Prerequisite Knowledge Required from O Level Add Mth ) Surds, Indices & Logrithms Rules for Surds. b= b =. 3. 4. b = b = ( ) = = = 5. + b n = c+ d n = c nd b = d Cution: + +, - Rtionlising the Denomintor (two tpes) () E.g. (b) E.g. 3 3 5 3 5 = = 5 5 5 5 (5+ ) (+ ) (+ ) = = = (5 ) (5 )(5+ ) (5 ) 3 Rules for Indices Sme bse = = Power Sme inde ( ) n = = () =( )n Zero inde = Negtive inde = ( ) - n = ( )n Frctionl inde = ( )
Rules for Logrithms For log to be defined, both nd must be positive. Logrithm log is written s lg Logrithm log is written s ln Product of Logrithm log = log + log Quotient of Logrithm log log log = Power Lw r log = r log Chnge of Bse log logc b b= log c Log of to n bse log = 0 Log of number to the sme bse s number log = Eponentil Grphs The grphs of = b re shown below for different rnge of vlues of & b. NOTE: MUST DRAW THE ASYMPTOTES FOR THESE CASES!
Logrithmic Grphs The grphs of = ln (b + c) re shown below for different rnge of vlues of & b. ) Qudrtic Equtions Qudrtic Eqution: + b + c = 0 Recll Qudrtic Formul: = ± ) For roots α nd β b) Useful: b Sum of roots = α + β = Product of roots = αβ = c Eqution is: α (sum of roots) + (product of roots) = 0 OR: ( α)( β ) = 0 α + β = ( α + β ) αβ 3
Discriminnt & tpe of roots D = b 4c Coefficient of > 0. hs min. vlue Coefficient of < 0. hs m. vlue Rel & distinct roots b 4c> 0 Rel & repeted (equl) roots b 4c= 0 No rel roots b 4c< 0 Question tpes Rel roots Line does not intersect curve Curve lies entirel bove -is (min. grph) e.g. > 0 for ll vlues of, 5 +3 > 0 lws Curve lies entirel below -is (m. grph) e.g. < 0 for ll vlues of, -5 5 +3 < 0 lws For rel roots, it cn be distinct or equl roots b 4c 0 4
Polnomils & Prtil Frction Recll how long division is done. Fctor Theorem A polnomil f() is divisible b ( ) => reminder = 0 => ( ) is fctor of f(). Prtil Frctions Proper rtionl lgebric epression, () () is when degree of numertor f() < degree of denomintor, g(). Note: To write s prtil frctions, first check tht the epression is proper. m+ n A B = + ( + b)( c+ d) + b c+ d p + q+ r A B C = + + ( + b)( c+ d) + b c+ d ( c+ d) p + q+ r A B+ c = + ( + b)( + c ) + b + c 3) Modulus Function The bsolute or modulus of rel number is denoted b nd defined s: The bove is useful when solving modulus eqution. Alterntivel, ou cn squre both sides to remove the modulus sign. Refer to propert 6 below. Properties of modulus:. 0. - = 3. b = b 4. = 5. n = n 6. = b implies tht = b Note: modulus is lws non-negtive. 5
4) Coordinte Geometr B(, ) θ A(, ). For points A(, ) nd B(, ), Length AB= ) + ( ) ( + + Mid-point of AB =, Grdient of AB= = tnθ. Line L hs grdient m. Line L hs grdient m. L is prllel to L m = km. L is perpendiculr to L m m =. 6
3. Ares of tringle: φ θ Are of ABC = = ( 3 3 OR Are of ABC = ( AB)( AC)sinθ (nti - clockwise direction) + 3 + 3 3 3 Eqution of stright line: = m + c, where m = grdient, c = -intercept Eqution of horizontl line: =, where is constnt. Eqution of verticl line: =, where is constnt. An -intercept is point where the grph cuts the -is. It is found b letting = 0. An -intercept is point where the grph cuts the -is. It is found b letting = 0. Sine Rule sinθ sinφ = BC AB Cosine Rule BC = AB + AC ( AB)( AC) cosθ 5) Liner Lw Re-write Non-Liner equtions into Liner equtions of form = m + c ) Non-liner equtions Liner Equtions Y-is X-is Grdient, m Y-intercept, C = m ln = m ln + ln ln ln m ln = b ln = ln b + ln ln ln b ln 7
6) Trigometric Function Trigonometric Rtio of Specil Angles 30 45 60 sin = 3 cos 3 = tn 3 = 3 3 3 Note: For ese of remembering, observe tht vlue of sin increses from to to. Alws write the vlue of the specil ngles in terms of the surd form insted of 0.866 for or 0.707 for Complementr Angles sin (90 - θ) = cos θ tn (90 - θ) = cot θ sec(90 - θ)= ( ) = = cosecθ cos (90 - θ) = sin θ cot (90 - θ) = tn θ cosec(90 - θ)= ( ) = = secθ Negtive Angles A positive ngle is n nti-clockwise rottion from the positive -is bout the origin. A negtive ngle is clockwise rottion from the positive -is bout the origin. For n ngle θ, cos (-θ) = sin (-θ) = tn (-θ) = cos θ - sin θ - tn θ 8
Smll Angle Approimtion Note: From the power series epnsions of sin, cos nd tn respectivel, when is smll nd mesured in rdins, sin cos tn.. Quotient Reltionships sinθ tn θ = cosθ cosθ cot θ = sinθ secθ = cosθ cos ecθ = sin b. Pthgors Trigonometric Identities c. R-Formul θ sin θ + cos θ = tn θ + = sec θ + cot θ = cos ec θ For > 0, b > 0 & cute ngle α, cos θ ± b sin θ R cos (θ α ) sin θ ± b cos θ R sin (θ ± α ) where R = +, tn α = The following re provided in the MF5: - Addition Formul - Double Angle Formul - Fctor Formul 9
d. Grphs of sin, cos nd tn functions = sin = cos = tn Amplitude = Amplitude = Amplitude undefined Period = 360º ( ) Period = 360º ( ) Period = 80º ( ) 7) Tngent nd Norml For curve = f(), d (i) Grdient of tngent = or f '( ) d d (ii) Grdient of tngent t (, b) = or f '( ). d (iii) Eqution of tngent t (, b): b= f '( )( ). (iv) Eqution of norml t (, b): b= ( ). f '( ) 8) Properties of Circle Smmetricl Properties of Circles A line through the centre of circle nd perpendiculr to chord will bisect the chord. = If OMQ = 90º then MP = MQ or vice vers Note: This propert is true for ll isosceles or equilterl tringle. Equl chords re equidistnt from the centre. If PQ = RS then OX = OY or vice vers 0
A tngent to circle is perpendiculr to the rdius. Tngents from n eternl point to circle re equl nd subtend equl ngles t the centre. OT bisects POQ & PTQ i.e. TP = TQ, POT = QOT & PTO = QTO Angle Properties of Circles Angle t Centre = Twice Angle t circumference POR = PQR Note: refer to three different digrms Angle in semi-circle PQR = 90º Note: PR is dimeter of circle
Angles in the sme segment PQR = QPR = PRS QSR Note: ngles t the circumference re subtended b the sme rc Opp. Angles in cclic qud SRQ + SPQ = 80º RSP + RQP = 80º Note: ll 4 sides must touch the circumference.