1. GCSE Further Maths (AQA) These slides can be used as a learning resource for students. Some answers are broken down into steps for understanding and some are “final answers” that need you to provide your own method for.

2. Coordinate Geometry GCSE FM Find the midpoint of the line segment from A(4, 1) to B(5, -2) Solution: M =

3. Coordinate Geometry GCSE FM Find the gradient of the line segment from C(- 3, 2) to D (0, 11) Solution: Write down the value of the gradient of line that is perpendicular to this line.

4. Coordinate Geometry GCSE FM Simplify the following surds (a) (c) (b)

5. Coordinate Geometry GCSE FM Express 0.45454545… as a fraction

6. Ratios, decimals & fractions GCSE FM (i)A is 40% of B. Express A in terms of B. (ii) C is 80% of A. Express C in terms of B Solution: A = 0.4 x B A = 0.4B (i) (ii) C = 0.8A C = 0.8(0.4B) C= 0.32B So this means C is 32% of B

7. Ratios, decimals & fractions GCSE FM C and D are in the ratio 2:5 (i)Express C in terms of D (ii)E is 60% of C. Write E in terms of D. Solution: (i) (ii) So this means E is 24% of D

8. Ratios, decimals & fractions GCSE FM Simplify the following. (i) (ii) (i) (ii)

9. Ratios GCSE FM Simplify the following ratio.

10. Algebraic fractions GCSE FM

11. Expand & Simplify….. Solution:

12. Sequences GCSE FM Find the nth term for this quadratic sequence…. 6, 15, 28, 45, 66…. nth term = 2n² + 3n + 1 9 13 1721.. 4 4 4 4.. nth term = 2n²….? Subtracting 2n² leaves 4, 7, 10, 13, 16….

13. Equations GCSE FM Solve the following equations : a) b)

14. Simultaneous Equations GCSE FM Solve the following equations :

15. Equations GCSE FM Solve the following equations : a) b)

16. Coordinate Geometry GCSE FM P is the point (a, b) and Q is the point (3a, 5b) Find in terms of a and b, (i) The gradient of PQ (iii) The midpoint of PQ (ii) The length of PQ

17. Rearanging formulae GCSE FM Make ‘t’ the subject of the following formulae (a) (b)

18. Solving inequalities GCSE FM Solve the following inequalities (a) (b)

19. Solving inequalities GCSE FM Solve the following inequality x² - 3x < 0 y = x² - 3x

20. Solving inequalities GCSE FM Given that Write out an inequality for (i) a + b(ii) a - b

21. Rearranging formulae GCSE FM Make r the subject (a) (b) Make l the subject

22. Equations GCSE FM Solve the following equation :

23. SURDS GCSE FM RATIONALISE the following (remove the surd from the denominator) : (a) (b)

24. SURDS GCSE FM RATIONALISE the following (remove the surd from the denominator) : (b)

25. Infinite sequences GCSE FM Find the first 5 terms using the following nth term : What is happening to the terms as n increases to ∞ (infinity) ? n12345 T As n ∞ The limiting value is 4

26. Infinite sequences GCSE FM Find the limit for the nth term as n ∞ : As n ∞ The limiting value is..

27. Infinite sequences GCSE FM Find the first 5 terms using the following nth term : What is happening to the terms as n increases to ∞ (infinity) ? n12345 T As n ∞ The limiting value is 2/3

28. Simultaneous equations GCSE FM Substitute y for x and hence solve the equations :

29. Simultaneous equations GCSE FM Substitute y for x and hence solve the equations :

30. Simultaneous equations GCSE FM Find the point of intersection for these two lines : y = 2x + 1 4 4 x + y = 4 y = 2x + 1 x + y = 4 x + (2x + 1) = 4 3x + 1 = 4 x = 1 So y = 3

31. Rearranging formulae GCSE FM Make a the subject of the following formulae….

32. Linear (straight line) graphs GCSE FM Find the equation of the line with gradient 3 and passing through (1, -2) (1, -2) Find the equation of the line that is perpendicular and passing through (1, -2)

33. Linear (straight line) graphs GCSE FM Distance between 2 points? We need PQ, QR and PR. Which two are the same? Use Pythagoras’ Theorem. Two equal lengths so isosceles.

