Solucionario II

7/25/2019 Solucionario II

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IIUN I DAD 1

OBRA COLECTIVA, DISEADA, CREADA Y PRODUCIDA

BAJO LA DIRECCIN DE:

ERLITA OJEDA ZAARTUDRA. EN CIENCIAS DE LA EDUCACIN

S o l u c i o n a r i o

S E C U N D A R I A


2/157

2 Matemtica II

Clave:a

Clave: e

Clave: c

Clave: c

Clave: a

Clave: b

Clave: d

Clave: d

Clave: c

Clave: e

Clave: c

1. II. La quiwicha es deliciosa y la maca tienegrandes propiedades

1. 800 1177 72,...3022 8

hay 72 nmeros mltiplos de 11

2. 200 1212 16,...8072 8hay 16 nmeros mltiplos de 12

2. p: 22+ 1 = 5 (V ) q: 32 1 = 6 (F)

(p q) (q p) (F F) (F V ) F V V

4. (p q) (r s) F

V F Luego: p F q F r V s V

5. (p q) (r s) F

V F

Luego: p V q F r V s F

6. (p q) (q r) V

~p q V ; q r V

p F

q Vr V

8. p: 23+ 32= 17 (V )q: 62= 36 (V )r: 32+ 42> 24 (V )

I. (p q) r

V V V

II. (p r) qV V

V

III. p (q r) V V V

9. (p q) (r p)

7. (p q) (r s) F

F F

Luego:p V r Fq V s F

3. p q

1U N I D A D

Pg. 10

Pg. 14

I. (p q) q F F F

II. (p q) (p s) V F

F

III. p (p r) V F F


3/157

Matemtica II

Clave: b

Clave: e

Clave: c

Clave: d

Clave: d

Clave: b

Clave: b

Clave: b

Clave: e

3. 187 < 6 < 794

187 < 6k < 794

31,1... < k < 132,3

k: 32, 33, 34, ...., 132

ktoma: 132 31 = 101 valoresLuego hay 101 nmeros

10. 120 = 233 5

S.D. = 23+1 12 1

31+1 13 1

51+1 15 1

S.D. = 15 4 6

S.D. = 360

11. 22n= 2n11n

C.D. = 25

(n + 1)(n + 1) = 25 n = 4

12. 3 + 1N 4 + 1 N = MCM(3; 4; 7) + 1

7 + 1 N = 84 + 1 N = 85

14. 4 a 7 2 5 8 = 11 + + +

4 + a 7 + 2 5 + 8 = 11

a 6 = 11

a = 6

15. x 2 y 3 y x = 45

Para que sea 5 x = 5 luego:

5 2 y 3 y 5 = 9

5 + 2 + y + 3 + y + 5 = 9

13. 4 a a 8 = 71 2 3 1

4 + 2a + 3a + 8 = 7

5a + 4 = 7

a = 2 a = 9

Luego "a" puede tomar 2 valores

4. 500 2346 21,... 40 23 17

hay 21 nmeros mltiplos de 23

5. Sean los nmeros impares (a 2); (a); (a + 2);luego:(a 2) + (a) + (a + 2) = 3a = 3

6. Sean los nmeros consecutivos (a 2); (a 1); a;(a + 1); (a + 2)

Luego:

a 2 + a 1 + a + a + 1 + a + 2 = 5a = 5

7. 12 = 223

hay 2 divisores primos

8. N = 243 55

C.D.(N) = (4 + 1)(1 + 1)(5 + 1)

= 60

9. 360 = 23325

C.D.compuestos= 24 3 1

= 20

Clave: dClave: c

Clave: c

5

9


4/157

4 Matemtica II

Clave: c

Clave: e

1. 50 175 5

10 35 5

2 7

MCD = 5 5 = 25

2. 16 48 96 15 2

8 24 48 15 2

4 12 24 15 2

2 6 12 15 2

1 3 6 15 2

1 3 3 15 3

1 1 1 5 5

1 1 1 1

3. MCM(7k; 3k) = 189

7 3 k = 189

k = 9

15 + 2y = 9 y = 6Piden:x + y = 11

Clave: dClave: a

Clave: e

Clave: aClave: c

Clave: c

Clave: d

16. C.D.(N) = 12

N = 2a 3b

C.D.(N) = (a+ 1)(b+ 1)

12 = (a+ 1)(b+ 1)

3 4 = (a+ 1)(b+ 1)

a= 2 b= 3

a= 3 b= 2

Por dato debe ser el menor posible nmero es:

N = 23 32

N = 72

Piden: 7 + 2 = 9

20. Sea n el nmero de veces que se debe mulplicar por 8 a 300, luego:

300 8n

= 223 5223n

= 23n+23 52

Por dato:

C.D. = 126

(3n + 3)(1 + 1)(2 + 1) = 126 n = 6

17. 7422222 = 8 + r

120 cifras

Luego:

222 = 8 + r

8 + 6 = 8 + r

r = 6

18. 10 + 20 + 30 + 40 + 50 + 5a = 37

150 + 5a = 37

7

a = 7

19. 4n

4n2

= 4n 1 1

16

= 4n1516

= 4n215

= 22n43 5

Por dato: C.D. = 28

(2n 3)(1 + 1)(1 + 1) = 28

2n 3 = 7

n = 5

Pg. 18

MCM = 25 3 5 = 48


5/157

Matemtica II

Clave: b

Clave: a

Clave: c

Clave: c

Clave: c

Clave: d

Clave: c

Clave: a

Clave: b

Clave: a

Clave: b

10. MCD(A; B) MCM(A; B) = A B

2 240 = 16 B

30 = B

11. 20 8 210 4 2

5 2 2

5 1 5

1 1

Piden: 200 : 40 = 5

12. MCD(180; 234)

180 234 2

90 117 3 30 39 3

10 13

Longitud del lado = 232 = 18

N lotes: = = 130

13. MCD(A; B) MCM(A; B) = A B

MCD(A; B) 350 = 1750

MCD(A; B) = 5

14. 2 6 1 2

1 3 1 3

1 1 1

N de ladrillos =6 6 6

= 36 2 3 1

15. MCD(A; B) = 4

4 3 2 2

A B 20 8 4

20 8 4 0

28 n lapiceros

4. MCD (135; 90; 225)

135 90 225 5

27 18 45 3

9 6 15 3

3 2 5Piden:

3 + 2 + 5 = 10

5. MCD (140; 168; 224)

140 168 224 4

35 42 56 7 5 6 8

n cajas: 5 + 6 + 8 = 19

6. MCD(12k; 18k; 30k) = 48

6k = 48

k = 8

7. A = 243654113

B = 23385272

MCM(A; B) = 24 385472113

8. A = 25337

B= 337 11

MCD(A; B) = 337

9. 40 + 3

N 30 + 3

16 + 3

N = 240 + 3

Nmnimo= 243

N = MCM(40, 30, 16) + 3

MCM = 23 5 = 40

6

AtotalAlote

180 234182


6/157

6 Matemtica II

Clave: c

Clave: e

Clave: e

Clave: b

Clave: e

Clave: cClave: b

Clave: b

Clave: a

6. 20 4 = 4 c

12 = c

324

= 40d

d = 5

Piden: 12 5 = 17

7.AB

= 29

A = 2k, B = 9k

Por dato:

9k 2k = 84

k = 12

Piden: B = 9(12) = 108

8.a8

= b10

= c18

= k

a = 8k, b = 10k, c = 18k

Por dato:

a + b + c = 180

8k + 10k + 18k = 180

k = 5

Piden: a = 8k

a = 40

9.AB

= 58

A = 5k, B = 8k

Por dato:

(5k)2+ (8k)2= 356

25k2+ 64k2= 356

89k

2

= 356 k = 2

Piden: A B = (5k)(8k)

= 40k2

= 160

B = 20 3+ 8

B = 68

A = 68 4 + 20A = 292

Clave: d

1. Total: 400 personas

Mujeres: 240

Varones: 160

VM

=160240

= 23

2.A = 3k, B = 4kPor dato:7k = 56k = 8

Piden: B = 4k = 32

3. A = 3k, B = 2k

Por dato:

3k 2k = 5

k = 5Piden : B = 2k = 10

4. Pedro=

180=

9

Jos 80 4

5. a = 3k

b = 7k

Por dato:

a + b = 80

10k = 80

k = 8

Piden: b = 7k = 56

Pg. 22


7/157

Matemtica II

Clave: a

Clave: a

Clave: c

Clave: e

10.Total: 400 personas

HM

= 3k2k

= 21

3k + 2k = 400

k = 80

V = 240; M = 160

Luego:240 x160 x

= 21

x = 80Luego, se retiraron 80 parejas

11. AlcoholAgua = 53 Alcohol = 5k; Agua = 3k

Por dato:

8k = 72

k = 9 Alcohol = 45

Luego:

4527 + x

= 910

x = 23

12.AB

= 811

= 79

8k + 1011k + 10

= 79

72k + 90 = 77k + 70

20 = 5k

4 = kPiden:

8k 411k 4

= 32 444 4

= 2840

= 710

Luego, la relacin era de 7 a 10

13.AB

= 5k8k

= 79

5k + 228k + 22

= 79

45k + 198 = 56k + 154

4 = k

A = 5k = 20 , B = 8k = 32

15.AB

= 7k5k

= 1

Luego:

7k 4 = 5k + 4

2k = 8

k = 4

A = 7k = 28; B = 5k = 20Piden:

A + B = 48

14.LA

= 95

L = 9k; A = 5k

Por dato:

2L + 2A = 336

18k + 10k = 336

28k = 336

k = 12

L A = (9k)(5k)

= 45(12)2

= 6 480

Clave: c

Clave: b

x+22

4

+22

+4

x

+ 10 aos

+ 10 aos


8/157

8 Matemtica II

Clave: cClave: c

Clave: aClave: c

Clave: c

Clave: c

Clave: c

Clave: b

Clave: a

Clave: d

Clave: d

Clave: d

1.15

100 60

1001 200 = 108

2.20

100

30100

60

1009 000 = 324

3.80

100

75100

100100

30 = 18

4. Tengo 100%

Vendo 20%

80% Vendo 40%

40%

Luego: 40% N = 200

N = 500

8. Au = 10 + 20 + 10 20100

%

Au = 32%

9. 100% 15% = 85%Sea N el precio de la grabadora, luego:

85% N = 170

N = 200

10.Tengo: 200

20% (200) = 40

Luego tengo:

200 + 40 = 240

Gast: 20% (240) = 48

Queda:

240 48 = 192

Piden: 200 192 = 8

11. Gano: 40% (400) = 160

Tengo:560

Luegogasto: 20% (560) = 112

Queda: 560 112 = 448

Al final gano (S/.)448 400 = 48

12. Nios: 40%

Nias: 60%

Ahora:

Nios: 20%

Nias: 60%

80%

80% 100%

60% x%

x = 100 6080

x = 75%

5. 200 100%

120 x%

buen estado

x = 120

100200

x = 60%

7. Du = 20 + 40 20 40100

%Du = 52%

6. 115 son papayas

345 no son papayas

460 100%

345 x%

x = 345 100460

x = 75%

Pg. 28

460


9/157

Matemtica II

1. V = 7k; M = 9k

Por dato:

16k = 160

k = 10

Luego: V = 70; M = 90

Entonces deben llegar 20 varones

2. N < 600

10 + 7

N 12 + 7 N = MCM(10; 12; 15) + 7 15 + 7

N = 60 + 7

N = 540 + 7

N = 547

Clave: c

Clave: b

Clave: c

Clave: a

6. V1= 10 ; D1= 15

V2= 5 ; D2= x

Se cumple:

V1D1= V2D210 15 = 5 D2

30 = D2 Clave: a

Clave: e

Clave: e

13. 540 15% = 81

Me deben S/. 81

Pagan: 20% 300 = 60

An me deben: 81 60 = 21

14. 210 100%

x 30%

x = 210 30100

x = 63

Piden:

210 + 63 = 273

15.

