MARIS STELLA HIGH SCHOOL PRELIMINARY EXAMINATION ONE SECONDARY FOUR MATHEMATICS PAPER /1

Transcription

1 Class Index Number Name MARIS STELLA HIGH SCHOOL PRELIMINARY EXAMINATION ONE SECONDARY FOUR MATHEMATICS PAPER /1 Additional Materials: NIL Candidates answer on the Question Paper. 13 May hours INSTRUCTIONS TO CANDIDATES Write your class, index number and name on all the work you hand in. Write in dark blue or black pen. You may use a pencil for any diagrams or graphs. Do not use 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. You are expected to use a scientific calculator to evaluate explicit numerical expressions. If the degree of accuracy is not specified in the question, and if the answer is not exact, give your answer to three significant figures. Give answer in degrees to one decimal place. For π, use either your calculator value or 3.142, unless the question requires the answer in terms of π. 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 number of marks for this paper is 80. For Examiner s Use 80 This document consists of 15 printed pages and 1 blank page. REPRODUCTION OF ANY PART OF THIS QUESTION PAPER WITHOUT PERMISSION IS STRICTLY PROHIBITED.

2 Mathematical Formulae Compound Interest Total amount = Mensuration Curved surface area of a cone = Surface area of a sphere = Volume of a cone = Volume of a sphere = Area of triangle ABC = Arc length =, where θ is in radians Sector area =, where θ is in radians Trigonometry Statistics Mean = Standard deviation = REPRODUCTION OF ANY PART OF THIS QUESTION PAPER WITHOUT PERMISSION IS STRICTLY PROHIBITED. 2

3 1. (a) Evaluate 5 2 decimal places , leaving your answer correct to ( ) 3.2 (b) Evaluate ( ) ( ). Give your answer in standard form. Answer: (a) [1] (b) [1] 2. Given that sin ABD = 5, express, as a fraction, the value of cos BAE. 13 E A B D C Answer: [2] REPRODUCTION OF ANY PART OF THIS QUESTION PAPER WITHOUT PERMISSION IS STRICTLY PROHIBITED. 3

4 3. The equation of a straight line, L, is 2x = 5y 5. (a) Find the gradient of the line L. (b) The straight line M is parallel to the line L and passes through the point (2,5). Find the equation of the line M. Answer: (a) [1] (b) [2] 4. A polygon has n sides. Two of its interior angles are 95 and 115. The remaining (n 2) exterior angles are 30 each. Calculate the value of n. Answer: [2] REPRODUCTION OF ANY PART OF THIS QUESTION PAPER WITHOUT PERMISSION IS STRICTLY PROHIBITED. 4

5 5. (a) On the Venn diagram shown in the answer space below, shade the set B ( A C ). (b) ε = {x : x is a positive integer and x 30} P = {x : x is a prime factor of 30} Q = {x : x is a perfect cube} R = {x : x is a multiple of 8} (i) (ii) List the elements of P using set notation. Find n( Q R). ε Answer: (a) A B C [1] (b) (i) [1] (ii) [1] 6. A bag contains ten cards which are numbered from 0 to 9. Two cards are drawn at random, one after another with replacement. Find the probability that (i) the sum of the two cards is 5, (ii) the numbers on the two cards are same, (iii) the numbers on the two cards are not prime numbers. Answer: (i) [1] (ii) [1] (iii) [1] REPRODUCTION OF ANY PART OF THIS QUESTION PAPER WITHOUT PERMISSION IS STRICTLY PROHIBITED. 5

6 If A = 2 3, B = 2 5 and C= 3 5, find the value of k if A 2 + B = C. 1 k 9 3 Answer: [3] 8. (a) Expand and simplify 5(2y 3) ( y + 4)(2 3y). (b) (i) Expand and simplify p 2 ( p + 2 q )( p 2 q ). (ii) Hence, find the value of Answer: (a) [1] (b)(i) [1] (ii) [2] REPRODUCTION OF ANY PART OF THIS QUESTION PAPER WITHOUT PERMISSION IS STRICTLY PROHIBITED. 6

