ม.5ตามผลการเร�ยนร กลมสาระการเร�ยนรคณตศาสตร (ฉบบปรบปรง พ.ศ. 2560)
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เลม 2
คณตศาสตร
ดร.อำพล ธรรมเจร�ญ
คณ
ตศาส
ตร ม
.5 เล
ม 2
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มเต
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ชนมธยมศกษาปท 5 เลม 2
แบบฝกหด รายว�ชาเพมเตม
คณตศาสตร
พฒนา ทกษะการคดขนสง และนำไปใชไดจร�ง
พรอมดวย แนวขอสอบเขาศกษาตอในระดบอดมศกษา
เพอใหผเร�ยนไดฝกทำกอนสอบจร�ง
เพม แบบทดสอบประจำหนวยการเร�ยนร ประจำภาคเร�ยนและแบบทดสอบพสจนศกยภาพ เพอวดความรความเขาใจของผเร�ยน
เพม สถานการณปญหา Financial Literacy เพอบรณาการ
คณตศาสตรกบความรทางการเง�น
พฒนา ทกษะกระบวนการทางคณตศาสตร
และทกษะแหงศตวรรษท 21 ดวยกจกรรมทหลากหลาย
11
1 3 3 11 4 6 4 1
1211
บฝ. คณตศาสตร ม.5 ล.2
78.-8 8 5 8 6 4 9 1 4 3 0 4 4
หนวยกำรเรยนรท 1จ�ำนวนเชงซอน 2
1.1 จ�ำนวนเชงซอน 2
1.2 สมบตเชงพชคณตของจ�ำนวนเชงซอน 5
1.3 กรำฟและคำสมบรณของ 16
จ�ำนวนเชงซอน
1.4 รำกทสองของจ�ำนวนเชงซอน 28
1.5 จ�ำนวนเชงซอนในรปเชงขว 33
1.6 รำกทnของจ�ำนวนเชงซอน 37
1.7 สมกำรพหนำม 43
Unit Test แบบทดสอบหนวยกำรเรยนรท 1 52
Challenging Test แบบทดสอบพสจนศกยภำพ 56
หนวยกำรเรยนรท 2หลกกำรนบเบองตน 58
2.1 กฎเกณฑเบองตนเกยวกบกำรนบ 58
2.2 กำรเรยงสบเปลยน 70
2.3 กำรจดหม 82
2.4 ทฤษฎบททวนำม 95
Math in Real Life คณตศำสตรในชวตจรง 100
Unit Test แบบทดสอบหนวยกำรเรยนรท 2 101
Challenging Test แบบทดสอบพสจนศกยภำพ 106
สำรบญ
หนวยกำรเรยนรท 3ควำมนำจะเปน 108
3.1 กำรทดลองสมและเหตกำรณ 108
3.2 ควำมนำจะเปนของเหตกำรณ 111
Math in Real Life คณตศำสตรในชวตจรง 143
Unit Test แบบทดสอบหนวยกำรเรยนรท 3 144
Challenging Test แบบทดสอบพสจนศกยภำพ 149
Financial Literacy 151
แบบทดสอบประจ�ำภำคเรยน 153
จานวนเชงซอนหนวยการเรยนรท 1
1.1 จานวนเชงซอน (Complex Number) 1.1 จานวนเชงซอน
ผลการเรยนร
• เขำใจจ�ำนวนเชงซอนและใชสมบตของจ�ำนวนเชงซอนในกำรแกปญหำ • หำรำกทnของจ�ำนวนเชงซอนเมอnเปนจ�ำนวนนบทมำกกวำ1 • แกสมกำรพหนำมตวแปรเดยวดกรไมเกนสทมสมประสทธเปนจ�ำนวนเตมและน�ำไปใชในกำรแกปญหำ
จ�ำนวนเชงซอนzคอจ�ำนวนทอยในรป
z=a+biหรอz=(a,b)
เมอa,b∊RโดยทRe(z)=aและIm(z)=b
• i=-1หรอi2=-1
•ถำaเปนจ�ำนวนจรงบวกแลว-a=ai
• เซตของจ�ำนวนเชงซอนเขยนแทนดวยสญลกษณC
ดงนนC={(a,b)∙a∊Rและb∊R }
• เซตของจ�ำนวนจรงเปนสบเซตของจ�ำนวนเชงซอนนนคอR⊂C • ระบบจ�ำนวนเชงซอนมสมบตกำรเทำกนของจ�ำนวนเชงซอนสองจ�ำนวน
แตไมมกำรเปรยบเทยบจ�ำนวนเชงซอนใดมำกกวำหรอนอยกวำ
•ควำมสมพนธระหวำงจ�ำนวนจรงและจ�ำนวนเชงซอน
ถำb=0 จะไดวำ zเปนจ�ำนวนจรง
ถำa=0 แตb≠0 จะไดวำ zเปนจ�ำนวนจนตภำพแท
•กำรหำคำin
in=1 เมอn÷4เหลอเศษ0(หำรลงตว)
in=i เมอn÷4เหลอเศษ1
in=-1เมอn÷4เหลอเศษ2
in=-i เมอn÷4เหลอเศษ3
2
1. บอกคาของRe(z)และIm(z)ของจ�านวนเชงซอนzตอไปน
1) 4-8i
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2) -5
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3) 3+8i
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4) 4i
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5) 35 -
52 i
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6) 3+33i
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2. หาคาของ
1) i89
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2) i635
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3) i46 +i87
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4) i83468
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5) i1286
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6) 4i92-6i7
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3. หาคาของ
1) i+i2+i4+i6+i7+i9+i11
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2) i2 •i3 •i4 •i5
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4. เขยนจ�านวนเชงซอนทก�าหนดใหอยในรปจ�านวนจนตภาพ
1) -4 ............................................................................................................................................................................................................
