Pierce Prelim 2009 Em1
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2 Mathematical Formulae Compound interest Total amount = P (1
r n ) 100
Mensuration Curved Surface area of a cone = r Surface area of a sphere = 4 r 2 Volume of a cone =
1 2 r h 3
Volume of a sphere =
Area of triangle ABC =
4 3 r 3 1 ab sin C 2
Arc length = r , where is in radians Sector area =
1 2 r , where is in radians 2
Trigonometry a b c sin A sin B sin C
a 2 b 2 c 2 2bc cos A
Statistics
Mean =
Standard deviation =
fx f
fx 2 fx f f
2
3 CALCULATORS CAN BE USED IN THIS PAPER Answer all questions 1
Use your calculator to find the value of
3.91 62.6 0.127
, giving your answer correct to two
significant figures.
Answer
2
[1]
The value of a car decreased by 32% between 2004 and 2008. In 2008 the car was valued at $58 208, find its original value in 2004.
Answer $
3
[2]
If Y exceeds X by 20% and Z exceeds Y by 20%, find the value of X : Z.
Answer X : Z =
:
[2]
4 4
Given that y is inversely proportional to x 2 and that y = 4 for a particular value of x. Find the value of y when x is doubled.
Answer y =
5
Express
[2]
8 2 as a single fraction in its simplest form. x 4 x2 2
Answer
[2]
5 6
The ordered stem-and-leaf diagram below records the height, in centimetres, of 40 students of Peirce Secondary School. Stem 14 15 16 17 18 19
7
Leaf 9 0 3 3 2 0
8 0 0 3 0 0
9 1 5 3 2 2
9 5 7 3 5 2
6 7 3 5
(a)
Write down the shortest height recorded.
(b)
Find the median height.
6 9 3 7
9 9 5 9
9 9 6
7
Answer (a)
cm [1]
(b)
cm [1]
The following dot diagram represents the number of goals scored in each of the 20 soccer matches.
0
1
2
3
4
5
No. of goals scored Find (a)
the modal number of goals scored,
(b)
the mean number of goals scored.
Answer (a)
goal(s) [1]
(b)
goal(s) [2]
6 8
The ratio of an interior angle to an exterior angle of a regular polygon is 13 : 2. Find
9
(a)
the number of sides of the polygon,
(b)
the sum of interior angles of the polygon.
Solve
Answer (a)
sides [2]
(b)
[1]
Answer x =
[3]
2 x 1 4 x 2 5x . 3 4 6
7 10
Consider the following sequence: Line 1 Line 2 Line 3
2+6 2+6+10 2+6+10+14
2
=2 2 2 =23 2 =2 4
=8 = 18 = 32
(a)
Write down Line 9 of the sequence.
(b)
State the sum 2 6 10 14 46 .
(c)
Write down a formula, in terms of n, for the sum, S, of Line n of the sequence.
Answer (a)
=
=
[2]
(b)
[1]
(c)
[1]
8 11
Solve the simultaneous equations
2 y 3x 5, 5 x 6 y 19.
Answer x =
12
(a)
Solve the inequality
,y=
2x 3 x 10 2 x . 5 2
Answer (a) (b)
[3]
[2]
Show your solution on the number line. Answer
–10
–5
0
5
10
[1]
9 13
A, B, C and D are points on a circle with centre O. BOC 54 and BAD 97 . D Calculate (a)
BDC ,
(b)
BCD ,
(c)
OCD .
A
97 O 54
C B
14
Answer (a)
[1]
(b)
[1]
(c)
[1]
(a)
Sketch y ( x 2) 2 5 in the axes provided below, showing the y-intercept and the turning point.
(b)
Write down the equation of the line of symmetry of the curve.