7 9. (i) Express 360 as a product of its prime factors. (ii) Find the lowest common multiple of 108 and 360, giving your answer as a product of its prime factors. (iii) Find the smallest positive integer k such that 360k is a cube number. Answer: (i) [1] (ii) [2] (iii) [1] 10. Factorise completely (a) 45x 2 81x, (b) 2( p q) 2 + 2q 2 p, (c) 8x 2 + 8x 6. Answer: (a) [1] (b) [2] (c) [1] REPRODUCTION OF ANY PART OF THIS QUESTION PAPER WITHOUT PERMISSION IS STRICTLY PROHIBITED. 7

8 11. The point (1,1) is marked on each diagram below. On these diagrams, sketch the graphs of (a) x = 2, (b) y = 2 x, (c) y = x 2, (d) y = 2x 3. Answer : (a) y (b) y O x O x [1] [1] y y (c) (d) O x O x [1] [1] REPRODUCTION OF ANY PART OF THIS QUESTION PAPER WITHOUT PERMISSION IS STRICTLY PROHIBITED. 8

9 12. (a) Solve 12 4x + 3 < 2x 2. (b) Write down the largest integer that satisfy 12 4x + 3 < 2x 2. Answer: (a) [3] (b) [1] 13. Find the total interest accumulated in 5 years if Edward were to invest $15000 in a savings account that pays (a) a simple interest at the rate of 2.2% per annum, (b) a compound interest at the rate of 2.2% per annum compounded half yearly. Answer: (a) $ [2] (b) $ [2] REPRODUCTION OF ANY PART OF THIS QUESTION PAPER WITHOUT PERMISSION IS STRICTLY PROHIBITED. 9

10 14. The difference of two numbers is 6. The larger number is 2 less than three times the smaller number. Find the two numbers. Answer: [3] 15. The masses (in kg) of 17 boys in a class are shown in the stem-and-leaf diagram below. Stem Leaf Key : 5 0 means 50 kg (a) Find the median of this distribution. (b) Find the standard deviation of this distribution. (c) Draw the box-and-whisker plot to represent the above distribution. Answer: (a) kg [1] (b) kg [2] (c) [3] REPRODUCTION OF ANY PART OF THIS QUESTION PAPER WITHOUT PERMISSION IS STRICTLY PROHIBITED. 10

11 16. The volumes of two geometrically similar open cylindrical oil tankers are 256 litres and 1372 litres. (a) Find the surface area of the bigger cylinder if the surface area of the smaller cylinder is 800 cm 2. (b) Find the mass of the smaller cylinder if the mass of the bigger cylinder is 3.5 kg. Answer: (a) cm 2 [3] (b) kg [2] REPRODUCTION OF ANY PART OF THIS QUESTION PAPER WITHOUT PERMISSION IS STRICTLY PROHIBITED. 11

12 17. Solve the following simultaneous equations 49 x 7 y+3 = y ( 1 27 )x = 9 Answer: x = y = [5] REPRODUCTION OF ANY PART OF THIS QUESTION PAPER WITHOUT PERMISSION IS STRICTLY PROHIBITED. 12

13 18. The diagram shows three points A, B and C with position vectors 2q - p, 3p - q and 5(3p - 2q) respectively. (a) Using vectors, show that A, B and C are collinear. (b) Find the ratio AB:BC. O A B C Answer: (a) [3] (b) [2] REPRODUCTION OF ANY PART OF THIS QUESTION PAPER WITHOUT PERMISSION IS STRICTLY PROHIBITED. 13

14 19. In the figure, ACD = 88, AEC = 46, CDE = 120 and TAB is a tangent to the circle with centre O. Find, giving reasons, (a) CAE, (b) OAC, (c) BAE. Answer: (a) o [1] (b) o [2] (c) o [3] REPRODUCTION OF ANY PART OF THIS QUESTION PAPER WITHOUT PERMISSION IS STRICTLY PROHIBITED. 14

15 20. The diagram shows the speed-time graph of a car s journey. Calculate (a) the acceleration when t = 15, (b) the speed when t = 15, (c) the time taken by the car to travel 1.6 km. Answer: (a) m/s 2 [2] (b) m/s [3] (c) s [3] REPRODUCTION OF ANY PART OF THIS QUESTION PAPER WITHOUT PERMISSION IS STRICTLY PROHIBITED. 15

