CARIBBEAN EXAMINATIONS COUNCIL SECONDARY EDUCATION CERTIFICATE EXAMINATION \IATHEMATICS Paper 02 - General Proficiiency. 2 hours 40 minutes

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1 FORlV TP TEST CODE JANUARY 2012 CARBBEAN EXAMNATONS COUNCL SECONDARY EDUCATON CERTFCATE EXAMNATON \ATHEMATCS Paper 02 - General Proficiiency 2 hours 40 minutes ( 04 JANUARY 2012 (a. m.) ) READ THE FOLLOWNG NSTRUCTONS CAREFULLY. 1. This paper consists of TWO sections. 2. There are EGHT questions in Section and THREE questions in Section Answer ALL questions in Section, and any TWO questions from Section Write your answers in the booklet provided. 5. All working must be clearly shown. 6. A list of formulae is provided on page 2 of this booklet. Required Examination Materials Electronic calculator Geometry set Graph paper (provided) īiiiiiiiiiiii '!!!!!!!!! - DO ~OT TVR~ THS PAGE UNTL YOU ARE TOLD TO DO SO. Copyright ( 2009 Caribbean Examinations Council ==- All rights reserved. o /JANUARYiF 2012 =

2 LST OF FORl"lULAE Volume of a pnsm f/=,117 where : is the mea of a cross-section and h is the perpendicular length. Volume of cylinder Volume of a right pyramid C ircum ference v = tir'h where r is the radius of the base and h is the perpendicular v = t Ah where A is the area of the base and h is the perpendicular C = 2nr where r is the radius of the circle. height. height. Arc length S = 3~0 x 2rrr where e is the angle subtended by the arc, measured in degrees. Area of a circle A = ttr' where r is the radius of the circle. Area of a sector A = 3~0 x ttr: where e is the angle of the sector, measured in degrees. Area of trapezium A = ~ (a -- b) h where a and b are the lengths of the parallel the perpendicular distance between the parallel sides. sides and his Roots of quadratic v equations fax 2 + bx + C = 0, -b±~b2-4ac then x = a Trigonometric ratios sin e cos O opposite side hypotenuse adjacent side hypotenuse Adjacent Opposite tan e = opposite adjacent side side Area of triangle Area of ~ = ~ bh where b is the length of the base and h is the perpendicular height. Area of :.'.vlbc = ~ ab sin C B Area of L.lA.BC = '1 1 1 (.1'-"1)(.1 -h)(.\ -c) C/h~ El ~. where s _Ll_+_h +_c 2 <b) Sine rule Cosine rule a b c sin A sin B sin C (/2 = h2 + c 2-2bc cos A o /JANUA.RY!F 2012 GO ON TO THE NEXT PAGE

3 Page J SECTON Answer ALL the questions in this section. All working must be clearly shown. 1. \a) LJsing a calculator, or otherwise, calculate the EXACT value of.», 3 \- j , expressing your answer as a fraction \. -+) 2 (ii).} expressing your answer in standard form. (b) A typist is paid a basic wage of $22.50 per hour for a 40-hour week. ( i) Calculate the typist's basic weekly wage. ( 1 mark) Overtime is paid at one and a half times the basic hourly rate. (ii) Calculate the overtime wage for ONE hour of overtime work. ( 1 mark) To earn some extra money, the typist decided to work overtime. Calculate (iii) the wage she would earn for overtime if she worked for a TOTAL of 52 hours during a given week. (iv) the number of overtime hours she must work during a given week to earn a TOTAL wage of $ marks O234020,'JANCARY/F 2012 GO ON TO THE NEXT PAGE

4 Pagt! 4-2. ( J.) Solve the pair of simultaneous equations (b) Factorise completely 3x + 2y = 13 x - 2).-' = -. (i i) 2X2-3x + 8x ( 1 mark) (c) Tickets for a football match are sold at $30 for EACH adult and S5 for EACH child. A company bought 28 tickets. (i ) lfx of these tickets were for adults, write in terms of.r, a) the number of tickets for chi ldren ( 1 mark) b) the amount spent on tickets for adults ( 1 mark) c) the amount spent on tickets for children. ( 1 mark) (ii) Show that the TOTAL amount spent on the 28 tickets is $(l5x + 420). ( 1 mark) (iii) Given that the cost of the 28 tickets was $660, calculate the number of adult tickets bought by the company. 12 marks 3. (a) A universal set, U, is defined as: U= {51, 52,53,54,55,56,57,58,59 } A and B are subsets of U, such that: // = { odd numbers} B = t prime numbers} List the members of the set A. ( 1 mark) (i i) (i ii) List the members of the set B. Draw a Venn diagram to represent the sets A, Band U. ( 1 mark) (b) ( i) Using a pair of compasses, a ruler and a pencil a) construct a triangle CDE in which DE = 10 cm, DC = X cm and L. CDE = 45. (.t marks) b) construct a line. CF. perpendicular to DE such that F lies on DE. (ii) Using a protractor, measure and state the size of L. DCE. ( 1 mark) 12 marks /JANUARY/F 2012 GO ON TO THE NEXT PAGE

