Tkss Prelim 2009 Em P2
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3 1.
2.
(a)
Given that 2 x
(b)
Given that the line
3m
xy , express y in terms of x. m
[3]
x y 1 passes through (0, 2) and (7, 3), a b find the value of a and b.
[3]
Squares are drawn to enclose numbers in the number array as shown in the diagram.
1
2
3
4
5
6
7
. . . .
2
4
6
8
10
12
14
. . . .
3
6
9
12
15
18
21
. . . .
4
8
12
16
20
24
28
. . . .
5
10
15
20
25
30
35
. . . .
The sums of the numbers in each square, Sn , are represented in the table below. Sum of numbers in square
Square, Sn S1
1
12 1
S2
1+2+2+4
32 9
S3 S4
(a) Copy and complete the table.
[2]
(b) Find a formula for Sn.
[2]
(c) The sum of numbers in the kth square is 44100. Find k.
[2]
[Turn over Tanjong Katong Secondary School
Prelim Mathematics P2
4
3.
Three football clubs, Rial, Bayern and Sporting, are playing in the annual soccer league. They played a few matches and their results can be represented in a matrix R as shown. Win Draw Lose Rial 6 2 2 Bayern 4 4 2 R Sporting 3 1 5 1 (a) The matrix G is 1 . Find RG. What do the elements in RG represent? 1
(b) (i) If a win earns 3 points, a draw earns 1 point and a loss earns no point, write a matrix P representing this information. (ii) Find RP and interpret the elements in the matrix.
4.
[2]
[1] [2]
Figure 1 shows a piece of thin metal sheet in the form of a sector of a circle 5 of radius 0.4 m and with reflex AOB . 3 The metal sheet is then shaped into a right circular cone with no overlapping as shown in Figure 2. P P A
5 3
O 0.4 m
B
A Figure 1
Figure 2
(a) Find the circumference of the cone, leaving your answer in terms of .
[2]
(b) Find the curved surface area of the cone.
[2]
(c) Find the radius and the height of the cone.
[4]
Tanjong Katong Secondary School
Prelim Mathematics P2
5 5.
In the diagram, A, B, C, D and M are points on level ground. C is due east of A, AM = 5 m, CM = 8 m, AD = 15 m, CD = 6 m, BM = BC and AMB 110 .
B N
5m
110
A
8m
M
C 6m 15 m D
(a)
(b)
Find (i)
the length BM,
[3]
(ii)
CAD ,
[3]
(iii) the bearing of D from A,
[1]
(iv) the area of ACD .
[2]
A vertical flag pole of height 6.5 m is located at point C. Find the smallest angle of elevation of the top of the pole when viewed from a point on path AB.
[2]
[Turn over
Tanjong Katong Secondary School
Prelim Mathematics P2
6 6.
The diagram below shows a swimming pool with a uniform width of 8 metres. The base of the pool is made up of a horizontal and a sloping plane. 10 m 1m
8m
3m
(a)
6m
Calculate the volume of the swimming pool.
[2]
` Pipe A can fill the swimming pool in x hours. Another pipe B can fill the pool in ( 2 x 1 ) hours. If both pipes A and B are turned on simultaneously, the pool will be filled in 4 hours. (b)
What fraction of the pool can be filled in one hour by using pipe A only?
[1]
(c)
What fraction of the pool can be filled in one hour by using pipe B only?
[1]
(d)
Form an equation in x and show that it simplifies to 2 x 2 13 x 4 0 .
[2]
(e)
Solve the above equation and find, to the nearest minutes, the time taken to fill the swimming pool by pipe B only. Explain why one of the answers is rejected.
(f)
[4]
The completely filled pool can then be drained by a pipe in a time of t hours. Copy the given axes and sketch the graph of depth of water against time.
[2]
depth (m) 3 2
O
Tanjong Katong Secondary School
t
time (h)
Prelim Mathematics P2
7 7.
In the figure, B, C, X, Y and Z are points on the circle centre O. AB and AC are tangents to the circle. It is given that BAC 60 , OCX m , OBX n and CZ and BZ are both equal to the radius of the circle.
(a) Find, stating your reasons clearly, (i)
BOC ,
[1]
(ii)
BZC ,
[2]
(iii)
BYZ .
[2]
(b) Identify, with reasons, the triangle which is congruent to triangle OBC.
[3]
(c) Deduce a relationship between m and n.
[2]
X C
m O Y
Z
n
60 A
B
[Turn over Tanjong Katong Secondary School
Prelim Mathematics P2
8 8.