16 Class Index Number Name MARIS STELLA HIGH SCHOOL PRELIMINARY EXAMINATION ONE SECONDARY FOUR MATHEMATICS PAPER /1 Additional Materials: NIL Candidates answer on the Question Paper. 13 May hours INSTRUCTIONS TO CANDIDATES Write your class, index number and name on all the work you hand in. Write in dark blue or black pen. You may use a pencil for any diagrams or graphs. Do not use 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. You are expected to use a scientific calculator to evaluate explicit numerical expressions. If the degree of accuracy is not specified in the question, and if the answer is not exact, give your answer to three significant figures. Give answer in degrees to one decimal place. For π, use either your calculator value or 3.142, unless the question requires the answer in terms of π. 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 number of marks for this paper is 80. For Examiner s Use 80 This document consists of 15 printed pages and 1 blank page. REPRODUCTION OF ANY PART OF THIS QUESTION PAPER WITHOUT PERMISSION IS STRICTLY PROHIBITED.

17 Mathematical Formulae Compound Interest Total amount = Mensuration Curved surface area of a cone = Surface area of a sphere = Volume of a cone = Volume of a sphere = Area of triangle ABC = Arc length =, where θ is in radians Sector area =, where θ is in radians Trigonometry Statistics Mean = Standard deviation = REPRODUCTION OF ANY PART OF THIS QUESTION PAPER WITHOUT PERMISSION IS STRICTLY PROHIBITED. 2

18 1. (a) Evaluate 5 2 decimal places , leaving your answer correct to ( ) 3.2 (b) Evaluate ( ) ( ). Give your answer in standard form. (a) ( ) 3.2 = (b) ( ) ( ) = ( 3) Answer: (a) 3.80 [1] (b) [1] 2. Given that sin ABD = 5, express, as a fraction, the value of cos BAE. 13 E A B D C cos BAE = - cos BAC = - sin ABC = - sin ABD = 5 13 Answer: 5 [2] 13 REPRODUCTION OF ANY PART OF THIS QUESTION PAPER WITHOUT PERMISSION IS STRICTLY PROHIBITED. 3

19 3. The equation of a straight line, L, is 2x = 5y 5. (a) Find the gradient of the line L. (b) The straight line M is parallel to the line L and passes through the point (2,5). Find the equation of the line M. (a) gradient = 2 5 (b) Equation of line M is y 5 = 2 (x 2) 5 y = 2 5 x or 5y = 2x + 21 alt mtd: subst (2, 5) into y = 2 5 x + c 5 = 2 5 (2)+ c c = 21 5 Equation of line M is y = 2 5 x or 5y = 2x + 21 Answer: (a) 2 5 [1] (b) y = 2 5 x or 5y = 2x + 21 [2] 4. A polygon has n sides. Two of its interior angles are 95 and 115. The remaining (n 2) exterior angles are 30 each. Calculate the value of n. (n 2)180 = (n 2)(150 ) 180 n 360 = n n = 270 n = 9 Answer: 9 [2] REPRODUCTION OF ANY PART OF THIS QUESTION PAPER WITHOUT PERMISSION IS STRICTLY PROHIBITED. 4

20 5. (a) On the Venn diagram shown in the answer space below, shade the set B ( A C ). (b) ε = {x : x is a positive integer and x 30} P = {x : x is a prime factor of 30} Q = {x : x is a perfect cube} R = {x : x is a multiple of 8} (i) List the elements of P using set notation. (ii) Find n( Q R). Answer: (a) ε A B C [1] (b) (i) {2, 3, 5} [1] (ii) 1 [1] 6. A bag contains ten cards which are numbered from 0 to 9. Two cards are drawn at random, one after another with replacement. Find the probability that (i) the sum of the two cards is 5, (ii) the numbers on the two cards are same, (iii) the numbers on the two cards are not prime numbers. (i) (ii) (iii) P(the sum of the two cards is 5)= 2xP(0,5) + 2xP(1,4) + 2xP(2,3) = 2( 1 10 )( 1 10 )+2( 1 10 )( 1 10 )+2( 1 10 )( 1 10 ) = 0.06 P(the numbers on the two cards are same) = P(0,0) + P(1,1) + P(2,2) +. +P(9,9) = 10( 1 10 )( 1 10 ) = 0.1 P(the numbers on the two cards are not prime numbers) = = 0.36 Answer: (i) 0.06 [1] (ii) 0.1 [1] (iii) 0.36 [1] REPRODUCTION OF ANY PART OF THIS QUESTION PAPER WITHOUT PERMISSION IS STRICTLY PROHIBITED. 5

