1. 1. S GD & T KHNH HO TR NG THPT L T TR NG THI TH I H C L N I - NM H C 2014-2015 Mn: TON Th i gian lm bi: 180 pht khng k th i gian giao Cu I: (2 ) Cho hm s 2 1 ( ) 1 x y f x x = = 1. Kh o st s bi n thin v v th (C) c a hm s . 2. G i I l giao i m c a hai ng ti m c n c a th (C) , hy tm trn th (C) i m M c honh dng sao cho ti p tuy n v i th (C) t i i m M c t ng ti m c n ng , ti m c n ngang l n l t t i A v B th a mn: 2IA2 + IB2 = 12 . Cu II: (2 ) Gi i cc phng trnh sau : 1. (1 sinx)(2sin 2 6cosx 2sinx 3) 2 2cos 1 x x + + + = + 2. 3 2 27 3 3 1 log log ( 4) log ( 2) 4 x x x+ + = Cu III: (1 ) Tnh tch phn : 1 2 0 ( ) 2 x x x x e I dx x e + = + Cu IV: (1 ) Trong m t ph ng v i h t a Oxy ,cho hai i m A(1;2); B(4;1) v ng th ng d: 3x-4y+5=0. Vi t phng trnh ng trn (C) i qua A,B v c t d t i C, D sao cho CD = 6. Cu V: (1 ) Trong m t chi c h p c ch a 6 vin bi , 5 vin bi vng v 4 vin bi tr ng. L y ng u nhin trong h p ra 4 vin bi. Tnh xc su t trong 4 vin bi l y ra khng c c 3 mu. Cu VI: (1 ) Cho hnh chp u S.ABCD c di c nh y b ng a, m t bn c a hnh chp t o v i m t y m t gc 600 . Mp(P) ch a AB v i qua tr ng tm G c a SAC c t SC , SD l n l t t i M,N . Tnh th tch kh i chp S.ABMN theo a Cu VII: (1 ) Gi i h phng trnh : 3 2 3 3 2 6 13 10 2 5 3 3 10 6 x x x y y x y x y x x y + = + + + + = + Cu VIII: (1 ) Cho 0,, zyx v 3=++ zyx .Tm gi tr nh nh t c a bi u th c: xzzyyx P ++ + ++ + ++ = )1ln(24 1 )1ln(24 1 )1ln(24 1 -----------------H T --------------- www.VNMATH.com
2. 2. Trang 1 P N MN TON THI TH L N I NM H C 2014-2015 - KH I 12 Cu p n i m 1. Kh o st s bi n thin v v th (C) : 2 1 ( ) 1 x y f x x = = : (1 ) + T p xc nh : D = R{1} + ( ) 2 1 ' 0; 1 y x D x = < : Hm s ngh ch bi n trn cc kho ng ( ;1) v (1; )+ 0.25 + 1 1 lim ; lim x x y y + = = + : TC x = 1 + lim 2 x y+ = : TCN y = 2 0.25 + B ng bi n thin: x - 1 + y' - - y 2 + - 2 0.25 + i m c bi t : (0;1) ; 1 ;0 2 + th : x y 2 1 I 1 0.25 2. (1 ) + I(1;2) . G i M(x0; 0 0 2 1 1 x x ) (C) , x0 > 0 ; 0 1x + Pttt v i (C) t i M : 0 02 0 0 2 11 : ( ) ( 1) 1 x d y x x x x = + 0.25 + A l giao i m c a d v TC A(1; 0 0 2 ) 1 x x + B l giao i m c a d v TCN B(2x0 -1; 2) 0.25 + Tnh c IA2 = ( ) 2 0 4 1x ; IB2 = 4(x0 1)2 + 2IA2 + IB2 = 12 2 4 2 0 0 02 0 2 ( 1) 3 ( 1) 3( 1) 2 0 ( 1) x x x x + = + = 0 0 2 0 00 2 0 00 0 0 1 1 2 1 1 0 (loai)( 1) 1 1 2 1 2( 1) 2 1 2 1 2 (loai) x x x xx x xx x x = = = = = = = + = = = 0.25 Cu I (2 ) + KL: V y c 2 i m c n tm: M1(2;3) ; M2 (1+ 2 ; 2+ 2 2 ) 0.25 www.VNMATH.com
3. 3. Trang 2 1. Gi i phng trnh: (1 sinx)(2sin 2 6cosx 2sinx 3) 2 2cos 1 x x + + + = + (1) + i u ki n: 1 2 cos 2 ; 2 3 x x k k Z + + 0.25 (1) (1 sinx)(4sin cos 6cos 2sin 3) 2 2cos 1 x x x x x + + + = + (1 sinx)(2sin 3)(2cosx 1) 2 2cos 1 x x + + = + 0.25 (1 sinx)(2sin 3) 2x + = 2 2sin sinx 1 0x + = 0.25 2 2 sinx 1 21 6sinx 2 5 2 6 x k x k x k = + = = + = = + th a mn i u ki n 0.25 2. Gi i phng trnh: 3 2 27 3 3 1 log log ( 4) log ( 2) 4 x x x+ + = (1) + K : 0 2x< V i K trn, (1) 3 3 3log log ( 4) log 2x x x + + = [ ]3 3log ( 4) log 2x x x + = ( 4) 2x x x + = 0.25 2 ( 4) 2 2 ( 4) 2 x x x x x x x x > + = < + = + 2 2 2 3 2 0 2 5 2 0 x x x x x x > + + = < + = 0.25 2 1 2 5 33 2 2 5 33 2 x x x xx x + + > = = =< = 0.25 Cu II: (2 ) i chi u k , nghi m c a pt : 5 33 2 x + = 0.25 1 2 0 ( ) 2 x x x x e I dx x e + = + = 1 2 0 ( 1) 2 x x x x e dx xe + + t t = xex dt = (x+1)ex dx 0.25 i c n: x = 0t =0 , x=1t=e 0.25 0 0 2 (1 ) 2 2 e e t I dt dt t t = = + + 0.25 Cu III: (1 ) = (t-2ln|t+2|) 0|e = e+2ln 2 2e + 0.25 www.VNMATH.com
