Mathematics WORKED SOLUTIONS FOR CSEC® EXAMINATIONS 2006 –2010
DEREK MCMONAGLE CSEC® is a registered trade mark(s) of the Caribbean Examinations Council (CXC). MATHEMATICS Worked Solutions for CSEC® Examinations 2006–2010 is an independent publication and has not been authorized, sponsored, or otherwise approved by CXC.
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Macmillan Education Between Towns Road, Oxford, OX4 3PP A division of Macmillan Publishers Limited Companies and representatives throughout the world www.macmillan-caribbean.com ISBN: 978-0-230-40738-1 Text © Derek McMonagle 2011 Design and illustration © Macmillan Publishers Limited 2011 All rights reserved; no part of this publication may be reproduced, stored in a retrieval system, transmitted in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of the publishers. Design by Tech Type Typeset by MPS Limited, a Macmillan Company Cover design by Mike Brain Graphic Design Ltd
Printed and bound in Malaysia 2015 2014 2013 2012 2011 10 9 8 7 6 5 4 3 2 1
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CONTENTS 1
Introduction
4
2
Time for revision
6
3
Mathematics – Paper 01 – Multiple choice questions
10
4
Mathematics – Paper 02 – General proficiency – May 2006
64
5
Mathematics – Paper 02 – General proficiency – May 2007
80
6
Mathematics – Paper 02 – General proficiency – May 2008
95
7
Mathematics – Paper 02 – General proficiency – May 2009
109
8
Mathematics – Paper 02 – General proficiency – May 2010
123
9
How did you do?
137
Table of topics from the Mathematics Syllabus
139
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1
INTRODUCTION
What is this book about? This book is your companion to the Caribbean Examinations Council (CXC) Secondary Education Certificate examination in Mathematics at General Proficiency level. It contains five sets of 60 multiple choice questions similar to those that will appear on Mathematics Paper 01, together with answers to these questions. It also contains complete answers to the questions set on the Mathematics Paper 02 in the May series of examinations between the years 2006 and 2010. In addition to the answer to each question, an appropriate explanation is also given so you don’t just get the right answer but, more importantly, you can see how it has been worked out! There is also an indication of how the marks are distributed so you can see how you might get partial credit for an answer even if it isn’t totally correct. How can I use this book? This book is designed to help you to increase your knowledge of mathematics and improve your chances of success in your forthcoming examination. One of the best ways for you to find out exactly what you know (or don’t know) and how well you can organise your knowledge is to try to answer actual examination questions taken from past papers. In addition to examination questions there is a chapter on how to revise. This will help you to draw up a revision timetable, and to stay focused on what you have to do. The chapter also includes tips from experienced examiners on how to avoid throwing away marks by making silly mistakes and how to squeeze those few extra marks by writing down what you know in the clearest possible way. Those few extra marks just might earn you a higher grade! This book is a very flexible revision aid and you can use it in different ways depending on what best suits your revision programme. •
At the end of your revision programme you could simply try to answer the questions on the examination papers to check how much mathematics you know by comparing your answers with those in this book.
However, this book allows you to make far better use of the examination questions as an actual part of your revision programme. At the back of the book there is a Table of Topics from the Mathematics Syllabus. This is a list of topics which together cover the entire content of the Mathematics syllabus. Alongside each topic there is a list of questions that appear in the multiple choice tests and in the 2006–2010 examination papers. •
A hard and daunting task, like revising for your mathematics examination, is often made easier by breaking it down into smaller parts. You may decide to organise your revision programme topic by topic and test yourself at the end of each topic. Each time you complete a topic you will have the satisfaction of knowing a little more and that will give you the confidence to carry on with your studies.
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•
You may be having trouble with particular topics. You can use the topic list to identify the questions about these topics very easily and concentrate your time on them. This might be useful at the end of your revision when time is short.
After completing the questions, you might like to compare your marks with the grade indicators provided by the examination board. This will give you some idea of what grade you are likely to get in your forthcoming examination. There is far more to this book than simply providing a set of correct answers. Read the explanation given for each question carefully, even if you got the question correct. It will help you to organise your answers in order to get all of the marks available. You will be able to apply much of the advice given on examination technique and organisation when you come to answer the questions in your examination.
