# Indices and Logarithms

Indices

$\text{Positive Index: }{{a}^{n}},a\ne 0$    $eg:{{2}^{3}},{{4}^{2}}$

$\text{Negative Index: }{{a}^{-n}}=\frac{1}{{{a}^{n}}}$   $eg:{{2}^{-3}}=\frac{1}{{{2}^{3}}}=\frac{1}{8}$

$\text{Zero Index: }{{a}^{0}}=1,a\ne 0$    $eg:{{2}^{0}}=1$

Fractional Indices:

${{a}^{\frac{1}{n}}}=\sqrt[n]{a},a\ne 0$       $eg:{{8}^{\frac{1}{3}}}=\sqrt[3]{8}=2$

${{a}^{\frac{m}{n}}}={{(\sqrt[n]{a})}^{m}},a\ne 0$   $eg:{{8}^{\frac{2}{3}}}={{(\sqrt[3]{8})}^{2}}={{(2)}^{2}}=4$

Law of Indices:

$\text{Rule 1: }{{a}^{m}}\times {{a}^{n}}={{a}^{m+n}}$   $eg:{{2}^{2}}\times {{2}^{3}}={{2}^{2+3}}={{2}^{5}}=32$

$\text{Rule 2: }{{a}^{m}}\div {{a}^{n}}={{a}^{m-n}}$   $eg:{{2}^{4}}\times {{2}^{2}}={{2}^{4-2}}={{2}^{2}}=4$

$\text{Rule 3: }{{({{a}^{m}})}^{n}}={{a}^{m\times n}}$      $eg:{{({{2}^{2}})}^{3}}={{2}^{2\times 3}}={{2}^{6}}=64$

# SPM Questions for Simultaneous Equations

At least one simultaneous equations question (5 marks) will be coming out in the SPM paper 2, section A.

1. Substitution (SPM 2005, SPM2007)
2. Using Formulae (SPM 2006, SPM 2009)

# Simultaneous Equation

Simultaneous equation problem could be solved by using

1. Substitution
2. Equating Coefficients
3. Using Formulae

# SPM Questions for Quadratic Functions

1. The Basic of quadratic functions (SPM 2010)
2. Determine max and min values of quadratic function (SPM2009)
3. How to sketch the graph of quadratic functions (SPM2010)
4. How to find the range of values of x in Quadratic inequalities (2007, 2006)

This topic is closely related to the topic of quadratic equations. We will discuss further  on 4 subtopics below:

1. The Basic of quadratic functions
2. Determine max and min values of quadratic function
3. How to sketch the graph of quadratic functions
4. How to find the range of values of x in Quadratic inequalities

# SPM Questions for Quadratic Equations

We will discuss some pass year SPM exam questions based on:

• Solve the quadratic equations – SPM 2003, SPM 2005
• Form a quadratic equation – SPM 2009
• Determine the conditions for the type of roots – SPM2010

## 3 Basic Techniques in Solving Quadratic Equation Questions

In this chapter we will learn 3 most basic techniques on how to:

3. Determine the conditions for the type of roots.

# SPM Questions for functions

## Frequently Asked Questions in SPM for Functions

We will discuss some pass year SPM exam questions based on the four concepts :

Relations – SPM 2010
Functions – SPM 2007
Composite Functions – SPM 2007
Composite & Inverse Functions – SPM 2010

# Functions

## 4  Basic Concepts You Need To Learn About Functions

Basically there are four concepts you need to master in this chapter:

1. Relations
2. Functions
3. Composite Functions
4. Inverse Functions