2006 Mathematics. Higher Paper 1. Finalised Marking Instructions

Transcription

1 Mathematics Higher Paper Finalised Marking Instructions The Scottish Qualifications Authority The information in this publication may be reproduced to support SQA qualifications only on a non-commercial basis. If it is to be used for any other purposes written permission must be obtained from the Assessment Materials Team, Dalkeith. Where the publication includes materials from sources other than SQA (secondary copyright), this material should only be reproduced for the purposes of eamination or assessment. If it needs to be reproduced for any other purpose it is the centre's responsibility to obtain the necessary copyright clearance. SQA's Assessment Materials Team at Dalkeith may be able to direct you to the secondary sources. These Marking Instructions have been prepared by Eamination Teams for use by SQA Appointed Markers when marking Eternal Course Assessments. This publication must not be reproduced for commercial or trade purposes.

2 Mathematics Higher: Instructions to Markers. Marks must be assigned in accordance with these marking instructions. In principle, marks are awarded for what is correct, rather than marks deducted for what is wrong.. Award one mark for each bullet point. Each error should be underlined in RED at the point in the working where it first occurs, and not at any subsequent stage of the working.. The working subsequent to an error must be followed through by the marker with possible full marks for the subsequent working, provided that the difficulty involved is approimately similar. Where, subsequent to an error, the working is eased, a deduction(s) of mark(s) should be made. This may happen where a question is divided into parts. In fact, failure to even answer an earlier section does not preclude a candidate from assuming the result of that section and obtaining full marks for a later section.. Correct working should be ticked ( ). This is essential for later stages of the SQA procedures. Where working subsequent to an error(s) is correct and scores marks, it should be marked with a crossed tick ( or X ). In appropriate cases attention may be directed to work which is not quite correct (e.g. bad form) but which has not been penalised, by underlining with a dotted or wavy line. Work which is correct but inadequate to score any marks should be corrected with a double cross tick.. The total mark for each section of a question should be entered in red in the outer right hand margin, opposite the end of the working concerned. Only the mark should be written, not a fraction of the possible marks. These marks should correspond to those on the question paper and these instructions.. It is of great importance that the utmost care should be eercised in adding up the marks. Where appropriate, all summations for totals and grand totals must be carefully checked. Where a candidate has scored zero marks for any question attempted, should be shown against the answer.. As indicated on the front of the question paper, full credit should only be given where the solution contains appropriate working. Accept answers arrived at by inspection or mentally where it is possible for the answer so to have been obtained. Situations where you may accept such working will normally be indicated in the marking instructions. 8. Do not penalise: working subsequent to a correct answer omission of units legitimate variations in numerical answers bad form correct working in the wrong part of a question

3 Mathematics Higher: Instructions to Markers 9. No piece of work should be scored through without careful checking - even where a fundamental misunderstanding is apparent early in the answer. Reference should always be made to the marking scheme - answers which are widely off-beam are unlikely to include anything of relevance but in the vast majority of cases candidates still have the opportunity of gaining the odd mark or two provided it satisfies the criteria for the mark(s).. If in doubt between two marks, give an intermediate mark, but without fractions. When in doubt between consecutive numbers, give the higher mark.. In cases of difficulty covered neither in detail nor in principle in the Instructions, attention may be directed to the assessment of particular answers by making a referal to the P.A. Please see the general instructions for P.A. referrals.. No marks should be deducted at this stage for careless or badly arranged work. In cases where the writing or arrangement is very bad, a note may be made on the upper left-hand corner of the front cover of the script. Transcription errors: In general, as a consequence of a transcription error, candidates lose the opportunity of gaining either the first ic mark or the first pr mark. Casual errors: In general, as a consequence of a casual error, candidates lose the opportunity of gaining the appropriate ic mark or pr mark. Do not write any comments on the scripts. A revised summary of acceptable notation is given on page. Working that has been crossed out by the candidate cannot receive any credit. If you feel that a candidate has been disadvantaged by this action, make a P.A. Referral. Throughout this paper, unless specifically mentioned, a correct answer with no working receives no credit. Summary Throughout the eamination procedures many scripts are remarked. It is essential that markers follow common procedures: Tick correct working. Put a mark in the outer right-hand margin to match the marks allocations on the question paper. Do not write marks as fractions. Put each mark at the end of the candidate s response to the question. Follow through errors to see if candidates can score marks subsequent to the error. Do not write any comments on the scripts.

