FORM TP J ANU AR Y 2009

Transcription

1 TEST CODE FORM TP J ANU AR Y 2009 CAR B B E A N E X A M N A T O N S C 0 U N C L SECONDARY EDUCATON CERTFCATE EXAiVlNA TON Paper 02 - MATHEwATCS General Proficiency 2 hours 40 minutes ( 06 JANUARY 2009 (a.mj) NSTRUCTONS TO CANDDATES. Answer ALL questions in Section, and ANY TWO in Section l. 2. Write your answers in the booklet provided. 3. All working must be shown clearly. 4. A list of formulae is provided on page :2of this booklet. Examination Materials Electronic calculator (non-programmable) Geometry set Graph paper (provided) DO NOT TURN THS PAGE UNTL YOU ARE TOLD TO DO SO Copyright 2007 Caribbean Examinations Councileu. A J rights reserved. o /1 AN LTAR Y F 2009

2 LST OF FORMULAE Volume of a prism V = Ah where A is the area of a cross-section and h is the perpcll i length. Volume of cylinder V= m.2h where r is the radius of the base and h is the perpendicular height. Volume of a right pyramid V = t,ih where A is the area of the base and h is the perpendicular height. Circumference Area of a circle C = 2nr where r is the radius of the circle. A = tir' where r is the radius of the circle. Area of trapezium A = ~ (a + b) h where a and b are the lengths of the parallel sides and his the perpendicular distance between the parallel sides. Roots of quadratic equations fax 2 + bx + c = 0, then x -b ± ~b2-4ac 2a Trigonometric ratios sin 8 = opposite side hypotenuse Opposite cos 8 tan 8 adjacent side hypotenuse. opposite side adjacent side Adjacent Area of triangle Areaof" = ~:: P::::;~:l:e :::~; ofthcllbaseand h is ~ Area of MBC = lab sin C 2 a C < b ) '1 i'l c Area of MBC = ~s(s - a)(s - b)(s - c) where s a + b+ c Sine rule Cl sin A b sin B c sin C Cosine rule -, a- b 2 + c 2-2bc cos A o /JANUAR YF 2009 GO ON TO THE NEXT PAC':

3 Page -+ SECTON 1. (a) Calculate the EXACT value of Answer ALL the questions in this section. All working must be clearly shown. 1 S giving your answer as a fraction. ( 3 marks) (b) BDS$ means Barbados dollars and EC$ means Eastern Caribbean dollars. (i) Karen exchanged BDS$ and received EC$ Calculate the value of one BDS$ in EC$. (ii) f Karen exchanged EC$ for BDS$, calculate the amount of BDS$ which Karen would receive. (Assume that the buying and selling rates are the same.] (c) A credit union pays 8% per annum compound interest on all fixed deposits. A customer deposited $ in an account. Calculate the TOTAL amount of money in the account at the end of two years. ( 4 marks) Total 11 marks 2. (a) Simplify, expressing your answer as Cl 'single fraction 2rn n Srn 3n ( 3 marks) (b) f a * b = a 2 - b, evaluate 5 * 2. ( 1mark) (c) Factorize completely 3x - 6y + x2-2xy. (d) A drinking straw of length 21 cm is cut into 3 pieces. The length of the first piece is x cm. The second piece is 3 cm shorter than the first piece. The third piece is twice as long as the first piece. (i) State, in terms of.r, the length of EACH of the pieces. (ii) Write an expression, in terms of.r, to represent the sum of the lengths of the three pieces of drinking straw. ( 1 mark) (i ii) Hence, calculate the value of.r. 3 marks) Total 12 marks ,O/JANUARY/F 2009 GO ON TO THE NEXT PAGE

4 Page :5 3. (a) (\ school has 90 students in Form S. 54 students study Physical Education. 42 students study Music. 6 students study neither Physical Education nor Music..\"students study both Physical Education and Music. (i) Copy the Venn diagram shown below. ( 1mark) p M u (ii) Show on your Venn diagram the information relating to the students in Form S. (iii) Calculate the number of students who study BOTH Physical Education and Music. ( 3 marks) (b) The diagram below, not drawn to scale, shows two straight wires, U and NK, supporting a vertical pole, MJ. LMN is a straight line on the horizontal plane. Angle.ML = 90, angle KNM= 30, U = 13 m, LM= 5 m and KN = 10 m. J K L Calculate. in metres, the length of (i) (ii)!'\llk.k. ( 3 marks) 3 marks) Total 12 marks JAN U AR Y F 2009 GO ON TO THE NEXT Pi\(JP

