C ( ; ) bo A C l BI AIAB vuong can (AB + CD).(IH + IK) c c e d = > C c;-2 - c • AB = X - V2 IK x - x^ = - A(-2;-6) yA=2yi-yH YA =-6 Goi M la trung diem canh AB Do HA la duong kinh nen HM AM Tam giac HAB c6 HM vua la trung tuye'n vua la duong cao nen AHAB can 20-3b'l tai H =^ HB = HA = 2R = 10, B e d B b;- doi xung ciia H qua I b = -4 (b-4)^ + 12-3b^ = 0 o b ^ - b - = o b = 12 Do X B > ^ B(l2;-4) lA = |x - 3| iL ie uO nT hi Da iH oc 01 / 3x + y - = x = H(4;2) + y ^ - x + y - = [y = Do I la true tam AABC va IH BC =i> A e IH Ket hop A e (C) => la diem r20-3b -2 HB = o ( b - ) ^ t = (khong thoa) w fa ce Cau 8.a: B e (A) B(t -1;6 - t;2t - 5), mat cau tam B tiep xuc voi hai mat phSng (Oxz) va (P) - t h i A eat ( C ) tai hai d i e m p h a n bi?t A , B c6 hoanh d p khacl Taco: A ( X , ; X , - m ) , B ( X ; X - m ) = > d ( A ; O x ) = X, - m , d ( B ; O y ) = Theo bai toan, ta c6: X j - m = X j , theo V i - et: A B C D ( A B // C D ) Bie't hai d i n h B ( ; ) va C ( ; - ) Giao d i e m I ciia h a i d u o n g 166 167 2m+1 ' " ^ 2m + l m = m - T u ta dugc Xj.Xj • Cau 2: cos^ x - I ^ 2 sin^ 6x = o x = Cau 6: v^b + ^ Chia hai vecho x^ + x +1, ta dugc : x-1 x - l < x ^ + x + lx^>-2 (luon dung) om Voi t < tuc c V a i t > tuc | - A - l - > c : > x - l > ( x ^ + x + l)4x2+3x + 5 I C = I B o l t + t - = c ^ t = - - (khong thoa t > ) hoac t = => I ( l ; l ) Phuong trinh duong thang I C : x + y - = ce Dat t = cosx=>dt = - s i n x d x 2t-3 1 ^.t = — I n 2t + / 12 2t + A Theo chi/orng t r m h chuan fa = AH^ => d [ A , ( S M P ) ] = A H 11 PHAN RIENG T h i sinh chi dupe chpn lam mpt hai phan (phan A hoac B) sinx = -(2cosx-3)(2cosx.3) w -sinx ^ sinx ww Cau 4: Tmh t.ch phan: sinx minP^Vs bo Vay, bat phuong trinh da cho c6 nghi?m x > (SMP) < ^ ( b + c) = ^ ( - a ) Xet g(b) = Vb + V ^ t < t ^ + < = > t < l hoac t > x-l => A H = SP bien voi mpi a e [ ; ] va g(a)>g(o) = \/b+ %/c = N/b + V - b voi t > 0, ta dug-c bat phuong trinh: 1 SP (SMP) Xet g(a) = a + V b + v/c voi a [ ; ] Ta c6: g'(a) = l > , suy g(a) dong -+2 Ta + X + • (SAP) ± MP => (SAP) maxP = — k h i a = —, b = c = — 2 s/ Vx^ x-l (SM,NP) = SMK = arccos K h i d o P a - c > - b Tim gia tri ww II PHAN RIENG Thi sinh chi dugc chpn lam mpt hai phan (phan A hoac B) Cau 7.a: Trong mat phing Oxy, cho hinh binh hanh ABCD c6 B(1;5) va duong cao A H CO phuong trinh x + 2y - = 0, voi H thuQC BC, duong phan giac cua goc ACB c6 phuong trinh la x - y - = Tim toa dinh A , C, D Cau 8.a: Trong