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Coimisiú a Scrúduithe Stáit State Examiatios Commissio Leavig Certificate 014 Markig Scheme Mathematics (Project Maths Phase 3) Higher Level Note to teachers ad studets o the use of published markig schemes

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Coimisiú a Scrúduithe Stáit State Examiatios Commissio Leavig Certificate 014 Markig Scheme Mathematics (Project Maths Phase 3) Higher Level Note to teachers ad studets o the use of published markig schemes Markig schemes published by the State Examiatios Commissio are ot iteded to be stadaloe documets. They are a essetial resource for examiers who receive traiig i the correct iterpretatio ad applicatio of the scheme. This traiig ivolves, amog other thigs, markig samples of studet work ad discussig the marks awarded, so as to clarify the correct applicatio of the scheme. The work of examiers is subsequetly moitored by Advisig Examiers to esure cosistet ad accurate applicatio of the markig scheme. This process is oversee by the Chief Examier, usually assisted by a Chief Advisig Examier. The Chief Examier is the fial authority regardig whether or ot the markig scheme has bee correctly applied to ay piece of cadidate work. Markig schemes are workig documets. While a draft markig scheme is prepared i advace of the examiatio, the scheme is ot fialised util examiers have applied it to cadidates work ad the feedback from all examiers has bee collated ad cosidered i light of the full rage of resposes of cadidates, the overall level of difficulty of the examiatio ad the eed to maitai cosistecy i stadards from year to year. This published documet cotais the fialised scheme, as it was applied to all cadidates work. I the case of markig schemes that iclude model solutios or aswers, it should be oted that these are ot iteded to be exhaustive. Variatios ad alteratives may also be acceptable. Examiers must cosider all aswers o their merits, ad will have cosulted with their Advisig Examiers whe i doubt. Future Markig Schemes Assumptios about future markig schemes o the basis of past schemes should be avoided. While the uderlyig assessmet priciples remai the same, the details of the markig of a particular type of questio may chage i the cotext of the cotributio of that questio to the overall examiatio i a give year. The Chief Examier i ay give year has the resposibility to determie how best to esure the fair ad accurate assessmet of cadidates work ad to esure cosistecy i the stadard of the assessmet from year to year. Accordigly, aspects of the structure, detail ad applicatio of the markig scheme for a particular examiatio are subject to chage from oe year to the ext without otice. Cotets Page Paper 1 Model Solutios... 3 Markig Scheme... 6 Structure of the markig scheme... 6 Summary of mark allocatios ad scales to be applied... 7 Detailed markig otes... 8 Paper Model Solutios Markig Scheme Structure of the markig scheme Summary of mark allocatios ad scales to be applied... 