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Name : ( ) Class : COMMONWEALTH SECONDARY SCHOOL PRELIMINARY EXAMINATION 009 SECONDARY FOUR EXPRESS / FIVE NORMAL ACADEMIC MATHEMATICS Paper 1 Date : 7 August 009 Candidates answer on the Question

1 Name : ( ) Class : COMMONWEALTH SECONDARY SCHOOL PRELIMINARY EXAMINATION 009 SECONDARY FOUR EXPRESS / FIVE NORMAL ACADEMIC MATHEMATICS Paper 1 Date : 7 August 009 Candidates answer on the Question Paper. 4016/01 Time : hours READ THESE INSTRUCTIONS FIRST Write your name, class and index number on all the work you hand in. Write in dark blue or black pen. You may use a soft 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 3.14, 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 the brackets [ ] at the end of each question or part question. The total number of marks for this paper is This question paper consists of 14 printed pages including this page. CSS/Prelim 009/Sec 4E5N/EMath P1/Chua IL/pg1 of 14

2 Mathematical Formulae Compound Interest Total amount = r P1 100 n Mensuration Curved surface area of a cone = πrl Surface area of a sphere = 4 r Volume of a cone = 1 3 r h Volume of a sphere = 4 r 3 3 Area of triangle ABC = 1 absin C Arc length = rθ, where θ is in radians Sector area = 1 r, where θ is in radians Trigonometry a sin A b sin B c sin C a = b + c bc cos A Statistics Mean = fx f Standard deviation = fx f fx f CSS/Prelim 009/Sec 4E5N/EMath P1/Chua IL/pg of 14

3 Answer all the questions. 1 Rearrange the formulae y 5 x 3 y to express y in terms of x. Answer : [] Solve the equation x x Answer : x = [] 3 Factorize fully the expression ab 4a ab c 4ac. Answer : [3] CSS/Prelim 009/Sec 4E5N/EMath P1/Chua IL/pg3 of 14

4 4 Mercury orbits around the Sun in 88 days, Venus does the same in 5 days and Earth takes 360 days. The last time an eclipse occurred (when the Sun, Mercury, Venus and Earth are set in a straight line) was in the year 199. By writing 88, 5 and 360 into the product of their prime factors, find the year in which the next eclipse would occur on Earth. Earth Venus Sun Mercury Answer : [3] 5 Mrs Cheah drove at 60kmh -1 for the first 1hour 0 minutes and 90kmh -1 for the rest of her journey. If the whole journey took hours, find the exact value of the average speed of Mrs Cheah s journey, leaving your answer in ms -1. Answer : [3] CSS/Prelim 009/Sec 4E5N/EMath P1/Chua IL/pg4 of 14

5 6(a) In the Venn diagram, shade the region (A ' B) A. [1] A B (b) Given that {x : x is an integer and 1 x p}, A {x : x is a multiple of } and B {x : x is a multiple of 3}. If values of p. n A B 5, find the largest and smallest possible Answer : largest p =, smallest p = [] 7 Write out the largest prime number satisfying the inequality 1 x 1x Answer : the largest prime number = [3] CSS/Prelim 009/Sec 4E5N/EMath P1/Chua IL/pg5 of 14

6 8 In one particular month, Hafizah gives her parents 15% of her salary, spends 5% on food, 1/5 on entertainment and 1/4 on rent. She uses the rest of her salary to invest in a structured deposit that pays compound interest of % per year. Her rent is $1600. (i) Find Hafizah s salary. (ii) Calculate the total interest she will receive in three years from her investment. Answer : (i) [1] (ii) [3] 9 The diagram shows a section of a regular 1-sided polygon which is cut from a circular piece of paper of radius 5cm. All the vertices of the polygon lie on the circumference of the circle. Find (i) one interior angle of the polygon, (ii) the amount of paper discarded, leaving your answer in terms of. Answer : (i) [1] (ii) [3] CSS/Prelim 009/Sec 4E5N/EMath P1/Chua IL/pg6 of 14

