1
TJC H2 JC1 MYA 2021
9758/01
1
A particular country ordered 11 million doses of vaccines to protect its population against
rubella, hepatitis A and chicken pox.
The country spent 244.25 million dollars in purchasing
r
million doses of rubella vaccines,
a
million doses of hepatitis A vaccines and
c
million
doses of chicken pox vaccines.
The cost of each dose of rubella vaccine is $6.50, hepatitis A
vaccine $21 and chicken pox vaccine $27.50.
Furthermore, it is known that the country had purchased 0.5 million more doses of hepatitis
A vaccines than of rubella vaccines.
Find how much more the country spent on chicken pox vaccines than on the other two types
of vaccines combined.
[4]
Solution
Marker’s Comments
1
r
+
a
+
c
= 11
 (1)
6.5
r
+ 21
a
+ 27.5
c
= 244.25
 (2)
a
=
r
+ 0.5
–
r
+
a
=
0.5
 (3)
From GC,
2,
2.5,
6.5
r
a
c
6.5 (27.5) – 2 (6.5) – 2.5 (21)
=
113.25
The country spent $113.25 million more on chicken pox
vaccines than on the other two types combined.
Note: If 2 000 000 or 2 million doses of
rubella vaccines has been purchased,
then
r
= 2, not 2x10
6
.
Need to answer to the question in
context.
2
TJC H2 JC1 MYA 2021
9758/01
2
(a)
Find the vector
u
which has a magnitude of 2 units and is in the same direction as
the vector
i + k
, where
i
,
j
and
k
are unit vectors along the
x
axis,
y
axis and
z
axis
respectively.
State the geometrical relationship between
u
and the
y
axis.
[2]
(b)
The position vectors of points
A
and
B
, relative to the origin
O
, are
a
and
b
respectively and the angle
AOB
is
.
The point
M
divides
AB
in the ratio
:1.
(i)
Write down
OM
in terms of
,
a
and
b
.
[1]
(ii)
The points
C
and
D
lie on
OA
and
OB
extended respectively such that
OC
=
pOA
and
OD
=
qOB
, where
p
and
q
are real numbers.
It is also given
that
a
and
b
are both unit vectors.
If
OM
is perpendicular to
CD
, show that
cos
, where
0.
p
q
q
p
q
p
[3]
Solution
Marker’s Comment
(a)
1
1
1
2
0
2
0
1
1
1
1
u
Since
1
2 0
1
u
is parallel to the
x

z
plane,
hence
u
is perpendicular to the
y
–axis.
Common errors:
1
1
2 0
or
2
1
1
0
u
Many students wrote
u
incorrectly
as vector
1. parallel to the
y
axis;
2. lies on plane contain
y
axis; or
3. make an angle of 45 degree with
the
y
axis.
(b)(i)
1
(
)
1
1
b
a
OM
b
a
Majority of the students answer this
correctly.
Common errors include
or
1
OM
b
a
b
a
Alternative Thinking
Unit vector along
y
axis
=
j
1
0
2
0
1
0
1
0
u
j
, hence
u
is perpendicular to the
y
–axis.
Did not convert
to unit vector.
Carelessly write
down the vector.
3
TJC H2 JC1 MYA 2021
9758/01
(b)(ii)
Given
OC
p OA
and
OD
q OB
Since
OM
is perpendicular to
CD
,
0
OM
CD
1
(
)
(
)
0
1
b
a
qb
pa
2
2
1
(
)
(
)
0
1
q b
p b a
q a b
p a
2
2
(
cos
)
(
cos
)
0
q b
p a b
q a b
p a
Since
1
(unit vectors)
a
b
(
)
(
)cos
0
q
p
p
q
cos
, where
0.