2009 Rv Em P2
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RIVER VALLEY HIGH SCHOOL 2009 PRELIMINARY EXAMINATION SECONDARY FOUR CANDIDATE NAME CLASS
4
INDEX NUMBER
___________________________________________________________________________
MATHEMATICS
4016/02
Paper 2
15 September 2009 2 hours 30 minutes
Additonal Materials:
Answer Paper Graph paper (1 sheet)
___________________________________________________________________________ READ THESE INSTRUCTIONS FIRST Write your class, index number and name on all the work you hand in. Write in dark blue or black pen on both sides of the paper. You may use a pencil for any diagrams or graphs. Do not use staples, paper clips, highlighters, glue or correction fluid. Answer all questions. If working is needed for any question it must be shown with the answer. Omission of essential working will result in loss of marks. Calculators should be used where appropriate. If the degree of accuracy is not specified in the question, and if the answer is not exact, give the answer to three significant figures. Give answers in degrees to one decimal place. For , use either your calculator value or 3.142, unless the question requires the answer in terms of . At the end of the examination, fasten all your work securely together. The number of marks is given in brackets [ ] at the end of each question or part question. The total of the marks for this paper is 100.
________________________________________________________________________ This document consists of 10 printed pages including this page
2009 RVHS Preliminary Examination
Page 1 of 10
O Level Mathematics (4016) P2
2 Mathematical Formulae Compound interest Total amount = P( 1 +
r n ) 100
Measuration Curved surface area of a cone = rl Surface area of a sphere = 4r2 1 Volume of a cone = r2h 3 4 Volume of a sphere = r3 3 1 Area of triangle ABC = ab sin C 2
Arc length = r, where is in radians 1 Sector area = r2, where is in radians 2
Trigonometry a b c = = sin A sin B sin C
a2 = b2 + c2 2bc cos A
Statistics Mean =
Standard deviation =
2009 RVHS Preliminary Examination
fx f fx 2 fx f f
Page 2 of 10
2
O Level Mathematics (4016) P2
3 Answer all the questions.
1
(a) Given that a 2, b 3 and c 1 , find the value of
2ab ac 2 . 5c
[2]
(b) By applying factorisation, simplify the expression
2 y2 8 . 2 y2 y 6
[3]
1 3 x . [3] x 1 4 6 x ___________________________________________________________________________
(c) Solve the equation
B
N
2 B
455 m
C
785 765 m
25 A
70
455 m
D
In the diagram, A, B, C and D are four points on a horizontal field. The bearing of B from A is 320, AB = AD = 455 m, AC = 765 m, BAC = 25 and CDA = 70. (a) Calculate (i) BC,
[2]
(ii) ACD,
[2]
(iii) the bearing of D from A,
[2]
(iv) area of triangle ABC.
[2]
(b) A tower of height 99 m is situated at the point B. Find the angle of elevation of the top of the tower from the point A.
[2]
___________________________________________________________________________ 2009 RVHS Preliminary Examination
Page 3 of 10
O Level Mathematics (4016) P2
4 3
(a) John inherits $120,000 from his grandparents. He decides to deposit the whole sum in the bank to earn interest. The bank offers him two plans. Plan A offers a compound interest of 3.5% per annum. Plan B offers a simple interest rate of 3.7% per annum. If John only intends to leave the money in the bank for 4 years, which plan would be a better choice? Explain your reasons. [4] (b) A digital camera can be purchased online at a price of US$234. A Singaporean shopkeeper decides to sell the same type of digital camera in his shop at a discounted price (in S$) that is equivalent to what a buyer can buy online. If the discount given is 15%, calculate the original selling price (in S$) of the digital camera in the Singaporean shop. [US$1 = S$1.70] [3]
(c) The value of a motorcycle bought in Jan 2007 decreased by 12% by the end of 2007. However, by end of 2008, its value increased by 15% from its value at the end of 2007. Express the eventual value of the motorcycle at the end of 2008 as a percentage of its original value in Jan 2007. [2] ___________________________________________________________________________ 4
The first four terms in a pattern of numbers, T1, T2, T3, T4, ………., are given below. T1 = 12 + 10 = 1 T2 = 32 + 21 = 11 T3 = 52 + 32 = 31 T4 = 72 + 43 = 61 (a) Write down an expression for T5 and show that T5 = 101.
