Amath notes

Additional Mathematics SA2 Overall Revision Notes Chapters 1 – 2 ( ) Simultaneous Equations, Indices, Surds, Logarithms Chapter 1: Simultaneous Equations There are 3 methods in solving simultaneous linear equations: 1.) Substitution Method 2.) Elimination Method 3.) Graphical Method There are several steps to follow: 1.) Express one unknown in terms of another unknown (avoid fractional expressions) 2.) Substitute this newly – formed equation into the non-linear equation 3.) Solve for the unknown 4.) Use the linear equation to find the other unknown. Chapter 2.1: Surds ( )( ) and . Rationalising Denominator: Multiply the square root to both numerator and denominator. m n mn mm nn a m b m a b m a b a b a b a b k c d k a c b d              Chapter 2.2: Indices mnn mn m n n n n mm m n m n m mm m mnn m n m n m a a a a xa x aa a b ab a a a a abb a a a a a a ) ( ) ( 1 1 ) ( ) ( ) ( 0                   When a > 1 xnaa xn   Chapter 2.3: Logarithms

Additional Mathematics SA2 Overall Revision Notes Chapters 3 - 4 Quadratic Functions and Inequalities Sum and Product of Roots 2In Sum of roots Product of roots ax bx c b a c a       We can use the sum and product of roots to write an equation. 2 (sum of roots) (product of roots) 0xx   Discriminant and Nature of Roots Intersection Terms Crosses / Cuts 2 points of intersection, 2 real/distinct roots/ discriminant more than 0. Touches / tangent 1 point of intersection, 2 real/equal roots/ discriminant = 0. Does not intersect / meet 0 points of intersection, no real roots, discriminant < 0. Meet Discriminant more than or equal to 0. Quadratic Inequality ( )( ) 0, or ( )( ) 0, x a x b x a x b x a x b a x b           Chapter 8: Linear Law The graph of a linear equation Y = mX + c is a straight line with gradient m and y intercept c. There are 2 parts to solving linear law questions: Draw a straight line graph to determine gradient and y-intercept, and to find the equation of the straight line. Key Steps: 1.) Force the equation into the form of Y = mX + c. 2.) Take some experimental values of x and y and compute the corresponding values of X and Y. 3.) Use these computed values to plot the points on a graph with X and Y axis. 4.) Draw a line passing through the plotted points. Always have more space at the lower end of graph for the line to cut the Y axis for Y-intercept. 5.) Obtain the Gradient and the Y-intercept. Note: In Y = mX + c (a): Y must not have any coefficient, (b): mX is part constant and part variable. (c): c must not contain any variable X and Y.

Additional Mathematics SA2 Overall Revision Notes Chapters 3 - 4 Polynomials/Partial Fractions Polynomial An expression that is a sum of terms in the form axn where n is non-negative and a is constant. To find unknown constants, either equate coefficients of like powers of x or substitute values of x. Remainder Theorem If a polynomial f(x) is divided by a linear divisor (x – a), the remainder is f(a). Factor Theorem If (x – a) is a factor of the polynomial f(x), f(a) = 0. Partial Fractions Basically, a linear factor that cannot be factorised is to be remained in the same form. A repeated linear factor like 2()ax b is to be split into 2: 2( ) ( ) AB ax b ax b .

Chapter 5: The Modulus Functions For a real number x, |x| represents the modulus / absolute value of x. It is always non- negative. To draw a modulus graph of the function, first draw the function then reflect the part of the function which is below the x axis upwards. Formulas: or ( ) ( ), ( ) 0 ( ) ( ) , ( ) ( ) x k x k x k f x g x g x f x g x f x g x ab a b aa bb           Chapter 6: Binomial Theorem 1 2 2( ) ...12 n n n n nnna b a a b a b b               23 1(1 ) 1 ...1 2 3 1 nnn n n n nx x x x x xn                               ( 1)( 2)...( 1) ! n n n n n r r r      Properties: 1.) Have n+1 terms 2.) Sum of powers of a and b = n. r+1th term: 1 nr r rnT a b r    or 1r rnTb r   Chapter 7: Coordinate Geometry

Additional Mathematics Chapter 9 Curves and Circles (Summary) Chapter 9.1: Graphs of ny ax When n is an even integer (-2, 0 and 2) Legend: Red: 2yx Black: 0 1yx Pink: 2 2 1yx x  1. Each curve is above or on the x axis. 2. Each curve is symmetrical about the x axis. 3. For the pink graph, it does not cut x or y axis. When n is an odd integer (-1, 1 and 3) Legend: Blue: 3yx Brown: 1y x x Black: 1 1yx x  4. Each curve is symmetrical about the origin. General Properties When a is constant, the graphs of ny ax are similar except that they differ in the steepness as seen in the graphs of 2yx  . If a < 0, then the graph of ny ax is a reflection of the graph of ny a x in the x axis.

2 Graphs of ny ax where n is a simple rational number 2. For yx or 1 2yx , x will be more or equal to 0 (x cannot be less than 0). y is also more than 0 as square root is taken to be positive. Legend: Black: yx . Brown: 3yx . 3 Graph of 2y kx 4 Equations of Circles Equation 2 2 2( ) ( )x a y b r    22 2 2 0x y gx fy c     Center of circle (a, b) (-g, -f) Radius r 22g f c 5 Linear Law (Revision) Always make an equation to Y = mX + c. (where m and c must be constant!) 2. Comparing concavity of curves. When yx , graph concaves downwards. When yx , graph is straight and constant. When 2yx , graph concaves upwards. 1. The graph of 2yx is actually a 90 degree clockwise rotation of the graph of 2yx about the origin O. 2. In general, the graphs of 2y kx have the same properties as that of 2yx except that they differ in the steepness. 3. Each graph passes through (0, 0) and is symmetrical about the x axis.

Additional Mathematics Chapter 11 and 12 Trigonometry Functions, Simple Trigonometric Identities/Equations Chapter 11.8: Graphs of the sine, cosine and tangent functions In general, the curves 360 or 2sin and cos have axis , amplitude and period y a bx c y a bx c y c a b      Graphs are shown on the next page. www.studgyuide.pk Chapter 11.1: Angle in Radian Measure 180 rad 1 rad180  1801rad 57.3 Chapter 11.2: Trigonometric Ratios for Acute Angles Just remember that the surd form of these numbers: 3 0.5773 2 0.7072 3 0.8062    Chapter 11.3: Trigonometric Ratios of Complimentary Angles sin(90 ) cos cos(90 ) sin 1tan(90 ) tan        sin cos2 cos sin2 1tan 2 tan              Chapter 11.4: Trigonometric Ratios of General Angles The acute angle formed when a line rotates about the origin is called the basic angle, denoted by . Always make the basic angle positive. 1st Quadrant 2nd Quadrant 3rd Quadrant 4th Quadrant 180      180      360 2         Chapter 11.5: Trigonometric Ratios of their General Angles and their Signs In the 1st quadrant, all 3 are positive. In the 2nd quadrant, only tangent is positive. In the 3rd quadrant, only sine is positive. In the 4th quadrant, only cosine is positive. If still turning anticlockwise after 4th quad, add 360 or 2 . A C S T Chapter 11.6: Trigonometric Ratios of Negative Angles sin( ) sin cos( ) cos tan( ) tan         Chapter 11.7: Solving Basic Trigonometric Equations 1.) By considering the sign of k, identify the possible quadrants where theta will lie. 2.) Find the basic angle alpha, the acute angle from e.g