Chuyên đề ôn thi Đại Học môn toán bộ 1

Chuyn n thi i hc (Phn i s) Quyn 1. Ti liu lu hnh ni b.

Nghim cm sao chp di mi hnh thc.

CHUYN N THI I HC (QUYN 1)(Phn 1: i s)

Ti liu c son theo nhu cu ca cc bn hc sinh khi trng THPT (c bit l khi 12).

Bin son theo cu trc cu hi trong thi tuyn sinh i hc Cao ng ca B GD&T.

Ti liu c chia ra lm 2 phn:

+ Phn 1: Phn i s (Chim khong 7 im) gm 2 quyn Mi quyn 5 chuyn . Trong phn ny c 10 chuyn :

Chuyn 1: Chuyn kho st hm s v cc cu hi ph trong kho st hm s.

Chuyn 2: Chuyn PT BPT i s.

Chuyn 3: Chuyn HPT HBPT i s.

Chuyn 4: Chuyn PT BPT HPT HBPT M v Logarit.

Chuyn 5: Chuyn Lng gic v PT Lng gic.

Chuyn 6: Chuyn Tch phn.

Chuyn 7: Chuyn T hp Xc sut.

Chuyn 8: Chuyn Nh thc Newtn.

Chuyn 9: Chuyn S phc.

Chuyn 10: Chuyn Bt ng thc.

+ Phn 2: Phn Hnh hc (Chim khong 3 im)

Trong phn ny c 5 chuyn :

Chuyn 1: Chuyn Th tch: Khi chp, Khi lng tr... Chuyn 2: Chuyn Hnh hc phng.

Chuyn 3: Chuyn Hnh hc khng gian.

Chuyn 4: Chuyn Phng trnh ng thng (*).

Chuyn 5: Chuyn Cc hnh c bit trong thi.

Cui cng, Phn tng kt v kinh nghim lm bi. Ti liu do tp th tc gi bin son:

1. Cao Vn T CN.Mng Ton Khoa CNTT Trng H CNTT&TT Thi Nguyn (Ch bin)

2. C Trn Th Ngc Loan CLB Gia S Thi Nguyn(ng ch bin).

3. Thy V Khc Mnh CLB Gia s Bc Giang (T vn).

4. Nguyn Th Kiu Trang SV Khoa Ton Trng HSP Thi Nguyn.

5. Nguyn Trng Giang Khoa CNTT Trng H CNTT&TT Thi Nguyn.

6. L Th Thanh Nga SVNC Khoa Ton Trng H SP Thi Nguyn.

7. Ng Th L Khoa CNTT Trng H CNTT&TT Thi Nguyn. Ti liu c lu hnh ni b - Nghim cm sao chp di mi hnh thc. Nu cha c s ng ca ban Bin son m t ng post ti liu th u c coi l vi phm ni quy ca nhm. Ti liu c b sung v chnh l ln th 2.Tuy nhm Bin son c gng ht sc nhng cng khng th trnh khi s sai xt nht nh.

Rt mong cc bn c th phn hi nhng ch sai xt v a ch email: [email protected] !

Xin chn thnh cm n!!!

Chc cc bn c mt k thi tuyn sinh i hc Cao ng nm 2015 an ton, nghim tc v hiu qu!!!

Thi Nguyn, thng 07 nm 2014

Trng nhm Bin son

Cao Vn T

CHUYN 1: KHO ST HM S V CC CU HI PH

Ch 1: KHO ST V V TH HM S

I. HM A THC:

* Hm s bc ba:

* Hm trng phng:

1. Tp xc nh: D=R

2. S bin thin:

a) Gii hn ti v cc:

a >0a 0a 0 (x ( R ( do 2eq \l(\o\ac(x, )) > 0 v ln2 > 0 )

( f(x) lun ng bin trn R, m f(1) = 0 nn phng trnh f(x) = 0 c nghim duy nht l x = 1.

b. 9eq \l(\o\ac(x, )) = 5eq \l(\o\ac(x, )) + 4eq \l(\o\ac(x, )) + 2.x, ))eq \l(\r(,20))

( HD gii: Bi ton trn c n 4 c s khc nhau, ta quyt nh chia cho c s ln nht 9eq \l(\o\ac(x, )). PT ( 1 = eq \b\rc\((\a\ar\vs7(,,,,))

eq \s\don1(\f(5,9))

eq \b\rc\)(\a\al\vs7(,,,,))

eq \o\al(\a\al\vs13(x,,)) +eq \b\rc\((\a\ar\vs7(,,,,))

eq \s\don1(\f(4,9))

eq \b\rc\)(\a\al\vs7(,,,,))

eq \o\al(\a\al\vs13(x,,))+ 2.eq \b\rc\((\a\ar\vs7(,,,,))

eq \s\don1(\f(,9))

eq \b\rc\)(\a\al\vs7(,,,,))

eq \o\al(\a\al\vs13(x,,)) ( Nhm nghim th ta thy x = 2 tha mn )

Do 0 < eq \s\don1(\f(5,9)) ; eq \s\don1(\f(4,9)) ; eq \s\don1(\f(,9)) < 1 nn ln eq \s\don1(\f(5,9)) < 0 , ln eq \s\don1(\f(4,9)) < 0 , ln eq \s\don1(\f(,9)) < 0.

Do f '(x) = eq \b\rc\((\a\ar\vs7(,,,,))

eq \s\don1(\f(5,9))

eq \b\rc\)(\a\al\vs7(,,,,))

eq \o\al(\a\al\vs13(x,,))ln eq \s\don1(\f(5,9)) +eq \b\rc\((\a\ar\vs7(,,,,))

eq \s\don1(\f(4,9))

eq \b\rc\)(\a\al\vs7(,,,,))

eq \o\al(\a\al\vs13(x,,))ln eq \s\don1(\f(4,9)) + 2.eq \b\rc\((\a\ar\vs7(,,,,))

eq \s\don1(\f(,9))

eq \b\rc\)(\a\al\vs7(,,,,))

eq \o\al(\a\al\vs13(x,,))ln eq \s\don1(\f(,9)) < 0 (x ( R

Nn hm s f(x) nghch bin trn R, m f(2) = 1 nn phng trnh f(x) = 1 c nghim duy nht x = 2.

