EDEXCEL STUDENT CONFERENCE 2006 A2 MATHEMATICS STUDENT NOTES

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1 EDEXCEL STUDENT CONFERENCE 006 A MATHEMATICS STUDENT NOTES South: Thursday 3rd March 006, Lodo

2 EXAMINATION HINTS Before the eamiatio Obtai a copy of the formulae book ad use it! Write a list of ad LEARN ay formulae ot i the formulae book Lear basic defiitios Make sure you kow how to use your calculator! Practise all the past papers - TO TIME! At the start of the eamiatio Read the istructios o the frot of the questio paper ad/or aswer booklet Ope your formulae book at the relevat page Durig the eamiatio Read the WHOLE questio before you start your aswer Start each questio o a ew page (traditioally marked papers) or Make sure you write your aswer withi the space give for the questio (o-lie marked papers) Draw clear well-labelled diagrams Look for clues or key words give i the questio Show ALL your workig - icludig itermediate stages Write dow formulae before substitutig umbers Make sure you fiish a prove or a show questio quote the ed result Do t fudge your aswers (particularly if the aswer is give)! Do t roud your aswers prematurely Make sure you give your fial aswers to the required/appropriate degree of accuracy Check details at the ed of every questio (e.g. particular form, eact aswer) Take ote of the part marks give i the questio If your solutio is becomig very legthy, check the origial details give i the questio If the questio says hece make sure you use the previous parts i your aswer Do t write i pecil (ecept for diagrams) or red ik Write legibly! Keep goig through the paper go back over questios at the ed if time At the ed of the eamiatio If you have used supplemetary paper, fill i all the boes at the top of every page

3 C3 KEY POINTS C3 Algebra ad fuctios Simplificatio of ratioal epressios (uses factorisig ad fidig commo deomiators) Domai ad rage of fuctios Iverse fuctio, f () [ ff () f f() ] Kowledge ad use of: domai of f rage of f ; rage of f domai of f Composite fuctios e.g. fg() The modulus fuctio Use of trasformatios (as i C) with fuctios used i C3 Trasformatio Descriptio 0 y f() + a a > 0 Traslatio of y f() through a a y f( + a) a > 0 Traslatio of y f() through 0 y af() a > 0 Stretch of y f() parallel to y-ais with scale factor a y f(a) a > 0 Stretch of y f() parallel to -ais with scale factor a y f() y f( ) For y 0, sketch y f() For y < 0, reflect y f() i the -ais For 0, sketch y f() For < 0, reflect [y f() for > 0] i the y-ais Also useful y f() Reflectio of y f() i the -ais (lie y 0) y f( ) Reflectio of y f() i the y-ais (lie 0) C3 Trigoometry sec cosec cos si cot cos ta si si + cos ; + ta sec ; + cot cosec si(a ± B) siacosb ± cosasib cos(a ± B) cosacosb m siasib ta A ± ta B ta(a ± B) m ta Ata B si si cos ; cos cos si cos si ; ta ta ta

4 Graphs of iverse trig. fuctios π / arcsi π / 0 arccos π π / < arcta < π / Epressig acosθ + bsiθ i the form rcos(θ ± α) or rsi(θ ± α) ad applicatios (e.g. solvig equatios, maima, miima) C3 Epoetials ad logarithms Graphs of y e ad y l ad use of trasformatios to sketch e.g. y e 3 + Solutios to equatios usig e ad l (e.g. e + 3) e la a C3 Differetiatio d(e ) e k d(e ) e k d(l ) d(l k) d(si k) d(cos k) k cos k d(ta k) k si k k sec k Differetiatio of other trig. fuctios: see formulae book Chai rule Product rule d y d y du du d d( uv) d v u d du + v d d y Quotiet rule dy d( u v ) du dv v u v C3 Numerical methods For a cotiuous fuctio, a chage i sig of f() i the iterval (a, b) a root of f() 0 i the iterval (a, b) Accuracy of roots by choosig a iterval (e.g..47 to d.pl. test f(.465) ad f(.475) for chage of sig) Iterative methods: rearragig equatios i the form + f( ) ad usig repeated iteratios

5 C4 KEY POINTS C4 Algebra ad fuctios Partial fractios: Methods for dealig with degree of umerator degree of deomiator, partial fractios of the form + 3 ( )( + ) A + B + C 5 A B C ad ( + )( 3) + 3 ( 3) C4 Coordiate geometry Chagig equatios of curves betwee Cartesia ad parametric form Use of y dt to fid area uder a curve dt C4 Sequeces are series Epasio of (a + b) for ay ratioal ad for < ab, usig ( + ) r + + C C where C r! r!( r)! r ( ) ( )( ) ! 3! C4 Differetiatio Implicit ad parametric differetiatio icludig applicatios to tagets ad ormals Epoetial growth ad decay d( a ) a l a Formatio of differetial equatios C4 Itegratio e e + c l + c e k k e k + c a a a a. l + c or l a + c a cos k k si k + c si k k cos k + c sec k k ta k + c Use of f ( ) l f ( ) + c f ( ) ad [f ( )] f ( )[f ( )] c Itegratio of other trig. fuctios: see formulae book Volume: use of π y whe rotatig about -ais Itegratio by substitutio Itegratio by parts

6 Use of partial fractios i itegratio Differetial equatios: first order separable variables dy g( y) h( ) e.g. f( )g( y) h( )k( y) y d k( y) f ( ) Trapezium rule applied to C3 ad C4 fuctios b a f ( ) h[y 0 + y + (y y )] where y i f(a + ih) ad h b a C4 Vectors If a i + yj + zk, a ( + y + z ) If a i + yj + zk, the uit vector i the directio of a is [(i + yj + zk) ( + y + z )] Scalar product: If OP p i + yj + zk ad OQ q ai + bj + ck ad POQ θ, the p.q p q cosθ ad p.q (i + yj + zk). (ai + bj + ck) a + by + cz If OP ad OQ are perpedicular, p.q 0 Vector equatio of lie where a is the positio vector of a poit o the lie ad m is a vector parallel to the lie: r a + λm Vector equatio of lie where a ad b are the positio vectors of poits o the lie: r a + λ(b a)

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PhysicsAndMathsTutor.com PhysicsAdMathsTutor.com physicsadmathstutor.com Jue 005 3. The fuctio f is defied by (a) Show that 5 + 1 3 f:, > 1. + + f( ) =, > 1. 1 (4) (b) Fid f 1 (). (3) The fuctio g is defied by g: + 5, R. 1 4 (c)