Mathematics Ordinary Level

L.16/19 Pre-Leaving Certificate Examination, 018 Mathematics Ordinary Level Marking Scheme Paper 1 Pg. Paper Pg. 36 Page 1 of 56

exams Pre-Leaving Certificate Examination, 018 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 B C D No. of categories 3 4 5 5 mark scale 0,, 5 0,, 4, 5 0,, 3, 4, 5 10 mark scale 0, 4, 7, 10 0, 4, 6, 8, 10 15 mark scale 0, 6, 10, 13, 15 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) 014 (LC-O1) Scale label A B C D No of categories 3 4 5 5 mark scale 0, 5 0,, 5 0,, 4, 5 10 mark scale 0, 5 10 0, 3, 7, 10 0,, 5, 8, 10 15 mark scale 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. 018.1 L.16/19_MS /76 Page of 75 exams

Summary of Marks 018 LC Maths (Ordinary Level, Paper 1) Section A Section B Q.1 (a) (i) 10C (0, 4, 7, 10) Q.7 (a) (i) 5C (0,, 4, 5) (ii) 10C (0, 4, 7, 10) (ii) 5C (0,, 4, 5) (b) 5D (0,, 3, 4, 5) (iii) 5C (0,, 4, 5) 5 (iv) 5C (0,, 4, 5) (b) (i) 10C (0, 4, 7, 10) (ii) 10D* (0, 4, 6, 8, 10) (c) (i) 5C* (0,, 4, 5) Q. (a) (i) 5C (0,, 4, 5) (ii) 5D* (0,, 3, 4, 5) (ii) 10D (0, 4, 6, 8, 10) 50 (b) (i) 5C (0,, 4, 5) (ii) 5B (0,, 5) 5 Q.8 (a) (i) 5C (0,, 4, 5) (ii) 5C* (0,, 4, 5) (b) 10D (0, 4, 6, 8, 10) Q.3 (a) (i) 10C (0, 4, 7, 10) (c) (i) 10C (0, 4, 7, 10) (ii) 5C (0,, 4, 5) (ii) 5D (0,, 3, 4, 5) (b) 10C (0, 4, 7, 10) (d) (i) 5D (0,, 3, 4, 5) 5 (ii) 5C* (0,, 4, 5) (iii) 5B (0,, 5) 50 Q.4 (a) (i) 5C (0,, 4, 5) (ii) 5D (0,, 3, 4, 5) (b) (i) 5C (0,, 4, 5) Q.9 (a) (i) 5C* (0,, 4, 5) (ii) 5C (0,, 4, 5) (ii) 5C (0,, 4, 5) 5 (b) (i) 10C (0, 4, 7, 10) (ii) 10D* (0, 4, 6, 8, 10) (iii) 10D (0, 4, 6, 8, 10) (iv) 10D (0, 4, 6, 8, 10) Q.5 (a) 15D* (0, 6, 10, 13, 15) 50 (b) 10D (0, 4, 6, 8, 10) 5 Q.6 (a) 5B (0,, 5) (b) (i) 5C (0,, 4, 5) (ii) 5D (0,, 3, 4, 5) (c) 10D (0, 4, 6, 8, 10) 5 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 papers(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. 018.1 L.16/19_MS 3/76 Page 3 of 75 exams

exams Pre-Leaving Certificate Examination, 018 Mathematics Ordinary Level Paper 1 Marking Scheme (300 marks) General 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. Marks will be lost if all necessary work is not clearly shown. Answers should include the appropriate units of measurement, where relevant. Answers should be given in simplest form, where relevant. Section A Concepts and Skills 150 marks Answer all six questions from this section. (5 marks each) Question 1 (5) 1(a) Joe earns a gross wage of 855 for a standard 38-hour working week. He pays income tax, universal social charge (USC) and pay-related social insurance (PRSI) on his gross wage. (i) Joe pays income tax at the rate of 0% on the first 670 he earns and 40% on the balance. He has weekly tax credits of 63. How much income tax does Joe pay weekly? Tax @ 0% 670 0% 0 670 100 134 Tax @ 40% (855 670) 40% 40 185 100 74 Gross tax Tax @ 0% + Tax @ 40% 134 + 74 08 Net tax Gross tax Tax credits 08 63 145 (10C) 018.1 L.16/19_MS 4/76 Page 4 of 75 exams

018 LC Maths [OL] Paper 1 Question 1 1(a) (i) Scale 10C (0, 4, 7, 10) Low partial credit: (4 marks) Some work of merit, e.g. sets up 0% of 670. Finds tax correctly @ 0% or @ 40% of some relevant figure and stops or continues incorrectly. High partial credit: (7 marks) Finds Gross tax per week correctly, but fails to find or finds incorrect Net tax per week. * No deduction applied for the omission of or incorrect use of units in questions involving currency. (ii) Joe also pays USC and PRSI on his gross wage. USC amounts to 9 10 each week and he pays PRSI on his gross wage. His net weekly take-home pay is 646 70. Find the percentage rate at which Joe pays PRSI. Take-home pay Gross pay Income tax USC PRSI 855 145 9 10 PRSI 646 70 855 145 9 10 PRSI 646 70 PRSI 855 145 9 10 646 70 34 0 % PRSI 34 0 855 0 04 100 4% 100 1 ** Accept students answers for Income tax paid from part (i) if not oversimplified. (10C) Scale 10C (0, 4, 7, 10) Low partial credit: (4 marks) Some work of merit, e.g. calculates 855 145 9 10 or 855 646 70 and stops or continues incorrectly. High partial credit: (7 marks) Finds PRSI correctly [ans. 34.0], but fails to find % PRSI or finds incorrect % PRSI. * No deduction applied for the omission of or incorrect use of symbol ( % ). 018.1 L.16/19_MS 5/76 Page 5 of 75 exams

