9FM0/01 & 02 Further Core Topics

​

​

Complex numbers

​

Note

     Basic

     Euler relation and DeMoirve theorem

     Use De Moirve to convert trig functions

     Roots and Loci

     Roots of unity

     Geometrical problem

     Complex no. / trig from & GP 

​

Exercise

     AS Book EX

     De Moivre's Theorem EX     Solutions

     Express series as GP & geometrical questions

     Solution   1E   1F Mixed

​

     AS Emporium 1   MS

     AS Emporium 2   MS

​

     Reviewed test 1   Solutions     

​

​

​

​

​

​

​

​

Series

Note

     Sum of a series

     Method of differences

     Maclaurin & Taylor series     video

     Visualization from graphs of adding extra terms for a Taylor series

​

Exercise

     Summation EX     Solutions

     Maclaurin & Taylor EX     Solutions

     Emporium   MS

​

     Series Homework    Solutions

     series homework 2    solutions

​

​

​

​

​

​

​

​

​

​

Root of polynomials

Note

     Root of polynomials

​

Exercise

     New book exercise

     Homework 1

     Practice 1    ANS

​

     

​

​

​

​

​

​

​

Matrices & Linear transformations

Note

     General rules for Matrix algebra

     Inverse a 3x3

     Inverse a 3x3 by row reduction *Not in syllabus, but interesting method

     more 3x3 inverse note

     Linear transformations

     Proof of linear transformations

     Proof of 3D rotation matrices

     Geometrical interpretation of 3 planes

     Invariant lines

     Determinant (old questions)    Solutions

​

Exercise

​

     New matrices practice    Answers

     invariant and transformations    Solutions

     Matrix homework    solutions

     Matrix Exam questions    solutions

     Matrix homework 2    solutions    (13 questions)

     Emporium 1   MS

     Emporium 2   MS

​

​

​

​

​

​

​

​

​

​

​

​

​

​

​

​

​

​

​

​

​

​

Vectors

​

​

​

Note

     Express a line equation

     Express a plane equation

     Finding normal of a plane without cross product

Problem solving examples

     Angle between line and plane

     Angle between two planes

​

     Shortest distance from a point to a line

     Distance from a point to a plane

     Distance between parallel planes
     Reflection of a point in a plane

     Shortest distance between two skew lines

     Angle between two adjacent planes of a tetrahedron 

     Shortest distance between a line and a plane that parallel

     line equation reflected by a plane

     Line of intersection between 2 planes  (using 2 points)

     Line equation between 2 planes  (convert Cartesian to vector form)

​

     More further vector notes & examples

​

Exercises

     Lines

     Planes     Solutions

​

     New book EX

     Exam practice    Solutions    *15questions

     Emporium   MS

​

     Extra class practice

​

     vector homework 2    solutions  *hard

     Exam questions    ANS

     Vector OCR     Solutions

     Vector homework 3    solutions  *easy

​

​

​

​

​

​

Proof by induction

​

Note

     Proof by induction

     Examples

​

Exercises

     Proof EX    Solutions

     Exam practice

     Emporium   MS

​

     Homework     solutions

     Homework 2    Solutions

​

​

​

​

​

​

​

​

​

​

​

​

​

​

​

​

​

​

Polar coordinates

Note

     Polar coordinates

​

Exercises

     Polar coordinates EX    Solutions

​

      Polar homework     Solutions      (19 questions)

      Polar homework 2    Solutions      (6 questions)

      Exam types    solutions    (56 questions) **tough!! >_<

​

​

​

​

​

Hyperbolic functions

Note

     Hyperbolic functions

     Differentiation

     Integration

     Differentiation Danny's note

     similarity with harmonic form

​

Exercises

     Hyperbolic functions EX   Solutions

​

     Homework    Solutions    (28 questions)

     Homework 2    solutions    (6 questions)

​

     

​

​

​

​

Methods in calculus

​

Note

     Basic rules for volumes of revolution

     Inverse trig proof examples

     Sketching inverse trig functions

​

Differentiating Inverse Trig/Hyperbolic functions

     Exercise   Solutions

​

​

Differential equations

Note

     Basic

     First order differential equations

     Second order differential equations

​

Exercises

     2nd order DE   Solutions

     1st order DE   Solutions

     1st order DE ver 2   Solutions     *very easy

     1st order DE modelling

     Coupled DE     Solutions

​

     DE homework 1     Solutions

     Application    Solutions

     DE homework 2     Solutions     (10 questions)

​

Integration

     Integration practice   ANS

     Hard integration ED's

     Improper integrals   solutions

     mean values of functions   solutions

     mean values of functions 2    solutions

​

     Integration Homework    solutions

     Volume & improper integrals    solutions

     calculus homework     Solutions     (13 questions)

     integration practice 2 (by substitution) solutions

    integration practice 3 (very hard) solutions 

    integration practice 4  solutions

​

​

​

​

Review exercises

​

     Review Exercises Book 1

     Review Exercises Book 2

​

Mixed practice A   Question paper   ANS

Mapping - Complex numbers, roots of polynomials, Matrices, Hyperbolic functions, Maclaurin series

​

Mixed practice B   Question paper   ANS

Mapping - Complex numbers, roots of polynomials, Matrices, Hyperbolic functions, Maclaurin series, Sum of series, Proof by inductions 

​

Mixed practice C   Question paper   ANS

Mapping - Differential equations, integrations

​

Mixed practice D   Question paper   Hint   ANS

Mapping - Complex numbers, Proof by induction, roots of polynomials, vectors and Sum of series

​

Mixed practice E   Question paper   ANS

Mapping - Complex numbers, Maclaurin series, Differential equations, Sum of series

​

Mixed practice J   Question paper   ANS

Mapping - Complex numbers, Vectors, proof by induction, Sum of series, Matrices.

​

Mixed practice K   Question paper   ANS

Mapping - Complex numbers, Vectors, proof by induction, Sum of series, Matrices, Maclaurin series. (Difficulty level: MEDIUM/HARD)

​

Mixed practice L   Question paper   ANS

Mapping -  Vectors, Matrices (Difficulty level: EASY)

​

Mixed practice M   Question paper   ANS

Mapping -  Complex numbers, roots of polynomial, matrices, proof by induction (Difficulty level: MEDIUM)

​

Mixed practice N   Question paper   ANS

Mapping -  Complex numbers, vectors, proof by induction 

(Difficulty level: HARD)

​

Mixed practice O   Question paper   ANS

Mapping -  inverse trig/hyperbolic functions, differential equations, polar coordinates, volumes. .  (Difficulty level: CHALLENGING)

​

​

Integral website

​

AS

complex number 1    solutions

complex number 2    solutions

roots of polynomials    solutions

proof    solutions

matrix 1   solutions

matrix 2   solutions

vector    solutions

volume    solutions

​

A2

first order DE    solutions

second order DE    solutions

calculus 1     solutions

calculus 2     solutions

complex number     solutions

hyperbolic functions     solutions

Maclaurin series     solutions

series     solutions

polar coordinates     solutions

​

Revison

complex number     solutions

polar coordinates     solutions

DE     solutions

integration     solutions

matrix vector     solutions

induction     solutions

Maclaurin series     solutions

​

​

​

​