2005 Mathematics. Higher. Finalised Marking Instructions

Transcription

1 Mathematics Higher Finalised Marking Instructions These Marking Instructions have been prepared by Eamination Teams for use by SQA Appointed Markers when marking Eternal Course Assessments.

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 marks(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 (eg 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. Page

3 . 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 the question 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 referral to the PA. Please see the general instructions for PA referrals.. No marks should be deducted at this stange 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 pd mark.. Casual errors: In general, as a consequence of a casual error, candidates lose the opportunity of gaining either the first ic mark or the first pd mark.. Do not write any comments on the scripts. A revised summary of acceptable notation is given on page. 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 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. Page

4 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 have been allotted may be shown on scripts margins dy d or The tick-cross. Use this to show correct work where you are following through subsequent to an error. y (, ) C ( ) m rad m m tgt tgt m The roof. Use this to show something is missing such as a crucial step in a proof of a condition etc. ( ) y The tilde. Use this to indicate a minor transgression which is not being penalised (such as bad form). 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. o sin invsin ( ) Remember No comments on the scripts. No abbreviations. 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

5 Find the equation of the line ST, where T is the Higher Mathematics Paper : Marking Scheme Version y point (, ) and angle STO is. S T (, ) O C G, G NC / ss use m tanθ pd use eact value ic interpret result Primary Method : Give mark for each m tan m y ( ) stated or implied by marks A candidate who states m tan( θ ), and does not go on to use it earns no marks. Incompletion m tan( ) y tan( )( ( )) award marks Common Error m sin( ) y ( ( )) Alternative Method OS tan( ) m cf y m + c y + Alternative Method cos( ) leading to ST ST and OS m y ( ( ) ) award marks

6 Higher Mathematics Paper : Marking Scheme Version Two congruent circles, with centres A and B, touch at P. Relative to suitable aes, their equations are B + y + + y and + y y +. (a) Find the coordinates of P. (b) Find the length of AB. A P a C G9, G CN / b C G9 CN ic ic interpret equ. of circle interpret equ. of circle pd process midpoint ss know how to find length pd process Primary Method : Give mark for each centre A (, ) centre B (, ) P (, ) AB AB ( ) + ( ( )) [Note ] [Note ] [CE ] marks marks Alternative Method for marks,, at, Each of the following may be awarded mark from the first two marks A (, ) and B (, ) A (, ) and B (, ) A (, ) and B (, ) p ( b+ a) b a P (, ) [Note ] At stage, some errors lead to unsimplified surds. DO NOT accept unsimplified square roots of perfect squares (up to ). e.g. would not gain. Treat P as bad form. Alternative Method for marks, Common Error for (b) + ( + ) AB + ( ) AB award mark for (b) r + ( ) or r ( ) + ( ) AB r Alternative Method for marks, AB AB

7 Higher Mathematics Paper : Marking Scheme Version D,OABC is a pyramid. A is the point (,, ), B is (,, ) and D is (,, 9). F divides DB in the ratio :. (a) Find the coordinates of the point F. z C D (,, 9) y F B (,, ) (b) Epress AF in component form. O A (,, ) a C G CN / b C G CN ss know to find DB ic interpret ratio pd process scalar times vector ic interpret vector and end points ic interpret coordinates to vector Primary Method : Give mark for each DB 9 DF DB DF 9 D ( 9,, ) so F (,, ) [Note ] marks AF mark Do not penalise candidates who write the coordinates of F as a column vector (treat as bad form). A correct answer to (a) with no working may be awarded one mark only. For guessing the coordinates of F, no marks should be awarded in (a). mark is still available in (b) provided the guess in (a) is geographically compatible with the diagram ie y z 9 In (a) Where the ratio has been reversed (ie :) leading to F(,, ) then marks may be awarded (,, ). In (b) Accept AF i + j + k for. Alternative Method [Marks -] DF FB s/i by f d b f f + 9 F (,, ) [Note ] Alternative Method [Marks -] AF AB + BF AF AB + BD AF + 9 AF ( A (,, so) F (,, ) Alternative Method [Marks -] mb+ nd f m + n m, n s/i by s/i by f + 9 F (,, ) [Note ] Alternative Method [Marks -]. y. z 9. so F (,, )

