Cambridge Assessment International Education Cambridge Pre-U Certificate. Published

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1 Cambridge Assessment International Education Cambridge Pre-U Certificate MATHEMATICS 979/0 Paper Pure Mathematics 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 acceptabilit of alternative answers. Mark schemes should be read in conjunction with the question paper and the Principal Eaminer Report for Teachers. Cambridge International will not enter into discussions about these mark schemes. Cambridge International is publishing the mark schemes for the series for most Cambridge IGCSE, Cambridge International A and AS Level and Cambridge Pre-U components, and some Cambridge O Level components. IGCSE is a registered trademark. This sllabus is approved for use in England, Wales and Northern Ireland as a Cambridge International Level 3 Pre-U Certificate. This document consists of printed pages. UCLES 08 [Turn over

2 Generic Marking Principles These general marking principles must be applied b all eaminers when marking candidate answers. The should be applied alongside the specific content of the mark scheme or generic level descriptors for a question. Each question paper and mark scheme will also compl with these marking principles. GENERIC MARKING PRINCIPLE : Marks must be awarded in line with: the specific content of the mark scheme or the generic level descriptors for the question the specific skills defined in the mark scheme or in the generic level descriptors for the question the standard of response required b a candidate as eemplified b the standardisation scripts. GENERIC MARKING PRINCIPLE : Marks awarded are alwas whole marks (not half marks, or other fractions). GENERIC MARKING PRINCIPLE 3: Marks must be awarded positivel: marks are awarded for correct/valid answers, as defined in the mark scheme. However, credit is given for valid answers which go beond the scope of the sllabus and mark scheme, referring to our Team Leader as appropriate marks are awarded when candidates clearl demonstrate what the know and can do marks are not deducted for errors marks are not deducted for omissions answers should onl be judged on the qualit of spelling, punctuation and grammar when these features are specificall assessed b the question as indicated b the mark scheme. The meaning, however, should be unambiguous. GENERIC MARKING PRINCIPLE : Rules must be applied consistentl e.g. in situations where candidates have not followed instructions or in the application of generic level descriptors. GENERIC MARKING PRINCIPLE 5: Marks should be awarded using the full range of marks defined in the mark scheme for the question (however; the use of the full mark range ma be limited according to the qualit of the candidate responses seen). GENERIC MARKING PRINCIPLE 6: Marks awarded are based solel on the requirements as defined in the mark scheme. Marks should not be awarded with grade thresholds or grade descriptors in mind. UCLES 08 Page of

3 < (3 ) ( if (3 ) < 0) Alternative (5+ 3) < or (3 ) or A < onl A < < 3 ( if 3 ) > 0 ) < < onl A B A Obtain at least the critical value (i) f() B Accept but not. (ii) 6 B (iii) 3 ( ) f ( ) 9 A Accept f(). (iv) B B for general shape of f() starting at (0, ) B for general shape of inverse starting at approimatel (, 0) and a suggestion of intersection. State Reflection in line B 3 z is a root implies (z ) is a factor B Must be stated or used. 3 z + z + z z ± 3 z z ± i 3 A Must reach a 3 term quadratic. Accept alternative methods e.g. coefficient matching Allow one error of substitution in correct quadratic formula if stated. If formula not stated and error present M0 UCLES 08 Page 3 of

4 0 (i) B Award for 3 branches of roughl correct shape with two asmptotes (ii) cosθ sinθ Allow equivalent methods tanθ π A or 5 5 π A A0 if more angles given or if in degrees 5(i) d 3 3 d + Differentiate b showing a decrease in power b in at least two terms. At m 7 Substitute in their derivative and a numerical result. 7 A 7( 6) 56 ( 6) 3 + ( 6) 3( 6) OR use A Confirm equalit UCLES 08 Page of

5 5(ii) (7 ) Intention shown to subtract curve tangent even as two separate integrals Main Alternative Show a complete method: (area of triangle from 6 to ) (integral from 5 to 6) + (integral from 5 to ) (area of integral from to ) b a 3 ( + 0+ )d OR OR b a b a 3 ( )d 3 ( )d OR b a b a A Integrate one or other cubic epressions F(b) F(a) Evaluate their integral with their a and b as F(b) F(a) onl. Convincing substitution of limits or evaluation of areas required. Show use of correct limits OR show correct calculations ( ) + ( ) A Accept correct limits also in A Accept 3 3 or eact equiv. UCLES 08 Page 5 of