4 Mathematics Higher: Instructions to Markers Higher Mathematics : A Guide to Standard Signs and Abbreviations Remember - No comments on the scripts. Please use the following and nothing else. Signs The tick. You are not epected to tick every line but of course you must check through the whole of a response. The cross and underline. Underline an error and place a cross at the end of the line. Bullets showing where marks are being allotted may be shown on scripts dy d = = = y = 8 margins The tick-cross. Use this to show correct work where you are following through subsequent to an error. C = (, ) m = m = rad m = m tgt tgt = y = The roof. Use this to show something is missing such as a crucial step in a proof or a 'condition' etc. = 8 = The tilde. Use this to indicate a minor transgression which is not being penalised (such as bad form). sin =. = invsin(. ) = 8. The double cross-tick. Use this to show correct work but which is inadequate to score any marks. This may happen when working has been eased. Remember - No comments on the scripts. No abreviations. No new signs. Please use the above and nothing else. All of these are to help us be more consistent and accurate. Note: There is no such thing as a transcription error, a trivial error, a casual error or an insignificant error. These are all mistakes and as a consequence a mark is lost. Page lists the syllabus coding for each topic. This information is given in the legend underneath the question. The calculator classification is CN(calculator neutral), CR(calculator required) and NC(non-calculator).

5 Year A determine range/domain A use the general equation of a parabola A8 use the laws of logs to simplify/find equiv. epression A recognise general features of graphs:poly,ep,log A solve a quadratic inequality A9 sketch associated graphs A sketch and annotate related functions A find nature of roots of a quadratic A solve equs of the form A = Be kt for A,B,k or t A obtain a formula for composite function A8 given nature of roots, find a condition on coeffs A solve equs of the form log b (a) = c for a,b or c A complete the square A9 form an equation with given roots A solve equations involving logarithms A interpret equations and epressions A apply A-A9 to solve problems A use relationships of the form y = a n or y = ab UNIT UNIT UNIT A determine function(poly,ep,log) from graph & vv A apply A8-A to problems A8 sketch/annotate graph given critical features A9 interpret loci such as st.lines,para,poly,circle A use the notation u n for the nth term A use Rem Th. For values, factors, roots G calculate the length of a vector A evaluate successive terms of a RR A solve cubic and quartic equations G calculate the rd given two from A,B and vector AB A decide when RR has limit/interpret limit A find intersection of line and polynomial G8 use unit vectors A evaluate limit A find if line is tangent to polynomial G9 use: if u, v are parallel then v = ku A apply A-A to problems A find intersection of two polynomials G add, subtract, find scalar mult. of vectors A confiirm and improve on appro roots G simplify vector pathways A apply A-A to problems G interpret D sketches of D situations G find if points in space are collinear G find ratio which one point divides two others G use the distance formula G9 find C/R of a circle from its equation/other data G given a ratio, find/interpret rd point/vector G find gradient from pts,/angle/equ. of line G find the equation of a circle G calculate the scalar product G find equation of a line G find equation of a tangent to a circle G use: if u, v are perpendicular then v.u= G interpret all equations of a line G find intersection of line & circle G8 calculate the angle between two vectors G use property of perpendicular lines G find if/when line is tangent to circle G9 use the distributive law G calculate mid-point G find if two circles touch G apply G-G9 to problems eg geometry probs. G find equation of median, altitude,perp. bisector G apply G9-G to problems G8 apply G-G to problems eg intersect.,concur.,collin. C differentiate sums, differences C find integrals of p n and sums/diffs C differentiate psin(a+b), pcos(a+b) C differentiate negative & fractional powers C integrate with negative & fractional powers C differentiate using the chain rule C epress in differentiable form and differentiate C epress in integrable form and integrate C integrate (a + b) n C find gradient at point on curve & vv C evaluate definite integrals C integrate psin(a+b), pcos(a+b) C find equation of tangent to a polynomial/trig curve C find area between curve and -ais C apply C-C to problems C find rate of change C find area between two curves C find when curve strictly increasing etc C8 solve differential equations(variables separable) C8 find stationary points/values C9 apply C-C8 to problems C9 determinenature of stationary points C sketch curvegiven the equation C apply C-C to problems eg optimise, greatest/least T use gen. features of graphs of f()=ksin(a+b), T solve linear & quadratic equations in radians T solve sim.equs of form kcos(a)=p, ksin(a)=q f()=kcos(a+b); identify period/amplitude T8 apply compound and double angle (c & da) formulae T epress pcos()+qsin() in form kcos(±a)etc T use radians inc conversion from degrees & vv in numerical & literal cases T find ma/min/zeros of pcos()+qsin() T know and use eact values T9 apply c & da formulae in geometrical cases T sketch graph of y=pcos()+qsin() T recognise form of trig. function from graph T use c & da formulaewhen solving equations T solve equ of the form y=pcos(r)+qsin(r) T interpret trig. equations and epressions T apply T-T to problems T apply T-T to problems T apply T-T to problems page

6 Higher Mathematics Paper : Marking Scheme Version Triangle ABC has vertices A(,), B(, ) A(, ) y and C(, ). (a) (b) (c) Find the equation of the median BD. Find the equation of the altitude AE. Find the coordinates of the point of intersection of BD and AE. O D C(, ) a,b,c,, C G, G8 CN / B(, ) E 8 9 ic interpret median ss find gradient ic state equation ss find gradient ss find perpendicular gradient ic state equation ss start to solve simultaneous equations pr solve for one variable pr process Primary Method : Give mark for each D = (, ) m = BD y = ( ) or y + = ( ) etc m = BC m = alt y = ( ( )) y = ( ) and y = ( ( )) 8 = 9 y = stated eplicitly or equivalent marks marks marks For candidates who find two medians,, and, 8, 9 are available. For candidates who find two altitudes,, and, 8, 9 are available. For candidates who find (a) altitude and (b) median see common error bo number. In (a) note that (, ) happens to lie on the median but does not qualify as a point to be used in. cont In (b) is only available as a consequence of attempting to find a perpendicular gradient. In (b) candidates who guess the coordinates for E and use these to find the equation AE, can earn no marks in this part. In (c) note that equating zeros is only a valid strategy when either the coefficients of or the coefficients of y are equal. 8 is a strategy mark for jutaposing the two required equations. 9 See general note at the foot of page. Common Error Finding two medians D = (, ) m = BD y = ( ) X X X y = & + y = 8 = 9 y = maimum of marks Common Error Finding two altitudes X X X m = BC m = alt y = ( ( )) y = & y = = 9 9 y = maimum of marks Common Error Finding (a) altitude and (b) median m = AC X m = BD y = ( ) X midpt of BC =, m = AC y = ( ( )) X y = & + y = 8 X = 9 X y = 9 maimum of marks

7 Higher Mathematics Paper : Marking Scheme Version A circle has centre C(, ) and passes through P(, ). y (a) (b) Find the equation of the circle. PQ is a diameter of the circle. Find the equation of the tangent to this circle at Q. C (, ) P (, ) a C G CN / b C G CN Q O ic enter coord. of centre in general equation ss find (radius) ss e.g. use PC = CQ to find Q pr find gradient of diameter ss know and use tangent perp. to diameter ic state equation Primary Method : Give mark for each ( a) + ( y b) = r ( ) + ( y ) r = 8 Q = (, ) m = diameter m = tangent y = stated or implied by marks marks In (a) ( 8) is not acceptable for. In (b) if the coordinates of Q are estimated (i.e. guessed) then can only be awarded if the coordinates are of the form (a, ) where a <. Alternative Method for (a) For answers of the form + y + g + fy + c = + y + y + c = c = In (b) is only available if an attempt has been made to find a perpendicular gradient. General applicable throughout the marking scheme There are many instances when follow throughs come into play and these will not always be highlighted for you. The following eample is a reminder of what you have to look out for when you are marking. eample At the stage a candidate start with the wrong coordinates for Q. Then X Q = (, ) X m = diameter X m = tangent X y = ( ) so the candidate loses but gains, and as a consequence of following through. Any error can be followed through and the subsequent marks awarded provided the working has not been eased. Any deviation from this will be noted in the marking scheme.