6 Detailed markig otes Marcaa breise as ucht freagairt trí Gaeilge [1] [] 014. M39 Coimisiú a Scrúduithe Stáit State Examiatios Commissio Leavig Certificate Examiatio 014 Mathematics (Project Maths Phase 3) Paper 1 Higher Level Friday 6 Jue Afteroo :00 4: marks Model Solutios Paper 1 Note: The model solutios for each questio are ot iteded to be exhaustive there may be other correct solutios. Ay examier usure of the validity of the approach adopted by a particular cadidate to a particular questio should cotact his / her advisig examier. [3] Istructios There are two sectios i this examiatio paper. Sectio A Cocepts ad Skills 150 marks 6 questios Sectio B Cotexts ad Applicatios 150 marks 3 questios Aswer all ie questios. Write your aswers i the spaces provided i this booklet. You may lose marks if you do ot do so. There is space for extra work at the back of the booklet. You may also ask the superitedet for more paper. Label ay extra work clearly with the questio umber ad part. The superitedet will give you a copy of the Formulae ad Tables booklet. You must retur it at the ed of the examiatio. You are ot allowed to brig your ow copy ito the examiatio. You will lose marks if all ecessary work is ot clearly show. Aswers should iclude the appropriate uits of measuremet, where relevat. Aswers should be give i simplest form, where relevat. Write the make ad model of your calculator(s) here: [4] Sectio A Cocepts ad Skills 150 marks Aswer all six questios from this sectio. Questio 1 (a) The graph of a cubic fuctio f (x) cuts the x-axis at x 3, x 1 ad x, ad the y-axis at (0, 6), as show. Verify that f (x) ca be writte as 3 f ( x) x x 5x 6. (5 marks) y f (x) 5 g(x) x x 3, x 1, x 3 ( x 3)( x 1)( x ) x x 5 6 f ( x) x 3 f ( x) x x 5x 6 f () ( x ) is OR f ( 3) ( x 3) is f ( 1) ( x 1) is a factor a factor a factor f ( x) ( x 3)( x 1)( x ) x 3 x 5x 6 [5] (b) (i) The graph of the fuctio g ( x) x 6 itersects the graph of the fuctio f ( x) above. Let f () x g() x ad solve the resultig equatio to fid the co-ordiates of the poits where the graphs of f ( x ) ad gx ( ) itersect. f ( x) g( x) x 3 x x 3 x( x x x ( x 1)( x 3) x 0, 5x 6 x 6 3x 0 x 3) 0 0 x 1, x 3 y 6, y 8, y 0 Poits: ( 3, 0), (0, 6), (1, 8) (ii) Draw the graph of the fuctio g ( x) x 6 o the diagram above. g( x) x 6 g( 3) ( 3) ( 3, 0) g(0) (0) 6 6 (0, 6) [6] Questio Let z1 1 i, where i 1. (a) The complex umber z1 is a root of the equatio z 7z 16z Fid the other two roots of the equatio. 3 (5 marks) z 1 i a root z 1 i a root. 1 1 ( z 1 i)( z 1 i) z z 5, a factor 3 Hece, ( z z 5)( az b) z 7z 16z 15 Equate coefficiets: a ad b a 7 b 3 3 Third factor: z 3 z 3 (z 7z 16z 15) ( z Or z 5) z 3 3 Third factor: z 3 z Other roots: 3 z 1 i, z3 (b) (i) Let w z1. z1, where z 1 is the cojugate of. 1 1 z 1 ad w o the Argad diagram ad label each poit. ( 1 i)( 1 ) w i 5 Im z 1 1 w Re z 1 [7] (ii) Fid the measure of the acute agle, zwz, formed by joiig 1 1 z1 to w to z 1 o the diagram above. Give your aswer correct to the earest degree. ta 1 1 z1wz1 z1wz z wz OR Im z 1 (1, ) 1 Re w (5, 0) - z 1 (1, ) zw 1 (0 ) (5 1) zw 0 zw 0 zz Cosie rule: 4 ( 0) ( 0) ( 0)( 0)cos 40cosθ 4 4 cosθ θ θ [8] Questio 3 (a) Prove, by iductio, that the sum of the first atural umbers, ( 1 ) 1 3, is. (5 marks) ( 1) To Prove: P ( ) 1 3 1(1 1) P(1) : 1 1, True Assume P() is true for k, ad prove P() is true for k 1. kk ( 1) k: 1 3 k ( k 1) To prove P(k1) ( k ) kk ( 1) kk ( 1) ( k 1) L.H.S. 1 3 k ( k 1) ( k 1) ( k 1) ( k ) R.H.S But P(1) is true, so P() is true etc. Hece, P() is true for all. (b) Hece, or otherwise, prove that the sum of the first eve atural umbers, 4 6, is. a ad d. S a ( 1) d 4 ( 1) ( ) ( ) ( ) OR S ( ) ( 1) ( 1) [9] [10] (c) Usig the results from (a) ad (b) above, fid a expressio for the sum of the first odd atural umbers i its simplest form. terms) 6 4 ( terms) 5 3 (1 1) ( 3 1 ) ( terms ) ( ) ( terms OR 1) ( ) ( 1) ( S S S S B A B A Questio 4 (a) Differetiate the fuctio x 3x 6 with respect to x from first priciples. (5 marks) f ( x) x 3x 6 f ( x h) ( x h) f ( x h) f ( x h) Limit h 0 h f ( x) 4xh h 3( x h) 6 x 3h f ( x) 4xh h Limit 0 h h 4xh h 3h 4x 3 3x 3h 6 (b) x Let f( x), x, x. Fid the co-ordiates of the poits at which the slope of the x taget to the curve y f (x) is 1. 4 x f ( x) x Let ux ( ) x u ( x) ad vx ( ) x v ( x) 1 ( x )() x(1) 4 f ( x) ( x ) ( x ) f ( x) 4 ( x ) 4 16 ( x ) x 4 or x 4 x or x 6 or x 4x 1 0 ( x )( x 6) 0 x 0 or x 6 0 x or x f( 6) 3 ad f() 1 6 Poits ( 6,3) ad (,1) [11] Questio 5 (a) Fid 5cos3x dx. (5 marks) 5cos3xdx si3x c 5 3 (b) The slope of the taget to a curve y f (x) at each poit ( x, y ) is x. The curve cuts the x-axis at (, 0). (i) Fid the equatio of f ( x ). dy (x ) dx y x x c At x, y c c 8 Hece, y x x 8 (ii) Fid the average value of f over the iterval 0 x 3, x. Average value: 1 b a b f ( xdx ) a x ( x x 8) dx x 8x [1] Questio 6 The th term of a sequece is T l a, where a 0 ad a is a costat. (a) (i) Show that T, T, ad 1 T 3 are i arithmetic sequece. (5 marks) T l a, T l a l a, T l a 3l a T T1 la la la T3 T 3l a l a l a T T T T. Hece, terms are i arithmetic sequece. 3 1 (ii) Prove that the sequece is arithmetic ad fid the commo differece. T l a l a, 1 T 1 l a ( 1) l a. T T 1 l a ( 1) l a l a, (a costat). Hece, the sequece is arithmetic. Commo differece: T T 1 l a (b) Fid the value of a for which T1 T T3 T98 T99 T T T T T T T l a l a 3l a 100 l a [ l a (100 1)l a] l [ a] 5050l a l a a e [13] (c) Verify that, for all values of a, ( T 1 T T3 T10 ) 100d ( T11 T1 T13 T0 ), where d is the commo differece of the sequece. ( T T T T ) d ( T T d) ( T T 1 T 13 10d) ( T T d) ( T 10 10d) OR ( T1 T T3 T10 ) 100d (l a l a 3l a 10l a) 100l a 10 ( l a (10 1)l a) 100l a 5( 11l a) 100l a ( ) l a T T T T 11l a 1 l a 13l a 0 l a 10 ( l a (10 1) l a) ( a) 531l 155l a Hece, L.H.S R.H.S [14] [15] Sectio B Cotexts ad Applicatios 150 marks Aswer all three questios from this sectio. Questio 7 (40 marks) (a) Three atural umbers a, b ad c, such that, a b c are called a Pythagorea triple. (i) Let. 1 ad 1, c b a Pick oe atural umber ad verify that the correspodig values of a, b ad c form a Pythagorea triple. 5 1 (1) (1) 1 4 (1) (1) 3 1 (1) 1 1: Let c c b b a a c b a (ii) Prove that 1, ad 1, a b c where, will always form a Pythagorea triple ) ( ) ( b a b a ) ( b a c (b) ADEC is a rectagle with AC 7 m ad AD m, as show. B is a poit o [AC] such that AB 5 m. P is a poit o [DE] such that DP x m. A B C 5 m m m x m D P E m A B C 5 m m m m x m (5x) m m D P M (7x) m E (i) Let f x PA PB PC ( ). Show that f( x) 3x 4x 86, for 0 x 7, x. PM PE ME (7 x) (5 x) f ( x) PA [ PD DA ] [ PM MB ] [ PE EC ] x ((5 x) ) ((7 x) ) x 3x PB x x 4x 86 PC x x 4 [16] (ii) The fuctio f (x) has a miimum value at x k. Fid the value of k ad the miimum value of f (x). f ( x) 3x 4x 86 f ( x) 6x 4 f ( x) 6 0 miimum f ( x) 0 6x 4 0 x 4 k f (4) 3(4) 4(4) OR f ( x) 3x 4x x 8x ( x 8x 16) ( x 4) 3 At x 4 miimum value for f( x) f x x (4) (4) 4(4) [17] Questio 8 I 011, a ew footbridge was opeed at Mize Head, the most south-westerly poit of Irelad. The arch of the bridge is i the shape of a parabola, as show. The legth of the spa of the arch, [ AB ], is 48 