CARIBBEAN EXAMINATIONS COUNCIL SECONDARY EDUCATION CERTIFICATE EXAMINATION MATHEMATICS. 2 hours 40 minutes

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1 . ( TEST coos FORM TP MA Y/JUNE 2004, :;,. CARBBEAN EXAMNATONS COUNCL SECONDARY EDUCATON CERTFCATE EXAMNATON Paper 02 - MATHEMATCS General Proficiency 2 hours 40 minutes NSTRUCTONS TO CANDDATES 1. Answer ALL questions in Section, and ANY TWO in Section Write your answers in the booklet provided. 3. All working must be shown clearly. 4. A list of formulae is provided on page 2 of this booklet. Examination Materials Electronic calculator (non-programmable) Geometry set Mathematical tables (provided) Graph paper (provided) [ DO NOT TURN TDS PAGE UNTL YOU ARE TOLD TO DO SO o f 2004 Copyright 2003 Caribbean Examinations All rights reserved. Council.

2 .i Page 2 LST OF FORMULAE Volume of a prism v = Ah where A is the area of a cross-section length. and h is the perpendicular Volume of a right pyramid v = 1 Ah where A is the area of the base and h is the perpendicular 3 height. Circumference C = 2nr where r is the radius of the circle. Area of a circle Area of trapezium A = n? where r is the radius of the circle. A = ~ (a,+ b) h where a and b are the lengths of the parallel sides and h is the perpendicular distance between the parallel sides. Roots of quadratic ". equations f af + bx + c = 0, -b ± ~b2-4ac then x = 2a Trigonometric ratios sin e opposi te side = hypotenuse Opposite cos e adjacent side = hypotenuse Adjacent. tan e opposite side = adjacent side Area of triangle Area of /). = 1bh where b is the length of the base and h is the perpendicular height~ Area of MBC = }ab sin C ~ Area of MBC =.s (s - a) (s - b) (s - c) ( b > where s = a + b + c 2 Sine rule Cosine rule o fF 2004 a sin A = b sin B c sin C a 2 = b 2 + c 2-2bc cos A C~------~~ ~A r

3 ;Pag~ j SECTON Answer ALL the questions in this section. All working must be clearly shown. 1. (a) Using a calculator, or otherwise, determine the exact value of (i) ~.: _ (iii) (6 marks) 5 (b) (i) Write your answer in Part (a) (i) correct to one significant figure. Write your answer in Part (a) in standard form. (c) (i) Mr Mitchell deposited $40000 in a bank and earned simple interest at 7% per annum for two years. Calculate the amount he will receive at the end of the two-year period. Mr Williams bought a plot of land for $ The value of the land appreciated by 7% each year. Calculate the value of the land after a period of two years. (4 marks) Total 12 marks 2. (a) Simplify: (i) x2- x- 4ab 2 + 2a 2 b ab (4 marks) (b) Express as a single fraction: 3p ""2 p +!L. (2 marks) Cc) Solve for x, given 3x 2-7x + 2 = 0 (4 marks) Total 10 marks o f 2004 GO ON TO THE NEXT PAGE

4 3. (a) A club has 160 members, some of whom play tennis (T) or cricket (C) or both. 97 play tennis, 86 play cricket and 10 play neither, x play both tennis and cricket. (i) Draw a Venn diagram to represent this information. How many members play both tennis and cricket? (5 marks) (b) n a beauty contest, the scores awarded by eight judges were: (i) Using the eight scores, determine: a) the mean! ' : : b) the median c) the mode. Only six scores are to be used. Which two scores may be omitted to leave the value of tlre median the same? (6 marks) "."i Total 11 marks 4. (a) (i) Using the formula A~ t ='V 12n calculate the value of t when m = 20 and n = 48. Express m as subject of the formula in (a) (i) above. (5 marks) (b) n the diagram below, not drawn to scale, EFGH is a rectangle. The point D on HG is. A 0 such that ED = DG = 12 cm and GDF = 43. Er ""7 F H, X----L--jr+-----'G D-+- 12cm ~ Calculate correct to one decimal place (i) the length of GF the length of HD (iii) the size of the angle HDE. (7 marks) Total 12 marks o F 2004 GO ON TO THE NEXT PAGE

