Mathematics
N2
Module 1: Exponents and logarithms
EXPONENTS (INDICES)
An exponent, also referred to as an index, is more or less a shorthand
notation for multiplying the same number by itself several times, for example:
2 × 2 × 2 = 23 = 8
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Module 1: Exponents and logarithms (continued)
LOGARITHMS
Logarithms are the ‘opposite’ of exponentials. The logarithm of a number is
the exponent by which another fixed value, the base, must be raised to
produce that number. Therefore a logarithm is an unknown exponent that will
be determined.
For example:
The logarithm of 100 to base 10 is 2, written as 𝑙𝑙𝑙𝑙𝑙𝑙10 100 = 2
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Module 2: Factorisation, HCF, LCM and algebraic
fractions
FACTORISATION
Factorisation is he process of writing a mathematical statement as a product
of its factors.
For example:
𝑎𝑎𝑎𝑎 + 𝑏𝑏𝑏𝑏 + 𝑐𝑐𝑐𝑐 = 𝑥𝑥(𝑎𝑎 + 𝑏𝑏 + 𝑐𝑐)
• 𝑥𝑥 and (𝑎𝑎 + 𝑏𝑏 + 𝑐𝑐) are factors of the three terms 𝑎𝑎𝑎𝑎 + 𝑏𝑏𝑏𝑏 + 𝑐𝑐𝑐𝑐
• By expanding 𝑥𝑥(𝑎𝑎 + 𝑏𝑏 + 𝑐𝑐) you get the following:
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𝑥𝑥(𝑎𝑎 + 𝑏𝑏 + 𝑐𝑐) = 𝑎𝑎𝑎𝑎 + 𝑏𝑏𝑏𝑏 + 𝑐𝑐𝑐𝑐
Module 2: Highest common factor and lowest common multiple
(continued)
HIGHEST COMMON FACTOR AND LOWEST COMMON MULTIPLE
The highest common factor (HCF) of two or more numbers is the common
factor of all those numbers with the greatest value.
The lowest common multiple (LCM) of two or more numbers is the smallest
possible number into which all the numbers can be divided exactly.
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Module 2: Highest common factor and lowest common multiple
(continued)
ALGEBRAIC FRACTIONS
An algebraic fraction is a fraction where the numerator and denominator are
algebraic expressions.
For example:
3𝑥𝑥𝑥𝑥
= 3𝑦𝑦
𝑥𝑥
In this example, the 𝑥𝑥 in the numerator cancels with the 𝑥𝑥 in the denominator,
which leaves the equation with 3𝑦𝑦.
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Module 3: Equations, word problems and manipulation
of technical formulae
LINEAR EQUATIONS
A linear equation is an algebraic equation in which each term is either a
constant or the product of a constant and (the first power of) a single
variable. The power of the highest variable is 1. Linear equations could have
one or more variables.
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Module 3: Equations, word problems and manipulation of technical
formulae (continued)
QUADRATIC EQUATIONS
A quadratic equation is any equation in the form 𝑎𝑎𝑎𝑎 2 + 𝑏𝑏𝑏𝑏 + 𝑐𝑐 = 0, where
𝑥𝑥 represents an unknown, and 𝑎𝑎, 𝑏𝑏, and 𝑐𝑐 are constants which are not equal
to 0.
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Module 3: Equations, word problems and manipulation of technical
formulae (continued)
SIMULTANEOUS LINEAR EQUATIONS
Sometimes there are two unknowns in one equation. For example:
In 2𝑥𝑥 + 3𝑦𝑦 = 8. If you substitute values of (𝑥𝑥 = 1) and (𝑦𝑦 = 2), they would
4
3
satisfy the equation. However, (𝑥𝑥 = 2) and (𝑦𝑦 = ), would also satisfy the
equation, as would (𝑥𝑥 =– 1) and (𝑦𝑦 = 3). There are an infinite number of
solutions. To avoid this, a second equation containing the same variables is
required, so that there is only one solution for 𝑥𝑥 and one solution for 𝑦𝑦.
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Module 3: Equations, word problems and manipulation of technical
formulae (continued)
WORD PROBLEMS
When an answer is needed to a mathematical problem that is described in
words as well as numbers, it is called a word problem. You find the solution
by setting up an equation that represents the problem, and then solve it.
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Module 3: Equations, word problems and manipulation of technical
formulae (continued)
MANIPULATION OF TECHNICAL FORMULAE
Manipulation of technical formula can also be described as changing the
subject of the formula.
