1 A- PHN CHUNG(7,0 im) Cu I (2,0 im) . Cho hm s: 4 22 3 = y x x(C). 1.Kho st s bin thin v v th (C) ca hm s. 2.Tm m ng thngy m =ct th (C) ti bn im phn bit M, N, P, Q ( sp th t t tri sang phi) sao cho di cc on thng MN, NP, PQ c gi s l di 3 cnh ca mt tam gic bt k. Cu II (2,0 im) 1.Gii phng trnh: 2sin .sin 4 2 2 cos 4 3cos .sin .cos 26x x x x x x | |= |\ . 2.Gii h phng trnh: ( ) ( )( )2 22 3 8 1 ,y8 3 13+ + =e+ + + =Rx y y xxx x y y. Cu III (1,0 im) .Tnh tch phn: I = 4211 4++}xxx edxx xe. Cu IV (1,0 im). Tnh th tch khi t din ABCD bit AB = a, AC = b, AD = c v
0BAC CAD DAB 60 = = = . Cu V(1,0 im). Chng minh phng trnh:( )11xxx x+= +lun c nghim thc dng duy nht. B- PHN RING(3,0 im). Th sinh ch c chn mt trong hai phn B.1. CHNG TRNH CHUN Cu VI a(2,0 im) 1. Trong mt phng Oxy, cho ng thng: 1 0 d x y + =v ng trn( )2 2: 2 4 0 C x y x y + + = . Tm ta im M thuc ng thng d m qua k c hai ng thng tip xc vi ng trn ( ) Cti A v B sao cho
060 AMB = . 2.TrongkhnggianOxyz,cho3im( ) ( ) ( ) ; 0; 0 ,B 0; ; 0 ,C 0; 0; Aa b c via,b,clccsdng thay i v tha mn 2 2 23 a b c + + = . Xc nh a, b, c sao cho khongcch tgc to O( ) 0; 0; 0n mt phng( ) ABCt gi tr ln nht. CuVIIa(1,0im).Tma,beRphngtrnh 2z az b 0 + + = cnhnsphcz 1 i = + lm nghim. B.2. CHNG TRNH NNG CAO Cu VI b(2,0 im) 1.Trong mt phng Oxy, cho prabol()2: P y x = . Vit phng trnh ng thng d i qua M(1; 3) sao cho din tch hnh phng gii hn bi (P) v d t gi tr nh nht. 2. Trong khng gian vi h to Oxyz, cho hai im( ) ( ) A 1; 5; 0 ,B 3; 3; 6v ng thng d:1 12 1 2x y z + = =. Xc nh v tr ca im Ctrn ngthng d din tch tam gic ABC t gi tr nh nht. Cu VII b (1,0 im).Gii phng trnh:
( ) ( ) ( )2 32 2 4 2 4 24 1 2221log 1 log 1 log 1 log 13x x x x x x x x + + + = + + + + . 01206008858 D: D: 2 A- PHN CHUNG(7,0 im) Cu I (2,0 im) . Cho hm s: 2 3
2xyx= (C). 1. Kho st s bin thin v v th (C) ca hm s. 2. Gi Ilgiao im ca hai tim cn. Tm im M thuc (C). Bit tip tuyn ca (C) ti M ct cc ng tim cn ti Jv Ksao cho ng trn ngoi tip tam gic IJK c din tch nh nht. Cu II (2,0 im) 1.Tm nghim0;2x | |e |\ . ca phng trnh sau y :2 234sin 3sin 2 1 2cos2 2 4xx x | | | | | | = + |||\ . \\ . .. 2.Gii h phng trnh: 3 32 28 27 184 6x y yx y x y+ =+ = . Cu III (1,0 im) . Tnh tch phn: I = 210 5 901 cos .sin .cos I x x xdx= }. Cu IV (1,0 im). Cho hnh chp S.ABC c y l tam gic ABC vung cn ti nh B, BA = BC = 2a, hnh chiu vung gc ca S trn mt phng y (ABC) l trung im E ca AB v SE = 2a. Gi I, J lnltltrungimcaEC,SC;MlimdingtrntiaicatiaBAsaocho ( )00 90 ECM = < e . B- PHN RING(3,0 im) B.1. CHNG TRNH CHUN Cu VI a(2,0 im) 1.TrongmtphngOxy,chohnhthoiABCDcphngtrnhhaicnhAB,ADthtl: 2 2 0 ;2x + y + 1= 0 x y + = . Cnh BD cha im M( ) 1; 2 . Tm to cc nh ca hnh thoi. 2.TrongkhnggianOxyz, cho ng thng 1 2:1 2 2x y zd += =. Vitphng trnh mt phng (P) bitrng (P) cha ng thng d v to vi mt phng (xOy) mt gc nh nht. Cu VII a (1,0 im).Tm tp hp im M m ta phc ca n tha mn iu kin:z 2 i 1 + = . B.2. CHNG TRNH NNG CAO Cu VI b(2,0 im) 1.TrongmtphngOxy,chotamgicABCcntiBeOx,phngtrnhcnhABcdng: 3 2 3 0 x y = ; tm ng trn ngoi tip tam gic l( ) 0; 2 I . Tm to cc nh ca tam gic. 2. Trongkhng gian Oxyz, cho hai im( ) A 2; 0; 0 v( ) J 2; 0; 0 . Gi s( ) l mt phngthay i, nhng lun i qua ng thng AJ v ct cc trc Oy, Oz ln lt ti cc im( ) B 0; b; 0,( ) C 0; 0; cvib, c 0 > . Chng minh rng: bcb c2+ =v tm b, c sao cho din tch tam gic ABC nh nht. Cu VII b (1,0 im).Tnh 0 0 1 1 2 2 3 3 2010 20102010 2010 2010 2010 20102 C 2 C 2 C 2 C 2 CP ...1.2 2.3 3.4 4.5 2011.2012= + + + . 01206008858 3 A- PHN CHUNG(7,0 im) Cu I (2,0 im) . Cho hm s: 3 21 54 43 2= y x mx mx(C). 1. Kho st s bin thin v v th (C) ca hm s khim 0 = . 2. Tm m hm s t cc tr ti 1 2, x xsao cho biu thc : 2 22 12 21 25 125 12x mx m mAx mx m m+ += ++ + t gi tr nh nht. Cu II (2,0 im) 1. Gii phng trnh:( ) tan tan 2sin 1 6cos 3 sin 1 tan tan2xx x x x x x| |+ + = + + |\ .. 2. Gii h phng trnh:6 2 65 26 2 62522 3322 33xyx x yx xxyy y xx y+ = + ++ = + + ( ) ,y x eR . Cu III (1,0 im) .Tnh tch phn:( )ln5ln 2.10 1 1x xdxIe e= } Cu IV (1,0 im). Cho hnh chp t gic S.ABCD c y ABCD l hnh vung tm O, cnh bng a. CnhbnSAvunggcviyhnhchpvSA a 2 = .GiHvKlnltlhnhchiucaA trn SB, SD. Chng minh( ) SC AHK v tnh th tch O.AHK. Cu V(1,0 im). Tm m phng trnh sau c nghim: ( ) ( )4 3 3 3 4 1 1 0 m x m x m + + + =B- PHN RING(3,0 im). Th sinh ch c chn mt trong hai phn B.1. CHNG TRNH CHUN Cu VI a(2,0 im) 1. Trong mt phng Oxy, cho hai ng trn:( ) ( ) ( ) ( )2 22 21 2C : x y 9 ;C : x 1 y 1 25 + = + = . Gi A, B l cc giao im ca( )1Cv( )2C . Vit phng trnh ng thng AB. Hy chng minh rng nuK AB ethKI KJ . 2.Gii h phng trnh: 2 2221xyx yx yx y x y+ + =++ = . Cu III (1,0 im)Tnh din tch hnh phng gii hn bi cc ng:, 0, 0,1 sinxy y x xx = = = =+ Cu IV (1,0 im). Cho hnh chp tam gic S.ABC c y l tam gic vung B,cnhSA (ABC) . T A k AD SB vAE SC . Bit AB = a, BC = b, SA = c.Tnh th tch ca khi chp S.ADE? Cu V(1,0 im).Cho, , a b cl cc s dng tha mn 1 1 12011a b c+ + = . Tm gi tr ln nht ca biu thc: 1 1 12 2 2Pa b c a b c a b c= + ++ + + + + + B- PHN RING(3,0 im). Th sinh ch c chn mt trong hai phn B.1. CHNG TRNH CHUN Cu VI a(2,0 im) 1.TrongmtphngOxy,chobnim( ) ( ) ( ) ( ) 1; 0 , 2; 4 , 1; 4 , 3;5 A B C D .TmtaimM thuc ng thng: 3 5 0 x y A =sao cho hai tam gic MAB v MCD c din tch bng nhau. 2.TrongkhnggianOxyz,chomtphng() : 2 1 0 P x y z + = vhaingthng 11 2 3:2 1 3x y zd + = = , 21 1 2:2 3 2x y zd+ = = .VitphngtrnhngthngAsongsongvi mt phng (P), vung gc vi ng thng 1dv ct ng thng 2dti im C c honh bng 3.Cu VII a (1,0 im) Tm phn thc ca s phc( ) 1 ,nz i n = + eN. Trong n tha mn( ) ( )4 5log 3 log 6 4 n n + + =B.2. CHNG TRNH NNG CAO Cu VI b(2,0 im) 1. Trong mt phng Oxy, cho elip( )2 2: 116 5x yE + =v hai im( ) ( ) 5; 1 , 1;1 A B . Tm mt ta im M nm trn (E) sao cho din tch tam gic MAB ln nht. 2.TrongkhnggianvihtaOxyz,chomtphng () ()2 2 2: 2 2 16 0, : 4 2 6 5 0 P x y z S x y z x y z + + = + + + + = . im M di ng trn (S), im N di ng trn (P). Tnh di ngn nht ca MN. Xc nh v tr ca MN tng ng. Cu VII b (1,0 im) . Gii h phng trnh sau: ( ) ( )22 22 2 2 02log 2 3log 1 4y xy y xx y y + + = + + =. 11 A- PHN CHUNG(7,0 im) Cu I (2,0 im) . Cho hm s: 2 12xyx+=+ c th( ) C1. Kho st s bin thin v v th( ) Cca hm s 2. Chng minh ng thng: d y x m = +lun ct th (C) ti hai im phn bit A, B. Tm m on AB c di nh nht. Cu II (2,0 im) 1.Gii phng trnh: 3os os os sin 2 02 6 3 2 2 6x xc c x c x | | | | | | | | + + + = ||||\ . \ . \ . \ .. 2.Gii h phng trnh: 3 2 2 36 9 4 02x xy xy yx y x y + = + + =. Cu III (1,0 im)Cho s thcln 2 a > .Tnh ln1032xxaeJ dxe=} v suy raln 2limaJ Cu IV (1,0 im). Cho hnhlng tr tam gic ABC.DEF c BE = a, gc gia ng thng BE vi mt phng (ABC) bng060 . Tam gic ABC vung ti C, gc
