4037 ADDITIONAL MATHEMATICS

Size: px

Start display at page:

Download "4037 ADDITIONAL MATHEMATICS"

Abner Harrell
5 years ago
Views:

1 CAMBRIDGE INTERNATIONAL EXAMINATIONS GCE Ordinary Level MARK SCHEME for the October/November 0 series 07 ADDITIONAL MATHEMATICS 07/ Paper, maximum raw mark 80 This mark scheme is published as an aid to teachers and candidates, to indicate the requirements of the examination. It shows the basis on which Examiners were instructed to award marks. It does not indicate the details of the discussions that took place at an Examiners 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 Examiner Report for Teachers. Cambridge will not enter into discussions about these mark schemes. Cambridge is publishing the mark schemes for the October/November 0 series for most IGCSE, GCE Advanced Level and Advanced Subsidiary Level components and some Ordinary Level components.

2 Page Mark Scheme Syllabus Paper GCE O LEVEL October/November 0 07 Mark Scheme Notes Marks are of the following three types: M Method mark, awarded for a valid method applied to the problem. Method marks are not lost for numerical errors, algebraic slips or errors in units. However, it is not usually sufficient for a candidate just to indicate an intention of using some method or just to quote a formula; the formula or idea must be applied to the specific problem in hand, e.g. by substituting the relevant quantities into the formula. Correct application of a formula without the formula being quoted obviously earns the M mark and in some cases an M mark can be implied from a correct answer. A Accuracy mark, awarded for a correct answer or intermediate step correctly obtained. Accuracy marks cannot be given unless the associated method mark is earned (or implied). B Accuracy mark for a correct result or statement independent of method marks. When a part of a question has two or more method steps, the M marks are generally independent unless the scheme specifically says otherwise; and similarly when there are several B marks allocated. The notation DM or DB (or dep*) is used to indicate that a particular M or B mark is dependent on an earlier M or B (asterisked) mark in the scheme. When two or more steps are run together by the candidate, the earlier marks are implied and full credit is given. The symbol implies that the A or B mark indicated is allowed for work correctly following on from previously incorrect results. Otherwise, A or B marks are given for correct work only. A and B marks are not given for fortuitously correct answers or results obtained from incorrect working. Note: B or A means that the candidate can earn or 0. B,, 0 means that the candidate can earn anything from 0 to. Cambridge International Examinations 0

3 Page Mark Scheme Syllabus Paper GCE O LEVEL October/November 0 07 (i) 6 6 C ( ) (px) or ( ) 0 p 60 p px [] Seen or implied, unsimplified for their coefficient of x 60 and attempt to solve (ii) coefficients of the terms needed ( ) 6 C () 5 p + ( 60) for realising that terms are involved for ( ) 6 C () 5 p or 9p, using their p. 8 [] lg y 5y + 60 lg0 for lg y lg y for lg0 or equivalent, allow when seen Or lg y lg0 (5y + 60) y 50y leading to y 0, 60 y must be positive so y 60 D [5] for use of loga logb loga/b or loga + logb log AB D for forming a term quadratic equation and an attempt to solve for y 60 only Cambridge International Examinations 0

4 Page Mark Scheme Syllabus Paper GCE O LEVEL October/November 0 07 sin tan sin sin cos sin sin cos cos sin ( cos ) sin cos cos Marks are awarded only if they can lead to a complete proof for the methods other than those shown below for dealing with tan and a fraction for factorising for use of identity cos + sin Alt solution sin sec for all correct Using tan sin sec LHS sin sec sin sin (sec ) sin tan Alt solution sin sec RHS sin sec sin sin cos sin ( cos ) cos sin sin cos cos sin sin cos cos cos tan sin use of tan x sin xsec x for factorising for use of identity for all correct for splitting sin and use of identity for multiplication for writing as two terms and cancelling for all correct Cambridge International Examinations 0

5 Page 5 Mark Scheme Syllabus Paper GCE O LEVEL October/November 0 07 (i) dy dx Alt solution x x ( x + ) e e ( x + ) ( x + ) ( ) ( ) e x x +, A x + A,, 0 for attempt at quotient rule for each error Must be convinced of correct simplification e.g. sight of (x + ) or (x + )(x + ) dy dx ( ) x ( + e x + ) e x ( x + ) for attempt at product rule x ( ) ( ) e x +, A x + A,,0 for each error Must be convinced of correct simplification e.g. sight of (x + ) or (x + )(x + ) (ii) x, y e, [] Accept /e 5 (i) f ( x ) f ( x ) ( ) x or for ( x ) or -5 [] For -5 only Alt method f f 5 for f of their f For -5 only (ii) f ' (x) g ' (x) 6x 0x for 6x for 0x Leading to (x ) (x + ) 0 x, for solution of quadratic equation obtained from differentiation of both for both Cambridge International Examinations 0