8 Higher Mathematics Paper : Marking Scheme Version Two functions f and g are defined on the set of real numbers by f () = + and g () =. (a) Find an epressions for (i) f g() ( ()). (ii) g f. (b) Determine the least possible value of f g() g f() a C A CN / b C A CN ic int. composition ic int. composition ic int. composition pr simplify all functions ic int. result Primary Method : Give mark for each f( g )= f( ) ( ) + g( f )= ( +) 9 min.value = 9 stated or implied by stated eplicitly marks marks In (a) marks are available for finding one of and the third mark is for f g() or g f() the other one. In (a) the finding of f f() and g g() earns no marks. is only available if has been awarded. In (b) for, no justification is necessary. Ignore any comments, rational or irrational. Alternative Marking [Marks -] g( f )= g( + ) ( + ) f( g )= ( )+ Common Error No. for (a) g and f transposed. X f( g )= f( + ) X ( + ) X g( f )= ( ) + Award out of Common Error No. for (a) X f( g )= f( + ) X ( + ) g( f )= ( + ) Award out of Common Error No. for (a) Repeated error f( g )= f( ) X ( + ) X g( f( ))= ( ) + Award out of 8

9 Higher Mathematics Paper : Marking Scheme Version A sequence is defined by the recurrence relation u 8. u, u. n+ n (a) State why the recurrence relation has a limit. (b) Find this limit. a C A NC /8 b C A NC Primary Method : Give mark for each sequence has limit since <. 8< mark ic state limit condition ss know how to find L pr process limt L = 8. L+ limit = marks For (a) Accept 8. < Alternative Method for (b) L =. 8 limit = <. 8< 8. lies between and 8. is a proper fraction Bad Form L =. limit = Do NOT accept award marks. 8 < a < 8. < unless a is clearly identifed/replaced by.8 anywhere in the answer. Common Error X L =. 8 X limit = In (b) L b = and nothing else gains no marks. a L = or or. etc does NOT gain. An answer of without any working gains NO marks. Any calculations based on wrong formulae gain NO marks. 9

10 Higher Mathematics Paper : Marking Scheme Version A function f is defined by f ()=.Find the coordinates of the stationary point on the graph with equation y = f() and determine its nature. C C8, C9 NC / B ss know to start to differentiate pr differentiate ss set derivative = pr solve pr evaluate ic justification ic state conclusion Primary Method : Give mark for each f =... ( ) f = = f( ) = nature table pt of infleion at (,) marks The = shown at must appear at least once somewhere in the working between and (but not necessarily at ). is only available as a consequence of solving f () =. A wrong derivative which eases the working will preclude at least from being awarded. Common Error No. f =... X ( ) f = X =, and are still available For marks and, a nature table is mandatory. The minimum amount of detail that is required is shown here: < > f () + + Candidates who use only f () = and try to draw conclusions from this cannot gain or. [ f () = is a necessary but not sufficient condition for identifying points of infleion]. is ONLY available subsequent to a correct nature table for the candidate s own derivative. Common Error No. f =... X ( ) f = X =, and are still available is lost in each of the following cases for the candidate s solution to the equation at. (i) (ii) (iii) = and = something else two wrong values for guess a value for Only one value for needs to be followed through for, and.

11 Higher Mathematics Paper : Marking Scheme Version The graph shown has equation y = + +. y The shaded area is bounded by the curve, the -ais, the y-ais and the line =. S O (,) (,) (a) Calculate the shaded area labelled S. (b) Hence find the total shaded area. a C C NC / b B C NC ss know to integrate pr integrate ic substitute limits pr evaluate ic use result from with new limits pr evaluate ss deal with the ve sign and evaluate total area for (a) Only a limited number of marks are available to candidates who differentiate see Common Error No.. In (a) candidates who transpose the limits can still earn if the deal with the ve sign appropriately. In (b) is lost for such statements as =. Primary Method : Give mark for each + + d + + ( ) or equivalent... d Alternative Method for (b) stated or implied by.. ( +. + ) ( )= 9 or equivalent... d... ( ) In (b) using...d earns no marks. Common Error No. + + d X + X (.. + ) X 9... d or equivalent X (.. + ) (.. + )= X Alternative Method for (b) ( ) Alternative Method for (b)... d... ( )

