− conversion between polar and rectangular forms of real and

complex numbers

− simplification of sums, differences, products and quotients of

surds or complex numbers expressed in rectangular form

− simplification of products or quotients of complex numbers

expressed in polar form

− use of De Moivre’s theorem in the simplification of

expressions such as (5cis Pi/2)^10

Manipulate different types of equations:

- Quadratic

-Cubic (rational roots only)

-Exponential such as 2^(3x+1)=5

-Logarithmic such as ln(x+5)=1.34 (any base)

Use the remainder and factor theorem

Complete a square

MERIT

I can...

Solve irrational equations such as x+2=2sqr(x)

Solve cubic equations with one integer root and two complex roots

Solve equations of the form z^n = a, z^n=r cis(theta)

EXCELLENCE

I can...

Solve a problem that requires an extended chain of reasoning

Prove a theorem

Solve a complicated equation

AS90644

ACHIEVE

I can...

Solve equations involving:

solving systems of three linear equations in three variables, where there is a unique solution (this may involve re-arrangement of equations and/or interpreting solutions).

solving a non-linear equation using the Newton-Raphson method with a given starting value, or the bisection method with a given starting interval (Newton-Raphson method includes derivatives of polynomials only)

optimising an objective function for a situation requiring techniques of linear programming, where the constraints and the objective function for the problem are given.

MERIT

I can...

Solve problems involving:

optimising an objective function for a linear programming problem, where you are expected to form your own constraints and objective function, and round the solution in relation to the context

using a suitable method to find an approximate solution to a non-linear equation (graphical, table, graphics calculator etc)

finding appropriate solutions to a non-linear equation using either the Newton-Raphson method or the bisection method to improve the approximation to a stated precision or for a specified number of iterations. Derivatives of functions other than polynomials will be given

forming and solving a 3x3 system of linear equations.

EXCELLENCE

I can...

Analyse or interpret the outcome, or the process used to solve equations or linear programming problems by:

discussing consistency or non-independence of 3x3 systems of linear equations, including geometric representatios

determine the effect of varying the constrains or objective function of a linear programming problem

considering the possibility of multiple solutions to a linear programming problems

giving advantages and disadvantages of the Newton-Raphson method or the bisection method for the problem

giving geometric description of the Newton-Raphson method or the bisection method.

AS90637

ACHIEVE

I can...

Solve straightforward problems that involve trigonometric funcstions like:

AsinB(x + C) + D

AcosB(x + C) + D

AtanB(x + C) + D, where C or D may be zero.

Solve problems that require knowledge of amplitude, period and frequency.

Solve equations such as:

AsinB(x + C) = K

AsinBx = K

AcosB(x + C) = K

AcosBx = K

AtanB(x + C) = K

AtanBx = K

MERIT

I can...

form an equation for the model and use the model to solve problems such us:

y = AsinB(x + C) + D

y = AcosB(x + C) + D

y = AtanB(x + C) + D

Work out the constants A, B, C from a worded problem

Manipulate trigonometric functions including:

reciprocal relationships

Pythagorean identities

compound angle formulae

double angle formulae

sum and product formulae , and combinations of these.

Solve equations providing a general solution or the solution within a specific domain.

EXCELLENCE

I can...

Solve problems that will require a chain of reasoning or may involve

a proof

developing a formula from a given starting points

rewriting a trigonometric expression in terms of a single trigonometric function

identifying and rectifying a flaw in reasoning

evaluation of the model (limitations, improvements, long-term accuracy)

solving more complex equations

solving 3-D trigonometric problems.

Candidates will be required to choose and apply appropriate trig relationships

AS90635

ACHIEVE

I can...

Differentiate functions such us:

polynomial ax^n

exponential (base e only)

logarithmic (base e only)

trigonometric, including reciprocals

(x^2+5x)^7

(2x-3)^(1/3)

7e^(2x)

ln(2x+7)

sin(5x)

Solve problems involving:

-optimization of a given function

-related rates of change, involving two directly related rates

-finding equations of tangents

-locating maxima and minima of polynomial functions

MERIT

I can...

Demonstrate knowledge of the following techniques:

differentiation from first principles of polynomial functions of degree less 3

differentiate products, quotients, implicit and parametric functions

Identify feature of a given graph including:

limits

differentiability

discontinuity

gradients

concavity

Sketch graphs of polynomials of degree more than 3 and identify features such as:

turning points

points of inflection

concavity.

Solve diffeerntiation problems involving:

interpretation of features of graph

optimisation

related rates of change, which may involve more than two directly related rates.

EXCELLENCE

I can...

Solve problems involving a combination of different techniques such as:

establishing a model

proving a theorem

AS90636

ACHIEVE

I can...

Integrate functions such as:

polynomials a{x ^ n , including negative powers of n and n=-1

exponentials ae^(bx+c), base e only

trig functions

rational functions (ax+b)/x

Solve problems involving the following techniques:

rates of change, eg, kinematics

differential equations of the forms y'=f(x)

separating variables that are easy to separate

finding areas under graphs

finding volumes of solids of revolution around x

finding areas using Simpson's Rule or the Trapezium Rule

Solve problems given the diagram for the area and volume

MERIT

I can...

Integrate functions that involve techniques such as:

products of trig functions

simple algebraic substitution

rational functions of the type f'(x)/f(x)

rational functions of the type (ax+b)/(cx+d)

Solve problems involving:

areas between graphs of polynomials

areas under graphs of combined

volumes of solids of revolution formed by around x or y axis

rates of change problems including

differential equations where required to write a differential including growth and decay, inflation

Newton's Law of Cooling and similar similar situations eg y'=ky

EXCELLENCE

I can...

Solve more complex integration problems involving techniques such as:

areas between graphs of functions, other than polynomials

volumes of solids of revolution formed by rotating around an axis parallel to either x or y

differential equations involving more difficult manipulation