Additional Mathematics

Transcription

1 Additioal Mathematics CSEC PAST PAPERS

2 Macmilla Educatio 4 Cria Street, Lodo, N1 9XW A divisio of Macmilla Publishers Limited Compaies ad represetatives throughout the world ISBN AER Caribbea Examiatios Coucil (CXC ) The author has asserted their right to be idetified as the author of this work i accordace with the Copyright, Desig ad Patets Act First published 2014 This revised editio published 2016 All rights reserved; o part of this publicatio may be reproduced, stored i a retrieval system, trasmitted i ay form, or by ay meas, electroic, mechaical, photocopyig, recordig, or otherwise, without the prior writte permissio of the publishers. Desiged by Macmilla Publishers Limited Cover desig by Macmilla Publishers Limited Cover photograph Caribbea Examiatios Coucil (CXC ) Cover photograph by Mrs Alberta Williams With thaks to the studets of the Sir Arthur Lewis Commuity College, St Lucia: Aki Ogulusi, Nechelle Joseph

3 CSEC Additioal Maths Past Papers LIST OF CONTENTS Paper 02 (03 May 2012) 3 Paper 032 (12 Jue 2012) 11 Paper 02 (07 May 2013) 13 Paper 032 (12 Jue 2013) 21 Paper 02 (06 May 2014) 23 Paper 032 (09 Jue 2014) 33 Paper 02 (05 May 2015) 38 Paper 032 (08 Jue 2015) 48 Paper 02 May/Jue Paper 032 May/Jue

4 TEST CODE FORM TP MAY/JUNE 2012 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 SECONDARY EDUCATION CERTIFICATE EXAMINATION ADDITIONAL MATHEMATICS Paper 02 Geeral Proficiecy 2 hours 40 miutes 03 MAY 2012 (p.m.) READ THE FOLLOWING INSTRUCTIONS CAREFULLY. 1. DO NOT ope this examiatio paper util istructed to do so. 2. This paper cosists of FOUR sectios. Aswer ALL questios i Sectio I, Sectio II ad Sectio III. 3. Aswer ONE questio i Sectio IV. 4. Write your solutios with full workig i the booklet provided. Required Examiatio Materials Electroic Calculator (o programmable) Geometry Set Mathematical Tables (provided) Graph Paper (provided) /F 2012 DO NOT TURN THIS PAGE UNTIL YOU ARE TOLD TO DO SO. Copyright 2011 Caribbea Examiatios Coucil All rights reserved.

5 - 2 - SECTION I Aswer BOTH questios. 1. (a) The fuctios f ad g are defied by f(x) = x 3 + 1, 0 < x < 3 g(x) = x + 5, x R where R is the set of real umbers. (i) Determie the compositio fuctio g(f(x)). (1 mark) (ii) State the rage of g(f(x)). (1 mark) (iii) Determie the iverse of g(f(x)). (2 marks) (b) If x + 2 is a factor of f(x) = 2x 3 3x 2 4x + a, fid the value of a. (2 marks) (c) Solve the equatio 3 2x 9(3 2x ) = 8. (5 marks) (d) (i) Express x 3 = 10 x 3 i the form log 10 x = ax + b. (2 marks) (ii) Hece, state the value of the gradiet of a graph of log 10 x versus x. (1 mark) Total 14 marks 2. (a) The quadratic equatio x 2 4x + 6 = 0 has roots α ad β. Calculate the value of α 2 + β 2. (5 marks) (b) Fid the rage of values of x for which 2x 5 > 0. (4 marks) 3x + 1 (c) A customer repays a loa mothly by icreasig the paymet each moth by $x. If the customer repaid $50 i the 5 th moth ad $70 i the 9 th moth, calculate the TOTAL amout of moey repaid at the ed of the 24 th moth. (5 marks) Total 14 marks /F 2012

6 - 3 - SECTION II Aswer BOTH questios. 3. (a) The equatio of a circle is give by x 2 + y 2 4x + 6y = 87. (i) A lie has equatio x + y + 1 = 0. Show that this lie passes through the cetre of the circle. (3 marks) (ii) Fid the equatio of the taget to the circle at the poit A ( 6, 3). (4 marks) (b) 1 Give OA = a, OB = b, AP = OA, 2 where a = 2 3 ad b = 1 2. (i) Write BP i terms of a ad b. (2 marks) (ii) Fid BP. (3 marks) Total 12 marks /F 2012

7 (a) The diagram shows a sector of a circle cetre O with a adjoiig square. The radius of the circle is 4 m. If the sector AOC subteds a agle at O, calculate, givig your aswer i terms of 3 π (i) the area of the shape OACMN (ii) the perimeter of the shape OACMN. (5 marks) (b) Give that si =, cos = ad si = cos =, evaluate without usig calculators, the exact value of cos. (3 marks) 12 (c) Prove the idetity 1 1 si θ. (4 marks) 1 cos θ cos θ + ta θ Total 12 marks /F 2012

8 - 5 - SECTION III Aswer BOTH questios. 5. (a) Differetiate the followig expressio with respect to x, simplifyig your aswer. 3x + 4 (4 marks) x 2 (b) The poit P (2, 10) lies o the curve y = 3x 2 + 5x 12. Fid equatios for (i) the taget to the curve at P (ii) the ormal to the curve at P. (5 marks) (c) The legth of the side of a square is icreasig at a rate of 4 cms 1. Fid the rate of icrease of the area whe the legth of the side is 5 cm. (5 marks) Total 14 marks (a) Evaluate (16 7x) 3 dx. (4 marks) (b) dy The poit Q (4, 8) lies o a curve for which = 3x 5. Determie the equatio of the dx curve. (3 marks) (c) Calculate the area betwee the curve y = 2 cos x + 3 si x ad the x-axis from x = 0 to. 3 (3 marks) (d) Calculate the volume of the solid formed whe the area eclosed by the curve y = x ad the x-axis, from x = 0 to x = 3, is rotated through 360 about the x-axis. [Leave your solutio i terms of ]. (4 marks) Total 14 marks /F 2012

9 - 6 - SECTION IV Aswer oly ONE questio. 7. (a) A survey carried out i a tow revealed that 25% of the households surveyed owed a laptop computer ad 70% owed a desktop computer. I additio, it was foud that 12% owed both a laptop ad a desktop computer. If a sample of households from the tow is selected at radom, determie the proportio that ow NEITHER a laptop NOR a desktop computer. (4 marks) (b) A bag cotais 4 red marbles, 3 black marbles ad 3 blue marbles. Three marbles are draw at radom without replacemet from the bag. Fid the probability that the marbles (i) draw are ALL of the SAME colour (3 marks) (ii) cotai EXACTLY 1 red marble. (3 marks) (c) The probability of hirig a taxi from garage A, B or C is 0.3, 0.5 ad 0.2 respectively. The probability that the taxi ordered will be late from A is 0.07, from B is 0.1 ad from C is 0.2. (i) (ii) Illustrate this iformatio o a tree diagram showig the probability o all braches. (3 marks) A garage is chose at radom, determie the probability that a) the taxi will arrive late (3 marks) b) the taxi will come from garage C give that it is late. (4 marks) Total 20 marks /F 2012

