Bridging the gap between GCSE and A level mathematics

Transcription

1 Bridging the gap between GCSE and A level mathematics This booklet is designed to help you revise important algebra topics from GCSE and make the transition from GCSE to A level a smooth one. You are advised to spend around an hour on each of the si sections, reading through the eamples carefully and completing questions from each eercise, on lined paper, showing well structured algebraic methods. Please mark your own work using the answers at the end of the booklet. You do not need to complete every question in every eercise but should make sure you are confident with all of these skills as they underpin a great deal of the A level maths content. The initial test (non-calculator) in September will be based on these topics. Your teacher will also want to see the work you have done over summer during the first week of lessons. Welcome to A Level mathematics! The object of these pages is to help you get started with the A Level course, and to smooth your path through it. Many students find A Level a challenge compared with GCSE. This is a recognised issue if you find this you are very far from being alone! I hope that these pages will help you. The main focus is on developing skills, as opposed to learning new material. So I suggest that you don t approach it with the mind-set what do I have to do to get full marks? but what can I learn that will help me in my future studies? You want to be fluent in a number of aspects of GCSE work not just able to get an answer that would score a mark in a GCSE eamination. For eample, you will want to get a result in a form that you yourself can go on and use readily. So there will be an emphasis on real fluency in algebra. Likewise, you should try to develop the ability to think of the shapes of graphs corresponding to algebraic formulae. Do not be afraid of any of this; over the years the vast majority of A Level students have succeeded with the course. But you will find the path easier, enjoy it more and be more confident if you are really on top of the basic vocabulary of the subject before you start putting it together into A Level sentences. One recurring theme here is; just because you can multiply out brackets, it doesn t mean that you should. Indeed, at this level it is often better to keep an epression in its factorised (bracketed) form. Good luck!

2 Contents Simple algebraic epressions... Eercise... 4 Algebraic Fractions & Rearranging... 5 Eercise... 0 Quadratic Epressions... Eercise Cancelling... 6 Eercise Simultaneous equations... 0 Eercise Fractional and negative powers, and surds... Eercise Further reading... 5 Answers, hints and comments... 6

3 Many people dislike algebra; for many it is the point at which they start switching off mathematics. But do persevere most of it is natural enough when you think about it the right way. Simple algebraic epressions Some very basic things here, but they should prove helpful. Are you fully aware that and are the same thing? 4 4 8(5 4) Eample Find the value of a for which 8 (5 4) is always true. a Solution Dividing 8 by and multiplying by (5 4) is the same as multiplying 8 by (5 4) and dividing by. So a =. You do not need to multiply anything out to see this! Remember that in algebraic fractions such as, the line has the same effect as a bracket round the denominator. You may well find it helpful actually to write in the bracket:. ( ) Eample Solve the equation. Solution Multiply both sides by ( ): = ( ) Multiply out the bracket: = 4 Add 4 to both sides: 7 = 7 Divide by : 4. A common mistake is to start by dividing by. That would give you will still have to multiply by ( ). 4 [not = 4] and Don t ever be afraid to get the -term on the right, as in the last line but one of the working. After all, 7 = means just the same as = 7!

4 Eample Make cos A the subject of the formula a = b + c bc cos A. Solution Here it is best to get the term involving cos A onto the left-hand side first, otherwise you are likely to get in a muddle with the negative sign. So: Add bc cos A to both sides: a + bc cos A = b + c Subtract a from both sides: bc cos A = b + c a Divide by bc: b c a cos A bc Rearranging the Cosine Formula is always a dangerous area, as you may well have found at GCSE. Some people actually prefer to memorise this formula for cos A. 7 Eample 4 Solve the equation ( ) (4 9) 5 5 Solution Do not multiply out the brackets to get fractions that leads to horrible numbers! Instead: 7 Multiply both sides by 5: 5 ( ) 5 (4 9) Cancel down the fractions: ( ) (4 9) Choose 5 as it gets rid of all the fractions. 9( ) 7(4 9) Now multiply out: = = 0 Hence the answer is = 9 This makes the working very much easier. Please don t respond by saying well, my method gets the same answer! You want to develop your fleibility and your ability to find the easiest method if you are to do well at A Level, as well as to be able to use similar techniques in algebra instead of numbers. It s not just this eample we are worried about it s more complicated eamples of a similar type.

