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‡√’¬ß≈”¥—∫®“°§π∑’Ë 1 ®“°´â“¬¡◊Õ‰ª¢«“¡◊Õ §√Ÿ§π∑’Ë 2 ¡’¢Õß‡≈àπ 51 ™‘Èπ ·®°„Àâπ—°‡√’¬π§π≈–

1 ™‘Èπ ‚¥¬·®°‡√’¬ß≈”¥—∫®“°§π∑’Ë 101 ®“°¢«“¡◊Õ¡“´â“¬¡◊Õ §√Ÿ§π∑’Ë 3 ¡’¢Õß‡≈àπ 45 ™‘Èπ

·®°„Àâπ—°‡√’¬π§π≈– 1 ™‘Èπ ‚¥¬·®°‡√’¬ß≈”¥—∫‡√‘Ë¡®“°§π∑’Ë 40 ®“°´â“¬¡◊Õ‰ª¢«“¡◊Õ

¡’π—°‡√’¬π°’Ë§π∑’Ë‰¥â√—∫¢Õß‡≈àπ 3 ™‘Èπ

5. ∂â“ a ‡ªìπ®”π«π‡μÁ¡∫«° ·≈â« (a + 4)(a + 2)(a › 2)(a › 4) + 36 ¡’§à“‡∑à“„¥

6. „ÀâÀ“®”π«π‡μÁ¡∫«° m ∑’Ë∑”„Àâ 7m2 + 7m + 7 ‡ªìπ‡®Á¥¬°°”≈—ß ’Ë

7. ∂â“ (a2 › a)3 + (2a2 › 4)3 = (3a2 › a › 4)3

·≈â«º≈§Ÿ≥¢Õß§”μÕ∫∑—ÈßÀ¡¥¢Õß ¡°“√π’È‡ªìπ‡∑à“„¥

8. + + +...+ ¡’§à“‡∑à“„¥12 + 1

13 + 2

14 + 3( )1

16 + 152

A 1-29 12/9/08, 10:23 AM4


9. ®”π«π‡μÁ¡∑’Ë¡“°∑’Ë ÿ¥∑’ËπâÕ¬°«à“À√◊Õ‡∑à“°—∫ ¡’§à“‡∑à“„¥

10. ∂â“ a = ·≈– b =

·≈â« ¡’§à“‡∑à“„¥a + b › 1a › b + 1

11. ∂â“ x + = ·≈â« x3 + x›3 ¡’§à“‡∑à“„¥

12. ®“°√Ÿª∂â“ AB = 3 Àπà«¬

AC = 7 Àπà«¬ AD = 4 Àπà«¬ ·≈â«

√Ÿª “¡‡À≈’Ë¬¡ ABC ¡’æ◊Èπ∑’Ë°’Ëμ“√“ßÀπà«¬

13. ∂â“ a Ó b = ab + a + b ·≈â«

1 Ó Ó Ó Ó...Ó ¡’§à“‡∑à“„¥12

13

14

14. ∂â“ a4x = 3 › 2 2 ·≈– a›4x = ·≈â« ¡’§à“‡∑à“„¥

15. „Àâ m ‡ªìπ®”π«π‡μÁ¡∫«°·≈– p ‡ªìπ®”π«π‡©æ“–∫«° ∂â“ m À“√ 777 ·≈– 910

·≈â«‡À≈◊Õ‡»… p ‡∑à“°—π·≈â« m2 + p2 ¡’§à“‡∑à“„¥

16. æ“√“‚∫≈“∑’Ëºà“π®ÿ¥°”‡π‘¥·≈–ºà“π®ÿ¥ (1, 12) ·≈– (3, 6) ¡’®ÿ¥¬Õ¥§◊Õ®ÿ¥„¥

17. 35 ‡¡◊ËÕ‡ª≈’Ë¬π‡ªìπ‡≈¢∞“πÀ°·≈â« μ—«‡≈¢ ÕßÀ≈—° ÿ¥∑â“¬‡ªìπ‡∑à“„¥

18. —¡ª√– ‘∑∏‘Ï ¢Õß x2 ®“°°“√°√–®“¬ (1 + x + x2)3 ‡ªìπ‡∑à“„¥

3 22

1x

5 + 110 + 1

10 + 510 › 1

A

B C

12549

a6x + a›6x

a2x + a›2x

13 › 2 2

1040

1035 + 3

734

A 1-29 12/9/08, 10:23 AM5


19. ‡≈¢‚¥¥ 1, 2, 3, 4, 5, 6, 7, 8, 9 π”¡“ √â“ß®”π«π∑’Ë¡’ ’ËÀ≈—° „Àâ¡’§à“¡“°°«à“ 6,000

‚¥¬‡≈¢‚¥¥·μà≈–À≈—°Àâ“¡´È” ¬°‡«âπμ—«‡≈¢ 4 ‡∑à“π—Èπ∑’Ë„™â´È”‰¥â ®– √â“ß‰¥â°’Ë®”π«π

20. ∂â“ m ·≈– n ‡ªìπ®”π«π‡μÁ¡∫«°´÷Ëß m2 › n4 = 19 ·≈â« m2 + n4 ¡’§à“‡∑à“„¥


21. ∂â“ a, b ‡ªìπ§”μÕ∫¢Õß ¡°“√ x2 › x › 1 = 0 ·≈â« a9 + b9 ¡’§à“‡∑à“„¥

22. „ÀâÀ“®”π«π„π·∂«∑’Ë 89 π—∫®“°´â“¬¡◊Õμ—«∑’Ë 3

·∂«∑’Ë 1 1

·∂«∑’Ë 2 2 3 4

·∂«∑’Ë 3 5 6 7 8 9

·∂«∑’Ë 4 10 11 12 13 14 15 16

μ—«Õ¬à“ß ‡™àπ ·∂«∑’Ë 4 ®“°´â“¬¡◊Õμ—«∑’Ë 3 §◊Õ 12

·∂«∑’Ë 3 ®“°´â“¬¡◊Õμ—«∑’Ë 3 §◊Õ 7

23. °”Àπ¥ ABC ‡ªìπ√Ÿª “¡‡À≈’Ë¬¡„¥ Ê ∑’Ë¡’æ◊Èπ∑’Ë 24 3 μ“√“ßÀπà«¬ ∂â“ a ‡ªìπ§«“¡¬“«

¥â“πμ√ß¢â“¡¡ÿ¡ A, b ‡ªìπ§«“¡¬“«¥â“πμ√ß¢â“¡¡ÿ¡ B ·≈– c ‡ªìπ§«“¡¬“«¥â“πμ√ß¢â“¡¡ÿ¡ C

·≈– a + b = 20 Àπà«¬ c = 16 Àπà«¬ ·≈â« a › b ¡’§à“‡∑à“„¥

24. π‘¬“¡ n! = 1 Ó 2 Ó 3 Ó 4 Ó ... Ó n ‡¡◊ËÕ n ‡ªìπ®”π«π‡μÁ¡

∂â“‡¢’¬π 20! „π√Ÿª A Ó 10n ‡¡◊ËÕ A ‡ªìπ®”π«π‡μÁ¡·≈â« n ¡’§à“‡∑à“„¥

25. º≈√«¡¢Õß‡≈¢‚¥¥ ´÷Ëß‡ªìπº≈≈—æ∏å¢Õß (333 ... 333)2 + 222 ... 222 ‡ªìπ‡∑à“‰√

} }¡’ 3 ®”π«π 2,006 μ—« ¡’ 2 ®”π«π 2,006 μ—«

A 1-29 12/9/08, 10:23 AM6


26. ∂â“ x1 = 2549, x

2 = , x

3 = , x

4 =

x5 = , x

6 = , x

7 = , x

8 = ·≈â«º≈§Ÿ≥

x1 x

2 x

3 x

4 x

5 x

6 x

7 x

8 ‡ªìπ‡∑à“„¥

27. ∂â“ x, y, z ‡ªìπ®”π«π®√‘ß∫«° ´÷Ëß xy + x + y = 11, yz + y + z = 14

zx + z + x = 19 ·≈â« xyz + x + y + z ¡’§à“‡∑à“„¥

28. ∂â“ x ·≈– y ‡ªìπ®”π«π®√‘ß ´÷Ëß x3 + y3 + (x + y)3 + 30xy = 2,000 ·≈â« x + y

¡’§à“‡∑à“„¥

29. ABC ‡ªìπ√Ÿª “¡‡À≈’Ë¬¡¡’ AB = 20 Àπà«¬ AC = 30 Àπà«¬ ¡ÿ¡ BAC = 120 Õß»“

∂â“ D ‡ªìπ®ÿ¥∫π BC ∑”„Àâ AD ·∫àß§√÷Ëß¡ÿ¡ BAC ·≈â« AD ¬“«°’ËÀπà«¬

5x4

6x5

7x6

8x7

2x1

3x2

4x3

30. ∂â“ x ‡ªìπ®”π«π®√‘ß‚¥¬∑’Ë 2 < x < 3 ·≈â«

x + 2 2x › 4 + x › 2 2x › 4 ¡’§à“‡∑à“„¥

31. ∂â“ + + + + + =

‚¥¬∑’Ë a ·≈– b ‡ªìπ®”π«π‡μÁ¡∑’Ë À.√.¡. ¢Õß a °—∫ b ‡∑à“°—∫ 1 ·≈â« a + b ¡’§à“‡∑à“„¥

32. ®“°√–∫∫ ¡°“√

4y = 12 › x2 .............................. ❶

4x = 12 › y2 .............................. ➋

∂â“ A ·≈– B ‡ªìπ®ÿ¥μ—¥¢Õß°√“ø¢Õß ¡°“√·≈â«

√–¬–√–À«à“ß®ÿ¥ A ·≈– B ‡∑à“°—∫°’ËÀπà«¬

17 ⋅ 9 ⋅ 11

19 ⋅ 11 ⋅ 13

15 ⋅ 7 ⋅ 9

11 ⋅ 3 ⋅ 5

13 ⋅ 5 ⋅ 7

111 ⋅ 13 ⋅ 15

ab

A 1-29 12/9/08, 10:23 AM7


33. √ŸªÀ°‡À≈’Ë¬¡¡ÿ¡‡∑à“ ´÷Ëß¡’§«“¡¬“« 4 ¥â“π ∑’Ë‡√’¬ßμ‘¥μàÕ°—π ‡ªìπ 6, 7, 8, 9 Àπà«¬

®–¡’§«“¡¬“«√Õ∫√Ÿª°’ËÀπà«¬

34. °”Àπ¥ ABC ‡ªìπ√Ÿª “¡‡À≈’Ë¬¡Àπâ“®—Ë« ¡’ AB = AC ¡ÿ¡ BAC = 80 Ì ∂â“ D ‡ªìπ®ÿ¥¿“¬„π

∑”„Àâ ¡ÿ¡ DAB = ¡ÿ¡ DBA = 10 Ì ·≈â«¡ÿ¡ ADC ¡’¢π“¥°’ËÕß»“

35. ABC ‡ªìπ√Ÿª “¡‡À≈’Ë¬¡ ¡’ P ‡ªìπ®ÿ¥¿“¬„π ≈“° AP æ∫ BC ∑’Ë D ≈“° BP

æ∫ CA ∑’Ë E ≈“° CP æ∫ AB ∑’Ë F ∑”„Àâ PD = PE = PF = 4 Àπà«¬

∂â“ AP + BP + CP = 6 Àπà«¬·≈â« (AP) Ó (BP) Ó (CP) ¡’§à“‡∑à“„¥

A 1-29 12/9/08, 10:23 AM8


μ—«Õ¬à“ß·π«§‘¥·∫∫∑¥ Õ∫§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3‡æ◊ËÕ°“√§—¥‡≈◊Õ°π—°‡√’¬π√–¥—∫‡¢μæ◊Èπ∑’Ë°“√»÷°…“

ªï æ.». 2549

§”™’È·®ß

1. ·∫∫∑¥ Õ∫©∫—∫π’È‡ªìπ·∫∫∑¥ Õ∫™π‘¥‡μ‘¡§”μÕ∫




3. ‡«≈“∑’Ë„™â„π°“√ Õ∫ „™â‡«≈“ 2 ™—Ë«‚¡ß

A 1-29 12/9/08, 10:23 AM9



1. 77777

À“√¥â«¬ 10 ‡À≈◊Õ‡»…‡∑à“„¥

·π«§‘¥

7 = 4 (1) + 3

77777

= 74n + 3

‡À≈◊Õ‡»… 3

μÕ∫ 3

μ—«Õ¬à“ß·π«§‘¥·∫∫∑¥ Õ∫§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3

‡æ◊ËÕ°“√§—¥‡≈◊Õ°π—°‡√’¬π√–¥—∫‡¢μæ◊Èπ∑’Ë°“√»÷°…“ ªï æ.». 2549

2. „ÀâÀ“§à“¢Õß (2 + 3)2 (2 › 3)3 + (2 › 3)2 (2 + 3)3 ¡’§à“‡∑à“„¥

·π«§‘¥

(2 + 3)(2 › 3) = 22 › ( 3)2

= 4 › 3

= 1

(2 + 3)2(2 › 3)3 + (2 › 3)2 (2 + 3)3= 12 (2 › 3) + 12(2 + 3)

= 4

μÕ∫ 4

710

4n + 3

A 1-29 12/9/08, 10:23 AM10


3. ªí®®ÿ∫—π‰¡μ√’Õ“¬ÿ 42 ªï ∂â“‰¡μ√’¡’Õ“¬ÿ‡∑à“°—∫¡πμ√’„πªí®®ÿ∫—π ·≈â«¡πμ√’®–¡’Õ“¬ÿ‡ªìπ

§√÷ËßÀπ÷Ëß¢Õß‰¡μ√’ ¥—ßπ—Èπ ªí®®ÿ∫—π¡πμ√’¡’Õ“¬ÿ°’Ëªï

·π«§‘¥

„Àâ ªí®®ÿ∫—π ¡πμ√’ Õ“¬ÿ x ªï ‰¡μ√’ Õ“¬ÿ 42 ªï

‡¡◊ËÕ‰¡μ√’ Õ“¬ÿ x ªï ¡πμ√’Õ“¬ÿ ªï

º≈μà“ß¢ÕßÕ“¬ÿ§ß∑’Ë

42 › x = x ›

42 = x

x = 28

μÕ∫ 28 ªï

x2

x2

4. ¡’π—°‡√’¬π 101 §π §√Ÿ§π∑’Ë 1 ¡’¢Õß‡≈àπ 75 ™‘Èπ ·®°„Àâπ—°‡√’¬π§π≈– 1 ™‘Èπ ‚¥¬·®°

‡√’¬ß≈”¥—∫ ®“°§π∑’Ë 1 ®“°´â“¬¡◊Õ‰ª¢«“¡◊Õ §√Ÿ§π∑’Ë 2 ¡’¢Õß‡≈àπ 51 ™‘Èπ ·®°„Àâπ—°‡√’¬π§π≈–

1 ™‘Èπ ‚¥¬·®°‡√’¬ß≈”¥—∫®“°§π∑’Ë 101 ®“°¢«“¡◊Õ¡“´â“¬¡◊Õ §√Ÿ§π∑’Ë 3 ¡’¢Õß‡≈àπ 45 ™‘Èπ

·®°„Àâπ—°‡√’¬π§π≈– 1 ™‘Èπ ‚¥¬·®°‡√’¬ß≈”¥—∫‡√‘Ë¡®“°§π∑’Ë 40 ®“°´â“¬¡◊Õ‰ª¢«“¡◊Õ

¡’π—°‡√’¬π°’Ë§π∑’Ë‰¥â√—∫¢Õß‡≈àπ 3 ™‘Èπ

·π«§‘¥

æ‘®“√≥“®“°μ“√“ß

π—°‡√’¬π§π∑’Ë

§π∑’Ë 11 75

§π∑’Ë 250 101

§π∑’Ë 340 84

π—°‡√’¬π∑’Ë‰¥â√—∫¢Õß‡≈àπ 3 ™‘Èπ §◊Õ§π∑’Ë 50 ∂÷ß§π∑’Ë 75 √«¡ 26 §π

μÕ∫ 26 §π

32

1 11 21 31 41 51 61 71 81 91 101

<

>

A 1-29 12/9/08, 10:23 AM11


5. ∂â“ a ‡ªìπ®”π«π‡μÁ¡∫«°·≈â« (a + 4)(a + 2)(a › 2)(a › 4) + 36 ¡’§à“‡∑à“„¥

·π«§‘¥

(a + 4)(a + 2)(a › 2)(a › 4) + 36 = (a2 › 16)(a2 › 4) + 36

= a4 › 20a2 + 64 + 36

= (a2 › 102)

= a2 › 10

μÕ∫ a2 › 10

6. „ÀâÀ“®”π«π‡μÁ¡∫«° m ∑’Ë∑”„Àâ 7m2 + 7m + 7 ‡ªìπ‡®Á¥¬°°”≈—ß ’Ë

·π«§‘¥

7m2 + 7m + 7 = 7(m2 + m + 1)

∴ ®–‰¥â 7(m2 + m + 1) = 74

m2 + m + 1 = 73 = 343

m2 + m › 342 = 0

(m › 18) (m + 19) = 0

m = 18

μÕ∫ 18

7. ∂â“ (a2 › a)3 + (2a2 › 4)3 = (3a2 › a › 4)3

·≈â«º≈§Ÿ≥¢Õß§”μÕ∫∑—ÈßÀ¡¥¢Õß ¡°“√π’È‡ªìπ‡∑à“„¥

·π«§‘¥

A = a2 › a

B = 2a2 › 4

C = 3a2 › a › 4

®–‰¥â A + B = C ¥—ßπ—Èπ A3 + B3 = C3 = A3 + B3 + 3AB (A + B)

®–‰¥â 3ABC = 0 π—Ëπ§◊Õ A = 0 À√◊Õ B = 0 À√◊Õ C = 0

A = 0 ®–‰¥â a2 › a = 0 ∴ a = 0, 1

B = 0 ®–‰¥â 2a2 › 4 = 0 ∴ a = 2, › 2

C = 0 ®–‰¥â 3a2 › a › 4 = 0 ∴ a = , ›1

º≈§Ÿ≥¢Õß§”μÕ∫∑—ÈßÀ¡¥‡∑à“°—∫ 0μÕ∫ 0

43

A 1-29 12/9/08, 10:23 AM12


9. ®”π«π‡μÁ¡∑’Ë¡“°∑’Ë ÿ¥∑’ËπâÕ¬°«à“À√◊Õ‡∑à“°—∫ ¡’§à“‡∑à“„¥

·π«§‘¥

=

=

= 105 ›

·μà < 1

∴ ®”π«π‡μÁ¡∑’Ë¡“°∑’Ë ÿ¥∑’ËπâÕ¬°«à“À√◊Õ‡∑à“°—∫ §◊Õ 99999

μÕ∫ 99999

8. „ÀâÀ“ + + +... + ¡’§à“‡∑à“„¥

·π«§‘¥

= ( 2 › 1 + 3 › 2 + 4 › 3 +...+ 16 › 15 )2

= (›1 + 16)2 = (›1 + 4)2 = (3)2

= 9

μÕ∫ 9

( )116 + 15

12 + 1

13 + 2

14 + 3

( )12 + 1

13 + 2

14 + 3

+ + +...+

2

21

16 + 15

1040

1035 + 3

105 (1035 + 3) › 105 (3)

1035 + 3

1040

1035 + 3

›105 (3)

1035 + 3

105 (1035 + 3)

1035 + 3

10. ∂â“ a = ·≈– b =

·≈â« ¡’§à“‡∑à“„¥

«‘∏’∑”

a = =

b = =

a + b › 1a › b + 1

5 + 110 + 1

10 + 510 › 1

50 › 5 + 10 › 19

5 + 110 + 1 ⋅ 10 › 1

10 › 1

⋅ 10 + 110 + 1

10 + 510 › 1

10 + 10 + 50 + 59

3(105)

1035+ 33(105)

1035+ 3

1040

1035 + 3

A 1-29 12/9/08, 10:23 AM13


a + b › 1 =

=

a › b + 1 =

=

=

=

= › 10

μÕ∫ › 10

10 2 + 2 109

50 › 5 + 10 › 19

› 10 + 50 + 5 + 109

+ 1

› 2 5 › 29

10 2 + 2 109 Ó 9

›2 5 › 2

a + b › 1a › b + 1

∴

10(2 5 + 2)›(2 5 + 2)

11. ∂â“ x + = ·≈â« x3 + x›3 ¡’§à“‡∑à“„¥

«‘∏’∑”

®“° x + =

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

= =

x3 + 3 x + =

x3 + 3 =

x3 =

=

μÕ∫

1x

( )

3 22

1x

3 22

1x

1x( ) 1

x

21x3

( )9(2)4 ( )3 2

2

( )1x

1x3+ 27 2

4

( )3 22 + 1

x327 2

4

1x3+ 27 2

4 › 2(9 2)4

9 24

9 24

3 22( )

50 › 5 + 10 › 19

+ 10 + 50 + 5 + 109

› 1

2

( )3 22

A 1-29 12/9/08, 10:23 AM14


12. ®“°√Ÿª∂â“ AB = 3 Àπà«¬

AC = 7 Àπà«¬ AD = 4 Àπà«¬ ·≈â«

√Ÿª “¡‡À≈’Ë¬¡ ABC ¡’æ◊Èπ∑’Ë°’Ëμ“√“ßÀπà«¬

·π«§‘¥ μàÕ AD ‰ª∂÷ß®ÿ¥ D′ ∑”„Àâ CD′ = AB = 3 Àπà«¬

æ◊Èπ∑’Ë ABC = æ◊Èπ∑’Ë ACD′

s =

ACD′¡’ s = = 9

æ◊Èπ∑’Ë ACD′ = s(s › a)(s › b)(s › c)

= 9(9 › 7)(9 › 3)(9 › 8)

= 9(2)(6)(1)

= 9 Ó 2 Ó 2 Ó 3

= 6 3 μ“√“ßÀπà«¬

μÕ∫ 6 3

∇ ∇

a + b + c2

∇

7 + 3 + 82

13. ∂â“ a Ó b = ab + a + b

·≈â« 1 Ó Ó Ó Ó ... Ó ¡’§à“‡∑à“„¥

·π«§‘¥

1 Ó = 1

1 Ó Ó = 2 Ó

= 2

= 3

12,549

12

14

13

A

B CD

∴12 ( )1

2+1

2+ 1 = 2

12

13

13

+ 13

+ 2( )13

34

7

A

B CD

34

7

D′

34 ∇

A 1-29 12/9/08, 10:23 AM15


14. ∂â“ a4x = 3 › 2 2 ·≈– a›4x = ·≈â« ¡’§à“‡∑à“„¥

·π«§‘¥ a4x =

a›4x =

=

=

=

=

= a4x › a4x ⋅ a›4x + a›4x

= 3 › 2 2 › 1 + 3 + 2 2

= 5

μÕ∫ 5

a6x + a›6x

a2x + a›2x

a6x + a›6x

a2x + a›2x

(a2x)3 + (a›2x)3

a2x + a›2x

(a2x + a›2x)(a4x › a4x ⋅ a›4x + a›4x)(a2x

+ a›2x)

13 › 2 2

1 Ó Ó Ó = 3 Ó

= 3

‚¥¬°“√§“¥‡¥“

1 Ó Ó Ó Ó ... Ó = 2549

μÕ∫ 2549

12

13

14

= 4

12549

12

14

13

3 › 2 2

Ó13 › 2 2

3 + 2 23 + 2 2

3 + 2 29 › 8

3 + 2 2

14

+ 3( )14 + 1

4

A 1-29 12/9/08, 10:23 AM16


15. „Àâ m ‡ªìπ®”π«π‡μÁ¡∫«°·≈– p ‡ªìπ®”π«π‡©æ“–∫«° ∂â“ m À“√ 777 ·≈– 910

·≈â«‡À≈◊Õ‡»… p ‡∑à“°—π·≈â« m2 + p2 ¡’§à“‡∑à“„¥

·π«§‘¥

777 = m(a) + p ‡¡◊ËÕ a ‡ªìπ®”π«π‡μÁ¡∫«° ................... ❶

910 = m(b) + p ‡¡◊ËÕ b ‡ªìπ®”π«π‡μÁ¡∫«° .................. ➋

➋ › ❶

910 › 777 = m(b) › m(a)

133 = m(b › a) b › a ‡ªìπ®”π«π‡μÁ¡∫«°

7 Ó 19 = m(b › a)

æ‘®“√≥“®“°‚®∑¬å

m = 19 ‡¡◊ËÕπ” 19 ‰ªÀ“√ 777 ·≈– 910 ·≈â« ®–‡À≈◊Õ‡»…‡∑à“°—π

§◊Õ 17 π—Ëπ§◊Õ p = 17

®–‰¥â m2 + p2 = 192 + 172 = 361 + 289 = 650

μÕ∫ 650

16. æ“√“‚∫≈“∑’Ëºà“π®ÿ¥°”‡π‘¥·≈–ºà“π®ÿ¥ (1, 12) ·≈– (3, 6) ¡’®ÿ¥¬Õ¥§◊Õ®ÿ¥„¥

·π«§‘¥

„Àâæ“√“‚∫≈“ π—Ëπ§◊Õ y = ax2 + bx + c.........➊ºà“π®ÿ¥ (0, 0) §◊Õ 0 = 0 + 0 + c

C = 0ºà“π®ÿ¥ (1, 12) 12 = a + b .....................❷ºà“π®ÿ¥ (3, 6) 6 = 9a + 3b

2 = 3a + b ...................❸

❷ › ❸ 10 = ›2a

a = ›5 ·∑π„π ❷

12 = ›5 + b

b = 17

∴ y = ›5x2 + 17x

®ÿ¥¬Õ¥§◊Õ

μÕ∫

( )›17›10

0 › 289›20

, = ,1710( )289

20

,1710( )289

20

A 1-29 12/9/08, 10:23 AM17


17. 35 ‡¡◊ËÕ‡ª≈’Ë¬π‡ªìπ‡≈¢∞“πÀ°·≈â« μ—«‡≈¢ ÕßÀ≈—° ÿ¥∑â“¬‡ªìπ‡∑à“„¥

·π«§‘¥

35 = 243

243 = 1.63 + 0.62 + 4.61 + 3.60 = (1,043)6

¥—ßπ—Èπ 35 ‡ª≈’Ë¬π‡ªìπ‡≈¢∞“πÀ°·≈â« μ—«‡≈¢ ÕßÀ≈—° ÿ¥∑â“¬‡ªìπ 43

μÕ∫ 43

∴

18. —¡ª√– ‘∑∏‘Ï ¢Õß x2 ®“°°“√°√–®“¬ (1 + x + x2)3 ‡ªìπ‡∑à“„¥

·π«§‘¥

„Àâ a = 1 + x, b = x2

(a + b)3 = a3 + 3a2b + 3ab2 + 3b3

(1 + x + x2)3 = 1 + 3x + 3x2 + x3 + 3x2 + 3x4 + ...

.ª. . ¢Õß x2 §◊Õ 3 + 3 = 6

μÕ∫ 6

19. ‡≈¢‚¥¥ 1, 2, 3, 4, 5, 6, 7, 8, 9 π”¡“ √â“ß®”π«π∑’Ë¡’ ’ËÀ≈—° „Àâ¡’§à“¡“°°«à“ 6,000

‚¥¬‡≈¢‚¥¥·μà≈–À≈—°Àâ“¡´È” ¬°‡«âπ‡≈¢ 4 ‡∑à“π—Èπ∑’Ë„™â´È”‰¥â ®– √â“ß‰¥â°’Ë®”π«π

·π«§‘¥

1. ∑ÿ°μ—«μà“ß°—π ___ ___ ___ ___

®”π«π 4 Ó 8 Ó 7 Ó 6 = 1,344 «‘∏’

2. ¡’ 4 ´È”°—π 2 μ—« ®”π«π 4 Ó 1 Ó 1 Ó 7 Ó 3 = 84 «‘∏’

3. ¡’ 4 ´È”°—π 3 μ—« ®”π«π 4 Ó 1 Ó 1 Ó 1 = 4 «‘∏’

®”π«π∑—ÈßÀ¡¥∑’Ë √â“ß‰¥â = 1,432 ®”π«π

μÕ∫ 1,432 ®”π«π

{

¡’¥’°√’¡“°°«à“ 2

A 1-29 12/9/08, 10:23 AM18


20. ∂â“ m ·≈– n ‡ªìπ®”π«π‡μÁ¡∫«°´÷Ëß m2 › n4 = 19 ·≈â« m2 + n4 ¡’§à“‡∑à“„¥

·π«§‘¥

m2 › n4 = (m › n2) (m + n2) = 19

19 ‡ªìπ®”π«π‡©æ“–®–‰¥â«à“

m › n2 = 1 .............. ❶m + n2 = 19 ............ ❷

❶ + ❷;2m = 20

m = 10

❷ › ❶;2n2 = 18

n2 = 9

n = 3

m2 + n4 = 100 + 81

= 181

μÕ∫ 181

A 1-29 12/9/08, 10:23 AM19



21. ∂â“ a, b ‡ªìπ§”μÕ∫¢Õß ¡°“√ x2 › x › 1 = 0 ·≈â« a9 + b9 ¡’§à“‡∑à“„¥

·π«§‘¥

a + b = 1

ab = ›1

a2 + b2 = (a + b)2 › 2ab

= 1 › 2(›1) = 3

a3 + b3 = (a + b)3 › 3ab(a + b)

= 13 › 3(›1)(1)

= 1 + 3 = 4

a4 + b4 = (a2 + b2)2 › 2a2b2

= 32 › 2(›1)2 = 7

®–‰¥â a + b = 1

a2 + b2 = 3

a3 + b3 = 4

a4 + b4 = 7

a5 + b5 = 11

a6 + b6 = 18

a7 + b7 = 29

a8 + b8 = 47

a9 + b9 = 76

À√◊Õ a9 + b9 = (a3)3 + (b3)3 = (a3 + b3)3 › 3a3b3(a3 + b3)

= (43) › 3(›1)3(4)

= 64 + 12

= 76

μÕ∫ 76

A 1-29 12/9/08, 10:23 AM20


22. „ÀâÀ“®”π«π„π·∂«∑’Ë 89 π—∫®“°´â“¬¡◊Õμ—«∑’Ë 3

·∂«∑’Ë 1 1

·∂«∑’Ë 2 2 3 4

·∂«∑’Ë 3 5 6 7 8 9

·∂«∑’Ë 4 10 11 12 13 14 15 16

μ—«Õ¬à“ß ‡™àπ ·∂«∑’Ë 4 ®“°´â“¬¡◊Õμ—«∑’Ë 3 §◊Õ 12

·∂«∑’Ë 3 ®“°´â“¬¡◊Õμ—«∑’Ë 3 §◊Õ 7

·π«§‘¥

μ—« ÿ¥∑â“¬¢Õß·∂«∑’Ë 1 §◊Õ 12 = 1

μ—« ÿ¥∑â“¬¢Õß·∂«∑’Ë 2 §◊Õ 22 = 4

μ—« ÿ¥∑â“¬¢Õß·∂«∑’Ë’ 3 §◊Õ 32 = 9

μ—« ÿ¥∑â“¬¢Õß·∂«∑’Ë’ 88 §◊Õ 882 = 7,744

∴ μ—«∑’Ë “¡¢Õß·∂«∑’Ë 89 §◊Õ 7,744 + 3 = 7,747

μÕ∫ 7,747

.

.

.

