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Anglican High School 2008 Sec 4 Mathematics SA2 Paper 1
1.
Answer all the questions. 5 2 Mr Pearson drafted a will, in which he will leave of his money to his wife, of the 12 3 remainder to his two children and the rest to be donated to a charitable organization. If the charitable organization receives $70 000, how much money does he have in total?
Answer 2.
$
Four Christmas light bulbs are made to flicker at intervals of 2 seconds, 10 seconds, 12 seconds and 20 seconds respectively. All the bulbs start to light up at 12 mid night. At what time will the bulbs light up together again after 12 mid night?
Answer 3.
[2]
[2]
The reciprocal of 3 −2 is 35m . Write down the value of m.
Answer
1
m=
[2]
Anglican High School 2008 Sec 4 Mathematics SA2 Paper 1
4.
Given that a = 3.9 × 10 −3 and b = 1.44 × 10 −2 , find the value of
4a b
, giving your answer
in standard form.
Answer 5.
[2]
Paul and Queen have two similar vases. Paul’s vase has a capacity of 200 cubic centimetres. Queen’s vase has a height three times that of Paul’s. Calculate the capacity of Queen’s vase.
cm3 [2]
Answer 6.
On a map, 5 cm represents an actual distance of 2 km. (a) Express the scale of the map in the form 1 : n. (b) Calculate the actual length of a road, in km, which is represented by a length of 12 cm on the map. (c) Given that the actual area of a park is 8.4 km2, find the area of the park on the map. Express your answer in cm2.
Answer (a) (b) (c)
2
[1] km [1] cm2 [1]
Anglican High School 2008 Sec 4 Mathematics SA2 Paper 1
7.
(a)
Write down the following numbers in descending order. 1.6 , − 1.62 , −0.4 , −1.62 , −
(b)
3 10
Find all the integers which satisfy both 8 2 x 13 and x 1 .
Answer (a)
[1]
(b)
8.
Given that p is 20% of q, find the value of
[2]
p , expressing your answer as a fraction in its 4q
lowest terms.
Answer 9.
[2]
A polygon has n sides. Two of its exterior angles are 55° and 75°, and the other (n − 2) interior angles are 157° each. Find the value of n.
Answer
3
n=
[2]
Anglican High School 2008 Sec 4 Mathematics SA2 Paper 1
10.
y is inversely proportional to the positive square root of x. It is known that y = 7 for a particular value of x. Find the value of y when this value of x is increased by 4 times.
Answer 11.
y =
[3]
In the diagram below, XD = 10 cm, DZ = 5 cm, ZY = 8 cm and XY = 17 cm. (a) Given that EXDZ is a straight line, explain why DYZ is a right-angled triangle. (b) Expressing your answer as a fraction in its simplest form, write down the value of tan ∠EXY . E
X
D 5 cm Z
10 cm
8 cm
17 cm
Y
Answer (a)
[1]
(b) tan ∠EXY =
4
[2]
Anglican High School 2008 Sec 4 Mathematics SA2 Paper 1
12.
Refer to the number pattern in the table below. Term (n) 1
A 1 4 1+ = 3 3 1 1 8 + = 3 5 15 1 1 12 + = 5 7 35 1 1 16 + = 7 9 63
2 3 4
B 4 + 32 = 5 2 2
8 2 + 15 2 = 17 2
16 2 + 63 2 = 65 2
10
(a) (b)
Fill in the blank in column B when n = 3. Fill in the blanks for columns A and B when n = 10.
3
13.
(a) (b)
(
)
[1] [2]
1
− 1 Simplify ÷ 64 y 6 2 . 5x Factorize 1 − a 2 b 2 + a 2 − b 2 completely.
Answer (a) (b)
5
[2] [2]
Anglican High School 2008 Sec 4 Mathematics SA2 Paper 1
14.
The figure shows OA = 10a, OB = 10b and X is the mid-point of OA. 1 Z is a point on OB produced such that OB = BZ . 2 A
10a
O (a)
X
Y
10b
B
Z
Express XZ in terms of a and b, as simply as possible.
The lines AB and XZ intersect at Y. (b) Given that XY = k XZ , where k is a real number, show that OY = 5(1 – k)a + 30kb.
Answer (a) XZ = Answer (b)
[2] [2]
6
Anglican High School 2008 Sec 4 Mathematics SA2 Paper 1
15.