21 If A = 2 3, B = 2 5 and C= 3 5, find the value of k if A 2 + B = C. 1 k 9 3 A 2 + B = C = k = k = k k = 3 k = -6 Answer: -6 [3] 8. (a) Expand and simplify 5(2y 3) ( y + 4)(2 3y). (b) (i) Expand and simplify p 2 ( p + 2 q )( p 2 q ). (ii) Hence, find the value of (a) 5(2 y 3) ( y + 4)(2 3y) = 10y 15 2y + 3y y (b) (i) p 2 ( p + 2 q )( p 2 q ) = p2 ( p 2 4 q 2 ) = 3y y 23 = 4 q 2 (b) (ii) = ( )( ) = = 100 Answer: (a) 3y y 23 [1] (b)(i) 4 [1] 2 q (ii) 100 [2] REPRODUCTION OF ANY PART OF THIS QUESTION PAPER WITHOUT PERMISSION IS STRICTLY PROHIBITED. 6

22 9. (i) Express 360 as a product of its prime factors. (ii) Find the lowest common multiple of 108 and 360, giving your answer as a product of its prime factors. (iii) Find the smallest positive integer k such that 360k is a cube number. (i) 360 = (ii) 108 = (ii) so, LCM = if 360k is a cube number, k = = 75 Answer: (i) [1] (ii) [2] (iii) 75 [1] 10. Factorise completely (a) 45x 2 81x, (b) 2( p q) 2 + 2q 2 p, (c) 8x 2 + 8x 6. (a) 45x 2 81x = 9x(5x 9) (b) 2( p q) 2 + 2q 2 p = 2( p q)2 2( p q) = 2( p q)[( p q) 1] = 2( p q)[ p q 1] (c) 8x 2 + 8x 6 = 2(4x 2 + 4x 3) = 2(2x + 3)(2x 1) Answer: (a) 9x(5x 9) [1] (b) 2( p q)[ p q 1]_ [2] (c) 2(2x + 3)(2x 1) [1] REPRODUCTION OF ANY PART OF THIS QUESTION PAPER WITHOUT PERMISSION IS STRICTLY PROHIBITED. 7

23 11. The point (1,1) is marked on each diagram below. On these diagrams, sketch the graphs of (a) x = 2, (b) y = 2 x, (c) y = x 2, (d) y = 2x 3. Answer : (a) y (b) y x -2 O O x x = -2 y = 2 x [1] [1] y y (c) (d) y = x 2 y = 2x 3 O x O x [1] [1] REPRODUCTION OF ANY PART OF THIS QUESTION PAPER WITHOUT PERMISSION IS STRICTLY PROHIBITED. 8

24 12. (a) Solve 12 4x + 3 < 2x 2. (b) Write down the largest integer that satisfy 12 4x + 3 < 2x 2. (a) 12 4x + 3 < 2x x + 3 and 4x + 3 < 2x x and x < x < 5 2 (b) largest integer = -3 Answer: (a) 15 4 x < 5 2 [3] (b) -3 [1] 13. Find the total interest accumulated in 5 years if Edward were to invest $15000 in a savings account that pays (a) a simple interest at the rate of 2.2% per annum, (b) a compound interest at the rate of 2.2% per annum compounded half yearly (a) simple interest = 100 = $1650 (b) compound interest = 15000( ) = $ Answer: (a) $ 1650 [2] (b) $ [2] REPRODUCTION OF ANY PART OF THIS QUESTION PAPER WITHOUT PERMISSION IS STRICTLY PROHIBITED. 9

25 14. The difference of two numbers is 6. The larger number is 2 less than three times the smaller number. Find the two numbers. Let x be the smaller number Then the larger number is x + 6 So, (x + 6) + 2 = 3x 2x = 8 x = 4 The 2 numbers are 4 and 10 Answer: 4, 10 [3] 15. The masses (in kg) of 17 boys in a class are shown in the stem-and-leaf diagram below. Stem Leaf Key : 5 0 means 50 kg (a) (b) (c) Find the median of this distribution. Find the standard deviation of this distribution. Draw the box-and-whisker plot to represent the above distribution. (a) median = 63 kg (b) standard deviation = (c) ( )2 7.36kg lower quartile 55.5kg, upper quartile = 68kg Answer: (a) 63 kg [1] (b) 7.36 kg [2] (c) [3] mass (kg) REPRODUCTION OF ANY PART OF THIS QUESTION PAPER WITHOUT PERMISSION IS STRICTLY PROHIBITED. 10

