Ah Sem 2
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Anglican High School 2008 Sec 4 Mathematics SA2 Paper 2
Answer all the questions. 1(a) Factorise completely (i)
.
(ii) (b)
It is given that
[2] .
_ N[3]
T _ , express c in terms of
[3] 298 _ m
(c)
_ B
Simplify
[3]
C _
137° _ _________________________________________________________________________________ 473m _ 2) The diagram shows three points, A, B and C 20° _ on a piece of horizontal land. It is given that AB = 473 m, BC = 298 m and ∠ABC =137º. BT is a tower crane standing vertically from B and the angle of elevation of T from A is 20º. Calculate the _ A (a) height of the tower crane. [1] (b) length of AC.
[2]
(c) bearing of C from A.
[3]
(d) shortest distance from AC to B.
[2]
(e) largest angle of depression from T to a point along AC.
[1]
__________________________________________________________________________________
3. (a) Find matrices C and D which satisfy the following matrix equations.
[3]
1
(b)
A wholesaler distributes three flavours of lollipops to two different stores. The number of packets delivered and the cost price per packet are shown in the table below. Flavours of lollipop Name of stores
Number of packets delivered Chocolate
Vanilla
Strawberry
A
40
50
30
B
30
60
20
Cost price per packet $3.60 $3.50 $3.20 Write down two matrices such that the product of these matrices will show the cost that each store will incur. Hence evaluate the cost incurred by store A and store B. [3] (c)
Given that ξ = { x : x is an integer, 12}, P and Q are two subsets of ξ where P = {x: x is a factor of 12} and Q = {x: x is a multiple of 4}. (i)
Illustrate the given information using a Venn diagram.
[2]
(ii)
List the members in the set
[1]
(iii)
Describe in words what the set
. represents.
[1]
__________________________________________________________________________________ 4. The price of petrol in January was y dollars per litre and Mr Tan paid $72 to fill up his car with petrol. He could cover a distance of 500 km with this amount of petrol. In August, the price of petrol increased by 20 cents. Mr Tan noticed that he was getting 2 litres of petrol less after paying $76. (a) Write down an expression, in terms of y, for the amount of petrol he received in January. [1] (b) Write down an expression, in terms of y, for the amount of petrol he received in August. [1] (c) Form an equation in y and show that it reduces to 5 y 2 + 11 y − 36 = 0 .
[3]
(d) Solve this equation and find the price of petrol in January.
[2]
(e) Calculate the distance travelled per litre of petrol for Mr Tan’s car.
[2]
_________________________________________________________________________________ 5. In the figure, O is the centre of the circle. AB and CB are tangents to the circle at F and G. Given that ∠ABC = 40°, ∠FEA = 66° and DE = EF, calculate, giving reasons (i)
∠GFB,
[1]
(ii)
∠FGD,
[1]
(iii)
obtuse angle ∠FOD,
[1] 2
(iv)
∠GDO,
[2]
(v)
∠EFA
[3] B
G
40 °
C
O
D F Di agram is not dra wn to scal e
E
66 ° A
6. A piece of paper is cut such that AB and CD are arcs of two concentric circles with centre at O as shown in Figure 1. The perimeter of ABCD is 40 cm, BC = AD = 2cm and ∠AOB = 1.2 radians. (a) Show that the length of OB = 14cm. [2] (b) Calculate the area of ABCD.
O
1.2
[2]
Fi gure 1
A
B
2
2
(c) Calculate the area of triangle AOB.
[2]
D
C
A hollow cone in Figure 2 is formed when OBC in Figure 1 is curled to coincide with the line OAD. (d) Find the radius of the base of the cone. [2] (e) Hence, calculate the volume of the cone.