mat p h i n g Oxyz, cho ba duong thang J 172 x+1 -1 = y-3 = z + -, J d, : x-1 I y z , = - = - v a d o : I 7^ 14 o m = - — 3 Cau 2: Bien doi phuong trinh ve dang cosl0x + l + cos8x=cosx+2|4cos''x-3cos3xjcosx A Theo chi/ofng trinh chuan d, : Hl/dfNG DAN GIAI s/ up BN Cau 9.b: Goi z, va Z2 la hai nghiem phuc ciia phuong trinh: ^x + 2y + 74x + y = '^r In^x + lnx , Cau 4: Tinh tich phan: I = J i(lnx + x + l ) , dai 4^/3 X y z+2 -==-r = —:;— - ^ 2cos9x.cosx + l=cosx + 2cos9x.cosx cosx = l x = k7t ( k e Z ) Vay, phuong trinh c6 hp nghiem Jiu 3: Dieu Ki?n: 11 cho viet lai: x + 2y^0 4x + y >0 x(3x-7y + l)=:-2y(y-l) (1) 7x + y + x + y =5 (2) 17.-^ Tuye'u chyn & Giin thiju tic thi Toan h(fc-Nguyen PMi Khanh , Nguyen Tat 77in (1) » X (3x - y + ) = - y (y - ) » Cty TNHH MTV DWH Khang Vift 4 V i —b > a - c > —b hay a < —b + c < b + c nen < 4x < z x 5 Bx^ _ ( y - ) x + ly'^ - y = c > ( x - y + l ) ( x - y ) = « y = 3x + l (S) hoac x = 2y (4) AXXQC: V7X + + N/TX + T = d i e u k i ? n : x > V49x^+21x + = 11 - 7x 11 ^ - y n-7x>0 o 24z = x+y 17 175x = 119 , ^ / \x va xet Q ( y ) = y+z z+x 24(z-x)(y2-zx) Ta c6: Q ' ( y ) = 76 50x + 24x T h a y (4) vao (2)tadu(?c: ^ In^x + l n x Cau 4: (Inx + x + l f +J9y = o y = l = > x = (i^i„^)3^ Inx - + l=>dt = + lnx dx ( l + Inx) s/ Ta t = + a^ Sa^ A ' K ^ Vx-Vz z+ x c ok bo ce « t + fa ww Vay, the tich k h o i t u d i ? n B A ' C K la : a^^/l5 12(b + c - a ) 12(c + a - b ) 25(a + b - c ) Cau 6: Ta c6: 6: 0Q = - P = —5^ -+ — - + c a b 2x=b+c-a a= y+ z Da,t 2y = c + a - b = > b = z + x 2z = a + b - c c= x+ y ^ 5Qz _ 48t V^ + V ^ z+ x 50 t+ t^+i voi af(t) = i « i ^ t+ t^+i o48t'*-100t^-104t2-loot+ 48 = 0 Y"Y«t = i.Tathay, f / Su Suy Q > Q ( V ^ ) > £ w 2a 48N/X (t + l ) ^ ( t ^ + l f Dvrng d u o n g cao B I ciia tarn giac A B C t h i B I ( C A ' K ) n e n B I la d u o n g cao cua k h o i chop B A ' C K va B I = , va Q ( V ^ ) = T a c o : f ( t ) t ^ - l O O t ^ - t ^ - l O O t + 48 om A ' K = Ir/lS A A - = a N / l , A C ' = A ' C = a V O A - = a V , O K = 50z 3'2 up A ' H ^ ro AA'2 48Vx /g C a u 5: A ' K H = 30° => A ' K = A ' H , >0 247 —->24+ ::^Q>64 y + z 50x 50 •>40 z+ x De thay k h i y -> -oo t h i l n x ( l + lnx) + lnxDat x+ y 24z 50x = + —— x+y y+z z+x va Q ' ( y ) = o y = ±Vzx f (x + y ) ( y + z) iL ie uO nT hi Da iH oc 01 / T h a y (S) vao (2) ta 24x Q = ^ ^1^ v2y = 56, f t v3y -25 t + l l - = Gi = 57 \ - = hay - P > o P < - I D a n g t h u c xay k h i a = 2c, 3b = 5c ^ A Vay, m a x ? = - k h i a = 2c,3b = 5c P H A N R I E N G T h i s i n h chi dugrc chpn lam mpt hai phan (phan A Hoac B) Theo chUtfng trinh chuan Cau 7.a: B C d i qua B ( ; ) va v u o n g goc A H nen B C : - 2x + y - = Toa d p C la n g h i ^ m cua h | : -2x + y - = x-y-l=0 >C(-4;-5) ^ 175 Tuyen chgn b Giai thifu dethi Todn hgc - Nguyen Phu Khdnh , Nguyen ~CfyTNlni Tat Thu x+y-6=0 [x-y-l =0 >K Phuang trinh AC: x - 2y - = 0, A = CA' n A H => A(4; -1) =2»b = l Trung diem l(0;-3) cua AC, dong thoi I la trung diem BD nen D(-1;-11) =:> A ( l + a;a;a), B(b;2b;b - 2) Aedj/Bedj AB = (b - - a;2b - a;b - - a ) AB.u[ = l + a - b + 2b-a + b - - a = A B = N/6 ^ [(b-l-af+(2b-af+(b-2-a)^=6 b= ] hoac Voi a = 3=^c = - | =>(P): f+ ( - | a = -5 ri.|iz-3| = | z - - i | ^{y => iz - = - y - + xi va z - - i = (x - 2) + (y -1) + 3f + +(y-if ={x-2f x = - y - (l) Cau 9,b: Co A' = 4(2 - i f + ( l + i)(5 + 3i) = 16 up , 1 + Z2 =9 /g ok bo Vol y = - l = > x = l = > z = l - i fa ce V26 C a u 7.b: Toa dp tarn duong tron la l(4;l);ban kinh R = I PHAN C H U N G C H O TAT CA CAC T H I S I N K C a u 1: Cho ham so y = x"* - 2mx^ + m c6 thj (C^) •s a) Khao sat su bien thien va ve thj (Cj) cua ham so' w ww B.Theo chuorng trinh nang cao DETHITHUfSd27 c , 15y^ + lOy - = o y = - hoac y = ^ Goi A la duong th5ng qua A va cat duong tron tai M, N phuang trinh cua A CO dang la: y = k(x - 9) + b) Tim tat ca cac gia trj thuc cua m de thj ham so (C^ ) y = x'* - 2mx^ + m C O ba diem eye trj t^o mpt tam giac c6 ban kinh vong tron npi tiep Ian Hon C a u 2: Giai phuong trinh: sin^ x=cos^ x+cos^ 3x GQI H la trung diem M N , ta c6: I H = +1 Vay phuang trinh c6 hai nghifm phuc: om Tir ( l ) va (2) suy 3(-2y -1)^ + 3y2 - y - = = .-i=:> R2- MN N2 = Vl7-12=>/5=d(l;A) •au 3: Giai h? phuong trinh: •k = : ^ y = x - 1 = ro < » x + ( y + f = x + ( y + l f c ^ x + y - y - = (2) |4k-l-9k + x + 3y - 4z - = V^y, matphSng c a n t i m ( P ) : j^ + y - z - z + 3i| = |2z + i| o |x + (y + 3)i| = l2x + (2y + l ) i Voiy = - ^ x = - - ^ z a= -l Voi a = - :r> - = (khong thoa) c b = -2 C a u 9.a: z = x + y i (x, y e =l « a=3 Ta fa = l Matkhac A M = v / s B N » A M ^ = 3BN^ « ( a - l ) ^ + + = s/ Ta c6: b iL ie uO nT hi Da iH oc 01 / C a u 8.a: dj c6 vecto chi phuong la u i = ( - l ; l ; l ) Voi K/.,),/x Vift -+-+-=1 x y z • (P): - + ^ + - = 1, vi (P) di qua M, N nen ta c6: a b c 1 a c >A'{6;0) 2'2 nvvil Caul8.b: Gia sir (P) cat Ox, Oy, Oz Ian lupt tai A(a;0;0), B{0;b;0), C(0;0;c) Cau Ggi A' la diem