2006 Mathematics. Higher Paper 2. 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 ). 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.. 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 = 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. = = The tilde. Use this to indicate a minor transgression which is not being penalised (such as bad form). sin( ) =. = invsin(. ) =. 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 A 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 A 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 A-A to problems A 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 G 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 G 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 G 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 C solve differential equations(variables separable) C find stationary points/values C9 apply C-C 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 T 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 PQRS is a parallelogram. P is the point (, ), S is (, ) y and Q lies on the -ais, as shown. The diagonal QS is perpendicular to the side PS. (a) Show that the equation of QS is + y =. (b) Hence find the coordinates of Q and R. O P(, ) S(, ) Q R a,b, C G CN / pr find gradient from two points ss use m m = ic state equation of the line ic completes proof ic interpret diagram ic interpret diagram Primary Method : Give mark for each m = PS m = QS y = ( ) completes proof Q = (, ) R = (, ) marks marks In (a) In the Primary method, is only available if an attempt has been made to find and use a perpendicular gradient. In the Primary method and the Alt. method, is only available for reaching the required equation. To gain, some evidence of completion needs to be shown e.g. y = ( ) ( y ) = ( ) + y = Alternative Method m = PS m = QS y = + c = + c completes proof Q = (, ) R = (, ) Sometimes candidates manage to find R first. Provided the coordinates of R are of the form (?, ), only then is available as a follow through. Alternative Method and are available to candidates who use their own erroneous equation for QS. 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 may switch the coordinates round so we have Q(, ) R(, ) repeated error so the candidate loses for switching the coordinates but gains 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. Let Q = (, q ) ( q ) = + + ( q ) + q = Q = (, ) and R = (, ) m = QS y = ( ) leading to y + = N.B. The coordinates of Q can also be arrived at by right-angled trig. Use the alt. method marking scheme with replaced by appropriate trig. work. The only acceptable value for q is.

7 Higher Mathematics Paper : Marking Scheme Version Find the value of k such that the equation k + k + =, k, has equal roots. C A CN /new ss know to use discriminant = ic interpret a,b,c pr substitute & factorise ic interpret solution Primary Method : Give mark for each " b ac" = a = k, b = k, c = kk ( ) k = and k = k = marks Alternative Method (completing the square) The evidence for and/or may not appear until the working immediately preceding the evidence for. i.e. a candidate may simply start, k k = kk ( ) ( ) + ( ) + = + + k equal roots + = k k = or k k, kk ( ) = The = has to appear at least once, at the stage or at the stage. In the Primary method, candidates who do not deal with the root k = cannot obtain. [see Common Errors and ] Minimum evidence for would be scoring out k = or k = underlined. Some candidates may start with the quadratic formula. Apply the marking scheme to the part underneath the square root sign. The use of any epression masquerading as the discriminant can only gain at most. Acceptable alternative for " b ac" = a = k, b = k, c = kk ( ) k = or Common Error at the stage " b ac" = a = k, b = k, c = kk ( ) k = or Common Error at the stage " b ac" = a = k, b = k, c = kk ( ) k = Common Error Division by k " b ac" = a = k, b = k, c = k k = k = k k =

8 Higher Mathematics Paper : Marking Scheme Version The parabola with equation y = + has a tangent at the point P(,). (a) Find the equation of this tangent. y O P(,) y (b) Show that the tangent found in (a) is also a tangent to the parabola with equation y = + and find the coordinates of the point of contact Q. O Q P(,) a C C CN / b C A CN 9 ss know to differentiate pr differentiate pr evaluate gradient ic state equation of tangent ss arrange in standard form ss substitute into quadratic pr process ic factorise & interpret ic state coordinates Primary Method : Give mark for each dy = d m = y = ( ) stated or implied by y = = + + = ( ) = equalroots so tgt 9 Q = (, ) marks marks In (a) is only available if an attempt has been made to find the gradient from differentiation. In (b) is only available for a numerical value of m. An = must occur somewhere in the working between and. is awarded for drawing a conclusion from the candidate s quadratic equation. Candidates may substitute the equation of the parabola into the equation of the line. This is a perfectly acceptable approach. Common Error dy = d = so = so m = y = ( ) y = = + = b ac = line is not tgt 9 so award marks Alternative Marking [Marks ] b ac = = line is a tangent Alternative Method for (b) = y + y = ( y + y + ) + y + y + + 9= ( y + ) = equal roots so tgt 9 Q= (, ) Alternative Method for (b) Find the equ. of the tgt to nd curve withgrad. + = Q= (, ) y ( ) = ( ) 9 y = which is the same equ. as ( a) stated or implied by stated eplicitly

