CORE 2 Summary Notes - Mathsbox

1 core 2 Summary Notes1 Indices Rules of Indicesxa xb =xa+bxa xb =xa b(xa)b=xabSolve 32x + 1 10 3x + 3 = 03 (3x)2 10 3x + 3 = 0 Let y = 3x3y2 10y + 3 = 0(3y 1)(y 3) = 0y =1gives 3x =133x = 1y= 3gives 3x = 3 x = 1 Solve 9x =12732x = 3 32x = 3x = 32 x0= 1 x a=1ax x n = n1x x nnmnmmx)x(==2 Graphs and TransformationsASYMPTOTESS traight lines that are approached by a graph which never actually meets 2 3 4 5 1 2 3 4 512345 1 2 3 4 5 Horizontal Asymptotey = 1 Vertical Asymptotex = 2y =1x 2+ 1 TRANSLATION- to find the equation of a graph after a translation of you replace xby (x-a) and y by (y - b) ab1yx1 2 3 4 5 1 2 3 4 512345 1 2 3 4 The graph of y = x2 -1 is translated through.

2 Write down the equation of 3- 2the graph formed.(y + 2) = (x-3)2 1y = x2 6x + 6y - b = f(x-a)ory = f(x-a) +by = x2 6x + 6y = = -f(x)REFLECTINGR eflection in the x-axis, replace y with yReflection in the y-axis, replace x with xSTRETCHINGS tretch of factor k in the x direction replace x by 1 xkStretch of factor k in the y direction replace y by1yky = f(-x)y = f(1 xk) Describe a stretch that will transform y = x2+ x -1 to thegraph of y = 4x2 + 2x -14x2 + 2x -1 = (2x)2 + (2x) -1So x has been replaced by of scale factor in the x directiony = kf(x)3 Sequences and Series 1 A sequence can be defined by the nth term such as un = n2 +1u1 = 12 +1 u2 = 22 +1 u3 = 32 +1 u4 = 42 +1= 2 = 5 =10 =17 An INDUCTIVE definition defines a sequence by giving the first term and a rule to findthe next +1 = 2un+ 1 u1=3 u2=7 u3=15 Some sequences get closer and closer to a value called the LIMIT these are knownas CONVERGING The sequence defined by Un+1 = + 2 u1=3 converges to a limit lFind the value of must satisfy the equation I = + = 2l = 2 SEQUENCEEach term is found by adding a fixed number (COMMON DIFFERENCE) to theprevious a u2= u1+ d u3= u2+ da is the first term ,d is the common difference the sequence isa, a + d , a + 2d, a + 3d.

3 Un= a + (n - 1)dSUM of the first n terms of an AP (Arithmetic progression)Sn =21n ( 2a + (n-1)d ) or21n(a + l) where l is the last term The sum of the first n positive integers is12n(n + 1) SIGMA notation Si= 05i3 =03 + 13 + 23 + 33 + 43 + thatSr= 120(3n 1) = 610= 2 + 5 + 8 + . 59a = 2d = 3S20 =202(2 2 + (20 1) 3)= 10 61= 6104 Geometric Sequences A geometric sequence is one where each term is found by MULTIPLYING theprevious term by a fixed number (COMMON RATIO) The nth term of a geometric sequence a, ar, ar2, ar3, ..is arn-1a is the first termr is the common ratio The sum of a geometric sequence a + ar + ar2 + ar3 ..+ arn-1 is a geometricseriesr 11)= a(rn -Sn- If -1 < r < 1 then the sum to infinity is1 ra-Find the sum to inifinty of the series 1 +23+49+827 Geometric Series first term = 1 common ratio =23 Sum to infinity =1= 331- 25 Binomial Expansion The number of ways of arranging n objects of which r are one type and (n-r) areanother is given byr!

4 (n r)!n!rn=- where n! = n(n-1)(n-2)..x 3 x 2 x 1 (1 +x)n = 1 + n 1 x+ n 2 x2+ n 3 x3+ ..xn(a + b)n = an + n 1 an 1b1 + n 2 an 2b2+ .. nn 1 1 a bn 1+ bnFind the coefficient of x3 in the expansion of (2+3x)9n = 9 , r = 3 9 (3x)3(2)63 Note 3 + 6 = 9 (n)= 84 27 x3 64 = 145152 Trigonometry Exact Values LEARN45 30 60sin 45 =12cos 45 =12tan 45 = 111 245sin 30 =12cos 30 =32tan 30 =131 12 2sin 60 =32cos 60 =12tan 60 =33060 3 COSINE RULEabcABCba2 =b2+c2 2bc cosA2 =a2 +c2 2ac cosBc2 =a2 +b2 2ab cosASINE RULE asinA=bsinB=csinCAREA of a TRIANGLE1ab sinC =12 2bc sinA =12ac Find the area of triangle PQRF inding angle P82 = 92 + 122 2 9 12 cos Pcos P =92+ 122 822 9 12= 0 745P = 41 8.

5 Area =12 9 12 sin 41 8 .= 35 99991 36 cm2 == 36 cm29cm12cm8cmRPQTry to avoid rounding until you reach your finalanswer180 =p radians 90 =p2radians 60 =p3radians45 =p4radians 30 =p6radiansRADIANS and ARCS360 = 2 OF AN ARC l = r lAREA of SECTOR = r2 rr7 Trigonometryy = sin sin ( ) = sin sin (180 ) = sin yx90 180 270 360 90 180 2701 1sin ( ) = sin (in radians)y = cos cos ( ) = cos cos (180 ) = cos yx90 180 270 360 90 180 2701 1cos ( ) = cos (in radians)y = tan yx90 180 270 360 90 180 270510 5 10tan x =sinxcos xTan ( ) = tan sin2x + cosx2 = 1 LEARN THIS IDENTITYSOLVING Solve the equation 2sin2x = 3 cos x for 0 < x < 2 22 sin2x = 3 cosx2(1 cos2x) = 3 cos xcos2x + 3cosx 2 = 0(2cos x 1)(cos x + 2)

6 = 0cos x = -2 has no x = - x =p3or2 p -3p = 5p3 PROVING IDENTITIESShow that (sin x + cos x )2 = 1 + 2sinxcos xLHS: (sin x + cos x)2 = (sin2 x + 2 sinxcos x + cos2 x)= sin2 x + cos2 x + 2 sinxcos x= 1 + 2 sin xcos xLHS = RHSTRANSFORMATIONSS tretch in they direction scale factor a y = a sin 6y = sin x y = sin (x+30)yx90 180 270 360 90 18012 1 2 AStretch in the direction scale factor1by = cos b y = sin x y = 4 sin xyx90 180 270 360 9080 112345 1 2 3 4y = cos y = cos 2 yx90 180 270 360 90 18012 1 Translation y = sin ( + c) - c y = cos + d 0 dy = cos y = cos +2yx90 180 270 360 90 18012345 1A8 Exponentials and Logarithms A function of the form y = ax is an exponential function The graph of y = ax is positive for all valuesof x and passes through the point (0,1)

7 A Logarithm is the inverse of anexponential functiony=ax fix = loga 32 = 5 because 25 = 32 LEARN THE FOLLOWING loga a = 1 loga 1 = 0loga ax =xal ogx =xLaws of Logarithmsloga m + loga n = loga mnloga m loga n = loga m n kloga m = loga mkSolve logx 4 + logx 16 = 3logx 64 = 3x3 = 64x = 4y= 3 x y = 4x y = 2xyx1 2 3 1 2 An equation of the form ax =b can be solved by taking logs of both sides9 Differentiation and Integration Ify=xn thendy=nxdxn 1 xn dx =xn + 1n+ 1+c for all values of n except n = 1 +x2x= x 12 +x32 = 2x12 +25x52 +cTRAPEZIUM RULEThe trapezium rule gives an approximation of the area under a the gap between the ordinates is h thenArea = h (end ordinates + twice sum of interior ordinates )Use the trapezium rule 4 strips to estimate the area under the graph ofy= 1 +x2fromx = 0tox = 2x y0 2 3 112 1 Area = x ( 1 + + 2( + 2))= (3sf)