Higher. MATHEMATICS GCSE for AQA. Student Book Answers Karen Morrison, Julia Smith, Pauline McLean, Rachael Horsman and Nick Asker

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1 Higher MATHEMATICS GCSE for AQA Student Book Answers Karen Morrison, Julia Smith, Pauline MLean, Rahael Horsman and Nik Asker

2 Contents Working with integers Colleting, interpreting and representing data Analsing data Properties of integers Working with frations Working with deimals Basi Algera Properties of polgons and D ojets Angles Perimeter Area Rounding and estimation Perentages Powers and roots Standard form Further algera Equations Funtions and sequenes Surds Basi proailit Units and measurement Formulae Volume and surfae area Further proailit Inequalities Ratio Proportion Graphs of linear funtions Interpreting graphs Cirles Vetor geometr Transformations in a plane Constrution and loi Similarit Congruene Pthagoras theorem Trigonometr Graphs of other funtions and equations Growth and dea Transformations of urves D ojets Camridge Universit Press

3 Student Book Answers Answers The answers to past paper questions are the work of the pulisher and authors and have een neither seen or verified AQA. Working with integers BEFORE YOU START a > > = d < a Mistake is just working left to right not doing operations in order. Corret answer is. Mistake is just working left to right not doing operations in order. Corret answer is. Order of operations is orret, ut alulation is wrong. Corret answer is. a C B B LAUNCHPAD a d a C D D d B e D = m EXERCISE A a i. ii. pakets. We need to know this so we an work out the total ost. Multipliation or repeated addition, eause there are paks whih eah ost.. d Divide the ost of one paket as there are pens in eah pak. e. would e a reasonale estimate for the prie of one pen, ut it is not the orret answer eause. =. not.. So. is too epensive.. a minutes a km km + = EXERCISE B a d e a d e a d e a d e. a and or and, and and d and This is an investigative task. The onl numers etween and that annot e made omining multiples of and are,,,,,, and. WORK IT OUT. Option A is orret. The mistake in option B is doing the addition first. Option B is orret. The mistake in option A is not doing the alulation in rakets first. Option B is orret. The mistake in option A is alulating + not. Option B is orret. The mistake in option A is doing the sutration efore the division. Option B is orret. The mistake in option A is not alulating in the numerator. Option A is orret. The mistake in option B is doing the addition efore the division. EXERCISE C a, and d are orret. e f a = ( ) = or ( ) = ( + ) ( ) = or ( + ) ( ) = or ( + ) ( ) = or ( + ) ( ) = or ( + ) ( ) = or ( + ) ( ) = or ( + ) ( ) = or ( + ) ( ) = or ( + ) ( ) = or ( + ) ( ) = or ( + ) ( ) = or ( + ) ( ) = a ( + ) ( ) ( ) d ( ) e ( + ) f ( ) g ( + ) ( ) h ( + ) ( ) i ( ) ( + ) j ( ) k ( + ) l no rakets needed m ( + ) n ( + ) o ( ) p ( ) q no rakets needed r no rakets needed a ( ) ( ) d + ( + ) a.. d. e. f g h. This is an investigation. Students should find their own methods and eplain their thinking. Camridge Universit Press

4 GCSE Mathematis for AQA (Higher) The formula n a + will work for an sequene (whether positive or negative), where n is the numer of terms, a is the lowest numer and is the highest numer in the set eause a + is the mean value of the set. There are man other methods of finding the answer though, and students ould researh these online if the are interested. EXERCISE D a d e f All alulations are orret. a h = m h = m, = m a Students own answers. Students own reasoning; should suggest that addition and multipliation (of whole numers) produes higher values. CHAPTER REVIEW a option B option D Students own answers. Students own answers. Some possile solutions are: + ; + ; + ; ;, and so on. Produt of olumns and and a C C Sum of rows Colleting, interpreting and representing data BEFORE YOU START Where h is height, ever m from. < h.,. < h. et, or ever m. < h. et a,,,, a From left to right:,, Measure student s drawing LAUNCHPAD Choose a representative sample, using a random seletion method to otain a mi of male and female students from different ears, and small enough so that ou have time to ask everone in our sample. Deide whether ou will let the students name their favourite song, or whether ou will make them hoose from a list of tpes of musi, or oth. Surve our sample. a Croatia, Greee Croatia & Greee; Russia & Portugal a to or (depends on aura of measurement of angle for setor) a The olumns are not the same width eause the data was surveed in age ategories (lasses) that do not eah span the same numer of ears, so the lass intervals in the data are not all the same. age <, i.e. ears. students WORK IT OUT. a Option D Option A will elude men, and elude women who have no hildren. Option B will, perhaps, tend to over-represent shoppers who are uing something on their wa to work. Option C will e restrited to shoppers who have hosen one kind of shop (it will e iased towards ook uers). EXERCISE A D and E are likel to give random results. D a person s name does not determine an other harateristi of that person E the hane of piking a given name is the same as piking an other name A, B and C are likel to give non-random results. A - the hosen street might e (for eample) ver wealth. B - eludes people who work or are out during the da. C - oung people are more likel to wear trainers. a An sensile suggestion where parents ma e found e.g. soft pla area; a park; nurser; a food setion of a supermarket. Approimatel. a would mean aout mahines. No, eause not all the memers are likel to e there at one. Sample the memers present at different times of the week. a. million. million WORK IT OUT. Option C is est: Option A wrongl uses a line graph to displa disrete data and has no sales; Option B does not show the frequenies, ut has one ar per test, and has no sales either. Camridge Universit Press

5 Student Book Answers EXERCISE B a Frequen a Visits Hair olour Blond Blak Brown Gre Frequen Frequen Frequen Numer of visits Favourite holida destination UK Spain Frane USA Destination USA has a frequen of. % of is. + % = + =. So Spain is hosen % more people. a C % d Greee a April mm Feruar d Appro mm e Wetter: saw over mm of rain a Jenn ould have spent a few minutes oserving the most popular snaks efore she egan her surve. Then she ould have written down a list of those snak names and used tall ounting to uild up a frequen tale ased on the list. This would save her from repeatedl writing the same snak names. Snak ought Frequen Chooar Juiear Cheese puffs NRG drink Gum Apple Crisps Fruit hews Snak ought Frequen Chooar Juiear Total ars: Cheese puffs Crisps Total savour snaks: NRG drink Gum Apple Fruit hews Total other: EXERCISE C a Average time spent wathing television; average time spent doing homework As students move into higher grades the wath less TV. As TV-wathing time dereases, the amount of time spent on homework inreases. d i minutes ii minutes a True Datalink and G-Commere True d G-Commere e G-Commere f Speedlink, eause the osts are greater than the inome g a No ou annot, as the values shown are perentages of a ompan s total sales, and not the atual value of the sales. C C d C e A third f Just over a quarter of its sales are diret from the shop; nearl % are internet sales; the remaining third are fairl evenl divided etween atalogue mail order and through an agent, with the latter aounting for slightl more. Camridge Universit Press

