F.3 MATHS

1)
3xy - (2x - y)(5x + y)
= 3xy - 2x(5x + y) + y(5x + y)
= 3xy - 10x² - 2xy + 5xy + y²
= 10x² + 6xy + y²

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2)
(3 + x)(x - 1) + (3 - x)(1 - x)
= 3(x - 1) + x(x - 1) + 3(1 - x) - x(1 - x)
= 3x - 3 + x² - x + 3 - 3x - x + x²
= 2x² - 2x

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3)
Formula: u³ - v³ = (u - v)(u² + uv + v²)

a³ - 64
= a³ - 4³
= (a - 4)(a² + 4a + 4²)
= (a - 4)(a² + 4a + 16)

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4.
Number of combination when throwing two dice = 6 x 6 = 36

Combinations of (sum = 2) : (1,1)
Combinations of (sum = 3) : (1,2), (2, 1)
Combinations of (sum = 4) : (1, 3), (3, 1), (2, 2)

P(sum not greater than 4)
= P(sum = 2) + P(sum = 3) + P(sum = 4)
= 1/36 + 2/36 + 3/36
= 6/36
= 1/6

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5)
There are 6 numbers formed: 258, 285, 528, 582, 852, 625

Among the 6 numbers, 3 of them are multiples of 6 : 258, 528, 852

P(the number is a multiple of 6) = 3/6 = 1/2

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6)
Among the 40 numbers from 1 to 40 :
11 numbers are multiples of 6 or 7 : 6, 7, 12, 14, 18, 21, 24, 28, 30, 35, 36
P(the number is a multiple of 6 or 7) = 11/40

(There is no number from 1 to 40 that is a multiple of both 6 and 7.)
(P(the number is a multiple of both 6 and 7) = 0)

2010-08-30 01:08:50 補充：
a³ + 64
= a³ + 4³
= (a + 4)(a² - 4a + 4²)
= (a + 4)(a² - 4a + 16)