0BAC 60 = , hnh chiu vung gc ca E ln (ABC) trng vi trng tm ca tam gic ABC. Tnh th tch ca t din D.ABC? Cu V(1,0 im).Cho ba s thc dng a, b, c tha mn: 3 3 32 2 2 2 2 21a b ca ab b b bc c c ca a+ + =+ + + + + +. Tm gi tr ln nht ca biu thcS a b c = + + . B- PHN RING(3,0 im). Th sinh ch c chn mt trong hai phn B.1. CHNG TRNH CHUN Cu VI a(2,0 im) 1.TrongmtphngOxy,choelip( )2 2: 125 16x yE + = .GiA,Blccimtrn(E)saocho 1 2AF 8 BF + =vi 1 2, FFl cc tiu im. Tnh 2 1AF BF + . 2. Trong khng gian Oxyz, cho hai ng thng:1 28 6 10: ;: 22 1 14 2x tx y zd d y tz t= + = = = = + Vit phng trnh ng thng d song song vi trc Ox v ct 1dti A, ct 2dti B. Tnh AB. Cu VII a (1,0 im) Gii phng trnh:( )22 2log 7 log 12 4 0 x x x x + + = . B.2. CHNG TRNH NNG CAO Cu VI b(2,0 im) 1. Trong mt phng Oxy, cho tam gic ABC cn c y l BC. nh A c ta l cc s dng, hai imB,CnmtrntrcOx,phngtrnhcnh( ) : 3 7 1 AB y x = .BitchuvicatamgicABC bng 18. Tm ta cc nh A, B,C. 2. Trong khng gian Oxyz, cho hnh thang cn ABCD vi( ) ( ) ( ) 3; 1; 2 , 1; 5;1 , 2;3; 3 A B C , trong AB l y ln, CD l y nh. Tm ta im D. Cu VII b (1,0 im)Chng minh rng nu( )na bi c di + = +th ( )2 2 2 2na b c d + = + . 12 A- PHN CHUNG(7,0 im) Cu I (2,0 im) . Cho hm s: 2 11xyx=+ c th( ) C1. Kho st s bin thin v v th( ) Cca hm s 2. Gi M l giao im hai ng tim cn ca (C). Tm trn th (C) im I c honh dng sao cho tip tuyn ti I vi th (C) ct hai ng tim cn ti A v B tha mn: 2 240 MA MB + = . Cu II (2,0 im) 1.Gii phng trnh:2 sin 2 3sin cos 24x x x | |+ = + + |\ .. 2.Gii h phng trnh: ( ) 2 322 2 2 2log 5log 21 3x y x yx y x y+ = ++ + =. Cu III (1,0 im). Tnh tch phn: 3221log1 3lnexI dxx x=+} Cu IV (1,0 im). Cho hnh t gic u ABCD.EFGH c khong cch gia hai ng thng AD v ED bng 2. di ng cho mt bn bng 5. Tnh th tch khi lng tr. Cu V(1,0 im).Cho, xyl hai s thc tha mn 2 22 x xy y + = . Tm gi tr ln nht v gi tr nh nht ca biu thc 2 22 3 M x xy y = + . B- PHN RING(3,0 im). Th sinh ch c chn mt trong hai phn B.1. CHNG TRNH CHUN Cu VI a(2,0 im) 1.TrongmtphngvihtaOxy,chohaingtrn( )2 21: 13 C x y + = v ( ) ( )222: 6 25 C x y + = . Gi A lgiao im ca( )1Cv( )2Cvi0Ay> . Vit phng trnh ng thng d i qua A v ct( ) ( )1 2, C Ctheo hai dy cung c di bng nhau. 2. Trong khng gian Oxyz, cho mt cu()2 2 2: 2 4 6 11 0 S x y z x y z + + + =v mt phng () : 2 2 17 0 x y z + + = . Vit phng trnh mt phng( ) song song vi( ) v ct (S) theo giao tuyn l ng trn c chu vi bng6 . Cu VII a (1,0 im). Cho 1 2, z zl cc nghim phc ca phng trnh 22 4 11 0 z z + = . Tnh gi tr ca biu thc ( )2 21 220121 2z zMz z+=+. B.2. CHNG TRNH NNG CAO Cu VI b(2,0 im) 1.TrongmtphngOxy,chohaingthng 1 2: 1 0, : 2 1 0 d x y d x y + + = = . Lpphngtrnh ng thng d i qua( ) 1;1 Mv ct 1 2, d dtng ng ti A, B sao cho2 0 MA MB + =, , ,. 2. Trong khng gian Oxyz, cho mt phng( ) cha ng thng 1:1 1 2x y z A = = v to vi mt phng( ) : 2 2 1 0 x y z + =gc 060 . Tm ta giao im M ca mt phng( ) vi trc Oz. Cu VII b (1,0 im) . Gii h phng trnh: ( ) ( )2 1,1x y x yx ye e xxye x y +++ = +e= + R . 13 A- PHN CHUNG(7,0 im) Cu I (2,0 im) . Cho hm s: 3 23 1 y x x = +c th( ) C1.Kho st s bin thin v v th( ) Cca hm s 2.TmhaiimA,Bthuc th( ) C saochotiptuynca(C)tiAvBsongsongvinhau ng thi4 2 AB = . Cu II (2,0 im) 1.Gii phng trnh:cot 2cot 2 tan 3 3 x x x + + + = . 2.Gii h phng trnh: 2012 2012 2011 20112 x yx y x y+ = + = + . Cu III (1,0 im).Cho hm s:( )( )31xaf x bxex= ++. Tm a, b bit( ) 0 22 f ' =v( )105 f x dx =} Cu IV (1,0 im).y ca khi lng tr ng ABC.DEF l tam gic u. Mt phng y to vi mt phng (DBC) mt gc 030 . Tam gic DBC c din tch bng 8. Tnh th tch khi lng tr ? Cu V(1,0 im).Cho hai s thc| | , 2011; 2012 x y e . Tm gi tr nh nht ca biu thc :( )( )2 22x y x yAxy+ +=B- PHN RING(3,0 im). Th sinh ch c chn mt trong hai phn B.1. CHNG TRNH CHUN Cu VI a(2,0 im) 1.TrongmtphngvihtaOxy,chongtrn( )2 2: 2 4 2 0 C x y x y + + + = .Vitphng trnh ng trn( ) C'tm( ) 5;1 Mbit( ) C'ct( ) Cti hai im, A B sao cho3 AB = . 2. Trong khng gian Oxyz, cho im( ) 2;1; 4 Mv ng thng 1 2 1:1 1 2x y xd = = . Tm im H thuc d sao cho 332HMOSA=bit4Hx > . Cu VII a (1,0 im)Cho 201311izi+ | |=|\ .. Chng minh rng: 1 2 3 *0,k k k kz z z z k+ + ++ + + = eN . B.2. CHNG TRNH NNG CAO Cu VI b(2,0 im) 1.TrongmtphngOxy,chotamgicABCc( ) ( ) ( ) 6; 3 , 4; 3 , 9; 2 A B C .TmimDthuc ng phn gic tronglca gc A t gic ABDC l hnh thang. 2.TrongkhnggianOxyz,chohngthng 1: ,0, 11 1mx y zd m mm m= = = = .Chngminh rng: mdnm trong mt mt phng c nh khim thay i. Cu VII b (1,0 im) .Tm m h phng trnh: 22 221xx x y mx y+ = + ++ = c nghim duy nht. 14 A- PHN CHUNG(7,0 im) Cu I (2,0 im) . Cho hm s: 21xyx= c th( ) C1. Kho st s bin thin v v th( ) Cca hm s 2. Tm hai im B, C nm trn hai nhnh ca th( ) Csao cho tam gic ABC cn tiA(2;0). Cu II (2,0 im) 1.Gii phng trnh: ( )22 3 os 2sin2 412cos 1xc xx | | |\ . =. 2.Gii h phng trnh: ( )( )2 2 3 3332 36x y x y xyx y+ = ++ =. Cu III (1,0 im). Tnh tch phn: 2 436ossin sin4c xI dxx x=| |+ |\ .}. Cu IV (1,0 im).Cho hnh chp tam gic S.ABC c y l tam gic u cnh a. CnhSA (ABC) ,SA = 2a. Gi M, N l hnh chiu vung gc ca A ln cc cnh SB, SC. Tnh th tch ca khi chp ABCMN? Cu V(1,0 im). Cho, , 0 a b c >tha 32a b c + + s . Chng minh rng: 1 1 1 152a b ca b c+ ++ + + > . B- PHN RING(3,0 im). Th sinh ch c chn mt trong hai phn B.1. CHNG TRNH CHUN Cu VI a(2,0 im) 1. Trong mt phngvih ta Oxy,cho elip( )2 2: 112 2x yE + = . Vit phng trnhhypebol (H) c hai ng tim cn l:2 y x = v c hai tiu im l hai tiu im ca (E). 2.TrongkhnggianOxyz,choim( ) 1; 0; 3 I vngthng 1 1 1:2 1 2x y zd + = = .Vitphng trnh mt cu (S) tm I v ctdti hai im, A B sao cho choIAB Avung ti I. Cu VII a (1,0 im) Gi s, , a b cl ba s thc sao chocos cos os 0 a bc c = .a) Hy tm phn o ca s phc( )( )( ) 1 tan 1 tan 1 tan z i a i b i c = + + + . b) Chng minh rng:( ) tan tan tan tan tan tan,a b c a b c a b c k k + + = + + = eZB.2. CHNG TRNH NNG CAO Cu VI b(2,0 im) 1. Trong mt phng Oxy, cho h ng thng ( ) ( )2 2: 4 6 3 4 0md m x my m + + = . Chng minh rng h ng thng mdtip xc vi mt cnic c nh. 2. Trong khng gian Oxyz, cho cc im( ) ( ) 4; 0; 0 , 0; 4; 0 A Bv mt phng() : 3 2 4 0 P x y z + + = . Gi I l trung im ca AB. Tm K m KI vung gc vi (P) ng thi K cch u gc O v (P). Cu VII b (1,0 im)Gii h phng trnh: ( )32log 32 12 3 81xx yy y y+ = + =. Vn Ph Quc, GV.Trng i hc Qung Nam D: 0982.333.443; 0934.825.925 15 A- PHN CHUNG(7,0 im) Cu I (2,0 im) . Cho hm s: 4 2 22 y x mx m m = + + + c th( )mC1. Kho st s bin thin v v th ca hm s khi2 m = . 2. Tm m th( )mCca hm s c ba im cc tr lp thnh mt tam gic c mt gc bng 0120 . Cu II (2,0 im) 1.Gii phng trnh: 1 1sin 2 sin 2cot 22sin sin 2x x xx x+ = . 2.Gii h phng trnh: 2 22 8 24+ + =+ = x y xyx y. Cu III (1,0 im). Tnh tch phn: 20sin I x xdx= }. Cu IV (1,0 im). Cho gc tam din vung Oxyz nh O trn Ox, Oy, Ozln lt ly cc im A, B, C sao cho OA + OB + OC + AB + AC + BC = L, gi V l th tch ca t din ABCD.Chng minh rng : 3( 2 1)162LVsCu V(1,0 im). Cho, , 0 a b c >tha3 ab a b + + = . Chng minh: 2 23 3 31 1 2a b aba bb a a b+ + s + ++ + + B- PHN RING(3,0 im). Th sinh ch c chn mt trong hai phn B.1. CHNG TRNH CHUN Cu VI a(2,0 im) 1. Trong mt phng vi h ta Oxy, cho ba ng thng 1 2 3: 2 3 0, : 3 4 5 0, : 4 3 2 0 d x y d x y d x y + = + + = + + = . Vit phng trnh ng trn c tm thuc 1dv tip xc vi 2 3, . d d2.TrongkhnggianOxyz,chohaiim( ) ( ) 0; 0; 4 , 2; 0; 0 A B vmtphng() : 2 5 0 P x y z + = = . Lp phng trnh mt cu (S) i qua, , OA B v c khong cch t tm I ca mt cu n mt phng (P) bng 56. Cu VII a (1,0 im). Gii phng trnh:( )3 22 1 3 1 0 z i z iz i + + + =trn tp s phcC. B.2. CHNG TRNH NNG CAO Cu VI b(2,0 im) 1. Trong mt phng Oxy, cho ng thng: 3 4 0 d x y =v ng trn( )2 2: 4 0 C x y y + = . Tm imMthuc d, imNthuc( ) Csao cho hai im ny i xng nhau qua( ) 3;1 A .2. Trong khng gian Oxyz, cho im( ) 0;1;1 Av hai ng thng: 1 211 2: ,:3 1 11xx y zd d y tz t= + = = = = +.Vit phng trnh ng thng d i qua imA , vung gc vi 1dv ct 2d. Cu VII b (1,0 im) .Tm m h phng trnh: ( ) ( )( )233 3222 5log 1 log 1 log 4log 2 5 log 2 5x xx xx x m ++ > + = c hai nghim thc phn bit. www.MATHVN.comwww.mathvn.com www.MATHVN.com Vn Ph Quc, GV.Trng i hc Qung Nam D: 0982.333.443; 0934.825.925 16 A- PHN CHUNG(7,0 im) Cu I (2,0 im) . Cho hm s: 2 41xyx=+c th( ) C1. Kho st s bin thin v v th( ) Cca hm s. 2. Tm trn th( ) C , hai im A v B i xng qua ng thng MN. Bit rng( ) ( ) 3; 0 , 1; 1 M N . Cu II (2,0 im) 1.Gii phng trnh: 41 3 74cos os2 os4 os2 4 2xx c x c x c + = . 2.Gii h phng trnh: 2 12 12 2 2011 12 2 2011 1yxx x xy y y + + = + + + = +. Cu III (1,0 im).Tnh tch phn: ( )2201211dxIx x=+}. Cu IV (1,0 im).Cho hnh chp. S ABCDc yABCD l hnh ch nht. Hai mt bnSABvSCD vung gc vi y. ng choACca y to vi cnhABmt gc . CnhSCc di bngav to vi mt phng( ) SABmt gc . Tnh th tch khi chp. S ABCD. Cu V(1,0 im)Cho, , a b cl ba s dng tha mn 34a b c + + = . Chng minh rng: 3 3 33 3 3 3 a b b c c a + + + + + s .Du = xy ra khi no? B- PHN RING(3,0 im). Th sinh ch c chn mt trong hai phn B.1. CHNG TRNH CHUN Cu VI a(2,0 im) 1. Trong mt phng vi h ta Oxy, cho parabol()2: P y x =v im( ) 0; 2 I . Tm ta hai im , A B thuc( ) Psao cho4 0 IA IB =, , ,. 