5 Eercise Find the values of the letters p, q and r that make the following pairs of epressions always equal. 7 ( ) ( 7 ) ( ) ( 7 ) 5 0 p q r Solve the following equations Make cos C the subject of the formula c = a + b ab cos C. 4 Multiply 5 4 by 8. Multiply ( + ) by. Multiply ( 7) by 6. (d) Multiply ( ) by Solve the following equations. 5 ( ) ( ) (5) ( 4) () ( ) 9 6 Make the subject of the following equations. a a a ( c d ) ( c d) ( d) b b b 7 Simplify the following as far as possible. aaaaa 5 bbbb b ccccc c (d) dddd 4

6 Algebraic Fractions & Rearranging Many people have only a hazy idea of fractions. That needs improving if you want to go a long way with maths you will need to be confident in handling fractions consisting of letters as well as numbers. Remember, first, how to multiply a fraction by an integer. You multiply only the top [what happens if you multiply both the top and the bottom of a fraction by the same thing?] Eample Multiply 4 9 by. Solution 4 =, so the answer is 9. Sometimes you can simplify the answer. If there is a common factor between the denominator (bottom) of the fraction and the number you are multiplying by, you can divide by that common factor. Eample Multiply 7 9 by. Solution 9 =, so the answer is 7. You will remember that when you divide one fraction by another, you turn the one you are dividing by upside down, and multiply. If you are dividing by a whole number, you may need to write it as a fraction. Eample Divide 7 8 by Solution 5, so the answer is But if you can, you divide the top of the fraction only. Eample 4 Divide 0 4 by Solution, so the answer is 4. Note that you divide 0 by Do not multiply out 5 4; you ll only have to divide it again at the end! Eample 5 Multiply 7y by. Solution = 6, so the answer is 6 7y. (Not 6 4y!)

7 Eample 6 Divide y 4 by y. Solution y y y y y, so the answer is y. [Don t forget to simplify.] 4 4 y 4y 4 4 PQR Eample 7 Divide by T. 00 Solution PQR 00 PQR PQR T 00 T 00T. PQR Here it would be wrong to say just 00, which is a mi (as well as a mess!) T Double fractions, or mitures of fractions and decimals, are always wrong. For instance, if you want to divide y z by, you should not say 0.5y z but y z. This sort of thing is etremely important when it comes to rearranging formulae. Eample 8 Make r the subject of the equation V = r h. Solution Multiply by : V = r h Don t divide by. Divide by and h: V h = r Square root both sides: V r. h You should not write the answer as V h or V h, as these are fractions of fractions. Make sure, too, that you write the answer properly. If you write V/h it s not at all clear that the whole epression has to be square-rooted and you will lose marks.

8 More complicated rearranging:

9 If you do get a compound fraction (a fraction in which either the numerator or the denominator, or both, contain one or more fractions), you can always simplify it by multiplying all the terms, on both top and bottom, by any inner denominators. Eample 9 Simplify. Solution Multiply all four terms, on both top and bottom, by ( ): ( ) ( ) ( ) ( ) ( ) ( ) You will often want to combine two algebraic epressions, one of which is an algebraic fraction, into a single epression. You will no doubt remember how to add or subtract fractions, using a common denominator. Eample 0 Simplify. Solution Use a common denominator. [You must treat ( ) and ( + ) as separate epressions with no common factor.] ( ) ( ) ( )( ) ( )( ) 4. ( )( ) Do use brackets, particularly on top otherwise you are likely to forget the minus at the end of the numerator (in this eample subtracting - gives +). Don t multiply out the brackets on the bottom. You will need to see if there is a factor which cancels out (although there isn t one in this case).