Answer (a) y
x
[2] (b)
[1]
10 15
The figure shows a piece of wood formed by a prism with a hole drilled through it. The diameter of the hole is 3 cm. It is given that AF = BG = CD = 5 cm, AB = 14 cm, BC = 12 cm, FG = 8 cm and the height of trapezium ABGFA is 4 cm. E
8 cm
F 5 cm
D
5 cm
G
33cm cm
C
5 cm
4 cm 12 cm
24 cm
A
14 cm
B
Taking 3.142 , find
16
(a)
the area of the shaded part,
(b)
the volume of the solid.
Answer (a)
cm2 [2]
(b)
cm2 [1]
5 3 Given that OP and QP , find 3 4 (a)
OQ ,
(b)
QP .
Answer (a) OQ (b)
[2] units [1]
11 17
18
Given two points A(5, 6) and B(2, 0) in the coordinate plane. Determine (a)
the length of the line AB,
(b)
the gradient of the line AB,
(c)
the equation of the straight line AB.
Answer (a)
units [1]
(b)
[1]
(c)
[2]
The numbers 84 and 96, written as the product of its prime factors, are 2 2 3 7 and 25 3 respectively. Find (a)
the highest common factor of 84 and 96,
(b)
the smallest integer value of n for which 84n is a multiple of 96,
(c)
the lowest number k such that 84k is a perfect square.
Answer (a)
[1]
(b) n =
[1]
(c) k =
[2]
12 1
19
20
x 2 4x 3
and give your answer in the form axn.
(a)
Simplify
(b)
Solve the equation 315 81 3x .
x
Answer (a)
[2]
(b)
[2]
(a)
Construct a triangle ABC where AB = 8 cm, AC = 9 cm and BC = 10 cm.
(b)
Construct the angle bisector of angle CAB.
(c)
Construct the perpendicular bisector of AC. Answer
[4]
13 21
(a)
The distance between the earth and the sun is estimated as 144 billion metres. Express this distance in standard form.
(b)
Nano carbon molecules, such as Fullerene, have a diameter of 0.7 nanometres. Express the diameter in picometres.
(c)
Given that x 2 10 2 and y 5 10 4 , express the following in standard form (i) (ii)
22
x , y x + 5y.
Answer (a)
m [1]
(b)
picometres [1]
(c) (i)
[1]
(ii)
[1]
On a map of a park, 2 cm represents 5 km. (a)
Express the map scale in the form 1 : n.
(b)
Find the length of a cycling track on the map which has an actual distance of 20 km.
(c)
Calculate the actual area of a lake in km2, which represents an area of 7 cm2 on the map.
Answer (a)
[1]
(b)
cm [1]
(c)
km2 [2]
14 The data below shows the attendance for a particular class of 40 pupils in ABC Secondary School in the month of August. The number of days each pupil was absent from school are recorded as follows: 2 1 0 3 (a)
0 2 1 1
4 0 1 0
2 1 0 3 4 1 0 0 0 0 0 2
5 0 0 2 1 0 0 0
0 2 0 1 2 3 1 0
Complete the frequency table. Answer Number of Days Absent Frequency
0
1
2
3
18
4
5
3 [1]
(b)
Represent the attendance data obtained in the histogram below. Answer
20 18
Frequency
23
16 14 12
10 8 6 4 2 0
0
1
2
3
4
5
Number of Days Absent for Month of August [2] (c)
How many pupils were absent for more than 3 days? Answer (c)
[1]
15 24
In the triangle ABC, AB = 12 cm, AC = 5 cm and BC = 13 cm. C
13 5
A (a)
12
B
D
Show that BAC is a right angle. Answer
[1] (b)
Find, as a fraction in the simplest form, (i)
sin ACB ,
(ii)
cos CBD .
Answer (b) (i)
[1]
(ii)
[2]
16 25
The diagram below shows the speed-time graph of a car which starts from rest. Speed (m/s)
v 20
Time, t (s) 40
15
0
60
(a)
Find the value of v, if the total distance covered in the first 40 seconds is 900 m.
(b)
Find the retardation at t = 50 s.