5 --...,' tr..... Page S 5. An answer sheet is provided for this question. (a) On the section of the answer sheet provided for 5 (a): (i) write down the coordinates of the point P (H) draw a line segment PQ through the point, P, such that the gradient of PQ is -3. (3 marks) 2 (b) On the section of the answer sheet provided for 5 (b): (i) draw the reflection of quadrilateral A in the mirror line, labelled M 1. Label its image B. (H) draw thereflection of quadrilateral B in the mirror line, labelled M 2. Label its image C.,- ''.(" (4 marks) (c) Complete the sentence in part (c) on your answer sheet, describing FULLY the single geometric transformation which maps quadrilateral A onto quadrilateral C. (3 marks) Total 10 marks o F 2004

6 Page 6 6. The amount a plumber charges for services depends on the time taken to complete the repairs plus a fixed charge. The graph below shows the charges in dollars (d) for repairs in terms of the number of minutes (t) taken to complete the repairs. i f! j'l ~ i! 1 ; r ' (a) (b) 10ij#.;J;;t -~i+ t-f'-d+- -+r1j-t+l-!' +,-h- -t-t"r"" iijii-;fiit#tjut;iit ++e--!-j+j ttt~l..ifh- i--t P -e-+-.,.+ -rt±+ 4 t-h- H..t... o ro w ~ ~ ~ ~ m t Number of minutes What was the charge for a plumbing job which took 20 minutes? (1 mark ) How many minutes were spent completing repairs that cost: (i) $38.00 $20.00? (2 marks) (c) (d) (e) What is the amount of the fixed charge? Calculate the gradient of the line. Write down the equation of the line in terms of d and t. (1 mark) (2 marks) (2 marks) (f) Determine the length of time taken to complete ajob for which the charge was $78_00. (3 marks) Total 11 marks o F 2004

7 ',Page 7 7. (a) A piece of wire is bent in the form of a circle and it encloses an area of 154 ern". (i) Calculate: a) the radius of the circle b) the circumference of the circle. 22 (Use rt = -) 7..' The same piece of wire is then bent in the form of a square. Calculate the area enclosed by the square. (6 marks) (b) The diagram below shows a map of Bay time drawn on a grid of 1 cm squares. The. scale of the map is 1:100'000. Sprin~ V ~ flail ~ ~ " Rose ( Hall / \ r-. / 11 / South Port '-.-/, (i) Find to the nearest km, the shortest distance between Rose Hall and South Port. Determine the bearing of South Port from Spring Hall (6 marks) Total 12 marks o f 2004 GO ON TO THE NEXT PAGE

8 Page 8 8. Two recipes for making chocolate drinks are shown in the table below. Cups of Milk Cups of chocolate Recipe A 3 2 Recipe B 2 1 (a) What percent of the mixture using Recipe A is chocolate? (2 marks) (b) By showing suitable calculations, determine which of the two recipes, A or B, is richer in chocolate. (2 marks) (c) f the mixtures from Recipe A and Recipe B are combined, what is the percent of chocolate in the new mixture? (2 marks) (d) A vendor makes chocolate drink using Recipe A. 3 cups of milk and 2 cups of chocolate can make 6 bottles of chocolate drink. A cup of milk costs $0.70 and a cup of chocolate costs $ (i) What is the cost of making 150 bottles of chocolate drink? What should be the selling price of each bottle of chocolate drink to make an overall profit of 20%? (6 marks) Total 12 marks o f 2004

9 -.: 'Page 9 SECTON 11 Answer TWO questions in this section ALGEBRA AND RELATONS, FUNCTONS AND GRAPHS ","'. 9. (a) The table below shows corresponding values for P and r. P m r 0.2 Given that P varies directly as r 3, calculate the values of m and n. (6 marks) (b) n the diagram below, not drawn to scale, AKLM and AST J are both rectangles, Ar- ~3x~,S~~~3--~K 2x Jr T 5 M : fL Given that AS = 3x cm, AJ = 2x cm, SK = 3 cm and JM = 5cm (i) Obtain an expression, in terms of x, for the area of rectangle AKLM. Given that the area ofrectangleaklmis oocm 2, show that 2x2 + 7x -'- 15 = 0 (iii) Hence, calculate the value of x and state the length of AK and AM. Total (9 marks) 15 marks o /f 2004

