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CON 12/9/08, 10:42 AM1
‡ √‘¡§‘¥§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550)
‡Õ° “√ π°.2/2551‚§√ß°“√æ—≤π“§ÿ≥¿“æ°“√‡√’¬π√Ÿâ Ÿà “°≈ ”π—°æ—≤π“π«—μ°√√¡°“√®—¥°“√»÷°…“ ”π—°ß“π§≥–°√√¡°“√°“√»÷°…“¢—Èπæ◊Èπ∞“π°√–∑√«ß»÷°…“∏‘°“√
CON 12/9/08, 10:42 AM2
™◊ËÕÀπ—ß ◊Õ : ‡ √‘¡§‘¥§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3
μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550)
ºŸâ√«∫√«¡‡√’¬∫‡√’¬ß : 𓬪√“‚¡∑¬å ¢®√¿—¬
: π“ßπ‘®«¥’ ‡®√‘≠‡°’¬√μ‘∫«√
: π“ßÕ√πÿ™ ¡—Ëß¡’ ÿ¢»‘√‘
: π“ß√—™π’ π“§π§√
: π“ß«√√≥«‘¿“ ÿ∑∏‡°’¬√μ‘
ISBN : 978-974-650-862-9
: ≈‘¢ ‘∑∏‘Ï ”π—°ß“π§≥–°√√¡°“√°“√»÷°…“¢—Èπæ◊Èπ∞“π
æ‘¡æå§√—Èß∑’Ë 1 : ¡‘∂ÿπ“¬π æ.». 2551
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æ‘¡æå∑’Ë : ‚√ßæ‘¡æå™ÿ¡πÿ¡ À°√≥å°“√‡°…μ√·Ààߪ√–‡∑»‰∑¬ ®”°—¥
79 ∂ππß“¡«ß»å«“π ·¢«ß≈“¥¬“« ‡¢μ®μÿ®—°√ °√ÿ߇∑æ¡À“π§√ 10900
‚∑√. 0-2561-4567 ‚∑√ “√ 0-2579-5101
𓬂™§¥’ ÕÕ ÿ«√√≥ ºâŸæ‘¡æåºâŸ‚¶…≥“ æ.». 2550
‚§√ß°“√æ—≤π“§ÿ≥¿“æ°“√‡√’¬π√Ÿâ Ÿà “°≈ ”π—°æ—≤π“π«—μ°√√¡°“√®—¥°“√»÷°…“ ”π—°ß“π§≥–°√√¡°“√°“√»÷°…“¢—Èπæ◊Èπ∞“π‚∑√. 0 2280 5562Website : http://www.inno.obec.go.th
: http://www.ilg.netE-mail : [email protected]
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¢—Èπæ◊Èπ∞“π ‚¥¬„™â°√–∫«π°“√·¢àߢ—π∑“ß«‘™“°“√ °√–∫«π°“√«‘®—¬æ—≤π“§ÿ≥¿“æ°“√‡√’¬π√Ÿâ
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¢—Èπæ◊Èπ∞“𠉥â√—∫‡™‘≠®“°°√–∑√«ß Àπ૬ߓπ∑“ß°“√»÷°…“μà“ߪ√–‡∑» „π°“√æ‘®“√≥“ àßπ—°‡√’¬π
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·≈–μ—«Õ¬à“ß·∫∫∑¥ Õ∫«‘∑¬“»“ μ√å „π°“√·¢àߢ—π∑“ß«‘™“°“√ ªï æ.». 2549-2550) ‡ªìπº≈º≈‘μ
®“°°“√»÷°…“«‘‡§√“–Àå‡π◊ÈÕÀ“ “√–°“√·¢àߢ—π∑“ß«‘™“°“√ √–¥—∫π“π“™“μ‘ ‡πâπ¥â“π°√–∫«π°“√∑“ß
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ªï æ.». 2549-2550 ‡Õ° “√™ÿ¥π’ȇº¬·æ√à‡ªìπ·π«∑“ßÀπ÷Ëß„π°“√æ—≤π“§ÿ≥¿“æ°“√‡√’¬π√Ÿâ·≈–
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„π°“√·¢àߢ—π∑“ß«‘™“°“√ ªï æ.». 2549 1
ë μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3
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ë 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 9
ë μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3
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ë μ—«Õ¬à“ß·π«§‘¥ ·∫∫∑¥ Õ∫§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3
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μ—«Õ¬à“ß·∫∫∑¥ Õ∫·≈–·π«§‘¥·∫∫∑¥ Õ∫§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3
„π°“√·¢àߢ—π∑“ß«‘™“°“√ ªï æ.». 2550 59
ë μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3
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ë μ—«Õ¬à“ß·π«§‘¥·∫∫∑¥ Õ∫§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3
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ë μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 (©∫—∫∑’Ë 1)
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ë 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 95
ë μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 (©∫—∫∑’Ë 2)
‡æ◊ËÕ°“√§—¥‡≈◊Õ°π—°‡√’¬π√–¥—∫ª√–‡∑» ªï æ.». 2550 119
ë μ—«Õ¬à“ß·π«§‘¥·∫∫∑¥ Õ∫§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 (©∫—∫∑’Ë 2)
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‰¥â®—¥∑”‡Õ° “√™ÿ¥‡ √‘¡§‘¥§≥‘μ»“ μ√å·≈–‡ √‘¡§‘¥«‘∑¬“»“ μ√å μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å
·≈–μ—«Õ¬à“ß·∫∫∑¥ Õ∫«‘∑¬“»“ μ√å ∑’Ë„™â„π°“√·¢àߢ—π∑“ß«‘™“°“√ ªï æ.». 2549-2550 ‡ªìπº≈º≈‘μ
®“°°“√¥”‡π‘πß“π‚§√ß°“√æ—≤π“§ÿ≥¿“æ°“√‡√’¬π√Ÿâ Ÿà “°≈ °‘®°√√¡°“√·¢àߢ—π∑“ß«‘™“°“√ √–¥—∫π“π“™“μ‘
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·≈–«‘∑¬“»“ μ√å ‡ªìπ°“√‡æ‘Ë¡¢’¥§«“¡ “¡“√∂¥â“π°√–∫«π§‘¥ °“√·°âªí≠À“ μ≈Õ¥®π “¡“√∂π” ‘Ëß∑’Ë
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‡≈à¡∑’Ë 1 ‡ √‘¡§‘¥§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 2
μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550)
‡≈à¡∑’Ë 2 ‡ √‘¡§‘¥§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3
μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550)
‡≈à¡∑’Ë 3 ‡ √‘¡§‘¥«‘∑¬“»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 2
μ—«Õ¬à“ß·∫∫∑¥ Õ∫«‘∑¬“»“ μ√å (ªï æ.». 2549-2550)
‡Õ° “√‡ √‘¡§‘¥§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ
∑’Ë 3 (ªï æ.». 2549-2550) ‡ªìπ°“√𔇠πÕ«‘∏’°“√ ¬ÿ∑∏»“ μ√å°“√·°â‚®∑¬åªí≠À“§≥‘μ»“ μ√å ‚¥¬„™â‡ªìπ
æ‘¡æ凢’¬«„π°“√ √â“ß·∫∫∑¥ Õ∫∑’Ë„™â„π°“√·¢àߢ—π∑“ß«‘™“°“√√Õ∫√–¥—∫‡¢μæ◊Èπ∑’Ë°“√»÷°…“·≈–
√Õ∫√–¥—∫ª√–‡∑» ª√–®”ªï æ.». 2549-2550 ‡æ◊ËÕ§—¥‡≈◊Õ°π—°‡√’¬π∑’Ë¡’§«“¡ “¡“√∂∑“ߧ≥‘μ»“ μ√å√–¥—∫
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μ—«Õ¬à“ß·∫∫∑¥ Õ∫·≈–·π«§‘¥·∫∫∑¥ Õ∫
§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3„π°“√·¢àߢ—π∑“ß«‘™“°“√ ªï æ.». 2549
‚¥¬ ”π—°ß“π§≥–°√√¡°“√°“√»÷°…“¢—Èπæ◊Èπ∞“π
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2 ✎ ‡ √‘¡§‘¥§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550)
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‡ √‘¡§‘¥§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550) ✎ 3
μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3
‡æ◊ËÕ°“√§—¥‡≈◊Õ°π—°‡√’¬π√–¥—∫‡¢μæ◊Èπ∑’Ë°“√»÷°…“ªï æ.». 2549
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1. ·∫∫∑¥ Õ∫©∫—∫π’ȇªìπ·∫∫∑¥ Õ∫™π‘¥‡μ‘¡§”μÕ∫
2. ·∫∫∑¥ Õ∫©∫—∫π’È¡’®”π«π 35 ¢âÕ ·∫à߇ªìπ 2 μÕπ §–·ππ‡μÁ¡ 100 §–·ππ
μÕπ∑’Ë 1 ¡’®”π«π¢âÕ Õ∫ 20 ¢âÕ §◊Õ ¢âÕ 1-20 ¢âÕ≈– 2 §–·ππ √«¡ 40 §–·ππ
μÕπ∑’Ë 2 ¡’®”π«π¢âÕ Õ∫ 15 ¢âÕ §◊Õ ¢âÕ 21-35 ¢âÕ≈– 4 §–·ππ √«¡ 60 §–·ππ
3. ‡«≈“∑’Ë„™â„π°“√ Õ∫ „™â‡«≈“ 2 ™—Ë«‚¡ß
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μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3
‡æ◊ËÕ°“√§—¥‡≈◊Õ°π—°‡√’¬π√–¥—∫‡¢μæ◊Èπ∑’Ë°“√»÷°…“ ªï æ.». 2549
μÕπ∑’Ë 1 ¡’®”π«π 20 ¢âÕ §◊Õ ¢âÕ 1-20 ¢âÕ≈– 2 §–·ππ
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2. (2 + 3)2 (2 › 3)3 + (2 › 3)2 (2 + 3)3 ¡’§à“‡∑à“„¥
3. ªí®®ÿ∫—π‰¡μ√’Õ“¬ÿ 42 ªï ∂Ⓣ¡μ√’¡’Õ“¬ÿ‡∑à“°—∫¡πμ√’„πªí®®ÿ∫—π ·≈â«¡πμ√’®–¡’Õ“¬ÿ
‡ªìπ§√÷ËßÀπ÷ËߢÕ߉¡μ√’ ¥—ßπ—Èπ ªí®®ÿ∫—π¡πμ√’¡’Õ“¬ÿ°’˪ï
4. ¡’π—°‡√’¬π 101 §π §√Ÿ§π∑’Ë 1 ¡’¢Õ߇≈àπ 75 ™‘Èπ ·®°„Àâπ—°‡√’¬π§π≈– 1 ™‘Èπ ‚¥¬·®°
‡√’¬ß≈”¥—∫®“°§π∑’Ë 1 ®“°´â“¬¡◊Õ‰ª¢«“¡◊Õ §√Ÿ§π∑’Ë 2 ¡’¢Õ߇≈àπ 51 ™‘Èπ ·®°„Àâπ—°‡√’¬π§π≈–
1 ™‘Èπ ‚¥¬·®°‡√’¬ß≈”¥—∫®“°§π∑’Ë 101 ®“°¢«“¡◊Õ¡“´â“¬¡◊Õ §√Ÿ§π∑’Ë 3 ¡’¢Õ߇≈àπ 45 ™‘Èπ
·®°„Àâπ—°‡√’¬π§π≈– 1 ™‘Èπ ‚¥¬·®°‡√’¬ß≈”¥—∫‡√‘Ë¡®“°§π∑’Ë 40 ®“°´â“¬¡◊Õ‰ª¢«“¡◊Õ
¡’π—°‡√’¬π°’˧π∑’ˉ¥â√—∫¢Õ߇≈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
·≈⫺≈§Ÿ≥¢Õߧ”μÕ∫∑—ÈßÀ¡¥¢Õß ¡°“√π’ȇªìπ‡∑à“„¥
8. + + +...+ ¡’§à“‡∑à“„¥12 + 1
13 + 2
14 + 3( )1
16 + 152
A 1-29 12/9/08, 10:23 AM4
‡ √‘¡§‘¥§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550) ✎ 5
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 Àπ૬
AC = 7 Àπ૬ AD = 4 Àπ૬ ·≈â«
√Ÿª “¡‡À≈’ˬ¡ ABC ¡’æ◊Èπ∑’Ë°’Ëμ“√“ßÀπ૬
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
·≈⫇À≈◊Õ‡»… p ‡∑à“°—π·≈â« m2 + p2 ¡’§à“‡∑à“„¥
16. æ“√“‚∫≈“∑’˺à“π®ÿ¥°”‡π‘¥·≈–ºà“π®ÿ¥ (1, 12) ·≈– (3, 6) ¡’®ÿ¥¬Õ¥§◊Õ®ÿ¥„¥
17. 35 ‡¡◊ËÕ‡ª≈’ˬπ‡ªìπ‡≈¢∞“πÀ°·≈â« μ—«‡≈¢ ÕßÀ≈—° ÿ¥∑⓬‡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
6 ✎ ‡ √‘¡§‘¥§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550)
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“߉¥â°’Ë®”π«π
20. ∂â“ m ·≈– n ‡ªìπ®”π«π‡μÁ¡∫«°´÷Ëß m2 › n4 = 19 ·≈â« m2 + n4 ¡’§à“‡∑à“„¥
μÕπ∑’Ë 2 ¡’®”π«π 15 ¢âÕ §◊Õ ¢âÕ 21-35 ¢âÕ≈– 4 §–·ππ
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≈’ˬ¡„¥ Ê ∑’Ë¡’æ◊Èπ∑’Ë 24 3 μ“√“ßÀπ૬ ∂â“ a ‡ªì𧫓¡¬“«
¥â“πμ√ߢⓡ¡ÿ¡ A, b ‡ªì𧫓¡¬“«¥â“πμ√ߢⓡ¡ÿ¡ B ·≈– c ‡ªì𧫓¡¬“«¥â“πμ√ߢⓡ¡ÿ¡ C
