Teddington School Sixth Form

Transcription

1 Teddington School Sith Form AS / A level Maths Induction and Key Course Materials

2 Introduction The Mathematics Department at Teddington School are delighted that you would like to continue your studies in mathematics by applying for A-Level at Teddington. This booklet to help support you from the start of the course, before you start in September, you need to feel confident about all the topics in this booklet and be able to complete the problems independently. You will have an assessment based on this pack in September. We will epect you to score at least 75% in that test. These are the basic skills you will need to succeed in A level Mathematics. We will be using them as building blocks as you study more comple topics during your A level course. Contained in this pack are : An assessment sheet Practice questions. Do these on paper, they will not be assessed. Thirteen assessment sheets. You need to complete all of these over the summer holidays. Write your answers in the booklet. You must bring this booklet to your first A-Level Mathematics lesson of the new term. We look forward to welcoming you in September. The Mathematics Department. If you are unsure about a method or want more practice, try these mymaths.co.uk (The Teddington School log in is "teddington" and the password is "pythagoras") Please fill this in as you complete each section of the assessed pieces. After you have attempted the work, marked it, corrected it, please hand it to your teacher to be verified. Sheet Page 7 Fractions 9 Algebra 1- Solving Linear Equations 11 Substitution 1 Algebra Rearranging equations 14 Factorising and Epanding Brackets 16 Indices 18 Surds 0 Algebra Quadratic Equations 4 Algebra 4 Simultaneous Equations 6 Straight line graphs 9 Trigonometry 1 Basic Trig Ratios and Pythagoras Trigonometry Sine and Cosine rule 4 Transformations of graphs Tick each bo Completed Marked Rate your Understanding You will be given the answers at the start of A Level course.

3 Practice Section Solve 1. Solve for : 10. Solve for : Solve for : Solve for : Solve for : Solve for : 7 7. Solve for : Solve for : Solve for : Solve for 1 1 d 1 1 d : 4d 1

4 Factorising 1. Factorise: 4y. Factorise: 8 1. Factorise: Factorise: Factorise: 4 Eponent Practice epand and simplify 1. a b. a b y y m m n Solving Inequalities

5 Geometry For these pairs of points, find the midpoint, distance, slope, and equation of the line. 1. (, 10 ), ( 4, 9 ). ( 1, 1 ), ( 4, 0 ). ( 4, 1 ), ( 0, 9 ) 4. ( 5, 15 ), ( 5, 9 ) 5. ( 0, 9 ), ( 6, 9 )

6 Fractions Assessment 1. Work these out : a) b) c) d) e) Write the reciprocals of these in a suitable form. a) 5 b) 6 1 c) 7 d) 0.1 e) 1. (Hint: The reciprocal of is ½ ). a) Evaluate giving your answer as a fraction 4 5 b) Evaluate. Give you answer in simplest form c) Evaluate 5

7 4. Write as single fraction 1 1

8 Algebra 1 Assessment - Solving Linear Equations 1. Solve the following equations : a) 4 ( ) b) (7 ) (5 4). Solve 6 5. Solve these equations : a) 7 b) 5 1 c) d) e) ( ) 7 f) ( ) 15 g) 10 5 h) 4 7

9 4. r 1 Write the inequality r 5. n is an integer. List the values of n such that: a) 6 n 1 b) 5 5n 0

10 Substitution into Formulae - Assessment 1. Given the formula v u s find s when a = 8 v = 1 u = 15.5 a. You are given the formula V 8 P. Work out the value of P when V=.85 and R = R 5. You are given the formula 9 v u at Work out the value of v when u=0, a= 6 and t= 5 4. v The braking distance, D metres, of a car is shown by the following formula D where v 5 f is its speed in kilometres per hour and f is the friction between the tyres and the road surface. a) Calculate D when v = 80 and f = 0.5 b) After an accident, it was shown from the skid marks that D = 50 and to calculate v. f. Use the formula 8

11 Algebra Assessment - Rearranging Equations 1. Make w the subject of the formula a) s w( r 14) b) p qw c) vw w. You are given the formula c( a 5) ( a). Rearrange the formula to give a in terms of c.. Rearrange the formula v u s to make u the subject. a 4. You are given the formula v u at. Rearrange the formula to give t in terms of v, u and a. 5. Rectangle A has length and width y. Rectangle B has length ( ) and width (y + 6)

12 Rectangle A y y + 6 Rectangle B The area of rectangle B is twice the area of rectangle A Find an epression for y in terms of.

13 Factorising and Epanding Brackets - Assessment 1. Epand and Simplify a) ( )( ) b) ( ). a) Factorise completely 4 b) Epand and simplify ( 5)( 5) c) Epand and simplify ( 7). a) Factorise completely 15 5 b) Epand and simplify ( 4 )( 4) c) Epand and simplify ( 6)

14 4. Simplify the epression 5 4 (Hint: factorise the top and bottom then cancel) 5. Factorise a) 4 1 b) 6 a) Simplify the epression 1 b) Simplify fully the following epression

15 Indices Assessment 1. Work out these and where appropriate leave the answer as a fraction a) 4 b) 64 c) 6 d) 9 1. a) Simplify h 4 b) Work out the value of 8 0. Evaluate 1 a) 15 b) 64 c) d) 4. a) Simplify t t 5 b) Simplify m m 5 c) Simplify y 5 y