C. 3eq \l(\o\ac(x, )) + 5eq \l(\o\ac(x, )) = 6x + 2

( HD gii: nhn xt 1 v ca phng trnh l " hm m ", cn v cn li l " hm a thc ". Khng th bin i nh cc dng cp trn ca chuyn nn ta quyt nh s PP hm s. Xt f(x) = 3eq \l(\o\ac(x, )) + 5eq \l(\o\ac(x, )) = 6x + 2 vi x ( R

Ta c f '(x) = 3eq \l(\o\ac(x, )) ln3 + 5eq \l(\o\ac(x, )) ln5 - 6 l hm s lin tc

V f '(0) = ln3 + ln5 - 6 < 0 , f '(1) = 3ln3 + 5ln5 - 6 > 0

Nn phng trnh f '(x) = 0 c nghim duy nht x = xeq \l(\o\ac( ,o)) Bng bin thin:

x xeq \l(\o\ac( ,o))

f '(x) - 0 +

f (x)

Da vo bng bin thin ta thy phng trnh f(x) = 0 c khng qu hai nghim phn bit.

M f(0) = f(1) = 0 nn mi nghim ca phng trnh cho l x = 0 hoc x = 1

c th ng dng PP hm s ny mt cch hiu qu trc tin bn nn " nhm nghim " PT cho trc. ng vi s nghim tm c ta s xut cch gii.

d. (2 - eq \l(\r(,3)))eq \l(\o\ac(x, )) + (2 + eq \l(\r(,3)))eq \l(\o\ac(x, )) = 4eq \l(\o\ac(x, ))

( HD gii: PT ( eq \b\rc\((\a\ar\vs9(,,,,))

eq \s\don1(\f(2-eq \l(\l( ))

eq \l(\r(,3)),4))

eq \b\rc\)(\a\al\vs9(,,,,))

eq \o\al(\a\al\vs13(x,,))+ eq \b\rc\((\a\ar\vs9(,,,,))

eq \s\don1(\f(2+eq \l(\l( ))

eq \l(\r(,3)),4))

eq \b\rc\)(\a\al\vs9(,,,,))

eq \o\al(\a\al\vs13(x,,)) = 1

Xr f(x) = eq \b\rc\((\a\ar\vs9(,,,,))

eq \s\don1(\f(2-eq \l(\l( ))

eq \l(\r(,3)),4))

eq \b\rc\)(\a\al\vs9(,,,,))

eq \o\al(\a\al\vs13(x,,))+ eq \b\rc\((\a\ar\vs9(,,,,))

eq \s\don1(\f(2+eq \l(\l( ))

eq \l(\r(,3)),4))

eq \b\rc\)(\a\al\vs9(,,,,))

eq \o\al(\a\al\vs13(x,,)) vi x ( R

V 0 < eq \s\don1(\f(2-eq \l(\l( ))

eq \l(\r(,3)),4)); eq \s\don1(\f(2+eq \l(\l( ))

eq \l(\r(,3)),4)) < 1 nn lneq \b\rc\((\a\ar\vs9(,,,,))

eq \s\don1(\f(2-eq \l(\l( ))

eq \l(\r(,3)),4))

eq \b\rc\)(\a\al\vs9(,,,,)) < 0 v lneq \b\rc\((\a\ar\vs9(,,,,))

eq \s\don1(\f(2+eq \l(\l( ))

eq \l(\r(,3)),4))

eq \b\rc\)(\a\al\vs9(,,,,)) < 0

Do , f'(x) = eq \b\rc\((\a\ar\vs9(,,,,))

eq \s\don1(\f(2-eq \l(\l( ))

eq \l(\r(,3)),4))

eq \b\rc\)(\a\al\vs9(,,,,))

eq \o\al(\a\al\vs13(x,,)).lneq \b\rc\((\a\ar\vs9(,,,,))

eq \s\don1(\f(2-eq \l(\l( ))

eq \l(\r(,3)),4))

eq \b\rc\)(\a\al\vs9(,,,,)) + eq \b\rc\((\a\ar\vs9(,,,,))

eq \s\don1(\f(2+eq \l(\l( ))

eq \l(\r(,3)),4))

eq \b\rc\)(\a\al\vs9(,,,,))

eq \o\al(\a\al\vs13(x,,)) lneq \b\rc\((\a\ar\vs9(,,,,))

eq \s\don1(\f(2+eq \l(\l( ))

eq \l(\r(,3)),4))

eq \b\rc\)(\a\al\vs9(,,,,)) < 0 (x ( R

Nn hm s f(x) lun nghch bin trn R, m f(1) = 1 nn phng trnh f(x) = 1 c nghim duy nht x = 1

e. 7eq \l(\o\ac(x-eq \l(\l( ))1, )) = 1 + 2logeq \l(\o\ac( ,7)) (6x - 5)eq \l(\o\ac(3, )) ( HD gii: iu kin 6x - 5 > 0 ( x > eq \s\don1(\f(5,6))

t y - 1 = logeq \l(\o\ac( ,7)) (6x - 5) th 7eq \l(\o\ac(y-eq \l(\l( ))1, )) = 6x - 5 (1) PT cho tr thnh 7eq \l(\o\ac(x-eq \l(\l( ))1, )) = 1 + 2logeq \l(\o\ac( ,7)) (6x - 5)eq \l(\o\ac(3, )) ( 7eq \l(\o\ac(x-eq \l(\l( ))1, )) = 1 + 6logeq \l(\o\ac( ,7)) (6x - 5)

( 7eq \l(\o\ac(x-eq \l(\l( ))1, )) = 1 + 6logeq \l(\o\ac( ,7)) 7eq \l(\o\ac(y-eq \l(\l( ))1, ))