6 6(i) λ 3µ 7λ µ 9 8λ µ λ 3µ 7λ µ 9 Equate components and form 3 simultaneous eqns. OR + 3µ 3 + λ + µ + 7λ 5 + µ 9 + 8λ Solve an pair of the simultaneous eqns λ 3, µ A Award for λ and µ correct 8( 3) ( ) so the lines intersect ( 5, 0, 33) or r 5i 0j 33k B A Substitute values into the third eqn and make eplicit conclusion e.g. using z component 9 + 8( 3) 33 and 5 + ( ) 33 Other components compared like this ield 5 and 0 OR substitute correct λ and µ into r and r to get correct intersection point in both. 6(ii) (i+ 7j+ 8 k) (3i+ j+ k) Use of correct formula θ cos θ cos 9 6 A Correct direction vector values used 8.3 A Award also for 0.9 UCLES 08 Page 6 of

7 7 d d d d d d 8 3 A Obtain term epression from implicit d differentiation but must include k. d d Set their 0 and solve to obtain a value for d + 3 Substitute their value for and obtain a quadratic in. 7 5,,, 3 3 d 8(8 ) 8 (6 8 ) d d d (8 ) 8 (8 ) A Obtain correct coordinates. A generous attempt at quotient rule, allowing some slips Accept alternative reasons and methods, e.g. one value is greater than the other with the same - coordinate and a mi and match of methods involving conics. Do not penalize a description of the ellipse as a circle d d d d 0.75 at 7, at 5, 3 so maimum A Final value must be shown so minimum A Final value must be shown Full credit can be obtained using conics: AA or ( ) ( ) A Ellipse centre, 3 A A (ma) and (min).5 UCLES 08 Page 7 of

8 8(i) sin3θ tan3θ cos3θ d(tan3 θ ) 3cos 3θ + 3sin 3θ dθ cos 3θ d(tan3 θ ) OR 3sec 3θ dθ 3+ tan 3θ ( ) 3+ 3tan 3θ A Dep on M marks Some attempt to show evidence for the use of (i) 8(ii) 3 tan 3θdθ tan3θ 3dθ + c or alternative approaches e.g. sec 3θ : 3 tan 3θdθ 3 sec 3θ 3dθ + c π 9 π tan 3θ dθ tan 3θ θ 3 π 9 π Award for atan 3θ + bθ. appearing loses the A tan π π tan π π π ( 3 ) (i) d k 0 d k d d and at, k 0 d k d < and hence maimum ln ln k k A A A B A For correct integral Award for sight onl of substitution of correct limits. Ma be awarded with incorrect integral i.e. after M0M0 aef Differentiate, put equal to 0 and attempt to solve aef Clear conclusion required. Accept < 0 k aef UCLES 08 Page 8 of

9 9(ii) ln k 0 ordinate stated either or 0. M0 if onl specific values used Accept a statement that if 0 there will be no real roots for B. k AG e A 9(iii) n+ ln 3 n 3 Correct formula 5, , , A Obtain at least correct iterates including 5.5 A Obtain answer correct to 3 s.f. 9(iv) If k < 0, k adds a positive quantit onto ln If k 0, the curve is ln which has no maimum and continuousl increases so crosses the -ais once and has one root B B Use of graphical arguments, e.g. e k and intersection points should be convincing Accept If k 0, the curve is ln which has one root. Arguments leading from ln 0 to should be convincing Hence the function ln k is continuousl increasing so crosses the -ais once onl. B Award if the two cases are considered and an overall conclusion drawn. 9(v) Should mention a maimum between 0 < k < e B Accept alternative arguments Conclude that the curve must cross the -ais twice giving two real roots. B UCLES 08 Page 9 of

10 0(i) d 3 d Some intention to separate variables k d e.g. sight of 3 k or 3 d d (k or ) A + B + C + A B + C or ± + A B + C or + Set up partial fractions in a reasonabl correct form e.g. ignore sign errors from an attempt to write the ( ) form For eample,, 0 or,, 0 or,, or,, A A Obtain at least one correct value Obtain an correct triplet for their partial fraction format ln ln( + ) ln ( ) + c Or ln ln( ) + C Or ln ln( ) + C Obtain k ln ± m ln( + ) ± n ln ( ) ± p ( + c ) Or k ln ± m ln( ) ± p (+ c) Or k ln ± m ln( ) ± p (+ c) ee ( + )( ) c Use at least one log law correctl Obtain an approimatel correct epression without logs, e.g. allow a coefficient slip but not omission of + c Accept e + c This is a fairl strict mark but allow reasonabl correct equivalent forms + Obtain their reasonable epression in the form f() b a reasonable method. A If the previous mark is M0, cannot be given here Obtain a correct epression for. Do not accept e + c for this mark UCLES 08 Page 0 of

11 0(ii) As becomes large and negative tends to 0 and lim f ( ) 0 B Dep (ii) requires a correct. Since 0 + Dep Accept minimal or no working. Hence lim f ( ) A + UCLES 08 Page of

Cambridge Assessment International Education Cambridge Ordinary Level. Published

Cambridge Assessment International Education Cambridge Ordinary Level ADDITIONAL MATHEMATICS 07/ Paper May/June 08 MARK SCHEME Maimum Mark: 80 Published This mark scheme is published as an aid to teachers