A 1-29 12/9/08, 10:23 AM21


23.°”Àπ¥ ABC ‡ªìπ√Ÿª “¡‡À≈’Ë¬¡„¥ Ê ∑’Ë¡’æ◊Èπ∑’Ë 24 3 μ“√“ßÀπà«¬ ∂â“ a ‡ªìπ§«“¡¬“«

¥â“πμ√ß¢â“¡¡ÿ¡ A, b ‡ªìπ§«“¡¬“«¥â“πμ√ß¢â“¡¡ÿ¡ B ·≈– c ‡ªìπ§«“¡¬“«¥â“πμ√ß¢â“¡¡ÿ¡ C

·≈– a + b = 20 Àπà«¬ c = 16 Àπà«¬ ·≈â« |a › b| ¡’§à“‡∑à“„¥

·π«§‘¥

æ◊Èπ∑’Ë ABC = s(s › a)(a › b)(s › c)

s =

æ◊Èπ∑’Ë ABC = 24 3

18(18 › a)(18 › 20 + a)(18 › 16) = 24 3

6 (18 › a)(a › 2) = 24 3

(18 › a)(a › 2) = 4 3

(18 ‹› a)(a › 2) = 48

18a › 36 › a2 + 2a = 48

a2 › 20a + 84 = 0

(a › 14)(a › 6) = 0

∂â“ a = 14, b = 6, À√◊Õ a = 6, b = 14

®–‰¥â |a › b| = |14 › 6| = 8

μÕ∫ 8

a + b + c2

= =20 + 16

2362 = 18

∇24.π‘¬“¡ n! = 1 Ó 2 Ó 3 Ó 4 Ó ... Ó n ‡¡◊ËÕ n ‡ªìπ®”π«π‡μÁ¡

∂â“‡¢’¬π 20! „π√Ÿª A Ó 10n ‡¡◊ËÕ A ‡ªìπ®”π«π‡μÁ¡·≈â« n ¡’§à“‡∑à“„¥

·π«§‘¥

æ‘®“√≥“ 20! ¡’ 5 §Ÿ≥°—πÕ¬Ÿà∑—ÈßÀ¡¥°’Ëμ—«

º≈§Ÿ≥¢Õß®”π«π‡μÁ¡∫«°μ—Èß·μà 1 ∂÷ß 20 ¡’ 5 „π 5, 10, 15 ·≈– 20

¥—ßπ—Èπ º≈§Ÿ≥¢Õß®”π«π‡μÁ¡∫«°μ—Èß·μà 1 ∂÷ß 20 ¡’ 10 §Ÿ≥°—πÕ¬Ÿà 4 μ—«

∴ n = 4

μÕ∫ 4

∇

a

b

A

C

c

B

A 1-29 12/9/08, 10:23 AM22


25.º≈√«¡¢Õß‡≈¢‚¥¥ ´÷Ëß‡ªìπº≈≈—æ∏å¢Õß (333 ... 333)2 + 222 ... 222 ‡ªìπ‡∑à“‰√

·π«§‘¥

32 + 2 = 11

332 + 22 = 1111

3332 + 222 = 111111

¡’ 3 ®”π«π 2,006 μ—« ¡’ 2 ®”π«π 2,006 μ—« ¡’ 1 ®”π«π 4,012 μ—«

∴ (333 ... 333)2 + 222 ... 222 = 111 ... 111

∴ º≈√«¡¢Õß‡≈¢‚¥¥ §◊Õ 4,012

μÕ∫ 4,012

} }

} } }26.∂â“ x

1 = 2,549, x

2 = , x

3 = , x

4 =

x5 = , x

6 = , x

7 = , x

8 = ·≈â«º≈§Ÿ≥

x1 x

2 x

3 x

4 x

5 x

6 x

7 x

8 ‡ªìπ‡∑à“„¥

·π«§‘¥

x1 x

2 = 2, x

3 x

4 = 4, x

5 x

6 = 6, x

7 x

8 = 8

x1 x

2 x

3 x

4 x

5 x

6 x

7 x

8= 2 Ó 4 Ó 6 Ó 8

= 384

μÕ∫ 384

2x1

3x2

4x3

5x4

6x5

7x6

8x7

27.∂â“ x, y, z ‡ªìπ®”π«π®√‘ß∫«° ´÷Ëß xy + x + y = 11, yz + y + z = 14

zx + z + x = 19 ·≈â« xyz + x + y + z ¡’§à“‡∑à“„¥

·π«§‘¥

xy + x + y + 1 = 12

(x + 1)(y + 1) = 12 ..................... ❶yz + y + z + 1 = 15

(y + 1)(z + 1) = 15 ..................... ❷zx + z + x + 1 = 20 ..................... ❸(z + 1)(x + 1) = 20

(y + 1)2 = = 912 Ó 1520

¡’ 3 ®”π«π 2,006 μ—« ¡’ 2 ®”π«π 2,006 μ—«

❶ Ó ❷;❸

A 1-29 12/9/08, 10:23 AM23


29.ABC ‡ªìπ√Ÿª “¡‡À≈’Ë¬¡ ¡’ AB = 20 Àπà«¬ AC = 30 Àπà«¬ ¡ÿ¡ BAC = 120 Õß»“

∂â“ D ‡ªìπ®ÿ¥∫π BC ∑”„Àâ AD ·∫àß§√÷Ëß¡ÿ¡ BAC ·≈â« AD ¬“«°’ËÀπà«¬

·π«§‘¥

≈“° DE // BA ®–‰¥â ¡ÿ¡ BAD = ¡ÿ¡ ADE = 60 Ì (¡ÿ¡·¬âß)

∴ DEA = 60 Ì ‰¥â AD = DE = EA = x

ABC ∼ EDC

=

=

y + 1 = 3

y = 2

·∑π§à“„π y = 2 „π ❶ x = 3

·∑π§à“ y = 2 „π ❷ z = 4

xyz + x + y + z = 24 + 9

= 33

μÕ∫ 33

28.∂â“ x ·≈– y ‡ªìπ®”π«π®√‘ß ´÷Ëß x3 + y3 + (x + y)3 + 30xy = 2,000 ·≈â« x + y

¡’§à“‡∑à“„¥

·π«§‘¥

x3 + 3x2y + 3xy2 + y3 + (x + y)3 › 3x2y › 3xy2 + 30xy = 2,000

(x + y)3 + (x + y)3 › 3xy(x + y › 10) › 2,000 = 0

2(x + y)3 › 2(10)3 › 3xy(x + y › 10) = 0

(x + y › 10)[2{(x + y)2 + 10(x + y) + 102} › 3xy] = 0

(x + y › 10)(2x2 + xy + 2y2 + 10x + 10y + 100) = 0

·μà 2x2 + xy + 2y2 + 10x + 10y + 100 > 0

¥—ßπ—Èπ x + y › 10 = 0

x + y = 10

μÕ∫ 10

∇ ∇

ABDE

CACE

20x

3030 › x

B

A E C

D

20

∧

30

A 1-29 12/9/08, 10:23 AM24


( )

31.∂â“ + + + + + =

‚¥¬∑’Ë a ·≈– b ‡ªìπ®”π«π‡μÁ¡∑’Ë À.√.¡. ¢Õß a °—∫ b ‡∑à“°—∫ 1 ·≈â« a + b ¡’§à“‡∑à“„¥

·π«§‘¥

2(30 › x) = 3x

x = 12

AD = 12 Àπà«¬

μÕ∫ 12 Àπà«¬

30.∂â“ x ‡ªìπ®”π«π®√‘ß ‚¥¬∑’Ë 2 < x < 3 ·≈â«

x + 2 2x › 4 + x › 2 2x › 4 ¡’§à“‡∑à“„¥

·π«§‘¥

„Àâ P = x + 2 2x › 4 + x › 2 2x › 4

P2 = x + 2 2x › 4 + 2 (x + 2 2x › 4)(x › 2 2x › 4) + x › 2 2x › 4

= 2x + 2 x2 › 4(2x › 4)

= 2x + 2 x2 › 8x + 16

= 2x + 2 (x › 4)2

= 2x + 2 x › 4

= 2x + 2(4 › x) ‡æ√“–«à“ x < 4 ®–‰¥â x ‹› 4 = 4 › x

= 2x + 8 › 2x

P2 = 8

P = 2 2

μÕ∫ 2 2

11 ⋅ 3 ⋅ 5

13 ⋅ 5 ⋅ 7

15 ⋅ 7 ⋅ 9

17 ⋅ 9 ⋅ 11

19 ⋅ 11 ⋅ 13

111 ⋅ 13 ⋅ 15

ab

11 ⋅ 3 ⋅ 5 =

( )1

1 ⋅ 31

3 ⋅ 5a14 › = 1

4 ( )5 › 11 ⋅ 3 ⋅ 5

11 ⋅ 3 ⋅ 5=

= 14

13 ⋅ 5

15 ⋅ 7›1

3 ⋅ 5 ⋅ 7

A 1-29 12/9/08, 10:23 AM25


¥—ßπ—Èπ ®“°‚®∑¬å

=

=

=

=

=

=

À.√.¡. 16 °—∫ 195 §◊Õ 1 ¥—ßπ—Èπ a + b = 16 + 195 = 211

μÕ∫ 211

( )64195

( )( )›1

413

1195

( )

( )

11 Ó 3 Ó 5 + +1

3 Ó 5 Ó 71

5 Ó 7 Ó 9 + +17 Ó 9 Ó 11 +1

9 Ó 11 Ó 131

11 Ó 13 Ó 15

› +14

11 Ó 3

13 Ó 5

13 Ó 5 › +1

5 Ó 71

5 Ó 7 › +17 Ó 9

17 Ó 9 ›

+

19 Ó 11

19 Ó 11 › +1

11 Ó 131

11 Ó 13 › 113 Ó 15

›14

11 Ó 3

113 Ó 15

14

65 ‹› 1195

14

16195

32.®“°√–∫∫ ¡°“√

4y = 12 › x2 ................................... ❶

4x = 12 › y2 ................................... ➋

∂â“ A ·≈– B ‡ªìπ®ÿ¥μ—¥¢Õß°√“ø¢Õß ¡°“√ ·≈â«√–¬–√–À«à“ß®ÿ¥ A ·≈– B

‡∑à“°—∫°’ËÀπà«¬

·π«§‘¥

4y = 12 › x2 .............................. ❶

4x = 12 › y2 .............................. ➋

®“° ❶ y = .............................. ➌

π”§à“ y ®“° ➌ ·∑π„π ➋

12 › x2

4

A 1-29 12/9/08, 10:23 AM26


4x = 12 ›

4x = 12 ›

4x =

x4 › 24x2 + 64x › 48 = 0

(x › 2)(x › 2)(x › 2)(x + 6) = 0

x = 2, x = ›6

∂â“ x = 2, y = = = 2

·≈– x = ›6, y = ›6

®ÿ¥μ—¥§◊Õ A (2, 2) ·≈– B (›6, ›6)

√–¬–∑“ß√–À«à“ß 2 ®ÿ¥ §◊Õ

(›6 › 2)2 + (›6 › 2)2 = 82 + 82 = 2 (82)

= 8 2 Àπà«¬

μÕ∫ 8 2 Àπà«¬

33.√ŸªÀ°‡À≈’Ë¬¡¡ÿ¡‡∑à“ ´÷Ëß¡’§«“¡¬“« 4 ¥â“π∑’Ë‡√’¬ßμ‘¥μàÕ°—π ‡ªìπ 6, 7, 8, 9 Àπà«¬

®–¡’§«“¡¬“«√Õ∫√Ÿª°’ËÀπà«¬

·π«§‘¥

( )( )12 › x2

2

4

144 › 24x2 + x4

16

192 › 144 + 24x2 › x4

16

12 › (2)24

84

9

8

y

B8 7 6

C

6

x

y

A

60 Ì

60 Ì

60 Ì60 Ì

120 Ì

120 Ì

120 Ì 120 Ì

60 Ì120 Ì

60 Ì

6

A 1-29 12/9/08, 10:23 AM27


ABC ‡ªìπ√Ÿª “¡‡À≈’Ë¬¡¥â“π‡∑à“¡’ BC = 21

BA = 21 ∴ y = 4

AC = 21, x + y + 6 = 21

x = 11

§«“¡¬“«√Õ∫√ŸªÀ°‡À≈’Ë¬¡ = 6 + 7 + 8 + 9 + 4 + 11 Àπà«¬

= 45 Àπà«¬


34.°”Àπ¥ ABC ‡ªìπ√Ÿª “¡‡À≈’Ë¬¡Àπâ“®—Ë« ¡’ AB = AC ¡ÿ¡ BAC = 80 Ì ∂â“ D ‡ªìπ®ÿ¥¿“¬„π

∑”„Àâ ¡ÿ¡ DAB = ¡ÿ¡ DBA = 10 Ì ·≈â«¡ÿ¡ ADC ¡’¢π“¥°’ËÕß»“

·π«§‘¥

√â“ß®ÿ¥ E „π ABC „Àâ¡ÿ¡ EAC = ¡ÿ¡ ECA = 10 Ì ≈“° ED

ABD ≅ ACE (¡.¥.¡.)

AD = AE

·μà¡ÿ¡ DAE = 60 Ì

∴ ADE ‡ªìπ√Ÿª “¡‡À≈’Ë¬¡¥â“π‡∑à“

¡ÿ¡ CED = 140 Ì ·≈– DE = CE

∴ ¡ÿ¡ EDC = 20 Ì

¡ÿ¡ ADC = 80 Ì

μÕ∫ 80 Ì∇

∇ ∇

∇

B

A

C

ED)

)

)

)

A 1-29 12/9/08, 10:23 AM28


35.ABC ‡ªìπ√Ÿª “¡‡À≈’Ë¬¡ ¡’ P ‡ªìπ®ÿ¥¿“¬„π ≈“° AP æ∫ BC ∑’Ë D ≈“° BP æ∫ CA

∑’Ë E ≈“° CP æ∫ AB ∑’Ë F ∑”„Àâ PD = PE = PF = 4 Àπà«¬ ∂â“ AP + BP + CP = 6

·≈â« (AP) Ó (BP) Ó (CP) ¡’§à“‡∑à“„¥

·π«§‘¥

„Àâ AP = x, BP = y, CP = z

®–‰¥â x + y + z = 6

=

= .................

=

= ................. ➋

=

= ................. ➌

➊ + ➋ +➌, 1 = + +

‰¥â (x + 4)(y + 4)(z + 4) = 4(x + 4)(y + 4) + 4(y + 4)(z + 4) + 4(z + 4)(x + 4)

xyz + 4xy + 4yz + 16x + 16y + 16z + 64 = 4(xy + 4x + 4y + 16 + yz + 4y + 4z + 16 +

zx + 4z + 4x + 16)

xyz = 16x + 16y + 16z + 128

= 16(x + y + z) + 128

= 96 + 128

= 224

μÕ∫ 224

æ.∑. BCP

æ.∑. ABC

∇ ∇

4x + 4

æ.∑. CAP

æ.∑. ABC

∇

4y + 4

Ó CA Ó PE

Ó CA Ó BE

1212

æ.∑. ABP

æ.∑. ABC

∇ ∇

4z + 4

Ó AB Ó PF

Ó AB Ó CF

1212

4x + 4

4y + 4

4z + 4

Ó BC Ó PD

Ó BC Ó AD

1212

∇ DC

E

B

y

F

A

x

P4

z4

A 1-29 12/9/08, 10:23 AM29


μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3

‡æ◊ËÕ°“√§—¥‡≈◊Õ°π—°‡√’¬π√–¥—∫ª√–‡∑»ªï æ.». 2549

§”™’È·®ß

1. ·∫∫∑¥ Õ∫©∫—∫π’È‡ªìπ·∫∫∑¥ Õ∫™π‘¥‡μ‘¡§”μÕ∫




3. ‡«≈“∑’Ë„™â„π°“√ Õ∫ „™â‡«≈“ 2 ™—Ë«‚¡ß

A 31-57 12/9/08, 10:25 AM31


μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3

‡æ◊ËÕ°“√§—¥‡≈◊Õ°π—°‡√’¬π√–¥—∫ª√–‡∑» ªï æ.». 2549

®ßÀ“§”μÕ∫∑ÿ°§”μÕ∫¢Õß ¡°“√

μÕπ∑’Ë 1 ¡’ 20 ¢âÕ ¢âÕ≈– 3 §–·ππ

1. ®”π«π‡μÁ¡∫«° 5 ®”π«π‡√’¬ß°—π ´÷Ëß¡’º≈§Ÿ≥‡ªìπ 6375600 º≈∫«°¢Õß®”π«π

∑’ËÕ¬Ÿà√–À«à“ß®”π«π∑’ËπâÕ¬∑’Ë ÿ¥°—∫®”π«π∑’Ë¡“°∑’Ë ÿ¥„π 5 ®”π«ππ’È‡ªìπ‡∑à“„¥

3. ®”π«π‡μÁ¡∫«°μ—Èß·μà 1 ∂÷ß 600 ∑’ËÀ“√¥â«¬ 3 À√◊Õ 5 À√◊Õ 7 ‰¡à≈ßμ—«¡’∑—ÈßÀ¡¥°’Ë®”π«π

4. „Àâ x ‡ªìπ®”π«π‡μÁ¡ ‚¥¬∑’Ë 1 ≤ x ≤ 65 ·≈– À.√.¡. ¢Õß x °—∫ 65 ‡∑à“°—∫ 1 ·≈â«º≈∫«°

¢Õß x ¡’§à“‡∑à“‰√

5. ∂â“‡¢’¬π®”π«π‡μÁ¡μ—Èß·μà 1 ∂÷ß 1000 μâÕß‡¢’¬πμ—«‡≈¢»Ÿπ¬å°’Ë§√—Èß

6. „Àâ A = 3x›1 › 2 Ó 9x ‡¡◊ËÕ x ‡ªìπ®”π«π®√‘ß §à“¡“°∑’Ë ÿ¥¢Õß A ‡ªìπ‡∑à“‰√

7. ®“°√Ÿª ABCD ‡ªìπ√Ÿª ’Ë‡À≈’Ë¬¡º◊πºâ“ M, N ‡ªìπ®ÿ¥°÷Ëß°≈“ß¢Õß¥â“π AD ·≈– AB

μ“¡≈”¥—∫ ®ßÀ“§«“¡¬“«¢Õß BC

D

M

Ay N

C

By

17

5x

2. ®ßÀ“§à“ 2548 Ó 2546 Ó 2544 Ó 2542 + 16 μÕ∫„π√Ÿª®”π«π‡μÁ¡

A 31-57 12/9/08, 10:25 AM32


10. °”Àπ¥æÀÿπ“¡ P(x) = P(P(x + 23)) ‡¡◊ËÕ x < 2,024 ·≈– P(x) = x › 17 ‡¡◊ËÕ x ≥ 2,024

·≈â«§à“¢Õß P(2,549) + P(2,006) ‡∑à“°—∫‡∑à“„¥

13. ∂â“ a + b + c = 1

ab + bc + ca = 2

abc = 3

®ßÀ“§à“ a3 + b3 + c3

15. √Ÿª “¡‡À≈’Ë¬¡√ŸªÀπ÷Ëß≈âÕ¡√Õ∫¥â«¬«ß°≈¡ ¥â“πÀπ÷Ëß¢Õß√Ÿª “¡‡À≈’Ë¬¡π—Èπ¬“«

12 ‡´πμ‘‡¡μ√ ·≈–¡ÿ¡μ√ß¢â“¡¥â“ππ—Èπ«—¥‰¥â 30 Ì «ß°≈¡π—Èπ¡’‡ âπºà“π»Ÿπ¬å°≈“ß¬“«‡∑à“„¥

8. ®ßÀ“®”π«π§ŸàÕ—π¥—∫ (x, y) ∑—ÈßÀ¡¥∑’Ë‡ªìπ§”μÕ∫¢Õß ¡°“√ › = 1 ‡¡◊ËÕ x, y

‡ªìπ®”π«π‡μÁ¡

9. °”Àπ¥ x, y ‡ªìπ®”π«π®√‘ß ‚¥¬∑’Ë ®ßÀ“§ŸàÕ—π¥—∫ (x, y) ∑—ÈßÀ¡¥

∑’Ë Õ¥§≈âÕß°—∫ ¡°“√π’È

17xy

y + 2x

+ =8 › x

y

11. ABC ‡ªìπ√Ÿª “¡‡À≈’Ë¬¡¡’ AB = 21 BC = 20 D ·≈– E ‡ªìπ®ÿ¥∫π CA ∑”„Àâ

CD = 8 DE = 12 ·≈– EA = 9 ¢π“¥¢Õß¡ÿ¡ DBE ‡ªìπ‡∑à“‰√

<

12. „Àâ ABC ‡ªìπ√Ÿª “¡‡À≈’Ë¬¡„¥ Ê μàÕ¥â“π AB ·≈– AC ÕÕ°‰ª∑“ß B ·≈– C ∂÷ß D

·≈– E μ“¡≈”¥—∫ OB ·≈– OC ‡ªìπ‡ âπ·∫àß§√÷Ëß¡ÿ¡ CBD ·≈–¡ÿ¡ BCE μ“¡≈”¥—∫

∂â“¡ÿ¡ BAE ¡’¢π“¥ 80 Ì®ßÀ“¢π“¥¢Õß BOC (∑’Ë‡ªìπ¡ÿ¡·À≈¡)

<

14. ∂â“ x ‡ªìπ®”π«π®√‘ß∫«° ·≈– ·≈â« ¡’§à“‡∑à“„¥1x

x + = 4 2 x3 › 1x3

3y

7x

A 31-57 12/9/08, 10:25 AM33


16. ®“°√Ÿª∂â“√Ÿª “¡‡À≈’Ë¬¡¡ÿ¡©“°¡’ MB + MA = BC + AC ∂â“ BC = 8 Àπà«¬ ·≈–

AC = 10 Àπà«¬ ®ßÀ“ MB

17. ‡ âπμ√ß L1 ºà“π®ÿ¥ (›2, 0) ·≈– (›1, 2) L

2 ‡ªìπ‡ âπμ√ß∑’Ëºà“π®ÿ¥°”‡π‘¥·≈–μ—Èß©“°°—∫ L

1

·≈â«æ◊Èπ∑’Ë√Ÿª “¡‡À≈’Ë¬¡∑’Ë≈âÕ¡√Õ∫¥â«¬·°π x ‡ âπμ√ß L1 ·≈–‡ âπμ√ß L

2 ‡ªìπ‡∑à“„¥

18. ®ßÀ“æ®πå∑’Ë 100 ‡¡◊ËÕ°”Àπ¥æ®πå∑’Ë 90 ¢Õß≈”¥—∫ 1, 2, 4, 7, 11, ... ¡’§à“‡ªìπ 4,006

20. „Àâ O ‡ªìπ®ÿ¥°”‡π‘¥≈“° à«π¢Õß‡ âπμ√ß OA ·≈– OB ∑’Ë∑”„Àâ®ÿ¥ A ·≈– B Õ¬Ÿà∫π‡ âπμ√ß

2x + y = 4 ‚¥¬¡’ OA = OB ·≈–¡ÿ¡ AOB ‡ªìπ¡ÿ¡©“° ®ßÀ“æ◊Èπ∑’Ë√Ÿª “¡‡À≈’Ë¬¡ OAB

μÕπ∑’Ë 2 ¡’ 10 ¢âÕ ¢âÕ≈– 4 §–·ππ

21. º≈∫«°¢Õß§”μÕ∫∑—ÈßÀ¡¥¢Õß ¡°“√ (x2 › 3x › 4)3 + (2x2 + x › 1)3

= (32x › 2x › 5)3 ‡ªìπ‡∑à“‰√

22. °”Àπ¥æÀÿπ“¡ P(x) = x4 › 6x3 › 5x2 › 8x › 6 ∂â“æÀÿπ“¡ P(x) À“√¥â«¬ x › 7 ≈ßμ—«

·≈â« x ¡’§à“‡ªìπ®”π«π‡μÁ¡∫«°°’Ë®”π«π

23. „Àâ p, g ‡ªìπ√“°¢Õß ¡°“√ x2 + 5x + 1 = 0 ·≈– r, s ‡ªìπ√“°¢Õß ¡°“√

x2 + 3x + 1 = 0 ®ßÀ“§à“ (p › r)(g › r)(p + s)(g + s) ¡’§à“‡∑à“„¥

19. . . . ¡’§à“‡ªìπ‡∑à“‰√( )1 + 34 ( )1 + 3

5 ( )1 + 36 ( )1 + 3

7... ( )1 + 3

50 ( )1 + 351

M

B

AC

A 31-57 12/9/08, 10:25 AM34


24. ®ß·¬°μ—«ª√–°Õ∫¢Õß (x2 + 5x + 6)(x2 + 20x + 96) › 4x2

25. ¡’®”π«π‡μÁ¡ x, y ∑’Ëμà“ß°—π °’Ë§Ÿà Õ¬Ÿà√–À«à“ß 1 ·≈– 100 ∑’Ë 49 ‰ªÀ“√ x2 + y2 ≈ßμ—«

∂â“ (x, y) °—∫ (y, x) ∂◊Õ«à“‡ªìπ§Ÿà‡¥’¬«°—π

26.

ABCD ‡ªìπ√Ÿª ’Ë‡À≈’Ë¬¡¥â“π¢π“π ®ÿ¥ E, F, G ·≈– H ‡ªìπ®ÿ¥°÷Ëß°≈“ß¥â“π AB, BC, CD ·≈– DA

μ“¡≈”¥—∫ ≈“° AG, CE, DF ·≈– BH μ—¥°—π‡°‘¥√Ÿª ’Ë‡À≈’Ë¬¡¥â“π¢π“π MNPQ

´÷Ëß¡’®ÿ¥ R, S, T ·≈– U ‡ªìπ®ÿ¥°÷Ëß°≈“ß¥â“π MN, NP, PQ ·≈– QM μ“¡≈”¥—∫

∂â“ à«π∑’Ë·√‡ß“¡’æ◊Èπ∑’Ë 15 μ“√“ßÀπà«¬ ®ßÀ“æ◊Èπ∑’Ë√Ÿª ’Ë‡À≈’Ë¬¡ ABCD

27. ®“°√Ÿª PA ·≈– QC —¡º— «ß°≈¡ O ∑’Ë®ÿ¥ A ·≈– C μ“¡≈”¥—∫

∂â“ QC = PA ®ßÀ“§à“ QA Ó QBPC Ó PB

34

C P

QA

OB

QT

P

MR

N

A

U

S

E

H

B

DG C

F

A 31-57 12/9/08, 10:25 AM35


29. „ÀâÀ“®”π«π„π·∂«∑’Ë 89 π—∫®“°´â“¬¡◊Õμ—«∑’Ë 3

·∂«∑’Ë 1 1

·∂«∑’Ë 2 2 3 4

·∂«∑’Ë 3 5 6 7 8 9

·∂«∑’Ë 4 10 11 12 13 14 15 16

μ—«Õ¬à“ß ‡™àπ ·∂«∑’Ë 4 ®“°´â“¬¡◊Õμ—«∑’Ë 3 §◊Õ 12

·∂«∑’Ë 3 ®“°´â“¬¡◊Õμ—«∑’Ë 3 §◊Õ 7

28. „Àâ L ‡ªìπ‡ âπμ√ß∑’Ë¡’§«“¡™—π‡ªìπ ºà“π®ÿ¥»Ÿπ¬å°≈“ß¢Õß«ß°≈¡ x2 + y2 › 4x + 2y › 4 = 0

·≈–μ—¥«ß°≈¡∑’Ë®ÿ¥ A ·≈– B ∂â“®ÿ¥ C ¡’æ‘°—¥‡ªìπ (›1, ›2) ·≈â«®ßÀ“æ◊Èπ∑’Ë

√Ÿª “¡‡À≈’Ë¬¡ ABC

30. √Ÿª “¡‡À≈’Ë¬¡ ABC ‡ âπ·∫àß§√÷Ëß¡ÿ¡ A æ∫ BC ∑’Ë®ÿ¥ D ®“°®ÿ¥ B ≈“°‡ âπμ—Èß©“°°—∫

AD ∑’Ë®ÿ¥ E ≈“° HG ºà“π®ÿ¥ E ·≈–¢π“π°—∫ AC æ∫ BC ∑’Ë®ÿ¥ G ·≈– AB ∑’Ë®ÿ¥ H

∂â“ AB = 26, BC = 28, AC = 30 ®ßÀ“§«“¡¬“« DG

43›

A 31-57 12/9/08, 10:25 AM36


μ—«Õ¬à“ß·π«§‘¥·∫∫∑¥ Õ∫§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3

‡æ◊ËÕ°“√§—¥‡≈◊Õ°π—°‡√’¬π√–¥—∫ª√–‡∑»ªï æ.». 2549

§”™’È·®ß

1. ·∫∫∑¥ Õ∫©∫—∫π’È‡ªìπ·∫∫∑¥ Õ∫™π‘¥‡μ‘¡§”μÕ∫




3. ‡«≈“∑’Ë„™â„π°“√ Õ∫ „™â‡«≈“ 2 ™—Ë«‚¡ß

A 31-57 12/9/08, 10:25 AM37


2. ®ßÀ“§à“ 2,548 Ó 2,546 Ó 2,544 Ó 2,542 + 16 μÕ∫„π√Ÿª®”π«π‡μÁ¡

·π«§‘¥

„Àâ a = 2,545

(2,548)(2,546)(2,544)(2,542) + 16

= (a + 3)(a + 1)(a › 1)(a › 3) + 16

= (a2 › 9)(a2 › 1) + 16

= a4 › 10a2 + 9 + 16

= a4 › 10a2 + 25

= (a2 › 5)2 = a2 › 5

= 2,5452 › 5

μÕ∫ 6,477,020

μ—«Õ¬à“ß·π«§‘¥·∫∫∑¥ Õ∫§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3

‡æ◊ËÕ°“√§—¥‡≈◊Õ°π—°‡√’¬π√–¥—∫ª√–‡∑» ªï æ.». 2549

®ßÀ“§”μÕ∫∑ÿ°§”μÕ∫¢Õß ¡°“√

μÕπ∑’Ë 1 ¡’ 20 ¢âÕ ¢âÕ≈– 3 §–·ππ

1. ®”π«π‡μÁ¡∫«° 5 ®”π«π‡√’¬ß°—π ´÷Ëß¡’º≈§Ÿ≥‡ªìπ 6375600 º≈∫«°¢Õß®”π«π

∑’ËÕ¬Ÿà√–À«à“ß®”π«π∑’ËπâÕ¬∑’Ë ÿ¥°—∫®”π«π∑’Ë¡“°∑’Ë ÿ¥„π 5 ®”π«ππ’È‡ªìπ‡∑à“„¥

·π«§‘¥

6375600 = 102 Ó 22 Ó 32 Ó 7 Ó 11 Ó 23

= 102 Ó 2 Ó 3 Ó 21 Ó 22 Ó 23

= 52 Ó 22 Ó 2 Ó 3 Ó 21 Ó 22 Ó 23

= 21 Ó 22 Ó 23 Ó 24 Ó 25

º≈∫«°¢Õß 22, 23 ·≈– 24 §◊Õ = 22 + 23 + 24 = 69

μÕ∫ 69

A 31-57 12/9/08, 10:25 AM38


3. ®”π«π‡μÁ¡∫«°μ—Èß·μà 1 ∂÷ß 600 ∑’ËÀ“√¥â«¬ 3 À√◊Õ 5 À√◊Õ 7 ‰¡à≈ßμ—«¡’∑—ÈßÀ¡¥°’Ë®”π«π

·π«§‘¥

À“√¥â«¬ 3 ≈ßμ—«¡’ = 200

À“√¥â«¬ 5 ≈ßμ—«¡’ = 120

À“√¥â«¬ 7 ≈ßμ—«¡’ = 85

À“√¥â«¬ 3 ·≈– 5 ≈ßμ—«¡’ = 40

À“√¥â«¬ 5 ·≈– 7 ≈ßμ—«¡’ = 17

À“√¥â«¬ 7 ·≈– 3 ≈ßμ—«¡’ = 28

À“√¥â«¬ 3, 5, 7 ≈ßμ—«¡’ = 5

¥—ßπ—Èπ ®”π«π‡μÁ¡μ—Èß·μà 1 ∂÷ß 600 À“√≈ßμ—«¥â«¬ 3 À√◊Õ 5 À√◊Õ 7 §◊Õ 200 + 120 +

85 › 40 › 17 › 28 + 5 = 325

¥—ßπ—Èπ ®”π«π‡μÁ¡μ—Èß·μà 1 ∂÷ß 600 ∑’ËÀ“√‰¡à≈ßμ—«¥â«¬ 3 À√◊Õ 5 À√◊Õ 7 §◊Õ

600 › 325 = 275

μÕ∫ 275

4. „Àâ x ‡ªìπ®”π«π‡μÁ¡ ‚¥¬∑’Ë 1 ≤ x ≤ 65 ·≈– À.√.¡. ¢Õß x °—∫ 65 ‡∑à“°—∫ 1 ·≈â«º≈∫«°

¢Õß x ¡’§à“‡∑à“‰√

·π«§‘¥

À.√.¡. ¢Õß x °—∫ 65 ‡∑à“°—∫ 1 · ¥ß«à“ x À“√¥â«¬ 5 À√◊Õ 13 ‰¡à≈ßμ—«

¥—ßπ—Èπ ®”π«π∑’ËÀ“√¥â«¬ 5 ≈ßμ—« §◊Õ 5, 10, 15, ..., 65

º≈∫«°‡ªìπ 5(1 + 2 + 3 + ... + 13) = 5

= 5 Ó 13 Ó 7 = 455

®”π«π∑’Ë 13 À“√≈ßμ—«§◊Õ 13, 26, 39, 52 ¡’§à“√«¡ 130

®”π«π∑’ËÀ“√¥â«¬ 5 À√◊Õ 13 ≈ßμ—«√«¡°—π‰¥â 585

º≈∫«°¢Õß 1 + 2 + 3 + ... + 65 = (65 + 1)

= 65(33)

= 2,145

¥—ßπ—Èπ º≈∫«°¢Õß x ¡’§à“ = 2,145 › 585 = 1,560

μÕ∫ 1,560

132( )(13 + 1)

652

6005

6007

6003 Ó 56005 Ó 76007 Ó 3

600105

6003

A 31-57 12/9/08, 10:25 AM39


( )

5. ∂â“‡¢’¬π®”π«π‡μÁ¡∫«°μ—Èß·μà 1 ∂÷ß 1,000 μâÕß‡¢’¬πμ—«‡≈¢»Ÿπ¬å°’Ë§√—Èß

·π«§‘¥

®”π«π 1 À≈—° ‰¡à¡’μ—«‡≈¢ 0 ‡¢’¬π 0 ‰¥â 0 §√—Èß

®”π«π 2 À≈—°

μ—«‡≈¢ 0 Õ¬ŸàÀ≈—°Àπà«¬ ‡¢’¬π 0 ‰¥â 9 Ó 1 = 9 §√—Èß

®”π«π 3 À≈—° μ—«‡≈¢ 0 Õ¬ŸàÀ≈—°Àπà«¬

9 Ó 10 Ó 1 = 90 §√—Èß

μ—«‡≈¢ 0 Õ¬ŸàÀ≈—° ‘∫

9 Ó 1 Ó 10 = 90 §√—Èß

®”π«π 4 À≈—° ¡’μ—«‡≈¢»Ÿπ¬å 3 μ—« §◊Õ ®”π«π 1,000 ‡¢’¬π 0 ‰¥â 3 §√—Èß

¥—ßπ—Èπμ—«‡≈¢ 0 ®–∂Ÿ°‡¢’¬π∑—ÈßÀ¡¥ 90 + 90 + 9 + 3 = 189 + 3 = 192

μÕ∫ 192

6. „Àâ A = 3x›1 › 2 Ó 9x ‡¡◊ËÕ x ‡ªìπ®”π«π®√‘ß §à“¡“°∑’Ë ÿ¥¢Õß A ‡ªìπ‡∑à“‰√

·π«§‘¥

3x›1 › 2 ⋅ 9x = › 2 ⋅ 32x

= ›2 ⋅ 32x +

= › 2 32x ›

= ›2(3x › )2 +

¥—ßπ—Èπ §à“¡“°∑’Ë ÿ¥§◊Õ

μÕ∫

3x

3 3x

3

112

2144

172

2144

=172

3x

6112

2

( )+ 112

2

( )›

A 31-57 12/9/08, 10:26 AM40


7. ®“°√Ÿª ABCD ‡ªìπ√Ÿª ’Ë‡À≈’Ë¬¡º◊πºâ“ M, N ‡ªìπ®ÿ¥°÷Ëß°≈“ß¢Õß¥â“π AD ·≈– AB

μ“¡≈”¥—∫ ®ßÀ“§«“¡¬“«¢Õß BC

D

M

Ay N

C

By

17

5x

·π«§‘¥

„π AMN; x2 + y2 = ( 5)2

x2 + y2 = 5 ...................... ➊

„π MDC; x2 + (2y)2 = ( 17)2

x2 + 4y2 = 17 ...................... ➋

➋ › ➊ 3y2 = 12

y2 = 4

„™â§à“ y ‡©æ“–§à“∫«°

y = 2 ·∑π„π ➊

x2 + 4 = 5

x = 1

¥—ßπ—Èπ BC = 2x = 2(1) = 2


∇

∇

8. ®ßÀ“®”π«π§ŸàÕ—π¥—∫ (x, y) ∑—ÈßÀ¡¥∑’Ë‡ªìπ§”μÕ∫¢Õß ¡°“√ › = 1 ‡¡◊ËÕ x, y

‡ªìπ®”π«π‡μÁ¡

·π«§‘¥

= 1 ........................... �

‡Õ“ xy §Ÿ≥ � ®–‰¥â 7y › 3x = xy

xy › 7y + 3x = 0

xy › 7y + 3x › 21 = ›21

3y

7x

3y

7x

›

A 31-57 12/9/08, 10:26 AM41


(x › 7)(y + 3) = ›21 ‡¡◊ËÕ x ≠ 0 ·≈– y ≠ 0

(x › 7, y + 3) = (›1, 21), (›3, 7), (›7, 3), (›21, 1), (1, ›21), (3, ›7), (7, ›3), (21, ›1)

‰¥â∑—ÈßÀ¡¥ 8 §ŸàÕ—π¥—∫ ·μà (x, y) = (0, 0) 1 §ŸàÕ—π¥—∫ §◊Õ

(x › 7, y + 3) = (›7, 3) ∑”„Àâ§à“ x = 0, y = 0 ®÷ß„™â‰¡à‰¥â

¥—ßπ—Èπ ®÷ß¡’§”μÕ∫ 7 §ŸàÕ—π¥—∫

μÕ∫ 7

9. °”Àπ¥ x, y ‡ªìπ®”π«π®√‘ß ‚¥¬∑’Ë ®ßÀ“§ŸàÕ—π¥—∫ (x, y) ∑—ÈßÀ¡¥

∑’Ë Õ¥§≈âÕß°—∫ ¡°“√π’È

·π«§‘¥

‡Õ“ xy §Ÿ≥∑—Èß Õß¢â“ß

y(y + 2) + 17 = x (8 › x)

y2 + 2y + 17 = 8x › x2

(y2 + 2y + 1) + (x2 › 8x + 16) = 0

(y +1)2 + (x › 4)2 = 0

y = › 1, x = 4

μÕ∫ 4, ›1

+ =8 › x

yy + 2

x

10. °”Àπ¥æÀÿπ“¡ P(x) = P(P(x + 23)) ‡¡◊ËÕ x < 2,024 ·≈– P(x) = x › 17 ‡¡◊ËÕ x ≥ 2,024

·≈â«§à“¢Õß P(2,549) + P(2,006) ‡∑à“°—∫‡∑à“„¥

·π«§‘¥

P(x) = P(P(x + 23)) ‡¡◊ËÕ x < 2,024

P(x) = x › 17 ‡¡◊ËÕ x ≥ 2,024

x = 2,549; P(2,549) = 2,549 › 17

= 2,532

x = 2,006; P(2,006) = P(P(2,006 + 23))

= P(P(2,029))

= (P(2,029 › 17))

= (P(,2012))

17xy

A 31-57 12/9/08, 10:26 AM42


( )( )

= P(P(2,012 + 23))

= P(P(2,035))

