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Toán học Phổ thông - Các phương pháp giải nhanh đề thi đại học - Hoàng Việt Quỳnh

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  1. Toặn hổc phưí thưng a WWW.MATHVN.COM
  2. Li nĩi đ u: o oo oo o oo o ơ o • ơ • ơ o ơo o o o o ơo ơ o • ơ o oo o oo oooo oơ ơơo ơoo oo o o oo o a a yya yyaye e WWW.MATHVN.COM
  3. Bài I: ơ ơ ư ươ  a y ey n yy  y  a n ey y y ươa  a y e a a a x = x0 + a1t  y = y0 + a2t  a a x = 3 + 2t  y = 4 + 3t yaa  ey n  e a a x = 2 + t   y = 2 − t x3 + 8 + 3 12 − x3 = 10 x3 + 8 12 − x3  ya     aay WWW.MATHVN.COM
  4. y ayaay 3 3 x + 8 + 312 −x = 10 X Y a aea X = 1+ 3t  Y = t-3 yyy yy 3 x+3 x+2 X Y  x + 3 = 1− t x + 3 = 1− 2t + t 2    3 3 x + 2 = t x + 2 = t ya  •  ư y y  x + 3 = u • a  y e 3 x + 2 = v yya • ayaa aa a x + y − xy = 3 (1)   x +1 + y +1 = 4 ()2  x +1 = 2 + t x +1 = t 2 + 4t + 4    2  y +1 = 2 − t y +1 = t − 4t + 4 x = t 2 + 4t + 3   y = t 2 − 4t + 3 (t 2 + 3 + 4t)( t 2 + 3 − 4t) WWW.MATHVN.COM
  5.  t 4 −10 t 2 + 9    ho ặc  y a f (x) = m a  x + 2m = 1+ 3t   3m − x = 3 − t x + 2m = 1+ 6t + 9t 2      3m − x = 9 − 6t + t 2   y 1) 2)  2x+ y +− 1 x += 11 3)  3x+ 2 y = 4 4) 1− sin()x ++ 1 cos() x = 1 WWW.MATHVN.COM
  6. Bài II: Các cách gi ải ph ươ ng trình và b ất ph ươ ng trình vơ t ỉ. ya ya aya yy ayeay a • • yaa • B ≥ 0  A = B   A = B 2 B ≥ 0  A B  B ≥ 0   2 A > B VD1. x ≥ 0  5 − x ≥ 0 0 ≤ x ≤ 5 0 ≤ x ≤ 5       2 2 10 − x ≥ 0 2 5x − x 2 = 5 − x  5(4 x − x ) = 25 −10 x + x  x + 5 − x + 2 x 5( − x) = 10 0 ≤ x ≤ 5    ∨ x 2 − 6x + 5 = 0 2 x − x + 3 x −1 x ≥ 1 x ≥ 1      x 2 + 2x − 3 > x 2 − 2x +1 x > 1 WWW.MATHVN.COM
  7. VD3.       VD4. − x 2 + x = 0   x − x 2 > 0 ⇔ x = 0 ∨ x = 1   2 x − x − 2 ≥ 0 ư yaya aayaaa a ea B = 0  B  B > 0 a  A ≥ 0 VD5.     3x − 5  2 −   ≥ 0   4    3x − 5 ≥ 0  x=3   3x − 5  2 x 2 − 8 =     4  WWW.MATHVN.COM
  8. ư aay yaayya yayay aaaa aayaaa ươ  f (u(x); n u(x))≥ 0 f ()u(x); n u(x) ≤ 0 n u(x)  f ()u(x); n u(x) = 0  VD1. 2 2  => t>0 ; t +2= x + x     VD2. t ≥ 0 x −1   t 2 +1 = x   t 2+1-(t+1)=2  t 2-t-2=0  t=2 ho ặc t=-1 x=5 VD3.  => WWW.MATHVN.COM
  9. ∨ t=-3   TH2: t=-3    f (n u(x) + n v(x)) n u(x) = u  a m v(x) = v 23 3x − 2 + 3 6 − 5x − 8 = 0 y 5 8 3 3x − 2 = u  u 3 + v 2 =   3 3  6 − 5x = v (v ≥ )0  2u + 3v − 8 = 0  2 5 3 2 8 5 3  8 − 2u  8 2  u + v =  u +   = (u + 2)(15 u − 26 u + 20 ) = 0 3 3 3  3  3        8 − 2u  8 − 2u  8 − 2u v = v = v =  3  3  3 u = −2    v = 4 A yaaya y ayy WWW.MATHVN.COM
  10.  1 1 x− = y − ()1  x y  3 2y= x + 1() 2 y 1   x= y a ()()1⇔x − y  1 +  = 0 ⇔  xy   xy = − 1  x= y = 1  xy=  xy = x= y  −1 + 5 ⇔  ⇔  ⇔==x y 2yx=+3 12 xx =+ 3 1 x−1 x2 +−= x 1 0  2   ()()   −1 − 5 x= y =  2  1  1  y = − 2 2 xy = − 1 x  y = − 4 2 1  1  3 xx x x xVN⇒ 3 ⇔  ⇔  x ++=2 −  ++  +>∀ 0, 2y= x + 1 2 3  4 2  2  2 − =x + 1 x+ x +2 = 0  x −+1515 −+  −− 1515 −−  ya ()()x; y = 1;1, ;  , ;  1 1  1 1  2 x++ 1 y(y + x) = 4y() 1  x,y∈ R . 2 () (x+ 1)(y +− x 2) = y() 2 (1) ⇔x2 ++ 1 yxy( +− 40*) = ( ) ux=2 +>1 0; vxy =+− 4 u− yv = 0( 3 ) ⇔  aya (3) ⇔+uuv( + 2.01) v =⇔ u + vv( + 20)  = u()() v+2 = y 4 ⇔v2 +2 v += 1 0 ⇔(v + 1)2 =⇔=−⇔+= 0 v 1 xy 3 2 ⇒ x+1 − y = 0 2 x=1 y = − 2 y ⇔ ⇔x +1 −() 3 − x = 0 ⇔  x=3 − y x= 2⇒ y = 5 x3− 8x = y 3 + 2y  x,y∈ R . 2 2 () x− 3 = 3(y + 1)() * 3 3 xy3− 3 =2() 4 xy + 3( xy−) = 6421( xy + ) ( ) ⇔ ⇔  yaya 2 2 2 2 x−3 y = 6 x−3 y = 6() 2 ⇔3( xy3322 −=−) ( x 34 y)( xy +⇔−) x 322 12 yxxy += 0 ⇔x( x2 +− xy12 y 2 ) = 0 yyayayya WWW.MATHVN.COM
