Ordinary Level Marking Scheme

[Pages:92]L.16/19

Pre-Leaving Certificate Examination, 2019

Mathematics

Ordinary Level

Marking Scheme

Paper 1 Paper 2

Pg. 2 Pg. 44

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Pre-Leaving Certificate Examination, 2019

Mathematics

Ordinary Level ? Paper 1 Marking Scheme (300 marks)

Structure of the Marking Scheme

Students' responses are marked according to different scales, depending on the types of response anticipated. Scales labelled A divide students' responses into two categories (correct and incorrect). Scales labelled B divide responses into three categories (correct, partially correct, and incorrect), and so on. These scales and the marks that they generate are summarised in the following table:

Scale label

A

No. of categories

2

5-mark scale

10-mark scale

15-mark scale

B 3 0, 2, 5

C 4 0, 2, 4, 5 0, 4, 7, 10

D 5 0, 2, 3, 4, 5 0, 4, 6, 8, 10

A general descriptor of each point on each scale is given below. More specific directions in relation to interpreting the scales in the context of each question are given in the scheme, where necessary.

Marking scales ? level descriptors

A-scales (two categories) incorrect response (no credit) correct response (full credit)

B-scales (three categories) response of no substantial merit (no credit) partially correct response (partial credit) correct response (full credit)

DEB 2014 (LC-O1)

Scale label A No of categories 2

5 mark scale 0, 5 10 mark scale 15 mark scale

B

C

D

3

4

5

0, 2, 5 0, 2, 4, 5

0, 5 10 0, 3, 7, 10 0, 2, 5, 8, 10

0, 5, 10, 15 0, 4, 7, 11, 1

C-scales (four categories)

response of no substantial merit (no credit) response with some merit (low partial credit) almost correct response (high partial credit) correct response (full credit)

D-scales (five categories)

response of no substantial merit (no credit) response with some merit (low partial credit) response about half-right (mid partial credit) almost correct response (high partial credit) correct response (full credit)

In certain cases, typically involving incorrect rounding, omission of units, a misreading that does not oversimplify the work or an arithmetical error that does not oversimplify the work, a mark that is one mark below the full-credit mark may also be awarded. Such cases are flagged with an asterisk. Thus, for example, scale 10C* indicates that 9 marks may be awarded.

The * for units is to be applied only if the student's answer is fully correct. The * is to be applied once only within each section (a), (b), (c), etc. of all questions. The * penalty is not applied for the omission of units in currency solutions.

Unless otherwise specified, accept correct answer with or without work shown.

Accept students' work in one part of a question for use in subsequent parts of the question, unless this oversimplifies the work involved.

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Summary of Marks ? 2019 LC Maths (Ordinary Level, Paper 1)

Section A

Q.1 (a) (b) (c)

5B* (0, 2, 5) 10C* (0, 4, 7, 10) 10D* (0, 4, 6, 8, 10)

25

Q.2 (a) (b)

(i) 5C (0, 2, 4, 5)

(ii) 5C (0, 2, 4, 5) (iii) 5B (0, 2, 5)

10D (0, 4, 6, 8, 10)

25

Q.3 (a) (i) 10C (0, 4, 7, 10) (ii) 5C (0, 2, 4, 5)

(b) (i) 5C (0, 2, 4, 5) (ii) 5D (0, 2, 3, 4, 5)

25

Q.4 (a) (i) 5C (0, 2, 4, 5)

(ii) 5C (0, 2, 4, 5)

(b) (i) (ii)

5C (0, 2, 4, 5)

(c)

10D (0, 4, 6, 8, 10)

25

Q.5 (a) (i) (ii)

10D (0, 4, 6, 8, 10)

(b)

5B* (0, 2, 5)

(c)

10C* (0, 4, 7, 10)

25

Q.6 (a) (i) 5C (0, 2, 4, 5)

(ii) 5C* (0, 2, 4, 5)

(b)

10D* (0, 4, 6, 8, 10)

(c)

5C* (0, 2, 4, 5)

25

Section B Q.7 (a)

(b) (c)

(i) 5C (0, 2, 4, 5) (ii) 5C (0, 2, 4, 5) (iii) 5C (0, 2, 4, 5)

(i) 10D (0, 4, 6, 8, 10) (ii) 10D (0, 4, 6, 8, 10) (iii) 5C (0, 2, 4, 5)

10D (0, 4, 6, 8, 10)

50

Q.8 (a) (b)

(c) (d) (e)

5C* (0, 2, 4, 5) (i) 5B (0, 2, 5) (ii) 5D* (0, 2, 3, 4, 5)

10D* (0, 4, 6, 8, 10) 5C (0, 2, 4, 5) (i) 10C (0, 4, 7, 10)

(ii) 5D (0, 2, 3, 4, 5) (iii) 5C* (0, 2, 4, 5)

50

Q.9 (a) (b) (c) (d)

(i) (ii)

10D* (0, 4, 6, 8, 10)

(i) (ii)