= P(2,035 › 17)

= P(2,018)

= P(P(2,018 + 23))

= P(P(2,041))

= P(2,041 › 17)

= P(2,024)

= 2,024 › 17

= 2,007

P(2,549) + P(2,006) = 2,532 + 2,007 = 4,539

μÕ∫ 4,539

11. ABC ‡ªìπ√Ÿª “¡‡À≈’Ë¬¡ ¡’ AB = 21, BC = 20, D ·≈– E ‡ªìπ®ÿ¥∫π CA ∑”„Àâ

CD = 8, DE = 12 ·≈– EA = 9 ¢π“¥¢Õß¡ÿ¡ DBE ‡ªìπ‡∑à“‰√

·π«§‘¥

‡π◊ËÕß®“°

AB2 + BC2 = (AC)2 ®–‰¥â ABC ‡ªìπ√Ÿª “¡‡À≈’Ë¬¡¡ÿ¡©“°

¡’ B = 90 Ì

AD = AB

∴ ADB = ABD = 90 Ì ›

CE = BC

∴ CEB = CBE = 90 Ì ›

x = 180 Ì

x =

=

μÕ∫ 45 Ì

< <

< <

A2

<

C2

<

A2

<

+90 Ì › +90 Ì ›

( )

<

A + C2

<

C2

<

90 Ì2

∇

A 21 B

<<

D20

C

E

12

8

9 x

A 31-57 12/9/08, 10:26 AM43


12. „Àâ ABC ‡ªìπ√Ÿª “¡‡À≈’Ë¬¡„¥ Ê μàÕ¥â“π AB ·≈– AC ÕÕ°‰ª∑“ß .B ·≈– .C ∂÷ß

.D ·≈– .E μ“¡≈”¥—∫ OB ·≈– OC ‡ªìπ‡ âπ·∫àß§√÷Ëß¡ÿ¡ CBD ·≈–¡ÿ¡ BCE μ“¡≈”¥—∫

∂â“¡ÿ¡ BAE ¡’¢π“¥ 80 Ì®ßÀ“¢π“¥¢Õß BOC (∑’Ë‡ªìπ¡ÿ¡·À≈¡)

·π«§‘¥

x + y = 100 Ì

x + 2a = y + 2b = 180 Ì

x + y + 2(a + b) = 360 Ì

2(a + b) = 260 Ì

a + b = 130 Ì

∴ m = 50 Ì

μÕ∫ 50 Ì

<

13. ∂â“ a + b + c = 1

ab + bc + ca = 2

abc = 3

®ßÀ“§à“ a3 + b3 + c3

·π«§‘¥

a3 + b3 + c3 › 3abc = (a + b + c)(a2 + b2 + c2 ‹› ab › bc › ca)

a3 + b3 + c3 › 3(3) = (1)((a2 + b2 + c2) › (ab + bc + ca))

a3 + b3 + c3 › 9 = a2 + b2 + c2 › 2

a3 + b3 + c3 = a2 + b2 + c2 + 7 ..................... ➊

®“° a + b + c = 1

a2 + b2 + c2 + 2ab + 2bc + 2ac = 1

a2 + b2 + c2 + 2(2) = 1

a2 + b2 + c2 = ›3 ·∑π„π ➊

a3 + b3 + c3 = ›3 + 7 = 4

μÕ∫ 4

)

)

C

E

bbyx

aa

O

D

B

A

80 Ì

m

A 31-57 12/9/08, 10:26 AM44


6

) (

15. √Ÿª “¡‡À≈’Ë¬¡√ŸªÀπ÷Ëß≈âÕ¡√Õ∫¥â«¬«ß°≈¡ ¥â“πÀπ÷Ëß¢Õß√Ÿª “¡‡À≈’Ë¬¡π—Èπ¬“«

12 ‡´πμ‘‡¡μ√ ·≈–¡ÿ¡μ√ß¢â“¡¥â“ππ—Èπ«—¥‰¥â 30 Ì«ß°≈¡π—Èπ¡’‡ âπºà“π»Ÿπ¬å°≈“ß¬“«‡∑à“„¥

·π«§‘¥

„π ABC; Sin 30 Ì =

x =

= 12

‡ âπºà“π»Ÿπ¬å°≈“ß¬“« 24 ‡´πμ‘‡¡μ√

μÕ∫ 24

( )( )

( )

14. ∂â“ x ‡ªìπ®”π«π®√‘ß∫«° ·≈– ·≈â« ¡’§à“‡∑à“„¥

·π«§‘¥

= (4 2)2

= 32

= 30

= 28

= 28

= ± 2 7

=

= ± 2 7 (31)

= 62 7

μÕ∫ 62 7

x + =1x

x2 + 2 + 1

x2

x2 › 2 + 1

x2

1x

x ›2

1x

x ›

x2 + 1 + 1

x2

x2 + 1

x2

x3 › 1

x3

1x

x +2

( )

1x

x ›

∇ 6x

612

A

x

B C

D

30 Ì60 Ì

4 2 x3 › 1x3

x3 › 1

x3

A 31-57 12/9/08, 10:26 AM45


16. ®“°√Ÿª∂â“√Ÿª “¡‡À≈’Ë¬¡¡ÿ¡©“°¡’ MB + MA = BC + AC

∂â“ BC = 8 Àπà«¬ ·≈– AC = 10 Àπà«¬ ®ßÀ“ MB

·π«§‘¥

MB + MA = BC + AC

∂â“ BC = 8 ·≈– AC = 10

®–‰¥â«à“ MB + MA = BC + AC

= 8 + 10

= 18

¥—ßπ—Èπ MA = 18 › MB

„™â∑ƒ…Æ’∫∑æ‘∑“‚°√—

(MA)2 = (MB + BC)2 + (AC)2

(18 › MB)2 = (MB + 8)2 + 102

324 › 36MB + MB2 = MB2 + 16MB + 164

®–‰¥â 52MB = 160

MB =

=

= 3

μÕ∫ 3

160524013

1131

13

M

B

AC

A 31-57 12/9/08, 10:26 AM46


17. ‡ âπμ√ß L1 ºà“π®ÿ¥ (›2, 0) ·≈– (›1, 2) L

2 ‡ªìπ‡ âπμ√ß∑’Ëºà“π®ÿ¥°”‡π‘¥·≈–μ—Èß©“°°—∫ L

1

·≈â«æ◊Èπ∑’Ë√Ÿª “¡‡À≈’Ë¬¡∑’Ë≈âÕ¡√Õ∫¥â«¬·°π x ‡ âπμ√ß L1 ·≈–‡ âπμ√ß L

2 ‡ªìπ‡∑à“„¥

·π«§‘¥

‡ âπμ√ß L1 ºà“π®ÿ¥ (›2, 0), (›1, 2) ·≈– (x, y)

§◊Õ ¡°“√ = = 2

y = 2(x + 2)

‡ âπμ√ß L2 ºà“π®ÿ¥ (0, 0) ·≈–¡’§«“¡™—π › ¡°“√§◊Õ y = › x

› x = 2(x + 2)

x = ›4x › 8

5x = ›8

x = ®–‰¥â y =

L1 μ—¥°—∫ L

2 ∑’Ë = ( , )

AOB ¡’∞“π¬“« 2 Àπà«¬ Ÿß Àπà«¬

®–μâÕßÀ“æ◊Èπ∑’Ë OAB = Ó OB Ó §«“¡ Ÿß

= (2)( ) μ“√“ßÀπà«¬

= 0.8 μ“√“ßÀπà«¬

μÕ∫ 0.8 μ“√“ßÀπà«¬

y › 0x + 2

0 › 2›2 + 1

12

12

45

12

12

45

›85

45

,( )

(›2, 0)

45

›85

∇

18. ®ßÀ“æ®πå∑’Ë 100 ‡¡◊ËÕ°”Àπ¥æ®πå∑’Ë 90 ¢Õß≈”¥—∫ 1, 2, 4, 7, 11, ... ¡’§à“‡ªìπ 4,006

·π«§‘¥

1 2 4 7 11

1 2 3 4

1 1 1

12

B 0

A

›85

45

∇

L2

L1

(›1, 2)

A 31-57 12/9/08, 10:26 AM47


®–‰¥âæ®πå∑’Ë n §◊Õ an2 + bn + c

an

= an2 + bn + c

a1

= a + b + c = 1

a2

= 4a + 2b + c = 2

a3

= 9a + 3b + c = 4

π”¡“‡¢’¬π ¡°“√; a + b + c = 1 .......................... ➊

4a + 2b + c = 2 .......................... ➋

9a + 3b + c = 3 .......................... ➌

➋ › ➊ 3a + b = 1 .......................... ➍

➌ › ➋ 5a + b = 2 .......................... ➎

➎ › ➍ 2a = 1

a =

·∑π§à“„π ➍ 3( ) + b = 1

b = 1 › =

·∑π§à“ a, b „π ➊

®–‰¥â › + c = 1

c = 1

∴ an

= n2 › n + 1

∑¥ Õ∫ n = 90 ®–‰¥â a90

= (90)2 › (90) + 1

= › 45 + 1 = 4,050 › 44

= 4,006

¥—ßπ—Èπ a100

®–‰¥â a100

= (100)2 › (100) + 1

= › 50 + 1

= 5,000 › 49

= 4,951

μÕ∫ 4,951

12

12

12

12

32

12

12

12

12

8,1002

12

12

›12

10,0002

A 31-57 12/9/08, 10:26 AM48


19. . . . ¡’§à“‡ªìπ‡∑à“‰√

·π«§‘¥

=

=

= 1,240.2

μÕ∫ 1,240.2

1 +( )1 + 34 ( )1 + 3

5 ( )1 + 36 ( )1 + 3

7...( )1 + 3

50 ( )351

74

85

96

107⋅ ⋅ ⋅ ...

52 Ó 53 Ó 544 Ó 5 Ó 6

13 Ó 53 Ó 95

20. „Àâ O ‡ªìπ®ÿ¥°”‡π‘¥≈“° à«π¢Õß‡ âπμ√ß OA ·≈– OB ∑’Ë∑”„Àâ®ÿ¥ A ·≈– B Õ¬Ÿà∫π‡ âπμ√ß 2x + y = 4

‚¥¬¡’ OA = OB ·≈–¡ÿ¡ AOB ‡ªìπ¡ÿ¡©“° ®ßÀ“æ◊Èπ∑’Ë√Ÿª “¡‡À≈’Ë¬¡ OAB

·π«§‘¥

æ◊Èπ∑’Ë Δ OAB = (OB)(OA)

À√◊Õæ◊Èπ∑’Ë Δ OAB = (AB)(OE)

OE = (√–¬– OE ®“°®ÿ¥ (0, 0) ‰ª¬—ß‡ âπμ√ß 2x + y › 4 = 0

Δ AOE ·≈– Δ BOE ‡∑à“°—π∑ÿ°ª√–°“√ ¡ÿ¡ OAE = OBE = 45 Ì ·≈– OE = AE = BE

æ◊Èπ∑’Ë Δ OAB = =

μÕ∫ À√◊Õ 3.2 μ“√“ßÀπà«¬

12

12

0 + 0 › 45

12

165

165

5350

5249⋅

5451⋅

y

OB

x

E

A

●

45( )( )8

5

< <

A 31-57 12/9/08, 10:27 AM49


μÕπ∑’Ë 2 ¡’ 10 ¢âÕ ¢âÕ≈– 4 §–·ππ

21. º≈∫«°¢Õß§”μÕ∫∑—ÈßÀ¡¥¢Õß ¡°“√

(x2 › 3x › 4)3 + (2x2 + x › 1)3 = (3x2 › 2x › 5)3 ‡ªìπ‡∑à“‰√

·π«§‘¥

„Àâ a = x2 › 3x › 4, b = 2x2 + x › 1, c = 3x2 › 2x › 5 ®–‡ÀÁπ«à“ a + b = c

®“°‚®∑¬å a3 + b3 = c3

(a + b)3 = c3 = a3 + 3ab(a + b) + b3

·μà a + b = c ®–‰¥â 3abc = 0

a3 + b3 = a3 + 3abc + b3

a = 0 ®–‰¥â x3 › 3x › 4 = 0 ∴ x = 4, ›1

b = 0 ®–‰¥â 2x2 › x › 1 = 0 ∴ x = , ›1

c = 0 ®–‰¥â 3x2 › 2x › 5 = 0 ∴ x = , ›1

§”μÕ∫¢Õß ¡°“√ ›1, , , 4

º≈∫«°¢Õß§”μÕ∫§◊Õ ›1 + + + 4 = 5

μÕ∫ 5

12

53

12

53

12

53

16

16

22. °”Àπ¥æÀÿπ“¡ P(x) = x4 › 6x3 › 5x2 › 8x › 6 ∂â“æÀÿπ“¡ P(x) À“√¥â«¬ x › 7 ≈ßμ—«

·≈â« x ¡’§à“‡ªìπ®”π«π‡μÁ¡∫«°°’Ë®”π«π

·π«§‘¥

x › 7 x4 › 6x3 › 5x2 › 8x › 6

x4 › 7x3

x3 › 5x2

x3 › 7x2

2x2 › 8x

2x2 › 14x

6x › 6

6x › 42

‡»… 36

x3 + x2 + 2x + 6)

A 31-57 12/9/08, 10:27 AM50


∴ x › 7 ®–μâÕßÀ“√ 36 ≈ßμ—«

36 = 22 Ó 32 ®”π«π‡μÁ¡À“√ 36 ≈ßμ—«¡’∑—ÈßÀ¡¥ 18 μ—«

§à“¢Õß x › 7 §◊Õ 1, ›1, 2, ›2, 3, ›3, 4, ›4, 6, ›6, 9, ›9, 12, ›12, 18, ›18, 36, ›36

§à“ x ‡ªìπ®”π«π‡μÁ¡≈∫‡¡◊ËÕ x › 7 = ›9, x › 7 = ›12

x › 7 = ›18 ·≈– x › 7 = ›36

§à“ x ‡ªìπ®”π«π‡μÁ¡∫«°¡’ 18 › 4 = 14

„™â‰¡à‰¥â 4 μ—« ®“°∑—ÈßÀ¡¥ 18 μ—«

®”π«π∑’Ë„™â‰¥â §◊Õ 14 μ—«

μÕ∫ 14 μ—«

23. „Àâ p, g ‡ªìπ√“°¢Õß ¡°“√ x2 + 5x + 1 = 0 ·≈– r, s ‡ªìπ√“°¢Õß ¡°“√

x2 + 3x + 1 = 0 ®ßÀ“§à“ (p › r) (g › r) (p + s) (g + s) ¡’§à“‡∑à“„¥

·π«§‘¥ p, g ‡ªìπ√“°¢Õß ¡°“√ x2 + 5x + 1 = 0

®–‰¥â p + g = ›5, pg = 1

r, s ‡ªìπ√“°¢Õß ¡°“√ x2 + 3x + 1 = 0

®–‰¥â r + s = ›3, rs = 1

(p › r)(g › r)(p + s) (g + s)

= (p › r)(g + s)(g › r)(p + s)

= [pg + ps › rg › rs][pg + gs › rp › rs]

= [1 + ps › rg › 1][1 + gs › rp › 1]

= (ps › rg)(gs › rp)

= pgs2 › p2sr › rsg2 + r2pg

= s2 › p2 › g2 + r2

= (s2 + r2) › (p2 + g2)

= [(s + r)2 › 2sr] › [(p + g)2 › 2pg]

= [(›3)2 › 2] › [(›5)2 › 2]

= (7) › (23) = ›16

μÕ∫ ›16

A 31-57 12/9/08, 10:27 AM51


24. ®ß·¬°μ—«ª√–°Õ∫¢Õß (x2 + 5x + 6)(x2 + 20x + 96) › 4x2

·π«§‘¥

(x + 3)(x + 2)(x + 12)(x + 8) › 4x2

= (x + 3)(x + 8)(x +2)(x + 12) › 4x2

= (x2 + 11x + 24)(x2 + 14x + 24) › 4x2

= (x2 + 24 + 11x)(x2 + 24 + 14x) › 4x2

= (x2 + 24)2 + 25x)(x2 + 24) + 154x2 › 4x2

= (x2 + 24)2 + 25x(x2 + 24) + 150x2

= (x2 + 24 + 15x)(x2 + 24 + 10x)

= (x2 + 15x + 24)(x2 + 10x + 24)

= (x2 + 15x + 24)(x + 4)(x + 6)

μÕ∫ (x2 + 15x+ 24)(x + 4 )(x + 6)

25. ¡’®”π«π‡μÁ¡ x, y ∑’Ëμà“ß°—π°’Ë§Ÿà ∑’ËÕ¬Ÿà√–À«à“ß 1 ·≈– 100 ∑’Ë 49 À“√ x2 + y2 ≈ßμ—«

∂â“ (x, y) °—∫ (y, x) ∂◊Õ«à“‡ªìπ§Ÿà‡¥’¬«°—π

·π«§‘¥

∂â“ x2 + y2 À“√≈ßμ—«¥â«¬ 49 ·≈â« x2 + y2 À“√≈ßμ—«¥â«¬ 7

·μà x2 À“√¥â«¬ 7 ®–‡À≈◊Õ‡»… 0, 1, 2 À√◊Õ 4 ‡∑à“π—Èπ

„π∑”πÕß‡¥’¬«°—π y2

´÷Ëß x2 + y2 À“√¥â«¬ 7 ≈ßμ—« ‡¡◊ËÕ ‡»…§◊Õ 0 + 0 = 0

º≈∑’Ëμ“¡¡“ x2 + y2 À“√¥â«¬ 7 ≈ßμ—« ‡¡◊ËÕ x2 ·≈– y2 À“√¥â«¬ 7 ≈ßμ—«

π—Ëπ§◊Õ ‡¡◊ËÕ x ·≈– y À“√¥â«¬ 7 ≈ßμ—« ·≈â« x2 + y2 À“√¥â«¬ 49 ≈ßμ—«

¥—ßπ—Èπ ‘Ëß∑’ËμâÕß°“√À“ §◊Õ ®”π«π§Ÿà≈”¥—∫∑’Ë·μ°μà“ß°—π ·≈–‡ªìπ®”π«π‡μÁ¡∫«°∑’Ë x ·≈– y

πâÕ¬°«à“ 100

100 = 7 Ó 14 + 2 ¡’æÀÿ§Ÿ≥¢Õß 7 √–À«à“ß 1 ∂÷ß 100 Õ¬Ÿà 14 ®”π«π

®”π«π§ŸàŸÕ—π¥—∫¡’ =

= =

= 7(15)

= 105

μÕ∫ 105

+ 14 142 › 14 + 2(14)2

142 › 142

142 + 142

14(14 + 1)2

A 31-57 12/9/08, 10:27 AM52


26.

ABCD ‡ªìπ√Ÿª ’Ë‡À≈’Ë¬¡¥â“π¢π“π ®ÿ¥ E, F, G ·≈– H ‡ªìπ®ÿ¥°÷Ëß°≈“ß¥â“π AB, BC, CD ·≈– DA

μ“¡≈”¥—∫ ≈“° AG, CE, DF ·≈– BH μ—¥°—π‡°‘¥√Ÿª ’Ë‡À≈’Ë¬¡¥â“π¢π“π MNPQ

´÷Ëß¡’®ÿ¥ R, S, T ·≈– U ‡ªìπ®ÿ¥°÷Ëß°≈“ß¥â“π MN, NP, PQ ·≈– QM μ“¡≈”¥—∫

∂â“ à«π∑’Ë·√‡ß“¡’æ◊Èπ∑’Ë 15 μ“√“ßÀπà«¬ ®ßÀ“æ◊Èπ∑’Ë√Ÿª ’Ë‡À≈’Ë¬¡ ABCD

·π«§‘¥

æ◊Èπ∑’Ë√Ÿª MNPQ ‡ªìπ 5 ‡∑à“¢Õß√Ÿª·√‡ß“ = 5 Ó 15 μ“√“ßÀπà«¬

æ◊Èπ∑’Ë√Ÿª ABCD ‡ªìπ 5 ‡∑à“¢Õßæ◊Èπ∑’Ë√Ÿª MNPQ = 5 Ó 5 Ó 15 μ“√“ßÀπà«¬

μÕ∫ 375 μ“√“ßÀπà«¬

QT

P

N

A

S

H

B

DG C

FM

R

U

E

A 31-57 12/9/08, 10:27 AM53


( )

27. ®“°√Ÿª PA ·≈– QC —¡º— «ß°≈¡ O ∑’Ë®ÿ¥ A ·≈– C μ“¡≈”¥—∫

∂â“ QC = PA ®ßÀ“§à“

·π«§‘¥ BCQ ∼ CAQ

®–‰¥â =

∴ BQ Ó AQ = (QC)2 ....................................... ➊

ABP ∼ CAP

®–‰¥â =

∴ CP Ó BP = AP2 ................................ ➋

∴ = =

μÕ∫

QA Ó QBPC Ó PB

34

∇ ∇

BQQC

CQQA

∇ ∇

APCP

BPAP

QA Ó QBPC Ó PB

➊

➋QCAP

2916

916

∴

∴

P

A Q

BO C●

A 31-57 12/9/08, 10:27 AM54


(x2 › 4x + 4) + y2 + 2y + 1 = 4 + 4 + 1

(x › 2)2 + (y + 1)2 = 9

«ß°≈¡¡’®ÿ¥»Ÿπ¬å°≈“ß∑’Ë (2, ›1) √—»¡’ 3

‡ âπμ√ß L ¡’§«“¡™—π ·≈–ºà“π®ÿ¥ (2, ›1)

§◊Õ y + 1 = (x › 2)

3y + 3 = ›4x + 8

4x + 3y › 5 = 0

À“ CD ®“°®ÿ¥ (›1, ›2)

‰ª¬—ß‡ âπμ√ß 4x + 3y › 5 = 0

CD = = = 3

¥—ßπ—Èπ æ◊Èπ∑’Ë√Ÿª “¡‡À≈’Ë¬¡ = Ó 3 Ó 6 = 9 (AB ‡ªìπ‡ âπºà“π»Ÿπ¬å°≈“ß¬“« 6 Àπà«¬)

μÕ∫ 9

28. „Àâ L ‡ªìπ‡ âπμ√ß∑’Ë¡’§«“¡™—π‡ªìπ ºà“π®ÿ¥»Ÿπ¬å°≈“ß¢Õß«ß°≈¡ x2 + y2 › 4x + 2y › 4 = 0

·≈–μ—¥«ß°≈¡∑’Ë®ÿ¥ A ·≈– B ∂â“®ÿ¥ C ¡’æ‘°—¥‡ªìπ (›1, ›2) ·≈â« ®ßÀ“æ◊Èπ∑’Ë

√Ÿª “¡‡À≈’Ë¬¡ ABC

·π«§‘¥

®“°«ß°≈¡ x2 + y2 › 4x + 2y › 4 = 0

›43›43

155

4(›1) + 3(›2) › 542 + 32

12

(›1, ›2)

43›

(2, ›1)

A

C

D

Bv

A 31-57 12/9/08, 10:27 AM55


29. „ÀâÀ“®”π«π„π·∂«∑’Ë 89 π—∫®“°´â“¬¡◊Õμ—«∑’Ë 3

·∂«∑’Ë 1 1·∂«∑’Ë 2 2 3 4·∂«∑’Ë 3 5 6 7 8 9·∂«∑’Ë 4 10 11 12 13 14 15 16μ—«Õ¬à“ß ‡™àπ ·∂«∑’Ë 4 ®“°´â“¬¡◊Õμ—«∑’Ë 3 §◊Õ 12

·∂«∑’Ë 3 ®“°´â“¬¡◊Õμ—«∑’Ë 3 §◊Õ 7·π«§‘¥∑’Ë 1

·∂«∑’Ë 1 ®”π«π¢«“ ÿ¥§◊Õ 12 = 1·∂«∑’Ë 2 ®”π«π¢«“ ÿ¥§◊Õ 22 = 4·∂«∑’Ë 3 ®”π«π¢«“ ÿ¥§◊Õ 32 = 9

.

.

.·∂«∑’Ë 88 ®”π«π∑’Ë¢«“ ÿ¥§◊Õ 882 = 7,744·∂«∑’Ë 89 ®”π«π∑’Ë “¡®“°´â“¬¡◊Õ§◊Õ 7,744 + 3 = 7,747·π«§‘¥∑’Ë 2

1, 2, 5, 10, 17, ... 1 3 5 7 2 2 2

√Ÿª∑—Ë«‰ª an

= an2 + bn + ca1

= a + b + c = 1............................... ➊a2

= 4a + 2b + c = 2............................... ➋a3

= 9a + 3b + c = 5............................... ➌

➌ › ➋, 5a + b = 3............................... ❹

➋ › ➊, 3a + b = 1............................... ➎

2a + (1) = 2a = 2

·∑π§à“ a = 1 „π ➎

®–‰¥â b = ›2·∑π§à“ a

= 1, b = ›2 „π ➊

®–‰¥â c = 2∴ a

n= n2 › 2n + 2

A 31-57 12/9/08, 10:27 AM56


30. √Ÿª “¡‡À≈’Ë¬¡ ABC ‡ âπ·∫àß§√÷Ëß¡ÿ¡ A æ∫ BC ∑’Ë®ÿ¥ D ®“°®ÿ¥ B ≈“°‡ âπμ—Èß©“°°—∫

AD ∑’Ë®ÿ¥ E ≈“° HG ºà“π®ÿ¥ E ·≈–¢π“π°—∫ AC æ∫ BC ∑’Ë®ÿ¥ G ·≈– AB ∑’Ë®ÿ¥ H

∂â“ AB = 26, BC = 28, AC = 30 ®ßÀ“ DG

·π«§‘¥

1) î = 2 (¡ÿ¡·¬âß)

2) î = 5 (‚®∑¬å)

3) 2 = 5

4) „π Δ AHE, AH = HE

5) √Ÿª Δ AEB ‡ªìπ√Ÿª “¡‡À≈’Ë¬¡¡ÿ¡©“°

4 ‡ªìπ¡ÿ¡ª√–°Õ∫¡ÿ¡©“°¢Õß 5

3 ‡ªìπ¡ÿ¡ª√–°Õ∫¡ÿ¡©“°¢Õß 2

·μà 2 = 5 ¥—ßπ—Èπ 3 = 4

Δ HEB ‡ªìπ Δ Àπâ“®—Ë« ¡’ BH = HE

¥—ßπ—Èπ AH = HE = HB

„π Δ ABC ‡π◊ËÕß®“° HG // AC ·≈– H ‡ªìπ®ÿ¥°÷Ëß°≈“ß¢Õß AB

G ‡ªìπ®ÿ¥°÷Ëß°≈“ß BC ®–‰¥â BG = 14

„π Δ ABC, AD ·∫àß§√÷Ëß¡ÿ¡ A

¥—ßπ—Èπ = ........................................... ➊

„Àâ BD = x ·≈â« DC = 28 › x ·∑π§à“„π ➊

®–‰¥â =

π—Ëπ§◊Õ x = 13 = BD

®–‰¥â DG = 1


BDBC

^ ^

^ ^

^ ^

^

^

^ ^ ^ ^

ABAC

2630

x28 › x

= (n › 1)2 + 1∴ a

89= (89 › 1)2 + 1= 882 + 1 = 7,744 + 7,745

∴®”π«π∑’ËÕ¬Ÿà·∂«∑’Ë 89 π—∫®“°´â“¬¡◊Õμ—«∑’Ë 3 §◊Õ 7,745 + 2 = 7,747

μÕ∫ 7,747

5

A

1

23

4

E

H

CGD

B

A 31-57 12/9/08, 10:27 AM57

μ—«Õ¬à“ß·∫∫∑¥ Õ∫·≈–·π«§‘¥·∫∫∑¥ Õ∫

§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3„π°“√·¢àß¢—π∑“ß«‘™“°“√ ªï æ.». 2550

‚¥¬ ”π—°ß“π§≥–°√√¡°“√°“√»÷°…“¢—Èπæ◊Èπ∞“π

A 59-88 12/9/08, 10:28 AM59


μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3

‡æ◊ËÕ°“√§—¥‡≈◊Õ°π—°‡√’¬π√–¥—∫‡¢μæ◊Èπ∑’Ë°“√»÷°…“ªï æ.». 2550

§”™’È·®ß

1. ·∫∫∑¥ Õ∫©∫—∫π’È‡ªìπ·∫∫‡μ‘¡§”μÕ∫


μÕπ∑’Ë 1 ®”π«π 20 ¢âÕ Ê ≈– 3 §–·ππ √«¡ 60 §–·ππ

μÕπ∑’Ë 2 ®”π«π 10 ¢âÕ Ê ≈– 4 §–·ππ √«¡ 40 §–·ππ

3. ‡«≈“∑’Ë„™â„π°“√ Õ∫ „™â‡«≈“ 1 ™—Ë«‚¡ß 30 π“∑’

A 59-88 12/9/08, 10:28 AM61


μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3

‡æ◊ËÕ°“√§—¥‡≈◊Õ°π—°‡√’¬π√–¥—∫‡¢μæ◊Èπ∑’Ë°“√»÷°…“ ªï æ.». 2550

§”™’È·®ß ·∫∫∑¥ Õ∫©∫—∫π’È‡ªìπ·∫∫‡μ‘¡§”μÕ∫ §–·ππ‡μÁ¡ 100 §–·ππ ¡’ 2 μÕπ ¥—ßπ’È

μÕπ∑’Ë 1 ®”π«π 20 ¢âÕ ¢âÕ≈– 3 §–·ππ

μÕπ∑’Ë 2 ®”π«π 10 ¢âÕ ¢âÕ≈– 4 §–·ππ

μÕπ∑’Ë 1

1. º≈∫«°∑—ÈßÀ¡¥¢Õß®”π«ππ—∫∑’ËÀ“√ 2550 ≈ßμ—« ¡’§à“‡∑à“„¥

2. ∂â“ 1 + = ·≈â« x ¡’§à“‡∑à“„¥1 6

1 + x51 +

4. ®ß‡√’¬ß®”π«πμàÕ‰ªπ’È®“°πâÕ¬‰ª¡“° 2514, 4258, 8171, 16128 ·≈– 32103

5. ¢“¬ ‘π§â“ 2 ™‘Èπ‰ª√“§“™‘Èπ≈– 9,999 ∫“∑ ÷Ëß™‘Èπ·√°‰¥â°”‰√ 10% à«π™‘Èπ∑’Ë Õß¢“¥∑ÿπ 10%

∂â“¢“¬‰ª∑—Èß Õß™‘Èπ ·≈â«‰¥â°”‰√À√◊Õ¢“¥∑ÿπ°’Ë∫“∑

6. ∂â“ x = ·≈– y = ·≈â« x6 › y6 › 3x2y2(x2 › y2) ¡’§à“‡∑à“„¥5 + 12

3 5 › 12

3

7. Àπ—ß ◊Õ‡≈à¡Àπ÷Ëß¡’®”π«πÀπâ“‰¡à‡°‘π 100 Àπâ“ ∂â“©’°Àπ—ß ◊ÕÕÕ° 1 ·ºàπ ·≈â«π”‡≈¢Àπâ“

¢ÕßÀπ—ß ◊Õ∫«°°—π®–‰¥â 2,195 ®ßÀ“«à“Àπ—ß ◊Õ‡≈à¡π’È¡’∑—ÈßÀ¡¥°’ËÀπâ“

4839

3. 1 + 2 ¡’§à“‡∑à“„¥+ 3 + ... + 1001 + 12( ) 1 + 1

3( ) +4 1 + 14

1 +( )1100( )

A 59-88 12/9/08, 10:28 AM62


9. ∂â“√Ÿª “¡‡À≈’Ë¬¡√ŸªÀπ÷Ëß¡’®ÿ¥¬Õ¥ “¡®ÿ¥§◊Õ ®ÿ¥ (›5, 0), ®ÿ¥ (0, 0) ·≈–®ÿ¥¬Õ¥¢Õß°√“ø

y = x2 + 5x + 7 ·≈â«√Ÿª “¡‡À≈’Ë¬¡√Ÿªπ’È¡’æ◊Èπ∑’Ë°’Ëμ“√“ßÀπà«¬

10. π”√ŸªÀâ“‡À≈’Ë¬¡¥â“π‡∑à“¡ÿ¡‡∑à“¡“μàÕ°—π¥—ß√Ÿª ‚¥¬„Àâ·μà≈–√Ÿª∑’Ë¡“μàÕ°—π¡’¥â“π√à«¡°—π

‡æ’¬ßÀπ÷Ëß¥â“π ∂â“„Àâ n ·∑π®”π«π√Ÿª∑’Ëπ”¡“μàÕ°—π ·≈– f(n) ·∑π®”π«π¥â“π¢Õß√Ÿª n

·≈â« f(2007) ¡’§à“‡∑à“„¥

11. ®“°√Ÿª ABCDEF ‡ªìπ√ŸªÀ°‡À≈’Ë¬¡¥â“π‡∑à“¡ÿ¡‡∑à“ ∂â“Õ—μ√“ à«π√–À«à“ßæ◊Èπ∑’Ë à«π∑’Ë

·√‡ß“°—∫æ◊Èπ∑’Ë√ŸªÀ°‡À≈’Ë¬¡ ABCDEF ‡∑à“°—∫ a : b ·≈– À.√.¡. ¢Õß a °—∫ b ‡∑à“°—∫ 1

·≈â« a2 + b2 ¡’§à“‡∑à“„¥

12. ¡’§à“‡∑à“„¥

8. ®“°√Ÿª Δ ABC ¡’æ◊Èπ∑’Ë 18 μ“√“ßÀπà«¬ ‚¥¬∑’Ë AB = BC, BE ⊥ AC ·≈– AD ⊥ BC

∂â“ AC = 4 Àπà«¬ ·≈â« AD ¬“«‡∑à“°—∫°’ËÀπà«¬

13. ÿà¡À¬‘∫ ≈“°∑’Ë¡’‡≈¢‚¥¥ 1 ∂÷ß 1000 ‡¢’¬π°”°—∫‰«â „∫≈– 1 ®”π«π ¡“ 1 „∫ ∂â“§«“¡πà“®–‡ªìπ

∑’Ë‰¥â ≈“°∑’Ë®”π«ππ—Èπ¬°°”≈—ß Õß·≈â«À“√¥â«¬ 2 ≈ßμ—« ‡∑à“°—∫ ‚¥¬∑’Ë b ≠ 0 ·≈–

À.√.¡. ¢Õß a °—∫ b ‡∑à“°—∫ 1 ·≈â« a + b ¡’§à“‡∑à“„¥

ab

2404 › 2400 + 452400 + 3

A

BD

E

C

A B

C

D

F

E

A 59-88 12/9/08, 10:28 AM63


14. ·¥ß´âÕ¡¬‘ßªóπ ‚¥¬«“ß°√–∫Õ°ªóπ∑”¡ÿ¡ 30 Ì °—∫·π«æ◊Èπ√“∫ ·≈–¡’‡ªÑ“∑’ËμâÕß°“√¬‘ß

Õ¬Ÿà Ÿß®“°æ◊Èπ√“∫ 12 ‡¡μ√ ∂â“®—∫‡«≈“‡¡◊ËÕ‡√‘Ë¡¬‘ß®π≈Ÿ°°√– ÿπ°√–∑∫‡ªÑ“ ®–„™â‡«≈“

3 «‘π“∑’ ·≈â«≈Ÿ°°√– ÿπ¡’§«“¡‡√Á«°’Ë‡¡μ√μàÕ«‘π“∑’

15. √Ÿª ’Ë‡À≈’Ë¬¡®—μÿ√— ¡’‡ âπ∑·¬ß¡ÿ¡¬“« 14 ‡´πμ‘‡¡μ√ ∫√√®ÿ„π«ß°≈¡ ‚¥¬¡’®ÿ¥¬Õ¥

∑—Èß ’ËÕ¬Ÿà∫π‡ âπ√Õ∫«ß ·≈â«æ◊Èπ∑’Ë«ß°≈¡∑’ËÕ¬ŸàπÕ°√Ÿª ’Ë‡À≈’Ë¬¡®—μÿ√— ‡ªìπ°’Ë‡∑à“¢Õßæ◊Èπ∑’Ë

