N2 Mathematics

Mathematics Downloaded from hlayiso.com 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 Downloaded from hlayiso.com www.futuremanagers.com

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 Downloaded from hlayiso.com www.futuremanagers.com

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 + 𝑏𝑏 + 𝑐𝑐 = (𝑎𝑎 + 𝑏𝑏 + 𝑐𝑐) 𝑏𝑏 𝑐𝑐 𝑥𝑥 • 𝑥𝑥 and (𝑎𝑎 + 𝑏𝑏 + 𝑐𝑐) are factors of the three terms 𝑎𝑎𝑎𝑎 + 𝑏𝑏 + 𝑐𝑐 𝑏𝑏 𝑐𝑐 • By expanding (𝑎𝑎 + 𝑏𝑏 + 𝑐𝑐) you get the following: 𝑥𝑥 Downloaded (𝑎𝑎 + 𝑏𝑏 + 𝑐𝑐)from 𝑥𝑥 hlayiso.com = 𝑎𝑎𝑎𝑎 + 𝑏𝑏 + 𝑐𝑐 𝑏𝑏 𝑐𝑐 www.futuremanagers.com

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. Downloaded from hlayiso.com www.futuremanagers.com

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: 3 𝑥𝑥 =3 𝑥𝑥 𝑦𝑦 𝑥𝑥 In this example, the 𝑥𝑥 in the numerator cancels with the 𝑥𝑥 in the denominator, which leaves the equation with 3 . Downloaded from hlayiso.com 𝑦𝑦 www.futuremanagers.com

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. Downloaded from hlayiso.com www.futuremanagers.com

Module 3: Equations, word problems and manipulation of technical formulae (continued) QUADRATIC EQUATIONS A quadratic equation is any equation in the form 𝑎𝑎𝑎𝑎 2 + 𝑏𝑏 + 𝑐𝑐 = 0, where 𝑏𝑏 𝑥𝑥 represents an unknown, and 𝑎𝑎, 𝑏𝑏, and 𝑐𝑐 are constants which are not equal to 0. Downloaded from hlayiso.com www.futuremanagers.com

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 2 + 3 = 8. If you substitute values of ( = 1) and ( = 2), they would 𝑥𝑥 𝑦𝑦 𝑥𝑥 𝑦𝑦 4 satisfy the equation. However, ( = 2) and ( = ), would also satisfy the 3 𝑥𝑥 𝑦𝑦 equation, as would ( =– 1) and ( = 3). There are an infinite number of 𝑥𝑥 𝑦𝑦 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 Downloaded from hlayiso.com www.futuremanagers.com

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. Downloaded from hlayiso.com www.futuremanagers.com

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 right- hand-side. This manipulation is described as making the wanted variable the subject of the formula. Downloaded from hlayiso.com www.futuremanagers.com

Module 4: Algebraic graphs LINEAR GRAPHS (STRAIGHT LINE GRAPHS) A linear equation can be represented graphically by a straight line, for example: 𝑦𝑦 = 3 + 2, or 𝑥𝑥 𝑦𝑦 = 𝑚𝑚 + 𝑐𝑐. 𝑚𝑚 A linear graph is a graph in which all the points representing the relationship between 2 quantities lie on a straight line. Downloaded from hlayiso.com www.futuremanagers.com

Module 4: Algebraic graphs (continued) PARABOLAS A parabola can be described by a quadratic equation of the following form: 𝑦𝑦 = 𝑎𝑎 2 + 𝑏𝑏 + 𝑐𝑐 𝑥𝑥 𝑏𝑏 Downloaded from hlayiso.com www.futuremanagers.com

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. Downloaded from hlayiso.com www.futuremanagers.com

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. Downloaded from hlayiso.com www.futuremanagers.com

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. Downloaded from hlayiso.com www.futuremanagers.com

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: 𝜃𝜃 𝑡𝑡ℎ𝑒𝑒 𝑠𝑠𝑠𝑠𝑠𝑠 𝑜𝑜𝑜𝑜 𝑡𝑡 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 𝑟𝑟 𝑟𝑟 ℎ𝑎𝑎 𝑚𝑚 𝜔𝜔 = = 𝑚𝑚𝑚𝑚𝑚𝑚 𝑖𝑖𝑖𝑖 𝑟𝑟 𝑎𝑎 𝑠𝑠 𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟 𝑖𝑖𝑖𝑖 𝑎𝑎 𝑡𝑡𝑡 𝑡𝑡 𝑡𝑡 ℎ𝑎𝑎 𝑡𝑡 𝑡𝑡 𝑠𝑠 𝑠𝑠 𝑠𝑠 Downloaded from hlayiso.com 𝑠𝑠𝑠𝑠𝑠𝑠 𝑠𝑠 𝑖𝑖𝑖𝑖 𝑡𝑡𝑡𝑡 𝑡𝑡 𝑎𝑎 𝑖𝑖𝑖𝑖 𝑡𝑡𝑡𝑡𝑡𝑡 www.futuremanagers.com

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.Downloaded from hlayiso.com www.futuremanagers.com

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. Downloaded from hlayiso.com www.futuremanagers.com

Module 6: Trigonometry TRIGONOMETRIC RATIOS For any angle 𝜃𝜃: 𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜 𝑂𝑂 sin 𝜃𝜃 = or 𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜 ℎ𝑦𝑦 𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦 𝑦𝑦 𝑦𝑦𝑦𝑦 𝐻𝐻 𝑎𝑎 𝑎𝑎𝑎𝑎 𝑎𝑎 𝐴𝐴 cos 𝜃𝜃 = or 𝑎𝑎 𝑎𝑎 𝑎𝑎𝑎𝑎 ℎ𝑦𝑦 𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦 𝑦𝑦 𝑦𝑦𝑦𝑦 𝐻𝐻 𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜 𝑂𝑂 tan 𝜃𝜃 = or 𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜 𝑎𝑎 𝑎𝑎𝑎𝑎 𝑎𝑎 𝑎𝑎 𝑎𝑎𝑎𝑎 𝑎𝑎 𝐴𝐴 Remember: SOH CAH TOA Downloaded from hlayiso.com www.futuremanagers.com

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 Downloaded a shape from occupies. hlayiso.com www.futuremanagers.com