Diagram A shows the drawing of a regular hexagonal nut and Diagram B shows its cross-section. The nut, ABCDEF, of thickness 2 cm is inscribed in a circle of radius 14 cm and centre O. The hole in the middle of the nut is a circle of radius 7 cm and centre O. B B
C
C
D 2 cm
A F
7 cm O
A
D
14 cm
E F
Diagram A
E Diagram B
(a) Show that EF = 14 cm.
[2]
(b) Calculate the shaded area in Diagram B and hence state the volume of the nut. [3] (c) How many lines of symmetry does ABCDEF has?
[1]
(d) A cylinder of length 1 m is used to store the nuts as shown below.
1m
(i) Calculate the minimum volume of the cylinder.
[2]
(ii) How many nuts can the cylinder hold?
[1]
Tanjong Katong Secondary School
Prelim Mathematics P2
9 9.
Andrew is planning for a holiday to Australia. He goes to a bank to exchange currency. The rate of exchange between Singapore dollars (S$) and Australian dollars (A$) is A$1 = S$1.33. (a) Andrew wants to change S$1500. (i) How much Australian dollars he will receive, giving your answer correct to the nearest dollars.
[2]
(ii) The bank charges a commission for every exchange. If Andrew has to pay $1527.75 altogether, calculate the commission rate as a percentage of the value of money exchanged. [2] (b) Andrew needs to buy batteries for his camera. If he buys them in Singapore, it will cost him S$5.70. If he buys them in Australia, it will cost him A$4.80. Calculate how much, in Singapore dollars, he can save by buying the batteries in Singapore. [2] (c) Andrew will be driving in Australia, so he compares the car hire rates from three car companies, Ride Best, Swift Co and Auto Mobile to get the best value. The table below shows the charges of the companies, based on the number of days for which the car is hired and the number of kilometers for which the car is driven. Company
Cost per day (A$)
Cost per kilometer (A$)
Ride Best
40
0.45
Swift Co
110
Nil
Auto Mobile
Nil
0.75
(i) Which company should Andrew hire the car from, if he intends to cover 700 km in 5 days?
[2]
(ii) By comparing the rental schemes offered by Swift Co and Auto Mobile, calculate the maximum number of kilometers, correct to the nearest kilometres, he can cover per day before it becomes more expensive to hire a car from Auto Mobile.
[2]
[Turn over
Tanjong Katong Secondary School
Prelim Mathematics P2
10 10.
Answer the whole of this question on a sheet of graph paper. The variables x and y are connected by the equation y
x 1 1, x 0 . x2
Some corresponding values of x and y are given in the following table.
x
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
y
5.00
1.00
0.11
a
0.44
0.56
0.63
0.69
(a) Calculate the value of a.
[1]
(b) Taking 4 cm to represent 1 unit on each axis, draw the graph of y
x 1 1 x2
for values of x in the range 0.5 x 4.0 .
[3]
(c) By drawing a suitable tangent to the graph, estimate the x-coordinate of the point at which the gradient of the tangent to the curve at that point is
3 . 4
[2]
(d) By adding a suitable straight line to the graph, find the root of the equation
1 3 x x 2 x 1 0 for 0.5 x 4.0 . 2
[3]
(e) By adding the line y 3 to the graph, find the range of values of x which satisfy the inequality x 2 x 1 4 x 2 .
Tanjong Katong Secondary School
[3]
Prelim Mathematics P2
11 11.
The weights of 80 athletes in Sports School A are distributed as shown in the cumulative frequency curve below. Weights of 80 athletes in Sports School A Cumulative Frequency 80 70 60 50 40 30 20 10 0 44
46
48
50
52
54
56
58
60
Weights (kg)
(a) Use your graph to estimate (i) the median weight,
[1]
(ii) the interquartile range.
[1]
(b) Athletes who weigh more than 55 kg are considered overweight. What is the percentage of overweight athletes in Sports School A?
[2]
[Turn over
Tanjong Katong Secondary School
Prelim Mathematics P2
12
The table below gives the same information in a different form. Weight in kg
Number of athletes
44 t 46
3
46 t 48
6
48 t 50
9
50 t 52
p
52 t 54
21
54 t 56
18
56 t 58
8
58 t 60
q
(c) Find the value of p and q.
[2]
(d) Hence, find the mean and standard deviation of the weights of 80 athletes in Sports School A.
[3]
The box and whisker diagram shows the weights of 80 athletes from Sports School B.
44
46
50
48
52
54
56
58
60
(e) Compare the weights of the athletes from Sports School A and B in two different ways. [2]
---------- End of Paper ----------
Tanjong Katong Secondary School
Prelim Mathematics P2
2.
