101 Chuyen de Luyen Thi Dai Hoc

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TRN ANH TUN TRNG I HC THNG MI

Cc chuyn

LUYN THI I HCWWW.VNMATH.COM

H NI - 2011



Mc lcWWW.VNMATH.COMI i s - Lng gic - Gii tchChng 1 Phng trnh, bt phng trnh, h i s

911

1.1. Phng trnh, bt phng trnh a thc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.1.1. Phng trnh, bt phng trnh bc hai . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.1.2. Phng trnh trnh bc ba . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.1.3. Phng trnh, bt phng trnh bc bn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.2. Phng trnh, bt phng trnh cha gi tr tuyt i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1.3. Phng trnh, bt phng trnh cha cn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 Vn 1 : Phng trnh, bt phng trnh c bn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 Vn 2 : Phng php t n ph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 Vn 3 : Phng php nhn lin hp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 Vn 4 : Phng php nh gi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 Vn 5 : Phng trnh, bt phng trnh c tham s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 1.4. H phng trnh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 1.4.1. Phng php th . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 1.4.2. Phng php phn tch thnh nhn t hoc coi mt phng trnh l phng trnh bc hai (ba) theo mt n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 1.4.3. Phng php t n ph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 1.4.4. Phng php hm s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 1.4.5. Phng php nh gi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 1.5. S nghim ca phng trnh, h phng trnh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 Vn 1 : Chng minh phng trnh c nghim duy nht . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 Vn 2 : Chng minh phng trnh c ng hai nghim phn bit . . . . . . . . . . . . . . . . . . . . . . 28 Vn 3 : Chng minh phng trnh c ng ba nghim phn bit . . . . . . . . . . . . . . . . . . . . . . 29 1.6. Phng trnh, bt phng trnh, h i s trong cc k thi tuyn sinh H . . . . . . . . . . . . . . . . . . . 29 1.7. Bi tp tng hp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 Chng 2 Bt ng thc 37

2.1. Phng php s dng bt ng thc Cauchy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 2.1.1. Bt ng thc Cauchy - So snh gia tng v tch . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 2.1.2. Mt s h qu trc tip . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 2.1.3. Bi tp ngh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 2.2. Bt ng thc hnh hc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 2.3. Phng php s dng iu kin c nghim ca phng trnh hoc h phng trnh . . . . . . . . . . . . . . 44 3

www.luyenthi24h.com www.luyenthi24h.com2.4. Bt ng thc trong cc k thi tuyn sinh H . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 2.5. Bi tp tng hp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 Chng 3 Lng gic 51

3.1. Phng trnh c bn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 3.2. Phng trnh dng a sin x + b cos x = c . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 3.3. Phng php t n ph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 3.4. a phng trnh v dng tch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 3.5. Phng php nh gi v phng php hm s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 3.6. Gi tr ln nht v nh nht ca biu thc lng gic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 3.7. Lng gic trong cc k thi tuyn sinh H . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 3.8. Bi tp tng hp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 Chng 4 T hp 69

4.1. Cc quy tc m. T hp, chnh hp, hon v . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 4.2. Gii phng trnh, bt phng trnh, h . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 4.3. H s ca xk trong khai trin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 4.4. H s ca xk trong khai trin nh thc (a + b)n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 4.5. H s ca xk trong khai trin (a + b)n (c + d)m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 4.6. H s ca xk trong khai trin (a + b + c)n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 4.7. Tnh tng cc h s t hp :n k ak Cn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

4.8. Phng php c bn vi ak ch l hm s m theo bin k . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 4.9. Phng php o hm vi ak l tch hm s m v a thc theo k . . . . . . . . . . . . . . . . . . . . . . . 78 4.10. Phng php tch phn vi ak l tch hm s m v phn thc theo k . . . . . . . . . . . . . . . . . . . . . 79 4.11. Bi tp tng hp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 Chng 5 Hm s 83

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5.1. Tnh n iu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 Vn 1 : Xt chiu bin thin ca hm s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 Vn 2 : Tm iu kin tham s hm s n iu trn mt min . . . . . . . . . . . . . . . . . . . . . . 84 Vn 3 : Gi tr ln nht, gi tr nh nht ca hm mt bin s . . . . . . . . . . . . . . . . . . . . . . . 87 Vn 4 : S dng tnh n iu chng minh bt ng thc . . . . . . . . . . . . . . . . . . . . . . . . . . 89 Vn 5 : ng dng s bin thin vo vic gii phng trnh, bt phng trnh, h . . . . . . . . . . . . . 91 Vn 6 : ng dng s bin thin vo bi ton s nghim phng trnh c tham s . . . . . . . . . . . . . 92 5.2. Cc tr ca hm s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 Vn 1 : S dng du hiu 1 v du hiu 2 xc nh cc im cc tr ca hm s . . . . . . . . . . . . 94 Vn 2 : iu kin ca tham s hm s t cc tr (cc i hoc cc tiu) ti x = x0 hoc th hm s t cc tr ti im (x0 ; y0 ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 Vn 3 : Tm iu kin hm s c cc tr v tha mn mt vi iu kin . . . . . . . . . . . . . . . . . 95 5.3. Tim cn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 Vn 1 : Tm tim cn ca th hm s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 Vn 2 : Cc bi ton v tim cn c tham s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 5.4. Tm i xng v trc i xng. im thuc th . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

www.luyenthi24h.com www.luyenthi24h.comVn 1 : Tm i xng, trc i xng . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 Vn 2 : Khong cch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 5.5. Bin lun s nghim ca phng trnh, bt phng trnh bng phng php th . . . . . . . . . . . . . . 103 5.6. Bi ton v s tng giao . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 5.7. S tip xc ca hai ng cong v tip tuyn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 Vn 1 : Vit phng trnh tip tuyn bit tip im . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 Vn 2 : Hai ng cong tip xc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 Vn 3 : Tip tuyn i qua mt im . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 Vn 4 : Tip tuyn c h s gc cho trc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 5.8. Hm s trong cc k thi tuyn sinh H . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 5.9. Bi tp tng hp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 Chng 6 M v lgart 127

6.1. Hm s m, hm s ly tha . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 6.2. Hm s logarit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 6.3. Phng trnh m v logarit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 Vn 1 : Phng trnh c bn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 Vn 2 : Phng php logarit hai v . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 Vn 3 : Phng php t n ph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 Vn 4 : Phng php phn tch thnh nhn t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 Vn 5 : Phng php nh gi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 6.4. Bt phng trnh m v logarit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 Vn 1 : Bt phng trnh c bn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 Vn 2 : Phng php t n ph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 Vn 3 : Phng php phn tch thnh nhn t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 6.5. H phng trnh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 6.6. Phng trnh m v lgarit trong cc k thi tuyn sinh H . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 6.7. Bi tp tng hp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 Chng 7 Tch phn 149

7.1. Cc dng ton c bn v nguyn hm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 Vn 1 : Chng minh mt hm s F(x) l mt nguyn hm ca hm s f (x) . . . . . . . . . . . . . . . . 149 Vn 2 : S dng bng nguyn hm c bn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 Vn 3 : Tm hng s C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 Vn 4 : Phng php nguyn hm tng phn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 Vn 5 : Phng php i bin s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 7.2. Cc dng ton tch phn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 Vn 1 : S dng tch phn c bn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 Vn 2 : Tch phn hm cha du tr tuyt i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 Vn 3 : Phng php tch phn tng phn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 Vn 4 : Phng php i bin s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 Vn 5 : Tch phn cc hm hu t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 Vn 6 : Tch phn mt s hm c bit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

www.luyenthi24h.com www.luyenthi24h.com7.3. ng dng tch phn tnh din tch hnh phng . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 7.4. ng dng tch phn tnh th tch vt th trn xoay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162 7.5. Tch phn trong cc k thi H . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 7.6. Bi tp tng hp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 Chng 8 S phc 167

II Hnh hcChng 9 Phng php ta trong trong mt phng

173175

9.1. Phng php ta trong mt phng . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 9.2. Phng trnh ca ng thng . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176 9.2.1. Cc bi ton thit lp phng trnh ng thng . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176 9.2.2. Cc bi ton lin quan n vic s dng phng trnh ng thng . . . . . . . . . . . . . . . . . . 176 9.2.3. Bi tp tng hp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 9.3. ng trn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180 9.4. ng elip . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 9.5. ng hypebol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184 9.6. ng parabol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186 9.7. Phng php ta trong mt phng qua cc k thi tuyn sinh H . . . . . . . . . . . . . . . . . . . . . . 187 9.8. Bi tp tng hp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188 Chng 10 M u v hnh hc khng gian. Quan h song song 191

10.1. i cng v ng thng v mt phng . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192 Vn 1 : Xc nh giao tuyn ca hai mt phng . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192 Vn 2 : Xc nh giao im ca ng thng a v mt phng (P) . . . . . . . . . . . . . . . . . . . . . 192 Vn 3 : Phng php chng minh ba im thng hng v ba ng thng ng quy . . . . . . . . . . . 193 Vn 4 : Tm thit din ca hnh chp ct bi mt phng . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 10.2. Hai ng thng song song . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 Vn 1 : Tm giao tuyn ca hai mt phng (dng quan h song song) . . . . . . . . . . . . . . . . . . . 195 Vn 2 : Chng minh hai ng thng song song . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196 Vn 3 : Chng minh hai ng thng cho nhau . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196 10.3. ng thng v mt phng song song . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 Vn 1 : Chng minh ng thng song song vi mt phng . . . . . . . . . . . . . . . . . . . . . . . . 197 Vn 2 : Tm giao tuyn ca hai mt phng. Dng thit din song song vi mt ng thng Vn 3 : Dng mt mt phng cha mt ng thng v song song vi ng thng khc Xc nh giao im ca ng thng vi mt phng . . . . . . . . . . . . . . . . . . . . . . . . . . 198 10.4. Hai mt phng song song . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199 Vn 1 : Chng minh hai mt phng song song . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199 Vn 2 : Tm giao tuyn ca hai mt phng Thit din ct bi mt mt phng song song vi mt mt phng cho trc . . . . . . . . . . . . . . 199 . . . . . . . 197

www.luyenthi24h.com www.luyenthi24h.comChng 11 Vect trong khng gian. Quan h vung gc 201

11.1. Vect trong khng gian. S ng phng ca cc vect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202 Vn 1 : Biu th mt vect qua ba vect khng ng phng . . . . . . . . . . . . . . . . . . . . . . . . . 202 Vn 2 : Chng minh cc ng thc vect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203 Vn 3 : Chng minh cc im thng hng v quan h song song . . . . . . . . . . . . . . . . . . . . . . 203 Vn 4 : Chng minh cc vect ng phng . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204 11.2. Hai ng thng vung gc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205 Vn 1 : Tnh gc gia hai vect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205 Vn 2 : Tnh gc gia hai ng thng a v b . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206 Vn 3 : Chng minh hai ng thng vung gc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207 11.3. ng thng vung gc vi mt phng . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207 Vn 1 : Chng minh ng thng a vung gc vi mt phng (P) . . . . . . . . . . . . . . . . . . . . . 207 Vn 2 : Chng minh hai ng thng vung gc vi nhau . . . . . . . . . . . . . . . . . . . . . . . . . 208 Vn 3 : Xc nh gc gia ng thng a v mt phng (P) . . . . . . . . . . . . . . . . . . . . . . . . 210 Vn 4 : Dng mt phng qua im M cho trc v vung gc vi mt ng thng d cho trc . . . . . 211 11.4. Hai mt phng vung gc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 Vn 1 : Xc nh gc gia hai mt phng . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 Vn 2 : Chng minh hai mt phng (P) v (Q) vung gc . . . . . . . . . . . . . . . . . . . . . . . . . 214 Vn 3 : Chng minh ng thng a vung gc vi mt phng (P) . . . . . . . . . . . . . . . . . . . . . 215 Vn 4 : Dng mt phng (Q) cha a v vung gc vi (P) (gi thit a khng vung gc vi (P)) . . . . . 216 11.5. Khong cch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217 Vn 1 : Tnh khong cch t im M n ng thng cho trc . . . . . . . . . . . . . . . . . . . . 217 Vn 2 : Dng ng thng i qua mt im A cho trc v vung gc vi mt mt phng (P) cho trc. Khong cch t im A n mt phng (P) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217 Vn 3 : on vung gc chung v khong cch gia hai ng thng cho nhau . . . . . . . . . . . . . 219 11.6. Khi a din v th tch khi a din . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222 Vn 1 : Phng php trc tip tm th tch khi chp . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222 Vn 2 : Tnh th tch hnh chp mt cch gin tip . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227 Vn 3 : Dng cng thc th tch gii mt s bi ton hnh hc . . . . . . . . . . . . . . . . . . . . . 228 11.7. Phn loi mt s hnh khi a din . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230 11.7.1. Hnh chp c cnh bn vung gc vi y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230 11.7.2. Hnh chp u . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231 11.7.3. Hnh chp c mt bn vung gc vi y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232 11.7.4. Hnh chp c hai mt vung gc vi y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233 11.7.5. Hnh chp c cc cnh bn bng nhau hoc cc cnh bn cng to vi y nhng gc bng nhau . . 233 11.7.6. Hnh hp - Hnh lng tr . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234 11.8. Bi tp tng hp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235 Chng 12 Mt cu v khi trn xoay 239

12.1. Mt cu, khi cu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239 12.2. Mt trn xoay. Mt tr, hnh tr v khi tr . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243

www.luyenthi24h.com www.luyenthi24h.comChng 13 Phng php khng gian to trong khng gian 249

13.1. H to trong khng gian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249 Vn 1 : Tm ta ca mt vect v cc yu t lin quan n vect tha mn mt s iu kin cho trc . 249 Vn 2 : ng dng ca tch v hng v tch c hng . . . . . . . . . . . . . . . . . . . . . . . . . . . 249 Vn 3 : Lp phng trnh ca mt cu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252 Vn 4 : Phng php ta gii hnh hc khng gian . . . . . . . . . . . . . . . . . . . . . . . . . . . 253 13.2. Phng trnh mt phng . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254 Vn 1 : Vit phng trnh mt phng i qua mt im v c mt vect php tuyn cho trc . . . . . . . 254 Vn 2 : V tr tng i ca hai mt phng . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255 Vn 3 : Khong cch t mt im n mt phng . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256 Vn 4 : Gc gia hai mt phng . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258 Vn 5 : V tr tng i gia mt phng v mt cu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258 13.3. Phng trnh ng thng . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260 Vn 1 : Phng trnh tham s v phng trnh chnh tc ca ng thng . . . . . . . . . . . . . . . . . 260 Vn 2 : Tm im trn ng thng tha mn iu kin cho trc . . . . . . . . . . . . . . . . . . . . . 260 Vn 3 : V tr tng i ca hai ng thng v trong khng gian . . . . . . . . . . . . . . . . . . 261 Vn 4 : V tr tng i gia ng thng v mt phng (P) . . . . . . . . . . . . . . . . . . . . . . . 262 Vn 5 : Khong cch t mt im n ng thng . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263 Vn 6 : V tr tng i gia ng thng v mt cu . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264 Vn 7 : Gc gia hai ng thng ; gc gia ng thng v mt phng . . . . . . . . . . . . . . . . . 266 Vn 8 : Phng trnh ng thng bit ng thng song song, hoc vung gc vi ng thng hoc mt phng khc, hoc nm trn mt phng khc . . . . . . . . . . . . . . . . . . . . . . . . . 267 Vn 9 : Phng trnh ng thng bit ct . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268 Vn 10 : Hnh chiu v tnh i xng . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 270 Vn 11 : Bi ton cc tr . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271 13.4. Hnh hc khng gian trong cc k thi tuyn sinh H . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273 13.5. Bi tp tng hp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278

III Hng dn v p s

287

8

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Phn I

i s - Lng gic - Gii tch

9


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Chng 1

Phng trnh, bt phng trnh, h i s1.1 Phng trnh, bt phng trnh a thc1.1.1 Phng trnh, bt phng trnh bc haiBi 1.1 : Gii v bin lun cc phng trnh sau : 1. (m 2)x2 2mx + m + 1 = 0 ; Bi 1.2 : Cho phng trnh : 1. Tm m phng trnh c hai nghim phn bit. 2. Tm m phng trnh c nghim duy nht. Bi 1.3 : Gi a, b, c l di ba cnh ca mt tam gic. Chng minh phng trnh sau v nghim : c2 x2 + (a2 b2 c2 )x + b2 = 0. Bi 1.4 : Cho phng trnh : x2 (2m + 3)x + m2 + 2m + 2 = 0. 1. Tm m phng trnh c hai nghim x1 , x2 . 2. Vit phng trnh bc hai c hai nghim 1 1 , . x1 x2 2. a 1 + = 2. x1 xa

(m2 4)x2 + 2(m + 2)x + 1 = 0.

3. Tm h thc gia x1 , x2 c lp vi tham s m. 4. Tm m phng trnh c hai nghim x1 , x2 tha mn x1 = 2x2 . Bi 1.5 : Cho phng trnh : x2 cos a.x + sin a 1 = 0. 1. Chng minh rng phng trnh lun c hai nghim x1 , x2 vi mi a. 2. Tm h thc gia x1 , x2 c lp vi a. 3. Tm gi tr ln nht, gi tr nh nht ca E = (x1 + x2 )2 + x2 x2 . 1 2 Bi 1.6 : Cho phng trnh : mx2 2(m 2)x + m 3 = 0. Tm m phng trnh c : 11

www.VNMATH.com1. hai nghim tri du ;

CHUYN LUYN THI I HC2. hai nghim dng phn bit ;


3. ng mt nghim m.

Bi 1.7 : Gii cc bt phng trnh sau : 1. x2 4x + 3 0 ; Bi 1.9 : Gii h bt phng trnh sau : Bi 1.10 : Tm m : 1. x2 mx + m + 3 0, x R ; x2 7x + 6 0 x2 8x + 15 0

4. x2 + (x + 1)2

2. (m + 1)x2 2(m 1)x + 3m 3 0 ;

2. mx2 + 4x + m > 0, x R ;

3. mx2 mx 5 < 0, x R.

Bi 1.11 : Tm m cc hm s sau xc nh vi mi x R : 1. y = m(m + 2)x2 + 2mx + 2 ; 2. y = 1 (1 m)x2 2mx + 5 9m ;

Bi 1.12 : Cho f (x) = (m + 1)x2 2(m 1)x + 3m 3. Tm m bt phng trnh : 1. f (x) < 0 v nghim. 2. f (x) 0 c nghim.

Bi 1.13 : Tm m cc bt phng trnh sau c tp nghim l R : 3x2 mx + 5 1. 1 0. Bi 1.16 : Tm m f (x) = 2x2 + mx + 3 0 vi mi x [1; 1]. Bi 1.17 : Tm m f (x) = x2 2mx m 0 vi mi x > 0. Bi 1.18 : Tm m f (x) = mx2 2(m + 1)x m + 5 > 0 vi mi x < 1. Bi 1.19 : Tm m f (x) = 2x2 (3m + 1)x (3m + 9) 0 vi mi x [2; 1].

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CHUYN LUYN THI I HC


1.1.2 Phng trnh trnh bc baBi 1.20 : Cho phng trnh : x3 (m2 m + 7)x (3m2 + m 6) = 0. 1. Tm m phng trnh c mt nghim l 1. 2. Vi m > 0 tm c cu trn, hy gii phng trnh . Bi 1.21 : Gii cc phng trnh sau : 1. x3 6x2 + 11x 6 = 0 ; 2. 2x3 + x + 3 = 0 ; 3. x3 5x2 + 7x 2 = 0 ;

4. x3 3 3x2 + 7x 3 = 0 ;

Bi 1.22 : Tm m cc phng trnh sau c ba nghim phn bit : 1. x3 (2m + 1)x2 + 3(m + 4)x m 12 = 0 ; Bi 1.23 : Tm m phng trnh : mx3 (3m 4)x2 + (3m 7)x m + 3 = 0 c ba nghim dng phn bit. 2. mx3 2mx2 (2m 1)x + m + 1 = 0 ;

1.1.3 Phng trnh, bt phng trnh bc bnBi 1.24 : Gii cc phng trnh sau : 1. x4 3x2 + 4 = 0 ; 2. (x 1)(x + 5)(x 3)(x + 7) = 297 ; 3. (x + 2)(x 3)(x + 1)(x + 6) = 36 ; 4. x4 + (x 1)4 = 97 ; 5. (x + 3)4 + (x + 5)4 = 16 ; Bi 1.25 : Tm cc gi tr ca m sao cho phng trnh x4 + (1 2m)x2 + m2 1 = 0. 1. V nghim ; 2. C hai nghim phn bit ; 3. C bn nghim phn bit. 6. 6x4 35x3 + 62x2 35 + 6 = 0 ; 7. x4 + x3 4x2 + x + 1 = 0 ; 8. x4 5x3 + 10x2 10x + 4 = 0 ; 9. x4 x2 + 6x 9 = 0 ; 10. 2x4 x3 15x2 x + 3 = 0.

Bi 1.26 : Tm cc gi tr ca a sao cho phng trnh (a 1)x4 ax2 + a2 1 = 0 c ba nghim phn bit. Bi 1.27 : Cho phng trnh : (m 1)x4 + 2(m 3)x2 + m + 3 = 0. Tm m phng trnh trn v nghim.

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www.VNMATH.comBi 1.28 : Cho phng trnh :

CHUYN LUYN THI I HC


x4 (2m + 1)x2 + m + 3 = 0. Bi 1.29 : Tm h phng trnh sau y c khng t hn hai nghim m khc nhau : x4 + hx3 + x2 + hx + 1 = 0. Bi 1.30 : Cho phng trnh : (x + 1)(x + 2)(x + 3)(x + 4) = m. Tm m phng trnh c bn nghim phn bit. Tm m phng trnh trn c bn nghim phn bit, trong mt nghim b hn 2 v ba nghim cn li ln hn 1.

1.2 Phng trnh, bt phng trnh cha gi tr tuyt i1. Phng trnh (bt phng trnh) | f (x)| + g(x) < 0 (hoc = , hoc > , hoc , hoc ) tng ng vi f (x) 0 hoc f (x) < 0 f (x) + g(x) < 0.

f (x) + g(x) < 0

Mt s phng trnh hoc bt phng trnh cha nhiu hn mt du gi tr tuyt i th vic ph du gi tr tuyt i s phc tp hn nhiu, phi chia thnh nhiu trng hp bng cch lp bng xt du cc biu thc trong du gi tr tuyt i. 2. Phng trnh (bt phng trnh) | f (x)| < |g(x)| (hoc = , hoc > , hoc , hoc ) phng php n gin l bnh phng hai v, chuyn v, phn tch thnh nhn t. 3. Mt s phng trnh v bt phng trnh thng dng (gi s a > 0).

|x| = a x = a hoc x = a. |x| < a a < x < a. |x| a a x a. |x| > a x < a hoc x > a. |x| a x a hoc x a.Bi 1.31 : Gii phng trnh |x2 8x + 15| = x 3.

Bi 1.32 : Gii cc phng trnh v bt phng trnh sau : 1. |x2 5x + 4| = x2 + 6x + 5; 2. |x 1| = 2x 1; Bi 1.33 : Gii cc phng trnh v bt phng trnh sau : x2 2 1. = 2; x+1 3x + 4 3; 2. x2 2x 3 1; 3. x3

3. | x2 + x 1| 2x + 5; 4. |x2 x| |x2 1|.

4. |2x + 3| = |4 3x|.

Bi 1.34 : Gii cc bt phng trnh sau :

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www.VNMATH.com1. |x2 5x + 4| x2 + 6x + 5;

CHUYN LUYN THI I HC


2. 4x2 + 4x |2x + 1| 5.

Bi 1.35 : Gii cc bt phng trnh sau : 1. 1

|x| 1 ; 1 + |x| 2 x2 4|x| |x| 7 0;

6. ||x| 1| < 1 x ;

2. log5 log

7. |x2 3x 7| + 2x 1 < x2 8x 5 ; 8. x2 |x2 3x 5| 5 < x + 1 ; 9. |x 1| + |x 2| > 3 + x ; 10. log3 |x2 4x| + 3 0; x2 + |x 5|

3. |x2 2x 8| > 2x ; 4. |x3 7x 3| < x3 + x2 + 3 ; 5. |x3 x2 + 4| + x3 x2 2x 2 0 ; Bi 1.36 : Gii cc bt phng trnh sau : 1. |3x + 2| + |2x 3| < 11 ; 2. |x2 3x 7| + |2x2 x 9| + |3x2 7x 5| < x + 15 ;

11. ||3x + 4x 9| 8| 3x 4x 1 ;

3. |x 1| + |2 x| > 3 + x ; 4. |x2 3x 17| |x2 5x 7| > 3.

Bi 1.38 : Gii v bin lun bt phng trnh sau theo tham s p :

Bi 1.37 : Tm m bt phng trnh : x2 + |x + m| < 2 c t nht mt nghim m. 2|x p| + 5|x 3p| + 4x + 6p + 12 0.

Bi 1.39 : Gii v bin lun bt phng trnh sau theo tham s p : |2x + 21p| 2|2x 21p| < x 21p. Bi 1.40 : Tm tt c cc gi tr thc ca tham s a sao cho bt phng trnh x2 |x a| |x 1| + 3 0 Bi 1.41 : Tm tt c cc gi tr ca a sao cho gi tr nh nht ca hm s y = x2 + 2x 1 + |x a| ln hn 2. Bi 1.42 : Tm tt c cc gi tr ca a sao cho gi tr nh nht ca hm s y = x2 + |x a| + |x 1| ln hn 2. Bi 1.43 : Tm tt c cc gi tr ca a sao cho gi tr nh nht ca hm s y = ax + |x2 4x + 3| ln hn 1. Bi 1.44 : Tm tt c cc gi tr ca a sao cho gi tr ln nht ca hm s y = 4x x2 + |x m| nh hn 4. ng vi mi x R.

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CHUYN LUYN THI I HC


1.3 Phng trnh, bt phng trnh cha cnVn 1 : Phng trnh, bt phng trnh c bn

Phng php chung l tm cch bnh phng hai v ( gim s cn, hoc mt cn) vi iu kin l hai v ca phng trnh phi khng m. 1. Phng trnh f (x) = g(x) f (x) 0 (hoc cng c th xt g(x) 0) f (x) = g(x). g(x) 0 f (x) = (g(x))2 . g(x) (hoc ) tng ng vi g(x) 0

2. Phng trnh

f (x) = g(x)

3. Bt phng trnh

f (x) >

f (x) > g(x).

4. Bt phng trnh

f (x) < g(x) (hoc ) tng ng vi

f (x) 0 g(x) 0 f (x) < (g(x))2 .

5. Bt phng trnh

f (x) > g(x) (hoc ) tng ng vi (I) f (x) 0 g(x) < 0 hoc (II) g(x) 0

f (x) > (g(x))2 .

Bi 1.45 : Gii phng trnh

x2 + 56x + 80 = x + 20. Bi 1.46 : Gii bt phng trnh x2 2x 15 < x 3. Bi 1.47 : Gii bt phng trnh x2 1 > x + 2. Bi 1.48 : Gii cc phng trnh sau : 1. 2. 2x2 + 4x 1 = x + 1; 4x2 + 101x + 64 = 2(x + 10);

3. 4.

x2 + 2x = 2x2 4x + 3; (x + 1)(x + 2) = x2 + 3x 4.

Bi 1.49 : Gii cc bt phng trnh: 1. 2. x2 + x 6 < x 1; 2x 1 2x 3; 3. 4. 2x2 1 > 1 x; x2 5x 14 2x 1.

Bi 1.50 : Tm tp xc nh ca mi hm s sau :

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www.VNMATH.com1. y = 2. y = x2 + 3x 4 x + 8;

CHUYN LUYN THI I HC


3. y = 4. y =

x2

x2 + x + 1 ; |2x 1| x 2

1 1 2 ; 7x + 5 x + 2x + 5

x2 5x 14 x + 3.

Bi 1.51 : Gii cc phng trnh sau : 1. 5x2 6x 4 = 2(x 1); 2. x2 + 3x + 12 = x2 + 3x.

Bi 1.52 : Gii cc bt phng trnh sau : 1. x2 + 6x + 8 2x + 3; 2x 4 > 1; 4. 5. x2 x 12 x 1;

2.

x2 3x 10 3. 6 (x 2)(x 3) x2 34x + 48 ;

x2 4x 12 > 2x + 3; x+5 6. < 1. 1x

Vn 2 : Phng php t n ph

Chng ta thng s dng mt s quy tc t n ph nh sau : 1. Nu phng trnh cha hai loi cn, c th ax + b, rt x, th vo phng trnh c phng trnh n u. (b) Hoc cng c th t u = n u(x), v = m v(x), ly tha rt ra rng buc gia u v v c 1 phng trnh theo u, v. Kt hp vi phng trnh ban u, ta c h hai n u, v. 2. t u = n u(x), ly tha hai v c phng trnh cha u, x. Kt hp vi phng trnh ban u, ta c h hai n (a) t u = n

u, x.Gii phng trnh bc hai (c l bnh phng mt s). 3. t n ph khng hon ton, t u = 4. Nu phng trnh cha a b v u(x), a v phng trnh bc hai theo u vi x coi nh l tham s. a b.

ab ta thng t u =

5. phng trnh ng cp, chng hn ng cp bc 2 : A.x2 + B.xy + C.y2 = 0. C cch gii nh sau : (a) Xt y = 0, rt c x; (b) Xt y x 0, chia c hai v cho y2 , t u = , a c v phng trnh bc hai theo u. y

Bi 1.53 : Gii cc phng trnh sau : 1. 3x2 + 21x + 18 + 2 x2 + 7x + 7 = 2 ; 2. x2 + x+1 =1; 4. 2x2 3x + 2 = x 3x 2 ; 4 97 x + 4 x = 5 ;

5. 6x2 10x + 5 (4x 1) 6x2 6x + 5 = 0 ;

3. 2(x2 + 2) = 5(x3 + 1) ;

6.

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CHUYN LUYN THI I HC


Bi 1.54 : Gii cc phng trnh sau : 1. 3x + 1 = 2 x + 2x + 2 ; 4 2. 2x2 + x + 6 + x2 + x + 2 = x + ; x x+3+

7. 8. 9. 10. 11.

3 3 3

x1+ x+ 3

3

3 x+1 = x 2; 3 x8;

1 = 3x + 1 ; x 4 4 4. 4 x + x + 1 = 2 2x + 1 ; 5. x2 + 4x + 3 + x2 + x = 3x2 + 4x + 1 ; 6. 3 x + 5 x 3 ; 3. x2 + 2x x Bi 1.55 : Gii cc phng trnh sau : 1. 1x+ 1 + x + 2 1 x2 = 4 ;

x 16 = 3

2x3 1 +

1 x3 = x ; x2 + x + 1 = 2 ;

x2 x + 1 +

2x2 + x + 9 + 2x2 x + 1 = x + 4.

3. x2 + 2x + 4. 2x2 + x +

x + 3 + 2x x + 3 = 9 ; x2 + 3 + 2x x2 + 3 = 9 ;

2. 2x +

x+1+

x + 2 x2 + x = 1 ;

Bi 1.56 : Gii cc phng trnh sau : 1. 2x2 + x + 3 = 3x x + 3 ; 2. 3. x+8= 3x2 + 7x + 8 ; 4x + 2 3x2 + 3x + 2 ; 3x + 1 x + 2 + x 2x + 1 4. = x+2; x + 2x + 1 5. ( x + 3 x + 1)(x2 + x2 + 4x + 3) = 2x.

x2 + x + 2 =

Bi 1.57 : Gii cc phng trnh sau : 1. 2. 3. 4. 5. 3 3 4 x+1+ x+1+ x+1+ 3 3 x+2 =1+ x2 = 3 x+ 4 3 x2 + 3x + 2 ; 6. x+3+ 4x =4 x; x+3

3

x2 + x ;

x =1+

x3 + x2 ;

x + 3 + 2x x + 1 = 2x + x2 + 4x + 3 ; x3 + x2 + 3x + 3 + 2x = x2 + 3 + 2x2 + 2x ;

3 7. 4 x + 3 = 1 + 4x + ; x 8. 2 x + 3 = 9x2 x 4 ; 9. 12 x + 2 x 1 = 3x + 9 ;

Bi 1.58 : Gii cc phng trnh sau : 1. 2. 4 x+3+ x+ 4 3 x=3; 4 x+1; 4. 2x + 1 + x x2 + 2 + (x + 1) x2 + 2x + 3 = 0 ; 5. x2 x + (x 5)2 5 x = 11( x + 5 x) ;3

x1=

3. 2 x2 = (2 x)2 ; Bi 1.59 : Gii cc phng trnh sau : 1. 8 1x+ 8 x=1;

6.

2x3

=1+

x+1 ; 2

4 2. 2 x + 1 2x = 1 ;

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CHUYN LUYN THI I HCVn 3 : Phng php nhn lin hp


Dng 1 : Phng trnh dng

u(x)

v(x) = f (x), trong f (x) v u(x) v(x) c cng nghim x = x0 .

Dng 2 : Phng trnh dng ( n u1 (x) n v1 (x)) + ( m u2 (x) m v2 (x)) = f (x), trong f (x); u1 (x) v1 (x); u2 (x) v2 (x) c cng nghim x = x0 ( y f (x) c th ng nht bng 0). Phng php gii loi ny l chng ta nhn lin hp theo tng cm, t (x x0 ) lm nhn t chung.

u(x) v(x) = f (x). u(x) v(x) (b) Chuyn v, t (x x0 ) lm nhn t chung. (a) Phng trnh tr thnh

Bi 1.60 : Gii cc phng trnh, cc bt phng trnh sau : 1. 3(2 + 2. x 2) = 2x + x + 6; 6. x2 + x 1 = (x + 2) x2 2x + 2; 7. 3 x + 24 + 12 x = 6;

x2 2 > x 4; 1+ 1+x 3. x 2 + 4 x = x2 6x + 11; 4. x 2 + 4 x = 2x2 5x 1;

8. 2 x2 7x + 10 = x + x2 12x + 20; 3 9. 2x2 11x + 21 = 3 4x 4; 10. 5x 1 + 3 9 x = 2x2 + 3x 1.

5.

1 x 2x + x2 = ; x 1 + x2

Bi 1.61 : Gii cc phng trnh sau : 1. x+4 2x + 3 = x 1 ;

5. 2 + 6. 1 + 7. 8. 4

4

x+6 =

2x + 5 + 2x ;

x+3;

2. x +

2x =

1 + x

x+

1 ; x

x+3= x+

3. (x 1) x + 1 + 2x + 1 = x + 2 ; 4. 1 1 + x + 5 = + 2x + 4 ; 2 x x

x+2+

x+6 = x+4 =

2x + 5 +

2x + 1 ;

x+8+

2x + 3 +

3x

Vn 4 : Phng php nh gi

C s ca phng php ny l chng ta s dng bt ng thc hoc phng php hm s nh gi. Cch 1 : C s nhn dng : (a) Nu hm s y = f (x) ng bin trn (a; b) v hm s y = g(x) nghch bin trn (a; b) th phng trnh f (x) = g(x) nu c nghim th nghim l duy nht. (b) Nu hm s y = f (x) ng bin (hoc nghch bin) trn (a; b) th phng trnh f (x) = c (vi c l hng s) nu c nghim th nghim l duy nht.

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www.VNMATH.comPhng php gii l :

CHUYN LUYN THI I HC


(a) Nhn thy x = x0 l mt nghim ca phng trnh cho. (b) Nu x > x0 , ta suy ra v tri ln hn v phi hoc ngc li. (c) Nu x < x0 , ta suy ra v tri ln hn v phi hoc ngc li. (d) Kt lun phng trnh cho c nghim duy nht x = x0 . Cch 2 : Nu hm s y = f (x) ng bin (hoc nghch bin) trn (a; b) th phng trnh f (u) = f (v) tng ng vi u = v. Cch 3 : Nu hm s y = f (x) tha mn f (x) = 0 c nhiu hn 1 nghim th chng ta lp bng bin thin suy ra phng trnh c ti ta bao nhiu nghim, ri nhm s nghim , dn n l tt c cc nghim ca phng trnh. Cch 4 : Nu f (x) c v g(x) c th phng trnh f (x) = g(x) tng ng vi f (x) = c g(x) = c.

Bi 1.62 : Gii cc phng trnh sau : 1. x=3; 2. x + 3 + x + x + 8 = 4 ; 3. x2 x + 1 + x2 + 7x + 1 = 4 x ; x+3+ 3 4. x+3 + 2x 1 = 2 ; 1+ 2x x2 x + 4 + 2x 1 = 5 ;

5.

Vn 5 : Phng trnh, bt phng trnh c tham s

1. S dng phng trnh, bt phng trnh c bn; 2. S dng t n ph, v t iu kin "cht" cho n; 3. S dng iu kin c nghim ca phng trnh bc hai; 4. S dng phng php hm s ch ra iu kin c nghim.

Bi 1.63 : Tm iu kin ca m phng trnh 1. c nghim thc ;

x2 + 2x m = 2x 1 : 3. c hai nghim thc phn bit.

2. c ng mt nghim thc ; 1 x+ + 2

Bi 1.64 : Tm iu kin ca m phng trnh x + Bi 1.65 : Tm iu kin ca m phng trnh

x+ m

1 = m c nghim thc. 4 4 = 0 c nghim thc.WWW.VNMATH.COM

16 x2

16 x2

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CHUYN LUYN THI I HC


x1 x+2 m + 2 = 0 c nghim thc. Bi 1.66 : Tm iu kin ca m phng trnh x+2 x1 4 Bi 1.67 : Tm iu kin ca m phng trnh x + 1 m x 1 + 2 x2 1 = 0 c nghim thc. Bi 1.68 : Tm iu kin ca m phng trnh x2 2x 3 = x + m

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www.VNMATH.com1. c nghim thc ;

CHUYN LUYN THI I HC


2. c hai nghim thc phn bit.

Bi 1.69 : Bin lun theo m s nghim thc ca phng trnh x + 1 + 1 x = m. Bi 1.70 : Tm iu kin m phng trnh x + 9 x = x2 + 9x + m c nghim thc. Bi 1.71 : Tm iu kin m phng trnh x + 4 x 4 + x + x 4 = m c nghim thc. x+m Bi 1.72 : Tm iu kin m phng trnh x + 6 x 9 + x 6 x 9 = c nghim thc. 6 4 Bi 1.73 : Tm m phng trnh x4 + 4x + m + x4 + 4x + m = 6 c nghim thc. 3 Bi 1.74 : Tm iu kin ca m phng trnh 1 x2 + 2 1 x2 = m : 1. c nghim thc duy nht ; 2. c nghim thc.

3x2 1 = 2x 1 + mx lun c nghim thc vi mi gi tr ca m. Bi 1.75 : Chng t rng phng trnh 2x 1 x+1 = m c nghim thc. Bi 1.76 : Tm m phng trnh (x 3)(x + 1) + 4(x 3) x3 3 3 Bi 1.77 : Tm m phng trnh 1 x + 1 + x = m c nghim thc. Bi 1.78 : Bin lun theo m s nghim thc ca phng trnh m x2 + 2 = x + m. Bi 1.79 : Tm m phng trnh x2 2x 3 = mx + m c nghim thc x 1. Bi 1.80 : Tm m phng trnh sau c nghim thc : Bi 1.81 : Tm m phng trnh x+ 1 x + 2m x(1 x) 24

x(1 x) = m.

x+

x2 x + 1 = m c nghim thc. 4

Bi 1.82 : Tm m cc phng trnh sau c nghim thc : 1. x2 + x + 1 x2 x + 1 = m ; 2. x2 + 1 x = m.

Bi 1.83 : Tm m phng trnh sau c nghim thc : x x + x + 12 = m 5x+ 4x . Bi 1.84 : Tm m phng trnh sau c nghim thc : m 4 4 x 2 + 2 x2 4 x + 2 = 2 x2 4.

Bi 1.85 : Tm m phng trnh sau c nghim thc : (4m 3) x + 3 + (3m 4) 1 x + m 1 = 0. Bi 1.86 : Tm m phng trnh sau c 4 nghim thc phn bit : m 1 + x2 1 x2 + 2 = 2 1 x4 + 1 + x2 1 x2 .

Bi 1.87 : Tm m phng trnh sau c nghim duy nht : x2 2x = mx + 1.WWW.VNMATH.COM

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CHUYN LUYN THI I HC


Bi 1.88 : Tm m phng trnh sau c nghim : x+ Bi 1.89 : Cho phng trnh : x2 + 2x + 4 1. Tm m phng trnh c nghim. 2. Tm m phng trnh c ng hai nghim phn bit. Bi 1.90 : Tm m phng trnh sau c nghim : Bi 1.91 : Cho phng trnh : x+1+ 3x (x + 1)(3 x) = m. (3 x)(x + 1) = m 3. 1 x2 = m.

|x + 1| + m|x 1| = (m + 1) x2 1.

1. Gii phng trnh khi m = 2 ; 2. Tm m phng trnh trn c nghim. Bi 1.92 : Tm m cc bt phng trnh sau c nghim : 1. 4x+ x+5 m; 2. mx x 3 m + 1.

Bi 1.93 : Tm m bt phng trnh m c nghim trong on 0; 1 +

3 .

x2 2x + 2 + 1 + x(2 x) 0

Bi 1.94 : Tm m bt phng trnh

(4 + x)(6 x) x2 2x + m nghim ng vi mi x [4; 6].

1.4 H phng trnh1.4.1 Phng php thBi 1.95 : Gii cc h phng trnh sau : x2 (y + 1)(x + y + 1) = 3x2 4x + 1 xy + x + 1 = x2 x 1 1 =y x y

1.

4.

2y = x3 + 1

2.

x3 y = 16 3x + y = 8

5.

x+y = xy =

3 3

x+y x y 12

3.

y(1 + x2 ) = x(1 + y2 ) x2 + 3y2 = 1

6.

x+y

xy =2 x2 y2 = 4 Trang 23

x2 + y2 +

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www.VNMATH.com7. x3 8x = y3 + 2y x2 3 = 3(y2 + 1) 8. |x2 2x| + y = 1 x2 + |y| = 1 9. x2 + y2 + x+y= 2xy =1 x+y x2 y 2x + y = 5

CHUYN LUYN THI I HC 11. 1 x+y 1 7y 1 x+y 3x 1 + x3 + 3xy2 = 49 =2


=4 2

12.

x2 8xy + y2 = 8y 17x 13. y( x + x + 3) = 3 x+ y= x+1 x+ xy + 1+ 1 = y

10.

7x + y +

14.

2x + y + x y = 2

y+1+

x y

1x=1

1.4.2 Phng php phn tch thnh nhn t hoc coi mt phng trnh l phng trnh bc hai (ba) theo mt nBi 1.96 : Gii cc h phng trnh sau : x3 + 3y2 x = 4 y3 + 3x2 y = 4; x2 + y + 1 = 0 x + y2 + 1 = 0; Bi 1.97 : Gii cc h phng trnh sau : x2 = 3x + 2y y2 = 3y + 2x 2. x2 2y2 = 2x + y y2 2x2 = 2y + x 3. x3 = 2x + y y3 = 2y + x 4. xy + x + y = x2 2y2 x 2y y x 1 = 2x 2y 8. 7. 6. y2 = (5x + 4)(4 x) y2 5x2 4xy + 16x 8y + 16 = 0 x3 + 1 = 2y y3 + 1 = 2x x+y+ xy=1+ x+ y=1 x2 y + 2x + 3y = 6 3xy + x + y = 5 x2 y2 3x3 = x2 + 2y2 3y3 = y2 + 2x2 ; x3 = 3x + 8y y3 = 3y + 8x; 4y x 4x y 3x = ; y x 3y = x3 = 5x + y y3 = 5y + x.

1.

3.

5.

2.

4.

6.

1.

5.

1.4.3 Phng php t n phBi 1.98 : Gii cc h phng trnh sau : x + xy + y = 11 x xy + y = 1;

1.

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www.VNMATH.comx2 y + xy2 = 20 1 1 5 + = ; x y 4 x2 + y2 = 2x2 y2 x + y + 1 = 3xy; x y + xy = 1 x2 + y2 = 2; x2 + y2 + x2 y2 = 1 + 2xy (x y)(1 + xy) = 1 xy;

CHUYN LUYN THI I HCx y 26 + = y x 5 x2 y2 = 24;

www.luyenthi24h.com www.luyenthi24h.com www.VNMATH.comx y + =4 y x x2 y2 x+y+ + = 4; y x x+y+ x + y + x2 y2 = 3xy 1 1 + xy = 1; x y x2 + y2 + xy = 3x2 y2 x2 + y2 xy = x2 y2 ;

2.

6.

9.

3.

7.

x2 + y2 + xy = 3 xy3 + yx3 = 2;

10.

4.

11. x y + =2 y x 1 1 + + x + y = 4; x y

5.

8.

12.

x + xy + y = 7 x2 + xy + y2 = 13;

Bi 1.99 : Gii cc h phng trnh sau : x + y + xy = 5 x2 + y2 + xy = 7 x y 13 + = y x 6 x+y =5 3. x2 + xy + y2 = 1 x y xy = 3 4. x2 + 1 + y(x + y) = 4y (x2 + 1)(y + x 2) = y Bi 1.100 : Gii cc h phng trnh sau : x2 + x + y + 1 + x +

1.

5.

4xy + 4(x2 + y2 ) + 2x + 1 =3 x+y

3 =7 (x + y)2

9.

x2 + y2 = 1 x+y+ xy =2 x2 + xy + y2 = 19(x y)2 x2 xy + y2 = 7(x y)

2.

6.

x2 3xy + y2 = 1 3x2 xy + 3y2 = 13

10.

7.

2x2 4xy + y2 = 1 3x2 + 2xy + 2y2 = 7

11.

x y + y x = 30 x x + y y = 35 x2 + y2 + 2xy = 8 2 x+ y =4

8.

y2 3xy = 4 x2 4xy + y2 = 1

12.

1.

y2 + x + y + 1 + y = 18 y2 + x + y + 1 y = 2;

6.

x+

x2 + x + y + 1 x +

y=4 x + 5 + y + 5 = 6; xy = 7

2.

x y 7 + = +1 y x xy x xy + y xy = 78; x2 + y2 + 2xy = 8 2 x + y = 4; x+y+ xy =4

7.

x+y

x2 + y2 + xy = 133; 8. (x y)(x2 y2 ) = 7 (x + y)(x2 + y2 ) = 175; 9. x x + y y = 2 xy x + y = 2; x+y+ xy =1+ x + y = 1;WWW.VNMATH.COM

3.

4.

x2 + y2 = 128; 5. x+ y=1 10.

x2 y2 Trang 25

|x| + |y| = 1;

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CHUYN LUYN THI I HC17.


11.

x + y

y 5 = x 2

3 + 2x2 y x4 y2 + x4 (2 2x2 ) = y4 1+ 1 + (x y)2 = x3 (x3 x + 2y2 );

x + y = 10 12. x y + y x = 30 x x + y y = 35; 2(x + y) = 3 3 x+ 3y = 63

18.3

x+

y = 10 x + 6 + y + 6 = 14 x2 y2

13.

x2 y +

xy2 x+ 19. x x2 y2 x 5 + 3x = y 30 6y x+y+ y

=

9x 5

14.

6x x+y 5 + = x+y 6x 2 x + y xy = 9; 7 x y + = y x 2 + xy x xy + y xy = 7; 5 3 y + 42x 5 3+ y + 42x 2y = 4 x=2

15.

20.

x2 y2 = 12

x2 y2 = 12. 20y = x+y+ xy x 16x = x+y xy 5y

16.

21.

Bi 1.101 : Gii cc h phng trnh sau : 2x + x2 y2 = 3 x2 + y2 = 1 2. 1 2 2x3 + 6y2 x = 1 x2 + y2 = x3 + 3y2 x = y x2 + 3y2 = 1 4. xy 1 3x = 1 xy 3x x+y 1 2y = 1 + xy 2y (x + y)(1 + xy) = 18xy (x2 + y2 )(1 + x2 y2 ) = 208x2 y2 11. 9. (x + y) 1 + (x2 + y2 ) 1 xy =62

1.

7.

y(x2 + 1) = 2x(y2 + 1) 1 (x2 + y2 ) 1 + 2 2 = 24 x y (x + y) 1 + 1 =4 xy + xy 1 xy =5

12.

2x + y +

1 =4 x 1 x2 + xy + = 3 x x2 y + 2y + x = 4xy 1 1 x + + =3 2 x xy y

8.

13.

3.

1 1+ xy

= 182

14.

x2 y + y = 2 1 x2 + 2 + x2 y2 = 3; x x2 + y2 + x + y = 4xy 1 1 y x + + 2 + 2 = 4; x y x y 2y(x2 y2 ) = 3x x(x2 + y2 ) = 10y.

10.

(x2 + y2 ) 1 +

5.

1 xy 3 + y3 ) 1 + 1 (x xy

=93

= 27

15.

6.

(x + y) 1 +

1 =4 xy 1 x2 + y2 xy + + =4 xy xy

x y + (x + y) = 15 y x x2 y2 + (x2 + y2 ) = 85 y2 x2

16.

Bi 1.102 : Gii cc h phng trnh : x2 + y2 3x + 4y = 1 3x2 2y2 9x 8y = 3WWW.VNMATH.COM

1.

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Trang 26

www.VNMATH.com1 xy 1 (x y) 2 + xy (x + y) 2 9 2 5 = 2 =

CHUYN LUYN THI I HC3. x xy y = 1 x2 y xy2 = 6 4. x(x + 2)(2x + y) = 9 x2 + 4x + y = 6

www.luyenthi24h.com www.luyenthi24h.com www.VNMATH.comy + xy2 = 6x2 1 + x2 y2 = 5x2 ;

2.

5.

6.

1 + x3 y3 = 19x3 y + xy2 = 6x2 .

Bi 1.103 : Cho h phng trnh :

x + xy + y = a + 1

x2 y + xy2 = a. Tm a h c t nht mt nghim (x; y) tha mn : x > 0 v y > 0. Bi 1.104 : Cho h phng trnh : x+1+ y+1 =3 x y + 1 + y x + 1 + y + 1 + x + 1 = m. 1. Gii h phng trnh vi m = 6. 2. Tm m h phng trnh trn c nghim.

1.4.4 Phng php hm sBi 1.105 : Gii cc h phng trnh sau : x3 5x = y3 5y x8 + y4 = 1 x+ y+ x2 = y2 = x2 2x + 2 = 3y1 + 1 y2 2y + 2 = 3x1 + 1 y 1 + 2x 1 x 1 + 2y 1 6. x+1+ y+1+ x+ 7y=4 ex ey = x y x log2 + log 2 4y3 = 10 2 ln(1 + x) ln(1 + y) = x y 2x2 5xy + y2 = 0 9. x+ y+ 2y= 2 2.

1.

4.

7x=4

7.

2.

5.

x+3=3 y y+ y+3 =3 x x3 3x = y3 3y x6 + y6 = 1

8.

3.

2x=

Bi 1.106 : Tm m h phng trnh sau c nghim : x+1+ y+1+ 3y=m

3x=m

Bi 1.107 : Chng minh rng vi mi m > 0, h phng trnh sau c nghim duy nht : 3x2 y 2y2 m = 0 3y2 x 2x2 m = 0

1.4.5 Phng php nh giBi 1.108 : Gii cc h phng trnh sau :

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www.VNMATH.com1. x+ y+ x+ y+1=1

CHUYN LUYN THI I HC1 1 + =4 x y 1 1 x2 + y2 + 2 + 2 = 4 x y x+y+ x2 + 2y2 = 3 x2 (y2 + 1) = 4 x3 y3 = 7 xy(x y) = 2 7. 3 4

www.luyenthi24h.com www.luyenthi24h.com www.VNMATH.comx+ x+ 3 4 y=1 y=1

y+

x+1=1 = x2 + y = y2 + x

4.

2.

x+ 3 y+3

2xy x2 2x + 9 2xy

5.

8.

x+ y+ 4

y2 2y + 9

2 y2 = 2 2 x2 = 2; 4 32 x y2 = 3 32 x + 6y = 24.

3.

y = x3 + 3x + 4 x = 2y3 6y 2

6.

9.

x+ x+

1.5 S nghim ca phng trnh, h phng trnhBi ton : Chng minh rng phng trnh f (x) = 0 c ng k nghim thc phn bit trong min D.1

Vn 1 : Chng minh phng trnh c nghim duy nht

Cch 1 : Lp bng bin thin ca hm s y = f (x) vi x D (tnh y cc gi tr ti u v cui mi tn), t suy ra c s nghim ca phng trnh. Cch 2 : Da vo hai nh l : nh l 1 : Nu hm s y = f (x) lun ng bin hoc nghch bin trn (a; b) th phng trnh f (x) = 0 c ti a mt nghim trong khong (a; b). nh l 2 : Nu hm s y = f (x) lin tc trn [a; b] v f (a). f (b) < 0 th phng trnh f (x) = 0 c t nht mt nghim trong khong (a; b).

Bi 1.109 : Chng minh rng cc phng trnh sau c nghim duy nht : 1. x5 + x4 + 2x3 + 2x2 + x + 1 = 0; Bi 1.110 : Chng minh rng phng trnh : x3 + 2. ex (x2 + 1) 4 = 0. x 1 = 0 c nghim duy nht.

Bi 1.111 : Chng minh rng phng trnh xx+1 = (x + 1) x c mt nghim dng duy nht. Bi 1.112 : Chng minh rng vi mi a > 0, h phng trnh sau c nghim duy nht : ex ey = ln(1 + x) ln(1 + y) yx=a

Vn 2 : Chng minh phng trnh c ng hai nghim phn bit

1

Nu k = 0 tc l phng trnh v nghim

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CHUYN LUYN THI I HC


Cch 1 : Lp bng bin thin ca hm s y = f (x) vi x D (tnh y cc gi tr ti u v cui mi tn), t suy ra c s nghim ca phng trnh. Cch 2 : Ch lp bng bin thin nhng khng tnh c ht tt c cc u mt (lc ny y=0 c nghim duy nht). T suy ra c phng trnh c ti a 2 nghim. Kt hp vi nh l 1 ta cng ch ra c phng trnh c t nht 2 nghim.

Bi 1.113 : Chng minh rng cc phng trnh sau c ng hai nghim thc phn bit : 1. x4 x2 2x 1 = 0; 2. x4 3x3 1 = 0; 3. 3x4 4x3 6x2 + 12x 20 = 0; 4. x3 2x x + 1 = 0.

Vn 3 : Chng minh phng trnh c ng ba nghim phn bit

Cch 1 : Lp bng bin thin ca hm s y = f (x) vi x D (tnh y cc gi tr ti u v cui mi tn), t suy ra c s nghim ca phng trnh. Cch 2 : Ch lp bng bin thin nhng khng tnh c ht tt c cc u mt (lc ny y=0 c ng 2 nghim). T suy ra c phng trnh c ti a 3 nghim. Kt hp vi nh l 1 ta cng ch ra c phng trnh c t nht 3 nghim.

Bi 1.114 : Chng minh rng cc phng trnh sau c ng ba nghim thc phn bit : 1. sin x x = 0; 2 2. 4x (4x2 + 1) = 1.

1.6 Phng trnh, bt phng trnh, h i s trong cc k thi tuyn sinh HBi 1.115 (C08) : Tm gi tr ca tham s m h phng trnh Bi 1.116 (C09) : Gii bt phng trnh Bi 1.117 (C10) : Gii h phng trnh x my = 1 mx + y = 3 x + 1 + 2 x 2 5x + 1. 2 2x + y = 3 2x y (x, y R). x2 2xy y2 = 2 1 1 =y x y c nghim (x; y) tha mn xy < 0.