10 Eample Simplify 5. Solution A common denominator may not be obvious, you should look to see if the denominator factorises first. is a common 5 5 factor, so the common ( ) denominator is ( )( ) ( )( ). ( ) 5 ( )( ) 5 ( )( ) 7 ( )( ) If one of the terms is not a fraction already, the best plan is to make it one. Eample Write as a single fraction. Solution ( ) 5 This method often produces big simplifications when roots are involved.

11 Eample Write as a single fraction. Solution ( ) ( ) Eercise Work out the following. Answers may be left as improper fractions. (e) (d) (f) 8 (g) 6 7 (h) (i) y (j) y y (k) 5 4y (l) 5 y 6y (m) 5 y 4 (n) y 4z (o) 6 y y (p) 5z 5a 6 z y Make the subject of the following formulae. A = V 4 W (u + v) = t (d) h Simplify the following compound fractions.

12 4 Write as single fractions. (e) (f) (g) (d) 4( ) 4 5 Write as single fractions. ( )

13 Quadratic Epressions You will no doubt have done much on these for GCSE. But they are so prominent at A Level that it is essential to make sure that you are never going to fall into any traps. First, a reminder that ( + ) is not equal to + 9 y is not equal to + y. It is terribly tempting to be misled by the notation into making these mistakes, which are really optical illusions. If you always remember that square means multiply by itself you will remember that ( ) ( )( ) From this it follows, of course, that 6 9 = ( + ), so 9 can t be +. In fact 9 does not simplify. Nor do 9 or 9. If you are tempted to think that they do, you will need to make a mental note to take care whenever one of these epressions comes up. You will certainly deal with many epressions such as ( + ) + (y 4) and you will need to be able to use them confidently and accurately. A related process is to write a quadratic epression such as 6 in the form ( a) b. This is called completing the square. It is often useful, because 6 is not a very transparent epression it contains in more than one place, and it s not easy either to rearrange or to relate its graph to that of. Completing the square for quadratic epressions in which the coefficient of is (these are called monic quadratics) is very easy. The number a inside the brackets is always half of the coefficient of.

14 Eample Write in the form ( + a) + b. Solution is a monic quadratic, so a is half of 6, namely. When you multiply out ( + ), you get [The -term is always twice a, which is why you have to halve it to get a.] isn t quite right yet; we need 4 at the end, not 9, so we can write = ( + ) = ( + ) 5. And we can solve the equation = 0 eactly without having to use the quadratic equation formula, to locate the roots of the function: = 0 ( + ) 5 = 0 ( + ) = 5 ( + ) = 5 [don t forget that there are two possibilities!] = 5 These are of course the same solutions that would be obtained from the quadratic equation formula not very surprisingly, as the formula itself is obtained by completing the square for the general quadratic equation a + b + c = 0. Non-monic quadratics Everyone knows that non-monic quadratic epressions are hard to deal with. Nobody really likes trying to factorise (although you should certainly be willing and able to do so for A Level, which is why some eamples are included in the eercises here). If you re unsure how to factorise non-monic quadratics, watch the video at before attempting the eercise. Eample Write + + in the form a( + b) + c. Solution First take out the factor of : + + = ( ) [you can ignore the.5 for now]

15 Now we can use the method for monic quadratics to write = ( + ) + (something) So we have, so far Half of = ( + ) + c [so we already have a = and b = ] Now ( + ) = ( ) = We want at the end, not 8, so: + + = ( + ) 8 + = ( + ) + 5. If the coefficient of is a perfect square you can sometimes get a more useful form. Eample Write in the form (a + b) + c. Solution It should be obvious that a = (the coefficient of a is 4). So = ( + b) + c If you multiply out the bracket now, the middle term will be b = 4b. So 4b must equal 0 and clearly b = 5. And we know that ( + 5) = So = ( + 5) = ( + 5) 6. Eercise Write without brackets. ( + 5) ( 4) ( + ) (d) ( ) (e) ( + )( ) (f) ( + 4)( 4)

