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Questions 1 to 10, 3 marks each
1.
The value of (4
× 5) ÷ (2 × 10) is
(A) 4
(B)
1
4
(C) 2
(D)
1
2
(E) 1
2.
In the diagram, the value of x is
(A) 20
(B) 90
(C) 30
(D) 80
(E) 60
... ... ... ... ... ... ... ... ... ... ... ... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...70
◦
x
◦
130
◦
3.
1 +
1
3 +
1
2
equals
(A)
6
5
(B)
7
6
(C)
9
2
(D)
3
2
(E)
9
7
4.
The straight line y = x + g passes throught the point (2,3). The value of g is
(A) 0
(B) 1
(C) 2
(D) 3
(E)
−1
5.
A two-digit number has tens digit t and its units digit u. If the digit 8 is placed
between these digits, the value of the three-digit number is
(A) t + u + 8
(B) 10t + 80 + u
(C) 10t + u + 8
(D) 100t + 10u + 8
(E) 100t + 80 + u
6.
P XQ is a right angled triangle with sides
of length 3 and 7 as shown. At P , P R is
drawn so that RP Q = 90
◦
and P R = P Q.
The area of
P RQ is
(A)
21
2
(B) 29
(C)
√
58
(D) 58
(E) 100
...... ...... ...... ...... ...... ...... ...... ......... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ... ... ... ... ... ... ... ... ... ... ... ... ...P
X
Q
R
... ... ... ...3
7
S 2
7.
In our school the average mark in Year 11 for a test was 70 and in Year 12 it was
80 for the same test. There were 20 students in Year 11 and 30 students in Year
12 who sat the test. The average mark for the two groups was
(A) 72
(B) 75
(C) 76
(D) 78
(E) 74
8.
Different tyres were fitted to a car, increasing the circumference of the wheels from
200 cm to 225 cm. On a journey of 1800 km, the number of revolutions of each
wheel was reduced by
(A) 50 000
(B) 1000
(C) 2000
(D) 100 000
(E) 7 200 000
9.
The sum of all but one of the internal angles of a pentagon is 400
◦
. The number
of degrees in the remaining angle is
(A) 40
(B) 120
(C) 140
(D) 160
(E) 400
10.
The value of
√
4
2
×
32
√
2 is
(A) 8
(B) 4
(C) 4
√
2
(D) 4
√
4
2
(E) 16
√
4
2
Questions 11 to 20, 4 marks each
11.
The difference between a positive fraction and its reciprocal is
9
20
. The sum of the
fraction and its reciprocal is
(A)
20
9
(B)
41
20
(C)
25
16
(D) 5
(E) not uniquely determined
12.
At time t = 0 a split forms in a balloon and the quantity Q of gas left in the
balloon at time t is given by
Q =
100
(1 + 2t)
2
.
The time taken for half the gas to escape is
(A)
√
2
− 1
2
(B)
1
2
(C)
1 +
√
2
2
(D)
√
2
(E)
10
√
2
− 1
10
13.
Two dice are thrown at random. The probability that the two numbers obtained
are the two digits of a perfect square is
(A)
1
9
(B)
2
9
(C)
7
36
(D)
1
4
(E)
1
3
14.
A square piece of paper has area 12 cm
2
. It
is coloured white on one side and shaded on
the other. One corner of the paper has been
folded over so that the sides of the triangle
formed are parallel to the sides of the square
as shown. The total visible area of the paper
is half shaded and half white. What is the
length, in centimetres, of the fold line U V ?
(A) 4
(B)
√
12
(C) 3
(D) 6
(E)
√
8
...... ...... ...... ...... ...... ...... ...... ...... ...... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... . .. .. . .. .. .. .. .. .. .. .. .. . .. .. .. . .. .. .. .. .. .. .. .. .. .. .. .. . .. .. .. .. . .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. . .. .. .. .. .. . .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. . .. .. .. .. .. .. . .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. . .. .. .. .. .. .. .. . .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. . .. .. .. .. .. .. .. .. . .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. . .. .. .. .. .. .. .. .. .. . .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. . .. .. .. .. .. .. .. .. .. .. . .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. . .. .. .. .. .. .. .. .. .. .. .. . .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. . .. .. .. .. .. .. .. .. .. .. .. .. . .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. . .. .. .. .. .. .. .. .. .. .. .. .. .. . .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. . .. .. .. .. .. .. .. .. .. .. .. .. .. .. . .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. . .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. . .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. . .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. . .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. . .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. . .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. . .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. . .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .P
Q
R
S
S
U
V
15.
In the triangle P QR shown, S and U are
points on QR and T is a point on P Q such
that T S
P R and U T
SP .
If QS = 4 cm and SR = 2.4 cm, then the
length of QU , in centimetres, is
(A) 2.4
(B) 2.5
(C) 3
(D) 3.2
(E) 4
... ... ... ... ... ... ... ... ... ... ... ...... ...... ...... ...... ...... ...... ...... ...... ...... ... ... ... ... ... ... ... ... ...... ...... ...... ...... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ...P
Q
R
T
U
S
16.
A train leaves Canberra for Sydney at 12 noon, and another train leaves Sydney
for Canberra forty minutes later. Both trains follow the same route and travel at
the same uniform speed, taking 3
1
2
hours to complete the journey. At what time
will they pass?
(A) 1:45 pm
(B) 2:00 pm
(C) 2:05 pm
(D) 2:15 pm
(E) 2:25 pm
17.
A spiral is formed by starting with an
isosceles right-angled triangle OX
1
X
2
,
where OX
1
is of length 1, then using
the hypotenuse OX
2
as a shorter side of
another isosceles right-angled triangle,
and so on. The first few steps are shown
in the diagram.
