F2 Maths難題

[匿名]

2013-09-14 7:11 pm

1)Simplify
(2)(4^x)/8^x

2)Kelvin uses a ruler to measure the width and the length of a card . The width of
the card is measured as 8.0 cm. It is given that the percentage error of the measurement is 3.125%.

Question: If the length of the card is measurede as 10.0cm , find the upper limit of the area of the card.

3)Factorize
(a) x^2-2x+1-y^2
(b)a^2-b^2-a+b

4)In the identity Ax(x-1)+B(x-1)(x-2)=x^2+Cx-4, find the values of A, B and C.

5)Simplify
2(3^x)+3^x/9^x

以上是我的難題, plz

回答 (1)

風平浪靜海闊天空

2013-09-15 3:03 am

✔ 最佳答案

1)
(2)(4^x)/8^x
=(2)[(2^2)^x]/(2^3)^x
=(2)[2^(2x)]/2^(3x)
=2^(1+2x-3x)
=2^(1-x)

2)
Upper limit of width
=8*(1+3.125%) cm
=8.25 cm

Maximum absolute error
=(8.25-8) cm
=0.25 cm

Upper limit of length
=10+0.25
=10.25

Upper limit of area
=8.25*10.25
=84.5625
=84.6 (corr to 3 sig fig)

3)
(a)x^2-2x+1-y^2
=(x+1)^2-y^2
=(x+1+y)(x+1-y)

(b)a^2-b^2-a+b
=a^2-b^2-a+b+1-1
=a^2-a+1-b^2+b-1
=(a^2-a+1)-(b^2-b+1)
=(a-1)^2-(b-1)^2
=(a-1+b-1)(a-1-b+1)
=(a+b-2)(a-b)

4)
Ax(x-1)+B(x-1)(x-2)=x^2+Cx-4
Ax^2-Ax+Bx^2-3Bx+2B=x^2+Cx-4
(A+B)x^2-(A+3B)x+2B=x^2+Cx-4

By comparing like terms,
A+B=1 ---(1)
-(A+3B)=C ---(2)
2B=-4 ---(3)

(3): 2B=-4
B=-2

Subst. B=-2 into (1):
A+(-2)=1
A=3

Subst. A=3, B=-2 into (2)
-(3+3(-2))=C
3=C
C=3

5)
2(3^x)+3^x/9^x
=2(3^x)+3^x/(3^2)^x
=2(3^x)+3^x/3^2x
=2(3^x)+3^(-x)

P.S. I doubt that there is a missing bracket in question #5. If that is true, the solution shall be:

[2(3^x)+3^x]/9^x
=3(3^x)/(3^2)^x
=3^(x+1)/3^2x
=3^(x+1-2x)
=3^(1-x)

參考: myself and myself only :D

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