√Ÿª ’Ë‡À≈’Ë¬¡®—μÿ√— (°”Àπ¥ π = )227

17. ®ßÀ“«à“®ÿ¥μ—¥¢Õß°√“ø¢Õß ¡°“√ x › 3y = 1 ·≈– 2x2 › 3xy › 20 = 0 ¡’√–¬–

Àà“ß°—π°’ËÀπà«¬

18. ®“°√Ÿª °”Àπ¥ ABC ‡ªìπ√Ÿª “¡‡À≈’Ë¬¡¡ÿ¡©“°∑’Ë¡’¡ÿ¡ BCA ‡ªìπ¡ÿ¡©“° BN ·≈– CM

‡ªìπ‡ âπ¡—∏¬∞“π ∂â“ BN ⊥ CM ·≈– BC ¬“« 6 Àπà«¬ ·≈â« BN ¬“«°’ËÀπà«¬

19. A ¢—∫√∂¬πμå¥â«¬§«“¡‡√Á« 80 °‘‚≈‡¡μ√μàÕ™—Ë«‚¡ß À≈—ß®“°ÕÕ°‡¥‘π∑“ß‰¥â 45 π“∑’

B ¢—∫√∂¬πμåÕÕ°®“°μ”·Àπàß‡¥’¬«°—∫ A ¥â«¬§«“¡‡√Á« 120 °‘‚≈‡¡μ√μàÕ™—Ë«‚¡ß

‡ªìπ‡«≈“ 20 π“∑’ ®“°π—ÈπÀ¬ÿ¥æ—° 10 π“∑’ ·≈â«‡¥‘π∑“ßμàÕ¥â«¬§«“¡‡√Á« 90 °‘‚≈‡¡μ√μàÕ™—Ë«‚¡ß

®ßÀ“«à“ B ®–‡¥‘π∑“ß∑—π A ‡¡◊ËÕ A ‡¥‘π∑“ß‰¥â°’Ë™—Ë«‚¡ß °’Ëπ“∑’

20. ‡»… à«π®”π«πÀπ÷Ëß ‡¡◊ËÕπ” 2 ¡“∫«°‡≈¢‚¥¥¢Õßμ—«‡»…·≈–μ—« à«π ®–∑”„Àâ‡»… à«ππ—Èπ‡ªìπ

·≈–‡¡◊ËÕπ” 4 ¡“≈∫‡≈¢‚¥¥¢Õßμ—«‡»…·≈–μ—« à«π ®–∑”„Àâ‡»… à«ππ—Èπ‡ªìπ ®ßÀ“«à“

‡»… à«π®”π«ππ—Èπ‡ªìπ‡∑à“„¥

710

16. π”≈«¥‡ âπÀπ÷Ëß¡“¢¥‡ªìπ«ß°≈¡‰¥âæ◊Èπ∑’Ë¿“¬„π«ß°≈¡‡∑à“°—∫ 1,386 μ“√“ß‡´πμ‘‡¡μ√ ·μàπ”¡“¢¥

‡ªìπ√Ÿª ’Ë‡À≈’Ë¬¡¡ÿ¡©“°‚¥¬„Àâ§«“¡¬“«¥â“π·μà≈–¥â“π‡ªìπ®”π«π‡μÁ¡∑’Ë¡“°°«à“ 30 ‡´πμ‘‡¡μ√

·≈â«√Ÿª ’Ë‡À≈’Ë¬¡¡ÿ¡©“°π’È¡’æ◊Èπ∑’Ë¡“°∑’Ë ÿ¥°’Ëμ“√“ß‡´πμ‘‡¡μ√ (°”Àπ¥ π = )227

58

B

C

M A

N

O

A 59-88 12/9/08, 10:28 AM64


μÕπ∑’Ë 2

21. ®ßÀ“§«“¡¬“«¥â“π¢Õß√Ÿª “¡‡À≈’Ë¬¡¥â“π‡∑à“∑’Ë¡’¢π“¥„À≠à∑’Ë ÿ¥ ∑’Ë∫√√®ÿÕ¬Ÿà„π√Ÿª ’Ë‡À≈’Ë¬¡

®—μÿ√— ∑’Ë¡’§«“¡¬“«¥â“π 1 Àπà«¬

22. ∂â“™“¬§πÀπ÷Ëß¬◊πÕ¬Ÿà∫πæ◊Èπ√“∫∑’Ë®ÿ¥ A ®–¡Õß‡ÀÁπ¬Õ¥‡ “∏ß¥â«¬¡ÿ¡‡ß¬ 15 Ì ·≈–∂â“‡¢“

‡¥‘π‡¢â“À“‡ “∏ß 50 ‡¡μ√ ‡¢“®–¡Õß‡ÀÁπ¬Õ¥‡ “∏ß¥â«¬¡ÿ¡‡ß¬ 45 Ì ®ßÀ“§«“¡ Ÿß¢Õß‡ “∏ßπ’È

μÕ∫„π√Ÿª∑»π‘¬¡ 2 μ”·Àπàß (°”Àπ¥ 3 = 1.732)

23. ∂â“ d ‡ªìπ À.√.¡. ¢Õß n › 1 °—∫ n2 + n + 1 ‡¡◊ËÕ n ‡ªìπ®”π«ππ—∫ ·≈â«º≈∫«°¢Õß d

∑’Ë‡ªìπ‰ª‰¥â∑—ÈßÀ¡¥¡’§à“‡∑à“„¥

24. ∂â“ ·≈â« A › B ¡’§à“‡∑à“„¥+ +

25. ∂â“ ABCD ‡ªìπ√Ÿª ’Ë‡À≈’Ë¬¡®—μÿ√— ¬“«¥â“π≈– 28 Àπà«¬ ‚¥¬¡’ M ‡ªìπ®ÿ¥∫π AD ∑’Ë∑”„Àâ

AM = 3MD, ¡’ I ‡ªìπ®ÿ¥∫π CD ∑’Ë∑”„Àâ¡ÿ¡ IBM ‡∑à“°—∫¡ÿ¡ ABM ·≈–¡’ N ‡ªìπ®ÿ¥∫π

CD ∑’Ë∑”„Àâ BN ·∫àß§√÷Ëß¡ÿ¡ IBC ·≈â«æ◊Èπ∑’Ë¢Õß√Ÿª “¡‡À≈’Ë¬¡ BMN ‡∑à“°—∫°’Ëμ“√“ßÀπà«¬

26. ∂â“ 60a = 3 ·≈– 60b = 5 ·≈â« 122(1 › b) ¡’§à“‡∑à“„¥

27. °”Àπ¥ ABCD ‡ªìπ√Ÿª ’Ë‡À≈’Ë¬¡®—μÿ√— ¬“«¥â“π≈– 900 Àπà«¬ ‚¥¬∑’Ë O ‡ªìπ®ÿ¥μ—¥¢Õß

‡ âπ∑·¬ß¡ÿ¡ ·≈–¡’ E, F ‡ªìπ®ÿ¥∫π¥â“π AD ‚¥¬∑’Ë®ÿ¥ E Õ¬Ÿà√–À«à“ß A °—∫ F

∂â“ DF = 400 Àπà«¬ ·≈–¡ÿ¡ EOF = 45 Ì ·≈â« AE ¬“«°’ËÀπà«¬

28. ABCDEFGH ‡ªìπ√Ÿª·ª¥‡À≈’Ë¬¡¡ÿ¡‡∑à“∑’Ë¡’¥â“π¬“« 2, 2 2, 4, 4 2, 6, 7, 7

·≈– 8 Àπà«¬ ®–¡’æ◊Èπ∑’Ë°’Ëμ“√“ßÀπà«¬

1 › a › b

19x › 82x2 › x ›21

A2x › 7

Bx + 3

A 59-88 12/9/08, 10:29 AM65


30. °”Àπ¥√Ÿª “¡‡À≈’Ë¬¡ ABC ¡’®ÿ¥¬Õ¥ A(1, 6), B(›2, 3) ·≈– C(3, ›4) ∂â“‡≈◊ËÕπ®ÿ¥ A

¢π“π‰ª∑“ß¢«“ 3 Àπà«¬ ¢÷Èπ‰ª¢â“ß∫π 2 Àπà«¬ –∑âÕπ®ÿ¥ B ¢â“¡·°π Y ·≈–À¡ÿπ

®ÿ¥ C √Õ∫®ÿ¥°”‡π‘¥∑«π‡¢Á¡π“Ãî°“‡ªìπ¡ÿ¡ 180 Ì ·≈â«√Ÿª “¡‡À≈’Ë¬¡∑’Ë‡°‘¥®“°°“√‡≈◊ËÕπ

¥—ß°≈à“«®–¡’æ◊Èπ∑’Ë‡ªìπ°’Ë‡∑à“¢Õß√Ÿª “¡‡À≈’Ë¬¡ ABC

29. °”Àπ¥ x, y ·≈– z ‡ªìπ®”π«π®√‘ß∫«° ´÷Ëß Õ¥§≈âÕß°—∫√–∫∫ ¡°“√

x2(y + z)2 = (3x2 + x + 1)y2z2

y2(z + x)2 = (4y2 + y + 1)z2x2

z2(x + y)2 = (5z2 + z + 1)x2y2 ·≈â« ¡’§à“‡∑à“„¥1x

+ 1y

+ 1z

A 59-88 12/9/08, 10:29 AM66


μ—«Õ¬à“ß·π«§‘¥·∫∫∑¥ Õ∫§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3‡æ◊ËÕ°“√§—¥‡≈◊Õ°π—°‡√’¬π√–¥—∫‡¢μæ◊Èπ∑’Ë°“√»÷°…“

ªï æ.». 2550

§”™’È·®ß

1. ·∫∫∑¥ Õ∫©∫—∫π’È‡ªìπ·∫∫‡μ‘¡§”μÕ∫


μÕπ∑’Ë 1 ®”π«π 20 ¢âÕ Ê ≈– 3 §–·ππ √«¡ 60 §–·ππ

μÕπ∑’Ë 2 ®”π«π 10 ¢âÕ Ê ≈– 4 §–·ππ √«¡ 40 §–·ππ

3. ‡«≈“∑’Ë„™â„π°“√ Õ∫ „™â‡«≈“ 1 ™—Ë«‚¡ß 30 π“∑’

A 59-88 12/9/08, 10:29 AM67


μ—«Õ¬à“ß·π«§‘¥·∫∫∑¥ Õ∫§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3

‡æ◊ËÕ°“√§—¥‡≈◊Õ°π—°‡√’¬π√–¥—∫‡¢μæ◊Èπ∑’Ë°“√»÷°…“ ªï æ.». 2550


μÕπ∑’Ë 1 ®”π«π 20 ¢âÕ ¢âÕ≈– 3 §–·ππ

μÕπ∑’Ë 2 ®”π«π 10 ¢âÕ ¢âÕ≈– 4 §–·ππ

μÕπ∑’Ë 1

1. º≈∫«°∑—ÈßÀ¡¥¢Õß®”π«ππ—∫∑’ËÀ“√ 2,550 ≈ßμ—« ¡’§à“‡∑à“„¥

·π«§‘¥

2,550 = 25(102)

= 52 Ó 2 Ó 51

= 52 Ó 2 Ó 3 Ó 17

= 2 Ó 3 Ó 52 Ó 17

º≈∫«° (1 + 2)(1 + 3)(1 + 5 + 52)(1 + 17)

= (3)(4)(31)(18)

= 6,696

μÕ∫ 6,696

A 59-88 12/9/08, 10:29 AM68


( )( ) ( ) 1 +

( )( ) ( ) 1 +

2. ∂â“ 1 + = ·≈â« x ¡’§à“‡∑à“„¥

·π«§‘¥

= 1 +

= 1 +

= 1 +

=

=

x = 4

μÕ∫ 4

1 6

1 + x51 +

4839

4839

939

1399 11 + 30

9 11 + 6

95

1 61 + 1 + 4

5

3. 1 + 2 ¡’§à“‡∑à“„¥

«‘∏’∑”

1 + 2

= 1 + 2( )

= 1 + 3 + 4 + 5 + ... + 101

= 1 + 2 + 3 + 4 + 5 + ... + 101 › 2

= ( )(101 + 1) › 2

= 5,149

μÕ∫ 5,149

+ 3( ) 4( )+ ... + 100( )101100

+

+ 13

1 + 12

1 +3 + ... + 100( )+ 14

4 1 + 1100

+ 13

1 + 12

1 +3 + ... + 100( )+ 14

4

32

43

54

1012

1 + 1100

A 59-88 12/9/08, 10:29 AM69


4. ®ß‡√’¬ß®”π«πμàÕ‰ªπ’È®“°πâÕ¬‰ª¡“° 2514, 4258, 8171, 16128 ·≈– 32103

·π«§‘¥

∑”∞“π„Àâ‡∑à“°—π ‡ªìπ 2

2514 = 2514

4258 = (22)258 = 2516

8171 = (23)171 = 2513

16128 = (24)128 = 2512

32103 = (25)103 = 2515

‡¡◊ËÕæ‘®“√≥“‡≈¢¬°°”≈—ß∑’Ë¡’∞“π‡¥’¬«°—π ∂â“‡≈¢™’È°”≈—ß„¥¡“°°«à“ ®”π«ππ—Èπ°Á¡“°°«à“¥â«¬

¥—ßπ—Èπ 2512 = 16128, 2513 = 8171, 2514, 2515 = 32103, 2516 = 4258

®”π«π∑’Ë„Àâ¡“‡√’¬ß®“°πâÕ¬‰ªÀ“¡“°§◊Õ 16128, 8171, 2514, 32103, 4258

μÕ∫ 16128, 8171, 2514, 32103, 4258

5. ¢“¬ ‘π§â“ 2 ™‘Èπ‰ª√“§“™‘Èπ≈– 9,999 ∫“∑ ÷Ëß™‘Èπ·√°‰¥â°”‰√ 10% à«π™‘Èπ∑’Ë Õß¢“¥∑ÿπ 10%

∂â“¢“¬‰ª∑—Èß Õß™‘Èπ ·≈â«‰¥â°”‰√À√◊Õ¢“¥∑ÿπ°’Ë∫“∑

·π«§‘¥

¢“¬™‘Èπ·√°‰¥â°”‰√ 10%

¢“¬ 110 ∫“∑ ®“°∑ÿπ 100 ∫“∑

¢“¬ 9,999 ∫“∑ ®“°∑ÿπ = 9,090 ∫“∑

™‘Èπ·√°‰¥â°”‰√ 909 ∫“∑

™‘Èπ∑’Ë 2 ¢“¥∑ÿπ 10%

¢“¬ 90 ∫“∑ ®“°∑ÿπ 100 ∫“∑

¢“¬ 9,999 ∫“∑ ®“°∑ÿπ = 11,110 ∫“∑

™‘Èπ∑’Ë 2 ¢“¥∑ÿπ 11,110 › 9,999 = 1,111 ∫“∑

¢“¬ 2 ™‘Èπ ¢“¥∑ÿπ 1,111 › 909 = 202 ∫“∑

μÕ∫ ¢“¥∑ÿπ 202 ∫“∑

100 Ó 9,999110

100 Ó 9,99990

A 59-88 12/9/08, 10:29 AM70


6. ∂â“ x = ·≈– y = ·≈â« x6 › y6 › 3x2y2(x2 › y2) ¡’§à“‡∑à“„¥

·π«§‘¥

x6 › y6 › 3x2y2(x2 › y2) = (x2)3 › (y2)3 › 3(x2)2y2 + 3x2(y2)2

= (x2 › y2)3

= [(x › y)(x + y)]3

=

= (1 Ó )3

= 5

μÕ∫ 5

5 + 12

3 5 › 12

3

7. Àπ—ß ◊Õ‡≈à¡Àπ÷Ëß¡’®”π«πÀπâ“‰¡à‡°‘π 100 Àπâ“ ∂â“©’°Àπ—ß ◊ÕÕÕ° 1 ·ºàπ ·≈â«π”‡≈¢Àπâ“

¢ÕßÀπ—ß ◊Õ∫«°°—π®–‰¥â 2,195 ®ßÀ“«à“Àπ—ß ◊Õ‡≈à¡π’È¡’∑—ÈßÀ¡¥°’ËÀπâ“

·π«§‘¥

∑¥≈Õß∫«°μ—«‡≈¢Àπâ“®“° 1 ∂÷ß ®”π«ππ—∫Õ◊Ëπ Ê

„Àâ‰¥âº≈≈—æ∏å„°≈â‡§’¬ß°—∫ 2,195

‡™àπ 1 + 2 + 3 + ... + 60 = (60 + 1)

= 30(61) = 1,830

À√◊Õ 1 + 2 + 3 + ... + 66 = (67) = 2,211

∴ ®–‰¥â«à“ 2,211 ‹› 2,195 = 16

´÷Ëß‰¡à “¡“√∂À“®”π«π∑’Ë¡’Õ¬Ÿàμ‘¥°—π√«¡°—π‰¥â 16

®÷ß∑¥≈Õß 1 + 2 + 3 +...+ 66 + 67 = 2,278

®–‰¥â 2,278 › 2,195 = 83 = 41 + 42

¥—ßπ—Èπ Àπ—ß ◊Õ‡≈à¡π’È¡’∑—ÈßÀ¡¥ 67 Àπâ“

μÕ∫ 67 Àπâ“

602

662

5 + 12

3[( › 5 › 12

3 ) 5 + 12

3( + 5 › 12

3 )]53

3

A 59-88 12/9/08, 10:29 AM71


8. ®“°√Ÿª Δ ABC ¡’æ◊Èπ∑’Ë 18 μ“√“ßÀπà«¬ ‚¥¬∑’Ë AB = BC, BE ⊥ AC ·≈– AD ⊥ BC

∂â“ AC = 4 Àπà«¬ ·≈â« AD ¬“«‡∑à“°—∫°’ËÀπà«¬

·π«§‘¥

‡π◊ËÕß®“° Δ ABC ¡’ AB = BC ¥—ßπ—Èπ Δ ABC ‡ªìπ√Ÿª “¡‡À≈’Ë¬¡Àπâ“®—Ë«

®–‰¥â BCA = BAC

·≈– AE = EC = 2 Àπà«¬ ( ¡∫—μ‘¢Õß Δ Àπâ“®—Ë«)

æ◊Èπ∑’Ë Δ ABC = Ó AC Ó BE

18 = Ó 4 Ó BE

BE = 9

Δ BCE; BC2 = 92 + 22

BC = 85

‡π◊ËÕß®“° Δ ADC ∼ Δ BEC (BCE = ACD, BEC = ADC, CAD = CBE)

®–‰¥â =

AD =

AD =

π—Ëπ§◊Õ AD ¬“« Àπà«¬

μÕ∫ Àπà«¬

12

12

AD9

485

3685

36 8585

36 8585

36 8585

^ ^^ ^ ^ ^

^ ^

A

BD

E

C

A 59-88 12/9/08, 10:29 AM72


9. ∂â“√Ÿª “¡‡À≈’Ë¬¡√ŸªÀπ÷Ëß¡’®ÿ¥¬Õ¥ “¡®ÿ¥§◊Õ®ÿ¥ (›5, 0), ®ÿ¥ (0, 0) ·≈–®ÿ¥¬Õ¥¢Õß°√“ø

y = x2 + 5x + 7 ·≈â«√Ÿª “¡‡À≈’Ë¬¡√Ÿªπ’È¡’æ◊Èπ∑’Ë°’Ëμ“√“ßÀπà«¬

·π«§‘¥

y = x2 + 5x + 7

= x2 + 5x + ( )2 + 7 › ( )2

= (x + )2 + 7 ›

= (x + )2 +

®ÿ¥¬Õ¥ Δ §◊Õ (› , ), (›5, 0), (0, 0)

æ∑. Δ = Ó (5) Ó ( ) μ“√“ßÀπà«¬

μÕ∫ μ“√“ßÀπà«¬

52

52

52

34

52

34

12

34

158

10. π”√ŸªÀâ“‡À≈’Ë¬¡¥â“π‡∑à“¡ÿ¡‡∑à“¡“μàÕ°—π¥—ß√Ÿª ‚¥¬„Àâ·μà≈–√Ÿª∑’Ë¡“μàÕ°—π¡’¥â“π√à«¡°—π

‡æ’¬ßÀπ÷Ëß¥â“π ∂â“„Àâ n ·∑π®”π«π√Ÿª∑’Ëπ”¡“μàÕ°—π ·≈– f(n) ·∑π®”π«π¥â“π¢Õß√Ÿª n √Ÿª

·≈â« f(2,007) ¡’§à“‡∑à“„¥

·π«§‘¥

®”π«π√Ÿª n 1 2 3 4 ... n

®”π«π¥â“π ƒ(n) 5 9 13 17 ... 4n + 1

ƒ(n) = 4n + 1

ƒ(2,007) = (4 Ó 2,007) + 1 = 8,029

μÕ∫ 8,029

52

254

A 59-88 12/9/08, 10:29 AM73


·π«§‘¥

2

= =

®–‰¥â a = 2, b = 3

μÕ∫ ¥—ßπ—Èπ 23 + 32 = 4 + 9 = 13

11. ®“°√Ÿª ABCDEF ‡ªìπ√ŸªÀ°‡À≈’Ë¬¡¥â“π‡∑à“¡ÿ¡‡∑à“ ∂â“Õ—μ√“ à«π√–À«à“ßæ◊Èπ∑’Ë à«π∑’Ë·√‡ß“

°—∫æ◊Èπ∑’Ë√ŸªÀ°‡À≈’Ë¬¡ ABCDEF ‡∑à“°—∫ a : b ·≈– À.√.¡. ¢Õß a °—∫ b ‡∑à“°—∫ 1

·≈â« a2 + b2 ¡’§à“‡∑à“„¥

1218

æ◊Èπ∑’Ë·√‡ß“æ◊Èπ∑’Ë√ŸªÀ°‡À≈’Ë¬¡

23

12. ¡’§à“‡∑à“„¥

·π«§‘¥ 24 ‹› 1

2400 + 3 2404 › 2400 + 45

2404 + 3.24

›2400 › 3.24 + 45 ›2400 › 3

› 48 + 48

24 › 1 = 15

μÕ∫ 15

2404 › 2400 + 452400 + 3

)

A B

C

D

F

E

A B

C

D

F

E

32

184

68910

1413

1716

115

5

11127

A 59-88 12/9/08, 10:29 AM74


13. ÿà¡À¬‘∫ ≈“°∑’Ë¡’‡≈¢‚¥¥ 1 ∂÷ß 1000 ‡¢’¬π°”°—∫‰«â „∫≈– 1 ®”π«π ¡“ 1 „∫ ∂â“§«“¡πà“®–‡ªìπ

∑’Ë‰¥â ≈“°∑’Ë®”π«ππ—Èπ¬°°”≈—ß Õß·≈â«À“√¥â«¬ 2 ≈ßμ—« ‡∑à“°—∫ ‚¥¬∑’Ë b ≠ 0

·≈– À.√.¡. ¢Õß a °—∫ b ‡∑à“°—∫ 1 ·≈â« a + b ¡’§à“‡∑à“„¥

·π«§‘¥

æ‘®“√≥“®”π«π§Ÿà∑’Ë¬°°”≈—ß Õß·≈â«‰¡à‡°‘π 1000

‰¥â·°à 22, 42, ..., (30)2 ´÷Ëß¡’∑—ÈßÀ¡¥ 15 ®”π«π

§«“¡πà“®–‡ªìπ∑’Ë‰¥â§◊Õ =

¥—ßπ—Èπ a = 3, b = 200

π—Ëπ§◊Õ a + b = 203

μÕ∫ 203

ab

151000

3200

14. ·¥ß´âÕ¡¬‘ßªóπ ‚¥¬«“ß°√–∫Õ°ªóπ∑”¡ÿ¡ 30 Ì °—∫·π«æ◊Èπ√“∫ ·≈–¡’‡ªÑ“∑’ËμâÕß°“√¬‘ß

Õ¬Ÿà Ÿß®“°æ◊Èπ√“∫ 12 ‡¡μ√ ∂â“®—∫‡«≈“‡¡◊ËÕ‡√‘Ë¡¬‘ß®π≈Ÿ°°√– ÿπ°√–∑∫‡ªÑ“ ®–„™â‡«≈“

3 «‘π“∑’ ·≈â«≈Ÿ°°√– ÿπ¡’§«“¡‡√Á«°’Ë‡¡μ√μàÕ«‘π“∑’

·π«§‘¥

®“°√Ÿª sin 30 Ì =

=

= 24

π—Ëπ§◊Õ „π‡«≈“ 3 «‘π“∑’ °√– ÿπªóπ‡§≈◊ËÕπ∑’Ë‰¥â√–¬–∑“ß 24 ‡¡μ√

Õ—μ√“‡√Á«¢Õß°√– ÿπªóπ 8 ‡¡μ√/«‘π“∑’

12x

12

12x

μÕ∫ 8

12X

30 Ì

A 59-88 12/9/08, 10:30 AM75


16. π”≈«¥‡ âπÀπ÷Ëß¡“¢¥‡ªìπ«ß°≈¡‰¥âæ◊Èπ∑’Ë¿“¬„π«ß°≈¡‡∑à“°—∫ 1,386 μ“√“ß‡´πμ‘‡¡μ√ ·μàπ”¡“¢¥

‡ªìπ√Ÿª ’Ë‡À≈’Ë¬¡¡ÿ¡©“°‚¥¬„Àâ§«“¡¬“«¥â“π·μà≈–¥â“π‡ªìπ®”π«π‡μÁ¡∑’Ë¡“°°«à“ 30 ‡´πμ‘‡¡μ√

·≈â«√Ÿª ’Ë‡À≈’Ë¬¡¡ÿ¡©“°π’È¡’æ◊Èπ∑’Ë¡“°∑’Ë ÿ¥°’Ëμ“√“ß‡´πμ‘‡¡μ√ (°”Àπ¥ π = )

·π«§‘¥

πr2 = 1,386 μ“√“ß‡´πμ‘‡¡μ√

r2 = 1,386 Ó μ“√“ß‡´πμ‘‡¡μ√

r2 = 63 Ó 7 μ“√“ß‡´πμ‘‡¡μ√

r = 21 ‡´πμ‘‡¡μ√

‡ âπ√Õ∫«ß 2πr = 2 Ó Ó 21 ‡´πμ‘‡¡μ√

= 132 ‡´πμ‘‡¡μ√

§«“¡°«â“ß + §«“¡¬“« = 66 ‡´πμ‘‡¡μ√

15. √Ÿª ’Ë‡À≈’Ë¬¡®—μÿ√— ¡’‡ âπ∑·¬ß¡ÿ¡¬“« 14 ‡´πμ‘‡¡μ√ ∫√√®ÿ„π«ß°≈¡‚¥¬¡’®ÿ¥¬Õ¥∑—Èß ’Ë

Õ¬Ÿà∫π‡ âπ√Õ∫«ß ·≈â«æ◊Èπ∑’Ë«ß°≈¡∑’ËÕ¬ŸàπÕ°√Ÿª ’Ë‡À≈’Ë¬¡®—μÿ√— ‡ªìπ°’Ë‡∑à“¢Õßæ◊Èπ∑’Ë√Ÿª

’Ë‡À≈’Ë¬¡®—μÿ√— (°”Àπ¥ π = )227

·π«§‘¥

®“°√Ÿª æ◊Èπ∑’Ë«ß°≈¡∑’ËÕ¬ŸàπÕ°√Ÿª ’Ë‡À≈’Ë¬¡®—μÿ√— = πr2 › ( Ó 14 Ó 14)

= ( Ó 7 Ó 7) › (7 Ó 14)

= 7(22 › 14) = 56

¥—ßπ—Èπ æ◊Èπ∑’Ë«ß°≈¡∑’ËÕ¬ŸàπÕ°√Ÿª ’Ë‡À≈’Ë¬¡®—μÿ√— ‡ªìπ = ‡∑à“¢Õßæ◊Èπ∑’Ë√Ÿª ’Ë‡À≈’Ë¬¡®—μÿ√—

μÕ∫

227

12

567 Ó 14

47

123456789012345678901123456789012345678901123456789012345678901123456789012345678901123456789012345678901123456789012345678901123456789012345678901123456789012345678901123456789012345678901123456789012345678901123456789012345678901123456789012345678901123456789012345678901123456789012345678901123456789012345678901123456789012345678901123456789012345678901123456789012345678901123456789012345678901123456789012345678901123456789012345678901

722

227

227

47

14

A 59-88 12/9/08, 10:30 AM76


17. ®ßÀ“«à“®ÿ¥μ—¥¢Õß°√“ø¢Õß ¡°“√ x › 3y = 1 ·≈– 2x2 › 3xy › 20 = 0 ¡’√–¬–

Àà“ß°—π°’ËÀπà«¬

·π«§‘¥

x › 3y = 1 ..............................�

2x2 › 3xy › 20 = 0 ..............................�

®“° �; x = 1 + 3y ·∑π„π �

®–‰¥â 2(1 + 3y)2 › 3(1 + 3y)y › 20 = 0

2(1 + 6y + 9y2) › (3y + 9y2) › 20 = 0

2 + 12y + 18y2 › 3y › 9y2 › 20 = 0

9y2 + 9y › 18 = 0

y2 + y › 2 = 0

(y › 1)(y + 2) = 0

y = 1 À√◊Õ ›2

·∑π§à“ y = 1 ®–‰¥â x = 1 + 3 = 4

y = ›2 ®–‰¥â x = 1 + 3(›2) = ›5

®ÿ¥μ—¥§◊Õ (4, 1) ·≈– (›5, ›2)

Y

§«“¡°«â“ß ¡“°°«à“ 30 ‡´πμ‘‡¡μ√

æ◊Èπ∑’Ë 31 Ó 35 = 1,085 μ“√“ß‡´πμ‘‡¡μ√

æ◊Èπ∑’Ë 32 Ó 34 = 1,088 μ“√“ß‡´πμ‘‡¡μ√

æ◊Èπ∑’Ë 33 Ó 33 = 1,089 μ“√“ß‡´πμ‘‡¡μ√

μÕ∫ 1,089 μ“√“ß‡´πμ‘‡¡μ√

(4, 1)3

X

9(›5, ›2)

A 59-88 12/9/08, 10:30 AM77


√–¬–√–À«à“ß (4, 1) ·≈– (›5, ›2) §◊Õ (›5 ›4)2 + (›2 ›1)2 Àπà«¬

= 81 + 9 Àπà«¬

= 90 Àπà«¬

= 3 10 Àπà«¬

μÕ∫ 3 10 Àπà«¬

18. ®“°√Ÿª °”Àπ¥ ABC ‡ªìπ√Ÿª “¡‡À≈’Ë¬¡¡ÿ¡©“°∑’Ë¡’¡ÿ¡ BCA ‡ªìπ¡ÿ¡©“° BN ·≈– CM

‡ªìπ‡ âπ¡—∏¬∞“π ∂â“ BN ⊥ CM ·≈– BC ¬“« 6 Àπà«¬ ·≈â« BN ¬“«°’ËÀπà«¬

·π«§‘¥

1. BCN = BOC μà“ß°Á‡∑à“°—∫ 90 Õß»“

2. CBN = OBC (¡ÿ¡√à«¡)

3. BNC = BCO (®“°¢âÕ 1, 2 ¡ÿ¡§Ÿà∑’Ë “¡¢Õß√Ÿª Δ)

4. Δ BCN ∼ Δ BOC (®“°¢âÕ 1, 2, 3)

π—Ëπ§◊Õ =

„Àâ BN = 3x ®–‰¥â BO = 2x

∴ =

6x2 = 6

x2 = 1

x = ±1 (§à“ ›1 „™â‰¡à‰¥â)

π—Ëπ§◊Õ BN ¬“« 3(1) = 3 Àπà«¬


BNBC

BCBO

3x6

^ ^

^ ^

^ ^

62x

6

B M A

N

C

O

A 59-88 12/9/08, 10:30 AM78


19. A ¢—∫√∂¬πμå¥â«¬§«“¡‡√Á« 80 °‘‚≈‡¡μ√μàÕ™—Ë«‚¡ß À≈—ß®“°ÕÕ°‡¥‘π∑“ß‰¥â 45 π“∑’

B ¢—∫√∂¬πμåÕÕ°®“°μ”·Àπàß‡¥’¬«°—∫ A ¥â«¬§«“¡‡√Á« 120 °‘‚≈‡¡μ√μàÕ™—Ë«‚¡ß

‡ªìπ‡«≈“ 20 π“∑’ ®“°π—ÈπÀ¬ÿ¥æ—° 10 π“∑’ ·≈â«‡¥‘π∑“ßμàÕ¥â«¬§«“¡‡√Á« 90 °‘‚≈‡¡μ√μàÕ™—Ë«‚¡ß

®ßÀ“«à“ B ®–‡¥‘π∑“ß∑—π A ‡¡◊ËÕ A ‡¥‘π∑“ß‰¥â°’Ë™—Ë«‚¡ß °’Ëπ“∑’

·π«§‘¥

A ‡¥‘π∑“ß x ™¡. ¥â«¬Õ—μ√“‡√Á« 80 °¡./™¡. ‰¥â∑“ß 80x °‘‚≈‡¡μ√

B ‡¥‘π∑“ß 120 °¡./™¡. ‡ªìπ‡«≈“ 20 π“∑’ ‰¥â∑“ß = 40 °‘‚≈‡¡μ√

B À¬ÿ¥æ—° 10 π“∑’ ·≈â«‡¥‘π∑“ßμàÕ¥â«¬§«“¡‡√Á« 90 °¡./™¡. „™â‡«≈“ x › ™¡.

