Math Exercise

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กลุ มสาระการเรียนรู คณิตศาสตร 1 ชั ้นมัธยมศึกษาปที ่ 6

โรงเรยีนเตรียมอดุมศึกษา

โรงเรียนเตรียมอดุมศึกษา

กลุ มสาระการเรยีนรู คณิตศาสตร วิชาคณิตศาสตรเสริม 5 (ค 30205)

แบบฝกหัดเสรมิทักษะคณิตศาสตรกลางภาค ชุดที ่ 1

ชื ่อ.............................................................................................................................หอง.........................เลขที ่.........................

คําสั ่ง จงแสดงวิธีทํา เขียนเปนขั ้นตอนตามหลักคณิตศาสตร ลงในชองวางซึ ่งเวนไว ให ในทายโจทยแตละขอ

1. บรษัิทแหงหนึ ่งผลิตลูกกวาด 2 ชนดิ คือ ชนิด A และ B โดยลูกกวาดชนดิ A ลงทุนเปนคาวัสดุ ในการผลติเมด็ละ 4 บาท คาจาง

ทําเมด็ละ 2 บาท และไดกําไรจากการขายเมด็ละ 3 บาท และลกูกวาดชนดิ B ลงทนุเปนคาวัสดุ ในการผลติเมด็ละ 2 บาท คาจางทํา

เมด็ละ 5 บาท และไดกําไรจากการขายเมด็ละ 6 บาท ถาใชเงนิลงทนุเปนคาวัสดุ ในการผลติลูกกวาดไมเกนิ 2,482 บาท และใชเงนิ

ลงทนุเปนคาจางทําไมเกนิ 3,801 บาท บรษัิทแหงนี ้ตองการผลดิลูกกวาดทั ้ง 2 ชนิดให ไดกําไรมากที ่สุด จะตองผลติลูกกวาดทั ้งชนิด

A และชนดิ B อยางละกี ่เมด็

ให P แทนกําไรในการผลติลูกกวาดทั ้ง 2 ชนิด มหีนวยเปนบาท

x แทนจํานวนเมลด็ที ่ผลิตชนิด A และ x > 0

y แทนจํานวนเมลด็ที ่ผลิตชนิด B และ y > 0

1.1 สมการจุดประสงค ...........................................................................................................................................................

1.2 อสมการขอจํากัด .............................................................................................................................................................

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เขียนกราฟของอสมการขอจํากัด เมื ่อ x และ y เปนจํานวนจริง พรอมทั ้งหาคําตอบ

Digitally signed by Perathorn Pooksombat

DN: cn=Perathorn Pooksombat, o, ou,

[email protected], c=TH

Date: 2011.10.06 02:57:09 +07'00'





2. มนีายแพทย 2 คน คุณครู 2 คน ทหาร 1 คน ตํารวจ 1 คน ถาตองการจัดคน 6 คนนี ้ยืนเรียงแถวเปนเสนตรงจะจัดไดกี ่วิธี โดยมี

เงื ่อนไขดังตอไปนี ้

2.1 ไมมเีงื ่อไขใดๆ..............................................................................................................................................................................................................

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2.2 นายแพทย 2 คน ยืนติดกันและคุณครูยนืแยกกัน

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2.3 คุณครู 2 คน ยนืติดกันและนายแพทยยนืแยกกัน

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2.4 คุณครู 2 คนตองยืนติดกันและนายแพทย 2 คนก็ตองยืนติดกัน

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2.5 คนอาชพีเดียวกันยืนไมติดกัน

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3. สรางคําที ่ ใชพยัญชนะทั ้งหมดในคําวา "นิมตฺิตํ สาธุ รูปาน ํกตฺญกตเวทติา" ( โดยไมคํานงึถงึความหมาย) โดยใหพยัญชนะ "ก"

ตองติดอยู กัน และ "น" ตองติดอยู กัน และ "ต" ตองอยู แยกกัน จะไดคําที ่แตกตางกันทั ้งสิ ้น x(y!) คํา จงหา x+y

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สภุาษติขางตนในภาษาไทยหมายความวา

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4. ลกูเสอืหมู หนึ ่งมี 8 คน ถาตองการเลอืกลกูเสือมา 6 คน เพื ่อนั ่งรอบกองไฟจะทําไดกี ่วิธี