O Fi gure 2
[2]
AB 2
2
DC
__ 7. (i)
In 2005, Shirley bought an apartment that costs $ 560 145 and she paid a 20% cash deposit. Give all your answers, correct to the nearest dollar for the following questions. Shirley used her Central Provident Fund (CPF) Ordinary Account to pay 10% of the 20% deposit. Calculate the amount used from this account. [2] Shirley intended to pay the remaining 80% of the purchase price through a bank loan. She was 3
offered loan packages from Bank C and Bank U. The table below shows the interest rates over 5 years. 1st year 2.50% * 1.40%
Bank C Bank U
2nd year 2.75% 2.40%
3rd year 2.75% 2.80%
4th year 3.75% 4.00%
5th year 3.75% 4.00%
*Comes with a 1.25% cash rebate. (ii)
Bank C offered a cash rebate of 1.25% of the loan amount to the customer in the first year. Calculate the interest that Shirley would have to pay Bank C, taking into account the cash rebate, at the end of the first year. Hence, calculate the total interest she would have to pay Bank C over three years. [3]
(iii)
Given that Bank U’s package reduces the loan amount by the previous year’s interest amount after each year, calculate the total interest she would have to pay at the end of the 3rd year.[4]
(iv)
Since the beginning of 2008, Shirley has been paying monthly instalments of $1,900 for her apartment using her savings from the CPF Ordinary Account. The interest she would have earned had her savings remained in her CPF account for the whole year is $570. Calculate the annual interest rate paid by the CPF Board. [2] _________________________________________________________________________________ 8.
Answer the whole of this question on a sheet of graph paper. The variables x and y are connected by the equation y =
1 2 7 x − − 12 . Some corresponding 4 x
values of x and y are given in the following table. x y
-7 1.25
-6 -1.83
-5 -4.35
-4 a
-3 -7.42
-2.5 -7.64
-2 -7.5
-1 -4.75
(a) Find the value of a.
-0.5 2.06 [1]
(b) Taking 2 cm to represent 1 unit on the horizontal axis and 2 cm to represent 1 unit 1 2 7 on the vertical axis, draw and label the graph of y = x − − 12 for the values of x 4 x in the range − 7 ≤ x ≤ −0.5 . [3] (c) By drawing a tangent, find the gradient of the curve y = . (d)
1 2 7 x − − 12 at the point where 4 x [2]
Using your graph, find the range of values of y for which − 4 ≤ x ≤ −1. 4
[1]
(e)
Solve the equation
1 2 7 x − x − − 13 = 0 by adding a suitable line. 4 x
[3] _____________________________________________________________________________ 9 (a) Two pieces of jelly, one cylindrical and the other hemispherical are shown in Figures 1 and 2 respectively. Each piece of jelly is cut into four equal parts. Given that the height of the cylindrical jelly is 3 cm and the diameters of the two jellies are both 10 cm, calculate the (i)
volume and surface area of one quarter of the cylindrical jelly.
[4]
(ii)
volume and surface area of one quarter of the hemispherical jelly.
[3]
Figure 1
Figure 2
(b) The figure below is a 3-dimensional solid. V is the vertex directly above D. ∠VDA = ∠GAE = ∠AEF = 90º and BC = FG are parallel. Given that ABCD is a square of 6cm, AE = 1.5 cm, EF = 5.4 cm and VD = 11cm, calculate (i)
the length of VA , [1] Cumulative Frequency
(ii)
∠BFE, 200
(iii)
the volume of the solid. [3]
V
[1]
11
160 D _________________________________________________________________
C 6
A 6 B 10 (a) The diagram 120 shows the cumulative frequency of fish caught by 200 participants in a contest. G
H
1.5
E
F
5.4
80
40 5 O
10
20
30
40
50
x fishes Use the graph, showing your working clearly, find the (i)
median [1]
,
(ii)
inter-quartile range, [1]
(iii)
95th percentile, [1]
(iv)
probability that a participant chosen at random caught 17 or more fish. [2]
If 128 participants each caught k fish or less, use the graph to estimate the value of k.
[1]
b) Copy and complete the following frequency distribution table using the information from the cumulative frequency curve. [2] Number of Fish Frequency 6 46 3 4 c) Using graph paper, draw a histogram to represent the frequency distribution in (b). [2] __________________________________________________________________________________ End of Paper. Answer Keys 1 (a) (i) (ii)
o.e. 6
(b)
o.e.
(c)
2 (a)
Height of Tower
(b)
Length of AC
(c)
Bearing of C from A is 016.4°
(d)
Shortest distance from AC to B
(e)
Largest angle of depression from T to AC ≈ 52.2°
3 (a) (i)
, D=
(ii) 3(b) (i)
Cost incur for Store A is $415 and Store B is $382 E Q
P 3
4 1
12
8
11
10
6 2
7
9
5
(ii) (iii)
Q’) = {1,2,3,6} or The members are 1,2,3,6. represents the set where the members are neither multiples of 4 nor factors of 12.
