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2 2 This book is under copyright to A-level Maths Tutor. However, it may be distributed freely provided it is not sold for profit. Contents indices 3 laws of logarithms 7 surds 12 inequalities 18 quadratic equations 22 partial fractions 28 polynomials 33 The Binomial Theorem 37 iteration 41 Sets theory 46 functions 50
3 3 Indices The Laws of Indices Indices - Multiplication remembering that: Examples
4 4 Indices - Division remembering that: Examples:
5 5 Indices - Powers remembering that: Examples: Indices - Roots and Reciprocals remembering that: and
6 6 Examples:
7 7 The Laws of Logarithms The Laws of Logarithms
8 8 Proofs
9 9 Changing the base Remember that the change of base occurs in the term where the base is 'x' or some other variable. Example
10 10 Simultaneous equations 'Substitution' simultaneous equations are common problems. First find what x is in terms of y. Then substitute for x in the other equation. Solve for y. Example
11 11 Variable in the index Take logs on both sides. Move the indices infront of the logs. Expand the equation. Collect x-terms to the left. Sum the numbers to the right. These problems can be tricky with the amount of arithmetic involved. So make sure you write everything down to make checking your working easier. Example
12 12 Surds Rules Surds are mathematical expressions containing square roots. However, it must be emphasized that the square roots are 'irrational' i.e. they do not result in a whole number, a terminating decimal or a recurring decimal. The rules governing surds are taken from the Laws of Indices. rule #1 examples rule #2
13 13 examples Some Useful Expressions expression #1
14 14 expression #2 - (the difference of two squares) Rationalising Surds - This is a way of modifying surd expressions so that the square root is in the numerator of a fraction and not in the denominator. The method is to multiply the top and bottom of the fraction by the square root.
15 15 Rationalising expressions using 'difference of two squares' Remembering that :......from 'useful expressions' above. Example #1 - simplify multiplying top and bottom by
16 16 Example #2 rationalise multiply top and bottom by Reduction of Surds - This is a way of making the square root smaller by examining its squared factors and removing them.
17 17 Rational and Irrational Numbers - In the test for rational and irrational numbers, if a surd has a square root in the numerator, while the denominator is '1' or some other number, then the number represented by the expression is 'irrational'. examples of irrational surds:
18 18 Inequalities Symbols The rules of inequalities (sometimes called 'inequations') These are the same as for equations i.e that whatever you do to one side of the equation(add/subtract, multiply/divide by quantities) you must do to the other. However, their are two exceptions to these rules. When you multiply each side by a negative quantity '<' becomes '>', or '>' becomes '<'. That is, the inequality sign is reversed. Similarly, when you divide each side by a negative quantity < becomes >, or > becomes<. That is, the inequality sign is reversed.
19 19 Examples Inequalities with one variable Example #1 - Find all the integral values of x where, The values of x lie equal to and less than 6 but greater than -5, but not equal to it. The integral(whole numbers + or - or zero) values of x are therefore: 6, 5, 4, 3, 2, 1, 0, -1, -2, -3, -4 Example #2 - What is the range of values of x where, Since the square root of 144 is +12 or -12(remember two negatives multiplied make a positive), x can have values between 12 and -12. In other words the value of x is less than or equal to 12 and more than or equal to -12. This is written:
20 20 Inequalities with two variables - Solution is by arranging the equation into the form Ax + By = C Then, above the line of the equation, Ax + By < C and below the line, Ax + By > C Consider the graph of -2x + y = -2 note - the first term A must be made positive by multiplying the whole equation by -1 The equation -2x + y = -2 becomes 2x - y =2 look at the points(red) and the value of 2x - y for each. The table below summarises the result. point(x,y) 2x - y value more than 2? above/below (1,1) 2(1)1(1) 1 no - less above (1,4) 2(1)-(4) -2 no - less above (2,3) 2(2)-(3) 1 no - less above (3,3) 2(3)-(3) 3 yes - more below (2,1) 2(2)-(1) 3 yes - more below (4,2) 2(4)-(2) 6 yes - more below
21 21 The Modulus The modulus is the numerical value of a number, irrespective of the sign it carries. hence l x l < 3 means -3 < x < 3 Example
22 22 Quadratic Equations Introduction The general form of a quadratic equation is: ax 2 +bx + c where a, b & c are constants The expression b 2-4ac is called the discriminant and given the letter (delta). All quadratic equations have two roots/solutions. These roots are either REAL, EQUAL or COMPLEX *. * complex - involving the square root of 1
23 23 Solution by factorising - This is best understood with an example. solve: You must first ask yourself which two factors when multiplied will give 12? The factor pairs of 12 are : 1 x 12, 2 x 6 and 3 x 4 You must decide which of these factor pairs added or subtracted will give 7? 1 : 12...gives 13, 11 2 : 6...gives 8, 4 3 : 4...gives 7, 1 Which combination when multiplied makes +12 and which when added gives -7? these are the choices: (+3)(+4), (-3)(+4), (+3)(-4) (-3)(-4) Clearly, (-3)(-4) are the two factors we want. therefore Now to solve the equation. factorising, as above either
24 24 or for the equation to be true. So the roots of the equation are: Completing the square This can be fraught with difficulty, especially if you only half understand what you are doing. The method is to move the last term of the quadratic over to the right hand side of the equation and to add a number to both sides so that the left hand side can be factorised as the square of two terms. e.g. However, there is a much neater way of solving this type of problem, and that is by remembering to put the equation in the following form:
25 25 using the previous example, Using the Formula - the two solutions of quadratic equations in the form are given by the formula:
26 26 Proof The proof of the formula is by using 'completing the square'.