Eventually we will reach for the first
time a situation where a side OX
k
of a
... ... ... ... ... ... ... ... ... ... ... ... ... ...... ...... ...... ...... ...... ... ... ... ... ... ... ... ...... ...... ...... ...... ...... ...... ...... ...... ...... ... ... ... ... ...
X
5
O
X
1
X
2
X
3
X
4
triangle overlaps OX
1
. What is the length of X
1
X
k
?
S 4
18.
The number of 5-digit numbers in which every two neighbouring digits differ by 3
is
(A) 40
(B) 41
(C) 43
(D) 45
(E) 50
19.
A ladder resting against a wall makes an angle
of 60
◦
with the wall. When the base of the
lad-der is moved 1 m further from the wall it makes
an angle of 45
◦
with the wall. The length of the
ladder, in metres, is
(A) 2
(B) 2(
√
2 + 1)
(C)
√
2 + 1
√
2
− 1
(D)
√
5
(E)
√
2
2 + 1
...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ... ......60
◦
45
◦
1 m
20.
A quarter circle is folded to form a cone.
... ... ... ... ... ... ... .... .... .... ... ... ... ... ... ... ... ... .. ... ... .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. . .. .. .. . .. .. .. . .. .. .. .. .. . .. . .. . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ... ... .. . . . . . . . . . . . . .... .. .. .. . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. ... ...... ... ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ... ... ... ... ... ... ... ... ... ... ... ... ... ... ...... ...... ...... ...... ...... ...... ...... ...... ...... .... ... ... ... ... ... . ... ... . ... ... . ... ... . ... ... . ... ... . ... ... . ... ... . ... ... .
θ
◦
If θ
◦
is the angle between the axis of symmetry and the slant height of the cone,
then sin θ
◦
equals
(A)
1
4
(B)
1
√
2
(C)
1
2
(D)
√
3
2
(E)
1
√
3
Questions 21 to 30, 5 marks each
21.
The number of real solutions of x +
x
2
+
√
x
3
+ 1 = 1 is
22.
The area of the shaded rectangle is
(A) between
1
4
and
5
16
(B) between
5
16
and
3
8
(C) between
3
8
and
7
16
(D) between
7
16
and
1
2
(E) more than
1
2
... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ...... ...... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ... ... ... . ... ...1
2
2
1
... . . . . . . . . . . . . ... ... ... ... ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..23.
When (1
− 2x)
3
(1 + kx)
2
is expanded, two values k
1
and k
2
of k give the coefficient
of x
2
as 40. The value of k
1
+ k
2
is
(A)
−1
(B) 8
(C) 10
(D) 12
(E) 14
24.
What is the area, in square units, enclosed by the figure whose boundary points
satisfy
|x| + |y| = 4?
(A) 2
(B) 4
(C) 8
(D) 16
(E) 32
25.
The number of digits in the decimal expansion of 2
2005
is closest to
(A) 400
(B) 500
(C) 600
(D) 700
(E) 800
For questions 26 to 30, shade the answer as an integer from 0 to 999 in
the space provided on the answer sheet.
26.
My name is Louis and my father has cooked me an L-shaped cake for my birthday.
He says that I must cut it into three pieces with a single cut, so that my brother
and sister can have a piece too. So, I have to cut it
like this
or this
but not like this.
10cm
10cm
10cm
20cm
30cm
...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... . ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... .He says that I have to be polite and let them have the first choice of the pieces,
but I just know they’ll be greedy and leave the smallest possible piece for me. So I
want to cut the cake so that my little piece will be as big as possible. If I do this,
how big, in square centimetres, will my piece be?
S 6
27.
The function y = f (x) is a function such that f (f (x)) = 6x
− 2005 for every real
number x. An integer t satisfies the equation f (t) = 6t
− 2005. What is this value
of t?
28.
A regular octahedron has eight triangular faces
and all sides the same length. A portion of a
regular octahedron of volume 120 cm
3
consists
of that part of it which is closer to the top vertex
than to any other one. In the diagram, the
out-side part of this volume is shown shaded, and it
extends down to the centre of the octahedron.
What is the volume, in cubic centimetres, of
this unusually shaped portion?
... ... ... ... ... ... ... ... ... ... ... ...... ...... ...... ...... ...... ...... ...... ... . .. . .. . . .. . .. . . .. . .. . .. . . .. . .. . . .. . .. . .. . . .. . .. . .. . . .. . .. . . .. . .. . .. . . .. . .. . . .. . .. . .. . . .. . .. . . .. . .. . .. . . .. . .. . . .. . .. . .. . . .. . .. . . .. . .. . .. . . .. . .. . . .. . .. . .. . . .. . .. . . .. . .. . .. . . .. . .. . .. . . .. . .. . . .. . .. . .. . . .. . .. . . .. . .. . .. . . .. . .. . . .. . .. . .. . . .. . .. . . .. . .. .. . .. . .. . .. . . .. . .. . . .. . .. . .. . . .. . .. . . .. . .. . .. . . .. . .. . . .. . .. . .. . . .. . .. . . .. . .. . .. . . .. . .. . .. . . .. . .. . . .. . .. . .. . . .. . .. . . .. . .. . .. . . .. . .. . . .. . .. . .. . . .. . .. . . .. . .. . .. . . .. . .. . . .. . .. . .. . . .. . .. . . .. . .. . .. . . .. . .. . . .. . .. . .. . . .. . .. . .. . . .. . .. . . .. . .. . .. . . .. . .. . . .. . ...... ...... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ...... ...... ...... ...... ...... ... ...... ...... ... ... ......... ... ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..