  11. x2+ xy −12 y 2 = 0 ( x−3 yx)( + 4 y ) = 0 ⇒ ⇔  x2−3 y 2 = 6 x2−3 y 2 = 6 xy−3 = 0  xy = 3  y= 1⇒ x = 3 ⇔  ⇔  x2−3 y 2 = 6  6 y 2 = 6  y= − 1⇒ x = − 3  78− 478  y= ⇒ x = xy=−4  xy =− 4 13 13 ⇔  ⇔  x2−3 y 2 = 6  13 y 2 = 6  78 478  y= − ⇒ x =  13 13 ya 78− 478  − 78478  ()()()x; y = 1;3,1;3, − −  ;  , ;  13 13  13 13   2 2 ()x− yx( + y ) = 13() 1  2 2 ()x+ yx() − y = 25() 2 aaay 2 2 2 2 2 2  ⇔+13(xyxy )()() −−− 25 xyxy( +=⇔−) 0()() xy 13 xy +− 25( xy +=)  0 ⇔−−( xy)( 12 x2 + 26 xyy − 12 2 ) =⇔−−− 0 2( xy)( 12 x2 + 26 xyy − 12 2 ) = 0 yya 3x= 2 y  y = − 3 25 2 −y  ⇔  3x= 2 y  y .  = 25 x = − 2 ()()32x− y 23 x − y = 0   9 3  ⇒ ⇔  2x= 3 y ⇔  2 2 2x= 3 y ()x+ yx() − y = 25  2  ()()x+ y x − y = 25   x = 3 252  1   y. y  = 25 ⇔   4 2  y = 2 aaa (−12x2 + 26 xy − 12 y 2 ) (3x− 2 y)( 2 x − 3 y ) yaayya ayyyae (−12x2 + 26 x − 12) = 0 y y a 3 2 x= ∨ x = y a y y a 2 3 3 2 x= yx ∨ = y y a 2 3 (−12x2 + 26 xy − 12 y 2 ) = 0 ⇔ (32x− y)( 23 x − y ) = 0 y (−+12x2 26 xy − 12 y 2 ) =−− 23( xyxy 2)( 2 −= 3) 0  aaa aaa aae WWW.MATHVN.COM
  12. ayya aey y x4− xy 3 + xy 22 = 1 x4+2 xyxy 3 + 22 =+ 2 x 9 Bài 1.  Bài 7.  x3 y− x 2 + xy = 1 x2 +2 xy = 6 x + 6 x2+ y 2 ++ xy = 4 xy+ x +1 = 7 y Bài 2.  Bài 8.   2 2 2 xxy()()+++1 yy += 1 2 x y+ xy +1 = 13 y 3 x2− xy + y 2 =3( x − y )  x+−1 y =− 8 x  Bài 9. Bài 3.   4 2 2 2 x+ xy + y =7() x − y ()x−1 = y 2 logx3+ 2 x 2 −− 3 xy 5 = 3  y + 2  x ( ) 3y = Bài 4.   x2 logy3+ 2 y 2 −− 3 yx 5 = 3 Bài 10.   y () x2 + 2 3x = x( x+ y +1) − 3 = 0  y2 Bài 5.   2 5  x y 1 1 ()+ −2 +=1 0 x− = y −  x Bài 11.  x y 9 9  3 x+ y = 1 2y= x + 1 Bài 6.   x25+ y 25 = x 16 + y 16 WWW.MATHVN.COM
  13. Bài III: ơ ư 22 21 2 1 1. sin x+ cos x =+ 1;1 tanx = ;1 += cot x . cos2x sin 2 x sinx cos x 1 2. tanx= ;cot x = ;tan x = . cosx sin x cot x sin(a± b ) = sin a cos b ± cos asinb cos(ab± ) = cos ab cos∓ sin ab sin o oo o 21+ cos2x 2 1 − cos2 x cosx= ;sin x = 2 2 oo o o 2tanx 1− tan2 x 2tan x sin 2x= ;cos2 x = ;tan 2 x = 1+ tan2x 1 + tan 2 x 1 − tan 2 x 1 cosab cos=() cos( ab −+ ) cos( ab + ) 2 1 sinab sin=() cos( ab −− ) cos( ab + ) 2 1 sinab cos=() sin( ab −+ ) sin( ab + ) 2 xy+ xy − sinx+ sin y = 2sin cos 2 2 xy+ xy − sinx− sin y = 2cos sin 2 2 xy+ xy − cosx+ cos y = 2cos cos 2 2 xy+ xy − cosx− cos y = − 2sin sin 2 2 WWW.MATHVN.COM
  14. ươư a yaya ye yya π  2sin2x−  + 4sin x += 1 0 6   3sin2x− cos2 x + 4sin x += 1 0  2sinx( 3cos2 x+ 2) − 2sin2 x = 0 sinx= 0 ⇔x = k π   2sinx( 3cos x− sin x + 2) = 0  1 π   3 cosxx− sin =−⇔ 1 cos x +  = cos x  2 6   x= k π  5π  x= + 2 k π  6  −7π x= + 2 k π  6 π a x 3π 4sin2 − 3 cos 2x =+ 1 2cos2 ( x − ) 2 4 ∈(0, π ) 2x 2  3 π  a 4sin− 3cos2x =+ 1 2cos x −  2 4  3π  ⇔−21() cosx − 3cos2x =++ 1 1 cos 2x −  2  ⇔−2 2cosx − 3cos2x =− 2 sin2x ⇔−2cosx = 3cos2x − sin2x aa 3 1 ⇔−cosx = cos2x − sin2x 2 2 π  5π 2 π 7 π ⇔cos2x +=  cos() π− x ⇔=+x k() ahayx =−+π h2() b 6  183 6 WWW.MATHVN.COM
  15. x∈( 0, π ) a a a 5π 17 π 5 π (0, π) x= ,x = ,x = 118 2 18 3 6 π 2 2 cos3 (x−− ) 3cos x − sin x = 0 4 3 π   ⇔2cosx −−   3cosx −= sinx 0 4   ⇔cosx + sinx3 − 3cosx −= sinx 0 () ⇔++cos332 x sin x 3cos xsinx + 3cosxsin 2 x −−= 3cosx sinx 0 cosx= 0 cosx≠ 0 ⇔  hay  sin3 x− sinx = 0 1+ 3tgx + 3tg23 x + tg x −− 3 3tg 2 x −− tgx tg 3 x = 0 2 π π ⇔sin x = 1 haytgx= 1 ⇔x = +π k ay x= + k π 2 4 π cos 2x − 1 tg(+ x ) − 3 tgx2 = 2 cos 2 x −2sin2 x ⇔−cot gx − 3tg2 x = cos2 x 1 π ⇔− −tgx02 =⇔ tgx 3 =−⇔ 1 tgx =−⇔=−+π∈ 1 x k,kZ tgx 4 ƯOƯƯ ∈ sin 2 x t 2 a cos 2 x 1− t 2 1− 2sin 2 x 3sinx− 4sin3 xtt = 3 − 4 3 sin2x=− 1 cos 2 x =− 1 t 2 cos 2x= 2 t 2 + 1 sin2x 1 − t 2 tan 2 x = = cos3x= 4cos3 x − 3cos xtt =− 4 3 3 cos 2x t 2 a WWW.MATHVN.COM
  16. 1 1 cot x = cos 2 x = t 1+ t 2 t 2 1− t 2 sin 2 x = cos 2 x = 1+ t 2 1+ t 2 1  2t sin2x=2t   t an2 x = 1+ t 2  1+ t 2 asin xb+ cos x a tan xb + atb + = = csin x+ d cos x c tan x + d ct + d   ∈ − 2; 2  t 2 −1 = ±t 2 + 1 ±2 ( ) t2−1  3 − t 3 sin33x+=+ cos x() sin xx cos() sin 22 x +− cos xxxt sin cos =−= 1  2  2 ƯƯ   y  ya      cos 2x 1 cotx−= 1 + sin2 x − sin2 x 1+ tanx 2 a 1− t 2    11+ t 2  t2 1 2 t −=1 + −()t ≠≠− 0; t 1 t1+ tt 1 +2 21 + t 2 π ⇔2t3 − 3 t 2 + 210 t −= ⇔t = 1 ⇔tanx =⇔=+ 1 x k π 4 cos3x+ cos2 x − cos x −= 1 0 ⇔4t3 − 32 tt + 2 −−−= 1 t 10 WWW.MATHVN.COM
  17. t = ± 1 cosx = ± 1 x= k π    ⇔−1 ⇔ 2π ⇔ 2π t = cosx = cos x= ± + 2 k π  2  3  3 1− sinx +− 1 cos x = 1 1−−+− sinxx cos 2 (1 sin x )(1 − cos x ) = 0 t 2 −1 ⇔ sin xcosx = 2 t 2 −1 121−+t + −= t 0 ⇔−+=+tt22142 t 2 −− 24 tt ⇔− (1) 2 =⇔= 0 t 1 2 π  π   π  2sinx +  = 1  sinx +  = sin   x= k π 4  4   4 cos 2 x sinx+ + 6tan2 x() 1sin −= x 2 1+ sin x t ∈[ − 1;1] 1−t2 t 2 t+ +6() 1 −=⇔−−= ttt 262 10 1+t 1 − t 2  π  x= + 2 k π  1  6 t =  1  2 sin x =  5π ⇔ ⇔2 ⇔ x= + 2 k π  −1   6 t = sinx = sin α   3 −1 x=arccos + 2 k π  3 1 sin6x+ cos 6 x = cos8 x 4 3 1 3 1− cos 4x  1 1− sin22 x = cos8 x ⇔− 1  = cos8 x 4 4 424  t ∈[ − 1;1]  2  π  πk π t=4 x = + k π x = + 3 1− t  1 2 2 4 16 4 1−  =() 2t − 1 ⇔ ⇔ ⇔  4 2  4  − 2  3π 3 πk π t = 4x=+ kπ  x =+  2  4 164 WWW.MATHVN.COM
  18. y 1 1  sin2x+− sinx − = 2cotg2x 2sinx sin2x  5x π   x π  3x  sin −  − cos −  = 2 cos  2 4   2 4  2 2  2 cos x+ 2 3 sin x cosx += 1 3(sin x + 3 cosx) sin2x cos2x  + = tgx − cot gx cosx sinx 1  ()()2cosx− 1 sin xx + sin2 − cos 2 x = 2  (2sinx+ 1)( 2cos x − 1) = 1  sin3xx+ cos 3 = 2( 1 − sin xx cos ) x  2sinx cos− cos x = 1 2 π  π  3  sin4xxx+ cos 4 + cos −  .sin 3 x −−=  0 4  4  2 2sinx+ cos x + 1  = a a sinx− 2cos x + 3 1 a 3 a x   tanxx+− cos cos2 xx = sin 1 + tan x tan  2  (2− sin2 2x) sin3 x  tan4 x + 1 = cos 4 x WWW.MATHVN.COM
  19. Bài IV: Tích Phân ay eee y O ươ A • f( ux( )). u ' ( xdx)  a π 2 sin 2 x I = ∫ 2 0 3+ cos x t=3 + cos 2 x ⇒ dt=2cos x( − sin x) dx ⇒ dt= − 2sin 2 xdx π 2 4 −dt 4 4 I=∫ = ln t⇒ I = ln 3 t 3 3 6 dx I = ∫ 2 2x+ 1 + 4x + 1 1 41x+ ⇒ t2 = 41 x + ⇒ tdtdx= 2 5(t+1 − 1 ) dt 5dt 5 dt 1  5 3 1 ∫2= ∫ − ∫ 2 =++lnt 1  =− ln 3()t+1 3t+1 3 () t + 1  t + 1212  3 π 4 dx I = ∫ 2 0 cosx 1+ tan x WWW.MATHVN.COM
  20. dx 1+ tanxt⇒ 2 = 1 + tan x⇒ 2 tdt = cos 2 x π 4 2 22tdt 2 2 I=∫ =2 ∫ dtt = 2 =− 222 1t 1 1 e 3− 2ln x I= ∫ dx. 1 x 1+ 2ln x dx 1+ 2lnx⇒ t2 = 1 + 2ln xtdt⇒ = x e 2 2 23−(t − 1 ) 2 10 2− 11 I=∫ tdt =−= ∫ ()4 tdt2 1t 1 3  aa  a ⇒  ⇒  du aa ∫ 2 ()u() x+ a 2  2 a2 + () u() x ⇒ aa 2 a2 − ()u() x ⇒ aa 3 dx ∫ 2 0 x + 9 a ⇒ dx=3( tan2 t + 1 ) dt π 4 WWW.MATHVN.COM
  21. π 2 π 4 3() tant+ 1 dt 1 π I=∫ = t 4 = 9 tan2 t + 1 3 12 0 () 0 5 2 dx I = ∫ 2 1 9−()x − 1 ⇒ dx= 3cos tdt 5 2 π 6 π π π π 63costdt 6 cos tdt 6 cos tdt π I =∫ = ∫ === ∫ t 6 2 2 cost 6 09− 9sint 0 1 − sin t 0 0 3 dx I = ∫ 2 2 1 x x + 3 3 tan t ⇒ dx=3( tan2 x + 1 ) dx π π 6 3 π1 π 2 dt 3() tant + 1 132 − 1 3 cos tdt I= dx =cos t = ∫2 2 ∫2 ∫ 2 3tant 3tan+ 3 3πsint 1 3sin π t 6cos2t cos 2 t 6 π π 13 d()sin t 13 623− I =−∫ 2 =− = 3π sint 3sin t π 9 6 6 WWW.MATHVN.COM
  22. Ư bb b ∫udv= uv − ∫ vdu aa a y aayaya  ∫ P( x)ln xdx a ln x y eax+ b     ∫ Px().sin( axb+ )  dx a   cos(ax+ b )  yyaya π 2 I=∫ (x + 1)sin2xdx. 0 u= x + 1⇒ du= dx π π  −()x + 1 1 2 π  −1 ⇒ I=cos 2 x2 + cos 2 xdx =+ 1 ⇒ ∫ dv= sin2 xdx v= cos 2 x 2 20 4  2 0 2 I=∫ (x − 2)lnx dx. 1  1 du= dx u= ln x  x x2  2 2  x  5 ⇒ ⇒ I=−2 x ln x −− 2 dx =−+ ln 4   2   ∫  dv=() x − 2 dx x 2  1  2  4  v= − 2 x 1  2 π 2 4 ∫ sin xdx 0 x⇒ t2 = x⇒ 2 tdt= dx π 2 4 π 2 WWW.MATHVN.COM
  23. π 2 B= 2∫ t sin tdt 0 π 2 I= ∫ tsin tdt 0 u= t  du = dt ⇒  dv=sin tdt  v = − cos t ππ π 2 π π Itt=−cos2 + cos tdt =− cos + 0cos 0 + sin t 2 = 1 ∫ 2 2 00 0 π 2 ∫ ex cos xdx 0 u= ex  du = e x dx ⇒  dv=−sin xdx  v =− cos x π π π π 2 π π 2 2 A=− excos x2 + e x cos xdx =− e2 cos + e0 cos 0 + ex cos xdx =+ 1 e x cos xdx ∫2 ∫ ∫ 0 0 0 0 π 2 K= ∫ ex cos xdx 0 u= ex  du = e x dx ⇒  dv=cos xdx  v = sin x π π 2 π K= exsin x2 −∫ e x sin xdx =− e2 A 0 0 π π π 1+ e 2 ay A=1 + eA2 − ⇒ 2 A= 1 + e 2 ⇒ A = 2 π ∫ xsin x cos 2 xdx 0 u= x du= dx ⇒ 2  2 dv= sin x cos xdx v= ∫sin x cos xdx v= ∫sin x cos 2 xdx t= cos x⇒ dt= − sin xdx WWW.MATHVN.COM
