DSE數,求解

1.
(a)
-9i * 5i
= -45i²
= -45 * (-1)
= 45

(b)
1/(5 - i)
= (5 + i)/(5 - i)(5 + i)
= (5 + i)/(25 - i²)
= (5 + i)/26


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2.
(y - 2)/(x + 1) = (-3 -2)/(5 + 1)
(y - 2)/(x + 1) = -5/6
6y - 12 = -5x - 5
5x + 6y - 7 = 0


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3.
x - 3y + 6 = 0
3y = x + 6
y = (1/3)x + 2

斜率 = 1/3
y 截距 = 2


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4.
(a)
x² - 12x + 36 = 0
(x - 6)² = 0
x = 6 (重根)


(b)
2x² + x + 3 = 0
x = [-1 ± √(1² - 4*2*3)]/(2*2)
x = (-1 + i√23)/4 或 x = (-1 - i√23)/4


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5.
x² - (-6)x + 7 = 0
x² + 6x + 7 = 0


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6.
F(x) = 2/(1 + x)
F(1) = 2/(1 + 1) = 1
F(-2) = 2/(1 - 2) = -2

3F(1) - 2F(-2)
= 3*1 - 2*(-2)
= 7


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7.
x² - 8x
= [x² - 8x + (8/2)²] - (8/2)²
= (x - 4)² - 16


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8.
所求的餘數
= f(-1/2)
= (-1/2)³ - 8(-1/2) - 6
= (-1/8) + 4 - 6
= -17/8


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9.
[(a^1/2)/b^3]^-1/3
= (a^1/2)^(-1/3)/(b^3)^(-1/3)
= [a^(-1/6)]/[b^(-1)]
= b/a^(1/6)


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10.
log4(2x + 1) = 2log5
4^(2log5) = 2x + 1
2x = 4^(2log5) - 1
x = [4^(2log5) - 1]/2
x ≈ 2.972


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11.
H.C.F = x - 1

L.C.M. = x²(x - 1)²(x + 1)²


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12.
兩斜率的積 = -1
(-3/2) * [-4/(-k)] = -1
-6/k = -1
k = 6


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13.
3x² - 4x + k = 3
3x² - 4x + (k - 3) = 0

判別式 Δ ≥ 0
(-4)² - 4*3*(k - 3) ≥ 0
16 - 12k + 36 ≥ 0
12k ≤ 52
k ≤ 13/3


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14.
把 x = 3 代入方程式中：
9(3)² + k(3) - 6 = 0
3k + 75 = 0
k = 25


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15.
x² - 3x + 4 = 0
兩根之和：α + β = 3
兩根之積：αβ = 4

α² + β²
= (α + β)² - 2αβ
= (3)² - 2(4)
= 1

(α - β)²
= (α² + β²) - 2αβ
= 1 - 2(4)
= -7

所求方程：
(x - 1)(x + 7) = 0
x² + 6x - 7 = 0


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16.
f(x) = x³ + 5x² + k

f(-1) = 1
(-1)³ + 5(-1)² + k = 1
k = -3

f(x) 除以 x - 1 時的餘數
= f(1)
= (1)³ + 5(1)² - 3
= 3


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17.
實際面積
= 2.5 * 6000² * (1/100)²
= 9000 m²


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18.
16^9 + 16^4 + 44
= 16^9 + 16^4 + 2*16 + 12
= 100001002C(16)

2012-04-17 18:32:08 補充：
14. 答案應為 k = -25，打字是漏了負號。

2012-04-17 18:32:28 補充：
在002 fighting gravity 的回答中，有以下錯誤：

1.(b) 的答案漏了括號，答案應是 (5 + i)/26 或 (5/26) + (i/26)

4.(b) 的公式， x = x = [-1 ± √(1² - 4*2*3)]/(2*2)，而不是 x = [-1 ± √(1² - 4*3)]/(2*2)。這導致答案錯誤。

6. 3(2/2)-2(2/-1) 等於 7 而非 -1。

17. 1/6000是長度的比例，面積比例是 (1/6000)²
農地實際面積應是 6000²*2.5/100² = 9000 m^2

1.(a)
-9i*5i
=45
1.(b)
1/5-i
=1/5-i * 5+i/5+i
=5+i/26
2.
P(-1,2),Q(5,-3)。
y=(-3-2)/(5+1)x+c
y=-5/6x+c i
代P(-1,2)入i
2=(-5/6)(-1)+c
c=7/6
y=-5/6x+7/6
6y=-5x+7
5x+6y-7=0

3.
x-3y+6 = 0
-3y=-x-6
y=(1/3)x+2
斜率=1/3
y截距=2

4.(a)
x^2-12x+36= 0
x=6
4.(b) 2x^2+x+3=0
x=(-1+11i)/4 or x=(-1-11i)/4
5.
X^2+6x+7=0

6.
F(x)= 2/1+x
3F(1)-2F(-2)=3(2/2)-2(2/-1)
=-1

7.
x^2-8x
h=-b/2a k=(4ac-b^2)/4a
h=8/(2*1) k=(4*1*0-64)/(4*1)
h=4 k=-16
x^2-8x=(x-4)^2-16

8.
餘數 =f(x)=x^3-8x-6 2x+1
f(-1/2)=(-1/2)^3-8(-1/2)-6 =-17/8

9.
[(a^1/2/)/b^3]^-1/3
=(a^-1/6)/(b^-3/3)
=(b^1)/(a^1/6)

10.
log4(2x+1)= 2log5
log(2x+1)=(log25)(log4)
2x+1=10^[(log25)(log4)]
x=2.97

11.
(x+1)^2(x-1)和x^2(x-1)^2
H.C.F=(x-1)
L.C.M=x^2(x+1)^2(x-1)^2

12.
3x+2y-1=0 i
4x-ky+1=0 ii
由i, 2y=-3x+1
y=-3/2x+1/2 iii
由ii -ky=-4x-1
y=4/kx+1/k iv
(-3/2)(4/k)=-1
4/k=2/3
2*3=k
k=6

13.
3x^2-4x+k=3
16-4(3)(k-3)>=0
12(k-3)=<16
k-3=<4/3
k=<13/3

14.
9(3)^2+k(3)-6=0
3k=-75
k=-25

15.
x^2-3x+4 =0
α+β=3 αβ=4
α^2+β^2
=(α+β)^2-2αβ
=3^2-2(4)=1
(α-β)^2
=α^2+β^2-2αβ
=1-2*4=-7
(x-1)(x+7)
=x^2+7x-x+7
=x^2+6x+7

16.
f(x) = x^3+5x^2+k
1=(-1)^3+5(-1)^2+k
k=-3
餘數=f(1)=(1)^3+5(1)^2-3
=3

17農地實際面積是=6000*2.5/100
=150m^2
18.
16^9+16^4+44
=16^9 + 16^4 + 2*16 + 12=100001002C(16)