4 (a)
(b) (d)
The price of petrol in January was $1.80 7
(e)
Distance travelled per litre of petrol =
5(i)
∠GFB = = 70°
(ii)
(iii) (iv) (v)
( FBG is isosceles because of tangents from external point P)
∠DEF = 180 – 66 = 114° (angle on a straight line) ∠FGD = 180 – 114 = 66° (angle in opposite seg) OR ∠FGD = 66° ( external angle = opposite angle of cyclic quad) Obtuse ∠FOD = 66 x 2 = 132°(∠ at centre = 2x∠ at circumference) ∠FGO = 20° ( FOG is isosceles , same radii OF and OG) ∠DGO = 66 – 20 = 46° ∠GDO = 46° ( DOG is isosceles , same radii OD and OG) ∠OFA = 90° (radius perpendicular to tangent) ∠OFE = (isosceles DOF and DEF) =57° ∠EFA = 90 – 57 = 33°
6(b)
Area of ABCD = 36cm2 Area of
(c)
Radius
(d)
Height of the cone
(e)
Volume of cone
7
(i)
She used $ 11 203 from her CPF Ordinary Account.
(ii)
Total interest paid over 3 years
(iii)
Total interest paid over the 3 years
(iv)
Interest rate =
$28952
8 x y
-7 1.25
-6 -1.83
-5 -4.35
-4 -6.25
-3 -7.42
8
-2.5 -7.64
-2 -7.5
-1 -4.75
-0.5 2.06
y=x+1
(a) a = - 6.25 ( Gradient at x = -2 is (d) (e)
9(a)(i)
58.9 cm3
Volume of one quarter cylindrical jelly
Surface area of one quarter of cylindrical jelly (ii)
Volume of one quarter hemispherical jelly ≈ 65.4 or 65.5 cm3 Surface area of one quarter of hemispherical jelly
(b)(i)
Length of VA =
(ii)
∠BFE ≈ 111.8°
(iii)
Volume of solid =
10(a)(i) (ii) (iii) (iv)
(exact value, rounding is not required)
Median = 25 Inter-quartile range = 10 Number of fish caught at the 95th percentile = 36 Participants who caught at least 17 fish = 172 Probability that a participant chosen caught 17 or more = 0.86 9
From the graph, value of k (b)
28
Number of Fishes
Frequency (Number of participants) 6 46 96 48 4
3 10 (b) Freq uency 110 100 96 90 80 70 60 50 48 40 30 20 10 6 4 0
10
20
30
40
50
10
No of Fishes
Answer all the questions. 1(a) Factorise completely (i)
.
(ii) (b)
It is given that
[2] .
_ N[3]
T _ , express c in terms of
[3] 298 _ m
(c)
_ B
Simplify
[3]
C _
137° _ _________________________________________________________________________________ 473m _ 2) The diagram shows three points, A, B and C 20° _ on a piece of horizontal land. It is given that AB = 473 m, BC = 298 m and ∠ABC =137º. BT is a tower crane standing vertically from B and the angle of elevation of T from A is 20º. Calculate the _ A (a) height of the tower crane. [1] (b) length of AC.
[2]
(c) bearing of C from A.
[3]
(d) shortest distance from AC to B.
[2]
(e) largest angle of depression from T to a point along AC.
[1]
__________________________________________________________________________________
3. (a) Find matrices C and D which satisfy the following matrix equations.
[3]
1
(b)
A wholesaler distributes three flavours of lollipops to two different stores. The number of packets delivered and the cost price per packet are shown in the table below. Flavours of lollipop Name of stores
Number of packets delivered Chocolate
Vanilla
Strawberry
A
40
50
30
B
30
60
20
Cost price per packet $3.60 $3.50 $3.20 Write down two matrices such that the product of these matrices will show the cost that each store will incur. Hence evaluate the cost incurred by store A and store B. [3] (c)
Given that ξ = { x : x is an integer, 12}, P and Q are two subsets of ξ where P = {x: x is a factor of 12} and Q = {x: x is a multiple of 4}. (i)
Illustrate the given information using a Venn diagram.
[2]
(ii)
List the members in the set
[1]
(iii)
Describe in words what the set
. represents.
[1]
__________________________________________________________________________________ 4. The price of petrol in January was y dollars per litre and Mr Tan paid $72 to fill up his car with petrol. He could cover a distance of 500 km with this amount of petrol. In August, the price of petrol increased by 20 cents. Mr Tan noticed that he was getting 2 litres of petrol less after paying $76. (a) Write down an expression, in terms of y, for the amount of petrol he received in January. [1] (b) Write down an expression, in terms of y, for the amount of petrol he received in August. [1] (c) Form an equation in y and show that it reduces to 5 y 2 + 11 y − 36 = 0 .
