ADDITIONAL MATHEMATICS 4037/12 Paper 1 October/November 2016 MARK SCHEME Maximum Mark: 80. Published

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1 Cambridge International Eaminations Cambridge Ordinary Level ADDITIONAL MATHEMATICS 07/ Paper October/November 06 MARK SCHEME Maimum Mark: 80 Published 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 before 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 Report for Teachers. Cambridge will not enter into discussions about these mark schemes. Cambridge is publishing the mark schemes for the October/November 06 series for most Cambridge IGCSE, Cambridge International A and AS Level components and some Cambridge O Level components. IGCSE is the registered trademark of Cambridge International Eaminations. This document consists of 6 printed pages. UCLES 06 [Turn over

2 Page Mark Scheme Syllabus Paper Cambridge O Level October/November Abbreviations awrt cao dep FT isw oe rot SC soi www answers which round to correct answer only dependent follow through after error ignore subsequent working or equivalent rounded or truncated Special Case seen or implied without wrong working (a) (i) 0 (iii) (b) (i) Q R P Q, or { } a, b, c B for each y + 5y 0, for 5y or 5log, for y, y,., 9, for correct attempt at the solution of their quadratic equation for dealing with one base logarithm correctly for each (i) B for each term, powers of must be 9 simplified Coefficients needed: 80 their + their 8 ( ) for dealing with terms Allow for 7 8 UCLES 06

3 Page Mark Scheme Syllabus Paper Cambridge O Level October/November (i) d y d ( + ) dy When, y 0, d Equation of normal: y + for correct derivative of log function for y 0 for attempt at a gradient of a perpendicular from differentiation and the equation of the normal Q 0, 9 or ( ) 0,0. or better R 0, ln or ( 0, 0.5 ) or better Area of PQR ln ft Follow through on their c from part (i) Allow (a) YX, XZ B B for both with no etras for correct with or without etras for both correct with etras B0 for anything else (b) (i) 7 8, for 8, for 7 C A - B , for pre-multiplication for any correct pair of elements, but must be from correct matrices UCLES 06

4 Page Mark Scheme Syllabus Paper Cambridge O Level October/November ( 0, ) or ( 0,.7 ) or better 7 (i) π, 6 or ( ) 0.5, or better, for each (iii) π cos 0 6 π oe or.09 or better for correct attempt to solve trigonometric equation (iv) π sin ( +c) 6 (v) Area π sin (i) 7 θ θ, so θ.97 θ.9 to dp CD θ sin CD awrt 9.6 or 9.7 π 0 or better (iii) Area of sector awrt 8 Area of triangle AOB awrt 67 or 68 Area of segment awrt 70 or 7 AD AB + segment area 5 leading to AD awrt 8. or 8.0 Alternative method: Area of sector awrt 8 Difference in length between BC (or AD) and OM where M is the midpoint of CD 6.88, allow awrt 6.9 Remaining area consists of two trapezia each of width 9.85 and each of area. ( BC 6.88 ) oe leading to AD awrt 8. or 8.0 for correct use of their limits, in radians, π into ksin. 6 for complete correct method to get θ must have evidence of working to more than dp, allow if.96 seen (truncated) for a complete method, may use cosine rule to get CD for sector area, allow unsimplified for a correct attempt at area for segment area (their sector area their triangle area) for complete method to find AD Allow for 8 for sector area for attempt to find difference between parallel sides for area of one trapezium ( BC their 6.88 ) their 9.85 oe for attempt to find either BC or AD UCLES 06