8 Higher Mathematics Paper : Marking Scheme Version Functions f and g + are defined on the set of real numbers. (a) Find h where h gf ( ). (b) (i) Write down the coordinates of the minimum turning point of y h. (ii) Hence state the range of the function h. a C A NC / b C A NC ic ic interpret comp. function build-up interpret comp. function build-up ic interpret function ic interpret function Primary Method : Give mark for each g( ) +, y stated or implied by [Note ] [Note ] marks marks For No justification is required for. Candidates may choose to dfferentiate etc but may still only earn one mark for a correct answer. For Accept y >, h, h >, h() >, h() Do not accept, > Common Error No. f ( + ) +, y award marks & apply.

9 Higher Mathematics Paper : Marking Scheme Version Differentiate + sin( ) with respect to. A C, C CN / pd start differentiation process pd use the chain rule Primary Method : Give mark for each ( + sin)... cos marks Common Error + sin sin cos award mark Common Error + sin sin cos award mark Common Error [miture of differentiating and integrating] + sin cos award marks Common Error + sin cos award mark 9

10 Higher Mathematics Paper : Marking Scheme Version (a) The terms of a sequence satisfy un+ kun +. Find the value of k which produces a sequence with a limit of. (b) A sequence satisfies the recurrence relation u mu +, u. n+ n (i) Epress u and u in terms of m. (ii) Given that u, find the value of m which produces a sequence with no limit. a C A CN / b B A, A CN ss pd know how to find limit process ic interpret rec. relation ic interpret rec. relation pd arrange in standard form pd process a quadratic ic use limit condition Primary Method : Give mark for each eg.. k + k u m + u m( m + ) + ( m( m + ) + ) m + m ( m )( m + ) m [,,] [Note ] [Note ] marks marks for (a) Guess and Check Guessing k and checking algebraically or iteratively that this does yield a limit of may be awarded mark. No working Simply stating that k earns no marks. Alternative Method for (a) b Using L a k k Wrong formula Work using an incorrect formula leading to a valid value of k (ie k <) may be awarded mark. for (b) If u is not a quadratic, then no further marks are available. An must appear at least once in working at the / stage. For candidates who make errors leading to no values outside the range < m <, or to two values outside the range, then they must say why they are accepting or rejecting in order to gain For, either crossing out the / or underlining the is the absolute minimum communication required for this i/c mark. [A statement would be preferable] Alternative Method for (a) L kl+ kl L L k L k Common Error Common Error u m + a u m + a m or equivalent m ( eased) award mark there are no values which do not yield a limit award marks

11 Higher Mathematics Paper : Marking Scheme Version The function f is of the form f () log b ( a). The graph of y f() is shown in the diagram. y (9, ) y f() (a) Write down the values of a and b. (b) State the domain of f. O (, ) a C A NC /9 b C A NC ic ic interpret the translation interpret the base ic interpret diagram Primary Method : Give mark for each a b domain is > a [Note ] [Note ] marks mark No justification is required for marks and. BUT simply stating ( a) log a and log 9 b with no further work earns no marks. b However log b 9 a and b 9 a may be awarded mark. Of course to gain the other mark, both values would need to be stated. Clearly > is correct but do not accept a domain of.

12 Higher Mathematics Paper : Marking Scheme Version A function f is defined by the formula f () + 9 where is a real number. (a) Show that ( ) is a factor of f (), and hence factorise f () fully. (b) Find the coordinates of the points where the curve with equation y f() crosses the - and y-aes. (c) Find the greatest and least values of f in the interval. a C A NC / b C A NC c B C NC ss know to use pd complete strategy ic interpret zero remainder ic interpret quadratic factor pd complete factorising Primary Method : Give mark for each eg eg remainder is zero so ( ) is a factor ( )( )( + ) stated eplicitly [Note ] marks In the Primary method, (a) Candidates must show some acknowledgement of the result of the synthetic division. Although a statement w.r.t. the zero is preferable, accept something as simple as underlining the zero. Candidates may use a second synthetic division to complete the factorisation. and are available. Alternative method (marks -) (linear factor by substitution) f f eg 9 ( )( )( + ) Alternative method (marks -) (quad factor by inspection) f f ( ) (...) ( ) ( )( )( + ) Alternative method (marks -) (long division) remainder is zero so ( - ) is a factor ( ) ( ) ( )( )( + )