[3]
(d) Solve this equation and find the price of petrol in January.
[2]
(e) Calculate the distance travelled per litre of petrol for Mr Tan’s car.
[2]
_________________________________________________________________________________ 5. In the figure, O is the centre of the circle. AB and CB are tangents to the circle at F and G. Given that ∠ABC = 40°, ∠FEA = 66° and DE = EF, calculate, giving reasons (i)
∠GFB,
[1]
(ii)
∠FGD,
[1]
(iii)
obtuse angle ∠FOD,
[1] 2
(iv)
∠GDO,
[2]
(v)
∠EFA
[3] B
G
40 °
C
O
D F Di agram is not dra wn to scal e
E
66 ° A
6. A piece of paper is cut such that AB and CD are arcs of two concentric circles with centre at O as shown in Figure 1. The perimeter of ABCD is 40 cm, BC = AD = 2cm and ∠AOB = 1.2 radians. (a) Show that the length of OB = 14cm. [2] (b) Calculate the area of ABCD.
O
1.2
[2]
Fi gure 1
A
B
2
2
(c) Calculate the area of triangle AOB.
[2]
D
C
A hollow cone in Figure 2 is formed when OBC in Figure 1 is curled to coincide with the line OAD. (d) Find the radius of the base of the cone. [2] (e) Hence, calculate the volume of the cone.
O Fi gure 2
[2]
AB 2
2
DC
__ 7. (i)
In 2005, Shirley bought an apartment that costs $ 560 145 and she paid a 20% cash deposit. Give all your answers, correct to the nearest dollar for the following questions. Shirley used her Central Provident Fund (CPF) Ordinary Account to pay 10% of the 20% deposit. Calculate the amount used from this account. [2] Shirley intended to pay the remaining 80% of the purchase price through a bank loan. She was 3
offered loan packages from Bank C and Bank U. The table below shows the interest rates over 5 years. 1st year 2.50% * 1.40%
Bank C Bank U
2nd year 2.75% 2.40%
3rd year 2.75% 2.80%
4th year 3.75% 4.00%
5th year 3.75% 4.00%
*Comes with a 1.25% cash rebate. (ii)
Bank C offered a cash rebate of 1.25% of the loan amount to the customer in the first year. Calculate the interest that Shirley would have to pay Bank C, taking into account the cash rebate, at the end of the first year. Hence, calculate the total interest she would have to pay Bank C over three years. [3]
(iii)
Given that Bank U’s package reduces the loan amount by the previous year’s interest amount after each year, calculate the total interest she would have to pay at the end of the 3rd year.[4]
(iv)
Since the beginning of 2008, Shirley has been paying monthly instalments of $1,900 for her apartment using her savings from the CPF Ordinary Account. The interest she would have earned had her savings remained in her CPF account for the whole year is $570. Calculate the annual interest rate paid by the CPF Board. [2] _________________________________________________________________________________ 8.
Answer the whole of this question on a sheet of graph paper. The variables x and y are connected by the equation y =
1 2 7 x − − 12 . Some corresponding 4 x
values of x and y are given in the following table. x y
-7 1.25
-6 -1.83
-5 -4.35
-4 a
-3 -7.42
-2.5 -7.64
-2 -7.5
-1 -4.75
(a) Find the value of a.
-0.5 2.06 [1]
(b) Taking 2 cm to represent 1 unit on the horizontal axis and 2 cm to represent 1 unit 1 2 7 on the vertical axis, draw and label the graph of y = x − − 12 for the values of x 4 x in the range − 7 ≤ x ≤ −0.5 . [3] (c) By drawing a tangent, find the gradient of the curve y = . (d)
1 2 7 x − − 12 at the point where 4 x [2]
Using your graph, find the range of values of y for which − 4 ≤ x ≤ −1. 4
[1]
(e)
Solve the equation
1 2 7 x − x − − 13 = 0 by adding a suitable line. 4 x
[3] _____________________________________________________________________________ 9 (a) Two pieces of jelly, one cylindrical and the other hemispherical are shown in Figures 1 and 2 respectively. Each piece of jelly is cut into four equal parts. Given that the height of the cylindrical jelly is 3 cm and the diameters of the two jellies are both 10 cm, calculate the (i)
volume and surface area of one quarter of the cylindrical jelly.
[4]
(ii)
volume and surface area of one quarter of the hemispherical jelly.