7 10(i) Write the expression x x 3 into the form a(x h) k. (ii) Hence sketch the graph of y x x 3, showing clearly the turning point and the x and y intercepts. [3] Answer : (i) [] CSS/Prelim 009/Sec 4E5N/EMath P1/Chua IL/pg7 of 14

8 A 11 In the diagram, ABC 90, AC = 9cm, BD = 15cm, DC = 6cm and AD = y cm. Calculate (i) the value of y, (ii) the value of tan ADC, without solving for any angles. 9cm y cm C 6cm D 15cm B Answer : (i) [] (ii) [] 1 The variables x, y and z are related. z varies directly as the square of x, y varies inversely as the cube root of z, and when x = 1, y = 1 and z = 7. (i) Find an expression for z in terms of x and y in terms of z. (ii) Hence show that y x /3. [1] Answer : (i) z =, y = [4] CSS/Prelim 009/Sec 4E5N/EMath P1/Chua IL/pg8 of 14

9 13 A piece of land on level ground is in the shape of an isosceles triangle ABC with the sides AB = AC. The diagram, drawn to a scale of 1cm : m, shows the side AC. Given that the bearing of B from A is 160, (i) draw the triangle ABC and write down the length of BC in metres. [1] (ii) A tree T is to be planted so that it is equidistant from points A and C and equidistant from lines AC and BC. Construct the perpendicular bisector of AC and the angle bisector of angle ACB and mark clearly with the point T the position of the tree. [3] N A C Answer : (i) BC = [1] CSS/Prelim 009/Sec 4E5N/EMath P1/Chua IL/pg9 of 14

10 14 The table below shows the scores obtained when a die is thrown a number of times. Score No. of times 3 4 x 1 3 (i) Write down the maximum value of x if the modal score is. (ii) Write down the minimum value of x if the median score is 3. (iii) Find the median score if the mean score is 3/7. Answer : (i) [1] (ii) [1] (iii) [3] 15 The equation of a line l is y x + 6 = 0. (i) Find the equation of the line parallel to line l and which passes through the point (1,-). (ii) Line l cuts the y-axis at A and the x-axis at B and B is the midpoint of the line AC. Find (a) the coordinates of the point C, (b) the length of AC. Answer : (i) [] (ii)a) [] (ii)b) [] CSS/Prelim 009/Sec 4E5N/EMath P1/Chua IL/pg10 of 14

11 16 The price of a ticket in each category at the Night Safari is given below: (i) The number of tickets sold on one weekend is given as follows. Adult Senior Citizen Child Saturday Sunday By putting the prices into a column matrix A and the number of tickets sold as matrix B, find the matrix C given by C = BA and describe what is represented by the elements of C. (ii) To improve the revenue during weekends, two plans are proposed : Plan 1 : Increase the price on Sunday only by 30%. Plan : Increase the price by 15% on each day. A 1x matrix P is such that PC gives the revenue for the weekend under Plan 1. Another 1x matrix Q is such that QC gives the revenue for the weekend under Plan. (a) Evaluate PC and QC. (b) State which plan would be more profitable., Answer : (i) C =. C represents [] (ii)a) PC = QC =. [] (ii)b) [1] CSS/Prelim 009/Sec 4E5N/EMath P1/Chua IL/pg11 of 14

12 17 Two open troughs X and Y are geometrically similar prisms with trapeziums and 3 rectangles making up their sides. The ratio of the sides of trough X to the sides of trough Y is 1 : 4. If the capacity of the trough Y is 100 cm 3, calculate (i) the ratio of the surface area of X to Y. (ii) the capacity of the trough X. (iii) the depth d cm of trough Y. X 5cm d cm 8cm Y 1cm Answer : (i) [1] (ii) [] (iii) [3] CSS/Prelim 009/Sec 4E5N/EMath P1/Chua IL/pg1 of 14