13 Higher Mathematics Paper : Marking Scheme Version A function f is defined by the formula f () + 9 where is a real number. (a) Show that ( ) is a factor of f (), and hence factorise f () fully. (b) Find the coordinates of the points where the curve with equation y f() crosses the - and y-aes. (c) Find the greatest and least values of f in the interval. Primary Method : Give mark for each ic interpret y-intercept ic interpret -intercepts ( 9, ) (, ),,,(, ) [Note ] marks ss set derivative to zero 9 pd solve ss evaluate function at an end point ic interpret results ic interpret results 9 or f( ) OR f() greatest value 9 least value [Note ] [Note ] marks In the Primary method (b) Only coordinates are acceptable for full marks. Simply stating the values at which it cuts the - and y- aes may be awarded mark (out of ). If all the coordinates are round the wrong way award mark. If the brackets are missing, treat as bad form. In the Primary method (c) Ignore any attempt to evaluate function at /. Alternative method (marks -) (nature table) 9 or nature table showing is ma. tp and thegreatest ( maimum) value is 9 f( ) OR f least value [Note ] [Note ] and are not available unless both end points and the st. points have been considered. In the Alt. method (c) is not available unless both end points have been considered. In (c) 9 Some candidates simply draw up a table using integer values from to and make conclusions from it. This earns 9 (Primary) ONLY, provided that one of the end points is correct.

14 Higher Mathematics Paper : Marking Scheme Version 9 If cos and < < π, find the eact values of cos and sin. 9 C T NC / ss pd use double angle formula process pd process pd process Primary Method : Give mark for each cos cos cos sin( ) marks Alternative Method In the event of cos sin being used, no marks are available until the equation reduces to a quadratic in either cos orsin( ). cos ±, sin ± loses. sin sin sin cos( ) and are only available as a consequence of attempting to apply the double angle formula. (This note does note apply to alt. method ) Guess and Check. For guessing that cos and sin, substituting them into any valid epression for cos and getting /, award mark only. Alternative Method +,, triangle a b a and angle bisector b a + a a (,, ) triangle t cos and sin t b a Common Error cos cos cos sin,, award mark only Common Incompletion cos cos cos sin award marks

15 Higher Mathematics Paper : Marking Scheme Version (a) Epress sin cos in the form ksin( a) where k > and a π. (b) Hence, or otherwise, sketch the curve with equation y + sin cos in the interval π. a C T NC / b A T NC ic ic epand compare coefficients pd process k pd process angle ic state equation ic completing graph ic completing graph ic completing graph 9 ic completing graph In the whole question Do not penalise more than once for not using radians. In (a) k sincos( a) cossin( a) No justification is required for is acceptable for is not available for an unsimplified sincos( a) cossin( a) is acceptabe for and or sincos( a) cossin( a) 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. is only available for an answer in radians. Treat ksincos( a) cossin( a) as bad form only if is gained. In (b) The correct sketch need not include annotation of ma, min or intercept for to be awarded but you would need to see the graph lying between y and y. 9 is available for one cycle of any sinusoidal curve of period π ecept y sin. Some evidence of a scale is required. For, accept in lieu of Primary Method : Give mark for each ksincos( a) kcossin( a) kcos( a), ksin( a) k a π y + sin π a sketch showing a sinusoidal curve y-intercept at, π ma at, π min at, Alternative marking for and 9 π/ π π/ π STATED EXPLICITLY STATED EXPLICITLY [ -] [ 9,] π π ma at and min at 9 graph lies between y and y Alternative method for to 9 (Calculus) dy cos + sin d tan π maat (, ) π min at, 9 y and annotated sketch. marks stated or implied by a correct sketch [Note ] and no -intercepts marks Do not penalise graphs which go beyond the interval...π.

