**01. Differentiation 1**

- Gradient of a curve
- First principles
- The dy/dx notation
- Tangents and Normals
- Velocity and Acceleration

**02. Differentiation **2

- d/dx and f'(x) notation
- Greatest and Least values
- Maxima and Minima
- Curve sketching
- Second derivative
- Small changes

**03. Integration**

- Reverse of differentiation
- Velocity and Acceleration
- Area under a curve
- Definite integrals
- Area cut off by a line
- Area enclosed by two curves
- Solids of revolution
- Center of gravity

**04. Further Differentiation**

- The Chain rule
- Rates of change
- Products and Quotients
- Implicit functions
- Parameters
- Second derivative

**05. Algebra 1**

- Rules of Surds
- Rules of Logarithm
**Quadratic equations**- Completing the square
- The quadratic formula
- The discriminant b²-4ac
- Mini and maxi values

- Sum and Product of roots

- The Remainder and Factor theorems

**06. Binomial theorem**

- The Pascal’s triangle
- Factorial notation
- Combinations
- The Binomial theorem
- The term independent of x
- Binomial theorem for any index

**07. Algebra 2**

**Inequalities**- Linear inequalities
- Quadratic inequalities

- Rational fraction inequalities

**Further equation methods**- Square root equations
- Equations reducing to quadratics

**Simultaneous equations in 3-unknowns**- Substitution and elimination

**08. Series**

**Arithmetical progression (AP)**- General term of an AP
- Sum of an AP

**Geometrical progression (GP)**- General term of a GP
- Sum of a GP

- Sum to infinity of a GP
- Proof by induction
- Further series

**09. Trigonometry **1

- Trigonometric ratios of angles
- Ratios of 30
**°**, 45**°**, and 60**°** - Secant, cosecant and cotangent
- Trigonometric equations
- Pythagoras theorem
- Compound angle formula
- Double angle formula
- The
*t*– formula - The form a
*cos(x)*+ b*sin(x)* - The factor formula
- Cosine and Sine rule
- Area of a triangle

**10. Trigonometry 2**

- Radian measure
- Length of an arc and area of a sector
**Differentiation of trig functions**- Small angles

- I
**ntegration of trig functions**- Reverse differentiation
- Even and odd powers
- Using factor formula

- Definite integrals
- Differentiation of inverse trig
- General solutions

**11. Further integration 1**

- Recognizing a function and its derivative
- Change of variable
- Changing limits
**Integration using inverse trig functions**- Using sin⁻¹(x)
- Using tan⁻¹(x)
- Quadratic denominators

- Definite integrals involving trig functions

**12. Exponential and Log functions**

- The Exponential function
- Deriving y = eˣ
- The natural logarithm function
- Recognizing a function and its derivative
- Logarithm of negative limits
- d/dx(aˣ) and ∫(aˣ) dx
- Logarithmic differentiation

**13. Partial fractions**

**CASE I: Denominator with only linear factors**- Cover-up method

- CASE II: Denominator with a quadratic factor
- CASE III: Denominator having repeated roots
- CASE IV: Improper fractions
- Differentiation after Partialization
- Integration after Partialization
- Binomial expansion after Partialization

**14. Further integration 2**

- Integration by parts
- Integration by parts with limits
- Taking dv/dx as 1
- Integration by parts more than once
- Integration by parts where the original integral appears again
- Change of variable t = tan(ˣ⁄₂)
- Change of variable t = tan(x)
- Splitting the numerator

**15. Coordinate geometry 1**

**The Circle**- Cartesian equation
- Parametric equation
- Parametric equation
- Circle from 3 points
- Circle from diameter ends
- Intersection of circle and line
- Gradient at a point
- Tangent from a point
- Intersection of 2 circles
- Orthogonal circles

**Conics 1 [The Parabola]**- The Parabola (e = 1)
- Form (y – k)² = 4a(x – h)
- Tangents / Normals
- Parametric equation
- Chord to parabola

**18. Differential equations**

**Solving differential equations**- By separating variables
- Inserting boundary conditions

- Exact differential equations
- Integrating factor method
- Homogenous equations
- Other useful substitutions
- Differential equations problems
- Forming differential equations

**19. Complex numbers**

- Algebra of complex numbers
- Conjugate complex
- Square root a complex number
- Further complex roots
- The Argand diagram
**Modulus and Argument**- Laws of Modulus
- Rules of Arguments

- Form r(cosθ + i sinθ)
- De Moivre’s theorem
- Identities in form
*cos***ⁿ**θ or*sin***ⁿ**θ - Identities in form cos
**ⁿ**θ or sin**ⁿ**θ - The nᵗʰ root of a complex number
- Complex loci
- Complex loci with inequalities

**20. Vectors in 3D**

- The Unit Vector
- Products of vectors
- Scalar product (Dot product)
- Vector product (Cross product)

**Vector equation of a line**- Intersection of 2 lines
- Angle between 2 lines
- Shortest distance of a point from a line

**The plane**- Area of a Parallelogram
- Area of a Triangle
- Intersection of planes
- Angle between planes
- Intersection of line and plane
- Shortest distance of a plane from a point