26 16. The volumes of two geometrically similar open cylindrical oil tankers are 256 litres and 1372 litres. (a) Find the surface area of the bigger cylinder if the surface area of the smaller cylinder is 800 cm 2. (b) Find the mass of the smaller cylinder if the mass of the bigger cylinder is 3.5 kg. (a) height of smaller cylinder height of bigger cylinder = = = 4 7 surface area of bigger cylinder = 800 ( 7 4 )2 = 2450 cm 2 (b) mass of small cyliner = 3.5 ( 4 7 ) kg Answer: (a) 2450 cm 2 [3] (b) kg [2] REPRODUCTION OF ANY PART OF THIS QUESTION PAPER WITHOUT PERMISSION IS STRICTLY PROHIBITED. 11

27 17. Solve the following simultaneous equations 49 x 7 y+3 = y ( 1 27 )x = 9 49 x 7 y+3 = x 7 y+3 = 7 3 2x ( y + 3) = 3 2x y = 6 81 y ( 1 27 )x = y 3 3x = 3 2 y = 2x 6...(1) 4 y 3x = 2...(2) subst. (1) into (2), 4(2x 6) 3x = 2 subst. x = 2 into (1), y = 2(2) 6 = -2 11x = 22 x = 2 Answer: x = 2 y = -2 [5] REPRODUCTION OF ANY PART OF THIS QUESTION PAPER WITHOUT PERMISSION IS STRICTLY PROHIBITED. 12

28 18. The diagram shows three points A, B and C with position vectors 2q - p, 3p - q and 5(3p - 2q) respectively. (a) Using vectors, show that A, B and C are collinear. (b) Find the ratio AB:BC. O A B C Answer: (a) AB = OB OA =3p q (2q p) =4p 3q BC = OC OB = 15p 10q (3p q) = 12p 9q Since AB = 1 BC, AB and BC are parallel, 3 And A, B, C are collinear [3] (b) AB = 1 3 BC so, AB BC = 1 3 (b) 1 [2] 3 REPRODUCTION OF ANY PART OF THIS QUESTION PAPER WITHOUT PERMISSION IS STRICTLY PROHIBITED. 13

29 19. In the figure, ACD = 88, AEC = 46, CDE = 120 and TAB is a tangent to the circle with centre O. Find, giving reasons, (a) CAE, (b) OAC, (c) BAE. (a) CAE = ( s in opp. segments) = 60 (b) (c) COA = 46 2 ( s in opp. segments) = OAC = 2 = 44 (base s of isos. Δ) BAO = 90 (tangent radius) OAE = (adj. s) = 16 BAE = (adj. s) = 106 Answer: (a) 60 o [1] (b) 44 o [2] (c) 106 o [3] REPRODUCTION OF ANY PART OF THIS QUESTION PAPER WITHOUT PERMISSION IS STRICTLY PROHIBITED. 14

30 20. The diagram shows the speed-time graph of a car s journey. Calculate (a) the acceleration when t = 15, (b) the speed when t = 15, (c) the time taken by the car to travel 1.6 km. (a) When t = 15, acceleration = (b) (c) let the speed when t = 15 be s s 0 60 = s = = 37.5 = 2.5 m/s 2 so, when t = 15, speed = 37.5 m/s let t be the time taken to travel 1.6 km (24)(60) = (t 24)60 2 t = so, time taken to travel 1.6 km is s Answer: (a) 2.5 m/s 2 [2] (b) 37.5 m/s [3] (c) 38 2 s [3] 3 REPRODUCTION OF ANY PART OF THIS QUESTION PAPER WITHOUT PERMISSION IS STRICTLY PROHIBITED. 15