34. Simultaneous equations GCSE FM Substitute y for x and hence solve the equations :

35. Simultaneous equations GCSE FM Substitute y for x and hence solve the equations :

36. Simultaneous equations GCSE FM

37. Substitute y for x and hence solve the equations :

38. Simultaneous equations GCSE FM Substitute y for x and hence solve the equations :

39. Simultaneous equations GCSE FM Substitute y for x and hence solve the equations : x = 3 and y = 2 x = 3 and y = -2 x = 3 and y = 3 x = -3 and y = -3 x = 1 and y = 2 x = -1 and y = 0

40. Pythagoras 3D GCSE FM Find :(i) AD(ii) CE(iii) AC A B C D E F

41. Linear (straight line) graphs GCSE FM 1. Find the equation of the line with gradient ½ and passing through (-4, 6) 2. Find the equation of the line perpendicular to y = 2x - 1 and passing through (2, 3)

42. Sequences GCSE FM Find the nth term for this quadratic sequence…. 1, 0, -3, -8, -15…. nth term = -n² + 2n -1 -3 -5 -7.. -2 -2 -2 -2.. nth term = -n²….? Subtracting -n² (adding n²) leaves 2, 4, 6, 8, 10….

43. Proof GCSE FM Show that f(x) = x² - 4x + 5 Hence explain why f(x) > 0 for all values of x. Let f(x) = (x – 2)² + 1 f(2) = (2 – 2)² + 1 = 0² + 1 = 1 f(-2) = = -4² + 1 = 17 (-2 – 2)² + 1

44. Proof GCSE FM Show that f(x) = x² - 4x + 5 Explain why f(x) > 0 for all values of x. Let f(x) = (x – 2)² + 1 (x – 2)² + 1 = x² - 4x + 4 + 1 = x² - 4x + 5 Squaring always makes a positive value so adding 1 is still positive.

45. Proof GCSE FM Show that f(x) = 9x² Hence explain why f(x) is a square number. Let f(x) = 2x³ - x²(2x – 9) 2x³ - 2x³ + 9x² 9x² = (3x)² so you are squaring 3x. This makes a square number = 9x²

46. Proof GCSE FM Show that f(n + 1) = n² + 2n + 1 Let f(n) = n² for all positive integer values of n f(n+1) = (n+1)² = (n+1)(n+1) = n² + 2n + 1

47. Proof GCSE FM Show that f(n + 1) + f(n – 1) is always even Let f(n) = n² for all positive integer values of n (n+1)² + (n-1)² = (n+1)(n+1)+(n-1)(n-1) = n² + 2n + 1 + n² - 2n + 1 = 2n² + 2

48. Proof GCSE FM Proving something is even means you have to show it is in the 2 times table (or a multiple of 2) 2n² + 2 2(n² + 1) Since we are multiplying by 2, this must be even

49. Functions 2f(x) = Let f(x) = x² f(2) = (2)² = 4 f(-2) == 4 (-2)² 2x² f(2x) == (2x)²

50. Using Functions 2f(x) = Let f(x) = 3 – x. Sketch this graph f(0) = = 3 3- 0 2(3-x) f(2x) = = 6–2x 3-(2x)= 3–2x 1 - f(x) = 1 - (3–x)= -2 + x = x - 2 What does this represent?

51. EXAM REVISION (FM) Review TEST 1Review TEST 2Recent Ratio and percentages. Ex 1A and 1D Equations of lines Ex 3D, 5B, 5C Simultaneous equations by substitution Ex 4B Midpoints, length of line segments Expanding brackets furtherCircles and lines (equation of a circle) Gradients of lines (including perpendicular lines) Ex 3C Proof (to be taught) Ex 4G 3D Pythagoras (including the diagonal of a cuboid) Ex 6E (part only!) Algebraic fractions Ex 1B, Ex 2C, 2D Linear Sequences Ex 4H Solving quadratic equations by factorising Ex 4A q. 1 only Factorising Ex 2A (including quadratics) Quadratic Sequences Ex 4I Quadratic graphs Ex 3E SURDS (including rationalising) Ex 1F and 1G Limiting value of a sequence Ex 4J Rearranging formulae Ex 2B Algebraic fractional equations Ex 1C, Ex 2E CHECK and REVIEW ALL HOMEWORK tasks. Area of a triangle Ex 7A Linear & Quadratic inequalities Ex 4D, 4E.

52. FOCUS YOUR EXAM REVISION (FM) TopicCHAPTER 1 Decimals, Fractions and percentages Ex 1 A Simplifying algebra Ex 1B Solving equations Ex 1C Ratios Ex 1D Further algebra – expanding brackets Ex 1E SURDS Ex 1F and Ex 1G TopicQuestions Factorising quadratics Ex 2 A Rearranging formulae Ex 2B and 2C Simplifying algebraic fractions Ex 2D Equations with fractions Ex 2E