16 Simplify the following equations into the form a + by + c = 0. ( + ) + (y + 4) = ( ) + (y ) ( + 5) + (y + ) = ( 5) + (y ) ( + ) + (y ) = ( + ) + (y + ) Simplify the following where possible (d) 9 (e) y (f) y y 4 Write the following in the form ( + a) + b (d) 4 (e) + (f) Write the following in the form a( + b) + c Write the following in the form (a + b) + c Factorise as fully as possible y 4 (d) 7 + (e) 5 + (f) (g) Multiply out and simplify.

17 4 Cancelling The word cancel is a very dangerous one. It means two different things, one safe enough and the other very likely to lead you astray. You can cancel like terms when they are added or subtracted. Eample Simplify ( y) + (y y ). Solution ( y) + (y y ) = y + y y = y. The y terms have cancelled out. This is safe enough. It is also usual to talk about cancelling down a fraction. Thus 0 =. However, this tends 5 to be very dangerous with anything other than the most straightforward numerical fractions. Consider, for instance, a fraction such as y. If you try to cancel this, you re almost y y certain not to get the right answer, which is in fact y (as we will see in Eample 4, below). Try instead to use the word divide. What happens when you cancel down 0 is that you 5 divide top and bottom by 5. If you can divide both the top and bottom of a fraction by the same thing, this is a correct thing to do and you will get a simplified answer. Contrast these two eamples: 4 8 y and 4 8 y. 4 4 In the first, you can divide both 4 and 8y by 4 and get + y, which is the correct answer (though it is rather safer to start by factorising the top to get 4( + y), after which it is obvious that you can divide top and bottom by 4.) In the second eample, you don t do the same thing. 4 8y = y. This can be divided by 4 to get 8y, which is the correct answer. Apparently here only one of the two numbers, 4 and 8, has been divided by 4, whereas before both of them were. That is true, but it s not a very helpful way of thinking about it. With problems like these, start by multiplying together any terms that you can (like the 4 and the 8y in the second eample). Then, if you can, factorise the whole of the top and/or the bottom of a fraction before doing any cancelling. Then you will be able to see whether you can divide out any common factors.

18 Eample Simplify 4 6 y. 6y Solution 4 6y ( y) y 6y 6( y) ( y) The top factorises as ( + y). The bottom factorises as 6( + y). and 6 have a common factor of, which can be divided out to give. But ( + y) and ( + y) have no common factor (neither nor divides into y or y, and neither nor y divides into ). So you can t go any further, and the answer is ( y ). ( y) Eample Eplain why you cannot cancel down y. Solution There is nothing that divides all four terms (, y, and ), and neither the top nor the bottom can be factorised. So nothing can be done. Eample 4 Simplify y. y y Solution Factorise the top as ( + y) and the bottom as y( + y): y ( y) y y y( y) Now it is clear that both the top and the bottom have a factor of ( + y). So this can be divided out to give the answer of y. Don t cancel down. Factorise if you can; divide all the top and all the bottom. Taking out factors I am sure you know that 7 + can be factorised as (7 + ). You should be prepared to factorise an epression such as 7( + ) + ( + ) in the same way.