·≈– a + b = 20 Àπ૬ c = 16 Àπ૬ ·≈â« a › b ¡’§à“‡∑à“„¥
24. 𑬓¡ n! = 1 Ó 2 Ó 3 Ó 4 Ó ... Ó n ‡¡◊ËÕ n ‡ªìπ®”π«π‡μÁ¡
∂Ⓡ¢’¬π 20! „π√Ÿª A Ó 10n ‡¡◊ËÕ A ‡ªìπ®”π«π‡μÁ¡·≈â« n ¡’§à“‡∑à“„¥
25. º≈√«¡¢Õ߇≈¢‚¥¥ ´÷Ë߇ªìπº≈≈—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
‡ √‘¡§‘¥§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550) ✎ 7
26. ∂â“ x1 = 2549, x
2 = , x
3 = , x
4 =
x5 = , x
6 = , x
7 = , x
8 = ·≈⫺≈§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≈’ˬ¡¡’ AB = 20 Àπ૬ AC = 30 Àπ૬ ¡ÿ¡ BAC = 120 Õß»“
∂â“ D ‡ªìπ®ÿ¥∫π BC ∑”„Àâ AD ·∫àߧ√÷Ëß¡ÿ¡ BAC ·≈â« AD ¬“«°’ËÀπ૬
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“°—∫°’ËÀπ૬
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
8 ✎ ‡ √‘¡§‘¥§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550)
33. √ŸªÀ°‡À≈’ˬ¡¡ÿ¡‡∑à“ ´÷Ëß¡’§«“¡¬“« 4 ¥â“π ∑’ˇ√’¬ßμ‘¥μàÕ°—𠇪ìπ 6, 7, 8, 9 Àπ૬
®–¡’§«“¡¬“«√Õ∫√Ÿª°’ËÀπ૬
34. °”Àπ¥ ABC ‡ªìπ√Ÿª “¡‡À≈’ˬ¡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≈’ˬ¡ ¡’ P ‡ªìπ®ÿ¥¿“¬„π ≈“° AP æ∫ BC ∑’Ë D ≈“° BP
æ∫ CA ∑’Ë E ≈“° CP æ∫ AB ∑’Ë F ∑”„Àâ PD = PE = PF = 4 Àπ૬
∂â“ AP + BP + CP = 6 Àπ૬·≈â« (AP) Ó (BP) Ó (CP) ¡’§à“‡∑à“„¥
A 1-29 12/9/08, 10:23 AM8
‡ √‘¡§‘¥§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550) ✎ 9
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’ȇªìπ·∫∫∑¥ Õ∫™π‘¥‡μ‘¡§”μÕ∫
2. ·∫∫∑¥ Õ∫©∫—∫π’È¡’®”π«π 35 ¢âÕ ·∫à߇ªìπ 2 μÕπ §–·ππ‡μÁ¡ 100 §–·ππ
μÕπ∑’Ë 1 ¡’®”π«π¢âÕ Õ∫ 20 ¢âÕ §◊Õ ¢âÕ 1-20 ¢âÕ≈– 2 §–·ππ √«¡ 40 §–·ππ
μÕπ∑’Ë 2 ¡’®”π«π¢âÕ Õ∫ 15 ¢âÕ §◊Õ ¢âÕ 21-35 ¢âÕ≈– 4 §–·ππ √«¡ 60 §–·ππ
3. ‡«≈“∑’Ë„™â„π°“√ Õ∫ „™â‡«≈“ 2 ™—Ë«‚¡ß
A 1-29 12/9/08, 10:23 AM9
10 ✎ ‡ √‘¡§‘¥§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550)
μÕπ∑’Ë 1 ¡’®”π«π 20 ¢âÕ §◊Õ ¢âÕ 1-20 ¢âÕ≈– 2 §–·ππ
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
‡ √‘¡§‘¥§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550) ✎ 11
3. ªí®®ÿ∫—π‰¡μ√’Õ“¬ÿ 42 ªï ∂Ⓣ¡μ√’¡’Õ“¬ÿ‡∑à“°—∫¡πμ√’„πªí®®ÿ∫—π ·≈â«¡πμ√’®–¡’Õ“¬ÿ‡ªìπ
§√÷ËßÀπ÷ËߢÕ߉¡μ√’ ¥—ßπ—Èπ ªí®®ÿ∫—π¡πμ√’¡’Õ“¬ÿ°’˪ï
·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“ߢÕßÕ“¬ÿ§ß∑’Ë
42 › x = x ›
42 = x
x = 28
μÕ∫ 28 ªï
x2
x2
4. ¡’π—°‡√’¬π 101 §π §√Ÿ§π∑’Ë 1 ¡’¢Õ߇≈àπ 75 ™‘Èπ ·®°„Àâπ—°‡√’¬π§π≈– 1 ™‘Èπ ‚¥¬·®°
‡√’¬ß≈”¥—∫ ®“°§π∑’Ë 1 ®“°´â“¬¡◊Õ‰ª¢«“¡◊Õ §√Ÿ§π∑’Ë 2 ¡’¢Õ߇≈àπ 51 ™‘Èπ ·®°„Àâπ—°‡√’¬π§π≈–
1 ™‘Èπ ‚¥¬·®°‡√’¬ß≈”¥—∫®“°§π∑’Ë 101 ®“°¢«“¡◊Õ¡“´â“¬¡◊Õ §√Ÿ§π∑’Ë 3 ¡’¢Õ߇≈àπ 45 ™‘Èπ
·®°„Àâπ—°‡√’¬π§π≈– 1 ™‘Èπ ‚¥¬·®°‡√’¬ß≈”¥—∫‡√‘Ë¡®“°§π∑’Ë 40 ®“°´â“¬¡◊Õ‰ª¢«“¡◊Õ
¡’π—°‡√’¬π°’˧π∑’ˉ¥â√—∫¢Õ߇≈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∑’ˉ¥â√—∫¢Õ߇≈àπ 3 ™‘Èπ §◊Õ§π∑’Ë 50 ∂÷ߧπ∑’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
12 ✎ ‡ √‘¡§‘¥§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550)
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
·≈⫺≈§Ÿ≥¢Õߧ”μÕ∫∑—ÈßÀ¡¥¢Õß ¡°“√π’ȇªìπ‡∑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≥¢Õߧ”μÕ∫∑—ÈßÀ¡¥‡∑à“°—∫ 0μÕ∫ 0
43
A 1-29 12/9/08, 10:23 AM12
‡ √‘¡§‘¥§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550) ✎ 13
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
14 ✎ ‡ √‘¡§‘¥§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550)
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
‡ √‘¡§‘¥§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550) ✎ 15
12. ®“°√Ÿª∂â“ AB = 3 Àπ૬
AC = 7 Àπ૬ AD = 4 Àπ૬ ·≈â«
√Ÿª “¡‡À≈’ˬ¡ ABC ¡’æ◊Èπ∑’Ë°’Ëμ“√“ßÀπ૬
·π«§‘¥ μàÕ AD ‰ª∂÷ß®ÿ¥ D′ ∑”„Àâ CD′ = AB = 3 Àπ૬
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“√“ßÀπ૬
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
16 ✎ ‡ √‘¡§‘¥§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550)
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
‡ √‘¡§‘¥§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550) ✎ 17
15. „Àâ m ‡ªìπ®”π«π‡μÁ¡∫«°·≈– p ‡ªìπ®”π«π‡©æ“–∫«° ∂â“ m À“√ 777 ·≈– 910
·≈⫇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“√“‚∫≈“∑’˺à“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
18 ✎ ‡ √‘¡§‘¥§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550)
17. 35 ‡¡◊ËÕ‡ª≈’ˬπ‡ªìπ‡≈¢∞“πÀ°·≈â« μ—«‡≈¢ ÕßÀ≈—° ÿ¥∑⓬‡ªìπ‡∑à“„¥
·π«§‘¥
35 = 243
243 = 1.63 + 0.62 + 4.61 + 3.60 = (1,043)6
¥—ßπ—Èπ 35 ‡ª≈’ˬπ‡ªìπ‡≈¢∞“πÀ°·≈â« μ—«‡≈¢ ÕßÀ≈—° ÿ¥∑⓬‡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“߉¥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“߉¥â = 1,432 ®”π«π
μÕ∫ 1,432 ®”π«π
{
¡’¥’°√’¡“°°«à“ 2
A 1-29 12/9/08, 10:23 AM18
‡ √‘¡§‘¥§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550) ✎ 19
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
20 ✎ ‡ √‘¡§‘¥§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550)
μÕπ∑’Ë 2 ¡’®”π«π 15 ¢âÕ §◊Õ ¢âÕ 21-35 ¢âÕ≈– 4 §–·ππ
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
‡ √‘¡§‘¥§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550) ✎ 21
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¥∑⓬¢Õß·∂«∑’Ë 1 §◊Õ 12 = 1
μ—« ÿ¥∑⓬¢Õß·∂«∑’Ë 2 §◊Õ 22 = 4
μ—« ÿ¥∑⓬¢Õß·∂«∑’Ë’ 3 §◊Õ 32 = 9
μ—« ÿ¥∑⓬¢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
22 ✎ ‡ √‘¡§‘¥§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550)
23.°”Àπ¥ ABC ‡ªìπ√Ÿª “¡‡À≈’ˬ¡„¥ Ê ∑’Ë¡’æ◊Èπ∑’Ë 24 3 μ“√“ßÀπ૬ ∂â“ a ‡ªì𧫓¡¬“«
¥â“πμ√ߢⓡ¡ÿ¡ A, b ‡ªì𧫓¡¬“«¥â“πμ√ߢⓡ¡ÿ¡ B ·≈– c ‡ªì𧫓¡¬“«¥â“πμ√ߢⓡ¡ÿ¡ C
·≈– a + b = 20 Àπ૬ c = 16 Àπ૬ ·≈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.𑬓¡ n! = 1 Ó 2 Ó 3 Ó 4 Ó ... Ó n ‡¡◊ËÕ n ‡ªìπ®”π«π‡μÁ¡
∂Ⓡ¢’¬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
‡ √‘¡§‘¥§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550) ✎ 23
25.º≈√«¡¢Õ߇≈¢‚¥¥ ´÷Ë߇ªìπº≈≈—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≈√«¡¢Õ߇≈¢‚¥¥ §◊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 = ·≈⫺≈§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
24 ✎ ‡ √‘¡§‘¥§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550)
29.ABC ‡ªìπ√Ÿª “¡‡À≈’ˬ¡ ¡’ AB = 20 Àπ૬ AC = 30 Àπ૬ ¡ÿ¡ BAC = 120 Õß»“
∂â“ D ‡ªìπ®ÿ¥∫π BC ∑”„Àâ AD ·∫àߧ√÷Ëß¡ÿ¡ BAC ·≈â« AD ¬“«°’ËÀπ૬
·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
‡ √‘¡§‘¥§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550) ✎ 25
( )
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 Àπ૬
μÕ∫ 12 Àπ૬
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
26 ✎ ‡ √‘¡§‘¥§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550)
¥—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“°—∫°’ËÀπ૬
·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
‡ √‘¡§‘¥§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550) ✎ 27
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 Àπ૬
μÕ∫ 8 2 Àπ૬
33.√ŸªÀ°‡À≈’ˬ¡¡ÿ¡‡∑à“ ´÷Ëß¡’§«“¡¬“« 4 ¥â“π∑’ˇ√’¬ßμ‘¥μàÕ°—𠇪ìπ 6, 7, 8, 9 Àπ૬
®–¡’§«“¡¬“«√Õ∫√Ÿª°’ËÀπ૬
·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
28 ✎ ‡ √‘¡§‘¥§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550)
ABC ‡ªìπ√Ÿª “¡‡À≈’ˬ¡¥â“π‡∑à“¡’ BC = 21
BA = 21 ∴ y = 4
AC = 21, x + y + 6 = 21
x = 11
§«“¡¬“«√Õ∫√ŸªÀ°‡À≈’ˬ¡ = 6 + 7 + 8 + 9 + 4 + 11 Àπ૬
= 45 Àπ૬
μÕ∫ 45 Àπ૬
34.°”Àπ¥ ABC ‡ªìπ√Ÿª “¡‡À≈’ˬ¡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≈’ˬ¡¥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
‡ √‘¡§‘¥§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550) ✎ 29
35.ABC ‡ªìπ√Ÿª “¡‡À≈’ˬ¡ ¡’ P ‡ªìπ®ÿ¥¿“¬„π ≈“° AP æ∫ BC ∑’Ë D ≈“° BP æ∫ CA
∑’Ë E ≈“° CP æ∫ AB ∑’Ë F ∑”„Àâ PD = PE = PF = 4 Àπ૬ ∂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 = + +