10 ,.. Page A vendor buys x kg of peanuts and y kg of cashew nuts. (a) (i) To get a good bargain, she must buy a minimum of 10 kg of peanuts and a minimum of 5 kg of cashew nuts. Write TWO inequalities which satisfy these conditions. She buys no more than 60 kg of nuts. Peanuts cost $4.00 per kg and cashew nuts cost $8.00 per kg and she spends at least $200.. Write TWO in9ualities which satisfy these conditions. (5 marks) (b) Using a scale of 2 cm to represent 10 kg on each axis, draw the graph of the FOUR in~ualities in (a) (i) and (a), On your graph, shade ONLY the region which satisfies all four inequalities. (6 marks) (c) The profit on the sale of 1 kg of peanuts is $2.00 and on 1 kg of cashew nuts is $5.00. (i) Using your graph, determine the number ofkilograms of each type of nut the vendor must sell in order to make the maximum profit. Calculate the maximum profit. (4 marks) Total 15 marks!i i o /F 2004

11 -Page 11 GEOMETRY AND TRGONOMETRY 11. Ca) n the diagram below, VWZ and WXYZ are two circles intersecting at Wand Z. svr is a. A " tangent to the circle at V, VWX and VZYare straight lines, TVY = 78 and SVX = 51. r! y (i) Calculate the size of EACH of the following angles, giving reasons for your answers.." a) VZW b) " Xyz (4 marks) (b) (i) Draw a diagram to represent the information given below. -, Show clearly the north line in your diagram. Town F is 50 km east of town G. Town H is on a bearing of 040 from town F. The distance from F to H is 65 km. Calculate, to the nearest kilometre, the actual distance GH. (iii) Calculate, to the nearest degree, the bearing of H from G. (11 marks) Total 15 marks o f 2004

12 !i Page (a) Given that sin e = -{3,0 ~ 0 ~ (i) Express in fractional or surd form the value of cos o. Show that the area of triangle CDE is r3square units, where CD = 30 units and DE = 20 units. ',, D E i 1.! (iii) Calculate the length of the side EC. (7 marks) (b) n this question, use n = 3.14 and assume the earth to bea sphere of radius 6370 km. The diagram below shows a sketch of the earth with the Greenwich Equator labelled. Meridian and the N! Greenwich Meridian ----'lr--~ ~~~~--Equator S The towns A and B are both on the circle of latitude 24 N. The longitude of A is ios' E and the longitude of B is 75 E. (i) Copy the sketch above of the earth and insert the points A and B on your diagram.!! i o F 2004

13 Calculate, correct to the Dearest kilometre, a) the radius of the circle of latitude 24 N b) the shortest distance between A and B, measured along the circle of latitude 24 N. (8 marks) Total 15 marks VECTORS AND MATRCES 13. The vertices of a quadrilateral, OABC, are (0, 0), (4" 2), (6, 10) and (2,8) respectively. Use a vector method to answer the questions which follow. (a) Write as a column vector, in the form [;], the vector (i) at. (b) ~ CB Calculate at ' the magnitude of at. (3 marks) (1 mark) (c) (i) \ State two geometrical relationships between the line segments OA and CB. Explain why OARC is a parallelogram. (4 marks) (d) f M is the midpoint of the diagonal OB, and N is the midpoint of the diagonal AC, determine the position vector (i) -) OM -) ON Hence, state one conclusion OABC. which can be made about the diagonals of the parallelogram (7 marks) TatallSmarks o F 2004

14 ~~;'.. Page An answer sheet is provided for this question. (a) F On the answer sheet provided, perform the following transformations: (i) Reflect triangle P in the y-axis. Label its image Q. Draw the line y = x and reflect triangle Q in this line. Label its image R. (5 marks) (iii) Describe, in words, the single geometric transformation which maps triangle P onto triangle R. (3 marks) (iv) Reflect triangle Q in the x-axis. Label its image S. (v) Write down the 2 x 2 matrix for the transformation which maps triangle P onto triangle S. (3 marks) Cb) (i) Write down the 2 x 2 matrices for a) a reflection in the y-axis b) a reflection in the line y = x. ~. :![ Using the two matrices in b (i) above, obtain a SNGLE matrix for a reflection in the y-axis followed by a reflection in the line y = x. (4 marks) Total 15 marks,., ENoOF'TEST o f 2004

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