AQA IGCSE Further Maths Revision Notes

[Pages:16]AQA IGCSE Further Maths Revision Notes

Formulas given in formula sheet:

Volume

of

sphere:

4 3

3

Surface area of sphere: 42

Volume

of

cone:

1 3

2

Curve surface area:

Area

of

triangle:

1 2

sin

Sine

Rule:

sin

=

sin

=

sin

Cosine Rule: 2 = 2 + 2 - 2 cos

Quadratic equation: 2 + + = 0 = -?2-4

2

Trigonometric Identities: tan sin

cos

sin2 + cos2 1

1. Number

Specification 1.1 Knowledge and use of numbers and the number system including fractions, decimals, percentages, ratio, proportion and order of operations are expected.

1.2 Manipulation of surds, including rationalising the denominator.

Notes

A few possibly helpful things:

If

:

=

:

(i.e.

the

ratios

are

the

same),

then

=

"Find the value of after it has been increased by %"

If say was 4, we'd want to ? 1.04 to get a a 4% increase.

Can

use

1

+

100

as

the

multiplier,

thus

answer

is:

(1 + 100)

"Show that % of is the same as % of "

GCSE recap:

100 ? = 100

100 ? = 100

Laws

of

surds:

?

=

and

=

But note that ? = not

Note also that ? = To simplify surds, find the largest square factor and put this

first:

12 = 43 = 23

75 = 253 = 53

52 ? 32 Note everything is being multiplied here. Multiply surd-ey things and non surd-ey things separately. = 15 ? 2 = 30

8 + 18 = 22 + 32 = 52 To `rationalise the denominator' means to make it a non-

surd. Recall we just multiply top and bottom by that surd:

6

3

6 3

?

3 3

=

63 3

=

23

The new thing at IGCSE FM level is where we have more

complicated denominators. Just multiply by the `conjugate':

this just involves negating the sign between the two terms:

3

3 6 + 2 3(6 + 2)

?

=

6 - 2

6 - 2 6 + 2

2

A trick to multiplying out the denominator is that we have

the difference of two squares, thus (6 - 2)(6 + 2) =

6 - 4 = 2 (remembering that 6 squared is 6, not 36!)

23-1 23-1 ? 33-4 = (23-1)(33-4) =

33+4

33+4 33-4

27-16

18-33-83+4 11

=

22-113 11

=

2

-

3

What can go ugly

I've seen students inexplicably reorder the terms in the denominator before they multiply by the conjugate, e.g.

(2 + 3)(3 - 2) Just leave the terms in their original order!

I've also seen students forget to negate the sign, just doing (2 + 3)(2 + 3) in the denominator. The problem here is that it won't rationalise the denominator, as we'll still have surds!

Silly error: 66 2 33

(rather than 36)



1

2. Algebra

Specification 2.2 Definition of a function

2.3 Domain and range of a function.

Notes

A function is just something which takes an input and uses some rule

to produce an output, i.e. () =

You need to recognise that when we replace the input with some

other expression, we need to replace every instance of it in the

output, e.g. if () = 3 - 5, then (2) = 32 - 5, whereas ()2 = (3 - 5)2. See my Domain/Range slides.

e.g. "If () = 2 + 1, solve (2) = 51

22 + 1 = 51 = ?5

The domain of a function is the set of possible inputs.

The range of a function is the set of possible outputs.

Use to refer to input and () to refer to output. Use "for all" if any

value possible. Note that < vs is important.

() = 2 Domain: for all Range: for all ()

() = 2 Domain: for all Range: () 0

() = Domain: 0 Range: () 0

() = 2 Domain: for all Range: () > 0

() = 1

Domain: for all except 0.

Range: for all () except 0.

() = 1

-2

Domain: for all except 2 (since we'd be dividing by 0)

Range: for all () except 0 (sketch to see it)

() = 2 - 4 + 7 Completing square we get ( - 2)2 + 3

The min point is (2,3). Thus range is () 3

You can work out all of these (and any variants) by a quick sketch and

observing how and values vary.

Be careful if domain is `restricted' in some way.

Range if () = 2 + 4 + 3, 1

When = 1, (1) = 8, and since () is increasing after

this value of , () 8.

To find range of trigonometric functions, just use a suitable

sketch, e.g. "() = sin()" Range: -1 () 1

However be careful if domain is restricted:

"() = sin() , 180 < 360". Range: -1 () 0

(using a sketch)

For `piecewise function',

fully sketch it first to find

range.

"The function () is

defined for all :

4 () = { 2

< -2 -2 2

12 - 4 > 2 Determine the range of ()."

From the sketch it is clear

() 4

You may be asked to construct a function given information

about its domain and range.

e.g. " = () is a straight line. Domain is 1 5 and

range is 3 () 11. Work out one possible expression

for ()."

We'd have this domain and range if line passed through

points (1,3) and (5,11). This gives us () = 2 + 1

What can go ugly

Not understanding what (2) actually means. Not sketching the graph! (And hence not being able to visualise what the range should be). This is particularly important for `piecewise' functions. Not being discerning between < and in the range. e.g. For quadratics you should have but for exponential graphs you should have ................
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