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Further Pure Mathematics 1 Course (9231)
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What I will learn?
 Use the relations between the roots and coefficients of polynomial equations e.g. to evaluate symmetric functions of the roots or to solve problems involving unknown coefficients in equations; restricted to equations of degree 2, 3 or 4 only.
 Use a substitution to obtain an equation whose roots are related in a simple way to those of the original equation
 Sketch graphs of simple rational functions, including the determination of oblique asymptotes, in cases where the degree of the numerator and the denominator are at most 2.
 Use the standard results for to find related sums and use the method of differences to obtain the sum of a finite series.
 Carry out operations of matrix addition, subtraction and multiplication, and recognise the terms zero matrix and identity (or unit) matrix.
 Understand the use of 2 X 2 matrices to represent certain geometric transformations in the xy plane.
 Understand the meaning of ‘invariant’ as applied to points and lines in the context of transformations represented by matrices, and solve simple problems involving invariant points and invariant lines.
 Understand the relations between Cartesian and polar coordinates, and convert equations of curves from Cartesian to polar form and vice versa.
 Use equations of lines and planes, together with scalar and vector products where appropriate, to solve problems concerning distances, angles and intersections.
 Use the method of mathematical induction to establish a given result.
 Recognise situations where conjecture based on a limited trial followed by inductive proof is a useful strategy, and carry this out in simple cases.
Course Curriculum
Roots of polynomial equations

Introduction
02:34 
Introduction to Quadratic Polynomials
04:46 
Forming new equations by algebraic manipulation
04:52 
Forming new equations by substitution
06:34 
Symmetric function of the roots of a quadratic equation
08:25 
Revision Exercise 1

Roots of Cubic polynomial equations
04:19 
08:30

Algebraic Manipulation (Worked Examples)
08:36 
Forming new Cubic polynomial equations by substitution
09:18 
Obtaining sum of squares formula from equation
10:21 
Revision Exercise 2

Roots of Quartic polynomial equations
02:27 
Forming new Quartic polynomial equations by substitution
05:10 
Recurrence relations of the roots
18:07 
Revision Exercise 3
Summation of Series

Series Introduction
02:36 
Sum and General Term of a series (Worked Example 1)
09:03 
Sum and General Term of a series (Worked Example 2)
05:39 
Standard Results
08:21 
Standard Results (Worked Example)
06:49 
Limits at Infinity
04:28 
Limits at Infinity (Worked Example)
11:19 
Method of differences
12:13 
Method of differences (Worked Example 1)
12:13 
Method of differences (Worked Example 2)
11:18 
Method of differences (Worked Example 3)
09:21 
Method of differences (Worked Example 4)
07:22 
Revision Exercise 1

Revision Exercise 2
Rational functions and graphs

Introduction
01:22 
How to find the asymptotes of a rational graph
16:27 
Determining the behaviour of a rational graph about the asymptotes
18:49 
Sketching modulus graphs
04:10 
Sketching modulus graphs (Worked example)
10:47 
Oblique asymptotes 1
08:46 
Oblique asymptotes 1 (worked example)
07:15 
Oblique asymptotes 2
13:00 
Oblique asymptotes 2 (worked example)
13:49 
Oblique asymptotes and Modulus graphs (worked example)
12:46 
Revision Exercise 1

Revision Exercise 2
Matrices & Transformations

Introduction
02:48 
Addition Subtraction and Scalar Multiplication
04:55 
Multiplication of Matrices
04:50 
Multiplication of Matrices (worked examples)
17:51 
Determinant and Inverse of a 2 x 2 Matrix
02:46 
Determinant of a 3 x 3 Matrix
04:14 
Inverse of a 3 x 3 Matrix
10:01 
Singular Matrices
03:46 
The product of a matrix and it’s inverse
10:10 
Inverse of the product of two matrices
05:00 
Transformations (Introduction)
02:54 
Reflection
03:57 
Rotation
04:42 
Enlargement
03:35 
Shear
04:37 
Stretch
04:11 
Successive transformations
02:23 
Area scale factor
01:58 
Invariant Points
05:54 
Invariant Points (Worked Example)
07:36 
11:14

Worked Example 1
13:34 
Worked Example 2
11:41 
Worked Example 3
12:04 
Revision Exercise 1

Revision Exercise 2
Polar Coordinates

Introduction
03:55 
Plotting points on a Polar graph
05:30 
Converting between Polar and Cartesian coordinates
12:55 
Converting equations from Cartesian to Polar form
04:58 
Converting equations Polar to Cartesian form
03:42 
Sketching Polar Graphs
02:02 
Sketching Circles
03:02 
Cardioid Graphs
02:23 
Area enclosed by a Polar graph
05:01 
Cardioid Graphs (worked example)
10:26 
Finding the area enclosed by two Polar Graphs
09:32 
Sketching Polar graphs for any given equation
09:31 
Greatest distance of a point from the pole
20:21 
Greatest distance of a point from the pole (worked example)
13:17 
The point furthest from the initial line
15:22 
The point furthest from the vertical line
22:16 
Revision Exercise 1

Revision Exercise 2
Vectors

Equation of a plane 1
04:20 
Vector Product
03:31 
Equation of a plane 2
09:30 
Equation of plane (worked examples)
08:39 
A line parallel to a plane
08:12 
02:15

Distance from a point to a plane 1
09:17 
Distance from a point to a plane 2
12:31 
Perpendicular distance from a point to a line
09:17 
Worked Example 1
11:34 
Shortest distance between two skew lines
03:41 
Angle between a line and a plane
05:45 
Worked Example 2
18:12 
Angle between two planes
03:53 
Worked Example 3
12:20 
Worked Example 4
25:42 
Worked Example 5
00:00 
Revision Exercise 1

Revision Exercise 2
Proof by Induction

Mathematical Induction – Introduction
02:45 
Matrices (worked example)
08:39 
Divisibility (worked example 1)
06:19 
Divisibility (worked example 2)
07:35 
Divisibility (worked example 3)
07:11 
Divisibility (worked example 4)
08:45 
Divisibility (worked example 5)
07:28 
Sequences (worked example 1)
06:24 
Sequences (worked example 2)
06:38 
Sequences (worked example 3)
11:28 
Sequences (worked example 4)
11:23 
Differentiation (worked example 1)
10:43 
Differentiation (worked example 2)
09:54 
Differentiation (worked example 3)
12:25 
Series (worked example 1)
15:20 
Series (worked example 2)
11:13 
Series (worked example 3)
07:23 
Revision Exercise 1

Revision Exercise 2
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$67.00

LevelIntermediate

Total Enrolled8

Duration16 hours

Last UpdatedSeptember 22, 2022
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A course by
Walter Chatyoka
Maths Tutor
Material Includes
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Requirements
 A desktop or laptop computer is recommended for optimum user experience. However, a phone or tablet can also be used as an alternative.
 A reliable internet connection for the duration of the course
 Have good foundational knowledge of Mathematics (IGCSE level).
 Ideally the student must have studied the majority of the Cambridge International AS & A Level Mathematics (9709).
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Target Audience
 Cambridge AS/A Level students