10 (a) A car startig from rest at a poit A, moves alog a straight lie reachig a velocity of 24 ms 1 by a costat acceleratio of 6 ms 2. The car maitais this costat velocity of 24 ms 1 for 5 secods ad is the brought to rest agai by a costat acceleratio of 3 ms 2. (i) Usig the graph sheet provided, draw a velocity-time graph to illustrate the motio of the car. (3 marks) (ii) Determie the TOTAL distace travelled by the car. (3 marks) (iii) A secod car, movig at a costat velocity of 32 ms 1 drives past poit A, 3 secods after the first car left poit A. Calculate the legth of time after the first car started that this secod car meets it. [Assume that the cars meet durig the time whe the first car is movig at a costat velocity.] (4 marks) (b) A particle moves i a straight lie with acceleratio give by a = (5t 1) ms 2 at ay time t secods. Whe t = 2 secods, the particle has velocity 4 ms 1 ad is 8 m from a fixed poit O. Determie (i) its velocity whe t = 4 (5 marks) (ii) its displacemet from O whe t = 3. (5 marks) Total 20 marks END OF TEST IF YOU FINISH BEFORE TIME IS CALLED, CHECK YOUR WORK ON THIS TEST /F 2012

11 TEST CODE FORM TP MAY/JUNE 2012 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 SECONDARY EDUCATION CERTIFICATE EXAMINATION ADDITIONAL MATHEMATICS Paper 02 Geeral Proficiecy Aswer Sheet for Questio 8. (a) (i) Cadidate Number... ATTACH THIS ANSWER SHEET TO YOUR ANSWER BOOKLET /F 2012

12 TEST CODE FORM TP MAY/JUNE 2012 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 SECONDARY EDUCATION CERTIFICATE EXAMINATION ADDITIONAL MATHEMATICS Paper 032 Geeral Proficiecy ALTERNATIVE TO SBA 90 miutes 12 JUNE 2012 (p.m.) Aswer all parts of the give questio /F 2012 DO NOT TURN THIS PAGE UNTIL YOU ARE TOLD TO DO SO. Copyright 2011 Caribbea Examiatios Coucil All rights reserved.

13 - 2 - (a) Two sports clubs, P ad Q, wish to use 600 m of fecig to eclose a court. They wish to determie which desig gives the maximum area. Sports Club P uses the 600 m of fecig to make a rectagular court measurig 3x m by 2y m as show i the diagram below. 2y m 3x m Sports Club Q uses the 600 m of fecig to make six equal-sized rectagular courts that are adjacet to each other as show i the diagram below. Each court measures x m by y m. y m y m x m x m x m For Sports Club P the mathematical problem is to maximize the area of eclosure to satisfy its perimeter ad the followig coditios: Maximize A = 6xy Subject to 6x + 4y = 600 (i) Formulate the mathematical problem for Sports Club Q. (2 marks) (ii) Determie the MAXIMUM area of the court for Sports Club Q. (3 marks) (iii) Show that Sports Club P has the maximum area whe a square eclosure is used ad determie the MAXIMUM possible area. (4 marks) (iv) Suggest which sports club desig should be used. (1 mark ) (b) The umbers log (a 3 b 7 ), log (a 5 b 12 ) ad log (a 8 b 15 ) are the first three terms of a arithmetic series. The 12 th term of the series is log b. Calculate the value of. (10 marks) Total 20 marks END OF TEST IF YOU FINISH BEFORE TIME IS CALLED, CHECK YOUR WORK ON THIS TEST /F 2012

14 TEST CODE FORM TP MAY/JUNE 2013 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 ADDITIONAL MATHEMATICS Paper 02 Geeral Proficiecy 2 hours 40 miutes 07 MAY 2013 (p.m.) READ THE FOLLOWING INSTRUCTIONS CAREFULLY. 1. DO NOT ope this examiatio paper util istructed to do so. 2. This paper cosists of FOUR sectios. Aswer ALL questios i Sectio 1, Sectio 2 ad Sectio Aswer ONE questio i Sectio Write your solutios with full workig i the booklet provided. 5. A list of formulae is provided o page 2 of this booklet. Required Examiatio Materials Electroic calculator (o programmable) Geometry Set Mathematical Tables (provided) Graph paper (provided) DO NOT TURN THIS PAGE UNTIL YOU ARE TOLD TO DO SO /F 2013 Copyright 2011 Caribbea Examiatios Coucil All rights reserved.

15 - 2 - LIST OF FORMULAE /F 2013

16 - 3 - SECTION 1 Aswer BOTH questios. All workig must be clearly show. 1. (a) Let f(x) = x 3 x 2 14x (i) Use the factor theorem to show that x + 4 is a factor of f(x). (2 marks) (ii) Determie the other liear factors of f(x). (3 marks) (b) A fuctio f(x) is give by f(x) = 2x 1 x + 2. (i) Fid a expressio for the iverse fuctio f 1 (x). (3 marks) (ii) The fuctio g is give by g(x) = x +1. Write a expressio for the composite fuctio, fg(x). Simplify your aswer. (2 marks) (c) Give that 5 3x 2 = 7 x + 2, show that 2(log5 + log7) x =. (4 marks) (log125 log7) Total 14 marks 2. (a) Let f(x) = 3x 2 + 6x 1. (i) Express f(x) i the form a(x + h) 2 + k where h ad k are costats. (3 marks) (ii) State the miimum value of f(x). (1 mark) (iii) Determie the value of x for which f(x) is a miimum. (1 mark) (b) Fid the set of values of x for which 2x 2 + 3x 5 0. (4 marks) (c) Fid the sum to ifiity of the followig series: Note: This series ca be rewritte as the sum of two geometric series. (5 marks) Total 14 marks /F 2013

17 - 4 - SECTION 2 Aswer BOTH questios. All workig must be clearly show. 3. (a) (i) A circle, C, has cetre with coordiates A (2,1) ad passes through the poit B (10,7). Express the equatio of the circle i the form x 2 + y 2 + hx + gy + k = 0, where h, g ad k are itegers to be determied. (3 marks) (ii) The lie l is a taget to the circle C at the poit B. Fid a equatio for l. (3 marks) (b) The positio vectors of two poits, P ad Q, relative to a fixed origi, O, are 10i 8j ad λi + 10j respectively, where λ is a costat. Fid the value of λ such that OP ad OQ are perpedicular. (3 marks) (c) The positio vectors of A ad B with respect to a fixed origi, O, are give by OA = 2i +5j ad OB = 3i 7j respectively. Fid the uit vector i the directio of AB. (3 marks) Total 12 marks 4. (a) π The diagram shows a sector cut from a circle of cetre O. The agle at O is. If the 6 perimeter of the sector is 5 (12 + π) cm, what is its area? (4 marks) 6 (b) Solve the equatio 2 cos 2 θ + 3 si θ = 0 for 0 θ 360. (5 marks) (c) 1 Give that ta (θ α) = ad that ta θ = 3, use the appropriate compoud agle 2 formula to fid the value of the acute agle α. (3 marks) Total 12 marks /F 2013