13 18 The diagram shows the speed time graph of speed (m/s) a cyclist over a period of T seconds. The cyclist sees a stretch of wet road ahead 6 and slows down uniformly from 6m/s to 3m/s in 0 seconds. He then progresses at constant 3 speed for 30 seconds, passing the stretch of wet road, before gaining speed uniformly T time (s) to 6m/s at T seconds. (i) Given that the cyclist s speed is 3.6m/s at t = 60s, find the value of T. (ii) Find the average speed of the particle for the first 50 seconds. (iii) On the axes in the answer space, sketch the corresponding distance-time graph for the period of T seconds, indicating the values of distance travelled clearly. Answer : (i) [] (ii) [] (iii) distance (m) [] T time (s) CSS/Prelim 009/Sec 4E5N/EMath P1/Chua IL/pg13 of 14

14 19 In the diagram, OPQR is a parallelogram. M is the midpoint of OQ, N is the midpoint of OM and L is the point on OR such that OL = LR. (a) Given that OP a and OR b, express as simply as possible in terms of a and b, (i) OM (ii) NP (iii) LM P Q (b) Explain why NP and LM are parallel. (c) Find the following ratios. a M (i) (ii) Area of OPN Area of PNQ Area of PNQ Area of OML O N b L R Answer : (a)(i) [1] (ii) [1] (iii) [1] (b) [1] (c)(i) [1] (ii) [1] End of Paper CSS/Prelim 009/Sec 4E5N/EMath P1/Chua IL/pg14 of 14

15 Answer Key 1) 3x 0 y ( x)( x) ) x = -1 3) a(b+)(b-)(1+c)(1-c) 4) 047 5) 4 19 m / s 9 6)a) 6)b) smallest p = 30, largest p = 35 7) 5 8)i) $6400 ii) $ )i) 150 ii) ) (x 1) 4 11)i) y = 5 ii) 1 1 1)i) 3 z 7x, y 3 3 z 13) BC = 17.6m 14)i) 3 ii) iii).5 15)i) y = x 4 ii) (6,6), 13.4 units 16)i) 309 C 3177 ii) PC = $ , QC = $ , Plan 1 is more profitable 17)i) 1/16 ii) iii) 6 18)i) T = 100 ii) ai) 1 (a b) ii) 1 (3a b) iii) 4 1 (3a b) b) 6 3 NP LM ci) 1/3 cii) 1 4 CSS/Prelim 009/Sec 4E5N/EMath P1/Chua IL/pg15 of 14

16 COMMONWEALTH SECONDARY SCHOOL PRELIMINARY EXAMINATION 009 SECONDARY FOUR EXPRESS/FIVE NORMAL MATHEMATICS 4016/0 Paper 7 August hours 30 minutes Additional Materials: Writing Paper Graph Paper (1 sheet) NAME: ( ) CLASS: READ THESE INSTRUCTIONS FIRST Write your name, index number and class 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 3.14, 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 question paper consists of 11 printed pages including the cover page.

17 Mathematical Formulae Compound Interest n r Total amount = P Mensuration Curved surface area of a cone = rl Surface area of a sphere = Volume of a cone = Volume of a sphere = 1 3 rh 4 3 r 4 r Area of triangle ABC = 1 sin ab C Arc length = r, where is in radians Sector area = 1 r, where is in radians 3 Trigonometry a b c sin A sin B sin C a b c bc cos A Statistics Mean = Standard deviation = fx f fx f fx f CSS/Prelim009/MATH/SEC4E5N/P/AGS/Page of 11

