Maths Department A Level Induction Booklet One of the most important things if you are to succeed at A Level Maths is to ensure you understand all the algebra you met at GCSE. Working through the eamples and questions in this booklet will help ensure you are prepared for your course. CONTENTS Chapter 1 Removing brackets page Chapter Linear equations 4 Chapter 3 Simultaneous equations 8 Chapter 4 Factors 10 Chapter Change the subject of the formula 13 Chapter 6 Solving quadratic equations 16 Chapter 7 Indices 18 Answers to Eercises 1 Name:..
Chapter 1: REMOVING BRACKETS To remove a single bracket, we multiply every term in the bracket by the number or the epression on the outside: Eamples 1) 3 ( + y) = 3 + 6y ) -( - 3) = (-)() + (-)(-3) = -4 + 6 To epand two brackets, we must multiply everything in the first bracket by everything in the second bracket. We can do this in a variety of ways, including * the smiley face method * FOIL (Fronts Outers Inners Lasts) * using a grid. Eamples: 1) ( + 1)( + ) = ( + ) + 1( + ) or ( +1)( + ) = + + + = + 3 + or 1 ( +1)( + ) = + + + = + 3 + ) ( - )( + 3) = ( + 3) - ( +3) = + 3 4-6 = 6 or ( - )( + 3) = 6 + 3 4 = 6 or - -4 3 3-6 ( +3)( - ) = + 3-4 - 6 = - - 6
EXERCISE A Multiply out the following brackets and simplify. 1. 7(4 + ). -3( - 7) 3. a 4(3a - 1) 4. 4y + y( + 3y). -3 ( + 4) 6. ( - 1) (3-4) 7. ( + )( + 3) 8. (t - )(t - ) 9. ( + 3y)(3 4y) 10. 4( - )( + 3) 11. (y - 1)(y + 1) 1. (3 + )(4 ) Two Special Cases Perfect Square: Difference of two squares: ( + a) = ( + a)( + a) = + a + a ( - a)( + a) = a ( - 3) = ( 3)( 3) = 4 1 + 9 ( - 3)( + 3) = 3 = 9 EXERCISE B Multiply out 1. ( - 1). (3 + ) 3. (7 - ) 4. ( + )( - ). (3 + 1)(3-1) 6. (y - 3)(y + 3) More help: http://www.mymaths.co.uk/tasks/library/loadlesson.asp?title=brackets/bracketsmovie&taskid=11 0
Chapter : LINEAR EQUATIONS When solving an equation, you must remember that whatever you do to one side must also be done to the other. You are therefore allowed to add the same amount to both side subtract the same amount from each side multiply the whole of each side by the same amount divide the whole of each side by the same amount. If the equation has unknowns on both sides, you should collect all the letters onto the same side of the equation. If the equation contains brackets, you should start by epanding the brackets. A linear equation is an equation that contains numbers and terms in. A linear equation does not 3 contain any or terms. More help on solving equations can be obtained by downloading the leaflet available at this website: http://www.mathcentre.ac.uk/resources/workbooks/mathcentre/web-simplelinear.pdf Eample 1: Solve the equation 64 3 = Solution: There are various ways to solve this equation. One approach is as follows: Step 1: Add 3 to both sides (so that the term is positive): 64 = 3 + Step : Subtract from both sides: Step 3: Divide both sides by 3: 39 = 3 13 = So the solution is = 13. Eample : Solve the equation 6 + 7 =. Solution: Step 1: Begin by adding to both sides 8 + 7 = (to ensure that the terms are together on the same side) Step : Subtract 7 from each side: 8 = - Step 3: Divide each side by 8: = -¼ Eercise A: Solve the following equations, showing each step in your working: 1) + = 19 ) = 13 3) 11 4 = 4) 7 = -9 ) 11 + 3 = 8 6) 7 + = 4
Eample 3: Solve the equation (3 ) = 0 3( + ) Step 1: Multiply out the brackets: 6 4 = 0 3 6 (taking care of the negative signs) Step : Simplify the right hand side: 6 4 = 14 3 Step 3: Add 3 to each side: 9 4 = 14 Step 4: Add 4: 9 = 18 Step : Divide by 9: = Eercise B: Solve the following equations. 1) ( 4) = 4 ) 4( ) = 3( 9) 3) 8 ( + 3) = 4 4) 14 3( + 3) = More help: http://www.mymaths.co.uk/tasks/library/loadlesson.asp?title=equations/solvingequationsmovie&t askid=118 EQUATIONS CONTAINING FRACTIONS When an equation contains a fraction, the first step is usually to multiply through by the denominator of the fraction. This ensures that there are no fractions in the equation. y Eample 4: Solve the equation + = 11 Solution: Step 1: Multiply through by (the denominator in the fraction): y + 10 = Step : Subtract 10: y = 1 Eample : Solve the equation 1 ( + 1) = 3 Solution: Step 1: Multiply by 3 (to remove the fraction) + 1 = 1 Step : Subtract 1 from each side = 14 Step 3: Divide by = 7 When an equation contains two fractions, you need to multiply by the lowest common denominator. This will then remove both fractions.
