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 =
 4  5 1  (2) 
,

  4.5,0.5
2 
 2
Coordinate Geometry GCSE FM
Find the gradient of the line segment
from C(-3, 2) to D (0, 11)
Solution:
11  2
9
m
 3
0  (3) 3
Write down the value of the gradient of
line that is perpendicular to this line.
1 1
m2 

m1
3
Coordinate Geometry GCSE FM
Simplify the following surds
(a)
72  36  2  6 2
(b)
3 18  5 2  9 2  5 2  4 2
(c)
3  2 2  2 
 63 2  2 2 2
 4 2
Coordinate Geometry GCSE FM
Express 0.45454545… as a fraction
x  0.4545454545....
100 x  45.4545454545....
99 x  45
45
x
99
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:
(ii)
(i)
C = 0.8A
A = 0.4 x B
A = 0.4B
C = 0.8(0.4B)
C= 0.32B
So this means C is 32% of B
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:
C 2
(i)

D 5
(ii) E  0.6C
5C  2 D
2D
C
5
E  0.6(0.4 D)
E  0.24 D
So this means E is 24% of D
Ratios, decimals & fractions GCSE FM
Simplify the following.
(i)
(i)
1 1

a b
(ii)
1 b

a ac
1 1 b
a
ab
 


a b ab ab
ab
1
b
1
ac
(ii)

 
a ac a b
c

b
Ratios GCSE FM
Simplify the following ratio.
48 : 108
16  3 : 36  3
4 3 :6 3
2:3
Algebraic fractions GCSE FM
x  6x  2 xx  5

x  5x  5 x  6
x x  2 
 x  5
Algebraic fractions GCSE FM
Expand & Simplify…..
x
2

 2 x x  3x  5
2

Solution:
x  3x  5 x
3
2
2 x  6 x  10 x
4
3
2
x  5 x  11x  10 x
4
3
2
Sequences GCSE FM
Find the nth term for this quadratic
sequence….
6, 15, 28, 45, 66….
9
4
13
4
17
21..
4
4..
nth term = 2n²….?
Subtracting 2n² leaves
4, 7, 10, 13, 16….
nth term = 2n² + 3n + 1
Equations GCSE FM
Solve the following equations :
a)
b)
3x  4
 2x 1
4
3
x 1
5
2x 1 2x  3

3
5
1
x3
2
Simultaneous Equations GCSE FM
Solve the following equations :
2x  y  8
2
5 x  2 y  21
4 x  2 y  16
5 x  2 y  21
5 x  2 y  21
4 x  2 y  16
x  5 y  2
Equations GCSE FM
Solve the following equations :
a)
b)
p p
 7
2 4
28
p
3
q q
 6
3 5
q  45
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
(ii) The length of PQ
(iii) The midpoint of PQ
Rearanging formulae GCSE FM
Make ‘t’ the subject of the following formulae
(a)
v  u  at
(b)
1 2
s  at
2
v u
t
a
2s
t
a
Solving inequalities GCSE FM
Solve the following inequalities
(a)
(b)
2n  5  7 n  25
n  6
3 n
 n 1
5
1
n
3
Solving inequalities GCSE FM
Solve the following inequality
x² - 3x < 0
2
y = x² - 3x
1
–4
–3
–2
–1
1
–1
–2
2
3
4
5
6
0 x3
7
Solving inequalities GCSE FM
Given that
1 a  4
3b7
Write out an inequality for
(i) a + b
1 a  4
3b 7
4  a  b  11
(ii) a - b
1 a  4
3b 7
6  a b 1
Rearranging formulae GCSE FM
(a) Make r the subject
3
(b) Make
3V
r
4
l the subject
2
 T 

 g l
 2 
4 3
V  r
3
l
T  2
g
Equations GCSE FM
Solve the following equation :
2x x
  x 1
3 4
3x
2x 
 3( x  1)
4
8 x  3 x  12( x  1)
x  12
SURDS GCSE FM
RATIONALISE the following
(remove the surd from the denominator) :
(a)
2
2
(b)
1 3
4 3
2 2

2 2
2 2

2
 2
SURDS GCSE FM
RATIONALISE the following
(remove the surd from the denominator) :
(b)
1 3
4 3

1  3 4  3 

4  3 4  3 
4 3 4 3 3

16  4 3  4 3  3
75 3

13
Infinite sequences GCSE FM
Find the first 5 terms using
the following nth term :
1
4
n
What is happening to the terms as n
increases to
∞
n
T
As n
(infinity) ?
1
2
3
∞
1
3
2
3
4
5
2
3
3
3
3
4
4
3
5
The limiting value is 4
Infinite sequences GCSE FM
Find the limit for the nth term
as n
∞:
1 3n