( 7eq \l(\o\ac(x-eq \l(\l( ))1, )) = 1 + 6(y - 1) ( 7eq \l(\o\ac(x-eq \l(\l( ))1, )) = 6y - 5 (2) Ly (1) tr (2) ta c: 7eq \l(\o\ac(y-eq \l(\l( ))1, )) - 7eq \l(\o\ac(x-eq \l(\l( ))1, )) = 6x - 6y

( 7eq \l(\o\ac(x-eq \l(\l( ))1, )) + 6(x - 1) = 7eq \l(\o\ac(y-eq \l(\l( ))1, ))+ 6(y - 1)( f(x - 1) = f(y - 1)

D thy f(t) = 7eq \l(\o\ac(t, )) + 6t l hm s ng bin trn R, m f(x - 1) = f(y - 1) ( x - 1 = y - 1 ( x = y Khi phng trnh cho c dng (1) ( 7eq \l(\o\ac(x-eq \l(\l( ))1, )) - 6x + 5 = 0 (3) ( nhm nghim x = 1, x = 2) Xt hm s g(x) = 7eq \l(\o\ac(x-eq \l(\l( ))1, )) - 6x + 5 (x ( R Ta c g'(x) = 7eq \l(\o\ac(x-eq \l(\l( ))1, )).ln7 - 6 nn g'(x) = 0 ( xeq \l(\o\ac( ,o)) = 1 + logeq \l(\o\ac( ,7)) eq \s\don1(\f(6,ln7))

Bng bin thin:

x xeq \l(\o\ac( ,o))

g'(x) - 0 +

g (x)

Da vo bng bin thin ta thy phng trnh f(x) = 0 ch c khng qu hai nghim phn bit

M f(1) = f(2) = 0 nn x = 1, x = 2 l cc nghim ca phng trnh.

V d 2: Gii cc phng trnh sau:

a.logeq \l(\o\ac( ,2)) x + logeq \l(\o\ac( ,3)) (2x - 1) + logeq \l(\o\ac( ,5)) (7x - 9) = 3

( HD gii: iu kin x > eq \s\don1(\f(9,7))

Xt hm s f(x) = logeq \l(\o\ac( ,2)) x + logeq \l(\o\ac( ,3)) (2x - 1) + logeq \l(\o\ac( ,5)) (7x - 9) vi x > eq \s\don1(\f(9,7))

Ta c f '(x) == eq \s\don1(\f(1,x.ln2)) + eq \s\don1(\f(2,-eq \l(\l( ))1)ln3)))) + eq \s\don1(\f(7,-eq \l(\l( ))9).ln5)))) > 0 (x > eq \s\don1(\f(9,7))

Vy hm s f(x) ng bin trn ( eq \s\don1(\f(9,7)) ; +() nn phng trnh f(x) = 3 nu c nghim s c nghim duy nht.

M f(2) = 3 nn phng trnh cho c nghim x = 2 b. xeq \l(\o\ac(3, )).logeq \l(\o\ac( ,3)) x = 27

( HD gii: x > 0

Vit phng trnh cho di dng logeq \l(\o\ac( ,3)) x - 3, ))eq \s\don1(\f(27,x)) = 0

Xt hm s f(x) = logeq \l(\o\ac( ,3)) x - 3, ))eq \s\don1(\f(27,x)) vi x > 0

Ta c f '(x) = eq \s\don1(\f(1,xln3)) + 4, ))eq \s\don1(\f(81,x)) > 0 (x > 0 nn hm s y = f(x) ng bin trn (0; +() nn phng trnh f(x) = 0 nu c nghim s c nghim duy nht. M f(3) = 0 nn phng trnh c nghim x = 3

c. 22, ))

eq \l(\l( ))eq \o(\a\ac\vs2(x+eq \l(\l( ))x, )) + logeq \l(\o\ac( ,2)) x = 2eq \l(\o\ac(x+eq \l(\l( ))1, )) ( HD gii: x > 0

PT ( 22, ))

eq \l(\l( ))eq \o(\a\ac\vs2(x+eq \l(\l( ))x, )) + logeq \l(\o\ac( ,2)) eq \s\don1(\f(+eq \l(\l( ))1))),xeq \l(\l( ))+eq \l(\l( ))1)) = 2eq \l(\o\ac(x+eq \l(\l( ))1, ))

( 22, ))

eq \l(\l( ))eq \o(\a\ac\vs2(x+eq \l(\l( ))x, )) + logeq \l(\o\ac( ,2)) (xeq \l(\o\ac(2, )) + x) - logeq \l(\o\ac( ,2)) (x + 1) = 2eq \l(\o\ac(x+eq \l(\l( ))1, ))

( 22, ))

eq \l(\l( ))eq \o(\a\ac\vs2(x+eq \l(\l( ))x, )) + logeq \l(\o\ac( ,2)) (xeq \l(\o\ac(2, )) + x) = 2eq \l(\o\ac(x+eq \l(\l( ))1, )) + logeq \l(\o\ac( ,2)) (x + 1)

t f(t) = 2eq \l(\o\ac(t, )) + logeq \l(\o\ac( ,2)) t ( t > 0)

Ta c f '(t) = 2eq \l(\o\ac(t, )) ln2 + eq \s\don1(\f(1,t.ln2)) > 0 (t > 0

Nn hm s y = f(t) lun ng bin trn (0; + () Li c f(xeq \l(\o\ac(2, )) + x) = f(x + 1) ( xeq \l(\o\ac(2, )) + x = x + 1 ( eq \b\lc\[(\a\al\vs0(=eq \l(\l( ))1eq \l(\l( ))(nhn))) ,eq \l(\l(x=eq \l(\l( ))-1eq \l(\l( ))(loi))))). Vy x = 1 l nghim phng trnh.