6 GCSE Mathematis for AQA (Higher) a Willhelm Susan Trevor d Their sore on the atual eam was worse than on their mok. WORK IT OUT. Option B shows the information est. Option A does not show the frequen of eah lateness, ut the atual minutes. Option C s ke has no units and the Up to wording is misleading. EXERCISE D Eletriit generation Gas Other fuels Coal Nulear Renewales a An two proportional responses, suh as Ireland has a smaller proportion of under s than Greee It isn t possile to sa, as we don t know the relative populations We an t sa whether it is true or false we don t know the numers, onl the proportions a An two of: Proportion of other vehiles has dereased. Proportion of light goods vehiles has inreased. Proportion of motorikes has inreased. Proportion of heav good vehiles has slightl dereased. Proportion of uses and oahes has slightl dereased.. million (answers within. million are aeptale) Aout % d Option B a TV ; students College B has a larger proportion of students who go on to further eduation. A larger proportion of College A students go on to emploment. Similar proportions go on to higher eduation. College B sees more students going on a gap ear. (Aept an tenale argument that auratel ites these oservations, as there is no lear winner.) College A EXERCISE E College B Class interval Cumulative Frequen ie reams < ie reams < ie reams < ie reams < ie reams < ie reams < Higher eduation Further eduation Emploment Gap ear Unemploment Cumulative frequen Camridge Universit Press Cumulative frequen urve showing numer of ie reams sold Numer of ie reams a Cumulative frequen urve showing maimum dail temperature Cumulative frequen Temperature in C a Cumulative frequen Time (minutes) a a Frequen Frequen Test results Length

7 Student Book Answers Frequen Length a % d % EXERCISE F a Temperature ( C) J F M A M J J A S O N D Month The temperature is mild all ear round, rising to the low twenties in summer (June to Septemer) and never falling elow an average of in winter Year a Teeth etrated J F M A M J J A S O N D Month Numer of teeth etrated The numer of teeth etrated peaked in April then fell off steadil to a low in August, reovering sharpl in the autumn ut falling again towards the end of the ear. In August man people are awa on holida and so not ale to e at the dentist, or unwilling to have a tooth etrated just efore the go awa. a League position Week Position The team fell from the numer slot at the start of the ear to a low of numer, slowl improving again as the season went on ut never quite regaining their former est position. a The third quarter. The fourth quarter, generall. In Quarter the were, Quarter and in Quarter. Given that this is the same quarter in eah of three ears, it is fair to sa that sales are improving. a Time series graph showing shed sales Numer of sheds Q Q Q Q Q Q Q Q Q Q Q Q Sales have steadil inreased ear-on-ear. Yes, shed sales are seasonal: the peak notal in the third quarter and are at their lowest in the first. a The numer of light vans has inreased steadil till, sine when it has remained fairl onstant. The numer of vehiles has remained largel the same The numer of heav goods vehiles might have dereased eause the are now larger, and eah one arries more goods, or the freight is eing arried the smaller light vans. (Aept an feasile answer.) No, this question annot e answered using the graph onl. a In June and in August There might have een some rainfall whih aused the level to rise. Between Feruar and Marh d mm e mm Camridge Universit Press

8 GCSE Mathematis for AQA (Higher) CHAPTER REVIEW a Taking the students whose names egin with a ertain letter, hoosing students whose irthda is in a partiular month, or an other suitale method. a Pie harts are est for omparison. Josh Ben Rent Food Transport Savings Josh spends a larger proportion of his mone on rent. Ben spends a larger proportion of his mone on food. Josh saves a greater proportion of his mone. Entertainment Ben spends a greater proportion of his mone on entertainment. Josh spends a greater proportion of his mone on transport. a Quarterl profits st quarter nd quarter rd quarter Compan A th quarter st quarter nd quarter Compan B rd quarter th quarter Generall Compan B makes higher profits and is therefore more suessful. The drop in profits for ompan B etween the rd quarter and th quarter of is the iggest hange. d Option B a Amount ( ) Frequen densit Distane, D (km) a Frequen densit Frequen densit.. Mahine A Sores Mahine B.... Sores. Mahine A has a smaller range of values. Mahine B makes more underweight pakets. Analsing data BEFORE YOU START a.,,,,, no mode, A(, ), B(, ), C (, ) a LAUNCHPAD = a ups. ( ) d a Between Marh and June the mean mark was unhanged, ut the performane of the middle half of all students worsened; however, the lowest and the highest marks oth improved. Not overall, no. Sale is misleading as it doesn t egin at zero a A negative orrelation Inreasing smoking dereases our life epetan Camridge Universit Press

9 Student Book Answers EXERCISE A option C a a a Das asent (d) Frequen Midpoint Midpoint frequen d <.. d <.. d <.. d <. d <. Total. d < mean:., median falls in lass: d <, range: s < s < s < s < s < mean:., median falls in lass: s <, range: No, as we have the data and an alulate it d s <. h <.. h <.. h <.. h <.. h <.. h <.. h <.. h <.. h <. Mean =., median falls in lass:. h <.. h <. Mean is., median falls in lass. h <.. The new group intervals lower the estimate of the mean, and the new lass holding the median value starts at a lower ottom limit than the previous one. a Other groupings are possile ut, starting at h and using half hour intervals gives si lasses. Time interval f h t < h h t < h h t < h h t < h h t < h h t < h Modal lass = h t < h Mean = h, median falls in lass : t < :, range = hours a. kg m < m < EXERCISE B median = ears; IQR = ears median = ; IQR = appro. median = minutes; IQR = minutes EXERCISE C a A: Mean, Median, Mode, Range. B: Mean., median., no mode, range. The results for test B are slightl higher on average, with a greater spread of results than in A. A s mode indiates a luster of lowerthan-average results. a Ahmed: mean., median., mode, range. Bill: mean., median, mode, range. Ahmed has had a etter season as his mean and median are higher, as is the total numer of runs sored Bus : mean., median, mode, range. Bus : mean., median, mode, range. Bus is etter on most measures, ut it has the greatest range so is potentiall less reliale. It does not matter that there is less data as all of these measures enale omparison of two samples even of differing sizes. Means: mahine A =., mahine B.. Both have same range (), median group ( < ) and modal group ( < ). Mahine B is the sensile hoie. a The median would e etter than the mean or the mode, as it takes into aount all values ut redues distortions aused etreme values, partiularl the high ones. Shoe size in m lass. (Aept answers in whih the data is unlikel to e skewed a few etremel small or large values; salaries in a firm would not e aeptale, for eample.) The most popular make of ar sold in the UK last ear. The numer of edrooms per house in a housing estate. (Aept eamples of disrete data in whih (a) nonnumeri data neessitates use of the mode; or () the most ommon ategor is what is of onern, and one is likel to dominate.) a Q =., median =, Q =. Camridge Universit Press