31 Common errors 2011 Prelims 1 Emaths Paper 1 Qn Common error Marks deducted? 1 a) Some students made the error of rounding off in intermediate steps; some students still unsure of the usage of = sign and sign b) Quite a number of students do not know that standard form is always expressed in 3 sig. fig. 2 Several students are still unsure of the Trigo ratios, without working out the angles. Those who are able to derive that cos BAE = cos BAC were still awarded method marks 3 a) Most students had no problems with this question. The very few who do, do not seem to know that to find gradient from y=mx+c, they must first make y the subject. b) Almost all had no difficulty in this part, but if they got part (a) wrong, it would affect this part. Method marks still awarded. 4 Many students were unable to derive that the sum of interior angles of the n sides polygon = (n x 2) x 180. The overall performance of this question is relatively bad. It s either they understood and got the full 2 marks or got no marks at all. 5 a) The large majority of students were able to shade correctly b) Some students do not know what prime factors are c) Some students do not know that n(x), rep. the no. of elements in X. They gave the element in X instead. 6 Many students used all sorts of tables to try and work out this question, and likely wasted a lot of time here. Overall question was poorly done likely because their poor understanding of the topic. Amongst those who were able to do the question right, they did not include essential working and statements. Only penalized based on presentation because each are 1 mark questions. 7 Square matrix; during comparison of terms 9k = 3 8 (a) Many will factorize again and some make mistake during factorization. (b) (i) Many fail to use bracket and make mistake in the final answer (-ve sign instead of +ve sign) (ii) Some used calculator 9 (a) In general ok (b) Many give 1080 instead of prime factors; Some confuse with HCF (c) Many forgot how to do this 0 marks given Marks deducted for presentation 0 marks 0.5 marks (method) No marks given 0.5 marks (method) No marks given 0 marks given 0 marks given Marks deducted for presentation (0.5 for each part) 0 mark given; 1 mark awarded 0.5 mark deducted 0.5 mark deducted no mark given 1 mark deducted 1 mark deducted 10 (a) Few solve the expression instead of 0 mark REPRODUCTION OF ANY PART OF THIS QUESTION PAPER WITHOUT PERMISSION IS STRICTLY PROHIBITED. 16

32 simplifying; Not factorize completely. (b) Some give +1 instead of 1; some cannot factorize (p q)**2 and (p q); some do not change the sign when changing p and q position. (c) Few solve the expression instead of simplifying; Not factorize completely. 11 (a to d)many students cannot do these questions correctly; No labeling of graphs. 12 (a) Missing number line; write or instead of and ; some do not know how to separate the expression. (b) Many write -2 instead of -3 as integer. Some give fraction. 13 (a) Generally ok; few used compound formula. (b) Many used 2.2% instead of 1.1%. 14 Generally ok; some did not define the variables/ term; Some did not construct the equations correctly. 15 a) in general ok b) students tend to find SD w/o working c) a lot of students do not know how to draw box and whisker plot. no no. line, but box and whisker correct no no. line and box and whisker not to scale/wrong not to scale 16 a) some do not know ratio of areas is sq. of ratio of lengths b) some do not know ratio of masses is proportionate to ratio of volumes 1 mark deducted 0 mark; 0.5 mark deducted 0 mark given; P deducted 0.5 mark deducted; no mark awarded no mark awarded No mark awarded 0 marks given 2.5 m awarded 0 marks awarded 0 marks awarded 0 marks deducted 0 marks deducted 17 Generally ok, with careless mistakes like 9=3 3 1 mark deducted 18 Badly done. At least half scored 0m a) No tilde 0 marks awarded Some think that because, A,B 1 marks deduced and C are collinear b) Students write division of vector instead of division of line seg/length of vector 0 marks awarded 19 In general ok. A lot of students write most reasons except adj. angles 0.5m deducted 20 Generally ok except c) A lot do not know how to do this qn 0 marks awarded REPRODUCTION OF ANY PART OF THIS QUESTION PAPER WITHOUT PERMISSION IS STRICTLY PROHIBITED. 17

33 Class Index Number Name MARIS STELLA HIGH SCHOOL PRELIMINARY EXAMINATION ONE SECONDARY FOUR MATHEMATICS PAPER /2 Additional Materials: 5 May 2011 Writing paper (6 sheets) 2 hour 30 minutes Graph paper (1 sheet) INSTRUCTIONS TO CANDIDATES Write your class, index number and name on all the work you hand in. Write in dark blue or black pen. You may use a pencil for any diagrams or graphs. Do not use 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. You are expected to use a scientific calculator to evaluate explicit numerical expressions. If the degree of accuracy is not specified in the question, and if the answer is not exact, give your answer to three significant figures. Give answer 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 number of marks for this paper is 100. For Examiner s Use 100 This document consists of 11 printed pages and 1 blank page. REPRODUCTION OF ANY PART OF THIS QUESTION PAPER WITHOUT PERMISSION IS STRICTLY PROHIBITED.