䴉 (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»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550) ✎ 31
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’ȇªìπ·∫∫∑¥ Õ∫™π‘¥‡μ‘¡§”μÕ∫
2. ·∫∫∑¥ Õ∫©∫—∫π’È¡’®”π«π 30 ¢âÕ ·∫à߇ªìπ 2 μÕπ §–·ππ‡μÁ¡ 100 §–·ππ
μÕπ∑’Ë 1 ¡’®”π«π¢âÕ Õ∫ 20 ¢âÕ §◊Õ ¢âÕ 1-20 ¢âÕ≈– 3 §–·ππ √«¡ 60 §–·ππ
μÕπ∑’Ë 2 ¡’®”π«π¢âÕ Õ∫ 10 ¢âÕ §◊Õ ¢âÕ 21-30 ¢âÕ≈– 4 §–·ππ √«¡ 40 §–·ππ
3. ‡«≈“∑’Ë„™â„π°“√ Õ∫ „™â‡«≈“ 2 ™—Ë«‚¡ß
A 31-57 12/9/08, 10:25 AM31
32 ✎ ‡ √‘¡§‘¥§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550)
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’ȇªìπ‡∑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 ·≈⫺≈∫«°
¢Õß x ¡’§à“‡∑à“‰√
5. ∂Ⓡ¢’¬π®”π«π‡μÁ¡μ—Èß·μà 1 ∂÷ß 1000 μâÕ߇¢’¬πμ—«‡≈¢»Ÿπ¬å°’˧√—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 ’ˇÀ≈’ˬ¡º◊πºâ“ M, N ‡ªìπ®ÿ¥°÷Ëß°≈“ߢÕߥâ“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
‡ √‘¡§‘¥§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550) ✎ 33
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≈’ˬ¡√ŸªÀπ÷Ëß≈âÕ¡√Õ∫¥â«¬«ß°≈¡ ¥â“πÀπ÷ËߢÕß√Ÿª “¡‡À≈’ˬ¡π—È𬓫
12 ‡´π쑇¡μ√ ·≈–¡ÿ¡μ√ߢⓡ¥â“ππ—Èπ«—¥‰¥â 30 Ì «ß°≈¡π—Èπ¡’‡ âπºà“π»Ÿπ¬å°≈“߬“«‡∑à“„¥
8. ®ßÀ“®”π«π§ŸàÕ—π¥—∫ (x, y) ∑—ÈßÀ¡¥∑’ˇªìπ§”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≈’ˬ¡¡’ 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≈’ˬ¡„¥ Ê μàÕ¥â“π AB ·≈– AC ÕÕ°‰ª∑“ß B ·≈– C ∂÷ß D
·≈– E μ“¡≈”¥—∫ OB ·≈– OC ‡ªìπ‡ âπ·∫àߧ√÷Ëß¡ÿ¡ CBD ·≈–¡ÿ¡ BCE μ“¡≈”¥—∫
∂â“¡ÿ¡ BAE ¡’¢π“¥ 80 Ì®ßÀ“¢π“¥¢Õß BOC (∑’ˇªìπ¡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
34 ✎ ‡ √‘¡§‘¥§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550)
16. ®“°√Ÿª∂â“√Ÿª “¡‡À≈’ˬ¡¡ÿ¡©“°¡’ MB + MA = BC + AC ∂â“ BC = 8 Àπ૬ ·≈–
AC = 10 Àπ૬ ®ßÀ“ MB
17. ‡ âπμ√ß L1 ºà“π®ÿ¥ (›2, 0) ·≈– (›1, 2) L
2 ‡ªìπ‡ âπμ√ß∑’˺à“π®ÿ¥°”‡π‘¥·≈–μ—Èß©“°°—∫ L
1
·≈â«æ◊Èπ∑’Ë√Ÿª “¡‡À≈’ˬ¡∑’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¢Õ߇ 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≈’ˬ¡ OAB
μÕπ∑’Ë 2 ¡’ 10 ¢âÕ ¢âÕ≈– 4 §–·ππ
21. º≈∫«°¢Õߧ”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
‡ √‘¡§‘¥§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550) ✎ 35
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 ’ˇÀ≈’ˬ¡¥â“π¢π“π ®ÿ¥ E, F, G ·≈– H ‡ªìπ®ÿ¥°÷Ëß°≈“ߥâ“π AB, BC, CD ·≈– DA
μ“¡≈”¥—∫ ≈“° AG, CE, DF ·≈– BH μ—¥°—π‡°‘¥√Ÿª ’ˇÀ≈’ˬ¡¥â“π¢π“π MNPQ
´÷Ëß¡’®ÿ¥ R, S, T ·≈– U ‡ªìπ®ÿ¥°÷Ëß°≈“ߥâ“π MN, NP, PQ ·≈– QM μ“¡≈”¥—∫
∂â“ à«π∑’Ë·√‡ß“¡’æ◊Èπ∑’Ë 15 μ“√“ßÀπ૬ ®ßÀ“æ◊Èπ∑’Ë√Ÿª ’ˇÀ≈’ˬ¡ 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
36 ✎ ‡ √‘¡§‘¥§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550)
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°≈“ߢÕß«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≈’ˬ¡ ABC
30. √Ÿª “¡‡À≈’ˬ¡ ABC ‡ âπ·∫àߧ√÷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»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550) ✎ 37
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’ȇªìπ·∫∫∑¥ Õ∫™π‘¥‡μ‘¡§”μÕ∫
2. ·∫∫∑¥ Õ∫©∫—∫π’È¡’®”π«π 30 ¢âÕ ·∫à߇ªìπ 2 μÕπ §–·ππ‡μÁ¡ 100 §–·ππ
μÕπ∑’Ë 1 ¡’®”π«π¢âÕ Õ∫ 20 ¢âÕ §◊Õ ¢âÕ 1-20 ¢âÕ≈– 3 §–·ππ √«¡ 60 §–·ππ
μÕπ∑’Ë 2 ¡’®”π«π¢âÕ Õ∫ 10 ¢âÕ §◊Õ ¢âÕ 21-30 ¢âÕ≈– 4 §–·ππ √«¡ 40 §–·ππ
3. ‡«≈“∑’Ë„™â„π°“√ Õ∫ „™â‡«≈“ 2 ™—Ë«‚¡ß
A 31-57 12/9/08, 10:25 AM37
38 ✎ ‡ √‘¡§‘¥§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550)
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’ȇªìπ‡∑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
‡ √‘¡§‘¥§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550) ✎ 39
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 ·≈⫺≈∫«°
¢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
40 ✎ ‡ √‘¡§‘¥§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550)
( )
5. ∂Ⓡ¢’¬π®”π«π‡μÁ¡∫«°μ—Èß·μà 1 ∂÷ß 1,000 μâÕ߇¢’¬πμ—«‡≈¢»Ÿπ¬å°’˧√—Èß
·π«§‘¥
®”π«π 1 À≈—° ‰¡à¡’μ—«‡≈¢ 0 ‡¢’¬π 0 ‰¥â 0 §√—Èß
®”π«π 2 À≈—°
μ—«‡≈¢ 0 Õ¬ŸàÀ≈—°Àπ૬ ‡¢’¬π 0 ‰¥â 9 Ó 1 = 9 §√—Èß
®”π«π 3 À≈—° μ—«‡≈¢ 0 Õ¬ŸàÀ≈—°Àπ૬
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
‡ √‘¡§‘¥§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550) ✎ 41
7. ®“°√Ÿª ABCD ‡ªìπ√Ÿª ’ˇÀ≈’ˬ¡º◊πºâ“ M, N ‡ªìπ®ÿ¥°÷Ëß°≈“ߢÕߥâ“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
μÕ∫ 2 Àπ૬
∇
∇
8. ®ßÀ“®”π«π§ŸàÕ—π¥—∫ (x, y) ∑—ÈßÀ¡¥∑’ˇªìπ§”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
42 ✎ ‡ √‘¡§‘¥§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550)
(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 Õߢâ“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
‡ √‘¡§‘¥§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550) ✎ 43
( )( )
= 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≈’ˬ¡ ¡’ 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≈’ˬ¡¡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
44 ✎ ‡ √‘¡§‘¥§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550)
12. „Àâ ABC ‡ªìπ√Ÿª “¡‡À≈’ˬ¡„¥ Ê μàÕ¥â“π AB ·≈– AC ÕÕ°‰ª∑“ß .B ·≈– .C ∂÷ß
.D ·≈– .E μ“¡≈”¥—∫ OB ·≈– OC ‡ªìπ‡ âπ·∫àߧ√÷Ëß¡ÿ¡ CBD ·≈–¡ÿ¡ BCE μ“¡≈”¥—∫
∂â“¡ÿ¡ BAE ¡’¢π“¥ 80 Ì®ßÀ“¢π“¥¢Õß BOC (∑’ˇªìπ¡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
‡ √‘¡§‘¥§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550) ✎ 45
6
) (
15. √Ÿª “¡‡À≈’ˬ¡√ŸªÀπ÷Ëß≈âÕ¡√Õ∫¥â«¬«ß°≈¡ ¥â“πÀπ÷ËߢÕß√Ÿª “¡‡À≈’ˬ¡π—È𬓫
12 ‡´π쑇¡μ√ ·≈–¡ÿ¡μ√ߢⓡ¥â“ππ—Èπ«—¥‰¥â 30 Ì«ß°≈¡π—Èπ¡’‡ âπºà“π»Ÿπ¬å°≈“߬“«‡∑à“„¥
·π«§‘¥
„π ABC; Sin 30 Ì =
x =
= 12
‡ âπºà“π»Ÿπ¬å°≈“߬“« 24 ‡´π쑇¡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
46 ✎ ‡ √‘¡§‘¥§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550)
16. ®“°√Ÿª∂â“√Ÿª “¡‡À≈’ˬ¡¡ÿ¡©“°¡’ MB + MA = BC + AC
∂â“ BC = 8 Àπ૬ ·≈– AC = 10 Àπ૬ ®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
‡ √‘¡§‘¥§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550) ✎ 47
17. ‡ âπμ√ß L1 ºà“π®ÿ¥ (›2, 0) ·≈– (›1, 2) L
2 ‡ªìπ‡ âπμ√ß∑’˺à“π®ÿ¥°”‡π‘¥·≈–μ—Èß©“°°—∫ L
1
·≈â«æ◊Èπ∑’Ë√Ÿª “¡‡À≈’ˬ¡∑’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 ¡’∞“𬓫 2 Àπ૬ Ÿß Àπ૬
®–μâÕßÀ“æ◊Èπ∑’Ë OAB = Ó OB Ó §«“¡ Ÿß
= (2)( ) μ“√“ßÀπ૬
= 0.8 μ“√“ßÀπ૬
μÕ∫ 0.8 μ“√“ßÀπ૬
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
48 ✎ ‡ √‘¡§‘¥§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550)
®–‰¥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
‡ √‘¡§‘¥§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550) ✎ 49
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¢Õ߇ 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≈’ˬ¡ OAB
·π«§‘¥
æ◊Èπ∑’Ë Δ OAB = (OB)(OA)
À√◊Õæ◊Èπ∑’Ë Δ OAB = (AB)(OE)
OE = (√–¬– OE ®“°®ÿ¥ (0, 0) ‰ª¬—߇ 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“√“ßÀπ૬
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
50 ✎ ‡ √‘¡§‘¥§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550)
μÕπ∑’Ë 2 ¡’ 10 ¢âÕ ¢âÕ≈– 4 §–·ππ
21. º≈∫«°¢Õߧ”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≈∫«°¢Õߧ”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
‡ √‘¡§‘¥§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550) ✎ 51
∴ 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
52 ✎ ‡ √‘¡§‘¥§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550)
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°’˧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∑”πÕ߇¥’¬«°—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
‡ √‘¡§‘¥§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550) ✎ 53
26.
