Percentages To find a % eg 5% then multiply by 0.05 To find a % increase (new value), x 1.05
To find a % eg 4.5% then multiply by 0.045 To find a % decrease (new value), x 0.955 Reversing the problem eg After a 10% rise in prices the house is worth £88,000. What was it worth before the rise? 1.10 x Value = 88,000
110% = 88,000 ÷110
÷110
1% = 800 100% = £80,000 x100
x100
or
Value = 88,000 ÷ 1.10
Value = £80,000
Area and Volume You must learn Area of rectangle = l x b Formula sheet Area of triangle = ½ x b x h Area = ½abSinC Area of circle = πr2 Circumference of circle = πd Volume of prism = Area of crosssection x length
The cylinder, cone & sphere are on the formula sheet!
Remember units cm2 , m2 etc for area cm3 , m3 etc for volume
1 litre = 1000cm3 1 ml = 1cm3 Composite shapes can be split into shapes for which you have a formula ! 1
Reversing the problem eg A cylinder of height 5cm has a volume of 100cm3 , find its radius
V = πr2h πr2h = V V r2 = h π V r = π h
or
100 = π x r2 x 5 100 = 3.14 x r2 x 5
100 = 15.7 x r2 r2 = 100 ÷ 15.7 r2 = 6.37 r = √6.37 = 2.5cm
100 r = πx5 = 2.5cm
Linear Relationships The standard form for the equation of a straight line is
m = gradient c = yintercept
y = mx + c
gradient =
y2 y1 y change in y = x x = change in x 2 1 x negative gradient
positive gradient
Algebra
Removing brackets: careful!
3(4x 5) = 12x 15
8 + 5(2x 3) = 8 + 10x 15 = 10x 7
2(5x 6) = 10x + 12
careful!
4 3(y 5) = 4 3y + 15 = 19 3y
4x + (2x + 3)(x 6)
(x 4)(x + 5)
= x2 + 5x 4x 20 = x2 + x 20
= 4x +2x212x+3x18 = 2x2 5x 18
careful!
6x (2x 5)(x + 7)
(x + 3)(x2 + 2x + 1)
= 6x (2x2+14x5x35)
= 6x (2x2 + 9x 35) = 6x 2x2 9x + 35 = 2x2 3x + 35
= x3 + 2x2 + x + 3x2+ 6x + 3 = x3 + 5x2 + 7x + 3
2
Algebra
Factorising: Common Factor
Difference of 2 squares
Trinomial
3m + 9m2
a2 b2 = (a + b)(a b)
x2 + 5x 6 = (x + 6)(x 1)
= 3m(1 + 3m)
Always look for a common factor first!
5a2 20b2 = 5(a2 4b2)
4a2 8a 12 = 4(a2 2a 3)
= 5(a + 2b)(a 2b)
= 4(a 3)(a + 1)
Harder Trinomials
Harder Trinomials
2
5w 2w 7 = (5w )(w )
= (5w + 7)(w 1)
2a2 + 3a 5 = (2a )(a )
+7 1 7 +1 +1 7 1 +7
+5 1 5 +1 +1 5 1 +5
= (2a + 5)(a 1)
Trial and error with factors
Trial and error with factors
Circles Length of arc
70 360 x πd
o
70
or
C = πd Arc = answer÷360 x 70
Area of sector
A = πr2 70 2 or Sector = answer÷360 x 70 360 x πr
o
70
bc
o
b
The radius of the circle is equal at all points on the circumference. Look for isosceles triangles!
c a
a
P
Look for right angles in the circle ng
t en
ta
T A o
rad ius
di o am et e
o r
B
3
Trigonometry
Rightangled Triangles Use Pythagoras sides only
SOHCAHTOA sides & angles Q
c
b
h
a
a
o
P
To find the hypotenuse
SinQ =
o h
CosQ =
a h
c2 = a2 + b2 To find a shorter side
a2 = c2 - b2
R
TanQ =
o a
Or use the SINE rule here
Isosceles Triangles bb
a
a
Isosceles (and equilateral) triangles split into two congruent rightangled triangles
Other triangles look at formula sheet! A
The Sine Rule side
c B
a
b
b c = = SinB SinA SinC
a
Use when you know 3 Rearrange to find a
C
A
The Sine Rule angle
SinA
b
c B
a
a
C
= SinB = SinC c b
Use when you know 3 Rearrange to find SinC
Remember: angles in a triangle add up to 180degrees If you know 2 angles you can find the third one! 4
The Cosine Rule side
B
a2 = b2 + c2 - 2bcCosA a
c A
b
Use this to find the third side when you know 2 sides and the included angle
C
The Cosine Rule angle
B
b2 + c2 - a2 CosA = 2bc
a
c
Use this to find the angle when you know all 3 sides
A
b
C
Area of a triangle
B
Area =
a
c Area A
b
1 2
bcSinC
Use this to find the third side when you know 2 sides and the included angle
C
(same as cosine rule side!)
5
Simultaneous Equations These arise when there are 2 unkowns eg x and y Set them out in the same order:
3x + 5y = 20 4x + 2y = 15
1 2
Arrange either x or y to have the same number but with opposite signs 1 x 4 and 2 x 3
Add to eliminate one unknown Solve to find the other Substitute back in and solve to find the second unkown Check your answers fit the other equation
Graphs, Charts & Tables You should be able to draw and interpret bar graphs, line graphs, scatter diagrams, stem & leaf diagrams & pie charts Cumulative Frequency
Keeps a running total
Medical records show the number of new cases of flu reported each week
Week Frequency Cumulative Frequency
1 2 3 4 Dot Plot
Used to look at the spread or to put data in order
Symmetrical
Uniform
14 52 117 143
14 38 65 26
Skewed to the right
Widely Spread
Skewed to the left
Tightly Clustered
6
Box Plots Lowest (L) = 2 2
Q1 = 7.5 Q2 = 14
L
7.5
14
Q1
Q2
20.5
24
H
Q3
Q3 = 20.5 A suitable scale
Highest (H) = 24
Statistics You must be able to find the mean, median, mode & quartiles Q3 - Q1 The semiinterquartile range = 2 It is useful for comparing data. A low SIQR means results are less variable Standard Devaition
Another measure of spread
Set data out in a table x
(x - x)
(x - x )2
Find the mean x then complete columns 2 and 3
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Formula sheet
31 40
s. d . =
27 29
Ʃ( x - x )
2
n - 1
2 Ʃ ( x - x)
Total
Take care to avoid calculation errors!!!!
Probability Probabilitly =
Number of favourable outcomes
Number of possible outcomes
Given the 10 spheres in the jar, if you were blindfolded what is the probablilty of picking (i) a blue (ii) a red
(i) P(blue) =
6 10 =
3 5
(ii) P(red) =
4 = 10
2 5
7
Algebraic Operations Adding & Subtracting Fractions
Find the lowest common denominator first
a + c b d ad cb + = bd bd
e.g.
= ad + cd bd Multiplying Fractions
e.g.
=
3a 4b x 9c 5b
12ab 45bc
Cancel first if you can
4a = 15c
Dividing Fractions
e.g.
9y2 5y ÷ 4 8 Change
= 5y x 8 2 9y 4 40y = 36y2
=
÷ to multiply Turn 2nd fraction upside down
Cancel first if you can
10 9y
Simplifying Fractions
e.g.
x2 - 25 x2 + 7x + 10
(x + 5)(x - 5) = (x + 5)(x + 2)
You can only cancel if terms are multiplied on top and bottom Factorise if you can!!!
= (x - 5) (x + 2) 8
Changing the subject of a formula LHS
e.g.