16 Higher Mathematics Paper : Marking Scheme Version (a) A circle has centre (t, ), t >, and radius units. Write down the equation of the circle. (b) Find the eact value of t such that the line y is y y O a tangent to the circle. (t, ) a C G CN / b A G CN Primary Method : Give mark for each ( t) + ( y ) mark ic state equ. of circle ss substitute pd rearrange in standard form. ss know to use "discriminant " ic identify " a"," b" and "" c pd process ( t) + t + t " b ac" a, b t, c t t ( t ) and t [Note ] [Note ] marks Subsequent to trying to use an epression masquerading as the discriminant e.g. a bc, only (from the last two marks) is still available. Treat t ± as bad form. Common Error No. a, b, c t ( t ) t t or. Alternative Method (for (b)) Let P be point of contact, C the centre of the circle. Consider triangle OPC. OPC 9 (tgt/radius) PC (radius) CP/OP tan(cop) (gradient of tgt) Hence OP and, by Pythagoras, t OC ( + ). Alternative Method (for (b)) y m and m tgt rad equ of radius is + y t ie t y ( y) + y y y t + y t

17 Higher Mathematics Paper : Marking Scheme Version S [] The boplot shows the salaries of male and female graduates working for a large company at the end of their third year of employment. Compare the salaries of these males and females. males females Graduate Salaries after years * * * Salaries ( ) Salaries ( ) S C../ NC / ic ic comment comment ic comment one comment from list one comment from list one comment from list marks males had higher salaries on average by range of salaries is broadly similar only females achieved same salary as top % males majority of males earned more than the average female any other reasonable comment S [] A bag contains blue and red counters. counters are drawn at random without replacement. The random variable X is the number of blue counters drawn. (a) Find the probability distribution for X. (b) Find E(X). S C../ NC / ss pd know to find P(X) etc process pd process pd process ss choose correct form pd process... + EX Σ p marks marks S A continuous random variable T has probability density function t t ft () < < otherwise. [] (a) Find the value of k. (b) Calculate P( < T < ). k, S B.. CN / ss know ftdt pd process pd process pd process ss use tdt pd integrate pd process limits k ftdt k tdt k + k tdt t k + t k + marks marks

18 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 marks(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 (eg 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. Page

19 . 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 the question 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 referral to the PA. Please see the general instructions for PA referrals.. No marks should be deducted at this stange 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 pd mark.. Casual errors: In general, as a consequence of a casual error, candidates lose the opportunity of gaining either the first ic mark or the first pd mark.. Do not write any comments on the scripts. A revised summary of acceptable notation is given on page. 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 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. Page

20 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 have been allotted may be shown on scripts margins dy d or The tick-cross. Use this to show correct work where you are following through subsequent to an error. y (, ) C ( ) m rad m m tgt tgt m The roof. Use this to show something is missing such as a crucial step in a proof of a condition etc. ( ) y The tilde. Use this to indicate a minor transgression which is not being penalised (such as bad form). 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. o sin invsin ( ) Remember No comments on the scripts. No abbreviations. 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

21 Higher Mathematics Paper : Marking Scheme Version Find d,. C C, C CN / ss: arrange in integrable form pd: integrate positive inde pd: integrate negative inde ic: complete including const. of int. Primary Method : Give mark for each + + c [Note ] marks If incorrectly epressed in integrable form, follow throughs must match the generic marking scheme. can only be awarded on follow through provided the integral involves a negative inde. can only be awarded if the constant of integration appears somewhere in the working. can only be awarded as a result of at least one valid integration at the or stage. Common Error Common Error Common Error [ seegeneric ] + c award marks + + c award marks Common Error + + c + c award marks Common Error ( )+ c award marks Common Error ( )+ c award marks award marks Throughout this paper, unless specifically mentioned, a correct answer with no working receives no credit.