34 Mathematical Formulae Compound Interest Total amount = r p n Mensuration Curved surface area of a cone rl 2 Surface area of a sphere 4 r 1 Volume of a cone r 2 h 3 4 Volume of a sphere r 3 1 Area of triangle ABC absin 2 Arc length r, where is in radians 3 C Sector area 1 r 2, where is in radians 2 Trigonometry a b sin A sin B c sin c a 2 = b 2 + c 2-2bc cos A Statistics Mean Standard deviation fx f 2 fx f fx f 2 REPRODUCTION OF ANY PART OF THIS QUESTION PAPER WITHOUT PERMISSION IS STRICTLY PROHIBITED. 2

35 1. Mary travelled by car from Singapore to Malacca, a distance of 230 km, at an average speed of x km/h. (a) Write down an expression for the time taken in hours, in terms of x. [1] (b) On the return journey, due to an accident, her average speed was reduced to ( x -12) km/h. Write down an expression for the time taken in hours, in terms of x. [1] (c) She took 40 minutes longer to travel back from Malacca. (i) Form an equation in x and show that it is reduces to x 2 12x [3] (ii) Solve the equation x 2 12x and hence, find the average speed of the car for the return journey. (d) The petrol consumption of the car is 12.3 km per litre. Find the amount of petrol used for the two-way journey, giving your answer correct to the nearest litre. [3] [2] 2. In the diagram, A, B and C represent three points on a horizontal field. B is 300 m from A on a bearing of 030 and C is 400 m from A on a bearing of 070 (a) Calculate the bearing of A from C. [2] (b) Calculate the distance of BC. [2] (c) An aircraft is at P, which is h m vertically above B and the angle of depression of A from P is Calculate the vertical height h of the aircraft. (d) A runner leaves A at 1240 and ran directly to C at a steady speed of 10 km/h. (i) When the runner is at D, he is due south of B. Calculate the distance AD. (ii) Hence, find the time, to the nearest minute, at which the runner reaches D. [2] [3] [3] REPRODUCTION OF ANY PART OF THIS QUESTION PAPER WITHOUT PERMISSION IS STRICTLY PROHIBITED. 3

36 3. (a) The cash price of a computer is $ John bought the computer by paying 10 % downpayment and monthly instalments of $75 over 15 months. Find (i) the total amount payable by John, (ii) the flat rate of interest of the instalments. [1] [3] (b) The tables show the exchange rates between Singapore dollars (S$) and US dollars (US$) given by banks A and B. Bank A does not charge any 1 commission and Bank B charges a commission of % for each transaction 2 or S$10 whichever is higher. Bank A Singapore Dollars (S$) US Dollars Selling Buying (US$) US$ (No commission required.) Bank B Singapore Dollars (S$) US Dollars Selling Buying (US$) US$ Commission: % or S$10, whichever is higher 2 Find out which bank Ah Beng should visit if (i) he wants to buy US$ 500, (ii) he wants to to sell US$2500? In each case, justify your answers. [3] [3] REPRODUCTION OF ANY PART OF THIS QUESTION PAPER WITHOUT PERMISSION IS STRICTLY PROHIBITED. 4

37 4. In the diagram, OABC is a parallelogram and D is the midpoint of BC. BE and OC produced intersect at the point F. BE : BF = 1: 3 and OC : OF = 1: 2. Given that OA = a and OC = c. F E B D C c A AA a O (a) Express and simplify the following vectors in term of a and c. (i) AC (ii) OD [1] (b) (i) Show that OE = kod, where k is a constant. (ii) Write down two facts about the vectors OD and (c) Calculate Area of ODF (i) Area of OEF, OE. [2] [3] [2] [2] (ii) Area of OCD Area of OABC. [2] REPRODUCTION OF ANY PART OF THIS QUESTION PAPER WITHOUT PERMISSION IS STRICTLY PROHIBITED. 5

38 5. The figure shows a wooden triangular prism where ABCD is a rectangular base, AF = FD, BE = EC and EG is perpendicular to BC. Given that AB 12 cm, EG 6 cm and BC 10 cm. Calculate (a) the length AG, [2] (b) the length AE, [2] (c) the angle EAG, [2] (d) the angle BEC, [2] (e) the volume of the prism. [2] REPRODUCTION OF ANY PART OF THIS QUESTION PAPER WITHOUT PERMISSION IS STRICTLY PROHIBITED. 6