‰¥â∑“ß 90(x › ) °‘‚≈‡¡μ√

√∂∑—Èß Õß∑—π°—π · ¥ß«à“√–¬–∑“ß∑’Ë√∂∑—Èß Õß«‘Ëß‡∑à“°—π

π—Ëπ§◊Õ 80x = 40 + 90(x › )

›10x =

x =

x = 7

∴ A ‡¥‘π∑“ß‰¥â 7 ™—Ë«‚¡ß 15 π“∑’

μÕ∫ A ‡¥‘π∑“ß‰¥â 7 ™—Ë«‚¡ß 15 π“∑’

120 Ó 2060

7560

54

54

›1452

14520

14

A 59-88 12/9/08, 10:30 AM79


20. ‡»… à«π®”π«πÀπ÷Ëß ‡¡◊ËÕπ” 2 ¡“∫«°‡≈¢‚¥¥¢Õßμ—«‡»…·≈–μ—« à«π ®–∑”„Àâ‡»… à«ππ—Èπ‡ªìπ

·≈–‡¡◊ËÕπ” 4 ¡“≈∫‡≈¢‚¥¥¢Õßμ—«‡»…·≈–μ—« à«π ®–∑”„Àâ‡»… à«ππ—Èπ‡ªìπ ®ßÀ“«à“

‡»… à«π®”π«ππ—Èπ‡ªìπ‡∑à“„¥

·π«§‘¥

„Àâ‡»… à«π®”π«ππ—Èπ§◊Õ

π” 2 ∫«°‡≈¢‚¥¥∑—Èßμ—«‡»…·≈–μ—« à«π‰¥â = ................. ➊

π” 4 ‰ª≈∫‡≈¢‚¥¥∑—Èßμ—«‡»…·≈–μ—« à«π‰¥â = ................. ➋

®“° ➊; 10x + 20 = 7y + 14

10x › 7y = ›6

50x › 35y = ›30

®“° ➋; 8x › 32 = 5y › 20 ........................... ➌

8x › 5y = 12

56x › 35y = 84 ................................... ➍

➍ › ➌ 6x = 114

x =

x = 19

·∑π x = 19 ®–‰¥â y =

y = 28

‡»… à«π®”π«ππ—Èπ§◊Õ

μÕ∫

710

xy

x + 2y + 2

710

x › 4y › 4

58

1146

1928

1928

12 › 152›5

58

A 59-88 12/9/08, 10:30 AM80


C

Q

P

B

μÕπ∑’Ë 2

21. ®ßÀ“§«“¡¬“«¥â“π¢Õß√Ÿª “¡‡À≈’Ë¬¡¥â“π‡∑à“∑’Ë¡’¢π“¥„À≠à∑’Ë ÿ¥ ∑’Ë∫√√®ÿÕ¬Ÿà„π√Ÿª ’Ë‡À≈’Ë¬¡

®—μÿ√— ∑’Ë¡’§«“¡¬“«¥â“π 1 Àπà«¬

·π«§‘¥

„Àâ PQR ‡ªìπ√Ÿª “¡‡À≈’Ë¬¡¥â“π‡∑à“∑’Ë¡’¢π“¥„À≠à∑’Ë ÿ¥

´÷Ëß¡’¥â“π¬“«¥â“π≈– y Àπà«¬

„Àâ CQ = RB = x Àπà«¬

®–‰¥â QA = AR = 1 › x Àπà«¬

®“° Δ ¡ÿ¡©“° PCQ ®–‰¥â y2 = 1 + x2

®“° Δ ¡ÿ¡©“° ARQ ®–‰¥â y2 = (1 › x)2 + (1 › x)2

¥—ßπ—Èπ 1 + x2 = 1 › 2x + x2 + 1 › 2x + x2

x2 › 4x + 1 = 0

x =

= 2 ± 3

®–‰¥â x = 2 › 3 ( 2 + 3 > 1)

®“° y2 = 1 + x2

y2 = 1 + 4 › 4 3 + 3

y = 8 › 4 3 „™â‡©æ“–§à“∫«°

π—Ëπ§◊Õ §«“¡¬“«¥â“π¢Õß√Ÿª “¡‡À≈’Ë¬¡¥â“π‡∑à“∑’Ë¡’¢π“¥„À≠à∑’Ë ÿ¥ = 2 2 › 3 Àπà«¬

μÕ∫ 2 2 › 3

4 ± 16 › 42

∴

1

y

1

RA

A 59-88 12/9/08, 10:30 AM81


22. ∂â“™“¬§πÀπ÷Ëß¬◊πÕ¬Ÿà∫πæ◊Èπ√“∫∑’Ë®ÿ¥ A ®–¡Õß‡ÀÁπ¬Õ¥‡ “∏ß¥â«¬¡ÿ¡‡ß¬ 15 Ì ·≈–∂â“‡¢“

‡¥‘π‡¢â“À“‡ “∏ß 50 ‡¡μ√ ‡¢“®–¡Õß‡ÀÁπ¬Õ¥‡ “∏ß¥â«¬¡ÿ¡‡ß¬ 45 Ì ®ßÀ“§«“¡ Ÿß¢Õß‡ “∏ßπ’È

μÕ∫„π√Ÿª∑»π‘¬¡ 2 μ”·Àπàß (°”Àπ¥ 3 = 1.732)

·π«§‘¥

∑’Ë®ÿ¥ E ≈“° DE ∑”„Àâ¡ÿ¡ DEB = 30 Ì

„Àâ DC = x = BC

„π Δ DEC; sin 30 Ì =

=

DE = 2x

„π Δ AED; ¡ÿ¡ DEA = 180 Ì › 30 Ì = 150 Ì

®–‰¥â¡ÿ¡ ADE = 15 Ì

Δ AED ‡ªìπ√Ÿª “¡‡À≈’Ë¬¡Àπâ“®—Ë«

AE = 2x

„π Δ DEC; tan 30 Ì =

=

EB = 3 x › x = ( 3 › 1)x = 50 › 2x

x =

= 25(1.732 › 1) = 25(0.732) = 18.30 ‡¡μ√

μÕ∫ 18.30 ‡¡μ√

12

xDE

xDE

13

503 + 1

⋅ =3 ‹› 13 › 1

50( 3 ‹› 1)2

xEB + x

xEB + x

) ) )x

CxBEA15 Ì 30 Ì 45 Ì

D

A 59-88 12/9/08, 10:30 AM82


23. ∂â“ d ‡ªìπ À.√.¡. ¢Õß n › 1 °—∫ n2 + n + 1 ‡¡◊ËÕ n ‡ªìπ®”π«ππ—∫ ·≈â«º≈∫«°¢Õß d

∑’Ë‡ªìπ‰ª‰¥â∑—ÈßÀ¡¥¡’§à“‡∑à“„¥

·π«§‘¥

‡π◊ËÕß®“° n2 + n + 1 = (n › 1)(n + 2) + 3

d ‡ªìπ À.√.¡. ¢Õß n › 1 °—∫ n2 + n + 1

¥—ßπ—Èπ d À“√ n › 1 ≈ßμ—« ·≈– d À“√ n2 + n + 1 ≈ßμ—«

®–‰¥â d À“√ (n › 1)(n + 2) + 3 ≈ßμ—«

·μà d À“√ (n › 1)(n + 2) ≈ßμ—« ¥—ßπ—Èπ d À“√ 3 ≈ßμ—«

®–‰¥â d §◊Õ 1 À√◊Õ 3

º≈∫«°¢Õß d ∑’Ë‡ªìπ‰ª‰¥â∑—ÈßÀ¡¥§◊Õ 1 + 3 = 4

μÕ∫ 4

24. ∂â“ ·≈â« A › B ¡’§à“‡∑à“„¥

·π«§‘¥

=

=

=

‚¥¬°“√‡∑’¬∫ —¡ª√– ‘∑∏‘Ï ®–‰¥â

A + 2B = 19 ........................................... ➊

3A › 7B = ›8 ........................................... ➋

➊ Ó 3, 3A + 6B = 57 ........................................... ➌

➌ › ➋, 13B = 65

B = 5

®–‰¥â A = 9

¥—ßπ—Èπ A › B = 9 › 5 = ›2

μÕ∫ ›2

=

+A2x › 7

Bx + 3

A(x + 3) + B(2x › 7)(2x › 7)(x + 3)

19x › 82x2 › x › 21

19x › 82x2 › x › 21

+ Bx + 3

A2x › 7

(A + 2B)x + (3A › 7B)2x2 › x › 21

A 59-88 12/9/08, 10:30 AM83


25. ∂â“ ABCD ‡ªìπ√Ÿª ’Ë‡À≈’Ë¬¡®—μÿ√— ¬“«¥â“π≈– 28 Àπà«¬ ‚¥¬¡’ M ‡ªìπ®ÿ¥∫π AD ∑’Ë∑”„Àâ

AM = 3MD, ¡’ I ‡ªìπ®ÿ¥∫π CD ∑’Ë∑”„Àâ¡ÿ¡ IBM ‡∑à“°—∫¡ÿ¡ ABM ·≈–¡’ N ‡ªìπ®ÿ¥∫π

CD ∑’Ë∑”„Àâ BN ·∫àß§√÷Ëß¡ÿ¡ IBC ·≈â«æ◊Èπ∑’Ë¢Õß√Ÿª “¡‡À≈’Ë¬¡ BMN ‡∑à“°—∫°’Ëμ“√“ßÀπà«¬

·π«§‘¥

À¡ÿπ√Ÿª “¡‡À≈’Ë¬¡ BCN √Õ∫®ÿ¥ B μ“¡‡¢Á¡π“Ãî°“‡ªìπ¡ÿ¡ 90 Ì

®–‰¥â Δ BN′M ≅ Δ BNM (¥.¡.¥.)

¥—ßπ—Èπ BN′M = BNM = BNC

Δ BSN ≅ Δ BCN

„Àâ CN = x ®–‰¥â ND = 28 › x ·≈– MN = 21 + x

√Ÿª “¡‡À≈’Ë¬¡¡ÿ¡©“° DMN; (28 › x)2 + 72 = (21 + x)2

∴ x = 4

æ◊Èπ∑’Ë Δ MDN = Ó 7 Ó 24 = 84 μ“√“ßÀπà«¬

æ◊Èπ∑’Ë Δ BMN = 282 › Ó 21 Ó 28 › Ó 4 Ó 28 › 84 μ“√“ßÀπà«¬

= 28(28 › ) › 84 μ“√“ßÀπà«¬

= › 84 μ“√“ßÀπà«¬

= 350 μ“√“ßÀπà«¬

μÕ∫ 350 μ“√“ßÀπà«¬

^ ^

12

12

)A

N´

x

S

21

M

7

DI N

B28

)

)

7

)

x C

^

12

28 Ó 312

21 + 42

A 59-88 12/9/08, 10:31 AM84


27. °”Àπ¥ ABCD ‡ªìπ√Ÿª ’Ë‡À≈’Ë¬¡®—μÿ√— ¬“«¥â“π≈– 900 Àπà«¬ ‚¥¬∑’Ë O ‡ªìπ®ÿ¥μ—¥

¢Õß‡ âπ∑·¬ß¡ÿ¡ ·≈–¡’ E, F ‡ªìπ®ÿ¥∫π¥â“π AD ‚¥¬∑’Ë E Õ¬Ÿà√–À«à“ß A °—∫ F

∂â“ DF = 400 Àπà«¬ ·≈–¡ÿ¡ EOF = 45 Ì ·≈â« AE ¬“«°’ËÀπà«¬

·π«§‘¥

„Àâ AE = x Àπà«¬

EF = 500 › x Àπà«¬

°”Àπ¥®ÿ¥ P ∫π CD

∑”„Àâ DP = AE

≈“° OP

®–‰¥â Δ AOE ≅ Δ DOP

≈“° FP ®–‰¥â

Δ OEF ≅ Δ OFP

FP = 500 › x

√Ÿª “¡‡À≈’Ë¬¡¡ÿ¡©“° FDP ¡’ (500 › x)2 = x2 + 4002

∴ x = 90 Àπà«¬


26. ∂â“ 60a = 3 ·≈– 60b = 5 ·≈â« 122(1 › b) ¡’§à“‡∑à“„¥

·π«§‘¥

12 = = = 601 › b

∴ 12 2(1 › b) = (601 › b)2(1 › b)

= (601 › a › b)

=

= 2

μÕ∫ 2

1 › a › b

605

6060b

1 › a › b

12

6060a Ó 60b

12( ) = )( 60

3 Ó 5

12

) ) A

x E

500 ‹› x

D P C

O

B

45 Ì 45 Ì

1 › a › b

F

A 59-88 12/9/08, 10:31 AM85


28. ABCDEFGH ‡ªìπ√Ÿª·ª¥‡À≈’Ë¬¡¡ÿ¡‡∑à“ ∑’Ë¡’¥â“π¬“« 2, 2 2, 4, 4 2, 6, 7, 7

·≈– 8 Àπà«¬ ®–¡’æ◊Èπ∑’Ë°’Ëμ“√“ßÀπà«¬

·π«§‘¥

æ∑. ABCDEFGH = (6 + )(10 + ) › (22 + 42 + 2( )2) μ“√“ßÀπà«¬

= 60 + 56 2 + › 10 ›

= 50 + 56 2 μ“√“ßÀπà«¬

μÕ∫ 50 + 56 2 μ“√“ßÀπà«¬

7 22

7 22

12

7 22

492

492

29. °”Àπ¥ x, y ·≈– z ‡ªìπ®”π«π®√‘ß∫«° ´÷Ëß Õ¥§≈âÕß°—∫√–∫∫ ¡°“√

x2(y + z)2 = (3x2 + x + 1)y2z2

y2(z + x)2 = (4y2 + y + 1)z2x2

z2(x + y)2 = (5z2 + z + 1)x2y2 ·≈â« ¡’§à“‡∑à“„¥

·π«§‘¥

= + 3 =

= + 4 =

= =

+ +1x

1y

1z

( )2 ++1y

1z

1x2 ( )2+1

x12 +

( )2 + 1y

+1z

1x

1y2 ( )2+1

y12 +

( )2 + 1z

+1x

1y

1z2 ( )2+1

z12 +

((

( (

(

( (

45 Ì 45 Ì

45 Ì

45 Ì

45 Ì

45 Ì 45 Ì

22

4 2

4

46

7

8

7

A B

11

G

F E

D

C

7 22

7 22

7 22

7 22

+ 5

114

154

194

4 2

2 2

1x

A 59-88 12/9/08, 10:31 AM86


30. °”Àπ¥√Ÿª “¡‡À≈’Ë¬¡ ABC ¡’®ÿ¥¬Õ¥ A(1, 6), B(›2, 3) ·≈– C(3, ›4) ∂â“‡≈◊ËÕπ®ÿ¥ A

¢π“π‰ª∑“ß¢«“ 3 Àπà«¬ ¢÷Èπ‰ª¢â“ß∫π 2 Àπà«¬ –∑âÕπ®ÿ¥ B ¢â“¡·°π Y ·≈–À¡ÿπ

®ÿ¥ C √Õ∫®ÿ¥°”‡π‘¥∑«π‡¢Á¡π“Ãî°“‡ªìπ¡ÿ¡ 180 Ì ·≈â«√Ÿª “¡‡À≈’Ë¬¡∑’Ë‡°‘¥®“°°“√‡≈◊ËÕπ

¥—ß°≈à“«®–¡’æ◊Èπ∑’Ë‡ªìπ°’Ë‡∑à“¢Õß√Ÿª “¡‡À≈’Ë¬¡ ABC

·π«§‘¥

= ................... ➊

= ................... ➋

= ................... ➌

➊ + ➋ + ➌

=

„Àâ a = ®–‰¥â a2 › a › =

a2 › a › 12 = 0

a = 4

= 4

μÕ∫ 4

( )+1y

1x›1

z › 12 ( )+1

x1z+1

y + 12

114

+1z

1y›1

x › 12

+1x

1z+1

y + 12

154

+1x

1z›1

y › 12

+1x

1z+1

y + 12

194

( )+1x

1z+1

y › 32 ( )+1

x1z+1

y + 12

454

34

454

+ +1x

1y

1z

( ) ( )

( ) ( )

+ +1x

1y

1z

A(1, 6)

C(3 ›4)

B(›2, 3)

‹›1

6

X

2 �·ª≈ß·≈â«

Y

4

‹›2‹›2

‹›4

1 2 3 C (›3, 4)

Y

B (2, 3)

4

6

8

2

A (4, 8)

›4 42X

›2 6 8

A 59-88 12/9/08, 10:31 AM87


®ÿ¥ A(1, 6) ‡≈◊ËÕπ‰ª∑“ß¢«“ 3 Àπà«¬ ¢÷Èπ‰ª¢â“ß∫π 2 Àπà«¬ ‡ªìπ A´(4, 8)®ÿ¥ B(›2, 3) –∑âÕπ¢â“¡·°π Y ‡ªìπ B(2, 3)

®ÿ¥ C(3, ›4) À¡ÿπ√Õ∫®ÿ¥°”‡π‘¥∑«π‡¢Á¡π“Ãî°“‡ªìπ¡ÿ¡ 180 Ì ‡ªìπ C´(›3, 4)æ∑. Δ ABC = (5 Ó 10) › {(3 Ó 3) + (2 Ó 10) + (7 Ó 5)}

= 50 › 32 = 18 μ“√“ßÀπà«¬

æ∑. Δ A´B´C´= 7 Ó 5 › (2 Ó 5 + 5 Ó 1 + 7 Ó 4)

= 35 › 21.5 = 13.5 μ“√“ßÀπà«¬

=

μÕ∫

12

12

æ∑. Δ A´B´C´æ∑. Δ ABC

13.518

34

A 59-88 12/9/08, 10:31 AM88


μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 (©∫—∫∑’Ë 1)‡æ◊ËÕ°“√§—¥‡≈◊Õ°π—°‡√’¬π√–¥—∫ª√–‡∑» ªï æ.». 2550

§”™’È·®ß

1. ·∫∫∑¥ Õ∫©∫—∫π’È‡ªìπ·∫∫‡μ‘¡§”μÕ∫ ¡’∑—ÈßÀ¡¥ 30 ¢âÕ §–·ππ‡μÁ¡ 100 §–·ππ


μÕπ∑’Ë 1 ®”π«π 20 ¢âÕ Ê ≈– 3 §–·ππ √«¡ 60 §–·ππ

μÕπ∑’Ë 2 ®”π«π 10 ¢âÕ Ê ≈– 4 §–·ππ √«¡ 40 §–·ππ

3. ‡«≈“∑’Ë„™â„π°“√ Õ∫ „™â‡«≈“ 1 ™—Ë«‚¡ß 30 π“∑’

A 89-118 12/9/08, 10:32 AM89


( ) ( ) ( ) ( )

μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 (©∫—∫∑’Ë 1)

‡æ◊ËÕ°“√§—¥‡≈◊Õ°π—°‡√’¬π√–¥—∫ª√–‡∑» ªï æ.». 2550


μÕπ∑’Ë 1 ®”π«π 20 ¢âÕ ¢âÕ≈– 3 §–·ππ

μÕπ∑’Ë 2 ®”π«π 10 ¢âÕ ¢âÕ≈– 4 §–·ππ

μÕπ∑’Ë 1

1. ∂â“ m, n, p ·≈– q ‡ªìπ®”π«π‡©æ“–∫«° ´÷Ëß mnpq = 27000001 ·≈â« m + n + p + q

¡’§à“‡∑à“„¥

4. ∂â“¡’‡ß‘π 3,440 ∫“∑ ‡ªìπ∏π∫—μ√©∫—∫≈– 20 ∫“∑ 50 ∫“∑ ·≈– 100 ∫“∑ √«¡ 72 ©∫—∫

·≈â«®–¡’∏π∫—μ√∑—Èß “¡™π‘¥„π‡ß◊ËÕπ‰¢∑’Ë°”Àπ¥„Àâ‰¥â°’Ë«‘∏’

7. ∂â“ A = 0.5(22,550 + 2›2,550), B = 0.5(22,550 › 2›2,550) ·≈â« A2 › B2 ¡’§à“‡∑à“„¥

3. ¡’‰¢àÕ¬Ÿà∑—ÈßÀ¡¥‰¡à‡°‘π 500 øÕß ∂â“π”‰ª„ à„πμ–°√â“„∫≈– 2, 3, 4, 5 À√◊Õ 6 øÕß ‡∑à“ Ê °—π

®–‡À≈◊Õ‰¢à 1 øÕß‡ ¡Õ ·μà∂â“„ àμ–°√â“„∫≈– 7 øÕß ‡∑à“ Ê °—π ®–‰¡à¡’‰¢à‡À≈◊Õ ®ßÀ“«à“

¡’‰¢à∑—ÈßÀ¡¥°’ËøÕß

5. ∂â“ a ‡ªìπ À.√.¡. ¢Õß 234 °—∫ 324 ·≈– b, c ‡ªìπ®”π«π‡μÁ¡„π™à«ß ›10 ∂÷ß 10 ∑’Ë∑”„Àâ

a = 234b + 324c ·≈â« a + b + c ¡’§à“‡∑à“„¥

2. ∂â“ x = 3 + ·≈â« x ¡’§à“‡∑à“„¥1 14 + 1

4 + ...3 +

6. ... ¡’§à“‡∑à“„¥1 › 122

1 › 132

1 › 142 1 › 1

(2,007)2

A 89-118 12/9/08, 10:32 AM90


10. ®“°√Ÿª Δ ABC ¡’ M ‡ªìπ®ÿ¥°÷Ëß°≈“ß¢Õß¥â“π BC ∂â“ AB = 4 Àπà«¬, AM = 3 Àπà«¬

·≈– AC = 8 Àπà«¬ ·≈â« BC ¬“«°’ËÀπà«¬

12. ∂â“°”Àπ¥æÀÿπ“¡ (x + 5)3 + (3x › 2)4 › (2x + 1)5 ·≈â« —¡ª√– ‘∑∏‘Ï¢Õß x3 ¡’§à“‡∑à“„¥

8. ∂â“ x ‡∑à“°—∫ x% ¢Õß y ·≈– y ‡∑à“°—∫ y% ¢Õß z ‚¥¬∑’Ë x, y, z ‡ªìπ®”π«π®√‘ß∫«°·≈â«

y + z ¡’§à“‡∑à“„¥

13. ∂â“ x + y = 11 ·≈– x2 + y2 = 16 ·≈â« x4 + y4 ¡’§à“‡∑à“„¥

14. °”Àπ¥„Àâ x ‡ªìπ®”π«π‡μÁ¡∫«° ·≈– n(x) ‡ªìπº≈∫«°¢Õß‡≈¢‚¥¥∑’Ë‡¢’¬π·∑π x

‡™àπ n(517) = 5 + 1 + 7 = 13 ·≈– n(3229) = 3 + 2 + 2 + 9 = 16 ‡ªìπμâπ

∂â“ y = (10k + 2 + 3.10k)2 ‡¡◊ËÕ k ‡ªìπ®”π«π‡μÁ¡∫«°·≈â« n(y) ¡’§à“‡∑à“„¥

15. ∂â“π”°√«¬‚≈À–ª≈“¬μ—¥∑’Ë¡’√—»¡’∞“π 2 ‡´πμ‘‡¡μ√ ·≈– 5 ‡´πμ‘‡¡μ√ ¡“À≈Õ¡‡ªìπ≈Ÿ°∫“»°å

¬“«¥â“π≈– 2 ‡´πμ‘‡¡μ√ ·≈â«®–‰¥â≈Ÿ°∫“»°å∑—ÈßÀ¡¥ 143 ≈Ÿ° ®ßÀ“«à“°√«¬‚≈À–

ª≈“¬μ—¥π’È¡’§«“¡ Ÿß°’Ë‡´πμ‘‡¡μ√ (°”Àπ¥ π = )227

9. §”μÕ∫¢Õß ¡°“√ x › x › x › x › 99 = 99 ¡’§à“‡∑à“„¥

11. ∂â“ N = 10296 › 10259 + 10222 › 10185 + 10148 › 10111 + 1074 › 1037 + 1 ·≈–

= 0.d1 d

2 d

3 ... d

m ‡ªìπ∑»π‘¬¡´È” ´÷Ëß¡’‡≈¢‚¥¥´È”„π·μà≈–™ÿ¥πâÕ¬∑’Ë ÿ¥ m ®”π«π

·≈â« m ¡’§à“‡∑à“„¥

⋅ ⋅1N

A

B MC

4 38

A 89-118 12/9/08, 10:32 AM91


17. °”Àπ¥ P ‡ªìπæ“√“‚∫≈“ y = x2 › 3x ∂â“ P′ ‰¥â®“°°“√‡≈◊ËÕπ P ¢π“π‰ª¢â“ß∫π

4 Àπà«¬ ·≈–‰ª∑“ß´â“¬ 3 Àπà«¬ ·≈– P′ ºà“π®ÿ¥ ›2, ·≈â« a ¡’§à“‡∑à“„¥

18. §«“¡‡√Á«¢ÕßπÈ”∑’Ë‰À≈ºà“π°äÕ°∑’ËÀπ÷Ëß·≈–°äÕ°∑’Ë Õß ‡ªìπÕ—μ√“ à«π 2 : 3 ∂â“‡ªî¥°äÕ°∑’Ë Õß

·≈–°äÕ°∑’Ë “¡æ√âÕ¡°—π ·≈â«πÈ”®–‰À≈‡¢â“∂—ß≈Ÿ°∫“»°å´÷Ëß°«â“ß¥â“π≈– 1.2 ‡¡μ√ ‚¥¬„™â

‡«≈“ 20 π“∑’ ®÷ß®–‡μÁ¡∂—ß ·≈–∂â“‡ªî¥°äÕ°∑’ËÀπ÷Ëß°—∫°äÕ°∑’Ë “¡æ√âÕ¡°—π ·≈â«πÈ”®–‰À≈

ÕÕ°®“°∂—ß¥—ß°≈à“«‚¥¬„π‡«≈“ 5 π“∑’ ®–‡À≈◊ÕπÈ” ¢Õß∂—ß ®ßÀ“«à“πÈ”‰À≈ºà“π°äÕ°∑’Ë “¡

¥â«¬§«“¡‡√Á«°’Ë≈‘μ√μàÕπ“∑’

19. °√–¥“πÀ¡“°√ÿ°°√–¥“πÀπ÷Ëß‰¥â√—∫°“√√–∫“¬ ’μ“¡™àÕß¬àÕ¬ Ê ¥â«¬ ’¢“«À√◊Õ ’πÈ”‡ß‘π

™àÕß≈–Àπ÷Ëß ’‡∑à“π—Èπ ‚¥¬æ∫«à“„π∑ÿ°√Ÿª ’Ë‡À≈’Ë¬¡º◊πºâ“∑’Ë¡’ 6 ™àÕß¬àÕ¬ (¢π“¥ 2 Ó 3

À√◊Õ 3 Ó 2) ®–¡’ 2 ™àÕß¬àÕ¬‡ªìπ ’πÈ”‡ß‘π à«π∑’Ë‡À≈◊Õ‡ªìπ ’¢“«‡ ¡Õ ∂â“°√–¥“ππ’È

¡’¢π“¥ 9 Ó 11 ·≈â«®–¡’™àÕß¬àÕ¬∑’Ë‡ªìπ ’πÈ”‡ß‘π∑—ÈßÀ¡¥°’Ë™àÕß

20. ®“°√Ÿª«ß°≈¡ O ·≈–«ß°≈¡ P μ—¥°—π∑’Ë®ÿ¥ C ·≈– D ‚¥¬¡’‡ âπμ√ß AB ‡ªìπ‡ âπ —¡º— √à«¡

∂â“ ACB = 107 Ì ·≈â« ADB ¡’¢π“¥°’ËÕß»“

1a

+ 1b

+ 1c

16. ∂â“ a + b + c = 4 ·≈– = 0 ·≈â« a2 + b2 + c2 ¡’§à“‡∑à“„¥

2a › 45

)(

45

^ ^

O P

D

C B

A

A 89-118 12/9/08, 10:32 AM92


μÕπ∑’Ë 2

21. ∂â“ ABC ‡ªìπ√Ÿª “¡‡À≈’Ë¬¡Àπâ“®—Ë«∑’Ë¡’ AB = AC, BAC = 66 Ì, D ‡ªìπ®ÿ¥∫π AB,

E ‡ªìπ®ÿ¥∫π AC ∑”„Àâ DE = BD + CE ·≈– I ‡ªìπ®ÿ¥∫π BC ∑’Ë∑”„Àâ DI ·∫àß§√÷Ëß

BDE ·≈â« DIE ¡’¢π“¥°’ËÕß»“

22. °”Àπ¥ ABC ‡ªìπ√Ÿª “¡‡À≈’Ë¬¡Àπâ“®—Ë«∑’Ë¡’ AB = AC, P ‡ªìπ®ÿ¥∫π¥â“π AC ·≈– Q

‡ªìπ®ÿ¥∫π¥â“π AB ∑”„Àâ AP = PQ = QB = BC ·≈â« BAC ¡’¢π“¥°’ËÕß»“

23. ∂â“ a, b, c ·≈– d ‡ªìπ®”π«π®√‘ß ´÷Ëß Õ¥§≈âÕß°—∫ ¡°“√

a = 82 › 58 › a

b = 82 + 58 › b

c = 82 › 58 + c

·≈– d = 82 + 58 + d ·≈â« abcd ¡’§à“‡∑à“„¥

24. ∂â“ ABC ‡ªìπ√Ÿª “¡‡À≈’Ë¬¡Àπâ“®—Ë«∑’Ë¡’ BAC = 100 Ì, M ‡ªìπ®ÿ¥¿“¬„π∑”„Àâ

MBA = 10 Ì ·≈– MCA = 5 Ì ·≈â« BMA ¡’¢π“¥°’ËÕß»“

25. ∂â“ √Ÿª ’Ë‡À≈’Ë¬¡ ABCD ¡’ BAC = 81 Ì, CAD = 27 Ì, ABD = 36 Ì ·≈– CBD = 30 Ì

·≈â« ADC ¡’¢π“¥°’ËÕß»“

26. √Ÿª “¡‡À≈’Ë¬¡ ABC ·π∫„π«ß°≈¡ O ‚¥¬¡’ P ‡ªìπ®ÿ¥°÷Ëß°≈“ß OA ·≈– Q ‡ªìπ

®ÿ¥°÷Ëß°≈“ß BC ∂â“ ABC ¡’¢π“¥‡ªìπ 4 ‡∑à“¢Õß¢π“¥¢Õß OPQ ·≈– ACB ¡’¢π“¥‡ªìπ

6 ‡∑à“¢Õß¢π“¥¢Õß OPQ ·≈â« OPQ ¡’¢π“¥°’ËÕß»“

27. °”Àπ¥„Àâ AB ‡ªìπ‡ âπºà“π»Ÿπ¬å°≈“ß«ß°≈¡∑’Ë¡’√—»¡’ 99 Àπà«¬ ·≈– P ‡ªìπ®ÿ¥¿“¬„π

®ßÀ“§«“¡πà“®–‡ªìπ∑’Ë APB ≤ 135 Ì (μÕ∫„π√Ÿª¢Õß π)

^

^ ^

^

^

^ ^

^ ^ ^ ^

^

^ ^ ^

^ ^

^

^

A 89-118 12/9/08, 10:32 AM93


( ) ( ) ( )

30. ∂â“°”Àπ¥√–∫∫ ¡°“√

a1 + a

2 + a

3 + ... + a

n = 96

a12 + a

22 + a

32 + ... + a

n2 = 144

a13 + a

23 + a

33 + ... + a

n3 = 216

‡¡◊ËÕ ai ‡ªìπ®”π«π®√‘ß∫«° ”À√—∫∑ÿ° i = 1, 2, 3, ..., n ·≈â« a

14 + a

24 + a

34 + ... + a

n4

¡’§à“‡∑à“„¥

29. ∂â“°”Àπ¥√–∫∫ ¡°“√

10x2 + 5y2 › 2xy › 38x › 6y + 41 = 0

3x2 › 2y2 + 5xy › 17x › 6y + 20 = 0

·≈â«§à“¢Õß x3 + y3 ‡ªìπ‡∑à“„¥

28. °”Àπ¥„Àâ (x1, y

1), (x

2, y

2) ·≈– (x

3, y

3) ‡ªìπ§”μÕ∫¢Õß√–∫∫ ¡°“√

x3 › 3xy2 = 1,999

y3 › 3x2y = 1,998

·≈â« 1 ¡’§à“‡∑à“„¥1 ›

x1

y1

1 ›x2

y2

1 ›x3

y3

A 89-118 12/9/08, 10:32 AM94


μ—«Õ¬à“ß·π«§‘¥·∫∫∑¥ Õ∫§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 (©∫—∫∑’Ë 1)‡æ◊ËÕ°“√§—¥‡≈◊Õ°π—°‡√’¬π√–¥—∫ª√–‡∑» ªï æ.». 2550

§”™’È·®ß

1. ·∫∫∑¥ Õ∫©∫—∫π’È‡ªìπ·∫∫‡μ‘¡§”μÕ∫ ¡’∑—ÈßÀ¡¥ 30 ¢âÕ §–·ππ‡μÁ¡ 100 §–·ππ


μÕπ∑’Ë 1 ®”π«π 20 ¢âÕ Ê ≈– 3 §–·ππ √«¡ 60 §–·ππ

μÕπ∑’Ë 2 ®”π«π 10 ¢âÕ Ê ≈– 4 §–·ππ √«¡ 40 §–·ππ

3. ‡«≈“∑’Ë„™â„π°“√ Õ∫ „™â‡«≈“ 1 ™—Ë«‚¡ß 30 π“∑’

A 89-118 12/9/08, 10:32 AM95


μ—«Õ¬à“ß·π«§‘¥·∫∫∑¥ Õ∫§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 (©∫—∫∑’Ë 1)

‡æ◊ËÕ°“√§—¥‡≈◊Õ°π—°‡√’¬π√–¥—∫ª√–‡∑» ªï æ.». 2550

§”™’È·®ß ·∫∫∑¥ Õ∫©∫—∫π’È‡ªìπ·∫∫‡μ‘¡§”μÕ∫ §–·ππ‡μÁ¡ 100 §–·ππ ¡’ 2 μÕπ ¥—ßπ’ÈμÕπ∑’Ë 1 ®”π«π 20 ¢âÕ ¢âÕ≈– 3 §–·ππ μÕπ∑’Ë 2 ®”π«π 10 ¢âÕ ¢âÕ≈– 4 §–·ππ

μÕπ∑’Ë 1

1. ∂â“ m, n, p ·≈– q ‡ªìπ®”π«π‡©æ“–∫«° ´÷Ëß mnpq = 27000001 ·≈â« m + n + p + q

¡’§à“‡∑à“„¥

·π«§‘¥

27000001 = 27000000 + 1

= (300)3 + 13

301 = 7 Ó 43

3002 › 300 + 1 = 9x4 › 3x2 + 1 ‡¡◊ËÕ x = 10

= 9x4 + 6x2 + 1 › 9x2

= (3x2 + 1)2 › (3x)2

= (3x2 › 3x + 1)(3x2 + 3x + 1)

= 271 Ó 331

27000001 = 7 Ó 43 Ó 271 Ó 331

m + n + p + q = 7 + 43 + 271 + 331

= 652

μÕ∫ 652

A 89-118 12/9/08, 10:32 AM96


2. ∂â“ x = 3 + ·≈â« x ¡’§à“‡∑à“„¥

·π«§‘¥

x = 3 +

x = 3 +

x › 3 = 3 + › 3

4x2 › 12x › 3 = 0

x =

x =

x =

®–‰¥â x =

μÕ∫

1 14 + 1

4 + ...3 +

1 14 + 1

4 + ...3 +

x4x + 1

1 1 x4 +

12 ± 144 + 488

12 ± 8 38

3 ± 2 32

3 ± 2 32

3 ± 2 32

A 89-118 12/9/08, 10:33 AM97


3. ¡’‰¢àÕ¬Ÿà∑—ÈßÀ¡¥‰¡à‡°‘π 500 øÕß ∂â“π”‰ª„ à„πμ–°√â“„∫≈– 2, 3, 4, 5 À√◊Õ 6 øÕß ‡∑à“ Ê °—π

®–‡À≈◊Õ‰¢à 1 øÕß‡ ¡Õ ·μà∂â“„ àμ–°√â“„∫≈– 7 øÕß ‡∑à“ Ê °—π ®–‰¡à¡’‰¢à‡À≈◊Õ ®ßÀ“«à“

¡’‰¢à∑—ÈßÀ¡¥°’ËøÕß

·π«§‘¥

«‘‡§√“–Àå n = 60k + 1 (§.√.π. ¢Õß 2, 3, 4, 5, 6 §◊Õ 60)

¥—ßπ—Èπ 7| (60k + 1)

≈Õß·∑π k ¥â«¬ 1 n = 61

≈Õß·∑π k ¥â«¬ 2 n = 121

≈Õß·∑π k ¥â«¬ 3 n = 181

≈Õß·∑π k ¥â«¬ 4 n = 241

≈Õß·∑π k ¥â«¬ 5 n = 301

À√◊ÕÀ“μ—«∑’Ë„°≈â‡§’¬ß°—∫ 60k ·μàÀ“√¥â«¬ 7 ≈ßμ—« §◊Õ 56k

¥—ßπ—Èπ 7| (60k + 1) › 56k

π—Ëπ§◊Õ 7| (4k + 1)

∂â“ k = 5 ®–∑”„Àâ 7| (4k + 1)

¥—ßπ—Èπ ®÷ß·∑π k ¥â«¬ 5 ≈ß„π 4k + 1 ®–‰¥â

7| ((60)(5) + 1)

7| (301)μÕ∫ 301 øÕß

A 89-118 12/9/08, 10:33 AM98


4. ∂â“¡’‡ß‘π 3,440 ∫“∑ ‡ªìπ∏π∫—μ√©∫—∫≈– 20 ∫“∑ 50 ∫“∑ ·≈– 100 ∫“∑ √«¡ 72 ©∫—∫

·≈â«®–¡’∏π∫—μ√∑—Èß “¡™π‘¥„π‡ß◊ËÕπ‰¢∑’Ë°”Àπ¥„Àâ‰¥â°’Ë«‘∏’

·π«§‘¥

«‘∏’∑’Ë 1

➀ ‡¡◊ËÕ¡’∏π∫—μ√©∫—∫≈– 20 ∫“∑ ®”π«π 2 ©∫—∫ ‡ªìπ‡ß‘π 40 ∫“∑

‡À≈◊Õ©∫—∫≈– 50 ∫“∑ ·≈– 100 ∫“∑ √«¡ 70 ©∫—∫ ‡ªìπ‡ß‘π 3,400 ∫“∑

©∫—∫≈– 50 ∫“∑ ®”π«π x ©∫—∫ ‡ªìπ‡ß‘π 50x ∫“∑

©∫—∫≈– 100 ∫“∑ ®”π«π 70 › x ©∫—∫ ‡ªìπ‡ß‘π 7,000 › 100x ∫“∑

√«¡ Õß©∫—∫‡ªìπ‡ß‘π 50x + 7,000 › 100 x = 3,400 ∫“∑

4,600 = 50 x

x = = 92 ©∫—∫

¡’∏π∫—μ√©∫—∫≈– 50 ∫“∑ 92 ©∫—∫ ‡ªìπ‰ª‰¡à‰¥â

➁ ‡¡◊ËÕ¡’∏π∫—μ√©∫—∫≈– 20 ∫“∑ ®”π«π 7 ©∫—∫ ‡ªìπ‡ß‘π 140 ∫“∑

‡À≈◊Õ©∫—∫≈– 50 ∫“∑ ·≈– 100 ∫“∑ √«¡ 65 ©∫—∫ ‡ªìπ‡ß‘π 3,300 ∫“∑

©∫—∫≈– 50 ∫“∑ ®”π«π x ©∫—∫ ‡ªìπ‡ß‘π 50x ∫“∑

©∫—∫≈– 100 ∫“∑ ®”π«π 65 › x ©∫—∫ ‡ªìπ‡ß‘π 6,500 › 100x ∫“∑

√«¡ Õß©∫—∫‡ªìπ‡ß‘π 50 x + 6,500 › 100 x = 3,300 ∫“∑

50 x = 3,200

x = 64

4,60050

®”π«π©∫—∫

(®”π«π‡ß‘π)

§”μÕ∫∑’Ë...