22 Triangles ACD and BCD are right-angled at D with angles p and q and lengths as shown in the diagram. (a) Show that the eact value of sin( p+ q) is. (b) Calculate the eact values of (i) cos( p+ q) (ii) tan(p + q). Higher Mathematics Paper : Marking Scheme Version A C p q D B a C T9 CN / b C T9 CN ic: interpret diagram ic: interpret diagram ss: epand sin(a+b) pd: sub. and complete ss: epand cos(a+b) pd: sub. and complete ic: use tan sin/cos() Primary Method : Give mark for each cos( p),sin( p) cos( q), sin( q) [Note ] sin( p)cos( q) + cos( p)sin( q) + & complete cos( p)cos( q) sin( p)sin( q) or equivalent fraction or equivalent fraction eg stated or implied by when written in the same order as eplicitly stated marks marks and may, if necessary, be awarded as follows sin( p), sin( q) cos( p),cos( q) For There has to be some working to show the completion. eg or + or + + Calculating appro angles using invsin and invcos can gain no credit at any point. Any attempt to use sin( p+ q) sin( p) + sin( q) loses and. Any attempt to use cos( p+ q) cos( p) + cos( q) loses and. This second option must not be treated as a repeated error. Alternative (for marks & ) sin( p+ q) sin( p+ q) and complete Alternative (for marks & ) + cos( p+ q).. Alternative (for marks & ) cos ( p+ q) cos( p+ q) with justification of the choice of negative sign eg..( + ) ( ) > + ( 9) or using the cosine rule

23 (a) A chord joins the points A(, ) and B(, ) on the circle as shown in the diagram. Show that the equation of the perpendicular bisector of chord AB is + y. (b) The point C is the centre of this circle. The tangent at the point A on the circle has equation + y. Find the equation of the radius CA. (c) (i) Determine the coordinates of the point C. (ii) Find the equation of the circle. a C G CN / b C G CN c C G CN To gain some evidence of completion needs to be shown Higher Mathematics Paper : Marking Scheme Version y O A (, ) Primary Method : Give mark for each m AB m ss: find perp. bisector midpoint (, ) pd: calc. perp. gradient y ( ) and complete [,] marks ss: find approp. mid-point y stated/implied by ic: complete proof m tgt ss: compare with y m + c m stated/implied by rad ic: state gradient y ( ) [Note ] ss: find gradient of radius marks ic: state equation of line 9 use + y 9 ss: solve sim. equations and y [,] pd: solve sim. equations, y ic: state equation of circle ( ) + ( y ) r pd: calculate radius r [Note ] marks y O C A (, ) B (, ) eg y ( ) y + y + is only available if an attempt has been made to find and use both a perpendicular gradient and a midpoint. is only available if an attempt has been made to find and use a perpendicular gradient. At the 9, stage Guessing (,) (from stepping) and checking it lies on perp. bisector of AB may be awarded 9 and Guessing (,) (with or without reason) and with no check gains neither 9 nor Solving y and + y leading to (,) will lose 9 and. to gain some evidence of use of the distance formula needs to be shown. At the and stage Subsequent to a guess for the coordinates of C, and are only available if the guess is such that << and <y<. Alternative [for 9 and ] 9 D(,) where D is intersection of the perp. to AB through B and the circle. C midpoint of AD (,) Common Error [ for to ] y + m m rad y ( ) eased award mark Common Error [ for to ] + y so m y ( ) award marks