10 GCSE Mathematis for AQA (Higher) a Q =, median =, Q =,,,,,,, a (aept lose to the numer as it s measurement) (aept lose to the numer as it s measurement) Team B d Team B e Team A f Team A s mean is higher and their IQR is overall higher. EXERCISE D There are no values on the vertial ais, and even if there were there is no definition of how the rime rate is measured and the aed should (ideall) e laelled to indiate what the numers mean. The horizontal sale is not onsistent. This is proal to make growth from onwards look etter than it is. The width of the ars hange, eaggerating the later sales a Vertial sale does not start at zero To give an eaggerated impression of the hanges a The are oth the same The D effet makes the sale diffiult to read Students own answers, ut a ar hart or a time-series graph would e est. EXERCISE E TV viewing Sattergraph omparing time spent viewing TV to time spent on homework Homework Graph shows a strong negative orrelation.. Engine size.. Sattergraph omparing fuel eonom to engine size Eonom (miles per gallon) a Temperature ( C) Sattergraph omparing ie ream sales and weather Ie ream sales Sales might also e affeted loation, da of the week or time of da a Sattergraph showing numer of residents in eah house Numer of residents No orrelation a Height jumped No orrelation D House numer Sattergraph omparing athletes heights to height jumped Athlete s height. Camridge Universit Press

11 Student Book Answers EXERCISE F B a. d e. a, Sattergraph omparing English and maths sores Maths sores English sores The point (, ) a. minutes Yes. The time of minutes is an outlier (mae due to an aident) so the ompan has alulated the mean from the remaining data to give. minutes. CHAPTER REVIEW a First lake Mean. Median Mode no mode Range Q Q IQR Seond lake Mean Median. Mode Range Q Q IQR The first lake is etter for eperts as the wind speed is more often higher, and if the one da of zero wind is disounted all measures of entral tenden are lose to preferred speed. The seond lake is etter for eginners with a slower wind speed and less variailit of wind speed. C a This and ever o plot shows the lowest and highest values, the range, the first and third quartiles, the interquartile range, the median. It shows the overall spread of heights in a diagram form whih makes omparison with other groups easier. a Time Frequen Midpoint Midpoint frequen t <. t <. t <. t <. Total Estimate of Chen s mean journe time is =. It is an estimate eause the alulation uses the midpoint of eah interval, not individual oserved times. Chen Frequen Lee Times, t (minutes) d Lee s journe generall takes longer than Chen s, and Lee has had the longest individual journe. Lee has a greater range of journe times in fat, Lee has on a few oasions got to ollege more quikl than Chen ever has. a The vertial sales are different Graph A, eause it makes the growth look igger filling up the whole graph with the lines, whereas the other has muh empt white spae suggesting growth that ould have happened ut didn t. An amitious manager of the ompan, eause he/ she might wish to show that the numer of susriers is still not large ompared to the potential numer of susriers. a Mass (grams) Sattergraph omparing mass of hoolate to prie Prie ( ) A strong positive orrelation Camridge Universit Press

12 GCSE Mathematis for AQA (Higher) Properties of integers BEFORE YOU START a a a,,,,,,,,,,, a D B C d A LAUNCHPAD a false false true d true e false f false a multiples of ; is inorret multiples of, is inorret fators of ; is inorret d multiples of, is inorret e fators of, is inorret f multiples of ; is inorret g primes to, is inorret a B A a C () D () EXERCISE A a,,,,,,,,,,,,,,,,,,, d,,,, e,, f,,,,,,, g,,,,, h,, i, option A a,,, various options, for eample,,,,,,, d, e,,,, et. (seletion is tehniall infinite) f,,,,,,,,, g,,,, h,,,,,,,, a even even even d odd e even f odd EXERCISE B a,,,,,,,,,,,, d,,, a d e f g h i option B a i ii i ii i ii EXERCISE C a,,,,,, option B a d e f g h a No. The produt of prime fators is unique for eah whole numer. Camridge Universit Press This is an investigation. The sieve of Eratosthenes is a grid of numers on whih multiples (of,, et.) are rossed out sstematiall, leaving onl primes unrossed. a The are named after the monk who first suggested them, although his initial thinking aout them was proven to e wrong with later disoveries. GIMPS aims to use the spare proessing power of linked omputers to runh greater and greater numers to see whether the are prime or not. a,, and, as well as,, and. All even numers are automatiall eluded, as are numers ending with (multiples of ); that leaves onl possile options, numers ending in,, and. As these ould e multiples of, or, man of these are not prime. EXERCISE D a option A option C a d e f g h Option C a,,, d, e, f, g, h, m shoppers students minutes a das times m. pm After seonds ( minutes). Fran laps, Aua laps and Claire laps. a m tiles people CHAPTER REVIEW Yes. You an deide trial division of prime numers up to half the size of the numer in question. option A a C M U L T I P L E S BE RI E V N S Q U A F D V I S O R d M e DD R E I CT X WO O P R O D U C T

13 Student Book Answers F:,,,,, F:,,, HCF = M:,,,,,, M:,,, LCM = a = a = = HCF =, LCM = = = HCF =, LCM = th step Marh and Marh os and girls per group Working with frations BEFORE YOU START a, and,, and and d,, and a C D B a A eause ou have to do the multipliation first. B eause ou have to do the multipliation and division efore addition. B eause ou have to do the multipliation efore sutration. LAUNCHPAD a a The mistake was adding the numerators and denominators. Corret answer: = The mistake was doing. Corret answer: The mistake was to add the numerators rather than multipling them. Corret answer: d The mistake was to multipl rather than divide. Corret answer: a of of of EXERCISE A a Option B Option C Option C a d e = f = g = h = a d Students own reasoning, ut the should realise that the an make an equation and solve for to find the unknown values. For eample, = a e f g i j k a,,,,,,,,, d h,,,, EXERCISE B a First find the fration etween and ; this is. l Then appl the rule to find the fration etween and, and so on. Three possile frations are:, and. Students an plae the results on a numer line, or ross multipl to show that the frations are larger/smaller than eah other. Students own researh. An internet searh for mediant frations will provide several interesting artiles and some proofs. EXERCISE C a e i f j m n o q = r s = u v a Option C Option B a e i f d g h k = l p = t w = = d = g h = j = k EXERCISE D Option C min m of a minute l Camridge Universit Press