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2
TIME FOR REVISION
School folklore abounds with stories of students who ‘never did any revision and got a grade I in the examination’. Do you believe them? Well, I suppose that every once in a while there might be a really gifted individual for whom this is true but for the vast majority of us lesser mortals the secret to examination success (and it isn’t really a secret – it’s just common sense) is revision. A long-term plan Be honest with yourself and realistic in your expectations. Do you really believe you can leave things until the last minute and then do well in your examination? Of course you can’t! In order to prepare yourself properly for an examination you need time. How much time depends on how hard you have worked over the period of the course, how much natural ability you have and how well you want to do in the examination. Revision is not about sitting down, opening your book at some random page and reading the text. Revision is about dividing the content of a syllabus into manageable sections and spending time specifically revising those sections, so that, over a period of several weeks, you cover all of the syllabus content. In order to revise efficiently, you will find it useful to keep a record of what you have done. The following table is a record of the revision carried out by a student in the first three weeks of preparation for their Mathematics examination. The topics in the table are taken from the syllabus. You will need a similar table for each of your other subjects.
SUBJECT Topic
MATHEMATICS GENERAL PROFICIENCY Week 1
Week 2
Sets Relations, functions and graphs
0.5 h 0.5 h
Week 6
Week 7
Exam week
0.5 h 0.5 h
0.5 h 0.5 h
Statistics Algebra
Week 5
1h
Measurement Consumer arithmetic
Week 4
0.5 h
Computation Number theory
Week 3
0.25 h 0.5 h
0.5 h 0.5 h
Geometry
0.5 h 1h
Trigonometry
1h
Vectors
0.5 h
Matrices
0.25 h
TOTAL TIME
3.75 h
0.5 h 4h
4.25 h
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What advantages does drawing up such a table have? •
It divides the syllabus content up into smaller parts so you can focus on each one individually.
•
It provides a visible record of what has been revised so that no topics are left out or neglected.
•
It provides a visible record of how long is being spent revising the subject so that time can be slowly increased as the examination week comes nearer.
•
It provides a visible record of what has been achieved to date which, in turn, increases confidence.
Notice that the amount of time spent revising a topic is between 15 minutes and 1 hour. •
Revising something for less than 15 minutes doesn’t really allow you enough time to get into the topic so you will achieve very little.
•
Revise for more than 1 hour and you will probably get very tired and stop being effective. However, we are all different; you might find that your tiredness threshold comes after 45 minutes or even sooner. You must decide on the maximum length of time you can revise effectively and organise your timetable accordingly.
Sitting reading notes for even a short time can be very boring, so try and make your revision time as interesting as possible by doing short bursts of different things. For example, in a 30-minute revision session you might spend the first 10 minutes reading, the second 10 minutes writing down key facts and the final 10 minutes attempting examination questions. The environment within which you revise is also very important. The ideal conditions for revision will vary from student to student. Some might be happiest sitting at a table somewhere cool and quiet, while others might prefer to sit in a comfortable chair, books on their lap, with quiet music playing in the background. You may need to experiment in order to find which conditions work best for you. However in doing this, be honest with yourself. Revising whilst you are watching your favourite television programmes might appear to be killing two birds with one stone but can you really say you are learning effectively like that? I don’t think so. The night before the examination • Never work late the night before an examination. You need a good night’s sleep before an examination. There is no problem with doing an hour or two of revision in the evening as long as you finish at least a couple of hours before you go to bed. This gives your mind time to unwind so that you don’t lie awake in bed worrying about knowing this or knowing that. The day of the examination • Make sure you have some breakfast, or if the examination is in the afternoon, have some lunch. Your body is like a machine; it needs fuel to work properly. You need to make sure that your blood sugar level is high and you have lots of energy. If you really can’t face eating a meal, suck a few glucose sweets. •
Don’t try to cram at the last minute. How much are you really going to learn while eating your breakfast or travelling on the bus to school? My guess is not a lot. If you have revised thoroughly you shouldn’t need to worry at this stage. 7
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By trying to cram at the last minute all that will happen is that you will start worrying about this topic or that. •
Make sure you have all of the equipment you are going to need. You can’t do a good job without the necessary tools. For this examination I recommend you have the following in your pencil case: 2 pens (1 spare) 2 pencils (1 spare) 1 pencil sharpener 1 eraser 1 ruler 1 protractor 1 set of compasses 1 calculator 1 spare set of calculator batteries Some students find it helps their concentration if they suck an occasional sweet during the examination so you might want to put a few of these in your pencil case as well.
•
Don’t discuss the examination with other students. Comments made by other students waiting to go into an examination can often be un-nerving. They might set you off worrying about whether you have done enough revision or whether you have revised particular topics thoroughly enough. Why worry about such things when it is too late to do anything about it? Some people have to chatter because they are nervous and if you let them, they will affect you in the same way. My advice is to keep your own counsel. If you revised thoroughly before the examination you have every right to feel quietly confident that you will do well. Don’t let anyone persuade you otherwise.