2.TrongkhnggianOxyz,chomtphng()2: 2 2 3 0 P x y m m + = vmtcu () ( ) ( ) ( )2 2 2: 1 1 1 9 S x y z + + + = .Tmm mt phng( ) P tipxcvimtcu() S .Vimtm c, hy xc nh ta tip im ca mt phng (P) v mt cu (S). Cu VII a (1,0 im) Cho, , , A B CDlbnimtrongmtphngphctheothtbiudinccs ( ) ( )4 3 3 ;2 3 3 ;1 3 ;3 i i i i + + + + + + .Chngminhrngbnim, , , A B CDcngnmtrnmt ng trn. B.2. CHNG TRNH NNG CAO Cu VI b(2,0 im) 1. Trong mt phng Oxy, cho im( ) 5; 0 B . im A nm trn gc phn t th nht sao cho tam gic OAB vung ti A v ng trn ni tip c bn knh1 r = . Tm ta nh A. 2. Trong khng gian Oxyz, cho hai mt cu( )2 2 21: 2 4 2 30 0 S x y z x y z + + + =( )2 2 22: 6 8 16 0 S x y z x y + + + = . Chng t rng hai mt cu( )1Sv( )2Stip xc trong vi nhau. Vit phng trnh tip din chung ca chng. Cu VII b (1,0 im) .Giiphng trnh: ( ) ( )3 3log log22012 2003 2012 20033x xx + > www.MATHVN.comwww.mathvn.com www.MATHVN.com Vn Ph Quc, GV.Trng i hc Qung Nam D: 0982.333.443; 0934.825.925 17 A- PHN CHUNG(7,0 im) Cu I (2,0 im) . Cho hm s: 22xyx=c th( ) C . 1. Kho st s bin thin v v th( ) Cca hm s. 2. Vit phng trnh tip tuyn vi (C), bit tip tuyn ct Ox, Oy ln lt ti, A B m tam gic OAB tha mn2 AB OA = . Cu II (2,0 im) 1.Gii phng trnh: 22tan tan 2sintan 1 2 4x xxx + | |= + |+\ .. 2.Gii h phng trnh: ( )( )22 25 4 45 4 16 8 16 0y x xy x xy x y= + + + = . Cu III (1,0 im). Tnh tch phn: ( )3ln 22302xdxIe=+}. Cu IV (1,0 im). Cho hnh chp S.ABCD c y ABCD l hnh ch nht,( ) ( ) SAB ABCD v SCD Au cnha , gc gia hai mt phng( ) SCDv( ) ABCDbng . Tnh th tch khi chp theoav . Tm th tch ln nht. Cu V(1,0 im). Cho s nguynn ( ) 2 n >v hai s thc khng m, xy . Chng minh 1 1 1 n n n n n nx y x y+ + ++ > + . Du = xy ra khi no? B- PHN RING(3,0 im). Th sinh ch c chn mt trong hai phn B.1. CHNG TRNH CHUN Cu VI a(2,0 im) 1. Trong mt phngvih ta Oxy, cho ng trn( ) ( )22: 4 4 C x y + = v im( ) 4;1 E . Tm ta cc imMtrn trc tung sao cho tMk c hai tip tuyn, MA MB n ng trn( ) Cvi, A B l cc tip im sao cho ng thngABi qua imE . 2. Trong khng gian Oxyz, cho im( ) 1; 1;1 A v hai ng thng: 11:1 2 3x y zd+= = 21 4:1 2 5x y zd = = .Chng minh hai ng thng 1 2, d dv imA cng nm trong mt mt phng. Cu VII a (1,0 im). Gii h phng trnh: log log2 2 3y xx yxy y=+ = . B.2. CHNG TRNH NNG CAO Cu VI b(2,0 im) 1.TrongmtphngOxy,chongtrn( )2 2: 12 4 36 0 C x y x y + + = .Vitphngtrnhng trn( ) C'tip xc vi hai trc ta Ox, Oyng thi tip xc ngoi vi ng trn() C . 2. Trong khng gian Oxyz, cho im( ) ( ) ( ) 2; 0; 0 , 2; 2; 0 , 0; 0; A B S m . Gi H l hnh chiu vung gc ca gc ta O trn ng thngSA. Chng minh rng vi mi0 m >din tch tam gicOBHnh hn 3. Cu VIIb(1,0 im).Chngminhrngvimis phcz,tnhtmttronghaibt ngthcsau xy ra: 112z + >hoc 21 1 z + > . www.MATHVN.comwww.mathvn.com www.MATHVN.com Vn Ph Quc, GV.Trng i hc Qung Nam D: 0982.333.443; 0934.825.925 18 A- PHN CHUNG(7,0 im) Cu I (2,0 im) . Cho hm s:( )31my x mx m C = + c th( ) C . 1. Kho st s bin thin v v th( ) Cca hm s khi3 m = . 2. Tm m tip tuyn ca th hm s cho ti im c honh 01 x = ct ng trn ( ) C : ( ) ( )2 22 3 4 x y + =theo mt dy cung c di nh nht. Cu II (2,0 im) 1.Gii phng trnh: 2sin 2 os42 2 sin 3sin3 os3 4x c xxx c x + | | | |= + + ||+\ . \ .. 2.Gii h phng trnh: ( ) ( )( )4 3 2 224 2 2 26 12 65 1 11 5x x x y y xx x y x + = = Cu III (1,0 im). Tnh tch phn: ( )231ln lnln 1ex xI dxx x+=+ +}. Cu IV (1,0 im). Trong mt phng (P) cho ng thngA v mt imA khng thucA. Trn ng thng vung gc vi (P) tiA, ly imSc nh khcA. Gc 090 xAy =xoay quanhA; hai tiaAx, AyctA ti, B C . ChoSA h =v( ) , d A a A = . Tnh. SABCVnh nht theohva . Cu V(1,0 im). Cho, , 0 xyz >thay i. Tm GTLN ca 2 2 23 3 3x y zQx yz y zx z xy= + ++ + +. B- PHN RING(3,0 im). Th sinh ch c chn mt trong hai phn B.1. CHNG TRNH CHUN Cu VI a(2,0 im) 1. Trong mt phng vi h ta Oxy, cho elip( )2 2: 19 4x yE + =v ng thng: 1 0md x my + =v im( ) 1; 0 C . Chng minh rng md lun ct( ) E tihai im phn bit, A B. TmmABC .c din tch ln nht. 2. Trong khng gian Oxyz, cho t dinABCD vi( ) ( ) ( ) 0; 0; 2 , 0; 2; 0 , 2; 0; 0 A B C , ( ) 2; 2; 2 D . Tm cc im c ta nguyn nm trong t din. Cu VII a (1,0 im). Tm s phcztha mn hai k:1 2 3 4 z i z i + = + + v 2 z iz i+ l mt s o. B.2. CHNG TRNH NNG CAO Cu VI b(2,0 im) 1. Trong mt phng Oxy, cho ng trn( )2 2: 2 3 0 C x y x + = . Gi, B Cl giao im ca ng thng vi ng trn( ) C . Hy tm cc imA trn ng trn( ) Csao choABC .c chu viln nht. 2. Trong khng gian Oxyz, cho mt cu()2 2 2: 4 4 2 7 0 S x y z x y z + + + =v ng thng mdl giao tuyn ca hai mt phng:() ( ) : 1 2 4 4 0 x my mz + + =v( ) ( ) : 2 2 1 8 0 x my m z + + = . Chng minh rng cc giao im ca mdv() Snm trn mt ng trn c nh khim thay i. Hy tm ta tm v bn knh ca ng trn . Cu VII b (1,0 im) Tmm phng trnh: ( )2 2 2 227 133log 2 2 4 log 2 0 x x m m x mx m + + + =c hai nghim 1 2, x xsao cho 2 21 21 x x + > . www.MATHVN.comwww.mathvn.com www.MATHVN.com Vn Ph Quc, GV.Trng i hc Qung Nam D: 0982.333.443; 0934.825.925 19 A- PHN CHUNG(7,0 im) Cu I (2,0 im) . Cho hm s: ( )3 2 2xy Cx+=+c th( ) C . 1. Kho st s bin thin v v th( ) Cca hm s. 2. ng thngy x =ct( ) Cti hai im phn bit, A B. Tmm ng thngy x m = +ct ( ) Cti hai im phn bit, CD sao cho tam gicABCD l hnh bnh hnh. Cu II (2,0 im) 1.Gii phng trnh: ( )2442 sin 2 sin 3tan 1osx xxc x+ = . 2.Gii h phng trnh: ( )( ) ( )2 4 2 2 2 422 4 2 23 2 1 21 1 2 2 1 0xy xy x x yx y x x x xy+ + = + + + + =. Cu III (1,0 im). Cho( ) Hl hnh gii hn bi th hm s: 2logxey x =, trcOxv ng thng c phng trnhx e = . Tnh th tch vt th trn xoay khi( ) Hquay quanhOx . Cu IV (1,0 im). Cho hnh chp t gic u. S ABCD c tt c cc cnh u bng a. Tnh theo a th tch khi chp . S ABCD v tnh bn knh mt cu tip xc vi tt c cc mt ca hnh chp . Cu V(1,0 im) Cho, , xyzl cc s dng. Tm gi tr nh nht ca biu thc: ( ) ( ) ( )3 3 3 3 3 33 3 32 2 24 4 4 2 .x y zP x y y z z xy z x| |= + + + + + + + + |\ . B- PHN RING(3,0 im). Th sinh ch c chn mt trong hai phn B.1. CHNG TRNH CHUN Cu VI a(2,0 im) 1. Trong mt phng vi h ta Oxy, cho ng thng( ) d c phng trnh : 0 x y =v im(2;1) M . Tm phng trnh ng thngA ct trc honh tiA ct ng thng ( ) dtiBsao cho tam gicAMBvung cn tiM2. Trong khng gian to cho ng thng d: 3 2 12 1 1x y z + += = v mt phng(P):2 0 x y z + ++ = . Gi M l giao im ca d v (P). Vit phng trnh ng thngA nm trong mt phng (P), vung gc vi d ng thi tho mn khong cch t M tiA bng42 .Cu VII a (1,0 im)Trong khai trin sau y c bao nhiu s hng hu t ( )43 5n+bit n tha mn 1 2 3 2 4964 1 4 1 4 1 4 1... 2 1nn n n nC C C C+ + + ++ + + + = . B.2. CHNG TRNH NNG CAO Cu VI b(2,0 im) 1. Trong mt phngvih ta Oxy cho ng trn(C)c phng trnh( ) ( )2 21 2 9 x y + + =v ng thng: 0 d x y m + + = . Tm m trn ng thng d c duy nht mt im A m t k c hai tip tuyn AB, AC ti ng trn (C)(B, C l hai tip im) sao cho tam gic ABC vung tiA. 2. Trong khng gian Oxyz, cho im A(3;2;3) v hai ng thng12 3 3:1 1 2x y zd = = v 21 4 3:1 2 1x y zd = =.Chng minh ng thng d1; d2 v im A cng nm trong mt mt phng. Xc nh to cc nh B v C ca tam gic ABC bit d1 cha ng cao BH v d2 cha ng trung tuyn CM ca tam gic ABC. Cu VII b (1,0 im) Gii bt phng trnh) 3 (log 5 3 log log242222 > x x x . www.MATHVN.comwww.mathvn.com www.MATHVN.com Vn Ph Quc, GV.Trng i hc Qung Nam D: 0982.333.443; 0934.825.925 20 A- PHN CHUNG(7,0 im) Cu I (2,0 im) .Cho hm s 3 2 33 4 y x mx m = +(m l tham s) c th l (Cm) 1. Kho st v v th hm s khi m = 1. 2. Xc nh m (Cm) c cc im cc i v cc tiu i xng nhau qua ng thngy x = . Cu II (2,0 im) 1.Gii phng trnh:1 cos 4 4 cos 3 24cos 22 2 = +|.|