018 LC Maths [OL] Paper 1 Question 1 1(b) Joe is paid time and a half for weekday overtime and weekend work. In addition to income tax and PRSI, he pays USC at the rate of 5% on his extra earnings. Find the minimum number of hours that Joe must work above his standard working week in order to receive a net weekly take-home pay in excess of 800. Additional pay 800 646 70 153 30 Normal pay per hour 855 38 50 Gross overtime pay per hour 50 1 5 33 75 Total deductions on each overtime hour 40% Income tax + 5% USC + 4% PRSI 49% of 33 75 Net overtime pay per hour 51% of 33 75 17 15 Overtime required for take-home pay to be above 800 153 30 17 15 8 906318... 9 hours (5D) ** Accept students answers for % PRSI from part (a)(ii) if not oversimplified. Scale 5D (0,, 3, 4, 5) Low partial credit: ( marks) Some work of merit, e.g. finds additional pay [ans. 153 30], Normal pay per hour [ans. 50] or Gross overtime pay per hour [ans. 33 75] and stops. Mid partial credit: (3 marks) Finds Total deductions on each overtime hour [ans. 49% of 33 75 or 16 5375] and stops or continues incorrectly. High partial credit: (4 marks) Finds Net overtime pay per hour [ans. 51% of 33 75 or 17 15], but fails to finish or finishes incorrectly. 018.1 L.16/19_MS 6/76 Page 6 of 75 exams

018 LC Maths [OL] Paper 1 Question (5) z 1 3 + 4i is a complex number, where i 1. (a) (i) Let z iz 1. Find z, in the form a + bi, where a, b R. (5C) z iz 1 i(3 + 4i) 6i + 8i 6i + 8( 1) 8 + 6i Scale 5C (0,, 4, 5) Low partial credit: ( marks) Some work of merit, e.g. substitutes correctly z 1 into iz 1 [ans. i(3 + 4i)] and stops or continues incorrectly. High partial credit: (4 marks) Correct multiplication, i.e. finds 6i + 8i, but fails to finish or finishes incorrectly. (ii) Let z 3 z 1. Find z 3, in the form a + bi, where a, b R. (10D) i z 3 z1 i 3 + 4i i 3 + 4i i i i 6i 8i 4i 6i 8( 1) 4( 1) 8 6i 4 4 3i 3 // i Scale 10D (0, 4, 6, 8, 10) Low partial credit: (4 marks) Some work of merit, e.g. substitutes z correctly z 1 into 3 + 4i [ans. ] i i and stops or continues incorrectly. Correctly identifies i. Mid partial credit: (6 marks) High partial credit: (8 marks) 3 + 4i i Finds, but fails to evaluate i i or evaluates incorrectly. Some multiplication above and below, even if by wrong conjugate. 6i 8i Finds or similar, but fails 4i to finish or finishes incorrectly. 018.1 L.16/19_MS 7/76 Page 7 of 75 exams

018 LC Maths [OL] Paper 1 Question (b) (i) Plot each of the points z 1, z and z 3 on the given Argand diagram and label each point clearly. (5C) z / iz 1 / 8 6i 6 4 Im( z) z 1 /34i Re( z) 8 6 4 4 6 z1 3 / / i z3/ iz i 1/ 8 6i 4 6 Scale 5C (0,, 4, 5) Low partial credit: ( marks) One point correctly plotted. Two points correctly plotted, but without labels. High partial credit: (4 marks) 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. (ii) Make one observation about the relative positions of the points you plotted on the diagram above. (5B) Observations: Any 1: z is twice as far from the origin as z 1 / modulus of z is twice the modulus of z 1 / z z 1 // z 1 is twice as far from the origin as z 3 / modulus of z 1 is twice the modulus of z 3 / z 1 z 3 // z iz 1 rotates z 1 through 90 anti-clockwise // z z 3 1 rotates z 1 through 90 clockwise i Scale 5B (0,, 5) Partial credit: ( marks) Some work of merit, e.g. states that three points are in different quadrants, z iz 1 z rotates z 1 anti-clockwise, z 3 rotates i z 1 clockwise or similar. 018.1 L.16/19_MS 8/76 Page 8 of 75 exams

018 LC Maths [OL] Paper 1 Question 3 (5) 3(a) (i) Solve for x: 4(5 + x) 5 5x 3(1 x), where x R. (10C) 4(5 + x) 5 5x 3(1 x) 0 + 8x 5 5x 3 + 6x 8x + 15 11x 3 8x 11x 3 15 3x 18 3x 18 x 6 Scale 10C (0, 4, 7, 10) Low partial credit: (4 marks) Some work of merit, e.g. any correct attempt at simplifying equation, e.g. 4(5 + x) 5 0 + 8x 5, etc. Simplifies one term correctly, i.e. 3x or 3x, 18 or 18. Correct answer, but with no work shown. High partial credit: (7 marks) Both multiplications done correctly, i.e. simplifies to 3x 18 or 3x 18, but fails to finish or finishes incorrectly. Error in expanding brackets, but finishes correctly. (ii) Verify your answer to part (i) above. (5C) 4(5 + x) 5 5x 3(1 x) @ x 6 4(5 + (6)) 5 5(6) 3(1 (6)) 4(5 + 1) 5 30 3(1 1) 4(17) 5 30 3( 11) 68 5 30 + 33 63 63 as 63 63, x 6 is a solution (of the equation) Scale 5C (0,, 4, 5) Low partial credit: ( marks) Some correct substitution of answer(s) from part (i) into equation. High partial credit: (4 marks) Finds 63 63, but no conclusion given (or incorrect conclusion if not equal). 018.1 L.16/19_MS 9/76 Page 9 of 75 exams

018 LC Maths [OL] Paper 1 Question 3 3(b) Solve the inequality: (1 + x) 8x 7, x Z, and show the solution set on the number line below. (10C) Solution (1 + x) 8x 7 + 4x 8x 7 4x 7 4x 7 4x 9 4x 9 9 x 4 1 x 4 x {, 1, 0, 1,, 3,...} Number line 3 1 0 1 3 Scale 10C (0, 4, 7, 10) Low partial credit: (4 marks) Some work of merit, e.g. any correct transposition. Mishandles inequality sign, but finishes 9 1 to x or. 4 4 Correct answer, but with no work shown. High partial credit: (7 marks) Finds x 4 9 or 4 1, but no solution set given or no number line shown. 9 1 Finds x or, but number line 4 4 shows x N or R. 018.1 L.16/19_MS 10/76 Page 10 of 75 exams