18 - 5 - SECTION 3 Aswer BOTH questios. All workig must be clearly show. 5. (a) Give that y = x 3 3x Fid (i) the coordiates of the statioary poits of y (5 marks) (ii) the secod derivative of y ad hece determie the ature of EACH of the statioary poits. (5 marks) (b) Differetiate y = (5x + 3) 3 si x with respect to x, simplifyig your result as far as possible. (4 marks) Total 14 marks 6. (a) Fid (5x 2 +4) dx. (2 marks) π (b) Evaluate 2 ((3 3six x 5 cos 5 cos x)dxx) dx. (4 marks) 0 dy (c) A curve passes through the poits P(0, 8) ad Q(4, 0) ad is such that = 2 2x. dx Fid the area of the fiite regio bouded by the curve i the first quadrat. (8 marks) Total 14 marks /F 2013

19 - 6 - SECTION 4 Aswer ONLY ONE questio. All workig must be clearly show. 7. (a) Of the persos buyig petrol at a service statio, 40 per cet are females. Of the females, 30 per cet pay for their petrol with cash, ad of the males, 65 per cet pay for their petrol with cash. (i) Copy ad complete the followig tree diagram, by puttig i all the missig probabilities, to show this iformatio. (2 marks) (ii) (iii) What is the probability that a customer pays for petrol with cash? Determie which is the more likely evet: (3 marks) Evet T: Customer is female, GIVEN that the petrol is paid WITH cash. Evet V: A male customer does NOT pay for petrol with cash. (4 marks) (b) The marks obtaied by 30 studets o a Eglish exam are give as (i) State ONE advatage of usig a stem ad leaf diagram versus a box ad whiskers plot to display the data. (1 mark) (ii) Costruct a stem ad leaf diagram to show the data. (3 marks) (iii) Determie the media mark. (2 marks) (iv) Calculate the semi iter-quartile rage of the marks. (3 marks) (v) Two studets are chose at radom from the class. Determie the probability that both scored less tha 50 o the exam. (2 marks) Total 20 marks /F 2013

20 (a) A particle starts from rest ad accelerates uiformly to 20 m s 1 i 5 secods. It cotiues at this velocity for 10 secods. It the accelerates agai uiformly to a velocity of 60 m s 1 i 5 secods. The particle the decelerates uiformly util it comes to rest, 15 secods later. (i) (ii) O the graph paper provided, draw a velocity-time graph to illustrate the motio of the particle. (3 marks) From your graph determie a) the total distace, i metres, travelled by the particle (4 marks) b) the average velocity of the particle for the etire jourey. (2 marks) (b) A particle travels i a straight lie i such a way that after t secods its velocity, v, from a fixed poit, O, is give by the fuctio v = 3t 2 18t Calculate (i) the values of t whe the particle is istataeously at rest (3 marks) (ii) (iii) the distace travelled by the particle betwee 1 secod ad 3 secods (3 marks) the value of dv whe dt a) t = 2 secods (2 marks) b) t = 3 secods. (1 mark) (iv) Give a iterpretatio for the value i a) 8 (b) (iii) a) (1 mark) b) 8 (b) (iii) b). (1 mark) Total 20 marks END OF TEST IF YOU FINISH BEFORE TIME IS CALLED, CHECK YOUR WORK ON THIS TEST /F 2013

21 TEST CODE FORM TP MAY/JUNE 2013 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 ADDITIONAL MATHEMATICS Paper 02 Geeral Proficiecy Aswer Sheet for Questio 8 (a) Cadidate Number... ATTACH THIS ANSWER SHEET TO YOUR ANSWER BOOKLET /F 2013

22 TEST CODE FORM TP MAY/JUNE 2013 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 ADDITIONAL MATHEMATICS Paper 032 Geeral Proficiecy Alterative Paper 1 hour 30 miutes 12 JUNE 2013 (p.m.) Aswer the give questios. DO NOT TURN THIS PAGE UNTIL YOU ARE TOLD TO DO SO /F 2013 Copyright 2011 Caribbea Examiatios Coucil All rights reserved.

23 - 2 - Aswer the give questios. 1. (a) A circle is draw with cetre at origi, O, ad radius 6 cm. Fid the coordiates of all itersectios of the circle with a origi cetred square of side legth 10 cm whose sides are parallel to the coordiate axes as illustrated i Figure 1. O Figure 1 (10 marks) (b) The cuboid show i Figure 2 has width x m ad its legth is twice its width. The volume of the cuboid is 720 m 3. x Figure 2 (i) Fid a expressio for the height, h, of the cuboid i terms of x. (3 marks) (ii) Show that a expressio for the surface area, A, of the cuboid is give by A = 2160 x + 4x 2. (3 marks) (iii) 3 Hece show that A has a statioary value whe x = (4 marks) Total 20 marks END OF TEST IF YOU FINISH BEFORE TIME IS CALLED, CHECK YOUR WORK ON THIS TEST /F 2013

24 TEST CODE FORM TP MAY/JUNE 2014 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 ADDITIONAL MATHEMATICS Paper 02 Geeral Proficiecy 2 hours 40 miutes 06 MAY 2014 (p.m.) READ THE FOLLOWING INSTRUCTIONS CAREFULLY. 1. This paper cosists of FOUR sectios. Aswer ALL questios i Sectio 1, Sectio 2 ad Sectio Aswer ONE questio i Sectio Write your solutios with full workig i the booklet provided. 4. A list of formulae is provided o page 2 of this booklet. Required Examiatio Materials Electroic Calculator (o programmable) Geometry Set Mathematical Tables (provided) Graph Paper (provided) DO NOT TURN THIS PAGE UNTIL YOU ARE TOLD TO DO SO /F 2014 Copyright 2011 Caribbea Examiatios Coucil All rights reserved.