metres. (50 marks) (0, 5) D C E A(0, 0) H(4, 0) B(48, 0) (a) Usig the co-ordiate plae, with A (0, 0) ad B (48, 0), the equatio of the parabola is y 0 013x 0 64 x. Fid the co-ordiates of C, the highest poit of the arch. y 0 013x dy dx 0 64x 0 06x x 4 y 0 013x 0 64 x 0 013(4) 0 64(4) C( 4, 7 488) OR Max height at C whe x 4 y 0013 x 064 x 0 013(4) (0 64)(4) C( 4, 7 488) [18] (b) The perpedicular distace betwee the walkig deck, [DE], ad [AB] is 5 metres. Fid the co-ordiates of D ad of E. Give your aswers correct to the earest whole umber. Equatio DE: y 5 Equatio of the parabola: y 0 013x 0 64 x x 0 013x 0 64x 0 64x ± (0 013)5 x (0 013) x 37 8 or x ± D( 10, 5), E(38, 5) [19] (c) Usig itegratio, fid the area of the shaded regio, ABED, show i the diagram below. Give your aswer correct to the earest whole umber. D(10,5) C E(38,5) A B Area ABED y dx Area of rectagle y dx ydx (38 10) ( ) 140 x x dx x 0 64x (10) 0 64(10) m 10 0 OR Cotd [0] Area uder curve betwee A ad B: 48 ( 0 013x 0 64 ) x 0 64x 3 3 x dx (48) 0 64(48) Traslate curve vertically dowwards ad fid area uder the curve betwee D ad E: 38 ( ) x x dx x 0 64x 5x (38) 0 64(38) 0 013(10) 0 64(10) 5(38) 5(10) Shaded area m [1] (d) Write the equatio of the parabola i part (a) i the form are costats. y k p x h ( ), where k, p, ad h y 0 013x x ( x 48x) ( x 48x ( 4) ( 4) ) ( x 4) ( x 4) y (e) Usig what you leared i part (d) above, or otherwise, write dow the equatio of a parabola for which the coefficiet of x is ad the co-ordiates of the maximum poit are ( 3, 4). Give fuctio: coefficiet of New fuctio: coefficiet of Fuctio: ( ) y 4 x 3 x, 0 013; maximum poit (4, 7 488) x, ; maximum poit (3, 4) OR Give fuctio: y ( x 4) y (max height) (coefficiet of x )( x x ) max New parabola: max height: 4 coefficiet of x : x max 3 y ( 4) ( x 3) y 4 ( x 3) [] Questio 9 (60 marks) kt Ciará is preparig food for his baby ad must use cooled boiled water. The equatio y Ae describes how the boiled water cools. I this equatio: t is the time, i miutes, from whe the water boiled, y is the differece betwee the water temperature ad room temperature at time t, measured i degrees Celsius, A ad k are costats. The temperature of the water whe it boils is 100 C ad the room temperature is a costat 3 C. (a) Write dow the value of the temperature differece, y, whe the water boils, ad fid the value of A. y at t 0 kt y Ae 77 Ae A 77 0 (b) After five miutes, the temperature of the water is 88 C. Fid the value of k, correct to three sigificat figures. At t 5, y kt y 77e 65 77e 5k l k k (c) Ciará prepares the food for his baby whe the water has cooled to 50 C. How log does it take, correct to the earest miute, for the water to cool to this temperature? y e t l t 77 7 t miutes [3] (d) Usig your values for A ad k, sketch the curve kt ( t) Ae for 0 t 100, t. f (0) t 0 y 77e 77 ; (0, 77) (10) t 10 y 77e 54 9 ; (10, 55) (0) t 0 y 77e 39 1; (0, 39) (30) t 30 y 77e 7 9 ; (30, 8) (40) t 40 y 77e 19 8 ; (40, 0) (50) t 50 y 77e 14 1; (50, 14) (60) t 60 y 77e 10 1 ; (60, 10) (70) t 70 y 77e 7 ; (70, 7) (80) t 80 y 77e 5 1 ; (80, 5) (90) t 90 y 77e 3 6 ; (90, 4) (100) t 100 y 77e 6 ; (100, 3) y g (t) 10 f (t) t [4] (e) (i) mt O the same diagram, sketch a curve g ( t) Ae, showig the water coolig at a faster rate, where A is the value from part (a), ad m is a costat. Label each graph clearly. (ii) Suggest oe possible value for m for the sketch you have draw ad give a reaso for your choice. Test m 0 0, m k ad m (10) m 0 0, t 10 y 77e 63 0 m k y 54 9 (from table) 0 05(10) m 0 05, t 10 y 77e 46 7 Ay value of m k for faster decay. (f) (i) Fid the rates of chage of the fuctio f (t) after 1 miute ad after 10 miutes. Give your aswers correct to two decimal places. dy y 77e 6103e dt t t t 1, dy dt 6103e dy t dt , 6103e 1 86 (ii) Show that the rate of chage of f (t) will always icrease over time. d y 00339t 0088e 0 dt dy is icreasig dt [5] Markig Scheme Paper 1, Sectio A ad Sectio B Structure of the markig scheme Cadidate resposes are marked accordig to differet scales, depedig o the types of respose aticipated. Scales labelled A divide cadidate resposes ito two categories (correct ad icorrect). Scales labelled B divide resposes ito three categories (correct, partially correct, ad icorrect), ad so o. The scales ad the marks that they geerate are summarised i this table: Scale label A B C D E No of categories mark scales 0, 5 0, 3, 5 0, 3, 4, 5 0,, 3, 4, 5 10 mark scales 0, 10 0, 5, 10 0, 5, 7, 10 0, 3, 7, 8, mark scales 0, 15 0, 7, 15 0, 7, 10, 15 0, 5, 9, 1, 15 0 mark scales 0, 0 0, 10, 0 0, 7, 13, 0 0, 5, 10, 15, 0 5 mark scales 0, 5 0, 1, 5 0, 8, 17, 5 0, 6, 1, 19, 5 0, 5, 10, 15, 0, 5 A geeral descriptor of each poit o each scale is give below. More specific directios i relatio to iterpretig the scales i the cotext of each questio are give i the scheme, where ecessary. Markig scales level descriptors A-scales (two categories) icorrect respose correct respose B-scales (three categories) respose of o substatial merit partially correct respose correct respose C-scales (four categories) respose of o substatial merit respose with some merit almost correct respose correct respose D-scales (five categories) respose of o substatial merit respose with some merit respose about half-right almost correct respose correct respose E-scales (six categories) respose of o substatial merit respose with some merit respose almost half-right respose more tha half-right almost correct respose correct respose I certai cases, typically ivolvig icorrect roudig, omissio of uits, a misreadig that does ot oversimplify the work or a arithmetical error that does ot oversimplify the work, a mark that is oe mark below the full-credit mark may also be awarded. Thus, for example, i scale 10C, 9 marks may be awarded. [6] Summary of mark allocatios ad scales to be applied Sectio A Sectio B Questio 1 (a) (b)(i) (b)(ii) Questio (a) (b)(i) (b)(ii) Questio 3 (a) (b) (c) Questio 4 (a) (b) Questio 5 (a) (b)(i) (b)(ii) 15C 5C 5C 5D 10C 10C 10D 10D 5B 15D 10D 5B 10C 10C Questio 7 (a)(i) (a)(ii) (b)(i) (b)(ii) Questio 8 (a) (b) (c) (d) (e) Questio 9 (a) (b) (c) (d) (e) (f)(i) (f)(ii) 10B 10D 5D 15C 15C 10C 10D 10C 5B 10C 10C 10C 15C 5C 5C 5C Questio 6 (a)(i) (a)(ii) (b) (c) 10C 5C 5C 5C [7] Detailed markig otes Sectio A Questio 1 (a) Scale 15C (0, 7, 10, 15) Oly oe value verified Recogisig oe factor Writig ( x 3)( x 1)( x ) Two relevat roots tested (b)(i) Scale 5C (0, 3, 4, 5) Equatios correct whe f ( x) g( x) Cubic equatio ot factorised Roots idetified (b)(ii) Scale 5C (0, 3, 4, 5) Oe poit foud i g(x) Oly oe poit idicated o graph Two poits idetified Two poits plotted but o graph draw [8] Questio (a) Scale 5D (0,, 3, 4, 5) Idetifies aother root Forms a equatio Mid Partial Credit: Works with correct quadratic factor Idicates