\| x x x 2.Tm m h phng trnh: 3 3 22 2 23 3 2 01 3 2 0x y y xx x y y m + =+ + = c nghim thc. Cu III (1,0 im).Cho 1, , , ;14xyz t| |e |\ .. Chng minh:1 1 1 1log log log log 84 4 4 4x y z ty z t x| | | | | | | | + + + > ||||\ . \ . \ . \ . Cu IV (1,0 im).Cho hnh lng tr tam gic ABC.ABC vi A.ABC l hnh chp tam gic u cnh yAB = a; cnh bn AA = b. Gi l gc gia hai mp(ABC) v mp(ABC). Tnhtanv th tch chp A.BCCB. Cu V(1,0 im).Tnh tch phn: 60tan( )4os2xxI dxc=} B- PHN RING(3,0 im). Th sinh ch c chn mt trong hai phn B.1. CHNG TRNH CHUN Cu VI a(2,0 im) 1. Trong mt phng vi h ta Oxy cho im A(1;1) v ng thng: 2 3 4 0 x y A + + = . Tm ta im B thuc ng thngA sao cho ng thng AB vA hp vi nhau gc 450. 2. Trong khng gian Oxyz, cho mt phng() : 2 4 0 P x y z + + =v mt cu (S): 2 2 22 4 2 3 0 x y z x y z + + + + = .Vit phng trnh tham s ng thng d tip xc vi (S) tiA(3;-1;1) v song song vi mt phng (P). Cu VII a (1,0 im) Gii phng trnh ( )1 2 3 23 7 ... 2 1 3 2 6480n n n nn n n nC C C C + + + + = trn tp *N . B.2. CHNG TRNH NNG CAO Cu VI b(2,0 im) 1. Trong mt phng to Oxy, cho Elip (E): 2 25 5 x y + = , Parabol()2: 10 P x y = . Hy vit phng trnh ng trn c tm thuc ng thng: 3 6 0 x y A + = , ng thi tip xc vi trc honh Ox v ct tuyn chung ca Elip (E) vi Parabol (P). 2. Trong khng gian Oxyz, cho mt phng() : 2 2 1 0 P x y z + + =v hai im( ) ( ) 1; 7; 1 , 4; 2; 0 A B . Lp phng trnh ng thngdl hnh chiu vung gc ca ng thng AB ln mt phng (P). Cu VII b (1,0 im)Gii h phng trnh: ( )( )3 2 3 223 5.6 4.2 02 2x y x x yx y y y x y x + = = + +. www.MATHVN.comwww.mathvn.com www.MATHVN.com Vn Ph Quc, GV.Trng i hc Qung Nam D: 0982.333.443; 0934.825.925 21 A- PHN CHUNG(7,0 im) Cu I (2,0 im) .Cho hm s:4 2 2( 10) 9 y x m x = + + . 1. Kho st s bin thin v v th ca hm s ng vi m = 0 2. Tm m th ca hm s ct trc honh ti 4 im pbit 1 2 3 4, , , x x x x tha mn iu kin:1 2 3 410 x x x x + + + = . Cu II (2,0 im) 1.Gii phng trnh: 2 22sin cos x 1 sin sin2x2 2t t | | | |= ||\ . \ .. 2.Gii h phng trnh: 2 23 3214 2 292 2xy y x y x y x yx y x y+ + = + + | | | |+ = || \ . \ .. Cu III (1,0 im). Tnh tch phn sau :ln3 2ln 21 2xx xe dxIe e= + } Cu IV (1,0 im).Mt hnh nn nhS, c tm ng trn y l. O , A B l hai im trn ng trn y sao cho khong cch tO n ng thngABbnga ,
060 ASO SAB = = . Tnh theoachiu cao v din tch xung quanh ca hnh nn Cu V(1,0 im).Tm gi tr ca m h phng trnh sau c ng hai nghim:( )88 82562x yx y m+ =+ = + B- PHN RING(3,0 im). Th sinh ch c chn mt trong hai phn B.1. CHNG TRNH CHUN Cu VI a(2,0 im) 1. Trong mt phng vi h ta Oxy cho ng thng d:cos sin 2cos 1 0. x t y t t + + + =Chng minh rng d lun tip xc vi mt ng trn c nh .2.TrongkhnggianOxyz,lpphngtrnhtngqutcamtphngiquaccim( ) 0; 0;1 M , ( ) 3; 0; 0 Nv to vi mt phng( ) Oxymt gc 3. Cu VII a (1,0 im)Cho n l mt s nguyn dng v( )0 1 2 21 ... ...nk nk nx a a x a x x x a x + = + + + + + + . Bit rng - s nguyn dng k( ) 1 1 k n s s sao cho 1 1.2 9 24k k ka a a += =Tnh ( ) 2011! 102012nM= . B.2. CHNG TRNH NNG CAO Cu VI b(2,0 im) 1. Trong mt phng to Oxy, choparabol:()2: P y x = v ng thngd:1 y mx = . Chng minh rng khi m thay i, ng thng() dlun ct parabol( ) Pti hai im phn bit M v N. Hy tm qu tch tm ng trn ngoi tip tam gic OMN khi m thay i. 2. Trong khng gian Oxyz, cho hai ng thng d v d ln lt c phng trnh : d :zyx ==12 vd : 15322+= = zyx. Vit phng trnh mt phng) (i qua d v to vid mt gc 030 Cu VII b (1,0 im) Gii phng trnh: ( )33 23log 1 2log x x x + + = . www.MATHVN.comwww.mathvn.com www.MATHVN.com Vn Ph Quc, GV.Trng i hc Qung Nam D: 0982.333.443; 0934.825.925 22 A- PHN CHUNG(7,0 im) Cu I (2,0 im) .Cho hm s:4 24 y x x m = + ( )mC1. Kho st s bin thin v v th ca hm s ng vi m = 0 2. Tm m th ca hm s ct trc honh ti 4 im phn bit sao din tch hnh phng gii hn bi( )mCv trc honh c phn trn bng phn di. Cu II (2,0 im) 1.Tm m phng trnh ( )4 42 sin cos cos 4 2sin 2 0 x x x x m + + + =c nghim trn0; .2( ( 2.Gii bt phng trnh: 21 2 1 2 2 x x x + + + > . Cu III (1,0 im). Tnh tch phn sau :230sin1 cos 2x xI dxx+=+} Cu IV (1,0 im). ): Cho t din u ABCD c cnh bng 1. Gi M, N l cc im ln lt di ng trn cc cnh AB, AC sao cho( ) ( ) DMN ABC . t AM = x, AN = y. Tnh th tch t din DAMN theo x v y. Chng minh rng: 3 x y xy + = . Cu V(1,0 im).Choa,b,clccsthcthomn3. a b c + + = Tmgitrnhnhtcabiu thc 4 9 16 9 16 4 16 4 9 .a b c a b c a b cM = + + + + + + + +B- PHN RING(3,0 im). Th sinh ch c chn mt trong hai phn B.1. CHNG TRNH CHUN Cu VI a(2,0 im) 1.TrongmtphngOxychongtrn(C)tmI(-1;1),bnknhR=1,Mlmtimtrn ( ) : 2 0 d x y + = . Hai tip tuyn qua M to vi (d) mt gc 450 tip xc vi(C) ti A, B. Vit phng trnh ng thng AB. 2. Trong khng gian Oxyz, vit phng trnh mt phng cch u hai ng thng d1 v d2 bit:12: 23x td y tz t= + = + =
21 2 1:2 1 5x y zd = = . Cu VII a (1,0 im) Trong cc s phc tha mn iu kin 3z 2 3i2 + = . Hy tm s phc c mun nh nht. B.2. CHNG TRNH NNG CAO Cu VI b(2,0 im) 1. Trong mt phng to Oxy, cho elip (E) c hai tiu im 1 2( 3; 0); ( 3; 0) F F v i qua im 13;2A| | |\ .. Lp phng trnh chnh tc ca (E) v vi mi im M trn elip, hy tnh biu thc: 2 2 21 2 1 23 . P FM F M OM FM F M = + . 2. Trong khng gian Oxyz cho t din ABCD bit A(0; 0; 2), B(-2; 2; 0), C(2; 0; 2),( ) DH ABC v3 DH =vi H l trc tm tam gic ABC. Tnh tan ca gc gia (DAB) v( ) ABC . Cu VII b (1,0 im) Gii h phng trnh:( )( )2 23 32 22 2log log4y x y x x xy yx y = ++ =. www.MATHVN.comwww.mathvn.com www.MATHVN.com Vn Ph Quc, GV.Trng i hc Qung Nam D: 0982.333.443; 0934.825.925 23 A- PHN CHUNG(7,0 im) Cu I (2,0 im) .Cho hm s:32xyx+=( ) C1. Kho st s bin thin v v th ( ) Cca hm s. 2. Tm m ng thng: 1 d y x m = + +ct( ) Cti hai im phn bit, A B sao cho
AOB nhn. Cu II (2,0 im) 1.Gii phng trnh: ( )338sin x 1 162sin x 27 0 + + = . 2.Gii h phng trnh: 3 22 1 34 1 9 8 52 4x yx x y x y xy += + = . Cu III (1,0 im). Tnh tch phn sau : ( ) 1ln2 ln 2 lnexdxIx x x=+ + }. Cu IV (1,0 im). Cho ng cao khi chp u S.ABC bng h khng i, gc y ca mt bn bngvi|.|
e2;4 .Tnh th tch ca khi chp theo h v .Vi gi tr no ca th th tch khi chp t gi tr ln nht . Cu V(1,0 im). Cho a, b, c l cc s dng thuc khong ( )0; 6va b c 3 3 + + = .Tm gi tr nh nht ca biu thc:2 2 21 1 1P6 a 6 b 6 c= + + . B- PHN RING(3,0 im). Th sinh ch c chn mt trong hai phn B.1. CHNG TRNH CHUN Cu VI a(2,0 im) 1. Trong mt phng Oxy cho hnh bnh hnhABCD c( ) 6; 6 D . ng trung trc ca onDCcphngtrnh 1: 2 3 17 0 x y A + + = vngphngicgc
BAC cphngtrnhl 2: 5 3 0 x y A + = . Xc nh ta cc nh cn li ca hnh bnh hnh. 2. Trong khng gian Oxyz, cho t din vi nh ( ) ( ) ( ) ( ) 2;0; 0 , 0; 4; 0 , 0;0; 6 , 2; 4;6 A B C DTm tp hp cc im M trong khng gian sao cho:40 MA MB MC MD + + + =, , , ,. Cu VII a (1,0 im) Gii phng trnh trnC:3 0 z z z i = . B.2. CHNG TRNH NNG CAO Cu VI b(2,0 im) 1. Trong mt phng to Oxy, cho tam gicABCc( ) 1; 5 Bv phng trnh ng cao : 2 2 0 ADx y + = , ng phn gic 2: 1 0 CC x y = . Tm ta cc nh, A C .2. Trong khng gian Oxyz cho ng thng 1 1:3 2 2x y zd += = v hai im( ) ( ) 3; 0; 2 , 1; 2;1 A B . K AA,BB ' 'vung gc vi ng thngd . Tnh diA B ' ' . Cu VII b (1,0 im)Tm m h phng trnh sau c nghim: 4(2 1)[ln(x + 1) lnx] = (2y + 1)[ln(y + 1) lny]3 1 2 ( 1)( 1) 1 0xy y x m x+ + + +=. www.MATHVN.comwww.mathvn.com www.MATHVN.com Vn Ph Quc, GV.Trng i hc Qung Nam D: 0982.333.443; 0934.825.925 24 A- PHN CHUNG(7,0 im) Cu I (2,0 im) .Cho hm s:24 262my x mx = + + ( )mC1. Kho st s bin thin v v th ca hm s khi2 m = . 2. Tm m ( )mCc ba im cc tr, , A B C( trong A thuc trc tung) sao cho t gicABOCl hnh bnh hnh. Cu II (2,0 im) 1.Gii phng trnh: 2 2 2 2os os 2 os 3 os 4 2 c x c x c x c x + + + = . 2.Gii h phng trnh: ( )4 42009 2013 2013 200920112 123xy x yx y x y+ + = + =. Cu III (1,0 im). Tnh tch phn sau : 3 202 11x xI dxx+ =+} Cu IV (1,0 im). Trong mt phng (P) cho tam gic u ABC cnh a, I l l trung im ca BC v D l im i xng ca Aqua I. Trn ng thngvunggcvi (P) ti Dlymt im S sao cho a 6SD2= .GiHlhnhchiucaItrnSA.Chngminhrng(SAB) (SAC) vtnhtheoath tch ca khi chp H.ABC. Cu V(1,0 im). Cho x, y, z l ba s tha0 x y z + + =Chng minh:3 4 3 4 3 4 6x y z+ + + + + > B- PHN RING(3,0 im). Th sinh ch c chn mt trong hai phn B.1. CHNG TRNH CHUN Cu VI a(2,0 im) 1. Trong mt phng ta Oxy, cho ngtrn( )2 2: 2 0 C x y x + + = . Vit phngtrnh tiptuyn ca( ) C , bit gc gia tip tuyn ny v trc tung bng30