ABCD ‡ªìπ√Ÿª ’ˇÀ≈’ˬ¡¥â“π¢π“π ®ÿ¥ E, F, G ·≈– H ‡ªìπ®ÿ¥°÷Ëß°≈“ߥâ“π AB, BC, CD ·≈– DA
μ“¡≈”¥—∫ ≈“° AG, CE, DF ·≈– BH μ—¥°—π‡°‘¥√Ÿª ’ˇÀ≈’ˬ¡¥â“π¢π“π MNPQ
´÷Ëß¡’®ÿ¥ R, S, T ·≈– U ‡ªìπ®ÿ¥°÷Ëß°≈“ߥâ“π MN, NP, PQ ·≈– QM μ“¡≈”¥—∫
∂â“ à«π∑’Ë·√‡ß“¡’æ◊Èπ∑’Ë 15 μ“√“ßÀπ૬ ®ßÀ“æ◊Èπ∑’Ë√Ÿª ’ˇÀ≈’ˬ¡ ABCD
·π«§‘¥
æ◊Èπ∑’Ë√Ÿª MNPQ ‡ªìπ 5 ‡∑à“¢Õß√Ÿª·√‡ß“ = 5 Ó 15 μ“√“ßÀπ૬
æ◊Èπ∑’Ë√Ÿª ABCD ‡ªìπ 5 ‡∑à“¢Õßæ◊Èπ∑’Ë√Ÿª MNPQ = 5 Ó 5 Ó 15 μ“√“ßÀπ૬
μÕ∫ 375 μ“√“ßÀπ૬
QT
P
N
A
S
H
B
DG C
FM
R
U
E
A 31-57 12/9/08, 10:27 AM53
54 ✎ ‡ √‘¡§‘¥§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550)
( )
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
‡ √‘¡§‘¥§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550) ✎ 55
(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¬—߇ âπμ√ß 4x + 3y › 5 = 0
CD = = = 3
¥—ßπ—Èπ æ◊Èπ∑’Ë√Ÿª “¡‡À≈’ˬ¡ = Ó 3 Ó 6 = 9 (AB ‡ªìπ‡ âπºà“π»Ÿπ¬å°≈“߬“« 6 Àπ૬)
μÕ∫ 9
28. „Àâ L ‡ªìπ‡ âπμ√ß∑’Ë¡’§«“¡™—π‡ªìπ ºà“π®ÿ¥»Ÿπ¬å°≈“ߢÕß«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≈’ˬ¡ 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
56 ✎ ‡ √‘¡§‘¥§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550)
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
‡ √‘¡§‘¥§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550) ✎ 57
30. √Ÿª “¡‡À≈’ˬ¡ ABC ‡ âπ·∫àߧ√÷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≈’ˬ¡¡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°≈“ߢÕß AB
G ‡ªìπ®ÿ¥°÷Ëß°≈“ß BC ®–‰¥â BG = 14
„π Δ ABC, AD ·∫àߧ√÷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
μÕ∫ 1 Àπ૬
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°“√·¢àߢ—π∑“ß«‘™“°“√ ªï æ.». 2550
‚¥¬ ”π—°ß“π§≥–°√√¡°“√°“√»÷°…“¢—Èπæ◊Èπ∞“π
A 59-88 12/9/08, 10:28 AM59
‡ √‘¡§‘¥§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550) ✎ 61
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’ȇªìπ·∫∫‡μ‘¡§”μÕ∫
2. ·∫∫∑¥ Õ∫©∫—∫π’È¡’®”π«π 30 ¢âÕ ·∫à߇ªìπ 2 μÕπ §–·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 59-88 12/9/08, 10:28 AM61
62 ✎ ‡ √‘¡§‘¥§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550)
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’ȇªìπ·∫∫‡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 Õߢ“¥∑ÿπ 10%
∂â“¢“¬‰ª∑—Èß Õß™‘Èπ ·≈⫉¥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«πÀπⓉ¡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
‡ √‘¡§‘¥§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550) ✎ 63
9. ∂â“√Ÿª “¡‡À≈’ˬ¡√ŸªÀπ÷Ëß¡’®ÿ¥¬Õ¥ “¡®ÿ¥§◊Õ ®ÿ¥ (›5, 0), ®ÿ¥ (0, 0) ·≈–®ÿ¥¬Õ¥¢Õß°√“ø
y = x2 + 5x + 7 ·≈â«√Ÿª “¡‡À≈’ˬ¡√Ÿªπ’È¡’æ◊Èπ∑’Ë°’Ëμ“√“ßÀπ૬
10. π”√ŸªÀⓇÀ≈’ˬ¡¥â“π‡∑à“¡ÿ¡‡∑à“¡“μàÕ°—π¥—ß√Ÿª ‚¥¬„Àâ·μà≈–√Ÿª∑’Ë¡“μàÕ°—π¡’¥â“π√à«¡°—π
‡æ’¬ßÀπ÷Ëߥâ“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≈’ˬ¡¥â“π‡∑à“¡ÿ¡‡∑à“ ∂â“Õ—μ√“ à«π√–À«à“ßæ◊Èπ∑’Ë à«π∑’Ë
·√‡ß“°—∫æ◊Èπ∑’Ë√ŸªÀ°‡À≈’ˬ¡ 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“√“ßÀπ૬ ‚¥¬∑’Ë AB = BC, BE ⊥ AC ·≈– AD ⊥ BC
∂â“ AC = 4 Àπ૬ ·≈â« AD ¬“«‡∑à“°—∫°’ËÀπ૬
13. ÿà¡À¬‘∫ ≈“°∑’Ë¡’‡≈¢‚¥¥ 1 ∂÷ß 1000 ‡¢’¬π°”°—∫‰«â „∫≈– 1 ®”π«π ¡“ 1 „∫ ∂ⓧ«“¡πà“®–‡ªìπ
∑’ˉ¥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
64 ✎ ‡ √‘¡§‘¥§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550)
14. ·¥ß´âÕ¡¬‘ߪóπ ‚¥¬«“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«°’ˇ¡μ√μàÕ«‘π“∑’
15. √Ÿª ’ˇÀ≈’ˬ¡®—μÿ√— ¡’‡ âπ∑·¬ß¡ÿ¡¬“« 14 ‡´π쑇¡μ√ ∫√√®ÿ„π«ß°≈¡ ‚¥¬¡’®ÿ¥¬Õ¥
∑—Èß ’ËÕ¬Ÿà∫π‡ âπ√Õ∫«ß ·≈â«æ◊Èπ∑’Ë«ß°≈¡∑’ËÕ¬ŸàπÕ°√Ÿª ’ˇÀ≈’ˬ¡®—μÿ√— ‡ªìπ°’ˇ∑à“¢Õßæ◊Èπ∑’Ë
√Ÿª ’ˇÀ≈’ˬ¡®—μÿ√— (°”Àπ¥ π = )227
17. ®ßÀ“«à“®ÿ¥μ—¥¢Õß°√“ø¢Õß ¡°“√ x › 3y = 1 ·≈– 2x2 › 3xy › 20 = 0 ¡’√–¬–
Àà“ß°—π°’ËÀπ૬
18. ®“°√Ÿª °”Àπ¥ ABC ‡ªìπ√Ÿª “¡‡À≈’ˬ¡¡ÿ¡©“°∑’Ë¡’¡ÿ¡ BCA ‡ªìπ¡ÿ¡©“° BN ·≈– CM
‡ªìπ‡ âπ¡—∏¬∞“π ∂â“ BN ⊥ CM ·≈– BC ¬“« 6 Àπ૬ ·≈â« BN ¬“«°’ËÀπ૬
19. A ¢—∫√∂¬πμå¥â«¬§«“¡‡√Á« 80 °‘‚≈‡¡μ√μàÕ™—Ë«‚¡ß À≈—ß®“°ÕÕ°‡¥‘π∑“߉¥â 45 π“∑’
B ¢—∫√∂¬πμåÕÕ°®“°μ”·Àπà߇¥’¬«°—∫ A ¥â«¬§«“¡‡√Á« 120 °‘‚≈‡¡μ√μàÕ™—Ë«‚¡ß
‡ªìπ‡«≈“ 20 π“∑’ ®“°π—ÈπÀ¬ÿ¥æ—° 10 π“∑’ ·≈⫇¥‘π∑“ßμàե⫬§«“¡‡√Á« 90 °‘‚≈‡¡μ√μàÕ™—Ë«‚¡ß
®ßÀ“«à“ B ®–‡¥‘π∑“ß∑—π A ‡¡◊ËÕ A ‡¥‘π∑“߉¥â°’Ë™—Ë«‚¡ß °’Ëπ“∑’
20. ‡»… à«π®”π«πÀπ÷Ëß ‡¡◊ËÕπ” 2 ¡“∫«°‡≈¢‚¥¥¢Õßμ—«‡»…·≈–μ—« à«π ®–∑”„À⇻… à«ππ—Èπ‡ªìπ
·≈–‡¡◊ËÕπ” 4 ¡“≈∫‡≈¢‚¥¥¢Õßμ—«‡»…·≈–μ—« à«π ®–∑”„À⇻… 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“√“߇´π쑇¡μ√ ·μàπ”¡“¢¥
‡ªìπ√Ÿª ’ˇÀ≈’ˬ¡¡ÿ¡©“°‚¥¬„À⧫“¡¬“«¥â“π·μà≈–¥â“π‡ªìπ®”π«π‡μÁ¡∑’Ë¡“°°«à“ 30 ‡´π쑇¡μ√
·≈â«√Ÿª ’ˇÀ≈’ˬ¡¡ÿ¡©“°π’È¡’æ◊Èπ∑’Ë¡“°∑’Ë ÿ¥°’Ëμ“√“߇´π쑇¡μ√ (°”Àπ¥ π = )227
58
B
C
M A
N
O
A 59-88 12/9/08, 10:28 AM64
‡ √‘¡§‘¥§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550) ✎ 65
μÕπ∑’Ë 2
21. ®ßÀ“§«“¡¬“«¥â“π¢Õß√Ÿª “¡‡À≈’ˬ¡¥â“π‡∑à“∑’Ë¡’¢π“¥„À≠à∑’Ë ÿ¥ ∑’Ë∫√√®ÿÕ¬Ÿà„π√Ÿª ’ˇÀ≈’ˬ¡
®—μÿ√— ∑’Ë¡’§«“¡¬“«¥â“π 1 Àπ૬
22. ∂â“™“¬§πÀπ÷Ë߬◊πÕ¬Ÿà∫πæ◊Èπ√“∫∑’Ë®ÿ¥ A ®–¡Õ߇ÀÁπ¬Õ¥‡ “∏ߥ⫬¡ÿ¡‡ß¬ 15 Ì ·≈–∂Ⓡ¢“
‡¥‘π‡¢â“À“‡ “∏ß 50 ‡¡μ√ ‡¢“®–¡Õ߇ÀÁπ¬Õ¥‡ “∏ߥ⫬¡ÿ¡‡ß¬ 45 Ì ®ßÀ“§«“¡ Ÿß¢Õ߇ “∏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—∫ ·≈⫺≈∫«°¢Õß d
∑’ˇªìπ‰ª‰¥â∑—ÈßÀ¡¥¡’§à“‡∑à“„¥
24. ∂â“ ·≈â« A › B ¡’§à“‡∑à“„¥+ +
25. ∂â“ ABCD ‡ªìπ√Ÿª ’ˇÀ≈’ˬ¡®—μÿ√— ¬“«¥â“π≈– 28 Àπ૬ ‚¥¬¡’ 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 ·∫àߧ√÷Ëß¡ÿ¡ IBC ·≈â«æ◊Èπ∑’Ë¢Õß√Ÿª “¡‡À≈’ˬ¡ BMN ‡∑à“°—∫°’Ëμ“√“ßÀπ૬
26. ∂â“ 60a = 3 ·≈– 60b = 5 ·≈â« 122(1 › b) ¡’§à“‡∑à“„¥
27. °”Àπ¥ ABCD ‡ªìπ√Ÿª ’ˇÀ≈’ˬ¡®—μÿ√— ¬“«¥â“π≈– 900 Àπ૬ ‚¥¬∑’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 Àπ૬ ·≈–¡ÿ¡ EOF = 45 Ì ·≈â« AE ¬“«°’ËÀπ૬
28. ABCDEFGH ‡ªìπ√Ÿª·ª¥‡À≈’ˬ¡¡ÿ¡‡∑à“∑’Ë¡’¥â“𬓫 2, 2 2, 4, 4 2, 6, 7, 7
·≈– 8 Àπ૬ ®–¡’æ◊Èπ∑’Ë°’Ëμ“√“ßÀπ૬
1 › a › b
19x › 82x2 › x ›21
A2x › 7
Bx + 3
A 59-88 12/9/08, 10:29 AM65
66 ✎ ‡ √‘¡§‘¥§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550)
30. °”Àπ¥√Ÿª “¡‡À≈’ˬ¡ ABC ¡’®ÿ¥¬Õ¥ A(1, 6), B(›2, 3) ·≈– C(3, ›4) ∂Ⓡ≈◊ËÕπ®ÿ¥ A
¢π“π‰ª∑“ߢ«“ 3 Àπ૬ ¢÷Èπ‰ª¢â“ß∫π 2 Àπ૬ –∑âÕπ®ÿ¥ B ¢â“¡·°π Y ·≈–À¡ÿπ
®ÿ¥ C √Õ∫®ÿ¥°”‡π‘¥∑«π‡¢Á¡π“Ãî°“‡ªìπ¡ÿ¡ 180 Ì ·≈â«√Ÿª “¡‡À≈’ˬ¡∑’ˇ°‘¥®“°°“√‡≈◊ËÕπ
¥—ß°≈à“«®–¡’æ◊Èπ∑’ˇªìπ°’ˇ∑à“¢Õß√Ÿª “¡‡À≈’ˬ¡ 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»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550) ✎ 67
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’ȇªìπ·∫∫‡μ‘¡§”μÕ∫
2. ·∫∫∑¥ Õ∫©∫—∫π’È¡’®”π«π 30 ¢âÕ ·∫à߇ªìπ 2 μÕπ §–·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 59-88 12/9/08, 10:29 AM67
68 ✎ ‡ √‘¡§‘¥§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550)
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 ·∫∫∑¥ Õ∫©∫—∫π’ȇªìπ·∫∫‡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“√ 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
‡ √‘¡§‘¥§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550) ✎ 69
( )( ) ( ) 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
70 ✎ ‡ √‘¡§‘¥§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550)
4. ®ß‡√’¬ß®”π«πμàÕ‰ªπ’È®“°πâÕ¬‰ª¡“° 2514, 4258, 8171, 16128 ·≈– 32103
·π«§‘¥
∑”∞“π„Àâ‡∑à“°—𠇪ìπ 2
2514 = 2514
4258 = (22)258 = 2516
8171 = (23)171 = 2513
16128 = (24)128 = 2512
32103 = (25)103 = 2515
‡¡◊ËÕæ‘®“√≥“‡≈¢¬°°”≈—ß∑’Ë¡’∞“π‡¥’¬«°—π ∂Ⓡ≈¢™’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 Õߢ“¥∑ÿπ 10%
∂â“¢“¬‰ª∑—Èß Õß™‘Èπ ·≈⫉¥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
‡ √‘¡§‘¥§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550) ✎ 71
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«πÀπⓉ¡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
„À≥âº≈≈—æ∏å„°≈⇧’¬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
´÷Ë߉¡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
72 ✎ ‡ √‘¡§‘¥§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550)
8. ®“°√Ÿª Δ ABC ¡’æ◊Èπ∑’Ë 18 μ“√“ßÀπ૬ ‚¥¬∑’Ë AB = BC, BE ⊥ AC ·≈– AD ⊥ BC
∂â“ AC = 4 Àπ૬ ·≈â« AD ¬“«‡∑à“°—∫°’ËÀπ૬
