EDEXCEL A LEVEL SAMPLE CHAPTER. This title has been selected for Edexcel s endorsement process

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1 EDEXCEL A LEVEL This title has been selected for Edecel s endorsement process SAMPLE CHAPTER S C I T A M E H T A M e For Y S A d n ar a SPHIE GLDIE, VAL HANRAHAN, CATH MRE, JEAN-PAUL MUSCAT, SUSAN WHITEHUSE SERIES EDITRS RGER PRKESS AND CATHERINE BERRY CNSULTANT EDITR KEITH PLEDGER Powered b

2 Features for great teaching and learning Each chapter opens with an activit to introduce the chapter, and an accompaning quotation to engage students. Each chapter is broken down into numbered sections, and subsections. Simultaneous equations Equations and inequalities Simultaneous equations both equations linear Discussion points Wh is one equation such as + 3 = 5 not enough to find the values of and? How man equations do ou need? How man equations would ou need to find the values of three variables, and z? Eample. Solution b substitution Solve the simultaneous equations + = = + Chapter Equations and inequalities There are man situations which can onl be described mathematicall in terms of more than one variable and to find the value of each variable ou need to solve two or more equations simultaneousl (i.e. at the same time). Such equations are called simultaneous equations. Discussion points designed for class discussion and to enhance individual understanding. Solution + = + ( + ) = Note Most of the fundamental ideas of science are essentiall simple and ma, as a rule, be epressed in a language comprehensible to everone. Albert Einstein ( ) This is a good method when is alread the subject of one of the equations. Saira and her friends are going to the cinema and want to bu some packets of nuts and packets of crisps. A packet of nuts costs 0 pence more than a packet of crisps. Two packets of nuts cost the same as three packets of crisps. What is the cost of each item? This is the tpe of question that ou ma find in a puzzle book. How would ou set about tackling it? You ma think that the following question appears ver similar. What happens when ou tr to find the answer? A packet of nuts cost 0 pence more than a packet of crisps. Four packets of nuts and three packets of crisps cost 0p more than three packets of nuts and four packets of crisps. What is the cost of each item? Eample. 3 = 3 = Take the epression for from the second equation and substitute it into the first. Substituting = into = + gives = 3 so the solution is =, = 3. Solution b elimination Solve the simultaneous equations Carefull worded eposition sets out the basics of each topic, with points for class discussion to enhance understanding = + 3 = 7 Solution Multipling the first equation b and the second b 3 will give two equations each containing the term 6: Multipling = b = To give two equations Multipling + 3 = 7 b = containing the term 6. Prior knowledge You need to be able to solve quadratic equations. This is covered in Chapter 3. Prior knowledge for specific sub-topics is highlighted. Subtracting = 3 Substituting this into either of the original equations gives =. The solution is =, = _0_MEI_Maths_Y_E_ indd -3 08//6 :8 pm Warning boes highlight common pitfalls to watch out for. Note boes include helpful hints and tips. Icons indicate the most straightforward questions bridging the GCSE-A Level divide. Each section ends with a banded eercise to test understanding. Answers are provided at the back of the book and online. Simultaneous equations Note Activities can be completed individuall or in groups. Multipling term b term Eercise 5. ① Find the equations of the following circles. centre (, 3), radius (ii) centre (, 3), radius (iii) centre (, 3), radius 3 (iv) centre (, 3), radius ② For each of the following circles state (a) the coordinates of the centre (b) the radius. + = (ii) + ( ) = (iii) ( ) + = 3 (iv) ( + ) + ( + ) = (v) ( ) + ( + ) = 5 ③ The equation of a circle is ( 3) + ( + ) = 6. Complete the table to show whether each point lies inside the circle, outside the circle or on the circle. ( + 3)( + ) = = (as required) Before starting the procedure for factorising a quadratic, ou should alwas check that the terms do not have a common factor as in, for eample This can be written as ( + 3) and then factorised to give ( 3) ( ). ACTIVITY 3. Each section contains several worked eamples to aid understanding. Factorise the following, where possible (ii) (iv) (v) (vii) (viii) () Which pairs give the same answers? (iii) (vi) Point (3, ) (, 5) (6, 6) (, 3) (0, ) (, 3) The following eamples show how ou can use quadratic factorisation to solve equations and to sketch curves. Eample 3. Solve = 0. (ii) Sketch the graph of =. Technolog boes provide suggestions on how to further understanding and embed learning with the use of graphing software, online resources, graphical calculators and spreadsheets. Solution First ou look for two numbers that can be added to give and multiplied to give : 6 + = 6 (+) =. The numbers are 6 and + and so the middle term,, is split into 6 +. = 0 TECHNLGY When working through this chapter, ou ma wish to use a graphical calculator or graphing software to check our answers where appropriate. 6 + = 0 ( 6) + ( 6) = 0 ( + )( 6) = 0 = or 6 When two brackets multipl to give zero, one or the other must equal zero. (ii) From part, = 0 when = and when = 6, so these are the points where the curve crosses the ais. Also, when = 0, =, so ou have the point where it crosses the ais. Inside utside n ④ Draw the circles ( ) + ( 5) = 6 and ( 3) + ( 3) =. In how man points do the intersect? ⑤ Sketch the circle ( + ) + ( 3) = 6, and find the equations of the four tangents to the circle which are parallel to the coordinate aes. ⑥ Find the coordinates of the points where each of these circles crosses the aes. + = 5 (ii) ( ) + ( + 5) = 5 (iii) ( + 6) + ( 8) = 00 ⑦ Find the equation of the circle with centre (, 7) passing through the point (, 5). ⑧ Show that the equation = 0 can be written in the form ( + ) + ( ) = r, where the value of r is to be found. Hence give the coordinates of the centre of the circle, and its radius. ⑨ Draw the circle of radius units which touches the positive and aes, and find its equation. ⑩ A(3, 5) and B(9, 3) lie on a circle. Show that the centre of the circle lies on the line with equation 3 + = 0. ⑪ For each of the following circles find (a) the coordinates of the centre (b) the radius = 0 (ii) = 0 (iii) = 0 ⑫ A circle passes through the points A(3, ), B(5, 6) and C(, 3). Calculate the lengths of the sides of the triangle ABC. (ii) Hence show that AC is a diameter of this circle. State which theorems ou have used, and in each case whether ou have used the theorem or its converse. (iii) Calculate the area of triangle ABC. ⑬ Find the midpoint, C, of AB where A and B are (, 8) and (3, ) respectivel. Find also the distance AC. (ii) Hence find the equation of the circle which has AB as a diameter. ⑭ A(, ) is a point on the circle ( 3) + ( + ) = 5. State the coordinates of the centre of the circle and hence find the coordinates of the point B, where AB is a diameter of the circle. (ii) C(, ) also lies on the circle. Use coordinate geometr to verif that angle ACB = 90. ⑮ The tangent to the circle + ( + ) = 5 at the point (, ) intersects the ais at A and the ais at B. Find the eact area of the triangle AB. ⑯ A circle passes through the points (, 0) and (8, 0) and has the ais as a tangent. Find the two possible equations for the circle. ⑰ A(6, 3) and B(0,) are two points on a circle with centre (, 8). Calculate the distance of the chord AB from the centre of the circle. (ii) Find the equation of the circle. ⑱ A(6, 6), B(6, ) and C(, ) are three points on a circle. Find the equation of the circle _0_MEI_Maths_Y_E_ indd -5 Chapter Equations and inequalities You would not epect to draw the lines and arrows in our answers. The have been put in to help ou understand where the terms have come from. 5 Icons show all eercise questions that involve problem-solving skills. 08//6 :8 pm