. 2.TrongkhnggianvihtaOxyzchoimA(10;2;-1)vngthngdcphngtrnh 1 21 3x ty tz t= += = +. Lp phng trnh mt phng (P) i qua A, song song vi d v khong cch t d ti (P) l ln nht. Cu VII a (1,0 im) Mt l hng c 10 sn phm, trong c 2 ph phm. Ly ty 6 sn phm t l hng . Hy tm xc sut trong 6 sn phm c khng qu 1 ph phm. B.2. CHNG TRNH NNG CAO Cu VI b(2,0 im) 1. Trong mt phng Oxy, cho hypebol( )2 2: 14 5x yH =v ng thng: 0 x y m A + = . Chng minh rngA lun ct( ) Hti hai im, MNthuc hai nhnh khc nhau ca( ) H ( )M Nx x < . Xc nhm 2 12 F N F M =( bit 1 2, F Fln lt l tiu im tri, phi ca( ) H ). 2. Trong khng gian Oxyz, cho cc mt phng: () : cos sin sin 6sin 5cos 0 P x t y t z t t t + + =;( ) : sin cos cos 2cos 5sin 0 Q x t y t z t t t + + = () : sin 2 cos 2 1 0 R x t y t z + = .( yt eR: tham s) Chng minh rng giao tuyn ca hai mt phng( ) Pv( ) Qsong song vi mt phng( ) R . Cu VII b (1,0 im) Tm cc gi tr ca tham s m phng trnh: 2 421xxm e e + = +c nghim thc . www.MATHVN.comwww.mathvn.com www.MATHVN.com Vn Ph Quc, GV.Trng i hc Qung Nam D: 0982.333.443; 0934.825.925 25 A- PHN CHUNG(7,0 im) Cu I (2,0 im). Cho hm s: 3 2113y x x x = + + + ( ) Cv ba im( ) ( )22 271;1 , 0; 2 , ;5 5A B C| | |\ .. 1. Kho st s bin thin v v th( ) Cca hm s. 2. Vit phng trnh tip tuynA vi th( ) Cbit rng giao im caA v ng thng : 1 d y x = +l trng tm caABC . . Cu II (2,0 im) 1.Gii phng trnh: 4 44sin 2 os 2os 4tan( ). tan( )4 4x c xc xx x += + 2.Gii h phng trnh:4 4 2 24 4 2 22 628 6 0x y x y x yy x y x y xx y x | |+ + + + = | \ .+ + =.Cu III (1,0 im). Tnh tch phn : 12 20.ln(1 ) I x x dx = +} Cu IV (1,0 im).Cho lng tr tam gic ABC.A1B1C1 c tt c cc cnh bng a, gc to bi cnh bn v mt phng y bng 300. Hnh chiu H ca im A trn mt phng (A1B1C1) thuc ng thng B1C1. Tnhkhong cch gia hai ng thng AA1 v B1C1 theo a. Cu V(1,0 im).Cho,b, c a eR. Chng minh rng :sin .sin .sin cos .cos .cos 1 a b c a b c + sB- PHN RING(3,0 im). Th sinh ch c chn mt trong hai phn B.1. CHNG TRNH CHUN Cu VI a(2,0 im) 1.TrongmtphngOxy,lpphngtrnhtngqutcangthngdbitngthngdiqua im M(1; 3) v chn trn cc trc ta nhngon thngc di bng nhau.2. Trong khng gian vi h ta Oxyz cho hai ng thng cho nhau : ( )11: 22x td y t tz t= = e= +R v11311:2==z y xd . Lp phng trnh mt phng song song v cch u hai ng thng d1 v d2. Cu VII a (1,0 im) Gii phng trnh: x xsin cos 1n n | | | |+ = ||\ . \ . vi 2 n < eN. B.2. CHNG TRNH NNG CAO Cu VI b(2,0 im) 1. Trong mt phng ta Oxy cho ng trn (C): x2 + y2 6x + 2y + 6 =0, v im A(1; 3). Vit phng trnh ng thng i qua A v ct (C),ti B, C sao cho BA = BC 2. Trong khng gian Oxyz, cho hai ng thng vi phng trnh: 11 1 1:1 2 2x y zd = =; 21 3:1 2 2x y zd+ = = .G iIl giao im ca 1dv 2d . Lp phng trnh ng thngdqua( ) 0; 1; 2 P ct 1 2, d dln lt ti, A B I =sao choAI AB = . Cu VII b (1,0 im) Gii h phng trnh: ( ) ( )2 2ln 1 ln 112 20 0x y x yx xy y + + = + = . www.MATHVN.comwww.mathvn.com www.MATHVN.com Vn Ph Quc, GV.Trng i hc Qung Nam D: 0982.333.443; 0934.825.925 26 A- PHN CHUNG(7,0 im) Cu I (2,0 im). Cho hm s: 4 22 4 y x mx = + ( )mC 1. Kho st s bin thin v v th ca hm s khi2 m = . 2. Tm cc gi tr cam tt c cc cc tr ca( )mCnm trn cc trc ta .Cu II (2,0 im) 1.Gii phng trnh: 5 os22cos3 2 tanc xxx+=+. 2.Gii h phng trnh:11 13xy xy xy y yx x x + + =+ = + Cu III (1,0 im). Tnh tch phn : ( )512 1 2 1 I x x x x dx = + + }. Cu IV (1,0 im). Cho hnh lng tr ABC.ABC c y l tam gic u cnh a, hnh chiu vung gc ca A ln mtphng (ABC) trng vi tmO ca tam gic ABC. Tnh th tch khi lng tr ABC.ABC bit khong cch gia AAv BC l a 34. Cu V(1,0 im). Cho, , 0 xyz > . Chng minh rng: ( )( )( )2 2 22 2 23 39xyz x y z x y zx y z xy yz zx+ + + + ++s+ + + +
B- PHN RING(3,0 im). Th sinh ch c chn mt trong hai phn B.1. CHNG TRNH CHUN Cu VI a(2,0 im) 1.TrongmtphngOxy,hytnhdintchtamgicunitipelip( )2 2: 116 4x yE + = ,nhnim ( ) 0; 2 Alm nh v trc tung lm trc i xng. 2. Trong khng gian vi h ta Oxyz , tm, , MNPln lt thuc cc ng thng 1 2 31 2 2 1 1: , : , :1 2 2 2 2 1 2 1 1x y z x y z x y zd d d = = = = = = sao cho, , MNPthng hng ng thi Nl trung im ca on thngMP . Cu VII a (1,0 im) . Gii h phng trnh: 3 55 33 5 log 5 log3 log 1 log 1y xx y = = B.2. CHNG TRNH NNG CAO Cu VI b(2,0 im) 1. Trong mt phng ta Oxy, tnh din tch tam gic u ni tip parabol()2: 2 P y x = , nhn nh ca parabol lm mt nh v trc honhOxlm trc i xng. 2. Trong khng gian Oxyz, cho mt cu()2 2 2: 2 2 2 0 S x y z x z + + + = . Tm imA thuc mt cu sao cho khong cch tA n mt phng() : 2 2 6 0 P x y z + + =ln nht. Cu VII b (1,0 im) Cho hm s ( )( )2 2 31 4 mmx m x m my Cx m+ + + +=+. Tmm mt im cc tr ca( )mCthuc gc phn t thI , mt im cc tr ca( )mCthuc gc phn t thIIIca h ta Oxy . www.MATHVN.comwww.mathvn.com www.MATHVN.com Vn Ph Quc, GV.Trng i hc Qung Nam D: 0982.333.443; 0934.825.925 27 A- PHN CHUNG(7,0 im) Cu I (2,0 im). Cho hm s: 2 11xyx+=+( ) C 1. Kho st s bin thin v v th( ) C ca hm s. 2. Tm ta imMthuc( ) Csao cho khong cch t giao im hai tim cn n tip tuynA ca( ) CtiMl ln nht. Cu II (2,0 im) 1.Gii phng trnh: ( )sin3sin 2 os2 tan sin oscosxx c x x x c xx + = + . 2.Gii h phng trnh:( )( )2 22011 20132011 201312014x yx y y x x y xy + = = + + + Cu III (1,0 im). Tnh th tch khi trn xoay to thnh khi hnh phng gii hn bi th hm s 1xxxeye=+, trc honh v ng thng1 x =quay quanh trc honh. Cu IV (1,0 im).Hnh chp t gic u SABCD c khong cch t A n mt phng( ) SBCbng 2. Vi gi tr no ca gco gia mt bn v mt y ca chp th th tch ca chp nh nht? Cu V(1,0 im).Cho x, y, z > 0 tha mn1 xyz = . Chng minh rng: 11 193xy yz zxyx zx y z+ +s . B- PHN RING(3,0 im). Th sinh ch c chn mt trong hai phn B.1. CHNG TRNH CHUN Cu VI a(2,0 im) 1. Trong mt phng Oxy, cho tam gicABCc phng trnh cha ng cao vng trung tuyn k t nhA ln lt c phng trnh l:2 13 0,13 6 9 0 x y x y = = . Tm ta , B Cbit tm ng trn ngoi tip tam gicABCl( ) 5;1 I .2. Trong khnggianvih taOxyz , cho im( ) ( ) ( ) 1; 0; 0 , 2; 1; 2 , 1;1; 3 A B C v ng thng 1 2:1 2 2x y x A = =. Vit phng trnh mt cu c tm thuc ng thngA, i qua imA v ct mt phng( ) ABCtheo mt ng trn c bn knh nh nht. Cu VII a (1,0 im)Cho s phc 1 2, z ztha mn 1 2 1 20 z z z z = = > . Tnh 4 41 22 1z zAz z| | | |= + ||\ . \ .. B.2. CHNG TRNH NNG CAO Cu VI b(2,0 im) 1. Trong mt phng ta Oxy,cho ng trn( )2 2: 1 C x y + =v ng thng: 0 d x y m + + = . Tm m dct( ) Cti, A B sao chABO .c din tch ln nht. 2. Trong khng gian Oxyz, cho im( ) 1; 2; 3 M . Vit phng trnh mt cu tmMv ct mt phng Oxytheo thit din l ng trn( ) Cc chu vi bng8 . Cu VII b (1,0 im) Gii bt phng trnh: ( ) ( )2 2514 3 1 log 2 8 6 1 05xx x x xx + + + + + s . www.MATHVN.comwww.mathvn.com www.MATHVN.com Vn Ph Quc, GV.Trng i hc Qung Nam D: 0982.333.443; 0934.825.925 28 A- PHN CHUNG(7,0 im) Cu I (2,0 im). Cho hm s: 42532 2xy x = + ( ) C 1. Kho st s bin thin v v th( ) C ca hm s. 2. ChoA l mt im nm trn( ) Cc honh lm. Tm cc gi tr thc cam tip tuyn ca ( ) Cct th( ) Cti hai im phn bit, B CkhcA sao cho3 AC AB =( B nm gia A v C ). Cu II (2,0 im) 1.Gii phng trnh: 2 22 sin sin 3 sin tan tan3 3 4 4x x x x x ( | | | | | | | |+ + = + + ||||(\ . \ . \ . \ . . 2.Gii bt phng trnh:( )24 1 3 1 x x x x + + s + . Cu III (1,0 im). Tnh tch phn: ( )165501dxIx=+}. Cu IV (1,0 im). Cho hnh chp S.ABCD c y l hnh thoi ,