2) -100 ............................................................................................................................................................................................................
3) 1+-9 ............................................................................................................................................................................................................
4) -4--1 ............................................................................................................................................................................................................
Exercise 1.1E
x e
r c i s
e 1.1
3
5. เขยนเครองหมาย✓ลงในชองเพอจ�าแนกวาจ�านวนทก�าหนดเปนจ�านวนชนดใด
จ�านวน
จ�านวนเชงซอน
จ�านวนจนตภาพแท
จ�านวนจรง
จ�านวนตรรกยะ
จ�านวนอตรรกยะ
จ�านวนเตม
-8-5i
2
-9i
3
2+i
14-3
0
6. หาเซตค�าตอบของสมการx2 - 1 = 0
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7. หาเซตค�าตอบของสมการx2 + 1 = 0
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E x
e r c i s
e 1.1
4
1.2 สมบตเชงพชคณตของจานวนเชงซอน(Algebraic Properties of Complex Number)
1.2 สมบตเชงพชคณตของจานวนเชงซอน
1. สมบตเชงพชคณตของจ�านวนเชงซอน
สมบตเชงพชคณตของจ�านวนเชงซอน
ก�ำหนดz1=a+biและz2=c+diเมอa,b,cและdเปนจ�ำนวนจรง
1. กำรเทำกน
z1=z2หรอa+bi=c+diกตอเมอa=cและb=d
2. กำรบวก
z1+z2=(a+bi)+(c+di)=(a+c)+(b+d)i
3. กำรคณจ�ำนวนเชงซอนกบคำคงตว
kz1=k(a+bi)=ka+kbiเมอkเปนคำคงตว
4. กำรคณจ�ำนวนเชงซอนกบจ�ำนวนเชงซอน
z1z2=(a+bi)(c+di)=(ac-bd)+(ad+bc)i
2. สมบตทเกยวของกบการบวกของจ�านวนเชงซอน
สมบตทเกยวของกบการบวกของจ�านวนเชงซอน
ก�ำหนดz,z1,z2และz3เปนจ�ำนวนเชงซอน
1. สมบตปดของกำรบวก
z1+z2เปนจ�ำนวนเชงซอน
2. สมบตกำรสลบทของกำรบวก
z1+z2=z2+z1 3. สมบตกำรเปลยนหมของกำรบวก
(z1+z2)+z3=z1+(z2+z3)
4. สมบตกำรมเอกลกษณของกำรบวก
มจ�ำนวนเชงซอน0ซงท�ำให0+z=z+0=z
5. สมบตกำรมตวผกผนของกำรบวก
มจ�ำนวนเชงซอน-zซงท�ำใหz+(-z)=(-z)+z=0
โดย-z=(-a,-b)
5
3. การลบจ�านวนเชงซอน
ก�ำหนดz1และz
2เปนจ�ำนวนเชงซอนใดๆ จะไดวำ
z1-z
2=z
1+(-z
2) = (a + bi) + [-(c + di)] = (a - c) + (b - d)i
4. สมบตทเกยวของกบการคณของจ�านวนเชงซอน
สมบตทเกยวของกบการคณของจ�านวนเชงซอน
ก�ำหนด z, z1, z2 และ z3 เปนจ�ำนวนเชงซอน
1. สมบตปดของกำรคณ
z1z2 เปนจ�ำนวนเชงซอน
2. สมบตกำรสลบทของกำรคณ
z1z2 = z2z1
3. สมบตกำรเปลยนหมของกำรคณ
(z1z2)z3 = z1(z2z3)
4. สมบตกำรมเอกลกษณของกำรคณ
มจ�ำนวนเชงซอน 1 ซงท�ำให 1 • z = z • 1 = z
5. สมบตกำรมตวผกผนของกำรคณ
มจ�ำนวนเชงซอน z-1 ซงท�ำให z • z-1 = z-1 • z = 1 โดยท z-1 = ( aa2 + b2, -
ba2 + b2)
6. สมบตกำรแจกแจง
z1(z2 + z3) = z1z2 + z1z3 และ (z1 + z2)z3 = z1z3 + z2z3
5. การหารจ�านวนเชงซอน
z1z2
= (a + bi) ( c - dic2 + d2) = ac + bdc2 + d2 + (bc - adc2 + d2 )i
6. สงยคของจ�านวนเชงซอน(ConjugateofaComplexNumber)
สงยคของจ�ำนวนเชงซอน คอ z = a - bi
สมบตของสงยคของจ�านวนเชงซอน
ก�ำหนด z, z1 และ z2 เปนจ�ำนวนเชงซอน
1. z = z
2. zz = a2 + b2
3. Re(z) = 12 (z + z)
4. Im(z) = 12i (z - z)
5. z1 ± z2 = z1 ± z2
6. z1 z2 = z1 z2
7. (z1z2
) = z1z2
เมอ z2 ≠ 0
8. ถำ z ≠ 0 แลว1z = (1z)6
1. หาจ�านวนจรงxและyในแตละขอตอไปน
1) 3x-4yi=9+12i ......................................................................................................
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2) 3x+yi=12 ......................................................................................................
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3) x+y+2xyi=7+20i ......................................................................................................
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4) 5x+(x+y)i=3+2i ......................................................................................................
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2. หาจ�านวนจรงaและbทท�าให(a+bi)+(2+5i)=3-i
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3. หาผลบวกของจ�านวนเชงซอนตอไปน
1) (-8,1)กบ(4,2)
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2) (2-4i)+(1-3i)+(-2+3i)
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E x
e r c i s
e 1.2
Exercise 1.2
7
4. หาผลคณของจ�านวนเชงซอนตอไปน
1) (-6,-5)(-6,5)
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2) (23+3i)(-63+5i)
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3) (2+5i)(1+i)(-4)
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5. หาผลลบของจ�านวนเชงซอนตอไปน
1) (4-5i)-(2+2i)
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2) (-3+2i)-(4+i)
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3) (-3-2i)-(-4-i)
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E x
e r c i s
e 1.2
8
6. หาผลหารของจ�านวนเชงซอนตอไปน
1) 5+9i2i
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2) (9-3i) ÷ (2+8i)
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7. ก�าหนดใหz1=2-iและz2=3+2iหาผลลพธของจ�านวนตอไปน
1) z1+z2 ........................................................................................................................................................................................................................................
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2) z1-z2 ........................................................................................................................................................................................................................................
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3) z1z2 ........................................................................................................................................................................................................................................
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E x
e r c i s
e 1.2
9
4)z1z2
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8. เขยนผลลพธใหอยในรปa+biเมอaและbเปนจ�านวนจรง
1) (5+i)-(5-i) .........................................................................................................................................................................
2) (4-5i)+(8-9i) .........................................................................................................................................................................
3) (4-i)-(-4-i) .........................................................................................................................................................................
4) (3- 2 i)+(4-8 i) .........................................................................................................................................................................
5) -(7-5i)+(2+3i) .........................................................................................................................................................................
6) 3(1-2i)-(4-5i) .........................................................................................................................................................................
7) i2(4-5i)
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8) i(4-2i)
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9) 5(i-4)-4(i-6)
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10) 13 (6-5i)-53 (3-i)
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11) -2(2i-3)
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12) 3i(2i+23)
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E x
e r c i s
e 1.2
10