n n
2n 1

n n
As n
∞
1  3n
2n  1
03

3


20
2
3
The limiting value is .. 2
Infinite sequences GCSE FM
Find the first 5 terms using
the following nth term :
2n  1
3n  2
What is happening to the terms as n
increases to
∞
n
T
As n
(infinity) ?
1
∞
2
3
5
3
5
8
7
11
4
9
14
5
11
17
The limiting value is 2/3
Simultaneous equations GCSE FM
Substitute y for x and hence
solve the equations :
2x  3y  8
y  2x
2 x  3(2 x)  8
2x  6x  8
8x  8
 x 1
y2
Simultaneous equations GCSE FM
Substitute y for x and hence
solve the equations :
x  2y  4
y  x 1
x  2( x  1)  4
x  2x  2  4
x6
 x  6
y  5
Simultaneous equations GCSE FM
Find the point of intersection for
these two lines :
4
y = 2x + 1
y = 2x + 1
x+ y= 4
x + (2x + 1) = 4
3x + 1 = 4
x = 1
4
So y = 3
Rearranging formulae GCSE FM
Make a the subject of the following
formulae….
2
2
c  a b
c  a b
2
c b  a
2
2
2
2
2
c b  a
2
2
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)
Linear (straight line) graphs GCSE FM
Distance between 2 points?
We need PQ , QR and PR.
Which two are the same?
Use Pythagoras’ Theorem.
PQ 
6  2  9  1
PR  65
2
QR  65
2
 80
Two equal lengths so isosceles.
Simultaneous equations GCSE FM
Substitute y for x and hence
solve the equations :
x  y  20
2
2
y2
x  (2 )  20
2
2
x  20  4
2
x  16  x  16
2
x  4
y2
Simultaneous equations GCSE FM
Substitute y for x and hence
solve the equations :
x  y 8
2
2
yx
x  (x )  8
2
2
2x  8
2
x 4
2
x  2
y  2
Simultaneous equations GCSE FM
x  y 8
2
2
yx
x  2
y  2
Simultaneous equations GCSE FM
Substitute y for x and hence
solve the equations :
x  y  10
2
2
y  x2
x  ( x  2)  10
2
2
x  ( x  4 x  4)  10
2
2
2x  4x  6  0
2
x  2x  3  0
2
( x  1)( x  3)  0
x  1
y  3
x3
y 1
Simultaneous equations GCSE FM
Substitute y for x and hence
solve the equations :
x  3 y  12
2
x y 2
x  3( x  2)  12
y  x2
2
x  3 x  6  12
2
x  3 x  18  0
2
( x  6)( x  3)  0
x  6
y  8
x3
y 1
Simultaneous equations GCSE FM
Substitute y for x and hence solve the
equations :
x3
x  y  13
2
2
x  y  18
2
2
yx
yx x
2
y  x 1
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
B
A
F
C
E
D
Linear (straight line) graphs GCSE FM
1. Find the equation of the line with
gradient ½ and passing through (-4, 6)
1
y  x 8
2
2. Find the equation of the line perpendicular
to y = 2x - 1 and passing through (2, 3)
1
y  x4
2
Sequences GCSE FM
Find the nth term for this quadratic
sequence….
1, 0, -3, -8, -15….
-1 -3
-5
-7..
-2
-2
-2
-2..
nth term = -n²….?
Subtracting -n² (adding n²) leaves
2, 4, 6, 8, 10….
nth term = -n² + 2n
Proof GCSE FM
Let f(x) = (x – 2)² + 1
f(2) = (2 – 2)² + 1
f(-2) = (-2 – 2)² + 1
= 0² + 1 = 1
= -4² + 1 = 17
Show that f(x) = x² - 4x + 5
Hence explain why f(x) > 0 for all values
of x.
Proof GCSE FM
Let f(x) = (x – 2)² + 1
Show that f(x) = x² - 4x + 5
(x – 2)² + 1 = x² - 4x + 4 + 1
= x² - 4x + 5
Explain why f(x) > 0 for all values of x.
Squaring always makes a positive
value so adding 1 is still positive.
Proof GCSE FM
Let f(x) = 2x³ - x²(2x – 9)
Show that f(x) = 9x²
2x³ - 2x³ + 9x² = 9x²
Hence explain why f(x) is a square
number.
9x² = (3x)² so you are squaring
3x. This makes a square number
Proof GCSE FM
Let f(n) = n²
for all positive integer values of n
Show that f(n + 1) = n² + 2n + 1
f(n+1) = (n+1)² = (n+1)(n+1)
= n² + 2n + 1
Proof GCSE FM
Let f(n) = n²
for all positive integer values of n
Show that f(n + 1) + f(n – 1) is
always even
(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
Let f(x) = x²
f(2) = (2)² =
4
f(-2) = (-2)² = 4
2f(x) = 2x²
f(2x) = = (2x)²
Using Functions
Let f(x) = 3 – x.
Sketch this graph
f(0) = 3- 0 = 3
What does this represent?
2f(x) = 2(3-x) = 6–2x
f(2x) = 3-(2x) = 3–2x
1 - f(x) = 1 - (3–x)
= -2 + x
=x -2
EXAM REVISION (FM)
Review TEST 1
Review TEST 2
Recent
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 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 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
Area of a triangle Ex 7A
Linear & Quadratic inequalities
Ex 4D, 4E.
CHECK and REVIEW ALL
HOMEWORK tasks.
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