In the triangle PQR, A and B are points on the sides PR and QR respectively. Given that PQ = 3x cm, QR = 2y cm, AB = 3 cm, AR = 3.5 cm, PQˆ R = BAˆ R and ∆PQR is similar to ∆BAR , find (a) the ratio of x : y, Q (b) the area of quadrilateral PQBA, in terms of x, if the area of ∆BAR = 4 cm2. 3x
P
2y B 3 A 3.5 R
Answer (a)
:
(b) 16.
cm2 [2]
Solve the equation 74 x = 5 − 9 x 2 , giving your answer correct to 2 decimal places.
Answer x = 17.
[2]
A straight line AC, whose equation is 2 x − 3 y + 6 = 0 , is drawn below. 7
or
[3]
Anglican High School 2008 Sec 4 Mathematics SA2 Paper 1
B is the point (3, 0). Find (a) the gradient of AC, (b) the coordinates of A, (c) the equation of the line through B which is parallel to AC, (d) the coordinates of C if the equation of BC is y +9 = 3x.
y C
A
B(3, 0)
0
x
Answer (a)
[1]
(b) A(
,
)
(c)
[2]
(d) C(
18.
[1]
,
)
[3]
On the grids in the answer space below, sketch the following graphs, indicating clearly all points of intersection on the x and y axes. Answer (i)
y =
1 x2
[1]
(ii)
y = 1 − x2
y 19.
[2]
y
(a) The distances of the school from the students’ homes are shown in the table below. 0
Find (i) (ii)
Distance from x home (km) No. of students
the mean, the mode, 8
1
2
3 0
4
12
8
4
6
x
Anglican High School 2008 Sec 4 Mathematics SA2 Paper 1
(iii)
(b)
the median. Answer (a)(i)
km
[2]
(ii)
km
[1]
(iii)
km
[1]
Each of the 300 Secondary One students was asked to name their favourite CCA. The table below shows the results of the choices made by the students.
CCA No. of Students
NCC 100
Girl Guides 95
Robotics Club x
Concert Band 55
Find the value of x and use the space below to draw a pie chart to illustrate the above results. [3] Answer x =
Company A
Company B
20.
The graphs show the charges of two telcommunications companies for telephone calls lasting up to 10 minutes.
9
Anglican High School 2008 Sec 4 Mathematics SA2 Paper 1
Company A charges 40 cents for calls of 3 minutes or less and then at the constant rate of 10 cents per minute. Company B charges a connection fee of p cents and all calls are charged at the constant rate of q cents per minute. Using these graphs, find 1 (a) the cost of a 3 minute call using Company A, 2 (b) the value of p and of q, (c) the range of times for which it would be cheaper to use Company A.
Answer (a) (b) p = (c) 21.
cents [1] ,q=
[2] [2]
There are 17 players vying for two places in one of the S-League Under-18 soccer teams. 12 are Malay and the remaining are Chinese. Two players are chosen at random. (a) Draw the probability tree diagram in the answer space below. Hence, calculate the probability that (b) the first player chosen is a Malay and the second player is a Chinese, (c) both players are of the same race, (d) the second player chosen is a Malay. 10
Anglican High School 2008 Sec 4 Mathematics SA2 Paper 1
Answer (a)
[2]
Answer (b)
22.
6v 2 + 6u 2 + 12uv . 3vx + 3ux + 4uw + 4vw
(a)
Simplify
(b)
If x 2 + y 2 = 73 and xy = 24, find the value of (i) (x + y)2 - 52 11
[1]
(c)
[2]
(d)
[2]
Anglican High School 2008 Sec 4 Mathematics SA2 Paper 1
(ii)
(5x – 5y)2
Answer (a)
End of paper. ANSWER KEY 1
$360 000
2
00 01 12
[3]
(b)(i)
[2]
(ii)
[2]
Anglican High School 2008 Sec 4 Mathematics SA2 Paper 1
3 4
2 5 1.3 × 10 −1
5
5400 cm 3
6(a)
1 : 40 000
m=
(b)
4.8 km
(c)
52.5 cm2.
7(a) (b)
−
3 , - 0.4, − 1.62 , -1.6, -1.62 10
Integers are -2, -1, 0 and 1.
