Learning Objectives These show clearly the purpose and extent of coverage for each topic.

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Howard Merritt
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1 Preface This book is prepared for students embarking on the study of Additional Mathematics. Topical Approach Examinable topics for Upper Secondary Mathematics are discussed in detail so students can focus on building their foundation in the subject. These show clearly the purpose and extent of coverage for each topic. Useful Notes Each chapter begins with a quick recap and presentation of the main focus and content with direct explanations to formulae and concepts. Worked Examples show best methods and sometimes, alternate ways of working out typical problems. Practice Questions Over 400 questions are provided so students learn to apply mathematical concepts confidently. Challenging Exercises are included for further application and learning. Worked Solutions Step-by-step solutions are included so students can learn independently. They also serve as a quick assessment of the work done. The Editorial Team

2 Con ten ts Quadratic Equations And Inequalities... 1 conditions for a quadratic equation to have (i) two real roots (ii) two equal roots (iii) no real roots and related conditions for a given line to (i) intersect a given curve (ii) be a tangent to a given curve (iii) not intersect a given curve solution of quadratic inequalities, and the representative of the solution set on the number line conditions for ax 2 + bx + c to be always positive (or always negative) relationships between the roots and coefficients of the quadratic equation ax 2 + bx + c = 0 Chapter 2 Indices And Surds four operations on indices and surds rationalising the denominator solving equations involving indices and surds Chapter 3 Polynomials multiplication and division of polynomials use of remainder and factor theorems factorisation of polynomials solving cubic equations Chapter 4 Simultaneous Equations In Two Unknowns solving simultaneous equations with at least one linear equation, by substitution. expressing a pair of linear equations in matrix form and solving the equations by inverse matrix method.

3 Chapter 5 Partial Fractions Include cases where the denominator is no more complicated than: (ax + b) (cx + d) (ax + b) (cx + d) 2 (ax + b) (x 2 + c 2 ) Chapter 6 Binomial Expansions Include: use of the Binomial Theorem for positive integer n use of the notations n! and ( n r ) use of the general term ( n r ) an r b r, 0 < r ñ n Chapter 7 Exponential, Logarithmic And Modulus Functions functions a x, e x, log a x, 1n x and their graphs laws of logarithms equivalence of y = a x and x = log a y change of base of logarithms function xxx and graph of xf(x)x, where xf(x)x is linear quadratic or trigonometric solving simple equations involving exponential, logarithmic and modulus functions Chapter 8 Trigonometric Functions, Identities And Equations six trigonometric functions for angles of any magnitude (in degrees or radians) principal values of sin 1 x, cos 1 x, tan 1 x exact values of the trigonometric functions for special angles (30º, 45º, 60º) or 6, 4, 3 amplitude, periodicity and symmetrices related to the sine and cosine functions graphs of y = a sin (bx) + c, y = a sin ( x b ) + c, y = a cos (bx) + c, y = a cos ( x b ) + c and y = a tan (bx), where a and b are positive integers and c is an integer

4 use of the following * sin A cos A cos A = tan A, sin A = cot A, sin2 A + cos 2 A = 1, sec 2 A = 1 + tan 2 A, cosec 2 A = 1 + cot 2 A * the expansions of sin (A ± B), cos (A ± B) and tan (A ± B) * the formulae for sin 2A, cos 2A and tan 2A * the formulae for sin A ± sin B and cos A ± cos B * the expression for a cos θ + b sin θ in the form R cos(θ ± α) or R sin(θ ± α) simplification of trigonometric expressions solution of simple trigonometric equations in a given interval proofs of simple trigonometric identities Chapter 9 Coordinate Geometry In Two Dimensions condition for two lines to be parallel or perpendicular midpoint of line segment finding the area of rectilinear figure given its vertices graphs of equations * y = ax n, where n is a simple rational number * y 2 = kx coordinate geometry of the circle with the equation (x a) 2 + (y b) 2 = r 2 and x 2 + y 2 + 2gx + 2 fy + c = 0 transformation of given relationships, including y = ax n and y = kb x, to linear form to determine the unknown constants from the straight line graph 0 Proofs In Plane Geometry symmetry and angle properties of triangles, special quadrilaterals and circles mid-point theorem and intercept theorem for triangles tangent-chord theorem (alternate segment theorem) use of above properties and theorems 1 Differentiation And Integration derivative of f(x) as the gradient of the tangent to the graph of y = f(x) at a point derivative as rate of change use of standard notations f '(x), f '(x), dy dx, d 2 y dx [ = d 2 dx ( dy dx ) ]

5 derivatives of x n, for any rational n, sin x, cos x, tan x, e x, and ln x, together with constant multiples, sums and differences derivatives of composite functions derivatives of products and quotients of functions increasing and decreasing functions stationary points (maximum and minimum turning points and stationary points of inflexion) use of second derivative test to discriminate between maxima and minima applying differentiation to gradients, tangents and normals, connected rates of change and maxima and minima problems integration as the reverse of differentiation integration of x n for any rational n, sin x, cos x, sec 2 x and e x, together with constant multiples, sums and differences integration of (ax + b) n (ax + b) for any rational n, sin(ax + b), cos(ax + b) and e definite integral as area under a curve evaluation of definite integrals finding the area of a region bounded by a curve and lines parallel to the coordinate axes finding areas of regions below the x-axis application of differentiation and integration to problems involving displacement, velocity and acceleration of a particle moving in a straight line with variable or constant acceleration Exclude: differentiation of functions defined implicity and parametrically finding the area of a region between a curve and an oblique line, or between two curves use of formulae for motion with constant acceleration 2 Challenging Exercise Solutions...S1 Calculators should be used when necessary. If the degree of accuracy is not specifi ed in any question, and if the answer is not exact, give the answer to three signifi cant fi gures. Answers in degrees should be given to one decimal place. For, use either the calculator value or 3.142, unless the question requires the answer in terms of.

Chap ter 1 QUADRATIC EQUATIONS AND INEQUALITIES Learning Objectives conditions for a quadratic equation to have (i) two real roots (ii) two equal roots (iii) no real roots and related conditions for

number line conditions for ax 2 + bx + c to be always positive (or always negative) relationships between the roots and coeffi cients of the quadratic equation ax 2 + bx + c = 0 USEFUL NOTES Nature

6 Chap ter 1 QUADRATIC EQUATIONS AND INEQUALITIES Learning Objectives conditions for a quadratic equation to have (i) two real roots (ii) two equal roots (iii) no real roots and related conditions for a given line to (i) intersect a given curve (ii) be a tangent to a given curve (iii) not intersect a given curve solution of quadratic inequalities, and the representative of the solution set on the number line conditions for ax 2 + bx + c to be always positive (or always negative) relationships between the roots and coeffi cients of the quadratic equation ax 2 + bx + c = 0 USEFUL NOTES Nature of Roots of Quadratic Equations For any quadratic equation y = ax 2 + bx + c = 0, x = b ± b 2 4ac. b 2 4ac is the discriminant of the equation. 1. If b 2 4ac < 0, the equation has no real roots. It has complex roots. If b 2 4ac ú 0, the equation has real roots. (i) If b 2 4ac = 0, the equation has equal (repeated, identical) roots which are equal to b. (ii) If b 2 4ac > 0, the roots are unequal (different, distinct). (a) If a, b and c are rational and b 2 4ac is non-zero and a perfect square, then the roots are rational. (b) If b 2 4ac is not a perfect square but positive, then the roots are irrational. 1

7 2. The x-coordinates of the points of intersection of the straight line y = mx + d and the quadratic curve y = ax 2 + bx + c can be obtained from: ax 2 + bx + c = mx + d Hence ax 2 + (b m) x + (c d) = 0 Discriminant = (b m) 2 4a(c d) If the discriminant is negative, the straight line and the curve have no common point. If the discriminant is zero, the straight line touches the curve at one and only one point. It is a tangent to the curve. If the discriminant is positive, the straight line intersects the curve at two distinct points. 3. If y = ax 2 + bx + c is either positive or negative for all values of x, then the equation has no real roots and b 2 4ac < 0. Quadratic Inequalities 1. y = ax 2 + bx + c = a [ x 2 + b a x + ( b ) 2 ] + c a ( b = a ( x + b ) 2 + 4ac b2 4a ) 2 4ac b2 (i) If a is positive, the minimum value of y is. 4a The corresponding value of x is b. 4ac b2 (ii) If a is negative, the maximum value of y is. 4a The corresponding value of x is b. 2. y = a(x α) (x β) where a > 0 (i) y > 0 for all real values of x. α and β are complex numbers. x 2

8 (ii) α α = β y > 0 if x α y = 0 if x = α x (iii) α β x y ú 0 if x ñ α or β ñ x y > 0 if x < α or β < x y = 0 if x = α or x = b y < 0 if α < x < β y ñ 0 if α ñ x ñ β The converse is also true. y is a minimum when x = α + β y = a(x α) (x β) where a < 0. (i) x y < 0 for all real values of x. α and β are complex numbers. (ii) α x α = β y < 0 if x α y = 0 if x = α 3

9 (iii) α β x y ñ 0 if x ñ α or β ñ x y < 0 if x < α or β < x y = 0 if x = α or x = β y > 0 if α < x < β y ú 0 if α ñ x ñ β The converse is also true. y is a maximum when x = α + β. 2 Conditions for a quadratic expression to be always positive or always negative If b 2 4ac is negative, the equation ax 2 + bx + c = 0 has no real roots. This means that the graph of y = ax 2 + bx + c is entirely above the x-axis if a is positive or entirely below the x-axis if a is negative. Therefore, if b 2 4ac is negative, ax 2 + bx + c is always positive if a is positive and is always negative if a is negative. Roots and Coefficients of Quadratic Equations 1. Let α and β be the roots of the quadratic equation ax 2 + bx + c = 0, where a, b and c are real numbers. Let α = b + b 2 4ac and β = b b 2 4ac. Sum of the roots = α + β = b + b 2 4ac + b b 2 4ac = 2b = b a = negative value of the coefficient of x divided by the coefficient of x 2 Product of the roots = αβ = b + b 2 4ac b b 2 4ac = ( b)2 ( b 2 4ac ) 2 () 2 = b2 b 2 + 4ac 4a 2 = a c = the constant term divided by the coefficient of x 2 If a = 1, the sum and product of roots are b and c respectively. 4

10 2. ax 2 + bx + c = 0 can also be expressed as x 2 + b a x + a c = 0, or x 2 (α + β) x + αβ = 0 where α and β are the roots of the quadratic equation. 3. A function of α and β is said to be symmetric if it remains unchanged when α and β are interchanged. α + β + αβ and α + β are symmetric functions of α and β. αβ They are the same as β + α + βα and β + α βα respectively. α(α β) But is not a symmetric function of α and β because β β(β α) it is not the same as α unless α = β. 4. Any symmetric function of α and β can be expressed in terms (α + β) and αβ. Hence, by using point 1, it can also be expressed as function of a, b and c. EXAMPLE (i) (α + β) 3 = α 3 + 3α 2 β + 3β 2 α + β 3 (ii) = α 3 + β 3 + 3αβ(α + β) ( b a ) 3 = (α 3 + β 3 ) + 3 ( c a ) ( b a ) α 3 + β 3 = b3 + 3bc a 3 a 2 3abc b3 = a 3 α β + β α 2 + β 2 α = αβ = (α + β)2 2αβ αβ ( = b ) 2 a 2 ( a c ) c a = b2 c ac 5. The sum, difference, product and quotient of any two symmetrical functions are also a symmetrical functions. 6. If p and q are the roots of x 2 (p + q)x + pq = 0, and (p + q) and pq are symmetric functions of α and β, then the coefficients of this equation can be expressed in terms of α + β and αβ and eventually in terms of a, b and c. 5

11 EXAMPLE If α and β are the roots of 2x 2 3x + 5 = 0, find the equation whose roots are 2α + β and α + 2β. Solution: α + β = 3 2 αβ = 5 2 Sum of the roots of the new equation = (2α + β) + (α + 2β) = 3(α + β) = = 9 2 Product of the roots of the new equation = (2α + β)(α + 2β) = 2α 2 + 4αβ + αβ + 2β 2 = 2α 2 + 4αβ + 2β 2 + αβ = 2(α 2 + 2αβ + β 2 ) + αβ = 2(α + β) 2 + αβ = 2 ( 3 2 ) = 7 The required equation is x x + 7 = 0 or 2x 2 9x + 14 = 0. 6

12 Practice 1 1. Solve the equation 3x x 6 = The sides of a right angled triangle are x 1, x + 2 and x + 3 m. Find the perimeter and area. 7

ADDITIONAL MATHEMATICS

ADDITIONAL MATHEMATICS GCE Ordinary Level (06) (Syllabus 4047) CONTENTS Page INTRODUCTION AIMS ASSESSMENT OBJECTIVES SCHEME OF ASSESSMENT 3 USE OF CALCULATORS 3 SUBJECT CONTENT 4 MATHEMATICAL FORMULAE