A Level Maths summer preparation work

Transcription

1 A Level Maths summer preparation work Welcome to A Level Maths! We hope you are looking forward to two years of challenging and rewarding learning. You must make sure that you are prepared to study A Level Maths and this study guide has been designed to help you to take responsibility for this. Success at Maths requires lots of practice and you will find it much easier in the first few weeks of the course if you have practised some of the algebra required. You must make sure you are confident with each GCSE topic in this guide. They are the key skills you will need to help you tackle harder problems at A Level and having them sorted before you start will make sure that you are in the best position to succeed. Maths rewards students who work hard and are prepared. You should try the for each topic and check your answers at the back. Before you start your A level you should be able to do these efficiently and accurately. If you are not confident with a topic then here are some tips on how to use the resources to improve. No idea how to start or getting the wrong: Watch the video at and make notes of the examples. Then try some of the practice and mark your answers from the website. Finally try the in this booklet again. Can remember a bit, but you re not confident yet: Try some practice for that topic at and mark your answers. If you are getting them correct then practice until you feel fluent and confident. If not, go back and watch the video and make notes on the examples. Can answer the confidently and are getting the answers correct: Tick that topic off and move on. Repeat for all the topics until you feel confident that you have mastered these skills and are ready to tackle A Level Maths. If you find there is something you are still not sure about then make a note of where you are getting stuck ready to ask your teacher in September. There are a few problems at the end of the key skills for a bit of summer fun! Complete your and notes on lined paper in a folder or in a large exercise book or notepad. You will need to show this to your teacher in September. Remember to show all of your working and to set your work out logically and clearly. Good luck

2 A. Algebraic manipulation 1. Expanding two brackets. Expanding three brackets. Factorising expressions 4. Using index laws Evaluate a) Expand (x + 5)( x) b) Expand and simplify (x + )(x 1) x(6 x) Expand and simplify (x + 6)(x 4)(x + 5) Factorise a) 6x y xy 5 b) x + 5x 14 c) x + x d) 4x 81y e) 9x 15x + 4 a) 16 b) 5 0 c) 7 1 Simplify x x a) x b) (x ) c) x 1 x x x Video 14 and practice Video 15 and practice Videos 117,118,119 & 10 and practice Videos 17, 17, 174 & 175 and practice 5. Algebraic fractions 6. Completing the square a) Simplify x 4x 1 x +9x+9 b) Write as a single fraction 5 x x+1 Write the following expressions in form (x + p) + q a) x + 4x + b) x x + 7 Write the following expressions in the form (p(x + q) + r a) 4x 8x 16 b) x + x + 6 Videos 1,, & 4 and practice Video 10 and practice B. Formulae 1. Substitution When x = 1, y = 4 and z = 9, evaluate the following expressions a) xy + z b) z x c) y + xz d) (y + z) e) yz. Changing the subject of a formula Change the subject of each formula to give the letter in brackets a) P = l + w [w] b) d = S [T] T c) y 7x = 7 y d) y = x+ 4 x [y] [x] Video 0 and practice Videos 7 & 8 and practice

3 C. Surds 1. Manipulating surds. Rationalising the denominator Simplify a) 45 b) ( + )( ) c) Rationalise and simplify a) b) 5 5 D. Quadratic equations 1. Solving by factorisation. Solving by completing the square. Solving by using the formula Solve by factorising a) 5x = 15x b) x + 7x + 10 = 0 c) x x = 10 d) x(x + 1) = x + 15 Solve by completing the square. Give your solutions in surd form. a) x + 8x 5 = 0 b) 5x + x 4 = 0 c) (x + )(x 4) = 5 Solve a) x + 6x + = 0 b) x 4x 7 = 0 Give your answers to significant figures and in surd form. E. Simultaneous equations 1. Solving simultaneous linear equations using elimination. Solving simultaneous linear equations using substitution. Solving simultaneous equations where one is quadratic Solve using elimination a) x + y = 11 x y = 9 b) x + y = 11 x + y = 4 Solve using substitution a) y = x + 1 5x + y = 14 b) x y = 16 4x + y = Solve a) y = x + 1 x + y = 1 b) y = x y xy = 8 Videos 05, 07 & 08 and practice Video 06 and practice Video 66 and practice Video 67a and practice Video 67 and practice Video 95 and practice Video 96 and practice Video 98 and practice