•
Don’t be tempted to cheat. If you have to resort to writing things on the back of your hand or on bits of paper hidden in your pencil case then you have not done sufficient revision. Are you really going to benefit from these illegal prompts? You might think you have invented a novel way of cheating but experienced examination invigilators will have seen it all before. Is it really worth the risk of being caught and bringing disgrace on yourself and your family? Of course it isn’t. Revise the subject thoroughly and you won’t even need to think of doing such a thing.
In the examination • Read through the examination paper at least twice. Spend the first 10 minutes reading through the paper. This is never time wasted. Look upon this first 10 minutes as an investment that could save you marks later on in the examination. •
Work out a rough time schedule in your mind. The Paper 02 examination last for 2 hours 40 minutes. Subtract 10 minutes reading time and that leaves you 2 hours 30 minutes, or 150 minutes and the examination paper has a total of 120 marks.
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150 5 1.25 minutes per mark. So for a Section I question worth 10 marks, This means that you have ____ 120 for example, you should be spending no more than 10 3 1.25 5 12.5 minutes answering it. All of the Section II questions are worth 15 marks so you should not be spending more than 15 3 1.25 5 18.75 minutes answering each one. You can only use this as a rough guide, since you are going to find some questions harder than others. However, if you find that you are spending significantly longer on a question than the number of marks justifies, perhaps you should move on and come back to it at the end of the examination if you have time. If you don’t, the danger is that you will not complete all of the questions you can answer and all those marks will be lost. •
Choose which questions to answer from Section II. Look carefully at the Section II questions and decide which you think you will find easiest to answer. It may be that you are stronger in some topics than others, so questions in your strong topics should be given particular consideration.
•
Read each question again before you attempt to answer it. Every experienced examiner will tell you that the biggest single mistake that students make is to answer questions that they are not asked. They don’t read the question properly, or they copy numbers down wrongly from the examination paper. Make sure you know exactly what is being asked of you before you start.
•
Take care with how you present your answers. Nothing alienates examiners more than scripts which are untidy and difficult to read. Examiners get paid by the script for marking so they will not waste time trying to decipher a horrible mess. Do yourself a favour and write your answers clearly, making sure that words and numbers can be understood, and that diagrams and graphs are drawn carefully with a sharp pencil.
•
Don’t leave questions unanswered. One of the certainties of an examination is that you will get no marks for leaving a question unanswered. You will not be penalised for giving the wrong answer so put something down on your answer sheet for each question. If you don’t know the answer to a multiple choice question then have a guess. At worst you have a one in four chance of being correct, and by giving the question some thought, you may be able to eliminate one or two of the wrong answers and improve the odds. In longer questions, marks are usually given for the method as well as the correct answers. Even if you are uncertain what to do and your answer is wrong you may score some marks for your method.
•
Check your working carefully. If you have some time at the end of the examination, don’t sit there looking out of the window feeling all smug and self-satisfied. Go back to the beginning and check through your answers. Check your arithmetic; many marks are lost because errors are made carrying out simple sums in the heat of the moment. Check that you have answered the question exactly as you have been told. For example, were you asked to give your answer in a particular format such as correct to 2 decimal places?
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SOLUTIONS
3
MATHEMATICS 2 PAPER 01 2 MULTIPLE CHOICE QUESTIONS
The questions in this section are not taken from actual examination papers because these are not available to the public. However, they are similar questions based on the curriculum content and examination style adopted in the General Proficiency Examination. The number of questions given over to each topic in a test reflects the content stated in the curriculum document. Number of questions
Topic Sets
4
Relations, functions and graphs
6
Computation
6
Number theory
4
Measurement
8
Consumer arithmetic
8
Statistics
6
Algebra
9
Geometry
9
TOTAL
Test A Paper 01
60
Paper 01 Test A
1 C The sequence increases by the set of odd numbers 3, 5, 7, 9 ... so the next number is 27 1 11 5 38.
1 The next number in the sequence 3, 6, 11, 18, 27 is
2 D 3 4 8 5 0.375
2 The fraction _38 written as a decimal is
A 34
A 0.305 3 B The second decimal place digit is a ‘4’ so the first decimal place digit is not rounded up.
B 36
B 0.325
C 38
C 0.350
D 40
D 0.375
3 The number 86.345 correct to one decimal place is A 86.0
B 86.3
C 86.4
D 86.5
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Use the following diagram to answer questions 4 and 5.