18 3 Answer all the questions. 1 (a) National Petroleum Company (NPC) provides 3 different grades of petrol. The price per litre of each grade of petrol is as follows: Petrol Grade Price per Litre ($) Grade Grade Grade (b) (i) (ii) Mr Soh pumped 4 litres of Grade 98 petrol for his car. Calculate the amount of money he paid for the petrol. [1] Mr Soh s car has a petrol consumption rate of 1.5 km per litre. Calculate the distance his car can travel with $50 worth of Grade 98 petrol. [] (iii) During a promotion month, the cost per litre of Grade 95 petrol was reduced by 15% but an instant rebate of $5 was given to car owners who pumped Grade 9 petrol. What is the maximum volume of Grade 9 petrol to be pumped before the total cost becomes more than the cost of pumping Grade 95 petrol? Give your answer in litres correct to 1 decimal place. [3] A shopkeeper sells two types of luxury handbags, Elegant and Convenient. Elegant handbags cost $7500 a piece and Convenient handbags cost $40 less. (i) (ii) Write down, in its simplest form, the ratio of the cost of Elegant handbags to Convenient handbags. [1] Given that the shopkeeper sold an Elegant handbag at a discount of 15% and a Convenient handbag at a discount of $50, calculate the total percentage discount given on the sale of the handbags. [] CSS/Prelim009/MATH/SEC4E5N/P/AGS/Page 3 of 11

19 4 Each diagram in the sequence below is made up of a number of dots. Diagram 1 Diagram Diagram 3 Diagram 4 (a) Draw the next diagram in the sequence. [1] (b) (c) The table shows the number of dots in each diagram. Diagram Number of dots p q Write down the values of p and of q. [] The formula for finding the number of dots in the nth diagram is An Bn C, where A, B and C are constants. Find the values of A, B and of C. [3] (d) Find the number of dots in Diagram 10. [1] (e) Which diagram has 53 dots? [] 3 (a) (i) Simplify (b) (ii) Solve a 6ab 9b 5a 45b. 6ac 3ad ac ad 6bc 3bd x x A box contains several red discs and green discs. A disc is randomly chosen and then placed back into the box and the process is repeated several times. The probability of choosing a red disc is p. [3] [] (i) (ii) Write down, in terms of p, the probability of choosing a green disc. [1] The process was repeated 8 times. Find the probability that (a) a red disc was chosen every time, [1] (b) at least one green disc was chosen. [1] CSS/Prelim009/MATH/SEC4E5N/P/AGS/Page 4 of 11

20 5 4 In the diagram, ACB 90, ABC 51, BEC 35, ACD 103, CD 4 cm, BC 4.6cm and CE 7.3 cm. Calculate (a) CBE, [] (b) the length of CA, [1] (c) the length of AD, [3] (d) the area of triangle BCE, [] (e) the shortest distance from E to CB produced. [] CSS/Prelim009/MATH/SEC4E5N/P/AGS/Page 5 of 11

21 6 5 An airplane is scheduled to fly to its destination 3500 km away. The speed of the airplane in still air is 600 km/h and the speed of wind, which is constant throughout, is x km/h. Due to a haze, the speed of the airplane in still air is reduced by 10%. (a) (b) (c) Write down an expression, in terms of x, for the time taken by the airplane, in hours, if it is flying in the direction of the wind. [1] Write down an expression, in terms of x, for the time taken by the airplane, in hours, if it is flying against the wind. [1] The difference in arrival time is 1 hour and 10 minutes. Write down an equation in terms of x, and show that it reduces to x 6000x [3] (d) (e) Solve the equation x 6000x [3] Hence, find the time taken by the airplane, in hours and minutes, if it is flying in the direction of the wind when there is no haze. [] CSS/Prelim009/MATH/SEC4E5N/P/AGS/Page 6 of 11

22 7 6 In the diagram, O is the centre of the circle and points P, S, T and R lie on the circumference of the circle. The tangent at P meets RT produced at Q. TS PS, TQ SQ and TRP 36. (a) Find (i) reflex angle POT, [] (ii) PTS, [] (iii) PQS. [3] (b) Show that PS bisects QPT. [3] CSS/Prelim009/MATH/SEC4E5N/P/AGS/Page 7 of 11