BI TP RN LUYN: Gii cc phng trnh sau:

1) 3eq \l(\o\ac(x, )) - 4 + x = 0

2) (0,5)eq \l(\o\ac(x, )) = 2x + 8

3) 3eq \l(\o\ac(x, )) + 4eq \l(\o\ac(x, )) = 5eq \l(\o\ac(x, )) 4) (eq \l(\r(,15)))eq \l(\o\ac(x, )) + 1 = 4eq \l(\o\ac(x, ))

5) 3eq \l(\o\ac(2x, )) + x - 66 = 0

6) 3eq \l(\o\ac(x, )) + 4eq \l(\o\ac(x, )) = 5x + 2

7) 2eq \l(\o\ac(2x, )) - 3eq \l(\o\ac(x, )) = 7

8) 9eq \l(\o\ac(x, )) = 8x + 1

9) 2eq \l(\o\ac(x, )) = 3x - 1

10) 4eq \l(\o\ac(x, )) - 2eq \l(\o\ac(x+eq \l(\l( ))1, )) + x - 1 = 0 11) 1 + 8eq \l(\o\ac(x, )) + 4eq \l(\o\ac(x, )) = 9eq \l(\o\ac(x, )) 12) 3eq \l(\o\ac(x, )) = 5 - 2x

13) 5eq \l(\o\ac(x, )) = 3eq \l(\o\ac(x, )) + 2

14) 1 + (3 + eq \l(\r(,8)))eq \l(\o\ac(x, )) + (3 - eq \l(\r(,8)))eq \l(\o\ac(x, )) = 7eq \l(\o\ac(x, ))

15) 7eq \l(\o\ac(6-eq \l(\l( ))x, )) = x + 2

16) 2eq \l(\o\ac(x, )) + 5eq \l(\o\ac(x, )) = 7eq \l(\o\ac(x, ))

17) 9.3eq \l(\o\ac(x, )) - 7eq \l(\o\ac(x, )) = 5.4eq \l(\o\ac(x, ))

18) 3eq \l(\o\ac(, )) = 2eq \l(\o\ac(x, )) - 1 19) 1 + 8eq \l(\o\ac(, )) = 3eq \l(\o\ac(x, ))

20) 2eq \l(\o\ac(x, )) + 5eq \l(\o\ac(x, )) + 3eq \l(\o\ac(x, )) = 10eq \l(\o\ac(x, )) 21) 25eq \l(\o\ac(x, )) + 10eq \l(\o\ac(x, )) = 2eq \l(\o\ac(2x+eq \l(\l( ))1, ))

22) 5eq \l(\o\ac(x+eq \l(\l( ))1, )) + 7eq \l(\o\ac(x+eq \l(\l( ))1, )) = 13eq \l(\o\ac(x+eq \l(\l( ))1, ))

23) 4.3eq \l(\o\ac(x, )) - 6eq \l(\o\ac(x, )) + 2 - x = 0 24) (eq \l(\r(,2+eq \l(\l( ))

eq \l(\r(,3)))))eq \l(\o\ac(x, )) + (eq \l(\r(,2+eq \l(\l( ))

eq \l(\r(,3)))))eq \l(\o\ac(x, )) = 2eq \l(\o\ac(x, ))

25) logeq \l(\o\ac( ,7)) (x + 2) = 6 - x

26) log(x - 2) = - xeq \l(\o\ac(2, )) + 2x + 3

27) x + log(xeq \l(\o\ac(2, )) - x - 6) = 4 + log(x + 2)

28) log(xeq \l(\o\ac(2, )) - 6x + 5) = log(x - 1) + 6 - x 29) x ,2))

eq \l(\l( ))eq \o(\a\ac\vs2(log9, )) = xeq \l(\o\ac(2, )) .3 ,2))

eq \l(\l( ))eq \o(\a\ac\vs2(logx, )) - x ,2))

eq \l(\l( ))eq \o(\a\ac\vs2(log3, )) 30) (1 + x)(2 + 4eq \l(\o\ac(x, ))) = 3.4eq \l(\o\ac(x, ))

31) logeq \l(\o\ac( ,2)) (1 + cosx) = 2cosx 32) 5eq \l(\o\ac(x, )) + 2eq \l(\o\ac(x, )) = 3eq \l(\o\ac(x, )) + 4eq \l(\o\ac(x, ))

33) x ,7))

eq \l(\l( ))eq \o(\a\ac\vs2(log11, )) + 3 ,7))

eq \l(\l( ))eq \o(\a\ac\vs2(logx, )) = 2x

34) logeq \l(\o\ac( ,2)) x, ))

eq \l(\l( ))eq \s\don1(\f(2-eq \l(\l( ))1,|x|)) = 1 + x - 2eq \l(\o\ac(x, )) 35) 5eq \l(\o\ac(x, )) + 3eq \l(\o\ac(x, )) + 2eq \l(\o\ac(x, )) = 28x - 18 36) (4eq \l(\o\ac(x, )) + 2)(2 - x) = 6

37) 5eq \l(\o\ac(x, )) + 2eq \l(\o\ac(x, )) = 2 - eq \s\don1(\f(x,3)) + 44logeq \l(\o\ac( ,2)) (2 - 5eq \l(\o\ac(x, )) + eq \s\don1(\f(131x,3)) ) 38) 4eq \l(\o\ac(x, )) + 2eq \l(\o\ac(x, )) = eq \s\don1(\f(14,3)) xeq \l(\o\ac(3, )) - 9xeq \l(\o\ac(2, )) + eq \s\don1(\f(25,3))x + 2

39) log(xeq \l(\o\ac(2, )) - x - 12) + x = log(x + 3) + 5 40) x(log 5 - 1) = log(2eq \l(\o\ac(x, )) + 1) - log6

41) 3xeq \l(\o\ac(2, )) - 2xeq \l(\o\ac(3, )) = logeq \l(\o\ac( ,2)) (xeq \l(\o\ac(2, )) + 1) - logeq \l(\o\ac( ,2)) x

42) (1 + eq \s\don1(\f(1,2x))).logeq \l(\o\ac( ,3)) + log2 = log(27 - 3eq \o(\a\ac\vs2(, )) )

43) logeq \o(\a\ac\vs0( ,2+eq \l(\l( ))

eq \l(\r(,3)))))) (xeq \l(\o\ac(2, )) - 2x - 2) = logeq \o(\a\ac\vs0( ,+eq \l(\l( ))

eq \l(\r(,3))))))) (xeq \l(\o\ac(2, )) - 2x - 3)