Answer (a) v =
[2] m/s2 [2]
(b) (c)
Use the diagram provided below to draw the distance-time graph of the car for the whole journey. Answer Distance (m) 1300 900
150 150 Time, t (s) 0
15
40
60
[2]
r n ) 100
Mensuration Curved Surface area of a cone = r Surface area of a sphere = 4 r 2 Volume of a cone =
1 2 r h 3
Volume of a sphere =
Area of triangle ABC =
4 3 r 3 1 ab sin C 2
Arc length = r , where is in radians Sector area =
1 2 r , where is in radians 2
Trigonometry a b c sin A sin B sin C
a 2 b 2 c 2 2bc cos A
Statistics
Mean =
Standard deviation =
fx f
fx 2 fx f f
2
3 CALCULATORS CAN BE USED IN THIS PAPER Answer all questions 1
Use your calculator to find the value of
3.91 62.6 0.127
, giving your answer correct to two
significant figures.
Answer
2
[1]
The value of a car decreased by 32% between 2004 and 2008. In 2008 the car was valued at $58 208, find its original value in 2004.
Answer $
3
[2]
If Y exceeds X by 20% and Z exceeds Y by 20%, find the value of X : Z.
Answer X : Z =
:
[2]
4 4
Given that y is inversely proportional to x 2 and that y = 4 for a particular value of x. Find the value of y when x is doubled.
Answer y =
5
Express
[2]
8 2 as a single fraction in its simplest form. x 4 x2 2
Answer
[2]
5 6
The ordered stem-and-leaf diagram below records the height, in centimetres, of 40 students of Peirce Secondary School. Stem 14 15 16 17 18 19
7
Leaf 9 0 3 3 2 0
8 0 0 3 0 0
9 1 5 3 2 2
9 5 7 3 5 2
6 7 3 5
(a)
Write down the shortest height recorded.
(b)
Find the median height.
6 9 3 7
9 9 5 9
9 9 6
7
Answer (a)
cm [1]
(b)
cm [1]
The following dot diagram represents the number of goals scored in each of the 20 soccer matches.
0
1
2
3
4
5
No. of goals scored Find (a)
the modal number of goals scored,
(b)
the mean number of goals scored.
Answer (a)
goal(s) [1]
(b)
goal(s) [2]
6 8
The ratio of an interior angle to an exterior angle of a regular polygon is 13 : 2. Find
9
(a)
the number of sides of the polygon,
(b)
the sum of interior angles of the polygon.
Solve
Answer (a)
sides [2]
(b)
[1]
Answer x =
[3]
2 x 1 4 x 2 5x . 3 4 6
7 10
Consider the following sequence: Line 1 Line 2 Line 3
2+6 2+6+10 2+6+10+14
2
=2 2 2 =23 2 =2 4
=8 = 18 = 32
(a)
Write down Line 9 of the sequence.
(b)
State the sum 2 6 10 14 46 .
(c)
Write down a formula, in terms of n, for the sum, S, of Line n of the sequence.
Answer (a)
=
=
[2]
(b)
[1]
(c)
[1]
8 11
Solve the simultaneous equations
2 y 3x 5, 5 x 6 y 19.
Answer x =
12
(a)
Solve the inequality
,y=
2x 3 x 10 2 x . 5 2
Answer (a) (b)
[3]
[2]
Show your solution on the number line. Answer
–10
–5
0
5
10
[1]
9 13
A, B, C and D are points on a circle with centre O. BOC 54 and BAD 97 . D Calculate (a)
BDC ,
(b)
BCD ,
(c)
OCD .
A
97 O 54
C B
14
Answer (a)
[1]
(b)
[1]
(c)
[1]
(a)
Sketch y ( x 2) 2 5 in the axes provided below, showing the y-intercept and the turning point.
(b)
Write down the equation of the line of symmetry of the curve.
Answer (a) y
x
[2] (b)
[1]
10 15
The figure shows a piece of wood formed by a prism with a hole drilled through it. The diameter of the hole is 3 cm. It is given that AF = BG = CD = 5 cm, AB = 14 cm, BC = 12 cm, FG = 8 cm and the height of trapezium ABGFA is 4 cm. E
8 cm
F 5 cm
D
5 cm
G
33cm cm
C
5 cm
4 cm 12 cm
24 cm
A
14 cm
B
Taking 3.142 , find
16
(a)
the area of the shaded part,
(b)
the volume of the solid.
