gcse further maths (aqa) these slides can be used as a learning resource for students. some answers...
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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
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.
33
9
)3(0
211
m
3
11
12
m
m
Coordinate Geometry GCSE FM
Simplify the following surds
72
25183
2223
(a)
(c)
(b)
236
242529
222236 24
26
Coordinate Geometry GCSE FM
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
Solve the following equations :
82 yx2125 yx
5x
1624 yx2125 yx
2
2125 yx1624 yx
2y
Equations GCSE FM
Solve the following equations :
a)
b)
742
pp
653
3
28p
45q
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
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
Solving inequalities GCSE FM
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
Solving inequalities GCSE FM
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
Infinite sequences GCSE FM
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
Infinite sequences GCSE FM
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
Simultaneous equations GCSE FM
Substitute y for x and hence solve the equations :
1
42
xy
yx
4)1(2 xx
422 xx
6
6
x
x5y
Simultaneous equations GCSE FM
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.
Simultaneous equations GCSE FM
Substitute y for x and hence solve the equations :
2
2022
y
yx
20)2( 22 x
16
4202
2
x
x
4x 2y
16 x
Simultaneous equations GCSE FM
Substitute y for x and hence solve the equations :
xy
yx
822
8)( 22 xx
4
822
2
x
x
2x 2y
Simultaneous equations GCSE FM
xy
yx
822
2x
2y
Simultaneous equations GCSE FM
Substitute y for x and hence solve the equations :
2
1022
xy
yx
10)2( 22 xx
10)44( 22 xxx
0642 2 xx 1x
0322 xx
0)3)(1( xx
3x3y 1y
Simultaneous equations GCSE FM
Substitute y for x and hence solve the equations :
2
1232
yx
yx
12)2(32 xx
12632 xx
01832 xx6x
0)3)(6( xx
3x8y 1y
2xy
Simultaneous equations GCSE FM
Substitute y for x and hence solve the equations :
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
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