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5. คนกลุ มหนึ ่งประกอบดวย ชาย 2 คน หญงิ 3 คนและเด็กผู หญงิ 4 คน จงหาจํานวนวิธีที ่จะจัดคนดังกลาวนั ่งรอบโตะวงกลม

โดยที ่หญงิทั ้งหมดนั ่งติดกัน และเด็กผู หญงิทั ้งหมดนั ่งติดกัน แตชายนั ่งไมติดกัน

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6. มจุีด 10 จุดที ่กําหนดใหอยู บนวงกลม ตองการสรางรูปเหลี ่ยมที ่มีจุดดังกลาวเปนจุดยอดจะสรางรูปเหลี ่ยมไดทั ้งหมดกี ่รูป

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7. ตองการสรางจํานวนนับที ่ประกอบดวยเลขโดด 5 ตัวจากเลข 10 ตัวดังตอไปนี ้ 1, 1, 2, 2, 2, 3, 3, 4, 5, 6จะไดจํานวนที ่แตกตาง

กันกี ่จํานวน

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8. บรจิาคหนังสอื 6 เลมใหเด็กชนบท 3 คน โดยที ่แตละคนจะไดรับหนังสอือยางนอยคนละ 1 เลม จะบรจิาคไดทั ้งหมดกี ่วิธี

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9. มเีหรยีญวงกลมพลาสติก 10 เหรียญ เปนเหรียญสีขาวเหมอืนกัน 5 เหรยีญ อกี 5 เหรียญมสีีแตกตางกัน คือ เขียว, เหลอืง, น ้ําเงนิ,

ฟา,สม จํานวนวิธีเลอืกเหรยีญมา 3 เหรยีญ จะเลอืกได x วิธี นําทั ้ง x วิธีมาเขียนใสกระดาษแข็งรูปสี ่เหลี ่ยมมมุฉากแผนละ 1 วิธี (

เขียนเปนรูปเหรียญวงกลม 3 เหรียญ) จะได x แผน แผนใดมีเหรียญสขีาวใหแตมสแีดงที ่มมุบนดานขวาของกระดาษ แผนใดไมมี

เหรียญสขีาวใหแตมสีมวงที ่มมุบนดานขวา ในที ่สุดก็จะไดกระดาษแตมสแีดงและสมีวงรวมกัน x แผน หลังจากนั ้นแบงกระดาษ

ทั ้งหมดออกเปน 2 กอง กองละเทาๆกัน โดยที ่แตละกองนั ้นตองมีกระดาษที ่แตมสแีดงเทากัน และกระดาษที ่แตมสีมวงเทากัน จง

หาจํานวนวิธีทั ้งหมดในการแบงกระดาษออกเปน 2 กอง

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10. สัมประสทิธิ ์ของ x2007 ซึ ่งไดจากการกระจาย 1 x x

2 x

2550 x−1

2550 เปนเทาใด

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11. จงหาผลสําเรจ็ของ i0i

1i

2i

3i

2008

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12. จงหาตัวผกผันการคูณของ 1

2−

i

2

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ชื ่อ.............................................................................................................................หอง.........................เลขที ่.........................

คําสั ่ง จงแสดงวิธีทํา เขียนเปนขั ้นตอนตามหลักคณิตศาสตร ลงในชองวางซึ ่งเวนไว ให ในทายโจทยแตละขอ

1. กําหนดสมการจุดประสงค คือ f(x,y) = a – x – 2y โดย a เปนจํานวนจริงบวก โดยมอีสมการขอจํากัดดังนี ้

x y10

y− x5

2x y30

ถาคาต่ ําสดุของ f(x,y) เทากับ 65 แลวคาสงูสดุของ f(x,y) จะเปนเทาใด

1) จากอสมการขอจํากัด เขียนกราฟไดดดังนี ้

2) จงเติมขอมลูในตารางตอไปนี ้

(x,y) f(x,y) = a – x – 2y

3) จงหาคาสงูสดุ

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2. กําหนด z เปนจํานวนเชงิซอนใดๆ โดยที ่ z =1i 1−i