8
1 20
9
12
10
7 2
11(a)
15 2 + 8 2 = 289 = 17 2 By the Pythagoras’ Theorem, ∆ XYZ is rightangled. Therefore, ∆ DYZ is also right-angled.
(b)
12(a) (b) 13(a)
(b) 14(a) (b)
−
8 15
12 2 + 35 2 = 37 2 1 1 40 + = ; 40 2 + 399 2 = 4012 19 21 399 8y3 o.e. 125 x 3 (1 + b)(1 − b)(1 + a 2 )
5(6b – a) XY = k XZ OY − OX = 5k(6b – a) 13
Anglican High School 2008 Sec 4 Mathematics SA2 Paper 1
OY = 30kb – 5ka + OX = 30kb – 5ka + 5a = 5(1 – k)a + 30kb (shown) 15(a) (b) 16 17(a) (b) (c) (d) 18(a)
(b)
∆PQR is similar to ∆BAR x:y=4:7 4(x2 – 1) m c2 0 .07 or - 8.29 2 y = x+2 3 (-3, 0) 2 y = x − 2 or 3 y = 2 x − 6 3 5 1 C( 4 , 5 ) 7 7 1 Graph of y = 2 x
Graph of y = 1 − x 2
14
Anglican High School 2008 Sec 4 Mathematics SA2 Paper 1
19(a)(i) (ii)
2 km 15 1 km
(iii)
2
(b)
2
x = 50
Robotics Club 60° Concert Band
66°
Girl Guides 114° 120° NCC
20(a) 45 cents (b)
p = 36, Gradient = q=5
(c)
86 − 36 50 = =5 10 − 0 10
0.8 < t < 5.2
21(a)
1st Player
2nd Player
(11) (16)
12 17
Malay player
(5) (16) (12) 3 = (16) 4
5 17
(c)
Chinese player
Malay player
Chinese player
( 4) 1 = (16) 4
(b)
Malay player
Chinese player
12 5 15 × = 17 16 68 12 11 5 4 19 × + × = 17 16 17 16 34
(d) 15
Anglican High School 2008 Sec 4 Mathematics SA2 Paper 1
12 11 5 12 12 × + × = 17 16 17 16 17 6(u + v) 3x + 4w
22(a) (b)(i) (ii)
96 625
16
1.
Answer all the questions. 5 2 Mr Pearson drafted a will, in which he will leave of his money to his wife, of the 12 3 remainder to his two children and the rest to be donated to a charitable organization. If the charitable organization receives $70 000, how much money does he have in total?
Answer 2.
$
Four Christmas light bulbs are made to flicker at intervals of 2 seconds, 10 seconds, 12 seconds and 20 seconds respectively. All the bulbs start to light up at 12 mid night. At what time will the bulbs light up together again after 12 mid night?
Answer 3.
[2]
[2]
The reciprocal of 3 −2 is 35m . Write down the value of m.
Answer
1
m=
[2]
Anglican High School 2008 Sec 4 Mathematics SA2 Paper 1
4.
Given that a = 3.9 × 10 −3 and b = 1.44 × 10 −2 , find the value of
4a b
, giving your answer
in standard form.
Answer 5.
[2]
Paul and Queen have two similar vases. Paul’s vase has a capacity of 200 cubic centimetres. Queen’s vase has a height three times that of Paul’s. Calculate the capacity of Queen’s vase.
cm3 [2]
Answer 6.
On a map, 5 cm represents an actual distance of 2 km. (a) Express the scale of the map in the form 1 : n. (b) Calculate the actual length of a road, in km, which is represented by a length of 12 cm on the map. (c) Given that the actual area of a park is 8.4 km2, find the area of the park on the map. Express your answer in cm2.
Answer (a) (b) (c)
2
[1] km [1] cm2 [1]
Anglican High School 2008 Sec 4 Mathematics SA2 Paper 1
7.
(a)
Write down the following numbers in descending order. 1.6 , − 1.62 , −0.4 , −1.62 , −
(b)
3 10
Find all the integers which satisfy both 8 2 x 13 and x 1 .
Answer (a)
[1]
(b)
8.
Given that p is 20% of q, find the value of
[2]
p , expressing your answer as a fraction in its 4q
lowest terms.
Answer 9.
[2]
A polygon has n sides. Two of its exterior angles are 55° and 75°, and the other (n − 2) interior angles are 157° each. Find the value of n.
