N3 Mathematics

Mathematics Downloaded from hlayiso.com N3

Module 1: Factors and fractions COMMON FACTOR AND GROUPING Common factors are factors shared by two or more numbers. The common factor can be a variable, a number, or a combination of numbers and variables. An expression with more than three terms can be factorised by grouping the terms to find a common factor. For example, the expression 2𝑎𝑎𝑎𝑎– 9𝑏𝑏 – 2𝑎𝑎𝑎𝑎 + 9𝑏𝑏 can be grouped to equal 𝑏𝑏 𝑏𝑏 Downloaded ( – from hlayiso.com )(2𝑎𝑎– 9𝑏𝑏) 𝑥𝑥 𝑦𝑦 www.futuremanagers.com

Module 1: Factors and fractions (continued) THE QUADRATIC TRINOMIAL A quadratic trinomial is an expression with three terms containing a variable of degree 2, for example: 𝑎𝑎 2 + 𝑏𝑏 + 𝑐𝑐 𝑥𝑥 𝑏𝑏 Downloaded from hlayiso.com www.futuremanagers.com

Module 1: Factors and fractions (continued) THE DIERENCE BETWEEN TWO SQUARES 2 The difference between two squares, such as 𝑥𝑥 – 𝑏𝑏 2 , can be identified as follows: • The expression consists of two terms only; • Both terms must be perfect squares; and • There must be a minus between the two terms. Downloaded from hlayiso.com www.futuremanagers.com

Module 1: Factors and fractions (continued) CUBIC FUNCTIONS 𝑓𝑓( ) = 𝑎𝑎 3 + 𝑏𝑏 2 + 𝑐𝑐 + 𝑑𝑑 is a function of the third degree, where 𝑎𝑎 ≠ 0. This 𝑥𝑥 𝑥𝑥 𝑥𝑥 𝑐𝑐 is also called a cubic function. There must be a term with 𝑥𝑥 3 for the polynomial to be cubic (𝑎𝑎 ≠ 0), but 𝑏𝑏, 𝑐𝑐 and 𝑑𝑑 can be equal to 0 (zero). Downloaded from hlayiso.com www.futuremanagers.com

Module 1: Factors and fractions (continued) ALGEBRAIC FRACTIONS Downloaded from hlayiso.com www.futuremanagers.com

Module 2: Exponents, surds and logarithms EXPONENTS An exponent is a mathematical way of representing a very large number or a very small number in a user-friendly format. It is basically a shorthand notation for multiplying the same number (the base) by itself several times. Example: This means thatDownloaded is raised to the power of 3. hlayiso.com from www.futuremanagers.com

Module 2: Exponents, surds and logarithms (continued) SURDS A surd is a square root of a whole number that produces an irrational number and cannot be written as decimal fractions. The square root of any prime number is a surd. For example: 3, 13 Downloaded from hlayiso.com www.futuremanagers.com

Module 2: Exponents, surds 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 3: Equations, word problems and manipulation of technical formulae EQUATIONS It is important to know that an equation has an equal sign. For example: 3 + 2(4– ) = 16 𝑥𝑥 𝑥𝑥 In order to solve an equation, is must be kept balanced. What is done to one side of the equation, must be done exactly the same to the other side. 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 (CHANGING THE SUBJECT OF A FORMULA) 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: Geometry of coordinates STRAIGHT LINES A linear equation can be represented graphically by a straight line, for example: 𝑦𝑦 = 3 + 2, or 𝑥𝑥 𝑦𝑦 = 𝑚𝑚 + 𝑐𝑐 𝑚𝑚 In a straight line equation: 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 𝑦𝑦 𝑚𝑚 = gradient (slope) = 𝑖𝑖𝑖𝑖 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 𝑖𝑖𝑖𝑖 𝑥𝑥 Downloaded from hlayiso.com www.futuremanagers.com

Module 5: Algebraic graphs SKETCH GRAPHS A graph is a diagram that shows either the relationship between some variable quantities, or the connections that exist between a set of points. These points are written as coordinates in relation to the 𝑥𝑥 − and 𝑦𝑦 −axis of the graph. In order to solve for the points, one can use the 𝑥𝑥 − and 𝑦𝑦 −intercepts: Downloaded from hlayiso.com www.futuremanagers.com

Module 5: Algebraic graphs (continued) GRAPHIC SOLUTIONS When solving two equations simultaneously, look for the ordered number pairs, ( ; ), that will satisfy both equations. ( ; ) is called the coordinates 𝑥𝑥 𝑦𝑦 𝑥𝑥 𝑦𝑦 of the point(s) of intersection of the graphs. Simultaneous equations can be solved totally accurate by using the algebraic method. Downloaded from hlayiso.com www.futuremanagers.com

Module 6: Differential calculus AVERAGE GRADIENT AND AVERAGE SPEED Gradient (slope) of a straight line: 𝑦𝑦 = 𝑚𝑚 𝑚𝑚 + 𝑐𝑐 In order to calculate the gradient (𝑚𝑚) of a straight line that passes through two points ( 𝑥; 𝑥𝑥 𝑦𝑦𝑦 ) and 𝑥𝑥𝑥 ; 𝑦𝑦𝑦 , calculate the change in the 𝑥𝑥 −value between the two points and the change in the 𝑦𝑦 −value between the same two points and then substitute the values into the formula for the gradient. 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 Speed = 𝑡𝑡 Downloaded from hlayiso.com 𝑡𝑡 𝑡𝑡𝑡𝑡 www.futuremanagers.com

Module 6: Differential calculus (continued) LIMITS When working with limits, you investigate what happens to the 𝑦𝑦 −value of a curve as you get closer to a specific 𝑥𝑥 −value on a curve. For example: if you start walking towards a wall, the wall will be your limit. Downloaded from hlayiso.com www.futuremanagers.com

Module 6: Differential calculus (continued) DIFFERENTIATION Differential calculus is the part of calculus concerning the rate at which a function is changing. The rules for differentiation are: Downloaded from hlayiso.com www.futuremanagers.com

Module 7: Trigonometry EQUATIONS AND EXPRESSIONS For any angle 𝜃𝜃: 𝐻𝐻 sec𝜃𝜃 = 𝐴𝐴 𝑂𝑂 sin 𝜃𝜃 = 𝐴𝐴 𝐻𝐻 cot 𝜃𝜃 = 𝑂𝑂 𝐴𝐴 cos 𝜃𝜃 = 1 𝐻𝐻 sin 𝜃𝜃 = csc 𝜃𝜃 𝑂𝑂 tan = 1 𝐴𝐴 co𝑠𝑠 𝜃𝜃 = 𝑠𝑠𝑠𝑠𝑠𝑠 𝜃𝜃 𝐻𝐻 csc𝜃𝜃 = 1 𝑂𝑂 𝑡𝑡 𝑡𝑡 𝜃𝜃 = Downloaded𝑡𝑡 from hlayiso.com 𝑐𝑐 𝑐𝑐 𝜃𝜃 𝑐𝑐 www.futuremanagers.com

Module 7: Trigonometry (continued) GRAPHS In order to draw a trigonometric graph: • Draw a graph by point- by-point plotting; and • Draw the graph of • 𝑦𝑦 = 𝑠𝑠in and know its 𝑥𝑥 characteristics. Downloaded from hlayiso.com www.futuremanagers.com