數學(from 4)

見到樓上個位好多都錯=.=我幫你由頭計過啦

1) 9.7=9.7x10/10=97/10
2) 64i*****/4i** = 16i*** = -16i
5) 1/(8-7i) = (8+7i)/[(8-7i)(8+7i)=(8+7i)/(64+49)=8/113+7i/113
so, A=8/113, B=7/113
6) (4-5i)-(-2+3i)(1-i)=(4-5i)-(-2+2i+3i+3)=(4+2-3)+(-5-2-3)i=3-10i
7) (6+5i)-(a+bi)=-2+3i
6-a=-2, 5-b=3,a=8,b=2
8) (4,-5)平行y軸的直線方程
x=4
9) (-3,1)和(5,2)的直線方程
(y-2)/(x-5)=(2-1)/(5+3)
(y-2)/(x-5)=1/8
8y-16=x-5
y=x/8+11/8
10) (-2,1)而斜率是5的直線方程
(y-1)/(x+2)=5
y-1=5x+10
y=5x+11

2009-12-19 13:30:53 補充：
11) 直線的x截距是-6而y截距是3,直線方程
for point (-6,0) & (0,3)
(y-0)/(x+6)=(3-0)/(0+6)
2y=x+6
y=x/2+3
12) 直線2x-y-4=0的x截距和y截距
for x=0, -y-4=0, y=-4(y截距)
for y=0, 2x-4=0, x=2(x截距)
13) 直線L過(0,1) 平行直線4x-3y-2=0,L的方程
4x-3y-2=0
3y=4x-2
y=4x/3-2/3
斜率=4/3
L: (y-1)/(x-0)=4/3
y-1=4x/3
y=4x/3+1

2009-12-19 13:31:21 補充：
14) 直線L1:8x+14y+3=0和L2:4x-7y+63=0的交點數目
L1: 14y=-8x-3
y=-4x/7-3/14 (斜率=-4/7)
L2: 7y=4x+63
y=4x/7+9(斜率=4/7)
由於都是直線, 斜率不同, 所以只有一個交點
15) 過(2,6)垂直直線x+5y-9=0的直線方程
x+5y-9=0
y=-x/5+9/5 (斜率= -1/5)
垂直線斜率=(-1)/(-1/5)=5
直線: (y-6)/(x-2)=5
y-6=5x-10
y=5x-4

2009-12-19 13:31:33 補充：
16) 因息分解:
a) 2x**-8x=0
x(2x-8)=0
x=0 or 4
b) y**-6y-16=0
(y-8)(y+2)=0
y=8 or -2
c) x**+2x-15=0
(x+5)(x-3)=0
x=-5 or 3
d) (2x-1)(x+1)=(x+1)
x+1=0 or 2x-1=1
x=-1 or 1

2009-12-19 13:31:58 補充：
17) 二次公息解
a) 12x**-x-35=0
(4x-7)(3x+5)=0
x=7/4 or -5/3
b) x**-3x+5=0
x = { -(-3)+√[ (-3)**-4(5) ]} / 2 or { -(-3) - √[ (-3)**-4(5) ]} / 2
D=[ (-3)**-4(5) ] =9-20=-11<0, no root
c) x(5x+3)=2(1+x)
5x**+3x-2x-2=0
5x**+x-2=0
D=[ (1)**-4(5)(-2) ]=41
x = (-1+√41)/10 or (-1-√41)/10

2009-12-19 13:32:21 補充：
17d) x(2x+3)=(2-x)(2x+3)
2x**+3x-4x-6+2x**+3x=0
4x**+2x-6=0
2x**+x-3=0
D=[ (1)**-4(2)(-6) ]=49
x=(-1+√49)/4 or (-1-√49)/4
x=3/2 or -2
18) 計算8x**-3x+5=0的判別息
D=[ (-3)**-4(8)(5) ]=9-160=-151
19) x**+8x+(k-1)=0有2個相等實根,k的值
D=[ (8)**-4(1)(k-1) ]=0
64-4k+4=0
k=-17

2009-12-19 13:32:30 補充：
20) x**+6x+k=0沒有實根,k的範圍
D=[ (6)**-4(1)(k) ] <0
36 < -4k
k< -9
21) 梯形的上低是1cm ,下低是x cm,高是(x+1)cm,梯形的面積是4cm**,x
(1+x)(x+1)/2 =4
x**+2x+1=8
x**+2x-7=0
D=[ (2)**-4(1)(-7) ] =32
x= { -(2)+2√8} / 2 or { -(2)-2√8} / 2
x = -1+√8 or -1-√8

2009-12-19 13:32:37 補充：
22) 若f(x)=2x-5,
c) f[f(1)]
f(1)=2(1)-5=-3
f[f(1)]=f(-3)=2(-3)-5=1
23) f(x)=x**+ax+2和f(2)=0
a) f(2)=4+2a+2=0
a=-3
b) n令f(n)=0,其中n不等於2
f(n)=n**-3n+2=0
(n-1)(n-2)=0
n=1 or 2(ignored)
n=1