Solucin de problemas Pg. 30

3. obreros h/d das zanja 3 4 6 12 x 6 2 16

3 4 612

= x 6 216

x = 84. n secretarias digitan da

2 350 7

x 600 42 7350

= x 4600

x = 6 Clave: d

5. Precio D.P. (tamao)2 I.P. Energa

P ET2

= k

360 M(14)2

=P

(21)2

360 M196

=

P

441

360 M 441 2

196 M= P

1 620 = P Clave: d

M4

M4

7. Lado = 3k ; Lado menor = 2k

A = A

(3k)2

= (2k) x 9k2= 2k x

4,5 k = x

PermetroPermetro

= 4(3k)2(2k) + 2(4,5k)

= 12k13k

= 1213

Clave: a

A1= abA2= (60%a)(140%b)A2= 84%abA2= 84% A1Luego, el rea disminuye en 16%

A1 A2 60%aa

b 140%b


10/157

10 Matemtica II

13. A = 47 + 5

B = 47 + 31

C = 47 + 41

A + B + C = 47 + 77 = 47 + (47 + 30)

= 47 + 30

residuo = 30

14. Pc = S/. 600

Pv = Pc 25% Pv

125% Pv = 600

Pv = 480

Pg. 33

10. Sea la cantidad de dinero "x", por dato:

Se pierde: 40% x queda 60% x

Se gana: 20%(60%x)

Se tiene: 120%(60%x)

Luego:

120%(60%x) 88 = 200

72x100

= 288

x = 400

8. 100% + 60% = 160%

3N 160% 100%

N 1603

% x

x =

1603

100160

x = 33,3%

9. El equipo gan 12 encuentros, luego:

12 + x 100%

x 85%

x = (12 + x) 85100

x = 68

1. (p q) r F

p V

q V

r F

2. I. V F V

II. F F V

III. V F F

IV. F F F

3. 999 = 9 111

= 323 37

= 3337

C.D.(999) = (3 + 1)(1 + 1)

= 8

Clave: a

Clave: a

Clave: c Clave: c

Clave: e

Clave: e

Clave: e

Clave: d

Clave: b

Clave: b

11. 36 = 720 30 t1 200

t = 2

12. Pv = 6 500

Pv = Pc + G

6 500 = 130% Pc

5 000 = Pc


11/157

Matemtica II

10. A B = 245

MCM(A; B) = 5MCD(A; B)

Sabemos que:

MCD(A; B) MCM(A; B) = A B

MCD(A; B) 5MCD(A; B) = 245MCD(A; B) = 7

MCM (A; B) = 35

A = 35

B = 7

Piden: 35 7 = 28

11.HP

= 35

= 57

3k + 205k + 20

= 57

21k + 140 = 25k + 100

40 = 4k

10 = k

Piden: H = 3k = 30

12. a24

= 24c

= 23

a

24= 2

3 a = 16

24c

= 23

36 = c

Piden:a + c = 52

13. rea das

72 8

142 x

x = 142 872

x = 32

4. 240 = 243 5

C.D.(240) = 5 2 2

= 20

C.D.(N) = C.D.S+ C.D.C

20 = 4 + CDC 16 = C.D.C

5. 9 999 = 9 11 101

= 3211 101

Piden:

3 + 11 + 101 + 1 = 116

6. 1 800 = 233252

C.D.(1 800) = 4(3)(3)

= 36

7. 820 = 225 41

= 4 (5 41)

C.D.(4) = (2)(2)

= 4

8. N = 412 410

N = 410 (42 1)

N = 220(16 1)

N = 2203 5

C.D.(N) = (21)(2)(2)

C.D.(N) = 84

9. AB = 6 A = 6B

MCM(A; B) = 72

MCM(6B; B) = 72

6B = 72

B = 12

Clave: c

Clave: c

Clave: c

Clave: c

Clave: cClave: b

Clave: e

Clave: e

Clave: a

Clave: d

+ 20 aos

+ 20 aos

Luego:


12/157

12 Matemtica II

1. A = 5k, B = 7k, C = 11k

Luego de sumar 130, 260 y n a cada uno deellos, respectivamente tenemos que:

(*)

5k + 130=

7k + 260=

11k + n

13 17 19

k = 195Reemplazamos en (*)

7(195) + 26017

= 11(195) + n19

Resolviendo n = 910

3. Sea : divisor comn de 2 520 y 2 000 y el myor posible (para usar menos trabajadores)

= MCD (2 520; 2 000) = 40 m

2 520 m

2 000 m

N murales

1 av: 2 52040

+ 1 = 63 + 1 = 64

2 av: 2 00040 + 1 = 50 + 1 = 51 115 murales

Como se necesitan 3 trabajadores por mural

115 3 = 345

2. Inicialmente: 8 personas / 10 das / 8 h/d

obra

Se hizo as: 8 personas (8 + x) personas 5 das 2 das

5 h/d 10 h/d

Se observa que las 8 personas han trabajado mitad del tiempo indicado para concluir la obpor lo tanto, solo han hecho la mitad del trabjo. En consecuencia ahora todos los obreros coel grupo que se incorpora debern terminar obra, es decir, debern realizar la mitad del trbajo.

Adems, recordemos que:

(N de obreros)(N de das)(N de h/d) = cte.

Reemplazando valores, tenemos que:

8 5 8 = (8 + x) 2 10

x = 8

14. Peones h/d das obra (m2)

14 7 15 150

21 8 x 240

14 7 15150

= 21 8 x240

x = 14

15. Pv = 2 600

Pv = Pc + G

Pv = 100% Pc + 30% Pc

2 600 = 130% Pc

2 000 = Pc

16. C = 4 000

r = 7,5% trimestral 30% anual

t = 15 meses

I= 4000 30 151 200

I= 1 500

Clave: b

Clave: a

Clave: aClave: c

Clave: c

Clave: c

Se contratarn x personasadicionales


13/157

Matemtica II

4. a + b + c = 12 abc = 12

bc = 4 (criterio por 4)

Como c es par:

c = 0 b = 4; 6; 8

c = 2 b = 1; 3; 5; 7; 9

c = 4 b = 2; 4; 6

c = 6 b = 1; 3; 5

c = 8 b = 0; 2

Luego:

3 + 5 + 4 + 3 + 2

= 17 soluciones


14/157

14 Matemtica II

10.ab

ba

= 1,805

a2 b2

ab=

1805 180900

a2 b2

ab= 65

36

ab = 9 4

a + b = 13

11. Gasta = 34

no gasta

G N

G + N = 49

34

N + N = 49

74

N = 49

Clave: a

Clave: d

Clave: d

Clave: b

Clave: bClave: b

Clave: b

Clave: d

Clave: d

Clave: a

1. Sea R el recorrido y F lo que falta recorrer, pordato:

15

R =35

F R + F = 12 ... (I)

R = 3FRemplazamos en (I) 4F = 12

F = 3 R = 9

2. Sea x el nmero de aves:

Palomas:4x5

Otros:x5

Gallinas:x5

56

Gallos16

x5

= 8

x = 2403. Sea x lo que me deben, luego:

x =78

960 = 840

Ahora:

Pagan:34

(840) = 630

Me deben: 840 630 = 210

4.12

100+

39

+582 58

900

108 + 300 + 524900 = 932900 = 233225

5. E =997 99

9+

29

E =898 + 2

9=

9009

= 100

Piden: 100 = 10

6. 23

910

56

+ x = 59

1 + 2x

2=

59

9 + 18x = 10

x =1

18

7. x =4 5

20= 1

20

y =5 4

20=

120

8. x +7

15= 10

15

x = 315

x =15

9. Tipo A Tipo B

Usan Queda Usan Queda

56

16

34

14

16

+ 14

=2 + 3

12= 5

12

2U N I D A D

Pg. 40

y > x


15/157

Matemtica II

12. 35

3= 27

125

1625

25

1125

=80 50 1

125=

29125

27125

+29

125=

56125

1

56

125 =

69

125

13.7098

= 5k7k

(5k)(7k) = 315

k2= 9

k = 3

Luego: 5(3)

7(3)

= 15

2121 15 = 6

15. M = 1 1

1 1 +

118

M = 1 1 +

187

M = 1 7

15=

815

16. E =3 + 1

12

1312

6 + 4 + 312

7413

E =

13123712

7413

E =1337

7413

= 2

17. P =1

+

2 +

159 73

90119

=1

+

2 +

159 73

110

P =1

+59 110293

=1

1465 + 9902637

P =1

24552637

=26372455

1,074

14. M = a + b

11a + 7b77

=844 155999 999

11a + 7b =77 844 155

999 999

11a + 7b = 65

4 3

a + b = 7

Clave: e

Clave: e

Clave: b

Clave: a

Clave: d

Clave: d

Clave: c

N = 28

G = 34

(28) = 2178

157


16/157

16 Matemtica II

18. E =321 32

900249 24

900

1

E = 289900

225900

1

E =1730

1530

1

=2

30

1

= 15

19. - Un litro de leche cuesta S/.17,50

- Un litro de vino cuesta S/.43,75

43,75 : 17,50 = 2,5

26.

Total: 5050

25

= 20

50 20 = 30

30 + 15 = 45

4523

= 30

Ahora:

45 30 = 15

15 + 35 = 50

20. 8431,24 : 0,97 = 8 692

21. 503,54 84,5 = 419,04

419,04 : 0,97 = 432

22. 846,40 : 32 = 26,45

El metro cuesta 26,45

Piden 20 m

26,45 20 = 529

23. 750 : 2 400 = 0,3125

0,3125 84 = 26,25

24. 2,60 12 = 31,2

31,2 : 13 = 2,4

25.x3

+ 40 = + 40x 2x3

x3

x3

+ 40 + 40 = 8425

Clave: e

Clave: d

Clave: a

Clave: b

Clave: c

Clave: c

Clave: c

Clave: b

Clave: b

Clave: d

Clave: c

3x + 360 = 1 260

x = 300

1. FFVF

3.

4.

2.12

= 0,5

34

= 0,75

3 = 1,73 p= 3,14

2 2 = 2,82

Ordenamos de mayor a menor

p; 2 2 ;12

; 34

; 3

Pg. 44

2,8 + 0 10 3 + 1523

p

2

B A = 3; +

+ 32

AB


17/157

Matemtica II

Clave: a

Clave: e

Clave: b

Clave: a

Clave: b

Clave: c

Clave: c

Clave: a

Clave: b

Clave: d

Clave:d

Clave: d

10. |x + 1| = 0

x + 1 = 0

x = 1

11. |3x 2| = 0

3x 2 = 0

x =23

12. |x 3| = 2

x 3 = 2 x 3 = 2

x = 5 x = 1

C.S. = {1; 5}

13. II y III

14.

15.

16.

17.

5.

6.

7.

8.

9.

A B = [1; 5]

A B = [1; 37; 9

B A = [2; 1

Q P = ; 5

M N = [8; 2]

M N = [4; +

A B =

A B = [2,3; 3,1

+

+

+

+

+

+

+

+

2

7 9

2

9

8

5

0

3

1

5

2

2

8

4

3

3,1

1 5

1

2

0

5

6

2

2,3

A

A

A

Q

M

M

A

A

B

B

B

P

N

N

B

B

1 + 2 3

2 p 3,777

4 5

e + 1 5 + 2

158


18/157

18 Matemtica II

Clave: b

Clave:

c

Clave: a

Clave: a Clave: b

Clave: b

Clave: e

Clave: e

18. 5 2x 1 < 4

+1 4 2x < 5

: 2 2 x 0)

y =7

x(x 4)

x 0 x 4 0

x 4

Dom = {0; 4} ...............(V )

Rpta: V F V

9. 2a + 1 = 5

2a = 6 a = 3

b = b + 4

2b = 4 b = 2

Piden:

3a 2b = 3(3) 2(2) = 9 + 4 = 5

8. g(x) = 7 + (3x 6)

3x 6 0

3x 6

x 2

Evaluamos para x = 2

7 + (3(2) 6))

7 + 0 = 7

Luego, Rango = [7, +

2

+ +

0 1

12

12


33/157

Matemtica II

12. f: funcin cuadrtica

f(x) = ax2+ bx + c

f(1) = 2

f(1) = 2f(2) = 4

Reemplazando:

a + b + c = 2 ...(I)

a b + c = 2 ...(II)

4a + 2b + c = 4 ...(III)

De (I) y (II)

2a + 2c = 0

a + c = 0 a = c

De (II) y (III)

2a 2b + 2c = 4

4a + 2b + c = 4

6a + 3c = 8

6(c) + 3c = 8

6c + 3c = 8

3c = 8

c =83

a = 83

En (I):

a + b + c = 2

83

+ b +83

= 2

b = 2

Reemplazamos en la funcin:

f(x) = 83

x2+ 2x + 8

Clave: b

Clave: a

Clave: a

Clave: a

13.f(x) = 42x 1 + 6 3x

Analizamos por partes:42x 1 2x 1 0

x 1

2 6 3x 6 3x 0

3x 6

3x 6

x 2

0 1/2 1 2

x [1/2; 2]

14.Ran(f ) = {3; 22; 59} , f(x) = x3 5

x3 5 = 3 x3 5 = 22 x3 5 = 5

x1= 2 x2= 3 x3= 4

Dom(f) = {2; 3; 4}

Piden: 2 + 3 + 4 = 9

15.f = {(2; 3), (3; 6), (2; 3a + b), (3; b)}

Se cumple:

3 = 3a + b b = 6

3 = 3a + 6

3 = 3a

a = 1Piden:

a + b = 1 + 6

= 5


34/157

34 Matemtica II

Clave: c

Clave: d

16.f(x) =1

x2 x

x2 x 0

x(x 1) 0

x 0 x 1 0 x 1

Dom f = {0, 1}

17. I. y = 2x + 7 (es funcin)

II. y = x2 2 (es funcin)

III. x2+ y2= 16 (no es funcin)

Rpta: Iy II

20. f(x) = x2+ x 2

f(3) = (3)2+ 3 2 = 10

f(4) = (4)2+ 4 2 = 18

f(5) = (5)2+ 5 2 = 28

f(2) = (2)2+ 2 2 = 4

f(1) = (1)2 + 1 2 = 0

f(0) = (0)2+ 0 2 = 2

Verificando:

I. f(3) + f(4) f(5) = 0

10 + 18 28 = 0 ... (V)

II. 2f(2) f(0) = f(3)

2(4) (2) = 10

8 + 2 = 10 ... (V )

III. 5f(2) 2f(3) = f(1)

5(4) 2(10) = 0

20 20 = 0 ... (V )

Rpta: VVV

18.x2 2x + 3 =x2

+ 2

2x2 4x + 6 = x + 4

2x2 5x + 2 = 0

x1= 2 x2 =12

Si x1= 2 y =22

+ 2 = 3P(2; 3)

Si x2=

1

2

y =

1

4 + 2 =

9

4

Q

1

2 ;

9

4

Suma de coordenadas:

2 + 3 +12

+94

=

8 + 12 + 2 + 94

=314

21.

Rpta: Forma un rectngulo

19.

