E X A M I N A T I O N S C O U N C I L

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1 C A R I B B E A N E X A M I N A T I O N S C O U N C I L CARIBBEAN SECONDARY EDUCATION CERTIFICATE EXAMINATION * Barcode Area * Front Page Bar Code FILL IN ALL THE INFORMATION REQUESTED CLEARLY IN CAPITAL LETTERS. TEST CODE SUBJECT MATHEMATICS Paper 02 PROFICIENCY GENERAL REGISTRATION NUMBER SCHOOL/CENTRE NUMBER NAME OF SCHOOL/CENTRE CANDIDATE S FULL NAME (FIRST, MIDDLE, LAST) * Barcode Area * Current Bar Code DATE OF BIRTH D D M M Y Y Y Y HOW MANY ADDITIONAL PAGES HAVE YOU USED IN TOTAL? SIGNATURE * Barcode Area * Sequential Bar Code

2 FORM TP /SPEC 2013 TEST CODE C A R I B B E A N E X A M I N A T I O N S C O U N C I L CARIBBEAN SECONDARY EDUCATION CERTIFICATE EXAMINATION MATHEMATICS SPECIMEN PAPER E-MARKING Paper 02 General Proficiency 2 hours 40 minutes INSTRUCTIONS TO CANDIDATES 1. Answer ALL questions in Section I, and ANY TWO in Section II. 2. Draw diagrams in HB2 pencil, but use black or dark-blue ink pen for all other writing. 3. You must answer the questions in the spaces provided. Answers written in the margins or on blank pages will not be marked. 4. All working must be shown clearly. 5. A list of formulae is provided on page 2 of this booklet. 6. If you need to re-write any answer and there is not enough space to do so on the original page, you must request extra lined pages from the invigilator. Remember to draw a line through your original answer and correctly number your new answer in the box provided. 7. If you use extra pages you MUST write your registration number and question number clearly in the boxes provided at the top of EVERY extra page. Required Examination Materials Electronic calculator (non-programmable) Geometry set DO NOT TURN THIS PAGE UNTIL YOU ARE TOLD TO DO SO. Copyright 2012 Caribbean Examinations Council. All rights reserved.

3 Page 2 LIST OF FORMULAE Volume of a prism Volume of cylinder Volume of a right pyramid Circumference Area of a circle Area of trapezium V = Ah where A is the area of a cross-section and h is the perpendicular length. V = πr 2 h where r is the radius of the base and h is the perpendicular height. 1 V = Ah where A is the area of the base and h is the perpendicular height. 3 C = 2πr where r is the radius of the circle. A = πr 2 where r is the radius of the circle. Roots of quadratic equations If ax 2 + bx + c = 0, 2 then x = b + b 4ac 2a opposite side Trigonometric ratios sin θ = hypotenuse Area of triangle 1 A = (a + b) h where a and b are the lengths of the parallel sides and h is 2 the perpendicular distance between the parallel sides. cos θ = tan θ = 1 Area of = bh where b is the length of the base and h is the 2 perpendicular height 1 Area of ABC = ab sin C 2 Area of ABC = s( s a)( s b)( s c) where s = Sine rule a = b = sin A sin B Cosine rule adjacent side hypotenuse opposite side adjacent side a + b + c 2 c sin C a 2 = b 2 + c 2 2bc cos A

4 Page 3 1. (a) Determine the EXACT value of: (i) (ii) SECTION I Answer ALL the questions in this section All working must be clearly shown giving your answer to 2 significant figures. (3 marks) (3 marks)

5 Page 4 (b) Mrs. Jack bought 150 T-shirts for $1 920 from a factory. (i) Calculate the cost of ONE T-shirt. The T-shirts are sold at $19.99 each. (ii) Calculate the amount of money Mrs. Jack received after selling ALL the T-shirts. (iii) Calculate the TOTAL profit made.

