Conquer
The
Academy
Welcome
back
Skip to content
Conquer
YOUR A LEVELS
Subjects
C
Cambridge
E
Pearson Edexcel
AQA
AQA
OCR
OCR
Competitive
Study space
Mistake Bank
AI Marking
Log in
Try for free
Study space
Mistake Bank
AI Marking
Subjects
C
Cambridge
E
Pearson Edexcel
AQA
AQA
OCR
OCR
Competitive
Try for free
Log in
Create free account
Revision Notes
Practice Questions
Topical Past Paper Questions
Past Papers
Resources
Revision Notes
Practice Questions
Topical Past Paper Questions
Past Papers
Further Pure Mathematics 1
/
Matrices 1
Contents
Invariant lines
Find origin lines, vertical lines and offset families without confusing fixed points.
Contents
All chapters
Further Mathematics · 9231
Further Pure Mathematics 1
Chapter 04 · Matrices 1
Invariant points and lines
Invariant lines
01
Roots of Polynomial Equations
Roots and coefficients
3
Roots, factors and multiplicity
Deriving quadratic root relations
Cubic and quartic root relations
Expressions in roots
2
Evaluating expressions in roots
Power sums from the original equation
New equations
5
Shifting and scaling roots
Reciprocal root transformations
Equations for squared roots
Equations for cubed roots
Combined roots and chapter review
Reference
1
Formula and method reference
02
Rational Functions
Rational graphs
4
Domains, intercepts and excluded points
Horizontal asymptotes and first sketches
Stationary points and quadratic denominators
Oblique asymptotes and parameters
Range and inequalities
2
The range of a rational function
Solving rational inequalities
Graph transformations
4
Reciprocal graphs
Output modulus and input modulus
Modulus graphs: asymptotes and solutions
Graphs of y² = f(x)
Review
1
Rational functions: review and method reference
03
Summation of Series
Finite sums
4
Sigma notation and linear sums
Sums of squares and cubes
Polynomial and patterned sums
Changing summation limits
Differences and limits
3
The method of differences
Recognising more complex differences
Convergence, remainders and bounds
Review
1
Summation of series: method selection
04
Matrices 1
Matrix algebra
6
Matrix entries and operations
Matrix multiplication
Inverse matrices of order 2
Determinants
Inverse matrices of order 3
Matrix equations and inverse products
Plane transformations
5
Matrices and point images
Stretches, enlargements and shears
Rotations and reflections
Combining matrix transformations
Area scale factors
Invariant points and lines
2
Invariant points
Invariant lines
Review
1
Matrices 1: mixed problems
05
Polar coordinates
Coordinates and curves
8
Polar coordinates and points
Polar and Cartesian equations
Tracing polar curves
Symmetry and cardioids
Petal curves
Spirals
Intersections of polar curves
Greatest distances
Areas
5
Areas from polar sectors
Areas of petals
Areas between curves
Common regions
Unfamiliar polar integrals
Review
1
Polar coordinates review
06
Vectors
Vector products
3
Vectors and projection
The vector product
Areas using vectors
Lines and planes
4
Lines and their intersections
Point-to-line distance
Equations of planes
Plane forms and construction
Intersections and angles
3
Line-plane intersections
Intersection of two planes
Angles with lines and planes
Distances and perpendiculars
3
Point-to-plane distance
Distance between skew lines
Common perpendiculars
Review
2
Vectors — Formula Summary
Vectors review
07
Proof by induction
Proof by induction
8
Why induction works
Proving sums
Proving recurrence formulas
Proving bounds
Proving derivative patterns
Proving matrix powers
Proving divisibility
Induction review
Search resources