Answer (a)
cm2 [2]
(b)
cm2 [1]
5 3 Given that OP and QP , find 3 4 (a)
OQ ,
(b)
QP .
Answer (a) OQ (b)
[2] units [1]
11 17
18
Given two points A(5, 6) and B(2, 0) in the coordinate plane. Determine (a)
the length of the line AB,
(b)
the gradient of the line AB,
(c)
the equation of the straight line AB.
Answer (a)
units [1]
(b)
[1]
(c)
[2]
The numbers 84 and 96, written as the product of its prime factors, are 2 2 3 7 and 25 3 respectively. Find (a)
the highest common factor of 84 and 96,
(b)
the smallest integer value of n for which 84n is a multiple of 96,
(c)
the lowest number k such that 84k is a perfect square.
Answer (a)
[1]
(b) n =
[1]
(c) k =
[2]
12 1
19
20
x 2 4x 3
and give your answer in the form axn.
(a)
Simplify
(b)
Solve the equation 315 81 3x .
x
Answer (a)
[2]
(b)
[2]
(a)
Construct a triangle ABC where AB = 8 cm, AC = 9 cm and BC = 10 cm.
(b)
Construct the angle bisector of angle CAB.
(c)
Construct the perpendicular bisector of AC. Answer
[4]
13 21
(a)
The distance between the earth and the sun is estimated as 144 billion metres. Express this distance in standard form.
(b)
Nano carbon molecules, such as Fullerene, have a diameter of 0.7 nanometres. Express the diameter in picometres.
(c)
Given that x 2 10 2 and y 5 10 4 , express the following in standard form (i) (ii)
22
x , y x + 5y.
Answer (a)
m [1]
(b)
picometres [1]
(c) (i)
[1]
(ii)
[1]
On a map of a park, 2 cm represents 5 km. (a)
Express the map scale in the form 1 : n.
(b)
Find the length of a cycling track on the map which has an actual distance of 20 km.
(c)
Calculate the actual area of a lake in km2, which represents an area of 7 cm2 on the map.
Answer (a)
[1]
(b)
cm [1]
(c)
km2 [2]
14 The data below shows the attendance for a particular class of 40 pupils in ABC Secondary School in the month of August. The number of days each pupil was absent from school are recorded as follows: 2 1 0 3 (a)
0 2 1 1
4 0 1 0
2 1 0 3 4 1 0 0 0 0 0 2
5 0 0 2 1 0 0 0
0 2 0 1 2 3 1 0
Complete the frequency table. Answer Number of Days Absent Frequency
0
1
2
3
18
4
5
3 [1]
(b)
Represent the attendance data obtained in the histogram below. Answer
20 18
Frequency
23
16 14 12
10 8 6 4 2 0
0
1
2
3
4
5
Number of Days Absent for Month of August [2] (c)
How many pupils were absent for more than 3 days? Answer (c)
[1]
15 24
In the triangle ABC, AB = 12 cm, AC = 5 cm and BC = 13 cm. C
13 5
A (a)
12
B
D
Show that BAC is a right angle. Answer
[1] (b)
Find, as a fraction in the simplest form, (i)
sin ACB ,
(ii)
cos CBD .
Answer (b) (i)
[1]
(ii)
[2]
16 25
The diagram below shows the speed-time graph of a car which starts from rest. Speed (m/s)
v 20
Time, t (s) 40
15
0
60
(a)
Find the value of v, if the total distance covered in the first 40 seconds is 900 m.
(b)
Find the retardation at t = 50 s.
Answer (a) v =
[2] m/s2 [2]
(b) (c)
Use the diagram provided below to draw the distance-time graph of the car for the whole journey. Answer Distance (m) 1300 900
150 150 Time, t (s) 0
15
40
60
[2]
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