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Core 1

1 Chapter 1 ~ Algebra and Functions

1.1 Like terms

1.2 Multiplying brackets

1.3 Rules of Indices

1.4 Factorising

1.5 Surds

1.5.1 √(ab)=√(a)x√(b)

1.5.2 √(a/b)=√(a)/√(b)

1.5.3 Rationalising

1.5.3.1 Multiply top and bottom
by denominator with
opposite central sign

2 Chapter 2 ~ Quadratic Functions

2.1 y = f(x) = ax^2 + bx +c

2.2 Solve quadratic equations by factorisation

2.3 Completing the square

2.3.1 1) Write formula >
y=x^2 + bx + c

2.3.1.1 2) Make a gap appear >
y=x^2 + bx ________________ + c

2.3.1.1.1 3) In the gap add on, then take off (0.5*b)^2 >
y=x^2 + bx + (0.5*b)^2 - (0.5*b)^2 + c

2.3.1.1.1.1 4) Add brackets > y=(x^2 + bx + (0.5*b)^2) - (0.5*b)^2 + c

2.3.1.1.1.1.1 5) Factorise large bracket and tidy up numbers at the end

2.4 x = [-b±√((b^2)-4ac))]/2a

2.5 Discriminant

2.5.1 b^2>4ac and a>0

2.5.1.1 Two different roots

2.5.1.2 u shape crosses x-axis twice

2.5.2 b^2=4ac and a>0

2.5.2.1 Two equal roots

2.5.2.2 u shape sits on x-axis

2.5.3 b^2<4ac and a>0

2.5.3.1 No real roots

2.5.3.2 u shape that doesn't touch x-axis

2.5.4 b^2>4ac and a<0

2.5.4.1 Two real roots

2.5.4.2 n shape crosses x-axis twice

2.5.5 b^2=4ac and a<0

2.5.5.1 Two equal roots

2.5.5.2 n shape sits on x-axis

2.5.6 b^2<4ac and a<0

2.5.6.1 No real roots

2.5.6.2 n shape that doesn't touch the x-axis

3 Chapter 3 ~ Equations and Inequalities

3.1 Solving simultaneous
linear equations

3.1.1 Elimination

3.1.2 Substitution

3.2 Solving simultaneous
equations where one
quadratic and one
linear equation

3.2.1 Substitution

3.3 Inequalities

3.3.1 Solve similar to equations

3.3.2 Number lines

3.3.3 Sketches

4 Chapter 4 ~ Sketching Curves

4.1 Cubic Curves

4.1.1 y = ax^3 + bx^2 + cx + d

4.1.2 Use factors of equation to work out where the curve crosses the x-axis

4.1.2.1 x-axis intersections are when x = 0

4.1.2.1.1 eg (x-2) > intersection at (2,0). (x+5) > intersection at (-5,0)

4.1.3 y = x^3

4.1.3.1 Smooth curve through (0,0)

4.2 Reciprocals

4.2.1 y = k/x

4.2.1.1 When k>0, curves appear in quadrants where
both values are either positive or negative

4.2.1.2 When k<0, curves appear in quadrants where
one value is positive and the other is negative

4.2.2 The further away k is from
0, the further away the
curves are from the axes

4.3 Transformations

4.3.1 f(x+a) > moves whole curve -a in the x-direction

4.3.2 f(x)+a > moves whole curve +a in y-direction

4.3.3 f(ax) > multiply x-coordinates by (1/a)

4.3.4 af(x) > multiply y-coordinates by a

5 Chapter 5 ~ Coordinate Geometry in the (x,y) Plane

5.1 y = mx + c

5.1.1 m is the gradient and c
is the y-intercept

5.2 ax + by +c = 0

5.2.1 a, b, and c are all integers

5.3 Gradient between two
points = (y2 - y1)/(x2 - x1)

5.3.1 Equation of a line using one point
and the gradient > y - y1 = m(x - x1)

5.3.2 Equation of a line between two points
> (y - y1)/(y2 - y1) = (x-x1)/(x2 - x1)

5.4 Two lines

5.4.1 Perpendicular

5.4.1.1 Gradient = -1/m

5.4.1.2 The product of two perpendicular lines is -1

5.4.2 Parallel

5.4.2.1 Same gradient

6 Chapter 6 ~ Sequences and Series

6.1 General term > nth term

6.1.1 a + (n-1)d

6.2 Un = 4n + 1

6.3 C1 only has arithmetic sequences

6.3.1 a > first term

6.3.2 d > common difference

6.4 Sum of an arithmetic sequence

6.4.1 Sn = (n/2)[2a + (n - 1)d]

6.4.2 sn = (n/2)(a + L)

6.4.2.1 L is the last term

6.4.3 ∑_(r=1)^10(5+2r) =7+9+...+25

7 Chapter 7 ~ Differentiation

7.1 Used to work out the gradient of a tangent

7.2 f(x) = x^n

7.2.1 f'(x) = nx^(n-1)

7.2.2 To get f'(x), multiply power by
number in front of x, then reduce
power by 1 for each part separately

7.3 (d^2y)/(dx^2) = f''(x)

8 Chapter 8 ~ Intergration

8.1 If dy/dx = x^n, the y = (1/(n+1))(x^(n+1)) + c

8.2 ∫x^n dx= x^(n+1)/(n+1)+c

8.3 Calculate c when given any point that the function of the curve passes through

8.4 Reverse of differentiation