5 Page 6 -t. The diagram below represents the graph of a straight line which passes through the points P and Q ' ' : '.~.;', y.... :ilz ~....,......,,' 3*.. ',p~ + ;,.,,,..,. : ', L....., ;.,t 2..;....":"':":":"".., ; ; +, + l.. ; ), : QV -3 /-2 v / ,r, ;..., ; i... x'" (a) State the coordinates of P and Q. (b) Determine (i) the gradient of the line segment PQ (i i) the equation of the line segment PQ. 2 marks) (c) The point (-8, t) lies on the line segment PQ. Determine the value of t. (2 marks) The line segment, AB, is perpendicular to PQ, and passes through (6,2). (d) Determine the equation of AB in the form y = mx + c. ( 3 marks) Total 11 marks () 12J4()2()/ J /\.NU AR Y F 2()09 GO ON TO THE NEXT PAGE

6 5. A company makes cereal boxes in the shape of a right prism. ----~ ",--"11....~m ~ cm 36cm EACH large box has dimensions 25 cm by 8 cm by 36 cm. (a) Calculate the volume in cubic centimetres, of ONE large cereal box. (b) Calculate the total surface area of ONE large cereal box. 4 marks) (c) The cereal from ONE large box can exactly fill six small boxes, each of equal volume. (i) Calculate the volume of ONE small cereal box. (ii) f the height of a small box is 20 cm, list TWO different pairs of values which the company can use for the length and width of a small box., Total 10 marks 6. (a) The diagram below, not drawn to scale, shows rectangle PQRS with PQ = 7.0 cm and QR = 5.5 cm. S u R 5.5 em! p.oem Q Using a ruler, a pencil and a pair of compasses only, construct the rectangle. ( 6 marks) o JANUARY/F 2009 GO ONTO THE NEXT PAGE

7 Page ~ An answer sheet is provided for this question. (h) The diagram below shows three triangles labelled L, M and N J ANU AR YF 2009 GO ON TO THE NEXT PAGE

8 Page l) L is mapped onto /V by a translation. (i) (ii) State, in the form (: J ' the vector which represents the translation. L is mapped onto N by an enlargement: a) Find and label on your answer sheet: the point G, the centre of the enlargement. Show your method clearly. b) State the coordinates of G. c) State the scale factor of the enlargement. ( 4 marks) Total 12 marks o JANUARY/P 2009 GO ON TO THE NEXT PAGE

9 Pa::.~() 7. The table below shows the distribution of marks on a test for a group of 70 students. :\ark 1-Frequency Cumulative Frequency ~ 1_1_-_2_0_~~ 5 ~ 7 _ ! (a) Copy and complete the table to show the cumulative frequency for the distribution. (b) (i) Using a scale of 1 cm to represent 5 marks on the horizontal axis and cm to represent 5 students on the vertical axis, draw the cumulative frequency curve for the scores. ( 5 marks) (ii) What assumption have you made in drawing your curve through the point (O,O)? ( 1 mark) (c) The pass mark for the test is 47. Use your graph to determine the number of students who passed the test. (d) What is the probability that a student chosen at random had a mark less than or equal to 30? Total 12 marks () /J :'\.N U i\f{ YiF :2009 (;0 ON TO THE NEXT PAGE

10 H.,\11 answer sheet is provided for this question. The figure below shows the first three diagrams in a sequence. Each diagram is nu, connected by line segments. n each diagram, there are cl dots and {line scgmcms.~ :1 J r:::=:;:::r.:::==+=:rtt+tj==r=r:::tt.:jj1'~:::n=:::::::l::=:::==r::=t::;::::q::;====r:::+:::::p:j=+~~'-'.f" -._.-.. ':::..:...: L::: f'""~"'"+''''''''' "'''' ;,..:..z: 1..'..;,,... '..1'1,' '."- ' 1 '..,.. ; cl..'.. ' '.t:' 1t,'. '.. ;~.T:.'.:..L....:.. ~...., f: ;;,;: ".,..:.:. N)... ~.'-..''~;:,oot;tvtj2,1.,itt...t.t.l.t~t..tl~~ '-.~.",n..as: J2,1?j;1S~'-T,tf,;.,xff;1 1 T.T::T::T;j. ~~;~,: ful~~" ~f~ /f_~ ":..,...,...;...:f,..,,.... ;'.~'-,..;..., +.,...,... ; ;...'...r..,..., 1.L..'L:'~_...:,.,...,f; _,."""..~T'...,,. ~... 1,:::1..,... ;....,...,.. '/".;'~'-,.,.f, N..':":", -"-i,. 'X.v: i'f"'t'tr;.:::r:.;l '.1 CTL:t ;:: 1 EEJEELElEr 1j :.., On the answer sheet provided: (a) Draw the fourth diagram in the sequence. ( 2marksj (b) Complete the table by inserting the missing values at the rows marked (i) and (ii) No. of dots Pattern connecting No. of line segments d and d 5 2x x xll-4 18 (i) (ii) 62 C ) ( ) ( ) ( ) 180 ( 2L;irks) Cc) (i) How many dots are in the sixth diagram of the sequence? (i i) How many line segments are in the seventh diagram or the sequence? ( 1 mark (i ii) Write the rule which shows how is related to d. o /J A N U AR Y F 2009 GO ON TO THE N