Clave: c

Clave: b

Clave: e

Clave: d

Dom (f) = [2;3]

Ran (f) = [4; 5]

Dom (f) Ran (f) = [2; 3]

5

10

y

y

x

x

f

3

6

4

0

2

y = 10 x = 6f(y) = 6

f(x) = 10


35/157

Matemtica II

22. f(x) = mx + b ... (I)

7 = f(4) = 4m + b

1 = f(3) = 3m + b

7 = 4m + b

1 = 3m b ()

6 = m

Reemplazamos en (I)

f(x) = 6x 17

Piden: f(2) = 6(2) 17

f(2) = 29

Piden Dom(f) Dom(g)

= [3; + {3}

= 3; +

A = b h2

A =10 5

2

A = 25

24. f(x) = x 3

x 3 0

x 3

Dom(f) = [3; +

g(x)=7 4x

x 3

x 3 0

x 3

Dom(g) = {3}

Clave: a

Clave: d

Clave: b

b = 17

5

y x = 5

(5; 5)

(5; 5)

x5

5

0

23.Sea : {(x, y} 2/x = |y| } y x = 5

Clave: d

Clave: c

1. Iy IV

2. 8x3y + 6x3y 10yx3= 12x3y

Pg. 77

Clave: a

Clave: a

Clave: c

Clave: a

3. {[7a2b3+ 8a3b2] 5a2b3} + 8a3b2

= {+7a2b3 8a3b2 5a2b3} + 8a3b2

= 2a2b3 8a3b2+ 8a3b2

= 2a2b3

4. P = 3x2 x4 + (2x2)3 100x12

P = 3x6+ 8x6 10x6

P = x6

Piden:

P2= (x6)2= x12

5. (3x6y5)2 81x24y20

9x12y10 9x12y10= 0

6.tpq10

33;

83

pt 10qx

Por trminos semejantes se cumple:

1 = t 10 t = 11x = 10

Luego los coeficientes son:1133

y83

Piden:1133

+83

= 3


36/157

36 Matemtica II

Clave: e

Clave: b

Clave:

b

Clave: e

Clave: d

10. P(x) = 3x + 5

P(P(x) = 3(P(x) 5) + x

P(3x + 5) = 3(3x + 5 5) + x

3(3x + 5) + 5 = 3(3x) + x

9x + 15 + 5 = 9x + x

20 = x

11. A =G.A. (P) + G.R. (z)

G.R. (x) G.R. (y)

A =12 + 5

4 3

A = 17

9.

P(1; 2) = 4(1)(2)

3

6(1)

2

(2) = 32 + 12 = 44

7. 9x5+ (30x13 : 5x10)2+ 4 28x10

7x83

+ x4

9x5+ (6x3)2+ 4 (4x2)3 + x4

9x5+ 36x6+ 4 64x6+ x4

Ordenando:28x6+ 9x5+ x4+ 4

8. Si P(x 2) = 3x + 4

x 2 = 5

x = 7

P(5) = 3(7) + 4 = 21 + 4 = 25

x 2 = 3

x = 5

P(3) = 3(5) + 4 = 15 + 4 = 19

x 2 = 4

x = 6

P(4) = 3(6) + 4 = 189 + 4 = 22

Piden:

P(5) + P(3) P(4)

= 25 + 19 22 = 22

Clave: c

Clave: e

Clave: b

Clave: c

Clave: d

12. 3x7yz6 + a

Por dato, G.A. = 16

7 + 1 + 6 + a = 16

Piden: a = 2

(2)3= 8

13. x6a + 4y3a + 6z4a + 2

Por dato, G.A. = 38

6a + 4 + 3a + 6 + 4a + 2 = 38

13a + 12 = 38

a = 2

Luego el monomio es: x16y12z10

G.R.(y) = 12

14. G.A. = 14 a + 6 = 14 a = 8

Piden:

a2 3a + 1 = 82 3(8) + 1 = 41

15. Por dato:

G.A. (M) = 23

2a + 3a 1 + 4a 2= 23

3 9a 3 = 69

a = 8

Piden: [2(8)]2= 256

16. Por dato:

G.R(x) = 14

4a + b = 14 ... (I)

G.A. (M) = 18

14 + 2a b = 18

2a b = 4 ... (II)

Sumamos (I) y (II)

6a = 18

a = 3 b = 2

Piden: a + b = 5


37/157

Matemtica II

Clave: cClave: d

Clave: b

Clave: a

Clave: b

Clave: e

Clave: c

Clave: c

17. Por dato:

P(2) = 7

2a + b = 7 ... (I)

P(3) = 8

3a + b = 8 ... (II) Restamos (I) y (II)

5a = 15

a = 3

21. P(x) = 5ax2+3bx 6 10x2+ 9x + 2c

= (5a 10)x2+ (3b + 9)x + (2c 6)

Como P(x)0 , se cumple:

5a 10 = 0 a = 2

3b + 9 = 0

b = 3 2c 6 = 0 c = 3

Piden :

b2+ c2=

(3)2+ 32= 9

a 2

22. Dato:

G.A.(P) G.R.(x) + 2G.R.(y) = 24

2n + 3 ( n + 4) + 2( n + 2) = 24

3n + 3 = 24

n = 7

Piden : G.R.(x) + G.R.(y) = n + 4 + n + 2= 2n + 6

= 2(7) + 6

= 20

23. R = P(40) P(P(P(2)))

R = 3(40) + 5 P(P(11))

R = 125 P(38)

R= 125 119

R = 6

24. P(x) = 4x3 6ax + (2b 5)

Por dato: T.I. = 7

2b 5 = 7

b = 6

Luego, P(x) = 4x3 6ax + 7

Por dato: P(2) P(1) = 18

(32 12a + 7) (4 + 6a + 7 ) = 18

39 12a 3 6a = 18 a = 3

Piden : ( 5b a)1/a= (30 3)1/3

= 3

18. Por dato: P(Q(Q(2))) = 1 ... (I)

Calculamos: Q(2) = 2 + 3

= 5

Reemplazamos en (I)

P(Q(5)) = 1

Calculamos Q(5) = 5 + 3 = 8

Reemplazamos: P(8) = 1

82 5(8) + m = 1

m = 23

19. 13x 15 = 2ax + 5a + bx 7b

13x 15 = (2a + b)x + (5a 7b)

2a + b = 13

(x7) 14a + 7b = 91 ...(I) 5a 7b = 15 ...(II)

Sumamos (I) y (II)

19a = 76

a = 4 b = 5

Piden: a b = 20

20. P(x) = 4x + 2x4+ 6mxm-5 3x3 4

Como el polinomio es completo, entonces:

m 5 = 2

m = 7

Piden:

4 + 2 + 6m 3 4

= 6m 1

= 6(7) 1

= 41


38/157

38 Matemtica II

Clave: a

1. {m + [( 4m + 8) 5m] + 9} + 8m

{m + [ 4m + 8 5m] + 9} + 8m

{m + [ 9m + 8] + 9} + 8m

{m 9m + 8 + 9} + 8m

{8m + 17} + 8m

8m + 17 + 8m = 17

Pg. 82

Clave: c

Clave: b

Clave: e

Clave: d

25. Por P. homogneo, se cumple:

a + 2 + 1 = 4 + b 2

a b = 1

Por dato:

b

2

a

2

= 5 (b a) (b + a) = 5

1

b + a = 5

a = 2 , b = 3

Luego: P(x; y) = 2x4y + 3x4y

= 5x4y

P(a; b) = P(2; 3)

= 5(2)4(3)

= 290

Piden: 290 = 5

48

28. Q(x) = m(x2 3x + 2) + n(x2 5x + 6)

+ p(x2 4x + 3)

Q(x) = (m + n + p)x2 (3m + 5n + 4p)x

+ (2m + 6n + 3p) Por dato: P(x) Q(x)

m + n + p = 7

3m + 5n + 4p = 6

2m + 6n + 3p = 1

Resolviendo:

m = 41 ; n = 7 ; p = 41

Piden:

m + n p = 41 + 7 (41)

= 89

26. Por P. homogneo, se cumple:

a2 + 3 + 2a = 3a +5

a2 a 2 = 0

a 2

a 1

a = 2 a = 1 ( no cumple)

Piden: (3a 1) (5a a3)

= a3 2a 1

= (2)3 2(2) 1

= 3

27. 8x 3 = 2ax a + bx + 3b

8x 3 = (2a + b)x + (3b a)

2a + b = 8 ... (I)

3b a = 3 (x2) 6b 2a = 6 ... (II)

Sumamos (I) y (II)

7b = 2 b =2

7

Reemplazamos en (I)

2a + 2

= 8 7

a =

27

7Piden : a + b =

27+

2=

29

7 7 7


39/157

Matemtica II

10. (5x2 6x 7) (x2+ 2x 1)

= 5x4+ 10x3 5x2+ 6x3 12x2+ 6x + 7x2

14x + 7

= 5x4+ 16x3 10x2 8x + 7

Piden:

5 + 16 10 8 + 7 = 0

2. A(x) = 4x2+ 3x 2

B(x) = 3x2+ 8x + 9

3A(x) 2B(x) 18x2+ 9x + 24

3(4x2+ 3x 2) 2(3x2+ 8x + 9) 18x2+ 9x + 24

12x2+ 9x 6 + 6x2 16x 18 18x2+ 9x + 24

18x2 7x 24 18x2+ 9x + 24 = 2x

3. 5x4 3x8 + 36x16 : 4x4 (2x3)4 8x12

15x12+ 9x12 16x12 8x12 = 0

4. [(2x3)4 : 8x10+14x5

7x3 3x2]8

[16x12 : 8x10+ 2x2 3x2]8

[2x2+ 2x2 3x2]8

[x2]8= x16

8. A = 8x 3x5+ 84x9 : 2x3 25x6 (3x2)3+ 2x6

A = 24x6+ 42x6 25x6 27x6+ 2x6

A = 16x6

Piden:

(A : 4x5)3= (16x6 : 4x5)3= (4x)3= 64x3

9. 2[P(x) + Q(x)] = 2[3x2+ x + 3)

= 6x2+ 2x + 66.

83

x6y5 72

x2y638 x

3y4

72

x6 + 2 3y5 6 + 4 = 72

x5y3

Reemplazando:

a = 7

2 , m = 5 , n = 3

Piden:

72

+ 15

13

= 105 + 6 1030

= 10130

7. A(x) = 2,3 + 0,4x2 + 3,8x

B(x) = 359

x + 2310

+ 25

x2

Usando generatriz:

A(x) = 25

x2 + 359

x + 2310

B(x) = 25

x2+ 359

x + 2310

Piden:

E = 4[A(x) B(x)]

E = 4 25

x2+ 3510

x + 2310

25

x2 3510

x 2310

E = 4[0] = 0

5.42xn + 8ym + 5z4 + p

6x6 + ny2 mz3 + p

= 7xn + 8 6 nym + 5 2 + mz4 + p 3 p

= 7x2y3z

Clave: c

Clave: c

Clave: b

Clave: b

Clave: e

Clave: e

Clave: e

Clave: a

Clave: d


40/157

40 Matemtica II

13.A = (5x + 2) (2x + 3) = 242

10x2+ 19x + 6 = 242

10x2+ 19x 236 = 0

10x + 59

x 4

(10x + 59) (x 4) = 0

x = 4

largo () = 5x + 2

= 5(4) + 2

= 22

ancho (a) = 2x + 3

= 2(4) + 3

= 11

Permetro = 2+ 2a = [2(22) + 2(11)]

= 66

17.(3x 4)(2x + 1) (3x + 2)(x 6) 3x(x 5)

6x2+ 3x 8x 4 (3x2 18x + 2x 12) = 3x2 + 1

6x

2

+ 3x 8x 4 3x

2

+ 18x 2x + 12 3x

2

+ 15x26x + 8

Clave: d

Clave: e

11. V = (7x + 3) (6x + 4) (5x + 2)

V = (42x2+ 46x + 12) (5x + 2)

V = 210x3 + 314x2 + 152x + 24

Para x = 1

V = 210 + 314 + 152 + 24

V = 700

12. A = (2x + y)(x y)

A = 2x2 2xy + xy y2

A = 2x2 xy y2

14. M(x) S(x) = D(x)

S(x) = M(x) D(x)

= 2x2+ 6x 3 (4x + 7)

= 2x2+ 2x 10

Clave: a

Clave: d

Clave: d

15. LO= 2(x 1)

= (2x 2)

16. A(x) B(x) = (2x + 5)(x 8)

ax2+ bx + c = 2x2 11x 40

a = 2 , b = 11 , c = 40

Piden : a + b c = 2 11 (40)

= 31

19. (3x + 2)(x 6) (2x 1 )(x 2)

= 3x2 18x + 2x 12 2x2+ 4x + x 2

= x2 11x 14

Piden: menor coeficiente = 14

20. Por dato:

G.R.(x) = 7

a + 5 = 7

a = 2

G.R.(y) = 6

b + 3 = 6

b = 3

18. A =3x2+ 2y2+ x2 y2

4x2y 2

A = (4x2+ y2)2x2y

A = 8x4y + 2x2y3

Clave: c

Clave: d

Clave: c

Clave: e


41/157

Matemtica II

22.R(x; y) = 8x2y4(5x4y2 6x3y4 x5y)

3x3y3(2x5y3+ 3xy6 2x4y5)

P(x; y) = 40x6y6 48x5y8 8x7y5+ 6x8y6

9x4y9+ 6x7y8

Luego G.A.(R) = 15

23.

A = [2(4x + 3y)] [3(5x 2y)]

A = (8x + 6y)(15x 6y)

A = 120x2 48xy + 90xy 36y2

A = 120x2+ 42xy 36y2

24.