3 At the end of each block of four chapters there are practice questions to help revision. Questions that test problem solving, technolog, modelling and mathematical proof skills are highlighted. A list of the knowledge is supplied in the learning outcomes at the end of each chapter. A link to the full eam-board learning objectives will be available on the Hodder Education website. Simultaneous equations PRACTICE QUESTINS MP T LEARNING UTCMES [3 marks] When ou have completed this chapter ou should be able to find length, midpoint and gradient of a straight line find the equation of a straight line given information on co-ordinates or gradient understand the relationship between a graph and its associated equation identif and understand the equation of a circle use algebra to solve problems involving lines and circles, including intersections sketch curves with equations of the form [3 marks] = kn =k Chapter Equations and inequalities M ① Solve the following pairs of simultaneous equations (a) b drawing accurate graphs (b) b an algebraic method. =7 (ii) + = 0 (iii) a + b = = = 0 a + b = ② Solve the following pairs of simultaneous equations: = 3 (ii) = + 5 (iii) + = 7 =+ +=8 = (iv) a 3b = 7 5a + b = 3 ③ Solve the following pairs of simultaneous equations: + 3 = 5 ii) 5 + = (iii) l 3m = 6 = 5 3 = 9 5l 7m = 9 (iv) 3r s = 7 5r 3s = 8 ④ Solve the following pairs of simultaneous equations: + 3 = 5 ii) 5 + = (iii) l 3m = 6 = 5 3 = 9 5l 7m = 9 (iv) 3r s = 7 5r 3s = 8 ⑤ Solve the following pairs of simultaneous equations: + 3 = 5 ii) 5 + = (iii) l 3m = 6 = 5 3 = 9 5l 7m = 9 (iv) 3r s = 7 5r 3s = 8 = a + b + c For a full list of learning objectives, go to [3 marks] KEY PINTS [3 marks] Some quadratic equations can be solved b factorising. You can sketch a quadratic graph b finding the points where the curve crosses the coordinate aes. 3 The verte and the line of smmetr of a quadratic graph can be found b completing the square. The quadratic formula = b ± b ac can be used to solve the a quadratic equation a + b + c = 0. [3 marks] 5 For the quadratic equation a + b + c = 0, the discriminant is given b b ac. If the discriminant is positive, the equation has two real roots. If the discriminant is a perfect square, these roots are rational. (ii) If the discriminant is zero, the equation has a repeated real root. (iii) If the discriminant is negative, the equation has no real roots. The ke points from the chapter are pulled out at the end for eas reference. FUTURE USES n n n n Solving simultaneous equations in which one equation is linear and one quadratic in Chapter. Finding points where a line intersects a circle in Chapter 5. Solving cubic equations and other polnomial equations in Chapter 7. You will use quadratic equations in man other areas of mathematics _0_MEI_Maths_Y_E_ indd 6-7 /0/6 7:35 pm Future uses indicates where material is utilised in subsequent chapters and/or in the A Level book. Problem solving Information collection Mountain modelling ridge Q Chapter Equations and inequalities Special activit pages are provided ever few chapters to help build problem solving skills. At this stage ou need the answers to two of the questions raised in stage : the steepest slope Chandra can walk along and how fast he can walk. Are the two answers related? 3 Processing and representation In this problem the processing and representation stage is where ou will draw diagrams to show Chandra s path. It is a three-dimensional problem so ou will need to draw true shape diagrams. There are a number of possible answers according to how man times Chandra crosses the slope. Alwas start b taking the easiest case, in this case when he crosses the slope times.this is shown as PR + RQ in Figure.8. The point R is half wa up the slope ST and the points A and B are directl below R and T. T 000 m Figure.7 R B Chandra is 75 ears old. He is hiking in a mountainous area. He comes to the point P in Figure.7. It is at the bottom of a steep slope leading up to a ridge between two mountains. Chandra wants to cross the ridge at the point Q. The ridge is 000 metres higher than the point P and a horizontal distance of 500 metres awa. The slope is 800 metres wide. The slope is much too steep for Chandra to walk straight up so he decides to zig-zag across it.you can see the start of the sort of path he might take. The time is noon. Estimate when Chandra can epect to reach the point Q. There is no definite answer to this question. That is wh ou are told to estimate the time.you have to follow the problem solving ccle.you will also need to do some modelling. Problem specification and analsis The problem has alread been specified but ou need to decide how ou are going to go about it. n A ke question is how man times Chandra is going to cross the width of the slope. Will it be, or, or 6, or...? n To answer this ou have to know the steepest slope Chandra can walk along. n What modelling assumption are ou going to make about the sloping surface? n To complete the question ou will also need to estimate how fast Chandra will walk _0_MEI_Maths_Y_E_ indd 8-9 A 500 m P 800 m S Figure.8 Draw the true shape diagrams for triangles SAR, R and PAR. Then use trigonometr and Pthagoras theorem to work out the angle RPA. If this is not too great, go on to work out how far Chandra walks in crossing the slope and when he arrives at Q. Now repeat this for routes with more crossings. Interpretation You now have a number of possible answers. For this interpretation stage, decide which ou think is the most likel time for 75 ear old Chandra to arrive at Q and eplain our choice. 3 9 /0/6 7:35 pm

4 Contents Getting the most from this book Prior knowledge Problem solving. Solving problems. Writing mathematics.3 Proof Surds and indices. Using and manipulating surds. Working with indices 3 Quadratic functions 3. Factorising quadratic epressions 3. The graphs of quadratic functions completing the square 3.3 The quadratic formula Equations and inequalities. Simultaneous equations both equations linear. Linear inequalities 5 Coordinate geometr 5. Working with straight lines 5. The equation of a straight line 5.3 The intersection of two lines 5. The circle 5.5 The intersection of a line and a curve 5.6 The intersection of two curves 6 Trigonometr 6. Trigonometric functions 6. Trigonometric functions for angles of an size 6.3 Solving equations using graphs of trigonometric functions 6. Triangles without right angles 6.5 The area of a triangle 7 Polnomials 7. Polnomial epressions 7. Dividing polnomials 7.3 Polnomial equations 8 Graphs and transformations 8. The shapes of curves 8. Function notation 8.3 Using transformations to sketch curves 8. Using transformations 8.5 Transformations and graphs of trigonometric functions 9 The binomial epansion 9. Binomial epansions 9. Selections 0 Differentiation 0. The gradient of the tangent as a limit 0. Differentiation using standard results 0.3 Tangents and normal 0. Increasing and decreasing functions, and turning points 0.5 Sketching the graphs of gradient functions 0.6 Etending the rule 0.7 Higher order derivatives 0.8 Practical problems 0.9 Finding the gradient from first principles Integration. Integration as the reverse of differentiation. Finding areas.3 Areas below the -ais. Further integration

5 Vectors 9 Kinematics. Vectors. Equal vectors.3 Vector geometr 3 Eponentials and logarithms 3. Eponential functions 3. Logarithms 3.3 The eponential function 3. The natural logarithm function 3.5 Modelling curves Data collection. Using statistics to solve problems. Sampling 5 Data processing, presentation and interpretation 5. Presenting different tpes of data 5. Ranked data 5.3 Discrete numerical data 5. Continuous numerical data 5.5 Bivariate data 5.6 Standard deviation 6 Probabilit 6. Working with probabilit 7 The binomial distribution 7. Introduction to binomial distribution 7. Using the binomial distribution 9. The language of motion 9. Speed and velocit 9.3 Acceleration 9. Using areas to find distances and displacement 9.5 The constant acceleration formulae 9.6 Further eamples 0 Forces and Newton s laws of motion 0. Force diagrams 0. Force and motion 0.3 Tpes of forces 0. Pulles 0.5 Appling Newton s second law along a line 0.6 Newton s second law applied to connected objects Variable acceleration. Using differentiation. Finding displacement from velocit.3 The constant acceleration formulae revisited Statistics dataset Answers Inde Contents 8 Statistical hpothesis testing using the binomial distribution 8. The principles and language of hpothesis testing 8. Etending the language of hpothesis testing Photo credits: Cover photo Baloncici/3RF.com p.6 polifoto/3rf.com 5

6 5 Coordinate geometr Most A place of for the everthing, fundamental ideas and everthing of science in are its essentiall place simple and Samuel ma, Smiles as a rule, (8 90) be epressed in a language comprehensible to everone. Albert Einstein Saira and her friends are going to the cinema and want to bu some packets The diagram of nuts shows and packets some of crisps. A packet of nuts costs 0 pence more than scaffolding a packet in which of crisps. some Two of the packets of nuts cost the same as three packets of horizontal crisps. pieces What are is the m long cost and of each item? others This are is the m. tpe All of the question vertical that ou ma find in a puzzle book. How would ou set pieces about are tackling m. it? You ma think that the following question appears ver similar. What happens An ant crawls along the when ou tr to find the answer? scaffolding from point P to point A packet Q, travelling of nuts either cost 0 pence more than a packet of crisps. Four packets of horizontall nuts and three or verticall. packets How of crisps cost 0p more than three packets of nuts far and does four the packets ant crawl? of crisps. What is the cost of each item? Q A mouse also goes from point P to point Q, travelling along horizontal or sloping pieces. How far does the mouse travel? A bee flies directl from point P to point Q. How far does the bee fl? P Figure 5. 6

7 Hint: When working through this chapter, ou ma wish to use a graphical calculator or graphing software to check our answers where appropriate. Eample 5. Working with straight lines Coordinates are a means of describing a position relative to a fied point, or origin. In two dimensions ou need two pieces of information; in three dimensions ou need three pieces of information. In the Cartesian sstem (named after René Descartes), position is given in perpendicular directions:, in two dimensions;,, z in three dimensions. This chapter concentrates eclusivel on two dimensions. The midpoint and length of a line segment When ou know the coordinates of two points ou can work out the midpoint and length of the line segment which connects them. Find: the coordinates of the midpoint, M (ii) the length AB. Solution You can generalise these methods to find the midpoint and length of an line segment AB. Let A be the point (, ) and B the point (, ). Find the midpoint of AB. The midpoint of two values is the mean of those values. The mean of the coordinates is +. Figure 5. The mean of the coordinates is + C has the same. coordinate as B So the coordinates of the midpoint, M are and the same + +,. coordinate as A. (ii) Find the length of AB. First find the lengths of AC and BC: AC = BC = B Pthagoras theorem: AB = AC + BC = ( ) + ( ) So the length AB is ( ) + ( ) Hint: Draw a right-angled triangle with AB as the hpotenuse and use Pthagoras theorem. A (, ) A (, ) Figure 5.3 M B (8, 5) B (, ) C (, ) 5 Chapter 5 Coordinate geometr 7

8 Working with straight lines Discussion point Does it matter which point ou call (, ) and which (, )? The gradient of a line A (, ) θ 7 = 3 B (6, 7) C 6 = 7 Gradient m = 3 6 = θ (theta) is the Greek letter th. α (alpha) and β (beta) are also used for angles. Figure 5. When ou know the coordinates of an two points on a straight line, then ou can draw that line. The slope of a line is given b its gradient. The gradient is often denoted b the letter m. In Figure 5., A and B are two points on the line. The gradient of the line AB is given b the increase in the coordinate from A to B divided b the increase in the coordinate from A to B. In general, when A is the point (, ) and B is the point (, ), the gradient is m = change in Gradient = change in When the same scale is used on both aes, m = tanθ (see Figure 5.). Parallel and perpendicular lines ACTIVITY 5. It is best to use squared paper for this activit. Draw the line L joining (0, ) to (, ). Draw another line L perpendicular to L from (, ) to (6, 0). Find the gradients m and m of these two lines. What is the relationship between the gradients? Is this true for other pairs of perpendicular lines? When ou know the gradients m and m, of two lines, ou can tell at once if the are either parallel or perpendicular see Figure 5.5. m Lines for which m m = will onl look perpendicular if the same scale has been used for both aes. m Figure 5.5 parallel lines: m = m perpendicular lines: mm = So for perpendicular lines: m = and likewise, m So m and m are the negative = m m reciprocal of each other. parallel lines: m = m m m perpendicular lines: m m = 8

9 Eample 5. A and B are the points (, 5) and (6, 3) respectivel (see Figure 5.6). Find: the gradient of AB (ii) the length of AB (iii) the midpoint of AB (iv) the gradient of the line perpendicular to AB. Solution A (, 5) B (6, 3) 5 Chapter 5 Coordinate geometr Figure 5.6 Gradient m = A A = = B B Gradient is difference in coordinates divided b difference in coordinates. It doesn t matter which point ou use first, as long as ou are consistent! (ii) Length AB = ( ) + ( ) = (6 ) + (3 5) = = (iii) Midpoint = A + B A + B, = + 6, 5 + ( 3 ) = (,) (iv) Gradient of AB: m AB = So gradient of perpendicular to AB is B A B A Check: = The gradient of the line perpendicular to AB is the negative reciprocal of m AB. 9

10 Working with straight lines Drawing a line, given its equation There are several standard forms for the equation of a straight line, as shown in Figure 5.7. (a) Equations of the form = a (b) Equations of the form = b = 3 Each point on the line has an coordinate of 3. (0, ) = Each point on the line has a coordinate of. All such lines are parallel to the ais. All such lines are parallel to the ais. (3, 0) (c) Equations of the form = m (d) Equations of the form = m + c (e) Equations of the form p + q + r = 0 = These are lines through the origin, with gradient m. = (0, ) = These lines have gradient m and cross the ais at point (0, c) (0, ) This is often a tidier wa of writing the equation = 0 Figure 5.7 (0, ) (, 0) (3, 0) = 3 + (3, 0) Eample 5.3 Sketch the lines (a) = and (b) 3 + = on the same aes. (ii) Are these lines perpendicular? Solution To draw a line ou need to find the coordinates of two points on it. (a) The line = passes through the point (0, ). Usuall it is easiest to find where the line cuts the and aes. The line is alread in the form = m + c. Substituting = 0 gives =, so the line also passes through (, 0). (b) Find two points on the line 3 + =. Substituting = 0 gives = = 6 substituting = 0 gives 3 = = 8. Set = 0 and find to give the -intercept. Then set = 0 and find to give the -intercept. 0

11 So the line passes through (0, 6) and (8, 0) (0, 6) (, 0) = 3 + = (0, ) Figure 5.8 (8, 0) (ii) The lines look almost perpendicular but ou need to use the gradient of each line to check. Rearrange the equation to make the subject so ou Gradient of = is. Gradient of 3 + = is 3 can find the gradient.. = 3 + Therefore the lines are not perpendicular as = ( ) 5 Chapter 5 Coordinate geometr Finding the equation of a line To find the equation of a line, ou need to think about what information ou are given. Given the gradient, m, and the coordinates = m ( ) (, ) of one point on the line. Take a general point (, ) on the line, as shown in Figure 5.9. (, ) (, ) Figure 5.9 The gradient, m, of the line joining (, ) to (, ) is given b m = This is a ver useful form of the m m ( ) = equation of a straight line.

12 Working with straight lines For eample, the equation of the line with gradient that passes through the point (3, ) can be written as ( ) = ( 3) which can be simplified to = 7. (ii) Given the gradient, m, and the -intercept (0, c) = m + c A special case of = m ( ) is when (, ) is the -intercept (0, c). The equation then becomes Substituting 0 = m + c = and = c into the equation as shown in Figure 5.0. When the line passes through the origin, the equation is = m The -intercept is (0, 0), so c = 0 as shown in Figure 5.. = m + c = m (0, c) Figure 5.0 Figure 5. (iii) Given two points, (, ) and (, ) The two points are used to find the gradient: m = = Discussion points How else can ou write the equation of the line? Which form do ou think is best for this line? This value of m is then substituted in the equation = m( ) This gives ( ) = r = ACTIVITY 5. (, ) Figure 5. (, ) (, ) A Show algebraicall that an equivalent form of = is = B Use both forms to find the equation of the line joining (, ) to (5, 3) and show the give the same equation.

13 Eample 5. Find the equation of the line perpendicular to + = which passes through the point P(, 5). Solution First rearrange + = into the form = m + c to find the gradient. = + = + 3 So the gradient is The negative reciprocal of is. So the gradient of a line perpendicular to = + 3 is. Using = m( ) when m = and (, ) is (, 5) ( 5) = ( ) + 5 = 8 = 3 For perpendicular gradients m m = So m = m Check: = 5 Chapter 5 Coordinate geometr Eercise 5. For the following pairs of points A and B, calculate: (a) the midpoint of the line joining A to B (b) the distance AB (c) the gradient of the line AB (d) the gradient of the line perpendicular to AB. A(, 5) and B(6, 8) (ii) A(, 5) and B( 6, 8) (iii) A(, 5) and B(6, 8) (iv) A(, 5) and B(6, 8) The gradient of the line joining the point P(3, ) to Q(q, 0) is. Find the value of q. 3 The three points X(, ), Y(8, ) and Z(, ) are collinear. Find the value of. The lie on the same straight line. For the points P(, ), and Q(3, 5), find in terms of and : the gradient of the line PQ (ii) the midpoint of the line PQ (iii) the length of the line PQ. 5 Sketch the following lines: = (ii) = (iii) = (iv) = + (v) = + 5 (vi) = 5 (vii) = 5 (viii) = 0 6 Find the equations of the lines (v) in Figure 5.3. Figure (iii) (v) (iv) (ii) 3

14 Working with straight lines 7 Find the equations of the following lines. parallel to = 3 and passing through (0, 0) (ii) parallel to = 3 and passing through (, 5) (iii) parallel to + 3 = 0 and passing through (, 5) (iv) parallel to 3 = 0 and passing through (5, ) (v) parallel to + = 3 and passing through (, 5) 8 Find the equations of the following lines. perpendicular to = 3 and passing through (0, 0) (ii) perpendicular to = + 3 and passing through (, 3) (iii) perpendicular to + = and passing through (, 3) (iv) perpendicular to = + 5 and passing through (, 3) (v) perpendicular to + 3 = and passing through (, 3) 9 Find the equations of the line AB in each of the following cases. A(3, ), B(5, 7) (ii) A( 3, ), B( 5, 7) (iii) A( 3, ), B( 5, 7) (iv) A(3, ), B(5, 7) (v) A(, 3), B(7, 5) 0 Show that the area enclosed b the lines = + 3, = 3, 3 + = 0 and = 0 forms + a rectangle. + = The perpendicular bisector is the line Find the equation at right angles to of the perpendicular AB (perpendicular) bisector of each of that passes though the following pairs of the midpoint of AB points. (bisects). A(, ) and B(3, 5) (ii) A(, ) and B (5, 3) (iii) A(, ) and B( 3, 5) (iv) A(, ) and B( 3, 5) (v) A(, ) and B(3, 5) A median of a triangle is a line joining one of the vertices to the midpoint of the opposite side. In a triangle AB, is at the origin, A is the point (0, 6), and B is the point (6, 0). Sketch the triangle. (ii) Find the equations of the three medians of the triangle. (iii) Show that the point (, ) lies on all three medians. (This shows that the medians of this triangle are concurrent.) 3 A quadrilateral ABCD has its vertices at the points (0, 0), (, 5), (0, 0) and ( 6, 8) respectivel. Sketch the quadrilateral. (ii) Find the gradient of each side. (iii) Find the length of each side. (iv) Find the equation of each side. (v) Find the area of the quadrilateral. The lines AB and BC in Figure 5. are equal in length and perpendicular. A gradient m gradient m θ E D B Figure 5. Show that triangles ABE and BCD are congruent. (ii) Hence prove that the gradients m and m satisf m m =. C

15 Eample 5.5 Discussion point The line l has equation = and the line m has equation = 3. What can ou sa about the intersection of these two lines? Eercise 5. The intersection of two lines The intersection of an two curves (or lines) can be found b solving their equations simultaneousl. In the case of two distinct lines, there are two possibilities: the are parallel, or (ii) the intersect at a single point. You often need to find where a pair of lines intersect in order to solve problems. The lines = 5 3 and + 3 = 0 intersect at the point P. Find the coordinates of P. Solution You need to solve the equations = 5 3 and + 3 = 0 simultaneousl. Substitute equation into : (5 3) + 3 = = 0 Multipl out the brackets. 3 6 = 0 Simplif 3 = 6 = Don t forget to find Substitute = into equation to find. the coordinate. = 5 3 = 3 So the coordinates of P are (, 3). 5 Chapter 5 Coordinate geometr For each pair of lines: Solve the equation simultaneousl (ii) Write down the coordinates of the point of intersection (a) = + 3 and = 6 + (b) = 3 and + = (c) 3 + = and 5 = 3 Find the coordinates of the points where the following pairs of lines intersect (a) = and = 7 (b) = + and = 7 The lines form three sides of a square. (ii) Find the equation of the fourth side of the square. (iii) Find the area of the square. 3 Find the vertices of the triangle ABC whose sides are given b the lines AB: = BC: = 53 and AC: 9 + =. (ii) Show that the triangle is isosceles. A(0, ), B(, ), C(, 3) and D(3, 0) are the vertices of a quadrilateral ABCD. Find the equations of the diagonals AC and BD. (ii) Show that the diagonals AC and BD bisect each other at right angles. (iii) Find the lengths of AC and BD. (iv) What tpe of quadrilateral is ABCD? 5

16 Straight line models 5 The line = 5 crosses the ais at A. The line = + crosses the ais at B. The two lines intersect at P. Find the coordinates of A and B. (ii) Find coordinates of the point of intersection, P. (iii) Find the eact area of the triangle ABP. 6 Triangle ABC has an angle of 90 at B. Point A is on the ais, AB is part of the line + 8 = 0, and C is the point (6, ). Sketch the triangle. (ii) Find the equations of the lines AC and BC. (iii) Find the lengths of AB and BC and hence find the area of the triangle. (iv) Using our answer to (iii), find the length of the perpendicular from B to AC. 7 Two sides of a parallelogram are formed b parts of the lines = 9 and = 9. Show these two lines on a graph. (ii) Find the coordinates of the verte where the intersect. Another verte of the parallelogram is the point (, ). (iii) Find the equations of the other two sides of the parallelogram. (iv) Find the coordinates of the other two vertices. 8 The line with equation 5 + = 0 meets the ais at A and the line with equation + = meets the ais at B. The two lines intersect at a point C. Sketch the two lines on the same diagram. (ii) Calculate the coordinates of A, B and C. (iii) Calculate the area of triangle BC where is the origin. (iv) Find the coordinates of the point E such that ABEC is a parallelogram. 9 A median of a triangle is a line joining a verte to the midpoint of the opposite side. In an triangle, the three medians meet at a point called the centroid of the triangle. Find the coordinates of the centroid for each triangle shown in Figure 5.5. (0, ) (0, 9) (6, 0) ( 5, 0) (5, 0) Figure Straight line models 0 Find the eact area of the triangle whose sides have the equations + =, = 8 and + =. Straight lines can be used to model real-life situations. ften simplifing assumptions need to be made so that a linear model is appropriate. Eample 5.6 The diameter of a snooker cue varies uniforml from 9 mm to 3 mm over its length of 0 cm. Varing uniforml means that the graph of diameter against distance from the tip is a straight line. Sketch the graph of diameter ( mm) against distance ( cm) from the tip. (ii) Find the equation of the line. (iii) Use the equation to find the distance from the tip at which the diameter is 5 mm. 6

17 Solution The graph passes through the points (0, 9) and (0, 3). diameter (mm) (0, 3) (0, 9) distance from tip (cm) Figure 5.6 (ii) Gradient = = = 0. Using the form = m + c, the equation of the line is = (iii) Substituting = 5 into the equation gives 5 = = 6 = 60 the diameter is 5 mm at a point 60 cm from the tip. 5 Chapter 5 Coordinate geometr Discussion points Which of these situations in Figure 5.7 could be modelled b a straight line? For each straight line model, what information is given b the gradient of the line? What assumptions do ou need to make that a linear model is appropriate? How reasonable are our assumptions? Interest earned on savings in a bank account against time Ta paid against earnings Mass of candle versus length of time it is burning Population of birds on an island against time Figure 5.7 Height of ball dropped from a cliff against time Cost of apples against mass of apples Distance travelled b a car against time Mobile phone bill against number of tets sent Profit of ice cream seller against number of sales Value of car against age of car Mass of gold bars against volume of gold bars Length of spring against mass of weights attached 7

18 Straight line models Eercise 5.3 The graph shows the distance-time graph for two walkers, Tom and Sam. The walkers set off from the same point and walk in a straight line. distance (m) 0 0 Tom Sam time (s) Figure 5.8 Who is walking fastest? Eplain how ou know. (ii) Find the speed of each walker. (iii) What is the distance between the walkers after 0 seconds? Two rival tai firms have the following fare structures: Firm A: fied charge of plus 0p per kilometre; Firm B: 60p per kilometre, no fied charge. Sketch the graph of price (vertical ais) against distance travelled (horizontal ais) for each firm (on the same aes). (ii) Find the equation of each line. (iii) Find the distance for which both firms charge the same amount. (iv) Which firm would ou use for a distance of 6 km? 3 A firm manufacturing jackets finds that it is capable of producing 00 jackets per da, but it can onl sell all of these if the charge to the wholesalers is no more than 0 per jacket. n the other hand, at the current price of 5 per jacket, onl 50 can be sold per da. Assuming that the graph of price P against number sold per da N is a straight line: sketch the graph, putting the number sold per da on the horizontal ais (as is normal practice for economists) (ii) find its equation. Use the equation to find: (iii) the price at which 88 jackets per da could be sold (iv) the number of jackets that should be manufactured if the were to be sold at 3.70 each. The freezing point of water is 0 C or 3 F and the boiling point of water is 00 C or F. Draw a conversion graph to convert degrees Celsius, C, to degrees Fahrenheit, F. Label the horizontal ais, degrees Celsius, and the vertical ais, degrees Fahrenheit. (ii) Give the equation of the line in the form F = ac + b where a and b are constants to be found. (iii) The lowest possible temperature (absolute zero) is -73 C. What is absolute zero in Fahrenheit? (iv) The point (k, k) lies on the line found in part iii). Find the value of k. What is the significance of k? 5 To clean the upstairs window on the side of a house, it is necessar to position the ladder so that it just touches the edge of the lean-to shed as shown in Figure 5.9. The coordinates represent distances from in metres, in the and directions shown. Figure 5.9 A ladder (.5, ) shed B (.5, 0) 8

19 Find the equation of the line of the ladder. (ii) Find the height of the point A reached b the top of the ladder. (iii) Find the length of the ladder to the nearest centimetre. 6 A ladder rests against the wall of a house. The foot of the ladder is a metres from the base of the wall and the top of the ladder is b metres up the wall. (0, b) Figure 5.0 (a, 0) Show that the equation of the line representing the ladder can be written + = a b 7 Figure 5. shows the suppl and demand of labour for a particular industr in relation to the wage paid per hour. Suppl is the number of people willing to work for a particular wage, and this increases as the wage paid increases. Demand is the number of workers that emploers are prepared to emplo at a particular wage: this is greatest for low wages. Find the equation of each of the lines. (ii) Find the values of L * and W * at which the market clears, i.e. at which suppl equals demand. (iii) Although economists draw the graph this wa round, mathematicians would plot wage rate on the horizontal ais. Wh? 8 When the market price p of an article sold in a free market varies, so does the number demanded, D, and the number supplied, S. In one case D = p and S = + p. Sketch both of these lines on the same graph. (Put p on the horizontal ais.) The market reaches a state of equilibrium when the number demanded equals the number supplied. (ii) Find the equilibrium price and the number bought and sold in equilibrium. 5 Chapter 5 Coordinate geometr W 6 suppl (500, 6) wage rate ( per hour) 5 3 (000, 5) (000, 3) (L*, W*) demand (500, 3) quantit of labour (person hours per week) L Figure 5. 9

20 The circle Prior knowledge Completing the square see Chapter 3 The circle You are of course familiar with the circle, and have done calculations involving its area and circumference. In this section ou are introduced to the equation of a circle. The circle is defined as the locus of all the points in a plane which are at a fied distance (the radius) from a given point (the centre). This definition allows ou to find the equation of a circle. Remember, the length of a line joining (, ) to (, ) is given b ( ) length = ( ) For a circle of radius 3, with its centre at the origin, an point (, ) on the circumference is distance 3 from the origin. So the distance of (, ) from (0, 0) is given b ( 0 ) + ( 0) = 3 + = 3 + = 9 This is the equation of the circle in Figure 5.. The circle in Figure 5.3 has a centre (9, 5) and radius, so the distance between an point on the circumference and the centre (9, 5) is. Locus means possible positions subject to given conditions. In two dimensions it can be a path or a region. This is just Pthagoras theorem. Figure 5. 3 (, ) + = 3 Squaring both sides. ( 9) + ( 5) = (, ) ( 5) (9, 5) ( 9) Figure 5.3 The equation of this circle in Figure 5.3 is: ( 9 ) + ( 5) = ( 9) + ( 5) = 6. 0

21 ACTIVITY 5.3 Sophie tries to draw the circle + = 9 on her graphical calculator. Eplain what has gone wrong for each of these outputs (ii) Chapter 5 Coordinate geometr Figure 5. Figure 5.5 ACTIVITY 5. Show that ou can rearrange ( + f ) + ( + g) = r to give + f g + c = 0 and find an epression for c in terms of f, g and r. These results can be generalised to give the equation of a circle as follows: centre (0, 0), radius r : + = r centre (a, b), radius r : ( a) + ( b) = r. Note In this form, the equation highlights some of the important characteristics of the equation of a circle. In particular: the coefficients of and are equal (ii) there is no term. In the full book, this chapter continues with the following sections: 5 The intersection of a line and a curve 6 The intersection of two curves

22 Learning outcomes LEARNING UTCMES Now ou have finished this chapter, ou should be able to: Use the equation of a straight line in the forms = m( ) and a + b + c = 0 Draw or sketch a line, given its equation Find the equation of a line Recall and use the relationships between gradients for parallel and perpendicular lines Solve problems with parallel and perpendicular lines Find the intersection of two lines Solve real-life problems that can be modelled b a linear function Find the centre and radius of a circle from its equation, when the equation of the circle is given in its standard form needs to be rewritten in completed square form Find the equation of a circle given the radius and the centre using circle theorems to find out the centre and radius Find the equation of the circumcircle of a triangle Find the equation of a tangent to a circle using the circle theorems Find the points of intersection of a line and a curve Understand the significance of a repeated root in the case of a line which is a tangent to the curve no roots in the case of a line which does not intersect the curve KEY PINTS For a line segment A(, ) and B(, ) (Figure 5.6) then: n The gradient of AB is B (, ) A (, ) n Figure 5.6 The midpoint is + +, n The distance AB is ( ) + ( ) Using Pthagoras theorem.

23 Two lines are parallel gradients are equal. 3 Two lines are perpendicular the product of their gradients is. The equation of a straight line ma take an of the following forms: n line parallel to the ais: = a n line parallel to the ais: = b n line through the origin with gradient m: = m n line through (0, c) with gradient m: = m + c n line through (, ) with gradient m: = m( ) 5 The equation of a circle is n centre (0, 0), radius r: + = r n centre (a, b), radius r: ( a) + ( b) = r. 6 The angle in a semicircle is a right angle (Figure 5.7). Figure Chapter 5 Coordinate geometr 7 The perpendicular from the centre of a circle to a chord bisects the chord (Figure 5.8). Figure The tangent to a circle at a point is perpendicular to the radius through that point (Figure 5.9). Figure To find the points of intersection of two curves, ou solve their equations simultaneousl. 3

24 Problem solving Integer point circles Figure Look at the circle in Figure Its equation is + = 00. It goes through the point (6, 8). Since both 6 and 8 are integers, this is referred to as an integer point in this question. This is not the onl integer point this circle goes through; another is ( 0, 0) and there are others as well. How man integer points are there inside the circle? (ii) How man circles are there with equations of the form + = 00, where 0 < N < 00 that pass through at least one integer point? How man of these circles pass through at least integer points? (iii) Devise and eplain at least one method to find the equation of a circle with radius greater than 0 units that passes through at least integer points.

25 Problem specification and analsis Parts and (ii) of the problem are well defined and so deal with them first. Start b thinking about possible strategies. There are several quite different approaches, based on geometr or algebra. You ma decide to tr more than one and see how ou get on. Part (iii) is more open ended. You have to devise and eplain at least one method. Leave this until ou get to the last stage of the problem solving ccle. B then our earlier work ma well have given ou some insight into how to go about it. Information collection In this problem there will probabl be a large amount of trial and error in our data collection. As well as collecting information, ou will be tring out different possible approaches. There are a number of cases that ou could tr out and so ou need to be on the lookout for patterns that will cut down on our work. You have to think carefull about how ou are going to record our findings sstematicall. 3 Processing and representation The work ou need to do at this stage will depend on what ou have alread done at the Information collection stage. You ma have alread collected all the information ou need to answer parts and (ii) b just counting up the numbers. Alternativel, however, ou ma have found some patterns that will help ou to work out the answers. You then need to find a good wa to present our answers. Think of someone who is unfamiliar with the problem. How are ou going to show such a person what ou have found in a convincing wa? Interpretation So far ou have been looking at parts and (ii) of the problem. The are well defined and all the answers are numbers. In part (iii), ou are now epected to interpret what ou have been doing b finding not just numbers but also a method, so that ou can continue the work with larger circles. To give a good answer ou will almost certainl need to use algebra but ou will also need to eplain what ou are doing in words. The wording of the questions suggests there is more than one method and that is indeed the case. So a reall good answer will eplore the different possibilities. 5 Chapter 5 Coordinate geometr 5

26 Practice questions PRACTICE QUESTINS FR CHAPTERS T 5 MP Prove that = 3 3 [] MP MP (ii) Show that 3+ + = [] Solve the equation 3 = +. [3] (ii) Find a value of which is a counter-eample to 0 >. [] 3 Do not use a calculator in this question. Figure 5.3 shows the curves = + and + 5 = 0. Find the coordinates of their points of intersection. Give our answers as simplified surds. [5] = = 0 Figure 5.3 (ii) Prove that = is a tangent to = + and state the coordinates of the point of contact. [] Do not use a calculator in this question. Write in the form ( + a) + b. [3] (ii) State the coordinates of the turning point of = and whether it is a minimum or maimum. [3] (iii) Sketch the curve = and solve the inequalit > 0. [] 5 Figure 5.3 shows a circle with centre C which passes through the points A (, ) and B (, ). A (, ) B (, ) C 6 Figure 5.3

27 AB is a chord of the circle. Show that the centre of the circle must lie on the line + = 3, eplaining our reasoning. [7] (ii) The centre of the circle also lies on the -ais. Find the equation of the circle. [5] 5 6 Figure 5.33 shows an equilateral triangle, ABC with A, B on the -ais and C on the -ais. G C A D E B Figure 5.33 Each side of triangle ABC measures units. Find the coordinates of points A, B and C in eact form. [] (ii) Show that the equation of line BC can be written as = 3( ). [] A rectangle DEFG is drawn inside the triangle, as shown. D, E lie on the -ais, G on AC and F on BC. (iii) Find the greatest possible area of rectangle DEFG. [7] F Chapter 5 Practice questions M T 7 Figure 5.3 shows a spreadsheet with the information about tpical stopping distances for cars from the Highwa Code. Figure 5.35 has been drawn using the spreadsheet Home A Speed (mph) Insert B Thinking distance (m) Page Laout f C Braking distance (m) 6 38 Formulas D Total stopping distance (m) Distance (m) Speed (mph) Thinking distance (m) Braking distance (m) Total stopping distance (m) Figure 5.3 Figure 5.35 (a) What feature of the scatter diagram suggests that the thinking distance is directl proportional to speed? [] (b) What does this tell ou about the thinking time for different speeds? Comment, with a brief eplanation, on whether this is a reasonable modelling assumption. [] (c) Write down a formula connecting the speed, mph and the thinking distance d m. [] 7

28 Practice questions (ii) The spreadsheet gives the following linear best fit model for the total stopping distance, m in terms of the speed mph. = (a) Use the model to find the total stopping distance for a speed of 0 mph. [] (b) Eplain wh this is not a suitable model for total stopping distance. [] (iii) The spreadsheet gives the following quadratic best fit model for the total stopping distance. = Speed (mph) Quadratic model (m) Figure 5.36 (a) Calculate the missing value for 0 mph. [] (b) Give one possible reason wh the model does not give eactl the same total stopping distances as those listed in the Highwa Code. [] 8

29 Integral A level Mathematics online resources Integral has been developed b MEI and supports teachers and students with high qualit teaching and learning activities including dnamic resources and self-marking tests and assessments that cover the new Edecel A level specifications. ur Student and Whiteboard etetbooks link seamlessl with Integral A level Mathematics online resources, allowing ou to move with ease between corresponding topics in the etetbooks and Integral resources. All ou need is to be subscribed to both the etetbooks and Integral. Student etetbooks are downloadable versions of the printed tetbook that teachers can assign and reassign to students Within the Student etetbook Integral resources are listed in the Integral resources tab You can also access the relevant Integral resources from within the individual pages of the Student etetbook

30 Whiteboard etetbooks are online, interactive versions of the printed tetbooks that are ideal for wholeclass discussion and annotation. Whiteboard etetbooks are full integrated with Integral so that all the relevant resources are readil accessible when ou need them whether ou are presenting at the front of the class, or lesson planning at home. Click on the resources within the etetbooks to be taken directl to a range of relevant Integral resources. To have full access to the etetbooks and Integral resources ou must be subscribed to both Dnamic Learning and Integral. To trial our etetbooks and/or subscribe to Dnamic Learning, visit: To view samples of the new Integral A level resources and/or subscribe to Integral, visit

31 Navigate the new Edecel A level Mathematics specifications confidentl with print and digital resources that support our planning, teaching and assessment needs alongside specialist-led CPD events to help inspire and create confidence in the classroom. The following print and digital resources will be entered into Edecel s endorsement process: March Edecel A level Mathematics Year Student Book Jul Edecel A level Further Mathematics Core Year (AS) Student Book June Edecel A level Further Mathematics Core Year Student Book ct Edecel A level Mathematics Year (AS) Student etetbook April 7 Edecel A level Mathematics Year Student etetbook Aug 7 Edecel A level Further Mathematics Core Year (AS) Student etetbook Jul 7 Edecel A level Further Mathematics Core Year Student etetbook Nov 7 7 per student for ears access.0 per student for ears access 6.80 per student for 3 ears access Edecel A level Further Mathematics Statistics Student Book Sept Edecel A level Further Mathematics Mechanics Student Book Sept Edecel A level Further Mathematics Statistics Student etetbook ct 7 Edecel A level Further Mathematics Mechanics Student etetbook ct 7 5 per student for ears access 8 per student for ears access per student for 3 ears access Visit to pre-order our class sets or to sign up for our Inspection Copies or einspection Copies. Download and view on an device or browser Add, edit and snchronise notes across an two devices Complete interactive, self-check questions Please see above for prices Whiteboard etetbooks are online, interactive versions of the printed tetbooks that enable ou to: Displa pages for whole-class teaching Add notes and highlight areas Insert double-page spreads into our lesson plans and homework activities 00 (small institution up to 900 students) 50 (large institution 90+ students) Publishing from March 07 To find out more about etetbooks visit: M Revision Notes Target success in Maths with this proven formula for effective, structured revision; ke content coverage is combined with eam-stle tasks and practical tips to create a revision guide that students can rel on to review, strengthen and test their knowledge. Pub from Jan 08 Sign up to our eupdates to keep updated: www. hoddereducation.co.uk/e-updates CPD training Ensure that ou are full prepared for the upcoming changes b attending one of our new specification courses. For more information and to book our place visit *Please see enclosed order form or visit for details. ci al o 3 fo ffer r! Boo on Stu one ks plu dent Stud -ear a s free c ent ete cess to tbo oks * Edecel A level Mathematics Year (AS) Student Book Student etetbooks provide a downloadable version of the printed tetbook that ou can assign to students so the can: Spe

32 EDEXCEL A LEVEL MATHEMATICS For Year and AS First teaching from September 07 This sample chapter is taken from Edecel A level Mathematics Year (AS) Student Book, which has been entered into Edecel s official endorsement process. Help students to develop their knowledge and appl their reasoning to mathematical problems with worked eamples, stimulating activities and assessment support tailored to the 07 Edecel specification. The content benefits from the epertise of subject specialist Keith Pledger and the support of MEI (Mathematics in Education and Industr). Prepare students for assessment with skills-building activities and practice questions tailored to the changed criteria. Develop a fuller understanding of mathematical concepts with real world eamples that help build connections between topics and develop mathematical modelling skills. Cement understanding of problem-solving, proof and modelling with dedicated sections on these ke areas. Confidentl teach the new statistics requirements with five dedicated chapters. Cover the use of technolog in Mathematics with a variet of questions based around topics such as the use of spreadsheets in statistics. Develop knowledge and map progress with bridging questions, graduated eercises and clear learning objectives. Provide clear paths of progression that combine pure and applied maths into a coherent whole. Series editors Roger Porkess has been involved with Hodder Education for over 0 ears and is the author of man books as well as reports on mathematics education. He has eamining eperience and is a member of ALCAB. Catherine Berr has etensive authoring eperience and is a developer of online resources for MEI s Integral platform. Consultant editor Keith Pledger was formerl a senior eaminer and is an eperienced and high-selling author with over a million books sold. MEI is our author partner for Edecel. MEI (Mathematics in Education and Industr) is at the forefront of improving maths education in the UK, and offers teachers of all GCSE, Core Maths and A Level specifications a range of resources and training to enhance mathematical skills. MEI provides advice and support for maths teachers in schools and colleges, and pioneers the development of innovative teaching and learning resources. MEI s resources are developed b subject eperts with a deep knowledge of maths and maths teaching, and are designed to support students to develop deep mathematical understanding. MEI s Integral online learning platform provides etensive, purpose-written resources for students and teachers of AS/A Level Mathematics, including innovative interactive materials and formative assessments with individualised student records. Tetbook subject to change based on fqual feedback. To request Inspection Copies or einspection Copies and pre-order our class sets visit