gcse further maths (aqa) these slides can be used as a learning resource for students. some answers...

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.

Coordinate Geometry GCSE FM

Find the midpoint of the line segment from A(4, 1) to B(5, -2)

Solution:

M = 5.0,5.4

2

)2(1,

2

54


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.

33

9

)3(0

211

m

3

11

12

m

m


Simplify the following surds

72

25183

2223

(a)

(c)

(b)

236

242529

222236 24

26


Express 0.45454545… as a fraction

....4545454545.0x....4545454545.45100 x

4599 x

99

45x

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 BSolution:

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

Ratios, decimals & fractions GCSE FMC 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:

CE 6.0

(i)

(ii) )4.0(6.0 DE DE 24.0

5

2

D

C DC 25

5

2DC

So this means E is 24% of D

Ratios, decimals & fractions GCSE FMSimplify the following.

(i)

(ii)

ba

11(i) (ii)

ac

b

a1

ab

a

ab

b

ba

11

ab

ba

b

ac

aac

b

a11

b

c

Ratios GCSE FM

Simplify the following ratio.

108:48

336:316

3:2

36:34

Algebraic fractions GCSE FM

6

5

x

xx 55

26

xx

xx

5

2

x

xx

Algebraic fractions GCSE FM

Expand & Simplify…..

532 22 xxxx

234 53 xxx Solution:

xxx 1062 23 xxxx 10115 234

Sequences GCSE FM

Find the nth term for this quadratic sequence….

6, 15, 28, 45, 66….

nth term = 2n² + 3n + 1

9 13 17 21.. 4 4 4 4..

nth term = 2n²….?Subtracting 2n² leaves

4, 7, 10, 13, 16….

Equations GCSE FM

Solve the following equations :

a)

b)

124

43

x

x

5

32

3

12

xx

5

31x

2

13x

Simultaneous Equations GCSE FM


82 yx2125 yx

5x

1624 yx2125 yx

2

2125 yx1624 yx

2y

Equations GCSE FM


a)

b)

742

pp

653

qq

3

28p

45q


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

Rearanging formulae GCSE FM

Make ‘t’ the subject of the following formulae

(a)

(b)

atuv

2

2

1ats

ta

uv

ta

s

2

Solving inequalities GCSE FM

Solve the following inequalities

(a)

(b)

25752 nn

15

3

n

n

6n

3

1n


Solve the following inequality

x² - 3x < 0

–4 –3 –2 –1 1 2 3 4 5 6 7

–2

–1

1

2

y = x² - 3x

30 x


Given that

41 a 73 b

Write out an inequality for

(i) a + b (ii) a - b

114 ba 16 ba

41 a73 b 73 b41 a

Rearranging formulae GCSE FM

Make r the subject

(a)

(b)

3

3

4rV

g

lT 2

Make l the subject

rV

3

4

3

lgT

2

2

Equations GCSE FM

Solve the following equation :

143

2 x

xx

)1(34

32 x

xx

)1(1238 xxx

12x

SURDS GCSE FM

2

2

34

31

RATIONALISE the following (remove the surd from the denominator) :(a)

(b)

22

22

2

22 2

SURDS GCSE FM

34

31

RATIONALISE the following (remove the surd from the denominator) :

(b) 3434

3431

3343416

33434

13

357

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) ?

n 1 2 3 4 5T

2

133

3

23

4

33

5

43

As n ∞ The limiting value is 4

n

14


Find the limit for the nth

term as n ∞:

As n ∞ The limiting value is ..

12

31

n

n

nnnnn

n12

31

02

30

2

3

2

3


Find the first 5 terms using the following nth term : 23

12

n

n

What is happening to the terms as n

increases to ∞ (infinity) ?

n 1 2 3 4 5T

8

5

5

311

7

14

9

17

11

As n ∞ The limiting value is 2/3

Simultaneous equations GCSE FM

Substitute y for x and hence solve the equations :

xy

yx

2

832

8)2(32 xx

862 xx

1

88

x

x2y



1

42

xy

yx

4)1(2 xx

422 xx

6

6

x

x5y


Find the point of intersection for these two lines :

y = 2x + 14

4

x + y = 4

y = 2x + 1

x + y = 4

x + (2x + 1) = 4

3x + 1 = 4

x = 1

So y = 3

Rearranging formulae GCSE FM

Make a the subject of the following formulae….

22 bac

222 bac 222 abc

abc 22

Linear (straight line) graphs GCSE FMFind 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)

Linear (straight line) graphs GCSE FM

Distance between 2 points?We need PQ , QR and PR.Which two are the same?Use Pythagoras’ Theorem.

801926 22 PQ

65PR 65QR Two equal lengths so isosceles.



2

2022

y

yx

20)2( 22 x

16

4202

2

x

x

4x 2y

16 x



xy

yx

822

8)( 22 xx

4

822

2

x

x

2x 2y


xy

yx

822

2x

2y



2

1022

xy

yx

10)2( 22 xx

10)44( 22 xxx

0642 2 xx 1x

0322 xx

0)3)(1( xx

3x3y 1y



2

1232

yx

yx

12)2(32 xx

12632 xx

01832 xx6x

0)3)(6( xx

3x8y 1y

2xy



13

322

yx

x

xy

yx

1822

1

2

xy

xxy

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

Pythagoras 3D GCSE FM

Find : (i) AD (ii) CE (iii) AC

A

B

C

D

E

F

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)

82

1 xy

42

1 xy

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….

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

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.

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

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

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

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

Functions

2f(x) =

Let f(x) = x²

f(2) = (2)² =

4

f(-2) =

= 4 (-2)²

2x²

f(2x) =

= (2x)²

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?

EXAM REVISION (FM)Review TEST 1 Review TEST 2 Recent

Ratio and percentages.Ex 1A and 1D

Equations of linesEx 3D, 5B, 5C

Simultaneous equations by substitution Ex 4B

Midpoints, length of line segments

Expanding brackets further Circles 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 fractionsEx 1B, Ex 2C, 2D

Linear SequencesEx 4H

Solving quadratic equations by factorising Ex 4A q. 1 only

Factorising Ex 2A(including quadratics)

Quadratic SequencesEx 4I

Quadratic graphsEx 3E

SURDS (including rationalising) Ex 1F and 1G

Limiting value of a sequenceEx 4J

Rearranging formulaeEx 2B

Algebraic fractional equationsEx 1C, Ex 2E

CHECK and REVIEW ALL HOMEWORK tasks.

Area of a triangle Ex 7A

Linear & Quadratic inequalitiesEx 4D, 4E.

FOCUS YOUR EXAM REVISION (FM)Topic CHAPTER 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

Topic Questions

Factorising quadratics Ex 2 A

Rearranging formulae Ex 2B and 2C

Simplifying algebraic fractions Ex 2D

Equations with fractions Ex 2E

gcse further maths (aqa) these slides can be used as a learning resource for students. some answers...

Documents

ratios gcse fm

inequalities gcse fm

surds gcse fm

c b slide

x slide

d slide

subject slide

infinite sequences gcse