Answer
3
n=
[2]
Anglican High School 2008 Sec 4 Mathematics SA2 Paper 1
10.
y is inversely proportional to the positive square root of x. It is known that y = 7 for a particular value of x. Find the value of y when this value of x is increased by 4 times.
Answer 11.
y =
[3]
In the diagram below, XD = 10 cm, DZ = 5 cm, ZY = 8 cm and XY = 17 cm. (a) Given that EXDZ is a straight line, explain why DYZ is a right-angled triangle. (b) Expressing your answer as a fraction in its simplest form, write down the value of tan ∠EXY . E
X
D 5 cm Z
10 cm
8 cm
17 cm
Y
Answer (a)
[1]
(b) tan ∠EXY =
4
[2]
Anglican High School 2008 Sec 4 Mathematics SA2 Paper 1
12.
Refer to the number pattern in the table below. Term (n) 1
A 1 4 1+ = 3 3 1 1 8 + = 3 5 15 1 1 12 + = 5 7 35 1 1 16 + = 7 9 63
2 3 4
B 4 + 32 = 5 2 2
8 2 + 15 2 = 17 2
16 2 + 63 2 = 65 2
10
(a) (b)
Fill in the blank in column B when n = 3. Fill in the blanks for columns A and B when n = 10.
3
13.
(a) (b)
(
)
[1] [2]
1
− 1 Simplify ÷ 64 y 6 2 . 5x Factorize 1 − a 2 b 2 + a 2 − b 2 completely.
Answer (a) (b)
5
[2] [2]
Anglican High School 2008 Sec 4 Mathematics SA2 Paper 1
14.
The figure shows OA = 10a, OB = 10b and X is the mid-point of OA. 1 Z is a point on OB produced such that OB = BZ . 2 A
10a
O (a)
X
Y
10b
B
Z
Express XZ in terms of a and b, as simply as possible.
The lines AB and XZ intersect at Y. (b) Given that XY = k XZ , where k is a real number, show that OY = 5(1 – k)a + 30kb.
Answer (a) XZ = Answer (b)
[2] [2]
6
Anglican High School 2008 Sec 4 Mathematics SA2 Paper 1
15.
In the triangle PQR, A and B are points on the sides PR and QR respectively. Given that PQ = 3x cm, QR = 2y cm, AB = 3 cm, AR = 3.5 cm, PQˆ R = BAˆ R and ∆PQR is similar to ∆BAR , find (a) the ratio of x : y, Q (b) the area of quadrilateral PQBA, in terms of x, if the area of ∆BAR = 4 cm2. 3x
P
2y B 3 A 3.5 R
Answer (a)
:
(b) 16.
cm2 [2]
Solve the equation 74 x = 5 − 9 x 2 , giving your answer correct to 2 decimal places.
Answer x = 17.
[2]
A straight line AC, whose equation is 2 x − 3 y + 6 = 0 , is drawn below. 7
or
[3]
Anglican High School 2008 Sec 4 Mathematics SA2 Paper 1
B is the point (3, 0). Find (a) the gradient of AC, (b) the coordinates of A, (c) the equation of the line through B which is parallel to AC, (d) the coordinates of C if the equation of BC is y +9 = 3x.
y C
A
B(3, 0)
0
x
Answer (a)
[1]
(b) A(
,
)
(c)
[2]
(d) C(
18.
[1]
,
)
[3]
On the grids in the answer space below, sketch the following graphs, indicating clearly all points of intersection on the x and y axes. Answer (i)
y =
1 x2
[1]
(ii)
y = 1 − x2
y 19.
[2]
y
(a) The distances of the school from the students’ homes are shown in the table below. 0
Find (i) (ii)
Distance from x home (km) No. of students
the mean, the mode, 8
1
2
3 0
4
12
8
4
6
x
Anglican High School 2008 Sec 4 Mathematics SA2 Paper 1
(iii)
(b)
the median. Answer (a)(i)
km
[2]
(ii)
km
[1]
(iii)
km
[1]
Each of the 300 Secondary One students was asked to name their favourite CCA. The table below shows the results of the choices made by the students.
CCA No. of Students
NCC 100
Girl Guides 95
Robotics Club x
Concert Band 55
Find the value of x and use the space below to draw a pie chart to illustrate the above results. [3] Answer x =
Company A
Company B
20.