V = (2x + 1)(x + 4) (3x 2)

V = (2x2 + 8x + x+ 4)(3x 2)

V = (2x2+ 9x + 4)(3x 2)

V = 6x3 4x2+ 27x2 18x + 12x 8

V = 6x3+ 23x2 6x 8

21.P(x) = 4x 3

Q(x) = 7x2 6x

R(x) = x3+ 5

5x3P(x) + 3x4R(x) + 2Q(x)

5x3 (4x 3) + 3x4(x3+ 5) + 2(7x2 6x)

20x4 15x3+ 3x7+ 15x4+ 14x2 12x

3x7+ 35x4 15x3+ 14x2 12x

G.R.(x) = 7

Clave: d

Clave: d

Clave: e

Clave: c

5x 2y

4x + 3y 2(4x + 3y)

3x 2

2x + 1x + 4

3(5x 2

Clave: b

Luego:

P(x; y) = x7y3+ 2x6y6+ 3x5y4

P(b; a) = P(3; 2)

= (37)(23) + 2(36)(26) + 3(35)(24)

= 17 496 + 93 312 + 11 664

= 122 472

P(a; b) = P(2; 3)

=(27)(33) + 2(26)(36) + 3(25)(34)

= 3 456 + 93 312 + 7 776

= 104 544

Piden:

122 472 104 544 429

= 17 499

Clave: b

Solucin de problemas

1. A R B

H. Valdizn Cercado de Lima

A. Loayza Miraflores

J.C. Ulloa AteVitarte

R = {(Hermilio Valdizn, Ate Vitarte),

(Arzobispo Loayza, Cercado de Lima),

(Jos Casimiro Ulloa, Miraflores)}

Pg. 84


42/157

42 Matemtica II

Clave: a

Clave: b

Clave: e

Clave: d

Clave: d

Clave: a

2. Distancia: d

Tiempo: t

d = 80t

Para t = 4,

d = 80(4)

d = 320

3. T: tarifa (S/.)

M: Masa (g)

Arequipa: T1= 3,50 + 0,75 m

Cusco: T2= 5,00 + 1,25 m

Trujillo: T3= 2.50 + 0,40 m

T1 = 3,50 + 0,75(7) = 8,75

T2= 5,00 + 1,25(13) = 21,25

T3= 2,50 + 0,40(4) = 4,10

Total = 8,75 + 21,25 + 4,10 = 34,10

5. Entrada: S/. 15 (por persona)

Souvenir: S/. 5

Sea g: Gasto total por persona

x: Cantidad de souvenirs

Luego, g(x) = 15 + 5x

6.

2p = 2(3x + 7y) + 2(2x + 3y)

2p = 6x + 14y + 4x + 6y

2p = 10x + 20y

7. y(t) = 2,68 + 0,01t

y es una funcion afn

Para t = 8, tenemos:

y(8) = 2,68 + 0,01(8) = 2,76

4. 1 S/. (2x2+ x + 1)

2 S/. (x2+ x 3)

3 S/. (5x + 2)

4 S/. (4x + 1)

5 S/. y

Luego se cumple:

(2x

2

+ x + 1) + (x

2

+ x 3) + (5x + 2)+ (4x + 1) + y = 3x2+ 14x + 1

3x2+ 11x 1 + y = 3x2+ 14x + 1

y = 3x + 2

2x + 3y

3x + 7y

Clave: b

Clave: b

9. 1: 4x + 5

2: 3(4x + 5) = 12x + 15

3: 12x + 15 + 50 = 12x + 65

Total: 4x + 5 + 12x + 15 + 12x + 65

Total = 28x + 85

8. V = (4x2y) (7x3y2)(5x2 + y3)

V = 28x5y3(5x2 + y3)

V = 140x7y3+ 28x5y6


43/157

Matemtica II

Clave: a

Clave:c

Clave: b

Clave: e

Clave: a

Clave: c

Clave: c

Clave: e

Taller de prctica

1. A = {5; 6; 7; 8} n(A) = 4

B = {4; 5; 6; 7; 8} n(B) = 5

n(A B) = 4 x 5 = 20

3. E = 23 + 1923 8 1

= 4214

= 3

4. a + b = 8

a b = 2

2a = 10

a = 5 b = 3

Luego:

H = 9 3 + 2(5)

H = 16

5. 3a 1 = 5 b + 1 = 7

3a = 6 b = 6

a = 2

T1= 4x5y7 ; T2= 33x5y7

Piden: 4 + 33 = 29

6. P(x; y) = 6xm+1ym2+ 4xm+3ym1

2m 1 2m + 2

G.A.(P) = 2m + 2

18 = 2m + 2

m = 8

2. AR

B

.11 .5

.12 .6

.13 .7

.14 .8

.15 .9

.10

Ran(R) = {5; 6}

Pg. 88

10.y(t) = 1 500 + 55 t2

4

t: 1 dia = 24 h

y(24) = 1 500 + 5(24) 242

4

y(24) = 1 500 + 120 144

y(24) = 1 476

11.y = 72 + 6x x2

Para x = 10, tenemos:

y = 72 + 6(10) 102

y = 32

Clave: d

Clave: b

7. A(x) = 3x2+ 5x 7

B(x) = 2x2 4x + 1

A(x) + B(x) = x2+ x 6

8. Q(Q(x)) = Q(2x 5)

= 2(2x 5) 5

= 4x 10 5

= 4x 15


44/157

44 Matemtica II

Clave: e

9. y = 4 3xx + 5

yx + 5y = 4 3x

yx + 3x = 4 5y

x = 4 5yy + 3

x y + 3 0

y 3

Ran (f ) = {3}

1. f(x) = (3 (x + 1)2 1)(3 (x + 1)2+ 3x + 1 + 1)

3x(3 x + 1 + 1)

f(x) =(3x + 1

2 1)(3x + 1

2+ 3x + 1 + 1)

3x(3 x + 1 + 1)

f(x) = (3

x + 1 + 1)(3

x + 1 1) + (3

x + 12

+3

x + 1 + 13x(3 x + 1 + 1)

f(x) =(3x + 1 1)(3x + 12 + 3x + 1+ 1)

3x

f(x) =(3x + 1)3 13

3x

f(x) =x + 1 1

3x=

x3x

=13

Es decir, f es una funcin constante, pues ndepende de la variable "x".

Luego, f(1010) =13

Clave: b

Clave: d

Clave: a

Clave: a10. m + 2n = 13

2m n = 6 ...(2)

m + 2n = 13

4m 2n = 12

5m = 25

m = 5 n = 4

f = {(3; 13), (5; 6); (6; 11), (2; 3); (4; 5)}

Dom(f ) = {3; 5; 6; 2; 4}

Ran(f ) = {13; 6; 11; 3; 5}

Dom(f) Ran(f) = {3; 5; 6}

11.p + q r = 8 ...(I)

q + r p = 2 ...(II)

p q + 2 = 6 ...(III)

De (I) y (II): 2q = 6 q = 3

En (III) : p 3 + 2 = 6 p = 7

En (I): 7 + 3 r = 8 r = 2

Piden:

pq 1+ 7r + qr + 1= 72 7(2) + 33

= 49 14 + 27 = 62

12.b2+ 2b 35 = 0

b 7 b = 7

b 5 b = 5

3c 7 = 0 c =73

2d2 72 = 0 d2= 36 d = 6

Piden:

F = 5(4) 3(5)

9(73

) 6=

515

=13

Pg. 89


45/157

Matemtica II

Clave: c

2.

Si la grfica de la funcin cuadrtica no inter-secta al eje de las abscisas, entonces < 0, esdecir, b2 4ac < 0.

Anlisis y procedimiento:

Como la grfica de la funcin

f(x) = 9x2 6nx + (n + 12) no interseca al eje de

las abscisas, entonces sus races no son reales,luego:

< 0

(6n)2 4(9)(n + 12) < 0

36n2 36(n + 12) < 0

36(n2 n 12) < 0

n2 n 12 < 0

n 4

n +3

(n 4)(n +3) < 0

Por el mtodo de los puntos crticos se tiene

n 3; 4

Entonces los valores enteros de "n" son 2; 1;0, 1; 2; 3.

Piden: (2) + (1) + (0) + (1) + (2) + (3) = 3

+ +

x

f(x) = ax2+ bx + c; a > 0

y

3 4

Clave: c

Clave: b

3. Tenemos las funciones inyectivas

f(x) = ax2+ bx + c; x [2; +

g(x) = ax2+ bx + d; x ; 2]

con a 0 d y c no necesariamente diferente

Consideremos a > 0

d < c, entonces las grcas de f y g son:

4. As: P(x)x 3 3

R = P(3 3)

Luego, aplicamos la regla de Ruffini.

1 3 3 3 9 3 0 5 7

3 3 3 3 9 3 0 0 15

1 3 0 0 5 22

Como el residuo es R = 22 3

entonces P(3 3) = 22 3

x

vrtice

Resto

fg

2

y

En ambas grficas, el vrtice tiene abscisa h =

Como h =x1+ x2

22 =

b2a

4a = b

4a + b = 0


46/157

46 Matemtica II

Clave: e

Clave: cClave: d

Clave: a

Clave: c

Clave: c

Clave: e

Clave: e Clave: a

Clave: e

Clave: c

Clave: c

1. A = b h

A = (x + 1)2(x + 1)

A = 2 (x + 1)

2

A = 2x2+ 4x + 2

2. V = (x2+ 1)3

V = (x2)3+ 3(x2)(1) + 3(x2)(1)2+ 13

V = x6+ 3x4+ 3x2+ 1

3. A = b h

A = (x + 1)(x 1) = x

2

1

4. A = (x + 3)2+ (x 2)2+ 2x2+ 8

A = x2+ 6x + 9 + x2 4x + 4 + 2x2+ 8

A = 4x2+ 2x + 21

5. P = (x + 2)2 (2x + 1)2+ x

P = x2+ 4x + 4 (4x2+ 4x + 1) + x

P = x2+ 4x + 4 4x2 4x 1 + x

P = 3x2+ x + 3

7. G = (x + 8)(x + 4) (x + 4)(x + 5) 3(4x 3) + 9x

G = x2+ 12x + 32 (x2+ 9x + 20) 12x + 9 + 9x

G = x2+ 12x + 32 x2 9x 20 12x + 9 + 9x

G = 21

6. J = (x 7)(x + 7) (x 7)2+ x2+ 49

J = x2 49 (x2 14x + 49) + x2+ 49

J = x2 49 x2+ 14x 49 + x2+ 49

J = x2+ 14x 49

8. N = (a + 2)(a 2)(a2+ 22) + 16

N = (a2 4)(a2+ 4) + 16

N = a4 16 + 16

N = a4

9. M = (5x + 4)(4x + 5) 20(x + 1)2

M = 20x2+ 16x + 25x + 20 20(x2+ 2x + 1)

M = 20x2+ 41x + 20 20x2 40x 20

M = x

10.E = ( 7 + 2)2 ( 7 2)2

Aplicamos la identidad de Legendre.

E = 4( 7)( 2)

E = 4 14

11.E = 32 1 + 3(22+ 1)(24+ 1)(28+ 1)

E = 32 1 + (22 1)(22+ 1)(24+ 1) (28+ 1)

E = 32 1 + (24 1)(24+ 1)(28+ 1)

E =32

1 + (28

1)(28

+ 1)

E = 32 1 + 216 1

E = 32 216= 2

12.F =(x + 9)2 (x + 13)(x + 5)

(x + 10)(x + 9) (x + 16)(x + 3)

F =x2+ 18x + 81 (x2+ 18x + 65)

x2+ 19x + 90 (x2+ 19x + 48)

F = x

2

+ 18x + 81 x

2

18x 65x2+ 19x + 90 x2 19x 48

16

42=

8

21

4U N I D A D

Pg. 97

8

21


47/157

Matemtica II

Clave: c

Clave: c

Clave: c Clave: c

Clave: a

Clave: a

Clave: b

Clave: b

Clave: d

13.P = (x + 3)(x + 5) (x 2)(x 6)

P = x2+ 8x + 15 (x2 8x + 12)

P = x2+ 8x + 15 x2+ 8x 12

P = 16x + 3

14.A = (x + 2)2+ (x + 4)2 2(x + 3)2

A = x2+ 4x + 4 + x2+ 8x + 16 2(x2+ 6x + 9)

A = 2x2+ 12x + 20 2x2 12x 18

A = 2

15.R = (x + a)(x a)(x2+ a2)(x4+ a4) + a8

R = (x2 a2)(x2+ a2)(x4+ a4) + a8

R = (x4 a4)(x4+ a4) + a8

R = x8 a8+ a8

R = x8

16.M = 5(2 + 2)3 14(1 + 2)3

M = 5[23+ 3(2)2( 2) + 3(2)( 2)2+ ( 2)3]

14[13+ 3(1)2( 2) + 3(1)( 2)2+ 23]

M = 5[8 + 12 2 + 6(2) + 2 2] 14[1 + 3 2 +

3(2) + 2 2]

M = 5(8 + 14 2 + 12) 14(1 + 5 2 + 6)

M = 5(20 + 14 2) 14(7 + 5 2)

M = 100 + 70 2 98 70 2

M = 2

17.A = (x2+ x + 6)(x2+ x 3) (x2+ x + 9)(x2+ x 6)

Hacemos cambio de variable, y = x2+ x

A = (y + 6)(y 3) (y + 9)(y 6)

A = y2+ 3y 18 (y2+ 3y 54)

A = y2+ 3y 18 y2 3y + 54

A = 54 18

A = 36

18.K = (x + 2)(x2 2x + 4) (x 3)(x2 + 3x + 9)

K = x3+ 23 (x3 33)

K = x3+ 8 x3+ 27

K = 35

19.M =(x + 2a)2+ (2x a)2

(x + a)(x a) + 2a2

M =x2+ 4ax + 4a2+ 4x2 4ax + a2

x2 a2+ 2a2

M =5x2+ 5a2

x2+ a2

M =5(x2+ a2)

x2+ a2

M = 5

20.A = [(xn+ 3)2 6xn]2 x2n(x2n+ 18)

A = [x2n+ 6xn+ 9 6xn]2 x4n 18x2n

A = [x2n+ 9]2 x4n 18x2n

A = x4n+ 18x2n+ 81 x4n 18x2n

A = 81

21.N = ( 13 2 13 + 2 x2)(x2+ 3) 9

N = ( 132 22 x2)(x2+ 3) 9

N = ( 13 4 x2)(x2+ 3) 9

N = ( 9 x2)(x2+ 3) 9

N = (3 x2)(3 + x2) 9

N = 9 x4 9

N = x4


48/157

48 Matemtica II

22.[(x + a)(x b) + ab][(x a)(x + b) + ab]+ (ax bx)2

= [x2+ (a b)x ab + ab][x2+ (b a)x ab + ab]

+ (a b)2 x2

= [x2+ (a b)x][x2 (a b)x] + (a b)2 x2

= (x2)2 (a b)2 x2+ (a b)2 x2

= x4

26.Por dato: x + y = 3

Elevamos al cubo

(x + y)3= (3)3

x3+ 3xy(x + y) + y3= 27

x3+ y3= 27 3(3)(3)

x3+ y3= 0

27.Por dato: a + b = 5; ab = 2

Elevamos al cuadrado

(a + b)2= (5)2

a2+ 2ab + b2 = 25

a2+ b2+ 2(2) = 25

a2+ b2 = 21

28.x 35 = 2 x 2 = 35

Elevamos al cubo

(x 2)3= 353

x3 3(2)x2+ 3x(2)2 23= 5

x3 6x2+ 12x 8 = 5

P = 5

29.x2+ 3x = 2 x(x + 3) = 2

A = x(x + 3)(x + 1)(x + 2) 2 2

A = 2 (x2+ 3x + 2) 2 2

A = 2( 2 + 2) 2 2

A = 2

30.x y = 45 , xy = 5

Por Legendre:

(x + y)2 (x y)2= 4xy

(x + y)2 (45)2= 4 5

(x + y)2= 5 5

x + y = 5 5

23.P = 3 (37 + 1)(349 + 1 37)

P = 3 (37)3+ 13

P = 37 + 1

P = 38

P = 2

24.K = ( 3 + 2)2 + ( 3 2)2

K = 2( 32+ 22)

K = 2(3 + 2)

K = 10

25.R = ( 7 + 3)2+ ( 7 3)2

( 5 + 3)( 5 3)

R =2( 72 + 32)

52 32

R = 2(7+3)5 3

R = 10

Clave: d

Clave: a

Clave: a

Clave:

c

Clave: e

Clave: b

Clave: c

Clave: c


49/157

Matemtica II

31.M = (x + 3)(x + 4)(x + 2)(x + 5) (x 2+ 7x + 11)2

M = (x2+ 7x + 12)(x2+ 7x + 10) (x2+ 7x + 11)2

Sea x2+ 7x = a, luego:

M = (a + 12)(a + 10) (a + 11)2

M = a2+ 22a + 120 a2 22a 121

M = 1

34. n +1

n

2

= 3

n +1

n= 3

Ahora elevamos al cubo:

n +1

n

3

= ( 3)3

n3+1

n3+ 3(n)

1

n n +

1

n= 3 3

n3+1

n3+ 3 3 = 3 3

n3+1

n3= 0

35.x2+ y2+ z2 = 29 ; x + y + z = 9

(x + y + z) = 9, elevamos al cuadrado

(x + y + z)2= (9)2

x2+ y2+ z2+ 2(xy + xz + yz) = 81

29 + 2(xy + xz + yz) = 81

2(xy + xz + yz) = 52

xy + xz + yz = 26

36.a + b = 5; a2+ b2= 17

De Legendre:

(a + b)2+ (a b)2= 2(a2+ b2)

52+ (a b)2= 2(17)

(a b)2= 34 25

(a b)2= 9

a b = 3

37.x + y + z = 0

x3+ y3+ z3= 3xyz

x2

yz+

y2

xz+

z2

xy

=x3+ y3+ z3

xyz=

3xyz

xyz= 3

32.(x + 2y)2+ (x 2y)2= 8xy

Por Legendre:

2(x2+ 4y2) = 8xy

x2 4xy + 4y2= 0

(x 2y)2= 0 x = 2y

Piden: A =2(2y)(y) y2

(2y)2

A =3y2

4y2

A =3

4

33.Del dato ab = 1, en M

M =ab4+ a

b3+ a+

ba4+ b

a3+ b

M = (ab)b3+ a

b3+ a+ (ba)a

3+ b

a3+ b

M =b3+ a

b3+ a+

a3+ b

a3+ b

M = 2

Clave: b

Clave: b

Clave: c

Clave: a

Clave: c

Clave: a

Clave: aClave: d

x + y = 5 45

Piden:

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

= ( 5 45)(4 5)

= 5


50/157

50 Matemtica II

Clave: d

Clave: d

Clave: c

Clave: a

Clave: b

38.a + b = 6 ; a2+ b2= 30

De a + b = 6, elevamos al cuadrado

(a + b)2= (6)2

a2+ 2ab + b2= 36

a2+ b2+ 2ab = 36

30 + 2ab = 36

ab = 3

Piden:

a2

b+

b2

a=

a3 + b3

ab=

(a + b)(a2+ ab + b2)

ab

=6[(a2+ b2) + ab]

ab

=6[30 + 3]

3

= 66

39.R = (x2+ 5x + 5)2 (x + 1)(x + 2)(x + 3)(x + 4)

R = (x2+ 5x + 5)2 (x2+ 5x + 4)(x2+ 5x + 6)

Sea x2+ 5x = a, luego:

R = (a + 5)2 (a + 4)(a + 6)R = a2+ 10a + 25 a2 10a 24

R = 1

42.Por Legendre:

(a + b)2+ (a b)2= 2(a2+ b2)

(a + b)2

+ (a b)2

= 2(8)a + b = 16 (a b)2

Para que (a + b) sea mximo, entonces

a b = 0, luego

(a + b)max.= 4

41.x

2y+

2y

x= 2

x2 + 4y2

2xy= 2

x2+ 4y2= 4xy

x2 4y2= 4xy

(x 2y)2= 0

x 2y = 0

x = 2y x

y= 2

Piden: x

y

8

= (2)8= 256

40. a b = 2; b c = 2

Elevamos al cuadrado a b = 2

(a b)2= (2)2

a2 2ab + b2= 4 ... (I)

Elevamos al cuadrado b c = 2

(b c)2= (2)2

b2 2bc + c2= 4 ... (II)

Luego: a b = 2

b c = 2

a c = 4

Ahora elevamos al cuadrado a c = 4

(a c)2

= 42

a2 2ac + c2= 16 ... (III)

Sumamos (I), (II), (III)

a2 2ab + b2= 4

b2 2bc + c2= 4

a2 2ac + c2= 16

2a2+ 2b2+ 2c2 2ab 2bc 2ac = 24

2(a2+ b2+ c2 ab bc ac) = 24

a2+ b2+ c2 ab bc ac = 12

+


51/157

Matemtica II

Clave: a

Clave: e

Clave: c

Clave:e

Clave: e

44.xn

yn+

yn

xn= 62

x2n+ y2n= 62xnyn

x2n

+ y2n

+ 2xn

yn

= 64xn

yn

(xn+ yn)2= 64xnyn

xn+ yn

xn yn= 8

Reemplazamos en A:

A = 38 = 2

45.T = (nn nn)2+ (nn+ nn)2 2(n2n n2n)

T = 2(n2n+ n2n) 2(n2n n2n)

T = 4n2n

46. Por dato:

x2+ 3x + 1 = 0

Piden:

47. Nos piden: x2+ y2

ax + by = 8

ay bx = 6

a2+ b2 = 5

Elevamos al cuadrado ax + by = 8

(ax + by)2= (8)2

a2x2 + 2abxy + b2y2= 64(I)

(ay bx)2= 62

a2y2 2abxy + b2x2= 36(II)

Sumamos (I) y (II):

a2x2+ 2abxy + b2y2= 64

a2y2 2abxy + b2x2= 36

x2(a2+ b2) + y2(a2+ b2) = 100

(x2+ y2)(a2+ b2) = 100

(x2 + y2) 5 = 100

x2+ y2= 20

43.E = x2+1

x2 4(x +

1

x) + 6

E =x4+ 1

x2

4x2+ 4

x+ 6

E =x4+ 1 4x3 4x + 6x2

x2

E =x4 4x3+ 6x2 4x + 1

x2

D(x) = x4 4x3+ 6x2 4x + 1

Para x = 1 x 1 = 0

R(x) = D(1)

R(x) = 1 4 + 6 4 + 1

R(x) = 0

Entonces buscamos los factores primos con

Ruffini

1 4 6 4 1

x = 1 1 3 3 1

1 3 3 1 0

x = 1 1 2 1

1 2 1 0

Q(x) = x2 2x + 1 = (x 1)2

D(x) = (x 1)(x 1)(x 1)2= (x 1)4

E =(x 1)4

x2=

(x 1)2

x

x

x + 1+ =

= = = 0

1

x + 2

x2+ 2x + x + 1

x2+ 3x + 2

x2+ 3x + 1

x2+ 3x + 2

0

0 + 1


52/157

52 Matemtica II

Clave: b

Clave: b

Clave: eClave: e

Clave: e

Clave: c

Clave: d

48. K =(ax + by)2+ (ay bx)2

x2+ y2

K=a2x2+ 2abxy + b2y2+ a2y2 2abxy + b2x2

x2+ y2

K= x

2

(a

2

+ b

2

) + y

2

(a

2

+ b

2

)x2+ y2

K=(x2+ y2) (a2+ b2)

x2+ y2

K = a2+ b2

1. 16x4 8x2 12x3+ 8x 4x2 2

16x

4

+ 8x

2

4x

2

3x 12x3+ 8x

12x3 6x

2x

Q(x) = 4x2 3x

R(x) = 2x

2. 1 1 4 6 7 2

2 2 1

2 4 2

1 1 2 1

1 2 1 11 1

R(x) = 11x + 1

3. Completamos:

3x5+ 0x4 5x3+ 0x2 3x + 7

x2 x 1

1 3 0 5 0 3 7

1 3 3

1 3 3 3

1 1 1

4 4 4

3 3 1 4 2 11

5. Completamos y ordenamos:

1 3 2 1 0 0 2

0 0 3 3 1 2 0 2 2

1 4 0 4 4

1 0 1

3 2 4 1 2 3

Piden: Q(x) = 3x3+ 2x2+ 4x 1

6.

3 12 2 1 5 91 4 8

2 6 2 4

9 3 6

4 2 3 2 3

Piden: (2)(3) = 6

4. 8x

3

15x

4

+ 6x

5

23x

2

+ 42x3 1 5x2

Ordenamos:

6x5 15x4+ 8x3 23x2+ 0x + 4

2x3 5x2+ 0x 1

Por Horner:

2 6 15 8 23 0 4

5 15 0 3

0 0 0 0 01 8 20 0 4

3 0 4 0 0 8

Q(x) = 3x2+ 4

R(x) = 8

Pg. 104

Q(x) = 3x3+ 3x2+ x + 4

El trmino lineal es x


53/157

Matemtica II

Clave: e

Clave: c

Clave: b

Clave: b

Clave: bClave: c

7. Ordenamos:

x3 x2+ 33x + 20

x + 6

Por Ruffini:

x + 6 = 0 1 1 33 20

x = 6 6 30 18

1 5 3 2

Q(x) = x2+ 5x + 3

R(x) = 2

8. Ordenamos:

x3 x2+ 10x + 28

x + 6

Por Ruffini:

x + 6 = 0 1 1 10 28

x = 6 6 30 120

1 5 20 148

Q(x) = x2+ 5x 20

9.x3 2x + 1

x + 3

Por el teorema del resto:

x + 3 = 0

x = 3

R(x) = D(3)R(3) = (3)3 2(3) + 1

R(3) = 27 + 6 + 1

R(3) = 20

10.Por el teorema del resto:

x + 4 = 0

x = 4

R(x) = D(4)

R(x) = (4)4

+ 4(4)3

2(4)2

5(4) + 12R(x) = 44 44 32 + 20 + 12

R(x) = 0


x + 1 = 0

x = 1

R(x) = D(1)

R(x) = 2(1)12+ (1)8+ (1)7+ 3(1)5 (1)

R(x) = 2 + 1 1 3 + 1 1

R(x) = 1


5x 1 = 0

x = 15

R(x) = D1

5

R(x) = 151

5

4

81

5

3

91

5

2

+ 71

5+ 1

R(x) =15

625

8

125

9

25+

7

5+ 1

R(x) =15 40 225 + 875 + 625

625

R(x) =1 250

625= 2

R(x) = 2


54/157

54 Matemtica II

13. Por el teorema del resto:

x + 1 = 0

x = 1

R(x) = D(1)

R(x) = (1)5+ (1)4+ (1)3+ (1)2+ (1) + 1

R(x) = 1 + 1 1 + 1 1 + 1

R(x) = 0


x a = 0

x = a

R(x) = P(a)

R(x) = a29+ 8a28+ 16a27

0 = a29+ 8a28+ 16a27

0 = a27(a2+ 8a + 16)

Como a 0

a2+ 8a + 16 = 0

(a + 4)2= 0

a = 4

17.Completamos:

3 3 0 7 A B

3 3 2

2 3 3 2

12 12 8

1 1 4 0 0

A 14 = 0 B + 8 = 0

A = 14 B = 8

Piden:A

B=

14

8=

7

4


x 2 = 0

x = 2

R(x) = D(2)

R(x) = 25+ 2(2)4 3(2)3 2(2) a

14. 1 1 5 10 10 5 1

3 3 3 1

3 2 6 6 2

1 1 3 3 1

1 2 1 0 0 0

Q(x) = x2+ 2x + 1

15. 6x5+ 4x4 26x3+ 33x2 24x + 62x3+ 0x2 3x + 1

2 6 4 26 33 24 6 0 0 9 3

3 4 0 6 2

1 17 0 51

2

17

2

3 2 17

2 36 26

51

2 6+

17

2

Clave: c

Clave: e

Clave: a

Clave: b

Clave: c

Q(x) = 3x2+ 2x 17

2Piden:

3 + 2 17

2=

7

2

Clave: c

0 = 32 + 32 24 4 a

a = 64 28

a = 36


55/157

Matemtica II

25.