RHS
4 V = 3 πr 3
m = p(x - k) x m = px - pk
3V = 4πr3
Common sense line
p x - pk = m
Common sense line
p x = m + pk
x = m + pk p Surds Simplify
e.g.
r
4πr3 = 3V
r3 = 3V 4π
What do you do now?
r = 3 3V 4π
Biggest square factor
Biggest square factor
√48 = √16 x √3
√20 = √4 x √5 = 2 x √5
= 4 x √3
= 2√5
= 4√3 also
√3 x √6 = √18
√18 = √9 x √2
Find a square factor
= 3√2
12 = 27 √60 = √5
3 x 4 √4 = = 3 x 9 9 √
2 3
60 5 = √12 = √4 x √3 = 2√3 9
Indices
e.g.
6p 4 x 4 p 3
= (6 x 4) x (p x p x p x p) x (p x p x p)
= 24p7 9a6 = 2 3a
Multiply the numbers first
9 3
a x a x a x a x a x a a x a
x
= 3a4
Divide the numbers first
Other rules of Indices
(2c4)3 = 23 x (c4)3
12x0 = 12 x 1 = 12 Since x0 = 1
= 8 x c12
= 8c12
1
2
x-2 = x2
( x-3 ) = x-6 =
1 x6
Fractional Indices
a
a
1 2
3
2
a
= √a 3
= (√a)
a
1 3
5 3
3
a
= √a 5
= (√a) 3
a
1 4
9 4
4 a = √
9
= (√a) 4
10
Quadratics Quadratic Equations y
y = (x + 1)(x 3)
Roots of (x + 1)(x 3) = 0 1
3
Either x = 1 or x = 3
x
x2 + 7x + 12 = 0
Solve algebraically
Factorise (x + 3)(x + 4) = 0 Either (x + 3) = 0 or (x + 4) = 0 x = 3 or x = 4
If you cannot factorise use the quadratic formula (on formula sheet) Every quadratic can be re-arranged to
ax2 + bx + c = 0
-b ± b2 - 4ac x = 2a Substitute taking care not to avoid calculator errors The turning point y
Any quadratic in the form
y = (x - a)2 + b ( a ,b ) x
has a min t.p. at (a,b)
x = a
Examples y
y
y
x = 6
(-3,2)
(4,1)
x
x
x
x = 4
(6,-2)
y = (x - 4)2 + 1
y = (x - 6)2 - 2
x = -3
y = (x + 3)2 + 2
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Trig Functions y
Period
1
x
360
0
y
Amplitude
Amplitude
Learn the shapes of sinx, cosx & tanx 1
y
Period
x
360
0
Period
0
360
180
x
-1
-1
y = sinx
o
o
y = tanx
y = cosx
o
o
o
Period = 360 Amplitude = 1
Period = 360 Amplitude = 1
o
Period = 180 Amplitude not measured
Amplitude
Identify the curve 5
y
y
o
y = 5sin3x
o
y = 3cos2x
3 0
360
x
0
x 1
180
x
-3
-5
y
360
y = 6sin5x o
y = tan6xo
y
6 0
360
x
0
-6
Trig Equations Solve
3cosx - 1 = 0
for 0 ≤ x ≤ 360
3cosx = 1
cos is positive
cosx = 0.3333... Base Angle = cos-1 0.333 = 70.5 o
x = 70.5
o
o
90
A
S o
or
x = 360 - 70.5 o x = 289.5
o
0 (360)
180
C
T o
270
12
1 Read the question carefully 2 Set out all your working carefully 3 Check you have answered the question
units, rounding etc 4 Never cross out anything unless you have
replaced it with something better 5 Would a diagram help? label what you
know. It sometimes helps to let a letter stand for an unknown length or angle. Is there a rightangled triangle Pythagoras? SOHCAHTOA? Non rightangled triangle Sine/cosine rule 6 Does the formula sheet help? 7 If you can't find what you are looking for can
you find something else does this help? Remember: sound mathematics which is leading towards the answer earns marks even if you don't manage to complete the questions 8 If you are really stuck move on and try to
come back to the question later 9 Keep an eye on the time
10 Keep calm if you have been studying, there
should be no surprises?
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