25 - 2 - LIST OF FORMULAE Arithmetic Series T = a + ( 1)d S = [2a + ( 1)d] 2 a(r 1) a Geometric Series T = ar 1 S = r 1 S =, 1 < r < 1 or r < 1 1 r Circle x 2 + y 2 + 2fx + 2gy + c = 0 (x + f) 2 + (y + g) 2 = r 2 Vectors v ^ v = cos θ = a b v = (x 2 + y 2 ) where v = xi + yj v a b Trigoometry si (A + B) si A cos B + cos A si B cos (A + B) cos A cos B si A si B ta A + ta B ta (A + B) 1 ta A ta B + + d Differetiatio (ax + b) = a(ax + b) dx 1 d si x = cos x dx d cos x = si x dx Σ x i Σ f Statistics i x i Σ (x i = 1 i = 1 x = =, S 2 i = 1 i x) 2 2 Σ f i x i i = 1 = = (x) 2 Σ f i Σ f i i = 1 i = 1 Probability P(A B) = P(A) + P(B) P(A B) 1 Kiematics v = u + at v 2 = u 2 + 2as s = ut + at /F 2014

26 - 3 - SECTION 1 Aswer BOTH questios. ALL workig must be clearly show. 1. (a) (i) The fuctio f is defied by f : x 1 x 2, x. Show that f is NOT oe-to-oe. 1 (ii) The fuctio g is defied by g : x x 3, x. 2 (1 mark) a) Fid fg(x), ad clearly state its domai. (2 marks) b) Determie the iverse, g 1, of g ad sketch o the same pair of axes, the graphs of g ad g 1. (3 marks) (b) Whe the expressio 2x 3 + ax 2 5x 2 is divided by 2x 1, the remaider is 3.5. Determie the value of the costat a. (3 marks) (c) The legth of a regtagular kitche is y m ad the width is x m. If the legth of the kitche is half the square of its width ad its perimeter is 48 m, fid the values of x ad y (the dimesios of the kitche). (5 marks) Total 14 marks 2. (a) Give that f(x) = 2x 2 12x 9. (i) Express f(x) i the form k + a (x + h) 2, where a, h ad k are itegers to be determied. (3 marks) (ii) State the maximum value of f(x). (1 mark) (iii) Determie the value of x for which f(x) is a maximum. (1 mark) (b) Fid the set of values of x for which 3 + 5x 2x 2 < 0. (4 marks) (c) A series is give by (i) Show that this series is geometric. (3 marks) (ii) Fid the sum to ifiity of this series, givig your aswer as a exact fractio. (2 marks) Total 14 marks /F 2014

27 - 4 - SECTION 2 Aswer BOTH questios. ALL workig must be clearly show. 3. (a) (i) Determie the value of k such that the lies x + 3y = 6 ad kx + 2y = 12 are perpedicular to each other. (3 marks) (ii) A circle of radius 5 cm has as its cetre the poit of itersectio of the two perpedicular lies i (i). Determie the equatio for this circle. (3 marks) (b) RST is a triagle i the coordiate plae. Positio vectors R, S, ad T relative to a origi, O, are 1 3 4, ad respectively. (i) Show that TRS ^ = 90. (4 marks) (ii) Determie the legth of the hypoteuse. (2 marks) [Hit: A rough drawig of RST might help]. Total 12 marks /F 2014

28 (a) Figure 1 shows the sector OAB of a circle with cetre O, radius 9 cm ad agle 0.7 radias. A 9 cm O 0.7 rad B H C Figure 1. (i) Fid the area of the sector OAB. (2 marks) (ii) Hece, fid the area of the shaded regio, H. (4 marks) (b) Give that si = 1 ad cos =, 3 show that cos x + = 1 ( 3 cos x si x), where x is acute. 6 2 (2 marks) (c) ta θ si θ 1 Prove the idetity cos θ cos θ (4 marks) Total 12 marks /F 2014

29 - 6 - SECTION 3 Aswer BOTH questios. ALL workig must be clearly show. 5. (a) The equatio of a curve is y = 3 + 4x x 2. The poit P (3, 6) lies o the curve. Fid the equatio of the taget to the curve at P, givig your aswer i the form ax + by + c = 0, where a, b, c,. (4 marks) (b) Give that f(x) = 2x 3 9x 2 24x + 7. (i) Fid ALL the statioary poits of f(x). (5 marks) (ii) Determie the ature of EACH of the statioary poits of f(x). (5 marks) Total 14 marks (a) Evaluate x (x 2 2) dx. (4 marks) (b) Evaluate 3 (4 cos x + 2 si x) dx, leavig your aswer i surd form. (4 marks) 0 dy (c) A curve passes through the poit P (2, 5) ad is such that = 6x 2 1. dx (i) Determie the equatio of the curve. (3 marks) (ii) Fid the area of the fiite regio bouded by the curve, the x-axis, the lie x = 3 ad the lie x = 4. (3 marks) Total 14 marks /F 2014

30 - 7 - SECTION 4 Aswer ONLY ONE questio. ALL workig must be clearly show. 7. (a) There are 60 studets i the sixth form of a certai school. Mathematics is studied by 27 of them, Biology by 20 of them ad 22 studets study either Mathematics or Biology. If a studet is selected at radom, what is the probability that the studet is studyig (i) both Mathematics ad Biology? (3 marks) (ii) Biology oly? (2 marks) (b) Two ordiary six-sided dice are throw together. The radom variable S represets the sum of the scores of their faces ladig uppermost. (i) Copy ad complete the sample space diagram below Sample space diagram of S (1 mark) (ii) Fid a) P (S > 9) (2 marks) b) P (S < 4). (1 mark) /F 2014

31 - 8 - (iii) Let D be the differece betwee the scores of the faces ladig uppermost. The table below gives the probability of each possible value of d. d P (D = d) 1 a 2 b 1 c Fid the values of a, b ad c. (3 marks) (c) The aptitude scores obtaied by 51 applicats for a supervisory job are summarized i the followig stem ad leaf diagram. 5 1 meas (i) Fid the media ad quartiles for the data give. (4 marks) (ii) Costruct a box-ad-whisker plot to illustrate the data give ad commet o the distributio of the data. (4 marks) Total 20 marks /F 2014

32 (a) Figure 2 below, ot draw to scale, shows the motio of a car with velocity, V, as it moves alog a straight road from Poit A to Poit B. The time, t, take to travel from Poit A to Poit B is 90 secods ad the distace from Poit A to Poit B is 1410 m. Figure 2. (i) What distace did the car travel from Poit A towards Poit B before startig to decelerate? (2 marks) (ii) Calculate the deceleratio of the car as it goes from 25 m s 1 to 10 m s 1. (5 marks) (iii) For how log did the car maitai the speed of 10 m s 1? (1 mark) (iv) From Poit B, the car decelerates uiformly, comig to rest at a Poit C ad coverig a further distace of 30 m. Determie the average velocity of the car over the jourey from Poit A to Poit C. (2 marks) /F 2014

33 (b) A particle travels alog a straight lie. It starts from rest at a poit, P, o the lie ad after 10 secods, it comes to rest at aother poit, Q, o the lie. The velocity v m s 1 at time t secods after leavig P is v = 0.72t t 3 for 0 < t < 5 v = 2.4t 0.24t 2 for 5 < t < 10 At maximum velocity the particle has o acceleratio. (i) Fid the time whe the velocity is at its maximum. (3 marks) (ii) Determie the maximum velocity. (2 marks) (iii) Fid the distace moved by the particle from P to the poit where the particle attais its maximum velocity. (5 marks) Total 20 marks END OF TEST IF YOU FINISH BEFORE TIME IS CALLED, CHECK YOUR WORK ON THIS TEST /F 2014

34 TEST CODE FORM TP MAY/JUNE 2014 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 ADDITIONAL MATHEMATICS Paper 032 Geeral Proficiecy ALTERNATIVE 1 hour 30 miutes 09 JUNE 2014 (p.m.) READ THE FOLLOWING INSTRUCTIONS CAREFULLY. 1. This paper cosists of ONE questio. Aswer the give questio. 2. Write your solutios with full workig i the booklet provided. 3. A list of formulae is provided o page 2 of this booklet. DO NOT TURN THIS PAGE UNTIL YOU ARE TOLD TO DO SO /F 2014 Copyright 2011 Caribbea Examiatios Coucil All rights reserved.