An equation can be manipulated in such a way that the required variable is
isolated on the left-hand-side, and everything else is written on the righthand-side. This manipulation is described as making the wanted variable the
subject of the formula.
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Module 4: Algebraic graphs
LINEAR GRAPHS (STRAIGHT LINE GRAPHS)
A linear equation can be represented graphically by
a straight line, for example: 𝑦𝑦 = 3𝑥𝑥 + 2, or
𝑦𝑦 = 𝑚𝑚𝑚𝑚 + 𝑐𝑐.
A linear graph is a graph in which all the points
representing the relationship between 2 quantities
lie on a straight line.
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Module 4: Algebraic graphs (continued)
PARABOLAS
A parabola can be described by a
quadratic equation of the following
form:
𝑦𝑦 = 𝑎𝑎𝑥𝑥 2 + 𝑏𝑏𝑏𝑏 + 𝑐𝑐
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Module 4: Algebraic graphs (continued)
GRAPHICAL SOLUTIONS OF SIMULTANEOUS EQUATIONS
Simultaneous equations can be solved exactly using an
algebraic method or approximately using the graphical method.
The first step is to draw the graphs and then read the
coordinates of the points where the two graphs cut each other.
These points are the solutions and are called the points of
intersection of the graphs.
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Module 5: Measuring of angles, angular and peripheral
velocity and sectors of circles
MEASURING OF ANGLES
There are several ways to measure the size of an angle. One way is to use
the unit degrees.
A 360° angle is a full rotation or a full circle. This is called one revolution.
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Module 5: Measuring of angles, angular and peripheral velocity and
sectors of circles (continued)
RADIANS
Radians are used as an alternative way of measuring angles, or the amount
of turn. A radian is defined as the angle made at the centre of a circle
between two radii when the arc length on the circumference between the two
radii (radiuses) is equal to the length of one radius.
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Module 5: Measuring of angles, angular and peripheral velocity and
sectors of circles (continued)
ANGULAR AND PERIPHERAL VELOCITY
Angular velocity, also called rotational velocity, is a quantitative expression of
the amount of rotation that a spinning object undergoes per unit time.
The angular velocity is the speed at the centre of a moving wheel or disc,
with which the revolutions are completed and is measured in radians per
time unit. The Angular Velocity Formula is as follows:
𝜔𝜔 =
𝜃𝜃
𝑡𝑡
=
𝑡𝑡ℎ𝑒𝑒 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑜𝑜𝑜𝑜 𝑡𝑡𝑡𝑡𝑡 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 𝑖𝑖𝑖𝑖 𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟 𝑖𝑖𝑖𝑖 ℎ𝑎𝑎𝑎𝑎 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚
𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑖𝑖𝑖𝑖 ℎ𝑎𝑎𝑎𝑎 𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑖𝑖𝑖𝑖 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠
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Module 5: Measuring of angles, angular and peripheral velocity and
sectors of circles (continued)
CIRCLE SECTORS
A circular sector or circle sector is the portion
of a disk enclosed by two radii and an arc,
where the smaller area is the minor sector and
the larger area is the major sector. In the
diagram, 𝜃𝜃 is the central angle in radians, 𝑟𝑟 the
radius of the circle, and 𝐿𝐿 is the arc length of
the minor sector.
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Module 5: Measuring of angles, angular and peripheral velocity and
sectors of circles (continued)
CHORDS IN A CIRCLE
A chord is a straight line drawn from one side of the circumference of a circle
to the other side, dividing the circle into two parts.
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Module 6: Trigonometry
TRIGONOMETRIC RATIOS
For any angle 𝜃𝜃:
sin 𝜃𝜃 =
cos 𝜃𝜃 =
tan 𝜃𝜃 =
𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜
ℎ𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦
𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎
ℎ𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦
𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜
𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎
or
or
or
𝑂𝑂
𝐴𝐴
𝑂𝑂
𝐻𝐻
𝐴𝐴
𝐻𝐻
Remember: SOH CAH TOA
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Module 7: Mensuration
MENSURATION
When calculating equations which have different units of measurement, we
need to remember:
• Perimeter: The total distance around a shape;
• Circumference: The total distance around a circle;
• Area: The total amount of surface space occupied by a shape;
• Surface area: The total outside area of a shape; and
• Volume: The total amount of space a shape occupies.
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