24 Higher Mathematics Paper : Marking Scheme Version The sketch shows the positions of Andrew(A), Bob (B) and B(,, 9) A(,, ) Tracy(T) on three hill-tops. Relative to a suitable origin, the coordinates (in hundreds of metres) of the three people are A(,, ), B(,, 9) and T(,, ). In the dark, Andrew and Bob locate Tracy using heat-seeking beams. (a) Epress the vectors TA andtb in component form. (b) Calculate the angle between these two beams. T(,, ) a C G CN / b C G Ca ic: state vector components ic: state vector components pd: find length of vector pd: find length of vector pd: find scalar product ss: use scalar product pd: evaluate angle In (a) For calculating AT and BT award mark out of. Treat column vectors written like (,, ) as bad form. In (b) For candidates who do not attempt, the formula quoted at must relate to the labelling in the question for to be awarded. Do not penalise premature rounding. Primary Method : Give mark for each TA TB TA TB 9 TA. TB cos( ˆ TA. TB ATB) TA TB ATB ˆ 9 OR. 9 OR. grads [,] c [Note ] [Note ] marks stated or implied by marks The use of tan( ATB ˆ ) TA. TB TA TB loses Alternative for to (Cosine Rule) TA The use of cos( ˆ TA. TB ATB ) are available. AB means that only and TB AB 9 cos( ˆ 9 + ATB). 9. ATB ˆ 9 stated or implied by Common Error No. Common Error No. TA t a TB t b award mark TA t + a TB t + b award mark Further common errors overleaf.

25 Higher Mathematics Paper : Marking Scheme Version The sketch shows the positions of Andrew(A), Bob (B) and Tracy(T) on three hill-tops. B(,, 9) A(,, ) Relative to a suitable origin, the coordinates (in hundreds of metres) of the three people are A(,, ), B(,, 9) and T(,, ). In the dark, Andrew and Bob locate Tracy using heat-seeking beams. (a) Epress the vectors TA andtb in component form. (b) Calculate the angle between these two beams. T(,, ) Common Error : Finding angle BOA using OB and 9 OB OB. OA OA and OA cos( ˆ OB. OA BOA ) OB OA BOA ˆ OR award mark perbullet c 9 Common Error : Finding angle BOT using OB and 9 OB OB. OT OT and OT cos( ˆ OB. OT BOT ) OB OT BOT ˆ OR c Common Error : Finding angle AOT using OA and OT OA 9 and OT OA. OT cos( ˆ OA. OT AOT ) OA OT AOT ˆ 9 OR c award mark perbullet Common Error : Finding angle ABT using BA and BT BA BA. BT and BT 9 cos( ˆ OA. OT ABT) OA OT ABT ˆ OR 9 c award mark per bullet Common Error : Finding angle BAT using AB and AB AB. AT AT and AT cos( ˆ AB. AT BAT ) AB AT BAT ˆ OR 9 c award mark per bullet award mark per bullet 9

26 The curves with equations y and y 9 intersect at K and L as shown. Calculate the area enclosed between the curves. Higher Mathematics Paper : Marking Scheme Version y y 9 y K L O C C CN /9 ss: find intersection pd: process quadratic equ. ss: upper lower ic: interpret limits pd: sub. & simplify Upper Lower pd: integrate ic: substitute limits pd: evaluate and complete Primary Method : Give mark for each 9 ± upper lower eg ( + 9 ) [,] [Note ] stated or implied by marks Alternative for to There is no penalty for working with + 9oreven 9 but in the latter case, the minus signs need to be dealt with correctly at some point for o be awarded. Candidates who attempt to find a solution using a graphics calculator earn no marks. The only acceptable solution is via calculus. is lost for subtracting the wrong way round and subsequently may be lost for such statements as square units so ignore the ve square units may be gained for statements such as so the area ( ) lower upper or lower upper are technically correct and hence all marks are available. L K For upper lower,,, and are available Differentiation loses, and. Using + 9 and 9d leading to zero can only gain and from the last marks. Candidates may attempt to split the area up. In Alt., for candidates who treat C as a triangle, the last three marks are not available. eg ( ) + Alternative for to y O y B A C y 9 9 d leading to B9 (.) ( ) d leading to A+C9 ( 9) d leading to C9 9 (. ) A 9 (.) Total area L [Note ]

27 The diagram shows the graph of y, >. Higher Mathematics Paper : Marking Scheme Version y Find the equation of the tangent at P, where. O P B C, C CN / ss : know to differentiate ic : epress in st. form pd : differentiate ve fractional inde pd : evaluate ve fractional inde pd : evaluate y-coord ic : state equ of tangent and are only available if an attempt to find the gradient is based on differential calculus. is not available to candidates who find and use a perpendicular gradient. is only available for a numerical value of m. Primary Method : Give mark for each y dy dy d d dy d y y ( ) nr y + nr not required [,,] marks Common Error Common Error y dy dy d d dy d 9 y y 9( ) eased award marks y d + c gradient 9 y y 9( ) Note Note award marks