[1]
(b) Write down an expression for T6 and evaluate it.
[1]
(c) Find an expression, in terms of n, for the nth term, Tn, of the pattern.
[3]
(d) Evaluate T15.
[1]
(e) (i) Simplify (2 n 1) 2 (2n 3) 2
[1]
(ii) Hence or otherwise, find and simplify, an expression, in terms of n, for Tn Tn1.
[2]
__________________________________________________________________________
2009 RVHS Preliminary Examination
Page 4 of 10
O Level Mathematics (4016) P2
5 5
Mr Lim bought some superior quality cooking oil for $800 for his mini-mart. For this, he has to pay $x for each litre of cooking oil. (a) Write down an expression, in terms of x, for the number of litres he bought.
[1]
(b) Due to leak in some containers , he lost 3 litres of cooking oil. He then sold each litre of the reminder of the cooking oil at a price of $2 more than what he has originally paid for each litre. Write down an expression, in terms of x, for the money he received from the sale of all the reminder of the cooking oil. [2] (c) Given further that he made a profit of $100. Write down an equation to represent all the above information, and show that it simplifies to
3 x 2 106 x 1600 0 .
[2]
(d) Solve the equation 3 x 2 106 x 1600 0 .
[3]
(e) Find, correct to the nearest whole number, the litres of cooking oil sold.
[2]
___________________________________________________________________________ 6 O
F
In the above diagram, the points A, B, C, D and E lie on the circumference of the two circles such that AEDC forms a cyclic quadrilateral and BCD is a straight line. The centre of the smaller circle is marked as O. The line AD and CE intersect at the point F. Given that AB = BC, AED = 65° and ACE = 70°. (a) Explain, with geometrical reason, why ECD = 45.
[2]
(b) Calculate (i) ACB,
[1]
(ii) ABC,
[1]
(iii) AOC.
[2]
(c) Find ADE and hence show that AFC is similar to EFD.
[2]
___________________________________________________________________________ 2009 RVHS Preliminary Examination
Page 5 of 10
O Level Mathematics (4016) P2
6 7
O
(a)
56
17 cm
17 cm
16 cm P
R Q
The diagram above shows part of a circle with centre O and radius 17 cm and the chord PR = 16 cm. (i) Show that angle POR = 0.980 radians.
[2]
(ii) Find the arc length PQR
[1]
(iii) Calculate the area of the shaded segment.
[3]
(b)
V
C
B
N
D
15
21
8 cm
A
6 cm
The diagram above shows a solid pyramid with a rectangular base ABCD and a vertical height VN. Given that AB = 8 cm, AD = 6 cm and VN = 12 cm, (i)
Show that VB = 13 cm,
[2]
(ii)
Hence, determine angle DVB.
[2]
___________________________________________________________________________ 2009 RVHS Preliminary Examination
Page 6 of 10
O Level Mathematics (4016) P2
7 8
18cm
6 cm
8 cm
6 cm A cone, a cylinder and a hemisphere are joined together to form a container as shown in the above diagram. The height and radius of the cone is 18 cm and 6 cm respectively and the height of the cylinder is 8 cm. (a) Calculate the volume of the whole container.
[4]
(b) The container is then filled with water up to a height level of 20 cm from the base tip of the hemisphere. The figure below shows the water level in the cone portion of the container in this situation with CA = 6 cm : V
18 cm C
D
6 cm A 6 cm
B
(i) Find the slant height VB of the cone.
[2]
(ii) By applying similar triangle property, find the length CD and VD.
[2]
(iii) Hence, find the total surface area of container that is in contact with water in this situation. [4] ___________________________________________________________________________
2009 RVHS Preliminary Examination
Page 7 of 10
O Level Mathematics (4016) P2
8 9
Answer the whole of this question on a sheet of graph paper. The variables x and y are related by the equation 4 y x . x Some corresponding values of x and y are given in the following table. X Y
0.5 8.5
1 5
1.5 p
2 4
2.5 4.1
3 4.33
4 5
5 5.8
6 6.67
(a) Calculate the value of p.
[1]
(b) Taking 2 cm to represent 1 unit on x-axis and 2 cm to represent 1 unit on y-axis, 4 draw the graph of y x for values of x in the range 0 x 6 . [3] x (c) Use your graph to determine the minimum value of y x
4 in the range x
0 x 6.
