What is the advantage of using standard form for very large or small numbers?

Standard form lets us represent very large or very small numbers in a concise and manageable way using powers of 10. This means we can write them more neatly, compare them more easily, and carry out calculations more efficiently.

True or False? There is always one non – zero digit before the decimal point in a standard form number.

True. There is always one (and only one ) non – zero digit before the decimal point in a standard form number.

True or False? Scientific notation is another term for standard form.

True. Scientific notation is another term for standard form.

True or False? The exponent k in standard form must always be positive .

False. The exponent k in standard form can be positive , negative , or zero .

What are bounds , in the context of estimation?

Bounds are the smallest ( lower bound ) and largest ( upper bound ) numbers that a rounded number can lie between .

Define the term lower bound .

The lower bound is the smallest possible value a number could have been before rounding .

True or False? A rounded number could have been equal to its upper bound before rounding.

False. A rounded number can not be equal to its upper bound before rounding. The upper bound of a rounded number is the smallest value that would have rounded up to the next highest rounded value.

True or False? The basic rule for finding bounds is " half up, half down ".

True. The basic rule for finding bounds is " half up, half down ". E.g. if a number is rounded to 31 to the nearest integer , then halfway down to the next lowest integer is 30.5 ( lower bound ), halfway up to the next highest integer is 31.5 ( upper bound ).

True or False? Percentage error should always be a positive number.

True. Percentage error should always be a positive number.

Define the term significant figures .

Significant figures are the digits in a number that carry meaning contributing to its precision. Significant figures start with the first non-zero digit .

True or False? When rounding to 3 significant figures , you always keep the first three non-zero digits .

False. When rounding to 3 significant figures , you keep the first three significant digits , which may include zeroes after the first non-zero digit.

What is the purpose of estimation ?

The purpose of estimation is to find approximate answers to difficult sums or to check if answers are about the right size (order of magnitude).

True or False? When dealing with currency , you should always round to the nearest whole number .

False. The appropriate rounding for currency depends on the specific currency being used. Some are rounded to whole numbers , others to 2 decimal places .

What is the rule for rounding up in problems involving minimum number of objects ?

In problems involving minimum number of objects , always round up to ensure there are enough objects to meet the requirement. E.g. if you calculate that 13.23 tins of paint are needed to complete a job, round up to 14 tins to make sure enough paint is bought.

Define the term system of linear equations .

A system of linear equations is where two or more linear equations work together , with variables that satisfy all equations simultaneously.

True or False? A system of linear equations may also be referred to as simultaneous equations .

True. A system of linear equations may also be referred to as simultaneous equations .

True or False? You can use a GDC to solve a system of linear equations .

True. You can use a GDC to solve a system of linear equations . Your GDC will have a simultaneous equations solving feature. However it is also useful to be able to solve simple simultaneous equations 'by hand'.

True or False? A polynomial equation of order five can have up to five solutions .

True. A polynomial equation of order five can have up to five solutions . In general, a polynomial equation of order n can have up to n (unique) solutions .

True or False? A polynomial equation of an odd degree will always have at least one solution .

True. A polynomial equation of an odd degree will always have at least one solution . (It is possible that a polynomial equation of an even degree will have no real-number solutions .)

List two ways that you can use a GDC to solve polynomial equations .

To use a GDC to solve polynomial equations you can: Use the equation solving feature(s) of your GDC to solve the equation directly. Use the graphing mode to visualise the equation, then use the analyse function to find the solutions (zeros or roots).

Define the term root in the context of polynomial equations .

In the context of polynomial equations , a root is a value of the variable that makes the polynomial equal to zero , also called a solution or zero of the equation.

True or False? A GDC will always show all solutions to a polynomial equation.

False. Some GDCs may only show the first solution to a polynomial equation, so it can important to graph the function to check how many solutions there should be.

What is the relationship between the degree of a polynomial and its graph ?

The degree of a polynomial determines the maximum number of times its graph can cross the x -axis . This corresponds to the maximum number of real roots .

Define the term zero in the context of polynomial equations .

In the context of polynomial equations , a zero is a value of the variable that makes the polynomial equal to zero , also called a root or solution .

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What is the advantage of using standard form for very large or small numbers?