10C* (0, 4, 7, 10)

(i) 5C* (0, 2, 4, 5)

(ii) 5B (0, 2, 5)

(i) 5C* (0, 2, 4, 5)

(ii) 5C* (0, 2, 4, 5)

(iii) 5B (0, 2, 5)

(iv) 5C* (0, 2, 4, 5)

50

Current Marking Scheme

Assumptions about these marking schemes on the basis of past SEC marking schemes should be avoided. While the underlying assessment principles remain the same, the exact details of the marking of a particular type of question may vary from a similar question asked by the SEC in previous years in accordance with the contribution of that question to the overall examination in the current year. In setting these marking schemes, we have strived to determine how best to ensure the fair and accurate assessment of students' work and to ensure consistency in the standard of assessment from year to year. Therefore, aspects of the structure, detail and application of the marking schemes for these examinations are subject to change from past SEC marking schemes and from one year to the next without notice.

Copyright

All rights reserved. This marking scheme and corresponding paper(s) are protected by Irish (EU) copyright law. Reproduction and distribution of these materials or any portion thereof without the written permission of the publisher is prohibited except for the immediate use within a classroom.

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Pre-Leaving Certificate Examination, 2019

Mathematics

Ordinary Level ? Paper 1 Marking Scheme (300 marks)

General Instructions

There are two sections in this examination paper.

Section A Section B

Concepts and Skills Contexts and Applications

150 marks 150 marks

6 questions 3 questions

Answer all nine questions.

Marks may be lost if answers do not include supporting work.

Marks may be lost if answers do not include the appropriate units of measurement, where relevant.

Marks may be lost if answers are not given in simplest form, where relevant.

Section A

Concepts and Skills

150 marks

Answer all six questions from this section. (25 marks each)

Question 1

(25)

The table below shows the population of Ireland according to the most recent censuses.

Census Year Population

2006 4,239,848

2011 4,588,252

2016 4,761,865

1(a) What is the population of Ireland according to the most recent census? Write this number in the form a ? 10n, where 1 a < 10 and n , correct to two significant figures.

(5B*)

4,761,865

4,800,000

= 4?8 ? 106

or

4,761,865

= 4?761865 ? 106

4?8 ? 106

Scale 5B* (0, 2, 5)

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Partial credit: (2 marks)

? Some work of merit, e.g. identifies 4,761,865 as the correct answer.

? Finds either a (unrounded) or n correctly. ? Writes down 4?2 ? 106 or 4?6 ? 106

as answer.

* Deduct 1 mark off correct answer only if final answer is incorrectly rounded or is not rounded - apply deduction only once to each section (a), (b), (c), etc. of question.

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Question 1 (cont'd.)

2019 LC Maths [OL] ? Paper 1

1(b) Show that the average annual increase in population between 2011 and 2016 was 0?757%, correct to three decimal places.

(10C*)

Population growth

= 4,761,865 ? 4,588,252

... part (a)

= 173,613

Average population growth per year

=

173,613 5

= 34,722?6

34,723

Average population growth rate per year

=

Change in population Original population

?

100 1

=

34,723 ? 100

4,588,252 1

= 0?756780...

0?757%

or

Population growth

= 4,761,865 ? 4,588,252

... part (a)

= 173,613

Overall population growth rate

=

Change in population Original population

?

100 1

=

173,613 4,588,252

?

100 1

= 3?783859...%

Average population growth rate per year

=

3 783859... 5

= 0?756771...

0?757%

** Accept students' answers from part (a) if not oversimplified.

Scale 10C* (0, 4, 7, 10)

Low partial credit: (4 marks) High partial credit: (7 marks)

? Some work of merit, e.g. writes down correct formula for population growth rate.

? Finds Average population growth per year [ans. 34,722?6 or 34,723] and stops or continues incorrectly.

? Some correct substitution into population growth rate formula and stops or continues incorrectly.

? Some correct substitution into population

growth rate formula and finishes to find

incorrect % growth.

?

Finds

34,723 4,588,252

or

173,613 4,588,252

?

100 1

,

but fails to finish or finishes incorrectly.

?

Finds

173,613 4,761,865

or

34,723 4,761,865

?

100 1

and finishes correctly [ans. 0?729].

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* Deduct 1 mark off correct answer only if final answer is incorrectly rounded

or is not rounded - apply deduction only once to each section (a), (b), (c), etc. of question.

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Question 1 (cont'd.)

2019 LC Maths [OL] ? Paper 1

1(c) Find the expected increase in the population of Ireland by the time of the next census in 2021, assuming the population grows at the same rate year-on-year.