39 6. The diagram shows a loaf of bread which has a uniform cross-section ABCDE, in which ABCE is a rectangle. CDE is an arc of a circle centred M, at the midpoint of AB. AB = 10 cm, BC = 6 cm and BQ = 45 cm. (a) Show that CME is 1.39 radians. [2] (b) Find the length of the arc CDE. [2] (c) Find the area of the sector MCDE. [2] (d) Find the total surface area of the loaf of bread. [3] 7. The diagram shows the first four of a sequence of figures. In Fig. 1, there are six vertices, seven edges and two squares. In Fig. 2, there are nine vertices, twelve edges and five squares. In Fig. 3, there are twelve vertices, seventeen edges and eight squares. The sequence continues as shown in Fig.4 and so on. (a) Find the value a, b and c. [3] REPRODUCTION OF ANY PART OF THIS QUESTION PAPER WITHOUT PERMISSION IS STRICTLY PROHIBITED. 7

40 Fig.1 Fig.2 Fig.3 Fig.4 Figure number, n Number of vertices, V Number of edges, E a Number of squares, X 2 b 8 c (b) Write down a formula, in terms of n, for (i) V, (ii) X. (d) (i) Using the results in (b), find out which n th figure which has twenty-six squares. (ii) Hence find the number of edges of this figure. [1] [1] [2] [1] 8. A fair blue die is numbered 2, 9, 10, 11, 12 and 13 and a fair red die is numbered 2, 3, 4, 14, 15 and 16. The possibility diagram when the two dice are thrown together is shown below. A cross X indicates that the number shown on the blue die is greater than the number shown on the red die. Red die Blue die X (a) Copy the diagram and indicate a cross X in each square where the number shown on the blue die is greater than the number shown on the red die. [2] REPRODUCTION OF ANY PART OF THIS QUESTION PAPER WITHOUT PERMISSION IS STRICTLY PROHIBITED. 8

41 (b) The two dice are thrown together, show that the probability that the number shown on the blue die is greater than the number shown on the red 5 die is 12. [1] (c) The two dice are thrown together twice. Calculate the probability that the number shown on the blue die is greater than the number shown on the red die (i) both times, [2] (ii) only once. [2] 9. The cumulative frequency curve shows the distribution of the long jump distances of 50 students. REPRODUCTION OF ANY PART OF THIS QUESTION PAPER WITHOUT PERMISSION IS STRICTLY PROHIBITED. 9

42 [2] (a) Find the values of p and q. Long Jump distance Number of students 3 x x 5 p 5 x x 7 q 7 x 8 4 (b) Using the frequency table in (a), estimate the mean and the standard deviation. (c) Students who could not jump beyond 5.5 m are required to attend a training course. Find the percentage of students who have to attend the training course. (d) If two students are selected at random from 50 students, find the probability that both of them jumped beyond 6 m. [4] [2] [2] 10. Answer the whole of this question on a sheet of graph paper. The table below gives some values of x and the corresponding values of y, correct two decimal places, for y 1 4 x 3 6x 2 8x. x REPRODUCTION OF ANY PART OF THIS QUESTION PAPER WITHOUT PERMISSION IS STRICTLY PROHIBITED. 10

43 y p (a) Calculate the value of p. [1] (b) Using the scale of 2 cm to represent 1 unit on each axis, draw a horizontal x-axis for 1 x 5 and and a vertical y-axis for 4 y 4. On your axes, plot the points given in the table and join them with a smooth curve. (c) By drawing a tangent, find the gradient of the curve at the point when x 1. [3] [2] (d) Use the graph to solve the equation x 3 6x 2 8x 8. [2] (e) (i) On the same axes, draw the graph 2y x 4 for 1 x 5. [2] (ii) Write down and simplify, a cubic equation which is satisfied by the [2] values of x at the points where the two graphs intersect. - THE END ERRATA FOR Q2 REPRODUCTION OF ANY PART OF THIS QUESTION PAPER WITHOUT PERMISSION IS STRICTLY PROHIBITED. 11

44 - N B C A D REPRODUCTION OF ANY PART OF THIS QUESTION PAPER WITHOUT PERMISSION IS STRICTLY PROHIBITED. 12