Core 1 Mathematics

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Surds

Rules of Surds

√ab = √a√b

√(a/b) = √a/√b

a√c ± b√c = (a±b)√c

Rationalising the denominator

Multiply the top and bottom of the fraction by the reverse of the denominator

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Sketching Quadratic Graphs

① UP/DOWN – Look at the coefficient of χ² - positive u shaped/negative n shaped

② AXES – Put χ=0 for the y-axis and solve Υ=0 for the x-axis (can use discriminant to see if it crosses x-axis)

③MAX/MIN – find the vertex by completing the square or finding the value of x at the line of symmetry (halfway between roots)

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Completing the Square

aχ²±bχ±c=0

① Halve the coefficient of χ

② Square the halved value

③ Put the halved value in the (χ±b) and take the squared value away from c

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The Quadratic Formula

aχ²±bχ±c=0

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The Discriminant

b²-4ac<0 – NO ROOTS

b²-4ac=0 – ONE REPEATED ROOT

b²-4ac>0 – TWO DISTINCT ROOTS

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Simulatenous Equations

Elimination

① Match the coefficients

② Eliminate and then solve for one variable

③ Find the other variable

④ Check the answer works

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Simultaneous Equations Cont.

One Quadratic and One Linear

① Isolate variable in linear equation

② Substitute into quadratic equation

③ Solve to get values for one variable

④ Put values into linear equation

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Geometric Interpretation

Two Solutions – Two points of intersection

One Solution – One point of intersection (line is a tangent to curve)

No Solutions – The graphs never meet


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Factorising Cubics

Ƒ(χ)= χ³±χ²±χ±c

① Find one factor

② Use grid method to find a quadratic

③ Factorise quadratic


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The Remainder Theorem

Ƒ(χ)= χ³±χ²±χ±c

① Substitute (χ±a) into ƒ(χ)

② Solve for remainder

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Coordinate Geometry

Finding the equation of a line

Υ-γ₁=m(χ-χ₁)

m - gradient

Finding the midpoint of a line

① (χ₁+χ₂) , (Υ₁+Υ₂)

② Then divide both values by 2

Distance between two points

D=√(χ₂-χ₁)+(Υ₂-Υ₁)

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Parallel & Perpendicular Lines

Parallel lines

have the same gradient

Perpendicular lines

product of gradient = -1

so m₂= -1/m₁

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Curve Sketching

Υ=kχⁿ

n positive and even – u-shape or n-shape

 

n positive and odd – ‘corner-to-corner’ shape

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Curve Sketching Cont.

Υ=kχⁿ

n negative and even – two bits next to each other


n negative and odd – two bits opposite each other


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Circles

Equation of a circle

(χ-a)²+(Υ-b)²=r²

Centre = (a,b)

Radius = r

Complete the square to find the equation of a circle

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Cicles Continued

Tangents/Normals

Tangents are perpendicular to the radius

Normals have the same gradient

Circle Properties

The angle in a semicircle is a right-angle

The perpendicular from the centre to a chord bisects the chord

A radius and tangent to the same point will meet at right angles

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Differentiation

dy/dx=nxⁿˉ¹

Finding Tangents/Normals

① Differentiate the function

② Find gradient (Tan= Same, Norm= -1/m)

③ Υ-Υ₁=m(χ-χ₁)

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Differentiation Cont.

Stationary Points

dy/dx=0

d²y/dx²<0 maximum

d²y/dx²>0 minimum

Increasing/Decreasing

Positive gradient - increasing

Negative gradient - decreasing

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Integration

ʃχⁿ dx = χⁿ⁺¹/n+1 +c

Area under curve

²ʃ₁ χ³ dx = [x⁴/4]²₁ = [2⁴/4]-[1⁴/4]

sometimes you have to find the limits by factorising the quadratic function

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Integration Cont.

Adding integrals

Diagrams

Diagrams

Diagrams

Subtracting integrals

Diagrams

Diagrams

Diagrams

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