5 rage :) (a) The following is an extract from a bus schedule. The bus begins its journey at Belleview, [ravels to Chagville and ends its journey at St. Andrews. Town Arrive Depart Belleview hall a.m. : Chagville 7:.35 a.m. 7:-+5 a.m. l St. Andrews, 8:00 a.m. (i ) How long did the bus spend at Chagville? ( 1 mark) (ii) How long did the bus take to travel from Belleview to Chagville? ( 1 mark) (iii) The bus travelled at an average speed of 54 km/hour tram Belleview to Chagville. Calculate, in kilometres, the distance from Belleview to Chagville. (b) Water is poured into a cylindrical bucket with a base area of 300 cm-. f 4.8 litres of water was poured into the bucket, what is the height of the water in the bucket? (c) The diagram below, not drawn to scale, shows a cuboid with length 13 cm, width -+ cm and height h cm. f---j------( /: /j._ hem ~/ / ( 13cm---~) - ~cm 1 (i ) State, in terms ofh, the area of the shaded face of the cuboid. 1 mark (ii) Write an expression, in terms ofh, for the volume of the cuboid. (1 mark) (iii) fthe volume of the cuboid is 286 cm' calculate the height. h. of the cuboid. 11 marks ()12]-+()2{)/L\~UARYT 2012 GO ON TO THE \JEXT PAGE

6 Page 6 5. (a) The diagram below, not drawn to scale, shows TWO triangles, JKL and,'vlp, with JK parallel to _HL. Lv{ = JP, KLP is a straight line, angle Jl.M = 22 and angle L.vlP = 36 _ J K p Calculate, giving reasons for your answers, the measure of EACH of the following: (ii) (iii) L lulp L L.K L.KL (iv) LKLJ ( 4 marks) (b) The diagram below shows a triangle, PQR, and its image, P'Q'R'_.!! Y, -_ ' 1--- r-----r--t---l j. 1! i! ~----l-----~i2 -;----'- i,! 1 :. :.!! ; ; ----, ---t "t---t----r-----, : '! R i! Q i i.!! ~ ~P! : ~ r i +-ti -4-2 o :----! ---: ~c;:;:;--j~----.r_-- --, ".. C --,----r--r---i------i 1 i - X _, j ~._-L_~_:_~~ -2! i t------i ~-_4-~-_+-~ ~ R" ;Q' ----L -'-- -'-- State the coordinates ofp and Q_ (ii) Describe fully the transformation that maps triangle PQR onto triangle P'Q'R'_ (iii) Write down the coordinates of the images of P and Q under the translation 10 marks lJANUARY/F 2012 GO ON TO THE NEXT PAGE

7 Page 7 6. The table below shows corresponding values of x and y for the function y = XC - 2x - 3, tor nteger values of.r from -2 to 4. 1 : 1 ~2 1 -\ 1 :J 1_ : 1 : 1 (a) Copy and complete the table. (b) Using a scale of 2 cm to represent 1 unit on the x-axis, and 1 cm to represent unit on the y-axis, plot the points whose x and y values are recorded in your table, and draw a smooth curve through your points. ( 4 marks) (c) Using your graph, estimate the value of y when x,= 3.5. Show on your graph how the value was obtained. (d) Without further calculations, write the equation of the axis of symmetry of the graph ( 1 mark) (ii) estimate the minimum value of the function y ( mark ) (iii) state the values of the solutions of the equation: X2-2x - 3 = O. (1 mark) 11 marks /JANUARYiF 2012 GO ON TO THE NEXT PAGE

8 7. The histogram below shows the distribution of heights of seedlings n Cl sample. f-frequency ~ JO,, - "..... " " ,...., l-... i....,.. ' "'" H' f.....,.,. '"... c., 1:..', , r.,......!", ,.. :... ~.':'.. ;-..,., ,....,it. 'll,,. :., i.i..,.,.,. '" r,l...,......,::......,. r...,.., , ,......i...' , _.,...,....,... "..,,... '1""'- " 20...;)U,_.;U, : r , Height in cm ".. it......,.' :.....;...,...,.'.',.i...i...~...,..-;..+...;....,... '.,.... "..... (~!) Copy and complete the frequency table for the data in the sample. Height in cm Mid-point Frequency S (b) Determine the: modal class interval mark) ( i i) the number of seedl ings in the sample 2 marks) (iii) the mean height of the seedlings.. marks) (iv) the probability that Cl seedling chosen at random has a height that is CJREATER than 30 cm. 12 marks () JANUARY F 2012 CO ON TO THE NEXT PACE