  24. −t3cos 3 x −t2 dt = +=− C + C ∫ 3 3 cos 3 x ⇒ v = − 3 cos3 x π 1π π 1 y A=− x +∫ cos 3 xdx =+ K 30 30 33 π π K=∫cos3 xdx = ∫ () 1 − sin 2 x cos xdx 0 0 ⇒ dt= cos xdx π 0 K=∫ ()1 − t2 dt = 0 0 π1 π ay A= + K = 3 3 3 π 2 x+ sin x D= ∫ dx π 1+ cos x 3  π u= x + sin x  2  du=()1 + cos x dx x+ sin x 1  D = ⇒  ∫ x dv= dx x π 2 2 x  v = tan 2cos 2cos  2 3 2  2 π π x2 2 x π   π 3  3 y Dxx=+()sin tan −+∫ () 1 cos xdx tan =+−+ 1    − K 2π π 22   323  3 3 π π π 2x 2 x x 2 K=+∫()1 cos x tan dx = ∫ 2cos2 tan dx = ∫ sin xdx π2 π 2 2 π 3 3 3 π 2 1 = −cos x = π 2 3 (9+ 2 3 )π aya 18 aa aaa ưư y WWW.MATHVN.COM
  25. y π 3  I= ∫sin2 xtgxdx . 0 7 x + 2  I= dx ∫ 3 0 x +1 e  I= ∫ x2 ln xdx 0 π 4  I=∫ ( tgxe + sin x cos xdx ) 0 π  I= ∫ cos x sin xdx 0 π 3  I=∫ tan2 x + cot 2 x − 2 dx π 6 π 2  I=∫ 2() 1 + cos 2 x dx −π 2 π 3 sin4x sin3 x  I= ∫ dx π tanx+ cot 2 x 6 10 dx  I = ∫ 5 x− 2 x − 1 e 3− 2ln x  I= ∫ dx. 1 x 1+ 2ln x π xsin x  I = ∫ 2 0 1+ sin x π 6 sinx+ sin 3 x  I = ∫ 0 cos 2 x  aa (P:y) = x2 − x + 3 d:y= 2x + 1. x2 27  ()()CyxCy1=2 ;2 = ;3() Cy = 27 x WWW.MATHVN.COM
  26. a a ya yay aaaaa yy aay A y= f( x ) fx( ) ≥0; ∀ xI ∈  fx( ) ≤0; ∀ xI ∈  1 yfx=() = xmx3 − 2 +() mm 2 +− 2 x 3 a (4; +∞ ) e (−∞ ;4 ) (2; +∞ ) D= R yx'2=−2 mxmm + 2 +− 2'2⇒ ∆ =− m + a  ∆≤'0 ⇔−m + 20 ≤ ⇔ m ≥ 2 y '( 0) ≤ 0 m2 + m −2 ≤ 0  ⇔  ⇔m ≤ 1 2 y '() 2≤ 0 m−3 m + 2 ≤ 0  ∆≤'0 ⇔−m + 20 ≤ ⇔ m ≥ 2 WWW.MATHVN.COM
  27.  ∆' > 0 m 0   −m +2 > 0  m 2 y '() 4≥ 0  2 ≥ ⇔ ⇔  m+9 m + 14 ≥ 0 ⇔    −2 ≤m ≤ 1  y '()− 2 ≥ 0 2    m−3 m + 2 ≥ 0 S   −4 < < 2 −4 <m < 2  2 −1 m2 y= x2 ++− mx 2() mmx 2 + 3 3 a (6; +∞ ) e (−∞ ;0 ) (6; +∞ ) y'=− x2 + 2 mx + m − m 2 ∆' = m a ⇔∆≤' 0 ⇔m ≤ 0 y '( 0) ≥ 0 −m2 + m ≥ 0 ⇔ ⇔  ⇔=m 1 2 y '() 2≥ 0 −m +5 m +≥ 4 0 (6; +∞ ) ∆' ≤ 0⇒ m ≤ 0 a (6; +∞ ) WWW.MATHVN.COM
  28.  ∆' > 0 m > 0   y'6() ≤⇔− 0  m2 + 13 m − 360 ≤   S m < 6  < 6   2 m ≤ 0 y ⇔ ⇔m ≤ 4 m∈[]0;4 2∆ ' ⇔−=⇔xx2 =⇔ 222 mm =⇔= 1 1 2 a e (−∞ ;0 ) (6; +∞ ) ∆≤' 0 ⇔m ≤ 0 ∆' ≥ 0  y '() 0≤ 0  y '() 6≤ 0 ⇔1 ≤m ≤ 4   S 0< < 6  2 m ≤ 0 y   1≤m ≤ 4 A 1 y= xmx32 − +()2 m 2 −+− 1 xmm 3 3 a y WWW.MATHVN.COM
  29. e a 3 3 a ( x1+ x 2 ) y'=− x2 2 mx + 21 m 2 − ∆=−'m2 + 1 ∆' > 0 −∞ +∞ a  y '( 0) = 0 2  2m − 1 = 0 2 ⇔S ⇔  ⇔=m 0 0 2  2   m 0 −m2 +1 > 0    2 m ⇔  2 m − 2 m > 0 ⇔   ⇒ −1 1 S m 0 m2 0 ⇔y'10() − > ⇔  2 m + 2 m > 0 ⇔   ⇔ 0 − 1 > − 1   m > − 1  2  ∆' > 0  m <1   y '()− 2 ≥ 0 2  2m+ 4 m +≥ 30 () ∀ m ⇔y '() 3≥ 0 ⇔  ⇔−<<1m 1   2m2 − 6 m +≥ 80 () ∀ m S  −2 ≤ ≤ 3 −2 ≤m ≤ 3  2 e WWW.MATHVN.COM
  30. −1 0 −1 0 m ≥ 0 0 a y '( 0) 0⇒ m 0 ⇔  3 P=()() xx1 + 2 −3 xxxx 121 + 2 → min 2 x x=2 m 2 − 1 −m +1 > 0  1 2 ya ⇔  x x m 3 2  1+ 2 = 2 Pm=()2 − 32() m − 1.2 m → min −1 0 WWW.MATHVN.COM
  31. S −∞ +∞ 2 y '(− 2) ≥ 0 a  y y '() 3≥ 0 yaaaa af (α ) ya aa f (α ) y aeaa S −b a ayay 2 2a S −b y a 2 2a ay  yy ayaye y aaayyy ya y ∞ yye aay eye yya a ax2 + bx + c  y = ax'2 + bx ' + c ' ab ac bc x2 +2 x + ab'' ac '' bc '' y ' = 2 yy ay ()ax'2 + bx ' + c a ≥ a y  ya ax+ b aaya y = a' x+ b ' y=− x33 mx 2 + 3( m −+ 1) x 4 a aa aa 2 5 aa ∆:y = 2 WWW.MATHVN.COM
  32. y '= 0 aa  y= f( x ) y y'= x2 − 2 xm − += 10 y=− x33 mx 2 + 3( m −+ 1) x 4 ⇒ y= x2 −−+2 xm 1 ( cxd +++=+) axbaxb () 0 ⇔=yx( 2 −−+2 xm 1(1)2) x −− mxm −+ 5 x2 +2 x − m += 101( ) ⇔  y=−2 mx − m + 5() 2 a  ⇒ ∆' ≥ 0 ⇒ m > 0 a aa ⇒ aa y=−2 mx − m + 5 ⇒ ∈ ⇔m = 2 y aa 2 5 2∆ ' ()1⇒ x− x = = 2 m 2 1 a 2 2 ()2⇒ yy21− =− 2 mxx() 21 − =− 4 mm ⇒ AB=()() x21 − x +− y 21 y = 2 5 m =1 2  ⇒16m+ 4 m = 20 ⇔ 5 ⇒ m =1 m = −  4 aa ∆:y = 2 ⇔dA( ; ∆=) dB( ; ∆ ) ∆:y = 2  y1−2 = y 2 − 2  y1= y 2 ⇔y1 −2 = y 2 − 2 ⇔ ⇔   y1−2 =−() y 2 − 2  yy1+ 2 = 4 ⇔−( 2mxm1 −++− 52) ( mxm 2 −+=⇔− 542) mxx( 12 +) − 2104 m += ⇔−2.22m − m + 104 =⇔= m 1 yx=+33 x 2 − 3( m − 1 ) x a aa ∆OAB aa aay ay e aaa 2 2 aa ()()x−1 + y − 1 = 4 aaaaa aaaaa WWW.MATHVN.COM
