Core One - complete set of revision notes.

Set of notes i typed up for my core one maths exam, which helped incredibly on OCR

Sorry the notes are seperated, we had two different teachers so everything is a little all over the place, but everything is pretty much there, hope it can help anyone, took me long enough to do!

Good Luck!

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Core 1
Laws of indices
For indices with the same base:
When multiplying the bases , add together the powers e.g. a2X a3 = a5
When dividing the bases, subtract the powers e.g. a2÷ a3 = a-1
When there is power to a power, multiply the powers e.g. (an)m = anm
Other Laws:
When any number is to the power of 0 it is always equal to 1 e.g. x0 = 1
A negative power becomes a fraction e.g. a-n = a1n
mn m
A fractional power turns into a root e.g. a n = ( a)
Solving indices
To solve an indices the base of the two need to be the same, once this is correct an equation can be
made of just the indices and then solved e.g.
5x-1 = 125 x-1 = 3
5x-1 = 53 x=4
Surds
Simplifying surds
To simplify a surd, 2 factors must be taken out of the original one of which is a square number e.g.
50 = 25 X 2 = 52
Rationalising surds (Rationalising the denominator)
When there is a surd on the bottom of a fraction the surd needs to be removed through multiplying
the whole fraction by another fraction that results in a cancelling out of the surd denominator e.g.
3 = 3 X 7 = 37
7 7 7 7
Geometry
Midpoint
To calculate the midpoint you must add the two co-ordinates together and divide by two
(x1+ x2) (y1+ y2 )
Generally: Midpoint = 2 , 2
For example : (0+8 2+4
2 , 2 ) = (4, 3)
Distance between two points (Length of a line segment)
Create a triangle and use Pythagoras' theorem
a2 + b2 = c2
Use these numbers to calculate the length of the hypotenuse : 42 + 32 = 16 + 9
25 = c2

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Gradient of a line between two co-ordinates
Use these figures to figure out the gradient by dividing :
For example = m = 4 2 =2
Straight Lines
Equations of Straight Lines
To find the equations of a straight line the minimum you need is a set of two co-ordinates and the
gradient, these must then be plugged into the following equation and rearranged into the suitable
form.…read more

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X
Y co-ordinates.
Differentiation
Differentiation is finding the gradient of curves. This can be asked in many ways and use in multiple
forms of questions,
All of these are ways of just asking you to differentiate the formulae given :
Differentiate with respect to X
Find dy dx
Find f'(x)
d(any letter)
Find d(another letter)
To differentiate a polynomial of the form xn you multiply by the power and take one off the power.
E.g.…read more

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The nature of stationary points are whether or not they are a maximum or a minimum turning point,
to find this out, once you have found the stationary points you must take the second derivative and
sub in both X's to that second derivative, once you have done so the final step is to see whether the
value of the second derivatives are positive or negative, if the value is <0 then the turning point is
maximum, if the value is >0 then the turning…read more

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±b2-4ac
Using the quadratic formulae 2a
A= number in front of the square
B = number in front of the X
C = number on its own
These numbers must be added into the formulae which can then be solved again leaving two
numbers (one where the square root is added and the other where it has been subtracted)
Completing the square
Take out the coefficient of x2 term out as a factor from the x2 and x terms, not from the end
number.…read more

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The discriminant is the part of the equation that determines what type of solutions you will get for
the formulae e.g. one/two/no answers.
Therefore the discriminant of the quadratic equation is b2 - 4ac :
If b2 - 4ac > 0 solving the equation will leave us with 2 distinct roots.
If b2 - 4ac = 0 solving the equation will leave us with 1 repeated root.
If b2 - 4ac < 0 solving the equation will leave us with no roots.…read more

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Solving a quadratic in x1 2
This disguised quadratic will be written with a square root of x in making it x1 2 . This therefore
makes the first power of x equal to two times the middle power of x.
To solve these equations you must let x = T so that x = t2 and rewrite the equation in terms of T
this should then give a normal looking quadratic for you to solve, e.g.…read more

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Linear inequalitites
as simple as rearranging equations, treat these as if there is an equal sign in the middle and solve
them in that way :
e.g.
10x - 3 5 - 6x
16x 8
x 1
2
Quadratic inequalities
To solve a quadratic inequality, one side must contain an x2 of some sort, and the other side must be
valued at 0. Once this is the form of the inequality then it can be solved in the way a normal quadratic
expression is.…read more

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If the question asks to sketch the graph of a curve such as y = (x + 2)(x - 1)2 the squaring of the last
bracket means that there is a repeated factor, this means that the curve touches the x axis at that
point, and therefore it is a turning point.…read more

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Tangents to Circles
Normal
Tangent
If asked to find the equations of either Tangent or Normal's to a circle (given its equation) at one
particular point, first you must find the co-ordinates of the centre by either using the equation given
or by completing the square to get an appropriate equation.…read more

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