F.4 maths questions (20分)

1.
By definition of logarithm:
If y = (10)^a, then a = logy

Simultaneous equations are set up:
(10)^a = y ... (1)
a = logy ... (2)

Substitute (2) into (1):
(10)^logy = y ...... (answer)


2.
Let a = logy

log(10)^logy
= (logy) x log(10)
= logy

Since log(10)^logy = logy,
then (10)^logy = y ...... (answer)


3.
Point-slope form of a st. line passing through (x1, y1) with slope m:
(y - y1)/(x - x1) = m

The required line:
(y + 1)/(x + 2) = 3/2
3(x + 2) = 2(y + 1)
3x + 6 = 2y + 2
3x - 2y + 4 = 0 ...... (answer)


4.
Point-slope form of a st. line passing through (x1, y1) with slope m:
(y - y1)/(x - x1) = m

(y - 1/3)/(x - 1/2) = -5/4
[(3y - 1)/3]/[(2x - 1)/2] = -5/4
[2(3y - 1)]/[3(2x - 1)] = -5/4
(6y - 2)/(6x - 3) = -5/4
4(6y - 2) = -5(6x - 3)
24y - 8 = -30x + 15
30x + 24y - 23 = 0 ...... (answer)

1.by simultaneous equations係咪即係想你用2條equation黎做?

2.log (10^log y)
= (log y)(log 10)
=log y
Therefore, 10^log y = y

3.y=3/2x + c
when y=-1, x=-2
-1=(3/2)(-2) + c
c=2
y=3/2x + 2

4. y=-5/4x +c
when y=1/3, x=1/2
1/3=(-5/4)(1/2) + c
c=23/24
y=-5/4x + 23/24