23 8 7 (a) The diagram shows the cross-section of a swing in a children s playground. The seat is suspended on a 1.8 m long rope. To oscillate the swing, the seat is pulled back to point A and released to swing an angle of 6 to point A. The seat makes one complete oscillation when it moves from point A to point A and back to point A again m A A (b) (i) (ii) Calculate the distance moved by the swing seat from point A to point A. [] Assuming that the swing oscillates regularly from point A to point A, find the speed of the swing, in metres per minute, if it makes 5 complete oscillations in minutes. [] The diagram shows the swing and a bench, 4 m away, in the children s playground. Both the bench seat and swing seat are at the same height above the ground. 1.8 m 4 m (i) (ii) Calculate the angle of depression of the edge of the bench seat from the top of the swing. [] A bird flies from the edge of the bench seat to the top of the swing. Calculate the distance the bird flies. [] CSS/Prelim009/MATH/SEC4E5N/P/AGS/Page 8 of 11

24 9 8 (a) A factory manufactures small decorative ornaments. Each decorative ornament is made up of two parts: a solid hemisphere with radius 7 cm and a solid cone with a height 10 cm, as shown in Diagram I. 7 cm 10 cm Diagram I (i) Calculate the volume of the hemisphere. [] (ii) (iii) The volume of the hemisphere is 3 times the volume of the cone. Find the base radius of the cone. [] Given that the solid cone is made with a light plastic material with a density of 0.9 g/cm 3, find the mass of the material used for the cone. [] The two pieces are joined together to form the decorative ornament as shown in Diagram II. (iv) Calculate the total external surface area of the ornament. Diagram II [4] (b) Given that the area of the major sector is 98 cm, find the value of and hence calculate the perimeter of the major sector. 6 cm rad. [] CSS/Prelim009/MATH/SEC4E5N/P/AGS/Page 9 of 11

z such that 77.5% of the male employees have a daily wage more than $ z. [] The box-and-whisker diagram represents the daily wages of 60 female employees in the same company.

25 10 9 The cumulative frequency curve below represents the daily wages of 80 male employees in a company Cumulative Frequency Daily Wages ($) Use the graph to estimate (a) the median daily wage, [1] (b) the interquartile range, [] (c) the value of z such that 77.5% of the male employees have a daily wage more than $ z. [] The box-and-whisker diagram represents the daily wages of 60 female employees in the same company Daily Wages ($) (d) (e) (f) Find the median daily wage of the female employees and the interquartile range. [3] Compare and comment briefly on the daily wages of the male and female employees in the company. [] Find the probability that an employee chosen at random from all the employees has a daily wage less than or equal to $38. [] CSS/Prelim009/MATH/SEC4E5N/P/AGS/Page 10 of 11

26 11 10 Answer the whole of this question on a sheet of graph paper. The following table gives the corresponding values of x and y, which 5 are connected by the equation y x 9, correct to 1 decimal x place. x y p -6.4 (a) Calculate the value of p correct to 1 decimal place. [1] (b) Using a scale of cm for 1 unit on the x -axis and 1 cm for 1 5 unit on the y -axis, draw the graph of y x 9 for the x values of x in the range 1x 8. [3] (c) Use your graph to find the value of y when x.5. [1] 5 (d) Use your graph to solve the equation x 1. x (e) (f) Find the coordinates of the point on the graph for which the gradient of the curve is -4. [] By drawing a suitable straight line, solve the equation 4x 6x 5 0 for 1x 8. [3] [] END OF PAPER CSS/Prelim009/MATH/SEC4E5N/P/AGS/Page 11 of 11

27 COMMONWEALTH SECONDARY SCHOOL PRELIMINARY EXAMINATION 009 SECONDARY FOUR EXPRESS/FIVE NORMAL MATHEMATICS 4016/0 1 (a) (i) Amt. of money paid = $ = $78.54 [B1] 50 (ii) Amt. of petrol = litres Dist. = km (iii) Let the max. volume be V litres V V 1.687V V V 5 V Max. volume is 7.0 litres. (b) (i) Elegant handbags : Convenient handbags = 7500 : 760 = 15 : 11 [B1] (ii) % discount = % % (a) [B1] (b) p 33 q 46 (c) no. of dots = n n 1 = n n [B1] [B1]