Bi 1.118 (A03) : Gii h phng trnh :

x

2y = x3 + 1. Bi 1.119 (A04) : Gii bt phng trnh : 2(x2 16) 7x + x3 > . x3 x3WWW.VNMATH.COM

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Bi 1.120 (A05) : Gii bt phng trnh : Bi 1.121 (A06) : Bi 1.122 (A07) : Bi 1.123 (A08) : Bi 1.124 (A08) :

5x 1 x 1 > 2x 4. x + y xy = 3 (x, y R). Gii h phng trnh : x+1+ y+1 =4 4 Tm m phng trnh sau c nghim thc : 3 x 1 + m x + 1 = 2 x2 1. 5 x2 + y + x3 y + xy2 + xy = 4 (x, y R). Gii h phng trnh : 4 + y2 + xy(1 + 2x) = 5 x 4 Tm cc gi tr ca tham s m phng trnh sau c ng hai nghim thc phn bit : 4 2x + 4 2x + 2 6 x + 2 6 x = m (m R).

3 Bi 1.125 (A09) : Gii phng trnh 2 3x 2 + 3 6 5x 8 = 0. x x 1. Bi 1.126 (A10) : Gii bt phng trnh 1 2(x2 x + 1) (4x2 + 1)x + (y 3) 5 2y = 0 Bi 1.127 (A10) : Gii h phng trnh 4x2 + y2 + 2 3 4x = 7 Bi 1.128 (B02) : Gii h phng trnh : xy= xy x + y = x + y + 2. 3y = y2 + 2 x2 x2 + 2 3x = . y2 3

(x, y R).

Bi 1.129 (B03) : Gii h phng trnh :

Bi 1.130 (B04) : Xc nh m phng trnh sau c nghim : m 1 + x2 1 x2 + 2 = 2 1 x4 + 1 + x2 1 x2 . x2 + mx + 2 = 2x + 1.

Bi 1.131 (B06) : Tm m phng trnh sau c hai nghim thc phn bit :

Bi 1.132 (B07) : Chng minh rng vi mi gi tr dng ca tham s m, phng trnh sau c hai nghim thc phn bit : x2 + 2x 8 = Bi 1.133 (B08) : Gii h phng trnh : m(x 2). (x, y R).

x4 + 2x3 y + x2 y2 = 2x + 9 x2 + 2xy = 6x + 6

Bi 1.134 (B09) : Gii h phng trnh

xy + x + 1 = 7y

x2 y2 + xy + 1 = 13y2 . Bi 1.135 (B10) : Gii phng trnh 3x + 1 6 x + 3x2 14x 8 = 0 (x R). Bi 1.136 (D02) : Gii bt phng trnh : (x2 3x) 2x2 3x 2 0. Bi 1.137 (D02) : Gii h phng trnh : 23x = 5y2 4y 4x + 2x+1 = y. 2x + 2 x+ y =1 x x + y y = 1 3m.WWW.VNMATH.COM

Bi 1.138 (D04) : Tm m h phng trnh sau c nghim :

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Bi 1.139 (D04) : Chng minh rng phng trnh sau c ng mt nghim : x5 x2 2x 1 = 0. Bi 1.140 (D05) : Gii phng trnh : 2 x + 2 + 2 x + 1 x + 1 = 4. Bi 1.141 (D06) : Gii phng trnh : 2x 1 + x2 3x + 1 = 0 (x R).

Bi 1.142 (D07) : Tm cc gi tr ca tham s m h phng trnh sau c nghim thc : 1 1 +y+ =5 x y 3 + 1 + y3 + 1 = 15m 10. x x3 y3 x+ Bi 1.143 (D08) : Gii h phng trnh : xy + x + y = x2 2y2 x 2y y x 1 = 2x 2y x(x + y + 1) 3 = 0 5 (x + y)2 2 + 1 = 0. x (x, y R).

Bi 1.144 (D09) : Gii h phng trnh

1.7 Bi tp tng hp x + 4 + x 4 = 2x 12 + 2 x2 16. Bi 1.146 : Gii bt phng trnh : x + 12 x 3 + 2x + 1. Bi 1.145 : Gii phng trnh : Bi 1.147 : Gii h phng trnh : x2 + y2 + x + y = 4 x(x + y + 1) + y(y + 1) = 2. Bi 1.148 : Gii h phng trnh : 2x + y + 1 x+y=1

3x + 2y = 4. Bi 1.149 : Gii bt phng trnh : 8x2 6x + 1 4x + 1 0. Bi 1.150 : Gii bt phung trnh : 2x + 7 5 x 3x 2. Bi 1.151 : Tm m h phng trnh sau c nghim : 72x+ x+1

72+

x+1

+ 2005x 2005

x2 (m + 2)x + 2m + 3 0. Bi 1.152 : Gii h phng trnh : (x2 + 1) + y(y + x) = 4y (x2 + 1)(y + x 2) = y Bi 1.153 : Gii h phng trnh : x3 8x = y3 + 2y x2 3 = 3(y2 + 1) Bi 1.154 : Gii h phng trnh : (x y)(x2 + y2 ) = 13 (x, y R).

(x, y R).

(x, y R). (x + y)(x2 y2 ) = 25 Bi 1.155 : Gii phng trnh : 3x 2 + x 1 = 4x 9 + 2 3x2 5x + 2, x R. Bi 1.156 : Gii h phng trnh : x2 xy + y2 = 3(x y) x2 + xy + y2 = 7(x y)3 (x, y R).WWW.VNMATH.COM

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Bi 1.157 : Gii phng trnh : x + 2 7 x = 2 x 1 + x2 + 8x 7 + 1, x R. Bi 1.158 : Tm m phng trnh :

m c nghim thuc on 0; 1 + 3 .

x2 2x + 2 + 1 + x(2 x) 0

Bi 1.159 : Gii h phng trnh :

x4 x3 y + x2 y2 = 1

x3 y x2 + xy = 1. 4 Bi 1.160 : Tm m phng trnh : x2 + 1 x = m c nghim. 4 Bi 1.161 : Tm m phng trnh : x4 13x + m + x 1 = 0 c ng mt nghim. Bi 1.162 : Tm m phng trnh : x 3 2 x 4 + x 6 x 4 + 5 = m c ng hai nghim thc. Bi 1.163 : Tm m h phng trnh : 2x y m = 0 x + xy = 1 c nghim duy nht.

Bi 1.164 : Vi gi tr no ca a th h c t nht mt nghim tha mn x, y > 0. Vi cc gi tr a tm c hy tm tt c cc nghim ca h cho : 1 1 + =4 x y 1 a2 + 1 2 + y2 + 1 + 1 = 2+ x 2a 2 2 + . x2 y2 a a x+y+ Bi 1.165 : Gii h phng trnh : y3 + y2 x + 3x 6y = 0 x2 + xy = 3. Bi 1.166 : Cho h phng trnh : x2 + y2 = m x + y = 6. 1. Gii h phng trnh vi m = 26 ; 2. Tm m h v nghim ; Bi 1.167 : Cho h phng trnh : x + xy + y = m + 2 x2 y + xy2 = m + 1. 1. Gii h phng trnh vi m = 3 ; Bi 1.168 : Cho h phng trnh : (x 2)2 + y2 = m x2 + (y 2)2 = m. Bi 1.169 : Cho h phng trnh : x = y2 y + m y = x2 x + m. 1. Gii h phng trnh vi m = 0 ; 2. Tm m h phng trnh c nghim ; 3. Tm m h phng trnh c nghim duy nht. 2. Xc nh m h c nghim duy nht. Tm m h c nghim duy nht. 3. Tm m h c nghim duy nht ; 4. Tm m h hai nghim phn bit.

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CHUYN LUYN THI I HC2|x| + |x| = y + x2 + a x2 + y2 = 1.


Bi 1.170 : Tm a h sau c nghim duy nht :

Bi 1.171 : Tm a h sau c nghim :

Bi 1.172 : Tm m phng trnh sau c nghim duy nht : 4x+

1a 1+a 2 + 10xy 5y2 2. 3x x2 + 2xy 7y2

x + 5 = m.

Bi 1.173 : Tm m phng trnh sau c nghim duy nht : 4 Bi 1.174 : Tm a h sau c nghim : x2 2xy 3y2 = 8 2x2 + 4xy + 5y2 = a4 4a3 + 4a2 12 + Bi 1.175 : Cho phng trnh x + 17 x2 + x 17 x2 = m. x+ 4 1x+ x+ 1 x = m.

105.

1. Gii phng trnh khi m = 9; 2. Tm m phng trnh c nghim thc; 3. Tm m phng trnh c nghim thc duy nht. x2 3 6 0. x Bi 1.177 : Chng t rng vi mi s m khng m th phng trnh sau lun c nghim thc Bi 1.176 : Gii bt phng trnh 2x2 5x 3x 3x2 + (3m2 7) x2 + 4 m3 + 6 = 0. x2 + 2 + x2 + 2 + y2 + 3 + x + y = 52

Bi 1.178 : Gii h phng trnh

+ 3 x y = 2.


x2 + y3 = 2y2

x + y3 = 2y. Bi 1.180 : Gii bt phng trnh 2 x 1 x + 2 > x 2. Bi 1.181 : Gii bt phng trnh 3x + 7 2x + 3 > x + 2. Bi 1.182 : Gii h phng trnh 2x2 + x + y2 = 7 (x + 3) 2x 1 + (y + 3) 2y 1 = 2 (x + 3)(y + 3) x + y = 2xy. Bi 1.184 : Gii h phng trnh 3x y =3 x2 + y2 x + 3y y 2 = 0. x + y2 x+WWW.VNMATH.COM

xy x + y = 3.


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CHUYN LUYN THI I HC


(x + 2)(2x 1) 3 x + 6 = 4 (x + 6)(2x 1) + 3 x + 2. xy x+y =2 Bi 1.186 : Gii h phng trnh x2 + y2 + x2 y2 = 4. Bi 1.187 : Gii h phng trnh x2 + xy + y2 = 7(x y)2 x2 xy + y2 = 3(x y). Bi 1.188 : Gii h phng trnh x+y+ y Bi 1.189 : Gii h phng trnh x2 y2 = 12

x2 y2 = 12.

(2x + 1)2 + y2 + y = 2x + 3

xy + x = 1. Bi 1.190 : Gii phng trnh (x2 + 1)2 = 5 x 2x2 + 4. Bi 1.191 : Gii h phng trnh x3 y3 + 2 = 0 x2 + y2 + x y = 0. Bi 1.192 : Gii phng trnh |x + 1 x2 | = 2(1 2x2 ). Bi 1.193 : Gii h phng trnh x2 + 6y = y + 3 x + y + x y = 4.

Bi 1.194 : Tm m phng trnh sau c nghim thc

x3 + x2 + x m(x2 + 1)2 = 0. Bi 1.195 : Gii bt phng trnh Bi 1.196 : Tm m phng trnh Bi 1.197 : Gii h phng trnh 1 2x2 2x2 + 3x 5

>

2x + y

mx + 13 = x 2 c nghim thc. 2y =3 x

1 . 2x 1

x y + xy = 3. Bi 1.198 : Gii h phng trnh x+1+ x+6+ y1=4 y + 4 = 6.

Bi 1.199 : Tm cc gi tr ca tham s m h phng trnh sau c nghim thc x2 + y2 + 2(x + y) = 2 xy(x + 2)(y + 2) = 2m (2m+1 1). Bi 1.200 : Chng minh rng vi mi m 2010 h phng trnh sau c khng qu mt nghim thc Bi 1.201 : Gii phng trnh x + 27 y + 27 y + 1 = (m 2010)y + 1 x + 1 = (m 2010)x + 1.

x + 1 + 1 = 4x2 +

3x.


x3 y(1 + y) + x2 y2 (2 + y) + xy3 30 = 0 x2 y + x(1 + y + y2 ) + y 11 = 0. Trang 34

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www.VNMATH.comBi 1.203 : Gii h phng trnh

CHUYN LUYN THI I HCx3 + 4y = y3 + 16x 1 + y2 = 5(1 + x2 ).



2 + 6y = x+

x x 2y y

x 2y = x + 3y 2. Bi 1.205 : Gii bt phng trnh x 2 x x2 x 2 2 x. Bi 1.206 : Gii h phng trnh x3 + y3 = 1 x2 y + 2xy2 + y3 = 2. Bi 1.207 : Tm cc gi tr ca tham s m bt phng trnh x(4 x) + m nghim ng vi mi gi tr ca x [2; 2 + Bi 1.208 : Gii h phng trnh 3]. x2 4x + 5 + 2 0

2x2 y + y3 = 2x4 + x6 (x + 2) y + 1 = (x + 1)2 . x 2y xy = 0 x 1 + 4y 1 = 2. x x8 y = x+y y x y = 5.



Bi 1.211 : Tm m h phng trnh

dng.

x3 y3 + 3y2 3x 2 = 0 c nghim thc. x2 + 1 x2 3 2y y2 + m = 0 Bi 1.212 : Gii phng trnh x + 3 + 2x x + 1 = 2x + x2 + 4x + 3. Bi 1.213 : Xc nh cc gi tr ca tham s m phng trnh 2x2 + 2mx + m + 1 = 1 x c ng mt nghim thc x2 + y2 + x2 y2 = 1 + 2xy

Bi 1.214 : Gii h phng trnh Bi 1.215 : Bi 1.216 : Bi 1.217 : Bi 1.218 :

x + x2 y + xy = xy2 + y + 1. Gii phng trnh 2x2 + 3x + 1 2x2 2 = x + 1. 6 Xc nh cc gi tr ca tham s m phng trnh x 1 m x + x3 x2 = 0 c nghim thc. x y 1 + = 3 + x4 3 + y4 8 Tm m h phng trnh sau c nghim thc xy = m. 4 + 3y4 + x4 y4 9 + 3x 1 1 Gii phng trnh + = 2. x 2 x2 x2 + y2 = 5 y 1(x + y 1) = (y 2) x + y. x2 + 1 + y2 + xy = 4y y x+y2 = 2 . x +1WWW.VNMATH.COM



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CHUYN LUYN THI I HC x+ x + 4 m 4 x = 3m c nghim thc.