©∫—∫≈– 20 ∫“∑

(®”π«π‡ß‘π)

©∫—∫≈– 50 ∫“∑

(®”π«π‡ß‘π)

©∫—∫≈– 100 ∫“∑

(®”π«π‡ß‘π)À¡“¬‡Àμÿ

μÕ∫ 8 «‘∏’

1 7(140) 64(3,200) 1(100)

2 12(240) 56(2,800) 4(400)

3 17(340) 48(2,400) 7(700)

4 22(440) 40(2,000) 10(1,000)

5 27(540) 32(1,600) 13(1,300)

6 32(640) 24(1,200) 16(1,600)

7 37(740) 16(800) 19(1,900)

8 42(840) 8(400) 22(2,200)

9 47(940) 0(0) 25(2,500)

À¡“¬‡Àμÿ ‡æ‘Ë¡∑’≈– 5 ©∫—∫ ≈¥∑’≈– 8 ©∫—∫ ‡æ‘Ë¡∑’≈– 3 ©∫—∫

‰¡à¡’∏π∫—μ√©∫—∫≈– 50 ∫“∑

A 89-118 12/9/08, 10:33 AM99


·π«§‘¥

«‘∏’∑’Ë 2

°”Àπ¥„Àâ x ·∑π®”π«π∏π∫—μ√©∫—∫≈– 20 ∫“∑

y ·∑π®”π«π∏π∫—μ√©∫—∫≈– 50 ∫“∑

·≈– z ·∑π®”π«π∏π∫—μ√©∫—∫≈– 100 ∫“∑

®–‰¥â«à“

x + y + z = 72 ......................... ➊

2x + 5y + 10z = 344 ......................... ➋

®“° ➊ z = 72 › x › y ·∑π„π ➋

2x + 5y + 10(72 › x › y) = 344

2x + 5y + 720 › 10x › 10y = 344

›8x › 5y = 344 › 720

›8x › 5y = ›376

8x + 5y = 376

8x = 376 › 5y

x = 47 ›

§”μÕ∫∑’Ë 1 y = 8, x = 47 › 5 = 42, z = 22

§”μÕ∫∑’Ë 2 y = 16, x = 47 › 10 = 37, z = 19

§”μÕ∫∑’Ë 3 y = 24, x = 47 › 15 = 32, z = 16

§”μÕ∫∑’Ë 4 y = 32, x = 47 › 20 = 27, z = 13

§”μÕ∫∑’Ë 5 y = 40, x = 47 › 25 = 22, z = 10

§”μÕ∫∑’Ë 6 y = 48, x = 47 › 30 = 17, z = 7

§”μÕ∫∑’Ë 7 y = 56, x = 47 › 35 = 12, z = 4

§”μÕ∫∑’Ë 8 y = 64, x = 47 › 40 = 7, z = 1

·μà y = 72, x = 47 › = 2, z = ›2 ‡ªìπ‰ª‰¡à‰¥â

μÕ∫ 8 «‘∏’

5y8

5(72)8

A 89-118 12/9/08, 10:33 AM100


( ) ( ) ( ) ( )

( ) ( )( )

( ) ( ) ( ) ( )

5. ∂â“ a ‡ªìπ À.√.¡. ¢Õß 234 °—∫ 324 ·≈– b, c ‡ªìπ®”π«π‡μÁ¡„π™à«ß ›10 ∂÷ß 10 ∑’Ë∑”„Àâ

a = 234b + 324c ·≈â« a + b + c ¡’§à“‡∑à“„¥

·π«§‘¥

‡π◊ËÕß®“° 234 = 2 Ó 32 Ó 13

·≈– 324 = 22 Ó 34

À.√.¡. ¢Õß 234 °—∫ 324 §◊Õ 2 Ó 32 = 18 = a

®“° a = 234b + 324c

18 = 2 Ó 32 (13b + 18c)

18 = 18(13(7) + 18(›5))

®–‰¥â b = 7, c = ›5

¥—ßπ—Èπ a + b + c = 18 + 7 › 5 = 20

μÕ∫ 20

6. ... ¡’§à“‡∑à“„¥

·π«§‘¥

=

=

=

=

=

μÕ∫

1 › 122

1 › 132

1 › 142 1 › 1

(2007)2

10042007

1 › 122

1 › 132

1 › 142 1 › 1

(2007)2

22 › 122

32 › 132

42 › 142

...

... 20072 › 1(2007)2

( )

(2)(3)2(4)2(5)2... (2005)2(2006)2(2007)(2008)22 32 42 52 ... (2007)2

...

12 Ó =2008

200710042007

(3)(1)(4)(2)(5)(3)(6)(4) ... (2006 + 1)(2005)(2008)(2006)22 32 42 52... (2007)2

(2 + 1)(2 › 1)22

(3 + 1)(3 › 1)32

(4 + 1)(4 › 1)42

(5 + 1)(5 › 1)52

(2007 + 1)(2007 › 1)20072Ó Ó Ó Ó Ó

A 89-118 12/9/08, 10:33 AM101


7. ∂â“ A = 0.5(22550 + 2› 2550), B = 0.5(22550 › 2› 2550) ·≈â« A2 › B2 ¡’§à“‡∑à“„¥

·π«§‘¥

A2 › B2 = (A › B)(A + B)

= (2›2550)(22550)

= 20

= 1

μÕ∫ 1

8. ∂â“ x ‡∑à“°—∫ x% ¢Õß y ·≈– y ‡∑à“°—∫ y% ¢Õß z ‚¥¬∑’Ë x, y, z ‡ªìπ®”π«π®√‘ß∫«°

·≈â« y + z ¡’§à“‡∑à“„¥

·π«§‘¥

®“° y = y% ¢Õß z ®–‰¥â y =

100y › yz = 0

y(100 › z) = 0

·μà y > 0 ¥—ßπ—Èπ 100 › z = 0

π—Ëπ§◊Õ z = 100

®“° x = x% ¢Õß y ®–‰¥â =

¥—ßπ—Èπ y = 100

®–‰¥â y + z = 100 + 100 = 200

μÕ∫ 200

yz100

x100

xy

9. §”μÕ∫¢Õß ¡°“√ x › x › x › x › 99 = 99 ¡’§à“‡∑à“„¥

·π«§‘¥

®“° x › x › x › x › 99 = 99

99 = x › x › x › x › 99

®–‰¥â x › 99 = 99

x › 99 = 992

x = 992 + 99

= 99(99 + 1)

= 99 Ó 100

μÕ∫ 9900

A 89-118 12/9/08, 10:33 AM102


·π«§‘¥

®“°√Ÿª„π Δ ABO; AO2 = 42 › (x › a)2 ........ ➊

„π Δ AOM; AO2 = 32› a2 ........➋

➊ = ➋ ®–‰¥â 42 › (x › a)2 = 32› a2

16 › x2 + 2ax ›a2 = 9 › a2

x2 › 2ax › 7 = 0 ....... ➌

„π Δ AOC; AO2 = 82› (a+x)2 .......➍

➊ = ➍ ®–‰¥â 16 › x2+ 2ax ›a2 = 64 ›a2› 2ax ›x2

4ax = 48

ax = 12 π”‰ª·∑π„π ➌

‰¥â x2 ›2(12) › 7 = 0

x2 = 31

x = 31

·μà BC = 2x

π—Ëπ§◊Õ ¥â“π BC ¬“« 2 31 Àπà«¬

μÕ∫ 2 31 Àπà«¬

10. ®“°√Ÿª Δ ABC ¡’ M ‡ªìπ®ÿ¥°÷Ëß°≈“ß¢Õß¥â“π BC ∂â“ AB = 4 Àπà«¬, AM = 3 Àπà«¬

·≈– AC = 8 Àπà«¬ ·≈â« BC ¬“«°’ËÀπà«¬

C

A

B O a x

2x

A

B MC

4 38

C

4 38

x - a

A 89-118 12/9/08, 10:33 AM103


11. ∂â“ N = 10296 › 10259 + 10222 › 10185 + 10148 › 10111 + 1074 › 1037 + 1 ·≈–

= 0.d1 d

2 d

3 ... d

m ‡ªìπ∑»π‘¬¡´È” ´÷Ëß¡’‡≈¢‚¥¥´È”„π·μà≈–™ÿ¥πâÕ¬∑’Ë ÿ¥ m ®”π«π

·≈â« m ¡’§à“‡∑à“„¥

·π«§‘¥

=

®–‰¥â N|(10m › 1)

(1037 + 1)N = 10333 + 1

(10333 › 1)(1037 + 1)N = 10666 › 1

®–‰¥â N|(10666 › 1)

¥—ßπ—Èπ m = 666

μÕ∫ 666

⋅ ⋅1N

1N

0.d1d

2d

3...d

m

10m › 1

12. ∂â“°”Àπ¥æÀÿπ“¡ (x + 5)3 + (3x › 2)4 › (2x + 1)5 ·≈â« —¡ª√– ‘∑∏‘Ï¢Õß x3 ¡’§à“‡∑à“„¥

·π«§‘¥

—¡ª√– ‘∑∏‘Ï¢Õß x3 ¢Õß (x + 5)3 §◊Õ 1

—¡ª√– ‘∑∏‘Ï¢Õß x3 ¢Õß (3x › 2)4 §◊Õ 4 Ó 33 Ó (›2) = ›216

—¡ª√– ‘∑∏‘Ï¢Õß x3 ¢Õß (2x + 1)5 §◊Õ 10 Ó 23 Ó 1 = 80

¥—ßπ—Èπ —¡ª√– ‘∑∏‘Ï¢Õß x3 ®“° (x + 5)3 + (3x › 2)4› (2x + 1)

§◊Õ 1 + (›216) › 80 = ›295

μÕ∫ ›295

A 89-118 12/9/08, 10:33 AM104


13. ∂â“ x + y = 11 ·≈– x2 + y2 = 16 ·≈â« x4 + y4 ¡’§à“‡∑à“„¥

·π«§‘¥

x4 + y4 = (x2 + y2)2 › 2x2y2

= (16)2 › 2x2y2 .................... ➊

®“° x + y = 11 .................... ➋

➋2 ®–‰¥â«à“ x2 + y2+ 2xy = 11

16 + 2xy = 11

2xy = ›5

xy = ·∑π„π ➊

x4 + y4 = (16)2 › 2( )2

= 256 › 2( )

= 256 › ( )

= 256 › 12.5 = 243.5

μÕ∫ 243.5

›52

›52

254

252

14. °”Àπ¥„Àâ x ‡ªìπ®”π«π‡μÁ¡∫«° ·≈– n(x) ‡ªìπº≈∫«°¢Õß‡≈¢‚¥¥∑’Ë‡¢’¬π·∑π x

‡™àπ n(517) = 5 + 1 + 7 = 13 ·≈– n(3229) = 3 + 2 + 2 + 9 = 16 ‡ªìπμâπ

∂â“ y = (10k + 2 + 3.10k)2 ‡¡◊ËÕ k ‡ªìπ®”π«π‡μÁ¡∫«°·≈â« n(y) ¡’§à“‡∑à“„¥

·π«§‘¥

®“° y = (10k + 2 + 3.10k)2

®–‰¥â y = (10k (10 2 + 3.))2

= (10k (100 + 3))2

= (10k(103))2

= 102k(10609)

¥—ßπ—Èπ n(y) = 1 + 0 + 6 + 0 + 9 = 16

μÕ∫ 16

A 89-118 12/9/08, 10:33 AM105


D

B

15. ∂â“π”°√«¬‚≈À–ª≈“¬μ—¥∑’Ë¡’√—»¡’∞“π 2 ‡´πμ‘‡¡μ√ ·≈– 5 ‡´πμ‘‡¡μ√ ¡“À≈Õ¡‡ªìπ≈Ÿ°∫“»°å

¬“«¥â“π≈– 2 ‡´πμ‘‡¡μ√ ·≈â«®–‰¥â≈Ÿ°∫“»°å∑—ÈßÀ¡¥ 143 ≈Ÿ° ®ßÀ“«à“°√«¬‚≈À–

ª≈“¬μ—¥π’È¡’§«“¡ Ÿß°’Ë‡´πμ‘‡¡μ√ (°”Àπ¥ π = )

·π«§‘¥

Δ ABC ~ Δ ADE

AB = 2x ¥—ßπ—Èπ BD = 3x

ª√‘¡“μ√°√«¬ª≈“¬μ—¥

π(5)2(5x) › π22(2x) = πx (125 › 8) ≈∫.´¡.

= ≈∫.´¡.

π”¡“À≈Õ¡‡ªìπ≈Ÿ°∫“»°å¬“«¥â“π≈– 2 ´¡. ‰¥â 143 ≈Ÿ°

ª√‘¡“μ√°√«¬ª≈“¬μ—¥ = 23 Ó 143 ≈∫.´¡.

x = ´¡.

3x = ´¡.

°√«¬‚≈À–ª≈“¬μ—¥¡’§«“¡ Ÿß 28 ‡´πμ‘‡¡μ√

μÕ∫ 28 ‡´πμ‘‡¡μ√

227

13

13

13

117πx3

117πx3

8 Ó 143 Ó 3 Ó 7117 Ó 22

28 Ó 33

1a

+ 1b

+ 1c

16. ∂â“ a + b + c = 4 ·≈– = 0 ·≈â« a2 + b2 + c2 ¡’§à“‡∑à“„¥

·π«§‘¥

®“° a + b + c = 4

®–‰¥â a2 + b2 + c2 + 2ab + 2ac + 2bc = 16 .....................➊

·≈– = 0

®–‰¥â bc + ac + ab = 0

2bc + 2ac + 2ab = 0 π”‰ª·∑π„π ➊

®–‰¥â a2 + b2+ c2 + 0 = 16 π—Ëπ§◊Õ a2 + b2 + c2 = 16

μÕ∫ 16

1a

+ 1b

+ 1c

C

E

2x

A

3x

5 ´¡.

A 89-118 12/9/08, 10:33 AM106


17. °”Àπ¥ P ‡ªìπæ“√“‚∫≈“ y = x2 › 3x ∂â“ P′ ‰¥â®“°°“√‡≈◊ËÕπ P ¢π“π‰ª¢â“ß∫π

4 Àπà«¬ ·≈–‰ª∑“ß´â“¬ 3 Àπà«¬ ·≈– P′ ºà“π®ÿ¥ › 2, ·≈â« a ¡’§à“‡∑à“„¥

·π«§‘¥

y = x2 › 3x ‡≈◊ËÕπ·°π‰ª‡ªìπ√–¬–∑“ß (›3, 4)

¥—ßπ—Èπ y › 4 = (x + 3)2 › 3(x + 3)

= x2 + 6x + 9 › 3x › 9

y = x2 + 3x + 4

ºà“π®ÿ¥ (›2, ) ¥—ßπ—Èπ y = = (›2)2 + 3(›2) + 4

2a › 4 = 5(4 + (›6) + 4)

2a = 5 Ó 2 + 4

a = = 7

μÕ∫ 7

2a › 45

)(

2a › 45

2a › 45

142

18. §«“¡‡√Á«¢ÕßπÈ”∑’Ë‰À≈ºà“π°äÕ°∑’ËÀπ÷Ëß·≈–°äÕ°∑’Ë Õß ‡ªìπÕ—μ√“ à«π 2 : 3 ∂â“‡ªî¥°äÕ°∑’Ë Õß

·≈–°äÕ°∑’Ë “¡æ√âÕ¡°—π ·≈â«πÈ”®–‰À≈‡¢â“∂—ß≈Ÿ°∫“»°å´÷Ëß°«â“ß¥â“π≈– 1.2 ‡¡μ√ ‚¥¬„™â

‡«≈“ 20 π“∑’ ®÷ß®–‡μÁ¡∂—ß ·≈–∂â“‡ªî¥°äÕ°∑’ËÀπ÷Ëß°—∫°äÕ°∑’Ë “¡æ√âÕ¡°—π ·≈â«πÈ”®–‰À≈

ÕÕ°®“°∂—ß¥—ß°≈à“«‚¥¬„π‡«≈“ 5 π“∑’ ®–‡À≈◊ÕπÈ” ¢Õß∂—ß ®ßÀ“«à“πÈ”‰À≈ºà“π°äÕ°∑’Ë “¡

¥â«¬§«“¡‡√Á«°’Ë≈‘μ√μàÕπ“∑’

·π«§‘¥

°äÕ°∑’Ë 1 πÈ”‰À≈ 2x ≈‘μ√/π“∑’

°äÕ°∑’Ë 2 πÈ”‰À≈ 3x ≈‘μ√/π“∑’

°äÕ°∑’Ë 3 πÈ”‰À≈ y ≈‘μ√/π“∑’

‡ªî¥°äÕ°∑’Ë 2 ·≈– 3 πÈ”‰À≈ 3x + y ≈‘μ√/π“∑’ „™â‡«≈“ 20 π“∑’

πÈ”‰À≈‡¢â“ 20(3x + y) = 1.2 Ó 1.2 Ó 1.2 Ó 1000 ≈‘μ√

3x + y = 86.4 ...................... ➊

‡ªî¥°äÕ°∑’Ë 1 ·≈– 3 æ√âÕ¡°—∫πÈ”‰À≈ 2x + y ≈‘μ√/π“∑’

„™â‡«≈“ 5 π“∑’ πÈ”‰À≈ 5(2x + y) = Ó 1.2 Ó 1.2 Ó 1.2 Ó 1000 ≈‘μ√

45

15

A 89-118 12/9/08, 10:34 AM107


2x + y = 69.12 ..................... ➋

x = 17.28 ≈‘μ√

y = 69.12 › 34.56 ≈‘μ√

= 34.56 ≈‘μ√

μ√«®§”μÕ∫ ‡ªî¥°äÕ° 2 ·≈– 3 πÈ”‰À≈ 3 Ó 17.28 + 34.56 = 86.4 ≈‘μ√/π“∑’

‡«≈“ 20 π“∑’ πÈ”‰À≈ 86.4 Ó 20 = 1728.0 = 12 Ó 12 Ó 12 ≈‘μ√

‡ªî¥°äÕ°∑’Ë 1 ·≈– 3 æ√âÕ¡°—∫πÈ”‰À≈ 2 Ó 17.28 + 34.56 = 69.12 ≈‘μ√/π“∑’

‡«≈“ 5 π“∑’ πÈ”‰À≈ 5 Ó 69.12 = 345.6 ≈‘μ√ ‡ªìπ = 0.2 ¢Õß∂—ß

‡À≈◊ÕπÈ” 1 › 0.2 = 0.8 = ¢Õß∂—ß

¥—ßπ—Èπ °äÕ°∑’Ë 3 πÈ”‰À≈¥â«¬§«“¡‡√Á« 34.56 ≈‘μ√/π“∑’

μÕ∫ 34.56 ≈‘μ√/π“∑’

345.612 Ó 12 Ó 12

45

19. °√–¥“πÀ¡“°√ÿ°°√–¥“πÀπ÷Ëß‰¥â√—∫°“√√–∫“¬ ’μ“¡™àÕß¬àÕ¬ Ê ¥â«¬ ’¢“«À√◊Õ ’πÈ”‡ß‘π

™àÕß≈–Àπ÷Ëß ’‡∑à“π—Èπ ‚¥¬æ∫«à“„π∑ÿ°√Ÿª ’Ë‡À≈’Ë¬¡º◊πºâ“∑’Ë¡’ 6 ™àÕß¬àÕ¬ (¢π“¥ 2 Ó 3 À√◊Õ

3 Ó 2) ®–¡’ 2 ™àÕß¬àÕ¬‡ªìπ ’πÈ”‡ß‘π à«π∑’Ë‡À≈◊Õ‡ªìπ ’¢“«‡ ¡Õ ∂â“°√–¥“ππ’È¡’¢π“¥

9 Ó 11 ·≈â«®–¡’™àÕß¬àÕ¬∑’Ë‡ªìπ ’πÈ”‡ß‘π∑—ÈßÀ¡¥°’Ë™àÕß

·π«§‘¥

μÕ∫ 33 ™àÕß

A 89-118 12/9/08, 10:34 AM108


·π«§‘¥

≈“° CD ®–‰¥â Δ ACD ·≈– Δ BCD ‡ªìπ√Ÿª “¡‡À≈’Ë¬¡∑’Ë·π∫„π«ß°≈¡

1. BAC = ADC ¡ÿ¡∑’Ë‡ âπ —¡º— ®√¥°—∫§Õ√å¥¡’¢π“¥‡∑à“°—∫¡ÿ¡„π

2. ABC = BDC à«π‚§âßμ√ß¢â“¡

3. ACB = 107 Ì °”Àπ¥„Àâ

4. BAC + ABC + ACB = 180 Ì º≈∫«°¡ÿ¡¿“¬„π√Ÿª “¡‡À≈’Ë¬¡

5. BAC + ABC = 73 Ì ®“°¢âÕ 3, 4

6. ·μà BAC + ABC = ADC + BDC ®“°¢âÕ 1, 2 ¡∫—μ‘°“√‡∑à“°—π

7. ¥—ßπ—Èπ ADC + BDC = 73 Ì ®“°¢âÕ 5, 6

μÕ∫ 73 Õß»“

20. ®“°√Ÿª«ß°≈¡ O ·≈–«ß°≈¡ P μ—¥°—π∑’Ë®ÿ¥ C ·≈– D ‚¥¬¡’‡ âπμ√ß AB ‡ªìπ‡ âπ —¡º— √à«¡

∂â“ ACB = 107 Ì ·≈â« ADB ¡’¢π“¥°’ËÕß»“^ ^

^ ^^

^

^ ^

^

^ ^ ^ ^

^ ^

}

O P

D

C B

A

^

^

^

107 Ì

O P

D

C B

A

A 89-118 12/9/08, 10:34 AM109


μÕπ∑’Ë 2

21. ∂â“ ABC ‡ªìπ√Ÿª “¡‡À≈’Ë¬¡Àπâ“®—Ë«∑’Ë¡’ AB = AC, BAC = 66 Ì, D ‡ªìπ®ÿ¥∫π AB,

E ‡ªìπ®ÿ¥∫π AC ∑”„Àâ DE = BD + CE ·≈– I ‡ªìπ®ÿ¥∫π BC ∑’Ë∑”„Àâ DI ·∫àß§√÷Ëß

BDE ·≈â« DIE ¡’¢π“¥°’ËÕß»“

·π«§‘¥

„Àâ M ‡ªìπ®ÿ¥∫π DE ∑”„Àâ BD = MD

≈“° IM

®–‰¥â Δ DBI ≅ Δ DMI (¥.¡.¥.)

¥—ßπ—Èπ DMI = 57 Ì ·≈– BID = MID

·≈– EM = EC

IME + ECI = 180 Ì

¥—ßπ—Èπ IMEC ·π∫„π«ß°≈¡

EIM = EIC

‡æ√“–©–π—Èπ DIE = MID + EIM = = 90 Ì

μÕ∫ 90 Õß»“

^

^ ^

180 Ì2

^ ^

^ ^

^ ^ ^

^ ^ ^

(

(

(( ((D

B I

E

C

M

A

66 Ì

57 Ì

57 Ì 57 Ì

A 89-118 12/9/08, 10:34 AM110


22. °”Àπ¥ ABC ‡ªìπ√Ÿª “¡‡À≈’Ë¬¡Àπâ“®—Ë«∑’Ë¡’ AB = AC, P ‡ªìπ®ÿ¥∫π¥â“π AC ·≈– Q

‡ªìπ®ÿ¥∫π¥â“π AB ∑”„Àâ AP = PQ = QB = BC ·≈â« BAC ¡’¢π“¥°’ËÕß»“

·π«§‘¥

≈“° QR æ∫ AC ∑’Ë R

„Àâ QP = QR ·≈– QAP = 4θ

®–‰¥â QPR = QRP = 8θ

BQR = 180 Ì › (4θ + 180 Ì › 16θ) = 12θ

‰¥â QBR = QRB = 90 Ì › 6θ

¥—ßπ—Èπ BRC = 180 Ì › (90 Ì › 6θ) › 8θ = 90 Ì › 2θ

·≈– BCR = = 90 Ì › 2θ

®–‰¥â BR = BC

¥—ßπ—Èπ Δ BQR ‡ªìπ√Ÿª “¡‡À≈’Ë¬¡¥â“π‡∑à“

12θ = 60 Ì, 4θ = 20 Ì

‡æ√“–©–π—Èπ BAC = 20 Ì

μÕ∫ 20 Õß»“

^

^

^ ^

^

^ ^

^

^ 180 Ì › 4θ2

^

23. ∂â“ a, b, c ·≈– d ‡ªìπ®”π«π®√‘ß ´÷Ëß Õ¥§≈âÕß°—∫ ¡°“√

a = 82 › 58 › a

b = 82 + 58 › b

c = 82 › 58 + c

·≈– d = 82 + 58 + d ·≈â« abcd ¡’§à“‡∑à“„¥

·π«§‘¥

a2 = 82 › 58 › a

(a2 › 82)2 = 58 › a

a4 › 164a2 + a + 6666 = 0

„π∑”πÕß‡¥’¬«°—π®–‰¥â b4 › 164b2 + b + 6666 = 0

c4 › 164c2 › c + 6666 = 0

d4 › 164d2 › d + 6666 = 0

P

Q

R

A

B C

4θ

A 89-118 12/9/08, 10:34 AM111


24. ∂â“ ABC ‡ªìπ√Ÿª “¡‡À≈’Ë¬¡Àπâ“®—Ë«∑’Ë¡’ BAC = 100 Ì, M ‡ªìπ®ÿ¥¿“¬„π∑”„Àâ

MBA = 10 Ì ·≈– MCA = 5 Ì ·≈â« BMA ¡’¢π“¥°’ËÕß»“

·π«§‘¥

≈“° AD ⊥ BC ∑’Ë D μ—¥ BM ∑’Ë E ≈“° CE

®–‰¥â CAD = 50 Ì

AEM = CEM = 60 Ì (¡ÿ¡¿“¬πÕ°√Ÿª “¡‡À≈’Ë¬¡¡’¢π“¥‡∑à“°—∫º≈∫«°

¢Õß¡ÿ¡¿“¬„π∑’Ë‰¡à„™à¡ÿ¡ª√–™‘¥)

ACM = ECM = 5 Ì

¥—ßπ—Èπ M ‡ªìπ®ÿ¥»Ÿπ¬å°≈“ß¢Õß«ß°≈¡·π∫„π√Ÿª “¡‡À≈’Ë¬¡ AEC

©–π—Èπ EAM = CAM = 25 ‡æ√“– AM ·∫àß§√÷Ëß EAC

·≈– AME = 180 Ì › 60 Ì › 25 Ì = 95 Ì

μÕ∫ 95 Õß»“

®–‰¥â a, b, ›c, ›d ‡ªìπ§”μÕ∫¢Õß ¡°“√

x4 › 164x2 › x + 6666 = 0

¥—ßπ—Èπ abcd = 6666

μÕ∫ 6666

^

^ ^ ^

^ ^

^ ^

^

^ ^ ^

^

B CD

E

A

M

30 Ì

10 Ì

30 Ì

5 Ì5 Ì

A 89-118 12/9/08, 10:34 AM112


25. ∂â“ √Ÿª ’Ë‡À≈’Ë¬¡ ABCD ¡’ BAC = 81 Ì, CAD = 27 Ì, ABD = 36 Ì ·≈– CBD = 30 Ì

·≈â« ADC ¡’¢π“¥°’ËÕß»“

·π«§‘¥

BDA = 36 Ì = ABD

∴ AB = AD

≈“° AE æ∫ BC ∑’Ë®ÿ¥ E „Àâ AEB = 66 Ì

≈“° ED ®–‰¥â AB = AE ·≈– BAE = 48 Ì

EAC = ECA = 33 Ì AE = CE

EAD = 60 Ì ·≈– AE = AD

®–‰¥â Δ ADE ‡ªìπ√Ÿª “¡‡À≈’Ë¬¡¥â“π‡∑à“

‰¥â AD = AE = ED

∴ EDC = ECD = = 63 Ì

ADC = 123 Ì

μÕ∫ 123 Õß»“

A

D

^ ^

^

^ ^ ^ ^

^

180 Ì › 54 Ì2^

^ ^

^

^

^ ^

81 Ì

66 Ì54 Ì

66 Ì

36 Ì 30 Ì

36 Ì

24 Ì C

B

E

A 89-118 12/9/08, 10:34 AM113


26. √Ÿª “¡‡À≈’Ë¬¡ ABC ·π∫„π«ß°≈¡ O ‚¥¬¡’ P ‡ªìπ®ÿ¥°÷Ëß°≈“ß OA ·≈– Q ‡ªìπ

®ÿ¥°÷Ëß°≈“ß BC ∂â“ ABC ¡’¢π“¥‡ªìπ 4 ‡∑à“¢Õß¢π“¥¢Õß OPQ ·≈– ACB ¡’¢π“¥‡ªìπ

6 ‡∑à“¢Õß¢π“¥¢Õß OPQ ·≈â« OPQ ¡’¢π“¥°’ËÕß»“

·π«§‘¥

„Àâ OPQ = x ®–‰¥â ABC = 4x ·≈– ACB = 6x

≈“° OC ®–‰¥â AOC = 8x

OAC = OCA = 90 Ì › 4x

¥—ßπ—Èπ OCQ = 10x › 90 Ì

QOC = 180 Ì › 10x

¥—ßπ—Èπ POQ = 180 Ì › 2x

OQP = x

∴ OP = OQ =

∴ OCQ = 30 Ì = 10x › 90 Ì

x = 12 Ì

μÕ∫ 12 Õß»“

^ ^ ^

^ ^

OC2

^ ^ ^

^

^ ^

^

^

^

^

A

B CQ

^

P

O

A 89-118 12/9/08, 10:34 AM114


( ) ( ) ( )

27. °”Àπ¥„Àâ AB ‡ªìπ‡ âπºà“π»Ÿπ¬å°≈“ß«ß°≈¡∑’Ë¡’√—»¡’ 99 Àπà«¬ ·≈– P ‡ªìπ®ÿ¥¿“¬„π

®ßÀ“§«“¡πà“®–‡ªìπ∑’Ë APB ≤ 135 Ì (μÕ∫„π√Ÿª¢Õß π)

·π«§‘¥

„Àâ C ‡ªìπ®ÿ¥»Ÿπ¬å°≈“ß√—»¡’ BC

‡¢’¬π à«π‚§âß ‡≈◊Õ°®ÿ¥ P ∫π à«π‚§âß

∑’Ë∑”„Àâ

APB = 135 Ì

BC = 99 2

æ◊Èπ∑’Ë‡´°‡¡πμå AB = π(99 2)2 › Ó (99 2)2

æ◊Èπ∑’Ë·√‡ß“ = π(99)2 › 2 æ◊Èπ∑’Ë‡´°‡¡πμå AB

= 2 Ó 992

n(S) = π Ó 992

n(E) = 2 Ó 992

∴ p(E) =

μÕ∫

^

14

12

2π

2π

28. °”Àπ¥„Àâ (x1, y

1), (x

2, y

2) ·≈– (x

3, y

3) ‡ªìπ§”μÕ∫¢Õß√–∫∫ ¡°“√

x3 › 3xy2 = 1999

y3 › 3x2y = 1998

·≈â« 1 ¡’§à“‡∑à“„¥1 ›

x1

y1

1 ›x2

y2

1 ›x3

y3

·π«§‘¥

x3 › 3xy2 › (y3 › 3x2y) = 0

x3 + .3x2y › 3xy2 › y3 = 0

19991998

19991998

19991998

^

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D

A

C

B

(P

A 89-118 12/9/08, 10:35 AM115


π” y3 À“√∑—Èß ¡°“√

( )3 + ( )2 › 3( ) › = 0

®–¡’ , , ‡ªìπ§”μÕ∫¢Õß ¡°“√

„Àâ P(a) = a3 + a2 › 3a ›

¡’ a = , ·≈–

P(1 › a) = (1 › a)3 + (1 › a)2 › 3(1 › a) ›

¡’§”μÕ∫¢Õß P(1 › a) = 0 §◊Õ 1 › , 1 › , 1 ›

®–‰¥â (1 › )(1 › )(1 › ) = 1 + › 3 ›

=

¥—ßπ—Èπ = 999

μÕ∫ 999

xy

59971998

xy

xy

19991998

59971998

19991998

x1

y1

x2

y2

x3

y3

59971998

19991998

x1

y1

x2

y2

x3

y3

x1

y1

x2

y2

x3

y3

59971998

19991998

1999

1

(1 › )(1 › )(1 › )x1

y1

x2

y2

x3

y3

x1

y1

x2

y2

x3

y3

A 89-118 12/9/08, 10:35 AM116


29. ∂â“°”Àπ¥√–∫∫ ¡°“√

10x2 + 5y2 › 2xy › 38x › 6y + 41 = 0

3x2 › 2y2 + 5xy › 17x › 6y + 20 = 0

·≈â«§à“¢Õß x3 + y3 ‡ªìπ‡∑à“„¥

·π«§‘¥

®“°√–∫∫ ¡°“√®–‰¥â

7x2 + 7y2 › 7xy › 21x + 21 = 0

x2 › xy + y2 › 3x + 3 = 0

( › + y)2 + ( › + 3) = 0

( › y)2 + 3( › 1)2 = 0

®”π«π®√‘ß a , a2 ≥ 0

∴ › y = 0 ·≈– › 1 = 0

x = 2, y = 1

¥—ßπ—Èπ x3 + y3 = 9

μÕ∫ 9

x2

x2

x2

x2

30. ∂â“°”Àπ¥√–∫∫ ¡°“√

a1 + a

2 + a

3 + ... + a

n = 96

a12 + a

22 + a

32 + ... + a

n2 = 144

a13 + a

23 + a

33 + ... + a

n3 = 216

‡¡◊ËÕ ai ‡ªìπ®”π«π®√‘ß∫«° ”À√—∫∑ÿ° i = 1, 2, 3, ..., n ·≈â« a

14 + a

24 + a

34 + ... + a

n4

¡’§à“‡∑à“„¥

·π«§‘¥

(a1 + a

2 + a

3 + ... + a

n)(a

13 + a

23 + a

33 + ... + a

n3) = 96 Ó 216 = (144)2

∴(a1 + a

2 + a

3 + ... + a

n)(a

13 + a

23 + a

33 + ... + a

n3) = (a

12 + a

22 +

a32 + ... + a

n2)2

®–‰¥â a1 = a

2 = a

3 = ... = a

n

x2

42xy2

6x2

3x2

4

∴

A 89-118 12/9/08, 10:35 AM117


¥—ßπ—Èπ na1

= 96

na12 = 144

na13 = 216

=

na14 = 324

∴ a14 + a

24 + a

34 + ... + a

n4 = 324

μÕ∫ 324

(na12 )(na

13)

na1

144 Ó 21696

A 89-118 12/9/08, 10:35 AM118


μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 (©∫—∫∑’Ë 2)‡æ◊ËÕ°“√§—¥‡≈◊Õ°π—°‡√’¬π√–¥—∫ª√–‡∑» ªï æ.». 2550