21i

31 i

41i

8

1i91−i

501i

551i

60

จงหา z ⋅ z

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3. ถา 2532i

43i=abi และ a,b เปนจํานวนจริงใดๆ แลว a+b เทากับเทาใด

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4. กําหนด z =∑n=1

2551

n1 2551−2008i

20082551i n

จงหา Re(z), Im(z)

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5. จงแสดงวา nr n

r 1=n1

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6. จํานวนคี ่ที ่อยู ระหวาง 1,000 และ 9,999 ซึ ่งเลขโดดในแตละหลักแตกตางกันทั ้งหมดหรือมเีลขโดดเพียง 2 ตําแหนงเทากัน มี

ทั ้งหมดก ี่จํานวน

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7. มคีน 10 คน ในจํานวนนี ้มนีาย ก, ข, ค รวมอยู ดวย ตองการจัด 10 คน มานั ่งรอบโตะกลมเพียง 6 คน นาย ก, ข, ค ถูกเลือกมาดวย

เสมอจะมวิีธีนั ่งแตกตางกันกี ่วิธี ถานาย ก นั ่งติดกับนาย ข แต ค ไมนั ่งติดกับ นาย ก

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8. มลีูกแกวทําดวยพลาสติกสแีดงเหมอืนกัน 3 ลูก สเีหลอืงเหมอืนกัน 2 ลูก และสสีม 1 ลกู พอคานําลกูแกวทั ้ง 6 ลกูมารอยเปน

พวงมาลัยซึ ่งเหน็เปนแบบตางๆกัน นายเลขตองการซื ้อพวงมาลัยทุกแบบจะตองซื ้อทั ้งหมดกี ่พวง จงวาดรูปประกอบใหครบทกุ

แบบ

9. กําหนดเลขโดด 1, 1, 1, 2, 2, 3, 3, 3, 4, 5ตองการเลอืกตัวเลขเหลานี ้อยางนอย 1 ตัว แต ไมเกนิ 3 ตัว จะเลอืกไดกี ่วิธี

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10. สามี – ภรรยา 7 คู ไดรับเชญิใหมารวมงานสําคัญอยางหนึ ่งเพียง 4 คน ตองเปนชาย 2 คน หญงิ 2 คน จงหาจํานวนวิธี ในการ

เลอืกชาย 2 คน และหญงิ 2 คน และมเีงื ่อนไขดังตอไปนี ้

10.1 เปนสามี – ภรรยากันทั ้ง 2 คู ..............................................................................................................................................................................................................

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10.2 เปนสามี – ภรรยา 1 คู อกี 2 คนไม ใชสามี – ภรรยากัน

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10.3 ไมมี ใครเปนสามี – ภรรยากัน

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11. คนกลุ มหนึ ่งจํานวน 8 คน ในจํานวนนี ้มีนายเอ, นายบี และนายซรีวมอยู ดวย ตองการจัดคนเหลานี ้ทั ้งหมดเขาในหองพัก 3 หอง

ซึ ่งตางกันโดยมี 2 หองบรรจุคนไดหองละ 3 คน สวนอกี 1 หองบรรจุ ได 2 คนโดยนายเอและนายบตีองอยู หองเดียวกัน สวนนายซี

ตองไมอยู หองเดียวกับนายเอ จงหาจํานวนวิธีจัดคนเหลานี ้เขาพักตายเงื ่อนไขดังกลาวขางตน

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12. กําหนดเซต A={a, b, c, d, e} , B={1, 2, 3}ถา X={ f ׀ f : A→ B และ f เปนฟงกชันที ่ ไม ใช "ทั ่วถึง"} แลวจงหาจํานวนสมาชกิ

ของเซต X

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13. ถาตองการสรางคําโดยเรยีงสับเปลี ่ยนตัวอักษรทั ้งหมดในคําวา “EARLYRETIRED” ( ไมคํานงึถึงความหมาย) โดยใหตัวอักษรที ่

ไมซ ้ํากันอยู ในตําแหนงที ่ ไมติดกัน แตจะมีตัวอกีษร 3 ตัวคือ A, D, I ซึ ่งถาอานจากซายไปขวาจะตองพบ A กอน D และพบ D กอน

I เสมอ จะมวิีธีจัดเรียงตัวอักษรทั ้งหมดตามเงื ่อนไขดังกลาวกี ่วิธี

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ชื ่อ.............................................................................................................................หอง.........................เลขที ่.........................

