Int 2 revision by andrew gibson

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

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

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

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

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

Circles Length of arc

70 360 x πd

C = πd Arc = answer÷360 x 70

Area of sector

A = πr2 70 2 or Sector = answer÷360 x 70 360 x πr

The radius of the circle is equal at all points on the circumference. Look for isosceles triangles!

c a

Look for right angles in the circle ng

t en

T A o

rad ius

di o am et e

o r

Trigonometry

Rightangled Triangles Use Pythagoras sides only

SOHCAHTOA sides & angles Q

To find the hypotenuse

SinQ =

o h

CosQ =

a h

c2 = a2 + b2 To find a shorter side

a2 = c2 - b2

TanQ =

o a

Or use the SINE rule here

Isosceles Triangles bb

Isosceles (and equilateral) triangles split into two congruent rightangled triangles

Other triangles look at formula sheet! A

The Sine Rule side

c B

b c = = SinB SinA SinC

Use when you know 3 Rearrange to find a

The Sine Rule angle

SinA

c B

= 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

a2 = b2 + c2 - 2bcCosA a

c A

Use this to find the third side when you know 2 sides and the included angle

The Cosine Rule angle

b2 + c2 - a2 CosA = 2bc

Use this to find the angle when you know all 3 sides

Area of a triangle

Area =

c Area A

1 2

bcSinC

Use this to find the third side when you know 2 sides and the included angle

(same as cosine rule side!)

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

Box Plots Lowest (L) = 2 2

Q1 = 7.5 Q2 = 14

7.5

20.5

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

Formula sheet

31 40

s. d . =

27 29

Ʃ( x - x )

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

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.

4πr3 = 3V

r3 = 3V 4π

What do you do now?

r = 3 3V 4π

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

= 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

x-2 = x2

( x-3 ) = x-6 =

1 x6

Fractional Indices

1 2

= √a 3

= (√a)

1 3

5 3

= √a 5

= (√a) 3

1 4

9 4

4 a = √

= (√a) 4

Quadratics Quadratic Equations y

y = (x + 1)(x 3)

Roots of (x + 1)(x 3) = 0 1

Either x = 1 or x = 3

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

x = 6

(-3,2)

(4,1)

x = 4

(6,-2)

y = (x - 4)2 + 1

y = (x - 6)2 - 2

x = -3

y = (x + 3)2 + 2

Trig Functions y

Period

360

Amplitude

Learn the shapes of sinx, cosx & tanx 1

Period

360

Period

360

180

-1

y = sinx

y = tanx

y = cosx

Period = 360 Amplitude = 1

Period = 180 Amplitude not measured

Amplitude

Identify the curve 5

y = 5sin3x

y = 3cos2x

3 0

360

x 1

180

-3

-5

360

y = 6sin5x o

y = tan6xo

6 0

360

-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

S o

x = 360 - 70.5 o x = 289.5

0 (360)

180

T o

270

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?