§”™’È·®ß

1. ·∫∫∑¥ Õ∫©∫—∫π’È‡ªìπ·∫∫‡μ‘¡§”μÕ∫


μÕπ∑’Ë 1 ®”π«π 20 ¢âÕ Ê ≈– 3 §–·ππ √«¡ 60 §–·ππ

μÕπ∑’Ë 2 ®”π«π 10 ¢âÕ Ê ≈– 4 §–·ππ √«¡ 40 §–·ππ

3. ‡«≈“∑’Ë„™â„π°“√ Õ∫ „™â‡«≈“ 1 ™—Ë«‚¡ß 30 π“∑’

A 119-152 12/9/08, 10:37 AM119


μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 (©∫—∫∑’Ë 2)

‡æ◊ËÕ°“√§—¥‡≈◊Õ°π—°‡√’¬π√–¥—∫ª√–‡∑» ªï æ.». 2550


μÕπ∑’Ë 1 ®”π«π 20 ¢âÕ ¢âÕ≈– 3 §–·ππ

μÕπ∑’Ë 2 ®”π«π 10 ¢âÕ ¢âÕ≈– 4 §–·ππ

μÕπ∑’Ë 1

4. ®ß‡¢’¬π§ŸàÕ—π¥—∫ (x, y) ∑’Ë ¡“™‘°·μà≈–μ—«Õ¬Ÿà„π√Ÿª‡≈¢¬°°”≈—ß ‚¥¬∑’Ë x ‡ªìπ®”π«π

∑’Ë¡“° ÿ¥¢Õß 235, 515 ·≈– 614 ·≈– y ‡ªìπ®”π«π∑’ËπâÕ¬∑’Ë ÿ¥¢Õß “¡®”π«ππ’È

7. ‡¢’¬π®”π«π 365 ®”π«π√Õ∫«ß°≈¡ ®”π«π∑’Ë 88 §◊Õ 7 ®”π«π∑’Ë 111 §◊Õ 2 ®”π«π∑’Ë 224

§◊Õ 1 ®”π«π∑’Ë 365 §◊Õ 5 ∂â“º≈∫«°¢Õß∑ÿ° 60 ®”π«π∑’Ë‡√’¬ßμ‘¥μàÕ°—π¡’§à“‡ªìπ 216 ‡ ¡Õ

·≈â«®”π«π∑’Ë 122 ¡’§à“‡∑à“„¥

3. ∂â“ m ∗ n = ·≈â« ((...(2550 ∗ 2549) ∗ 2548) ∗ ... ∗ 1) ∗ 0 ¡’§à“‡∑à“„¥

5. ‡≈¢‚¥¥À≈—ß®ÿ¥∑»π‘¬¡μ”·Àπàß∑’Ë 8884 ¢Õß ‡ªìπ‡∑à“„¥

2. °”Àπ¥ x ‡ªìπ®”π«π‡μÁ¡´÷Ëß Õ¥§≈âÕß°—∫ ¡°“√ x + 4 x › 4 + x › 4 x › 4 = 4

º≈∫«°¢Õß§à“ x ∑—ÈßÀ¡¥‡ªìπ‡∑à“„¥

6. ∂â“æÀÿπ“¡ P(x) ∑’Ë¡’¥’°√’ 4 ·≈–¡’ ¡∫—μ‘¥—ßμàÕ‰ªπ’È

P(0) = 0, P(1) = P(›1) = P(2) = P(›2) = 1 ·≈â« P(5) ¡’§à“‡∑à“„¥

m + nmn + 4

8897

1. ... ¡’§à“‡∑à“„¥( )20072

20072 › 1( )22

22 › 1( )32

32 › 1( )42

42 › 1

A 119-152 12/9/08, 10:37 AM120


10. ®“°√Ÿª BO ·∫àß§√÷Ëß¡ÿ¡ CBA, CO ·∫àß§√÷Ëß¡ÿ¡ ACB ·≈– MN ºà“π O ¢π“π°—∫ BC

∂â“ AB = 12 Àπà«¬ BC = 24 Àπà«¬ ·≈– AC = 18 Àπà«¬ ·≈â«§«“¡¬“«‡ âπ√Õ∫√Ÿª

√Ÿª “¡‡À≈’Ë¬¡ AMN ¡’§à“°’ËÀπà«¬

12. °”Àπ¥ x1, x

2, x

3, ..., x

100 ‡ªìπ®”π«π®√‘ß∫«° ´÷Ëß∑”„Àâ

x1 + = 1, x

2 + = 4, x

3 + = 1, ... , x

99 + = 1 ·≈– x

100 + = 4

·≈â« x1 + x

2 + x

3 + ... + x

100 ¡’§à“‡∑à“„¥

8. ∂â“ ABC ‡ªìπ√Ÿª “¡‡À≈’Ë¬¡Àπâ“®—Ë«∑’Ë¡’ AB = AC, BAC = 80 Ì, M ‡ªìπ®ÿ¥¿“¬„π

∑’Ë∑”„Àâ MAC = 20 Ì ·≈– MCA = 30 Ì ·≈â« MBA ¡’¢π“¥°’ËÕß»“

13. ∂â“ x1 + 4x

2 + 9x

3 + 16x

4 + 25x

5 + 36x

6 + 49x

7= 4

4x1 + 9x

2 + 16x

3 + 25x

4 + 36x

5 + 49x

6 + 64x

7= 44

9x1 + 16x

2 + 25x

3 + 36x

4 + 49x

5 + 64x

6 + 81x

7= 444

·≈â« 16x1 + 25x

2 + 36x

3 + 49x

4 + 64x

5 + 81x

6 + 100x

7 ¡’§à“‡∑à“„¥

9. °”Àπ¥ ABC ‡ªìπ√Ÿª “¡‡À≈’Ë¬¡¥â“π‡∑à“∑’Ë¡’ P ‡ªìπ®ÿ¥¿“¬„π∑”„Àâ PA = 6 Àπà«¬

PB = 8 Àπà«¬ ·≈– PC = 10 Àπà«¬ √Ÿª “¡‡À≈’Ë¬¡ ABC ¡’æ◊Èπ∑’Ë°’Ëμ“√“ßÀπà«¬

11. °”Àπ¥ P(x) ‡ªìπæÀÿπ“¡ ‚¥¬∑’Ë P(x) = x4 + ax3 + bx2 + cx + d

∂â“ P(1) = 6, P(›2) = 3, P(3) = ›2 ·≈– P(›4) = ›9 ·≈â« |P(4) + 4)| ¡’§à“‡∑à“„¥

1x2

1x3

1x4

1x100

1x1

^

^ ^^

A

B C

M NO

A 119-152 12/9/08, 10:37 AM121


14. ®“°√Ÿª ABC, DEF ·≈– PQR ‡ªìπ√Ÿª “¡‡À≈’Ë¬¡¥â“π‡∑à“ ‚¥¬∑’Ë AD = BD ·≈â«æ◊Èπ∑’Ë

√Ÿª “¡‡À≈’Ë¬¡ ABC ‡ªìπ°’Ë‡∑à“¢Õßæ◊Èπ∑’Ë√Ÿª “¡‡À≈’Ë¬¡ PQR

15. ∂â“√Ÿª ’Ë‡À≈’Ë¬¡ ABCD ¡’®ÿ¥¬Õ¥Õ¬Ÿà∫π‡ âπ√Õ∫«ß¢Õß«ß°≈¡∑’Ë¡’ O ‡ªìπ®ÿ¥»Ÿπ¬å°≈“ß

‚¥¬∑’Ë AC ‡ªìπ‡ âπºà“π»Ÿπ¬å°≈“ß¢Õß«ß°≈¡ ·≈– CAD = 18 Ì, BDC = 50 Ì ·≈â« OBD

¡’¢π“¥°’ËÕß»“

16. ∂â“ 7x + y = 21 ·≈– 32x + y = 1 ·≈â« 3(7x + 1 + 7y › 2) ¡’§à“‡∑à“„¥

17. °”Àπ¥ ¡°“√ 15x + 14y = 7 ∂â“ x ∑’Ë‡ªìπ®”π«π‡μÁ¡∑’Ë¡“°∑’Ë ÿ¥ ÷Ëßª√–°Õ∫¥â«¬ ¡“™‘°

4 À≈—° ∑’Ë∑”„Àâ (x, y) ‡ªìπ§”μÕ∫¢Õß ¡°“√π’È ‡¡◊ËÕ y ‡ªìπ®”π«π‡μÁ¡ ·≈â« x ¡’§à“‡∑à“„¥

18. °”Àπ¥„Àâ n ·≈– d ‡ªìπ®”π«π‡μÁ¡∫«° ∂â“ d À“√ 13n + 6 ·≈– 12n + 5 ≈ßμ—«

·≈â«º≈∫«°¢Õß§à“ d ∑’Ë‡ªìπ‰ª‰¥â∑—ÈßÀ¡¥ ¡’§à“‡∑à“„¥

19. √ŸªÀ≈“¬‡À≈’Ë¬¡¥â“π‡∑à“¡ÿ¡‡∑à“√ŸªÀπ÷Ëß ¡’‡ âπ∑·¬ß¡ÿ¡∑—ÈßÀ¡¥ 20 ‡ âπ ·≈–·π∫„π

«ß°≈¡√—»¡’ 2 Àπà«¬ √ŸªÀ≈“¬‡À≈’Ë¬¡π’È¡’æ◊Èπ∑’Ë‡∑à“„¥

20. √Ÿª ’Ë‡À≈’Ë¬¡ ABCD ¡’æ◊Èπ∑’Ë 2,550 μ“√“ßÀπà«¬ ‚¥¬∑’Ë P, Q, R ·≈– S ‡ªìπ®ÿ¥°÷Ëß°≈“ß¥â“π

∂â“√Ÿª ’Ë‡À≈’Ë¬¡ BNMP ·≈–√Ÿª ’Ë‡À≈’Ë¬¡ DTOR ¡’æ◊Èπ∑’Ë 123 ·≈– 456 μ“√“ßÀπà«¬

μ“¡≈”¥—∫ ·≈â«æ◊Èπ∑’Ë¢Õß√Ÿª ’Ë‡À≈’Ë¬¡ MNOT ‡∑à“°—∫°’Ëμ“√“ßÀπà«¬

C

A

B

P FD

QR

^^ ^

E

A 119-152 12/9/08, 10:37 AM122


μÕπ∑’Ë 2

21. ·∫∫®”≈Õß CH4 ·π∫„π∑√ß‡À≈’Ë¬¡ ’ËÀπâ“ ‚¥¬∑’Ë·μà≈–Àπâ“‡ªìπ√Ÿª “¡‡À≈’Ë¬¡¥â“π‡∑à“

∑’Ë¬“«¥â“π≈– 6 6 Àπà«¬ ·μà≈–Àπâ“‡∑à“°—π∑ÿ°ª√–°“√ ·≈–¡’ H Õ¬Ÿà∑’Ë®ÿ¥¡ÿ¡ ·≈â«·¢π

C › H ¬“«‡∑à“„¥

22. ∂â“ (x2 › x + 1)3(x3 + 2x2 + 2x + 1)5 = a21

x21 + a20

x20 + a19

x19+ ... + a2x2 + a

1x + a

0

‡¡◊ËÕ ai ‡ªìπ§à“§ß∑’Ë ·≈â« a

1 +

a2 +

a3 +

... +

a10

¡’§à“‡∑à“„¥

23. ∂â“ = 2 ·≈â« x ¡’§à“‡∑à“„¥

24. ∂â“ x, y ·≈– z ‡ªìπ®”π«π®√‘ß ∑’Ë Õ¥§≈âÕß°—∫√–∫∫ ¡°“√ x + 2y + 3z = 13

x2 + 4y2 + 9z2 + 3x › 2y + 15z = 82 ·≈â« xyz + x + y + z ¡’§à“‡∑à“„¥

25. ∂â“ a2 › 2a = ›1, b2 › 3b = 1 ·≈– c2 › 4c = ›1 ·≈â« 3a3 › b3 + c3 + + +

+ 200 ¡’§à“‡∑à“„¥

26. °”Àπ¥„Àâ a, b, c ‡ªìπ®”π«π‡μÁ¡∫«° ∂â“ a + b + c = 20 = ab + bc › ca › b2 ·≈â«

º≈∫«°¢Õß§à“ abc ∑’Ë‡ªìπ‰ª‰¥â∑—ÈßÀ¡¥ ¡’§à“‡∑à“„¥

27. √Ÿª “¡‡À≈’Ë¬¡ ABC ¡’ D, E ·≈– F ‡ªìπ®ÿ¥∫π¥â“π BC ∑”„Àâ AD ⊥ BC, AE ·∫àß§√÷Ëß

BAC ·≈– BF = CF ∂â“ BAD = DAE = EAF = FAC ·≈â« BAC + 2ABC + 4ACB

¡’§à“‡∑à“„¥

2 + x2 + 2 + x

+ 2 › x2 › 2 › x

3a3

1b3

1c3

^^ ^^ ^^ ^^

28. ∂â“°”Àπ¥√–∫∫ ¡°“√

abc + ab + bc + ca + a + b + c = 71

bcd + bc + cd + db + b + c + d = 191

cda + cd + da + ac + c + d + a = 95

dab + da + ab + bd + d + a + b = 143

·≈â« abcd + a + b + c + d ¡’§à“‡∑à“„¥

A 119-152 12/9/08, 10:37 AM123


30. ∂â“ ABCD ‡ªìπ√Ÿª ’Ë‡À≈’Ë¬¡ ¡’ BAC = CAD = 66 Ì, BCA = 15 Ì ·≈– ACD = 9 Ì

·≈â« 2ABD + ADB ¡’§à“‡∑à“„¥

29. ∂â“°”Àπ¥√–∫∫ ¡°“√

x1 + x

2 + x

3 = 4, x

2 + x

3 + x

4 = 6, x

3 + x

4 + x

5 = 8,

x4 + x

5 + x

6 = 12, x

5 + x

6 + x

7 = 15, x

6 + x

7 + x

8 = 19,

x7 + x

8 + x

9 = 23, x

8 + x

9 + x

10 = 27, x

9 + x

10 + x

1 = 30,

x10

+ x1 + x

2 = 36

·≈â« 3x1 + 4x

10 ¡’§à“‡∑à“„¥

^ ^ ^

^ ^

^

A 119-152 12/9/08, 10:37 AM124


μ—«Õ¬à“ß·π«§‘¥·∫∫∑¥ Õ∫§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 (©∫—∫∑’Ë 2)‡æ◊ËÕ°“√§—¥‡≈◊Õ°π—°‡√’¬π√–¥—∫ª√–‡∑» ªï æ.». 2550

§”™’È·®ß

1. ·∫∫∑¥ Õ∫©∫—∫π’È‡ªìπ·∫∫‡μ‘¡§”μÕ∫


μÕπ∑’Ë 1 ®”π«π 20 ¢âÕ Ê ≈– 3 §–·ππ √«¡ 60 §–·ππ

μÕπ∑’Ë 2 ®”π«π 10 ¢âÕ Ê ≈– 4 §–·ππ √«¡ 40 §–·ππ

3. ‡«≈“∑’Ë„™â„π°“√ Õ∫ „™â‡«≈“ 1 ™—Ë«‚¡ß 30 π“∑’

A 119-152 12/9/08, 10:37 AM125


2,007 ⋅ 2,0072,006 2,008

1. ... ¡’§à“‡∑à“„¥

·π«§‘¥

‡π◊ËÕß®“° = ®–‰¥â

... = ...

=

=

μÕ∫

§”™’È·®ß ·∫∫∑¥ Õ∫©∫—∫π’È‡ªìπ·∫∫‡μ‘¡§”μÕ∫ §–·ππ‡μÁ¡ 100 §–·ππ ¡’ 2 μÕπ ¥—ßπ’ÈμÕπ∑’Ë 1 ®”π«π 20 ¢âÕ ¢âÕ≈– 3 §–·ππ

μÕπ∑’Ë 2 ®”π«π 10 ¢âÕ ¢âÕ≈– 4 §–·ππ

μÕπ∑’Ë 1

( )32

32 › 142

42 › 122

22 › 1

22

22 › 132

32 › 142

42 › 12,0072

2,0072 › 1

22 + 1

22 › 1

μ—«Õ¬à“ß·π«§‘¥·∫∫∑¥ Õ∫§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 (©∫—∫∑’Ë 2)

‡æ◊ËÕ°“√§—¥‡≈◊Õ°π—°‡√’¬π√–¥—∫ª√–‡∑» ªï æ.». 2550

( )( )

22

22 › 1( ) ( )

( )( )( ) ( )2 ⋅ 21 3

( )3 ⋅ 32 4

( )4 ⋅ 43 5

( )21 ( )

2,0072,008

2,0071,0042,007

1,004

( )2,0072

2,0072 › 1

( ) ( )

A 119-152 12/9/08, 10:38 AM126


2. °”Àπ¥ x ‡ªìπ®”π«π‡μÁ¡´÷Ëß Õ¥§≈âÕß°—∫ ¡°“√ x + 4 x › 4 + x › 4 x › 4 = 4

º≈∫«°¢Õß§à“ x ∑—ÈßÀ¡¥‡ªìπ‡∑à“„¥

·π«§‘¥

(x › 4) + 4 x › 4 + 4 + (x › 4) › 4 x › 4 + 4 = 4

⏐ x › 4 + 2⏐+⏐ x › 4 › 2⏐= 4

®–‰¥â x ≥ 4 ·≈– x › 4 + 2 + 2 › x › 4 = 4 ®√‘ß

¥—ßπ—Èπ 2 › x › 4 ≥ 0

®–‰¥â x ≤ 8

©–π—Èπ x = 4, 5, 6, 7, 8

º≈∫«°§◊Õ 4 + 5 + 6 + 7 + 8 = 30

μÕ∫ 30

3. ∂â“ m ∗ n = ·≈â« (...(2550 ∗ 2549) ∗ 2548) ∗ ... ∗ 1) ∗ 0 ¡’§à“‡∑à“„¥

·π«§‘¥

m ∗ 2 = =

(...(2550 ∗ 2549) ∗ 2548) ∗ ... ∗ 1) ∗ 0 = ∗ 0

= ∗ 0

=

=

μÕ∫

m + nmn + 4

m + 22m + 4

12

+ 112

+ 412

)(13134

112

112

A 119-152 12/9/08, 10:38 AM127


4. ®ß‡¢’¬π§ŸàÕ—π¥—∫ (x, y) ∑’Ë ¡“™‘°·μà≈–μ—«Õ¬Ÿà„π√Ÿª‡≈¢¬°°”≈—ß ‚¥¬∑’Ë x ‡ªìπ®”π«π

∑’Ë¡“° ÿ¥¢Õß 235, 515 ·≈– 614 ·≈– y ‡ªìπ®”π«π∑’ËπâÕ¬∑’Ë ÿ¥¢Õß “¡®”π«ππ’È

·π«§‘¥

‡æ√“–«à“ 235 = (25)7 = 327

614 = (62)7 = 367

®–‰¥â 614 > 235

‡æ√“–«à“ 515 = (53)5 = 1255

235 = (27)5 = 1285

®–‰¥â 235 > 515

π—Ëπ§◊Õ 614 ¡’§à“¡“°∑’Ë ÿ¥ ·≈ 515 ¡’§à“πâÕ¬∑’Ë ÿ¥

μÕ∫ (614, 515)

5. ‡≈¢‚¥¥À≈—ß®ÿ¥∑»π‘¬¡μ”·Àπàß∑’Ë 8,884 ¢Õß ‡ªìπ‡∑à“„¥

·π«§‘¥

®–‰¥â ‡ªìπ∑»π‘¬¡´È” ´÷Ëß¡’μ—«‡≈¢‚¥¥´È”‡ªìπ™ÿ¥ Ê ≈– 96 μ—«

‡π◊ËÕß®“° 8,884 = 96(92) + 52

¥—ßπ—Èπ μ—«‡≈¢‚¥¥„πμ”·Àπàß∑’Ë 8,884 ¢Õß ®–μ√ß°—∫μ—«‡≈¢‚¥¥„πμ”·Àπàß

∑’Ë 52 ¢Õß

„Àâ = 0.d1 d

2 d

3 ... d

96

®–‰¥â d1 + d

49= 9

d2 + d

50= 9

d3 + d

51= 9

d4 + d

52= 9

d4

= 2

¥—ßπ—Èπ d52

= 7

μÕ∫ 7

8897

8897

8897

8897

8897

⋅ ⋅

A 119-152 12/9/08, 10:38 AM128


6. ∂â“æÀÿπ“¡ P(x) ∑’Ë¡’¥’°√’ 4 ·≈–¡’ ¡∫—μ‘¥—ßμàÕ‰ªπ’È

P(0) = 0, P(1) = P(›1) = P(2) = P(›2) = 1 ·≈â« P(5) ¡’§à“‡∑à“„¥

·π«§‘¥

‡π◊ËÕß®“°æÀÿπ“¡ P(x) ¡’¥’°√’ 4

¥—ßπ—Èπ P(X) Õ¬Ÿà„π√Ÿª P(X) = ax4 + bx3 + cx2 + dx + e

„Àâ Q(x) ‡ªìπæÀÿπ“¡∑’Ë°”Àπ¥‚¥¬ Q(1) = Q(›1) = Q(2) = Q(›2) = 0

¥—ßπ—Èπ Q(x) = c(x › 1)(x + 1)(x › 2)(x + 2) ‡¡◊ËÕ c ‡ªìπ§à“§ßμ—«

®“°‡ß◊ËÕπ‰¢ ®–‰¥â«à“ P(x) = Q(x) + 1

·≈– P(0) = 0 = c(›1)(1)(›2)(2) + 1

®–‰¥â«à“ 4c + 1 = 0 ¥—ßπ—Èπ c = ›

©–π—Èπ P(x) = › (x › 1)(x + 1)(x › 2)(x + 2) + 1

¥—ßπ—Èπ P(5) = › (5 › 1)(5 + 1)(5 › 2)(5 + 2) + 1

= › (4)(6)(3)(7) + 1

= ›126 + 1

= ›125

μÕ∫ ›125

14

14

14

14

A 119-152 12/9/08, 10:38 AM129


8. ∂â“ ABC ‡ªìπ√Ÿª “¡‡À≈’Ë¬¡Àπâ“®—Ë«∑’Ë¡’ AB = AC, BAC = 80 Ì, M ‡ªìπ®ÿ¥¿“¬„π

∑’Ë∑”„Àâ MAC = 20 Ì ·≈– MCA = 30 Ì ·≈â« MBA ¡’¢π“¥°’ËÕß»“

·π«§‘¥

√â“ß ≈“° AO ⊥ BC ∑’Ë X ∑”„Àâ BCO = CBO = 10 Ì

«‘∏’À“ 1. BAO = CAO = 40 Ì

2. ABC = ACB = = 50 Ì

3. ACM = MCO = 30 Ì ·≈– MAC = MAO = 20 Ì

4. M ‡ªìπ®ÿ¥»Ÿπ¬å°≈“ß«ß°≈¡·π∫„π Δ AOC

5. MO ·∫àß§√÷Ëß AOC ·≈– AOC = 180 Ì › 40 Ì › 60 Ì = 80 Ì

6. AOM = 40 Ì ·≈– BAM = BAO + MAO = 40 Ì + 20 Ì = 60 Ì

7. ΔOXB ≅ ΔOXC ∑”„Àâ AOB = AOC = 80 Ì

8. MOB = MOA + AOB = 40 Ì + 80 Ì

9. � ABOM ·π∫„π«ß°≈¡∑”„Àâ MBA = AOM = 40 Ì

μÕ∫ 40 Ì

7. ‡¢’¬π®”π«π 365 ®”π«π√Õ∫«ß°≈¡ ®”π«π∑’Ë 88 §◊Õ 7 ®”π«π∑’Ë 111 §◊Õ 2 ®”π«π∑’Ë 224

§◊Õ 1 ®”π«π∑’Ë 365 §◊Õ 5 ∂â“º≈∫«°¢Õß∑ÿ° 60 ®”π«π∑’Ë‡√’¬ßμ‘¥μàÕ°—π¡’§à“‡ªìπ 216 ‡ ¡Õ

·≈â«®”π«π∑’Ë 122 ¡’§à“‡∑à“„¥

·π«§‘¥

À.√.¡. ¢Õß 60 °—∫ 365 ‡ªìπ 5

μ“¡∑ƒ…Æ’√—ßπ°æ‘√“∫ ®–‰¥âº≈∫«°¢Õß®”π«π 5 ®”π«π∑’Ë‡√’¬ßμ‘¥°—π∑ÿ°™ÿ¥¡’§à“§ß∑’Ë

®”π«π∑’Ë 88 ‡∑à“°—∫®”π«π∑’Ë 3 §◊Õ 7

®”π«π∑’Ë 111 ‡∑à“°—∫®”π«π∑’Ë 1 §◊Õ 2

®”π«π∑’Ë 224 ‡∑à“°—∫®”π«π∑’Ë 4 §◊Õ 1

®”π«π∑’Ë 365 ‡∑à“°—∫®”π«π∑’Ë 5 §◊Õ 5

º≈∫«°¢Õß®”π«π 5 ®”π«π∑’Ë‡√’¬ßμ‘¥μàÕ°—π‡ªìπ = 18

¥—ßπ—Èπ ®”π«π∑’Ë 2 §◊Õ 3 ·≈–®”π«π∑’Ë 122 §◊Õ 3

μÕ∫ 3

21612

^

^ ^^

^ ^

180 Ì› 80 Ì2

^

^

^

^

^^ ^^

^

^

^

^

^

^

^

^

^ ^

^

^

^

A

B

O

C

M

X

A 119-152 12/9/08, 10:38 AM130


9. °”Àπ¥ ABC ‡ªìπ√Ÿª “¡‡À≈’Ë¬¡¥â“π‡∑à“∑’Ë¡’ P ‡ªìπ®ÿ¥¿“¬„π∑”„Àâ PA = 6 Àπà«¬

PB = 8 Àπà«¬ ·≈– PC = 10 Àπà«¬ √Ÿª “¡‡À≈’Ë¬¡ ABC ¡’æ◊Èπ∑’Ë°’Ëμ“√“ßÀπà«¬

·π«§‘¥

Δ ADC ‰¥â®“°°“√À¡ÿπ Δ APB ∑«π‡¢Á¡π“Ãî°“ 60 Õß»“ √Õ∫®ÿ¥ A

Δ BEA ‰¥â®“°°“√À¡ÿπ Δ BPC ∑«π‡¢Á¡π“Ãî°“ 60 Õß»“ √Õ∫®ÿ¥ B

Δ APC ‰¥â®“°°“√À¡ÿπ Δ BFC ∑«π‡¢Á¡π“Ãî°“ 60 Õß»“ √Õ∫®ÿ¥ C

2 (æ◊Èπ∑’Ë Δ ABC) = (62 + 82 + 102) + 3 Ó 24

= 50 3 + 72

æ◊Èπ∑’Ë Δ ABC = 36 + 25 3 μ“√“ßÀπà«¬

μÕ∫ 36 + 25 3 μ“√“ßÀπà«¬

A

B C

E

F

D

P

34

10. ®“°√Ÿª BO ·∫àß§√÷Ëß¡ÿ¡ CBA, CO ·∫àß§√÷Ëß¡ÿ¡ ACB ·≈– MN ºà“π O ¢π“π°—∫ BC

∂â“ AB = 12 Àπà«¬ BC = 24 Àπà«¬ ·≈– AC = 18 Àπà«¬ ·≈â«§«“¡¬“«‡ âπ√Õ∫√Ÿª

√Ÿª “¡‡À≈’Ë¬¡ AMN ¡’§à“°’ËÀπà«¬

·π«§‘¥

1. BO ·∫àß§√÷Ëß ABC (‚®∑¬å°”Àπ¥)

2. MBO = OBC (®“°¢âÕ 1)

3. NM // BC (‚®∑¬å°”Àπ¥)

4. MOB = OBC (®“°¢âÕ 3 ¡ÿ¡·¬âß)

5. Δ BMO ‡ªìπ Δ Àπâ“®—Ë« (®“°¢âÕ 4)

6. MB = MO (®“°¢âÕ 5)

7. OC ·∫àß§√÷Ëß BCA (‚®∑¬å°”Àπ¥)

^

^ ^

^ ^

^

A

B C

M NO

A 119-152 12/9/08, 10:38 AM131


8. ®–‰¥â NCO = OCB (®“°¢âÕ 7)

9. NOC = OCB (®“°¢âÕ 3 ¡ÿ¡·¬âß)

10. Δ ONC ‡ªìπ√Ÿª “¡‡À≈’Ë¬¡Àπâ“®—Ë« (®“°¢âÕ 9)

11. NO = NC (®“°¢âÕ 3 ¡∫—μ‘¢Õß√Ÿª “¡‡À≈’Ë¬¡Àπâ“®—Ë«)

12. ‡ âπ√Õ∫√Ÿª Δ AMN = AM + MN + NA

= AM + MO + ON + AN

= AM + MB + NC + AN

= AB + AC

∴ ‡ âπ√Õ∫√Ÿª Δ AMN = 12 + 18 Àπà«¬


11. °”Àπ¥ P(x) ‡ªìπæÀÿπ“¡‚¥¬∑’Ë P(x) = x4 + ax3 + bx2 + cx + d

∂â“ P(1) = 6, P(›2) = 3, P(3) = ›2 ·≈– P(›4) = ›9 ·≈â« |P(4) + 4| ¡’§à“‡∑à“„¥

·π«§‘¥

„Àâ P(x) = Q(x) + 7 › x2

®–‰¥â P(1) = Q(1) + 7 › 12

6 = Q(1) + 6 ¥—ßπ—Èπ Q(1) = 0

P(›2) = Q(›2) + 7 › (›2)2

3 = Q(›2) + 3 ¥—ßπ—Èπ Q(›2) = 0

P(3) = Q(3) + 7 › 32

›2 = Q(3) › 2 ¥—ßπ—Èπ Q(3) = 0

P(›4) = Q(›4) + 7 › (›4)2

›9 = Q(›4) › 9 ¥—ßπ—Èπ Q(›4) = 0

· ¥ß«à“

(x › 1), (x + 2), (x › 3), (x + 4) ‡ªìπμ—«ª√–°Õ∫¢Õß Q(x)

π—Ëπ§◊Õ P(x) = (x › 1)(x + 2)(x › 3)(x + 4) + 7 › x2

P(4) = (4 › 1)(4 + 2)(4 › 3)(4 + 4) + 7 › 42

= (3)(6)(1)(8) + 7 › (4)2

= 144 + 7 › 16

= 135

¥—ßπ—Èπ |P(4) + 4| = 135 + 4 = 139

μÕ∫ 139

^ ^

^^

A 119-152 12/9/08, 10:38 AM132


12. °”Àπ¥ x1, x

2, x

3, ..., x

100 ‡ªìπ®”π«π®√‘ß∫«° ´÷Ëß∑”„Àâ

x1 + = 1, x

2 + = 4, x

3 + = 1, ..., x

99 + = 1 ·≈– x

100 + = 4

·≈â« x1 + x

2 + x

3 + ... + x

100 ¡’§à“‡∑à“„¥

·π«§‘¥

x1 + ≥ 2

x2 + ≥ 2

...x

99 + ≥ 2

x100

+ ≥ 2

®–‰¥â ≥ 2100

®–‰¥â 450 ≥ 2100

2100 = 2100

∴ ®–‰¥â x1 + = 2 ; x

1 =

x2 + = 2 ; x

2 =

...x

100 + = 2 ; x

100 =

·∑π§à“·≈â«·°â ¡°“√®–‰¥â

x1 = x

3 = x

5 = ... = x

99 =

x2 = x

4 = x

6 = ... = x

100 = 2

x1 + x

2 + x

3 + ... + x

100 = 125

μÕ∫ 125

1x1

1x2

1x2

x1

x2

1x3

1x3

x2

x3

1x100

1x100

x99

x100

1x1

x100x1

( )x1 + 1

x2

( )x2 + 1

x3

... ( )x99

+1

x100 ( )x

100 +

1x2

x1

x2

1x2

1x3

x2

x3

1x3

1x1

x100x1

1x1

12

1x4

1x1

A 119-152 12/9/08, 10:38 AM133


13. ∂â“ x1 + 4x

2 + 9x

3 + 16x

4 + 25x

5 + 36x

6 + 49x

7= 4

4x1 + 9x

2 + 16x

3 + 25x

4 + 36x

5 + 49x

6 + 64x

7= 44

9x1 + 16x

2 + 25x

3 + 36x

4 + 49x

5 + 64x

6 + 81x

7= 444

·≈â« 16x1 + 25x

2 + 36x

3 + 49x

4 + 64x

5 + 81x

6 + 100x

7 ¡’§à“‡∑à“„¥

·π«§‘¥

x1 + 4x

2 + 9x

3 + 16x

4 + 25x

5 + 36x

6 + 49x

7= 4.........................➊

4x1 + 9x

2 + 16x

3 + 25x

4 + 36x

5 + 49x

6 + 64x

7= 44 ...................... ➋

9x1 + 16x

2 + 25x

3 + 36x

4 + 49x

5 + 64x

6 + 81x

7= 444 .................... ➌

➋ › ➊ 3x1 + 5x

2 + 7x

3 + 9x

4 + 11x

5 + 13x

6 + 15x

7= 40 ...................... ➍

➌ › ➋ 5x1 + 7x

2 + 9x

3 + 11x

4 + 13x

5 + 15x

6 + 17x

7= 400 .................... ➎

➎ › ➊ 2x1 + 2x

2 + 2x

3 + 2x

4 + 2x

5 + 2x

6 + 2x

7= 360 .................... ➏

➏ + ➎ 7x1 + 9x

2 + 11x

3 + 13x

4 + 15x

5 + 17x

6 + 19x

7 = 760 ................... ➐

➐ › ➌ 16x1 + 25x

2 + 36x

3 + 49x

4 + 64x

5 + 81x

6 + 100x

7 = 1,204

μÕ∫ 1,204

A 119-152 12/9/08, 10:39 AM134


14. ®“°√Ÿª ABC, DEF ·≈– PQR ‡ªìπ√Ÿª “¡‡À≈’Ë¬¡¥â“π‡∑à“ ‚¥¬∑’Ë AD = BD æ◊Èπ∑’Ë

√Ÿª “¡‡À≈’Ë¬¡ ABC ‡ªìπ°’Ë‡∑à“¢Õßæ◊Èπ∑’Ë√Ÿª “¡‡À≈’Ë¬¡ PQR

·π«§‘¥

„Àâ AC = 4 ®–‰¥â AF = 2

„Àâ AP = x, PR = y ·≈– FR = z

¥—ßπ—Èπ Δ AFR ∼ Δ ARC

1. Δ AFR ∼ Δ ARC

= = ®–‰¥â x + y = 2 2

¥—ßπ—Èπ = π—Ëπ§◊Õ 2 z = x

2. Δ QRE ∼ Δ CRF

= =

= =

z2 = 4 › 2z

z2 + 2z › 4 = 0

z = = 1 + 5 (§«“¡¬“«¥â“π‡ªìπ®”π«π®√‘ß∫«°)

=

zx

x + y4 z

x2

2 2

QECF

ERFR

RQRC

z2

yx

›2 ± 4 + 162

z2

y2 z

A

B

P FD

QR

A

F

R

2

x + y

z

A

R

C

x + y

4

x

E C

2x + y

2 › zz

A 119-152 12/9/08, 10:39 AM135


y = = = 2(3 › 5)

= = = 7 + 3 5

μÕ∫ 7 + 3 5

2z2

2

162(14 › 6 5)

47 › 3 5

15. ∂â“√Ÿª ’Ë‡À≈’Ë¬¡ ABCD ¡’®ÿ¥¬Õ¥Õ¬Ÿà∫π‡ âπ√Õ∫«ß¢Õß«ß°≈¡∑’Ë¡’ O ‡ªìπ®ÿ¥»Ÿπ¬å°≈“ß

‚¥¬∑’Ë AC ‡ªìπ‡ âπºà“π»Ÿπ¬å°≈“ß¢Õß«ß°≈¡ ·≈– CAD = 18 Ì, BDC = 50 Ì ·≈â« OBD

¡’¢π“¥°’ËÕß»“

·π«§‘¥

‡æ√“–«à“ ADC = 90 Ì ·≈– BDC = BAC = 50 Ì

®–‰¥â ADB = 40 Ì ∑”„Àâ AOB = 80 Ì

‡æ√“–«à“ DAC = CBD = 18 Ì

·μà ABC = 90 Ì

¥—ßπ—Èπ OBD = 90 Ì › 50 Ì › 18 Ì = 22 Ì

μÕ∫ 22 Ì(

18 Ì

16. ∂â“ 7x + y = 21 ·≈– 32x + y = 1 ·≈â« 3(7x + 1 + 7y › 2) ¡’§à“‡∑à“„¥

·π«§‘¥

®“° 32x + y = 1

®–‰¥â 32x + y = 30

¥—ßπ—Èπ 2x + y = 0

y = ›2x

®“° 7x + y = 21

∂â“ y = ›2x ®–‰¥â 7(x + (›2x)) = 21

7›x = 21

7x = 21›1 =

7y = 7 ›2x = = (21)2

121

2 (6 › 2 5)2

æ◊Èπ∑’Ë Δ ABCæ◊Èπ∑’Ë Δ PQR

^ ^ ^

^ ^ ^

^ ^

^ ^

^

^

121

›2

) (

A B

DCP

O

A 119-152 12/9/08, 10:39 AM136


π—Ëπ§◊Õ 3(7x + 1 + 7y › 2) = 3 [(7x . 7) + 7y .