19 Eample 5 Factorise 7( + ) + ( + ) Solution 7( + ) + ( + ) = ( + ) (7 + ( + )) = ( + ) ( + ). The only differences between this and 7 + are that the common factor is ( + ) and not ; and that the other factor, here (7 + ( + )), can be simplified. If you multiply out the brackets you will get a cubic and you will have great difficulty in factorising that. Don t multiply out brackets if you can help it! Epressions such as those in the net eercise, question 4 parts and (d) and question 5 parts (e) (h), occasionally arise in two standard techniques, the former in Further Mathematics (Mathematical Induction) and the latter in A Mathematics (the Product and Quotient Rules for differentiation). They may look a bit intimidating at this stage; feel free to omit them if you are worried by them. Eercise 4 Simplify the following as far as possible. 5 + y + 7 y + 4y + y + 4y y. 4 6 (d) 46 (e) y (f) y (g) 40y 8 6y (h) 6y 9 y (i) 4 9y y (j) 4 6y 6 9y (k) 5y 6y 0 y (l) 4y 6 8y (m) (n) y y y y Make the subject of the following formulae. a b py qz a 4y b qz

20 Simplify the following. ab r h rb 4 hr 4 Simplify into a single factorised epression. ( ) + 5( ) 4( + ) + 5( + ) 4 k( k ) ( k ) (d) k( k )(k ) ( k ) 6 5 Simplify as far as possible ( ) ( ) (d) ( ) ( ) ( ) (e) (f) (g) (h) ( )

21 5 Simultaneous equations I am sure that you will be very familiar with the standard methods of solving simultaneous equations (elimination and substitution). You will probably have met the method for solving simultaneous equations when one equation is linear and one is quadratic. Here you have no choice; you must use substitution. Eample Solve the simultaneous equations + y = 6 + y = 0 Solution Make one letter the subject of the linear equation: = 6 y Substitute into the quadratic equation (6 y) + y = 0 Solve 0y 6y + 6 = 0 (y )(5y ) = 0 to get two solutions: y = or.6 Substitute both back into the linear equation = 6 y = or.8 Write answers in pairs: (, y) = (, ) or (.8,.6) You can t just square root the quadratic equation. [Why not?] You could have substituted for y instead of (though in this case that would have taken longer try to avoid fractions if you can). It is very easy to make mistakes here. Take great care over accuracy. It is remarkably difficult to set questions of this sort in such a way that both pairs of answers are nice numbers. Don t worry if, as in this eample, only one pair of answers are nice numbers. Questions like this appear in many GCSE papers. They are often, however, rather simple (sometimes the quadratic equations are restricted to those of the form + y = a) and it is important to practice less convenient eamples.

22 Eercise 5 Solve the following simultaneous equations. + y = 4 + y = + y = 0 y = + y + y = 4 y + y = + y = y = 5 c + d = y = 5 c + 4d = y = y + y = y + 6y = 4 + 4y = + y = y = 7 0 y + y = 0 + y = 5 + y = 9 + y + 5y = 5 y + + y = 7 y = y = 5 + y + 5y = y y = 0 y = y = 0 5 y = 6 y y = + y = 7

23 6 Fractional and negative powers, and surds This may seem a rather difficult and even pointless topic when you meet it at GCSE, but you will soon see that it is etremely useful at A Level, and you need to be confident with it. Negative powers give reciprocals ( over the power). Fractional powers give roots (such as ). 0 = for any (apart from 0 0 which is undefined). Eamples 0 = (d) 4 7 7/4. The easiest way of seeing this is to write it as 7 ( ) 4. There is a particularly nice way of understanding the negative powers. Consider the following: Every time you move one step to the right you multiply by. Now consider the sequence continuing, right-to-left: Each time you move one step to the left you divide by. Take particular care when there are numbers as well as negative powers. Eample 0 0 but 0 0 or (0).

24 The usual rules of powers and brackets tell you that 0 is not the same as (0). You will make most use of the rules of surds when checking your answers! An answer that you give as 6 will probably be given in the book as, and as 7. Before 7 worrying why you have got these wrong, you should check whether they are equivalent! Indeed, they are, as and ( 7) ( 7) ( 7) 9 7 The first of these processes is usually signalled by the instruction write in surd form and the second by rationalise the denominator. Remember also that to put a square root in surd form you take out the biggest square factor you can. Thus 48 = 6 = 4 (noting that you should take out 6 and not 4).