44) eq \s\don1(\f(-eq \l(\l( ))2eq \l(\o\ac(x, ))))),3eq \l(\l( ))-eq \l(\l( ))x)) = 1

45) (2 + eq \l(\r(,2))) ,2))

eq \l(\l( ))eq \o(\a\ac\vs2(logx, )) + x(2 - eq \l(\r(,2))) ,2))

eq \l(\l( ))eq \o(\a\ac\vs2(logx, )) = 1 + xeq \l(\o\ac(2, )) 46) 5eq \l(\o\ac(logx, )) - 3eq \l(\o\ac(logx-eq \l(\l( ))1, )) = 3eq \l(\o\ac(logx+eq \l(\l( ))1, )) - 5eq \l(\o\ac(logx-eq \l(\l( ))1, ))

47) logeq \l(\o\ac( ,4)) (logeq \l(\o\ac( ,2)) x) + logeq \l(\o\ac( ,2)) (logeq \l(\o\ac( ,4)) x) = 2

48) logeq \l(\o\ac( ,2)) x + logeq \l(\o\ac( ,3)) x + logeq \l(\o\ac( ,4)) x = logeq \l(\o\ac( ,20)) x

49) logeq \l(\o\ac( ,2)) (x - 2, ))

eq \l(\l( ))eq \l(\r(,x-eq \l(\l( ))1))).logeq \l(\o\ac( ,3)) (x + 2, ))

eq \l(\l( ))eq \l(\r(,x+eq \l(\l( ))1))) = logeq \l(\o\ac( ,6)) (x - 2, ))

eq \l(\l( ))eq \l(\r(,x-eq \l(\l( ))1)))

DNG 6: TUYN TP CC DNG BI TP NNG CAO - C BIT.

chng trnh trung hc ph thng hin hnh th 5 dng ton cp trn l ph hp vi hc sinh nht t nhng dng n gin n phc tp. i vi dng 6, chuyn dnh mt cht " ton gii tr " v m mang " t duy " cho cc bn hc sinh bng nhng phng php gii " khng ging ai " ! Mi cc bn th sc.

S dng phng php i lp ( nh gi 2 v ca phng trnh )

V d 1: Gii phng trnh 2, ))

eq \l(\l( ))eq \l(\r(,3x+eq \l(\l( ))6xeq \l(\l( ))+eq \l(\l( ))7)) + 2, ))

eq \l(\l( ))eq \l(\r(,5x+eq \l(\l( ))10xeq \l(\l( ))+eq \l(\l( ))21)) = 5 - 2x - xeq \l(\o\ac(2, ))

( HD gii: iu kin (x ( R

Ta c V Tri = 2, ))

eq \l(\l( ))eq \l(\r(,3x+eq \l(\l( ))6xeq \l(\l( ))+eq \l(\l( ))7)) + 2, ))

eq \l(\l( ))eq \l(\r(,5x+eq \l(\l( ))10xeq \l(\l( ))+eq \l(\l( ))21))

Trong 2, ))

eq \l(\l( ))eq \l(\r(,3x+eq \l(\l( ))6xeq \l(\l( ))+eq \l(\l( ))7)) = eq \l(\r(,+eq \l(\l( ))1)eq \l(\o\ac(2, ))

eq \l(\l( ))+eq \l(\l( ))4)))) ( eq \l(\r(,4)) = 2

v 2, ))

eq \l(\l( ))eq \l(\r(,5x+eq \l(\l( ))10xeq \l(\l( ))+eq \l(\l( ))21)) = eq \l(\r(,+eq \l(\l( ))1)eq \l(\o\ac(2, ))

eq \l(\l( ))+eq \l(\l( ))16)))) ( eq \l(\r(,16)) = 4

Vy V Tri ( 2 + 4 = 6

Mt khc, v phi = 5 - 2x - xeq \l(\o\ac(2, )) = 6 - (x + 1)eq \l(\o\ac(2, )) ( 6

Vy v tri ch bng v phi ( VT = VP = 6 ( x = -1

V d 2: Gii phng trnh 3eq \l(\o\ac(2x+eq \l(\l( ))2, )) + 4, ))

eq \l(\l( ))eq \l(\r(,3x-eq \l(\l( ))6xeq \l(\o\ac(2, ))

eq \l(\l( ))+eq \l(\l( ))7)) = 1 + 2.3eq \l(\o\ac(x+eq \l(\l( ))1, )) ( HD gii: iu kin (x ( R

Ta c pt ( 3eq \l(\o\ac(2x+eq \l(\l( ))2, )) + 4, ))


eq \l(\l( ))+eq \l(\l( ))7)) = 1 + 2.3eq \l(\o\ac(x+eq \l(\l( ))1, ))

( 4, ))


eq \l(\l( ))+eq \l(\l( ))7)) = 1 + 2.3eq \l(\o\ac(x+eq \l(\l( ))1, )) - 3eq \o(\a\ac\vs2(+eq \l(\l( ))1))), ))

Ta c V Tri = 4, ))


eq \l(\l( ))+eq \l(\l( ))7)) = 2, ))

eq \l(\l( ))eq \l(\r(,-eq \l(\l( ))1)eq \l(\o\ac(2, ))

eq \l(\l( ))+eq \l(\l( ))4)))) ( 2

V Phi = 1 + 2.3eq \l(\o\ac(x+eq \l(\l( ))1, )) - 3eq \o(\a\ac\vs2(+eq \l(\l( ))1))), )) = 2 - (3eq \l(\o\ac(x+eq \l(\l( ))1, )) - 1)eq \l(\o\ac(2, )) ( 2

Vy phng trnh ch c nghim ( VT = VP = 2 ( 2, ))

eq \l(\l( ))eq \b\lc\{(\a\al\vs0(x-eq \l(\l( ))1eq \l(\l( ))=eq \l(\l( ))0 ,3eq \l(\o\ac(x+eq \l(\l( ))1, ))

eq \l(\l( ))-eq \l(\l( ))1eq \l(\l( ))=eq \l(\l( ))0)) ( x = -1 V d 3: Gii phng trnh logeq \l(\o\al(2,2)) (x - 1) + 4, ))


eq \l(\l( ))+eq \l(\l( ))247)) = logeq \l(\o\ac( ,2)) (2xeq \l(\o\ac(2, )) - 4x + 2)