16 5. a) Write down the value of 0 5 b) Find the value of 7 c) Simplify

17 Surds Assessment 1. a) Simplify as much as possible b) Find the value of ( m p) when m 5 and p 0. a) Write 6 in the form a b where a and b are prime numbers. b) Simplify fully 5 8. K is an integer K 1 Find the value of K

18 4. a) Evaluate b) Evaluate 9 8 c) Simplify 1

19 Algebra Assessment - Quadratic Equations 1. Epand and simplify a) ( )( 6) b) ( 4 )(6 ). Factorise each Quadratic Equation a) 6 b) 9 70 c) d) 5 e) 1 15 f) Factorise the following a) 4 b) 6 4 c) 10

20 4. b b 4ac a) Use the quadratic formula to solve the equation a Give your answer to decimal places. You must show your working. 5. b b 4ac a) Use the quadratic formula to solve the equation 6 0. a Give your answer to decimal places. You must show your working. b b 4ac b) Use the quadratic formula to solve the equation a Give your answer to decimal places. You must show your working. Completing the Square 6. a) Find the values of p and q such that 6 14 ( p) q

21 b) Find the values of p and q such that q p ) ( 14 8 c) Find the values of p and q such that q p ) ( a) Solve the equation b) Solve the equation Give you answer correct to decimal places.

22 9. A rectangular lawn has a path of width on three sides as shown. 5 m m The lawn is 5m long and m wide. The total area of the lawn and path is 9m² m a) By forming an epression, in terms of, for the total area of the lawn and path, show that m b) By solving the equation find the value of m

23 1. Solve the simultaneous equations y y. Solve the simultaneous equations y y. Solve the simultaneous equations y y Algebra 4 Assessment - Simultaneous Equations

24 Solving Simultaneous Quadratic Equations. The equation 7 y y represents a straight line. The equation 61 represents a circle. The line and the circle intersect at two points. Show that the -coordinates of the two points are given by the solution to the equation Solve the equation (give to dp). y 7

25 Straight Line Graphs - Assessment 1. (a) Make y the subject of the equation y 6 (b) On the grid, draw the line with equation y (c) On the grid, shade the region for which y 6, 0 4 and y 0. a) On the aes below draw the graph of y 4 1 b) Write down the equation of the line which is already drawn on the graph. y

26 . Find the equation of the straight line perpendicular to 1 y 7 that passes through the point 4 (1,) 4. An unknown line passes through the point (7,4) and is perpendicular to the line 1 y. a) State the gradient of the line perpendicular to 1 y b) Find the equation of the unknown line 5. line b y line a a) Calculate the gradient of the line A. b) Calculate the gradient of the line B. c) Write down the equation of the line A. d) Write down the equation of the line B. e) Write down the equation of the line which also passes through the point (0,) but is perpendicular to the line B.

27 6. y a) Write down the equation of the line shown. b) Write down the equation of the line which also passes through the point (0,) but is perpendicular to the line shown. 7. Use the graphs below to solve the following sets of sumultaneous equations y y y 6 y 1 a) y 6 y 18 b) y y 18 c) y y 1 =. =. =. y=. y=. y=.

28 Trigonometry 1 Assessment Basic Trig Ratios and Pythagoras 1. Work out the perimeter of each triangle. (a) (b) Diagrams NOT 5.6 cm accurately drawn 9.1 cm 10. cm 8. cm. The diagram show a right-angled triangle, ABC. A B C Angle ABC = 90 a) Calculate sin tan 4 ABC and PQR are similar triangles. P Q 15 y R PQ is the shortest side of triangle PQR. Angle PRQ=90 and PR = 15 cm b) (i) What is the value of cos y? (ii) What is the length of PQ?

29 . Find the area of triangle ABC to significant figures. C 5cm A 6 B 4. ABCDEFGH is a cuboid with sides 10cm, 8cm and 6cm as shown. H G E F 6cm D A 10cm Find the angle CAG B 8cm C 5. The graph of y = sin is shown below. a) Solve the equation sin 0. for values of between 0 and 60 b) Sketch on the graph above y cos

30 6. A D 0cm 1cm E 0cm B The diagram show a kite ABCD. AE=1cm, DE = EB = 0cm and BC = 8cm. a) Calculate the size of triangle EBC 8cm C b) Calculate the length of ECand hence find the area of this kite.

31 Trigonometry Assessment - Sine and Cosine Rule 1. B C ABCD is a quadrilateral with diagonal AC. 4. m m AB = 4. m and CD = 5. m. Angle BAC = 48, angle BCA = 19 and angle ACD = 79. A D Calculate the length of AD.. A buoy, B, is 80 km from port A on a bearing of 060. A lighthouse, L, is 100 km from the buoy, B, on a bearing of 15. A ship sails from port A to lighthouse L. Calculate the distance between the port and the lighthouse. 80 km B 15 Diagram NOT accurately drawn km A L

32 . 4 cm C 100 In triangle ABC, AC = 4 cm, AB = 7 cm and angle ACB = 100 Calculate the area of the triangle. A 7 cm B 4. North P km R Q is 14. km due south of P. R is 18.5 km from P The bearing of R from P is 068. Calculate the distance from Q to R. Q

33 1. Transformations of Graphs Assessment

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