Getting started with A Level Maths - South Devon College

Getting started with A Level Maths

This booklet will help you bridge the gap between GCSE and A level Maths and provide you with the best possible start with your A level studies.

Essential tools for A level Maths: You will need;

• Calculator – Casio fx-CG50 Advanced colour graphic calculator This is the most advanced graphics calculator approved by exam boards for the UK market and will be invaluable in your A Level Mathematics

• Textbook – AQA A-Level Mathematics (For A level Year 1 and AS) Published by Hodder Education ISBN 9781471852862

Congratulations on choosing to study A level Maths at South Devon College! To help you prepare, this booklet will enable you to brush up on some of the key skills that you have learned at GCSE. You are going to need to use them from Day 1, so if you don’t have a good grasp of the basics then you will need to work on them over the first week so that you can start A level Maths with confidence.

• Do the questions in the booklet. Check your answers with those at the back and mark your work.

• If there is anything you have got wrong, be sure to go back and revise that topic – the aim of this booklet is to get 100% correct.

• Resilience and problem solving ability are two of the key skills you will need in A level Maths – it is likely that you will get stuck when working through some of the problems in this booklet, what is more important is that are resourceful and determined enough to find the answers.

• There are many great exam resources on the internet to help you on the higher level maths topics. Consider looking at sites like or mathsgenie.co.uk or examsolutions.net for video tutorials.

• Go back over any errors and correct your mistakes.

There will be a test after your first week at college based on the topics in this booklet. Please bring the booklet with you to show us that you have completed and marked it – this is NOT optional.

How does A level Maths differ from GCSE Maths? GCSE Maths A Level Maths If you are naturally good at maths you will understand the concepts without difficulty and can do well without much extra studying.

Understanding the topics can be more challenging even for talented mathematicians and you will need to do a LOT of extra study outside maths classes.

The answer is what matters, you only get a few marks for showing your working out.

The method matters more than the answer. Often you are given the answer and the question is more about showing the steps that you need to take to get there.

You are given an exercise book and your teacher will tell you when to take notes and what to write.

You will need workbooks, a folder and dividers and a rough book to separate out your “neat” notes from your rough working. You are responsible for making and managing your own notes, organising your work and making revision notes.

Nobody minds how you set out your workings provided you get the answer.

How you present your work, using the correct notation and showing logical steps in your workings is critical. An examiner needs to be able to understand your methods.

Fractions You need to be really confident with numerical fractions so that you know what to do with algebraic ones.

Multiplication 23

× 45

= 2×43×5

= 815

and 2 × 35

= 21

× 35

= 65 NOT

610

So, using algebra:

2𝑥𝑥 �3𝑥𝑥4 � = �

2𝑥𝑥1 ��

3𝑥𝑥4 � =

6𝑥𝑥2

4 = 3𝑥𝑥2

2

Always simplify fractions by dividing numerator and denominator by any common factors

Division 83

÷ 23

= 83

× 32

= 8×33×2

= 82

= 4 So, using algebra:

5𝑥𝑥 ÷1𝑥𝑥 = �

5𝑥𝑥1 ��

𝑥𝑥1� = 5𝑥𝑥2

Addition and Subtraction Start by making the denominators the same

54

+ 32

= 54

+ 64

= 114

NB: In A level Maths an “improper” fraction is preferred, rather than using mixed numbers or decimals So, using algebra:

2𝑥𝑥5 −

12 =

2𝑥𝑥 × 25 × 2 −

1 × 52 × 5 =

4𝑥𝑥10 −

510 =

4𝑥𝑥 − 510

Without using a calculator, work out these giving your answer as a single, simplified fraction.

1. 34

×25

2. 2 + 35

3. 32

÷14

4. 2

7�4

5. 3𝑥𝑥5

× 4

6.1𝑥𝑥

+ 2𝑥𝑥

7. 5

32�

8. 2

3�3

4�

9. �38

÷14� ×

2𝑥𝑥3

10.3𝑥𝑥

+ 2𝑥𝑥2

11. �32

×14� + 3

12. 2𝑥𝑥 + 7

2−

35

Indices You will need to be able to manipulate indices all the time in A level Maths so ensure that you are confident with all your index rules. You need to know these;

𝑎𝑎𝑚𝑚𝑎𝑎𝑛𝑛 = 𝑎𝑎𝑚𝑚+𝑛𝑛 𝑎𝑎𝑚𝑚

𝑎𝑎𝑛𝑛= 𝑎𝑎𝑚𝑚 ÷ 𝑎𝑎𝑛𝑛 = 𝑎𝑎𝑚𝑚−𝑛𝑛 (𝑎𝑎𝑚𝑚)𝑛𝑛 = (𝑎𝑎𝑛𝑛)𝑚𝑚 = 𝑎𝑎𝑚𝑚×𝑛𝑛

(𝑎𝑎𝑎𝑎)𝑛𝑛 = 𝑎𝑎𝑛𝑛𝑎𝑎𝑛𝑛 (𝑎𝑎𝑏𝑏

)𝑛𝑛 = 𝑎𝑎𝑛𝑛

𝑏𝑏𝑛𝑛 𝑎𝑎0 = 1 𝑎𝑎1 = 𝑎𝑎 𝑎𝑎−1 = 1

𝑎𝑎

A negative power means a reciprocal 3−2 = 132

= 19 or (1

2)−2 = (2

1)2 = 4

A fractional power indicates a root 813 = √83 = 2 (since 2x2x2=8)

Example Without a calculator, simplify the following – leave your answer in the form 𝑎𝑎𝑛𝑛

13. 𝑎𝑎4 × 𝑎𝑎3

14. 𝑎𝑎5 ÷ 𝑎𝑎3 15. (𝑥𝑥3)2

Evaluate the following to find a numerical value (no calculators)

16. (25)3

17. 2713 18. 9

32

19. 81−14 20. (2

3)−2

21. �49

Practice working with indices and make sure you know all the index rules

1632 = (√16 )3 = 43 = 64

18−43 = (

18

)43 = (

1√83 )4 = �

12�4

= 1

16

Hint – do the negatives/reciprocals first, then the root, then the top power

(9𝑥𝑥4𝑦𝑦3)12 = 3𝑥𝑥2𝑦𝑦

32

Hint – EVERYTHING in the brackets needs to be square rooted.

Indices – Expressing in the form 𝒂𝒂𝒂𝒂𝒏𝒏 It is important to be able to write expressions in the form 𝑎𝑎𝑥𝑥𝑛𝑛 and for this it is vital to understand the rules regarding numerical multipliers in indices and fractions. An important technique is the ability to separate the numbers from the 𝑥𝑥 terms Example You can split the numerator of a fraction to make 2 separate terms but you can NEVER do this with a denominator. Example Be careful though This is WRONG – this fraction cannot be

simplified

22. 5√𝑥𝑥

23. 2𝑥𝑥3

24. 3√𝑥𝑥

25. √𝑥𝑥5

26. (2𝑥𝑥3

)2 27. 1√𝑥𝑥3

28. (2√𝑥𝑥)3 29. 4

3𝑥𝑥5

30. √𝑥𝑥3𝑥𝑥

31. 3𝑥𝑥2

√𝑥𝑥

32. 𝑥𝑥−2𝑥𝑥2

33. 12𝑥𝑥2𝑦𝑦

34. (27𝑥𝑥6𝑦𝑦5)13

35. (16𝑥𝑥2

𝑦𝑦)−

14

Common Mistakes 13𝑥𝑥2

= 3𝑥𝑥−2 √4𝑥𝑥 = 4𝑥𝑥12 Both WRONG!

13𝑥𝑥2

= �13� � 1

𝑥𝑥2� = 1

3𝑥𝑥−2 √4𝑥𝑥 = √4 × √𝑥𝑥 = 2𝑥𝑥

12 Correct

2𝑥𝑥

= 2 × 1𝑥𝑥

= 2𝑥𝑥−1

65𝑥𝑥2

= �65� �

1𝑥𝑥2� =

65𝑥𝑥−2

2 + 𝑥𝑥√𝑥𝑥

= 2√𝑥𝑥

+ 𝑥𝑥√𝑥𝑥

= 2 �1√𝑥𝑥� +

𝑥𝑥1

𝑥𝑥12

= 2𝑥𝑥−12 + 𝑥𝑥

12

𝑥𝑥2

𝑥𝑥 + 1≠𝑥𝑥2

𝑥𝑥+

𝑥𝑥2

1

Surds A surd is an irrational root eg √2, √3 but not √4 or √9 who have whole number answers. Simplifying surds: √𝑎𝑎𝑎𝑎 = √𝑎𝑎√𝑎𝑎 √20 = √4 × 5 = √4 √5 = 2√5

�𝑎𝑎𝑏𝑏

= √𝑎𝑎√𝑏𝑏

�34 = √3

√4= √3

2

Example √75 + 2√12 = √25 × 3 + 2√4 × 3 = √25√3 + 2√4√3 = 5√3 + 4√3 = 9√3 Rationalising the denominator: This means re-writing a fraction so that there is no surd on the bottom. Where there is only one term in the denominator we do this by multiplying both the top and the bottom by the surd that is on the bottom. Where there is more than one term on the bottom we need to use the difference of 2 squares to find a multiplier that will get rid of the surd on the bottom.

Example 1√5

= 1√5

× √5√5

= 1×√5√5×√5

= √55

3

1+√2= 3

1+√2× 1−√2

1−√2= 3 (1−�2)

�1+√2�(1−√2)= 3−3√2

1−2= 3−3√2

−1= −3 + 3√2

Write 13−√3

in the form 𝑎𝑎 + 𝑎𝑎√3

13−√3

× 3+√33+√3

= 3+√39−3√3+3√3−√3√3

= 3+√36

= 36

+ √36

= 12

+ 16 √3

Write in the form 𝑎𝑎√𝑎𝑎

36. √27 37. √48 38. √122

39. √20− 3√45 40. √200 + √18 − 2√50

Rationalise the denominator

41. 2√3

42. 11+√2

43. 34−√2

Remember √5 × √5 = 5 NOT 25

Quadratics Quadratics turn up EVERYWHERE in A level Maths – the good news is that the basic techniques are ones that you already know for GCSE maths. Factorisation Ensure you are good at factorising into double brackets 𝑥𝑥2 − 5𝑥𝑥 + 6 = (𝑥𝑥 − 3)(𝑥𝑥 − 2) Remember that not all quadratics can be factorised or even solved! The quadratic formula You need to learn this and use it with confidence. Be aware that your answer may contain

surds. If 𝑎𝑎𝑥𝑥2 + 𝑎𝑎𝑥𝑥 + 𝑐𝑐 = 0 then 𝑥𝑥 = −𝑏𝑏∓√𝑏𝑏2−4𝑎𝑎𝑎𝑎

2𝑎𝑎

Difference of 2 squares 𝑎𝑎2 − 𝑎𝑎2 = (𝑎𝑎 + 𝑎𝑎)(𝑎𝑎 − 𝑎𝑎)

Examples 9 − 𝑥𝑥2 = (𝑥𝑥 + 3)(𝑥𝑥 − 3) 4𝑥𝑥2 − 25 = (2𝑥𝑥 + 5)(2𝑥𝑥 − 5) Factorise the following quadratics

44. 𝑥𝑥2 + 2𝑥𝑥 − 15 ( 𝑥𝑥 − 3 )( ) 45. 𝑥𝑥2 − 9𝑥𝑥 − 10 46. 6𝑥𝑥2 + 2𝑥𝑥 47. 49 − 4𝑥𝑥2 48. 2𝑥𝑥2 + 5𝑥𝑥 − 3 49. 4𝑥𝑥2 + 4𝑥𝑥 + 1

Solve using the quadratic formula without a calculator (leave in surd form if necessary)

50. 𝑥𝑥2 − 5𝑥𝑥 + 4 = 0

51. 3𝑥𝑥2 + 2𝑥𝑥 − 1

52. 𝑥𝑥2 = 3𝑥𝑥 + 2

Factorise and solve

53. 10𝑥𝑥2 − 2𝑥𝑥 = 0

54. 9𝑥𝑥 − 27𝑥𝑥2 + 0

55. 14𝑥𝑥2 − 21𝑥𝑥 = 0

Completing the square Some quadratics are “perfect squares” eg 𝑥𝑥2 + 4𝑥𝑥 + 4 = (𝑥𝑥 + 2)(𝑥𝑥 + 2) = (𝑥𝑥 + 2)2 Most quadratics are not, however it can be useful to write them as square that is “adjusted” slightly. Example 𝑥𝑥2 + 4𝑥𝑥 + 7 = (𝑥𝑥 + 2)2 − 4 + 7 = (𝑥𝑥 + 2)2 + 3 In general 𝒂𝒂𝟐𝟐 + 𝒃𝒃𝒂𝒂 + 𝒄𝒄 = (𝒂𝒂+ 𝒉𝒉𝒂𝒂𝒉𝒉𝒉𝒉 𝒐𝒐𝒉𝒉 𝒃𝒃)𝟐𝟐 – (𝒉𝒉𝒂𝒂𝒉𝒉𝒉𝒉 𝒐𝒐𝒉𝒉 𝒃𝒃)𝟐𝟐 + 𝒄𝒄 Examples 𝑥𝑥2 + 6𝑥𝑥 + 2 = (𝑥𝑥 + 3)2 − 32 + 2 = (𝑥𝑥 + 3)2 − 7

𝑥𝑥2 − 4𝑥𝑥 + 3 = (𝑥𝑥 − 2)2 − (−2)2 + 3 = (𝑥𝑥 − 2)2 − 1

𝑥𝑥2 + 5𝑥𝑥 − 2 = �𝑥𝑥 +52�

2

− �52�

2

− 2 = �𝑥𝑥 +52�

2

−334

Complete the square leaving these expressions in the form (𝒂𝒂+ 𝒑𝒑)𝟐𝟐 + 𝒒𝒒 56. 𝑥𝑥2 + 8𝑥𝑥 + 7

57. 𝑥𝑥2 − 2𝑥𝑥 − 15

58. 𝑥𝑥2 + 6𝑥𝑥 + 10

59. 𝑥𝑥2 + 12𝑥𝑥 + 100

60. 𝑥𝑥2 − 3𝑥𝑥 − 1

61. 𝑥𝑥2 − 12𝑥𝑥 + 1

Solving equations by completing the square Many quadratics can be solved using this technique. Example Solve 𝑥𝑥2 − 4𝑥𝑥 − 5 = 0 Complete the square (𝑥𝑥 − 2)2 - 9 = 0 Put the number on the right (𝑥𝑥 − 2)2 = 9 Square root both sides (remember ± signs) 𝑥𝑥 − 2 = ±3 Add 2 to both sides to get TWO answers 𝑥𝑥 = 2 ± 3 so 𝑥𝑥 = 5 or 𝑥𝑥 = −1 Solve these by completing the square

62. 𝑥𝑥2 + 6𝑥𝑥 − 7 = 0 63. 𝑥𝑥2 − 2𝑥𝑥 − 3 = 0 64. 𝑥𝑥2 + 5𝑥𝑥 = −6

Triangles and Trigonometry Right angled triangles

For any triangle

Find the missing side or angle

65.

66.

67.

68.

69. 70.

Or for any triangle

Area = 12

𝑎𝑎𝑎𝑎 sin𝐶𝐶

Sine Rule 𝑎𝑎

sin𝐴𝐴 = 𝑎𝑎

sin𝐵𝐵 = 𝑐𝑐

sin 𝐶𝐶

Or sin𝐴𝐴𝑎𝑎

= sin𝐵𝐵𝑏𝑏

= sin𝐶𝐶𝑎𝑎

Cosine Rule 𝑎𝑎2 = 𝑎𝑎2 + 𝑐𝑐2 – 2bc cos A

Pythagoras’s Theorem 𝑎𝑎2 + 𝑎𝑎2 = ℎ2

Trigonometric Rations

sin 𝜃𝜃 = 𝑜𝑜𝑜𝑜𝑜𝑜ℎ𝑦𝑦𝑜𝑜

cos𝜃𝜃 = 𝑎𝑎𝑎𝑎𝑎𝑎ℎ𝑦𝑦𝑜𝑜

tan𝜃𝜃 = 𝑜𝑜𝑜𝑜𝑜𝑜𝑎𝑎𝑎𝑎𝑎𝑎

Practice Test Are you ready for A Level Maths yet? Try this test in exam conditions with a time limit of 1 hour. Use lined paper and show all working out. Mark it using the answers at the back (2 marks per question) and convert your score to a percentage. You should be getting above 80%. If you get less than 60% this is cause for concern and you will need to go over all the topics in this booklet again carefully, brush up your skills and try the test again. Between 60-80% shows there are areas where you will need to do extra work to ensure a smooth transition to A level Maths – focus on the questions you got wrong and do any corrections diligently.

1. Write as a single fraction:

a) 325� b) �3𝑥𝑥

2 ÷ 5

3� × 1

3

2. Evaluate: Simplify fully:

a) 16−14 b) (16

𝑥𝑥3)−

12

3. Write in the form 𝑎𝑎𝑥𝑥𝑛𝑛:

a) 23𝑥𝑥

b) 4√𝑥𝑥5

4. Simplify:

a) √32 b) √20 + 2√45 − 3√80 5. Rationalise the denominator:

a) 1√2

b) 5

2−√3

6. Solve the quadratics by factorising: a) 𝑥𝑥2 − 5𝑥𝑥 − 24 = 0 b) 9𝑥𝑥2 − 4 = 0

7. Solve these quadratics using the formula (leave your answer in surd form if necessary) a) 6𝑥𝑥2 + x – 1 = 0 b) 𝑥𝑥2 − 7𝑥𝑥 + 9 = 0

