ZNOTES.ORGUPDATED TO 2020-22 SYLLABUSCAIE A2 LEVELMATHS (9709)SUMMARIZED NOTES ON THE PURE 3 SYLLABUSCAIE A2 LEVEL MATHS(9709)1.Algebra1×21×2×3Factor case:if constant is not 1,pull out a factor from1.1.The Modulus Functionbrackets to make it 1 use general equation.Do notforget the indices.Substitution case:if bracket contains more than oneIt gives the absolute value of a number.term(e.g.(2-z+22))then make the last part u,The modulus of a value gives the distance of the valuefrom the origin.expand and then substitute back in.Finding the limit of z **in expansion:No line with a modulus ever goes under the x-axis.Any line that does go below the x-axis,when modulated is**E.g.(1+ax)",limit can be found by substituting axreflected above it.between the modulus sign in |1 and altering it tohave only in the modulus(S15-P31}Question 3:Show that,for small values of2,리=a2=a2where the value of the constant k is to be determined.Solution:Expand (1-2x2 until the term·Graph of y=|x1×2=1-2x+3x2(1+2x2)-2=1-2(2x2)+3(2x2)2=1-42+12x41×2=1+1.2.Polynomialsthose of (1-222)-2To find unknowns in a given identity,either(1-4x2+12x4)-(1+4x2-4x4).Substitute suitable values of=-8x2+16x4Factor theorem:If (-t)is a factor of the function p(z)The value ofk is:thenp(t)=016Remainder theorem:If the function f(z)is divided by(x-t)then the remainder:R=f(t)1.4.Partial FractionsDividend Divisor x Quotient+RemainderA B1.3.Binomial SeriesExpanding (1+x)"where |<1WWW.ZNOTES.ORGCAIE A2 LEVEL MATHS(9709)Multiply(rx+s),substitute =and find B15B=-1Multiply(pa),substitute=and find AThus,the partial fraction is:Multiply (r+s)2,substitute=and find C-2-12+Substitute any constant e.g.z =0 and find B2.Logarithmic ExponentialFunctions.Taketo the other side,subtract and simplify.y=ax台loga y=xLinear eqn.left at top is equal to Bx+Cloga 1 =0loga a=1Improper fraction case:if numerator has a to the degreeloga b"三nlogabof power equivalent or greater than the denominatorloga b+loga c■loga bcthen another constant is present.This can be found byloga b-loga c▣loga cdividing denominator by numerator and using remainderloga(S12-P33}Question 8:Express the following in partial fractions:2.2.Graphs of ln(x)and ex4x2-7x-1(x+1)(2x-3)Solution:Expand the brackets4x2-7x-12x2-x-32Greatest power ofa same in numerator and denominator,thus is an improper fraction case3Making into proper fraction:22x2-x-34x2-7x-14x2-2x-6-5x+5This is written as:2+(x+1)(2x-3)Now proceed with nommal case for the fraction:3.Trigonometry3.1.RatiosA(2x-3)+B(x+1)=5-5xWhen =-11sec=-5A=5+5cos1A=-2sinWWW.ZNOTES.ORG