Pure Mathematics

Chapters for Download

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 acos(x) + bsin(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
  • Integration 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 cosⁿθ or sinⁿθ
  • Identities in form cosⁿθ or sinⁿθ
  • 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