14 GCSE Mathematis for AQA (Higher) meals m m m m a pages EXERCISE E a e f a Option C Option D g d = h = a up = ups h d = h e h f = h g h h = min i seonds a d e f g h min General referene, tehnolog, engineering, omputers. Week : m, week : m and week : m. first lass, usiness lass, eonom and low-ost. EXERCISE F a + d e + (fators,, ) Yes. Students an investigate this and ma find the theorem that proves this if the are interested. CHAPTER REVIEW Option D a m a ottles ( full ottles) plots m deep ups = d Working with deimals BEFORE YOU START = a... d. e. a < > < d > e = a d e f LAUNCHPAD There are man possile answers. For eample: a... a a. a C C C d B a e A f C EXERCISE A option B a e i f j g d h a... d. e. f. g. h. i. j. k.. l. m. n.. o. a repeating digits of the numerator reurring non terminating deimals.,..,.,..,.. d...,... e predition is... and... ased on multiples of a.,.,.,.,..,.,.,.,..,.,.,.,. d.,.,.,.,. e.,,,,.,. a,,,,,, a = d = e g h = = f = i = = f.,,.,., a < < < d = e > f < g > h > i < Camridge Universit Press

15 Student Book Answers There are man possile answers. a... a Steel Dragon California Sreaming shorter d Steel Dragon and The Ultimate e. km,. km,. km,. km,. km.. km EXERCISE B, student disussion EXERCISE C option B a... d. e. f. a... d. e. f. g. h. i j. k. l. m. n. o a. s. s. s d Unlikel. The first runner will start from still so will proal take longer than the other runners who start from a running position. es, Nadia has. litres. mg vitamin C,. mg oron and. mg alium a. kwh. kwh EXERCISE D option A. km ups (with some juie left over) posts (there are approimatel gaps of.m ut there will e a post at eah end). a. per da. per week.. Dail rate weekl rate eause there are five da weeks in a ear (as opposed to seven da weeks). EXERCISE E Nazeem s hpothesis is orret as long as the original fration is in its simplest terms. is not in its simplest terms; when redued to it fits Nazeem s original hpothesis. EXERCISE F a..... d.. e... f.. g... h... a d e f g h a d e f g h m EXERCISE G a and and and Students should realise that the reurring version of the terminating fration has a denominator of n. Students own reasoning and disussion. CHAPTER REVIEW a.,.,.,.,..,,,.,.,.,,,.,. a A. m A + B + C a > > > a a.. d. a.. d. e. f.. Kate, she has. more. a d Basi Algera BEFORE YOU START a B A A a i and ii and a a C B D d A LAUNCHPAD a n + (n ) n + a a + + a + a a mn mp + z + z a + = [ ]( + ) + = [ ]( + ) = [ ]( ) d a a = a( [ ] [ ] ) e + = ( [ ] [ ] [ ] ) a ( + ) ( + ) ( + ) EXERCISE A ( + ) a + ( ) d ( + ) e or f ( ) Option C a a d d a e d f mn g pq h a i m j a k a l m m a n de o a Camridge Universit Press

16 GCSE Mathematis for AQA (Higher) a B G I d H e C f D g F h A i E a a d p e f g h a i n a a d e a f g p h i a j k Option C a Perimeter: +, area: + Perimeter: +, area: + + Perimeter: +, area: + ( + ) or d Perimeter: +, area: + a + a (C ) C/ EXERCISE B a d e f g h a d e f g h i j k l e.g. + = and = a if = and =, + = if = and =, = EXERCISE C a unlike like like d like e unlike f unlike g like h like i unlike j unlike k like l like a + d d a + a + a e f + g f a g mn a a + [ a ] = a [ ] = mn + [ mn ] = mn d pq [ pq ] = pq e + [ ] = f m [ m ] = m g a [ a ] = a h st + [ st ] = st a option B option D option D d Option A a a [ ] = a [ ] = a [ ] = a d m [ n ] = mn e a [ a ] = a f p [ p ] = p g [ ] = h m [ mn ] = m n a z a d a d e f g h i a e EXERCISE D option C f a m g p d h a inorret; a + inorret; a + a orret d inorret; p + e inorret; a + a f orret g inorret; + h inorret; a a i orret j inorret; a + a + + d e + e f f a + a g h a + a i a + a + d e + f p p g z a h + p + q p + q p q p q p + q p q a Let a = : + = and = (note if ou use, it will e equal) Let = : () + + = and () + = + = Let m = : ( + ) = and + = d Let = : + = and + = EXERCISE E a ( + ) (m n) (a ) d ( z) e ( z) f (a ) g p(q r) h ( ) i a( ) j ( ) k ( ) l (a + ) m (a + ) n ( + ) o ( ) a ( + ) ( + z + ) ( + z) d ( )( + ) e (a )(a ) f ( ) g ( + ) h ( ) i a( + ) EXERCISE F a false true true d false e true all total m so, es, it is a magi square. a (a + ) (a + ) = a + + = + P = a + +. units a dimensions ( + ) Students own answers. Must sum to + (two sides = half the perimeter). Camridge Universit Press

17 Student Book Answers A = + (a ) = + a A = a + (a )( ) = a + a a + = a + a a + a + + a a a a a a a a a + a a m n q mn m n q mn mn m m m n q mn m a Let the numer e. Doule is. Add si gives +. Halve it gives +. Sutrat the original numer () gives. Students own ideas. a =, = a g +, are full simplified d e a (a + ) ( a) + d a pq + p p CHAPTER REVIEW a + e, = g, = a Let =. (p + ) d + ( + ) = and () + =, so this is not an identit. Let m =. ( ) = and () =, so this is not an identit. Let =. ( ) + ( + ) = and () =, so this suggests an identit. Left hand side = ( ) + ( + ) = + + = = right hand side. /t + /w = (w + t)/tw n + (n + ) = n +. n + is alwas an odd numer sine n is alwas an even numer. (n + ) n = n + n + n (the differene etween squares of two onseutive numers) = n + = n + (n + ) (the sum of two onseutive numers). If the sides are equal then = ( + ) leading to = and ( ) = + also leading to =. Sine these values of are the same, the sides of the quadrilateral ould all e. Properties of polgons and D ojets BEFORE YOU START a Point Verte Edge d Fae e Right angle f Aute angle g Base h Height a Retangle ABCE; triangle ADE; trapezium ABCD Otahedron LAUNCHPAD a Parallelogram Lines are parallel PQ // SR d PSR (or RSP) Two lines of refletive smmetr and rotational smmetr of order a Right-angled Isoseles Equilateral d Otuse-angled isoseles a Parallelogram, retangle, square, rhomus Rhomus, square Parallelogram, retangle, square, rhomus d Retangle, square e Trapezium f Quadrilateral Solid Mathematial name EXERCISE A Numer of faes Numer of edges Numer of verties uoid ue square-ased pramid a Equilateral triangle Regular pentagon Regular heagon Man eamples ould e provided a Stop sign d Regular otagon Plaing die Sheet of paper d Shape of the side of a house Camridge Universit Press

