Download Week 7: Non-Linear Equations and review of topic.

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Topic 1: Algebra
Achievement Standard 91577: Apply the algebra of complex numbers in solving problems
This standard is an External worth 5 credits.
Week 7: Non-Linear Equations and review of topic.to be completed by Monday 21 March
This week we are finishing off the Algebra standard. The types of questions and equations are not covered well in the
textbook so I have put together a booklet of exam questions. Worked solutions to all of these will be on the webpage.
Learning Objectives
1.
2.
3.
4.
To be able to solve irrational equations involving surds.
To be able to use log properties to solve equations involving logs and indices.
To use algebraic manipulation to solve complex problems.
To complete a review of the topic and work through a sample assessment.
Topic
Surd equations
Review of logs
Log equations
Notes
Delta Mathematics 2nd
Edition Old Book
Delta Maths NCEA Level 3
New Book
I have put together a booklet of exam
questions involving non-linear equations and
other sundry manipulation questions. These
questions range from Achieved to
Excellence.
These equations involve the use of squaring
both sides to remove the square root of an
expression.
Worksheet: Non
Linear Equations
Worksheet: Non Linear
Equations
Question 1 and 2 on
the worksheet.
There are no examples
in the textbook
Page 43
Ex 3.2 Q 2 and 3
Page 43
Ex 3.3 Q 1-25
Question 1 and 2 on the
worksheet.
There are no examples
in the textbook
All the exercises below
are from the scanned
pages.
Page 1
Ex 3.2 Q 2 and 3
Page 2
Ex 3.3 Q 1-25
Page 43
Ex 3.2 Q 4 and 5
Page 44
Ex 3.3 Q 26
Worksheet: Questions
3, 4, 5 and 6
Page 45
Ex 3.4 Q 1 to 10
Worksheet: Questions
7 and 8
Worksheet: Questions
9, 10, 11, 12 and 13
Page 1
Ex 3.2 Q 4 and 5
Page 2
Ex 3.3 Q 26
Worksheet: Questions
3, 4, 5 and 6
Page 3
Ex 3.4 Q 1 to 10
Worksheet: Questions 7
and 8
Worksheet: Questions
9, 10, 11, 12 and 13
This work is not covered in the new textbook
so I have scanned some pages for those
using the new book.
Review your knowledge of logs by reading
pages 42 and 43.
You need to understand the link between
logs and indices (exponents or powers). You
also need to be able to use the 3 log rules
which are on page 43.
See also my sheet called Log Rules which
shows how the log rules are derived from
your knowledge of indices.
You need to use the properties of logs to
solve these types of equations.
Exponential
(Index) equations
Solving these equations involve the use of
logs.
Read the worked examples on Page 44.
Algebraic
manipulation
There are a selection of questions that
require the use of algebraic manipulation to
either solve equations or show a
mathematical statement is true.
Review of topic
Read the summary on the next page and
complete the sample assessment.
Review of Topic 1: Algebra
Achievement Standard 91577: Apply the algebra of complex numbers in solving problems
This standard is an External worth 5 credits.
Learning Objectives


To review the content for the algebra standard.
To complete a sample assessment for algebra.
The content of standard 91577 can be summarised into 3 sections.
Apply the algebra of complex numbers in solving problems
Review of standard 91577:
Algebraic manipulation
 Indices
 Expanding brackets
 Linear factorising
 Quadratic factorising
 Rearranging formulae
 Algebraic fractions
 Linear equations and inequations
 Quadratic equations and inequations.
Real Numbers
 Working with surds
 Simplifying surds, addition and
multiplication
 Rationalising the denominator, division.
Complex Numbers





be able to add, subtract, multiply and
divide complex numbers in both
rectangular and polar form.
Convert from rectangular to polar form
and vice versa
Plot complex numbers on an Argand
diagram
De Moivres theorem for powers
Finding equations to describe loci where
the points satisfy conditions linked to
modulus, argument, real or imaginary
parts of z=x+iy.
Solving Equations
Quadratic
 use the quadratic formula and give solutions, either real or complex, in exact form using surds
 complex solutions come in conjugate pairs when coefficients are all real.
 nature of roots
 sum and product of roots
Polynomial
 use of the remainder and factor theorem to factorise and solve equations
 use of long division
 to be able to write a factor given a root.
 complex solutions come in conjugate pairs when coefficients are all real.
Non-linear equations
 equations with surds and the use of squaring to solve these
Log equations
 use of the properties of logs to help solve equations
Exponential equations
 use of logs to solve equations with x in the power
Finding complex roots
 use of de Moivre’s theorem to find all solutions
 understand the symmetrical nature of complex roots.
The table above gives you an overview of the standard.
Going back through the work outlines will give you more detail of each item.
Other resources are linked from the webpage. These include YouTube videos and past NCEA assessments with answers.