5 Page 5 Mark Scheme Syllabus Paper Cambridge O Level October/November a 9 b 8 9 p a 8 + b 0 leading to 9a+ b + 0 oe and 7a+ b 8 0 oe 9 (i) p : ( 0) leading to a, b 5 ( ) for attempt at p for differentiation and attempt at p for solution of simultaneous equations, to get either a or b for both ( )( ) + oe, for attempt at long division or factorisation (iii) ( )( ) , ( ) leading to, Must be using ( + ) correctly using part to get for solution of the quadratic equation 0 (a) (i) U 0U + U leading to U 6 D for realising that area under the graph is needed and attempt to find an area for equating their area to 65 and attempt to solve Gradient of line: 0., for use of the gradient, must be negative (b) (i) 7 t 8ln t. or better for a correct attempt to solve t e 8 (iii) acceleration t t t e 8 e 8 8, for a correct attempt to differentiate using the chain rule When t, a 6.98, for use of t in their acceleration UCLES 06

6 Page 6 Mark Scheme Syllabus Paper Cambridge O Level October/November (i) ln y ln A+ ln b 0. Gradient: ln b, b Intercept: ln A 0.6 A.0 Alternative ln y ln A+ ln b 0. ln b+ ln A 0.08 ln b+ ln A A.0 and b Alternative. Ab.08 Ab A.0 and b When 6, ln y 0.7 y.9 (iii) When y., ln y D D,, may be implied, if equation not seen specifically, by correct values for A and b for use of gradient to obtain ln b Allow for e 0.05 for use of one of the given points correctly Allow for 0.6 e or. for one correct equation for attempt to obtain either lna or lnb from simultaneous equations 0.05 Allow for b e 0.6 and a e or. for correct attempt to obtain b or A, must already have B 0.05 Allow for b e 0.6 and a e or. for ln y their ln A + 6 their ln b or y their A ( their b) 6 allow awrt.8 to.0 for ln. their ln A + their ln b or ( theirb). theira allow 0.5 to.5 UCLES 06

7 Cambridge International Eaminations Cambridge Ordinary Level ADDITIONAL MATHEMATICS 07/ Paper October/November 06 MARK SCHEME Maimum Mark: 80 Published 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 before 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 Report for Teachers. Cambridge will not enter into discussions about these mark schemes. Cambridge is publishing the mark schemes for the October/November 06 series for most Cambridge IGCSE, Cambridge International A and AS Level components and some Cambridge O Level components. IGCSE is the registered trademark of Cambridge International Eaminations. This document consists of 8 printed pages. UCLES 06 [Turn over

8 y Page Mark Scheme Syllabus Paper Cambridge O Level October/November Abbreviations awrt answers which round to cao correct answer only dep dependent FT follow through after error isw ignore subsequent working oe or equivalent rot rounded or truncated SC Special Case soi seen or implied www without wrong working for symmetrical shape as in the diagram with curved maima of equal height and cusps on the -ais for a complete curve with all low points on the -ais and all high points on y for a complete curve meeting the o o o -ais at 0, 90, 50 only. m 9 m + ( )( ) m m+ m m + Alternative Method 9 ( m m ) m ( m + ) (Am + B) m Comparing coefficients A, A + B 0, B 9 for multiplying each term by m, using a common denominator of m or for multiplying numerator and denominator by m m for a correct epression that will cancel ( m )( m+ ) ( m 9 )( m ), m + (m 9) ( m )( m+ )(m ), or equivalents (m+ )(m ) for m or A, B for correct epansion for correct comparisons to obtain A and B for m or A, B UCLES 06

9 Page Mark Scheme Syllabus Paper Cambridge O Level October/November (i) p+ ( p+ ) 0 ( + ) ( p) p 0 oe D for obtaining a -term quadratic in the form a + b + c ( 0) for correct substitution of their a, b and c into b ac and use of discriminant. p ( p+ ) or p p 9 0 p p 6 0 for full correct working, the only sign used, used before division by and used in answer line and penultimate line. Correct method of solution p p 9 0 leading to critical values ± 5 p for correct substitution in the quadratic formula or for correct attempt to complete the square. (allow sign error in either method) for both correct critical values 5 5 p, p + for correct range (i) B for each correct term ( '6') + ( ' 8' ) + ( '5') for correctly obtaining three products using their coefficients in (i) for two correct out of three products (unsimplified) cao 05 cao for 05 selected as final answer 5 (i) log9 y log9 + log9 y for use of log AB loga + logb log log y + log 9 log 9 log log y 5 + log + log y 5 Alternative method 5 log9 y 5 y 9 log y 5 log + log y 5 5 for correct method for change of base. Division by log 9 should be seen and not implied. for dealing with correctly and finishing off for obtaining y as a power of for correct use of log for using law for logs and arriving at correct answer UCLES 06

10 Page Mark Scheme Syllabus Paper Cambridge O Level October/November ( ) log 5 log 6 ( ) log + 5log 6 ( ) log 5log 6 0 leading to log 6, log 79, y, y 79 D for substitution, correct epansion of brackets and manipulation to get a term quadratic for a correct quadratic equation in the form a + b + c 0 for both solutions for method of solution of log k or log y k for all and y correct 6 (i) 6 m for p FT for 6 p m ln ln ln 6 6 (iii) ( a ) ( a ) 6 ln ln a 6 a 5 only D D for correct use of limits in ln ( ) implied by following equation for dealing with logs correctly for solution of a k for 5 obtained from an eact method p May be UCLES 06

11 Page 5 Mark Scheme Syllabus Paper Cambridge O Level October/November (i) ln y ln A+ b Gradient: b 0.8 Intercept or use of equation: ln A.7 A 0 Alternative Method.5 ln A+.5b and.5 ln A + b leading to b 0.8 ln A.7 and A 0 Alternative Method e.5 Ae b e.5 Ae.5b leading to b 0.8 and A 0 for equation, may be implied, must be using ln unless recovered for b 0.8 oe for ln A.7 oe, allow.65 to.75 for A 0, allow 05 to 6 Allow A in terms of e for one equation for b 0.8 for ln A.7 for A 0 or e.7 for e.5 Ae b or.8 Ae b for e.5 Ae.5b or. Ae.5b for b 0.8 for A 0 or e.7 When 0.,.5, ln y. y 9 ( allow 8.5 to 9.5) or e. for a complete method to obtain y, using either the graph, using their values in the equation for lny or using their values in the equation for y. (iii) When y 0, ln y,.5 so 0.7 ( allow 0.5 to 0.9) for a complete method to obtain, using either the graph, using their values in the equation for lny or using their values in the equation for y. UCLES 06

12 Page 6 Mark Scheme Syllabus Paper Cambridge O Level October/November (a) (i) cosecθ sinθ cosecθ sinθ sinθ sinθ sin θ sinθ ( sin θ ) sinθ or D for using cosecθ and either attempt to sinθ multiply top and bottom by sin θ or an attempt to combine terms in denominator. for correct use of -sin θ cos θ cos θ sec θ for completing the proof Alternative Method using cosec cosecθ cosecθ cosecθ sinθ cosecθ cosecθ cosec θ cosec θ + cot θ cot θ D for using sinθ and an attempt to cosecθ combine terms in denominator. for use of + cot θ cosec θ tan θ + sec θ for completing the proof cos θ, cosθ ± or tan θ, tanθ ± or sin θ, sinθ ± o o o o θ 60,0, 0, 00 for using (i) to obtain a value for cos θ, tan θ or sin θ and then taking the square root. for two correct values for two further correct values and no etras in range. (b) π tan + π π 7π π π π,, π π π,,, for correct order of operations, can be implied by π for for π π If there are etra solutions in range in addition to the two correct ones then A0 UCLES 06

13 Page 7 Mark Scheme Syllabus Paper Cambridge O Level October/November (a) (i) 8 C mmm Either 0 8 C C C C C C C C 700 Total 860 B,,0 for a correct plan B correct numbers with no etras correct numbers (out of or ) for correct total Or their 8 C ( C5 + C 5) 8568 ( ) Total 860 for correct plan for 5 subtracted for 56 subtracted for correct total (b) (i) 0 P P 060 for correct unsimplified for correct numerical answer (iii) Answer to (i) - 7 P for correct plan for correct unsimplified for correct numerical answer Alternative: symbol: 560 symbols: symbols: 500 Total: 660 B,,0 B for all correct for correct (out of or ) for correct sum UCLES 06