18 GCSE Mathematis for AQA (Higher) EXERCISE B a d e f g h i Column A A shape that has two fewer sides than an otagon. A shape that has two sides more than a triangle. A shape with four sides. An -sided shape. A figure that has length and height. A losed plane shape with all sides m long and all angles the same size. A -sided figure Another name for a regular -sided polgon. The more ommon name for a regular -sided polgon. Column B heagon pentagon quadrilateral otagon two-dimensional regular polgon deagon square equilateral triangle a true true true d true e false f false g true h true a B A C A B D D C AB // DC and is interseted CA. DA // CB and is interseted CA. G E F P Q As alternate interior angles are equal ACD = CAB and DAC = ACB. This means ΔDAC and ΔBCA have two angles equal and the side CA in ommon, so the triangles are ongruent, and hene AB = DC and DA = CB. EXERCISE C a none all lines (AB, CD, EF, GH) CD, HG d AB option B Shape Numer of lines of smmetr S R Order of rotational smmetr square retangle isoseles triangle equilateral triangle parallelogram regular heagon regular otagon regular deagon A propeller with three lades. H Lines of smmetr: vertial and horizontal aes through the entre of the image, and oth diagonal aes through the image; rotational smmetr of order. Students own answers. Students own answers. EXERCISE D Equilateral triangle. Otuse-angled isoseles triangle. (a This is possile. ) The angles would sum to more than. Angles do not sum to. d Angles of an equilateral triangle are all. e Isoseles triangle has two sides of equal length. Option A a Isoseles a =, =, =, d =, e =, f =, g =, h =, i = a. mm. mm mm WORK IT OUT. Option B is orret. In option A, and do not meet at the same point on a straight line. In option C, the seond shape is not a rhomus. EXERCISE E Option A a Retangle, square Retangle, square Parallelogram, retangle, square, rhomus d Quadrilateral e Square f parallelogram, retangle, square, rhomus Shape Diagonals are equal in length Diagonals iset eah other Diagonals are perpendiular rhomus parallelogram square kite retangle Interior angles of a rhomus do not all equal. No, it ould e a rhomus. Two possiilities: and, or and. a Alwas true Sometimes true Sometimes true d Alwas true e Sometimes true a True False False d True = mm AD = BC and AD//BC (as ABCD is a parallelogram). AD = FE and AD//FE (as ADEF is a parallelogram). Therefore, FE = BC and FE//BC. So, FECB is a parallelogram (as one pair of opposite sides are equal and parallel). No, it ould e an isoseles trapezium. Camridge Universit Press

19 Student Book Answers EXERCISE F Answers here are for (a) a small linder mounted on the flat fae of the larger linder, () pramid full on top of a ue, () pentagonal pramids joined at ase and (d) triangular prism full on top of one of the flat surfaes of the uoid. (Students ma have other orret answers; hek against their skethes.) D Shape Polhedron Faes Verties Edges a Large and small linder Cue and square pramid Two idential pentagonal pramids d Triangular prism and uoid D shape Faes Verties Edges ue uoid triangular pramid square pramid triangular prism heagonal prism a E = F + V faes It works for oth ojets Trunated pramid: F =, E =, V = ; F + V = = E Pramid on top of uoid: F =, E =, V = ; F + V = = E d No. F + V =, E =, a m. m m CHAPTER REVIEW a False True False d True e True Heagon has lines of smmetr (lines onneting opposite verties and lines onneting midpoints of opposite sides) and rotational smmetr of order. =, =, d = No, the missing angle is eause angles in a quadrilateral total. a =, = a =, =, = Yes eause oth pairs of opposite sides are parallel. No, the diagonals of oth a rhomus and a kite interset at right angles a edges faes, verties faes, edges d fae, edges, verties a parallel perpendiular trapezium d right angle e otuse angle f refletive smmetr a CB//DE ED, DC and DC, CB d a d a. g m Angles BEFORE YOU START a d a right-angled isoseles triangle rhomus retangle = as the triangle formed the diagonals is isoseles and ase angles of an isoseles triangle are equal a LAUNCHPAD a ; angles on a straight line sum to. ; angles round a point sum to. ; angles round a point sum to. no, angles on a line are various ominations eg a and, and g, and f = f onl if the red line is perpendiular to the two parallel lines, ie and f are ; angle BCA = ; angle CAB = angle BCA =, so = = EXERCISE A option A option B a d refle; otuse; otuse a =, = ( + = and + = ) = (the angles are on a straight line) p = (the angles are round a point) angle ABE = + + = ; if AE is a straight line, angle ABE must e a = (straight line), = (vertiall opposite), z = (vertiall opposite) = (straight line), = (vertiall opposite), w = (vertiall opposite), z = (vertiall opposite) z = a =, =, a =, = =, =, a =, = =, =, a =, = d =, =, a =, = a a read the wrong sale a = (vertiall opposite) = (straight line), = (angles in a triangle), z = (straight line) = (angles round a point) Camridge Universit Press

20 GCSE Mathematis for AQA (Higher) EXERCISE B option C a =, =, =, d = option A = a =, =, = =, = a =, =, =, d =, e =, f =, g = CEG = DCF = a = = a AB is parallel to DC (alternate angles) AB is not parallel to DC (o-interior angles do not sum to ) AB is parallel to DC (o-interior angles sum to ) EXERCISE C option B a = = =, = a =, = =, =, z = a = (ase angles isoseles triangle), = (angles in a triangle), = (straight line) = Either two angles of and one of ; or two angles of and one of ECD = (orresponding angles), ACE = (alternate angles) So ACD = + a = (angles in a triangle), = (orresponding angles), z = (orresponding angles) z = (angles in a triangle) = ( ACB = (isoseles triangle) BCE = (alternate) ACD = (straight line) so, = (angles in triangle)) d = (angles in a triangle) e = (angles in a triangle) option A EXERCISE D Students drawings triangles; triangles Numer of sides in polgon Numer of triangles Angle sum of interior angles numer of triangles is two less than the numer of sides n a times two less than n (n ) interior angle = sum of interior angles = = WORK IT OUT. Option A is orret. In the other solutions the pentagon has not een divided into triangles orretl. EXERCISE E option C a a d a otagon =, =, z =, No. The sum of its eterior angles must equal, so n =. This gives n =., n is not an integer so there is no regular polgon with interior angles of. a CHAPTER REVIEW a a = = = d d = e e = f f = a (orresponding) (o-interior) pentagon =, = z = a deagon I = sum of interior angles, E = sum of eterior angles We know I = (n ) (formula for sum of interior angles) I + E = n the sum of eah interior and eterior angle is and there are n angles So E = n I = n (n ) = n n + = =, = = Sum of interior angles =, adding all angles gives =, so = ABC =, BCD = These are o-interior angles and the sum to, so AB and DC are parallel; ABCD is therefore a trapezium. a =. Perimeter BEFORE YOU START a Pentagon Heagon Otagon a m m. m d mm a True False False l = p w p r = p + Camridge Universit Press