  33. y '= 0 aa  y= f( x )  y ' 2  =x +2 x − m += 1 0 ⇔  3  3 2 2 yx=+331 x −() m − xx =+−+() 2 xm 112() x +− mxm +− 1 x2 +2 x − m += 101( ) ⇔  y=−2 mx + m − 1 () ∆ a  ⇒ ∆' ≥ 0 ⇒ m > 0 a aa ∆OAB   ⇔OA ⊥ OB    OA= ( xA; y A ) ⇔ OAOB.  OB= () xB; y B ⇔xxyy1212 + =⇔0 xx 12 +−( 2 mxm 1 +−− 12)( mxm 2 +−= 10) 2 2 2 ⇔+xx124 mxx 12 +−+( 22 m mxx)()() 12 ++−= m 10 2 ⇔ ()()−++m14 mm2 −++− 1( 2 mm 2 + 2.2) −+−() m 10 = ⇔−4m3 + 9 m 2 − 7 m += 20 ⇔−( 4m2 + 52 m −)( m −= 10) ⇔m = 1 a  VN vì ∆=− 7 y CD aa ⇔y1. y 2 ⇔  4 m ≠ 1 aay ⇔x1 x 2 > 0 x1 x2 ⇔−m +>1 0 ⇔ m < 1 ay x+ x   y a 1 2 ;− 2mx + m − 1  2  ⇒ M(−1;3 m − 1 ) Ycbt⇔=531 m −⇔ m = 2 ay e aaa ∆:2y =− mx +−⇔∆ m 1:2 mx +−+= y m 10 WWW.MATHVN.COM
  34. 2m .0+ 0 − m + 1 2 2 ⇔d( O ; ∆) = 1 ⇔ = 1 ⇔−+=()()m1 2 m +⇔ 132 mm 2 + 20 = ()2m 2 + 1 2 m = 0  ⇔ −2 ayay m =  3 2 2 aa ()()x−1 + y − 1 = 4 ⇔d( I; ∆) = R ∆:2mx ++−= y m 10 2m .1+ 1 − m + 1 2 ⇒ = 2 ⇔+()m2 = 16 m2 +⇔− 4 15 mm 2 + 4 = 0 ()2m 2 + 1 m = 0  4 ⇔ 4 a m = m = 15  15 aaaaa −2mx + m −= 1 0 m −1  aa ∆ ⇒ ⇒ M ;0  y = 0 2m  y=−2 m .0 + m − 1 aa ∆ y ⇒ ⇒ N()0; m − 1 x = 0  m =1  m −1 1  1  ⇔xyM = N ⇔ =−⇔ m1 − 1.10  m −= ⇔m = 2m 2 m   2  −1 m =  2 −1 1 y ∆ aa aya m = 2 2 aaaaa 1 1 1 ⇔S = OM. ON ⇔= x y ∆OMN 2 8 2 M N  m = 2  2 m  2 m−2 m + 1 = ⇔ 1 1m − 1 1 ()m −1  2 m = ⇔=.m −⇔= 1 ⇔   2 42m 42 m  −m m2 −2 m + 1 = () VN  2 yaa WWW.MATHVN.COM
  35. ƯAOAA Cy1:= fx( ) ; C 2 : y = gx( ) aa aa fx( ) = gx( )  fx( ) = gx( )   f'()() x= gx ' ưuaa aya 2mx− 3 m − 2 ()C: y=() m ≠ − 2 (d) : y= x − 1 m x −1 a a [−2;3 ] e aa 2mx− 3 m − 2 =−x1 ⇔ gxx()() :213302 − mxm +++= x −1 S −∞ x x +∞ 1 2 2 g( x )  m > 2 ∆' > 0 ⇔  aa   m 0  −6 −6  12+()m ++ 13 m +> 30  m >  − 1    2 m > − 2 m > 2 g (2) > 0  44−()m ++ 13 m +> 30 −+<m3 0  m < 3 S ⇔ ⇔  ⇔  < 2 m +1 < 2 m<1  m < 1  2 WWW.MATHVN.COM
  36. m 0  330m+ > ⇔ m >− 1 ⇔S ⇔  0 ≤ m+≥1 0 ⇔ m ≥− 1  2 aya e g(00330) 2020) m m 0 WWW.MATHVN.COM
  37. y ưu y ∆ aa ∆ x a aa x1 0 m2 +8 m 0 m. g () 1> 0 −2m > 0  m 0 k>0  k > 0 ⇔  ⇔  a g ()2≠ 0 44+−+≠k 10  k ≠ 9 Bxy( 11;) ; Cxy( 22 ; ) a 2 2 2 x+2 x − k += 101( ) x  y= kx −2 k + 4() 2 2∆ ' ⇔−=x x = 2 k 2 1 a Hình 3 ⇔−=yy21 kxx( 21 −) = 2 kk  2 2 BC=()() x21 − x +− y 21 y = 2 2 ⇔44kk +3 = 224 ⇔ kk 3 +−=⇔= 480 k 1 WWW.MATHVN.COM
  38. y ∆:y = 1( x −+ 24) yfx=( ) =− x33 x 2 + 2 y a ay ayya a ay Aa(;2)− ∈ dy : =− 2 ∆ a A( a ;− 2) y= kx( −− a ) 2 ( ∆ ) ∆ a x3−32 x 2 += kxa( −−) 21( )  2 3x− 6 x = k () 2 aya xx3−3236 2 +=( x 2 − xxa)( −−) 23( ) ⇔2x3 − 3( a + 1) x 2 + 6 ax −= 40 3 ⇔−( x22)  x −−+=( 31 ax) 20  x = 2 ⇔  2 gx()()()=2 x − 31 a − x += 20 4 ay   2  5 ∆g > 0 ()3a − 1 − 16 > 0  a ⇔ ⇔  ⇔  3 ()* g ()2≠ 0 2.22 − 3a − 1.2 +≠ 2 0 () a ≠ 2 ayya x0= 2; x 1 ; x 2 aay 2 2 kf0=='20;( ) kfx 11112222 = '( ) =− 36; xxkfx = '( ) =− 36 xx  k0 = 0 2 2 22  ⇔−(3636x1 xx 12)( −=−⇔ x 2) 19 xx 12 − 2 xxxx 1212( ++) 4 xx 12  =− 1 ( ) ea 3a − 1 x+ x = x x =1 1 2 x 1 2  3a − 1   55 ⇔912 −  +=− 4  1 ⇔a = a 2   27 55  y A;− 2  27  WWW.MATHVN.COM
  39. OƯOƯ ươ (Cm ) y (Cm ) (Cm ) ya n a y= f( xm; ) y=±++()()() ax bm gx nnguyn: ê ≥ 2 (C) : y= gx( ) n ±+()()()ax bm + gx = gx ∀m  n−1 ±naaxbm()()() + + g' x = g ' x (Cm ) y ∆ a  c axb++ = kxx()() −+0 y 0 1  x+ d   c a−2 = kxa()() ≠ 2  ()x+ d a a c axad+− = kxd()() + 3 x+ d 2c 2c bad−+ =−−+ kxdy() ⇔ =−−+++k()() x d y adb 4 x+ d 0 0 x+ d 0 0 ayeya ∀m ax+ b cx+ d 3 2 2 (Cm ) : yx=+ 221 x +( m +) xm ++ 2 3 2 2 2 3 2 a (Cm ) : yx=+ 221 x +( m +) xm ++ 2 ⇔()xm + ++++ x x x 2 (Cyx) :=3 + x 2 ++ x 2 (Cm ) a 2 ()xm+ ++++= xxx322 xxx 32 +++ 2  1 2 2 () 2()xmx+ + 321321 ++= x x ++ x WWW.MATHVN.COM