BAD = . Hai mt bn (SAB) v (SAD) cng vung gc vi mt y, hai mt bn cn li hp vi y mt gc . Cnh SA = a. Tnh din tch xung quanh v th tch khi chp S.ABCD. Cu V(1,0 im).Cho h phng trnh: ( ) ( )2 2 22 22 1 2 2 02 9 0m mx m y m mx y x+ + + =+ + = . Chng minh h phng trnh trn lun c hai nghim phn bit( )1 1, x yv( )2 2, x y .Tm m ( ) ( )2 21 2 1 2P x x y y = + t gi tr nh nht. B- PHN RING(3,0 im). Th sinh ch c chn mt trong hai phn B.1. CHNG TRNH CHUN Cu VI a(2,0 im) 1.TrongmtphngOxy,chotamgicABC c: 2 3 0 d x y = lngphngictronggcA. Bit( ) ( )1 16; 0 , 4; 4 B C lnltlhnhchiuvunggccanh, B C trnccngthng , ACAB . Xc nh ta cc nh, , A B Cca tam gicABC . 2. Trong khng gian vi h ta Oxyz , cho hai ng thng:1 21 2: , :1 1 1 1 1 2x y z x y z A = = A = = v im( ) 1; 0;1 A . Xc nh 1M eA , 2N eAsao cho6 MN =v. 3 AMAN =, ,. Cu VII a (1,0 im). Tm s phcztha mn h phng trnh: ( ) ( )( )21 2 1 2 62 3 0i z i zz i z z + + =+ + =. B.2. CHNG TRNH NNG CAO Cu VI b(2,0 im) 1. Trong mt phng ta Oxy,cho ba ng thng: 1: 3 0 d x y =, 2: 2 5 0 d x y + = , 3: 0 d x y = . Tm ta cc im 1 2 3, , , d B d C A D d e e e t gicABCD l mt hnh vung.2. Trong khng gian Oxyz, cho bn im( ) ( ) ( ) ( ) 1; 0; 0 , 0;1; 0 , 1;1; 0 , 0; 0; A B C D mvi0 m > . Gi , EFtheo th t l hnh chiu vung gc ca gc ta O ln cc ng thngAD vBD. Vit phng trnh mt phng( ) Pcha cc ng thngOEvOF . Tmm
0EOF 45 = . Cu VII b (1,0 im) Gii h phng trnh: ( ) ( )3 3log log2 24 4 44 2 21log 4 4 log log 32xy xyx y x x y =+ = + + +. www.MATHVN.comwww.mathvn.com www.MATHVN.com Vn Ph Quc, GV.Trng i hc Qung Nam D: 0982.333.443; 0934.825.925 29 A- PHN CHUNG(7,0 im) Cu I (2,0 im). Cho hm s: 12 1xyx+=+( ) C1. Kho st s bin thin v v th( ) C ca hm s. 2. Tmm : 2 2 1 0 d mx y m + + =ct( ) Cti hai im phn bit, A B sao cho biu thc 2 2P OA OB = +t gi tr nh nht.Cu II (2,0 im) 1.Gii phng trnh: 462cos 23 1 tan 7cosxxx| |+ + = |\ . 2.Gii h phng trnh:( ) ( ) ( )2 22 23 9 10 3 313 63x y x y x yx yx y + = ++ + = Cu III (1,0 im).Cho 8 8: cos sin; y = 0 ; x = 0 ; x = 2S y x x = + ` ).Tm xVkhi S quay quanh Ox. Cu IV (1,0 im).Cho hnh chpS.ABCD c yABCD l hnh ch nhtviAB a, AD 2a, = =cnhSAvunggc viy,cnhSBtovimtphngymtgc o60 . TrncnhSAlyimMsaocho a 3AM3= .Mt phng( ) BCMct cnhSD ti imN. Tnh th tch khi chp. . S BCNMCu V(1,0 im).Cho ba s thc khng m, , xyztha mn 2 2 23 x y z + + = . Hy tm gi tr ln nht ca biu thc 5A xy yz zxx y z= + + ++ +.B- PHN RING(3,0 im). Th sinh ch c chn mt trong hai phn B.1. CHNG TRNH CHUN Cu VI a(2,0 im) 1. Trong mt phng Oxy, cho elip( )2 2: 18 2x yE + = . Vit phng trnh ng thngdct( ) Eti hai im phn bit c ta l cc s nguyn.2. Trong khng gian vi h ta Oxyz , cho hnh thoiABCD c din tch bng12 2 , nhA thuc Oz , nhC thuc mt phngOxy , hai nhBvDthuc ng thng 1:1 1 2x y zd+= =vBc honh dng. Tm ta cc im, , , A B CD. Cu VII a (1,0 im)Gi 1 2, z zl cc nghim phc ca phng trnh: 24 5 0 z z + = .Tnh( ) ( )2011 20111 21 1 z z + . B.2. CHNG TRNH NNG CAO Cu VI b(2,0 im) 1. Trong mt phng ta Oxy,cho parabol()2: 4 P y x =c tiu imF . GiMl im tha iu kin:3 FM FO = , , ,dl ng thng bt k quaM ,dct( ) Pti hai im phn bit, A B. Chng minh tam gicOABl tam gic vung.2. Trong khng gian Oxyz, cho mt phng() : 2 2 9 0 P x y z + + =v hai im( ) ( ) 3; 1; 2 , 1; 5; 0 A B . Tm ta imMthuc( ) Psao cho. MAMB, , t gi tr nh nht. Cu VII b (1,0 im) Gii bt phng trnh: ( ) ( )2 21 5 3 13 5log log 1 log log 1 x x x x + + > + . www.MATHVN.comwww.mathvn.com www.MATHVN.com Vn Ph Quc, GV.Trng i hc Qung Nam D: 0982.333.443; 0934.825.925 30 A- PHN CHUNG(7,0 im) Cu I (2,0 im). Cho hm s: 21x mymx=+( )mC1. Kho st s bin thin v v th ca hm s khi1 m =2. Chng minh rng vi mi0 m = th( )mCct : 2 2 d y x m = ti hai im phn bit, A B thuc mt ng( ) Hc nh. ng thngdct, Ox Oyln lt ti cc im, MN . Tmm 3OAB OMNS S =. .. Cu II (2,0 im) 1.Tm( ) 0; x etha mn phng trnh:2os2 1cot 1 sin sin 21 tan 2c xx x xx = + + 2.Gii h phng trnh: ( ) ( )( )3 2 2 32 21 2 301 11xy y x y y xyxy x y y y+ + + + =+ + + + = ( , xy eR). Cu III (1,0 im). Tnh tch phn: 324tanos 1 osxI dxc x c x=+}. Cu IV (1,0 im). Cho hnh chp S.ABCD c y ABCD l hnh vung cnh a, SA = h vung gc mt phng (ABCD), M l im thay i trn CD. K SH vung gc BM. Xc nh v tr M th tch t din S.ABH t gi tr ln nht. Tnh gi tr ln nht . Cu V(1,0 im).Tmm h phng trnh sau c nghim: 231 2 2 13 3 2x y xy xx x xy m+ = = + . B- PHN RING(3,0 im). Th sinh ch c chn mt trong hai phn B.1. CHNG TRNH CHUN Cu VI a(2,0 im) 1.TrongmtphngOxy,chohnhbnhhnhABCDcdintchbng6vhainh ( ) ( ) 1; 2 , 2; 3 A B . Tm ta hai nh cn li, bit giao im ca hai ng cho hnh bnh hnh nm trn trcOxv c honh dng.2. Trong khng gian vi h ta Oxyz , cho hai ng thng: 1 1 1:2 1 2x y zd = = v 2 3 4:1 2 3x y z A = = . Bit rngdvA ct nhau. Hyvitphng trnh mt phng( ) Pcha A sao cho gc giadv( ) Pln nht. Cu VII a (1,0 im) Tm s phcztha mn: 222 . 8 z z z z + + =v2 z z + = . B.2. CHNG TRNH NNG CAO Cu VI b(2,0 im) 1. Trong mt phng ta Oxy,cho hnh thoiABCD c tm( ) 2;1 Iv2 AC BD = . im 10;3M | | |\ . thuc ng thngAB , im( ) 0; 7 Nthuc ng thngCD. Tm ta nhBbit0Bx > . 2. Trong khng gian Oxyz, cho hai ng thng 121 2:4x y zdn m = = ; 21:1 2 1x m y zd = = Tm, m n 1 2, d dsong song v khi tnh khong cch gia 1 2, d d . Cu VII b (1,0 im) Gii bt phng trnh: ( )22 5 3 3 1 2.5 223 .5 1xxx x xx + + + + 0 ta c :29(1 )(1 )(1 ) 256yxx y+ + + > .B- PHN RING(3,0 im). Th sinh ch c chn mt trong hai phn B.1. CHNG TRNH CHUN Cu VI a(2,0 im) 1.TrongmtphngOxy,chohnhvungABCDc( ) 2; 6 A ,nhB thucngthng : 2 6 0 d x y + = . Gi, MNln lt l hai im nm trn hai cnh, BC CD sao choBM CN = . Bit AMctBNti 1 14;2 5I | | |\ .. Xc nh ta nhC. 2.TrongkhnggianvihtaOxyz,chongthng 3 2 1:2 1 1x y zd + += =vmtphng () : 2 0 P x y z + + + = . Lp phng trnh ng thngA nm trong mt phngd , ctdv to vidgc ln nht. CuVIIa(1,0im)TrnmtphngtaOxy tmtphpccimbiudinsphc ( )1 3 2 z i z ' = + +trong zl s phc tha:1 2 z = . B.2. CHNG TRNH NNG CAO Cu VI b(2,0 im) 1. Trong mt phng ta Oxy, cho ng trn( )2 2: 16 C x y + = . Vit phng trnh chnh tc ca elip( ) Ec tm sai 12e = bit( ) Ect( ) Cti bn im, , , A B CD sao choABsong song vi trc honh v2 AB CD = . 2. Trong khng gian Oxyz, cho h mt phng ( ), ,: 1 0a b cP ax by cz + + =,( ) , , 0 a b c >v 1 1 112 3 a b c+ + = . Tm, , a b c ( ), , a b cPct cc trc ta , , Ox Oy Ozln lt ti, , A B Csao cho OABC c th tch ln nht. Cu VII b (1,0 im) Gii h phng trnh: ( ) ( )2 2 23 33 3 27 9log 1 log 1 1x y x y x yx y+ + + + + = ++ + + = www.MATHVN.comwww.mathvn.com www.MATHVN.com Vn Ph Quc, GV.Trng i hc Qung Nam D: 0982.333.443; 0934.825.925 32 A- PHN CHUNG(7,0 im) Cu I (2,0 im). Cho hm s:( )3 22 2 3 y x x m x m = + + ( )mC1. Kho st s bin thin v v th ca hm s khi1 m = .2. Tmm tip tuync h s nh nht ca( )mCi qua im 551;27A| | |\ .. Cu II (2,0 im) 1. Gii phng trnh: 3 3sin .sin3 os cos3 18tan . tan6 3x x c x xx x += | | | | + ||\ . \ . 2. Gii bt phng trnh: ( )2 4 26 3 1 1 0 x x x x + + + +sCu III (1,0 im). Tnh tch phn: ( )121022 9 3 2xx xI dx= } Cu IV (1,0 im).Cho lng tr ng. ABC A B C '''c 0, 2 , 120 AC a BC a ACB = = =v ng thngA C 'to vi mt phng( ) ABB A ' 'gc 030 . GiMl trung im caBB' . Tnh th tch khi lng tr cho v khong cch gia hai ng thngAMvCC'theoa . Cu V(1,0 im). nh m h phng trnh sau c nghim: ( )( )( )3 3 2 4 23 3 3 3 8 2 2 4 411 1 2mx x x xym x x x m x y x+ + + = + + + + = B- PHN RING(3,0 im). Th sinh ch c chn mt trong hai phn B.1. CHNG TRNH CHUN Cu VI a(2,0 im) 1.TrongmtphngvihtaOxy ,chongtrn( ) ( ) ( )2 2: 1 2 9 C x y + = .Bittamgic ABCu ni tip( ) Cc( ) 2; 2 A . Tm ta cc nh, B C .2.TrongkhnggianOxyz choim( ) ( ) ( ) 1; 2; 1 , 1;1; 2 , 2; 1; 2 , A B C D lnhthtcahnh bnh hnhABCD. Tm imSthuc trc cao sao cho th tch khi chp. S BCD bng 4.Cu VII a (1,0 im)Gii phng trnh: ( ) ( )2 2log log23 1 3 1 1x xx x + + = + . B.2. CHNG TRNH NNG CAO Cu VI b(2,0 im) 1. Trong mt phngOxy , cho im( ) 3; 0 Av elip( ) Ec phng trnh: 2219xy + = . Tm ta cc im, B Cthuc( ) Esao cho tam gicABCvung cn tiA. 2. Trong khng gianOxyzcho im( ) ( ) ( ) ( ) 5; 3;1 , 4; 1; 3 , 6; 2; 4 , 2;1; 7 A B C D . Tm tp hp cc imMsao cho3 2 MA MB MC MD MA MB + + = , , , , , ,. Cu VII b (1,0 im)Tm s thcm bnh phng s phc 31m izi+= l mt s thc. www.MATHVN.comwww.mathvn.com www.MATHVN.com Vn Ph Quc, GV.Trng i hc Qung Nam D: 0982.333.443; 0934.825.925 33 A- PHN CHUNG(7,0 im) Cu I (2,0 im). Cho hm s: 4 22 2 y x mx = + ( )mC1. Kho st s bin thin v v th ca hm s khi1 m = .2. Tmm ( )mC c ba im cc tr to thnh mt tam gic vung c ng trn ngoi tip i qua im 3 9;5 5D| | |\ ..Cu II (2,0 im) 1. Gii phng trnh: 2421 tan8 os sin 4 24 1 tanxc x xx | |+ + = |+\ . 