21 Student Book Answers LAUNCHPAD m a mm mm. mm mm a. m. m m. m (to dp) EXERCISE A m. m Option D a a + (or + ) z Option C m m m a All involve the produt of a ase a perpendiular height. For the retangle and square, the perpendiular height happens to e on of the sides of the shape. Beause there is no general relationship etween the lengths of the four sides. a m posts needed with the etras. a Team A: m Team B: m Team C: m. km/hr WORK IT OUT. Option A is orret eause there are sides to the perimeter eah of. m. EXERCISE B retangle width mm parallelogram length mm rhomus side length and width. m square length and width. m a m. m. m d. m e mm a m m m d m e. m f m Option C. m EXERCISE C a. mm. m. m d. m e. m Option C. m. mm a. m d i. m. m ii. m a. mm. m. m A square plate with side lengths. m m EXERCISE D a. m. m. m d. m a. m. m. m d. m e. m f. m g. m h. m i. m j. m WORK IT OUT. Option B is orret. Option A is wrong eause the whole irumferene of eah irle was alulated rather than just. Option C is wrong eause the wrong formula for working out the irumferene was used. EXERCISE E a. m. m a Disus. m. m outer and. m. m Line A is. m long. Line B is. m long.. m and. m. km Option B. m. m a. m. m. m d. m/s e. km f i. million people per annum ii. million CHAPTER REVIEW Option B m Yes.. m is the irumferene and as the rion is m, she has plent spare to tie the ow. m. m No. Cled. km. mm. m m. m a Main. m, stage. m, stage. m Main. m, stage. m, stage. m; Total. m. m Camridge Universit Press

22 GCSE Mathematis for AQA (Higher) Area BEFORE YOU START a Parallelogram Trapezium Retangle a a. a m = m m = mm km = m LAUNCHPAD a A = A = A = d A = e A = a Area (A) oeffiient radius. m ( dp) m m (to dp) EXERCISE A a Option C Option C a. m. mm m. m. m m. m a. m red triangle:. m, green and lue triangles:. m. m d i. mm ii mm ased on white triangle eing approimatel of height of large middle triangle. e i mm ii mm iii mm f Red.%, white.%, lue.% WORK IT OUT. Option C is orret. In Option A oth dimensions are inorret; in Option B the height is inorret. WORK IT OUT. Option A is orret. In Option B the area should e for a retangle not a triangle. In Option C there are m in m. EXERCISE B a m mm m d. m a. mm. m m d. m m Option B a m m m d m e m a m kg of soil, kg of ompost m d i. ii a ( + ) + + d e + f EXERCISE C Option A a. m. m. m d. m a. mm. m. m d. mm. m a. m. m a. m. m a. m. m Cirumferene = πr =. mm. So r = mm. Area = πr =. mm a mm diameter. mm, irumferene. mm diameter. mm, irumferene. mm EXERCISE D a + = m.. +. (...) =. m.... =. m d. +. = m e.. =. m f.... (..) =. m g. π. +. π. =. m h =. m a. m. m. m d. m e. m f. m g. m h. m Option B a perimeter =. m, area =. m perimeter =. m, area =. m perimeter =. m, area =. m EXERCISE E (assuming an ut to fit). m a. m. m. m. m. m. m. a. m. m a possile dimensions: retangles.. m, parallelograms.. m no as area = ase height Camridge Universit Press

23 Student Book Answers Possile answers are: Parallelograms ould e etter for ar parks that do not have edges at right angles. The ould also e useful if spae is tight as, sine vehiles will not have to turn through when reversing, it ma e possile to redue the gap etween rows of parking as. d Students skethes and reasoning CHAPTER REVIEW. m m Option C. m. m. m. m m Rounding and estimation BEFORE YOU START a Corret Inorret Inorret a True True False d True a... LAUNCHPAD a. d. e f. m a.. Around Around litres a. m m <. m a. (a + ) <.. (a) <.. a + a <. EXERCISE A a A A B d A e B a i ii iii iv i ii iii iv i ii iii iv d i ii iii iv e i ii iii iv f i ii iii iv a d m e no, it is. million to the nearest and (or and ) EXERCISE B Option C a i. ii. iii. i. ii. iii. i. ii. iii. d i. ii. iii. e i. ii. iii. a... d. e. f. g. h. i. There will e a variet of justifiations for answers. a. kg. km per litre d. ±. a Rounded to the nearest million The value ould have een as low as as this rounds up to million and the value ould have een just short of, as values just elow this would round down to million. EXERCISE C a i ii iii iv. i ii iii. iv. i ii iii. iv d i ii, iii. iv e i. ii, iii. iv Rounding. to two deimal plaes will give. whih doesn t tell us anthing. Rounding to two signifiant figures is a more aurate wa to round ver small numers. a Option B. kg/m m/s d. m/s a ±. Rounding to two deimal plaes is more aurate than rounding to two signifiant figures. Rounding in two different was would mean that the answer is less aurate than if ou alwas rounded to two deimal plaes. EXERCISE D a... d. a.. d. e. f. Rounding to two deimal plaes is the most useful wa to approimate as this is to the nearest penn. (Rounding up would also alwas ensure there is enough mone to over the ill.) WORK IT OUT. Estimate A is the losest estimate to the atual ost, ut students ma have justifiation for hoosing a different estimate. (For eample, alwas rounding up so ou have an overestimate ma e good for udgeting.) EXERCISE E a = = = d. = e = f = g = h. = Option D a Answer C, = Answer B, = Camridge Universit Press