[1]
(d) By drawing a tangent, find the gradient of the curve at the point (4, 5).
[2]
(e) By adding appropriate straight lines to your graph, find the solutions of the following equations in the range 0 x 6 for 4 6, x 4 (ii) 2 x 8 . x
(i) x
[2] [2]
(f) Describe the change in y-coordinate value of points on the curve y x the corresponding x-coordinate value decreases from 0.5 towards 0.
4 when x [1]
_______________________________________________________________________
2009 RVHS Preliminary Examination
Page 8 of 10
O Level Mathematics (4016) P2
9 10 (a) The cumulative frequency curve below illustrates the marks obtained, out of 60, by 120 students in a Mathematics test. 120
Cumulative frequency
100
80
60
40
20
0
10
20
30
40
50
60
Marks (x) (i)
Use the graph to determine (a) the median mark,
[1]
(b) the upper quartile,
[1]
(c) the passing mark if
1 of the sudent failed the test. 6
[1]
(ii) The following is the grouped frequency table of marks of the 120 students. 0 < x 10 10 < x 20 20 < x 30 30 < x 40 40 < x 50 50 < x 60 Marks (x) Frequency 4 11 m N 31 7 Find the value of m and n. [1] (iii)
Using your grouped frequency table, calculate an estimate of (a) the mean mark, (b) the standard deviation.
2009 RVHS Preliminary Examination
Page 9 of 10
[3] O Level Mathematics (4016) P2
10 10 (b) A bag contains six identical balls numbered 1, 2, 3, 4, 5 and 6. Two balls are drawn at random, one after the other, from the bag without replacement. Copy and complete the following possibility diagram and use to find the probability that the sum of the numbers drawn is a prime number.
2 nd drawn number
(i)
(ii)
+ 1 2 3 4 5 6
1
1st drawn number 2 3 4 5 3 4 5
3 4 5
5
6
[2]
Copy and complete the following probability tree diagram and use it to find the probability that 1 even and 1 odd numbered ball are drawn. 2nd draw
1st draw
even (
)
(
)
even
3 6
odd
even (
)
odd 2 5
odd
[3]
___________________________________________________________________________ 2009 RVHS Preliminary Examination
Page 10 of 10
O Level Mathematics (4016) P2
4
INDEX NUMBER
___________________________________________________________________________
MATHEMATICS
4016/02
Paper 2
15 September 2009 2 hours 30 minutes
Additonal Materials:
Answer Paper Graph paper (1 sheet)
___________________________________________________________________________ READ THESE INSTRUCTIONS FIRST Write your class, index number and name on all the work you hand in. Write in dark blue or black pen on both sides of the paper. You may use a pencil for any diagrams or graphs. Do not use staples, paper clips, highlighters, glue or correction fluid. Answer all questions. If working is needed for any question it must be shown with the answer. Omission of essential working will result in loss of marks. Calculators should be used where appropriate. If the degree of accuracy is not specified in the question, and if the answer is not exact, give the answer to three significant figures. Give answers in degrees to one decimal place. For , use either your calculator value or 3.142, unless the question requires the answer in terms of . At the end of the examination, fasten all your work securely together. The number of marks is given in brackets [ ] at the end of each question or part question. The total of the marks for this paper is 100.
________________________________________________________________________ This document consists of 10 printed pages including this page
2009 RVHS Preliminary Examination
Page 1 of 10
O Level Mathematics (4016) P2
2 Mathematical Formulae Compound interest Total amount = P( 1 +
r n ) 100
Measuration Curved surface area of a cone = rl Surface area of a sphere = 4r2 1 Volume of a cone = r2h 3 4 Volume of a sphere = r3 3 1 Area of triangle ABC = ab sin C 2
Arc length = r, where is in radians 1 Sector area = r2, where is in radians 2
Trigonometry a b c = = sin A sin B sin C
a2 = b2 + c2 2bc cos A
Statistics Mean =
Standard deviation =
2009 RVHS Preliminary Examination
fx f fx 2 fx f f
Page 2 of 10
2
O Level Mathematics (4016) P2
3 Answer all the questions.
1
(a) Given that a 2, b 3 and c 1 , find the value of
2ab ac 2 . 5c
[2]
(b) By applying factorisation, simplify the expression
2 y2 8 . 2 y2 y 6
[3]
1 3 x . [3] x 1 4 6 x ___________________________________________________________________________
(c) Solve the equation
B
N
2 B
455 m
C
785 765 m
25 A
70
455 m
D
In the diagram, A, B, C and D are four points on a horizontal field. The bearing of B from A is 320, AB = AD = 455 m, AC = 765 m, BAC = 25 and CDA = 70. (a) Calculate (i) BC,
[2]
(ii) ACD,
[2]
(iii) the bearing of D from A,
[2]
(iv) area of triangle ABC.