2. Gii bt phng trnh:( )24 , ,16 2 3x y x y x yxyx y x+ = e = + RCu III (1,0 im). Tnh tch phn: ( )52ln 1 11 1xI dxx x += + } Cu IV (1,0 im). Cho hnh chp. S ABC c yABCl tam gic vung cn tiC , cnh huyn bng3 . aGiGl trng tm tam gicABC ,( ) SG ABC , 142aSB = . Tnh th tch khi chp . S ABC v khong cch tB n mt phng( ) SAC . Cu V(1,0 im). Cho, 0 xy >tha 3 3x y x y + = . Chng minh rng: 2 24 1 x y + < . B- PHN RING(3,0 im). Th sinh ch c chn mt trong hai phn B.1. CHNG TRNH CHUN Cu VI a(2,0 im) 1. Trong mt phngOxy , cho hnh thang vungABCD vung tiA vD c y ln lCD, ng thngAD c phng trnh3 0 x y = , ng thngBD c phng trnh2 0 x y = , gc to bi ng thngBCvABbng 045 . Vit phng trnh ng thngBCbit din tch hnh thang bng 24 v imBc honh dng. 2. Trong khng gianOxyz , cho im( ) ( ) 0; 2; 0 , 0; 0; 1 A B vC Ox e . Vit phng trnh mt phng ( ) ABCbit khong cch tCn mt phng( ) : 2 2 0 P x y z + =bng khong cch tCn ng thng 1 2:1 2 2x y z +A = =. Cu VII a (1,0 im) TmmeR phng trnh:( )22 2 1 2 1 0 z m z m + + + =c 2 nghim phn bit 1 2, z z eC tha mn 1 210 z z + = . B.2. CHNG TRNH NNG CAO Cu VI b(2,0 im) 1. Trong mt phngOxycho cc im( ) ( ) 0;1 , 2; 1 A B v cc ng thng: ( ) ( )1: 1 2 2 0 d m x m y m + + =;( ) ( )2: 2 1 3 5 0 d mx m y m + + = . Chng minh 1dv 2dlun ct nhau. Gi 1 2P d d = , tmmsao choPA PB +ln nht. 2.Trong khng gianOxyz , cho hai im 5 51; 2; , 4; 2;2 2A B| | | | ||\ . \ .. Tm ta imMtrn mt phng ( ) Oxysao cho tam gicABMvung tiMv c din tch nh nht. Cu VII b (1,0 im) Gii h phng trnh: ( ) ( )2 23 53log log 1x yx y x y =+ = www.MATHVN.comwww.mathvn.com www.MATHVN.com Vn Ph Quc, GV.Trng i hc Qung Nam D: 0982.333.443; 0934.825.925 34 A- PHN CHUNG(7,0 im) Cu I (2,0 im). Cho hm s:( )3 212 3 1 13y x mx m x = + + +( )mC1. Kho st s bin thin v v th ca hm s khi0 m = . 2. Vit phng trnh tip tuynA ca( )mCti im c honh bng 1. Tmm giao im caA v: 2 d y x =cch u gc ta . Cu II (2,0 im) 1. Gii phng trnh: 5 3sin os 2 os2 4 2 4 2x x xc c | | | | = ||\ . \ .. 2. Gii h phng trnh: ( ) ( )( ) ( )22 21 1 4 312 2 3 7 1 12 3 5x y x y x yx x y xy y x+ + + = + + ++ + = +,, xy eR. Cu III (1,0 im). Tnh tch phn: ( )40tan .ln oscosx c xI dxx=} Cu IV (1,0 im). Cho hnh chp. S ABCD c yABCD l hnh vung cnh bnga ,3 SA a =v SA vung gc vi mt phng y. Tnh theoath tch khi t din. S ACD v tnh cosin ca gc gia hai ng thng, SBAC . Cu V(1,0 im). Cho, , 0 a b c >tha1 ab bc ca + + = . Chng minh rng: 2 223101 11a b ca bc+ + s+ ++. B- PHN RING(3,0 im). Th sinh ch c chn mt trong hai phn B.1. CHNG TRNH CHUN Cu VI a(2,0 im) 1. Trong mt phngOxy , cho ng trn( ) C : ( ) ( )2 2 251 22x y + + =v ng thng : 3 4 20 0 d x y + = . Lp phng trnh cc cnh hnh vungABCD ngoi tip( ) CbitA d e . 2. Trong khng gianOxyz , cho cc im( ) ( ) ( ) ( ) ; 0; 0 , ; ; 0 , 0; ; 0 , 0; 0; 2 Ba Ca a D a S a . Gi sNl trung im ca cnhSD. Tm gi tr nguyn dng ln nht caa khong cch gia hai ng thngSBvCNln hn 27a. Cu VII a (1,0 im) Vit dng lng gic ca s phc ( )81 3 z i = + . Trong cc acgumen ca s phcz , hy tm acgumen c s o dng nh nht. B.2. CHNG TRNH NNG CAO Cu VI b(2,0 im) 1. Trong mt phngOxycho tam gicABCni tip ng trn( )2 2: 4 2 8 0 C x y x y + = . nh A thuc tiaOy , ng cao v tCnm trn ng thng: 5 0 d x y + = . Tm ta cc nh, , A B Cbit rng imCc honh l mt s nguyn. 2.Trong khng gianOxyz , cho ng thng 1 4:2 1 2x y zd+ = = v cc im( ) 1; 2; 7 A , ( ) ( ) 1; 5; 2 , 3; 2; 4 B C . Tm ta imMthucdsao cho 2 2 2MA MB MC t gi tr ln nht. Cu VII b (1,0 im) Cho hm s: ( )222x m x myx+ = ( )mC . Tm m ( )mCc cc tr ti cc im A, B sao cho ng thng AB i qua gc ta . www.MATHVN.comwww.mathvn.com www.MATHVN.com Vn Ph Quc, GV.Trng i hc Qung Nam D: 0982.333.443; 0934.825.925 35 A- PHN CHUNG(7,0 im) Cu I (2,0 im).Cho hm s: 21xyx=+ ( ) C1. Kho st s bin thin v v th ( ) Cca hm s. 2.VitphngtrnhtiptuynAca( ) Cbittiptuyntovihaingtimcnca( ) C mt tam gic c bn knh ng trn ni tip ln nht. Cu II (2,0 im) 1. Gii phng trnh: 29 62cos os 110 5x xc = 2. Gii h phng trnh: 222 2 1 34 22 2 1 34 2x x y xy xy x y xy y+ + = ++ + = + ,, xy eR. Cu III (1,0 im).Cho( ) ( ) { }2 2 2: 2;: 8 P y x C x y = + = .( ) Pchia( ) Clm 2 phn. Tm t s din tch ca hai phn ? Cu IV (1,0 im).Chohnhchp. S ABCcyltamgicvungcntiAvAB AC a = = .Mtphng( ) SBCvung gc vi mt phng y, gc gia mi mt bn cn li vi mt phng y bng 045 . Tnh theoath tch khi chp. S ABC. Cu V(1,0 im). Gi s hai s thc( ) , 0;1 xy ev1 x y + = . Tm gi tr nh nht ca biu thc.x yx y +B- PHN RING(3,0 im). Th sinh ch c chn mt trong hai phn B.1. CHNG TRNH CHUN Cu VI a(2,0 im) 1.TrongmtphngOxy ,choelip( )2 2: 125 16x yE + = chaitiuim 1 2, FF .Tm imM trn( ) Esao cho 1 21 21 1P MF MFMF MF= + + +t gi tr ln nht. 2. Trong khng gianOxyz , cho bn ng thng: 11 2:1 2 2x y zd = = ; 22 2:2 4 4x y zd = = ; 31:2 1 1x y zd= =; 42 1:2 2 1x y zd = = Chngminh 1 2, d d cngthucmtmt phng( ) P .Vitphngtrnhmtphng( ) P vchng minh c mt ng thngA ct c bn ng thng trn. Vit phng trnh ng thngA . Cu VII a (1,0 im) Cho hai s phc 1 2 123 6 ,3iz i z z = + = c cc im biu din trong mt phng phc tng ng l, A B. Chng minh rng tam gicOABvung tiO. B.2. CHNG TRNH NNG CAO Cu VI b(2,0 im) 1. Trong mt phngOxy , vit phng trnh ng thngA i qua( ) 1;8 M , ct chiu dng ca cc trc, Ox Oyti, A B sao choABnh nht.2.TrongkhnggianOxyz ,chohnhvungABCDcnh( ) 1; 1; 2 C vngcho 1 1 1:4 1 1x y zBD+ += =. Tm ta cc nh, , A BD bit imBc honh dng. Cu VII b (1,0 im) Gii h phng trnh: ( )( )2331 42 1 log 1log 31 log 1 2 2xxyxyy+ = + = www.MATHVN.comwww.mathvn.com www.MATHVN.com Vn Ph Quc, GV.Trng i hc Qung Nam D: 0982.333.443; 0934.825.925 36 A- PHN CHUNG(7,0 im) Cu I (2,0 im). Cho hm s:( )3 23 3 2 1 y x x mm x = + (1) 1. Kho st s bin thin v v th ca hm s khi0 m =2. Tm cc gi tr cam hm s (1) c hai cc tr cng du. Cu II (2,0 im) 1. Gii phng trnh: 3 3 2tan ot cot 2 55 x c x x + + = . 2. Gii h phng trnh: ( )( )( )4 42 2 21 1221 13 32y xx yy x x yx y2 = + = + + ,, xy eR Cu III (1,0 im). Tnh tch phn: 220os2 cos 21 cos cos osc x xI dxx x c x+ +=+ + } Cu IV (1,0 im). Chohnhlngtrtamgic. ABC A B C ' '' cBB a ' = ,gcgiangthngBB' vmtphng ( ) ABCbng 060 ; tam gicABCvung tiCv
060 BAC = . Hnh chiu ca imB'ln mt phng ( ) ABCtrng vi trng tm ca tam gicABC . Tnh th tch khi t dinA ABC 'theoa . Cu V(1,0 im).Chng minh rng h phng trnh: 222012120121xyyeyxex+ =+ = c ng hai nghim phn bit( ) , xytha mn1, 1 x y > > . B- PHN RING(3,0 im). Th sinh ch c chn mt trong hai phn B.1. CHNG TRNH CHUN Cu VI a(2,0 im) 1.TrongmtphngOxy ,lpphngtrnhngtrn( ) C bit( ) C ctmnmtrnngthng : 2 3 0 x y A =v ct hai trc ta theo hai dy cung c di bng nhau v bng 2.2. TrongkhnggianOxyz , cho mt phng( ) : 2 2 12 0 P x y z =vhai im( ) ( ) 1;1;3 , 2;1; 4 A B . Tm tp hp tt c cc im( ) C P esao cho tam gicABCc din tch nh nht.Cu VII a (1,0 im) Chn ngu nhin ba s t tp { }2: 12 11 0 X x x x = e + s N . Tnhxcsut ba s c chn ra c tng l mt s chn.B.2. CHNG TRNH NNG CAO Cu VI b(2,0 im) 1.TrongmtphngOxy ,chotamgicABC c( ) ( ) ( ) 1; 3 , 1;1 , 3; 0 A B C .Lpphngtrnhng thngA bitA quaA v cng vi ng thng' Acng quaA chia tam gicABCthnh ba phn c din tch bng nhau. 2.Trong khng gianOxyz , cho ng thng( )3 2: 13x ty t tz t= + A = + e =+Rv mp ( ) : 2 5 0 x y z + + = . Gi( ) A = A . Tm im( ) , B C eA esao cho2 6 BA BC = =v