·π«§‘¥
‡π◊ËÕß®“° Δ ABC ¡’ AB = BC ¥—ßπ—Èπ Δ ABC ‡ªìπ√Ÿª “¡‡À≈’ˬ¡Àπâ“®—Ë«
®–‰¥â BCA = BAC
·≈– AE = EC = 2 Àπ૬ ( ¡∫—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 ¬“« Àπ૬
μÕ∫ Àπ૬
12
12
AD9
485
3685
36 8585
36 8585
36 8585
^ ^^ ^ ^ ^
^ ^
A
BD
E
C
A 59-88 12/9/08, 10:29 AM72
‡ √‘¡§‘¥§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550) ✎ 73
9. ∂â“√Ÿª “¡‡À≈’ˬ¡√ŸªÀπ÷Ëß¡’®ÿ¥¬Õ¥ “¡®ÿ¥§◊Õ®ÿ¥ (›5, 0), ®ÿ¥ (0, 0) ·≈–®ÿ¥¬Õ¥¢Õß°√“ø
y = x2 + 5x + 7 ·≈â«√Ÿª “¡‡À≈’ˬ¡√Ÿªπ’È¡’æ◊Èπ∑’Ë°’Ëμ“√“ßÀπ૬
·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“√“ßÀπ૬
μÕ∫ μ“√“ßÀπ૬
52
52
52
34
52
34
12
34
158
10. π”√ŸªÀⓇÀ≈’ˬ¡¥â“π‡∑à“¡ÿ¡‡∑à“¡“μàÕ°—π¥—ß√Ÿª ‚¥¬„Àâ·μà≈–√Ÿª∑’Ë¡“μàÕ°—π¡’¥â“π√à«¡°—π
‡æ’¬ßÀπ÷Ëߥâ“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
74 ✎ ‡ √‘¡§‘¥§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550)
·π«§‘¥
2
= =
®–‰¥â a = 2, b = 3
μÕ∫ ¥—ßπ—Èπ 23 + 32 = 4 + 9 = 13
11. ®“°√Ÿª ABCDEF ‡ªìπ√ŸªÀ°‡À≈’ˬ¡¥â“π‡∑à“¡ÿ¡‡∑à“ ∂â“Õ—μ√“ à«π√–À«à“ßæ◊Èπ∑’Ë à«π∑’Ë·√‡ß“
°—∫æ◊Èπ∑’Ë√ŸªÀ°‡À≈’ˬ¡ 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≈’ˬ¡
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
‡ √‘¡§‘¥§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550) ✎ 75
13. ÿà¡À¬‘∫ ≈“°∑’Ë¡’‡≈¢‚¥¥ 1 ∂÷ß 1000 ‡¢’¬π°”°—∫‰«â „∫≈– 1 ®”π«π ¡“ 1 „∫ ∂ⓧ«“¡πà“®–‡ªìπ
∑’ˉ¥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∑’ˬ°°”≈—ß Õß·≈⫉¡à‡°‘π 1000
‰¥â·°à 22, 42, ..., (30)2 ´÷Ëß¡’∑—ÈßÀ¡¥ 15 ®”π«π
§«“¡πà“®–‡ªìπ∑’ˉ¥â§◊Õ =
¥—ßπ—Èπ a = 3, b = 200
π—Ëπ§◊Õ a + b = 203
μÕ∫ 203
ab
151000
3200
14. ·¥ß´âÕ¡¬‘ߪóπ ‚¥¬«“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«°’ˇ¡μ√μàÕ«‘π“∑’
·π«§‘¥
®“°√Ÿª sin 30 Ì =
=
= 24
π—Ëπ§◊Õ „π‡«≈“ 3 «‘π“∑’ °√– ÿπªóπ‡§≈◊ËÕπ∑’ˉ¥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
76 ✎ ‡ √‘¡§‘¥§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550)
16. π”≈«¥‡ âπÀπ÷Ëß¡“¢¥‡ªìπ«ß°≈¡‰¥âæ◊Èπ∑’Ë¿“¬„π«ß°≈¡‡∑à“°—∫ 1,386 μ“√“߇´π쑇¡μ√ ·μàπ”¡“¢¥
‡ªìπ√Ÿª ’ˇÀ≈’ˬ¡¡ÿ¡©“°‚¥¬„À⧫“¡¬“«¥â“π·μà≈–¥â“π‡ªìπ®”π«π‡μÁ¡∑’Ë¡“°°«à“ 30 ‡´π쑇¡μ√
·≈â«√Ÿª ’ˇÀ≈’ˬ¡¡ÿ¡©“°π’È¡’æ◊Èπ∑’Ë¡“°∑’Ë ÿ¥°’Ëμ“√“߇´π쑇¡μ√ (°”Àπ¥ π = )
·π«§‘¥
πr2 = 1,386 μ“√“߇´π쑇¡μ√
r2 = 1,386 Ó μ“√“߇´π쑇¡μ√
r2 = 63 Ó 7 μ“√“߇´π쑇¡μ√
r = 21 ‡´π쑇¡μ√
‡ âπ√Õ∫«ß 2πr = 2 Ó Ó 21 ‡´π쑇¡μ√
= 132 ‡´π쑇¡μ√
§«“¡°«â“ß + §«“¡¬“« = 66 ‡´π쑇¡μ√
15. √Ÿª ’ˇÀ≈’ˬ¡®—μÿ√— ¡’‡ âπ∑·¬ß¡ÿ¡¬“« 14 ‡´π쑇¡μ√ ∫√√®ÿ„π«ß°≈¡‚¥¬¡’®ÿ¥¬Õ¥∑—Èß ’Ë
Õ¬Ÿà∫π‡ âπ√Õ∫«ß ·≈â«æ◊Èπ∑’Ë«ß°≈¡∑’ËÕ¬ŸàπÕ°√Ÿª ’ˇÀ≈’ˬ¡®—μÿ√— ‡ªìπ°’ˇ∑à“¢Õßæ◊Èπ∑’Ë√Ÿª
’ˇÀ≈’ˬ¡®—μÿ√— (°”Àπ¥ π = )227
·π«§‘¥
®“°√Ÿª æ◊Èπ∑’Ë«ß°≈¡∑’ËÕ¬ŸàπÕ°√Ÿª ’ˇÀ≈’ˬ¡®—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 ’ˇÀ≈’ˬ¡®—μÿ√— ‡ªìπ = ‡∑à“¢Õßæ◊Èπ∑’Ë√Ÿª ’ˇÀ≈’ˬ¡®—μÿ√—
μÕ∫
227
12
567 Ó 14
47
123456789012345678901123456789012345678901123456789012345678901123456789012345678901123456789012345678901123456789012345678901123456789012345678901123456789012345678901123456789012345678901123456789012345678901123456789012345678901123456789012345678901123456789012345678901123456789012345678901123456789012345678901123456789012345678901123456789012345678901123456789012345678901123456789012345678901123456789012345678901123456789012345678901
722
227
227
47
14
A 59-88 12/9/08, 10:30 AM76
‡ √‘¡§‘¥§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550) ✎ 77
17. ®ßÀ“«à“®ÿ¥μ—¥¢Õß°√“ø¢Õß ¡°“√ x › 3y = 1 ·≈– 2x2 › 3xy › 20 = 0 ¡’√–¬–
Àà“ß°—π°’ËÀπ૬
·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 ‡´π쑇¡μ√
æ◊Èπ∑’Ë 31 Ó 35 = 1,085 μ“√“߇´π쑇¡μ√
æ◊Èπ∑’Ë 32 Ó 34 = 1,088 μ“√“߇´π쑇¡μ√
æ◊Èπ∑’Ë 33 Ó 33 = 1,089 μ“√“߇´π쑇¡μ√
μÕ∫ 1,089 μ“√“߇´π쑇¡μ√
(4, 1)3
X
9(›5, ›2)
A 59-88 12/9/08, 10:30 AM77
78 ✎ ‡ √‘¡§‘¥§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550)
√–¬–√–À«à“ß (4, 1) ·≈– (›5, ›2) §◊Õ (›5 ›4)2 + (›2 ›1)2 Àπ૬
= 81 + 9 Àπ૬
= 90 Àπ૬
= 3 10 Àπ૬
μÕ∫ 3 10 Àπ૬
18. ®“°√Ÿª °”Àπ¥ ABC ‡ªìπ√Ÿª “¡‡À≈’ˬ¡¡ÿ¡©“°∑’Ë¡’¡ÿ¡ BCA ‡ªìπ¡ÿ¡©“° BN ·≈– CM
‡ªìπ‡ âπ¡—∏¬∞“π ∂â“ BN ⊥ CM ·≈– BC ¬“« 6 Àπ૬ ·≈â« BN ¬“«°’ËÀπ૬
·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 Àπ૬
μÕ∫ 3 Àπ૬
BNBC
BCBO
3x6
^ ^
^ ^
^ ^
62x
6
B M A
N
C
O
A 59-88 12/9/08, 10:30 AM78
‡ √‘¡§‘¥§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550) ✎ 79
19. A ¢—∫√∂¬πμå¥â«¬§«“¡‡√Á« 80 °‘‚≈‡¡μ√μàÕ™—Ë«‚¡ß À≈—ß®“°ÕÕ°‡¥‘π∑“߉¥â 45 π“∑’
B ¢—∫√∂¬πμåÕÕ°®“°μ”·Àπà߇¥’¬«°—∫ A ¥â«¬§«“¡‡√Á« 120 °‘‚≈‡¡μ√μàÕ™—Ë«‚¡ß
‡ªìπ‡«≈“ 20 π“∑’ ®“°π—ÈπÀ¬ÿ¥æ—° 10 π“∑’ ·≈⫇¥‘π∑“ßμàե⫬§«“¡‡√Á« 90 °‘‚≈‡¡μ√μàÕ™—Ë«‚¡ß
®ßÀ“«à“ B ®–‡¥‘π∑“ß∑—π A ‡¡◊ËÕ A ‡¥‘π∑“߉¥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“∑’ ·≈⫇¥‘π∑“ßμàե⫬§«“¡‡√Á« 90 °¡./™¡. „™â‡«≈“ x › ™¡.
‰¥â∑“ß 90(x › ) °‘‚≈‡¡μ√
√∂∑—Èß Õß∑—π°—π · ¥ß«à“√–¬–∑“ß∑’Ë√∂∑—Èß Õß«‘Ë߇∑à“°—π
π—Ëπ§◊Õ 80x = 40 + 90(x › )
›10x =
x =
x = 7
∴ A ‡¥‘π∑“߉¥â 7 ™—Ë«‚¡ß 15 π“∑’
μÕ∫ A ‡¥‘π∑“߉¥â 7 ™—Ë«‚¡ß 15 π“∑’
120 Ó 2060
7560
54
54
›1452
14520
14
A 59-88 12/9/08, 10:30 AM79
80 ✎ ‡ √‘¡§‘¥§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550)
20. ‡»… à«π®”π«πÀπ÷Ëß ‡¡◊ËÕπ” 2 ¡“∫«°‡≈¢‚¥¥¢Õßμ—«‡»…·≈–μ—« à«π ®–∑”„À⇻… à«ππ—Èπ‡ªìπ
·≈–‡¡◊ËÕπ” 4 ¡“≈∫‡≈¢‚¥¥¢Õßμ—«‡»…·≈–μ—« à«π ®–∑”„À⇻… à«ππ—Èπ‡ªìπ ®ßÀ“«à“
‡»… à«π®”π«ππ—Èπ‡ªìπ‡∑à“„¥
·π«§‘¥
„À⇻… 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
‡ √‘¡§‘¥§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550) ✎ 81
C
Q
P
B
μÕπ∑’Ë 2
21. ®ßÀ“§«“¡¬“«¥â“π¢Õß√Ÿª “¡‡À≈’ˬ¡¥â“π‡∑à“∑’Ë¡’¢π“¥„À≠à∑’Ë ÿ¥ ∑’Ë∫√√®ÿÕ¬Ÿà„π√Ÿª ’ˇÀ≈’ˬ¡
®—μÿ√— ∑’Ë¡’§«“¡¬“«¥â“π 1 Àπ૬
·π«§‘¥
„Àâ PQR ‡ªìπ√Ÿª “¡‡À≈’ˬ¡¥â“π‡∑à“∑’Ë¡’¢π“¥„À≠à∑’Ë ÿ¥
´÷Ëß¡’¥â“𬓫¥â“π≈– y Àπ૬
„Àâ CQ = RB = x Àπ૬
®–‰¥â QA = AR = 1 › x Àπ૬
®“° 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≈’ˬ¡¥â“π‡∑à“∑’Ë¡’¢π“¥„À≠à∑’Ë ÿ¥ = 2 2 › 3 Àπ૬
μÕ∫ 2 2 › 3
4 ± 16 › 42
∴
1
y
1
RA
A 59-88 12/9/08, 10:30 AM81
82 ✎ ‡ √‘¡§‘¥§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550)
22. ∂â“™“¬§πÀπ÷Ë߬◊πÕ¬Ÿà∫πæ◊Èπ√“∫∑’Ë®ÿ¥ A ®–¡Õ߇ÀÁπ¬Õ¥‡ “∏ߥ⫬¡ÿ¡‡ß¬ 15 Ì ·≈–∂Ⓡ¢“
‡¥‘π‡¢â“À“‡ “∏ß 50 ‡¡μ√ ‡¢“®–¡Õ߇ÀÁπ¬Õ¥‡ “∏ߥ⫬¡ÿ¡‡ß¬ 45 Ì ®ßÀ“§«“¡ Ÿß¢Õ߇ “∏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≈’ˬ¡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
‡ √‘¡§‘¥§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550) ✎ 83
23. ∂â“ d ‡ªìπ À.√.¡. ¢Õß n › 1 °—∫ n2 + n + 1 ‡¡◊ËÕ n ‡ªìπ®”π«ππ—∫ ·≈⫺≈∫«°¢Õß d
∑’ˇªìπ‰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 ∑’ˇªìπ‰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
84 ✎ ‡ √‘¡§‘¥§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550)
25. ∂â“ ABCD ‡ªìπ√Ÿª ’ˇÀ≈’ˬ¡®—μÿ√— ¬“«¥â“π≈– 28 Àπ૬ ‚¥¬¡’ 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 ·∫àߧ√÷Ëß¡ÿ¡ IBC ·≈â«æ◊Èπ∑’Ë¢Õß√Ÿª “¡‡À≈’ˬ¡ BMN ‡∑à“°—∫°’Ëμ“√“ßÀπ૬
·π«§‘¥
À¡ÿπ√Ÿª “¡‡À≈’ˬ¡ 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≈’ˬ¡¡ÿ¡©“° DMN; (28 › x)2 + 72 = (21 + x)2
∴ x = 4
æ◊Èπ∑’Ë Δ MDN = Ó 7 Ó 24 = 84 μ“√“ßÀπ૬
æ◊Èπ∑’Ë Δ BMN = 282 › Ó 21 Ó 28 › Ó 4 Ó 28 › 84 μ“√“ßÀπ૬
= 28(28 › ) › 84 μ“√“ßÀπ૬
= › 84 μ“√“ßÀπ૬
= 350 μ“√“ßÀπ૬
μÕ∫ 350 μ“√“ßÀπ૬
^ ^
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
‡ √‘¡§‘¥§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550) ✎ 85
27. °”Àπ¥ ABCD ‡ªìπ√Ÿª ’ˇÀ≈’ˬ¡®—μÿ√— ¬“«¥â“π≈– 900 Àπ૬ ‚¥¬∑’Ë O ‡ªìπ®ÿ¥μ—¥
¢Õ߇ âπ∑·¬ß¡ÿ¡ ·≈–¡’ E, F ‡ªìπ®ÿ¥∫π¥â“π AD ‚¥¬∑’Ë E Õ¬Ÿà√–À«à“ß A °—∫ F
∂â“ DF = 400 Àπ૬ ·≈–¡ÿ¡ EOF = 45 Ì ·≈â« AE ¬“«°’ËÀπ૬
·π«§‘¥
„Àâ AE = x Àπ૬
EF = 500 › x Àπ૬
°”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≈’ˬ¡¡ÿ¡©“° FDP ¡’ (500 › x)2 = x2 + 4002
∴ x = 90 Àπ૬
μÕ∫ 90 Àπ૬
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
86 ✎ ‡ √‘¡§‘¥§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550)
28. ABCDEFGH ‡ªìπ√Ÿª·ª¥‡À≈’ˬ¡¡ÿ¡‡∑à“ ∑’Ë¡’¥â“𬓫 2, 2 2, 4, 4 2, 6, 7, 7
·≈– 8 Àπ૬ ®–¡’æ◊Èπ∑’Ë°’Ëμ“√“ßÀπ૬
·π«§‘¥
æ∑. ABCDEFGH = (6 + )(10 + ) › (22 + 42 + 2( )2) μ“√“ßÀπ૬
= 60 + 56 2 + › 10 ›
= 50 + 56 2 μ“√“ßÀπ૬
μÕ∫ 50 + 56 2 μ“√“ßÀπ૬
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
‡ √‘¡§‘¥§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550) ✎ 87
30. °”Àπ¥√Ÿª “¡‡À≈’ˬ¡ ABC ¡’®ÿ¥¬Õ¥ A(1, 6), B(›2, 3) ·≈– C(3, ›4) ∂Ⓡ≈◊ËÕπ®ÿ¥ A
¢π“π‰ª∑“ߢ«“ 3 Àπ૬ ¢÷Èπ‰ª¢â“ß∫π 2 Àπ૬ –∑âÕπ®ÿ¥ B ¢â“¡·°π Y ·≈–À¡ÿπ
®ÿ¥ C √Õ∫®ÿ¥°”‡π‘¥∑«π‡¢Á¡π“Ãî°“‡ªìπ¡ÿ¡ 180 Ì ·≈â«√Ÿª “¡‡À≈’ˬ¡∑’ˇ°‘¥®“°°“√‡≈◊ËÕπ
¥—ß°≈à“«®–¡’æ◊Èπ∑’ˇªìπ°’ˇ∑à“¢Õß√Ÿª “¡‡À≈’ˬ¡ 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
88 ✎ ‡ √‘¡§‘¥§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550)
®ÿ¥ A(1, 6) ‡≈◊ËÕπ‰ª∑“ߢ«“ 3 Àπ૬ ¢÷Èπ‰ª¢â“ß∫π 2 Àπ૬ ‡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“√“ßÀπ૬
æ∑. Δ A´B´C´= 7 Ó 5 › (2 Ó 5 + 5 Ó 1 + 7 Ó 4)
= 35 › 21.5 = 13.5 μ“√“ßÀπ૬
=
μÕ∫
12
12
æ∑. Δ A´B´C´æ∑. Δ ABC
13.518
34
A 59-88 12/9/08, 10:31 AM88
‡ √‘¡§‘¥§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550) ✎ 89
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’ȇªìπ·∫∫‡μ‘¡§”μÕ∫ ¡’∑—ÈßÀ¡¥ 30 ¢âÕ §–·ππ‡μÁ¡ 100 §–·ππ
2. ·∫∫∑¥ Õ∫©∫—∫π’È¡’®”π«π 30 ¢âÕ ·∫à߇ªìπ 2 μÕπ §–·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
90 ✎ ‡ √‘¡§‘¥§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550)
( ) ( ) ( ) ( )
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 ·∫∫∑¥ Õ∫©∫—∫π’ȇªìπ·∫∫‡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. ∂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¥„À≥Ⱂ˫‘∏’
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“𔉪„ à„πμ–°√â“„∫≈– 2, 3, 4, 5 À√◊Õ 6 øÕß ‡∑à“ Ê °—π
®–‡À≈◊Õ‰¢à 1 øÕ߇ ¡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
‡ √‘¡§‘¥§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550) ✎ 91
10. ®“°√Ÿª Δ ABC ¡’ M ‡ªìπ®ÿ¥°÷Ëß°≈“ߢÕߥâ“π BC ∂â“ AB = 4 Àπ૬, AM = 3 Àπ૬
·≈– AC = 8 Àπ૬ ·≈â« BC ¬“«°’ËÀπ૬
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≈∫«°¢Õ߇≈¢‚¥¥∑’ˇ¢’¬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 ‡´π쑇¡μ√ ·≈– 5 ‡´π쑇¡μ√ ¡“À≈Õ¡‡ªìπ≈Ÿ°∫“»°å
¬“«¥â“π≈– 2 ‡´π쑇¡μ√ ·≈â«®–‰¥â≈Ÿ°∫“»°å∑—ÈßÀ¡¥ 143 ≈Ÿ° ®ßÀ“«à“°√«¬‚≈À–
ª≈“¬μ—¥π’È¡’§«“¡ Ÿß°’ˇ´π쑇¡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
92 ✎ ‡ √‘¡§‘¥§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550)
17. °”Àπ¥ P ‡ªìπæ“√“‚∫≈“ y = x2 › 3x ∂â“ P′ ‰¥â®“°°“√‡≈◊ËÕπ P ¢π“π‰ª¢â“ß∫π
4 Àπ૬ ·≈–‰ª∑“ߴ⓬ 3 Àπ૬ ·≈– P′ ºà“π®ÿ¥ ›2, ·≈â« a ¡’§à“‡∑à“„¥
18. §«“¡‡√Á«¢ÕßπÈ”∑’ˉÀ≈ºà“π°äÕ°∑’ËÀπ÷Ëß·≈–°äÕ°∑’Ë Õß ‡ªìπÕ—μ√“ à«π 2 : 3 ∂Ⓡªî¥°äÕ°∑’Ë Õß
·≈–°äÕ°∑’Ë “¡æ√âÕ¡°—π ·≈â«πÈ”®–‰À≈‡¢â“∂—ß≈Ÿ°∫“»°å´÷Ëß°«â“ߥâ“π≈– 1.2 ‡¡μ√ ‚¥¬„™â
‡«≈“ 20 π“∑’ ®÷ß®–‡μÁ¡∂—ß ·≈–∂Ⓡªî¥°äÕ°∑’ËÀπ÷Ëß°—∫°äÕ°∑’Ë “¡æ√âÕ¡°—π ·≈â«πÈ”®–‰À≈
ÕÕ°®“°∂—ߥ—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÷Ë߉¥â√—∫°“√√–∫“¬ ’μ“¡™àÕ߬àÕ¬ Ê ¥â«¬ ’¢“«À√◊Õ ’πÈ”‡ß‘π
™àÕß≈–Àπ÷Ëß ’‡∑à“π—Èπ ‚¥¬æ∫«à“„π∑ÿ°√Ÿª ’ˇÀ≈’ˬ¡º◊πºâ“∑’Ë¡’ 6 ™àÕ߬àÕ¬ (¢π“¥ 2 Ó 3
À√◊Õ 3 Ó 2) ®–¡’ 2 ™àÕ߬àÕ¬‡ªìπ ’πÈ”‡ß‘π à«π∑’ˇÀ≈◊Õ‡ªìπ ’¢“«‡ ¡Õ ∂â“°√–¥“ππ’È
¡’¢π“¥ 9 Ó 11 ·≈â«®–¡’™àÕ߬àÕ¬∑’ˇªìπ ’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
‡ √‘¡§‘¥§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550) ✎ 93
μÕπ∑’Ë 2
21. ∂â“ ABC ‡ªìπ√Ÿª “¡‡À≈’ˬ¡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 ·∫àߧ√÷Ëß
BDE ·≈â« DIE ¡’¢π“¥°’ËÕß»“
22. °”Àπ¥ ABC ‡ªìπ√Ÿª “¡‡À≈’ˬ¡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≈’ˬ¡Àπâ“®—Ë«∑’Ë¡’ BAC = 100 Ì, M ‡ªìπ®ÿ¥¿“¬„π∑”„Àâ
MBA = 10 Ì ·≈– MCA = 5 Ì ·≈â« BMA ¡’¢π“¥°’ËÕß»“
25. ∂â“ √Ÿª ’ˇÀ≈’ˬ¡ ABCD ¡’ BAC = 81 Ì, CAD = 27 Ì, ABD = 36 Ì ·≈– CBD = 30 Ì
·≈â« ADC ¡’¢π“¥°’ËÕß»“
26. √Ÿª “¡‡À≈’ˬ¡ 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 Àπ૬ ·≈– 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
94 ✎ ‡ √‘¡§‘¥§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550)
( ) ( ) ( )
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»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550) ✎ 95
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’ȇªìπ·∫∫‡μ‘¡§”μÕ∫ ¡’∑—ÈßÀ¡¥ 30 ¢âÕ §–·ππ‡μÁ¡ 100 §–·ππ
2. ·∫∫∑¥ Õ∫©∫—∫π’È¡’®”π«π 30 ¢âÕ ·∫à߇ªìπ 2 μÕπ §–·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
96 ✎ ‡ √‘¡§‘¥§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550)
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’ȇªìπ·∫∫‡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
‡ √‘¡§‘¥§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550) ✎ 97
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
98 ✎ ‡ √‘¡§‘¥§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550)
3. ¡’‰¢àÕ¬Ÿà∑—ÈßÀ¡¥‰¡à‡°‘π 500 øÕß ∂â“𔉪„ à„πμ–°√â“„∫≈– 2, 3, 4, 5 À√◊Õ 6 øÕß ‡∑à“ Ê °—π
®–‡À≈◊Õ‰¢à 1 øÕ߇ ¡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„°≈⇧’¬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
‡ √‘¡§‘¥§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550) ✎ 99
4. ∂â“¡’‡ß‘π 3,440 ∫“∑ ‡ªìπ∏π∫—μ√©∫—∫≈– 20 ∫“∑ 50 ∫“∑ ·≈– 100 ∫“∑ √«¡ 72 ©∫—∫
·≈â«®–¡’∏π∫—μ√∑—Èß “¡™π‘¥„π‡ß◊ËÕπ‰¢∑’Ë°”Àπ¥„À≥Ⱂ˫‘∏’
·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
100 ✎ ‡ √‘¡§‘¥§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550)
·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
‡ √‘¡§‘¥§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550) ✎ 101
( ) ( ) ( ) ( )
( ) ( )( )
( ) ( ) ( ) ( )
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
102 ✎ ‡ √‘¡§‘¥§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550)
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
‡ √‘¡§‘¥§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550) ✎ 103
·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 𔉪·∑π„π ➌
‰¥â x2 ›2(12) › 7 = 0
x2 = 31
x = 31
·μà BC = 2x
π—Ëπ§◊Õ ¥â“π BC ¬“« 2 31 Àπ૬
μÕ∫ 2 31 Àπ૬
10. ®“°√Ÿª Δ ABC ¡’ M ‡ªìπ®ÿ¥°÷Ëß°≈“ߢÕߥâ“π BC ∂â“ AB = 4 Àπ૬, AM = 3 Àπ૬
·≈– AC = 8 Àπ૬ ·≈â« BC ¬“«°’ËÀπ૬
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
104 ✎ ‡ √‘¡§‘¥§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550)
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
‡ √‘¡§‘¥§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550) ✎ 105
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≈∫«°¢Õ߇≈¢‚¥¥∑’ˇ¢’¬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
106 ✎ ‡ √‘¡§‘¥§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550)
D
B
15. ∂â“π”°√«¬‚≈À–ª≈“¬μ—¥∑’Ë¡’√—»¡’∞“π 2 ‡´π쑇¡μ√ ·≈– 5 ‡´π쑇¡μ√ ¡“À≈Õ¡‡ªìπ≈Ÿ°∫“»°å
¬“«¥â“π≈– 2 ‡´π쑇¡μ√ ·≈â«®–‰¥â≈Ÿ°∫“»°å∑—ÈßÀ¡¥ 143 ≈Ÿ° ®ßÀ“«à“°√«¬‚≈À–
ª≈“¬μ—¥π’È¡’§«“¡ Ÿß°’ˇ´π쑇¡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 ‡´π쑇¡μ√
μÕ∫ 28 ‡´π쑇¡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 𔉪·∑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
‡ √‘¡§‘¥§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550) ✎ 107
17. °”Àπ¥ P ‡ªìπæ“√“‚∫≈“ y = x2 › 3x ∂â“ P′ ‰¥â®“°°“√‡≈◊ËÕπ P ¢π“π‰ª¢â“ß∫π
4 Àπ૬ ·≈–‰ª∑“ߴ⓬ 3 Àπ૬ ·≈– 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”∑’ˉÀ≈ºà“π°äÕ°∑’ËÀπ÷Ëß·≈–°äÕ°∑’Ë Õß ‡ªìπÕ—μ√“ à«π 2 : 3 ∂Ⓡªî¥°äÕ°∑’Ë Õß
·≈–°äÕ°∑’Ë “¡æ√âÕ¡°—π ·≈â«πÈ”®–‰À≈‡¢â“∂—ß≈Ÿ°∫“»°å´÷Ëß°«â“ߥâ“π≈– 1.2 ‡¡μ√ ‚¥¬„™â
‡«≈“ 20 π“∑’ ®÷ß®–‡μÁ¡∂—ß ·≈–∂Ⓡªî¥°äÕ°∑’ËÀπ÷Ëß°—∫°äÕ°∑’Ë “¡æ√âÕ¡°—π ·≈â«πÈ”®–‰À≈
ÕÕ°®“°∂—ߥ—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
108 ✎ ‡ √‘¡§‘¥§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550)
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÷Ë߉¥â√—∫°“√√–∫“¬ ’μ“¡™àÕ߬àÕ¬ Ê ¥â«¬ ’¢“«À√◊Õ ’πÈ”‡ß‘π
™àÕß≈–Àπ÷Ëß ’‡∑à“π—Èπ ‚¥¬æ∫«à“„π∑ÿ°√Ÿª ’ˇÀ≈’ˬ¡º◊πºâ“∑’Ë¡’ 6 ™àÕ߬àÕ¬ (¢π“¥ 2 Ó 3 À√◊Õ
3 Ó 2) ®–¡’ 2 ™àÕ߬àÕ¬‡ªìπ ’πÈ”‡ß‘π à«π∑’ˇÀ≈◊Õ‡ªìπ ’¢“«‡ ¡Õ ∂â“°√–¥“ππ’È¡’¢π“¥
9 Ó 11 ·≈â«®–¡’™àÕ߬àÕ¬∑’ˇªìπ ’πÈ”‡ß‘π∑—ÈßÀ¡¥°’Ë™àÕß
·π«§‘¥
μÕ∫ 33 ™àÕß
A 89-118 12/9/08, 10:34 AM108
‡ √‘¡§‘¥§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550) ✎ 109
·π«§‘¥
≈“° CD ®–‰¥â Δ ACD ·≈– Δ BCD ‡ªìπ√Ÿª “¡‡À≈’ˬ¡∑’Ë·π∫„π«ß°≈¡
1. BAC = ADC ¡ÿ¡∑’ˇ âπ —¡º— ®√¥°—∫§Õ√奡’¢π“¥‡∑à“°—∫¡ÿ¡„π
2. ABC = BDC à«π‚§âßμ√ߢⓡ
3. ACB = 107 Ì °”Àπ¥„Àâ
4. BAC + ABC + ACB = 180 Ì º≈∫«°¡ÿ¡¿“¬„π√Ÿª “¡‡À≈’ˬ¡
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
110 ✎ ‡ √‘¡§‘¥§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550)
μÕπ∑’Ë 2
21. ∂â“ ABC ‡ªìπ√Ÿª “¡‡À≈’ˬ¡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 ·∫àߧ√÷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
‡ √‘¡§‘¥§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550) ✎ 111
22. °”Àπ¥ ABC ‡ªìπ√Ÿª “¡‡À≈’ˬ¡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≈’ˬ¡¥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∑”πÕ߇¥’¬«°—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
112 ✎ ‡ √‘¡§‘¥§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550)
24. ∂â“ ABC ‡ªìπ√Ÿª “¡‡À≈’ˬ¡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≈’ˬ¡¡’¢π“¥‡∑à“°—∫º≈∫«°
¢Õß¡ÿ¡¿“¬„π∑’ˉ¡à„™à¡ÿ¡ª√–™‘¥)
ACM = ECM = 5 Ì
¥—ßπ—Èπ M ‡ªìπ®ÿ¥»Ÿπ¬å°≈“ߢÕß«ß°≈¡·π∫„π√Ÿª “¡‡À≈’ˬ¡ AEC
©–π—Èπ EAM = CAM = 25 ‡æ√“– AM ·∫àߧ√÷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
‡ √‘¡§‘¥§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550) ✎ 113
25. ∂â“ √Ÿª ’ˇÀ≈’ˬ¡ 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≈’ˬ¡¥â“π‡∑à“
䴉 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
114 ✎ ‡ √‘¡§‘¥§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550)
26. √Ÿª “¡‡À≈’ˬ¡ 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
‡ √‘¡§‘¥§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550) ✎ 115
( ) ( ) ( )
27. °”Àπ¥„Àâ AB ‡ªìπ‡ âπºà“π»Ÿπ¬å°≈“ß«ß°≈¡∑’Ë¡’√—»¡’ 99 Àπ૬ ·≈– 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∑’ˇ´°‡¡πμå AB = π(99 2)2 › Ó (99 2)2
æ◊Èπ∑’Ë·√‡ß“ = π(99)2 › 2 æ◊Èπ∑’ˇ´°‡¡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
116 ✎ ‡ √‘¡§‘¥§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550)
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
‡ √‘¡§‘¥§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550) ✎ 117
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
118 ✎ ‡ √‘¡§‘¥§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550)
¥—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»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550) ✎ 119
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’ȇªìπ·∫∫‡μ‘¡§”μÕ∫
2. ·∫∫∑¥ Õ∫©∫—∫π’È¡’®”π«π 30 ¢âÕ ·∫à߇ªìπ 2 μÕπ §–·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 119-152 12/9/08, 10:37 AM119
120 ✎ ‡ √‘¡§‘¥§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550)
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 ·∫∫∑¥ Õ∫©∫—∫π’ȇªìπ·∫∫‡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
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 ∂⓺≈∫«°¢Õß∑ÿ° 60 ®”π«π∑’ˇ√’¬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
‡ √‘¡§‘¥§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550) ✎ 121
10. ®“°√Ÿª BO ·∫àߧ√÷Ëß¡ÿ¡ CBA, CO ·∫àߧ√÷Ëß¡ÿ¡ ACB ·≈– MN ºà“π O ¢π“π°—∫ BC
∂â“ AB = 12 Àπ૬ BC = 24 Àπ૬ ·≈– AC = 18 Àπ૬ ·≈⫧«“¡¬“«‡ âπ√Õ∫√Ÿª
√Ÿª “¡‡À≈’ˬ¡ AMN ¡’§à“°’ËÀπ૬
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≈’ˬ¡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≈’ˬ¡¥â“π‡∑à“∑’Ë¡’ P ‡ªìπ®ÿ¥¿“¬„π∑”„Àâ PA = 6 Àπ૬
PB = 8 Àπ૬ ·≈– PC = 10 Àπ૬ √Ÿª “¡‡À≈’ˬ¡ ABC ¡’æ◊Èπ∑’Ë°’Ëμ“√“ßÀπ૬
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
122 ✎ ‡ √‘¡§‘¥§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550)
14. ®“°√Ÿª ABC, DEF ·≈– PQR ‡ªìπ√Ÿª “¡‡À≈’ˬ¡¥â“π‡∑à“ ‚¥¬∑’Ë AD = BD ·≈â«æ◊Èπ∑’Ë
√Ÿª “¡‡À≈’ˬ¡ ABC ‡ªìπ°’ˇ∑à“¢Õßæ◊Èπ∑’Ë√Ÿª “¡‡À≈’ˬ¡ PQR
15. ∂â“√Ÿª ’ˇÀ≈’ˬ¡ 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°≈“ߢÕß«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 ∑’ˇªìπ®”π«π‡μÁ¡∑’Ë¡“°∑’Ë ÿ¥ ÷Ëߪ√–°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—«
·≈⫺≈∫«°¢Õß§à“ d ∑’ˇªìπ‰ª‰¥â∑—ÈßÀ¡¥ ¡’§à“‡∑à“„¥
19. √ŸªÀ≈“¬‡À≈’ˬ¡¥â“π‡∑à“¡ÿ¡‡∑à“√ŸªÀπ÷Ëß ¡’‡ âπ∑·¬ß¡ÿ¡∑—ÈßÀ¡¥ 20 ‡ âπ ·≈–·π∫„π
«ß°≈¡√—»¡’ 2 Àπ૬ √ŸªÀ≈“¬‡À≈’ˬ¡π’È¡’æ◊Èπ∑’ˇ∑à“„¥
20. √Ÿª ’ˇÀ≈’ˬ¡ ABCD ¡’æ◊Èπ∑’Ë 2,550 μ“√“ßÀπ૬ ‚¥¬∑’Ë P, Q, R ·≈– S ‡ªìπ®ÿ¥°÷Ëß°≈“ߥâ“π
∂â“√Ÿª ’ˇÀ≈’ˬ¡ BNMP ·≈–√Ÿª ’ˇÀ≈’ˬ¡ DTOR ¡’æ◊Èπ∑’Ë 123 ·≈– 456 μ“√“ßÀπ૬
μ“¡≈”¥—∫ ·≈â«æ◊Èπ∑’Ë¢Õß√Ÿª ’ˇÀ≈’ˬ¡ MNOT ‡∑à“°—∫°’Ëμ“√“ßÀπ૬
C
A
B
P FD
QR
^^ ^
E
A 119-152 12/9/08, 10:37 AM122
‡ √‘¡§‘¥§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550) ✎ 123
μÕπ∑’Ë 2
21. ·∫∫®”≈Õß CH4 ·π∫„π∑√߇À≈’ˬ¡ ’ËÀπâ“ ‚¥¬∑’Ë·μà≈–ÀπⓇªìπ√Ÿª “¡‡À≈’ˬ¡¥â“π‡∑à“
∑’ˬ“«¥â“π≈– 6 6 Àπ૬ ·μà≈–ÀπⓇ∑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 ∑’ˇªìπ‰ª‰¥â∑—ÈßÀ¡¥ ¡’§à“‡∑à“„¥
27. √Ÿª “¡‡À≈’ˬ¡ ABC ¡’ D, E ·≈– F ‡ªìπ®ÿ¥∫π¥â“π BC ∑”„Àâ AD ⊥ BC, AE ·∫àߧ√÷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
124 ✎ ‡ √‘¡§‘¥§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550)
30. ∂â“ ABCD ‡ªìπ√Ÿª ’ˇÀ≈’ˬ¡ ¡’ 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»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550) ✎ 125
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’ȇªìπ·∫∫‡μ‘¡§”μÕ∫
2. ·∫∫∑¥ Õ∫©∫—∫π’È¡’®”π«π 30 ¢âÕ ·∫à߇ªìπ 2 μÕπ §–·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 119-152 12/9/08, 10:37 AM125
126 ✎ ‡ √‘¡§‘¥§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550)
2,007 ⋅ 2,0072,006 2,008
1. ... ¡’§à“‡∑à“„¥
·π«§‘¥
‡π◊ËÕß®“° = ®–‰¥â
... = ...
=
=
μÕ∫
§”™’È·®ß ·∫∫∑¥ Õ∫©∫—∫π’ȇªìπ·∫∫‡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
‡ √‘¡§‘¥§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550) ✎ 127
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
128 ✎ ‡ √‘¡§‘¥§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550)
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
‡ √‘¡§‘¥§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550) ✎ 129
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°”À𥂥¬ 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
130 ✎ ‡ √‘¡§‘¥§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550)
8. ∂â“ ABC ‡ªìπ√Ÿª “¡‡À≈’ˬ¡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 ·∫àߧ√÷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 ∂⓺≈∫«°¢Õß∑ÿ° 60 ®”π«π∑’ˇ√’¬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∑’ˇ√’¬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∑’ˇ√’¬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
‡ √‘¡§‘¥§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550) ✎ 131
9. °”Àπ¥ ABC ‡ªìπ√Ÿª “¡‡À≈’ˬ¡¥â“π‡∑à“∑’Ë¡’ P ‡ªìπ®ÿ¥¿“¬„π∑”„Àâ PA = 6 Àπ૬
PB = 8 Àπ૬ ·≈– PC = 10 Àπ૬ √Ÿª “¡‡À≈’ˬ¡ ABC ¡’æ◊Èπ∑’Ë°’Ëμ“√“ßÀπ૬
·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“√“ßÀπ૬
μÕ∫ 36 + 25 3 μ“√“ßÀπ૬
A
B C
E
F
D
P
34
10. ®“°√Ÿª BO ·∫àߧ√÷Ëß¡ÿ¡ CBA, CO ·∫àߧ√÷Ëß¡ÿ¡ ACB ·≈– MN ºà“π O ¢π“π°—∫ BC
∂â“ AB = 12 Àπ૬ BC = 24 Àπ૬ ·≈– AC = 18 Àπ૬ ·≈⫧«“¡¬“«‡ âπ√Õ∫√Ÿª
√Ÿª “¡‡À≈’ˬ¡ AMN ¡’§à“°’ËÀπ૬
·π«§‘¥
1. BO ·∫àߧ√÷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 ·∫àߧ√÷Ëß BCA (‚®∑¬å°”Àπ¥)
^
^ ^
^ ^
^
A
B C
M NO
A 119-152 12/9/08, 10:38 AM131
132 ✎ ‡ √‘¡§‘¥§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550)
8. ®–‰¥â NCO = OCB (®“°¢âÕ 7)
9. NOC = OCB (®“°¢âÕ 3 ¡ÿ¡·¬âß)
10. Δ ONC ‡ªìπ√Ÿª “¡‡À≈’ˬ¡Àπâ“®—Ë« (®“°¢âÕ 9)
11. NO = NC (®“°¢âÕ 3 ¡∫—μ‘¢Õß√Ÿª “¡‡À≈’ˬ¡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 Àπ૬
μÕ∫ 30 Àπ૬
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
‡ √‘¡§‘¥§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550) ✎ 133
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