35 - 2 - LIST OF FORMULAE Arithmetic Series T = a + ( 1)d S = [2a + ( 1)d] 2 a(r 1) a Geometric Series T = ar 1 S = r 1 S =, 1 < r < 1 or r < 1 1 r Circle x 2 + y 2 + 2fx + 2gy + c = 0 (x + f) 2 + (y + g) 2 = r 2 Vectors v ^ v = cos θ = a b v = (x 2 + y 2 ) where v = xi + yj v a b Trigoometry si (A + B) si A cos B + cos A si B cos (A + B) cos A cos B si A si B ta A + ta B ta (A + B) 1 ta A ta B + + d Differetiatio (ax + b) = a(ax + b) dx 1 d si x = cos x dx d cos x = si x dx Σ x i Σ f Statistics i x i Σ (x i = 1 i = 1 x = =, S 2 i = 1 i x) 2 2 Σ f i x i i = 1 = = (x) 2 Σ f i Σ f i i = 1 i = 1 Probability P(A B) = P(A) + P(B) P(A B) 1 Kiematics v = u + at v 2 = u 2 + 2as s = ut + at /F 2014

36 (a) A studet has to compute the area uder the graph of a fuctio. He reasos that he ca do so by subdividig the area ito a ifiitely large umber of rectagles. To help himself, he ivestigates by fidig the area uder the graph of the fuctio f(x) = x over the iterval [0,1], usig the method of circumscribed rectagles as show i Figure 1. f(x) f(x) = x 0 x 1 x 2 x 3... x -1 1 x Figure 1. Circumscribed Rectagles (i) (ii) The studet subdivides the iterval [0, 1] ito equal subitervals. Calculate the width, x, of each subiterval. (1 mark) Let the poits of subdivisio be x 0 = 0, x 1, x 2, x 3,..., x 1, x = 1 as show i Figure 1. Fid the values of x 1, x 2, x 3,..., x 1 i terms of. (1 mark) (iii) (iv) (v) Determie the heights h 1, h 2, h 3,..., h of the circumscribed rectagles over each of the respective subitervals. (2 marks) Determie the area A 1, A 2, A 3,..., A of the respective circumscribed rectagles. (2 marks) Show that the sum, S, of the areas of these circumscribed rectagles is give by + 1 S =. 2 (3 marks) (Hit: You will eed to evaluate the sum of a series. State ay theorem used.) (vi) a) Compute S() for = 10, 20, 50 ad 100, givig your aswers to three decimal places. (2 marks) b) What umber does S() approach as gets larger? (1 mark) /F 2014

37 - 4 - (b) The variables x ad y are related by a law of the form y = ax, where a ad are itegers. The approximate values for y, correspodig to the give values of x are show i Table 1. Table 1 x y (i) (ii) (iii) Use logarithms to reduce this relatio to a liear form, givig your values of lg x ad lg y correct to two decimal places where appropriate. (2 marks) Usig the graph paper provided ad a scale of 2 cm to represet 0.1 uits o the x-axis, ad 1 cm to represet 0.2 uits o the y-axis, plot a suitable straight lie graph of lg y agaist lg x. (2 marks) Use your straight lie graph to estimate the value of the costat a ad the value of the costat. (4 marks) Total 20 marks END OF TEST IF YOU FINISH BEFORE TIME IS CALLED, CHECK YOUR WORK ON THIS TEST /F 2014

38 TEST CODE FORM TP MAY/JUNE 2014 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 ADDITIONAL MATHEMATICS Paper 032 Geeral Proficiecy ALTERNATIVE Graph Sheet for Questio 1 (b) (ii) Cadidate Number /F 2014 ATTACH THIS ANSWER SHEET TO YOUR ANSWER BOOKLET

39 TEST CODE FORM TP MAY/JUNE 2015 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 ADDITIONAL MATHEMATICS Paper 02 Geeral Proficiecy 2 hours 40 miutes 05 MAY 2015 (p.m.) READ THE FOLLOWING INSTRUCTIONS CAREFULLY. 1. This paper cosists of FOUR sectios. Aswer ALL questios i Sectio I, Sectio II ad Sectio III. 2. Aswer ONE questio i Sectio IV. 3. Write your solutios with full workig i the booklet provided. 4. A list of formulae is provided o page 2 of this booklet. Required Examiatio Materials Electroic Calculator (o-programmable) Geometry Set Mathematical Tables (provided) Graph Paper (provided) DO NOT TURN THIS PAGE UNTIL YOU ARE TOLD TO DO SO /F 2015 Copyright 2013 Caribbea Examiatios Coucil All rights reserved.

40 - 2 - LIST OF FORMULAE Arithmetic Series T = a + ( 1)d S = [2a + ( 1)d] 2 Geometric Series T = ar 1 a(r 1) a S = r 1 S =, 1 < r < 1 or r < 1 1 r Circle x 2 + y 2 + 2fx + 2gy + c = 0 (x + f) 2 + (y + g) 2 = r 2 Vectors v ^ v = cos θ = a b v = (x 2 + y 2 ) where v = xi + yj v a b Trigoometry si (A + B) si A cos B + cos A si B cos (A + B) cos A cos B si A si B ta A + ta B ta (A + B) 1 ta A ta B + + d Differetiatio (ax + b) = a(ax + b) 1 dx d si x = cos x dx d cos x = si x dx Σ x i Σ f Statistics i x i Σ (x i = 1 i = 1 x = =, S 2 i = 1 i x) 2 2 Σ f i x i i = 1 = = (x) 2 Σ f i Σ f i i = 1 i = 1 Probability P(A B) = P(A) + P(B) P(A B) Kiematics v = u + at v 2 = u as s = ut + at /F 2015

41 - 3 - SECTION I Aswer BOTH questios. ALL workig must be clearly show. 1. (a) The fuctios f ad g are defied by f (x) = x 2 + 5, x > 1 g (x) = 4x 3, x R where R is the set of real umbers. Fid the value of g (f (2)). (2 marks) (b) The fuctio h is defied by h(x) = 3x + 5 x 2 where x R, x 2. Determie the iverse of h(x). (3 marks) (c) (d) Give that x 2 is a factor of k(x) = 2x 3 5x 2 + x + 2, factorize k(x) completely. (3 marks) Solve the followig equatios: (i) 16 x+2 = 1 4 (2 marks) (ii) log 3 (x + 2) + log 3 (x 1) = log 3 (6x 8) (4 marks) Total 14 marks /F 2015

