Mathematics 3 Outcome 1. Vectors (9/10 pers) Lesson, Outline, Approach etc. This is page number 13. produced for TeeJay Publishers by Tom Strang

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1 Vectors (9/0 pers) Mathematics 3 Outcome / Revise positio vector, PQ = q p, commuicative, associative, zero vector, multiplicatio by a scalar k, compoets, magitude, uit vector, (i, j, ad k) as well as directio vectors ad directio cosies. Revise sectio formula (ot compulsory i S5) Page 44 Ex Page 4 Ex Qu 6-8 Page 46 Ex Qu,, 9 Revise scalar product ad its usage i calculatig agles ab. => cosθ = a b Go over the fact that scalar product is distributive => a. (b + c) = a. b + a. c ad the fact that if a. b = 0 => a is perpedicular to b Go over example of fidig a uit vector u which is at 45 to vector a ad is 60 to vector b. 3 Itroduce the Right Haded Vector system (ad L.H.) Defie a x b as a ew vector c, i.e. a x b = c where b θ a x b = a b si θ a = (area of parallelogram) ad where [ a, b, (a x b )] forms a right haded system Show how to calculate a x b usig :- i j k a b = a a a 3 b b b 3 => a x b = ( ab ba ) i ( ab ba ) j + ( ab ba ) k Do a few examples Page 5 Ex 4 Qu, a, b, 5, 8, 9 a, b, 3-4 examples o fidig a x b 4 Show that (a x b) = (b x a) Show that a x (b x c) = (a.c)b (a.b)c (vect. triple product) Page 5 Ex 4 Page 9 Ex Qu 3, 4, 6, 7, 4, a, b Calculate the area of triagle ABC where A(,3, ), B(4,3,0) ad C(,,) usig vector product (ad area = AB AC ) be able to fid a uit vector perpedicular to vectors a & b 5 Equatio(s) of a lie i 3 dimesios:- (i) Vector form r = a + td (ii) Parametric form x = a + tl y = b + tm z = c + t A d a O R r Page 66 Ex 9A Page Ex 3 Qu, a, b,, a, 3, a, c, e, 5 (iii) Symmetric form x a y b z c = = = l m t cot d... This is page umber 3

2 Vectors (9/0 pers) (cot d...) Mathematics 3 Outcome... cot d 5 Poit out that to fid the equatio(s) of a lie, you require two thigs: a poit o the lie ad the directio of the lie. Example:- Fid lie joiig A(,0,) to B(,,0) Show that equatio of a lie is ot uique i how it appears. Page 67 Ex 9B Qu 6 Equatio of a plae ( ) Discuss the NORMAL () to a plae If A is a kow poit o the plae ( ), the ormal ad X is ay other poit o the plae, the :-. x =. a A O a = x l m X Page 57 Ex 6 Page 5/6 Ex 4 Qu, a, b,, a, 3 4, a, c, 5, a, 9, 0 => lx + my + z = la + mb + c or lx + my + z = k (Cartesia form) Show also parametric form r = a + λu + µv Establish that to fid the equatio of a plae you require two facts: a poit A o the plae ad the ormal () to it. 7/8 (i) Agles betwee two lies = agle betwee the two directio ratios => use the scalar product L θ L Page 70 Ex page Ex 5 Qu Qu, (ii) Agle betwee two plaes = agle betwee the two ormals. Page 59 Ex 7A Page Ex 5 Qu Qu,, 3 => use. (etc.) (iii) Agle (θ) betwee a lie (L) ad a plae ( ) => θ = (90 φ) where φ is the agle betwee the lie L ad the ormal to the plae φ θ L Page 68 Ex 0 Page Ex 5 Qu 3 Qu (ii) a, b, c Qu a, b, 3, 4, a => simply fid φ from l. 9/0 cot d... This is page umber 4

3 Vectors (9/0 pers) (cot d...) Mathematics 3 Outcome 9/0 (i) Itersectio of two lies Study the set of 3 simultaeous equatios, solve ay two ad check the solutio i the third. Page 70 Ex Page Ex 5 Qu 4 Qu, (ii) Lie of itersectio of two plaes (a) use x to get directio of the lie of itersectio Page 7 Ex Qu, Page Ex 5 Qu 5. a/b (b) set x (or y or z) = ay val;ue ad use this to fid the values of the other two variables, givig the coordiates of a poit o the lie. etc. (iii) Itersectio of three plaes (a) sigle poit } (b) meet o a lie (use Gaussia elimiatio) Page 78 Ex 5 Qu, a, c,, a, c Page Ex 5 Qu 5. c (c) do t meet at all (iv) Itersectio of a lie ad a plae x = a + tl, y = b + tm, z = c + t () ad px + qy + rz = k () => substitute equatios () ito () ad solve Page 68 Ex 0 Page Ex 5 Qu 6- Qu (i) d, e, f Qu, c, 4, b, c Cumulative total = periods (+ two weeks for prelims ) This is page umber 5

4 Matrices (7 pers) (or codesed to 5/6)* Mathematics 3 Outcome Explai Matrix, term/elemet, row, colum, order, equatig matrices ad traspose, square, zero matrix. Page 4 Ex, Qu,, 3a Page 7, Ex 4 a, c, e, 0 Go over addig, subtractig, ad multiplyig by a real umber ad establish that A + B = B + A, (A ) = A (A + B) = A + B Page 5 Ex Qu 6 g, i, p, Page 8, Ex, Page 9 r, t, 7a, f, 9, 0 Ex 3 & page 30 Ex 4 3/4 Show how to multiply two matrices which are coformable ad fid A etc ad establish rules AB BA, (AB)C = A(BC), A(B + C) = AB + AC, (AB) = B A 5 Defie det A = A for x ad 3 x 3 matrices ad show that det (AB) = deta x detb page 0 Ex 3 Page 33, Ex 5 Qu a, c,, a, c, k, m, o 3 a, 4, 5, a, c Page Ex 4A, 6, 7, 8 Page 6 Ex 5 Qu, b, d, h Page 35, Ex 6 Page 5, Ex 7, Qu 4, 5a, b 6 Fid iverse A - of a x matrix ad show that A - exists iff det A 0. Show that (AB) - = B - A - 7. Fid the iverse of a 3 x 3 matrix usig elemetary row operatios (possibly covered i Maths ) Page 9 Ex 6A, Page 38, Ex 8 Qu,, 4,, 8, 9 (some) Page 8 Ex 8, Page 4 Ex 9 Qu 8 Solve simultaeous equatios i ad 3 ukow usig matrix iverses. 9 Use matrices to represet simple geometric trasformatios such as reflectio, rotatio, ad dilatatio (dilatio) Page 8 Ex 8 Page 4 Ex 0 Qu 3 page 4 Ex Page 3 ex 9a Page 44 Ex Qu,, 5 (some) 6 * ote if the work o Matrices i Maths has bee exteded, this ca be shorteed to 5-6 periods Cumulative total = 0 periods (+ two weeks for prelims 30) This is page umber 6

5 Further Sequeces ad Series (4 periods) Mathematics 3 Outcome 3 Defie a Power Series ad the Maclauri series:- f ( 0) f ( 0) f ( 0) 3 f( x) = f( 0) + x + x + x +...!! 3! 3 x x x x Show that e = x!! 3! 3 x x l( + x) = x < x <! 3! x x x x six = x! 3! 5! 7! 4 6 x x x cos x = x! 4! 6! x x x ta x = x < x < ( + x ) =... + x + x + x Page 9 Ex 4 Page 53 Ex Qu a, f Page 95 Ex 6 Qu a, f,, a, d Qu 3 a, 5 a, 6 a (i) Combie expasios x e.g. show how to expad e si 3x, l(cosx), etc 3/4 Iteratio:- Remid of simple Iteratio techiques leared i S4/5 Show how to rearrage the fuctio, x 4x 8 = 0 to x = x => x x 4 = 4, ( g( x) = x ) 4 x = 8 => x x 4 = 8, ( g( x) = 8 x 4 x 4 ) x = ( x + ) => x = ( x + ), ( g( x) = ( x + ) ) Page 96 Ex 6 Page 54 Ex Qu 7, 8 Page 99 ex 8 Page 6 Ex 3 page 0 Ex 9 Qu,, 4 (hard) ad use a calculator (computer) to fid solutios to the above recurrece relatios. Go over idea of staircase ad cobweb diagrams ad the test for covergece (if a root at α, the will coverge to iff g ( α) < ) Go over -3 examples of testig for covergece ad homig i o solutios. Cumulative total = 4 periods (+ two weeks for prelims 34) This is page umber 7

6 Further Differetial Equatios (5/6 periods) Mathematics 3 Outcome 4 First Order Liear Differetial Equatios (FOLDE s) Go over where itegratig factor comes from & its usage dy i solvig:- + P( x) y = Q( x) => ( ) µ( x) = e P x (itegratig factor) d => ( µ ( x) y) = µ ( x) Q( x) => y = µ ( xqx ) ( ) µ ( x ) Go over example such as :- dy 3 x + ( x ) y = x /3 Go over -3 more examples like (a) dy y = 6 e x dy dy (b) ( + x ) xy = x( + x ) (c) + y = x + 4 icludig Geeral ad Particular solutios 4/5 Secod order Liear Differetial Equatios :- a d y b dy cy f x + + = ( ) Start with homogeeous case where f(x) = 0 Defie big D method => Auxiliary Equatio :- => ad + bd + c = 0 ad go over types of sol s α x (i) solutios x = α, β => y = Ae + Be (ii) solutio x = α => y = ( Ax + B) e αx (iii) complex solutios => y = e ( Asiβx + B cos βx) (x = α ± βi) αx βx Page 4 ex page 67 Ex Qu (+ some of Qu 3) page 6 Ex Qu Page 9 Ex 3 Page 69 Ex Qu a, b, a, b Page 0 Ex 4 Qu a, b, a, b Page Ex 5A Qu a, b, a, b (defie this as the Complemetary Fuctio (C.F.) Go over example of each type) 5/6 Go over o-homogeeous case where f(x) 0. (f(x) is simple polyl or asix or bcosx ) ** could exted here to iclude expoetial fuctios) ** Solutio to this called the Particular Itegral (P.I.) Page 6 Ex 7A As may as possible Page 74 Ex * assesses beyod poly l ad trig work Be able to fid the Geeral Solutio y = P.I. + C.F. ad particular Solutios give boudary coditios. Go over -3 examples Cumulative total = 30 periods (+ two weeks for prelims 40) This is page umber 8

7 Further Number Theory & Proofs (4 periods) Mathematics 3 Outcome 5 Much may already have bee covered i Maths. Revise method of Direct proof, by Cotradictio (Cotrapositive as well??) A mixture from Ex, Page 8 Ex 4 A ad B /3 Expad Iductio Prove that r= 3 r = r = ( + ) 4 r= = r r = Page 4 Ex 3A Page 85 Ex 5 As much as possible (+ further examples) 4 Go over Divisio Algorithm. Itroduce the Euclidia Algorithm Show that if the g.c.d (a,b) = d the d = xa + yb for uique itegers x ad y. (Possibly some base work if time) 5 Review Page 45 Ex 4 page 87 Ex 7 Qu a, c, e, g, i Page 88 Ex 8 Page 47 Ex 5 Qu - 4 (page 5 Ex 7 - some) (page 89 Ex 9) For Sessio Ed of Mathematics 3 Total = 3 periods for Mathematics 3 Cumulative Total for M, M & M3 = 34 p s Assumig Maths 3 started aroud Mar 4th, ad 8 period doe before prelims, this uit should ed about Thursday th April =>3 periods revisio (icludig specime NAB3) + period for test = 36 periods TEST aroud Wed 7th April Actual Test Date = FINAL Exam i May TeeJay Revisio Booklet (haded out Earlier) alog with TeeJay Specime papers ad Last year s paper to be used i ru-up to Exam This is page umber 9