11 Page 12 SECTON n Answer TWO questions in this section. ALGEBRA AND RELATONS, FUNCT](ONS AND GRAPHS 9. ( a) Make t the subject of the formula ~~t;'. ( 3 marks) (b) (i) Express the furi'ctionfix) = 2x2-4x - 13 in the formj(x) = a(x + h)2 + k. ( 3 marks) Hence, or otherwise, determine (ii) the values of x at which the graph cuts the x-axis ( 4 marks) (iii) the interval for which fix) ~ 0 (iv) (v) the minimum value of fix) the value of x at which fix) is a minimum. ( 1 mark) Total 15 marks 01?J40201 J ANU AR Y F 2009 GO ON TO THE NEXT PAGE

12 - Page (a) Two functions are defined as follows: 1:X---7x-3, g :.r ---7.r -- (i) Calculate 1(6). 1 mark) (iil FindF1(;t). ( 1 mark) (ii i) Show thatfg (2) = fg (-2) = O. ( 3 marks) (b) An answer sheet is provided for this question. The distance-time graph below describes the journey of a train between two train stations, A and B. Answer the questions below on the answer sheet. Distance in km,/n ~uu 120 B / 0 vu /,/ '" / -.v A/ o / Time in minutes (i) For how many minutes was the train at rest at B? ( 1 mark) (ii) Determine the average speed of the train, in kmlh, on its journey from A to B. ( 3 marks) The train continued its journey away from stations A and B to another station C, which is 50 km from B. The average speed on this journey was 60 km/h. (iii) Calculate the time, in minutes, taken for the train to travel from B to C. ( 3 marks) (iv) On your answer sheet, draw the line segment which describes the journey of the train from B to C. ( 3 marks) Total 15 marks ANU AR YF 2009 GO ON TO THE NEXT PAGE

13 Pagt: l,..j. GEOMETRY AND TRGONOMETRY it. (<) (i) Copy and complete the table for the graph of.. v = tan x. tor 10 ::; x ::; 60 v. ) l i \' ~~~--~--~--~--~-----~--~ ( 1 mark) (ii) Using a scale of2 cm to represent 10 (l0 degrees) on the horizontal axis and 10 cm to represent unit on the vertical axis, plot the values from the table and draw a smooth curve through your points. ( 4 marks) (iii) Use your graph to estimate the value of x when y = 0.7. ( 1 mark) (b) The diagram below, not drawn to scale, shows a circle with centre, O. EA and EB are tangents to the circle, and angle AEB = 48. c E Calculate, giving reasons for your answer, the size of EACH of the following angles: (i) LOAE 2 marks) (i i) LAOB ( 3 marks) (iii ) L>\CB 2 marks) (iv) L/\DB 2 marks) Total 15 marks 01 n...j.020january/f 2009 GO ON TO THE NEXT PAGE

14 12. (a) On the diagram below. not drawn to scale. TU = 8 m. TV = 10 ll. V\ angle {7nV= 60" and angle WUV= 40. u 8m v T w Calculate (i) the length of UW (ii) the size of the angle UVW (iii) the area of triangle TUW. o 12_-:q020/JANUARY/F 2009 GO ON TO THE NEXT PAGE