2009-12-19 13:32:45 補充：
24) 若g(x)=kx**-2x-3和g(3)=g(3分1),k=
g(3)=9k-6-3=9(k-1)
g(1/3)=k/9-2/3-3=1/9(k-6-27)=1/9(k-33)
g(3)=g(1/3)
9(k-1)=1/9(k-33)
81(k-1)=(k-33)
81k-81 - (k-33) = 0
80k-48=0
5k-3=0
k=3/5

2009-12-19 13:32:51 補充：
25) 若f(x)=x**+2x-3,則f(x+1)+f(-x)=
f(x+1)=(x+1)**+2(x+1)-3
=x**+2x+1+2x+2-3
=x**+4x

f(-x)=(-x)**+2(-x)-3
=x**-2x-3

f(x+1)+f(-x) = (x**+4x) + (x**-2x-3)
=2x**+2x-3

(我唔sure岩唔岩...)
/ = 分數線

1) 97/100

2) 64i*****=64i --- 4i**=4(-1)=-4
64i*****/4i**
= 64i/-4
= -16i

5) 8-7i分1 = 8-7i = 8+(-7i)

6) (4-5i)-(-2+3i)(1-i) = (4-5i)-[1(-2+3i)+i(-2+3i)]
= (4-5i)-(-5+i)
=9-6i OR 3(3-2i)

7 (6+5i)-(a+bi) = -2+3i
-(a+bi) = -2+3i-6-5i
-(a+bi) = -(4+2i)
∴a=4 and b=2

8) (4,-5)平行y軸的直線方程
x=4

9) (-3,1)和(5,2)的斜率 = (1-2)/(-3-5) = 1/8
方程 = (y-2)/(x-5) = (1-2)/(-3-5)
(y-2)/(x-5)=1/8
8y-16 =x-5
∴方程 : x-8y+11=0

10) (-2,1)而斜率是5的直線方程︰
(y-1) / [(x-(-2)] = 5
y-1 = 5x+10
∴5x-y+11 = 0


11) 直線的x截距是-6而y截距是3,直線方程
∵直線通過(-6,0) 和 (0,3)
斜率 : (0-3)/(-6-3) = 1/3
方程 : (y-3)/(x-0) = 1/3
3y-9 = x
∴x-3y+9=0

12) 直線2x-y-4=0的x截距和y截距
代x=0入2x-y-4 = 0
y-4 = 0
y = 4

代y=0入2x-y-4 = 0
2x-4 = 0
x = 2

13) 直線L過(0,1) 平行直線4x-3y-2=0, L的方程
4x-3y-2=0的斜率 : (-4)/(-3) = 4/3
∵L⊥直線4x-3y-2 = 0
∴(y-1)/(x-0) = 4/3
3y-3 = 4x
L的方程 : 4x-3y+3 = 0

14) 無交點(L1∥L2)

15) 過(2,6)垂直直線x+5y-9=0的直線方程
直線x+5y-9=0斜率 :　(-1)/5
∵方程⊥直線x+5y-9=0
∴斜率 = -1(-1/5) = 5
(y-6)/(x-2) = 5
5y-30 =x-2
方程 : x-5y+28=0

16) 因息分解:
a) 2x**-8x = 0
(2x)(x-4) = 0
b) y**-6y-16 = 0
(y+2)(y-8) = 0

c) x**+2x-15=0
(x+5)(x-3) = 0

d) (2x-1)(x+1) = (x+1)
(2x-1+1)(x+1) = 0
2x(x+1) = 0

17) 二次公息解(答案<根息>):

a) 12x**-x-35 = 0
(x+3/5)(x-7/4) = 0
根是 -3/5 , 7/4

b) x**-3x+5=0 (用配方法)
x = {-(-3)√[(-3)**-4(1)(5)]} / 2(1)
=[3√(11)i] / 2
(無實根)


c) x(5x+3) = 2(1+x) (用配方法)
5x**+x-2 = 0
……
x = (-1√41)/10

d) x(2x+3) = (2-x)(2x+3)
……
4x**+4x-6 = 0
……
x = (-1√7)/2

18) 計算8x**-3x+5=0的判別息 (配方法)
△ = (-3)**-4(8)(5)
= 9-160
= -151
19) x**+8x+(k-1)=0有2個相等實根,k的值
∵x**+8x+(k-1)=0有2個相等實根
∴△ = 8**-4(1)(k-1) = 0
k-1 = -64/-4
k = 17

20) x**+6x+k=0沒有實根,k的範圍
∵x**+6x+k=0沒有實根
∴△ = 6**-4(1)(k) ＜ 0
k ＜ -36/-4
k ＜ 9

21) 梯形的上低是1cm ,下低是x cm,高是(x+1)cm,梯形的面積是4cm**,x
(1+x)(x+1)2 = 4
(1+x)** = 8
1+x = √8
x = (√8)-1

22) 若f(x)=2x-5,

c) f[f(1)]
= f[2(1)-5]
= f(-3)
= 2(-3)-5
= -11

23) f(x)=x**+ax+2和f(2)=0
a) a的值;
f(2) = (2)**+a(2)+2 = 0
2a = -6
a = -3

b) n令f(n)=0,其中n不等於2
f(n) = (n)**-3(n)+2 = 0
n**-3n+2 = 0 (配方法or十字相乘)
(n-1)(n-2) = 0
n = 2 , 1