Eample 6: Solve the equation + 1 + + = 4 Solution: Step 1: Find the lowest common denominator: The smallest number that both 4 and divide into is 0. Step : Multiply both sides by the lowest common denominator Step 3: Simplify the left hand side: 0( + 1) 0( + ) + = 40 4 4 0 ( + 1) 0 ( + ) + = 40 4 ( + 1) + 4( + ) = 40 Step 4: Multiply out the brackets: + + 4 + 8 = 40 Step : Simplify the equation: 9 + 13 = 40 Step 6: Subtract 13 9 = 7 Step 7: Divide by 9: = 3 Eample 7: Solve the equation 3 + = 4 6 Solution: The lowest number that 4 and 6 go into is 1. So we multiply every term by 1: 1( ) 1(3 ) 1 + = 4 4 6 Simplify 1 + 3( ) = 4 (3 ) Epand brackets 1 + 3 6 = 4 6 + 10 Simplify 1 6 = 18 + 10 Subtract 10 6 = 18 Add 6 = 4 Divide by = 4.8 Eercise C: Solve these equations 1) 1 ( 3) + = ) 1 = + 4 3 3 y y 3) + 3 = 4) 4 3 3 = + 7 14
Eercise C (continued) ) 7 1 13 y 1 y + 1 y + = 6) + = 3 6 7) 1 + 3 + = 8) 3 10 = 1 More help: http://www.mymaths.co.uk/tasks/library/loadlesson.asp?title=equations/solvingequationswithfrac tions&taskid=1183 FORMING EQUATIONS Eample 8: Find three consecutive numbers so that their sum is 96. Solution: Let the first number be n, then the second is n + 1 and the third is n +. Therefore n + (n + 1) + (n + ) = 96 3n + 3 = 96 3n = 93 n = 31 So the numbers are 31, 3 and 33. Eercise D: 1) Find 3 consecutive even numbers so that their sum is 108. ) The perimeter of a rectangle is 79 cm. One side is three times the length of the other. Form an equation and hence find the length of each side. 3) Two girls have 7 photographs of celebrities between them. One gives 11 to the other and finds that she now has half the number her friend has. Form an equation, letting n be the number of photographs one girl had at the beginning. Hence find how many each has now.