  40. 2 ()x+ m = 0 a ()1 ⇔  2()x+ m = 0 3 2 y ∀m (Cyx) := + x ++ x 2 (m−2) xm −( 2 − 2 m + 4 ) ()C: y = a m x− m 2 (m−2) xm −( − 2 m + 4 ) 4 ()C: y = ⇔y =() m −2 − m x− m x− m (∆) : y = ax + b ⇔ a ∀m  4 ()m−−2 =+ axb () 1  x− m  ()I  4 2 = a ()2 ()x− m a 4 ⇒ =a()() x − m 3 x− m y 8− 8 ⇔−−()m2 =+⇔ bam =−()() amb 124 ++ xm− xm − ay 2 ⇔()()amb −1 ++ 2  = 16 a 2 2 ⇔−()()()()()am12 +− 21 abmb + 2 +−−= 2160* a ∀m ⇔ (*) ∀m ()a −12 = 0  a =1 ⇔2()()a − 1 b + 20 = ⇔  b=2 ∨ b =− 6  2 ()b+2 − 16 a = 0 y yy WWW.MATHVN.COM
  41. y 1 1 y= x3 −+() m1 x 2 + 2 () mmx 2 + − 3 3 a 3 e 322 3 (Cm ) : yx=+ 33 mx +−++( mm 1) xm ++ 1 a x1 x 2 −14 a + = x2 x 1 5 3 (Cm ) : yx= − 3 x + 2 a y ya y ay ay aya y ay ay e y 4 2 (Cm ) : y=− x 2 mx + 21 m − y x3+ mx 2 −1 = 0 32 2 3 (Cm ) : yx=− 3 mx + 3( m −− 1 ) xm 3 (Cm ) : yx= + kx( ++ 1) 1 (∆) :y = x + 1 3 2 3 (Cm ) : y= x − 3 mx + 4 m (d) : y= x a 2x + 1 ()C: y = ya m x + 2 (3m+ 1 ) xm −2 + m ()C: y = () 1 a m x+ m a a ∀m a 3 2 (Cymm ) :=+( 331) x −( mx +) −( 61 mxm +++) 11( ) a WWW.MATHVN.COM
  42. yy yyay yy 5−x3 −3 x 2 + 7 lim x→1 x2 −1 5−−+xx323 7 52 −− xx 323 +− 72  a lim= lim −  () 1 x→12 x → 1  2 2  x−1 x − 1 x − 1  2 5−x3 − 2 1 − x 3 −( x + x + 1) −3 lim2 = lim lim= () 2 x→1x −1 x → 1 ()x2−1() 5 − x 3 + 2 x→1 ()x+1() 5 − x 3 + 2 8 3x 2 +7 − 2 x 2 − 1 1 1 lim2 = lim lim= () 3 x→1x −1 x → 1 2  x→1 22 3 2 12 ()()x2−13 x 2 ++ 723 x 2 ++ 74  3 ()x+7 + 2 x ++ 74   −3 1 11 ay A = − = 8 12 24 ư aa a a ayaya a 5−−xc33 x 2 +− 7 c  ∀c ∈ R f x = −  () 2 2  x−1 x − 1  3 aa fx1 () =5 − xc − 3 2 fx2 () = x +7 − c yaay   f1 (1) = 0  c = 2  f2 ()1= 0   ⇔c = 6 ⇔c = 2 a    f1 ()−1 = 0   c = 2  f2 ()−1 = 0 yyy aaa f( x ) 0 F() x = g() x 0 WWW.MATHVN.COM
  43. fx( ) + c fx( ) − c f() x =1 + 2 gx() gx() αi (i =1;2; ) a  f1 (αi ) + c = 0 a  ()i =1;2;  f1 ()αi − c = 0 f( x) + c f( x) − c lim 1 lim 2 e x→αi g() x x→αi g() x ay 33x 2− 1 + 2 x 2 + 1 lim x→0 1− cos x 12+x −3 13 + x lim x→0 x2 ơơo yyy ayaa yyay a ay • a log x b • a log a x = log b a m logxn = log x an n a  • t= log a f( x ) f( x ) t= a  • 2 2 231x2− x + 81.4231xx−+− 78.6 231 xx −+ + 16.9 ≤ 0() 1 2 2 2 2 6231xx−+  9 231 xx −+ 3231xx−+  3 2.231( xx −+ ) ()1⇔− 81 78 + 16  ≤ 0 ⇔−81 78 + 16  ≤ 0 4  4 2  2 2 3  2x− 3 x + 1 t =   2  3 27  16t2 − 78 t + 81 ≤ 0 ⇔t ∈ ; 2 8  2 3 3  2x− 3 x + 1 27 ⇔≤  ≤⇔≤−+≤12x2 3 x 13 2 2  8 WWW.MATHVN.COM
  44.  3 x ≥  2 2 2 x ≥ 2 2xx−+≥ 311  2 xx −≥ 30   x ≤ 0  ⇔  ⇔  ⇔ 1 2xx2−+≤ 313  2 xx 2 −−≤ 320  1 x ≤ x ≤  2  2  x ≥ 2 ex+− x1− e 1 +− x 1 ≤ x − 1 u= x + x − 1  ⇔u − v = x − 1 v=1 + x − 1 eu− e v = uv − ⇔fu( ) ≤ fv( ) fx( ) = ex − x; x ≥ 1 ⇒ f'( x) = e x + 1 > 0 ⇒ f( x ) u≤ v ⇔+xx −≤+11 x −⇔≤− 1 x 1 log2( 1+x) = log 3 x t log3 x= t ⇔ x = 3 t t t log2 ( 1+x) =⇔+ t 1 x =⇔+ 2 1( 3) = 2 t t 2 13t  131 t   2  3 + =⇔1  +  =  +  22  2222    t 1  t  3  ⇔f( t) = f (2) ⇔t = 2 f() x =  +   2   2  ⇔t =2 ⇔ x = 9 x+1  logx  log2 ( 4− 8)  ≥ 1() 1 x−12()x− 1 3 5 4−>⇔ 802 >⇔ 2 2()x −>⇔> 13 x 2 x+1  x−1 x − 1 x ()1⇔ logx log2 ( 4 − 8)  ≥ log x x ⇔log42( −≥⇔ 8) x log4 2( −≥ 8) log2 2 4x 2x ≤ 0 (loai ) ⇔482x−1 −≥⇔−−≥⇔ x 280 x  ⇔≥x 3 4 2x ≥ 8  x−+12 − y = 1() 1  2 3 3log99()x− log 3 y = 3() 2 WWW.MATHVN.COM
  45. x ≥1  0 1 x + 2  2 2x2 − x − 1 ⇔ =2v à x > 1 x + 2 5 ⇔2xx2 −−= 350à vx >⇔= 1 x 2 2 2 2 log2 ( x+ 6 x − 7 ) logxx+− 15log xx −−= 45 2 ≥ 2 3( ) 5 ( ) 1  2 1+ log 4 x − x +  log( x−+ 2) log x ≥ log( x + 2 ) 4  0.2 3 5 2 ( x+3) log2( x +++ 2) 4( x 2) log 3 ( x += 2) 16 log2x+ log 3 x ≥ log 23 xx log xlog2 3+ x2 = x log 2 5 1 log312 ()x−+ =+ 2log2 () x + 1 logx+3 2 aay eex−= y ln( 1 +− x) ln( 1 + y )  x− y = a  2 2 log2( x+ y) = 1 + log 2 () xy  x, y∈ R 2 2 () 3x− xy + y = 81 WWW.MATHVN.COM
