TABLE OF CONTENTS 3 CHAPTER 1


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2 TABLE OF CONTENTS 3 CHAPTER 1 Set Lnguge & Nottion 3 CHAPTER 2 Functions 3 CHAPTER 3 Qudrtic Functions 4 CHAPTER 4 Indices & Surds 4 CHAPTER 5 Fctors of Polynomils 4 CHAPTER 6 Simultneous Equtions 4 CHAPTER 7 Logrithmic & Exponentil Functions 5 CHAPTER 8 Stright Line Grphs 5 CHAPTER 9 Circulr Mesure 5 CHAPTER 10 Trigonometry
3 CHAPTER 11 Permuttions & Combintions CHAPTER 12 Binomil Expnsions CHAPTER 13 Vectors in 2 Dimensions CHAPTER 14 Mtrices CHAPTER 15 Differentition & Integrtion
4 1. SET LANGUAGE & NOTATION A welldefined collection of objects is clled set nd ech object is clled member or element of the set A set is denoted by cpitl letter nd is expressed by: o Listing its elements, e.g. V = {, e, i, o, u} o A set builder nottion R set of rel numbers R + set of positive rel numbers N set of nturl numbers Z set of integers Z + set of positive integers o e.g. {x: x is prime number nd x < 30} For ny finite set P, n(p) denotes the number of elements in P A null or empty set is denote by { } or For ny two sets P nd Q: o P = Q if they hve the sme elements o P Q if x P x Q o P Q = {x: x P nd x Q} o P Q = then P nd Q re disjoint sets o P Q = {x: x P or x Q} For ny set P nd universl set ξ o P ξ nd 0 n(p) n(ξ) o P = {x: x ξ nd x P} o P P = o P P = ξ 2. FUNCTIONS Onetoone functions: ech x vlue mps to one distinct y vlue e.g. f(x) = 3x 1 Mnytoone functions: there re some f(x) vlues which re generted by more thn one x vlue e.g. f(x) = x 2 2x + 3 Domin = x vlues Rnge = y vlues Nottion: f(x) cn lso be written s f: x To find rnge: o Complete the squre x 2 2x + 3 (x 1) o Work out min/mx point Minimum point = (1,2) ll y vlues re greter thn or equl to 2. f(x) 2 Onetomny functions do not exist Domin of g(x) = Rnge of g 1 (x) Solving functions: o f(2): substitute x = 2 nd solve for f(x) o fg(x): substitute x = g(x) o f 1 (x): let y = f(x) nd mke x the subject Trnsformtion of grphs: o f( x): reflection in the yxis o f(x): reflection in the xxis o f(x) + : trnsltion of units prllel to yxis o f(x + ): trnsltion of units prllel to xxis o f(x): stretch, scle fctor 1 prllel to xxis o f(x): stretch, scle fctor prllel to yxis Modulus function: o Denoted by f(x) o Modulus of number is its bsolute vlue o Never goes below xxis o Mkes negtive grph into positive by reflecting negtive prt into xxis Solving modulus function: o Sketch grphs nd find points of intersection o Squre the eqution nd solve qudrtic Reltionship of function nd its inverse: o The grph of the inverse of function is the reflection of grph of the function in y=x 3. QUADRATIC FUNCTIONS To sketch y = x 2 + bx + c 0 o Use the turning point: Express y = x 2 + bx + c s y = (x h) 2 + k by completing the squre x 2 + nx (x + n 2 2 ) ( n 2 2 ) (x + n) 2 + k Where the vertex is ( n, k) > 0 ushped minimum point < 0 nshped mximum point Find the xintercept: o Fctorize or use formul Type of root by clculting discriminnt b 2 4c o If b 2 4c = 0, rel nd equl roots o If b 2 4c > 0, rel nd distinct roots o If b 2 4c < 0, no rel roots PAGE 3 OF 7