14 Page 8 Mark Scheme Syllabus Paper Cambridge O Level October/November (i) ( ) f e ( + c ) passing through ( 0, ) 0 0 e +c f e + ( ) f ( 0) 8 Normal: 8y+ y y+ 8 D D for one correct term for one correct term or e for a second correct term with no etras for correct method to find c. for correct equation for m 8 for equation of normal using m for solving normal equation simultaneously with y to get a value of 8 (i) 8 leads to oe 5 8 Area. oe 5 a 8t 8 When t, a 6 8 for,.6 oe 5 FT for a numerical answer equal to their for 8t 8 for 6 0.5,.5, for each (iii) s t t + t when t, s when t, s 0 D D for at least two terms correct all correct for calculating displacement when either or t for calculating displacement at doubling. t and t total distance travelled for oe allow. Alternative method D D As before Dfor calculating displacement when t 0.5 or for calculating distance travelled between t 0.5 and t.5 D for doubling distance travelled between t 0.5 and t.5 or for adding that distance to displacement at t 0.5 for oe allow. UCLES 06

15 Cambridge International Eaminations Cambridge Ordinary Level ADDITIONAL MATHEMATICS 07/ Paper October/November 06 MARK SCHEME Maimum Mark: 80 Published 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 before 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 Report for Teachers. Cambridge will not enter into discussions about these mark schemes. Cambridge is publishing the mark schemes for the October/November 06 series for most Cambridge IGCSE, Cambridge International A and AS Level components and some Cambridge O Level components. IGCSE is the registered trademark of Cambridge International Eaminations. This document consists of 6 printed pages. UCLES 06 [Turn over

16 Page Mark Scheme Syllabus Paper Cambridge O Level October/November Abbreviations awrt answers which round to cao correct answer only dep dependent FT follow through after error isw ignore subsequent working oe or equivalent rot rounded or truncated SC Special Case soi seen or implied www without wrong working 0.6 OR ( ) ( )( ) and 0.6 a( ) + b( + ) ( )( )( + ) ( ) oe a+ b a+ b 6 b and a D D www use of or ( ) but not both. solve correct term quadratic www Common denominator or ( )( + ) equate constant terms and terms. both correct solve two linear equations to obtain a or b both correct lg lg lg0 soi anywhere soi anywhere + 0 lg lg lg oe + 0 ( )( ) only soi division; logs may be removed obtain correct term quadratic equation and attempt to solve 5 must not remain. UCLES 06

17 Page Mark Scheme Syllabus Paper Cambridge O Level October/November (i) t 0 N e 8 or Do not accept non integer responses. 0.05t N e 0.05t 500 e 000 ln t ln0.5 t (days) insert and make -0.05t e subject take logs and make t the subject awrt 7.7 (iii) dn 0.05t 00e dt dn t 8 ± 67 (.0) dt 0.05t ke where k is a constant k 00 or awrt ± 67 mark final answer 5 (i) dy 7 d + dy 8 7 d Equation of tangent : y 6 y insert into their gradient and use (, 6) and their gradient of tangent in equation of line. Tangent cuts curve again ( )( )( ), y equate curve and their linear answer from (i). factorise: ( ± ) and a two or three term quadratic is sufficient. Allow long division withhold final if (, ) not clearly identified as their sole answer. 6 (i) cos sin cos sin + tan + cot sin cos + + cos sin cos sin cos +sin cos +sin ( cos sin )( cos +sin ) cos +sin ( ) sin cos tan and cot cos sin Attempt to multiply by cos and sin AG sin + cos sin cos 5cos sin 5 tan 5., 8.7 equate and collect sin and cos oe FT from tan k UCLES 06