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What is the advantage of using standard form for very large or small numbers?
Standard form lets us represent very large or very small numbers in a concise and manageable way using powers of 10. This means we can write them more neatly, compare them more easily, and carry out calculations more efficiently.
True or False?
In standard form, numbers are always written in the form $a \times 10^{k}$ , where $1 \leq a \leq 10$ and $k$ is an integer.
False.
In standard form, numbers are always written in the form $a \times 10^{k}$ , where $1 \leq a < 10$ (not $1 \leq a \leq 10$ ) and $k$ is an integer.
In standard form, the value of a must be greater than or equal to 1 and less than 10 .
True or False?
There is always one non – zero digit before the decimal point in a standard form number.
True.
There is always one (and only one) non – zero digit before the decimal point in a standard form number.
What does $a E n$ mean on a calculator display?
On a calculator display, $a E n$ means $a \times 10^{n}$ in standard form.
(Some calculators use that form of notation instead of the usual standard form notation.)
True or False?
Scientific notation is another term for standard form.
True.
Scientific notation is another term for standard form.
True or False?
The exponent k in standard form must always be positive.
False.
The exponent k in standard form can be positive, negative, or zero.
What are bounds, in the context of estimation?
Bounds are the smallest (lower bound) and largest (upper bound) numbers that a rounded number can lie between.
Define the term lower bound.
The lower bound is the smallest possible value a number could have been before rounding.
True or False?
A rounded number could have been equal to its upper bound before rounding.
False.
A rounded number can not be equal to its upper bound before rounding.
The upper bound of a rounded number is the smallest value that would have rounded up to the next highest rounded value.
True or False?
The basic rule for finding bounds is "half up, half down".
True.
The basic rule for finding bounds is "half up, half down".
E.g. if a number is rounded to 31 to the nearest integer, then
- halfway down to the next lowest integer is 30.5 (lower bound),
- halfway up to the next highest integer is 31.5 (upper bound).
What is the formula for percentage error?
The formula for percentage error is $ε = |\frac{v_{A} - v_{E}}{v_{E}}| \times 100 %$
Where:
- $ν_{A}$ is the approximate value
- $ν_{E}$ is the exact value
This formula is contained in the exam formula booklet.
True or False?
Percentage error should always be a positive number.
True.
Percentage error should always be a positive number.
What are exact values?
Exact values are forms that represent the full and precise value of a number, often involving fractions, roots (or surds), logarithms, or mathematical constants (e.g. $π$ ).
Define the term significant figures.
Significant figures are the digits in a number that carry meaning contributing to its precision.
Significant figures start with the first non-zero digit.
True or False?
When rounding to 3 significant figures, you always keep the first three non-zero digits.
False.
When rounding to 3 significant figures, you keep the first three significant digits, which may include zeroes after the first non-zero digit.
What is the purpose of estimation?
The purpose of estimation is to find approximate answers to difficult sums or to check if answers are about the right size (order of magnitude).
True or False?
When dealing with currency, you should always round to the nearest whole number.
False.
The appropriate rounding for currency depends on the specific currency being used. Some are rounded to whole numbers, others to 2 decimal places.
What is the rule for rounding up in problems involving minimum number of objects?
In problems involving minimum number of objects, always round up to ensure there are enough objects to meet the requirement.
E.g. if you calculate that 13.23 tins of paint are needed to complete a job, round up to 14 tins to make sure enough paint is bought.
Define the term absolute value.
Absolute value is the non-negative value of a number without regard to its sign, denoted by vertical bars | | around the number.
Absolute value is sometimes also referred to as the modulus of a number.
E.g. the absolute value of 3 is 3, i.e. $|3| = 3$ .
The absolute value of -7 is 7, i.e. $|- 7| = 7$ .
What is a linear equation?
A linear equation is an equation of the first order (degree 1), usually written in the form $a x + b y + c = 0$ where $a$ , $b$ , and $c$ are constants.
Define the term system of linear equations.
A system of linear equations is where two or more linear equations work together, with variables that satisfy all equations simultaneously.
True or False?
A system of linear equations may also be referred to as simultaneous equations.
True.
A system of linear equations may also be referred to as simultaneous equations.
True or False?
You can use a GDC to solve a system of linear equations.
True.
You can use a GDC to solve a system of linear equations.
Your GDC will have a simultaneous equations solving feature. However it is also useful to be able to solve simple simultaneous equations 'by hand'.
What is a polynomial?
A polynomial is an algebraic expression consisting of a finite number of terms, with non-negative integer indices only.
E.g. $3 x^{2} - 5 x + 17$ is a polynomial.
What is a polynomial equation?
A polynomial equation is an equation where a polynomial is set equal to zero.
E.g. $3 x^{2} - 5 x + 17 = 0$ is a polynomial equation.
True or False?
A polynomial equation of order five can have up to five solutions.
True.
A polynomial equation of order five can have up to five solutions.
In general, a polynomial equation of order n can have up to n (unique) solutions.
True or False?
A polynomial equation of an odd degree will always have at least one solution.
True.
A polynomial equation of an odd degree will always have at least one solution.
(It is possible that a polynomial equation of an even degree will have no real-number solutions.)
List two ways that you can use a GDC to solve polynomial equations.
To use a GDC to solve polynomial equations you can:
1. Use the equation solving feature(s) of your GDC to solve the equation directly.
2. Use the graphing mode to visualise the equation, then use the analyse function to find the solutions (zeros or roots).
Define the term root in the context of polynomial equations.
In the context of polynomial equations, a root is a value of the variable that makes the polynomial equal to zero, also called a solution or zero of the equation.
True or False?
A GDC will always show all solutions to a polynomial equation.
False.
Some GDCs may only show the first solution to a polynomial equation, so it can important to graph the function to check how many solutions there should be.
What is the relationship between the degree of a polynomial and its graph?
The degree of a polynomial determines the maximum number of times its graph can cross the x-axis.
This corresponds to the maximum number of real roots.
Define the term zero in the context of polynomial equations.
In the context of polynomial equations, a zero is a value of the variable that makes the polynomial equal to zero, also called a root or solution.

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1. Number & Algebra

Number Toolkit

Exponentials & Logs

Sequences & Series

Financial Applications

Complex Numbers

Further Complex Numbers

Matrices

Eigenvalues & Eigenvectors

2. Functions

Linear Functions & Graphs

Further Functions & Graphs

Modelling with Functions

Functions Toolkit

Transformations of Graphs

Modelling with Logarithmic, Logistic & Piecewise Functions

3. Geometry & Trigonometry

Geometry Toolkit

Geometry of 3D Shapes

Trigonometry

Trigonometric Identities & Equations

Voronoi Diagrams

Matrix Transformations

Vector Properties

Vector Equations of Lines

Modelling with Vectors

Graph Theory

4. Statistics & Probability

Statistics Toolkit

Correlation & Regression

Non-linear Regression

Probability

Probability Distributions

Random Variables

Binomial Distribution

Normal Distribution

Combinations of Normal Distributions & Sample Mean Distributions

Poisson Distribution

Hypothesis Testing using the Chi-squared Distribution

Hypothesis Testing for Population Parameters

Transition Matrices & Markov Chains

5. Calculus

Differentiation

Further Differentiation

Integration

Further Integration

Kinematics

Differential Equations

Coupled & Second Order Differential Equations