[2]
(b) A tower of height 99 m is situated at the point B. Find the angle of elevation of the top of the tower from the point A.
[2]
___________________________________________________________________________ 2009 RVHS Preliminary Examination
Page 3 of 10
O Level Mathematics (4016) P2
4 3
(a) John inherits $120,000 from his grandparents. He decides to deposit the whole sum in the bank to earn interest. The bank offers him two plans. Plan A offers a compound interest of 3.5% per annum. Plan B offers a simple interest rate of 3.7% per annum. If John only intends to leave the money in the bank for 4 years, which plan would be a better choice? Explain your reasons. [4] (b) A digital camera can be purchased online at a price of US$234. A Singaporean shopkeeper decides to sell the same type of digital camera in his shop at a discounted price (in S$) that is equivalent to what a buyer can buy online. If the discount given is 15%, calculate the original selling price (in S$) of the digital camera in the Singaporean shop. [US$1 = S$1.70] [3]
(c) The value of a motorcycle bought in Jan 2007 decreased by 12% by the end of 2007. However, by end of 2008, its value increased by 15% from its value at the end of 2007. Express the eventual value of the motorcycle at the end of 2008 as a percentage of its original value in Jan 2007. [2] ___________________________________________________________________________ 4
The first four terms in a pattern of numbers, T1, T2, T3, T4, ………., are given below. T1 = 12 + 10 = 1 T2 = 32 + 21 = 11 T3 = 52 + 32 = 31 T4 = 72 + 43 = 61 (a) Write down an expression for T5 and show that T5 = 101.
[1]
(b) Write down an expression for T6 and evaluate it.
[1]
(c) Find an expression, in terms of n, for the nth term, Tn, of the pattern.
[3]
(d) Evaluate T15.
[1]
(e) (i) Simplify (2 n 1) 2 (2n 3) 2
[1]
(ii) Hence or otherwise, find and simplify, an expression, in terms of n, for Tn Tn1.
[2]
__________________________________________________________________________
2009 RVHS Preliminary Examination
Page 4 of 10
O Level Mathematics (4016) P2
5 5
Mr Lim bought some superior quality cooking oil for $800 for his mini-mart. For this, he has to pay $x for each litre of cooking oil. (a) Write down an expression, in terms of x, for the number of litres he bought.
[1]
(b) Due to leak in some containers , he lost 3 litres of cooking oil. He then sold each litre of the reminder of the cooking oil at a price of $2 more than what he has originally paid for each litre. Write down an expression, in terms of x, for the money he received from the sale of all the reminder of the cooking oil. [2] (c) Given further that he made a profit of $100. Write down an equation to represent all the above information, and show that it simplifies to
3 x 2 106 x 1600 0 .
[2]
(d) Solve the equation 3 x 2 106 x 1600 0 .
[3]
(e) Find, correct to the nearest whole number, the litres of cooking oil sold.
[2]
___________________________________________________________________________ 6 O
F
In the above diagram, the points A, B, C, D and E lie on the circumference of the two circles such that AEDC forms a cyclic quadrilateral and BCD is a straight line. The centre of the smaller circle is marked as O. The line AD and CE intersect at the point F. Given that AB = BC, AED = 65° and ACE = 70°. (a) Explain, with geometrical reason, why ECD = 45.
[2]
(b) Calculate (i) ACB,
[1]
(ii) ABC,
[1]
(iii) AOC.
[2]
(c) Find ADE and hence show that AFC is similar to EFD.
[2]
___________________________________________________________________________ 2009 RVHS Preliminary Examination
Page 5 of 10
O Level Mathematics (4016) P2
6 7
O
(a)
56
17 cm
17 cm
16 cm P
R Q
The diagram above shows part of a circle with centre O and radius 17 cm and the chord PR = 16 cm. (i) Show that angle POR = 0.980 radians.