4. 4. Trang 3 y x C D A D O I B I Nh n xt A thu c d nn A trng v i C hay D . (Gi s A trng C) G i I(a;b) l tm ng trn (C), bn knh R>0. (C) i qua A,B nn IA=IB=R 2 2 2 2 (1 ) (2 ) (4 ) (1 )a b a b R + = + = 3 6b a = 0.25 Suy ra I(a;3a-6) v R = 2 10 50 65a a + (1) G i H l trung i m CD IH CD v IH = d(I;d) = 9 29 5 a + R=IC= ( ) 2 2 2 9 29 9 25 a CH IH + = + (2) 0.25 T (1) v (2) , c: 2 10 50 65a a + = ( ) 2 9 29 9 25 a + 2 1 13 56 43 0 43 13 a a a a = + = = 0.25 Cu IV (1 ) + a=1 (1; 3); 5I R = . Pt ng trn (C): (x-1)2 +(y+3)2 =25 + 43 13 a = 43 51 5 61 ( ; ); 13 13 13 I R = . Pt ng trn (C): 2 2 43 51 1525 13 13 169 x y + = 0.25 S cch ch n 4 vin bi b t k trong h p : 4 15 1365C = cch 0.25 + Ch n 2 bi , 1 bi tr ng , 1 bi vng: 2 1 1 6 5 4. .C C C + Ch n 1 bi , 2 bi tr ng , 1 bi vng: 1 2 1 6 5 4. .C C C +Ch n 1 bi , 1 bi tr ng , 2 bi vng: 1 1 2 6 5 4. .C C C S cch ch n 4 vin bi c 3 mu : 2 1 1 6 5 4. .C C C + 1 2 1 6 5 4. .C C C + 1 1 2 6 5 4. .C C C = 720 cch 0.25 S cch ch n 4 vin bi khng c 3 mu : 1365-720 = 645 cch 0.25 Cu V (1 ) Xc su t c n tm : P = 645 43 1365 91 = 0.25 www.VNMATH.com
5. 5. Trang 4 JI N MG O C A D B S G i O l giao i m c a AC v BD S.ABCD l hnh chp t gic u nn SO (ABCD) G i I, J l n l t l trung i m c a AB v CD , xc nh c gc gi a m t bn (SCD) v m t y (ABCD) l 0 60SJI = . 0.25 Nh n xt SIJ u ; SO = 3 2 a ; VS.ABCD = 3 1 3 . 3 6 ABCD a SO S = ( vtt) 0.25 Trong (SAC) , AG c t SC t i M , M l trung i m c a SC C/minh c MN// AB v N l trung i m c a SD . 1 1 2 4 SABM SABM S ABCD SABC V SM V V V SC = = = . 1 1 4 8 SAMN SAMN S ABCD SACD V SM SN V V V SC SD = = = 0.25 Cu VI (1 ) 3 . . . . 3 3 8 16 S ABMN S ABM S AMN S ABCD a V V V V = + = = ( vtt) 0.25 Gi i h phng trnh : 3 2 3 3 2 6 13 10 (1) 2 5 3 3 10 6 (2) x x x y y x y x y x x y + = + + + + = + (1 ) 3 3 (1) ( 2) ( 2)x x y y + = + Xt hm s f(t) = t3 +t , t R c f (t) = 3t2 +1>0, t R f(t) ng bi n trn R v (1) 2x y = (3) 0.25 Thay (3) vo (2): 3 2 5 3 3 5 2 3 10 26 (4); 1 2 x x x x x x+ = + 0.25 + Ch ng minh g(x) = 3 3 5 2x x+ ng bi n trn o n 5 1; 2 + Ch ng minh h(x) = 3 2 3 10 26x x x + ngh ch bi n trn o n 5 1; 2 g(2) = h(2) = 2 x=2 l nghi m duy nh t c a pt (4) 0.25 Cu VII (1 ) p s ( ) ( ); 2;0x y = 0.25 Cho 0,, zyx v 3=++ zyx .Tm gi tr nh nh t c a bi u th c: xzzyyx P ++ + ++ + ++ = )1ln(24 1 )1ln(24 1 )1ln(24 1 Cu VIII (1 ) V i cba ,, >0,p d ng b t ng th c Csi ta c : 1 1 1 1 1 1 9 ( ) 9a b c a b c a b c a b c + + + + + + + + (1) D u = x y ra cba == p d ng (1) ta c zzyyxx P ++++++ )1ln(2)1ln(2)1ln(212 9 0.25 www.VNMATH.com
6. 6. Trang 5 Xt [ ]3;0,)1ln(2)( += ttttf t t t tf + = + = 1 1 1 1 2 )(' ; 10)(' == ttf 32ln4)3(,14ln)1(,0)0( === fff 14ln)(32ln4 tf 0.25 12ln 2 9 ( ) ( ) ( ) 3ln 4 3 12ln 2 3 ( ) ( ) ( ) 12 9 3ln 4 f x f y f z f x f y f z + + + + + + + 9 9 3 12 ( ) ( ) ( ) 9 3ln 4 3 ln 4 P f x f y f z = + + + + + V y 3 1 3 ln 4 MinP x y z= = = = + 0.25 M i cch gi i khc ng c a hs u cho i m tng ng v i m i ph n c a cu www.VNMATH.com