doi xung B qua duong phan giac (d): x - y - = 0, BA n (d) = K Duong thSng KB di qua B va vuong goc (d) nen KB c6 phuong trinh x + y - = Toa dp diem K la nghi^m cua h?: MIV 21 J9x+y V x W ] 2x \ y > y -9 = 18 1x2 ) Tuyen cht?n & GiaithJQU aethi , Toan HQC - Nguyen Phu RhAnh , Ni;,nf,'„ Tnf r IHU VxV''+3xe''+e''+K Cau 4: Tinh tich phan: I = I dx n y I J V H H MIV UVVH Khang VI HlTtifNGDANGlAl I PHAN CHUNG CHO TAT CA CAC THI SINH Cau 1: a) Danh cho ban dpc b) m > thi thj ham so da cho eo cue trj xe^+l Cau 5: Cho tu di^n deu SABC Gpi (P) la mat phSng di qua duong cao SO ciia tu di^n; mat ph3ng (P) cat cac mat phSng (SBC), (SCA) va (SAB) Ian lupt theo A ( ; m ) , B(-Vii^;m-m2),c(>A^;m-m2) =^S.^^^ = (ABC) cac goc a, p, y Chung minh: tan^ a + tan^ (3 + tan^ y = 12 p = vm^ +m + V m iL ie uO nT hi Da iH oc 01 / cac giao tuyen SM, SN, SP Cac giao tuyen Ian lupt tao voi mat phing Cau 6: Cho cac so thuc duong a, b, c doi mot khac thoa man 2a < c va Si L^i'^o r = ^ > l « V m - % m >m^+ m c \ ab + be = 2c^ Tim gia tri Ion nhat cua bieu thuc: P = + •; + • a-b b-c c-a II PHAN RIENG Thi sinh chi dug^c chpn lam mpt hai phan (phan A o (cos 3x + cOS5x) cos 3x=0 o cos2x.cos x.cos 3x=0 Cau 7.a: Trong mat phang tpa Oxy, cho parabol (P): y = x^ + 2x - Xet eos2x = hinh binh hanh ABCD A { - ; - ) , B(2;5) thuoc (P) va tam I cua hinh binh cos X up ro Cau 8.a: Trong mat phing tpa Oxyz, cho diem A(l; 2; -1), B(2; 1; I ) ; C(0; I ; 2) i = ^^-^ = ^-i^ Hay lap Cau 3: Nhan thay om phuong trinh duong thMng ( A ) di qua true tam cua tam giac ABC, nam 2x=r-+k7t cos2x = x=-+k4 2 cos3x = 2x 9x + ^ Cau 4: I = Ij w Cau 7.b: Trong mat phSng tpa dp Oxy, tam giac ABC can tai A, c6 dinh B va C Q = 9xy + x=-+k6 18x2 y2 + Z_ + , v ^ I^, do: I, = /(xe" + l)dx va Tinhli= ww thupc duong thing di: x + y + = Duong cao di qua dinh B la d2: x - 2y - = 0, = f^^^il^x '^e" +1 I= X ffxe' voi V t e ; - ok NO 180 ro NM Xet f ( t ) = l ^' /g , , om >/3cosm-3sinm NO s/ a>/3cosm-3asinm 23 N NM O sinm.- c , +- ^-^ c c Ta A p dung djnh h' Menelauyt cho AOHM c6 CM H N NM O H AO m Vay, ( ) dung Do c6 dieu phai chiing minh sinm a ^2sm^ a^ (3) aVScosm (3cosm - V S s i n m f IS^sin^m + cos^mj Mr CM = H M - H C = Op2; (3cosm + >/3sinm)^ ta chung minh (2) Khg^w H K h a t , tuc I la tiep diem cua tiep tuye'n (d)//AB cua (P) Phuong trinh duong thang A B : y = 3x - 11=> ( d ) : y = 3x+ c iV3 r/3cosm + sinm 6sinm 3cosm+ ^/3sinm (d) tiep xuc (P) tai diem I I 2'~4 •C - ; - ) , D 181