Chapter 3: SIMULTANEOUS EQUATIONS An eample of a pair of simultaneous equations is 3 + y = 8 + y = 11 In these equations, and y stand for two numbers. We can solve these equations in order to find the values of and y by eliminating one of the letters from the equations. In these equations it is simplest to eliminate y. We do this by making the coefficients of y the same in both equations. This can be achieved by multiplying equation by, so that both equations contain y: 3 + y = 8 10 + y = = To eliminate the y terms, we subtract equation from equation. We get: 7 = 14 i.e. = To find y, we substitute = into one of the original equations. For eample if we put it into : 10 + y = 11 y = 1 Therefore the solution is =, y = 1. Remember: You can check your solutions by substituting both and y into the original equations. Eample: Solve + y = 16 3 4y = 1 Solution: We begin by getting the same number of or y appearing in both equation. We can get 0y in both equations if we multiply the top equation by 4 and the bottom equation by : 8 + 0y = 64 1 0y = As the SIGNS in front of 0y are DIFFERENT, we can eliminate the y terms from the equations by ADDING: 3 = 69 + i.e. = 3 Substituting this into equation gives: 6 + y = 16 y = 10 So y = The solution is = 3, y =. If you need more help on solving simultaneous equations, you can download a booklet from the following website: http://www.mathcentre.ac.uk/resources/workbooks/mathcentre/web-simultaneous1.pdf http://www.mymaths.co.uk/tasks/library/loadlesson.asp?title=simultaneous/simequmoviehard&ta skid=1174
Eercise: Solve the pairs of simultaneous equations in the following questions: 1) + y = 7 ) + 3y = 0 3 + y = 9 3 + y = -7 3) 3 y = 4 4) 9 y = + 3y = -6 4 y = 7 ) 4a + 3b = 6) 3p + 3q = 1 a 4b = 43 p + q = 14
Chapter 4: FACTORISING Common factors We can factorise some epressions by taking out a common factor. Eample 1: Factorise 1 30 Solution: 6 is a common factor to both 1 and 30. We can therefore factorise by taking 6 outside a bracket: 1 30 = 6( ) Eample : Factorise 6 y Solution: is a common factor to both 6 and. Both terms also contain an. So we factorise by taking outside a bracket. 6 y = (3 y) Eample 3: Factorise 9 3 y 18 y Solution: 9 is a common factor to both 9 and 18. The highest power of that is present in both epressions is. There is also a y present in both parts. So we factorise by taking 9 y outside a bracket: 9 3 y 18 y = 9 y(y ) Eample 4: Factorise 3( 1) 4( 1) Solution: There is a common bracket as a factor. So we factorise by taking ( 1) out as a factor. The epression factorises to ( 1)(3 4) Eercise A Factorise each of the following 1) 3 + y ) 4 y 3) pq p q 4) 3pq - 9q ) 3 6 6) 8a b 1a 3 b 4 7) y(y 1) + 3(y 1)
Factorising quadratics Simple quadratics: Factorising quadratics of the form + b + c The method is: Step 1: Form two brackets ( )( ) Step : Find two numbers that multiply to give c and add to make b. These two numbers get written at the other end of the brackets. Eample 1: Factorise 9 10. Solution: We need to find two numbers that multiply to make -10 and add to make -9. These numbers are -10 and 1. Therefore 9 10 = ( 10)( + 1). General quadratics: Factorising quadratics of the form a + b + c The method is: Step 1: Find two numbers that multiply together to make ac and