  46. a a x x sin2x cos 2 x ( 51++) 2m( 51 −=) 2 x 9+ 9 = m log2 ( 3−x) > 1 73x+ 9.5 22 xx = 5 + 9.7 3 x 3x− x x−3 x − 3 x x 7 5 16+−( x 64) +−= 82 x 0 ()()5= 7 x x −10 5 10 3+ 3 − 84 = 0 logx2 + 1 < log axa + 1 ( ) ( ) 2( ) 2 ()() x−3 x − 3 16+−( x 64) +−= 82 x 0 a a aa x2 − x x−3 =− 9 6 x + x 2 254( x+−) 53 x −≤ 53 x + 3.25x−2+( 3x − 10) .5 x − 2 +−= 3 x 0 log2 ( xy++) logm ( xy −=) 1  x2− y 2 = m (m+316) x +( 2 m − 14) x ++= m 10 9x−m .3 x + 2 m += 10  3x− 1    x + 1     log34 log   ≤ log 11 log  x 15 x+1  3 x − 1 log2 log 0.5  2−   ≤ 2   3 4  16   WWW.MATHVN.COM
  47. A ea U 1 y=()() x2 − mx 2 + 1 a 4 ay 7 cos23x+ sin 2 x = 2sin x 2 xxx+−( 44) −= x 4( x − 2 ) x2 + log( cos x ) lim 2 x→0 2xsin x −x 2 + 1 aaaa ya ∆ 2 2 2 y ∈[0;1 ] Axy=−()()() +− yz +− zx ươu a y ∆ C∈ dx: − 3 y += 10 a a x y−1 z − 1 y ∆ a d : = = 1 −1 1 1 x+1 y − 2 z − 4 d = = = 2 2 1− 1 n n −1 n axn()()()−+1 ax n −1 − 1 ++ ax 1 −+=∀∈ 1 axxR 0 , a2+ a 3 + a 1 = 231 ươa x2 y 2 a y M∈() E : + = 1 y 6 3 yaya 1 1 1 1 + + = OA2 OB 2 OC 2 OM 2 2n n nπ a ()1+ i y CCC024− + −+− () 1 C 2 n = 2cos n 222nnn 2 n 2 (n∈ N, n > 0 ) WWW.MATHVN.COM
  48. U 2 y= − x4 + x 2 9 ya (1− 3 cosxx) sin +−( 3 cos xx) cos = 1  x+2 y − x − 2 y = 2  2 2  3 x++3 x − 4 y = 5 e dx ∫ 2 1 x+ x1 − ln x a ∆ (a< b < a 2 ) x, y ∈[ 1;2] . 11   11 2 2 Axy=+()2 ++− 2 4() xy  − xy   xy ươu 2 2 x y a y M∈() E : + = 1 6 3 ya ∆ ya x−2 y z − 1 x−1 y + 1 z d : = = d : = = a 1 −2 1 1 2 1− 2 1 e ươa a yyeaa yy 2 3 + i a zn y WWW.MATHVN.COM
  49. U mx + 2 y = x+ m yay 1 cos2x+ cos x + 3sin x = 2 logxx+−2 12.log xx −−= 2 12 2 2( ) 3 ( ) π 2 sin 3 xdx ∫ 4 0 ()1+ cos x ya a  y2 1 x2 − xy + = a  2 4  2 x+ x − y = m ươu a aaya 4 2 y x− 3 y z ya d : = = 4− 5 3 a a b 5i a, b∈ R + = 3 z = z+1 z − 5 1 1+ 2 i ươa a a ∆ y A∉ Ox; B ∈ Oy C∈ dx: +−= y 1 0 aaay y a WWW.MATHVN.COM
  50. U x3 y= − + x 2 3 aaa ya 4tanx+ 2 tan xx+ sin 2 = 2 1 + 2sin 2 x 2+ 2x − 3 x ≥ x 2− 2x − 5 x π 4 sin 4 x ∫ 4 4 dx 0 sinx+ cos x yaa a2+ b 2 + c 2 aa Aa=3 + b 3 + c 3 − 3 ươu yy aa  a AI+2 BI = 0 M∈( P ) a yaa ươa ayaya ya a ∆ x2 −( m +1) x + 2 m − 1 y = x− m WWW.MATHVN.COM
  51. U x − 3 y = x +1 a 3 cos2x− cos xx cos3 + cos 2 3 x = 4  1 1  x2+ + y 2 + = 3  x2 y 2   1 1 + = 1  x+ y xy 4 3 xln x dx ∫ 2 3 1+ x 4 ⊥ ∆ ∆ a 2 1 1 a b25 ab a + = 1 A = + + a b a−1 b − 1 4()a2+ b 2 ươu yy a aa ∆ M∈( P ) a a b a, b∈ R Ziiii=−2 + 3 − 4 + + i 2009 a + = 1 1+z 1 − z ươa x= t  x−1 y z y d1 : y= 1 + 2 t d2 : = =  1− 1 1 2 + t a A∈ d 1 6 a x  log3 log 3 2 2+y = 6 logy+ log x = 1  x y  x3 WWW.MATHVN.COM
  52. U x4 y= − mx2 ++ m 1 a 4 aa π  π  π sin2x++  cos2 x +=  tan x + 6  3  4  3 3 11   11 ()xy+3 +++ 3 () xy  += 8  xy   xy   x y log log= 1  22 3 3 3 xdx I = ∫ 1+x2 0 e aa 1 1 1 bc+ ca + ab + a + + = 1 A = + + a b c bc33+ ca 3 + 3 ab 33 + ươu a y ∆ Ad∈1 =2 xy −+= 1 0 BC,∈ d2 : x +−= y 20 a ya (Q) : xy− − 2 z = 0 1 a C3− C 2 = A 3 n+1 n + 1 7 n ươa 2 2 2 2 a ya (Cm ) : x+ y − 2 mx − my +−= m 20 (Cx) :+ y − 310 x += yaa y ( Px) :+ 2 yz + −= 10 x−2 y + 1 z ∆: = = a 2 1− 1 z4+ z 2 +1 = 0 z∈ C WWW.MATHVN.COM
  53. U y= x3 −3 ax 2 + b (a, b > 0 ) a aa ∆ x  2 tan2x 1+ tan x .tan  = 2  sin3 x  1 1 1  + =  x+ y xy 2  5 2 1  − =  x2+ y 2 xy 22 2 ex − x + 1 lim x→0 ln() 1+ sin x aayaa aaaa 1 +ln()x +−+ xm 2 xm − ≤ 1 2 mx− x 2 ươu a y A(7;1) , B(− 3; − 4) , C ( 1;4 ) ∆ x−1 y + 1 z − 2 yaa d : = = 1 2− 1 x+1 y − 2 z ∆: = = 2 1 1 6 a x3 aaa ( x2 + x − 1) ươa a y (Cx) :2+ y 2 − 650 x += aaa yaaay x−1 y + 1 z y M (2;1;0 ) = = −2 1 − 1 aa 5 a x3 aaa ( x2 + x − 1) WWW.MATHVN.COM
  54. U mx +1 y = x +1 ya sin2x+ sin2 xx .sin4 = cos 2 2 x 223+x+ 2 23 − x − 722( xx + − ) ≤ 15 1 1 aya ()C: y = 1 + + 1 − , x x aya aa d; d a   1 2 y M∈ d1, N ∈ d 2 a AM, CN a  12 1 xy+ = x +  x1+ y ln y  1 1 xy+ = y 2 +  y1+ x ln x ươu a ya (P) : y2 = x a ∆ a ( yA < 0) x−1 y − 1 z − 2 yaa d : = = 2 2 1 aaaa a ươa a yeaa MF1. MF 2 yaa (Qxz) :+−= 1 0 và ( Rxyz) : +−+= 2 1 0 Z=++++(1 2 ii 32 2009 i 2008)( 12 −+−++ iii 3 2 4 3 2009 i 2008 ) WWW.MATHVN.COM