134 ✎ ‡ √‘¡§‘¥§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550)
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
‡ √‘¡§‘¥§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550) ✎ 135
14. ®“°√Ÿª ABC, DEF ·≈– PQR ‡ªìπ√Ÿª “¡‡À≈’ˬ¡¥â“π‡∑à“ ‚¥¬∑’Ë AD = BD æ◊Èπ∑’Ë
√Ÿª “¡‡À≈’ˬ¡ ABC ‡ªìπ°’ˇ∑à“¢Õßæ◊Èπ∑’Ë√Ÿª “¡‡À≈’ˬ¡ 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
136 ✎ ‡ √‘¡§‘¥§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550)
y = = = 2(3 › 5)
= = = 7 + 3 5
μÕ∫ 7 + 3 5
2z2
2
162(14 › 6 5)
47 › 3 5
15. ∂â“√Ÿª ’ˇÀ≈’ˬ¡ 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°≈“ߢÕß«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
‡ √‘¡§‘¥§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550) ✎ 137
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 ∑’ˇªìπ®”π«π‡μÁ¡∑’Ë¡“°∑’Ë ÿ¥ ÷Ëߪ√–°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÷ËߢÕß ¡°“√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 μâÕ߇ªìπæÀÿ§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
138 ✎ ‡ √‘¡§‘¥§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550)
18. °”Àπ¥„Àâ n ·≈– d ‡ªìπ®”π«π‡μÁ¡∫«° ∂â“ d À“√ 13n + 6 ·≈– 12n + 5 ≈ßμ—«
·≈⫺≈∫«°¢Õß§à“ d ∑’ˇªìπ‰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≈’ˬ¡¥â“π‡∑à“¡ÿ¡‡∑à“√ŸªÀπ÷Ëß ¡’‡ âπ∑·¬ß¡ÿ¡∑—ÈßÀ¡¥ 20 ‡ âπ ·≈–·π∫„π
«ß°≈¡√—»¡’ 2 Àπ૬ √ŸªÀ≈“¬‡À≈’ˬ¡π’È¡’æ◊Èπ∑’ˇ∑à“„¥
·π«§‘¥
√Ÿª 4 ‡À≈’ˬ¡¡’‡ âπ∑·¬ß¡ÿ¡ 2 ‡ âπ
√Ÿª 5 ‡À≈’ˬ¡¡’‡ âπ∑·¬ß¡ÿ¡ 5 ‡ âπ
√Ÿª 6 ‡À≈’ˬ¡¡’‡ âπ∑·¬ß¡ÿ¡ 9 ‡ âπ
√Ÿª 7 ‡À≈’ˬ¡¡’‡ âπ∑·¬ß¡ÿ¡ 14 ‡ âπ
√Ÿª 8 ‡À≈’ˬ¡¡’‡ âπ∑·¬ß¡ÿ¡ 20 ‡ âπ
√Ÿª “¡‡À≈’ˬ¡ 1 √Ÿª ¡’¡ÿ¡¬Õ¥°“ß = 45 Ì
æ◊Èπ∑’Ë√Ÿª “¡‡À≈’ˬ¡ = ( 2)( 2) sin 45 Ì = μ“√“ßÀπ૬
¥—ßπ—Èπ √Ÿª 8 ‡À≈’ˬ¡®–¡’æ◊Èπ∑’Ë 8 = 4 2 μ“√“ßÀπ૬
μÕ∫ 4 2 μ“√“ßÀπ૬
360 Ì8
12
22
22 )(
A 119-152 12/9/08, 10:39 AM138
‡ √‘¡§‘¥§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550) ✎ 139
20. √Ÿª ’ˇÀ≈’ˬ¡ ABCD ¡’æ◊Èπ∑’Ë 2,550 μ“√“ßÀπ૬ ‚¥¬∑’Ë P, Q, R ·≈– S ‡ªìπ®ÿ¥°÷Ëß°≈“ߥâ“π
∂â“√Ÿª ’ˇÀ≈’ˬ¡ BNMP ·≈–√Ÿª ’ˇÀ≈’ˬ¡ DTOR ¡’æ◊Èπ∑’Ë 123 ·≈– 456 μ“√“ßÀπ૬
μ“¡≈”¥—∫ ·≈â«æ◊Èπ∑’Ë¢Õß√Ÿª ’ˇÀ≈’ˬ¡ MNOT ‡∑à“°—∫°’Ëμ“√“ßÀπ૬
·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“√“ßÀπ૬
æ◊Èπ∑’Ë � MNOT = 1,275 › (123 + 456) = 696 μ“√“ßÀπ૬
μÕ∫ 696 μ“√“ßÀπ૬
12
NO
R C
Q
BP
M
T
D
S
A
A 119-152 12/9/08, 10:39 AM139
140 ✎ ‡ √‘¡§‘¥§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550)
μÕπ∑’Ë 2
21. ·∫∫®”≈Õß CH4 ·π∫„π∑√߇À≈’ˬ¡ ’ËÀπâ“ ‚¥¬∑’Ë·μà≈–ÀπⓇªìπ√Ÿª “¡‡À≈’ˬ¡¥â“π‡∑à“∑’ˬ“«
¥â“π≈– 6 6 Àπ૬ ·μà≈–ÀπⓇ∑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 Àπ૬
μÕ∫ 9 Àπ૬
HP
H4
H2
H1
T
C
H3
A 119-152 12/9/08, 10:39 AM140
‡ √‘¡§‘¥§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550) ✎ 141
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
142 ✎ ‡ √‘¡§‘¥§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550)
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
‡ √‘¡§‘¥§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550) ✎ 143
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
144 ✎ ‡ √‘¡§‘¥§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550)
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
‡ √‘¡§‘¥§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550) ✎ 145
26. °”Àπ¥„Àâ a, b, c ‡ªìπ®”π«π‡μÁ¡∫«° ∂â“ a + b + c = 20 = ab + bc › ca › b2 ·≈â«
º≈∫«°¢Õß§à“ abc ∑’ˇªìπ‰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“∑’ˇªìπ‰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 ∑’ˇªìπ‰ª‰¥â§◊Õ 112 °—∫ 154
º≈∫«°®–‡∑à“°—∫ 112 + 154 = 266
μÕ∫ 266
A 119-152 12/9/08, 10:40 AM145
146 ✎ ‡ √‘¡§‘¥§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550)
27. √Ÿª “¡‡À≈’ˬ¡ ABC ¡’ D, E ·≈– F ‡ªìπ®ÿ¥∫π¥â“π BC ∑”„Àâ AD ⊥ BC, AE ·∫àߧ√÷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
‡ √‘¡§‘¥§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550) ✎ 147
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
148 ✎ ‡ √‘¡§‘¥§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550)
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
‡ √‘¡§‘¥§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550) ✎ 149
30. ∂â“ ABCD ‡ªìπ√Ÿª ’ˇÀ≈’ˬ¡ ¡’ 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
150 ✎ ‡ √‘¡§‘¥§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550)
§≥–∑”ß“π
∑’˪√÷°…“1. §ÿ≥À≠‘ß°…¡“ «√«√√≥ ≥ Õ¬ÿ∏¬“ ‡≈¢“∏‘°“√§≥–°√√¡°“√°“√»÷°…“¢—Èπæ◊Èπ∞“π2. π“¬¡—ß°√ °ÿ≈«“π‘™ √Õ߇≈¢“∏‘°“√§≥–°√√¡°“√°“√»÷°…“¢—Èπæ◊Èπ∞“π3. 𓬠¡‡°’¬√μ‘ ™Õ∫º≈ √Õ߇≈¢“∏‘°“√§≥–°√√¡°“√°“√»÷°…“¢—Èπæ◊Èπ∞“π4. 𓬫‘π—¬ √Õ¥®à“¬ √Õ߇≈¢“∏‘°“√§≥–°√√¡°“√°“√»÷°…“¢—Èπæ◊Èπ∞“π5. π“߇∫≠®≈—°…≥å πÈ”øÑ“ ºŸâÕ”π«¬°“√ ”π—°«‘™“°“√·≈–¡“μ√∞“π°“√»÷°…“6. π“ßÕ√∑—¬ ¡Ÿ≈§” ºŸâÕ”π«¬°“√ ”π—°æ—≤π“π«—μ°√√¡°“√®—¥°“√»÷°…“7. 𓬥‘≈° 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. 𓬻—°¥‘Ï ‘π ™àÕߥ“√“°ÿ≈ π—°«‘™“°“√»÷°…“ ”π—°æ—≤π“π«—μ°√√¡°“√®—¥°“√»÷°…“4. π“ß “««√πÿ™ √ÿà߇√◊Õ߇®√‘≠°ÿ≈ ‡®â“Àπâ“∑’Ë∫√‘À“√ß“π∏ÿ√°“√ √—°…“°“√„πμ”·Àπàßπ—°«‘™“°“√»÷°…“
5. 𓬪√“‚¡∑¬å ¢®√¿—¬ »÷°…“π‘‡∑»°å
”π—°ß“π‡¢μæ◊Èπ∑’Ë°“√»÷°…“°√ÿ߇∑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. 𓬪Ø≈ ‡ª√¡ª√’¥‘Ï »÷°…“π‘‡∑»°å ”π—°ß“π‡¢μæ◊Èπ∑’Ë°“√»÷°…“‡æ™√∫ÿ√’ ‡¢μ 1
9. 𓬫 —πμå «√‡™…∞å »÷°…“π‘‡∑»°å ”π—°ß“π‡¢μæ◊Èπ∑’Ë°“√»÷°…“√âÕ¬‡ÕÁ¥ ‡¢μ 2
10. 𓬻ÿ¿°√ æß»å∑Õß¡’ »÷°…“π‘‡∑»°å ”π—°ß“π‡¢μæ◊Èπ∑’Ë°“√»÷°…“∫ÿ√’√—¡¬å ‡¢μ 1
11. π“¬∫√√∑—¥ «¿—°¥‘χæ™√ »÷°…“π‘‡∑»°å ”π—°ß“π‡¢μæ◊Èπ∑’Ë°“√»÷°…“ °≈π§√ ‡¢μ 1
12. 𓬇©≈‘¡æ≈ ‡ ¢–æ—π∏å »÷°…“π‘‡∑»°å ”π—°ß“π‡¢μæ◊Èπ∑’Ë°“√»÷°…“™ÿ¡æ√ ‡¢μ 1
13. 𓬮‘√«—≤πå √—°æà«ß »÷°…“π‘‡∑»°å ”π—°ß“π‡¢μæ◊Èπ∑’Ë°“√»÷°…“μ“° ‡¢μ 1
14. 𓬙àÕ©—μ√ ‰™¬ ¡π÷° »÷°…“π‘‡∑»°å ”π—°ß“π‡¢μæ◊Èπ∑’Ë°“√»÷°…“‡æ™√∫ÿ√’ ‡¢μ 1
15. 𓬫—≈≈¿ æ÷Ëßæ—π∏ÿå »÷°…“π‘‡∑»°å ”π—°ß“π‡¢μæ◊Èπ∑’Ë°“√»÷°…“ ÿ‚¢∑—¬ ‡¢μ 1
16. π“ß “«®”‡√‘≠ ‡®’¬«À«“π ‚√߇√’¬π¡À‘¥≈«‘∑¬“πÿ √≥å Õߧå°√¡À“™π
A 119-152 12/9/08, 10:40 AM150
‡ √‘¡§‘¥§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550) ✎ 151
17. π“¬≈ÕÕ ‡æ‘Ë¡ ¡∫—μ‘ ‚√߇√’¬π∫¥‘π∑√‡¥™“ ( ‘ßÀå ‘ßÀ‡ π’)
”π—°ß“π‡¢μæ◊Èπ∑’Ë°“√»÷°…“°√ÿ߇∑æ¡À“π§√ ‡¢μ 2
18. π“¬∫—≥±‘μ ΩÕ¬∑Õß ‚√߇√’¬π‡μ√’¬¡Õÿ¥¡»÷°…“
”π—°ß“π‡¢μæ◊Èπ∑’Ë°“√»÷°…“°√ÿ߇∑æ¡À“π§√ ‡¢μ 1
19. π“ß≈–‡¡’¬¥ °√∫ß°™¡“» ‚√߇√’¬π «π°ÿÀ≈“∫«‘∑¬“≈—¬
”π—°ß“π‡¢μæ◊Èπ∑’Ë°“√»÷°…“°√ÿ߇∑æ¡À“π§√ ‡¢μ 1
20. π“ß«√√≥«‘¿“ ÿ∑∏‡°’¬√μ‘ ‚√߇√’¬π “¡‡ π«‘∑¬“≈—¬
”π—°ß“π‡¢μæ◊Èπ∑’Ë°“√»÷°…“°√ÿ߇∑æ¡À“π§√ ‡¢μ 1
21. 𓬉¡μ√’ »√’∑Õß·∑â ‚√߇√’¬π‡μ√’¬¡Õÿ¥¡»÷°…“
”π—°ß“π‡¢μæ◊Èπ∑’Ë°“√»÷°…“°√ÿ߇∑æ¡À“π§√ ‡¢μ 1
22. π“ß√—™π’ π“§π§√ ‚√߇√’¬πæ√À¡¡“πÿ √≥å ®—ßÀ«—¥‡æ™√∫ÿ√’
”π—°ß“π‡¢μæ◊Èπ∑’Ë°“√»÷°…“‡æ™√∫ÿ√’ ‡¢μ 1
23. π“ß —߇«’¬π Õ‘π∑√ª√– ß§å ‚√߇√’¬π«—¥¥à“π ”‚√ß
”π—°ß“π‡¢μæ◊Èπ∑’Ë°“√»÷°…“ ¡ÿ∑√ª√“°“√ ‡¢μ 1
24. 𓬫‘ ÿ∑∏‘Ï §ß°—≈ªá ‚√߇√’¬π§«π‡π’¬ß«‘∑¬“
”π—°ß“π‡¢μæ◊Èπ∑’Ë°“√»÷°…“ ߢ≈“ ‡¢μ 2
25. π“ß “«æ√æ√√≥ Õ‘π∑√ª√–‡ √‘∞ ºŸâÕ”π«¬°“√‚√߇√’¬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‘¡æ奒¥ ”π—°æ—≤π“π«—μ°√√¡°“√®—¥°“√»÷°…“
2. π“ß “«æ—∑¬“ ∑‘»‡πμ√ æπ—°ß“πæ‘¡æ奒¥ ”π—°æ—≤π“π«—μ°√√¡°“√®—¥°“√»÷°…“
ª°
1. 𓬪√–¡ÿ¢ ∫ÿ≠ ‘√‘ √ÕߺŸâÕ”π«¬°“√‚√߇√’¬π«—¥· πμÕ
”π—°ß“π‡¢μæ◊Èπ∑’Ë°“√»÷°…“°“≠®π∫ÿ√’ ‡¢μ 2
2. π“ß»√’‡¡◊Õß ∫ÿ≠·æ∑¬å ‚√߇√’¬πÕπÿ∫“≈μ“°øÑ“
”π—°ß“π‡¢μæ◊Èπ∑’Ë°“√»÷°…“π§√ «√√§å ‡¢μ 3
3. π“¬∏√√¡√—μπå ∫ÿ≠·æ∑¬å ‚√߇√’¬π»√’π¿“‡¢μ«‘∑¬“
”π—°ß“π‡¢μæ◊Èπ∑’Ë°“√»÷°…“π§√ «√√§å ‡¢μ 3
A 119-152 12/9/08, 10:40 AM151
152 ✎ ‡ √‘¡§‘¥§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550)
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™ √ÿà߇√◊Õ߇®√‘≠°ÿ≈ ‡®â“æπ—°ß“π∏ÿ√°“√ ”π—°æ—≤π“π«—μ°√√¡°“√®—¥°“√»÷°…“
∫√√≥“∏‘°“√
1. 𓬪√“‚¡∑¬å ¢®√¿—¬ »÷°…“π‘‡∑»°å
”π—°ß“π‡¢μæ◊Èπ∑’Ë°“√»÷°…“°√ÿ߇∑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‘ ‚√߇√’¬π “¡‡ π«‘∑¬“≈—¬
”π—°ß“π‡¢μæ◊Èπ∑’Ë°“√»÷°…“°√ÿ߇∑æ¡À“π§√ ‡¢μ 1
5. π“ß√—™π’ π“§π§√ ‚√߇√’¬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
‡ √‘¡§‘¥§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550) ✎ 153
A 119-152 12/9/08, 10:41 AM153