The graphs show the charges of two telcommunications companies for telephone calls lasting up to 10 minutes.
9
Anglican High School 2008 Sec 4 Mathematics SA2 Paper 1
Company A charges 40 cents for calls of 3 minutes or less and then at the constant rate of 10 cents per minute. Company B charges a connection fee of p cents and all calls are charged at the constant rate of q cents per minute. Using these graphs, find 1 (a) the cost of a 3 minute call using Company A, 2 (b) the value of p and of q, (c) the range of times for which it would be cheaper to use Company A.
Answer (a) (b) p = (c) 21.
cents [1] ,q=
[2] [2]
There are 17 players vying for two places in one of the S-League Under-18 soccer teams. 12 are Malay and the remaining are Chinese. Two players are chosen at random. (a) Draw the probability tree diagram in the answer space below. Hence, calculate the probability that (b) the first player chosen is a Malay and the second player is a Chinese, (c) both players are of the same race, (d) the second player chosen is a Malay. 10
Anglican High School 2008 Sec 4 Mathematics SA2 Paper 1
Answer (a)
[2]
Answer (b)
22.
6v 2 + 6u 2 + 12uv . 3vx + 3ux + 4uw + 4vw
(a)
Simplify
(b)
If x 2 + y 2 = 73 and xy = 24, find the value of (i) (x + y)2 - 52 11
[1]
(c)
[2]
(d)
[2]
Anglican High School 2008 Sec 4 Mathematics SA2 Paper 1
(ii)
(5x – 5y)2
Answer (a)
End of paper. ANSWER KEY 1
$360 000
2
00 01 12
[3]
(b)(i)
[2]
(ii)
[2]
Anglican High School 2008 Sec 4 Mathematics SA2 Paper 1
3 4
2 5 1.3 × 10 −1
5
5400 cm 3
6(a)
1 : 40 000
m=
(b)
4.8 km
(c)
52.5 cm2.
7(a) (b)
−
3 , - 0.4, − 1.62 , -1.6, -1.62 10
Integers are -2, -1, 0 and 1.
8
1 20
9
12
10
7 2
11(a)
15 2 + 8 2 = 289 = 17 2 By the Pythagoras’ Theorem, ∆ XYZ is rightangled. Therefore, ∆ DYZ is also right-angled.
(b)
12(a) (b) 13(a)
(b) 14(a) (b)
−
8 15
12 2 + 35 2 = 37 2 1 1 40 + = ; 40 2 + 399 2 = 4012 19 21 399 8y3 o.e. 125 x 3 (1 + b)(1 − b)(1 + a 2 )
5(6b – a) XY = k XZ OY − OX = 5k(6b – a) 13
Anglican High School 2008 Sec 4 Mathematics SA2 Paper 1
OY = 30kb – 5ka + OX = 30kb – 5ka + 5a = 5(1 – k)a + 30kb (shown) 15(a) (b) 16 17(a) (b) (c) (d) 18(a)
(b)
∆PQR is similar to ∆BAR x:y=4:7 4(x2 – 1) m c2 0 .07 or - 8.29 2 y = x+2 3 (-3, 0) 2 y = x − 2 or 3 y = 2 x − 6 3 5 1 C( 4 , 5 ) 7 7 1 Graph of y = 2 x
Graph of y = 1 − x 2
14
Anglican High School 2008 Sec 4 Mathematics SA2 Paper 1
19(a)(i) (ii)
2 km 15 1 km
(iii)
2
(b)
2
x = 50
Robotics Club 60° Concert Band
66°
Girl Guides 114° 120° NCC
20(a) 45 cents (b)
p = 36, Gradient = q=5
(c)
86 − 36 50 = =5 10 − 0 10
0.8 < t < 5.2
21(a)
1st Player
2nd Player
(11) (16)
12 17
Malay player
(5) (16) (12) 3 = (16) 4
5 17
(c)
Chinese player
Malay player
Chinese player
( 4) 1 = (16) 4
(b)
Malay player
Chinese player
12 5 15 × = 17 16 68 12 11 5 4 19 × + × = 17 16 17 16 34
(d) 15
Anglican High School 2008 Sec 4 Mathematics SA2 Paper 1
12 11 5 12 12 × + × = 17 16 17 16 17 6(u + v) 3x + 4w
22(a) (b)(i) (ii)
96 625
16