Expected population:

P(2017)

= P(2016) ? 1?00757

= 4,761,865 ? 1?00757

P(2018)

= P(2017) ? 1?00757 = 4,761,865 ? 1?00757 ? 1?00757 = 4,761,865 ? (1?00757)2

P(2019)

= P(2018) ? 1?00757 = 4,761,865 ? (1?00757)2 ? 1?00757 = 4,761,865 ? (1?00757)3

P(2020)

= P(2019) ? 1?00757 = 4,761,865 ? (1?00757)3 ? 1?00757 = 4,761,865 ? (1?00757)4

P(2021)

= P(2020) ? 1?00757 = 4,761,865 ? (1?00757)4 ? 1?00757 = 4,761,865 ? (1?00757)5 = 4,761,865(1?038427...)

= 4,944,851?107410... 4,944,851

or

Expected population in 2021:

F

= P (1 + i)t

P

= 4,761,865

t

= 2021 ? 2016

= 5

i

= 0?757%

F

= 4,761,865(1 + 0?00757)5 = 4,761,865(1?00757)5

= 4,761,865(1?038427...)

= 4,944,851?107410...

4,944,851

Expected increase in population = 4,944,851 ? 4,761,865

= 182,986

(10D*)

Scale 10D* (0, 4, 6, 8, 10)

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Low partial credit: (4 marks)

? Some work of merit, e.g. writes down

`P(2017) = P(2016) ? 1?00757' or similar [method ] or P = 4,761,865, t = 5 and i = 0?757%' or similar [method ]. ? Finds correct value for P(2017) [ans. 4,797,912] [method ] and stops or continues incorrectly.

? Writes down correct relevant formula or some correct substitution into formula (not stated) [method ] and stops or continues incorrectly.

Mid partial credit: (6 marks)

? Finds correct value for P(2017), P(2018), and P(2019) [method ] and stops or continues incorrectly.

? Substitutes correctly into formula, i.e. 4,761,865(1 + 0?00757)5 or (1?00757)5, and stops or continues incorrectly.

High partial credit: (8 marks)

? Finds correct value for P(2021) [method ] or F [method ] [ans. 4,944,851], but

fails to finish or finishes incorrectly.

* Deduct 1 mark off correct answer only if final answer is incorrectly rounded

or is not rounded - apply deduction only once to each section (a), (b), (c), etc.

of question.

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Question 2

2019 LC Maths [OL] ? Paper 1 (25)

2(a) z1 = 2 ? 3i is a complex number, where i2 = ?1.

(i) Let z2 = iz1 and z3 = iz2.

Find z2 and z3, in the form a + bi, where a, b .

(5C)

z2

=

iz1

= i(2 ? 3i)

= 2i ? 3i2

= 2i ? 3(?1)

= 3 + 2i

or

Scale 5C (0, 2, 4, 5)

z3

=

iz2

= i(3 + 2i)

= 3i + 2i2

= 3i + 2(?1)

= ?2 + 3i

z3

=

iz2

=

i(iz1)

=

i2z1

= ?1(2 ? 3i)

= ?2 + 3i

Low partial credit: (2 marks) High partial credit: (4 marks)

? Some work of merit, e.g. substitutes correctly z1 into iz1 [ans. i(2 ? 3i)] and stops or continues incorrectly.

? Correct multiplication, i.e. finds 2i ? 3i2, but fails to finish or finishes incorrectly.

? Writes down z3 = i(iz1) or i2z1 and stops or continues incorrectly.

? Finds one correct value of z2 or z3, but fails to find or finds incorrect other value.

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Question 2 (cont'd.)

2019 LC Maths [OL] ? Paper 1

2(a) (cont'd.)

(ii) Plot z1, z2 and z3 on the Argand diagram.

Label each point clearly.

(5C)

Im(z) 6

z3/iz2/-2 + 3i 4 2

-6 -4 -2 -2 -4

z2/iz1/3 + 2i Re(z)

246

z31/2i-z13/-i 8 + 6i

-6

Scale 5C (0, 2, 4, 5)

Low partial credit: (2 marks) High partial credit: (4 marks)

? One point correctly plotted. ? Two points correctly plotted, but without

labels.

? Two points correctly plotted and labelled. ? Three points correctly plotted, but without

labels. ? All points correctly plotted and labelled,

but real and imaginary axes interchanged.

(iii) Use your diagram to describe what happens when a complex number

is multiplied by i.

(5B)

Explanation:

Any 1:

? rotates z / complex number through 90? anti-clockwise about the origin //

? z2 = iz1 rotates z1 through 90? anti-clockwise about the origin //

? z3 = iz2 rotates z2 through 90? anti-clockwise about the origin //

? both numbers are the same distance from the origin / moduli of both numbers are equal //

? z2 is the same distance from the origin as z1 / modulus of z2 is equal to the modulus of z1 / | z2 | = | z1 | //

? z3 is the same distance from the origin as z2 / modulus of z3 is equal to the modulus of z2 / | z3 | = | z2 |

Scale 5B (0, 2, 5)

Partial credit: (2 marks)

? Some work of merit, e.g. states that three points are in different quadrants.

? Incomplete explanation, e.g. `z rotated

anti-clockwise' (no mention of 90? angle) or `z rotated through an angle of 90?' (no mention of anti-clockwise) or similar.

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