add to make b. Step : Split up the b term using the numbers found in step 1. Step 3: Factorise the front and back pair of epressions as fully as possible. Step 4: There should be a common bracket. Take this out as a common factor. Eample : Factorise 6 + 1. Solution: We need to find two numbers that multiply to make 6-1 = -7 and add to make 1. These two numbers are -8 and 9. Therefore, 6 + 1 = 6-8 + 9 1 = (3 4) + 3(3 4) (the two brackets must be identical) = (3 4)( + 3) Difference of two squares: Factorising quadratics of the form a Remember that a = ( + a)( a). Therefore: 9 = 3 = ( + 3)( 3) 16 = ( ) = ( + )( ) Also notice that: and 8 = ( 4) = ( + 4)( 4) 3 3 48y = 3 ( 16 y ) = 3 ( + 4 y)( 4 y) Factorising by pairing We can factorise epressions like + y y using the method of factorising by pairing: + y y = ( + y) 1( + y) (factorise front and back pairs, ensuring both brackets are identical) = ( + y)( 1)
If you need more help with factorising, you can download a booklet from this website: http://www.mathcentre.ac.uk/resources/workbooks/mathcentre/web-factorisingquadratics.pdf http://www.mymaths.co.uk/tasks/library/loadlesson.asp?title=factorising/factorisehigher&taskid= 116 Eercise B Factorise 1) ) 3) 6 + 6 16 + + 4) ) 3 (factorise by taking out a common factor) + 3 6) y + 17 y + 1 7) 7 y 10 y + 3 8) + 10 30 9) 4 10) 3 y + 3y 11) 4 1 + 8 1) 16m 81n 13) 3 4 y 9a y 14) 8( + 1) ( + 1) 10
Chapter : CHANGING THE SUBJECT OF A FORMULA We can use algebra to change the subject of a formula. Rearranging a formula is similar to solving an equation we must do the same to both sides in order to keep the equation balanced. Eample 1: Make the subject of the formula y = 4 + 3. Solution: y = 4 + 3 Subtract 3 from both sides: y 3 = 4 Divide both sides by 4; y 3 = 4 y 3 So = is the same equation but with the subject. 4 Eample : Make the subject of y = Solution: Notice that in this formula the term is negative. y = Add to both sides y + = (the term is now positive) Subtract y from both sides = y Divide both sides by y = Eample 3: ( F 3) The formula C = is used to convert between Fahrenheit and Celsius. 9 We can rearrange to make F the subject. ( F 3) C = 9 Multiply by 9 9C = ( F 3) (this removes the fraction) Epand the brackets 9C = F 160 Add 160 to both sides 9C + 160 = F Divide both sides by 9C + 160 = F 9C + 160 Therefore the required rearrangement is F =. Eercise A Make the subject of each of these formulae: 1) y = 7 1 ) y = + 4 3) 4y = 4) 3 4(3 ) y = 9
Rearranging equations involving squares and square roots Eample 4: Make the subject of + y = w Solution: Subtract y from both sides: Square root both sides: + y = w = w y (this isolates the term involving ) = ± w y Remember that you can have a positive or a negative square root. We cannot simplify the answer any more. Eample : Make a the subject of the formula 1 a t = 4 h Solution: Multiply by 4 Square both sides Multiply by h: Divide by : 1 a t = 4 h a 4t = h a 16t = h 16t h = a 16t h = a Eercise B: Make t the subject of each of the following 1) wt P = ) 3r wt P = 3r 3) V 1 3 = π t h 4) P = t g ) w( v t) Pa = 6) g r = a + bt More help: http://www.mymaths.co.uk/tasks/library/loadlesson.asp?title=simplify/rearrangehigher&taskid=11 70
More difficult eamples Sometimes the variable that we wish to make the subject occurs in more than one place in the formula. In these questions, we collect the terms involving this variable on one side of the equation, and we put the other terms on the opposite side. Eample 6: Make t the subject of the formula a t = b + yt Solution: a t = b + yt Start by collecting all the t terms on the right hand side: Add t to both sides: a = b + yt + t Now put the terms without a t on the left hand side: Subtract b from both sides: a b = yt + t Factorise the RHS: a b = t( y + ) Divide by (y + ): So the required equation is a b = t y + a b t = y + Wa Eample 7: Make W the subject of the formula T W = b Solution: This formula is complicated by the fractional term. We begin by removing the fraction: Multiply by b: bt bw = Wa Add bw to both sides: bt = Wa + bw (this collects the W s together) Factorise the RHS: bt = W ( a + b) Divide both sides by a + b: W bt = a + b If you need more help you can download an information booklet on rearranging equations from the following website: http://www.mathcentre.ac.uk/resources/workbooks/mathcentre/web-formulae-tom.pdf Eercise C Make the subject of these formulae: 1) a + 3 = b + c ) 3( + a) = k( ) 3) + 3 y = 4) = 1+ a b
Chapter 6: SOLVING QUADRATIC EQUATIONS A quadratic equation has the form a b c + + = 0. There are two methods that are commonly used for solving quadratic equations: * factorising * the quadratic formula Note that not all quadratic equations can be solved by factorising. The quadratic formula can always be used however. Method 1: Factorising Make sure that the equation is rearranged so that the right hand side is 0. It usually makes it easier if the coefficient of is positive. Eample 1 : Solve 3 + = 0 Factorise ( 1)( ) = 0 Either ( 1) = 0 or ( ) = 0 So the solutions are = 1 or = Note: The individual values = 1 and = are called the roots of the equation. Eample : Solve = 0 Factorise: ( ) = 0 Either = 0 or ( ) = 0 So = 0 or = http://www.mymaths.co.uk/tasks/library/loadlesson.asp?title=factorising/solvequadsbyfactoring&t askid=1181 Method : Using the formula Recall that the roots of the quadratic equation a + b + c = 0 are given by the formula: b ± = b 4ac a Eample 3: Solve the equation = 7 3 Solution: First we rearrange so that the right hand side is 0. We get We can then tell that a =, b = 3 and c = -1. Substituting these into the quadratic formula gives: 3 3 4 ( 1) 3 10 + 3 1 = 0 ± ± = = (this is the surd form for the solutions) 4 If we have a calculator, we can evaluate these roots to get: = 1.81 or = -3.31 If you need more help with the work in this chapter, there is an information booklet downloadable from this web site: http://www.mathcentre.ac.uk/resources/workbooks/mathcentre/web-quadraticequations.pdf http://www.mymaths.co.uk/tasks/library/loadlesson.asp?title=quadraticformula/formulamovie&tas kid=1160
EXERCISE 1) Use factorisation to solve the following equations: a) + 3 + = 0 b) 3 4 = 0 c) = 1 ) Find the roots of the following equations: a) + 3 = 0 b) 4 = 0 c) 4 = 0 3) Solve the following equations either by factorising or by using the formula: a) 6-4 = 0 b) 8 4 + 10 = 0 4) Use the formula to solve the following equations to 3 significant figures. Some of the equations can t be solved. a) +7 +9 = 0 b) 6 + 3 = 8 c) 4 7 = 0 d) 3 + 18 = 0 e) 3 + 4 + 4 = 0 f) 3 = 13 16