ตอนที ่ 1 จงเติมเฉพาะคําตอบที ่ถกูตอง

1.

1) นักเรียนกลุ มหนึ ่งมี 8 คน เปนนักเรยีนชาย 3 คน และครูที ่ปรึกษา 2 คน จัดนักเรียนกลุ มนี ้พรอมครมูายนืถายรูปในแนว

เสนตรง โดยที ่ครูที ่ปรกึษา 2 คน ยืนติดกันเสมอ และนักเรยีนชายไมยนืติดกันเลยจะมกีี ่วิธี (ตอบในรปูแฟคทอเรยีล)

ตอบ .......................................................................................................................................................................................

2) นําอกัษร a, a, b, b มาเรยีงเปนวงกลม ไดทั ้งหมดกี ่วิธี

ตอบ .......................................................................................................................................................................................

3) โยนเหรยีญบาทที ่แตกตางกัน 10 เหรียญ จะมกีี ่วิธีที ่หงายหัว 3 เหรยีญ และหงายกอย 7 เหรยีญ

ตอบ .......................................................................................................................................................................................

4) ชาย 6 คน และหญงิ 6 คน นํามาจัดเปนแนววงกลม โดยชายหญงิยืนสลับเพศกันทลีะ 3 คน จะจัดไดทั ้งหมดกี ่วิธี

ตอบ .......................................................................................................................................................................................

5) แบงนักเรียน 12 คน เปน 3 กลุ ม กลุ มละ 4 คน ไปรวมงานนอกโรงเรียน 3 งานที ่แตกตางกัน จะทําไดทั ้งหมดกี ่วิธี โดย

ไปงานละ 1 กลุ ม

ตอบ .......................................................................................................................................................................................





ตอนที ่ 2 จงแสดงวิธีทํา เขียนเปนขั ้นตอนตามหลักคณิตศาสตร ลงในชองวางซึ ่งเวนไว ให ในทายโจทยแตละขอ

2. รานทําเฟอรนิเจอรแหงหนึ ่งผลติโตะและเกาอี ้ออกจําหนาย โดยมชีางที ่ทํางานอยู 5 คน แตละคนทํางานไมเกิน 40 ชั ่วโมงตอ

สัปดาห โดยใชเวลาในการผลติโตะและเกาอี ้แคตัวละ 20 ชั ่วโมง และ 10 ชั ่วโมงตามลําดับ ถาในแตละสัปดาหผลิตโตะอยางมาก 5

ตัว และผลติเกาอี ้อยางมาก 12 ตัว จงหาวา

2.1 ถาทางรานขายโตะไดกําไรตัวละ 1,200 บาท ขายเกาอี ้ ไดกําไรตัวละ 800 บาท เขาควรผลติและเกาอี ้อยางละกี ่ตัวตอ

สัปดาห จึงจะไดกําไรมากที ่สดุ

กําหนดให x แทน .......................................................................................................................................................

y แทน .......................................................................................................................................................

และ P แทน .......................................................................................................................................................

อสมการขอจํากัด คือ .............................................................................................................................................................

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เขียนกราฟของอสมการขอจํากัด เมื ่อ x และ y เปนจํานวนจริง

ฟงกชันจุดประสงค คือ ..........................................................................................................................................................

ดังนั ้นทางรานควรผลิตโตะจํานวน....................ตัว และเกาอี ้จํานวน....................ตัว จึงจะไดกําไรสงูสดุ

2.2 ถาผลกําไรจากการขายโตะแตละตัวเปนสามเทาของผลกําไรจากการขายเกาอี ้แตละตัว ทางรานจะตองผลิตโตะและ

เกาอี ้สัปดาหละกี ่ตัวจึงจะไดกําไรมากที ่สุด

ฟงกชันจุดประสงค คือ ..........................................................................................................................................................

ดังนั ้นทางรานควรผลิตโตะจํานวน....................ตัว และเกาอี ้จํานวน....................ตัว จึงจะไดกําไรสงูสดุ





3. จงหาจํานวนวิธีในการทาสรีปูวงกลม 4 วงที ่คลองกันดังรูป(ที ่พลิกไม ได) ดวยสแีดง สีเหลือง สนี ้ําเงนิ สีฟา และสสีม โดยมี

เงื ่อนไขวาตองมีอยางนอย 2 วง ที ่มสีีเหมอืนกัน

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4. ตองการสรางจํานวนนับที ่มคีามากกวา 3 ลาน จากตัวเลข ดังตอไปนี ้ 0, 1, 1, 3, 3, 3, 5จะไดจํานวนที ่แตกตางกันกี ่จํานวน

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5. กระดาษแข็งรปูวงกลม ดานหนามรีูปดาว 5 แฉกในวงกลมดังรูป โดยดานหลังเปนพื ้นขาวไมมรีูปใดๆ จะระบายสเีฉพาะดานหนา

ไดกี ่วิธีถาตองระบายตามเงื ่อนไข 3 ขอ ตอไปนี ้

1.ระบายพื ้นวงกลม 5 บรเิวณที ่อยู นอกรูปดาวดวยสดํีา หรือ สีขาว สี ใดสีหนึ ่งเทานั ้น

2. ระบายกลางรปูดาวซึ ่งเปนรูป 5 เหลี ่ยมดานเทา มุมเทา ดวยสี ใดสีหนึ ่งจากแมสคืีอ สแีดง สเีหลอืง หรอืสนี ้ําเงนิ

3. รูปดาวในสวนที ่เปนรูปสามเหลี ่ยมแตละรปูระบายสตีางกันทุกรปู ดวยสีของรุ งกนิน ้ํา (มวง, คราม, น ้ําเงนิ, เขียว,

เหลอืง, แสด, แดง) และตองสตีางกันกับรปูหาเหลี ่ยมดานเทา มมุเทา ตรงกลางดวย

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6. คนกลุ มหนึ ่งประกอบดวย ผู ชาย 4 คน ผู หญงิ 4 คน และเด็ก 1 คน ใหทกุคนนั ่งรอบโตะกลม ถาผู ชายและผู หญงินั ่งขนาบเด็กคน

นี ้และถาผู ชายและผู หญงินั ่งสลับเพศกัน เพศละ 1 คน จะไดกี ่วิธี

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7. คนกลุ มหนึ ่งมี 7 คน ประกอบดวย นิด และ หนอย กับเพื ่อนอกี 5 คน จะเขาหองพักในบานหลังหนึ ่งซึ ่งมี 3 หองนอน หองแรก

พักได 3 คน หองที ่สองกับหองที ่สามพักไดหองละ 2 คน จะจัดคนทั ้งหมดเขาพักไดกี ่วิธี ถาตองจัดใหนดิและหนอยพักหองเดียวกัน

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8. จงกระจาย a2b−3c4d 5 อยู ในรูปผลสําเรจ็ จํานวนพจนที ่ ไดจากการกระจายมกีี ่พจน

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9. จากการกระจาย x3

2–

1

x3

14

จงแสดงวิธีทําเพื ่อหาพจนที ่มตัีวแปร x−3

2

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10. กําหนดให k = 3

2 5

−5 3

2 4

1

2 10 3

2 3

1

2 2

−10 3

2 2

1

2 3

5 3

2 1

2 4

– 1

2 5

จงแสดงวิธีทํา เพื ่อหาคา k 3−10

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11. จงแสดงวิธีหาจํานวนในแตละขอตอไปนี ้ในรูป abi เมื ่อ a , b∈ℝ

11.1 ถา z 1=2−i และ z 2=−86i จงหา z 2 z

1

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11.2 ถา 2z2−2z5=0 แลว จงหา z

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11.3 ถา z 2=3−4i แลว จงหา z

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ชื ่อ.............................................................................................................................หอง.........................เลขที ่.........................

ตอนที ่ 1 จงเติมเฉพาะคําตอบที ่ถกูตอง

1. คาของ n จากสมการ n n2−1n2−43!−1=11!22!33!77!

ตอบ .....................................................................................................................................................................................................

2. สรางจํานวนที ่มคีามากกวา 300 แต ไมเกนิ 5 หลัก จากเลขโดด 0 ถึง 5 โดยเลขโดดในแตละหลักไมซ ้ํากัน ไดทั ้งหมดกี ่วิธี

ตอบ .....................................................................................................................................................................................................