28 Higher Mathematics Paper : Marking Scheme Version Solve the equation log ( ) log ( ), <. A A CN ss: use the log laws ss: know to convert from log to epo pd: process conversion pd: find valid solution Primary Method : Give mark for each log use log ( b) c b a a c stated or implied by See Cave marks For Accept answer as a decimal. Common Error No. log log() 9 award marks Common Error No. log which is notavalidsol. Alternative log log Cave log leading to award marks BUT stated or implied by log leading to,,, award marks award marks Common Error No. log log. log award marks

29 Higher Mathematics Paper : Marking Scheme Version Two functions, f and g, are defined by f ksin and g sin where k >. y A y g() y f() The diagram shows the graphs of y f and y g intersecting at O, A, B, C and D. Show that, at A and C, cos k. O π B C D π A T CN / ss: equate for intersection ss: use double angle formula pd: factorise pd: process two solutions ic: complete proof Primary Method : Give mark for each ksin sin k sincos sin ( kcos ) sin and cos k sin, π, π i.e. at (O),B and D and cos() is for A and C. k [Note ] [Note ] marks Only is available for candidates who substitute a numerical value for k at the start. is only available if a suitable comment regarding points (O), B and D is made. If all the terms are transposed to one side, then an needs to appear at least once. For Alternative and are not available unless has been awarded. Common Error ksin sin k sincos sin( ) sin kcos kcos cos at A and C. k award marks Common Error ksin sin k sincos sin k cos cos at A and C. k award marks Alternative for and at (O), B and D, sin( ) so at A and C, kcos cos k. Alternative for and at A and C, sin so at A and C, kcos cos k. Alternative for to ksin sin k sincos sin at A and C, sin so at A and C, k cos cos k

30 Higher Mathematics Paper : Marking Scheme Version 9 The value V (in million) of a cruise ship t years after launch is given by the formula t V e. (a) What was its value when launched? (b) The owners decide to sell the ship once its value falls below million. After how many years will it be sold? 9 a B A CN / b A A Ca Primary Method : Give mark for each V ( m) t mark pd: evaluate at t ic: substitute V pd: process ic: epo to log conversion pd: solve a logarithmic equation t e t e t ln t [Note ] marks in (b) For accept any correct answer which rounds to. An answer obtained by trial and improvement which rounds to may be awarded a ma. of mark (out of ) but only if they have checked 9 as well. In following through from an error, is only available for a positive answer. Common Error t log( e ) log. t log log log. t log t. award mark Solution via graphics calculator t e choose to graph y e asketch[ see below] t y t y e solution t Alternative for (b) (takings logs of both sides) t e t e tlog ( e) log k k where k e or k t Alternative t [Note ] e log. tloge log.. t. 99 t Alternative t e. t ln + lne ln. tlne ln ln t t Note t You could also graph, for eample, y e and y

31 Higher Mathematics Paper : Marking Scheme Version Vectors a and c are represented by two sides of an equilateral triangle with sides of length units, as shown in the diagram. Vector b is units long and b is perpendicular to both a and c. Evaluate the scalar product a. ( a+ b+ c). b a c A G9 CN / ss: use distributive law pd: process scalar product pd: process scalar product pd: process scalar product & complete Primary Method : Give mark for each a.a + a.b + a.c a.a 9 9 a.c see CAVE a.b and atotal of [,] [Note ] marks Treat a.a written as a as bad form. Treat a.b written as ab as an error unless it is subsequently evaluated as a scalar product. Similarly for a.c. Using p.q p q sinθ consistently loses mark. (ie ma. available is ) CAVE a.(a + b + c) a.a + a.b + a.c followed by a.a 9 earns and. When attaching the components c b,, a When attaching the components, all marks are available. c b,, a,only is available. but a.(a + b + c) a.a + a.b + a.c followed by a.a 9, a. c 9, a.b earns only.