9 Higher Mathematics Paper : Marking Scheme Version The circles with equations ( ) + ( y ) = and + y k y k =. have the same centre. Determine the radius of the larger circle. C G9 CN / ic state centre of circle ss equate -coordinates, find k. ic find radius of circle ic substitute into the radius formula ic process radius formula and compare. Primary Method : Give mark for each C, k = R = R = ( ) + ( ) ( ) = ( ) > or " nd circle " or equivalent marks requires no justification. Evidence for may appear for the first time at the stage. If R = is clearly stated at the stage, then it does not have to appear at the stage for the conclusion to be drawn. For any formula masquerading as the radius formula (e.g. see Common Error ), and are NOT available. Alternative Method + y y + = k = R = R = ( ) + ( ) ( ) > or " nd circle " or equivalent Common Error C =(, ) k = R = R = ( ) + ( ) < or " st circle " Common Error C =(, ) k = R = R = ( ) + ( ) + ( ) > or " nd circle " 9

10 Higher Mathematics Paper : Marking Scheme Version The curvey = f() is such that dy =. The curve passes through the d point (, 9). Epress y in terms of. C/B C CN / ss know to integrate pr integrate ic substitute values pr process constant Primary Method : Give mark for each y =... 9= ( ) ( ) + c y = + stated or implied by stated eplicitly marks The equation y = must appear somewhere in the solution. Common Error Missing equation y =... d 9= ( ) ( ) + c c = award marks Common Error : Not using (, 9) y =... d ( ) ( ) + c = y = award marks Alternative Marking y =... y = + c and 9= ( ) ( ) + c c = stated eplicitly

11 Higher Mathematics Paper : Marking Scheme Version P is the point (,, ) and Q is (,, ). (a) Write down PQ in component form. (b) Calculate the length of PQ. (c) Find the components of a unit vector which is parallel to PQ. a C G CN /9 b C G c B G ic state vector components pr find the length of a vector ic state unit vector Primary Method : Give mark for each PQ = PQ = mark mark mark Note In (a) It is perfectly acceptable to write the components as a row vector eg PQ Treat PQ = ( ). = (,, ) as bad form (i.e. not penalised). In (b) is not awarded for an unsimplified. Beware of misappropriate use of the scalar product where, by coincidence, p.q =. In (c) Accept for.

12 Higher Mathematics Paper : Marking Scheme Version The diagram shows the graph of a function y = f(). Copy the diagram and on it sketch the graphs of () a y = f( ) () b y = + f( ) Q(,) y O P(, a) y = f() a C A CN /new b C A ic know translate parallel to -ais, +ve dir. ic annotate points ic know translate parallel to y-ais, +ve dir. ic annotate points Primary Method : Give mark for each translate units right and annotate one point annotate the other point [ P'(, a) Q'(, )] translate (a) units up and annotate one point marks annotate the other point [P"(, a + ) Q"(, )] marks For (a) A translation of earns a maimum of mark with both points clearly annotated and f() retaining its shape. Any other translation gains no marks. Alternative Method translate units right and annotate one point annotate the other point [ P'(, a) Q'(, )] translate original and annotate one point annotate the other point [P"(, a + ) Q"(, )] In the Primary method For (b) and are only available for applying the translation to the resultant graph from (a). A translation of earns a maimum of mark with both points clearly annotated and the resultant graph from (a) retaining its shape. Any other translation gains no marks. In the Alternative method For (b) A translation of, or applied to the original graph earns a maimum of mark with both points clearly annotated and the resultant graph retaining its original shape. Any other translation gains no marks. In either method For (a) and (b) For the annotated points, accept a superimposed grid or clearly labelled aes. 9 A candidate may choose to use two separate diagrams. This is acceptable.

13 Higher Mathematics Paper : Marking Scheme Version The diagram shows a right-angled triangle with height unit, base units and an angle of a at A. () a Find the eact values of () i sina A ( ii) sina. () b By epressing sina as sin( a + a), find the eact value of sin a. a a C T9 CN / b B T CN Note ic interpret diagram for sin(a ) ss use double angle formula for sin(a) ic interpret diagram for cos(a ) pr substitute and complete ss use compound angle formula pr use double angle formula for cos(a) ic substitute pr complete Calculating approimate angles using arcsin and arccos gains no credit. Primary Method : Give mark for each sin( a ) = sin( a ) = sin( a )cos( a ) cos( a ) = sin( a ) = marks sin( a ) = sin( a )cos( a + ) cos( a )sin( a ) cos( a ) = sin( a ) =. +. sin( a ) = marks There are processing marks, and. None of these are available for an answer >. sin(a) =. and cos(a) =. are the only two decimal fractions which may receive any credit. Some candidates may double the height of the triangle and then call the base angle a. This error is equivalent to Common Error illustrated on the right. Common Error An eample based on a numerical error in Pythagoras sin( a ) = sin( a ) = sin( a )cos( a ) cos( a ) = sin( a ) = sin( a ) = sin( a )cos( a ) + cos( a )sin( a ) cos( a ) = cos ( a ) = or equivalent sin( a ) =. +. sin( a ) = Common Error An eample of Incorrect formulae sin( a ) = sin( a ) = sin( a ) sin( a ) = sin( a ) = sin( a )cos( a ) + cos( a )sin( a ) cos( a ) = cos( a ) = sin( a ) =. +. sin( a ) =

14 Higher Mathematics Paper : Marking Scheme Version 9 dy y = cos,, find. d C/B C,C CN /9 Primary Method : Give mark for each ss epress in differentiable form pr differentiate a term with a negative power pr start to process a compound derivative pr complete compound derivative + sin marks For clearly integrating, correctly or otherwise, only is available. If you cannot decide whether a candidate has attempted to differentiate or integrate, assume they have attempted to differentiate.