18 Page Mark Scheme Syllabus Paper Cambridge O Level October/November (i) h 9 9 A ( ) 9 ( 7 ) B//0 Must be clear that 9 is the height of the trapezium. + oe must be seen AG da d ( ) ( ) 0.5 da d (i) f' ( ) A 6 or 8 8 or 5 or.6 ( + ) ( ) ( + ) 9 ( + ) A//0 product rule on correct function minus each error, allow unsimplified. equate to 0 and simplify to a linear or quadratic equation. correct three term quadratic obtained Etra positive answer loses penultimate. ignore negative solution. quotient rule or product rule all correct 6 6 www beware 9 9 gets A0 ( + ) d + c + FT c their D top limit bottom limit in their integral. or 0.0 or 0.96 or 0. (iii) y y + y + make y or the subject f ( ) Domain : FT take cube root (as long as y or equals a fraction with terms in or y only) oe FT change and y can be done at any time Allow upper limit of.86. Do not isw UCLES 06

19 Page 5 Mark Scheme Syllabus Paper Cambridge O Level October/November (i) tangent touches circle + k k 8 ( ) ( ) k + 8k k+ 6 0 or better Equal roots as tangent touches circle : b ac ( k) ( k ) k 6 k + only D eliminate y or allow unsimplified use of discriminant on term quadratic soi oe any inequality loses last b 0 a 5 9 y 5 5 OR tangent y cuts radius y + at 5 y 5 OR Obtain oe ( )( ) y 5 5 use b a find equation of radius and attempt to solve with tangent obtain any term quadratic using their non zero k and reach (iii) TP ( 0.) + ( + 0.8) for using their T and ( 0, ) Signs must be correct.. UCLES 06

20 Page 6 Mark Scheme Syllabus Paper Cambridge O Level October/November (i) rj 5000 cos0 + t 000 p cos50 coordinate oe y coordinate oe.5tcos tcos t.5cos70 + cos0 095 awrt or 090 or 00.5cos0 cos p ( ) p. awrt (i) Free choice : no. of ways 6 5 C C D equate their values (must be terms) make t the subject allow one sign error equate their y values(must be terms) and insert their t or t. 6 C another n Cr term only 5 C and answer or vice versa Both Mr and Mrs Coldicott 5 C C C another n Cr term only C and answer or vice versa (iii) Mr C and not Mrs C 5 C C( 60) Not Mr C and Mrs C 5 C C ( ) Total 80 OR Total (i) neither Neither 5 C C 0 Total An incorrect final answer does not affect the awarding of the first two marks. www UCLES 06

21 Cambridge International Eaminations Cambridge Ordinary Level ADDITIONAL MATHEMATICS 07/ Paper October/November 06 MARK SCHEME Maimum Mark: 80 Published 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 before 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 Report for Teachers. Cambridge will not enter into discussions about these mark schemes. Cambridge is publishing the mark schemes for the October/November 06 series for most Cambridge IGCSE, Cambridge International A and AS Level components and some Cambridge O Level components. IGCSE is the registered trademark of Cambridge International Eaminations. This document consists of 6 printed pages. UCLES 06 [Turn over

22 Page Mark Scheme Syllabus Paper Cambridge O Level October/November Abbreviations awrt cao dep FT isw oe rot SC soi www answers which round to correct answer only dependent follow through after error ignore subsequent working or equivalent rounded or truncated Special Case seen or implied without wrong working Question Answer Mark Part Marks ( ) ( ) ( ) ( ) rationalise with ( 5 ) numerator ( or terms) denominator and completion lne ln6e ln6e ln6 + lne ln6 + ln6 or ln 6 or one law of indices/logs second law of indices/logs www oe in base 0 (i) ( ) ( ) d sin + cos cos +sin sin d +cos +cos cos +cos +sin ( +cos ) +cos +cos ( ) Quotient Rule (or Product Rule from (sin)( + cos) - ) correct unsimplified use of sin +cos oe completion sin d 0 +cos +cos awrt.56 0 correct integrand UCLES 06

