4727 Further Pure Mathematics 3

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1 hysicsandmathstutor.com 477 Mark Scheme June Further Pure Mathematics 6 6 B For arg z seen or imlied i cos isin cos isin 8 8 cos isin, 8 8 cos 5 isin 5 8 8, For dividing arg z by i A A For any one correct root For other roots and no more in range 0 (i) 5 e B For stating correct inverse in the form r e i (ii) i i i( ) re re rre For stating distinct elements multilied A For showing roduct of correct form (iii) i Z e B i For e seen or imlied i i e B For correct answer. aef 5 (i) [6 4, 7 8, 0 7 ] on ( 6 4 ) 4( 7 8 ) ( 0 7 ) 8 B For oint on l seen or imlied For substituting into equation of (,, ) A For correct oint. Allow osition vector (ii) METHOD n [ 4,8,7] [, 4, ] * (*de) n k [,, 8] A For correct vector For direction of l and normal of seen For attemting to find n n (,, ) OR (6, 7, 0) For finding scalar roduct of their oint on l with their attemt at n, or equivalent xy8z 6 A 5 For correct equation, aef cartesian METHOD r [,, ] OR [6, 7, 0] For stating eqtn of lane in arametric form (may be [ 4,8,7] [, 4, ] A imlied by next stage), using [,, ] (ft from (i)) Or [6, 7, 0], and n (as above) n x 4 For writing as linear equations y 84 z 7 For attemting to eliminate and xy8z 6 A For correct equation aef cartesian METHOD (6 ) 4( 74 ) ( 0 ) 8 For finding foot of erendicular from oint on l to (0,, 6) A For correct oint or osition vector From oints (,, ), (6, 7, 0), (0,, 6), n = vector roduct of of [,0,], [6, 8, 4], [ 4,8,7] Use vector roduct of vectors in lane n k [,, 8] (,, ) OR (6, 7, 0) For finding scalar roduct of their oint on l with their attemt at n, or equivalent xy8z 6 A For correct equation aef cartesian 8 4

2 hysicsandmathstutor.com 477 Mark Scheme June (i) IF e x dx ln x x d x dx x e (ii) y x x y x x x x c (0, ) c c 4 4 x y x x x A For IF stated or imlied. Allow and omission of dx For integration and simlification to AG (intermediate ste must be seen) * For multilying both sides by IF A (*de) (*de) A 6 8 For integrating RHS to k x For correct equation (including + c) In either order: For substituting (0, ) into their GS (including +c) For dividing solution through by IF, including dividing c or their numerical value for c For correct solution aef (even unsimlified) in form n y f( x) 5 (i) m 6m9( 0) m A For attemting to solve correct auxiliary equation For correct m CF = ( A Bx)e x A For correct CF (ii) ke x and kx e x both aear in CF B For correct statement (iii) y kx e x y kxe kx e x For differentiating kx e x twice A For correct y aef x x x y ke kxe 9kx e A For correct y aef x ke x9x x8x 9x x e For substituting y, y, y into DE k A 5 For correct k 9 5

3 hysicsandmathstutor.com 477 Mark Scheme June (i) METHOD n [,, 0] [, 5, ] For attemting to find vector roduct of the air of direction vectors [,, 6] k[,, ] A For correct n Use (,,) For substituting a oint into equation r. [,, 6] 6 r. [,, ] A 4 For correct equation. aef in this form METHOD x y 5 z For writing as linear equations For attemting to eliminate and x yz A For correct cartesian equation r. [,, ] A For correct equation. aef in this form (ii) For r atb METHOD b [,, ] [7, 7, ] k [,, ] A For attemting to find n n For a correct vector. ft from n in (i) x yz e.g. x, y or z 0 in 7x7yz For attemting to find a oint on the line a 0,, OR, 0, 0 OR [,, ] A For a correct vector. ft from equation in (i) SR a correct vector may be stated without working Line is (e.g.) r [,,] t [,, ] A 5 For stating equation of line ft from a and b SR for a [,,] stated award M0 METHOD x yz In either order: Solve 7x7yz For attemting to solve equations by eliminating one variable (e.g. z) Use arameter for another variable (e.g. x) to find other variables in terms of t For attemting to find arametric solution (eg) y t, z t A A For correct exression for one variable For correct exression for the other variable ft from equation in (i) for both Line is (eg) 0,, t [,, ] solutions METHOD x yz eg x, y or z 0 in 7x7yz For attemting to find a oint on the line a 0,, OR, 0, 0 OR [,, ] A For a correct vector. ft from equation in (i) SR a correct vector may be stated without working SR for a [,,] stated award M0 eg [, 0, 0] [,, ] For finding another oint on the line and using it with the one already found to find b b k [,, ] A For a correct vector. ft from equation in (i) Line is (eg) r [,,] t [,, ] A For stating equation of line. ft from a and b 6

4 hysicsandmathstutor.com 477 Mark Scheme June (ii) contd METHOD 4 A oint on is [, 5, ] On [, 5, ] [7,7, ] For using arametric form for and substituting into. A For correct unsimlified equation A For correct equation Line is (e.g.) r [,,] ()[,, 0] [, 5, ] For substituting into for or [,,] or 7,, t [,, ] A For stating equation of line 9 7 (i) cos isin c i c scs i s For using de Moivre with n cos c cs and A For both exressions in this form (seen or imlied) sin c ss SR For exressions found without de Moivre M0 A0 c s s tan sin For exressing in terms of c and s c cs cos tan tan tan ( tan ) tan A 4 For simlifying to AG tan tan (ii) (a) tan (iii) t t( t ) t t t0 B For both stages correct AG (b) ( t)( t 4t) 0 For attemt to factorise cubic A For correct factors ( t ), t A For correct roots of quadratic sign for smaller root A 4 For choice of sign and correct root AG tan d t ( t )d B For differentiation of substitution and use of sec tan 0 tan d B For integral with correct θ limits seen ln sec ln sec 0 6 ln ln 4 For integrating to k ln sec OR k ln cos A 5 4 For substituting limits and sec OR cos seen 4 For correct answer aef 4 7

5 hysicsandmathstutor.com 477 Mark Scheme June (i) (ii) a a aa a a B For use of given roerties to obtain AG a aa aa B For use of given roerties to obtain AG SR allow working from AG to obtain relevant roerties 4 e order = B For correct order with no incorrect working seen a e order a = 4 B For correct order with no incorrect working seen a 4 a 4 e order a = 4 B For correct order with no incorrect working seen a a a a. a. a OR a a. a a a 6 a a order a = 4 (iii) METHOD a, a a {, ea,, a} {, eaa,, a} which is a cyclic grou METHOD e a a e e a a a a a e a e a a a e a Comleted table is a cyclic grou METHOD e a a e e a a a a a e a e a a a e a Identity = e Inverses exist since EITHER: e is in each row/column OR: is self-inverse; a, a form an inverse air A 5 M A A 4 A B A B B For relevant use of (i) or given roerties For correct order with no incorrect working seen For use of the given roerties to simlify and a For obtaining a and a For justifying that the set is a grou For attemting closure with all 9 non-trivial roducts seen For all 6 roducts correct For justifying that the set is a grou For attemting closure with all 9 non-trivial roducts seen For all 6 roducts correct For stating identity For justifying inverses ( e e may be assumed) 8

6 hysicsandmathstutor.com 477 Mark Scheme June 009 (iv) METHOD e.g. a. a a a. a commutative not METHOD Assume commutativity, so (eg) a B A 4 For attemting to find a non-commutative air of elements, at least one involving a (may be embedded in a full or artial table) For simlifying elements both ways round For a correct air of non-commutative elements For stating Q non-commutative, with a clear argument a For setting u roof by contradiction (i) a. a a. a a For using (i) and/or given roerties But and are distinct B For obtaining and stating a contradiction Q is non-commutative A For stating Q non-commutative, with a clear argument 5 9

physicsandmathstutor.com 4727 Mark Scheme June 2010

physicsandmathstutor.com 4727 Mark Scheme June 2010 477 Mark Scheme June 00 Direction of l = k[7, 0, 0] Direction of l = k[,, ] EITHER n = [7, 0, 0] [,, ] For both directions [ x, y, z]. [7,0, 0] = 0 7x 0z = 0 OR [ x, y, z]. [,, ] = 0 x y z = 0 n = k[0,,