018 LC Maths [OL] Paper 1 Question 4 (5) The diagram shows the graph of the function f(x) x + in the domain 3 x 4, x R. 4(a) (i) On the same diagram, draw the graph of the function g(x) x + 5, x R. (5C) 1 y 10 8 gx () 6 4 f() x 3 1 1 3 4 x g(x) x + 5 @ x 0 g(0) (0) + 5 5 (0, 5) g @ x 1 g(1) (1) + 5 7 (1, 7) g @ x g() () + 5 9 (, 9) g @ y 0 0 x + 5 x 5 x 5 1 // ( 1, 0) g ** Accept co-ordinates of any two points on g. Scale 5C (0,, 4, 5) Low partial credit: ( marks) Some work of merit, e.g. finds correctly y co-ordinate of any point on g. Finds correctly co-ordinates (x, y) of one point on g. Plots correctly co-ordinates (x, y) of one point on g (no work shown). High partial credit: (4 marks) Finds correctly co-ordinates (x, y) of two points on g, but fails to plot points or plots points incorrectly. Plots correctly co-ordinates (x, y) of two points on g (no work shown), but fails to join points. 018.1 L.16/19_MS 11/76 Page 11 of 75 exams

018 LC Maths [OL] Paper 1 Question 4 4(a) (i) Hence, use the graphs to find the two values of x for which g(x) f(x). (5C) Values of x for which g(x) f (x) Using graph: x 1 x 3 ** Accept students graphs from part (a)(i) if not oversimplified. Scale 5C (0,, 4, 5) Low partial credit: ( marks) Some work of merit, e.g. marks one or both points of intersection on graph. Finds correctly x co-ordinate of one point of intersection, i.e. 1 or 3. Finds correctly y co-ordinate of one point of intersection, i.e. 3 or 11. High partial credit: (4 marks) Finds correctly ( 1, 3) and (3, 11). Finds correctly y co-ordinate of both points of intersection, i.e. 3 and 11. (ii) Verify your answer to part (i) above by using algebra to solve g(x) f(x). (5D) g(x) f (x) x + 5 x + x x + 5 0 x x 3 0 (x 3)(x + 1) 0 x 3 0 x 3 x + 1 0 x 1 Scale 5D (0,, 3, 4, 5) Low partial credit: ( marks) Some work of merit, e.g. states relevant b formula. Equates x + 5 x + and stops or continues incorrectly. Mid partial credit: (3 marks) Finds correct quadratic equation, i.e. x x 3 0 and stops or continues incorrectly. High partial credit: (4 marks) Finds correct quadratic equation, but finds only one correct value of x. Finds correct quadratic equation and substitutes correctly into b formula, but fails to finish or finishes incorrectly. 018.1 L.16/19_MS 1/76 Page 1 of 75 exams

018 LC Maths [OL] Paper 1 Question 4 4(b) (i) Find f (x), the derivative of f(x). Hence, find the value of x at which the tangent to the graph of f(x) is parallel to g(x). (5C) f (x) x + f (x) x tangent to the graph of f(x) is parallel to g(x) slope of tangent slope of g(x) g(x) x + 5 mx + c m g m tangent f (x) m g x x 1 Scale 5C (0,, 4, 5) Low partial credit: ( marks) Some work of merit, e.g. states that slope of tangent equal to f (x) or similar. f (x) correctly differentiated [ans. x] and stops or continues incorrectly. Finds correct slope of g(x) [ans. ] and stops or continues incorrectly. High partial credit: (4 marks) Finds f (x) and m g / m tangent correctly and equates f (x) m g / m tangent, but fails to finish or finishes incorrectly. (ii) Hence, find the equation of this tangent. (5C) f (x) x + f (1) (1) + 3 point of contact (1, 3) m tangent y y 1 m(x x 1 ) y 3 (x 1)... award full marks x x y 3 + x y 1 or y mx + c 3 (1) + c 3 + c c 3 1 y x + 1... award full marks x y 1 Scale 5C (0,, 4, 5) Low partial credit: ( marks) Some work of merit, e.g. writes down correct relevant formula. Finds correct value of f (1). High partial credit: (4 marks) Some correct substitution into relevant formula, but fails to finish or finishes incorrectly. 018.1 L.16/19_MS 13/76 Page 13 of 75 exams

018 LC Maths [OL] Paper 1 Question 5 (5) 5(a) Find the two values of x for which x + 4x 5 0. Give your answer correct to one decimal place. (15D*) x x b ± b 4ac a 4 ± (4) 4()( 5) () 4 ± 16 + 40 4 4 ± 56 4 4 ± 7 483314... 4 x 4 7 483314... 4 11 483314... 4 87088... 87 x 4 + 7 483314... 4 3 483314... 4 0 87088... 0 87 Scale 15D* (0, 6, 10, 13, 15) Low partial credit: (6 marks) Some work of merit, e.g. writes down correct relevant b formula. a, b and c explicitly identified. Some correct substitution into b formula and stops or continues incorrectly. Attempt at factorising quadratic equation. Mid partial credit: (10 marks) Substitutes correctly into b formula and stops or continues incorrectly. 4 ± 4 Finds x 4 and stops or continues. High partial credit: (13 marks) Substitutes correctly into b formula, 4 ± 56 i.e. x, but fails to finish 4 or finishes incorrectly. Finds one correct value of x only. Finds incorrect solutions using appropriate method (allow up to three minor errors in total). * Deduct 1 mark off correct answer only if final answer is incorrectly rounded or not rounded - apply only once to each section (a), (b), (c), etc. of question. 018.1 L.16/19_MS 14/76 Page 14 of 75 exams

018 LC Maths [OL] Paper 1 Question 5 5(b) When two numbers are added together, the sum is equal to 46. When the smaller number is subtracted from the larger number, the result is equal to 1. By writing two equations to represent this information, or otherwise, find the values of both numbers. (10D) Two equations: Let a be the larger number and b be the smaller number a + b 46 a b 1 Values of both numbers: a + b 46 a b 1 a 58 a 9 a + b 46 9 + b 46 9 + b 46 9 17 or a + b 46 ( 1) a b 1 ( 1) a + b 46 a + b 1 b 34 b 17 a + b 46 a + 17 46 a 46 17 9 or a b 1 9 b 1 b 1 9 17 b 17 or a b 1 a 17 1 a 1 + 17 9 Scale 10D (0, 4, 6, 8, 10) Low partial credit: (4 marks) Some work of merit, e.g. writes down one equation in terms of a and b, i.e. a + b 46 or a b 1. Mid partial credit: (6 marks) Finds both equations in terms of a and b, and stops or continues incorrectly. Multiplies equation(s) by appropriate constant(s) to facilitate the cancellation of a or b term. Finds either variable (a or b) correctly by trial and error, but fails to verify in both equations or verifies incorrectly. High partial credit: (8 marks) Finds first variable (a or b), but fails to find second variable or finds incorrectly. Finds both variables (a and b) correctly with no work shown. Finds both variables (a and b) correctly by graphical means. Finds both variables (a and b) by trial and error, but does not verify in both equations or verifies incorrectly. 018.1 L.16/19_MS 15/76 Page 15 of 75 exams