5 Intersections of line nd curve: if the simultneous equtions of the line nd curve leds to simultneous eqution then: o If b 2 4c = 0, line is tngent to the curve o If b 2 4c > 0, line meets curve in two points o If b 2 4c < 0, line does not meet curve Qudrtic inequlity: o (x d)(x β) < 0 d < x < β o (x d)(x β) > 0 x < d or x > β 4. INDICES & SURDS Definitions: o for > 0 nd positive integers p nd q 0 = 1 1 p p = p = 1 p p q = ( Rules: o for > 0, b > 0 nd rtionl numbers m nd n m n = m+n n b n = (b) n m n n = m n n b n = ( b ) ( m ) n = mn 5. FACTORS OF POLYNOMIALS To find unknowns in given identity o Substitute suitble vlues of x OR o Equlize the given coefficients of like powers of x Fctor Theorem: If (x t) is fctor of the function p(x) then p(t) = 0 Reminder Theorem: If function f(x) is divided by (x t) then: Reminder = f(t) The formul for reminder theorem: Dividend = Divisor Quotient + Reminder 6. SIMULTANEOUS EQUATIONS Simultneous liner equtions cn be solved either by substitution or elimintion Simultneous liner nd nonliner equtions re generlly solved by substitution s follows: o Step 1: obtin n eqution in one unknown & solve it o Step 2: substitute the results from step 1 into the liner eqution to find the other unknown The points of intersection of two grphs re given by the solution of their simultneous equtions p ) q PAGE 4 OF 7 7. LOGARITHMIC & EXPONENTIAL FUNCTIONS Definition o for > 0 nd 1 y = x x = log y For log y to be defined y > 0 nd > 0, 1 When the logrithms re defined log 1 = 0 log b + log c log bc b log = 1 log b log c log c log b log b log b n n log b log When solving logrithmic equtions, check solution with originl eqution nd discrd ny solutions tht cuses logrithm to be undefined Solution of x = b where 1, 0, 1 If b cn be esily written s n, then x = n x = n Otherwise tke logrithms on both sides, i.e. log x = log b nd so x = log log 10 ln log e Logrithmic & Exponentil Grphs log b log 8. STRAIGHT LINE GRAPHS Eqution of stright line: y = mx + c y y 1 = m(x x 1 ) Grdient: m = y 2 y 1 x 2 x 1 Length of line segment: Length = (x 2 x 1 ) 2 + (y 2 y 1 ) 2
6 Midpoint of line segment: ( x 1 + x 2 2, y 1 + y 2 ) 2 Prllelogrm: o ABCD is prllelogrm digonls AC nd BD hve common midpoint o Specil prllelogrms = rhombuses, squres, rectngles Specil grdients: o Prllel lines: m 1 = m 2 o Perpendiculr lines: m 1 m 2 = 1 Perpendiculr bisector: line psses through midpoint To work out point of intersection of two lines/curves, solve equtions simultneously 9. CIRCULAR MEASURE Rdin mesure: π = 180 2π = 360 Degree to Rd = π 180 Arc length: Are of sector: s = rθ A = 1 2 r2 θ Rd to Degree = 180 π TANGENT CURVE CAST DIAGRAM Trigonometric rtios: sec θ = 1 cosec θ = 1 cot θ = 1 cos θ sin θ tn θ Trigonometric identities: sin θ tn θ = sin 2 θ + cos 2 θ = 1 cos θ cot 2 θ + 1 = cosec 2 θ tn 2 θ + 1 = sec 2 θ Sketching trigonometric grphs: 10. TRIGONOMETRY Trigonometric rtio of specil ngles: SINE CURVE COSINE CURVE 11. PERMUTATIONS & COMBINATIONS Bsic Counting Principle: to find the number of wys of performing severl tsks in succession, multiply the number of wys in which ech tsk cn be performed: e.g Fctoril: n! = n (n 1) (n 2) o NOTE: 0! = 1 Permuttions: o The number of ordered rrngements of r objects tken from n unlike objects is: o Order mtters Combintions: n P r = n! (n r)! o The number of wys of selecting r objects from n unlike objects is: n n! C r = r! (n r)! Order does not mtter PAGE 5 OF 7