27 27 Example find the two values of x that satisfy the following quadratic equation:
28 28 Partial Fractions some definitions: Proper Fraction When the degree(index) of the function is higher in the denominator than the numerator. Improper Fraction When the degree(index) of the function is higher in the numerator than the denominator. Partial Fractions Factorising the denominator of a proper fraction means that the fraction can be expressed as the sum(or difference) of other proper fractions. Simple addition/subtraction of algebraic fractions As with simple fraction arithmetic, a common denominator is found from the denominators of either fraction and the numerators altered to be fractions of the new denominator. Equations & Identities Equations are satisfied by discrete values of the variable involved. Example: Identities are satisfied by any value of the variable used. Note the equals sign '=' is modified to reflect this.
29 29 Example: When we make partial fractions(below) we are creating an identity from the original expression. Denominator with only 'linear factors' By 'linear' we mean that x has a power no higher than '1'. In other words, this method does not work with x 2, x 3, x 4 etc. For each linear factor of the type: there is a partial fraction: Example: where x is a variable and A,B,a,b,c,d are constants, where 'a' is not equal to 'b'.
30 30 Example Denominator with 'repeated' linear factors For each 'repeated' linear factor of the type: there is a partial fraction: Example:
31 31 Example Denominator with a quadratic factor For each quadratic factor of the type: there is a partial fraction:
32 32 Example: Example
33 33 Polynomials Introduction A polynomial is an expression which: consists of a sum of a finite number of terms has terms of the form kx n (x a variable, k a constant, n a positive integer) Every polynomial in one variable (eg 'x') is equivalent to a polynomial with the form: Polynomials are often described by their degree of order. This is the highest index of the variable in the expression. eg: containing x 5 order 5, containing x 7 order 7 etc. These are NOT polynomials: 3x 2 +x 1/2 +x second term has an index which is not an integer(whole number) 5x -2 +2x -3 +x -5 indices of the variable contain integers which are not positive examples of polynomials: 5 2 x +5x +2x+3 7 (x +4x 2 )(3x-2) x+2x 2 3-5x +x 4-2x 5 +7x 6
34 34 Algebraic long division If f(x) the numerator and d(x) the denominator are polynomials and the degree of d(x) <= the degree of f(x) and d(x) does not =0 then two unique polynomials q(x) the quotient and r(x) the remainder exist, so that: Note - the degree of r(x) < the degree of d(x). We say that d(x) divides evenly into f(x) when r(x)=0. Example
35 35 The Remainder Theorem If a polynomial f(x) is divided by (x-a), the remainder is f(a). Example Find the remainder when (2x 3 +3x+x) is divided by (x+4). The reader may wish to verify this answer by using algebraic division.
36 36 The Factor Theorem ( a special case of the Remainder Theorem) (x a) is a factor of the polynomial f(x) if f(a) = 0 Example
37 37 The Binomial Theorem Introduction This section of work is to do with the expansion of (a+b) n and (1+x) n. Pascal's Triangle and the Binomial Theorem gives us a way of expressing the expansion as a sum of ordered terms. Pascal's Triangle This is a method of predicting the coefficients of the binomial series. Coefficients are the constants(1,2,3,4,5,6 etc.) that multiply each variable, or group of variables. Consider (a+b) n variables a, b. The first line represents the coefficients for n=0. (a+b) 0 = 1 The second line represents the coefficients for n=1. (a+b) 1 = a + b The third line represents the coefficients for n=2. (a+b) 2 = a 2 + 2ab + b 2 The sixth line represents the coefficients for n=5. (a+b) 5 = a 5 + 5a 4 b + 10a 3 b a 2 b 3 + 5ab 4 + b 5 The Binomial Theorem builds on Pascal's Triangle in practical terms, since writing out triangles of numbers has its limits.
38 38 The General Binomial Expansion ( n 1 ) This is a way of finding all the terms of the series, the coefficients and the powers of the variables. The coefficients, represented by n C r, are calculated using probability theory. For a deeper understanding you may wish to look at where n C r comes from; but for now you must accept that: where 'n' is the power/index of the original expression and 'r' is the number order of the term minus one If n is a positive integer, then: Example #1
39 39 Example #2 It is suggested that the reader try making similar questions, working through the calculations and checking the answer here (max. value of n=8)
40 40 The Particular Binomial Expansion This is for (1+x) n, where n can take any value positive or negative, and x is a fraction ( - 1<x<1 ). Example Find the first 4 terms of the expression (x+3) 1/2.