12 Higher Mathematics Paper : Marking Scheme Version Solve the equation sin sin = in the interval. C T NC / ss know to use double angle formula pr factorise pr solve ic know eact values An = must appear somewhere between the start and evidence. The inclusion of etra answers which would have been correct with a larger interval should be treated as bad form and NOT penalised. The omission of a correct answer (e.g. ) means the candidates loses a mark ( in the Primary Method). Candidates may embark on a journey with the wrong formula for sin( ). With an equivalent level of difficulty it may still be worth a maimum of marks. See Common Error No.. Candidates who draw a sketch of y = sin( ) and y = sin( ) giving,8, may be awarded and. Primary Method : Give mark for each sin( ) sin( )cos( = ) sin( )( cos( ))= sin( ) = or cos( ) =. =, 8,,, Alternative Marking Method (Cross marking for and ) sin( ) sin( )cos( = ) sin( )( cos( ))= sin( ) = and =, 8, cos( ) =. and =, Alternative Method Division by sin() sin( ) sin( )cos( = ) either sin( ) = or sin( ) sin( ) = =, 8, cos( ) =. =, Common Error No. X sin( ) ( sin ( ))= sin ( + ) sin( ) = X ( sin( ) ) ( sin( + ) )= X sin( ) = or sin( ) = X =,, = award marks Common Error No. X sin( ) sin ( = ) X sin( ) sin( ) = X sin( ) = or sin( ) = X =, 8,, 9 award marks Common Error No. sin sin = sin =, sin = etc gainsnomarks

13 Higher Mathematics Paper : Marking Scheme Version 8 (a) Epress + in the form a( + b) + c. (b) Write down the coordinates of the turning point on the parabola with equation. y = + 8 a B A NC / b C A NC ss know how to complete (deal with the a ) pr process the value of b pr process the value of c ic interpret equation of parabola Primary Method : Give mark for each a = b = c = (, ) Note Alternative Method should be used for assessing part marks/follow throughs. For, no justification is required. Candidates may choose to differentiate etc. but may still earn only one mark for the correct answer. For, accept ( b, c). Alternative Method for (a) ( + ) ( + ) ( + ) (, ) Alternative Method for (a) : Comparing coefficients + = a + ab + ab + c a = ab = ab + c = b = c = (, )

14 Higher Mathematics Paper : Marking Scheme Version k 9 u and v are vectors given by u = and v = k, where k >. k + (a) If u.v = show that k + k k =. (b) Show that (k + ) is a factor of k + k k and hence factorise k + k k fully. (c) Deduce the only possible value of k. (d) The angle between u and v is θ. Find the eact value of cos θ. θ u v marks marks mark marks 8 a C G CN / b C A NC c C A CN d C G8 NC pr find scalar product ic complete proof ss know to use k = pr complete evaluation and conclusion ic start to find quadratic factor ic complete quadratic factor pr factorise completely 8 ic interpret k 9 ic interpret vectors pr find magnitudes ss use formula No eplanation is required for k but the chosen value must follow from the working for or. Do not accept. In primary method ( ) and alternative ( ) candidates must show some acknowledgement of the resulting zero. Although a statement w.r.t. the zero is preferable, accept something as simple as underlining the zero. Only numerical values are acceptable for 9, and ; answers Primary Method : Give mark for each stated or implied by u.v = k. +. ( k )+ k +. before completion k + k k = and complete marks know to use k = + ( ) = + is a factor ( k + )( k...) ( k + ) k 8 ( k + )( k + )( k ) k = 9 u =,v = u = and v = cosθ = N.B. 9 and may be cross-marked. stated eplicitly stated or implied by marks mark marks are acceptable in unsimplified form eg cosθ = Alternative method (marks ) Long Division k + k + k k k + k k k k remainder iszeroso( k+ ) isa factor k ( k + )( k + )( k ) stated eplicitly Alternative method (marks ) Synthetic Division " f( )" = so ( k + ) is a factor k ( k + )( k + )( k ) stated eplicitly

15 Two variables, and y, are connected by the law y = a. A graph of Higher Mathematics Paper : Marking Scheme Version log against is a straight line passing through the origin and the point A(,). Find the value of a. A(, ) log y A A NC /9 O ss know to take logarithms ic substitute known point pr solve pr solve Primary Method : Give mark for each log ( y) = log ( a ) = log ( a ) a = a = marks Note m = and nothing else gains no marks. Alternative Method For, a correct answer without any legitimate evidence gains NO marks. For, ignore the inclusion of a negative answer. log ( y) = log ( a ) = log ( a) log ( a) = a = Alternative Method log ( y) = m + c m =, c = y = y = = a = Common Error log ( y) = log ( a ) X X X log = log ( a ) = a a = Alternative Method At A log ( y) = y = a = a = Common Error X log ( y) = X X y = X a = Alternative Method log ( y) = log ( a ) log ( y) = log ( a) log ( a) = a = =