6 Page 5 (iv) Calculate the profit made as a percentage of the cost price, giving your answer correct to the nearest whole number. 2. (a) Given that a = 1, b = 2 and c = 3, find the value of: (i) a + b + c (ii) b 2 c 2 Total 11 marks

7 Page 6 (b) (c) Write the following phrases as algebraic expressions: (i) seven times the sum of x and y (ii) the product of TWO consecutive numbers when the smaller number is y Solve the pair of simultaneous equations: 2x + y = 7 and x 2y = 1 (3 marks)

8 Page 7 (d) Factorise completely: (i) 4y 2 z 2 (ii) 2ax 2ay bx + by (iii) 3x x 8 Total 12 marks

9 Page 8 3. (a) A survey was conducted among 40 tourists. The results were: 28 visited Antigua (A) 30 visited Barbados (B) 3x visited both Antigua and Barbados x visited neither Antigua nor Barbados (i) (ii) Complete the Venn diagram below to represent the information given above. x A B U Write an expression, in x, to represent the TOTAL number of tourists in the survey. (iii) Calculate the value of x.

10 Page 9 (b) The diagram below, not drawn to scale, shows a wooden toy in the shape of a prism, with cross section ABCDE. F is the midpoint of EC, and BAE = CBA = 90. E A Calculate (i) (ii) (iii) D 5 cm 5 cm F 6 cm the length of EF the length of DF B C the area of the face ABCDE. 10 cm 5 cm (3 marks) Total 12 marks

11 Page (a) When y varies directly as the square of x, the variation equation is written y = kx 2, where k is a constant. (i) Given that y = 50 when x = 10, find the value of k. (ii) Calculate the value of y when x = 30. (b) (i) Using a ruler, a pencil and a pair of compasses, construct triangle EFG with EG = 6 cm, FEG = 60 and EGF = 90. Draw the triangle on the following page. (5 marks) (ii) Measure and state a) the length of EF b) the size of EFG Total 11 marks /PT 2012

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13 Page (a) The diagram below shows the graph of y = x 2 + 2x 3 for the domain 4 x 2. y 5 y = x 2 + 2x x Use the graph to determine (i) the scale used on the x axis (ii) the value of y for which x = 1.5

14 Page 13 (iii) the values of x for which y = 0 (iv) the range of values of y, giving your answers in the form a y b, where a and b are real numbers.

15 Page 14 (b) The functions f and g are defined as f(x) = 2x 5 and g(x) = x (i) Calculate the value of a) f (4) b) gf (4) c) Find f 1 (x). Total 12 marks

16 Page (a) The diagram below, not drawn to scale, shows two straight lines, PQ and RS, intersecting a pair of parallel lines, TU and VW. T V Q R y Determine, giving a reason for EACH of your answers, the value of (i) x (ii) y. x S P U W

17 Page 16 (b) The diagram below shows triangle LMN, and its image, triangle L M N, after undergoing a rotation. (i) L M N y N L M Describe the rotation FULLY by stating a) the centre b) the angle c) the direction. x

18 Page 17 (ii) (iii) State TWO geometric relationships between triangle LMN and its image, triangle L M N. Triangle LMN is translated by the vector Determine the coordinates of the image of the point L under this transformation. Total 11 marks

19 Page A class of 24 students threw the cricket ball at sports. The distance thrown by each student was measured to the nearest metre. The results are shown below. (a) Complete the frequency table for the data shown above. Distance (m) Frequency (b) State the lower boundary for the class interval (c) (3 marks) Determine the probability that a student, chosen at random, threw the ball a distance recorded as 50 metres or more.

20 Page 19 (d) Using a scale of 1 cm on the x-axis to represent 5 metres, and a scale of 1 cm on the y-axis to represent 1 student, draw a histogram to illustrate the data. (5 marks) Total 11 marks

21 Page The diagram below shows the first three figures in a sequence of figures. Each figure is made up of squares of side 1 cm. Fig. 1 Fig. 2 Fig. 3 (a) On the grid below, draw the FOURTH figure (Fig. 4) in the sequence.