7 12. BINOMIAL EXPANSIONS The binomil theorem llows expnsion of ny expression in the form ( + b) n (x + y) n n = C 0 x n n + C 1 x n 1 n y + C 2 x n 2 y 2 n + + C n y n e.g. Expnd (2x 1) 4 (2x 1) 4 4 = C 0 (2x) C 1 (2x) 3 ( 1) 4 + C 2 4 (2x) 2 ( 1) 2 + C 3 (2x) ( 1) 3 + C 4 ( 1) 4 = 1(2x) 4 + 4(2x) 3 ( 1) +6(2x) 2 ( 1) 2 + 4(2x) ( 1) 3 + 1( 1) 4 = 16x 4 32x x 2 8x + 1 The powers of x re in descending order 13. VECTORS IN 2 DIMENSIONS Position vector: position of point reltive to origin, OP Forms of vector: ( b ) AB p i bj Prllel vectors: sme direction but different mgnitude Generlly, AB = OB OA Mgnitude = i 2 + j 2 Unit vectors: vectors of mgnitude 1 o Exmples: consider vector AB AB = 2i + 3j AB = 13 Unit vector = 1 (2i + 3j) 13 Colliner vectors: vectors on the sme line Dot product: (i + bj). (ci + dj) = (ci + bdj) Angle between two diverging vectors:. b cos A = b Reltive Velocity Motion in the wter: V w = true velocity of wter V P/W = velocity of P reltive to W still wter Course tken by P is direction of V P/W Motion in the ir: V w = true velocity of wind or ir V P/W = velocity of P reltive to W still wind/ir Course tke by P is direction of V P/W V P/Q = V P V Q MATRICES Order of mtrix: mtrix with m rows nd n columns, Order = m n Adding/subtrcting mtrices: dd/subtrct ech corresponding element Sclr multipliction: to multiply mtrix by k, multiply ech element by k Multiplying mtrices: multiply row by column Identity mtric: I = ( 1 0 ) IA = A nd AI = I 0 1 Clculting the determinnt: A = ( b ) A = (d bc) c d Inverse of 2 by 2 mtrix: o Switch leding digonl, negte secondry digonl o Multiply by 1 A A = ( b c d ) A 1 = 1 ( d b d bc c ) A 1 A = AA 1 = I Solving simultneous liner equtions by mtrix method: x + by = h cx + = k Eqution cn be written s: b ( c d ) (x y ) = (h k ) Rerrnge it nd solve: ( x y ) = 1 d bc ( d b c ) (h k ) For mtrix to give unique solutions: b ( c d ) DIFFERENTIATION & INTEGRATION 15.1 Differentition FUNCTION 1ST DERIVATIVE 2 ND DERIVATIVE y = x n = d 2 y nxn 1 = n(n 1)xn 2 2 INCREASING FUNCTION DECREASING FUNCTION > 0 < 0 Sttionry point: equte first derivtive to zero = 0 PAGE 6 OF 7
8 2 nd Derivtive: finds nture of the sttionry point o If vlue +ve, min. point negtive sttionry point o If vlue ve, mx. point positive sttionry point Chin rule: = du du Product rule: Quotient rule: = v Specil Differentils dv du = u + v du dv u v 2 of sin x = cos x of cos x = sin x of tn x = sec2 x of ex+b = e x+b of ln x = 1 x of ln(f(x)) = f (x) f(x) Relted rtes of chnge: o If x nd y re relted by the eqution y = f(x), then the rtes of chnge nd re relted by: dt dt dt = dt Smll chnges: o If y = f(x) nd smll chnge δx in x cuses smll chnge δy in y, then δy ( ) δx x=k Logs Algebr Trig e To find re under the grph (curve nd xxis): o Integrte curve o Substitute boundries of x o Subtrct one from nother (ignore c) y c To find volume under the grph (curve nd xxis): o Squre the function o Integrte nd substitute o Multiply by π d πy 2 c To find re/volume between curve nd yxis: o Mke x subject of the formul o Follow bove method using yvlues insted of xvlues Specil Integrls d sin(x + b) = 1 cos(x + b) + c cos(x + b) = 1 sin(x + b) + c sec 2 (x + b) = 1 tn(x + b) + c 1 x + b = 1 ln x + b + c 15.3 Kinemtics e x+b = 1 ex+b + c 15.2 Integrtion xn+1 x n = (n + 1) + c (x + b) n (x + b)n+1 = + c (n + 1) Definite integrl: substitute coordintes/vlues & find c Integrting by prts: u dv du = uv v o Wht to mke u: LATE Prticle t instntneous rest, v = 0 Mximum displcement from origin, v = 0 Mximum velocity, = 0 PAGE 7 OF 7
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