6 Page 6 Mark Scheme Syllabus Paper GCE O LEVEL October/November Area under the curve: x d 0 x x x 0 for attempt to integrate ( 0) D D for application of limits 0 Area of trapezium ( + )( ) 0 Shaded area for area of trapezium, allow unsimplified for subtraction of the two areas Shaded area [6] Must be in this form Or: Equation of chord: y x for the equation of the chord unsimplified Shaded area x + xdx 0 for subtraction for attempt to integrate x x 0 D [6] x x for m or equivalent, where m is the gradient of their chord D for application of limits Cambridge International Examinations 0

7 Page 7 Mark Scheme Syllabus Paper GCE O LEVEL October/November (i) t (t t + ) Correct determinant seen unsimplified Leading to, t [] for simplification and solution for solution of det Aonly, not /det A (ii) A 6 7, A 7 6, for, for matrix 6 7 x 0 y for dealing correctly with the factor of x y for pre-multiplying their by their A to obtain a column matrix x, leading to x, y y [5] Allow x for y 8 (i) ( ) sin 7. 5 for attempt to find the area of the triangle and equate to sin,.5 6 [] for solution to obtain the given answer Solution must include.5. or.5 (ii) sin CD, (CD.567) for attempt to find CD Arc length 6(.5) Perimeter + + 6(.5) + their CD for arc length for sum of appropriate lengths awrt 5.9 (iii) Area 6 (.5) (awrt) [] for sector area for subtraction of the areas Cambridge International Examinations 0

8 Page 8 Mark Scheme Syllabus Paper GCE O LEVEL October/November (a) (i) 6 ( cos x) 5 + cos x 6 cos x + cos x 0 ( cos x ) ( cos x + ) 0 x 70.5 x 0 (ii) cos x sin y sin y only so y 9.5, 60.5 (b) cot z ( cot z ) 0, D, [] for use of sin x ( cos x) correctly for solution of a term quadratic in cos and attempt at solution of a trig equation for each correct solution D for relating cos x and sin y or other correct method of solution for attempt to use a factor cot z 0, π z for π (.57) cot z, tan z so z 0.97 dealing with cot and attempt at solution 0 (i) lg s (ii) lgs lgt (iii) No marks in this part unless lgt v lgs graph is used Gradient : n (allow..9) Intercept : log k, or other method k 00 (allow 90 0 ) Alt method Using simultaneous equations, points used must lie on the plotted line. [] D [], M, Allow in table or on graph if no contradiction No marks for graph unless lgt against lgs (or lnt against lns) for or more points correct D for a line through or correct points all points correct with a straight line extending at least from first point to last point calculates gradient for n for use of intercept and dealing with logarithm correctly (can use another point) Must attempt to solve valid equations. k 00 and n (iv) When t, lg t 0.6 so lg s 0.69 s.9 (allow.8 5. ) [] for valid method using either the correct graph or using lgt nlgs + lgk or t ks n using their n and their k Cambridge International Examinations 0

9 Page 9 Mark Scheme Syllabus Paper GCE O LEVEL October/November 0 07 (i) k x 5 x e + e 0, for each term integrated correctly, allow unsimplified e k + 5 e k 5 + for application of limits to an integral of the form Ae x + Be -x e k + 5 e k 0 for equating to and attempt to rearrange to obtain a term equation. Must be using an integral of the form Ae x + Be -x e k e k [5] Answer given, so must be convinced (ii) y y for solution of quadratic equation leading to e k 5, k 0.58, 0.7 e k, for solving equations involving exponentials for each Cambridge International Examinations 0

0606 ADDITIONAL MATHEMATICS

CAMBRIDGE INTERNATIONAL EXAMINATIONS International General Certificate of Secondary Education MARK SCHEME for the October/November 0 series 0606 ADDITIONAL MATHEMATICS 0606/ Paper, maximum raw mark 80