(a)
Given that 2 x
(b)
Given that the line
3m
xy , express y in terms of x. m
[3]
x y 1 passes through (0, 2) and (7, 3), a b find the value of a and b.
[3]
Squares are drawn to enclose numbers in the number array as shown in the diagram.
1
2
3
4
5
6
7
. . . .
2
4
6
8
10
12
14
. . . .
3
6
9
12
15
18
21
. . . .
4
8
12
16
20
24
28
. . . .
5
10
15
20
25
30
35
. . . .
The sums of the numbers in each square, Sn , are represented in the table below. Sum of numbers in square
Square, Sn S1
1
12 1
S2
1+2+2+4
32 9
S3 S4
(a) Copy and complete the table.
[2]
(b) Find a formula for Sn.
[2]
(c) The sum of numbers in the kth square is 44100. Find k.
[2]
[Turn over Tanjong Katong Secondary School
Prelim Mathematics P2
4
3.
Three football clubs, Rial, Bayern and Sporting, are playing in the annual soccer league. They played a few matches and their results can be represented in a matrix R as shown. Win Draw Lose Rial 6 2 2 Bayern 4 4 2 R Sporting 3 1 5 1 (a) The matrix G is 1 . Find RG. What do the elements in RG represent? 1
(b) (i) If a win earns 3 points, a draw earns 1 point and a loss earns no point, write a matrix P representing this information. (ii) Find RP and interpret the elements in the matrix.
4.
[2]
[1] [2]
Figure 1 shows a piece of thin metal sheet in the form of a sector of a circle 5 of radius 0.4 m and with reflex AOB . 3 The metal sheet is then shaped into a right circular cone with no overlapping as shown in Figure 2. P P A
5 3
O 0.4 m
B
A Figure 1
Figure 2
(a) Find the circumference of the cone, leaving your answer in terms of .
[2]
(b) Find the curved surface area of the cone.
[2]
(c) Find the radius and the height of the cone.
[4]
Tanjong Katong Secondary School
Prelim Mathematics P2
5 5.
In the diagram, A, B, C, D and M are points on level ground. C is due east of A, AM = 5 m, CM = 8 m, AD = 15 m, CD = 6 m, BM = BC and AMB 110 .
B N
5m
110
A
8m
M
C 6m 15 m D
(a)
(b)
Find (i)
the length BM,
[3]
(ii)
CAD ,
[3]
(iii) the bearing of D from A,
[1]
(iv) the area of ACD .
[2]
A vertical flag pole of height 6.5 m is located at point C. Find the smallest angle of elevation of the top of the pole when viewed from a point on path AB.
[2]
[Turn over
Tanjong Katong Secondary School
Prelim Mathematics P2
6 6.
The diagram below shows a swimming pool with a uniform width of 8 metres. The base of the pool is made up of a horizontal and a sloping plane. 10 m 1m
8m
3m
(a)
6m
Calculate the volume of the swimming pool.
[2]
` Pipe A can fill the swimming pool in x hours. Another pipe B can fill the pool in ( 2 x 1 ) hours. If both pipes A and B are turned on simultaneously, the pool will be filled in 4 hours. (b)
What fraction of the pool can be filled in one hour by using pipe A only?
[1]
(c)
What fraction of the pool can be filled in one hour by using pipe B only?
[1]
(d)
Form an equation in x and show that it simplifies to 2 x 2 13 x 4 0 .
[2]
(e)
Solve the above equation and find, to the nearest minutes, the time taken to fill the swimming pool by pipe B only. Explain why one of the answers is rejected.
(f)
[4]
The completely filled pool can then be drained by a pipe in a time of t hours. Copy the given axes and sketch the graph of depth of water against time.
[2]
depth (m) 3 2
O
Tanjong Katong Secondary School
t
time (h)
Prelim Mathematics P2
7 7.
In the figure, B, C, X, Y and Z are points on the circle centre O. AB and AC are tangents to the circle. It is given that BAC 60 , OCX m , OBX n and CZ and BZ are both equal to the radius of the circle.
(a) Find, stating your reasons clearly, (i)
BOC ,
[1]
(ii)
BZC ,
[2]
(iii)
BYZ .
[2]
(b) Identify, with reasons, the triangle which is congruent to triangle OBC.
[3]
(c) Deduce a relationship between m and n.
[2]
X C
m O Y
Z
n
60 A
B
[Turn over Tanjong Katong Secondary School
Prelim Mathematics P2
8 8.
Diagram A shows the drawing of a regular hexagonal nut and Diagram B shows its cross-section. The nut, ABCDEF, of thickness 2 cm is inscribed in a circle of radius 14 cm and centre O. The hole in the middle of the nut is a circle of radius 7 cm and centre O. B B
C
C
D 2 cm
A F
7 cm O
A
D
14 cm
E F
Diagram A
E Diagram B
(a) Show that EF = 14 cm.
[2]
(b) Calculate the shaded area in Diagram B and hence state the volume of the nut. [3] (c) How many lines of symmetry does ABCDEF has?
[1]
(d) A cylinder of length 1 m is used to store the nuts as shown below.
1m
(i) Calculate the minimum volume of the cylinder.
[2]
(ii) How many nuts can the cylinder hold?
[1]
Tanjong Katong Secondary School
Prelim Mathematics P2
9 9.
Andrew is planning for a holiday to Australia. He goes to a bank to exchange currency. The rate of exchange between Singapore dollars (S$) and Australian dollars (A$) is A$1 = S$1.33. (a) Andrew wants to change S$1500. (i) How much Australian dollars he will receive, giving your answer correct to the nearest dollars.
[2]
(ii) The bank charges a commission for every exchange. If Andrew has to pay $1527.75 altogether, calculate the commission rate as a percentage of the value of money exchanged. [2] (b) Andrew needs to buy batteries for his camera. If he buys them in Singapore, it will cost him S$5.70. If he buys them in Australia, it will cost him A$4.80. Calculate how much, in Singapore dollars, he can save by buying the batteries in Singapore. [2] (c) Andrew will be driving in Australia, so he compares the car hire rates from three car companies, Ride Best, Swift Co and Auto Mobile to get the best value. The table below shows the charges of the companies, based on the number of days for which the car is hired and the number of kilometers for which the car is driven. Company
Cost per day (A$)
Cost per kilometer (A$)
Ride Best
40
0.45
Swift Co
110
Nil
Auto Mobile
Nil
0.75
(i) Which company should Andrew hire the car from, if he intends to cover 700 km in 5 days?
[2]
(ii) By comparing the rental schemes offered by Swift Co and Auto Mobile, calculate the maximum number of kilometers, correct to the nearest kilometres, he can cover per day before it becomes more expensive to hire a car from Auto Mobile.
[2]
[Turn over
Tanjong Katong Secondary School
Prelim Mathematics P2
10 10.
Answer the whole of this question on a sheet of graph paper. The variables x and y are connected by the equation y
x 1 1, x 0 . x2
Some corresponding values of x and y are given in the following table.
x
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
y
5.00
1.00
0.11
a
0.44
0.56
0.63
0.69
(a) Calculate the value of a.
[1]
(b) Taking 4 cm to represent 1 unit on each axis, draw the graph of y
x 1 1 x2
for values of x in the range 0.5 x 4.0 .
[3]
(c) By drawing a suitable tangent to the graph, estimate the x-coordinate of the point at which the gradient of the tangent to the curve at that point is
3 . 4
[2]
(d) By adding a suitable straight line to the graph, find the root of the equation
1 3 x x 2 x 1 0 for 0.5 x 4.0 . 2
[3]
(e) By adding the line y 3 to the graph, find the range of values of x which satisfy the inequality x 2 x 1 4 x 2 .
Tanjong Katong Secondary School
[3]
Prelim Mathematics P2
11 11.
The weights of 80 athletes in Sports School A are distributed as shown in the cumulative frequency curve below. Weights of 80 athletes in Sports School A Cumulative Frequency 80 70 60 50 40 30 20 10 0 44
46
48
50
52
54
56
58
60
Weights (kg)
(a) Use your graph to estimate (i) the median weight,
[1]
(ii) the interquartile range.
[1]
(b) Athletes who weigh more than 55 kg are considered overweight. What is the percentage of overweight athletes in Sports School A?
[2]
[Turn over
Tanjong Katong Secondary School
Prelim Mathematics P2
12
The table below gives the same information in a different form. Weight in kg
Number of athletes
44 t 46
3
46 t 48
6
48 t 50
9
50 t 52
p
52 t 54
21
54 t 56
18
56 t 58
8
58 t 60
q
(c) Find the value of p and q.
[2]
(d) Hence, find the mean and standard deviation of the weights of 80 athletes in Sports School A.
[3]
The box and whisker diagram shows the weights of 80 athletes from Sports School B.
44
46
50
48
52
54
56
58
60
(e) Compare the weights of the athletes from Sports School A and B in two different ways. [2]
---------- End of Paper ----------
Tanjong Katong Secondary School
Prelim Mathematics P2
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