42 (a) Give that f (x) = 3x 2 9x + 4: (i) Express f (x) i the form a (x + b) 2 + c, where a, b ad c are real umbers. (3 marks) (ii) State the coordiates of the miimum poit of f (x). (1 mark) (b) The equatio 3x 2 6x 4 = 0 has roots α ad β. Fid the value of 1 α + 1 β. (4 marks) (c) Determie the coordiates of the poits of itersectio of the curve 2x 2 y + 19 = 0 ad the lie y + 11x = 4. (3 marks) (d) A employee of a compay is offered a aual startig salary of $ which icreases by $2 400 per aum. Determie the aual salary that the employee should receive i the ith year. (3 marks) Total 14 marks /F 2015

43 - 5 - SECTION II Aswer BOTH questios. ALL workig must be clearly show. 3. (a) The equatio of a circle is give by x 2 + y 2 12x 22y = 0. (i) Determie the coordiates of the cetre of the circle. (2 marks) (ii) Fid the legth of the radius. (1 mark) (iii) Determie the equatio of the ormal to the circle at the poit (4, 10). (3 marks) (b) The positio vectors of two poits, A ad B, relative to a origi O, are such that OA = 3i 2j ad OB = 5i 7j. Determie (i) the uit vector AB (3 marks) (ii) the acute agle AOB, i degrees, to oe decimal place. (3 marks) Total 12 marks /F 2015

44 (a) The followig diagram shows a circle of radius r = 4 cm, with cetre O ad sector AOB which subteds a agle, θ = π 6 radias at the cetre. If the area of the triagle AOB = 1 2 r2 si θ, the calculate the area of the shaded regio. (4 marks) (b) Solve the followig equatio, givig your aswer correct to oe decimal place. 8 si 2 θ = 5 10 cos θ, where 0 o < θ < 360 o (4 marks) (c) Prove the idetity si θ + si 2θ 1 + cos θ + cos 2θ ta θ. (4 marks) Total 12 marks /F 2015

45 - 7 - SECTION III Aswer BOTH questios. ALL workig must be clearly show. 5. (a) Differetiate the followig expressio with respect to x, simplifyig your aswer. (2x 2 + 3) si 5x (4 marks) (b) (i) Fid the coordiates of all the statioary poits of the curve y = x 3 5x 2 + 3x + 1. (3 marks) (ii) Determie the ature of EACH poit i (i) above. (2 marks) (c) A spherical balloo of volume V = 4 3 π r 3 is beig filled with air at the rate of 200 cm 3 s 1. Calculate, i terms of π, the rate at which the radius is icreasig whe the radius of the balloo is 10 cm. (5 marks) Total 14 marks /F 2015

46 π (a) Evaluate 3 cos θ dθ. (4 marks) π 6 (b) A curve has a equatio which satisfies dy dx = kx(x 1) where k is a costat. Give that the value of the gradiet of the curve at the poit (2, 3) is 14, determie (i) the value of k (2 marks) (ii) the equatio of the curve. (4 marks) (c) Calculate, i terms of π, the volume of the solid formed whe the area eclosed by the curve y = x ad the x-axis, from x = 0 to x = 1, is rotated through 360 o about the x-axis. (4 marks) Total 14 marks /F 2015

47 - 9 - SECTION IV Aswer oly ONE questio. ALL workig must be clearly show. 7. (a) There are three traffic lights that a motorist must pass o the way to work. The probability that the motorist has to stop at the first traffic light is 0.2, ad that for the secod ad third traffic lights are 0.5 ad 0.8 respectively. Fid the probability that the motorist has to stop at (i) ONLY ONE oe traffic light (4 marks) (ii) AT LEAST TWO traffic lights. (4 marks) (b) Use the data i the followig table to estimate the mea of x. x f (4 marks) (c) Research i a tow shows that if it rais o ay oe day the the probability that it will rai the followig day is 25%. If it does ot rai oe day the the probability that it will rai the followig day is 12%. Startig o a Moday ad give that it rais o that Moday: (i) (ii) Draw a probability tree diagram to illustrate the iformatio, ad show the probability o ALL of the braches. (4 marks) Determie the probability that it will rai o the Wedesday of that week. (4 marks) Total 20 marks /F 2015

48 (a) 1 A particle movig i a straight lie has a velocity of 3 m s at t = 0 ad 4 secods later 1 its velocity is 9 m s. (i) O the aswer graph sheet, provided as a isert, draw a velocity time graph to represet the motio of the particle. (3 marks) (ii) Calculate the acceleratio of the particle. (3 marks) (iii) Determie the icrease i displacemet over the iterval t = 0 to t = 4. (4 marks) (b) A particle moves i a straight lie so that t secods after passig through a fixed poit 2 O, its acceleratio, a, is give by a = (3t 1)m s. The particle has a velocity, v, of 1 4 m s whe t = 2 ad its displacemet, s, from O is 6 metres whe t = 2. Fid (i) the velocity whe t = 4 (5 marks) (ii) the displacemet of the particle from O whe t = 3. (5 marks) Total 20 marks END OF TEST IF YOU FINISH BEFORE TIME IS CALLED, CHECK YOUR WORK ON THIS TEST /F 2015

49 TEST CODE FORM TP MAY/JUNE 2015 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 ADDITIONAL MATHEMATICS Paper 032 Geeral Proficiecy 5038 ALTERNATIVE 90 miutes 08 JUNE 2015 (p.m.) READ THE FOLLOWING INSTRUCTIONS CAREFULLY. 1. This paper cosists of ONE questio. Aswer the give questio. 2. Write your solutios with full workig i the booklet provided. 3. A list of formulae is provided o page 2 of this booklet DO NOT TURN THIS PAGE UNTIL YOU ARE TOLD TO DO SO. Copyright 2013 Caribbea Examiatios Coucil All rights reserved /F 2015

50 - 2 - LIST OF FORMULAE Arithmetic Series T = a + ( 1)d S = [2a + ( 1)d] 2 a(r 1) a Geometric Series T = ar 1 S = r 1 S =, 1 < r < 1 or r < 1 1 r Circle x 2 + y 2 + 2fx + 2gy + c = 0 (x + f) 2 + (y + g) 2 = r 2 Vectors v ^ v = cos θ = a b v = (x 2 + y 2 ) where v = xi + yj v a b Trigoometry si (A + B) si A cos B + cos A si B cos (A + B) cos A cos B si A si B ta A + ta B ta (A + B) 1 ta A ta B + + d Differetiatio (ax + b) = a(ax + b) dx 1 d si x = cos x dx d cos x = si x dx Σ x i Σ f Statistics i x i Σ (x i = 1 i = 1 x = =, S 2 i = 1 i x) 2 2 Σ f i x i i = 1 = = (x) 2 Σ f i Σ f i i = 1 i = 1 Probability P(A B) = P(A) + P(B) P(A B) 1 Kiematics v = u + at v 2 = u 2 + 2as s = ut + at /F 2015