22 Page 21 (b) Study the patterns in the table shown below, and complete the table by inserting the appropriate values in the rows numbered (i), (ii), (iii) and (iv). (i) 4 (ii) 5 (iii) 15 (iv) Area of Perimeter Figure Figure of Figure (cm 2 ) (cm) = = = 16 n (8 marks) Total 10 marks

23 Page 22 SECTION II Answer TWO questions in this section. ALGEBRA AND RELATIONS, FUNCTIONS AND GRAPHS 9. (a) The diagram below shows the speed-time graph of the motion of an athlete during a race. Speed in m/s Time in seconds (i) Using the graph, determine a) the MAXIMUM speed b) the number of seconds for which the speed was constant

24 Page 23 (ii) c) the TOTAL distance covered by the athlete during the race. During which time-period of the race was a) the speed of the athlete increasing b) the speed of the athlete decreasing c) the acceleration of the athlete zero?

25 Page 24 (b) A farmer supplies his neighbours with x pumpkins and y melons daily, using the following conditions: First condition : y 3 Second condition : y x Third condition : the total number of pumpkins and melons must not exceed 12. (i) (ii) (iii) (iv) Write an inequality to represent the THIRD condition. Using a scale of 1 cm to represent one pumpkin on the x-axis and 1 cm to represent one melon on the y-axis, draw the graphs of the lines associated with the THREE inequalities. Use the graph paper on the following page. (4 marks) Identify, by shading, the region on your graph which satisfies the THREE inequalities. Determine, from your graph, the minimum values of x and y which satisfy the conditions. Total 15 marks

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27 Page 26 MEASUREMENT, GEOMETRY AND TRIGONOMETRY 10. (a) In the diagram below, not drawn to scale, PQ is a tangent to the circle PTSR, so that angle RPQ = 46 angle RQP = 32 and TRQ is a straight line. T P S Calculate, giving a reason for EACH step to your answer, (i) angle PTR R Q

28 Page 27 (ii) (iii) angle TPR angle TSR (3 marks)

29 Page 28 (b) The diagram below, not drawn to scale, shows a vertical flagpole, FT, with its foot, F, on the horizontal plane EFG. ET and GT are wires which support the flagpole in its position. The angle of elevation of T from G is 55, EF = 8 m, FG = 6 m and EFG = 120. E T F m 6 m Calculate, giving your answer correct to 3 significant figures (i) the height, FT, of the flagpole 55 G

30 Page 29 (ii) the length of EG (iii) the angle of elevation of T from E. (3 marks) (3 marks) Total 15 marks

31 Page 30 VECTORS AND MATRICES 11. (a) A and B are two 2 2 matrices such that A = and B = 2 1 (i) Find AB. (ii) Determine B 1, the inverse of B.

32 Page 31 (iii) Given that 5 2 x 2 = 2 1 y 3, x write as the product of TWO matrices. y (iv) Hence, calculate the values of x and y.

33 Page 32 (b) The diagram below, not drawn to scale, shows triangle JKL. J M and N are points on JK and JL respectively, such that 1 1 JM = JK and JN = JL. 3 3 (i) On the diagram above show the points M and N. (ii) Given that JM = u and JN = v, write, in terms of u and v, an expression for a) JK b) MN c) KL K L

34 Page 33 (iii) Using your findings in (b) (ii), deduce TWO geometrical relationships between KL and MN. END OF TEST Total 15 marks

35 DO NOT WRITE ON THIS PAGE

36 CANDIDATE S RECEIPT INSTRUCTIONS TO CANDIDATE: 1. Fill in all the information requested clearly in capital letters. TEST CODE: SUBJECT: MATHEMATICS Paper 02 PROFICIENCY: GENERAL REGISTRATION NUMBER: FULL NAME: (BLOCK LETTERS) Signature: Date: 2. Ensure that this slip is detached by the Supervisor or Invigilator and given to you when you hand in this booklet. 3. Keep it in a safe place until you have received your results. INSTRUCTION TO SUPERVISOR/INVIGILATOR: Sign the declaration below, detach this slip and hand it to the candidate as his/her receipt for this booklet collected by you. I hereby acknowledge receipt of the candidate s booklet for the examination stated above. Signature: Supervisor/Invigilator Date: