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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;
β’ Calculator β 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
β’ Textbook β 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 donβt 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.
β’ Do the questions in the booklet. Check your answers with those at the back and mark your work.
β’ If there is anything you have got wrong, be sure to go back and revise that topic β the aim of this booklet is to get 100% correct.
β’ Resilience and problem solving ability are two of the key skills you will need in A level Maths β 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.
β’ 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.
β’ 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 β 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 βneatβ 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
Γ 45
= 2Γ43Γ5
= 815
and 2 Γ 35
= 21
Γ 35
= 65 NOT
610
So, using algebra:
2π₯π₯ οΏ½3π₯π₯4 οΏ½ = οΏ½
2π₯π₯1 οΏ½οΏ½
3π₯π₯4 οΏ½ =
6π₯π₯2
4 = 3π₯π₯2
2
Always simplify fractions by dividing numerator and denominator by any common factors
Division 83
Γ· 23
= 83
Γ 32
= 8Γ33Γ2
= 82
= 4 So, using algebra:
5π₯π₯ Γ·1π₯π₯ = οΏ½
5π₯π₯1 οΏ½οΏ½
π₯π₯1οΏ½ = 5π₯π₯2
Addition and Subtraction Start by making the denominators the same
54
+ 32
= 54
+ 64
= 114
NB: In A level Maths an βimproperβ fraction is preferred, rather than using mixed numbers or decimals So, using algebra:
2π₯π₯5 β
12 =
2π₯π₯ Γ 25 Γ 2 β
1 Γ 52 Γ 5 =
4π₯π₯10 β
510 =
4π₯π₯ β 510
Without using a calculator, work out these giving your answer as a single, simplified fraction.
1. 34
Γ25
2. 2 + 35
3. 32
Γ·14
4. 2
7οΏ½4
5. 3π₯π₯5
Γ 4
6.1π₯π₯
+ 2π₯π₯
7. 5
32οΏ½
8. 2
3οΏ½3
4οΏ½
9. οΏ½38
Γ·14οΏ½ Γ
2π₯π₯3
10.3π₯π₯
+ 2π₯π₯2
11. οΏ½32
Γ14οΏ½ + 3
12. 2π₯π₯ + 7
2β
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;
ππππππππ = ππππ+ππ ππππ
ππππ= ππππ Γ· ππππ = ππππβππ (ππππ)ππ = (ππππ)ππ = ππππΓππ
(ππππ)ππ = ππππππππ (ππππ
)ππ = ππππ
ππππ ππ0 = 1 ππ1 = ππ ππβ1 = 1
ππ
A negative power means a reciprocal 3β2 = 132
= 19 or (1
2)β2 = (2
1)2 = 4
A fractional power indicates a root 813 = β83 = 2 (since 2x2x2=8)
Example Without a calculator, simplify the following β leave your answer in the form ππππ
13. ππ4 Γ ππ3
14. ππ5 Γ· ππ3 15. (π₯π₯3)2
Evaluate the following to find a numerical value (no calculators)
16. (25)3
17. 2713 18. 9
32
19. 81β14 20. (2
3)β2
21. οΏ½49
Practice working with indices and make sure you know all the index rules
1632 = (β16 )3 = 43 = 64
18β43 = (
18
)43 = (
1β83 )4 = οΏ½
12οΏ½4
= 1
16
Hint β do the negatives/reciprocals first, then the root, then the top power
(9π₯π₯4π¦π¦3)12 = 3π₯π₯2π¦π¦
32
Hint β EVERYTHING in the brackets needs to be square rooted.
Indices β Expressing in the form ππππππ It is important to be able to write expressions in the form πππ₯π₯ππ 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 π₯π₯ 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 β this fraction cannot be
simplified
22. 5βπ₯π₯
23. 2π₯π₯3
24. 3βπ₯π₯
25. βπ₯π₯5
26. (2π₯π₯3
)2 27. 1βπ₯π₯3
28. (2βπ₯π₯)3 29. 4
3π₯π₯5
30. βπ₯π₯3π₯π₯
31. 3π₯π₯2
βπ₯π₯
32. π₯π₯β2π₯π₯2
33. 12π₯π₯2π¦π¦
34. (27π₯π₯6π¦π¦5)13
35. (16π₯π₯2
π¦π¦)β
14
Common Mistakes 13π₯π₯2
= 3π₯π₯β2 β4π₯π₯ = 4π₯π₯12 Both WRONG!
13π₯π₯2
= οΏ½13οΏ½ οΏ½ 1
π₯π₯2οΏ½ = 1
3π₯π₯β2 β4π₯π₯ = β4 Γ βπ₯π₯ = 2π₯π₯
12 Correct
2π₯π₯
= 2 Γ 1π₯π₯
= 2π₯π₯β1
65π₯π₯2
= οΏ½65οΏ½ οΏ½
1π₯π₯2οΏ½ =
65π₯π₯β2
2 + π₯π₯βπ₯π₯
= 2βπ₯π₯
+ π₯π₯βπ₯π₯
= 2 οΏ½1βπ₯π₯οΏ½ +
π₯π₯1
π₯π₯12
= 2π₯π₯β12 + π₯π₯
12
π₯π₯2
π₯π₯ + 1β π₯π₯2
π₯π₯+
π₯π₯2
1
Surds A surd is an irrational root eg β2, β3 but not β4 or β9 who have whole number answers. Simplifying surds: βππππ = βππβππ β20 = β4 Γ 5 = β4 β5 = 2β5
οΏ½ππππ
= βππβππ
οΏ½34 = β3
β4= β3
2
Example β75 + 2β12 = β25 Γ 3 + 2β4 Γ 3 = β25β3 + 2β4β3 = 5β3 + 4β3 = 9β3 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 1β5
= 1β5
Γ β5β5
= 1Γβ5β5Γβ5
= β55
3
1+β2= 3
1+β2Γ 1ββ2
1ββ2= 3 (1βοΏ½2)
οΏ½1+β2οΏ½(1ββ2)= 3β3β2
1β2= 3β3β2
β1= β3 + 3β2
Write 13ββ3
in the form ππ + ππβ3
13ββ3
Γ 3+β33+β3
= 3+β39β3β3+3β3ββ3β3
= 3+β36
= 36
+ β36
= 12
+ 16 β3
Write in the form ππβππ
36. β27 37. β48 38. β122
39. β20β 3β45 40. β200 + β18 β 2β50
Rationalise the denominator
41. 2β3
42. 11+β2
43. 34ββ2
Remember β5 Γ β5 = 5 NOT 25
Quadratics Quadratics turn up EVERYWHERE in A level Maths β 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 π₯π₯2 β 5π₯π₯ + 6 = (π₯π₯ β 3)(π₯π₯ β 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 πππ₯π₯2 + πππ₯π₯ + ππ = 0 then π₯π₯ = βππββππ2β4ππππ
2ππ
Difference of 2 squares ππ2 β ππ2 = (ππ + ππ)(ππ β ππ)
Examples 9 β π₯π₯2 = (π₯π₯ + 3)(π₯π₯ β 3) 4π₯π₯2 β 25 = (2π₯π₯ + 5)(2π₯π₯ β 5) Factorise the following quadratics
44. π₯π₯2 + 2π₯π₯ β 15 ( π₯π₯ β 3 )( ) 45. π₯π₯2 β 9π₯π₯ β 10 46. 6π₯π₯2 + 2π₯π₯ 47. 49 β 4π₯π₯2 48. 2π₯π₯2 + 5π₯π₯ β 3 49. 4π₯π₯2 + 4π₯π₯ + 1
Solve using the quadratic formula without a calculator (leave in surd form if necessary)
50. π₯π₯2 β 5π₯π₯ + 4 = 0
51. 3π₯π₯2 + 2π₯π₯ β 1
52. π₯π₯2 = 3π₯π₯ + 2
Factorise and solve
53. 10π₯π₯2 β 2π₯π₯ = 0
54. 9π₯π₯ β 27π₯π₯2 + 0
55. 14π₯π₯2 β 21π₯π₯ = 0
Completing the square Some quadratics are βperfect squaresβ eg π₯π₯2 + 4π₯π₯ + 4 = (π₯π₯ + 2)(π₯π₯ + 2) = (π₯π₯ + 2)2 Most quadratics are not, however it can be useful to write them as square that is βadjustedβ slightly. Example π₯π₯2 + 4π₯π₯ + 7 = (π₯π₯ + 2)2 β 4 + 7 = (π₯π₯ + 2)2 + 3 In general ππππ + ππππ + ππ = (ππ+ ππππππππ ππππ ππ)ππ β (ππππππππ ππππ ππ)ππ + ππ Examples π₯π₯2 + 6π₯π₯ + 2 = (π₯π₯ + 3)2 β 32 + 2 = (π₯π₯ + 3)2 β 7
π₯π₯2 β 4π₯π₯ + 3 = (π₯π₯ β 2)2 β (β2)2 + 3 = (π₯π₯ β 2)2 β 1
π₯π₯2 + 5π₯π₯ β 2 = οΏ½π₯π₯ +52οΏ½
2
β οΏ½52οΏ½
2
β 2 = οΏ½π₯π₯ +52οΏ½
2
β334
Complete the square leaving these expressions in the form (ππ+ ππ)ππ + ππ 56. π₯π₯2 + 8π₯π₯ + 7
57. π₯π₯2 β 2π₯π₯ β 15
58. π₯π₯2 + 6π₯π₯ + 10
59. π₯π₯2 + 12π₯π₯ + 100
60. π₯π₯2 β 3π₯π₯ β 1
61. π₯π₯2 β 12π₯π₯ + 1
Solving equations by completing the square Many quadratics can be solved using this technique. Example Solve π₯π₯2 β 4π₯π₯ β 5 = 0 Complete the square (π₯π₯ β 2)2 - 9 = 0 Put the number on the right (π₯π₯ β 2)2 = 9 Square root both sides (remember Β± signs) π₯π₯ β 2 = Β±3 Add 2 to both sides to get TWO answers π₯π₯ = 2 Β± 3 so π₯π₯ = 5 or π₯π₯ = β1 Solve these by completing the square
62. π₯π₯2 + 6π₯π₯ β 7 = 0 63. π₯π₯2 β 2π₯π₯ β 3 = 0 64. π₯π₯2 + 5π₯π₯ = β6
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
ππππ sinπΆπΆ
Sine Rule ππ
sinπ΄π΄ = ππ
sinπ΅π΅ = ππ
sin πΆπΆ
Or sinπ΄π΄ππ
= sinπ΅π΅ππ
= sinπΆπΆππ
Cosine Rule ππ2 = ππ2 + ππ2 β 2bc cos A
Pythagorasβs Theorem ππ2 + ππ2 = β2
Trigonometric Rations
sin ππ = ππππππβπ¦π¦ππ
cosππ = ππππππβπ¦π¦ππ
tanππ = ππππππππππππ
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 β focus on the questions you got wrong and do any corrections diligently.
1. Write as a single fraction:
a) 325οΏ½ b) οΏ½3π₯π₯
2 Γ· 5
3οΏ½ Γ 1
3
2. Evaluate: Simplify fully:
a) 16β14 b) (16
π₯π₯3)β
12
3. Write in the form πππ₯π₯ππ:
a) 23π₯π₯
b) 4βπ₯π₯5
4. Simplify:
a) β32 b) β20 + 2β45 β 3β80 5. Rationalise the denominator:
a) 1β2
b) 5
2ββ3
6. Solve the quadratics by factorising: a) π₯π₯2 β 5π₯π₯ β 24 = 0 b) 9π₯π₯2 β 4 = 0
7. Solve these quadratics using the formula (leave your answer in surd form if necessary) a) 6π₯π₯2 + x β 1 = 0 b) π₯π₯2 β 7π₯π₯ + 9 = 0
8. Complete the square and write in the form (π₯π₯ + ππ)2 + ππ
a) π₯π₯2 + 2π₯π₯ β 6 b) π₯π₯2 + 3π₯π₯ + 14
9. Find the side marked π₯π₯ or ππ to 1.d.p
10. a) Simplify 33 Γ 3ππ b) Hence write 2ππππ7 Γ (3ππ+1) in the form 3ππ
Quadratic formula
π₯π₯ = βππ β βππ2 β 4ππππ
2ππ
Cosine rule ππ2 = ππ2 + ππ2 β 2ππππ cosπ΄π΄
Score /40
Solutions 1. 3
10
2. 135
3. 6
4. 114
5. 12π₯π₯5
6. 3π₯π₯
7. 103
8. 89
9. X
10. 3π₯π₯+2π₯π₯2
11. 278
12. 10π₯π₯+2910
13. ππ7 14. ππ2 15. π₯π₯6 16. 8
125
17. 3 18. 27
19. 13
20. 94
21. 23
22. 5π₯π₯12
23. 2π₯π₯β3
24. 3π₯π₯β12
25. 15π₯π₯12
26. 49π₯π₯2
27. π₯π₯β13
28. 8π₯π₯32
29. 43π₯π₯β5
30. 13π₯π₯β
12
31. 3π₯π₯ 32
32. π₯π₯β1 β 2π₯π₯β2
33. 12π₯π₯β2π¦π¦β1
34. 3π₯π₯2π¦π¦53
35. 12π₯π₯β
12π¦π¦
14
36. 3β3 37. 4β3 38. β3 39. β7β5 40. 3β2
41. 2β33
42. β1 + β2
43. 12+3β214
44. (π₯π₯ β 3)(π₯π₯ +5)
45. (π₯π₯ β 10)(π₯π₯ +1)
46. 2π₯π₯(3π₯π₯ + 1) 47. (7 + 2π₯π₯)(7β
2π₯π₯) 48. (2π₯π₯ β 1)(π₯π₯ +
3) 49. (2π₯π₯ + 1)2 50. π₯π₯ = 4 ππππ π₯π₯ =
1
51. π₯π₯ = 13
ππππ π₯π₯ =β1
52. π₯π₯ = 3 Β±β172
53. 2π₯π₯(5π₯π₯ β1), π π ππ π₯π₯ =0 ππππ π₯π₯ = β 1
5
54. 9π₯π₯(1β3π₯π₯)π π ππ π₯π₯ =0 ππππ π₯π₯ = 1
3
55. 7π₯π₯(2π₯π₯ β3), π π ππ π₯π₯ =0 ππππ π₯π₯ = 3
2
56. (π₯π₯ + 4)2 β 9 57. (π₯π₯ β 1)2 β 16 58. (π₯π₯ + 3)2 + 1
59. (π₯π₯ + 6)2 + 64
60. οΏ½π₯π₯ β 32οΏ½2β 13
4
61. (π₯π₯ β 14)2 + 15
16
62. π₯π₯ =β7 ππππ π₯π₯ = 1
63. π₯π₯ = 3 ππππ π₯π₯ =β1
64. π₯π₯ = β2 ππππ π₯π₯ = β3
65. π₯π₯ = 9 66. π₯π₯ = 15.6 67. πππ π = 29.1Λ 68. ππ = 45.5Λ 69. ππ = 48.8Λ 70. ππ + 4.7
Answers β 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.