( HD gii: x - 1 > 0 ( x > 1

Ta c PT ( 4, ))


eq \l(\l( ))+eq \l(\l( ))247)) = logeq \l(\o\ac( ,2)) (2xeq \l(\o\ac(2, )) - 4x + 2) - logeq \l(\o\al(2,2)) (x - 1) Ta c VT = 4, ))


eq \l(\l( ))+eq \l(\l( ))247)) = 2, ))

eq \l(\l( ))eq \l(\r(,-eq \l(\l( ))9)eq \l(\o\ac(2, ))

eq \l(\l( ))+eq \l(\l( ))4)))) ( 2

VP = logeq \l(\o\ac( ,2)) (2xeq \l(\o\ac(2, )) - 4x + 2) - logeq \l(\o\al(2,2)) (x - 1)

= logeq \l(\o\ac( ,2))[2(x - 1)eq \l(\o\ac(2, ))] - logeq \l(\o\al(2,2)) (x - 1)

= 1 + 2logeq \l(\o\ac( ,2))(x - 1) - logeq \l(\o\al(2,2)) (x - 1)

= 2 - [logeq \l(\o\ac( ,2)) (x - 1) - 1]eq \l(\o\ac(2, )) ( 2

Do phng trnh cho ch c nghim ( VT = VP = 2

( 2, ))

eq \l(\l( ))eq \b\lc\{(\a\al\vs0(x-eq \l(\l( ))9eq \l(\l( ))=eq \l(\l( ))0 , ,2))

eq \l(\l( ))eq \l(\l(log(xeq \l(\l( ))-eq \l(\l( ))1)eq \l(\l( ))-eq \l(\l( ))1eq \l(\l( ))=eq \l(\l( ))0)))) ( eq \b\lc\{(\a\al\vs0(=eq \l(\l( ))(eq \l(\l( ))3)) ,xeq \l(\l( ))-eq \l(\l( ))1eq \l(\l( ))=eq \l(\l( ))2)) ( x = 3 (nhn v x > 1)

V d 4: Gii phng trnh 2eq \l(\o\ac(x-eq \l(\l( ))1, )) - 22, ))

eq \l(\l( ))eq \o(\a\ac\vs2(x-eq \l(\l( ))x, )) = (x - 1)eq \l(\o\ac(2, ))

( HD gii: Ta c VP = (x - 1)eq \l(\o\ac(2, )) ( 0 ( xeq \l(\o\ac(2, )) - 2x + 1 ( 0 ( xeq \l(\o\ac(2, )) - x ( x - 1 Mt khc VT = 2eq \l(\o\ac(x-eq \l(\l( ))1, )) - 22, ))

eq \l(\l( ))eq \o(\a\ac\vs2(x-eq \l(\l( ))x, )) ( 0 (do 2 > 1, hm ng bin v xeq \l(\o\ac(2, )) - x ( x - 1 )

Do phng trnh cho ch c nghim ( VT = VP = 0 ( x = 1 Dng aeq \l(\o\ac(u, )) - aeq \l(\o\ac(v, )) = v - u ( aeq \l(\o\ac(u, )) + u = aeq \l(\o\ac(v, )) + v ( dng tnh n iu ca hm s.

V d 1: Gii phng trnh 52, ))

eq \l(\l( ))eq \o(\a\ac\vs2(x+eq \l(\l( ))3xeq \l(\l( ))+eq \l(\l( ))2, )) - 52, ))

eq \l(\l( ))eq \o(\a\ac\vs2(2x+eq \l(\l( ))5xeq \l(\l( ))+eq \l(\l( ))3, )) = (x + 1)eq \l(\o\ac(2, ))

( HD gii: t u = xeq \l(\o\ac(2, )) + 3x + 2 ; v = 2xeq \l(\o\ac(2, )) + 5x + 3 th v - u = (x + 1)eq \l(\o\ac(2, ))

PT thnh 5eq \l(\o\ac(u, )) - 5eq \l(\o\ac(v, )) = v - u ( 5eq \l(\o\ac(u, )) + u = 5eq \l(\o\ac(v, )) + v.

Xt f(t) = 5eq \l(\o\ac(t, )) + t (t ( R c f '(t) = 5eq \l(\o\ac(t, )) ln5 + 1 > 0 (t ( R

( f(t) lun ng bin trn R, m f(u) = f(v) ( u = v ( (x + 1)eq \l(\o\ac(2, )) = 0 ( x = -1 Dng logeq \l(\o\ac( ,a)) u - logeq \l(\o\ac( ,a)) v = v - u ( logeq \l(\o\ac( ,a)) u + u = logeq \l(\o\ac( ,a)) v + v ( dng tnh n iu ca hm s.

V d 2: Gii phng trnh logeq \l(\o\ac( ,3)) 2, ))

eq \l(\l( ))eq \s\don1(\f(x+eq \l(\l( ))xeq \l(\l( ))+eq \l(\l( ))3,2xeq \l(\o\ac(2, ))

eq \l(\l( ))+eq \l(\l( ))4xeq \l(\l( ))+eq \l(\l( ))5)) = xeq \l(\o\ac(2, )) + 3x + 2

( HD gii: iu kin 2, ))


eq \l(\l( ))+eq \l(\l( ))4xeq \l(\l( ))+eq \l(\l( ))5)) > 0 ( (x ( R

t u = xeq \l(\o\ac(2, )) + x + 3; v = 2xeq \l(\o\ac(2, )) + 4x + 5 th v - u = xeq \l(\o\ac(2, )) + 3x + 2

PT thnh logeq \l(\o\ac( ,3)) u - logeq \l(\o\ac( ,3)) v = v - u ( logeq \l(\o\ac( ,3)) u + u = logeq \l(\o\ac( ,3)) v + v

Xt f(t) = logeq \l(\o\ac( ,3)) t + t (t > 0 c f '(t) = eq \s\don1(\f(1,t.ln3)) + 1 > 0 (t > 0