Athletics
Baseball
Cricket In the Venn diagram a dot (•) represents one pupil. The diagram shows the pupils in a form who take part in athletics, baseball and cricket. 4 How many pupils in the form take part in athletics? A 5
B 6
C 8
D 11
5 How many pupils in the form take part in two of the sports but not in all three? A 2
B 9
C 11
D 22
6 If M 5 {1, 3, 5, 7, 9}, N 5 {2, 5, 7} and O 5 {5, 6} then M ∩ N ∪ O 5 A {5}
B {5, 6, 7}
C {1, 2, 5, 6, 7, 9}
D {1, 2, 9}
4 D Pupils taking part in athletics include those who take part in athletics only plus those who take part in athletics and other sports, i.e. 5 1 3 1 2 1 1 5 11. 5 B The different combinations are athletics and baseball (3), baseball and cricket (5) and cricket and athletics (1) giving a total of 9. 6 B (M ∩ N) ∪ O is the set of numbers in both M and N or in O.
11
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Use the following diagram to answer questions 7 and 8.
5 cm
4 cm
8 cm 3 cm
7 A Volume 5 area of cross-section 3 length 5 _12 3 3 3 4 3 8 5 48 cm3
s
d
7 The volume of the prism is A 48 cm3
B 60 cm3
C 80 cm3
D 96 cm3
8 D Total surface area 5 sum of the areas of all faces 5 2 3 s _12 3 3 3 4 d 1 (4 3 8) 1 (3 3 8) 1 (5 3 8) 5 108 cm2
8 The total surface area of the prism is
9 C 21.45 hrs to midnight 5 2 h 15 min 2 h 15 min 1 6 h 55 min 5 8 h 70 min 5 9 h 10 min
9 The flight from Kingston to London leaves at 21.45 hrs and arrives at 6.55 hrs Jamaica time. How long does the flight take?
10 C 400 1 400 3 0.065 5 $426
A 40 cm2
B 72 cm2
C 96 cm2
A 8 h 10 m
B 8 h 55 m
C 9 h 10 m
D 9 h 55 m
D 108 cm2
10 A sales tax of 6.5% is charged on a bill of $400. The bill after tax is A $374.00 B $406.50 C $426.00 D $462.00
11 A He must take home 100 2 35 5 65% of his weekly wage 100 5 $210 Weekly wage 5 136.50 3 ____ 65
11 A man loses 35% of his weekly wage in stoppages. If he takes home $136.50 how much does he earn each week? A $210
12 D 18 1 18 3 0.45 5 $26.10
C $310
D $390
12 A shopkeeper buys a box of candles for $18.00. If she wants to make a profit of 45% how much should she resell them for? A $8.10
13 B The decimal point is moved 3 places to the right; the fourth significant figure is less than 5 therefore the third significant figure is not rounded up.
B $230
B $9.90
C $25.90
D $26.10
13 0.001 951 1 in standard form correct to 3 significant figures is A 1.95 3 1022
B 1.95 3 1023
C 1.96 3 1022
D 1.96 3 1023
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11 _ 7 14 The set of fractions { __ , 2 , _1 , __ written in 20 3 2 12 }
descending order of magnitude is A
7 __ , 11 , _1 { _23 , __ 12 20 2 }
B
7 __ , 11 , _2 { _12 , __ 12 20 3 }
C
11 __ , 7 , _2 { _12 , __ 20 12 3 }
D
11 __ , 7 , _1 { _23 , __ 20 12 2 }
15 Which of the following is a prime number? A 21
B 23
C 25
B 9
C 12
{
}
15 B 23 has no factors except itself and 1.
D 27
16 What is the H.C.F. of the numbers 36, 54 and 90? A 6
14 A If we express these fractions over a common 33 , ___ 40 , ___ 30 , ___ 35 it is easier to denominator: ___ 60 60 60 60 arrange them in descending order.
D 18
17
16 D 18 is the largest number that will divide exactly into these numbers. 17 B There are 9 fully shaded and 16 partially shaded squares; the partially shaded squares are approximately equal to 7 fully shaded squares so the shaded area is about 9 1 7 5 16 squares.
Which of the following is the nearest estimate of the area of the shaded figure above? A 12 cm2
B 16 cm2
C 20 cm2
D 26 cm2
18 If f : x → 3x2 2 1 then f (2) is A 3 19
B 9
C 11
18 C 3 3 (2)2 2 1 5 12 2 1 5 11
D 13 19 D f : x → 2x 1 1 is the only function that correctly maps all the given values.
1→3 2→5 3→7 4→9 Which of the following could describe the mapping above? A f : x → 3x 2 1
B f:x→x15
C f : x → 3x
D f : x → 2x 1 1
20 A van is worth $14 000 today. If its value depreciates by 12% per annum, how much will it be worth in one year? A $1680
20 B It will be worth 100 2 12 5 88% of its original value 5 14 000 3 0.88 5 $12 320
B $12 320 C $13 832 D $15 680 13
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