15 Higher Mathematics Paper : Marking Scheme Version A curve has equation y = sin cos. (a) Epress sin cos in the form ksin( a) where k > and a π. (b) Hence find, in the interval π, the -coordinate of the point on the curve where the gradient is. a C T CR /9 b A/B T CR ss epand ic compare coefficients pr process k pr process a ic state result ss set derivative = gradient pr process from the derivative Primary Method : Give mark for each ksin( )cos( a) kcos( )sin( a) kcos() a =, ksin( a) = k = a = 9. sin(. 9) dy = cos(. 9) = d =. stated eplicitly stated eplicitly marks marks In (a) ( ) is acceptable for. k sin( )cos( a) cos( )sin( a) Treat ksin( )cos( a) cos( )sin( a) as bad form if is gained. No justification is required for. is not available for an unsimplified. ( ) is acceptable evidence for sin( )cos( a) cos( )sin( a) and. Candidates may use any form of the wave equation to start with as long as their final answer is in the form ksin( a). If it is not, then is not available. Common Error Working in degrees ( sin( )cos( a) cos( )sin( a) ) kcos( a) =, ksin( a) = k = a =. sin(.) dy = cos(. ) = d =. is only available for (i) an answer in radians which rounds to. OR (ii) an answer given as a multiple of π eg.. π 9. Award () a marks and () b marks kcos( a) = and ksin( a) = leading to a =.99 can only gain if a comment intimating that this answer is not in the given interval is given. In (b) 9 In (b) candidates have a choice of two starting points. They can either start from y = sin(. 9) as shown in the Primary method OR they can start from dy = cos( ) + sin( ). Either of d these starting positions may be awarded. Candidates who work in degrees will lose for attempting to differentiate. is only available as a consequence of solving dy d =. Do not penalise etra solutions at the stage (e.g..).

16 It is claimed that a wheel is made from wood which is over years old. To test this claim, carbon dating is used. Higher Mathematics Paper : Marking Scheme Version The formula At () = Ae. t is used to determine the age of the wood, where A is the amount of carbon in any living tree, At ()is the amount of carbon in the wood being dated and t is the age of the wood in years For the wheel it was found that At () was % of the amount of carbon in a living tree. Is the claim true? A/B A CR / ic interpret information ic substitute ss take logarithms pr process ic interpret result Primary Method : Give mark for each At ( ) =. A. t e =.. t ln( e )= ln (. ). t = ln(. ) t = yearsso claimvalid stated or implied by stated or implied by marks Candidates may choose a numerical value for A at the start of their solution. Accept this situation. is only available if has been awarded. In following through from an error, is only available for a positive value of t. Alternative Method Graph and Calculator Solution. A( ) = Ae. A and year old piece of wood contains.% carbon. try a point where t > eg.. A ( ) getting. A. sketch of y= Ae t showing. a monotonic decreasing function. points representing eg (,.%) etc observation that the point lies between the two plotted values for t and so claim valid.

17 Higher Mathematics Paper : Marking Scheme Version PQRS is a rectangle formed according to the following conditions: it is bounded by the lines = and y = y = y (, ) Q R (, ) P lies on the curve with equation y R is the point (, ). = between (, ) and (, ) (a) (i) Epress the lengths of PS and RS in terms of, the -coordinate of P. = (ii) Hence show that the area, A square units, of PQRS is given by A =. (b) Find the greatest and least possible values of A and the corresponding values of for which they occur. O P, (, ) S y = a A C CN / b 9 A/B C ic interpret diagram to find PS ic interpret diagram to find RS ic complete proof ic epress in differentiable form ss know to set derivative to zero pr differentiate pr process equation pr evaluate area at the turning point 9 pr evaluate area at the end point pr evaluate area at the end point ic state conclusion Primary Method : Give mark for each PS = RS = Area = ( ) and complete da = d + = A() = 9 A( ) = A( ) = ma. A= at = and min. A= at = or = marks marks For there needs to be clear evidence that candidates have multiplied out the brackets in order to complete the proof. An = must appear somewhere in the working between and. At the stage, ignore the omission or inclusion of =. has to be as a consequence of solving da d =. is only available if both end points have been considered.