  55. U y=−( xmx)( 2 −+ x 1) yaaya 1− sinx + cos x tan x = 1+ sinx + cos x 2 log2 log 2x+ log 2 x 0.25 ()752+x =() 322 − (Cy) := x2 − 2 x − 3 d: y= x + 1 aaya ⊥ a 1 1  +2 = 3 a  x y y  2 2x+ y = m ươu a y ∆ ay hA =2 x + y += 4 0 mA = y −2 = 0 y mB : 3 x+ 11 y + 21 = 0 x= t x−2 y − 1 z − 2  y d1 := = ,d2 : y = 2 t a 1 2 1  z=1 + t a a x, y∈ R a 1 1 1 − = x+−()2 y i 2 ++ y xi ()1+ i 2 ươa x2 y 2 a y ()H:− = 1 () a , b > 0 a F ; F a ; F a2 b 2 1 2 2 a ∆F1 MN aa ∆F1 MN = 4 3 yya ∆ 3x 3 y = + 4 x ∆ WWW.MATHVN.COM
  56. U −x4 y= ++()() mxm12 − , 1 a 2 ay a π  4sin2sinxx+=+  1 3sin2 xx − cos2 3  x2 − xy +4 y = 8  xy+ y2 +3 x = 12 x y = a e x+ln x a aaa   log242( xm−+=+) 1log 3  mxx −−( 1)( − 3 )  y y ươu x+2 y z − 1 y d : = = y 2 1 1 a aaa aa a 6 3  x3  1 log323  logx− log  = + log 2 x x  3  2 ươa x= t  yaa d1 : y= 1 + 2 t y  2 + t a ∆ a a 2π 2 x2 +(2 + cosϕ) x + 3 sin ϕ φ ∈(0;2 π ) y = a x −1 a WWW.MATHVN.COM
  57. U 2x y = , () 1 x −1 a yaa 16sin2 x+ 4cos 4 x = 3 cos xx + sin 3 x−5( x −− x 5) = 2 2 2 aya ()()()C:3 x− + y − 11 = aya y a a 2 a a a2+ b 2 + c 2 = 1. a3++− b 3 c 3 3 abc ≤ 1 ươu a yay a∆ y ∈ay π 012x− 2 CCC2++++ 3 4 Cx = 120 , x ∈ N ươa yy   a ∈a AM+ BM aeay  x xy2− 2 xy −  y xy xy 4+ 2 = 5.4  log3x+ log 5 y = log 53 xy .log WWW.MATHVN.COM
  58. U −x3 16 y= + mx2 −2() m − 2 x + ()1 3 3 sin3x+ sin x = 3cos( x − 1 ) 4 log+ log 0, 25 ≥ log x2 2x2 2 x 0.5 1 4 x I= ∫ dx 0 1− 2 x ayayy aaaa 1 1 a2+ b 2 ab a + = 1 P = + a+ b ab ab a+ b ươu a y ∆ ∈y 3x− y − 10 = 0 a yA>0 > y B yya 2 2 4z 4 + 1 = 0 ươa a y (P) : y2 = 2 x a F aya F 1 1 + MF NF yaa ⊥ y n+1 n a x6 aaa Px=+()1( xx2 ++ 1 ) a x10 WWW.MATHVN.COM
  59. aa ay yaaay ayay yyeeyy 231x3−( a +) x 2 + 6 ax −= 40( 1 ) y 2x3− 3( A + 1) x 2 + 6 Ax −= 40 aya aa yayayy ya y ayayy ya ya a yx=−+331( m) x 2 + 2( m 2 ++− 414 m) xmm( + 1 ) x3−+31( Ax) 2 + 2( A 2 ++− 414 AxAA) ( += 10) y yaaa a yaayya a ya yay a ay a sin2x+ cos2 x − cos x + 3sin x = 2( 1 ) yaa ay y π π π a ±; ± ; ± 3 6 2 π π và yaaa 2 6 WWW.MATHVN.COM
  60. 1 y a 2 (sinx − 1 ) ay (2sinx − 1 ) ae (sinx − 1 ) (1) ⇔ 3sinx −+− 3 1 cos xxx + sin2 + cos2 = 0 ⇔3(sinx −++− 1) 1( 1 2sin2 xxx) + sin2 − cos = 0 ⇔3sin( x −+− 1) 2(1 sin2 xxx ) + sin2 − cos = 0 ⇔−−(sinx 1)( 1 2sin x) + 2sin xxx cos −= cos 0 (sinx−− 11)( 2sin xxx) + cos( 2sin −=⇔−− 1) 0( sin x 11)( 2sin xx + cos) = 0 yaa y OE x=2 + 2 t x =1   y d1 : y= − 1 + t d2 : y= 1 + t   z =1 z=3 − t a a a eayae eyae A aeayae e aaae  a M(2;− 1;0) ∈ dN ;( 1;1;3 ) ∈ d ⇒ MN (−1;2;3 ) yay 1 2  e  aae MN (−1;2;3 )           d1; d 2  . MN A; B  . C a e d =     ey ()d1; d 2     d1; d 2  A; B    aae A& B a     ea  y A& B aa        yaya 11 y 3 a WWW.MATHVN.COM
  61. aeya (α ) ey         a⊥ d 1 ay a   ()d1= Ad; 2 = B a⊥ d 2       yaa a d1; d 2  a     a e y y a =( − 1;2;2 ) (α ) a (α ) :−−+( x 2212) ( y ++) ( z) =⇔−+ 0 xyz 2230 + += yyay aaya yaa OE n n −1 aa Px( ) = ax0 + ax 1 ++ a n ( x− c ) a n−1 n − 2 Px( ) =−( xcbx)( 0 + bx 1 ++ bxbn− 1 + n ) bi ( i= 0;1;2;3; ; n ) e a a a a a a a a a Px( ) =2 xx4 +− 3 8 xx 2 −+ 6 ae y Px( ) =+( x22)( x3 − 3 x 2 −++ 2 x 30) yaaeaa yaaye aay 231x3−( a +) x 2 + 6 ax −= 40( 1 ) 2x3− 3( a + 1) x 2 + 6 ax −= 40 ay a a a 3 y ⇔−( x22)  x −−+=( 31 ax) 20  ya mx3−(34 m −) x 2 +( 37 m −) xm −+= 30 ( A ) WWW.MATHVN.COM
  62. aaa ⇒ e 2 ( A) ⇔−( x1)  mx − 22( m −) xm +−= 30  ⇔gx( ) = mx2 −2( m − 2) xm +−= 30 a  m ≠ 0  2 ∆='()()m − 2 − m m −> 30  m − 2 ⇔S = > 0 ⇔m ∈( −∞;0) ∪ ( 3;4 )  m  m − 3 P = > 0  m g()()1=− m 2 m −+−≠ 2 m 30 x3 −−1 m( x − 101) = ( ) (1) ⇔x3 − mx +− m 1 ya e y ⇔−( x1)( xx2 ++− 1 m ) = 0 gx()= x2 ++− x 1 m = 0 a  3 ∆=4m − 3 > 0  m > 3 ⇔ ⇔⇔<≠  4 m 3 g()1=++− 111 m ≠ 0 4 m ≠ 3 ea y aa a x3−4 mx 22 + m x + 6 m 3 = 0 y (Q) =2 x5 + 3 xx 43 −− 7 x 2 + 11 x + 9 a aa aaa WWW.MATHVN.COM
  63. AO WWW.MATHVN.COM