9 Page 9 8. An answer sheet is provided for this question. Sarah is making a pattern of squares using straws. She uses four straws for the sides and two longer straws for the diagonals. The first three figures in her sequence of shapes are shown below: Figure Figure 2 Figure J (a) On your answer sheet, draw Figure 4, the FOURTH shape in the pattern. (b) On your answer sheet, complete the TWO rows in the table for Figure 4 (ii ) Figure 10. Number of Straws Figure Formula Number 1(6) (6) - 3 3(6) ~ ~ (ii) ( 4 marks) (c) Which figure in the sequence uses 106 straws"? (d) Obtain an expression in n, for the total number of straws used in the 11th pattern. 10 marks o /JANUARY/F 2012 GO ON TO THE NEXT PAGE

10 Page () SECTON 11.vnswer T\VO questions in this section. ALGEBRA AND RELATONS, FUNCTONS AND GRAPHS 9. (a) ( i) Make x the subject of the formula x-4 (ii) Hence, determine the inverse of (x) 2x+3 x-4 where.r j:. 4. (iii) Find the value ofx for whichf(x) = O. (b) The diagram below shows the graphs of three lines and a shaded region defined by three inequalities associated with these lines. The inequality associated with the line 3y = x S 3y ~ x....:::±± -- ~....[LT::... l..+v-!...,.,...,.."...,... "... " ". 1.. '... "...LL... :'\'. ""'...1' ,...;,-~ 1.,, + + /... "'". '~ """'f' o ~o -x State the other TWO inequalities which define the shaded region. Th~ function p = 4x + 3y satisfies the solution set represented by the closed triangular region. (ii) dentify the three pairs Of(X -.1') values for whichp has a maximum or a minimum value. (iii) Which pair of (x. ') values makes p a maximum? Justify your answer. ( 4 marks) 15 marks () ;./A.NUARY F 2012 GO ON TO THE NEXT PAGE

11 Page 11 \EASURE1VENT, GEOMETRY AND TR[GONOMETRY 10. (a) The diagram below, not drawn to scale. shows a regular hexagon with centre. 0, and.-10 = 5 cm. \,,,, B Determine the size of angle AGB. (ii) Calculate, to the nearest whole number, the area of the hexagon. (b) The diagram below, not drawn to scale, shows a vertical pole, PL, standing on a horizontal plane, KL1H. The angle of elevation of P from K is 28, KL = 15 m, LM= 19 m and L KLM= 115. p K M Copy the diagram. Show the angle of elevation, 28 and ONE right angle. (ii ) Calculate. giving your answer to 2 significant figures. the measure of a) PL 2 marks) b) KJf 3 marks) c) the angle of elevation of P from.h. 15 marks iJANUARYiF 2012 GO ON TO THE NEXT PAGE

12 t 11. (a) The diagram below shows two position vectors 0.1 am! OB. --'" --'" Page :.. :" : f ! ' / , l-"+-,.:j rl:7" ::-..:t,~ 1... '.,;: :::'... =,'J1. T=-:. -: 1.=..=.. :..... T. t.ll''':.. 1:E=1==t.. -.s-..,1: ~;~fg.~~=,~~, \:' 3""......~.....,...; , f;;,...tL:.~ie::..::, T':: T,.~:t, :::f.:. f. ':: x Write as a column vector, in the form (.: J :... ~:o\,'. ~y ~ml.. ;.. ~.. a) OA b) c) Os --'" BA ( mark ( mark) (ii) Given that C is the mid-point of the line AB, write as a column vector in the form ( XJ y : a) b) ~ BC --'" OC 1 mark mark) ( b) Land.\..1are two matrices where Evaluate ( i i ) L\f L..;... 2M - and M = (.0 ( c) The matrix. Q. is such that Q (4, 2 2 marks) ( i i) Find Q. Using a matrix method. find the values ofv andv in the equation! 4 21 (.\"1 = isi. \\1 1)(.1) (3) E~D OFTEST 15 marks F YOU F:\,SH BEFORE TME S CALLED, CHECK YOCR \VORK O~ THS TEST.

13 FORM TP rest CODE JANUARY :::012 CARBBEA~ EXAMNATO:'lS COU~CL SECONDARY EDUCATON CERTFCATE EXAMNATON 'ATHE\lATCS Paper 02 - General Proficiency Answer Sheet for Question 8 Candidate Number (a) Draw the FOLRTH shape in the pattern: (b) Complete the table for Figure 4. and Figure 10. Number of Straws Figure Formula Number 1(6) :2 2( 6)- 1 1 } 3(6) - : ~ ~ (i i ) (C) \Vhich figure in the sequence uses 1()6 straws') (d) Obtain ~111 expression in 11. for the total number of straws used ill the "ill pattern.,\tl\ch THS,\:\SWER SHEET TO YOl'R,,\,\jSWER BOOKLET