4 F. Graphs of functions 1. Quadratic graphs a) On squared paper plot the graph of y = + 4x x for x from - to 4. Use your graph to find estimate for the solutions of x 4x = b) On separate axes sketch these graphs showing where they cross the axes. i) y = x 5x + 6. Cubic, reciprocal & exponential graphs. Trigonometric graphs 4. Equations of circles 5. Applying the transformations of y = f(x)+a, y = f(x + a), y = -f(x) and y = f(-x) to the graph of y = f(x) ii) y = x + 4x Sketch these graphs, showing any intercepts. a) y = x b) y = 1 x c) y = x Sketch these graphs for x from 0 to 60 a) y = sin x b) y = cos x c) y = tan x a) Sketch the graph of x + y = 100 showing where it crosses the axes. b) What is the equation of a circle centred on the origin with a radius of 5? a) If you know the graph of y = f(x) explain how you would draw i) y = f(x) + 5 ii) y = f(x + 5) iii) y = f( x) iv) y = f(x) b) The coordinate A, (,) lies on y = f(x). Where will A move to on the graph i) y = f( x) ii) y = f(x) iii) y = f(x) iv) y = f(x ) G. Functions 1. Finding composite functions. Finding the inverse of a function f(x) = x + 1, g(x) = x Find a) f( ) d) fg(x) b) g ( 1 ) e) fg( 4) c) gf(x) f) gf(1) Find the inverse function, f 1 (x) for a) f(x) = x + b) f(x) = 4 x c) f(x) = x x Videos 64, 65 & 71 and practice Videos 44, 45 & 46 and practice Videos 8, 9 & 40 and practice Video 1 and practice Video and practice Video 70 and practice Video 69 and practice

5 Just for fun: A. Algebraic manipulation 1. Expanding two brackets a) 10 4x 6x b) 4x + 4x (1x + x ) = x 8x. Expanding three brackets x 11x 8x 10. Factorising expressions a) xy 4 (x + 5y) b) (x + 7)(x ) c) (x + )(x 1) d) (x + 9y)(x 9y) (difference of two squares) e) (x 1)(x 4) 4. Using index laws Evaluate a) 64 b) 1 c) 1 Simplify a) b) 8x 6 c) x 5. Algebraic fractions a) Simplify x 7 x+ b) Write as a single fraction x 17 x (x )(x+1) 6. Completing the square Write the following expressions in form (x + p) + q a) (x + ) 1 b) (x 1) + 6 Write the following expressions in the form (p(x + q) + r a) 4(x 1) 0 b) (x + 4 ) + 9 8

6 B. Formulae 1. Substitution When x = 1, y = 4 and z = 9, evaluate the following expressions a) xy + z = 7 b) z x = c) y + xz = 0.5 d) (y + z) =5. Changing the subject of a formula yz =-4 Change the subject of each formula to give the letter in brackets a) w = P l b) T = S d c) y = x + x = 4y y+ C. Surds 1. Manipulating surds Simplify a) 45 = 5 b) ( + )( ) = -1. Rationalising the denominator D. Quadratic equations = Rationalise and simplify 5 a) 5 b) Solving by factorisation Solve by factorising a) x = 0 or x = b) x = or x = 5 c) x = 5 or x = d) x = 5 or x =. Solving by completing the square. Solving by using the formula Solve by completing the square. Give your solutions in surd form. a) x = 4 ± 1 b) x = ± c) x = 1 ± 14 Solve a) x = 1 + = -0.4 or x = 1 = 1.58 b) x = 1 + =.1 or x= 1 = 1.1 E. Simultaneous equations 1. Solving simultaneous linear equations using elimination Solve using elimination a) x = 6, y = 1 b) x =, y = 5

7 . Solving simultaneous linear equations using substitution. Solving simultaneous equations where one is quadratic a) x = 1, y = b) x = 4.5, y = 7 Solve a) When x =, y = and when x =, y = b) When x =, y = 4 and when x =, y = 4 F. Graphs of functions 1. Quadratic graphs Use your graph to find estimate for the solutions of x 4x = Solutions where graph crosses x axis. x = -0.6 or.6. On separate axes sketch these graphs showing where they cross the axes. i) Crosses at (0,5) (,0) (,0) ii) Crosses at (0,0) and (4,0). Cubic, reciprocal & exponential graphs. Trigonometric graphs 4. Equations of circles b What is the equation of a circle centred on the origin with a radius of 5? x + y = 5

8 5. Applying the transformations of y = f(x)+a, y = f(x + a), y = -f(x) and y = f(-x) to the graph of y = f(x) G. Functions 1. Finding composite functions. Finding the inverse of a function a) If you know the graph of y = f(x) explain how you would draw i) y = f(x) + 5 Translate up 5 units ii) y = f(x + 5) Translate left 5 units iii) y = f( x) Reflect in the y axis iv) y = f(x) Reflect in the x axis b) The coordinate A, (,) lies on y = f(x). Where will A move to on the graph i) y = f( x) -> (-,) ii) y = f(x) -> (, -) iii) y = f(x) -> (, 0) iv) y = f(x ) -> (4, ) f(x) = x + 1, g(x) = x a) 4 d) ( x ) + 1 = x b) 9 e) = c) x +1 f) a) f 1 (x) = x b) f 1 (x) = 4 (self inverse) x c) f 1 (x) = = x 1 1 x