8. Complete the square and write in the form (𝑥𝑥 + 𝑜𝑜)2 + 𝑞𝑞

a) 𝑥𝑥2 + 2𝑥𝑥 − 6 b) 𝑥𝑥2 + 3𝑥𝑥 + 14

9. Find the side marked 𝑥𝑥 or 𝑎𝑎 to 1.d.p

10. a) Simplify 33 × 3𝑛𝑛 b) Hence write 2𝜃𝜃𝜃𝜃7 × (3𝑛𝑛+1) in the form 3𝑝𝑝

Quadratic formula

𝑥𝑥 = −𝑎𝑎 ∓ √𝑎𝑎2 − 4𝑎𝑎𝑐𝑐

2𝑎𝑎

Cosine rule 𝑎𝑎2 = 𝑎𝑎2 + 𝑐𝑐2 − 2𝑎𝑎𝑐𝑐 cos𝐴𝐴

Score /40

Solutions 1. 3

10

2. 135

3. 6

4. 114

5. 12𝑥𝑥5

6. 3𝑥𝑥

7. 103

8. 89

9. X

10. 3𝑥𝑥+2𝑥𝑥2

11. 278

12. 10𝑥𝑥+2910

13. 𝑎𝑎7 14. 𝑎𝑎2 15. 𝑥𝑥6 16. 8

125

17. 3 18. 27

19. 13

20. 94

21. 23

22. 5𝑥𝑥12

23. 2𝑥𝑥−3

24. 3𝑥𝑥−12

25. 15𝑥𝑥12

26. 49𝑥𝑥2

27. 𝑥𝑥−13

28. 8𝑥𝑥32

29. 43𝑥𝑥−5

30. 13𝑥𝑥−

12

31. 3𝑥𝑥 32

32. 𝑥𝑥−1 − 2𝑥𝑥−2

33. 12𝑥𝑥−2𝑦𝑦−1

34. 3𝑥𝑥2𝑦𝑦53

35. 12𝑥𝑥−

12𝑦𝑦

14

36. 3√3 37. 4√3 38. √3 39. −7√5 40. 3√2

41. 2√33

42. −1 + √2

43. 12+3√214

44. (𝑥𝑥 − 3)(𝑥𝑥 +5)

45. (𝑥𝑥 − 10)(𝑥𝑥 +1)

46. 2𝑥𝑥(3𝑥𝑥 + 1) 47. (7 + 2𝑥𝑥)(7−

2𝑥𝑥) 48. (2𝑥𝑥 − 1)(𝑥𝑥 +

3) 49. (2𝑥𝑥 + 1)2 50. 𝑥𝑥 = 4 𝑜𝑜𝑜𝑜 𝑥𝑥 =

1

51. 𝑥𝑥 = 13

𝑜𝑜𝑜𝑜 𝑥𝑥 =−1

52. 𝑥𝑥 = 3 ±√172

53. 2𝑥𝑥(5𝑥𝑥 −1), 𝑠𝑠𝑜𝑜 𝑥𝑥 =0 𝑜𝑜𝑜𝑜 𝑥𝑥 = − 1

5

54. 9𝑥𝑥(1−3𝑥𝑥)𝑠𝑠𝑜𝑜 𝑥𝑥 =0 𝑜𝑜𝑜𝑜 𝑥𝑥 = 1

3

55. 7𝑥𝑥(2𝑥𝑥 −3), 𝑠𝑠𝑜𝑜 𝑥𝑥 =0 𝑜𝑜𝑜𝑜 𝑥𝑥 = 3

2

56. (𝑥𝑥 + 4)2 − 9 57. (𝑥𝑥 − 1)2 − 16 58. (𝑥𝑥 + 3)2 + 1

59. (𝑥𝑥 + 6)2 + 64

60. �𝑥𝑥 − 32�2− 13

4

61. (𝑥𝑥 − 14)2 + 15

16

62. 𝑥𝑥 =−7 𝑜𝑜𝑜𝑜 𝑥𝑥 = 1

63. 𝑥𝑥 = 3 𝑜𝑜𝑜𝑜 𝑥𝑥 =−1

64. 𝑥𝑥 = −2 𝑜𝑜𝑜𝑜 𝑥𝑥 = −3

65. 𝑥𝑥 = 9 66. 𝑥𝑥 = 15.6 67. 𝜃𝜃𝑠𝑠 = 29.1˚ 68. 𝜃𝜃 = 45.5˚ 69. 𝜃𝜃 = 48.8˚ 70. 𝑎𝑎 + 4.7

Answers – Test For each question give yourself; 2 marks for a perfect answer and perfect working out. 1 mark for either a correct answer with an error/omission in the calculations or a correct method and working with an error in the answer. 0 marks for errors in answer and no working out.