24 GCSE Mathematis for AQA (Higher) a d. =. + =.. =.. = =. = a.. = =.. a m seonds a = Not sensile = Not sensile = Sensile d = Not sensile e =. Sensile f. =. Not sensile.. m ±.. EXERCISE F a. n <.. n <.. n <. d. n <. e. n <. f. n <. g. n <. h. n <. i. n <. a. m <. m. m <. m. m <. m d. mm <. mm a Option B No.. m length of wood <. m Least weight. kg; greatest weight. kg (weight <.) a. and.,. and.,. and. In the eample, there is nearl a saving for gold and a saving for platinum. a Lower ound:. m Upper ound:. m Lower ound:. seonds Upper ound:. seonds. m L <. m EXERCISE G Upper ound =. kg. kg = kg Lower ound =. kg. kg = kg a LB:. m, UB:. m and LB:. m, UB. m LB:. m, UB:. m LB:. m, UB:. m a. km area <. km.% ( dp) This error represents aout half a square km of land and if land pries are high, this ould e quite signifiant... = a i m ii. m ( sf). m. π <. ( sf). m/s s <. m/s ( dp) g. m/s v <. m/s CHAPTER REVIEW. = p and Option C No, a ar ould weigh. g and this is an error of.% whih eeeds the tolerane.. ohms LB:.%, UB:.% ( dp) Perentages BEFORE YOU START a.. d. =, =, =, =, =, = a false false false d true e false LAUNCHPAD a a %,.,, %,.%,.%,.,., %, %.%.%.. EXERCISE A a Option D Option D Option B a % % % d % e % f.% g.% h % i % j % k % l % m % n % o.% p.% a d e f g h a... d. e. f. g. h. i. j k. l. a.%.%.% % a.%,,., %,., %,,., e.%, %,, %,.,, %,.,. d.,,.,, % Camridge Universit Press

25 Student Book Answers % % a out of.% WORK IT OUT. A is orret. B Answer is too ig. is not a perentage, it is a mied numer ( ) C. is whih is onl.%. D The % sign is missing from the alulation. E is not %, it is onl.%. EXERCISE B a. d e f g h i a. kg. m d. kg e f. min g h. m i litres out of a option D option C., so phones a. a. m. m a t =.% gold and t = % gold. g. g d Student s own reasoning, ut researh will show that even if arat is onl.% pure gold, gold remains the largest omponent of the allo. EXERCISE C a The village Students own reasoning, ut given the numers, the it is proal still the most risk in terms of rime. WORK IT OUT. Student B is orret. Student A is wrong. Part of a rae annot e greater than the whole (%), so % an t e right. EXERCISE D a %.% % d % e % f.% g.% h.% i.% j.% k.% l.% m.% n.% o.% p.% q % r.% s.% t % option C Angelika % % %.%.% a.%.% % WORK IT OUT. Answers here will var from student to student. EXERCISE E a.. d e. f. option D a.. d e. f. option D.,. a It means that ompared with the average amount of rainfall (in mm) over the past period, the rainfall inreased almost. You d need to know the average rainfall so ou ould work out how muh more rain atuall fell in.. EXERCISE F a g. kg d. option C a.. d. option A a students students g. kg runners CHAPTER REVIEW a option D option B a a %.%.% d % e % f.% Camridge Universit Press

26 GCSE Mathematis for AQA (Higher) % inrease. million. hours ( h min) a.%.%. to nearest pene Powers and roots BEFORE YOU START a < = > d = a B C C a LAUNCHPAD a d e ( ) f ( ) a = = = d = e = = f = EXERCISE A Option D a Inde Base = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = The negative powers are unit frations of the mathing positive powers. Ever seond power of two is equivalent to a power of four. d If it does not end in it annot e a power of. a T F T d F e T f F g F h T i T j T k F l T a d e f g h i j k l EXERCISE B a d e f g h i j k l m n o p q r a d e f g h a d e f a > < > d < e < f > EXERCISE C Option C a d e f g h i j k l m a d e f g h i j k l m n o a = = = d e = f = g = h = i j k l a d e f g h i EXERCISE D a d e f g h i j k l a d e f g h i i k l a d e f g h i j k l a T F: T d T e T f T g F: h T i T EXERCISE E a d e f () g () h () a e f g d h ( ) Camridge Universit Press

27 Student Book Answers a d e f g h i EXERCISE F Students estimates ma var slightl. a... d. e. f. Answers will depend on students solutions to question. Students estimates ma var slightl. a... d = e. f. a =, m = a =, m = a =, m = a =, m =. mm. mm EXERCISE G a Option A Option D a Suki is orret. Marie entered the power without rakets, and as eponents get preferene in operations, the alulator finds the ue and then divides the total.. mg. mg d.% a.. F = P(.) a. seonds d. i Matt s as its falling a shorter distane ii. seonds a Students own answers. Elephant:. seonds, Human:. seonds. Elephant s lood takes seonds longer to irulate.. kg d Mouse:. m ; Cat. m e Approimatel alories f The roots and powers do not give eat rational answers. Plus the formulae are ased on mean values and all animals are unique. a. Jupiter. das; Uranus. das. Uranus takes das longer (appro.. ears longer). Merur. It is losest, so has the smallest orit. das. CHAPTER REVIEW Option C a d a,,,,,,,,, a ( ),,, a d e f a. d e f g h i. m. = = volts a Aout seond The longer the pendulum, the longer it takes to omplete one swing. Yes the will fit. Container has r =., so d =.. Bisuit diameter of will fit. a, Standard form BEFORE YOU START a... d a.. a True True False: should e d True LAUNCHPAD D C B E.. EXERCISE A Option D a... d. e f g. h. i. j. k. l. m. n. o p. q. r. a d e. f g h. i j. k. l. m. n. o. a.. d e. f g. h. a.. d e. f g h. kg Camridge Universit Press

28 GCSE Mathematis for AQA (Higher) EXERCISE B Displa will var depending on alulator used. Option A a... d. e. f. g. h. i. j. EXERCISE C Option D a... d. e. f. g. h. i. j. a... d. e. f. g. h. i. EXERCISE D Option D a.. d. e f g h a.. d. e f g h. a... d. e f. ( sf) g. h a m m m d m a. times; times. links EXERCISE E a.. d. e. f Option C a The Paifi. km Total area =. (. ) =. = km. miles a Virus A. m. mm EXERCISE F Option C a.. a piels. m piels d m e m a. nm. nm a seonds = seonds. seonds =. seonds ( sf) a. ells. ells ( sf) Student s individual prolems. CHAPTER REVIEW a.. d. e. f a d. Option C a... d. e. ( sf) f. ( sf) a... d. ( sf) a.. a m m m a The Sun. a. km. km. km Further algera BEFORE YOU START a and are not like terms as the have different powers. + Yes / + / a a a d a B C D d A LAUNCHPAD e a a (a + ) (a + ) ( ) ( ) (p ) (p + ) d ( + )( ) a + ( + ) ( + ) EXERCISE A Option D a d + e f + + a d e a a + f + g + h i + a a z + + z + z a = EXERCISE B a a d ( + )( ) a Eah of the two terms will itself e a perfet square and the sign etween the terms will e a negative. Camridge Universit Press