( f(t) lun ng bin trn (0; +() m f(u) = f(v) ( u = v ( xeq \l(\o\ac(2, )) + 3x + 2 = 0 ( eq \b\lc\[(\a\al\vs0(x=eq \l(\l( ))-1 ,xeq \l(\l( ))=eq \l(\l( ))-2))

BI TP RN LUYN: Gii cc phng trnh sau:

a) logeq \l(\o\ac( ,2)) 2, ))


eq \l(\l( ))+eq \l(\l( ))xeq \l(\l( ))+eq \l(\l( ))4)) = xeq \l(\o\ac(2, )) - 5

b) 2logeq \l(\o\al(2,9)) x = logeq \l(\o\ac( ,3)) x.logeq \l(\o\ac( ,3)) (eq \l(\r(,2x+eq \l(\l( ))1)) - 1 c) 2eq \o(\a\ac\vs2(-eq \l(\l( ))xeq \l(\o\ac(2, )),xeq \l(\o\ac(2, )))), )) - 2eq \o(\a\ac\vs2(-eq \l(\l( ))2x,xeq \l(\o\ac(2, )))), )) = eq \s\don1(\f(1,2)) - eq \s\don1(\f(1,x))

d) logeq \l(\o\ac( ,3)) eq \s\don1(\f(2x-eq \l(\l( ))1,eq \l(\l((x-eq \l(\l( ))1)eq \l(\o\ac(2, )))))) = 3xeq \l(\o\ac(2, )) - 8x + 5 e) 2 ,5))

eq \l(\l( ))eq \o(\a\ac\vs2(logxeq \l(\o\ac(3, )), )) + 2 ,5))

eq \l(\l( ))eq \o(\a\ac\vs2(logxeq \l(\o\ac(2, )), )) = x + x ,5))

eq \l(\l( ))eq \o(\a\ac\vs2(log7, ))

f) logeq \l(\o\ac( ,2)) eq \s\don1(\f(2x+eq \l(\l( ))1,eq \l(\l((x-eq \l(\l( ))1)eq \l(\o\ac(2, )))))) = 2xeq \l(\o\ac(2, )) - 6x + 2

g) logeq \l(\o\ac( ,3)) (2, ))

eq \l(\l( ))eq \l(\r(,x-eq \l(\l( ))3xeq \l(\l( ))+eq \l(\l( ))2)) + 2) + (0,2)eq \o(\a\ac\vs2(3x-eq \l(\l( ))xeq \l(\o\ac(2, ))

eq \l(\l( ))-eq \l(\l( ))1, )) = 2 H PHNG TRNH M LOGARIT

Bi tp 1: Gii h phng trnh sau:

Hng dn

t (iu kin u, v>0), ta c h phng trnh:

Theo nh l viete o th hai s u v v l nghim ca phng trnh bc hai:

(n y cc bn t gii qut nhe..!)

Bi tp 2: Gii h phng trnh:

Hng dn.

Nhn hai v phng trnh cho , ta c phng trnh:

t Khi ta c phng trnh:

Gii phng trnh ta c hai nghim t=1 v t=-2. V t>0 nn nhn nghim t=1

Vi t=1 th

Vy h cho tng ng vi:

Kt lun: Tp nghim ca h phng trnh l


Hng dn.

iu kin xc nh ca h phng trnh l x, y > 0. Vi iu kin ta c:

(Cc bn t gii qut tip nhe..!)

p s: H phng trnh cho c nghim


Hng dn.

iu kin xc nh ca h phng trnh l x, y > 0. Vi iu kin ta c:

Theo nh l viete o ta c hai s x, y l nghim ca phng trnh:

(cc bn t gii quyt tip nhe..!)

p s: H pt c nghim:


Hng dn.

iu kin xc nh ca h phng trnh l . Vi iu kin ta c:

Tip theo ta t Khi ta c h phng trnh:

(Cc bn t gii quyt tip nhe..!)



Hng dn.

iu kin xc nh ca h phng trnh l x, y >0. Vi iu kin ta c:

Kt lun: H phng trnh cho c nghim


Hng dn.

iu kin xc nh ca h phng trnh l x>y>0. Vi iu kin ta c:




Hng dn.





(Trch thi H khi D 2002)

Hng dn.

Ta c:

( Ch ).

Kt lun: Tp nghim ca h phng trnh:


(Trch thi H khi A nm 2004)


(Trch thi H khi B nm 2005)


Hng dn.



p s: H phng trnh c nghim


Hng dn.

H phng trnh:

Kt lun: H pt c nghim


Hng dn.


Kt lun: H phng trnh c nghim


Hng dn.

Cch 1:

H phng trnh:

t khi ta c phng trnh:


Cch 2:

H phng trnh:

p dng nh l vite ta c: v l hai nghim ca phng trnh bc hai


p s: H phng trnh c nghim hoc


Hng dn.

Cch 1:

H phng trnh:

t khi ta c phng trnh:


Cch 2:

H phng trnh:

p dng nh l vite ta c: v l hai nghim ca phng trnh bc hai


p s: H phng trnh c nghim l v


Hng dn.


t Khi ta c h phng trnh




Hng dn.

iu kin xc nh ca h phng trnh l x>y>0. Vi iu kin ta c:

Xt h phng trnh:

Ta c:

Xt h phng trnh

Ta c:

Kt lun: H phng trnh c nghim l v


Hng dn.


Ly logarit c s 10 hai v ca hai phng trnh trong h ta c:

Tip theo ta t (Cc bn t gii tip nhe...!)



1.

2.

3.

4.


1.

2.

3.

4.


1.

2.

3.

4.

5.


1.

2.

3.

4.

5.

CHUYN 5: PHNG TRNH LNG GIC

L thuyt c bn:

1. Phng trnh

* Nu th phng trnh v nghim.* Nu th

c bit:

2. Phng trnh

* Nu th phng trnh v nghim.* Nu th

c bit:

3. Phng trnh

*

*

4. Phng trnh

*

*

Cc gi tr c bit cn nh:

.

