CAPE - 2007 - Pure MathematicsUnit 2 - Paper 03-2
Section A (Module 1)
1 The rate of decay of a radioactive substance after t hours was found to be directly proportional to the amount A of the substance remaining
a( ) Form a differential equation to model this situation [3 marks]
b( ) Given that when t = 0 A = A0 show that A A0 e ktkt [8 marks]
c( ) If the time taken for the remaining substance to be12
A0 is T show that
A A0 e
t ln. 2.
Tt ln. 2.
TA
t
[4 marks]
d( ) Prove that the time taken for the amount remaining to be reduced to
130
A0 is 4.91T [5 marks]
Total 20 marks
a( )dAdt
kAdAdt
b( )
A0
AA
1A
d0
ttkd
A0
AA
1A
dt
k lnAA0
. ktlnAA0
. kt
A A0 e ktA kt
c( )12
A0 A0 e kT12
A0 A kT kT ln12
.kT kln 2.
Tk
ln 2.
T
A A0 e
t ln. 2.
Tt ln. 2.
TA
t
d( )130
A0 A0 e
t ln. 2.
Tt ln. 2.
TA
t
ln130
. t ln. 2.
Tln
130
. tt
ln 30.
ln 2.T
ln 30.
ln 2.T t = 4.91T
1
Section B - Module 2
2 a( ) The common ratio r of a geometric series is given by r7 x
x2 6
x
x
Find the values of x for which the series is convergent [9 marks]
b( ) An oil well produced 100 000 barrels of oil in its first year but output fell by 7.5% each year thereafter
i( ) Determine to 2 sig fig the number of barrels of oil that could be produced
a( ) from the well as the maximum total output [5 marks]
b( ) by the well in the first 15 years [4 marks]
ii( ) What percentage of the maximum total output of oil would be left after 15 years?
[2 marks]
Total 20 marks
a( )7 x
x2 61< 49 x2 x4 12 x2 36<
x4 37 x2 36 0> x2 1 x2 36 0>
x 6< 1 x< 1< x 6>
b( ) i( ) production is GP with a = 100 000 and r = (1 - 0.075) = 0.925
max output: S∞100000
1 0.925∞ productionmax 1300000 2 sig fig.( )sig fig
ii( ) S15100000 1 0.92515.
1 0.925S15 920000 2 sig fig.( ). sig fig
iii( ) percent =1300000 920000
1300000100 29.231= 29 % 2 sig fig.( ).
2
Section C - Module 3
3 a( ) Ten students 8 girls and 2 boys are to be seated in a row
i( ) Determine the number of different ways these students can be arranged if there are no restrictions
[2 marks]
ii( ) Find the probability that the two boys do not sit next to each other [6 marks]
b( ) The cost of 2 computers 2 printers and 1 scanner is $5 950
The cost of 4 computers 1 printer and 1 scanner is $11 450
The cost of 5 computers 3 printers and 2 scanners is $14 600
Let x, y and z represent the cost in dollars of a computer a printer and a scanner respectively
i( ) Use the given information to form three equations [3 marks]
ii( ) Write down the system of equations in matrix form [1 mark]
iii( ) Hence calculate the values of x y and z [8 marks]
a( ) i( ) 10 ! 3628800 ways10 ! ways
ii( ) boys next to each other: 9 ! 2 !
P(boys next to each other) = 9 ! 2 !10 !
15
9 ! 2 !10 !
P(boys not next to each other) = 19 ! 2 !10 !
45
19 ! 2 !10 !
b( ) i( ) 2x + 2y + z = 59504x + y + z = 114505x + 3y + 2z = 14600
3
ii( )
2
4
5
2
1
3
1
1
2
x
y
z
.5950
11450
14600
2
4
5
2
1
3
1
1
2
x
y
z
.
2
4
5
2
1
3
1
1
2
1
yields
1
3
7
1
1
4
1
2
6
x
y
z
1
3
7
1
1
4
1
2
6
5950
11450
14600
. x 2800 y 100 z 150
4