Bi 1.221 : Tm m phng trnh Bi 1.222 : Gii h phng trnh

x3 y = 24 3 2 x3 + y = 6 3.

x1+ y1=3 Bi 1.223 : Gii h phng trnh x + y (x 1)(y 1) = 5. 4 4 Bi 1.224 : Tm m phng trnh m x 2 + 2 x2 4 x + 2 = 2 x2 4 c nghim. Bi 1.225 : Gii h phng trnh x2 + y2 + x + y = 18 x(x + 1)y(y + 1) = 72. Bi 1.226 : Gii h phng trnh 7x + y + 2x + y = 5

2x + y + 20x + 5y = 38.


xy + x2 = 1 + y xy + y2 = 1 + x. 2 x x2 = x + 1 x c nghim. 3

Bi 1.228 : Tm cc gi tr ca tham s m phng trnh m + 5 1 Bi 1.229 : Gii bt phng trnh 5 x + 2x + + 5. 2x 2 x

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Chng 2

Bt ng thc2.1 Phng php s dng bt ng thc Cauchy2.1.1 Bt ng thc Cauchy - So snh gia tng v tchCho ba s khng m a, b, c, ta c : 1. a+b ab, du bng xy ra khi a = b ; 2 a+b+c 3 2. abc, du bng xy ra khi a = b = c. 3

2.1.2 Mt s h qu trc tipH qu 1 : So snh gia tng nghch o v tng. Cho ba s dng a, b, c c : 1. 1 1 4 + ; a b a+b 2. 1 1 1 9 + + . a b c a+b+c

H qu 2 : So snh gia tng bnh phng v tng. Cho ba s thc a, b, c c : 1. 2(a2 + b2 ) (a + b)2 ; H qu 3 : So snh gia tng, tng bnh phng v tch. Cho ba s thc a, b, c c : 1. (a + b + c)2 3(ab + bc + ca) ; 2. a2 + b2 + c2 ab + bc + ca. 2. 3(a2 + b2 + c2 ) (a + b + c).

2.1.3 Bi tp nghBi 2.1 : Cho a, b, > 0. Chng minh rng : ab(a + b) a+b 2 23

(a + b)(a2 + ab + b2 ) a3 + b3 (a2 + b2 )3 . 6 2 (a + b)3

Bi 2.2 : Cho a, b > 0 v a + b 1. Chng minh rng :

37

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www.VNMATH.com1. 1 1 + 4; a b

CHUYN LUYN THI I HC2. 1 1 + + a + b 5. a b


Bi 2.3 : Cho cc s khng m a, b, c c a + b + c 3. Chng minh rng : 1. a + b + c ab + bc + ca ; 2. a + b + c ab + bc + ca.

Bi 2.4 : Cho x, y > 0. Chng minh rng : (1 + x)(1 + y) (1 + xy)2 . 1 1 Bi 2.5 : Cho x, y > 0. Chng minh rng : x2 + y2 + + 2( x + y). x y 1 1 + . Bi 2.6 : Cho x, y > 0 v x + y = 1. Tm gi tr nh nht ca P = 2 2 x +y xy x y z Bi 2.7 : Cho x, y, z > 0 v x + y + z = 1. Tm gi tr ln nht ca P = + + . x+1 y+1 z+1 b2 1 a2 + . a+1 b+1 3 Bi 2.9 : Cho cc s thc dng a, b, c. Chng minh rng : Bi 2.8 : Cho a, b > 0 v a + b = 1. Chng minh rng : 1 1 1 1 1 1 + + + + . a + 3b b + 3c c + 3a 2a + b + c 2b + c + a 2c + a + b Bi 2.10 : Chng minh rng vi mi a, b, c > 0 u c : 1. 1 1 1 27 + + ; a(b + c) b(c + a) c(a + b) 2(a + b + c)2 2. 1 1 1 27 + + . a(a + b) b(b + c) c(c + a) 2(a + b + c)2

1 . ab a+b ab Bi 2.12 : Cho a, b > 0. Tm gi tr nh nht ca biu thc S = + . ab a + b 3 1 1 1 Bi 2.13 : Cho a, b, c > 0 v a + b + c . Tm gi tr nh nht ca biu thc S = a + b + c + + + . 2 a b c 2 + y2 + z2 2(xy + yz). Bi 2.14 : Chng minh rng vi mi s dng x, y, z u c : x Bi 2.11 : Cho a, b > 0 v a + b 1. Tm gi tr nh nht ca S = ab + Bi 2.15 : Cho a, b, c > 0 v a + b + c = 4. Chng minh rng : ab bc ca + + 1. a + b + 2c b + c + 2a c + a + 2b Bi 2.16 : Cho a, b, c > 0. Chng minh rng : ab bc ca a+b+c + + . a + 3b + 2c b + 3c + 2a c + 3a + 2b 6 Bi 2.17 : Cho a, b, c > 0. Chng minh rng : 1. 2. a+b b+c c+a + + 6; c a b a b c 3 + + ; b+c c+a a+b 2 3. 4. a2 b2 c2 a+b+c + + ; b+c c+a a+b 2 a3 b3 c3 a2 + b2 + c2 + + . b+c c+a a+b 2

Bi 2.18 : Cho a, b, c > 0 v abc = 1. Tm gi tr nh nht ca cc biu thc sau : a2 b2 c2 1. P = + + ; b+c c+a a+b a3 b3 c3 + + ; b+c c+a a+b TRN ANH TUN - 0974 396 391 - (04) 66 515 343 2. Q = a2 a b2 b c2 c 3. R = + + ; b+c c+a a+b bc ca ab 4. S = 2 + 2 + 2 ; 2c 2a a b+a b c+b c a + c2 bWWW.VNMATH.COM

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Bi 2.19 : Cho x, y, z, t > 0 v xyzt = 1. Tm gi tr nh nht ca biu thc : P= x3 (yz 1 1 1 1 + 3 + 3 + 3 . + zt + ty) y (zt + tx + xz) z (tx + xy + yt) t (xy + yz + zx)

Bi 2.20 : Cho a, b, c > 0. Tm gi tr nh nht ca biu thc sau : 1. P = b c a + + . b + 2c c + 2a a + 2b 2. Q = a b c + + , m N, m > 2.1 b + mc c + ma a + mb

Bi 2.21 : Cho a, b, c > 0. Chng minh rng : 1. (a + b)(b + c)(c + a) 8abc ; 2. bc ca ba + + a + b + c. a b c

Bi 2.22 : Cho a, b, c l di ba cnh ca mt tam gic. Chng minh rng : 1. a b c + + 3; b+ca c+ab a+bc 2. a2 b2 c2 + + a + b + c. b+ca c+ab a+bc

Bi 2.23 :

1. Cho a, b, c l di ba cnh ca mt tam gic, p l na chu vi ca tam gic. Chng minh rng : (p a)(p b)(p c) abc . 8

2. Cho tam gic ABC c chu vi bng 3 v di ba cnh ca tam gic l a, b, c. Chng minh rng : 4(a3 + b3 + c3 ) + 15abc 27. Bi 2.24 : Cho a, b, c, d > 0 v a + b + c + d = 1. Chng minh rng : 1 1 1 1 1 1 1 1 81. a b c d Bi 2.25 : Cho a, b 1. Chng minh rng : a b 1 + b a 1 ab. Bi 2.27 : Cho a, b, c > 0. Chng minh rng : Bi 2.28 : Bi 2.29 : Bi 2.30 : Bi 2.31 :

Bi 2.26 : Cho a, b, c 0 v a + b + c = 1. Chng minh rng : ab + bc + ca + abc a2

10 . 27

2 1 1 1 + . + bc 2 ab ac 3 2 Cho a, b > 0 v a + b = 1. Chng minh rng : + 2 16. ab a + b2 1 1 1 1 Cho a, b, c > 0 v + + 2. Chng minh rng : abc . 1+a 1+b 1+c 8 2 + b2 a Cho a > b > 0 v ab = 1. Chng minh rng : 2 2. ab 1 1 Tm gi tr nh nht ca A = (1 + x) 1 + + (1 + y) 1 + vi x, y > 0 tha mn x2 + y2 = 1. y x

Bi 2.32 : Cho x, y, z > 1 tha mn x + y + z = xyz. Tm gi tr nh nht ca : P= Bi 2.33 : Cho a, b, c > 1. Chng minh rng : 3 alogb c + blogc a + cloga b 3 abc.1

y2 z2 x2 + 2 + 2 . x2 y z

Mt cch tng qut, tm gi tr nh nht ca R =

a b c + + vi a, b, c, x, y l nhng s dng xb + yc xc + ya xa + yb

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Bi 2.34 : Cho a, b, c > 0 v a + b + c = 1. Chng minh rng : 1+ 1 a 1+ 1 b 1+2

1 64. c

Bi 2.35 : Cho a, b > 0. Chng minh rng : (a + b)2 + Bi 2.36 : Cho a, b, c > 0. Chng minh rng : a2 b

1 1 + a b

8.

bc ca ab 1 1 1 1 + 2 + 2 + + . 2c 2a 2b +a b c+b c a+c 2 a b c

Bi 2.37 : Cho a, b, c > 0. Chng minh rng : bc ca a+b+c ab + + . a+b b+c c+a 2 1 Bi 2.38 : Cho a 3. Tm gi tr nh nht ca biu thc S = a + . a 1 Bi 2.39 : Cho a 2. Tm gi tr nh nht ca biu thc S = a + 2 . a Bi 2.40 : Cho a, b, c 0 tha mn a2 + b2 + c2 = 1. Tm gi tr nh nht ca biu thc S =a+b+c+ 1 . abc x y + . 1y 1x

Bi 2.41 : Cho x, y > 0 v x + y = 1. Tm gi tr nh nht ca biu thc S = Bi 2.42 : Cho a, b, c 0 v a + b + c = 1. Tm gi tr ln nht ca biu thc S = 3 a+b+ 3 b+c+ 3 c + a.

Bi 2.43 : Cho a, b, c > 0 v a + b + c = 3. Tm gi tr ln nht ca biu thc S =3

a(b + 2c) +

3

b(c + 2a) +

3

c(a + 2b).

Bi 2.44 : Cho a 2; b 6; c 12. Tm gi tr ln nht ca biu thc 3 4 bc a 2 + ca b 6 + ab c 12 S = . abc a b c 2 3 a+b b+c c+a + + + + vi mi a, b, c > 0. b c a 2 c a b Bi 2.46 : Cho a, b, c > 0 v a + b + c = 3. Chng minh rng : Bi 2.45 : Chng minh rng : a3 b3 c3 3 + + . (a + b)(a + c) (b + c)(b + a) (c + a)(c + b) 4 Bi 2.47 : Cho a, b, c > 0 v a + b + c = 3. Chng minh rng : a3 b3 c3 + + 1. b(2c + a) c(2a + b) c(2b + c) Bi 2.48 : Cho a, b, c > 0 v a2 + b2 + c2 = 1. Chng minh rng : a3 b3 c3 1 + + . b + 2c c + 2a a + 2b 3 Bi 2.49 : Cho a, b, c > 0 v a2 + b2 + c2 = 1. Chng minh rng : a3 b3 c3 1 + + . a+b b+c c+a 2

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Bi 2.50 : Cho a, b, c > 0 v ab + bc + ca = 1. Chng minh rng : a 1 + a2 + b 1 + b2 + 3 . 1 + c2 2 c

Bi 2.51 : Cho a, b, c > 0 v ab + bc + ca = 1. Chng minh rng : 1 1 9 1 + + . a(a + b) b(b + c) c(c + a) 2 Bi 2.52 : Cho a, b, c > 0 v a + b + c = 1. Chng minh rng : b c 9 a + + . 2 2 2 (b + c) (c + a) (a + b) 4 Bi 2.53 : Cho a, b, c > 0 v a2 + b2 + c2 = 3. Chng minh rng : Bi 2.54 : Cho a, b, c > 0 v a + b + c = 1. Chng minh rng : ab bc ca + + 3. c a b

ca ab 1 bc + + . a + bc b + ca c + ab 2

Bi 2.55 : Cho a, b, c > 0 v a + b + c = 2. Chng minh rng : ca ab bc + + 1. 2a + bc 2b + ca 2c + ab

Bi 2.56 : Cho a, b, c > 0 v abc = 1. Chng minh rng : a3 b3 c3 3 + + . (1 + b)(1 + c) (1 + c)(1 + a) (1 + a)(1 + b) 4 Bi 2.57 : Cho a, b, c > 0 v abc = 1. Chng minh rng : 1 1 1 3 + + . a3 (b + c) b3 (c + a) c3 (a + b) 2 Bi 2.58 : Cho a, b, c > 0. Chng minh rng : Bi 2.59 : Bi 2.60 : Bi 2.61 : Bi 2.62 : 1 1 1 1 1 1 + + 2 + + . a b c a+b b+c c+a 1 1 1 Cho a, b, c > 0 v a + b + c 1. Chng minh rng : 2 + + 9. a + 2bc b2 + 2ca c2 + 2ab 1 1 Cho a, b > 0 v a + b 1. Chng minh rng : 2 + 6. a + b2 ab 1 1 Cho a, b > 0 v a + b 1. Chng minh rng : 2 + + 4ab 7. 2 a +b ab Cho a, b, c > 0 v ab + bc + ca = abc. Chng minh rng : 1 1 1 3 + + < . a + 2b + 3c b + 2c + 3a c + 2a + 3b 16 a b c + + vi a, b, c > 0 v a + b + c = 1. 1+ba 1+cb 1+ac Bi 2.64 : Cho x, y, z > 0 v x2 + y2 + z2 = 1. Tm gi tr nh nht ca biu thc : Bi 2.63 : Tm gi tr nh nht ca : A = P= y2 x y z + 2 + 2 . 2 2 +z z +x x + y2

Bi 2.65 : Cho x, y l hai s thc thay i. Tm gi tr ln nht v gi tr nh nht ca biu thc : P= (x + y)(1 xy) . (1 + x2 )2 (1 + y2 )2WWW.VNMATH.COM

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Bi 2.66 : Cho x, y, z l ba s thc tha mn x + y + z = 0. Tm gi tr nh nht ca P= 2x + 3 + 2y + 3 + 2z + 3.

Bi 2.68 : Cho 0 < a b c d e v a + b + c + d + e = 1. Chng minh rng : a(bc + be + cd + de) + cd(b + e a) 1 . 25

Bi 2.67 : Cho cc s thc x, y, z tha mn x + y + z = 6. Chng minh rng : 8x + 8y + 8z 4x+1 + 4y+1 + 4z+1 .