15 Page f> n this question use T = 27 2, and assume that the earth is a sphere of radius km. (b) The diagram below, not drawn to scale, represents the surface of the earth. The points Nand S represent the North and South Poles respectively. The Equator and Greenwich Meridian are labelled. The following are also shown on the diagram but they are not labelled: circles of latitude 30 0 N and 200S circle of longitude 40 E. N Greenwich Meridian s (i) Copy the diagram and label it ts:;show a) circles of latitude 30 0 N and 20 0 S b) circle of longitude 40 E c) ( 5 marks) (ii) The position of the point Q is 20 Sand 40 E. Calculate, correct to 3 significant figures, a) the SHORTEST distance from P to Q measured along the common circle of longitude b) the radius of the circle of latitude on which Q lies. Total 15 marks o 12J4020JANUARY F 2009 GO ON TO THE NEXT PAGE

16 Page 17 VECTORS AND MATRCES --'! --'1 13. The diagram below shows position vectors OP and OQ. (a) Write as a column vector, in the form CJ (i) (ii) ~ OP ~ OQ ( 1 mark) ( 1 mark) (b) The point R has coordinates (8,9). (i) Express QR ~ as a vector in the form (~y~,j, (ii) ~ ~ Using a vector method, show that OP is parallel to QR. ( 1 mark) ~ (i ii) Determine the magnitude of the vector PR. <. 2 marks) (c) The point S has coordinates (a, /J). ~ (i) Write QS as a column vector, in terms of a and h. ~ ~ (ii) Given that QS = OP, calculate the value of (/ and the value of b. 3 marks) (iii) Using a vector method, show that OPSQ is a parallelogram. (3 marks) Total 15 marks Ol2J4020JANUARY/F 2009 CO ON TO THE NEXT PACE

17 14. Cl) Calculate the matrix prod uct 3AB, w here A = (~ ~) and B = (~ ; ). ( 3 marks) (b) The diagram below, not drawn to scale, shows a triangle, ABC whose coordinates are stated. y :\ (1, 2) ~ B (1, 1) C (2,1) 0 x Triangle ABC undergoes two successive transformations, V followed by W, where V = (2 0) and W = (-1 0). o 2 1 (i) State the effect of Von triangle ABC. (ii) Determine the 2 x 2 matrix that represents the combined transformation of V followed by W. ( 3 marks) (iii) Using your matrix in (b) (ii), determine the coordinates of the image of triangle ABC under this combined transformation. ( 3 marks) (c) (i) Write the following simultaneous equations 10 the form AX = B where A, X and B are matrices: l l.v + 6\" = 6 91" + 5v = 7 2 marks) (ii ) Hence. write the solution (or.r and y as Cl product of two matrices. Total 2 marks) 15 marks END OFTEST JANUARY/F ~009

18 FORM rfp TEST CODE J AJ.'JUAR Y 2009 CARlBBEAN EXAiVNATONS COUNCL SECONDARY EDUCATON CERTFCATE EXAl'VfNATON Paper 02 - MATHE~'lATCS General Proficiency Answer Sheet for Question 6 (h) Candidate Number M / 1 / [7 x ,.., ; +./~f..r------'~--~~ 1 j- ;., LV /1'3... N., /... ; ;...., ; ;7, _ -4,... i""""""" ".../ , ; ;....!,, '!,,f....,..!, ;.... "..., T7;.1, i r ,, 1'. : :......, -8 " /.!,.. f 1. j ATTACH THS A~SWER SHEET TO YOUR ANSWER BOOKLET /1\i\UARY/F 20U9

19 FO-RM TP TEST CODE J AN U ARY 2U09 C.\ RB B E A N E X A M N A T ON S C 0 U N C L SECONDARY EDUCATON CERTFCATE EXAJ\UNATON Paper 02 - \-ATHEMATCS General Proficiency Answer Sheet tor Question 8 Candidate Number. (a)..,.. (b) No. of dots d Pattern connecting landd No. of line segments 5 2x5-4 6 (i) (i i) 8 L 2xR-4 11 i -2-X-ll l t C._~ (----../) --)[ ) 18_0_---' 12 (i) (ii) (i ii) ATTACH THS ANSWER SHEET TO YOUR ANS\VER BOOKLET

20 FORM TP TEST CODE U.l"""""'+U~u JANUARY 200Y CAR B B E A N E X A :V N A T ON S C 0 U N C L SECONDARY EDUCAT[ON CERTFCATE EXAMNATON Paper 02 - ;vathelyla TCS General Proficiency Answer Sheet for Question 10 (b) Candidate Number. (i) (ii) (iii) (iv) Distance in km : 120 B so H,\ i/! V, /1 /,/. ; :,,,, : i '." " '"... "..." ! i,...! o so Time in minutes ATTACH THS ANSWER SHEET TO YOUR Al'l'S\VERBOOKLET o 23-:J-020iJ/\NUARY/F 2009,...,