1 3 1 2 a a

1 3 3

1 4 4 4

9 9 93 4 9 0 b

a 13 = 0 ; a + 9 = b

a = 13 ; 13 + 9 = b

22 = b


x + 2b = 0

x = 2b R(x) = (2b + 3b)7 [(2b)7 11b7]

R(x) = b7 (128b7 11b7)

R(x) = b7+ 139b7

R(x) = 140b7

24.


x 1 = 0

x = 1

R(x) = D(1)

R(x) = [2(1) 1]16+ k(1)5+ 5(1)2 8

R(x) = [1 + k + 5 8]

0 = k 2

k = 2

R(x) = 62

3

3

192

3

2

+ 192

3 16

R(x) = 68

27 19

4

9+

38

3 16

R(x) =16

9

76

9+

38

3 16

R(x) =16 76 + 114 144

9

R(x) = 90

9

R(x) = 1020.1 2 1 0 a b

1 2 4

1 1 2

2 5 5 10

2 1 5 0 0

a + 7 = 0 10 b = 0

a = 7 10 = b

a b = 7 10

= 70

21.

5 5 11 10 a b

1 1 22 10 2 4

10 2 4

1 2 2 0 0

a 4 + 2 = 0 b + 4 = 0

a = 2 b = 4

Rpta.: 2 ; 4

Clave: a

Clave: a

Clave: e

Clave: a

Clave: c

Clave: c

Clave: d

x(x 1) + (x 2) (x + 1) + 4

(x + 2) (x 3)

Por el teorema del resto:x2 x 6 = 0

x2 x = 6

Reemplazamos en el dividendo:

R(x) = 6 + 6 + 2

= 14

(x2 x) + (x2 x) + 2

x2 x 6

22.Por el teorema del resto

3x 2 = 0

x =2

3

R(x) = D2

3


56/157

56 Matemtica II

Clave: e

Clave: b

Clave: a

Clave: b

26. R(x) = 5x2 3x + 7

2 8 0 4 m n p

1 4 0 12

0 4 2 0 6

3 2 1 0 3

4 2 1 5 3 7

m 11 = 5 n + 6 = 3 p + 3 = 7

m = 16 n = 9 p = 4

Piden:

m + n + p = 16 + (9) + 4 = 11

27.

3 6 0 1 a b

3 6 4

6 6 4

2 9 9 6

2 2 3 0 0

a + 13 = 0 ; b + 6 = 0

a = 13 b = 6

Piden: b a = 6 (13)

= 7

1.n + 2

4=

n + 4

5

5n + 10 = 4n + 16

n = 6

Clave: d

5. n = 7 ; k = 4

tk= xn k , yk 1

t4= x3 y3

Clave: a

4. 2p

2= 3q

1= 12

p = 12 ; q = 4

Piden: p + q = 16

Clave: c

6.155

5= 31 ;

93

3= 31 n = 31

tk= (1)k 1 xn k , yk 1 , k = 3

t3= x28 y2

Clave: e

7. n = 25 tc =

n + 1

2 = 13

t13= (x4)12 (y4)12

t13= x48 y48

2.xn+ ym

x3+ y4

n

3=

m

4= 21

n = 3(21) = 63

m = 4(21) = 84

Pg. 110

m + n = 84 + 63

m + n = 147

Clave: c

3.3a + 6

3=

a + 12

2

6a + 12 = 3a + 36

3a = 24 a = 8

N trminos = 3(8) + 63

= 10


57/157

Matemtica II

Clave: a

8.(3x)4 (1)4

3x 1n = 4 ; k = 3

t3= (3x)1(1)2

t3= 3x

Clave: b

9.(x + 5)5 25

(x + 5)1 21 n = 5 ; k = 4

t4= (x + 5)1 (2)3

t4= (x + 5)8

t4= 8x + 40

Clave: a

12. (x2

)41

141

x2 1

n = 41 ; k = 3

t3= (x2)41 3(1)3 1

t3= x76

Clave: b

14.a75 b30

a15 b6=

(a15)5 (b6)5

a15 b6

n = 5 ; k = 5

t5= (a15)5 5 (b6)5 1

t5= b24

G.A. = 24

Clave: b

16.(x2)2 (3xy2)2

x2 (3xy2)

x3+ (2y2)3

x + 2y2

= x2+ 3xy2 (x2 x(2y2) + (2y2)2)

= x2+ 3xy2 x2+ 2xy2 4y4

= 5xy2 4y4

Clave: e

13.(2x)5 y5

2x yn = 5

tc : k =n + 1

2=

5 + 1

2= 3

t3= (2x)5 3y3 1

t3= 4x2y2

Clave: d

10.y8 x8

y + x

n = 8 ; k = 6

tk= (1)k 1yn kxk 1

t6= y8 6x6 1

t6= y2x5

Clave: e

11.(x3)7+ (y3)7

x3+ y3

n = 7 ; k = 5

t5= (1)5 1 (x3)7 5 (y3)5 1

t5= x6y12

Clave: a

15. x3+ 1x + 1

+ x 1

x2 x + 1 + x 1 = x2


58/157

58 Matemtica II

17.x 3x 81

3x 3

x

1 + 34

x 3

=x 34

x 3

n = 4 y k = 3

t3= x4 3

(3)3 1

t3= 9x

13

13

13

13

43

13

Clave: c

Clave: e

Clave: c

Clave: a

Clave: e Clave: e

18.x64 y48

x4 y3

=(x4)16 (y3)16

x4 y3

n = 16 ; k = 6

t6= (x4)16 6 (y3)6 1

t6= x40y15

G.A. = 40 + 15G.A. = 55

19.a38 b57 c19

a2 b3 c

(a2)19 (b3c)19

a2 b3c

n = 19 ; k = 10

t10= (a2)19 10(b3c)10 1

t10= a18b27c9

G.A. = 18 + 27 + 9

G.A. = 54

20.(x + 3)36 x36

3

=(x + 3)36 x36

(x + 3) x

n = 36 ; k = 29

t29= (x + 3)36 29 (x)29 1

Para: x = 1:

t29= (1 + 3)7 (1)28

t29= 128

21. (x + 4)3 64x

=(x + 4)3 43

(x + 4) 4

n = 3 ; k = 2

t2= (x + 4)3 2 (4)2 1

t2= 4(x + 4)

22.P =x102+ x96+ x90+ + 1

x90+ x72+ x54+ + 1

P =

(x6)18 118

x6 1

=

x108 1

x6 1

P = x18

1x6 1

= (x6

)3

13

x6 1

P = (x6)2+ x6+ 1

P = x12+ x6+ 1

(x18)6 16

x18 1

x108 1

x18 1

x

4

34

x 3

13

13


59/157

Matemtica II

25.x75 yb

xc y2

75

c=

b

2= n

75

c= n

b

2= n

tk= (xc)

k

(y2)k 1

xay24= x(75 kc)y(2k 2)

a = 75 kc 24 = 2k 2

13 = k

Por termino central:

13 =n + 1

2

25 = n

Clave: e

Clave: b

75c

Clave:a

Clave: a

Clave: e

24.xa yb

x3 y7, se cumple:

a

3=

b

7= 9

a = 27 ; b = 63

(x3)9 (y7)9

x3 y7

t6= (x3)

9 6 (y7)6 1

t6= x9y35

26.x

4n 2

x

4n 4

+ x

4n 6

+ x

2

1

=x4n + 1

x2+ 1

27.xa + yb

x + y2 ; Por dato: tk= x

4y10

a

1=

b

2a = r ; b = 2r

xr + (y2)r

x + y2

tk= (1)k 1xr k (y2)k 1= x4y10

Se cumple:

r k = 4 ; 2k 2 = 10

k = 6

r = 6 + 4

r = 10

Piden:

a b = r 2r

= 2r2

= 2(10)2

= 200

23.xm yn

x5 y7 ;

m

5=

n

7m = 5r ; n = 7r

x5r y7r

x5 y7=

(x5)r (y7)r

x5 y7

t16= (x5)r 16 (y7)16 1

x95y105= x5r 80y105

Luego, se cumple:

5r 80 = 95

5r = 175

r = 35

Piden:

m + n = 5r + 7r

= 12r

= 12(35)

= 420

Luego:75

c= 25 c = 3

b

2= 25 b = 50

De: a = 75 kc

a = 75 (13)(3)

a = 36

Piden:

a + b + c = 36 + 3 + 50

a + b + c = 89


60/157

60 Matemtica II

28.xa yb

x3 y4

a

3=

b

4a= 3r ; b= 4r

t6= (x3

)r 6

(y4

)6 1

t6= x3r 18y20

t9= (x3)r 9 (y4)9 1

t9= x3r 27y32

t7= (x3)r 7 (y4)7 1

t7= x3r 21y24

Por dato:

t6 t9

t7= x12y28

x3r 18y20 x3r 27y32x3r 21y24

= x12y28

x3r 18 + 3r 27 3r + 21 y20 + 32 24= x12y28

x3r 24y28 = x12y28

3r 24 = 12

r = 12

Piden:

a+ b= 7r

= 7(12) = 84

Clave: b

Clave: d

Clave: e

Clave: c

Clave: e

Clave: c

Clave: a

Miscelnea Pg. 112 y 113


x 1 = 0

x = 1

R(x) = D(1)

R(x) = (1)3 2(1) + 1

R(x) = 0

2. (x + 8)(x 8) = x2 64

3. (x 3)

3

= x

3

3(x

2

)(3) + 3x(3)

2

3

3

= x3 9x2+ 27x 27

4. Como contamos a partir del final:

x3n yn

x3 y=

yn x3n

y x3

t8= (y)n 8 (x3)8 1

t8= yn 8x21

G.A. = n 8 + 21

38 = n + 13

25 = n

5. a + b = 6 ; ab = 3

Elevamos al cubo: a + b = 6

(a + b)3= 63

a3+ b3+ 3ab(a + b) = 216

a3+ b3+ 3(3)(6) = 216 a3+ b3= 162

6. A = pr2

A = p(2x + 1)2

A = p(4x2+ 4x + 1)

7.xm yn

x5 y7 ;

m

5=

n

7 m = 5r ; n = 7r

t8= (x5)r 8 (y7)8 1

t8= x5r 40y49


61/157

Matemtica II

Clave: e

Clave: e

Clave: d

Clave: e

Clave: b

Clave: a

8. a + b = 3 ; ab = 4

Elevamos al cuadrado a + b = 3

(a + b)2= (3)2

a2+ 2ab + b2= 9

a2+ b2+ 2(4) = 9

a2+ b2= 1

10.A=b h

2

A= (x 3)(x2

+ 3x + 9)2

A=x3 33

2

A=x3 27

2

12.

Averde= (x + 2)2+ 4 2 + (x + 1)2

= x2+ 4x + 4 + 8 + x2+ 2x + 1

= 2x2+ 6x + 13

11.

2 a 4 e 11 q r

1 = b 2 c = 6 4

3 d = 2 1 3 2 2 6 n = 3 9 6

2 m = 1 3 p = 1 3 10

Luego:

A = (m + n + p + q + r) (a + b + c + d + e)

A = (1 + 3 1 + 8 + 4) (4 + 1 6 2 + 13)

A = 13 10 = 3

9. A = pr2

D = (x 2)(x + 2)

D = (x2 4)

r =x2 4

2

Nos piden el rea:

A = px2 4

2

2

A = px2 8x + 16

4

G.A. = 5r 40 + 49

59 = 5r + 9

10 = r

Piden:

m + n = 5r + 7r

= 12r

= 12(10)

= 120

= 13: = 8 = 4=

4

x + 1

x + 3

x + 6

x + 2

2x + 7

3x + 10

4x

2

x + 1

x + 2

x + 1

2x + 3

13.Aazul= (x + 3)(3x + 4) + (x + 6)(x + 1)

= 3x2+ 13x + 12 + x2+ 7x + 6

= 4x2+ 20x + 18

Clave: c


62/157

62 Matemtica II

Clave: e

Clave: b

Clave: a

Clave: c

14.

AT= 2[(x + 4)(x + 2) + (x + 4)(x + 1) + (x + 2)(x + 1)]

AT= 2(x2+ 6x + 8 + x2+ 5x + 4 + x2+ 3x + 2)

AT= 2(3x2+ 14x + 14)

AT= 6x2+ 28x + 28

x + 4

x + 1

x + 2

Clave: e

Clave: e

Clave: e

Clave: e

Pg. 115

1. P = (x + 2)(x 2) (x 1)2 6x 1

P = x2 4 (x2 2x + 1) 6x 1

P = x2 4 x2+ 2x 1 6x 1

P = 4x 6

15.V = (x + 4)(x + 2)(x + 1)

V = (x2+ 6x + 8)(x + 1)

V = x3 + 7x2+ 14x + 8

4. E = ( 5 + 2)2+ ( 5 2)2

E = 2[( 5)2+ ( 2)2]

E = 2(5 + 2)

E = 14

Clave: b


x + 2 = 0

x = 2

R(x) = D(2)

R(2) = 3(2)3+ (2)2 5(2) + 20

R(2) = 24 + 4 + 10 + 20

R(2) = 10

Clave: e

5. a2+ b2= 3ab

(a + b)2 (a b)2

(a + b)2+ (a b)2

4ab

2(a2+ b2)=

4ab

2(3ab)=

2

3


x 1 = 0 x = 1

R(x) = D(1)

R(x) = (1)5+ (1)4+ (1)3 (1)2+ 6

R(x) = 8


x 1 = 0 ; R(x) = 0

x = 1

R(x) = D(1)

R(1) = (1)3+ m(1)2+ n(1) + 1

0 = 1 + m + n + 1

2 = m + n

2. M = (x 3)(x + 3) + (x 2) 2+ 2x2+ 2x + 13

M = x2 9 + x2 4x + 4 + 2x2+ 2x + 13

M = 4x2 2x + 8

3. A = 2x + 13 + (2x + 5)2 (2x 5)2

A = 2x + 13 + 4(2x)(5)

A = 2x + 13 + 40x

A = 42x + 13


63/157

Matemtica II

12.( 2 + 3 + 5)( 2 + 3 5)

= [( 2 + 3) + 5)][ 2 + 3) 5]

Por diferencia de cuadrados.

= ( 2 + 3)2

52

= 22+ 2 2 3 + 32 5

= 2 + 2 6 + 3 5

= 2 6

13.

1 6 b 12 e

2 a 8 c 4

1 4 5 d 0

e 4 = 0 ; 2d = 4 ; 12 + c = 2

e = 4 d = 2 c = 10

b 8 = 5 6 + a = 4

b = 13 a = 2

Nos piden:

a + b c + d e = 2 + 13 (10) + 2 4

a + b c + d e = 19

Clave: d

Clave: d

10.x5 32

x 2

=(x)5 (2)5

x 2 ; n = 5 ; k = 3

t3= x5 323 1

t3= 4x2

Clave: e

14.32x5 + 243y5

2x + 3y

=(2x)5 (3y)5

2x+ 3y

t4= (1)41(2x)5 4(3y)4 1

t4= (2x)(3y)3 t4= 54xy

3

Clave: e

11.x35 y49

x5 y7

=(x5)7 (y7)7

x5

y7 ; n = 7

El trmino central es t4, luego:

t4= (x5)7 4 (y7)4 1

t4= x15y21

Clave: e

Clave: e

9.x2 5x + 6 + 25

x2 5x + 1

x2 5x + 31

x2 5x + 1

y + 31

y + 1

y

y

Por el teorema del resto:

y + 1 = 0 y = 1

R(y) = D( 1)

= 1 + 31

= 30

R(x) = 30


64/157

64 Matemtica II

Clave: e

Clave:b

Clave: a

Clave: d

1. M =

[(a + b)2+ (a b)2][(a + b)2 (a b)2]

2a2+ 2b2

M =2(a2+ b2) (4ab)

2(a2+ b2)

M = 4ab

2. E =an

bn+

bn

an=

a2n+ b2n

an bn ... (I)

Por dato:

an+ bn= 3 anbn

Elevamos al cuadrado

(an+ bn)2= (3 an bn)2

a2n+ 2anbn+ b2n= 9anbn

a2n+ b2n= 7anbn

Reemplazamos en (I)

E =a2n+ b2n

a

n

b

n =7anbn

a

n

b

n = 7

3. x3y4+ x2y2= x2y2(xy2+ 1) = 3xy

xy xy2+ 1 = 3xy

Cancelamos xy en ambos miembros y elevam

al cuadrado.

xy2+ 1 = 9 xy2= 8

Luego: 3xy2= 38 = 2

4. xm+1

xm= 2

x2m+ 1 = 2xm

x2m 2xm+ 1 = 0

(xm 1)2= 0

Luego: xm= 1

Piden: x3m+ x-3m

= (xm)3 + (xm)3

= 13+ 13= 2

Pg. 116


65/157

Matemtica II

Clave: d

Clave: a

Clave: c

Clave: c

Clave: bClave: e

Clave: e

Clave: e

Clave: e

Clave: e

Clave: a

1. P(x) = (x + 3)(x2 2x + 1)

P(x) = (x + 3)(x 1)2

Luego un factor es (x 1)

8. F(a; b) = 5a9b3+ 15a6b7

F(a; b) = 5a6b3 (a3+ 3a4)

3 factores primos

9. T(a; b) = a3+ a2b + ab2+ b3

T(a; b) = (a3+ ab2) + (a2b + b3)

T(a; b) = a(a2+ b2) + b(a2+ b2)

T(a; b) = (a2+ b2)(a + b)

2. 2bx3 3abx2+ 2cx 3ac

bx2(2x 3a) + c(2x 3a)

(2x 3a)(bx2+ c)

2 factores primos

3. Es directa por producto notable.

4. P(x) = ax4 bx3+ x2

P(x) = x2 (ax2 bx + 1)

2 factores primos

5. P(x) = a(x 3) x + 3

P(x) = a(x 3) (x 3)

P(x) = (x 3)(a 1)

6. P(x) = x3+ 8

P(x) = x3+ 23

P(x) = (x + 2)(x2 2x + 4)

Calculamos la suma de los trminos indepen-dientes:

2 + 4 = 6

7.

P(x; y) = y

2

n

2

+ x

2

m

2

+ y

2

m

2

+ x

2

n

2

Agrupamos:

(y2n2+ y2m2) + (x2m2+ x2n2)

y2 (n2+ m2) + x2(m2+ n2)

(n2+ m2)(x2 + y2)

10. P(x) = x3+ x2 x 1

Agrupamos:

P(x) = (x3 x) + (x2 1)P(x) = x(x2 1) + (x2 1)

P(x) = (x2 1)(x + 1)

P(x) = (x + 1)(x 1)(x + 1)

P(x) = (x + 1)2 (x 1)

11. a(1 b2) + b(1 a2)

a ab2+ b a2b

(a a2b) + (ab2+ b)

a(1 ab) + b(ab + 1)

a(1 ab) + b(1 ab)

(1 ab)(a + b)

12. P(x; y) = (x + 1)2 (y 2)2

P(x; y) = [(x + 1) + (y 2)] [(x + 1) (y 2)]

P(x; y) = (x 1 + y)(x + 3 y)

13. R(x) = 8x3+ 27

= (2x)3+ 33

= (2x + 3)[(2x)2 (2x)(3) + 32]

5U N I D A D

Pg. 122


66/157

66 Matemtica II

Clave: c

Clave: c

Clave: a

Clave: d

Clave: d

Clave:

e

Clave: c Clave: c

14.T(x; y) = (xy + 1)2 (x + y)2

T(x; y) = (xy)2+ 2xy + 1 x2 2xy y2

T(x; y) = (xy)2+ 1 x2 y2

= (x2y2 x2) (y2 1)

= x2(y2 1) (y2 1)

= (y2 1) (x2 1)

= (y + 1)(y 1)(x + 1)(x 1)

Calculamos la suma de los factores primos:= y + 1 + y 1 + x + 1 + x 1

= 2y + 2x

= 2(x + y)

17. P(x) = x7+ c3x4 c4x3 c7

Agrupamos:

(x7+ c3x4) (c4x3+ c7)

x4 (x3+ c3) c4(x3+ c3)

(x3+ c3)(x4 c4)

(x3+ c3)(x2+ c2)(x2 c2)

(x3+ c3)(x2+ c2)(x + c)(x c)

(x + c)(x2 xc + c2)(x2+ c2)(x + c)(x c)

5 factores primos

18.

P(x; y) = a

2

b

2

+ x

2

y

2

+ 2(ax by)P(x; y) = a2 b2+ x2 y2+ 2ax 2by

P(x; y) = (a2+ 2ax + y2) (b2+ 2by + y2)

P(x; y) = (a + x)2 (b + y)2

P(x; y) = (a + x + b + y)(a + x b y)

19. 4x4+ 81y4+ 36x2y2

4x4+ 36x2y2+ 81y4

(2x2)2+ 36(xy)2+ (9y2)2

(2x2+ 9y2)2

20. E(x) = (x 3)(x 2)(x 1) + (x + 2)(x 1) (x 1

E(x) = (x 1)[(x 3)(x 2) + x + 2 1]

E(x) = (x 1)(x2 5x + 6 + x + 1)

E(x) = (x 1)(x2 4x + 7)

Factor primo lineal: x 1

15. P(x) = x3 125

P(x) = x3 53

P(x) = (x 5)(x2+ 5x + 25)

2 factores primos

16. P(a) = a3+ 2a2 a 2

Los divisores del T.I. 2: {1; 2}

1 2 1 2

a = 1 1 3 2

1 3 2 0

(a 1)(a2+ 3a + 2)

(a 2) (a 1)

(a 1)(a + 2)(a + 1)

El factor primo con mayor trmino indepen-diente es (a + 2).

= (2x + 3)(4x2 6x + 9)

F.P.: 2x + 3 la suma de coeficientes es 2 + 3 = 5

F.P.: 4x2 6x + 9 la suma de coeficientes es

4 6 + 9 = 7


67/157

Matemtica II

21. P(a; b; c) = a(b2+ c2) + b(c2+ a2)

P(a; b; c) = ab2+ ac2+ bc2+ ba2

P(a; b; c) = (ab2+ ba2) + (ac2+ bc2)

P(a; b; c) = ab(b + a) + c2(a + b)

P(a; b; c) = (a + b) (ab + c2)

Suma de

coeficientes

22. x2 5xy 14y2 41y + 2x 15

x 2y 5

x 7y 3

(x + 2y + 5)(x 7y 3)

24. P(x) = 4x2 4x 3

2x 3

2x +1

P(x) = (2x 3)(2x + 1)

Suma de coeficientes 2 3 = 1

2 + 1 = 3

25. x2+ 2x 120

x +12

x 10

(x + 12)(x 10)

Suma de factores primos: 2x + 2

26. 6x2 13xy + 2y2+ 5y 8x + 2

6x y 2

x 2y 1

(6x y 2)(x 2y 1)

27. 2x2+ 3y2+ 7xy y + 3x 2

Ordenamos:

2x2+ 7xy + 3y2 y + 3x 2

2x 1y 1

x 3y 2

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

28. x2 y2+ 10y 25x2+ 0xy y2+ 0x + 10y 25

x y 5

x y 5

(x y + 5)(x + y 5)

29. x3 4x2 13x 8

Posibles ceros: {1, 2, 4, 8}

1 4 13 8

x = 1 1 5 8

1 5 8 0

(x + 1)(x2 5x 8)

= 1 + 1 = 2

Clave: b

Clave: e

Clave: a

Clave: b

Clave: a

Clave: a

Clave: b

Clave: c

Clave: b

23. x3 2x2+ 3x + 6

Los divisores del trmino independiente

6: (1, 2, 3, 6)

1 2 3 6

x = 1 1 3 6 1 3 6 0

(x + 1)(x2 3x + 6)


68/157

68 Matemtica II

30. x3 14x2+ 47x + 8


1 14 47 8

x = 8 8 48 8

1 6 1 0

(x 8)(x2 6x 1)

31. P(a; b) = a(b2+ b + 1) + b(a2+ a + 1) + a2+ b2

P(a; b) = ab2+ ab + a + a2b + ab + b + a2+ b2

P(a; b) = (a2b + ab2) + (a + b) + (a2+ 2ab + b2)

P(a; b) = ab(a + b) + (a + b) + (a + b)2

P(a; b) = (a + b) [(a(1 + b) + (1 + b)]

P(a; b) = (a + b)(1 + b)(a + 1)

36.mn4 5m2n3 4m3n2+ 20m4n

(mn4 5m2n3) (4m2n2 20m4n)

mn3 (n 5m) 4m3n (n 5m)

(n 5m)(mn3 4m3n)

mn(n 5m)(n2 4m2)

mn(n 5m)(n + 2m)(n 2m)

37. P(x; y; z) = x2+ y2+ x(y + z) + y(x + z)

P(x; y; z) = x2+ y2+ xy + zx + yx + zy

P(x; y; z) = (x2+ xy) + (y2+ yx) + (zx + zy)

P(x; y; z) = x(x + y) + y(y + x) + z(x + y)

P(x; y; z) = (x + y)(x + y + z)

38. P(x) = x20 (x27+ x20+ 1) + x7(x20+ 1) + 1

P(x) = x47+ x40+ x20+ x27+ x7+ 1

P(x) = (x47+ x40) + (x27+ x20) + x7+ 1

P(x) = x40(x7+ 1) + x20(x7+ 1) + (x7+ 1)

P(x) = (x7+ 1) (x40 + x20+ 1)

32.A(x) = x4+ 6x2+ 9

x2 3

x2 3

(x2+ 3)2

33. P(x) = 4x4+ 26x2+ 36

2x2 4

2x2 9

(2x2+ 4)(2x2+ 9)

34. x2+ 6y2 5xy x 6

x2 5xy + 6y2 x + y 6

x 3y 3

x 2y +2

(x 3y 3)(x 2y + 2)

35.M(x) = x3+ x2 x 1

Posibles ceros = {1}

1 1 1 1

x = 1 1 2 1

1 2 1 0

(x 1)(x2+ 2x + 1)Clave: c

Clave: d

Clave: d

Clave: e

Clave: a

Clave: a

Clave: d

Clave: b

Clave: c

(x 1) (x + 1)2

Piden:

1 1 = 0 1 + 1 = 2


69/157

Matemtica II

40. x2 4xy + 3y2 8y + 4x + 4

x 3y 2

x y 2

(x 3y + 2)(x y + 2)