[3]
Figure 1
Figure 2
(b) The figure below is a 3-dimensional solid. V is the vertex directly above D. ∠VDA = ∠GAE = ∠AEF = 90º and BC = FG are parallel. Given that ABCD is a square of 6cm, AE = 1.5 cm, EF = 5.4 cm and VD = 11cm, calculate (i)
the length of VA , [1] Cumulative Frequency
(ii)
∠BFE, 200
(iii)
the volume of the solid. [3]
V
[1]
11
160 D _________________________________________________________________
C 6
A 6 B 10 (a) The diagram 120 shows the cumulative frequency of fish caught by 200 participants in a contest. G
H
1.5
E
F
5.4
80
40 5 O
10
20
30
40
50
x fishes Use the graph, showing your working clearly, find the (i)
median [1]
,
(ii)
inter-quartile range, [1]
(iii)
95th percentile, [1]
(iv)
probability that a participant chosen at random caught 17 or more fish. [2]
If 128 participants each caught k fish or less, use the graph to estimate the value of k.
[1]
b) Copy and complete the following frequency distribution table using the information from the cumulative frequency curve. [2] Number of Fish Frequency 6 46 3 4 c) Using graph paper, draw a histogram to represent the frequency distribution in (b). [2] __________________________________________________________________________________ End of Paper. Answer Keys 1 (a) (i) (ii)
o.e. 6
(b)
o.e.
(c)
2 (a)
Height of Tower
(b)
Length of AC
(c)
Bearing of C from A is 016.4°
(d)
Shortest distance from AC to B
(e)
Largest angle of depression from T to AC ≈ 52.2°
3 (a) (i)
, D=
(ii) 3(b) (i)
Cost incur for Store A is $415 and Store B is $382 E Q
P 3
4 1
12
8
11
10
6 2
7
9
5
(ii) (iii)
Q’) = {1,2,3,6} or The members are 1,2,3,6. represents the set where the members are neither multiples of 4 nor factors of 12.
4 (a)
(b) (d)
The price of petrol in January was $1.80 7
(e)
Distance travelled per litre of petrol =
5(i)
∠GFB = = 70°
(ii)
(iii) (iv) (v)
( FBG is isosceles because of tangents from external point P)
∠DEF = 180 – 66 = 114° (angle on a straight line) ∠FGD = 180 – 114 = 66° (angle in opposite seg) OR ∠FGD = 66° ( external angle = opposite angle of cyclic quad) Obtuse ∠FOD = 66 x 2 = 132°(∠ at centre = 2x∠ at circumference) ∠FGO = 20° ( FOG is isosceles , same radii OF and OG) ∠DGO = 66 – 20 = 46° ∠GDO = 46° ( DOG is isosceles , same radii OD and OG) ∠OFA = 90° (radius perpendicular to tangent) ∠OFE = (isosceles DOF and DEF) =57° ∠EFA = 90 – 57 = 33°
6(b)
Area of ABCD = 36cm2 Area of
(c)
Radius
(d)
Height of the cone
(e)
Volume of cone
7
(i)
She used $ 11 203 from her CPF Ordinary Account.
(ii)
Total interest paid over 3 years
(iii)
Total interest paid over the 3 years
(iv)
Interest rate =
$28952
8 x y
-7 1.25
-6 -1.83
-5 -4.35
-4 -6.25
-3 -7.42
8
-2.5 -7.64
-2 -7.5
-1 -4.75
-0.5 2.06
y=x+1
(a) a = - 6.25 ( Gradient at x = -2 is (d) (e)
9(a)(i)
58.9 cm3
Volume of one quarter cylindrical jelly
Surface area of one quarter of cylindrical jelly (ii)
Volume of one quarter hemispherical jelly ≈ 65.4 or 65.5 cm3 Surface area of one quarter of hemispherical jelly
(b)(i)
Length of VA =
(ii)
∠BFE ≈ 111.8°
(iii)
Volume of solid =
10(a)(i) (ii) (iii) (iv)
(exact value, rounding is not required)
Median = 25 Inter-quartile range = 10 Number of fish caught at the 95th percentile = 36 Participants who caught at least 17 fish = 172 Probability that a participant chosen caught 17 or more = 0.86 9
From the graph, value of k (b)
28
Number of Fishes
Frequency (Number of participants) 6 46 96 48 4
3 10 (b) Freq uency 110 100 96 90 80 70 60 50 48 40 30 20 10 6 4 0
10
20
30
40
50
10
No of Fishes