51 - 3 - Aswer this questio. 1. (a) The daily profit, f (x), i hudreds of dollars, for a compay that maufactures x radios daily is give by f (x) = x x 300. Determie the maximum daily profit of the compay. (6 marks) (b) (c) The combied area of a square ad a rectagle is 128 cm 2. The width of the rectagle is 3 cm more tha the legth of a side of the square, ad the legth of the rectagle is 3 cm more tha its width. Detemie the dimesios of the square ad the rectagle. (5 marks) A maufacturer of taks makes a tak, with a rectagular base, as show i the diagram below, where h is the height of the isosceles triagle ad x the width of the rectagular base. x 4 4 h 4 (i) Obtai a expressio for the area A of the shaded sectio (top surface) as a fuctio of the distace x betwee the parallel sides. (5 marks) (ii) Hece, fid the domai of A (that is, the rage of values of x ). (4 marks) Total 20 marks END OF TEST IF YOU FINISH BEFORE TIME IS CALLED, CHECK YOUR WORK ON THIS TEST /F 2015

52 TEST CODE FORM TP MAY/JUNE 2016 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 ADDITIONAL MATHEMATICS Paper 02 Geeral Proficiecy 2 hours 40 miutes READ THE FOLLOWING INSTRUCTIONS CAREFULLY. 1. This paper cosists of FOUR sectios. Aswer ALL questios i Sectio I, Sectio II ad Sectio III. 2. Aswer ONE questio i Sectio IV. 3. Write your aswers i the spaces provided i this booklet. 4. Do NOT write i the margis. 5. A list of formulae is provided o page 4 of this booklet. 6. If you eed to rewrite ay aswer ad there is ot eough space to do so o the origial page, you must use the extra page(s) provided at the back of this booklet. Remember to draw a lie through your origial aswer. 7. If you use the extra page(s) you MUST write the questio umber clearly i the box provided at the top of the extra page(s) ad, where relevat, iclude the questio part beside the aswer. Required Examiatio Materials Electroic calculator (o programmable) Geometry set Mathematical tables (provided) Graph paper (provided) DO NOT TURN THIS PAGE UNTIL YOU ARE TOLD TO DO SO /F 2016 Copyright 2013 Caribbea Examiatios Coucil All rights reserved. * Barcode Area * Sequetial Bar Code

53 - 4 - LIST OF FORMULAE Arithmetic Series T = a + ( 1)d S = [2a + ( 1)d] 2 a(r 1) a Geometric Series T = ar 1 S = r 1 S =, 1 < r < 1 or r < 1 1 r Circle x 2 + y 2 + 2fx + 2gy + c = 0 (x + f) 2 + (y + g) 2 = r 2 Vectors v ^ v = cos θ = a b v = (x 2 + y 2 ) where v = xi + yj v a b Trigoometry /F 2016 si (A + B) si A cos B + cos A si B cos (A + B) cos A cos B si A si B ta A + ta B ta (A + B) 1 ta A ta B d Differetiatio (ax + b) = a(ax + b) dx 1 d si x = cos x dx d cos x = si x dx Σ x i Σ f Statistics i x i Σ (x i = 1 i = 1 x = =, S 2 i = 1 i x) 2 2 Σ f i x i i = 1 = = (x) 2 Σ f i Σ f i Probability P(A B) = P(A) + P(B) P(A B) 1 Kiematics v = u + at v 2 = u 2 + 2as s = ut + at i = 1 + * Barcode Area * Sequetial Bar Code i = 1 DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA

54 - 5 - DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA /F 2016 SECTION I Aswer BOTH questios. ALL workig must be clearly show. 1. (a) The domai for the fuctio f (x) = 2x 5 is { 2, 1, 0, 1}. (i) (ii) Determie the rage of the fuctio. (2 marks) Fid f 1 (x). (1 mark) * Barcode Area * Sequetial Bar Code

55 - 6 - (iii) (iv) /F 2016 Sketch the graphs of f (x) ad f 1 (x) o the same axes. Commet o the relatioship betwee the two graphs. * Barcode Area * Sequetial Bar Code (2 marks) (1 mark) (b) Solve the equatio 2 2x (2 x ) 3 = 0. (4 marks) DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA

56 - 7 - DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA (c) (i) Give that T = k p, make c the subject of the formula. (ii) /F 2016 (2 marks) Solve the equatio h c log (x + 1) + log (x 1) = 2 log (x + 2). (2 marks) * Barcode Area * Sequetial Bar Code Total 14 marks

57 (a) (i) Determie the ature of the roots of the quadratic equatio 2x 2 + 3x 9 = 0. (ii) /F 2016 (1 mark) Give that f (x) = 2x 2 + 3x 9, sketch the graph of the quadratic fuctio, clearly idicatig the miimum value. * Barcode Area * Sequetial Bar Code (5 marks) DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA

58 - 9 - DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA (b) Evaluate 3. (c) /F (3 marks) A ma ivested $x i a compay i Jauary 2010, o which he ears quarterly divideds. At the ed of the secod, third ad fourth quarter i 2011, he eared $100, $115 ad $130 respectively. Calculate the total divideds o his ivestmet by the ed of (5 marks) * Barcode Area * Sequetial Bar Code Total 14 marks

59 /F 2016 SECTION II Aswer BOTH questios. ALL workig must be clearly show. 3. (a) (i) The poits M (3, 2) ad N ( 1, 4) are the eds of a diameter of circle C. Determie the equatio of circle C. (5 marks) (ii) Fid the equatio of the taget to the circle C at the poit P ( 1, 6). (3 marks) * Barcode Area * Sequetial Bar Code DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA

60 DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA (b) /F 2016 The positio vector of two poits A ad B, relative to a fixed origi, O, are a ad b respectively, where a =, ad b =. P lies o AB such that PB = AB Fid the coordiates of OP. (4 marks) * Barcode Area * Sequetial Bar Code Total 12 marks

61 (a) The followig diagram (ot draw to scale) shows two sectors, AOB ad DOC. OB ad OC are x cm ad (x + 2) cm respectively ad agle AOB = θ. (b) /F π If θ = radias, calculate the area of the shaded regio i terms of x. 9 (4 marks) 3 2 Give that cos 30 = ad si 45 =, without the use of a calculator, evaluate 2 2 cos 105, i surd form, givig your aswer i the simplest terms. (5 marks) * Barcode Area * Sequetial Bar Code DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA

62 DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA si (θ + α) (c) Prove that the idetity ta θ + ta α. cos θ cos α /F 2016 (3 marks) * Barcode Area * Sequetial Bar Code Total 12 marks