3. นําตัวอักษรทั ้งหมดในคําวา “PROPORTION” มาเรยีงสับเปลี ่ยน โดยให P แยกกัน แต O อยู ติดกัน 3 ตัวไม ได จะเรยีงสับเปลี ่ยน

ไดทั ้งหมดกี ่วิธี

ตอบ .....................................................................................................................................................................................................

4. คนกลุ มหนึ ่งเปนชาย 7 คน หญงิ 5 คน และเด็ก 3 คน ถาตองการเลอืกชาย 4 คน หญงิ 4 คนและ เด็ก 2 คน มานั ่งลอมเปนวงกลม

โดยใหชายและหญงินั ่งสลับกันทีละ 1 คนและเด็ก 2 คน นั ่งไมติดกัน จะทําไดกี ่วิธี

ตอบ .....................................................................................................................................................................................................

5. วิธีจัดคน 6 คนเขาพักในหองซึ ่งเหมอืนกัน 3 หอง โดยแตละหองมีคนอยู อยางนอย 1 คน มทัี ้งหมดกี ่วิธี

ตอบ .....................................................................................................................................................................................................

6. มหีนังสือคณิตศาสตรเหมอืนกัน 5 เลม หนังสอืเคมีตางกัน 3 เลม หนังสือฟสกิสเหมือนกัน 2 เลม ตองการนําหนังสอืทั ้งหมดจัด

เรยีงบนชั ้นหนังสอืตามแนวเสนตรงโดยหนังสอืฟสิกสตองอยู ติดกันและหนังสอืเคมตีองไมอยู ติดกัน จะเรยีงหนังสอืทั ้งหมดไดกี ่

วิธี

ตอบ .....................................................................................................................................................................................................

7. นักเรียนชั ้นมธัยมศึกษาตอนปลายของโรงเรยีนแหงหนึ ่งมีจํานวน 240 คน จะแบงนักเรียนทั ้งหมดออกเปน 5 กลุ ม กลุ มละ 40

คนมจํีานวน 3 กลุ ม และกลุ มละ 60 คนมจํีานวน 2 กลุ ม จะแบงไดทั ้งหมดกี ่วิธี (ตอบในรปูแฟคทอเรยีล)

ตอบ .....................................................................................................................................................................................................





8. มนี ้ําอดัลมขนาด 16 ออนซ ชนดิเดียวกันและเหมอืนกันจํานวน 12 ขวด นําไปแจกหมดใหเด็ก 3 คน โดยแบงน ้ําอดัลมเปน 3 ขวด

4 ขวด และ 5 ขวด จะทําไดกี ่วิธี

ตอบ .....................................................................................................................................................................................................

9. จงหาพจนที ่ไมมี x จากการระจาย 3

2x– 4x

26

ตอบ .....................................................................................................................................................................................................

10. สัมประสทิธิ ์ของ x9 จากการกระจาย 1− x 2− x

23x−4x35 เทากับเทาใด

ตอบ .....................................................................................................................................................................................................

11. กําหนดใหN =a

3a

2a

1a

0

เปนจํานวนเต็มบวก 4 หลัก ซึ ่งสําหรับ i=0, 1, 2, 3 และ0a

i9

ถา N มคีาไมมากกวา

5,000 และ a0a1a2a3 แลวมี N ที ่แตกตางกันกี ่จํานวน

ตอบ .....................................................................................................................................................................................................

12. ถา z เปนจํานวนเชงิซอนซึ ่ง z =i

2−i

3−4i

2iแลวคาของ Re(z)-Im(z) เทากับเทาใด

ตอบ .....................................................................................................................................................................................................

13. ให z เปนจํานวนเชิงซอน ซึ ่ง z =12i3i24i350i49 เขียน z ในรปู a+bi เทากับเทาใด

ตอบ .....................................................................................................................................................................................................

ตอนที ่ 2 จงแสดงวิธีทํา เขียนเปนขั ้นตอนตามหลักคณิตศาสตร ลงในชองวางซึ ่งเวนไว ให ในทายโจทยแตละขอ

14. ให z เปนจํานวนเชิงซอนซึ ่ง Im(z)=2553 ถา n เปนจํานวนจรงิซึ ่งz

z n=3i แลว จงแสดงวิธีหาคา n

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