018 LC Maths [OL] Paper 1 Question 6 (5) The first three patterns in a sequence of patterns of tiles are shown below. Pattern 1 Pattern Pattern 3 6(a) Draw the next two patterns in the sequence. (5B) Pattern 4 Pattern 5 Scale 5B (0,, 5) Partial credit: ( marks) Some work of merit, e.g. correct number of vertical tiles in both patterns or correct number of horizontal tiles in both patterns (i.e. 6 in Pattern 4 and 7 in Pattern 5). Draws one correct pattern. 6(b) (i) Based on the patterns shown, complete the table below. (5C) Pattern number (n) Number of White Tiles Number of Grey Tiles 1 7 6 10 3 1 13 4 0 16 5 30 19 Scale 5C (0,, 4, 5) Low partial credit: ( marks) One or two correct entries. High partial credit: (4 marks) Three, four or five correct entries. 018.1 L.16/19_MS 16/76 Page 16 of 75 exams

018 LC Maths [OL] Paper 1 Question 6 6(b) (ii) Show that the number of white tiles in each pattern forms a quadratic sequence. (5D) Pattern Number (n) 1 3 4 5 Number of White Tiles 6 1 0 30 Change (1st difference) 4 6 8 10 Change of change (nd difference) Conclusion: as the nd difference is constant, the number of white tiles in each pattern forms a quadratic sequence Scale 5D (0,, 3, 4, 5) Low partial credit: ( marks) Some work of merit, e.g. writes down one or two 1st differences and stops. Mid partial credit: (3 marks) Finds all 1st differences for the pattern sequence given, i.e. 4, 6, 8, 10, and stops or continues incorrectly. High partial credit: (4 marks) Finds nd differences for the pattern sequence given, but no conclusion given or incorrect conclusion. 018.1 L.16/19_MS 17/76 Page 17 of 75 exams

018 LC Maths [OL] Paper 1 Question 6 6(c) Assuming the pattern continues, the number of white tiles in the nth pattern of the sequence is given by the formula W n n + bn + c, where b, c Z. Find the value of b and the value of c. (10D) W n n + bn + c W 1 (1) + b(1) + c 1 + b + c b + c 1 b + c 1 W () + b() + c 6 4 + b + c b + c 6 4 b + c b + c 1 ( 1) b + c ( 1) b c 1 b + c b 1 b + c 1 or 1 + c 1 c 1 1 0 or b + c 1 ( ) b + c ( 1) b + c b c c 0 b + c 1 or b + 0 1 b 1 b + c (1) + c + c c 0 b + c b + 0 b 1 Scale 10D (0, 4, 6, 8, 10) Low partial credit: (4 marks) Some work of merit, e.g. finds one or both equations in terms of b and c for two patterns. [In the case of W 1 and W, b + c 1 and/or b + c.] Mid partial credit: (6 marks) Multiplies equation(s) by appropriate constant(s) to facilitate cancellation of b or c terms. Finds either variable (b or c) correctly by trial and error, but fails to verify in both equations or verifies incorrectly. High partial credit: (8 marks) Finds first variable (b or c), but fails to find second variable or finds incorrectly. Finds both variables (b and c) correctly with no work shown. Finds both variables (b and c) correctly by graphical means. Finds both variables (b and c) by trial and error, but does not verify in both equations or verifies incorrectly. 018.1 L.16/19_MS 18/76 Page 18 of 75 exams

018 LC Maths [OL] Paper 1 Section B Contexts and Applications 150 marks Answer all three questions from this section. (50 marks each) Question 7 (50) Fiona wishes to save up to buy a car. She joins a savings scheme in her local credit union. She plans to save 50 per month and to increase this amount by 5 every month. 7(a) (i) Complete the table below to show Fiona s monthly contributions for the period shown. (5C) Month (n) 1 3 4 1 4 Contribution ( ) 50 55 60 65 305 365 Monthly Contribution: Month 50 + 5 55 Month 3 55 + 5 60 Month 4 60 + 5 65 Month 1 50 + 5(11) 50 + 55 305 Month 4 50 + 5(3) 50 + 115 365 Scale 5C (0,, 4, 5) Low partial credit: ( marks) One or two correct answers. High partial credit: (4 marks) Three or four correct answers. * No deduction applied for the omission of or incorrect use of units in questions involving currency. (ii) Show that Fiona s monthly contributions form an arithmetic sequence. (5C) Month (n) 1 3 4 Contribution ( ) 50 55 60 65 Change (1st difference) 5 5 5 Conclusion: Any 1: as the 1st difference is constant, the monthly contributions form an arithmetic sequence // an arithmetic sequence is one in which the difference between any two successive terms is a constant // etc. Scale 5C (0,, 4, 5) Low partial credit: ( marks) Some work of merit, e.g. writes down one or two 1st differences and stops. High partial credit: (4 marks) Finds all 1st differences for the sequence of monthly contributions given, but no conclusion given or incorrect conclusion. 018.1 L.16/19_MS 19/76 Page 19 of 75 exams

018 LC Maths [OL] Paper 1 Question 7 7(a) (iii) Find, in terms of n, a formula that gives Fiona s monthly contribution in the nth month of the scheme. T n a + (n 1)d a T 1 50 d T T 1 55 50 5 T n 50 + (n 1)5 50 + 5n 5 5n + 45 (5C) Scale 5C (0,, 4, 5) Low partial credit: ( marks) Some work of merit, e.g. writes down correct relevant formula for T n. Identifies a or d correctly. Some correct substitution into T n. High partial credit: (4 marks) Identifies both a and d correctly and substitutes correctly into relevant formula, i.e. T n 50 + (n 1)5, but fails to finish or finishes incorrectly. (iv) Using your formula, or otherwise, find in which month Fiona contributes 400. (5C) T n 5n + 45 400 5n + 45 400 5n 400 45 155 n 155 5 31 // 31st month ** Accept students answers from part (iii) if not oversimplified. Scale 5C (0,, 4, 5) Low partial credit: ( marks) Some work of merit, e.g. writes down T n 400 and stops. Attempts to find T n 400 using list method (terms to at least T 10 ). High partial credit: (4 marks) Finds T 40 correctly using list method. Equates 5n + 45 400 correctly, but fails to finish or finishes incorrectly. 018.1 L.16/19_MS 0/76 Page 0 of 75 exams