Chapter 7: INDICES Basic rules of indices y 4 means y y y y. 4 is called the inde (plural: indices), power or eponent of y. There are 3 basic rules of indices: 1) ) 3) ( a ) m n m n a a = a + e.g. 3 3 = 3 m n m n a a = a 8 6 e.g. 3 3 = 3 m n mn 4 9 = a e.g. ( 3 ) = 3 10 Further eamples y y = y 4 3 7 3 4a 6a = 4a (multiply the numbers and multiply the a s) 6 8 c 3c = 6c (multiply the numbers and multiply the c s) ( ) 7 7 4d 4d 3d = = 8d (divide the numbers and divide the d terms i.e. by subtracting 3d the powers) Eercise A Simplify the following: 1) ) 3) 4) ) 6) b b = (Remember that 3c c = 3 b c bc = n n = 6 ( 6 ) 8 3 8n n = d d = 11 9 b 1 = b ) 3 7) ( a ) = 4 8) ( ) 3 d = Help: http://www.mymaths.co.uk/tasks/library/loadlesson.asp?title=powers/indicespart&taskid=104
More comple powers Zero inde: Recall from GCSE that 0 a = 1. This result is true for any non-zero number a. Therefore ( ) Negative powers 0 3 0 0 = 1 = 1.304 = 1 4 A power of -1 corresponds to the reciprocal of a number, i.e. Therefore = 1 1 0. = = 4 0. 1 1 1 4 = 4 a = a 1 1 (you find the reciprocal of a fraction by swapping the top and bottom over) This result can be etended to more general negative powers: This means: 3 4 1 1 = = 3 9 1 1 = = 4 16 1 1 1 4 = = = 16 4 4 1 a n 1 =. n a Fractional powers: Fractional powers correspond to roots: 1 / a = a 1/ 3 3 a = a 1/ 4 4 a = a In general: 1/ n a n = a Therefore: 1/ 3 3 8 = 8 = 1 / = = 1 / 4 4 10000 = 10000 = 10 m / n 1 / n A more general fractional power can be dealt with in the following way: a = ( a ) 3 / 3 4 = 4 = = 8 So ( ) 3 / 3 1/ 3 8 8 4 = = = 7 7 3 9 3 / 3 / 3 3 36 36 6 16 36 = = = = 1 Help: http://www.mymaths.co.uk/tasks/library/loadlesson.asp?title=powers/indicespart3&taskid=1301 m
Eercise B: Find the value of: 1) ) 1 / 4 1/ 3 7 3) ( 1 ) 1/ 9 4) ) 6) 7) 8) 9) 0 18 7 1 / 3 7 3 8 / 3 10) ( 0.04 ) 1/ 11) 1) 8 7 1 16 / 3 3 / Simplify each of the following: 13) 14) a 3a 1/ / 3 1) ( ) 1/ 4 y
SOLUTIONS TO THE EXERCISES CHAPTER 1: E A 1) 8 + 3 ) -1 + 1 3) -7a + 4 4) 6y + 3y ) 4 6) 7 1 7) + + 6 8) t 3t 10 9) 6 + y 1y 10) 4 + 4 4 11) 4y 1 1) 1 + 17 E B 1) + 1 ) 9 + 30 + 3) 49 8 + 4 4) 4 ) 9-1 6) y 9 CHAPTER E A 1) 7 ) 3 3) 1½ 4) ) -3/ 6) -7/3 E B 1).4 ) 3) 1 4) ½ E C 1) 7 ) 1 3) 4/7 4) 3/3 ) 3 6) 7) 9/ 8) E D 1) 34, 36, 38 ) 9.87, 9.6 3) 4, 48 CHAPTER 3 1) = 1, y = 3 ) = -3, y = 1 3) = 0, y = - 4) = 3, y = 1 ) a = 7, b = -6) p = 11/3, q = 4/3 CHAPTER 4 E A 1) (3 + y) ) ( y) 3) pq(q p) 4) 3q(p 3q) ) ( - 3) 6) 4a 3 b (a 3b ) 7) (y 1)(y + 3) E B 1) ( 3)( + ) ) ( + 8)( ) 3) ( + 1)( + ) 4) ( 3) ) (3-1 )( + ) 6) (y + 3)(y + 7) 7) (7y 3)(y 1) 8) ( 3)( + ) 9) ( + )( ) 10) ( 3)( y) 11) 4( )( 1) 1) (4m 9n)(4m + 9n) 13) y(y 3a)(y + 3a) 14) (4 + )( 4) CHAPTER E A y + 1 9y + 0 1) = ) = 4y 3) = 3(4 y + ) 4) = 7 1 E B 3rP 3rP 3V P g Pag 1) t = ) t = ± 3) t = ± 4) t = ) t = v 6) t = ± w w π h w E C c 3 3a + k y + 3 ab 1) = ) = 3) = 4) a b k 3 y = b a r a b CHAPTER 6 1) a) -1, - b) -1, 4 c) -, 3 ) a) 0, -3 b) 0, 4 c), - 3) a) -1/, 4/3 b) 0.,. 4) a) -.30, -1.70 b) 1.07, -0.699 c) -1.0, 1.4 d) no solutions e) no solutions f) no solutions CHAPTER 7 E A 1) b 6 ) 6c 7 3) b 3 c 4 4) -1n 8 ) 4n 6) d 7) a 6 8) -d 1 E B 1) ) 3 3) 1/3 4) 1/ ) 1 6) 1/7 7) 9 8) 9/4 9) ¼ 10) 0. 11) 4/9 1) 64 13) 6a 3 14) 1) y