Further Pure 1

 

Chapter 1 - Roots and Coefficients of a Quadratic Equation
 

1.1 Roots of a Quadratic Equation

1.2 Symmetrical functions of roots

1.3 Equations with related roots

The connection between roots and coefficients of a quadratic equation.

Manipulating expressions involving a + b and ab.

Chapter 2 - Series
 

2.1 Summation of Series

2.2 Sums of the squares of natural numbers

2.3 Sums of the cubes of natural numbers

Use of formulae for the sum of of the squares and the sum of cubes of the natural numbers

Chapter 3 - Matrices
 

3.1 Notation

3.2 Addition & Subtraction of Matrices

3.3 Multiplication of Matrices

3.4 The Identity Matrix

2 X 2 and 2 X 1 matrices; addition and subtraction, multiplication by a scalar.

Multiplying a 2 x 2 matrix by a 2 x 2 matrix or by a 2 x 1 matrix.

The identity matrix I for a 2 x 2 matrix.

Chapter 4 - Matrices and transformations
 

4.1 Linear Transformations

4.2 Finding the Matrix representing a transformation

4.3 Exact values of trigonometric ratios

4.4 Identifying the geometrical transformation

4.5 Combining two transformations

Transformations of points in the x/y plane represented by 2 x 2 matrices.

Transformations will be restricted to rotations about the origin, reflections in a line through the origin, stretches parallel to the x- and y- axes, and enlargements with centre the origin.

Use of the standard transformation matrices given in the formula booklet.

Combinations of these transformations.


Chapter 5 - Graphs of Rational Functions
 
 

5.1 Linear numerator and denominator

5.2 Two distinct linear factors in the denominator

5.3 Repeated factor in the denominator

5.4 Irreducible quadratic in the denominator

5.5 Stationary Points

5.6 Inequalities

Sketching Graphs 1

Sketching Graphs 2

Sketching Graphs 3

Graphs of rational functions of the form

Sketching the graphs.

Finding the equations of the asymptotes which will always be parallel to the coordinate axes.

Finding points of intersection with the coordinate axes or other straight lines.

Solving associated inequalities.

Using quadratic theory (not calculus) to find the possible values of the function and the coordinates of the maximum or minimum points on the graph.

Eg for which has real roots if 16k2 +8km 8 > 0, ie k > -1 or k < , stationary points are (1, -1) and (-2, ).


Chapter 6 - Conics and Algebra
 
 

6.1 The parabola

6.2 The ellipse

6.3 The hyperbola

6.4 The rectangular hyperbola

 

GeoGebra Files

Graphs of parabolas. Ellipses and hyperbolas with equations

and .


Chapter 7 - Complex numbers
 
 

7.1 What is a complex number

7.2 Calculating with complex numbers

7.3 Roots of Quadratic Equations

7.4 Linear Equations with complex coefficients

Non-real roots of quadratic equations.

Sum, difference and product of complex numbers in the form x + iy.

Comparing real and imaginary parts.

Complex conjugates awareness that non-real roots of quadratic equations with real coefficients occur in conjugate pairs.

Including solving equations eg 2z +z*=1 + i

where z* is the conjugate of z.


Chapter 8 - Calculus
 
 

8.1 Differentiating from first principles

8.2 Improper Integrals

Finding the gradient of the tangent to a curve at a point, by taking the limit as h tends to zero of the gradient of a chord joining two points whose x- co-ordinates differ by h.

Evaluation of simple improper integrals.


Chapter 9 - Trigonometry
 
 
9.1 General solutions of equations involving cosines
9.2 General solutions of equations involving sines
9.3 General solutions of equations involving tangents
General solution of trigonometric equations including use of exact values for the sine, cosine and tangent of pie/6, pie/4, pie/3.

Chapter 10 - Numerical Solutions of Equations
 
 

10.1 Location of root within an interval

10.2 Interval bisection

10.3 Linear interpolation

10.4 The Newton-Raphson method

10.5 Step-by-step solution of differential equations

Finding roots of equations by interval bisection, linear interpolation and the Newton-Raphson method.

Solving differential equations of the form .

Reducing a relation to a linear law.

Graphical illustration of these methods.

Using a step-by-step method on the linear approximations with given values for x0, y0 and h.

E.g. ; y2 = ax3 + b; y = axn;

y = abx.

Use of logarithms to base 10 (or e ) where appropriate.

Given numerical values of (x,y), drawing a linear graph and using it to estimate the values of the unknown constants.


Chapter 11 - Linear Laws
 
 

11.1 Reducing a relation to a linear law

11.2 Use of logarithms