018 LC Maths [OL] Paper 1 Question 7 7(b) (i) Find, in terms of n, a formula that gives the sum of Fiona s total contributions from the first to the nth month of the scheme. (10C) S n n {a + (n 1)d} a 50 d 5... part (a)(iii) S n n {(50) + (n 1)(5)} n {500 + 5n 5} n 5n 495n (5n + 495) // + ** Accept students answers from part (a)(iii) if not oversimplified. Scale 10C (0, 4, 7, 10) Low partial credit: (4 marks) Some work of merit, e.g. writes down correct relevant formula for S n. Some correct substitution into S n formula. High partial credit: (7 marks) Substitutes correctly into relevant formula, i.e. S n n {(50) + (n 1)(5)}, but fails to finish or finishes incorrectly. (ii) Fiona is guaranteed to receive a fixed return of 14,000 at the end of three years if she does not take any money out of the savings scheme until that time. Find the percentage return that Fiona will receive on her savings. Give your answer correct to two decimal places. (10D*) 3 years 3 1 36 months S n n 5n 495n (5n + 495) // +... part (b)(i) Sum of contributions over 3 years, S 36 S 36 36 (5(36) + 495) 18(180 + 495) 18(675) 1,150 Guaranteed return 14,000 1,150 1,850 % Return 1,850 1,150 100 1 15 6337... 15 3% 018.1 L.16/19_MS 1/76 Page 1 of 75 exams

018 LC Maths [OL] Paper 1 Question 7 7(b) (ii) ** Accept students answers from part (b)(i) if not oversimplified. Scale 10D* (0, 4, 6, 8, 10) Low partial credit: (4 marks) Some work of merit, e.g. writes down 3 years 36 contributions or similar. Some correct substitution into S n formula. Substitutes correctly into T 36 [ans. 45]. Mid partial credit: (6 marks) Substitutes correctly into relevant formula, 36 i.e. S 36 (5(36) + 495), but fails to finish or finishes incorrectly. High partial credit: (8 marks) Finds correct value for Guaranteed return [ans. 1,850], but fails to finish or finishes incorrectly. * Deduct 1 mark off correct answer only if final answer is incorrectly rounded or not rounded - apply only once to each section (a), (b), (c), etc. of question. * No deduction applied for the omission of or incorrect use of symbol ( % ). 7(c) Another option is for Fiona to borrow 14,000 from a finance company to buy a car. The loan is to be repaid in equal monthly repayments over the term of the loan. Interest is charged monthly at an annual percentage rate (APR) of 13 5% on the amount borrowed for the entire term of the loan. (i) Using the formula (1 + r) 1 1 + i, where r is the monthly rate and i is the annual rate of interest, find the rate of interest charged monthly which corresponds to an APR of 13 5%, correct to two decimal places. (1 + r) 1 1 + i (1 + r) 1 1 + 0 135 1 135 1 + r 1 1 135 1 1 010608... r 1 010608... 1 0 010608... r 1 060859...% 1 06% (5C*) Scale 5C* (0,, 4, 5) Low partial credit: ( marks) Some work of merit, e.g. identifies i 0 135 or 1 + i 1 + 0 135 1 135. Divides or multiplies 1 135 by 1. Some correct substitution into formula and stops or continues. High partial credit: (4 marks) 1 Finds 1 + r 1 135 1 or 1 010608..., but fails to finish or finishes incorrectly. * Deduct 1 mark off correct answer only if final answer is incorrectly rounded or not rounded - apply only once to each section (a), (b), (c), etc. of question. * No deduction applied for the omission of or incorrect use of symbol ( % ). 018.1 L.16/19_MS /76 Page of 75 exams

018 LC Maths [OL] Paper 1 Question 7 7(c) (ii) Fiona decides to borrow the money and repay the loan over 3½ years. Calculate, correct to the nearest euro, the amount of her monthly repayments. 3½ years 3½ 1 4 months Total amount repayable over the term of loan F P(1 + i) t 14,000(1 + 0 0106) 4 14,000(1 0106) 4 14,000(1 557149...) 1,800 094668... 1,800 Amount of monthly repayments 1,800 4 519 047619... 519 (5D*) Scale 5D* (0,, 3, 4, 5) Low partial credit: ( marks) Some work of merit, e.g. writes down 3½ years 4 repayments or similar. Writes down correct relevant formula and stops. Some correct substitution into relevant formula and stops or continues. Mid partial credit: (3 marks) Substitutes correctly into formula, i.e. 14,000(1 0106) 4 or 14,000(1 557149...), and stops or continues incorrectly. High partial credit: (4 marks) Finds correct value of F [ans. 1,800], but fails to finish or finishes incorrectly. * Deduct 1 mark off correct answer only if final answer is incorrectly rounded or not rounded - apply only once to each section (a), (b), (c), etc. of question. * No deduction applied for the omission of or incorrect use of units in questions involving currency. 018.1 L.16/19_MS 3/76 Page 3 of 75 exams

018 LC Maths [OL] Paper 1 Question 8 (50) A company seeking new investors predicts that the projected value of their investment will follow the function: v(t) 5t 180t + 10,000, where v is the value of the investment, in euro, and t is the time, in months, after the investment is made. 8(a) (i) How much are potential investors asked to invest initially? (5C) v(t) 5t 180t + 10,000 @ t 0 v(0) 5(0) 180(0) + 10,000 0 + 0 + 10,000 10,000 Scale 5C (0,, 4, 5) Low partial credit: ( marks) Some work of merit, e.g. writes down value of investment @ t 0 or v(0). Some correct substitution into v(t) equation and stops or continues. Substitutes a value for t ( 0) into v(t) equation and evaluates correctly. High partial credit: (4 marks) Substitutes correctly into v(t) equation, i.e. v(0) 5(0) + 180(0) + 10,000, but fails to finish or finishes incorrectly. * No deduction applied for the omission of or incorrect use of units in questions involving currency. (ii) How long will it take before their investment is profitable? (5C*) v(t) 5t 180t + 10,000 investment is profitable when v(t) > 10,000 5t 180t + 10,000 > 10,000 5t 180t > 10,000 10,000 > 0 t(5t 180) > 0 t > 0... not applicable 5t 180 > 0 5t > 180 t > 180 5 > 36 months Scale 5C* (0,, 4, 5) Low partial credit: ( marks) Some work of merit, e.g. writes down profitable when v(t) > 10,000. Substantially correct approach to solving quadratic equation. High partial credit: (4 marks) Solves quadratic equation correctly, i.e. finds t 0 and 36, but fails to identify correct answer. Correct answer, but no work shown. * Deduct 1 mark off correct answer only for the omission of or incorrect use of units ( months ) - apply only once to each section (a), (b), (c), etc. of question. 018.1 L.16/19_MS 4/76 Page 4 of 75 exams

018 LC Maths [OL] Paper 1 Question 8 8(b) Use calculus to find how much the investment is worth when it reaches its minimum value. (10D) dv Minimum value when 0 dt v(t) 5t 180t + 10,000 dv dt 10t 180 0 10t 180 0 10t 180 t 18 months Value of investment @ t 18 v(18) 5(18) 180(18) + 10,000 1,60 3,40 + 10,000 8,380 Scale 10D (0, 4, 6, 8, 10) Low partial credit: (4 marks) Some work of merit, e.g. writes down dv Minimum value when 0 and stops. dt One term correctly differentiated. Mid partial credit: (6 marks) Both terms correctly differentiated and stops or continues incorrectly. High partial credit: (8 marks) Equates linear equation correctly, i.e. v (t) 10t 180 0, but fails to find correct value of t. Finds correct value of t [18 months], but fails to finish or finishes incorrectly. Correct answer, but no work shown. * No deduction applied for the omission of or incorrect use of units in questions involving currency. 8(c) (i) Use the value function, v(t), to complete the table, showing the projected value of the investment over time. (10C) Time (t) 6 1 18 4 30 36 4 Value ( ) 9,100 8,560 8,380 8,560 9,100 10,000 11,60 v(t) 5t 180t + 10,000 @ t 18 v(18) 5(18) 180(18) + 10,000 1,60 3,40 + 10,000 8,380 @ t 4 v(4) 5(4) 180(4) + 10,000,880 4,30 + 10,000 8,560 @ t 30 v(30) 5(30) 180(30) + 10,000 4,500 5,400 + 10,000 9,100 018.1 L.16/19_MS 5/76 Page 5 of 75 exams

018 LC Maths [OL] Paper 1 Question 8 8(c) (i) @ t 36 v(36) 5(36) 180(36) + 10,000 6,480 6,480 + 10,000 10,000 @ t 4 v(4) 5(4) 180(4) + 10,000 8,80 7,560 + 10,000 11,60 Scale 10C (0, 4, 7, 10) Low partial credit: (4 marks) One or two correct entries. High partial credit: (7 marks) Three or four correct entries. * No deduction applied for the omission of or incorrect use of units in questions involving currency. (ii) Use the data in the table to draw a graph to represent the projected value of the investment over the period of time indicated. Label your graph clearly. 10001 (5D) Value of investment () 110001 100001 9000 wx () vx () 80001 x 6 10 1 18 4 30 36 40 4 Time (months) ** Accept students answers from part (c)(i) if not oversimplified. Scale 5D (0,, 3, 4, 5) Low partial credit: ( marks) Some work of merit, e.g. plots correctly one or two points. Draws correct general shape of graph. Mid partial credit: (3 marks) Three or four points plotted correctly with correct general shape of graph. High partial credit: (4 marks) Five or six points plotted correctly with correct general shape of graph. All points plotted correctly, but not joined or joined by straight lines. 018.1 L.16/19_MS 6/76 Page 6 of 75 exams

018 LC Maths [OL] Paper 1 Question 8 8(d) The projected value of another investment opportunity is predicted to follow the function: w(t) 70t + 8,000, where w is the value, in euro, and t is the time, in months, after the investment is made. (i) On the same axes above, draw a graph to show the projected value of this investment for 0 t 4, t R. Label your graph clearly. w(t) 70t + 8,000 @ t 0 w(0) 70(0) + 8,000 0 + 8,000 8,000 (0, 8,000) w @ t 6 w(6) 70(6) + 8,000 40 + 8,000 8,40 (6, 8,40) w @ t 1 w(1) 70(1) + 8,000 840 + 8,000 8,840 (1, 8,840) w @ t 18 w(18) 70(18) + 8,000 1,60 + 8,000 9,60 (18, 9,60) w @ t 4 w(4) 70(4) + 8,000 1,680 + 8,000 9,680 (4, 9,680) w @ t 30 w(30) 70(30) + 8,000,100 + 8,000 10,100 (30, 10,100) w @ t 36 w(36) 70(36) + 8,000,50 + 8,000 10,50 (36, 10,50) w @ t 4 w(4) 70(4) + 8,000,940 + 8,000 10,940 (4, 10,940) w (5D) Scale 5D (0,, 3, 4, 5) Low partial credit: ( marks) Some work of merit, e.g. finds correctly co-ordinates of one point on graph of w(t). Plots correctly one point on graph of w(t) (no work shown). Mid partial credit: (3 marks) Finds correctly one point on graph of w(t) and plots point correctly. Finds correctly two points on graph of w(t), but fails to plot either point. Plots correctly two points on graph of w(t) (no work shown). High partial credit: (4 marks) Finds and plots two points correctly, but points not joined or not joined by straight line. 018.1 L.16/19_MS 7/76 Page 7 of 75 exams

018 LC Maths [OL] Paper 1 Question 8 8(d) (ii) Use your graphs to estimate the time interval for which the projected value of this investment is higher than that of the other investment. Using graph: Time interval 40 (±1) 10 (±1) 30 (±) months (5C*) ** Accept answer based on students graph in part (c)(ii) if not oversimplified. Scale 5C* (0,, 4, 5) Low partial credit: ( marks) Some work of merit, e.g. marks clearly one or both points of intersection on graph (with or without labels), but no figures. Finds correctly x co-ordinate of one point of intersection, i.e. 10 or 40. Finds correctly y co-ordinate of one point of intersection, i.e. 8,700 or 10,800. High partial credit: (4 marks) Both points of intersection on graph with figures indicated, but fails to find or finds incorrect time interval. Finds correctly y co-ordinate of both points of intersection, i.e. 8,700 and 10,800. Finds correct answer for time interval, but no work shown on graph. Final answer outside of tolerance, but work shown on student s graph. * Deduct 1 mark off correct answer only for the omission of or incorrect use of units ( months ) - apply only once to each section (a), (b), (c), etc. of question. (iii) Which of the above investment opportunities would you recommend? Give a reason for your answer. (5B) First investment / v(t) better return from this investment in the long term / after first 3 years // etc. ** Accept any other appropriate reason. Second investment / w(t) Any 1: return on this investment is always positive // if you have to withdraw from the investment at any time, you will not have lost any money // smaller initial investment required // etc. ** Accept any other appropriate reason. Scale 5B (0,, 5) Partial credit: ( marks) Answer incomplete, but with some merit. * No marks for recommendation if reason omitted. 018.1 L.16/19_MS 8/76 Page 8 of 75 exams

018 LC Maths [OL] Paper 1 Question 9 (50) A club proposes to erect two floodlighting towers at opposite corners of its rectangular playing field of length 80 m and width 60 m. A power cable is required between the diagonal corners, A and B. The cost of digging a trench to lay the power cable is 0 per m, while the cost of boring underground to install the cable is 40 per m. The club does not wish to dig a trench across the playing field. 9(a) One option is to dig a trench for 0 m from A along the edge of the playing field and then to bore underground from this point diagonally across the field to B, as shown. 60 m A 0 m (i) Use the Theorem of Pythagoras to find y, the distance required to be bored underground. Give your answer correct to one decimal place. (5C*) 80 m y 80 0 60 m Theorem of Pythagoras: Hyp Opp + Adj y (80 0) + (60) (60) + (60) 3,600 + 3,600 7,00 y 7, 00 84 85813... 84 9 m B 60 m Scale 5C* (0,, 4, 5) Low partial credit: ( marks) Some work of merit, e.g. writes down correct formula for Pythagoras theorem. Some correct substitution into formula for Pythagoras theorem. High partial credit: (4 marks) Finds y 7,00 or y 7, 00, but fails to finish or finishes incorrectly. * Deduct 1 mark off correct answer only if final answer is incorrectly rounded or not rounded or for the omission of or incorrect use of units ( m ) - apply only once to each section (a), (b), (c), etc. of question. (ii) Hence, find the total cost of this option. (5C) Total cost of option (0 m 0) + (84 9 m 40) 400 + 3,396 3,796 Scale 5C (0,, 4, 5) Low partial credit: ( marks) Some work of merit, e.g. writes down correct formula for Pythagoras theorem. Finds cost of one part correctly, i.e. 0 m 0 or 84 9 m 40. High partial credit: (4 marks) Finds (0 m 0) + (84 9 m 40), but fails to finish or finishes incorrectly. * No deduction applied for the omission of or incorrect use of units in questions involving currency. 018.1 L.16/19_MS 9/76 Page 9 of 75 exams

018 LC Maths [OL] Paper 1 Question 9 9(b) To investigate other options, a general formula for y (distance to be bored underground) is derived for different values of x (length of trench to be dug). 60 m A x (i) Show that the general formula for y is ( 80 x ) + 60. (10C) Hyp Opp + Adj y (80 x) + (60) 80 m y 80 x y ( 80 x ) + 60 B 60 m Scale 10C (0, 4, 7, 10) Low partial credit: (4 marks) Some work of merit, e.g. writes down correct formula for Pythagoras theorem. Finds 80 x and stops or continues incorrectly. Some correct substitution into formula for Pythagoras theorem. High partial credit: (7 marks) Finds y (80 x) + (60), but fails to finish or finishes incorrectly. (ii) Find the value of y for each value of x given in the table below. Give your answers correct to one decimal place. (10D*) x (metres) y (metres) Total Cost ( ) 10 9 3,888 0 84 9 30 78 1 40 7 1 50 67 1 60 63 70 60 8 80 60 0 y ( 80 x ) + 60 @ x 0 y ( 80 0) + 60 3,600 + 3, 600 7, 00 84 85813... 84 9 m @ x 30 y ( 80 30) + 60,500 + 3, 600 6, 100 78 10496... 78 1 m 018.1 L.16/19_MS 30/76 Page 30 of 75 exams

018 LC Maths [OL] Paper 1 Question 9 9(b) (ii) @ x 40 y ( 80 40) + 60 1,600 + 3, 600 5, 00 7 11105... 7 1 m @ x 50 y ( 80 50) + 60 900 + 3, 600 4, 500 67 08039... 67 1 m @ x 60 y ( 80 60) + 60 400 + 3, 600 4, 000 63 45553... 63 m @ x 70 y ( 80 70) + 60 100 + 3, 600 3, 700 60 8765... 60 8 m @ x 80 y ( 80 80) + 60 0 + 3, 600 3, 600 60 m Scale 10D* (0, 4, 6, 8, 10) Low partial credit: (4 marks) One or two correct entries. Mid partial credit: (6 marks) Three or four correct entries. High partial credit: (8 marks) Five or six correct entries. * Deduct 1 mark off correct answers only if final answers are incorrectly rounded or not rounded - apply only once to each section (a), (b), (c), etc. of question. 018.1 L.16/19_MS 31/76 Page 31 of 75 exams

018 LC Maths [OL] Paper 1 Question 9 9(b) (iii) Write a formula in x and y for the total cost of installing the power cable. (10D) Cost 0x + 40y Hence, complete the table above to show the total cost for each option given. x (metres) y (metres) Total Cost ( ) 10 9 3,888 0 84 9 3,796 30 78 1 3,74 40 7 1 3,484 50 67 1 3,684 60 63 3,78 70 60 8 3,83 80 60 0 4,000 Cost 0x + 40y @ x 0 Cost 0(0) + 40(84 9) 400 + 3,396 3,796 @ x 30 Cost 0(30) + 40(78 1) 600 + 3,14 3,74 @ x 40 Cost 0(40) + 40(7 1) 600 +,884 3,484 @ x 50 Cost 0(50) + 40(67 1) 1,000 +,684 3,684 @ x 60 Cost 0(60) + 40(63 ) 1,00 +,58 3,78 @ x 70 Cost 0(70) + 40(60 8) 1,400 +,43 3,83 @ x 80 Cost 0(80) + 40(60 0) 1,600 +,400 4,000 018.1 L.16/19_MS 3/76 Page 3 of 75 exams