32 Higher Mathematics Paper : Marking Scheme Version (a) Show that is a solution of the cubic equation + p + p+. (b) Hence find the range of values of p for which all the roots of the cubic equation are real. a C A CN / b A A CN pd: evaluate the function at ss: strategy for finding other factors ic: quadratic factor ss: strategy for real roots ic: substitute pd: process ss: starts to solve inequation ic: complete Primary Method : Give mark for each f( ) + p p+ p p p p + ( p ) + " b ac" and " " ( p ) ( p )( p + ) p, p p, p mark [Note ] [,] [Note ] marks For alternative method, (as is also) is for interpreting the result of a synthetic division. Candidates must show some acknowledgement of the result of the synthetic division. Although a statement w.r.t. the zero is preferable, accept something as simple as underlining the zero. Alternative method for marks, (starting with synth. division) p p p p Treat missing at as Bad Form is only available as a consequence of obtaining a quadratic factor from a division of the cubic. f( ) etc [Note ] Using b ac > loses An must appear at least once somewhere between and Where errors occur at the / stages, then,, are still available for solving a -term quadratic inequation. Evidence for may be a table of values or a sketch b± b ac For candidates who start with, all marks a are available (subject to working being equivalent to the Primary Method). Wrong disciminant: Using b + ac only (out of the last marks) is available. Any other epression masquerading as the discriminant loses all of the last marks. Marks should still be recorded as out of and Alternative method for marks, (quad. factor obtained by inspection) f( ) + p p+ f ( + )( ) etc Common Error (marks to ) ( p ) ( p ) p p award marks out of last

33 Higher Mathematics Paper : Marking Scheme Version [] The scatter diagram shows pairs of data values for and y where y Σ, Σy, Σ, Σy and Σy. (a) Find the equation of the regression line. (b) Estimate the value of y when. O S C.. Ca / b C.. CN THE PRIMARY MWETHOD OR ANY ALTERNATIVE METHOD pd: calculate S pd: calculate S pd: calculate b pd: calculate a & state equ. ic: use equ. of regression line y Primary Method : Give mark for each S S b 9 a and y 9 y. y marks mark [] The diagram represents the probability density function for a continuous random variable X. (a) Find the value of k. (b) Find the median. probability density k S a A.. CN / b A.. CN THE PRIMARY MWETHOD OR ANY ALTERNATIVE METHOD ss: state total area ic: find epression for total area pd: process ss: know total area. pd: process Primary Method : Give mark for each area + + k. k + ( m ). m marks marks

34 Higher Mathematics Paper : Marking Scheme Version [9] (a) Eplain briefly the difference between sample standard deviation and range as measures of spread. (b) In statistics mode, a calculator shows the summary statistics for a certain data set. One data value,, is shown to be erroneous and is deleted. Calculate the sample standard deviation of the new data set of 9 values correct to decimal places. S a B../ CN /9 b B.. Ca. S n Σ. σ. 9 Σ. min ma... THE PRIMARY MWETHOD OR ANY ALTERNATIVE METHOD ic: eplanation pd: find new pd: find new ss: use formula for S pd: process Primary Method : Give mark for each SD is a measure of spread about mean whereas ( Σ. ma Σ. 9 S. 9 min. 9 ) is a measure of range. mark marks [] A large organisation decides to run a mini-lottery for charity. Each participant selects any three different numbers from to inclusive. Every Friday the three winning numbers are drawn at random from the. Each participant with these winning numbers shares the jackpot. (a) Find the number of possible combinations and hence find the probability of a particular combination winning a share of the jackpot. (b) Find the probability that someone chooses the winning combination eactly twice within successive weeks. S a B..,.. Ca / b A.. Ca THE PRIMARY MWETHOD OR ANY ALTERNATIVE METHOD ss: find combination pd: calculate probability ic: interpret p(win) ss: find combination pd: process Primary Method : Give mark for each No. of outcomes prob 9 pl p( wins in ). 9 marks marks