41 41 Iteration Introduction Repeatedly solving an equation to obtain a result using the result from the previous calculation, is called 'iteration'. The procedure is used in mathematics to give a more accurate answer when the original data is only approximate. Problems usually involve finding the root of an equation when only an approximate value is given for where the curve crosses an axis. Direct/Fixed Point Iteration method: 1. rearrange the given equation to make the highest power of x the subject 2. find the power root of each side, leaving x on its own on the left 3. the LHS x becomes x n+1 4. the RHS x becomes x n The equation is now in its iterative form. We start by working out x 2 from the given value x 1. x 3 is worked out using the value x 2 in the equation. x 4 is worked out using the value x 3 and so on.
42 42 Example Find correct to 3 d.p. a root of the equation f(x) = x 3-2x + 3 given that there is a solution near x = -2
43 43 Iteration by Bisection method: 1. reduce the interval where the root lies into two equal parts 2. decide in which part the solution resides 3. repeat the process until a consistent answer is achieved for the degree of accuracy required Example Find correct to 3 d.p. a root of the equation f(x) = 2x 2-2x + 7 given that there is a solution near x = -2
44 44 Newton-Raphson Method This uses a tangent to a curve near one of its roots and the fact that where the tangent meets the x-axis gives an approximation to the root. The iterative formula used is:
45 45 Example Find correct to 3 d.p. a root of the equation f(x) = 2x 2 + x - 6 given that there is a solution near x = 1.4
46 46 Set Theory Introduction A set is a collection of objects, numbers or characters. {abcdef...wxyz} {1,2,3,4,...45, 46, 47} etc. Note how the set is enclosed in brackets {...} A definite set is one in which all its members are known. Sets are given uppercase letters: A, B, C, etc. The elements of sets are given lowercase letters: a, b, c,..etc. An element x belonging to the set A is written: A constraint bar {......} is for setting a property that all members satisfy. A{x l x has the colour blue} - all elements of A are blue Common Sets
47 47 Venn Diagrams Venn diagrams are used to visualise sets and their relations to one another. Above is a diagramatic representation of set A. The set can be represented mathematically as: A{1,3,5,7,9}. Note that set A(the circle) is a subset of the Universal set(the rectangle). A' (A-dash)is called the complement of A. It contains all elements which are not members of A. A and A' together make up the Universal set.
48 48 The union of sets A and B contains all of the elements from both sets. The intersection of sets A and B contains a particular group of elements that exist in set A and in set B.
49 49 Subsets If B is a subset of A. Then all of the elements of B are also in A.
50 50 Functions Introduction To thoroughly understand the terms and symbols used in this section it is advised that you visit 'sets of numbers' first. Mapping(or function) This a 'notation' for expressing a relation between two variables(say x and y). Individual values of these variables are called elements eg x 1 x 2 x 3... y 1 y 2 y 3... The first set of elements ( x) is called the domain. The second set of elements ( y) is called the range. A simple relation like y = x 2 can be more accurately expressed using the following format: The last part relates to the fact that x and y are elements of the set of real numbers R(any positive or negative number, whole or otherwise, including zero) One-One mapping Here one element of the domain is associated with one and only one element of the range. A property of one-one functions is that a on a graph a horizontal line will only cut the graph once.
51 51 Example R + the set of positive real numbers Many-One mapping Here more than one element of the domain can be associated with one particular element of the range. Example Z is the set of integers(positive & negative whole numbers not including zero)
52 52 Complete function notation is a variation on what has been used so far. It will be used from now on. Inverse Function f -1 The inverse function is obtained by interchanging x and y in the function equation and then rearranging to make y the subject. If f -1 exists then, ff -1 (x) = f -1 f(x) = x It is also a condition that the two functions be 'one to one'. That is that the domain of f is identical to the range of its inverse function f -1. When graphed, the function and its inverse are reflections either side of the line y = x. Example Find the inverse of the function(below) and graph the function and its inverse on the same axes.
53 53 Composite Functions A composite function is formed when two functions f, g are combined. However it must be emphasized that the order in which the composite function is determined is important. The method for finding composite functions is: find g(x) find f[g(x)]
54 54 Example For the two functions, find the composite functions (i fg (ii g f
55 55 Exponential & Logarithmic Functions Exponential functions have the general form: where 'a' is a positive constant However there is a specific value of 'a' at (0.1) when the gradient is 1. This value, or 'e' is called the exponential function. The function(above) has one-one mapping. It therefore possesses an inverse. This inverse is the logarithmic function.
56 56 Notes This book is under copyright to A-level Maths Tutor. However, it may be distributed freely provided it is not sold for profit.
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