28 A 1 B C (d) no. of dots = = 118 (e) n n 53 n n 55 0 n15 n n 15 or n 17 (N.A) Diagram 15 has 53 dots. (other methods are acceptable) [B1] [B1] [B1] [B1] 3 (a) (i) (ii) a 6ab 9b 5a 45b 6ac 3ad ac ad 6bc 3bd a 3b a c d 3b c d 3a c d 5 a 3b a 3b a 3b c d a 3b 3a c d 5 a 3b a 3b a 3b 15a x x x x x x (b) (i) P(choosing a green disc) = 1 p [B1] (ii) (a) P(red disc chosen each time) = 8 p [B1] (b) P(at least one green disc chosen) = 1 P(red disc chosen each time) = 1 p 8 [B1] 4 (a) By Sine Rule, sincbe sin CSS/Prelim009/MATH/SEC4E5N/P/AGS/Page of 7

29 3 5 (b) (c) CBE CA tan CA cm By Cosine Rule, AD (4)cos103 AD cm (d) BCE ( s sum of ) Area of BCE sin cm 1 (e) 4.6 shortest dist Shortest dist cm (a) Speed of plane 90 = = 540 km/h Time taken (with wind) = 3500 h 540 x [B1] [M] [B1] (b) Time taken (against wind) = 3500 h 540 x [B1] (c) x 540 x x 540 x x x 540 x 540 x x x x x 6000x (Shown) (d) x 6000x CSS/Prelim009/MATH/SEC4E5N/P/AGS/Page 3 of 7

30 x or (N.A.) 48. (e) Time taken 3500 = h 5 h 4 min 6 (a) (i) TOP 36 ( at ctr. = at circumference) 7 Reflex POT ( s at a pt.) 88 7 (ii) TSP (opp. s of cyclic quad.) PTS (base of isos. ) 18 (iii) PTR 90 ( in a semicircle) QTS ( s on a st. line) 7 SQT ( s sum of isos. ) 36 QPR 90 (tangent rad.) PQR ( s sum of ) 54 PQS (b) SPT 18 (isos. ) TPO (base of isos. ) 54 QPS Since QPS SPT 18, PS bisects QPT. 6 (a) (i) dist. moved = m CSS/Prelim009/MATH/SEC4E5N/P/AGS/Page 4 of 7

31 5 8 (ii) No. of oscillations in 1 min =.5 Dist. travelled in 1 min = m Speed is 9.74 m/min. (b) (i) Let the angle of depression be a. tana a Angle of depression is 4.. (ii) dist (by Pythagoras' Thm.) dist m cm (a) (i) Vol. of hemisphere = 7 (ii) Let the base radius of the cone be r cm. 1 r Base radius of cone is 4.78 cm. 1 (iii) Mass = g (iv) Let the slanted height of the cone be l cm. l (by Pythagoras' Thm.) l Total external surface area = cm [M] r (b) rad. 9 CSS/Prelim009/MATH/SEC4E5N/P/AGS/Page 5 of 7

32 6 4 Perimeter of major sector = cm 3 9 (a) Median daily wage $43 [B1] (b) Interquartile range $49 $36 = $13.5 (c) 77.5% of the male employees = = 18 z 35 (d) Median daily wage $38 Interquartile range $46 $18 = $8 (e) The median wage of males ($43) is higher than that of females ($38). The interquartile range for wage of males (13) is smaller than that of females (8) and thus, the wage of males is less widespread as compared to that of females. (f) P(wage less than or equal to $38) = [B1] [B1] [B1] CSS/Prelim009/MATH/SEC4E5N/P/AGS/Page 6 of 7

33 7 CSS/Prelim009/MATH/SEC4E5N/P/AGS/Page 7 of 7

EAST VIEW SECONDARY SCHOOL SECOND SEMESTRAL EXAMINATION 2017 SECONDARY ONE EXPRESS

EAST VIEW SECONDARY SCHOOL SECOND SEMESTRAL EXAMINATION 017 SECONDARY ONE EXPRESS CANDIDATE NAME CLASS INDEX NUMBER MATHEMATICS 4048/0 Paper 11 October 017 Total Marks: 50 1Hours15Minutes Additional Materials: