4037 ADDITIONAL MATHEMATICS

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1 CAMBRIDGE INTERNATIONAL EXAMINATIONS GCE Ordinary Level MARK SCHEME f the May/June 04 series 407 ADDITIONAL MATHEMATICS 407/ Paper, maimum raw mark 80 This mark scheme is published as an aid to teachers and candidates, to indicate the requirements of the eamination. It shows the basis on which Eaminers were instructed to award marks. It does not indicate the details of the discussions that took place at an Eaminers meeting befe marking began, which would have considered the acceptability of alternative answers. Mark schemes should be read in conjunction with the question paper and the Principal Eaminer Rept f Teachers. Cambridge will not enter into discussions about these mark schemes. Cambridge is publishing the mark schemes f the May/June 04 series f most IGCSE, GCE Advanced Level and Advanced Subsidiary Level components and some Ordinary Level components.

2 Page Mark Scheme Syllabus Paper GCE O LEVEL May/June rationalise the denominat to get ( + ) ( + ) squaring to get ( )( + ) their4 9 + oe isw 4 4 Crectly eliminate y better ( k 9) + [ 0] + = Use b 4ac oe better oe Reach ( k 9 = ±4) their solves their ( k 8k + 6) = 0 k = and cao + ( ) squaring to get better rationalising the denominat to get their( )( + ) better crect simplification Allow k + = etc oe allow even if terms not collected; condone = y provided later wk implies it should be 0 must be applied to a term quadratic epression containing k as a coefficient; condone < 0 etc. condone 9 k = ± 4 ; condone an inequality at this stage mark final answer, do not isw; A0 if inequalities f final answers Cambridge International Eaminations 04

3 Page Mark Scheme Syllabus Paper GCE O LEVEL May/June (i) ( ) 4( ) 7( ) + = 0 d with completion to d = 0 B at least d = 0, d = 0; N.B. = 0 must be seen implied by = d = d, may be seen in following step. convincingly showing ( ) 4( ) 7( ) + 0 = 0 ; at least = 0 crect synthetic division at least as far as isw a =, b = 7, c = 0 isw B,, 0 each err; must be seen referenced in even if found in (i) (iii) (iii) ( + )( )( ) f factising quadratic ft crect; condone omission of ( + ) f ft crect use of fmula ft crect completing the square,, If M0 then SC f all three roots stated without wking verified/found by trials Cambridge International Eaminations 04

4 Page 4 Mark Scheme Syllabus Paper GCE O LEVEL May/June (i) 7 + isw 4 4 B,,,0 one mark f each of p, q, r crect in a crectly fmatted epression; allow crect equivalent values; 7 If B0 then SC f SC f crect values seen in increct fmat e.g f a crect completed square fm of the iginal epression in a different but crect fmat. e.g their 7 4 their 0. Bft strict ft ; their 7 4 must be a proper fraction decimal rounded to sig figs me truncated to 4 figs me their = oe 4 Bft strict ft ; must be crectly attributed (i) B,, 0 each err if B0 then f crect terms seen; may be unsimplified e.g. 4, ( 4), ( 4) ( 0) = a + their soi condone sign errs only; must be their 0 from (i) a = validly obtained ( their 0 ) a + ( 60) = b + their soi condone sign errs only ; must be their 0 and their60 from (i) and their a if used b = validly obtained Cambridge International Eaminations 04

5 Page Mark Scheme Syllabus Paper GCE O LEVEL May/June (a) (i) B = B + B as final answers (b) log log a seen implied B* may be implied by log log log log = seen implied log log = log B Bdep* not from wrong wking log 4 cao B must be 4 not e.g. ; with no wrong wking seen 7 (i) 4 ( e ) 4 e + isw + cos ( sin ) isw B + B B each term of the sum crect; must be a sum of two terms B f ( k ± sin ) + cos and k a constant (iii) d d ( sin ) cos = soi B d d ( + ) = soi B ( + ) their cos their ( + ) sin isw Bft f crect fm of quotient rule ft their cos and their ; allow crect use of product and chain rules to obtain sin ( + ) + cos ( + ) oe Cambridge International Eaminations 04

6 Page 6 Mark Scheme Syllabus Paper GCE O LEVEL May/June Substitution of either y + into equation of curve and brackets epanded 8 0[ 0] + y[ = 0] = Solving their quadratic (, 6) oe and (, 0) oe isw 7 6 cao isw y obtained *+* Bdep* condone one sign err in either equation of curve epansion of brackets; condone omission of = 0, BUT y + must be crect dep on a valid substitution attempt f crect pair of codinates crect pair of y codinates 9 (i) [ ] ( + y ) = ( + c) oe 0 = ( ( 4) + ) + c oe 6 B + B f ( ) + f valid attempt to find c; condone slips e.g. omission of power sign err ( + ) y = + c seen and c = ( + ) y = + isw ( + ) + = ( + ) d + ( + const) ( + ) +. 0 = ( (.) + ) + (.) ( ( 0) + ) + 0 B + B Bft must have y =.; condone f() =... + B f ( + ), B f ( + ) B ft their c from (i) provided c 0 f a genuine attempt to find F(.) F(0) in an attempt to integrate their y; if their F(0) is 0 must see at least their F(.) 0; condone + c as long as their c is not numerical oe isw 0 if decimal.7 me accurate e.g..66 Cambridge International Eaminations 04

7 Page 7 Mark Scheme Syllabus Paper GCE O LEVEL May/June (i) Taking logs of both sides log y = log A + log b any base; must be an eplicitly crect statement crect fm; any base; no recovery from increct method steps b: awrt to one sf isw awrt 4 to one sf isw B gradient f b = e their soi; their gradient must be crectly evaluated as rise/run A: awrt 0. to one sf B B f A = e SC f A = e = 0. 7 (giving an awrt 0.7) (iii) Evidence of graph used at ln y =.4 soi 0 their0. = ( their4).9 = their(.4) + their 0.6 ln( 0) = ln( their4) + ln( their0.) awrt 4.4 to two sf Cambridge International Eaminations 04

8 Page 8 Mark Scheme Syllabus Paper GCE O LEVEL May/June (i) f ( ) > [ f ( ) ](, ) B condone y > + = y ( ) = log ( + ) f y + = mark final answer log ( y + ) = and f ( ) = log( + ) log f ( ) ( ) f + = log (any base f this fm) Domain > Bft ft their range of f provided mathematically valid inequality interval Range f () > B condone f() > y > (iii) ( ) oe isw B e.g. ( ) + ( ) + + ( ) 0 = leading to = 0, impossible oe B = 0 of gf which is outside domain = = 0 ( ) = = 0 0 is not in the domain (and so gf ( ) = 0 has no solutions) [ = ] = 0 Cambridge International Eaminations 04

9 Page 9 Mark Scheme Syllabus Paper GCE O LEVEL May/June (i) d y = d Solving their [ 0 by factising quadratic fmula completing the square B attempt at differentiation resulting in quadratic epression with two terms crect; allow = Y < > [ 0 omitted here. Critical values and 4 Y, [ 4 A0 if spurious attempt to combine; mark final answer Evaluating their dy at = d Use of m m = to get m nmal = their ( ) must be eplicit statement of gradient of nmal ; may be seen in equation y = 8 soi B y their8 = their ( ) y = their + c and c = their7 isw P(0, 7) cao ft B ft their m provided a genuine attempt at m ; no ft if m = their m nmal tangent Cambridge International Eaminations 04

4037 ADDITIONAL MATHEMATICS

4037 ADDITIONAL MATHEMATICS CAMBRIDGE INTERNATIONAL EXAMINATIONS Cambridge Ordinary Level MARK SCHEME f the October/November 05 series 07 ADDITIONAL MATHEMATICS 07/, maximum raw mark 80 This mark scheme is published as an aid to