63 /F 2016 SECTION III Aswer BOTH questios. ALL workig must be clearly show. dy 5. (a) Fid give that y = 5x 2 4, simplifyig your aswer. dx (b) (4 marks) The poit P (1, 8) lies o the curve with equatio y = 2x (x + 1) 2. Determie the equatio of the ormal to the curve at the poit P. (5 marks) * Barcode Area * Sequetial Bar Code DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA

64 DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA (c) /F 2016 Obtai the equatio for EACH of the two tagets draw to the curve y = x 2 at the poits where y = 16. (5 marks) * Barcode Area * Sequetial Bar Code Total 14 marks

65 (a) (i) Fid (3 cos θ 5 si θ) dθ /F 2016 (3 marks) (ii) Evaluate x 3 dx. x 2 (4 marks) * Barcode Area * Sequetial Bar Code DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA

66 DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA (b) /F 2016 The followig figure shows the fiite regio R bouded by the lies x = 1, x = 2 ad the arc of the curve y = (x 2) Calculate the area of the regio R. (4 marks) * Barcode Area * Sequetial Bar Code

67 dy (c) The poit P (1, 2) lies o the curve which has a gradiet fuctio give by = 3x 2 6x. dx Fid the equatio of the curve /F 2016 (3 marks) * Barcode Area * Sequetial Bar Code Total 14 marks DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA

68 DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA /F 2016 SECTION IV Aswer oly ONE questio. ALL workig must be clearly show. 7. (a) Use the data set provided below to aswer the questios which follow. (i) Costruct a stem-ad-leaf diagram to represet the give data. * Barcode Area * Sequetial Bar Code (3 marks)

69 (ii) (iii) (iv) (v) /F 2016 State a advatage of usig the stem-ad-leaf diagram to represet the give data. (1 mark) Determie the mode. (1 mark) Determie the media. (2 marks) Determie the iterquartile rage. * Barcode Area * Sequetial Bar Code (3 marks) DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA

70 DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA (b) Two evets, A ad B, are such that P(A) = 0.5, P(B) = 0.8 ad P(A B) = 0.9. (i) (ii) (iii) /F 2016 Determie P(A B). (2 marks) Determie P(A B). (2 marks) State, givig a reaso, whether or ot A ad B are idepedet evets. (2 marks) * Barcode Area * Sequetial Bar Code

71 (c) /F 2016 A bag cotais 3 red balls, 4 black balls ad 3 yellow balls. Three balls are draw at radom with replacemet from the bag. Fid the probability that the balls draw are all of the same colour. (4 marks) * Barcode Area * Sequetial Bar Code Total 20 marks DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA

72 DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA 8. (a) A motorist starts from a poit, X, ad travels 100 m due North to a poit, Y, at a costat speed of 5 m s 1. He stays at Y for 40 secods ad the travels at a costat speed of 10 m s 1 for 200 m due South to a poit, Z. Displacemet (metres) (i) /F 2016 O the followig grid, draw a displacemet time graph to display this iformatio. Time (secods) * Barcode Area * Sequetial Bar Code (5 marks)

73 (ii) (iii) /F 2016 Calculate the average speed for the whole jourey. (3 marks) Calculate the average velocity of the whole jourey. (3 marks) * Barcode Area * Sequetial Bar Code DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA

74 DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA (b) A particle startig from rest, travels i a straight lie with a acceleratio, a, give by a = cos t where t is the time i secods. (i) Fid the velocity of the particle i terms of t. (3 marks) (ii) Calculate the displacemet of the particle i the iterval of time t = π to t = 2π. (6 marks) END OF TEST Total 20 marks IF YOU FINISH BEFORE TIME IS CALLED, CHECK YOUR WORK ON THIS TEST. * Barcode Area * Sequetial Bar Code /F 2016

75 TEST CODE FORM TP MAY/JUNE 2016 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 ADDITIONAL MATHEMATICS Paper 032 Geeral Proficiecy ALTERNATIVE TO SCHOOL-BASED ASSESSMENT 1 hour 30 miutes READ THE FOLLOWING INSTRUCTIONS CAREFULLY. 1. This paper cosists of ONE compulsory questio. Aswer the give questio. 2. Write your aswers i the spaces provided i this booklet. 3. Do NOT write i the margis. 4. A list of formulae is provided o page 4 of this booklet. 5. If you eed to rewrite ay aswer ad there is ot eough space to do so o the origial page, you must use the extra page(s) provided at the back of this booklet. Remember to draw a lie through your origial aswer. 6. If you use the extra page(s) you MUST write the questio umber clearly i the box provided at the top of the extra page(s) ad, where relevat, iclude the questio part beside the aswer. DO NOT TURN THIS PAGE UNTIL YOU ARE TOLD TO DO SO /F 2016 Copyright 2013 Caribbea Examiatios Coucil All rights reserved. * Barcode Area * Sequetial Bar Code

76 - 4 - LIST OF FORMULAE Arithmetic Series T = a + ( 1)d S = [2a + ( 1)d] 2 a(r 1) a Geometric Series T = ar 1 S = r 1 S =, 1 < r < 1 or r < 1 1 r Circle x 2 + y 2 + 2fx + 2gy + c = 0 (x + f) 2 + (y + g) 2 = r 2 Vectors v ^ v = cos θ = a b v = (x 2 + y 2 ) where v = xi + yj v a b Trigoometry /F 2016 si (A + B) si A cos B + cos A si B cos (A + B) cos A cos B si A si B ta A + ta B ta (A + B) 1 ta A ta B d Differetiatio (ax + b) = a(ax + b) dx 1 d si x = cos x dx d cos x = si x dx Σ x i Σ f Statistics i x i Σ (x i = 1 i = 1 x = =, S 2 i = 1 i x) 2 2 Σ f i x i i = 1 = = (x) 2 Σ f i Σ f i Probability P(A B) = P(A) + P(B) P(A B) 1 Kiematics v = u + at v 2 = u 2 + 2as s = ut + at i = 1 + * Barcode Area * Sequetial Bar Code i = 1 DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA

77 - 5 - DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA A maufacturig plat wats to miimize the amout of material used to produce pait cotaiers. The project team has the task of desigig a cotaier to hold 19 litres (5 gallos) of pait. The team suggests three desigs, Desig I, Desig II ad Desig III. (a) Express the volume of pait i m (1 mark) /F 2016 * Barcode Area * Sequetial Bar Code

78 - 6 - Desig I A cylidrical cotaier of height, h, ad radius, r. (b) /F 2016 Surface area of a cylider = 2 π r (h + r) Take π = Usig the required volume, formulate the appropriate equatio for Desig I to show that the miimum surface area is m (5 marks) * Barcode Area * Sequetial Bar Code DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA

79 - 7 - DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA Desig II A rectagular cotaier of height, h, with a top ad a base of area x 2. (c) Usig the required volume, formulate the appropriate equatio for Desig II to determie the miimum surface area (7 marks) /F 2016 * Barcode Area * Sequetial Bar Code