Bi 2.69 : Cho a, b, c l ba s dng tha mn iu kin ab + bc + ca = abc. Chng minh rng : a2 b2 c2 a+b+c + + . a + bc b + ca c + ab 4 Bi 2.70 : Cho a, b, c l cc s thc dng, chng minh rng : b+c a+3

4(b3

+

c3 )

+

c+a b+3

4(c3

+

a3 )

+

a+b c+3

4(a3 + b3 )

2.

Bi 2.71 : Cho a, b, c l cc s thc dng, chng minh rng : 1 1 1 1 + + . a3 + b3 + abc b3 + c3 + abc c3 + a3 + abc abc Bi 2.72 : Cho a, b, c l cc s thc dng tha mn abc = 1. Chng minh rng : a3 + b3 b3 + c3 c3 + a3 + 2 + 2 2. a2 + ab + b2 b + bc + c2 c + ca + a2 Bi 2.73 : Cho ba s thc dng a, b, c. Chng minh rng : 2 a 2 b 2 c 1 1 1 + 3 + 3 2 + 2 + 2. 3 + b2 2 2 a b +c c +a a b c Bi 2.74 : Cho a, b, c > 0. Chng minh rng : 1 1 1 a+b+c + + . a2 + bc b2 + ca c2 + ab 2abc Bi 2.75 : Cho a, b, c l ba s dng sao cho ab + bc + ca 1. Chng minh rng : a3 b3 c3 3 + + . b2 + 1 c2 + 1 a2 + 1 4

2.2 Bt ng thc hnh hcBi 2.76 : Cho a, b, c R. Chng minh rng : a2 + b2 + 4c2 + 4ac + a2 + b2 + 4c2 4ac 2 a2 + b2 .

Bi 2.77 : Vi mi a, b, c, d R. Chng minh rng : a2 + b2 + c2 + d2 + 2ac + 2bd

a2 + b2 +

c2 + d2 .

Bi 2.78 : Cho x, y, z > 0. Chng minh rng : x+2 y+3 z 14(x + y + z).WWW.VNMATH.COM

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Bi 2.79 : Cho bn s a, b, c, d R tha mn a2 + b2 = 1 v c + d = 3. Chng minh rng : 9+6 2 . ac + bd + cd 4

Bi 2.80 : Vi mi a, b, c R. Chng minh rng : a2 + ab + b2 + a2 + ac + c2 b2 + bc + c2 .

Bi 2.81 : Vi mi x, y R. Chng minh rng : 4 cos2 x cos2 y + sin2 (x y) + Bi 2.82 : Vi mi x, y R. Chng minh rng : 4x2 + y2 + 12x + 9 + Bi 2.83 : Cho a + b + c = 1, ax + by + cz = 4 vi a, b, c 9a2 + a2 x2 + 4x2 + y2 4x 6y + 10 5. 0. Chng minh rng : 9c2 + c2 z2 5. 4 sin2 x sin2 y + sin2 (x y) 2.

9b2 + b2 y2 +

Bi 2.84 : Cho a, b, c > 0. Chng minh rng : a2 ab 2 + b2 + b2 bc 3 +

c2

a2 ac

2

3 + c2 .

Bi 2.85 : Cho a, b, c > 0 v abc + bc + ca = abc. Chng minh rng : b2 + 2a2 c2 + 2b2 a2 + 2c2 + + 3. ab bc ac Bi 2.86 : Cho x2 + y2 = 1. Chng minh rng : x2 5 + 2xy y2 5 6. Bi 2.87 : Cho x2 + xy + y2 = 3 y2 + yz + z2 = 16 v x, y, z l cc s thc dng. Chng minh rng : xy + yz + zx 8.

Bi 2.88 : Cho x, y, z l nhng s dng. Chng minh rng : x2 + xy + y2 + y2 + yz + z2 + z2 + zx + x2 3(x + y + z).

Bi 2.89 : Cho a + b + c = 12. Chng minh rng : 3a + 2 a + 1 + 3b + 2 b + 1 + 3c + 2 c + 1 3 17.

Bi 2.90 : Cho cc s dng x, y, z v x + y + z 2. Chng minh rng : 4x2 1 + 2+ x 4y2 1 + 2+ y 4z2 1 + 2 z

145 . 2

Bi 2.91 : Gi s x, y, u, v R tha mn : x2 + y2 = 1; u2 + v2 + 16 = 8u + 4v. Tm gi tr ln nht ca biu thc P = 8u + 4v 2(ux + vy). Bi 2.92 : Cho x, y, z l cc s dng tha mn xy + yz + zx = 5. Tm gi tr nh nht ca P = 3x2 + 3y2 + z2 .

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2.3 Phng php s dng iu kin c nghim ca phng trnh hoc h phng trnh - phng php min gi trBi 2.93 : Tm gi tr ln nht v nh nht ca hm s : f (x) = 2x2 + 7x + 23 . x2 + 2x + 10 x2 (x 4y)2 Bi 2.94 : Tm gi tr ln nht v gi tr nh nht ca biu thc P = , vi x2 + y2 > 0. x2 + 4y2

Bi 2.95 : Cho x l s dng, y l s thc ty . Tm gi tr ln nht, gi tr nh nht (nu c) ca biu thc : P= xy2 (x2 + 3y2 ) x + x2 + 12y2 .

Bi 2.96 : Tm gi tr nh nht ca biu thc P = x2 + y2 , vi 2x2 + y2 + xy 1. Bi 2.97 : Cho cc s thc x, y tha mn iu kin : 3 x( 3 x 1) + 3 y( 3 y 1) = 3 xy. Tm gi tr ln nht, nh nht ca biu thc : P = 3 x + 3 y + 3 xy. Bi 2.98 : Cho x, y tha mn iu kin : x2 xy + y2 = 3. Tm gi tr ln nht v nh nht ca biu thc : P = x2 + xy 2y2 . Bi 2.99 : Cho hai s thc x, y tha mn iu kin : x 3 x + 1 = 3 y + 2 y. Tm gi tr ln nht, nh nht ca biu thc P = x + y. Bi 2.100 : Cho hai s thc x, y tha mn : x2 + y2 = 2(x + y) + 7. Tm gi tr ln nht, gi tr nh nht ca biu thc P = 3 x(x 2) + 3 y(y 2). Bi 2.101 : Cho cc s thc x, y tha mn : 4x2 3xy + 3y2 = 6. Tm gi tr ln nht, gi tr nh nht ca biu thc P = x2 + xy 2y2 . Bi 2.102 : Cho cc s thc x, y tha mn : x + 1 + y + 9. x+ y = 4. Tm gi tr ln nht, gi tr nh nht ca biu thc P =

Bi 2.103 : Cho cc s thc x, y tha mn : xy + x + y = 3. Tm gi tr ln nht, gi tr nh nht ca biu thc P = 3x 3y + x2 y2 . y+1 x+1 Bi 2.104 : Cho a, b 0 v a2 + b2 + ab = 3. Tm gi tr nh nht v gi tr ln nht ca biu thc P = a4 + b4 + 2ab a5 b5 . Bi 2.105 : Cho cc s thc x, y tha mn x + y = 2. Tm gi tr ln nht ca P = (x3 + 2)(y3 + 2).

2.4 Bt ng thc trong cc k thi tuyn sinh HBi 2.106 (C08) : Cho hai s thc x, y thay i v tho mn x2 + y2 = 2. Tm gi tr ln nht v gi tr nh nht ca biu thc: P = 2(x3 + y3 ) 3xy. Bi 2.107 (C10) : Cho hai s thc dng thay i x, y tha mn iu kin 3x + y 1. Tm gi tr nh nht ca biu thc 1 1 A= + . x xy Bi 2.108 (A03) : Cho x, y, z l ba s dng v x + y + z 1. Chng minh rng : x2 + 1 + x2 y2 + 1 + y2 z2 + 1 82. z2WWW.VNMATH.COM

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Bi 2.109 (A05) : Cho x, y, z l cc s dng tho mn :

1 1 1 + + = 4. Chng minh rng : x y z

1 1 1 + + 1. 2x + y + z x + 2y + z x + y + 2z Bi 2.110 (A06) : Cho hai s thc x 0, y 0 thay i v tho mn iu kin : (x + y)xy = x2 + y2 xy. Tim gi tr ln 1 1 nht ca biu thc A = 3 + 3 . x y Bi 2.111 (A07) : Cho x, y, z l cc s thc dng thay i v tho mn iu kin xyz = 1. Tm gi tr nh nht ca biu thc : x2 (y + z) y2 (z + x) z2 (x + y) + P= + . y y + 2z z z z + 2x x x x + 2y y

Bi 2.112 (A09) : Chng minh rng vi mi s thc dng x, y, z tha mn x(x + y + z) = 3yz ta c : (x + y)3 + (x + z)3 + 3(x + y)(y + z)(z + x) 5(y + z)3 . Bi 2.113 (B05) : Chng minh rng vi mi x R, ta c : Khi no ng thc xy ra. 12 5x

+

15 4

x

+

20 3

x

33 + 4x + 5x .

Bi 2.114 (B06) : Cho x, y l cc s thc thay i. Tm gi tr nh nht ca biu thc : A= (x 1)2 + y2 + (x + 1)2 + y2 + |y 2|.

Bi 2.115 (B07) : Cho x, y, z l ba s thc dng thay i. Tm gi tr nh nht ca biu thc : P=x x 1 + 2 yz +y y z 1 1 + +z + . 2 xz 2 xy

Bi 2.116 (B08) : Cho hai s thc x, y thay i v tho mn h thc x2 + y2 = 1. Tm gi tr ln nht v gi tr nh nht 2(x2 + 6xy) . ca biu thc P = 1 + 2xy + 2y2 Bi 2.117 (B09) : Cho cc s thc x, y thay i tha mn (x + y)3 + 4xy 2. Tm gi tr nh nht ca biu thc : A = 3(x4 + y4 + x2 + y2 ) 2(x2 + y2 ) + 1. Bi 2.118 (B10) : Cho cc s thc khng m a, b, c tha mn a + b + c = 1. Tm gi tr nh nht ca biu thc M = 3(a2 b2 + b2 c2 + c2 a2 ) + 3(ab + bc + ca) + 2 a2 + b2 + c2 . Bi 2.119 (D05) : Cho cc s dng x, y, z tho mn xyz = 1. Chng minh rng : 1 + x3 + y3 1 + y3 + z3 1 + z3 + x3 + + 3 3. xy yz zx Bi 2.120 (D07) : Cho a b > 0. Chng minh rng : 2a + 1 b 1 a 2b + b . 2a 2 Bi 2.121 (D08) : Cho x, y l hai s thc khng m thay i. Tm gi tr ln nht v gi tr nh nht ca biu thc : P= (x y)(1 xy) . (1 + x)2 (1 + y)2

Bi 2.122 (D09) : Cho cc s thc khng m x, y thay i v tha mn x + y = 1. Tm gi tr ln nht v gi tr nh nht ca biu thc : S = (4x2 + 3y)(4y2 + 3x) + 25xy. Bi 2.123 (D10) : Tm gi tr nh nht ca hm s y = x2 + 4x + 21 x2 + 3x + 10.

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2.5 Bi tp tng hp5 Bi 2.124 : Gi s x, y l hai s dng thay i tho mn iu kin x + y = . Tm gi tr nh nht ca biu thc : 4 1 4 S = + . x 4y a c b2 + b + 50 Bi 2.125 : Gi s a, b, c, d l bn s nguyn thay i tho mn 1 a < b < c < d 50. Chng minh + b d 50b a c v tm gi tr nh nht ca biu thc : S = + . b d Bi 2.126 : Cho x, y, z l ba s tho mn x + y + z = 0. Chng minh rng : 3 + 4x + 3 + 4y + 3 + 4z 6. Bi 2.127 : Chng minh rng vi mi x, y > 0 ta c : (1 + x) 1 + ng thc xy ra khi no. Bi 2.128 : Cho a, b, c l ba s dng tho mn a + b + c = 3 . Chng minh rng : 4 3 3 3 a + 3b + b + 3c + c + 3a 3. y x 9 1+ y2

256.

Khi no ng thc xy ra? 1 Bi 2.129 : Chng minh rng 0 y x 1 th x y y x . ng thc xy ra khi no ? 4 Bi 2.130 : Cho x, y, z l ba s dng v xyz = 1. Chng minh rng : x2 y2 z2 3 + + . 1+y 1+z 1+x 2 Bi 2.131 : Cho x, y l cc s thc tho mn iu kin x2 + xy + y2 3. Chng minh rng : 4 3 3 x2 xy 3y2 4 3 3.

Bi 2.132 : Cho cc s thc x, y, z tho mn iu kin 3x + 3y + 3z = 1. Chng minh rng : 9x 9y 9z 3x + 3y + 3z + y + z . 3x + 3y+z 3 + 3z+x 3 + 3x+y 4 Bi 2.133 : Cho hai s dng x, y thay i v tho mn iu kin x + y 4. Tm gi tr nh nht ca biu thc A = 3x2 + 4 2 + y3 + . 4x y2 11 7 Bi 2.134 : Tm gi tr nh nht ca hm s : y = x + + 4 1 + 2 , x > 0. 2x x Bi 2.135 : Cho x, y, z l cc s thc dng. Tm gi tr nh nht ca biu thc : P=3

4(x3 + y3 ) +

3

4(y3 + z3 ) +

3

4(z3 + x3 ) + 2

x y z + 2+ 2 . 2 y z x

Bi 2.136 : Cho a, b l cc s dng tho mn ab + a + b = 3. Chng minh rng : 3a 3b ab 3 + + a2 + b2 + . b+1 a+1 a+b 2 Bi 2.137 : Cho x, y > 0 v xy = 100. Hy xc nh gi tr nh nht ca biu thc P = x2 + y2 . xy Trang 46

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Bi 2.138 : Gi s phng trnh ax2 + bx + c c hai nghim thuc on [0; 1]. Xc nh a, b, c biu thc P c gi tr nh (a b)(2a c) nht, gi tr ln nht, trong P = . a(a b + c) 1 1 y2 + 2 . Bi 2.139 : Cho x, y > 0 tha mn x + y = 1. Tm gi tr nh nht ca biu thc P = x2 + 2 y x Bi 2.140 : Chng minh cc bt ng thc sau vi a, b, c l cc s nguyn khng m : 1+ a 1+ b 1+ c + + 3 + a + b + c. 3 1+ b 1+ c 1+ a Bi 2.141 (*) : Cho 6 s thc x1 , x2 , . . . , x6 [0; 1]. Chng minh rng : (x1 x2 )(x2 x3 )(x3 x4 )(x4 x5 )(x5 x6 )(x6 x1 ) Bi 2.142 : Cho x, y, z > 0. Chng minh rng : x6 1 . 16

2y 2z 1 1 1 2x + 6 + 6 4 + 4 + 4. 4 4 4 +y y +z z +x x y z4 4 x i=1 4 i=1 i

Bi 2.143 : Cho x1 , x2 , x3 , x4 > 0 tha mn

4 i=1

x1 = 1. Hy tm gi tr nh nht ca T =

. x y + 2 . 2 +y x + y4

x3 i x4

Bi 2.144 : Cho x, y l hai s dng thay i tha mn xy = 1. Tm gi tr ln nht ca biu thc A =

Bi 2.145 : Cho hai s thc x, y tha mn x2 + y2 = x + y. Tnh gi tr ln nht, gi tr nh nht ca biu thc A = x3 + y3 + x2 y + xy2 . Bi 2.146 : Cho ba s thc dng x, y, z tha mn iu kin x2 + y2 + z2 3. Tm gi tr nh nht ca biu thc : P= 1 1 1 + + . xy + z2 yz + x2 zx + y2

Bi 2.147 : Cho ba s dng a, b, c tha mn iu kin ab + bc + ca = 2abc. Chng minh rng 1 1 1 1 + + . 2 2 2 a(2a 1) b(2b 1) c(2c 1) 2 Bi 2.148 : Cho x, y l cc s thc dng thay i tha mn iu kin xy y 1. Tm gi tr nh nht ca biu thc x2 y3 P = 2 + 9 3. y x Bi 2.149 : Cho cc s thc khng m x, y, z tha mn x2 + y2 + z2 = 3. Tm gi tr ln nht ca biu thc A = xy + yz + 5 zx + . x+y+z Bi 2.150 : Cho cc s thc dng x, y, z tha mn x + y + z = 1. Tm gi tr nh nht ca biu thc P = z2 z3 . + xy x3 y3 + 2 + x2 + yz y + zx

Bi 2.151 : Cho x, y, z > 0 tha mn 13x + 5y + 12z = 9. Tm gi tr ln nht ca biu thc A =

xy 3yz 6xz + + . 2x + y 2y + z 2z + x

Bi 2.152 : Cho cc s thc x, y, z tha mn iu kin x2 + y2 + z2 = 3. Tm gi tr ln nht ca A = x3 (y + z) + y3 (z + x) + z3 (x + y). Bi 2.153 : Gi s x, y, u, v R tha mn iu kin x2 + y2 = 1, u2 + v2 + 16 = 8u + 4v. Tm gi tr ln nht ca biu thc M = 8u + 4v 2(ux + vy).

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www.luyenthi24h.com www.luyenthi24h.com www.VNMATH.com 3. Tnh gi tr nh nht ca

Bi 2.154 : Cho a, b, c l cc s thc dng thay i v tha mn iu kin a + b + c = P= a2 + ab + b2 + b2 + bc + c2 + c2 + ca + a2 .

Bi 2.155 : Cho x, y R tha mn x2 + y2 2x 4y + 4 = 0. Chng minh rng

x2 y2 + 2 3xy 2(1 + 2 3)x + (4 2 3)y 5 4 3.

Bi 2.156 : Gi s x, y, z l cc s thc tha mn x + y + z = 6. Chng minh rng 8x + 8y + 8z 4x+1 + 4y+1 + 4z+1 . Du ng thc xy ra khi no ? Bi 2.157 : Cho cc s thc dng x, y, z tha mn x + y + z = 1. Tm gi tr nh nht ca biu thc : P= x2 (y + z) y2 (z + x) z2 (x + y) + + . yz zx xy

Bi 2.158 : Cho a, b, c l cc s dng tha mn abc = 8. Hy tm gi tr ln nht ca biu thc P= 1 1 1 + + . 2a + b + 6 2b + c + 6 2c + a + 6 x 1 x2 + 2 . 3 1 y2 y

Bi 2.159 : Cho x, y > 0 v tha mn x + y = 1. Chng minh rng

Bi 2.160 : Cho hai s thc khng m x, y tha mn x2 + y2 + xy = 3. Tm gi tr ln nht, gi tr nh nht ca biu thc P = x3 + y3 (x2 + y2 ). Bi 2.161 : Cho a, b, c l cc s thc khng m, khc nhau tng i mt v tha mn iu kin ab + bc + ca = 4. Chng minh rng 1 1 1 + + 1. 2 2 (a b) (b c) (c a)2

Bi 2.162 : Cho x, y, z l ba s thc thuc (0; 1]. Chng minh rng

1 1 1 5 + + . xy + 1 yz + 1 zx + 1 x + y + z Bi 2.163 : Cho a, b, c l di ba cnh ca mt tam gic. Chng minh rng a 1 1 2 b c + + + + < 2. 3a + b 3a + c 2a + b + c 3a + c 3a + b

Bi 2.164 : Cho a, b, c l nhng s dng tha mn a2 + b2 + c2 = 3. Chng minh rng 1 1 1 4 4 4 + + + + . a + b b + c c + a a2 + 7 b2 + 7 c2 + 7 Bi 2.165 : Cho x, y R, chng minh rng |x y| |x| |y| + . 1 + |x y| 1 + |x| 1 + |y|

Bi 2.168 : Cho a, b, c l cc s thc dng tha mn a + b + c = abc. Chng minh rng : 1. c ab 1 + 1 + c2 ;

4x + y 2x y + . xy 4 (1 + a)(1 + b)(1 + c) Bi 2.167 : Cho a, b, c l cc s thc dng tha mn a + b + c = 1. Tm gi tr nh nht ca P = . (1 a)(1 b)(1 c) Bi 2.166 : Cho hai s dng x, y tha mn x + y = 5. Tm gi tr nh nht ca biu thc P =

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www.VNMATH.com2. ab + bc + ca 3 +

CHUYN LUYN THI I HC b2 + 1 + c2 + 1.


a2 + 1 +

Bi 2.169 : Cho a, b, c l cc s thc dng tha mn ab + bc + ca = 3. Chng minh rng : 1 1 1 1 + + . 1 + a2 (b + c) 1 + b2 (c + a) 1 + c2 (a + b) abc Bi 2.170 : Cho x, y, z l cc s thc dng tha mn xyz = 1. Chng minh rng : 1 1 1 + + 1. x+y+1 y+z+1 z+x+1 2 y 2 x 2 z 1 1 1 Bi 2.171 : Cho x, y, z > 0. Chng minh rng 3 + 3 + 3 2 + 2 + 2. 2 2 2 x +y y +z z +x x y z Bi 2.172 : Cho a, b, c l di ba cnh ca mt tam gic. Chng minh rng : 4a 9b 16c + + 26. b+ca c+ab a+bc 3 4 3 1 2a + 4 2 1 . 2

Bi 2.173 : Cho cc s thc khng m a, b. Chng minh rng : a2 + b + b2 + a +

2b +

Bi 2.174 : Cho cc s thc dng a, b, c thay i tha mn a + b + c = a. Chng minh rng : a2 + b b2 + c c2 + a + + 2. b+c c+a a+b Bi 2.175 : Cho cc s thc dng x, y, z. Chng minh rng : x2 xy y2 yz z2 zx + + 0. x+y y+z z+x Bi 2.176 : Cho cc s dng a, b, c tha mn abc = 1. Chng minh rng 1 1 1 + 2 + 2 + 3 2(a + b + c). 2 a b c yz 2 33 Bi 2.177 : Cho ba s dng x, y, z tha mn x + y + z = . Chng minh rng x (y + z). 3x 6 Bi 2.178 : Cho cc s thc x, y tha mn 0 x v 0 y . Chng minh rng cos x + cos y 1 + cos(xy). 3 3 Bi 2.179 : Cho s nguyn n (n > 2) v hai s thc khng m x, y. Chng minh rng n xn + yn n+1 xn+1 + yn+1 . ng thc xy ra khi no?

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Chng 3

Lng gic3.1 Phng trnh c bnBi 3.1 : Gii cc phng trnh sau : a) sin x = 3 ; 2

b) 3 cos 2x +

=1; 6

c) sin 3x = cos 2x ;

Bi 3.2 : Tm tt c cc nghim thuc ; ca phng trnh : 2 tan 3x + Bi 3.3 : Gii phng trnh : cos (. sin x) = cos sin 2x = 0. 1 + sin x cos 2x cos x Bi 3.5 : Gii phng trnh : = 0. cos x Bi 3.6 : Gii phng trnh : cos x cot 2x = sin x. Bi 3.4 : Gii phng trnh : Bi 3.7 : Gii phng trnh : =0; 4 Bi 3.8 : Gii phng trnh : 32 x2 . cos 2x = 0. cos 8x Bi 3.9 : Gii phng trnh : = 0. sin 4x sin 3x Bi 3.10 : Gii phng trnh : = 1. sin 2x a) cos2 2x + sin2 x + b) cos 2x. sin x + =0; 4 . sin x . 2 1 = . 3 3

3 Bi 3.11 : Gii phng trnh : cos3 x sin 3x + sin3 x cos 3x = . 8 3 3 x sin 3x = sin3 4x. Bi 3.12 : Gii phng trnh : sin x cos 3x + cos Bi 3.13 : Gii phng trnh : cos 10x + 2 cos2 4x + 6 cos 3x cos x = cos x + 8 cos x cos3 3x. 1 Bi 3.14 : Gii phng trnh : cos x cos 2x cos 4x cos 8x = . 16 2 x tan x tan 3x = 2. Bi 3.15 : Gii phng trnh : tan 11 Bi 3.16 : Gii phng trnh : tan2 x + cot2 x + cot2 2x = . 3 51WWW.VNMATH.COM

CHUYN LUYN THI I HC www.VNMATH.comcot2 x tan2 x = 16(1 + cos 4x). cos 2x 7 cot x . Bi 3.18 : Gii phng trnh : sin4 x + cos4 x = cot x + 8 3 6 3(sin x + tan x) Bi 3.19 : Gii phng trnh : 2(1 + cos x) = 0. tan x sin x Bi 3.20 : Gii phng trnh : cos 3x tan 5x = sin 7x. Bi 3.17 : Gii phng trnh : Bi 3.21 : Gii phng trnh : sin4 x + cos4 x 1 = (tan x + cot 2x). sin 2x 2


3.2 Phng trnh dng a sin x + b cos x = cBi 3.22 : Tm nghim ca phng trnh : cos 7x 3 sin 7x = 2

6 2 0 v x2 + y2 = 1. Tm GTLN ca P = x3 + y3 . Bi 3.224 (C-2008) : Cho hai s thc x, y thay i v tho mn x2 + y2 = 2. Tm gi tr ln nht v gi tr nh nht ca biu thc: P = 2(x3 + y3 ) 3xy. Bi 3.225 (H-KB2008) : Cho hai s thc x, y thay i v tho mn x2 + y2 = 1. Tm gi tr ln nht v gi tr nh nht 2(x2 + 6xy) ca biu thc: P = . 1 + 2xy + 2y2

3.7 Lng gic trong cc k thi tuyn sinh HBi 3.227 (C09) : Gii phng trnh : (1 + 2 sin x)2 cos x = 1 + sin x + cos x. 5x 3x Bi 3.228 (C10) : Gii phng trnh 4 cos cos + 2(8 sin x 1) cos x = 5. 2 2 Bi 3.229 (A02) : Tm nghim thuc khong (0; 2) ca phng trnh : 5 sin x + Bi 3.230 (A03) : Gii phng trnh : cot x 1 = ca tam gic ABC. Bi 3.232 (A05) : Gii phng trnh : cos2 3x cos 2x cos2 x = 0. cos 3x + sin 3x = cos 2x + 3. 1 + 2 sin 2x Bi 3.226 (C08) : Gii phng trnh : sin 3x 3 cos 3x = 2 sin 2x.

cos 2x 1 + sin2 x sin 2x. 1 + tan x 2 Bi 3.231 (A04) : Cho tam gic ABC khng t, tho mn iu kin : cos 2A + 2 2 cos B + 2 2 cos C = 3. Tnh ba gc

2 cos6 x + sin6 x sin x cos x Bi 3.233 (A06) : Gii phng trnh : = 0. 2 2 sin x Bi 3.235 (A08) : Gii phng trnh : Bi 3.236 (A09) : Gii phng trnh : 1 1 + sin x sin x = 4 sin7 4

Bi 3.234 (A07) : Gii phng trnh : 1 + sin2 x cos x + 1 + cos2 x sin x = 1 + sin 2x.3 2

x .

(1 2 sin x) cos x = 3. (1 + 2 sin x)(1 sin x) (1 + sin x + cos 2x) sin x + 1 4 = cos x. Bi 3.237 (A10) : Gii phng trnh 1 + tan x 2

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CHUYN LUYN THI I HC www.VNMATH.comBi 3.238 (B02) : Gii phng trnh : sin2 3x cos2 4x = sin2 5x cos2 6x. 2 Bi 3.239 (B03) : Gii phng trnh : cot x tan x + 4 sin 2x = . sin 2x Bi 3.240 (B04) : Gii phng trnh : 5 sin x 2 = 3(1 sin x) tan2 x.


Bi 3.241 (B05) : Gii phng trnh : 1 + sin x + cos x + sin 2x + cos 2x = 0. x = 4. Bi 3.242 (B06) : Gii phng trnh : cot x + sin x 1 + tan x tan 2 Bi 3.243 (B07) : Gii phng trnh : 2 sin2 2x + sin 7x 1 = sin x. Bi 3.244 (B08) : Gii phng trnh : sin3 x 3 cos3 x = sin x cos2 x 3 sin2 x cos x. Bi 3.245 (B09) : Gii phng trnh : sin x + cos x sin 2x + 3 cos 3x = 2(cos 4x + sin3 x). Bi 3.246 (B10) : Gii phng trnh (sin 2x + cos 2x) cos x + 2

101 Chuyen de Luyen Thi Dai Hoc

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