[2]
(ii) Find the arc length PQR
[1]
(iii) Calculate the area of the shaded segment.
[3]
(b)
V
C
B
N
D
15
21
8 cm
A
6 cm
The diagram above shows a solid pyramid with a rectangular base ABCD and a vertical height VN. Given that AB = 8 cm, AD = 6 cm and VN = 12 cm, (i)
Show that VB = 13 cm,
[2]
(ii)
Hence, determine angle DVB.
[2]
___________________________________________________________________________ 2009 RVHS Preliminary Examination
Page 6 of 10
O Level Mathematics (4016) P2
7 8
18cm
6 cm
8 cm
6 cm A cone, a cylinder and a hemisphere are joined together to form a container as shown in the above diagram. The height and radius of the cone is 18 cm and 6 cm respectively and the height of the cylinder is 8 cm. (a) Calculate the volume of the whole container.
[4]
(b) The container is then filled with water up to a height level of 20 cm from the base tip of the hemisphere. The figure below shows the water level in the cone portion of the container in this situation with CA = 6 cm : V
18 cm C
D
6 cm A 6 cm
B
(i) Find the slant height VB of the cone.
[2]
(ii) By applying similar triangle property, find the length CD and VD.
[2]
(iii) Hence, find the total surface area of container that is in contact with water in this situation. [4] ___________________________________________________________________________
2009 RVHS Preliminary Examination
Page 7 of 10
O Level Mathematics (4016) P2
8 9
Answer the whole of this question on a sheet of graph paper. The variables x and y are related by the equation 4 y x . x Some corresponding values of x and y are given in the following table. X Y
0.5 8.5
1 5
1.5 p
2 4
2.5 4.1
3 4.33
4 5
5 5.8
6 6.67
(a) Calculate the value of p.
[1]
(b) Taking 2 cm to represent 1 unit on x-axis and 2 cm to represent 1 unit on y-axis, 4 draw the graph of y x for values of x in the range 0 x 6 . [3] x (c) Use your graph to determine the minimum value of y x
4 in the range x
0 x 6.
[1]
(d) By drawing a tangent, find the gradient of the curve at the point (4, 5).
[2]
(e) By adding appropriate straight lines to your graph, find the solutions of the following equations in the range 0 x 6 for 4 6, x 4 (ii) 2 x 8 . x
(i) x
[2] [2]
(f) Describe the change in y-coordinate value of points on the curve y x the corresponding x-coordinate value decreases from 0.5 towards 0.
4 when x [1]
_______________________________________________________________________
2009 RVHS Preliminary Examination
Page 8 of 10
O Level Mathematics (4016) P2
9 10 (a) The cumulative frequency curve below illustrates the marks obtained, out of 60, by 120 students in a Mathematics test. 120
Cumulative frequency
100
80
60
40
20
0
10
20
30
40
50
60
Marks (x) (i)
Use the graph to determine (a) the median mark,
[1]
(b) the upper quartile,
[1]
(c) the passing mark if
1 of the sudent failed the test. 6
[1]
(ii) The following is the grouped frequency table of marks of the 120 students. 0 < x 10 10 < x 20 20 < x 30 30 < x 40 40 < x 50 50 < x 60 Marks (x) Frequency 4 11 m N 31 7 Find the value of m and n. [1] (iii)
Using your grouped frequency table, calculate an estimate of (a) the mean mark, (b) the standard deviation.
2009 RVHS Preliminary Examination
Page 9 of 10
[3] O Level Mathematics (4016) P2
10 10 (b) A bag contains six identical balls numbered 1, 2, 3, 4, 5 and 6. Two balls are drawn at random, one after the other, from the bag without replacement. Copy and complete the following possibility diagram and use to find the probability that the sum of the numbers drawn is a prime number.
2 nd drawn number
(i)
(ii)
+ 1 2 3 4 5 6
1
1st drawn number 2 3 4 5 3 4 5
3 4 5
5
6
[2]
Copy and complete the following probability tree diagram and use it to find the probability that 1 even and 1 odd numbered ball are drawn. 2nd draw
1st draw
even (
)
(
)
even
3 6
odd
even (
)
odd 2 5
odd
[3]
___________________________________________________________________________ 2009 RVHS Preliminary Examination
Page 10 of 10
O Level Mathematics (4016) P2
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