= 3 Ó 7 + 212 Ó = 28

μÕ∫ 28

172)( ]

[( 121 ) 1

72( )]

17. °”Àπ¥ ¡°“√ 15x + 14y = 7 ∂â“ x ∑’Ë‡ªìπ®”π«π‡μÁ¡∑’Ë¡“°∑’Ë ÿ¥ ÷Ëßª√–°Õ∫¥â«¬ ¡“™‘°

4 À≈—° ∑’Ë∑”„Àâ (x, y) ‡ªìπ§”μÕ∫¢Õß ¡°“√π’È ‡¡◊ËÕ y ‡ªìπ®”π«π‡μÁ¡ ·≈â« x ¡’§à“‡∑à“„¥

·π«§‘¥

‡æ√“–«à“ 15(7) + 14(›7) = 7

¥—ßπ—Èπ (7, ›7) ‡ªìπ§”μÕ∫Àπ÷Ëß¢Õß ¡°“√π’È

‡π◊ËÕß®“° 15x + 14y = 7 ®–‰¥â y =

∂â“„Àâ x = 7 + a ®–‰¥â y = = ›7 ›

‡æ√“–«à“ y ‡ªìπ®”π«π‡μÁ¡ ¥—ßπ—Èπ 14 À“√ 15a ≈ßμ—«

·μà À.√.¡. ¢Õß 15 °—∫ 14 ‡∑à“°—∫ 1 ¥—ßπ—Èπ a μâÕß‡ªìπæÀÿ§Ÿ≥¢Õß 14

¥—ßπ—Èπ ¡’®”π«π‡μÁ¡ b ∑’Ë∑”„Àâ a = 14b

π—Ëπ§◊Õ x = 7 + 14b

·μà x = 7 + 14b ≥ 9,999

14b ≥ 9,992

b ≥ 713.7

®–‰¥â b = 713

π—Ëπ§◊Õ x = 7 + 14(713) = 9,989

μÕ∫ 9,989

7 › 15x14

7 › 15(7 + a)14

15a14

A 119-152 12/9/08, 10:39 AM137


18. °”Àπ¥„Àâ n ·≈– d ‡ªìπ®”π«π‡μÁ¡∫«° ∂â“ d À“√ 13n + 6 ·≈– 12n + 5 ≈ßμ—«

·≈â«º≈∫«°¢Õß§à“ d ∑’Ë‡ªìπ‰ª‰¥â∑—ÈßÀ¡¥ ¡’§à“‡∑à“„¥

·π«§‘¥

‡æ√“–«à“ d À“√ 13n + 6 ≈ßμ—« ·≈– d À“√ 12n + 5 ≈ßμ—«

¥—ßπ—Èπ d À“√ 12(13n + 6) ≈ßμ—« ·≈– d À“√ 13(12n + 5) ≈ßμ—«

®–‰¥â d À“√ 12(13n + 6) › 13(12n + 5 ≈ßμ—«¥â«¬

‡æ√“–«à“ 12(13n + 6) › 13(12n + 5) = 156n + 72 › 156n › 65 = 7

®–‰¥â d À“√ 7 ≈ßμ—« ®“° d ‡ªìπ®”π«π‡μÁ¡ ¥—ßπ—Èπ d = 1 À√◊Õ d = 7

º≈∫«°‡∑à“°—∫ 1 + 7 = 8

μÕ∫ 8

19. √ŸªÀ≈“¬‡À≈’Ë¬¡¥â“π‡∑à“¡ÿ¡‡∑à“√ŸªÀπ÷Ëß ¡’‡ âπ∑·¬ß¡ÿ¡∑—ÈßÀ¡¥ 20 ‡ âπ ·≈–·π∫„π

«ß°≈¡√—»¡’ 2 Àπà«¬ √ŸªÀ≈“¬‡À≈’Ë¬¡π’È¡’æ◊Èπ∑’Ë‡∑à“„¥

·π«§‘¥

√Ÿª 4 ‡À≈’Ë¬¡¡’‡ âπ∑·¬ß¡ÿ¡ 2 ‡ âπ

√Ÿª 5 ‡À≈’Ë¬¡¡’‡ âπ∑·¬ß¡ÿ¡ 5 ‡ âπ

√Ÿª 6 ‡À≈’Ë¬¡¡’‡ âπ∑·¬ß¡ÿ¡ 9 ‡ âπ

√Ÿª 7 ‡À≈’Ë¬¡¡’‡ âπ∑·¬ß¡ÿ¡ 14 ‡ âπ

√Ÿª 8 ‡À≈’Ë¬¡¡’‡ âπ∑·¬ß¡ÿ¡ 20 ‡ âπ

√Ÿª “¡‡À≈’Ë¬¡ 1 √Ÿª ¡’¡ÿ¡¬Õ¥°“ß = 45 Ì

æ◊Èπ∑’Ë√Ÿª “¡‡À≈’Ë¬¡ = ( 2)( 2) sin 45 Ì = μ“√“ßÀπà«¬

¥—ßπ—Èπ √Ÿª 8 ‡À≈’Ë¬¡®–¡’æ◊Èπ∑’Ë 8 = 4 2 μ“√“ßÀπà«¬

μÕ∫ 4 2 μ“√“ßÀπà«¬

360 Ì8

12

22

22 )(

A 119-152 12/9/08, 10:39 AM138


20. √Ÿª ’Ë‡À≈’Ë¬¡ ABCD ¡’æ◊Èπ∑’Ë 2,550 μ“√“ßÀπà«¬ ‚¥¬∑’Ë P, Q, R ·≈– S ‡ªìπ®ÿ¥°÷Ëß°≈“ß¥â“π

∂â“√Ÿª ’Ë‡À≈’Ë¬¡ BNMP ·≈–√Ÿª ’Ë‡À≈’Ë¬¡ DTOR ¡’æ◊Èπ∑’Ë 123 ·≈– 456 μ“√“ßÀπà«¬

μ“¡≈”¥—∫ ·≈â«æ◊Èπ∑’Ë¢Õß√Ÿª ’Ë‡À≈’Ë¬¡ MNOT ‡∑à“°—∫°’Ëμ“√“ßÀπà«¬

·π«§‘¥

≈“° BD ®–‰¥â æ◊Èπ∑’Ë Δ DAP = æ◊Èπ∑’Ë Δ DBP ·≈–æ◊Èπ∑’Ë Δ BCR = æ◊Èπ∑’Ë Δ BDR

æ◊Èπ∑’Ë � BPDR = æ◊Èπ∑’Ë � ABCD = 1,275 μ“√“ßÀπà«¬

æ◊Èπ∑’Ë � MNOT = 1,275 › (123 + 456) = 696 μ“√“ßÀπà«¬

μÕ∫ 696 μ“√“ßÀπà«¬

12

NO

R C

Q

BP

M

T

D

S

A

A 119-152 12/9/08, 10:39 AM139


μÕπ∑’Ë 2

21. ·∫∫®”≈Õß CH4 ·π∫„π∑√ß‡À≈’Ë¬¡ ’ËÀπâ“ ‚¥¬∑’Ë·μà≈–Àπâ“‡ªìπ√Ÿª “¡‡À≈’Ë¬¡¥â“π‡∑à“∑’Ë¬“«

¥â“π≈– 6 6 Àπà«¬ ·μà≈–Àπâ“‡∑à“°—π∑ÿ°ª√–°“√ ·≈–¡’ H Õ¬Ÿà∑’Ë®ÿ¥¡ÿ¡ ·≈â« ·¢π C › H

¬“«‡∑à“„¥

·π«§‘¥

H4P = (6 6)2 › (3 6)2

= 3 6 ⋅ 3= 9 2

∴ H4T = 6 2 = H

3T

„Àâ·¢π C › H ¬“« x ®–‰¥â CH3 = x

CH1

= x

Δ H1TH

3, H

1T = (6 6)2 › (6 2)2

H1T = 12

Δ CTH3, CT = 12 › x

CH3

= x

x2 = (12 › x)2 + (6 2)2

x2 = 144 › 24x + x2 + 72

24x = 216

x = 9 Àπà«¬


HP

H4

H2

H1

T

C

H3

A 119-152 12/9/08, 10:39 AM140


22. ∂â“ (x2 › x + 1)3(x3 + 2x2 + 2x + 1)5 = a21

x21 + a20

x20 + a19

x19+ ... + a2x2 + a

1x + a

0

‡¡◊ËÕ a1 ‡ªìπ§à“§ß∑’Ë ·≈â« a

1 +

a2 +

a3 +

... +

a10

¡’§à“‡∑à“„¥

·π«§‘¥

®“° x2 › x + 1 ·≈– x3 + 2x2 + 2x + 1 ‡ªìπ palindrome

®–‰¥âº≈§Ÿ≥‡ªìπ palindrome ¥â«¬

¥—ßπ—Èπ a1 +

a2 +

a3 +

... +

a10

= (a1 +

a2 +

a3 +

... +

a20

)

= (a0 +

a1 +

a2 +

... +

a21

› a0 › a21

)

= (a0 +

a1 +

a2 +

... +

a21

› 2)

®“° (x2 › x + 1)3(x3 + 2x2 + 2x + 1)5 = a21

x21 + a20

x20 + ... + a1x + a

0

„Àâ x = 1 ®–‰¥â (13)(65) = a21

+ a20

+ ... + a1+ a

0

a0 +

a1 +

a2 +

... +

a21

= 65 = 7,776

a1 +

a2 +

a3 +

... +

a10

= (7,776 › 2)

= 3,887

μÕ∫ 3,887

12

12

12

12

A 119-152 12/9/08, 10:39 AM141


23. ∂â“ = 2 ·≈â« x ¡’§à“‡∑à“„¥

·π«§‘¥

a = 2 + x, b = 2 › x

a2 + b2 = 4

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

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

4 2 › a2b + ab2 = 2 2 + 2a › 2b › 2ab

0 = a2b › ab2 + 2a › 2b › 2ab › 2 2

0 = ab(a › b) + 2(a › b) › 2(ab + 2)

0 = (ab + 2)(a › b › 2)

‡π◊ËÕß®“° ab + 2 ≠ 0 ·≈– a › b › 2 = 0

¥—ßπ—Èπ 2 + x › 2 › x = 2

¬°°”≈—ß Õß 2 + x › 2 (2 + x)(2 › x) + 2 › x = 2

2 = 2 4 › x2

1 = 4 › x2

1 = 4 › x2

x2 = 3

x = ± 3

μ√«®§”μÕ∫®–‰¥â x = ± 3

μÕ∫ ± 3

2 + x2 + 2 + x

+ 2 › x2 › 2 › x

A 119-152 12/9/08, 10:39 AM142


24. ∂â“ x, y ·≈– z ‡ªìπ®”π«π®√‘ß ∑’Ë Õ¥§≈âÕß°—∫√–∫∫ ¡°“√ x + 2y + 3z = 13

x2 + 4y2 + 9z2 + 3x › 2y + 15z = 82 ·≈â« xyz + x + y + z ¡’§à“‡∑à“„¥

·π«§‘¥

x2 + 4y2 + 9z2 + 3x + 15z = 82 + 2y

x2 + 4y2 + 9z2 + 4x + 18z = 82 + x + 2y + 3z

x2 + 4x + 4 + 4y2 + 9z2 + 18z + 9 = 82 + 13 + 4 + 9

(x + 2)2 + (2y)2 + (3z + 3)2 = 108

®“° x + 2y + 3z = 13

(x + 2) + 2y + (3z + 3) = 18

„Àâ a = x + 2, b = 2y, c = 3z + 3 ®–‰¥â

a2 + b2 + c2 = 108 ........................... ➊

a + b + c = 18

(a + b + c)2 = 324

®–‰¥â ab + bc + ca = 108 ........................... ➋

2 Ó ➊ › 2 Ó ➋, 2a2 + 2b2 + 2c2 › 2ab › 2bc › 2ca = 0

(a › b)2 + (b › c)2 + (c › a)2 = 0

a = b = c = 6

¥—ßπ—Èπ x = 4, y = 3 ·≈– z = 1

xyz + x + y + z = 20

μÕ∫ 20

A 119-152 12/9/08, 10:40 AM143


25. ∂â“ a2 › 2a = ›1, b2 › 3b = 1 ·≈– c2 › 4c = ›1 ·≈â« 3a3 › b3 + c3 + + +

+ 200 ¡’§à“‡∑à“„¥

·π«§‘¥

a + = 2, b › = 3, c + = 4

a3 + = (a + )3 › 3(a + )

= 8 › 6

= 2

b3 › = (b › )3 + 3(b › )

= 27 + 9

= 36

c3 + = (c + )3 › 3(c + )

= 64 › 12

= 52

¥—ßπ—Èπ 3a3 › b3 + c3 + + + + 200 = 222

μÕ∫ 222

3a3

1b3

1c3

1a

1b

1c

1a3

1a

1a

1b3

1b

1b

1c3

1c

1c

3a3

1b3

1c3

A 119-152 12/9/08, 10:40 AM144


26. °”Àπ¥„Àâ a, b, c ‡ªìπ®”π«π‡μÁ¡∫«° ∂â“ a + b + c = 20 = ab + bc › ca › b2 ·≈â«

º≈∫«°¢Õß§à“ abc ∑’Ë‡ªìπ‰ª‰¥â∑—ÈßÀ¡¥ ¡’§à“‡∑à“„¥

·π«§‘¥

®“° ab + bc › ca › b2 = 20

®–‰¥â (a › b)(b › c) = 20

‚¥¬‰¡à‡ ’¬π—¬∑—Ë«‰ª „Àâ a ≥ b ≥ c

æ‘®“√≥“§à“∑’Ë‡ªìπ‰ª‰¥â

a › b b › c a b c a + b + c = 20 abc

1 20 22 21 1 - -

2 10 13 11 1 - -

4 5 11 7 2 20 154

5 4 12 7 3 - -

11 6 2

10 2 14 4 2 20 112

20 1 22 2 1 - -

§à“ abc ∑’Ë‡ªìπ‰ª‰¥â§◊Õ 112 °—∫ 154

º≈∫«°®–‡∑à“°—∫ 112 + 154 = 266

μÕ∫ 266

A 119-152 12/9/08, 10:40 AM145


27. √Ÿª “¡‡À≈’Ë¬¡ ABC ¡’ D, E ·≈– F ‡ªìπ®ÿ¥∫π¥â“π BC ∑”„Àâ AD ⊥ BC, AE ·∫àß§√÷Ëß

BAC ·≈– BF = CF ∂â“ BAD = DAE = EAF = FAC ·≈â« BAC + 2ABC + 4ACB

¡’§à“‡∑à“„¥

^^ ^^ ^^ ^^

·π«§‘¥

„Àâ BAD = x ®–‰¥â ABD = 90 Ì › x ·≈– ACB = 90 Ì › 3x

√â“ß ≈“° FT „Àâ TF ⊥ BC æ∫ AC ∑’Ë T ≈“° BT

1. Δ BFT ≅ Δ CFT (¥.¡.¥.)

2. BTF = CTF = 3x Ì

3. TBF = TCF = 90 Ì › 3x Ì

4. ®–‰¥â TBA = TFA = 2x Ì

¥—ßπ—Èπ ∴ ABFT ·π∫„π«ß°≈¡

4x = 90 Ì

x = 22.5 Ì

BAC = 90 Ì, ABC = 67.5 Ì ·≈– ACB = 22.5 Ì

∴ BAC + 2ABC + 4ACB = 315 Ì

μÕ∫ 315 Ì

^^ ^

^^

^^

^^

^^

^

^

^^

))))

)) ))C

FB

A

T

ED

xxxx

x2x

3x 3x

90 Ì › 3x90 Ì2x Ì

A 119-152 12/9/08, 10:40 AM146


28. ∂â“°”Àπ¥√–∫∫ ¡°“√

abc + ab + bc + ca + a + b + c = 71

bcd + bc + cd + db + b + c + d = 191

cda + cd + da + ac + c + d + a = 95

dab + da + ab + bd + d + a + b = 143

·≈â« abcd + a + b + c + d ¡’§à“‡∑à“„¥

·π«§‘¥

π” 1 ∫«° ∑ÿ° ¡°“√ ®–‰¥â

(a + 1)(b + 1)(c + 1) = 72 = 23 Ó 32 ......................................➊

(b + 1)(c + 1)(d + 1) = 192 = 26 Ó 3 ......................................➋

(c + 1)(d + 1)(a + 1) = 96 = 25 Ó 3 ......................................➌

(d + 1)(a + 1)(b + 1) = 144 = 24 Ó 32 ......................................➍

➊ Ó ➋ Ó ➌ Ó ➍; (a + 1)3(b + 1)3(c + 1)3(d + 1)3 = 218 Ó 36

(a + 1)(b + 1)(c + 1)(d + 1) = 26 Ó 32................➎

➎ ÷ ➊ d + 1 = 23

d = 7

➎ ÷ ➋ a + 1 = 3

a = 2

➎ ÷ ➌ b + 1 = 2 Ó 3

b = 5

➎ ÷ ➍ c + 1 = 22

c = 3

§à“¢Õß abcd + a + b + c + d = (2 Ó 5 Ó 3 Ó 7) + (2 + 5 + 3 + 7)

= 210 + 17

= 227

μÕ∫ 227

A 119-152 12/9/08, 10:40 AM147


29. ∂â“°”Àπ¥√–∫∫ ¡°“√

x1 + x

2 + x

3 = 4, x

2 + x

3 + x

4 = 6, x

3 + x

4 + x

5 = 8,

x4 + x

5 + x

6 = 12, x

5 + x

6 + x

7 = 15, x

6 + x

7 + x

8 = 19,

x7 + x

8 + x

9 = 23, x

8 + x

9 + x

10 = 27, x

9 + x

10 + x

1 = 30,

x10

+ x1 + x

2 = 36

·≈â« 3x1 + 4x

10 ¡’§à“‡∑à“„¥

·π«§‘¥

π” ¡°“√∑—ÈßÀ¡¥¡“∫«°°—π

3x1 + 3x

2 + 3x

3 + 3x

4 + 3x

5 + 3x

6 + 3x

7 + 3x

8 + 3x

9 + 3x

10= 180

x1 + x

2 + x

3 + x

4 + x

5 + x

6 + x

7 + x

8 + x

9 + x

10= 60

(x1 + x

2 + x

3) + (x

4 + x

5 + x

6) + (x

7 + x

8 + x

9) + x

10= 60

4 + 12 + 23 + x10

= 60

39 + x10

= 60

x10

= 21

x1 + (x

2 + x

3 + x

4) + (x

5 + x

6 + x

7) + (x

8 + x

9 + x

10) = 60

x1 + 6

+ 15 + 27 = 60

x1 + 48 = 60

x1

= 12

3x1 + 4x

10= 36 + 84

= 120

μÕ∫ 120

A 119-152 12/9/08, 10:40 AM148


30. ∂â“ ABCD ‡ªìπ√Ÿª ’Ë‡À≈’Ë¬¡ ¡’ BAC = CAD = 66 Ì, BCA = 15 Ì ·≈– ACD = 9 Ì

·≈â« 2ABD + ADB ¡’§à“‡∑à“„¥

·π«§‘¥

μàÕ AD ∂÷ß E

„Àâ DCE = 24 Ì

≈“° BE

®–‰¥â EAB + BCE = 180 Ì

¥—ßπ—Èπ � ABCE ·π∫„π«ß°≈¡

®–‰¥â EBC = CEB = 66 Ì

®–‰¥â Δ BCT ≅ Δ CET

∴®–‰¥â Δ BDT ≅ Δ DET

·≈–®–‰¥â ACB = AEB = DBE = 15 Ì

¥—ßπ—Èπ ®–‰¥â ABD = 18 Ì ·≈– ADB = 30 Ì

‡æ√“–©–π—Èπ 2ABD + ADB = 66 Ì

μÕ∫ 66 Ì

^ ^ ^

^ ^

^

^^

^

^

^

^

^

^ ^

^ ^

^

B

A

C

E

240

90

150

660

150660

660

150

660

D T

A 119-152 12/9/08, 10:40 AM149


§≥–∑”ß“π

∑’Ëª√÷°…“1. §ÿ≥À≠‘ß°…¡“ «√«√√≥ ≥ Õ¬ÿ∏¬“ ‡≈¢“∏‘°“√§≥–°√√¡°“√°“√»÷°…“¢—Èπæ◊Èπ∞“π2. π“¬¡—ß°√ °ÿ≈«“π‘™ √Õß‡≈¢“∏‘°“√§≥–°√√¡°“√°“√»÷°…“¢—Èπæ◊Èπ∞“π3. π“¬ ¡‡°’¬√μ‘ ™Õ∫º≈ √Õß‡≈¢“∏‘°“√§≥–°√√¡°“√°“√»÷°…“¢—Èπæ◊Èπ∞“π4. π“¬«‘π—¬ √Õ¥®à“¬ √Õß‡≈¢“∏‘°“√§≥–°√√¡°“√°“√»÷°…“¢—Èπæ◊Èπ∞“π5. π“ß‡∫≠®≈—°…≥å πÈ”øÑ“ ºŸâÕ”π«¬°“√ ”π—°«‘™“°“√·≈–¡“μ√∞“π°“√»÷°…“6. π“ßÕ√∑—¬ ¡Ÿ≈§” ºŸâÕ”π«¬°“√ ”π—°æ—≤π“π«—μ°√√¡°“√®—¥°“√»÷°…“7. π“¬¥‘≈° æ—≤πå«‘™—¬‚™μ‘ ¢â“√“™°“√∫”π“≠ °√–∑√«ß»÷°…“∏‘°“√8. π“ß “«»√’ ¡√ æÿà¡ –Õ“¥ ¢â“√“™°“√∫”π“≠ °√–∑√«ß»÷°…“∏‘°“√

§≥–∑”ß“π1. π“ßπ‘®«¥’ ‡®√‘≠‡°’¬√μ‘∫«√ π—°«‘™“°“√»÷°…“ ”π—°æ—≤π“π«—μ°√√¡°“√®—¥°“√»÷°…“2. π“ßÕ√πÿ™ ¡—Ëß¡’ ÿ¢»‘√‘ π—°«‘™“°“√»÷°…“ ”π—°æ—≤π“π«—μ°√√¡°“√®—¥°“√»÷°…“3. π“¬»—°¥‘Ï ‘π ™àÕß¥“√“°ÿ≈ π—°«‘™“°“√»÷°…“ ”π—°æ—≤π“π«—μ°√√¡°“√®—¥°“√»÷°…“4. π“ß “««√πÿ™ √ÿàß‡√◊Õß‡®√‘≠°ÿ≈ ‡®â“Àπâ“∑’Ë∫√‘À“√ß“π∏ÿ√°“√ √—°…“°“√„πμ”·Àπàßπ—°«‘™“°“√»÷°…“

5. π“¬ª√“‚¡∑¬å ¢®√¿—¬ »÷°…“π‘‡∑»°å

”π—°ß“π‡¢μæ◊Èπ∑’Ë°“√»÷°…“°√ÿß‡∑æ¡À“π§√ ‡¢μ 1

6. π“ß∑‘æ¬å«√√≥ «—≤π“ß°Ÿ√ ¡À“«‘∑¬“≈—¬∏√√¡»“ μ√å

7. π“¬∫ÿ≠∏√√¡ ∑—Ëß∑Õß »÷°…“π‘‡∑»°å ”π—°ß“π‡¢μæ◊Èπ∑’Ë°“√»÷°…“™—¬π“∑

8. π“¬ªØ≈ ‡ª√¡ª√’¥‘Ï »÷°…“π‘‡∑»°å ”π—°ß“π‡¢μæ◊Èπ∑’Ë°“√»÷°…“‡æ™√∫ÿ√’ ‡¢μ 1

9. π“¬« —πμå «√‡™…∞å »÷°…“π‘‡∑»°å ”π—°ß“π‡¢μæ◊Èπ∑’Ë°“√»÷°…“√âÕ¬‡ÕÁ¥ ‡¢μ 2

10. π“¬»ÿ¿°√ æß»å∑Õß¡’ »÷°…“π‘‡∑»°å ”π—°ß“π‡¢μæ◊Èπ∑’Ë°“√»÷°…“∫ÿ√’√—¡¬å ‡¢μ 1

11. π“¬∫√√∑—¥ «¿—°¥‘Ï‡æ™√ »÷°…“π‘‡∑»°å ”π—°ß“π‡¢μæ◊Èπ∑’Ë°“√»÷°…“ °≈π§√ ‡¢μ 1

12. π“¬‡©≈‘¡æ≈ ‡ ¢–æ—π∏å »÷°…“π‘‡∑»°å ”π—°ß“π‡¢μæ◊Èπ∑’Ë°“√»÷°…“™ÿ¡æ√ ‡¢μ 1

13. π“¬®‘√«—≤πå √—°æà«ß »÷°…“π‘‡∑»°å ”π—°ß“π‡¢μæ◊Èπ∑’Ë°“√»÷°…“μ“° ‡¢μ 1

14. π“¬™àÕ©—μ√ ‰™¬ ¡π÷° »÷°…“π‘‡∑»°å ”π—°ß“π‡¢μæ◊Èπ∑’Ë°“√»÷°…“‡æ™√∫ÿ√’ ‡¢μ 1

15. π“¬«—≈≈¿ æ÷Ëßæ—π∏ÿå »÷°…“π‘‡∑»°å ”π—°ß“π‡¢μæ◊Èπ∑’Ë°“√»÷°…“ ÿ‚¢∑—¬ ‡¢μ 1

16. π“ß “«®”‡√‘≠ ‡®’¬«À«“π ‚√ß‡√’¬π¡À‘¥≈«‘∑¬“πÿ √≥å Õß§å°√¡À“™π

A 119-152 12/9/08, 10:40 AM150


17. π“¬≈ÕÕ ‡æ‘Ë¡ ¡∫—μ‘ ‚√ß‡√’¬π∫¥‘π∑√‡¥™“ ( ‘ßÀå ‘ßÀ‡ π’)

”π—°ß“π‡¢μæ◊Èπ∑’Ë°“√»÷°…“°√ÿß‡∑æ¡À“π§√ ‡¢μ 2

18. π“¬∫—≥±‘μ ΩÕ¬∑Õß ‚√ß‡√’¬π‡μ√’¬¡Õÿ¥¡»÷°…“

”π—°ß“π‡¢μæ◊Èπ∑’Ë°“√»÷°…“°√ÿß‡∑æ¡À“π§√ ‡¢μ 1

19. π“ß≈–‡¡’¬¥ °√∫ß°™¡“» ‚√ß‡√’¬π «π°ÿÀ≈“∫«‘∑¬“≈—¬

”π—°ß“π‡¢μæ◊Èπ∑’Ë°“√»÷°…“°√ÿß‡∑æ¡À“π§√ ‡¢μ 1

20. π“ß«√√≥«‘¿“ ÿ∑∏‡°’¬√μ‘ ‚√ß‡√’¬π “¡‡ π«‘∑¬“≈—¬

”π—°ß“π‡¢μæ◊Èπ∑’Ë°“√»÷°…“°√ÿß‡∑æ¡À“π§√ ‡¢μ 1

21. π“¬‰¡μ√’ »√’∑Õß·∑â ‚√ß‡√’¬π‡μ√’¬¡Õÿ¥¡»÷°…“

”π—°ß“π‡¢μæ◊Èπ∑’Ë°“√»÷°…“°√ÿß‡∑æ¡À“π§√ ‡¢μ 1

22. π“ß√—™π’ π“§π§√ ‚√ß‡√’¬πæ√À¡¡“πÿ √≥å ®—ßÀ«—¥‡æ™√∫ÿ√’

”π—°ß“π‡¢μæ◊Èπ∑’Ë°“√»÷°…“‡æ™√∫ÿ√’ ‡¢μ 1

23. π“ß —ß‡«’¬π Õ‘π∑√ª√– ß§å ‚√ß‡√’¬π«—¥¥à“π ”‚√ß

”π—°ß“π‡¢μæ◊Èπ∑’Ë°“√»÷°…“ ¡ÿ∑√ª√“°“√ ‡¢μ 1

24. π“¬«‘ ÿ∑∏‘Ï §ß°—≈ªá ‚√ß‡√’¬π§«π‡π’¬ß«‘∑¬“

”π—°ß“π‡¢μæ◊Èπ∑’Ë°“√»÷°…“ ß¢≈“ ‡¢μ 2

25. π“ß “«æ√æ√√≥ Õ‘π∑√ª√–‡ √‘∞ ºŸâÕ”π«¬°“√‚√ß‡√’¬π«—¥∑“ßÀ≈«ß‚æ∏‘Ï∑Õß

”π—°ß“π‡¢μæ◊Èπ∑’Ë°“√»÷°…“ππ∑∫ÿ√’ ‡¢μ 1

√Ÿª‡≈à¡

1. π“ß “« “¬æ‘≥ Ÿ≠¬’Ë¢—π æπ—°ß“πæ‘¡æå¥’¥ ”π—°æ—≤π“π«—μ°√√¡°“√®—¥°“√»÷°…“

2. π“ß “«æ—∑¬“ ∑‘»‡πμ√ æπ—°ß“πæ‘¡æå¥’¥ ”π—°æ—≤π“π«—μ°√√¡°“√®—¥°“√»÷°…“

ª°

1. π“¬ª√–¡ÿ¢ ∫ÿ≠ ‘√‘ √ÕßºŸâÕ”π«¬°“√‚√ß‡√’¬π«—¥· πμÕ

”π—°ß“π‡¢μæ◊Èπ∑’Ë°“√»÷°…“°“≠®π∫ÿ√’ ‡¢μ 2

2. π“ß»√’‡¡◊Õß ∫ÿ≠·æ∑¬å ‚√ß‡√’¬πÕπÿ∫“≈μ“°øÑ“

”π—°ß“π‡¢μæ◊Èπ∑’Ë°“√»÷°…“π§√ «√√§å ‡¢μ 3

3. π“¬∏√√¡√—μπå ∫ÿ≠·æ∑¬å ‚√ß‡√’¬π»√’π¿“‡¢μ«‘∑¬“

”π—°ß“π‡¢μæ◊Èπ∑’Ë°“√»÷°…“π§√ «√√§å ‡¢μ 3

A 119-152 12/9/08, 10:40 AM151


ºŸâ√—∫º‘¥™Õ∫‚§√ß°“√æ—≤π“§ÿ≥¿“æ°“√‡√’¬π√Ÿâ Ÿà “°≈

1. π“ßπ‘®«¥’ ‡®√‘≠‡°’¬√μ‘∫«√ π—°«‘™“°“√»÷°…“ ”π—°æ—≤π“π«—μ°√√¡°“√®—¥°“√»÷°…“

2. π“ßÕ√πÿ™ ¡—Ëß¡’ ÿ¢»‘√‘ π—°«‘™“°“√»÷°…“ ”π—°æ—≤π“π«—μ°√√¡°“√®—¥°“√»÷°…“

3. π“ß “««√πÿ™ √ÿàß‡√◊Õß‡®√‘≠°ÿ≈ ‡®â“æπ—°ß“π∏ÿ√°“√ ”π—°æ—≤π“π«—μ°√√¡°“√®—¥°“√»÷°…“

∫√√≥“∏‘°“√

1. π“¬ª√“‚¡∑¬å ¢®√¿—¬ »÷°…“π‘‡∑»°å

”π—°ß“π‡¢μæ◊Èπ∑’Ë°“√»÷°…“°√ÿß‡∑æ¡À“π§√ ‡¢μ 1

2. π“ßπ‘®«¥’ ‡®√‘≠‡°’¬√μ‘∫«√ π—°«‘™“°“√»÷°…“ ”π—°æ—≤π“π«—μ°√√¡°“√®—¥°“√»÷°…“

3. π“ßÕ√πÿ™ ¡—Ëß¡’ ÿ¢»‘√‘ π—°«‘™“°“√»÷°…“ ”π—°æ—≤π“π«—μ°√√¡°“√®—¥°“√»÷°…“

4. π“ß«√√≥«‘¿“ ÿ∑∏‡°’¬√μ‘ ‚√ß‡√’¬π “¡‡ π«‘∑¬“≈—¬

”π—°ß“π‡¢μæ◊Èπ∑’Ë°“√»÷°…“°√ÿß‡∑æ¡À“π§√ ‡¢μ 1

5. π“ß√—™π’ π“§π§√ ‚√ß‡√’¬πæ√À¡¡“πÿ √≥å ®—ßÀ«—¥‡æ™√∫ÿ√’

”π—°ß“π‡¢μæ◊Èπ∑’Ë°“√»÷°…“‡æ™√∫ÿ√’ ‡¢μ 1

A 119-152 12/9/08, 10:40 AM152


A 119-152 12/9/08, 10:41 AM153

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