Bi tp:

I. PHNG TRNH BC NHT I VI SINX V COSX

Bi 1. Gii cc phng trnh sau :

a.

b.

c.

d.

Gii

a.

b. . iu kin :

Khi :

c.

EMBED Equation.DSMT4

d.


a.

b.

c.

d.

Gii

a.

b.

Ta c : . Do :

. Phng trnh v nghim .

c.

d.


a.

b.

c.

Gii

a.

b.

Ta c :

Cho nn (1) :

V :

c.

Do :

Cho nn (c) tr thnh :


a.

b.

c.

d.

Gii

a.

Chia hai v hw[ng trnh cho 2 ta c :

b.

Chia hai v phng trnh cho 2 ta c kt qu :

c.

T cng thc nhn ba : cho nn phng trnh (c) vit li :

d.

II. PHNG TRNH : BC NHT - BC HAI

I VI MT HM S LNG GIC


a.

b.

c.

d.

Gii

a. . iu kin : (*)

Phng trnh (a) tr thnh :

EMBED Equation.DSMT4 Cho nn (a)

Vy : . Kim tra iu kin :

- . Cho nn nghim phng trnh l

- Vi phm iu kin , cho nn loi .

Tm li phng trnh c mt h nghim :

b.

Do :

c.

EMBED Equation.DSMT4 d.

- Vi sinx =0

- Do : . Cho nn phng trnh v nghim .

Bi 2. Gii cc phng trnh sau

a.

b.

c.

d.

Gii

a.

b. . iu kin : cosx khc khng .

Khi phng trnh tr thnh :

c. . iu kin :

Phng trnh (c)

Nghim ny tha mn iu kin .

d. . iu kin :

d

Vy phng trnh c nghim : ( Tha mn diu kin )


a.

b.

c.

d.

Gii

a. . iu kin :

Khi :

Cc h nghim ny tha mn iu kin .

b. . iu kin : (*)

Khi :

Nhng do iu kin (*) Ta ch c nghim : , tha mn . cng l nghim

c.

d. .

Do Phng trnh c nghim :


a.

b.

c.

d. Cho : . Hy gii phng trnh : f'(x)=0.

Gii

a.

EMBED Equation.DSMT4 b. . iu kin :

Chia hai v phng trnh cho : . Khi phng trnh c dng :

t :

Do phng trnh c nghim :

c. . iu kin :

Khi :

. Nhng nghim : vi phm iu kin .

Vy phng trnh c nghim :

d. Cho : . Hy gii phng trnh : f'(x)=0.

Ta c :

- Trng hp : cosx=0

- Trng hp :


a.

b.

c.

d.

Gii

a.

t : . Khi phng trnh tr thnh : (2)

Nhan hai v vi 2cost ta c :

b.

iu kin : . Khi phng trnh tr thnh :

Cc nghim tha mn iu kin .

c. . t : . Khi phng trnh c dng :

Ch xy ra khi : . Nu phng trnh c nghim th tn ti k,l thuc Z sao cho h c nghim chung . C ngha l :

d.

iu kin : . Khi phng trnh tr thnh :

Nghim ny tha mn iu kin (*)


a.

b.

c.

d.

Gii

a. .

Do : . Cho nn mu s khc khng .

Phng trnh tr thnh :

Vy : .

i chiu vi iu kin c ngha th ta phi b i cc nghim ng vi k l l : . Do phng trnh ch c nghim ng vi k l chn : x=

b. . iu kin : (*)

Phng trnh

Do : . Tha mn iu kin (*)

c.

- Trng hp :

- Trng hp :


EMBED Equation.DSMT4 t :

Chng t f(t) ng bin . Khi ti f(-1)=1 v f(1)=9 cho nn vi mi

Vy phng trnh v nghim .

d. . iu kin :

Phng trnh tr thnh :

Do Phng trnh ch c nghim :


a.

b.

c.

d.

Gii

a. . iu kin : . Khi phng trnh vit li :

Vy phng trnh c nghim l :

b. . iu kin :

Phng trnh

.

Tha mn (*)

c. . iu kin :

Khi phng trnh tr thnh :

. Tha mn iu kin (*).

d. . iu kin : (*)

C 2 phng php gii :

Cch 1.

Cch 2.

. ( Nh kt qu trn )


a.

b.

c.

d.

Gii

a.


Vy phng trnh c nghim :

b. . iu kin : sin2x khc 1 (*)

Phng trnh tr thnh :

i chiu vi iu kin (*) th vi vi phm iu kin . Cho nn phng trnh ch cn nghim :

c.

d. . Do :


a.

b.

c.

d. 3tan2x-4tan3x=

Gii

a. .

b.

c.

d. 3tan2x-4tan3x=

iu kin : Phng trnh tr thnh :

i chiu vi iu kin ta thy nghim . Vi phm iu kin , nn b loi .

Vy phng trnh cn c nghim l :


a.

b

c.

d.

Gii

a.

b.

t :

Do phng trnh cho tr thnh :

c. . iu kin : .

Khi PTd/ tr thnh :

. Phng trnh v nghim .

d.

III. PHNG TRNH I XNG THEO SINX, COSX


a.

b.

c.

d.

Gii

a. .

Do :

b. (1)

t :

. Do phng trnh :

c. . iu kin : . Khi phng trnh (c) tr thnh :

t : . Thay vo phng trnh ta c :

Tha mn iu kin .

d. . iu kin : .

Khi :

Trng hp :

Trng hp : sinx+cosx-sinx cosx=0 .

t : Cho nn phng trnh :


a.

b.

c.

Gii

a. . iu kin : cosx khc 0 . Khi phng trnh tr thnh :

V sinx=1 lm cho cosx=0 vi phm iu kin . Do

Trng hp : sinx+cosx-sinx cosx=0 .

t : Cho nn phng trnh :

Vy nghim ca phng trnh l :

b.

Trng hp :

Trng hp : sinx+cosx+sinxcosx+1=0

Do phng trnh c nghim :

c.

. ( b nghim t=-3

Chuyên đề ôn thi Đại Học môn toán bộ 1

Documents