divisio of quadratic ito cubic Fids third factor (b)(i) Scale 10C (0, 5, 7, 10) Plots oe poit correctly Fids z1 Poits plotted but ot labelled or labelled icorrectly Two poits plotted ad labelled Calculates w (b)(ii) Scale 10C (0, 5, 7, 10) Legth of ay oe side of triagle calculated correctly Correct defiitio of trig ratio Correct cos rule Recogises the half-agle cos value calculated but agle ot foud ta value of half-agle calculated [9] Questio 3 (a) Scale 10D (0, 3, 7, 8, 10) Oe correct step i iductio Statemet P(1) true Mid Partial Credit: Uses ( k 1) term o LHS Some work with ( k 1) term Correct RHS No coclusio (b) Scale 10D (0, 3, 7, 8, 10) Recogisig as commo factor ( 1) Mid Partial Credit: Correct equatio from (a) Takig out of series Work ot completed OR (b) Scale 10D (0, 3, 7, 8, 10) [whe series tested as a AP] Recogitio of a Recogitio of d Correct AP formula oly Mid Partial Credit: Some substitutio ito correct formula Work ot fully simplified Aswer ot i required form S missig (c) Scale 5B (0, 3, 5) Partial Credit: S S idicated B A S B S A Use of correct series from (b) Note: Must use result from (a) ad (b) here [30] Questio 4 (a) Scale 15D (0, 5, 9, 1, 15) Itroduces f ( x h) Mid Partial Credit: f ( x h) f ( x) expressed (eed ot be simplified) RHS oly f ( x h) f( x) Limit h 0 h (eed ot be simplified) (b) Scale 10D (0, 3, 7, 8, 10) du dv Either or correct dx dx No differetiatio but writes f '( x) 1 4 Mid Partial Credit: f '( x) correct but ot simplified Correct values of x from studets work [31] Questio 5 (a) Scale 5B (0, 3, 5) Partial Credit: Some correct itegratio Itegrad does ot cotai c c oly (b)(i) Scale 10C (0, 5, 7, 10) Some correct itegratio Itegrad does ot cotai c c oly dy dy x or slope of taget dx dx Substitutes (, 0) but c ot simplified Note: must have c i equatio to get high partial marks (b)(ii) Scale 10C (0, 5, 7, 10) Correct formula oly Some correct itegratio Idicatio of itegratio with correct limits If oly values used e.g. f (0), f (1), f () etc. whe 0 x 3, give Low Partial Credit for two or more values Limits iserted ito fuctio but ot calculated 1 missig from formula ( b a) Note: NO CREDIT differetiatio NO CREDIT o itegratio [3] Questio 6 NOTE: Whe particular values are used i ALL sectios give Low Partial Credit at most each time (a)(i) Scale 10C (0, 5, 7, 10) Oly oe term correct Either T ) or T ) correct ( T1 ( 3 T (a)(ii) Scale 5C (0, 3, 4, 5) Uses two cosecutive geeral terms Recogitio of commo differece ad o more Shows series arithmetic but does ot state commo differece (b) Scale 5C (0, 3, 4, 5) Writes three or more terms i form of ad Correct AP formula stated Correct T formula Correct substitutio ito formula l a ad does ot fiish Note: accept a e for full marks l a (c) Scale 5C (0, 3, 4, 5) Recogisig T T 10d or similar work 11 1 LHS correct i terms of l a RHS correct i terms of l a Note: log is ot eeded i first solutio box [33] Questio 7 (a)(i) Scale 10B (0, 5, 10) Partial Credit: Correct substitutio of chose value Not squarig values Note: Allow 10 marks for 0 ad correct work i (a)(i) (a)(ii) Scale 10D (0, 3, 7, 8, 10) a or b or c expressed i terms of Mid Partial Credit: Ay two terms Three terms fully squared ( a b ) fully worked out i terms of Notes for (a)(i) ad (a)(ii): - Mark particular case with scheme for (a)(i) wherever it occurs - Mark geeral case

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