29 Student Book Answers i No. The numer of s and s is not the same in eah raket. ii Yes. Notie that the epression must first e divided the ommon fator,. iii Yes. EXERCISE C a d + + a d a ( + )( )( ) + + These are a perfet square. EXERCISE D a ( + )( + ) ( + )( + ) ( + )( + ) d ( + )( + ) e ( + )( + ) f ( + )( + ) a ( )( ) ( )( ) ( )( ) d ( )( ) e ( )( ) f ( )( ) a ( + )( + ) ( )( ) ( )( + ) d ( + )( + ) e ( )( + ) f ( + )( ) Option B WORK IT OUT. Option B is orret. Option A put in the raket, not. Option C has a negative in oth rakets. EXERCISE E a ( + )( ) (p + )(p ) (w + )(w ) d (p + q)(p q) e (s + )(s ) f (h + g)(h g) Option D a ( + )( ) ( + z)( z) ( )( ) d ( + )( + ) e ( + )( ) f ( + + )( + ) a ( )( + ) = = ( )( + ) = = ( )( + ) = = d ( )( + ) = = e ( )( + ) = = f ( )( + ) = = a ( )( + ) = = a = = ( )( + ) = = a = ( )( + ) = = a = d (.)( +.) =.. =. a =. EXERCISE F a ( + )( + ) ( + )( + ) ( )( + ) d ( + )( + ) e ( )( ) f ( )( + ) g ( )( + ) h ( )( + ) i ( + )( ) j ( + )( ) + fatorises into ( )( ), so the length is ( ) m. + + fatorises into ( + )( + ). ( + ) is half the ase, so ( + ) is the height. a Let a = ( + ). The epression eomes a + a +, whih fatorises into (a + )(a + ). Replaing a with ( + ): ( + ) + ( + ) + [( + ) + ]( + + ). i ( )( ) ii ( + )( + ) WORK IT OUT. In all ases, parts of the epressions onl have een anelled. : annot e simplified ( ) ( + ) ( ) : = ( + )( + ) ( + ) : ( + ) ( ) ( + ) ( ) = ( )( ) ( + ) ( + ) EXERCISE G a d + e + f g + + Option B a d e a + ( )( ) + ( )( + ) d p + p(p + ) EXERCISE H f + ( + )( + ) e p + p + (p )(p + ) ( ) + ( + )( + ) ( )( + ) f ( )( )( ) a T ( + )( + ) ( + ) d T e T f ( )( + ) = ( )( + ) and ( ) ( + ) ( ) = ( + + ) = ( + )( + ), so ( ) is not a fator ( ) ( + )( ) = ( ) ( ) = + a a... d. i + + = ( + ) + as ( + ) ii + + = ( + ) + as ( + ) Triangle is right-angled if ( + ) + ( ) = ( + ) Epanding and simplifing gives = Fatorising gives ( )( + ) = >, so = is the onl solution a If perimeter = m and width is w, length is ( w)/ = w, so area = w( w) w w = (w ) + As (w ), area Camridge Universit Press

30 GCSE Mathematis for AQA (Higher) ( / ) ( / ) = + / So + / and + / a + + CHAPTER REVIEW (a + ) a (a ) a + + Option B a ( + )( ) ( + )( + ) Area = ( )( + ) + ( ) = + = = a z ( ) ( + ) ( + )( + ) d ( + )( ) a (p q) d e (p + q)(p q) Equations BEFORE YOU START + ( + )( ) ( )( ) f a D B A d C A: + = represents the statement. a + [ ] = [] = a + [a] = d [ ] = e [] = f [ ] = a C ( )( ) A ( +) B ( + )( ) LAUNCHPAD d D (+ )( ) a D = B = A = d E = e C = ( + )( ) f You an hek whether a solution is orret sustituting it ak into the equation. a a = = a = d =. or =. Yes (In the seond equation, taking from oth sides produes the first equation) + = = = or = = ± ( ) a =, =, = a + = has solutions: =, = ; =, = ; =, = ; =, = ; =, = = and = are the onl pair that satisf the pair of equations simultaneousl. a = = solutions; approimate solution is =. EXERCISE A a d e f a d e f Option C EXERCISE B a d e f g h. i j Option D EXERCISE C a = ; = = ; = + = ; = d = ; = e + + = ; = ; numers are and. f = ; = g + = ; a + = ; Melissa is + = ;. + = ; d ( + ) = ( + ) + ; Daughter is e + + = ; = ; Gina is f + =.( + ); = ; woman is a m ( +) m + = ; = ; side of square is m + = ; = ; length = m, width =. m A =, B =, C = and eah of apple and orange juie artiles at and artiles at tikets at and tikets at EXERCISE D Option C a = or = = or = = or = d = or = e = or = f = or = g = or = h = ± i = ± No, it s a sum rather than a differene of two squares. a = or = = or = = or = d = or = e = or = f = or = g = or = h = or = Camridge Universit Press

31 Student Book Answers EXERCISE E a +, ( ) + +, ( + ) + +, ( + ) d + +, ( + ) e +, ( ) f +, ( ) g + +, ( + ) h +, ( ) i + +, ( + ) j +, ( ) Option D a =. or. =. or. =. or. d =. or. e =. or. f =. or. g =. or -. h =. or. i =. or -. a =. or. =. or. =. or. d =. or. e =. or. f =. or. ± a = = ± ± = ± d = ± e = f = ± EXERCISE F a ( + ) = ; = or = ( ) = ; = or = =; = or = d ( + ) = ; = ; numers are and. = m m = ; dimensions of retangles are m m and m m, giving an area of m for eah retangle m and m EXERCISE G a =, = =, = =, = d =, = e =, = f =, = Option D a =, = =, = =, = a =, = =, = =, = d =, = e =, = f =, = EXERCISE H a =, = =, = =, = d =, = e =, = f =, = g =, = h =, = i =, = a =, = =, = =, = a =, = =, = =, = d =, = e =, = f =, = a =, = =, = =, = d =, = e =, = f =, = EXERCISE I Fizzers ost p; toffees ost p Option B Three p piees and fifteen p piees and a =, = and. hard drive, flash and EXERCISE J a =, = =, = or =, = no solution d The two lines will not interset so there is no solution to the pair of equations. a (, ) (, ) or (, ) (, ) or (, ) d (, ) or (, ) e (, ) or (, ) f (, ) or (, ) Option C (, ) and (, ) EXERCISE K a miles minutes mph a a ~ minutes km km/hour a litres minutes Students hosen points and eplanations a When units have een sold (Costs = revenue = ). It tells the usiness owner how man units must e sold in order to make a profit. =, = a -ais is distane along the ground; -ais is height = and =. The are the roots d (.,.) a The roots of the equation are the values where the graph rosses the -ais. = and = = + a Appro =., =. =., =. The aura is limited how aurate the graph is and how well the values an e read from the sale. a m a. kg. m. to. m Camridge Universit Press