F.4 MATHS ANS + WORKING STEP

k, k1 and k­2 below are all constant.


1.
y = (k1/x) + k2x

When x = 1, y = 5 :
5 = k1 + k2 ... [1]

When x = 4, y = 25/2 :
50 = k1 + 16k2 ... [2]

[2] - [1] :
15k2 = 45
k2 = 3

Put k2 = 3 into [1] :
5 = k1 + 3
k1 = 2

Hence, y = (2/x) + 3x

When x = 2 :
y = (2/2) + 3*2
y = 7


2.
x ∝ y²/z
x = ky²/z

When y = yo and z = zo, x = xo :
xo = kyo²/zo

When y = yo(1 + 20%) and z = zo(1 - 20%) :
x = k[yo(1 + 20%)]²/[zo(1 - 20%)]
x = 1.8xo

% increase of x
= [(1.8xo - xo)/xo] * 100%
= 80%


3.
x ∝ 1/√y
x = k/√y

When y = yo, x = xo :
xo = k/√yo

When y = yo(1 + 69%) :
x = k/√[yo(1 + 69%)]
x = x­o/1.3

% decrease of x
= {[xo - (xo/1.3)]/xo} * 100%
= (23又1/13)%


4.
x ∝ y/z
x = ky/z

When y = 2 and z = 3, x = 7 :
7 = k*2/3
k = 21/2
Hence, x = 21y/2z

When y = 6 and z = 7 :
x = 21*6/2*7
x = 9


5.
y ∝ 1/x³
y = k/x³

When x = xo, y = yo :
yo = k/xo³

When x = xo(1 + 100%) :
y = k/[xo(1 + 100%)]³
y = 0.125yo

% decrease of y
= [(yo - 0.125yo)/yo] * 100%
= 87.5%


6.
y ∝ x²/√z
y = kx²/√z

When x = 2 and z = 9, y = 1 :
1 = k*2²/√9
k = 3/4
Hence, y = 3x²/4√z

When x = 1 and z = 4 :
y = 3*1²/4√4
y = 3/8


7.
y = k1 + (k2/x)

When x = 1, y = 7 :
7 = k1 + k2 ... [1]

When x = 3, y = 3 :
3 = k1+ (k2/3) ... [2]

[1] - [2] :
2k2/3 = 4
k2 = 6

Put k2 = 6 into [1] :
7 = k1 + 6
k1 = 1

Hence, y = 1 + (6/x)

When x = 2 :
y = 1 + (6/2)
y = 4


8.
y ∝ 1/x
y = k/x


When x = xo, y = yo :
yo = k/xo ... [1]

When x = xo(1 + 50%) :
y = k/[xo(1 + 50%)]
y = (1/1.5)yo

% decrease of y
= {[yo - (1/1.5)yo]/yo} * 100%
= (33又1/3)%


9.
z ∝ x²/y
z = kx²/y

When x = 4 and y = 3,z = 2 :
2 = k*4²/3
k = 3/8
Hence, z = 3x²/8y

When x = 2 and z = 3 :
3 = 3*2²/8y
y = 1/2


10.
y = k1x + (k2/√x)

When x = 1, y = 4 :
4 = k1 + k2 ... [1]

When x = 4, y = 10 :
10 = 4k1 + (k2/√4)
20 = 8k1 + k2 ... [2]

[2] - [1] :
7k1 = 16
k1 = 16/7

Put k1 = 16/7 into [1] :
4 = (16/7) + k2
k2 = 12/7

Hence, y = (16x/7) + (12/7√x)

When x = 16 :
y = (16*16/7) + (12/7√16)
y = 37