41. x3+ 2x2 4x 8


1 2 4 8

x = 2 2 8 8

1 4 4 0

(x 2)(x2+ 4x + 4) = (x 2)(x + 2) 2

Rpta.: x + 2

Clave: e

Clave: c

Clave: d

39. x3+ 6x2+ 14x + 15


1 6 14 15

x = 3 3 9 15

1 3 5 0

(x + 3)(x2+ 3x + 5)

Clave: c

Clave: a

1. Sea T la longitud inicial de la tela

Se vende:34

T

Queda: 1

4T

Luego 14

T = 30

T = 120

2. Guillermo en 1 dia har1

12de la obra

Guillermo y Jos en 1 dia harn18

de la obra

Jos en un da har:

18

1

12=

12 896

=4

96=

124

Luego:

En 1 da1

24obra

En x das 1 obra

(toda la obra)

x =1

24

1 1

x = 24

3. Seaab

la fraccin, luego:

a + 5b + 5

=45

5(a + 5) = 4(b + 5)

5a + 25 = 4b + 20

5a 4b = 5 (I)

a 2b 2

=58

8(a 2) = 5(b 2)

8a 16 = 5b 10

8a 5b = 6 (II)

Multiplicamos (I) por 5 y (II) por 4

(5a 4b = 5) (5)

(8a 5b = 6) (4)

25a + 20b = 25(+)

32a 20b = 24

7a = 49

a = 7

Pg. 129


70/157

70 Matemtica II

Clave: b

Clave: b

Clave: b

Clave: d

Clave: a

Clave: a

Clave: a

4. 4 (3x 2) = 12

4 3x + 2 = 12

6 3x = 12

x = 2

8.3 x + 7 = 93 x = 2

x = 8

9. 3x 4y = 41 (I)

11x + 6y = 47 (II)


(3x 4y = 41) (6)

(11x + 6y = 47) (4)

18x 24y = 246+

44x + 24y = 188

62x = 434 x = 7

En (I): 3(7) 4y = 41

21 4y = 41

4y = 20

y = 5

Luego: xy = 35

5. 3x 4(x + 3) = 8x + 6

3x 4x 12 = 8x + 6

x 12 = 8x + 6

9x = 18

x = 2

6. 8x 15x 30x 51x = 53 + 31x 172

88x = 119 + 31x

119x = 119

x = 1

7. (5 3x) (4x + 6) = (8x + 11) (3x 6)

5 3x + 4x 6 = 8x + 11 3x + 6

1 + x = 5x + 17 4x = 18

x = 92

En (I): 5(7) 4b = 5

35 4b = 5

4b = 40

b = 10

Piden: a + b = 17

10. 9x + 11y = 14 .(I)

6x 5y = 34 .(II)


(9x + 11y = 14) (5)

(6x 5y = 34) (11)

45x + 55y = 70+

66x 55y = 374

111x = 444

x = 4

En (I): 9(4) + 11y = 14

36 + 11y = 14

11y = 22


71/157

Matemtica II

y = 2

Luego: x + 2y = 0

Clave: e

Clave: a

Clave: b

Clave: a

Clave: e

11. 10x 3y = 36 (I)

2x + 5y = 4 .(II)


(10x 3y = 36) (5)

(2x + 5y = 4) (3)

50x 15y = 180+

6x + 15y = 12

56x = 168

x = 3

En (I): 10(3) 3y = 36

30 3y = 36

y = 2

Luego: x = 3 ; y = 2

Piden:

x y = 3 (2) = 5

13.x6

+ 2x + 1 = 5x12

34

MCM(6; 12; 4) = 12

Multiplicamos por 12 a cada una de los trmino

12 x6

+ 12 2x + 12 1 = 12 5x12

12 34

2x + 24x + 12 = 5x 9

21x = 21

x = 1

14.4x9

13

= 2x + 15

MCM (9; 3; 5) = 45

Multiplicamos por 45 a cada uno de los trmnos

45 4x9

45 13

= 45 2x + 15

5 4x 15 = 9 (2x + 1)

20x 15 = 18x + 9

x = 12

15.12x +

14 =

110x+

15

MCM (2; 4; 10; 5) = 20

Multiplicamos por 20 a cada uno de los trmnos

12.(m 3)

2 (2m + 1)

3+ m

5+ 3 = 2m 1

3

MCM (2; 3; 5) = 30

Multiplicamos por 30 a cada uno de los trminos

30 (m 3)2

30 (2m + 1)3

+ 30 m5

+ 30 2m

= 30 2m 30 13

15(m 3) 10(2m + 1) + 6m + 90 = 60m 10

15m 45 20m 10 + 6m + 90 = 60m 10

m + 35 = 60m 10

59m = 45

m = 4559


72/157

72 Matemtica II

22.a2x + b2= b2x + a2+ a b

x(a2 b2) = a2 b2+ a b

x =(a + b)(a b) + a b

(a + b)(a b)

x =(a b)(a + b + 1)

(a + b)(a b)

Clave: c

Clave: e

Clave: a

Clave: b

Clave: e

Clave: e

Clave: c

16. 3x x3

= 48

8x3

= 48

x = 18

18.

Falta Sobra

Nmero de hijos =10 + 48 6

= 7

Nmero de caramelos = 7 6 + 4 = 46

19.

Falta Sobra

Nmero de carpetas =4 + 12 1

= 5

Nmero de alumnos = 5 1 + 4 = 9

20. 5x 6y = 60

7x + 3y = 27 (2)

5x 6y = 60(+)

14x + 6y = 54

19x = 114

x = 6

21. MCM = 15

15x + 9x 27 = 60 25x + 60

24x 27 = 120 25x

49x = 147

x = 3

17. Sean D y d los nmeros:

D nmero mayor

d nmero menor

D d D = 3d + 9 (I)

9 3

d + 300 D72 3 1

d + 300 = (3 1) D + 72

d + 300 = 2D + 72

Luego de (I):

d + 300 = 2(3d + 9) + 72

d + 300 = 6d + 18 + 72

5d = 210

d = 42De (I): D = 3(42) + 9

D = 135

Luego: D = 135, d = 42

20 12x

+ 20 14

= 20 110x

+ 20 15

10x

+ 5 = 2x

+ 4

1 = 8x

x = 8

Luego: 3 x = 38 = 2


73/157

Matemtica II

23. Sobra Sobra

Nmero de hijos =37 11

8 6= 13

Nmero de caramelos = 13 8 + 11 = 115

Suma de cifras: 1 + 1 + 5 = 7

27.(I) 3x 2y = 12 (4)

(II) 15x + 8y = 20

12x 8y = 48(+)

15x + 8y = 20

27x = 28 x =

28

27

Reemplazamos en (I):

32827

2y = 12

289

12 = 2y

809

= 2y y = 409

Piden:

x + y =2827

409

=28 120

27=

9227

24.Si Rosa tiene S/. 80 y Mara tiene S/. 60, enton-ces lo que han ganado es:

340 80 60 = 200Luego cada una gan = 100

Nmero de das =10020

= 5

25. Pedro: x

Juana: 4x + 20

Luego: x + 4x + 20 = 920

x = 180 Pedro tiene S/. 180

26.Prez: P

Quispe: Q

P Q = 38 (I)

P 7 = 2Q

P = 2Q + 7 (II)

Reemplazamos (II) en (I):2Q + 7 Q = 38

Q = 31

Luego el seor Quispe tiene 31 aos

Clave: a

Clave: c

Clave: e

Clave: b

Clave: b

Clave: d

Clave: e

1.

Sea m el nmero de manzanas:3m 1 < m + 3

2m < 4

m < 2

Entonces:

m = 1

Pg. 133

x =a + b + 1

a + b

Clave: e

2. x: N de canicas

x + 8 > 18 x > 10 (I)x 5 < 7 x < 12 (II)

De (I) y (II):

10 < x < 12 x = 11


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74 Matemtica II

Clave: e

Clave: d

Clave: cClave: d

Clave: a

Clave: e

Clave: c

3. Sea x el nmero de hijos:

50x > 320

x > 6,4 (I)

40x < 320

x < 8 .(II)

De (I) y (II) tenemos

6,4 < x < 8

El nico valor entero es 7

8.13

16

x > 1

Nmeros enteros positivos y menores que 5:

2, 3, 4 (tres nmeros)

4. 5x 1 < 6x + 7

8 < x

x > 8

9.x 1

4+ x 2

5 4

x > 1

6. 3x 1 20

3x 21

x 7

C.S. = [7; +

7. MCM (3; 2; 6) = 6

2(3x 2) + 3(x 3) 5

6x 4 + 3x 9 5

9x 18

x 2

Mayor valor de x es 2


75/157

Matemtica II

Clave: d

Clave: b

Clave: b

Clave: d

Clave: c

Clave: a

Clave: c

10.5(x + 1) > 3(x 1)

5x + 5 > 3x 3

2x > 8

x > 4

No puede tomar el valor de 6

11.x: edad de Juan

2x 17 < 35

2x < 52

x < 26 (I)

x2

+ 3 > 15

x2 > 12

x > 24 (II)

De (I) y (II)

24 < x < 26 x = 25

14.2(x 3) + 3(x 2) > 4(x 1)

2x 6 + 3x 6 > 4x 4

5x 12 > 4x 4

x > 8

El menor valor entero x es 9

15.x + 1

2+ x 1

36

MCM (2; 3) = 6

multiplicamos por 6 a cada trmino

6 x + 12

+ 6 x 13

6 6

3(x + 1) + 2(x 1) 36

5x + 1 36

x 7

C.S. = 7; +

12.Sea x la cantidad de patos

x 35 >x

2 2x 70 > x

x > 70 (I)

(x 35) + 3 18 < 22

x 50 < 22

x < 72 (II)

De (I) y (II): 70 < x < 72

Luego: x = 71

16.x 3 < 2x + 2 < x + 5

x 3 < 2x + 2

5 < x

x > 5 ... (I)

2x + 2 < x + 5

x < 3 ... (II)

De (I) y (II): 5 < x < 3

C.S. = 5; 3

13.5x + 3

2+ 2x + 1

4> 1

Multiplicamos por 4 a cada trmino

4 5x + 32

+ 4 2x + 14

> 4 1

10x + 6 + 2x + 1 > 4

12x > 3

x > 14

C.S. = 1/4; +


76/157

76 Matemtica II

Clave: c

Clave: b

Clave: b

Clave: d

17.11 6x 1 x < 7 2x

11 6x 1 x

10 5x

2 x

x 2 (I) 1 x < 7 2x

x < 6 (II)

De (I) y (II): 2 x < 6

Valores enteros de x : 2, 3, 4, 5

cuatro

19.Sea x un nmero impar y mltiplo de 3

x x4

< 120

3x4

< 120

x < 160 (I)

20.Sea ab el nmero de 2 cifras:

a + b > 10

b > 10 a (I)

a 2b > 4

a 4 > 2b

a 4

2> b

b 156

De (I) y (II): 156 < x < 160

Luego x impar y mltiplo de 3 es: 159


77/157

Matemtica II

Clave: a

Clave: e

Clave: c

21.x + 2

3+ x + 6

5+ x + 3

75

MCM (3; 5; 7) = 105

Multiplicamos a cada trmino por 105

105 x + 2

3 + 105 x + 6

5 + 105 x + 3

7

105 5

35(x + 2) + 21(x + 6) + 15(x + 3) 525

35 x + 70 + 21x + 126 + 15x + 45 525

71x + 241 525

71 x 289

x 4

C.S. = ; 4]

Intervalo no solucin: 4; + 24. x + 14

+ x 12

+ x + 36

10

MCM (4; 2; 6) = 12


12 x + 14

+ 12 x 12

+ 12 x + 36

12

3(x + 1) + 6(x 1) + 2(x + 3) 120

3x + 3 + 6x 6 + 2x + 6 120

11x + 3 120

x 11711

... (I)

2x 1

5+ 3x 1

82

MCM (5; 8) = 40


40 2x 15

+ 40 3x 18

40 2

8(2x 1) + 5(3x 1) 80

16x 8 + 15x 5 80

31x 13 80

31x 93

x 3 ...(II)

23.2x + 5

3< x 1

2

4x + 10 < 3x 3

x < 13 (I)

22.x 3

2< x < x + 1

3

x 3

2< x

x 3 < 2x

3 < x ...(I)

x 0

x2 5x < 0

x(x 5) < 0

P.C.: x = 0 x 5 = 0

x = 0 x = 5

C.S. = 0; 5

12. 42 x x2> 0

x2+ x 42 < 0

x +7

x 6

(x + 7)(x 6) < 0

P.C.: x + 7 = 0 x 6 = 0

x = 7 x = 6

C.S. = 7; 6

13. x(x 8) + 8 > 4(1 x)

x2 8x + 8 > 4 4x

x2 4x + 4 > 0

(x 2)(x 2) > 0

(x 2)2

> 0x 2

C.S. = {2}

14. x2+ 2x + 1 + x2+ 4x + 4 = x2+ 6x + 9

x2 4 = 0

(x + 2)(x 2) = 0

x + 2 = 0 x = 2

x 2 = 0 x = 2Mayor valor: 2

15. x = 2

2(2)2+ (2m 1)(2) + m 21 = 0

8 + 4m 2 + m 21 = 0

5m = 15

m = 3

16. 40 3x x20

x2+ 3x 40 0

x 5

x + 8

(x 5)(x + 8) 0

P.C.: x 5 = 0 x + 8 = 0

x = 5 x = 8

C.S. = [8; 5]

Clave: c

Clave: d

Clave: a

Clave: d

Clave: e

Clave: e

+

+

+

+

2

0

+

+

0

5

+ +

7 +6


81/157

Matemtica II

17. x2 3x 2x

x2 5x 0

x(x 5) 0

P.C.: x

Solucionario II

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Transcript of Solucionario II