23 Page Mark Scheme Syllabus Paper Cambridge O Level October/November Question Answer Mark Part Marks (i) ( ) a ( a+ b 6) () ( a+ b ) p b 0 p 0 + a+ b 0 a 5 and b ( ) + ( ( ) + 7+ ) ( )( + )( + ) p 5 ( ) p 0,,. 5 (i) AB ( + ) + ( ) ( + )( ) cos60 solve their linear equations for a or b find quadratic factor correct quadratic factor soi factorise quadratic factor and write as product of linear factors if 0 scored, SC for roots only use cosine rule at least 7 terms correct completion AG sina sin60 6 ( ) sin60 6 sin A oe or or sine rule (or cosine rule) correct eplicit epression for sina AG (iii) Area ( + )( ) sin60 correct substitution into sin ab C 6 (i) dy sec d π dy π sec d y 8 y 8 Equation of tangent π ( y π 6, y + 6.9, π ) evaluated UCLES 06

24 Page Mark Scheme Syllabus Paper Cambridge O Level October/November Question Answer Mark Part Marks sec tan+ 7 tan tan 6 0 oe (tan )(tan + ) 0 tan or tan.5,.0 use sec + tan to obtain a term quadratic in tan solve three term quadratic for tan etras in range lose final 7 (i) ( ) r + h 0.5h+ oe r 0.5h + h+ h r h+ 0.75h correct epansion and r subject and completion www AG π r h π ( h + h 0.75 h ) V dv π h.5 h dh + ( ) any correct form in terms of h only differentiate V correct differentiation dv 0.5h h 0 dh h.9 only equate to 0 and solve term quadratic cao V h h d π.5 when.9 d (iii) ( h) ( 7.55 ) < 0 so maimum differentiate their term d V and substitute dh their h draw correct conclusion www 8 (i) 6 costoa 0 TOA 0.97 any method area of major sector 6 ( π their 0.97 ) ( 79.7) area of half kite ( 6 ) 0 6 () area of kite (8) complete correct plan awrt 8 (iii) arc length 6 (π their0.97) ) awrt.6 M D D or for 6 ( their 0.97) D for π 6 6 ( their 0.97) any method their major sector + their kite complete correct method UCLES 06

25 Page 5 Mark Scheme Syllabus Paper Cambridge O Level October/November Question Answer Mark Part Marks 9 (i) p tanα ± or ± or 8. or 7.6 seen 08 could use cos or sin (iii) (iv) their p r A + t 5 q r B + t 5 (v) 5 t 5 t t 0 r A r B and equate y/j and solve for t (vi) 5 only (vii) q only 0 (i) ( ) ( ) fg ln e + + isw ( ) ( ) ff ln ln + + isw y (iii) e + y e g - ( ) ln oe change and y and make e y the subject (iv) e or 7.9 (v) ( ) ( ln + ) gf e + 0 gf correct and equation set up correctly ln e e + 0 e 7 7 or.5 e one law of indices/logs second law of indices/logs www if 0 scored, SC for 7. UCLES 06

26 Page 6 Mark Scheme Syllabus Paper Cambridge O Level October/November Question Answer Mark Part Marks (i) q q + pq q+ q A p p p+ p pq+ 9 B,,0 each error A 5A I + pq 0 or 9 + pq 5 pq 8 equate top left or bottom right elements 8 8 accept p, q q p det A 6 pq 6 pq p and solve p q their det A p and use their pq k oe to solve for p or q FT from their pq k UCLES 06

Cambridge International Examinations Cambridge International General Certificate of Secondary Education. Published

Cambridge International Examinations Cambridge International General Certificate of Secondary Education. Published Cambridge International Eaminations Cambridge International General Certificate of Secondary Education ADDITIONAL MATHEMATICS 0606/ Paper October/November 06 MARK SCHEME Maimum Mark: 80 Published This