Coimisin na Scrduithe Stit State Examinations Commission

Leaving Certificate 2012

Marking Scheme

Higher Level

Design and Communication Graphics

Coimisin na Scrduithe StitState Examinations Commission

Leaving Certificate 2015

Marking Scheme

Mathematics

Higher Level

Note to teachers and students on the use of published marking schemes

Marking schemes published by the State Examinations Commission are not intended to be standalone documents. They are an essential resource for examiners who receive training in the correct interpretation and application of the scheme. This training involves, among other things, marking samples of student work and discussing the marks awarded, so as to clarify the correct application of the scheme. The work of examiners is subsequently monitored by Advising Examiners to ensure consistent and accurate application of the marking scheme. This process is overseen by the Chief Examiner, usually assisted by a Chief Advising Examiner. The Chief Examiner is the final authority regarding whether or not the marking scheme has been correctly applied to any piece of candidate work.

Marking schemes are working documents. While a draft marking scheme is prepared in advance of the examination, the scheme is not finalised until examiners have applied it to candidates work and the feedback from all examiners has been collated and considered in light of the full range of responses of candidates, the overall level of difficulty of the examination and the need to maintain consistency in standards from year to year. This published document contains the finalised scheme, as it was applied to all candidates work.

In the case of marking schemes that include model solutions or answers, it should be noted that these are not intended to be exhaustive. Variations and alternatives may also be acceptable. Examiners must consider all answers on their merits, and will have consulted with their Advising Examiners when in doubt.

Future Marking Schemes

Assumptions about future marking schemes on the basis of past schemes should be avoided. While the underlying assessment principles remain the same, the details of the marking of a particular type of question may change in the context of the contribution of that question to the overall examination in a given year. The Chief Examiner in any given year has the responsibility to determine how best to ensure the fair and accurate assessment of candidates work and to ensure consistency in the standard of the assessment from year to year. Accordingly, aspects of the structure, detail and application of the marking scheme for a particular examination are subject to change from one year to the next without notice.

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Contents Page Paper 1 Model Solutions ........................................................................................................................ 3 Marking Scheme ........................................................................................................................ 21 Structure of the marking scheme ...................................................................................... 21

Summary of mark allocations and scales to be applied .................................................... 22

Detailed marking notes ..................................................................................................... 23

Paper 2 Model Solutions ........................................................................................................................ 33 Marking Scheme ........................................................................................................................ 52 Structure of the marking scheme ...................................................................................... 52

Summary of mark allocations and scales to be applied .................................................... 53

Detailed marking notes ..................................................................................................... 54

Marcanna breise as ucht freagairt tr Ghaeilge ........................................................................... 63

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2015. M29

Coimisin na Scrduithe Stit State Examinations Commission

Leaving Certicate Examination 2015

Mathematics

Paper 1 Higher Level

Friday 5 June Afternoon 2:00 4:30

300 marks

Model Solutions Paper 1 Note: The model solutions for each question are not intended to be exhaustive there may be other correct solutions. Any examiner unsure of the validity of the approach adopted by a particular candidate to a particular question should contact his / her advising examiner.

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Instructions

There are two sections in this examination paper. Section A Concepts and Skills 150 marks 6 questions

Section B Contexts and Applications 150 marks 3 questions

Answer all nine questions.

Write your answers in the spaces provided in this booklet. You may lose marks if you do not do so. You may ask the superintendent for more paper. Label any extra work clearly with the question number and part.

The superintendent will give you a copy of the Formulae and Tables booklet. You must return it at the end of the examination. You are not allowed to bring your own copy into the examination.

You will lose marks if all necessary work is not clearly shown.

You may lose marks if the appropriate units of measurement are not included, where relevant. You may lose marks if your answers are not given in simplest form, where relevant.

Write the make and model of your calculator(s) here:

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Section A Concepts and Skills 150 marks Answer all six questions from this section. Question 1 (25 marks) Mary threw a ball onto level ground from a height of 2 m. Each time the ball hit the ground it bounced back up to 4

3 of the height of the previous bounce, as shown.

(a) Complete the table below to show the maximum height, in fraction form, reached by the ball

on each of the first four bounces.

Bounce 0 1 2 3 4

Height (m) 21

23

89

3227

12881

(b) Find, in metres, the total vertical distance (up and down) the ball had travelled when it hit the ground for the 5th time. Give your answer in fraction form.

13643 9 27 81 525 6532 2 2 2 102 8 32 128 128 64

+ + + + = + = =

m

or

( )

4

43 32 4

34

1364

3 9 27 812 2 2 22 8 32 128

1 (2 2

1

525 6532 10 m64 64

S + + + + = +

= +

= + = =

Bounce

Hei

ght

2 m

0

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(c) If the ball were to continue to bounce indefinitely, find, in metres, the total vertical distance it would travel.

32

34

3 92 2 ... 2 22 8 1

2 21

2 12 14 m

ar

+ + + = +

= + = + =

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Question 2 (25 marks)

Solve the equation 3 23 9 11 0.x x x + = Write any irrational solution in the form ,a b c+ where , , .a b c

3 2

3 2

( ) 3 9 11(1) 1 3(1) 9 11 0

f x x x xf

= += + =

1x = is a solution. (x 1) is a factor or or

2x x2 11 x 3x 22x x11 1 2x x2 11

Hence, other factor is 2 2 11x x

22 ( 2) 4(1)( 11) 2 48 2 4 3 1 2 32(1) 2 2

x = = = =

Solutions: {1, 1 2 3+ , 1 2 3 }

2 2 11x x 3 21 3 9 11x x x x +

3 2x x 22 9 11x x + xx 22 2 + 11 11x + 11 11x +

( )( )2 3 21 11 3 9 11x x Ax x x x + = + 3 2 2 3 21 3 9 11

1 32

x Ax x x Ax x x xAA

+ + = + = =

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Question 3 (25 marks)

Let += xxxxf ,2712)( 2 . (a) (i) Complete Table 1 below.

Table 1

x 3 4 5 6 7 8 9

)( xf 0 5 8 9 8 5 0

(ii) Use Table 1 and the trapezoidal rule to find the approximate area of the region bounded by the graph of f and the x-axis.

( )

( )

1 2 3 1221 0 0 2 5 8 9 8 52

n nhA y y y y y = + + + + +

= + + + + + +

35 = square units

(b) (i) Find 9

3

.)( dxxf

( )9

2

3

12 27x x dx +

( ) ( )

93 2

3

12 273 2

243 486 243 9 54 81

x x x = +

= + +

= 36

(ii) Use your answers above to find the percentage error in your approximation of the area,

correct to one decimal place.

%82100

361 =

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Question 4 (25 marks)

(a) The complex numbers 1 2 3, a n d z z z are such that 1 2 3

2 1 1 ,z z z

= + = and

,233 iz = where 2 1.i = Write in the form ,a b i+ where , .a b

iizzz 23

132

1112

321 +

+=+=

( )( ) ii

iiii

5125

23323223

++=

+++=

iz

iii

ii

iiz

+=

+=

++=

++=

526

136555

5512

5512

2