060 ABC = . Cu VII b (1,0 im) Tmm bt PT: ( ) ( )1226 1 6 2 1602012x xxx m mex x ( + + ( > + ng | | 0;1 x e . www.MATHVN.comwww.mathvn.com www.MATHVN.com Vn Ph Quc, GV.Trng i hc Qung Nam D: 0982.333.443; 0934.825.925 37 A- PHN CHUNG(7,0 im) Cu I (2,0 im) . Cho hm s: 21xyx+= (C). 1. Kho st s bin thin v v th (C) ca hm s. 2. GiAl ng thng i qua( ) 1; 0 Av c h s gcm. Tmm Act( ) Cti hai im phn bit , MNthuc hai nhnh ca th sao cho2 AM AN = .Cu II (2,0 im) 1. Gii phng trnh:5 os2 6 3sin 2 c x x = . 2. Gii bt phng trnh: 3 3 2 24 6 7 12 6 2 x x x x x + + + + > Cu III (1,0 im) . Tnh tch phn: 20sin1 sin 2xe xI dxx=+} Cu IV (1,0 im). Cho hnh chp. S ABCc y l tam gic u cnha ,SA vung gc vi mt phng y, gc gia mt phng( ) SBCv mt phng( ) ABCbng 060 . Gi, HKln lt l hnh chiu vung gc ca imA lnSBvSC ,Il trung imBC . Tnh th tch khi t dinAIHK . Cu V(1,0 im).Cho hm s( ) : 0; f + R tha mn iu kin:( )441tan 2 tantanf x xx= +0;4x | | e |\ ..Chng minh rng:( ) ( ) sin cos 1960;2f x f x x | |+ > e |\ .. B- PHN RING(3,0 im). Th sinh ch c chn mt trong hai phn B.1. CHNG TRNH CHUN Cu VI a(2,0 im) 1. Trong mt phng Oxy, cho hypebol( )2 22 2: 13x yHa a = , nhA thuc nhnh phi ca( ) Hv tiu im 1Fthuc nhnh tri. Mt ng trn di ng i quaA v 1Fct( ) Hti, , MNPkhcA. Chng minh tam gicMNPl tam gic u.2. Trong khng gian Oxyz, cho tam gicABCvung cn tiA c trng tm( ) 3; 6;1 Gv( ) 4;8; 1 M l trung im caBC. ng thngBC nm trong mt phng2 2 14 0 x y z + + = . Tm ta cc nh , , A B C . Cu VII a (1,0 im). Tm cc cn bc hai caos2 sin 2 z c i + = . B.2. CHNG TRNH NNG CAO Cu VI b(2,0 im) 1.TrongmtphngOxy, chongtrn( ) ( ) ( )2 2: 1 3 9 C x y + + = vhaiim( ) ( ) 1;1 , 2; 2 A B . Tm ta cc im, CD nm trn ng trn( ) Csao choABCD l hnh bnh hnh. 2.Trongkhnggianvihto Oxyz,chohaingthng: 13 3 3:2 2 1x y zd = = ; 2d lgiao tuyn ca hai mt phng:5 6 6 13 0 x y z + =v6 6 7 0 x y z + = . Chng minh rng 1d v 2dct nhau. GiIlgiao im ca 1dv 2d . Tm ta cc im, A B ln lt thuc 1 2, d dsao cho tam gicIAB vung cn tiIv c din tch bng 4142. Cu VII b (1,0 im).Gii h phng trnh: ( )( )2012 22012 2log 2 3 5 2012log 2 3 5 2012xyx x x yy y y x+ =+ =. www.MATHVN.comwww.mathvn.com www.MATHVN.com Vn Ph Quc, GV.Trng i hc Qung Nam D: 0982.333.443; 0934.825.925 38 A- PHN CHUNG(7,0 im) Cu I (2,0 im) . Cho hm s: 2 3 mxyx m+= (mC ). 1. Kho st s bin thin v v th ca hm s khi1 m =2.GiIl giao im hai tim cn. Tmm tip tuyn bt k ca hm s ct hai tim cn ti, A B sao cho din tch tam gicIAB bng42 . Cu II (2,0 im) 1. Gii phng trnh: 88cot tan 8sin3x x x | | = |\ .. 2. Gii h phng trnh: ( )( )2 22 272 1 2 127 6 14 0x y xyx y xy x y =+ + + = ,, xy eR. Cu III (1,0 im) .Tnh tch phn: 236tan t antan tanx x x xI dxx x x x (+ | | | |= + ( || \ . \ . ( } CuIV(1,0im).Chohnhchpu. S ABCccccnhybnga ,ngcaohnhchpl 3 a .Mtphng( ) P quacnhyBC vvunggcvicnhbnSA.Himtphng( ) P chia hnh chp thnh 2 phn c t s th tch l bao nhiu? Cu V(1,0 im). Cho, 0 xy > . Tm gi tr nh nht ca biu thc: ( ) ( )2 24 2 2 47 7 x x y y y xAxy xy+ + +=+. B- PHN RING(3,0 im). Th sinh ch c chn mt trong hai phn B.1. CHNG TRNH CHUN Cu VI a(2,0 im) 1. Trong mt phng Oxy, cho ng thng: 2 4 0 d x y + =v imBchy trnd . Trn tiaOBly imA tha mn. 1 OAOB = . Hy tm tp hp cc imA.2. Trong khng gian Oxyz, cho im( ) 4; 3; 2 M v hai ng thng: 12 3 1:1 2 2x y zd += = ; 22 1 2:1 2 1x y zd + = =. Vit phng trnh ng thngdquaMct 1 2, d dln lt ti, A B sao cho2 MA MB = . Cu VII a (1,0 im). Cho cc s phc( ) , 0 p q q = . Chng minh rng cc nghim ca phng trnh 2 20 z pz q + + =c mun bng nhau th pq l s thc. B.2. CHNG TRNH NNG CAO Cu VI b(2,0 im) 1. Trong mt phng Oxy, cho 1 2: 2 2 0, : 2 1 0 d x y d x y + = + = . Gi, , A B Clnlt l hnh chiu vung gc ca im 5 12;13 13M | | |\ . xung 1 2, d d vOx . Chng minh ba im, , A B Cthng hng. 2. Trong khng gian vi h to Oxyz, cho ng thng 2: 2 33 2x td y tz t= + = + = v mt cu ()2 2 2: 4 4 8 1 0 S x y z x y z + + + = . Chng minhdct() Sti hai im phn bit , A B. Vit phng trnh mt phng( ) i qua, A B v ct () Stheo giao tuyn l mt ng trn ln nht.Cu VII b (1,0 im).Gii h phng trnh: ( )292 44 log 4log 3 1 log 32 3 10x yx y + =+ = . www.MATHVN.comwww.mathvn.com www.MATHVN.com Vn Ph Quc, GV.Trng i hc Qung Nam D: 0982.333.443; 0934.825.925 39 A- PHN CHUNG(7,0 im) Cu I (2,0 im) . Cho hm s: 33 y x x = +(C). 1. Kho st s bin thin v v th (C) ca hm s. 2. Tm trn th( ) Cc bao nhiu b im, , , A B CD sao cho t gicABCD l hnh vung tm O. Cu II (2,0 im) 1. Tm nghim nguyn dng ca phng trnh: ( )2sin 3 9 16 80 04x x x( = ( . 2. Gii h phng trnh: ( )( )2 21 221 1 3 1y xx y xy x x+ = ++ = +( ) , xyeR . Cu III (1,0 im) . Tnh tch phn: { }20min 3 , 4xI x dx = }. Cu IV (1,0 im). Chohnhchp. S ABC,giG ltrngtmtamgicSBC .MtphngquayquanhAGctcnh , SBSCtheo th t ti, MN . Gi 1Vl th tch t dinSAMN ; Vl th tch t dinSABC . Tm gi tr ln nht, gi tr nh nht ca t s 12VV. Cu V(1,0 im). Cho 2,5, a b c > v3 a b c + + = . Chng minh rng:2 2 226 5 26 5 26 595 2 5 2 5 2a b ca b c + + >+ + +. B- PHN RING(3,0 im). Th sinh ch c chn mt trong hai phn B.1. CHNG TRNH CHUN Cu VI a(2,0 im) 1. Trong mt phng Oxy, cho ng thng: 2 0 x y A + + =v ng trn( )2 2: 4 2 0 C x y x y + = .GiI ltmca( ) C ,M limthucA.QuaM kcctiptuyn , MA MB n( ) C( A v B l cc tip im). Tm ta imMbit t gicMAIBc din tch bng 10. 2. Trong khng gian Oxyz, cho ba im( ) ( ) ( ) 2; 0; 0 , 0; 4; 0 , 0; 0; 4 A C S . Tm ta imBthuc mt phngOxysao cho t gicOABC l hnh ch nht. Vit phng trnh mt cu() Si qua bn im , , , OB CS . Tm ta 1Ai xng viA quaSC .Cu VII a (1,0 im). Cho s phcztha mn1 z = . Chng minh: 3 21 1 1 5 z z z s + ++ + s . B.2. CHNG TRNH NNG CAO Cu VI b(2,0 im) 1. Trong mt phng Oxy, hy vit phng trnh chnh tc ca elip( ) Ebit rng( ) Ec tm sai bng 53 v hnh ch nht c s ca( ) Ec chu vi bng 20. 2. Trong khng gian vi h to Oxyz, cho mt cu()2 2 2: 10 2 6 10 0 S x y z x y z + + + =v mt phng ( ) : 2 2 5 0 P x y z + + + = . T mt imMtrn mt mt phng( ) Pk mt ng thngA tip xc vi mt cu() StiN . Tm v tr imM11 MN = . Cu VII b (1,0 im).Gii bt phng trnh: ( ) ( )2011 2012log 2010 log 2011 2x xe + + + s . www.MATHVN.comwww.mathvn.com www.MATHVN.com Vn Ph Quc, GV.Trng i hc Qung Nam D: 0982.333.443; 0934.825.925 40 A- PHN CHUNG(7,0 im) Cu I (2,0 im) . Cho hm s: ( )22 1xyx+= (C). 1. Kho st s bin thin v v th (C) ca hm s. 2. Tm tt c cc gi tr ca tham smeR ng thng: d y x m = +ct th( ) Cti hai im phn bit, A B sao cho 2 2372OA OB + = . Cu II (2,0 im) 1.Gii phng trnh:5sin 2cos 3 1 5cos3 2sin 1 x x x x = . 2.Gii h phng trnh: 62 3 32 3 3 6 3 4xx y yyx x y x y = ++ = + . Cu III (1,0 im) . Tnh tch phn: 22111xI dxx x+=+ }. Cu IV (1,0 im). Chohnhlpphng. ABCD A B C D '''' ccnhbnga .Gi, MN lnltltrungimcccnh , A B B C '' '' . Tnh theoath tch khi t dinAD MN 'v khong cch tA n ng thngD N ' . Cu V(1,0 im). Cho, , a b cl cc s dng tha mn 12a b c + + = . Tnh gi tr ln nht ca biu thc:( )( )( )( )( )( )( )( )( )( )( )( )a b b c b c a c a c a bPa b b c a c b c a c a b a c a b b c+ + + + + += + ++ + + + + + + + + + + + B- PHN RING(3,0 im). Th sinh ch c chn mt trong hai phn B.1. CHNG TRNH CHUN Cu VI a(2,0 im) 1.TrongmtphngOxy,chongtrn( )2 2: 16 C x y + = .Vitphngtrnhchnhtccaelipc tmsai 12e = bitelipctngtrn( ) C tibnim, , , A B CDsaochoAB songsongvitrc honh v2 AB CD = . 2.TrongkhnggianOxyz,choccim( ) ( ) 2; 0; 0 , 1;1;1 A H .Vitphngtrnhmtphng( ) P i qua, A Hsao cho( ) Pct, Oy Ozln lt ti, B Ctha mn din tch tam gicABCbng4 6 .Cu VII a (1,0 im). Cho s phc 1 32 2z i = + . Hy tnh 21 z z + + . B.2. CHNG TRNH NNG CAO Cu VI b(2,0 im) 1. Trong mt phng Oxy, cho parabol( )2: 16 P y x =v im( ) 1; 4 A . Hai im phn bit, B C( khc A) di ng trn( ) Psao cho
090 BAC = . Chng minh rng ng thngBClun i qua mt im c nh.2.TrongkhnggianvihtoOxyz,chomtphng( ) : 2 3 6 18 0 x y z + + = .Gi, , A B C ln lt l giao im ca( ) vi cc trc, , Ox Oy Ozv giHl trc tm tam gicABC . Chng minh rngvimiM thucmtphng( ) khngtrngviccim, , , A B CH talunc: 2 2 2 22 2 2 22MA MB MC MHOA OB OC OH+ + = + . Cu VII b (1,0 im).Gii bt phng trnh: ( )2 3 4 2 22 25 6 log log 5 5 6 x x x x x x x x x x + + > ++ + . www.MATHVN.comwww.mathvn.com www.MATHVN.com