25 Eercise 6 Write the following as powers of. 5 5 (d) 5 (e) (f) Write the following without negative or fractional powers. 4 0 /6 (d) /4 (e) / Write the following in the form a n. 4 5 (d) (e) 6 4 Write as sums of powers of Write the following in surd form (d) 5 (e) 6 Rationalise the denominators in the following epressions (d) 5 (e) Simplify

26 Further reading There are not many books designed for the sort of transition that this booklet represents, but an outstanding eception is: Fyfe, M. T., Jobbings, A. and Kilday, K. (007) Progress to Higher Mathematics, Arbelos. ISBN Alternatively, also published by Arbelos is an epansion of the same book which is more specifically aimed at the transition to A Level. You may not need quite so much as this: Fyfe, M. T., and Kilday, K. (0) Progress to Advanced Mathematics, Arbelos. ISBN

27 Answers, hints and comments Eercise p = 7 q = 5 r = 0 = = 5 = 4 a b c cosc ab 4 ( + 5) 4( + ) ( + 7) (d) ( ) 5 = 4 = b ad ac b (or ad b ) b ac d(4 b) bc (or db ( 4) ) bc 7 a 4 c 4 4 (d) d 4 Eercise (d) 4 5 (e) (f) 8 (g) 7 (h) 6 5 (i) y (j) y (k) 5 4 y (l) 5 6 y (m) 5 y 4 (n) 4 y z (o) y 5z (p) 5a yz A V 4 u v (d) t W h 4 5 ( )( ) 7 ( )( ) ()()

28 (d) 5 (e) (f) (g) 6 4()( ) 5 Eercise (d) (e) 4 (f) y + = y = 0 + y = 0 impossible ( ) impossible (d) impossible (e) impossible (f) + y 4 ( + 4) + ( 5) ( + ) 5 (d) ( ) 7 (e) ( ½) ¼ (f) ( ½) ¼ 5 ( + ) + 4 5( ) ( + ½) + ½ 6 ( + ) + 5 ( ) 5 (4 + 5) 7 ( 5)( + 5) 4( )( + ) ( y )( + y )

29 (d) ( )( ) (e) ( )( ) (f) ( + )( ) (g) (4 + 5)( ) 8 6 Eercise (d) (e) + y (f) y (g) 5y 4 y (h) y y (i) can t be simplified (j) (k) y (l) can t be simplified (m) (n) bpy aqz 4by aqz 6 abr h br 4 [see Eample 5] ( ) (5 4) ( + ) (4 + 5) ( k )( k ) (d) 6 ( k )( k )( k ) (d) () (e) (f) (g) (h) ( )

30 Eercise 5 The answer to the question why not? in eample (page 6) is that + y has no simple square root. In particular it is not + y. [Remember that ( + y) = + y + y.] (, 4), (, ) (6, ), (7, 0) (, ), (, 0) 4 (, ), (, ) 8 5 (, ), ( 5, 4 ) 6 (7, 4), (8, 7 ) 5 7 ( 5, 4), ( 9, 5 5 ) 8 (, ), ( 5, 7 9 ) 9 (, ), (, 4) 0 (, 6), ( 9, 9 ) 5 4 (, ), ( 9, 9 ) (, ), ( 8 ( 5, ), ( 4 5, 5 ) 4 (, ) (only) 5 (6, 5) (only) 6 (6, ), ( 4, 7 ) ), 9 Eercise 6 5 /5 (d) /5 (e) / (f) / 4 6 (d) 4 (e) 4 / 5 / (d) ½ (e) (d) 5 5 (e) (7 ) (d) 4 7 In this question apply the method of 5(e) to each separate part. ( ) + ( ) + (4 ) + + (00 99) = 00 = 9. ( 5) (e) 6 + 5