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Topical Past Paper Questions
Past Papers
Further Pure Mathematics 1
/
Vectors
Contents
Distance between skew lines
Use the common normal to measure the shortest separation.
Practise Vectors
Contents
All chapters
Further Mathematics · 9231
Further Pure Mathematics 1
Chapter 06 · Vectors
Distances and perpendiculars
Distance between skew 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
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