018 LC Maths [OL] Paper 1 Question 9 9(b) (iii) Scale 10D (0, 4, 6, 8, 10) Low partial credit: (4 marks) Some work of merit, e.g. writes down cost of one part correctly, i.e. 0x or 40y. One or two correct entries, but no evidence of formula. Mid partial credit: (6 marks) Correct formula for cost, i.e. 0x + 40y, but fails to evaluate any for value. Three or four correct entries, but no evidence of formula. High partial credit: (8 marks) Correct formula for cost, i.e. 0x + 40y, but not all values evaluated (at least two). Five or six correct entries, but no evidence of formula. * No deduction applied for the omission of or incorrect use of units in questions involving currency. (iv) Use the formula from part (i) to find the shortest route between A and B. Hence, state whether this is the cheapest option. Justify your answer by calculation. (10D) y ( 80 x ) + 60 Shortest route when x 0 y ( 80 0) + 60 80 + 60 6,400 + 3, 600 10, 000 100 m Cost 0x + 40y Cost 0(0) + 40(100 0) 0 + 4,000 4,000 Conclusion: the shortest route is not the cheapest option Scale 10D (0, 4, 6, 8, 10) Low partial credit: (4 marks) Some work of merit, e.g. writes down formula for y and states Shortest route when x 0 or similar. Some correct substitution into formula for y and stops or continues incorrectly. Mid partial credit: (6 marks) Finds y 10, 000 or 100 and stops or continues incorrectly. High partial credit: (8 marks) Finds cost of this option [ans. 4,000], but no conclusion given or incorrect conclusion. 018.1 L.16/19_MS 33/76 Page 33 of 75 exams

018 LC Maths [OL] Paper 1 Notes: 018.1 L.16/19_MS 34/76 Page 34 of 75 exams

018 LC Maths [OL] Paper 1 Notes: 018.1 L.16/19_MS 35/76 Page 35 of 75 exams

. exams Pre-Leaving Certificate Examination, 018 Mathematics Ordinary Level Paper 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 B C D No. of categories 3 4 5 5 mark scale 0,, 5 0,, 4, 5 0,, 3, 4, 5 10 mark scale 0, 4, 7, 10 0, 4, 6, 8, 10 15 mark scale 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) 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. 014 (LC-O1 & O) Accept students work in one part of a question for use in subsequent parts of the question, unless this oversimplifies the work involved. Scale label A B C D No of categories 3 4 5 5 mark scale 0, 5 0,, 5 0,, 4, 5 10 mark scale 0, 5 10 0, 3, 7, 10 0,, 5 15 mark scale 0, 5, 10, 15 0, 4, 7, 0 mark scale 5 mark scale 0, 6, 1 018.1 L.16/19_MS 36/76 Page 36 of 75 exams

Summary of Marks 018 LC Maths (Ordinary Level, Paper ) Section A Section B Q.1 (a) (i) 5C (0,, 4, 5) Q.7 (a) (i) 5C (0,, 4, 5) (ii) 5C (0,, 4, 5) (ii) 10D (0, 4, 6, 8, 10) (iii) 5C (0,, 4, 5) (b) 10D* (0, 4, 6, 8, 10) (b) (i) 5C (0,, 4, 5) (c) (i) 5C (0,, 4, 5) (ii) 5C (0,, 4, 5) (ii) 5D (0,, 3, 4, 5) 5 (iii) (iv) 5B (0,, 5) (d) (i) 5D (0,, 3, 4, 5) Q. (a) (i) 5C (0,, 4, 5) (ii) 5C* (0,, 4, 5) (ii) 5C (0,, 4, 5) 50 (b) (i) 10D (0, 4, 6, 8, 10) (ii) 5C (0,, 4, 5) 5 Q.8 (a) (i) 5B (0,, 5) (ii) 5C (0,, 4, 5) Q.3 (a) (i) 5B* (0,, 5) (iii) 10D* (0, 4, 6, 8, 10) (ii) 5C* (0,, 4, 5) (b) (i) 5C (0,, 4, 5) (b) (i) 10D* (0, 4, 6, 8, 10) (ii) 10D (0, 4, 6, 8, 10) (ii) 5D* (0,, 3, 4, 5) (iii) 5C (0,, 4, 5) 5 (c) 10D* (0, 4, 6, 8, 10) 50 Q.4 (a) (i) 10C (0, 4, 7, 10) (ii) 5B (0,, 5) (b) (i) Q.9 (a) (i) 10C* (0, 4, 7, 10) (ii) 10D* (0, 4, 6, 8, 10) (ii) 10C* (0, 4, 7, 10) 5 (iii) 5C* (0,, 4, 5) (iv) 5C* (0,, 4, 5) (b) (i) 10C* (0, 4, 7, 10) Q.5 (a) 10D* (0, 4, 6, 8, 10) (ii) 5C* (0,, 4, 5) (b) (i) 10C* (0, 4, 7, 10) (iii) 5C (0,, 4, 5) (ii) 5C* (0,, 4, 5) 50 5 Q.6 (a) 5C* (0,, 4, 5) (b) 5C (0,, 4, 5) (c) (i) 5C* (0,, 4, 5) (ii) 5C (0,, 4, 5) 5 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 papers(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. 018.1 L.16/19_MS 37/76 Page 37 of 75 exams

exams Pre-Leaving Certificate Examination, 018 Mathematics Ordinary Level Paper Marking Scheme (300 marks) General 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. Marks will be lost if all necessary work is not clearly shown. Answers should include the appropriate units of measurement, where relevant. Answers should be given in simplest form, where relevant. Section A Concepts and Skills 150 marks Answer all six questions from this section. (5 marks each) Question 1 (5) A survey of 136 students was carried out. They were asked whether they were studying French (F) or Spanish (S). Of those surveyed, 87 study French, 85 study Spanish and 54 study both languages. 1(a) (i) Represent this information on the Venn Diagram. (5C) F [ 87 54] [ 33] [ 54] S [ 85 54] [ 31] [ 136] [ 18] Scale 5C (0,, 4, 5) Low partial credit: ( marks) Some work of merit, e.g. writes down 87 54 or 85 54 in the correct position on the Venn diagram. One correct/consistent element calculated or inserted on the Venn diagram. High partial credit: (4 marks) Two correct/consistent elements calculated or inserted on the Venn diagram. 018.1 L.16/19_MS 38/76 Page 38 of 75 exams

018 LC Maths [OL] Paper Question 1 1(a) (ii) A student is chosen at random from those surveyed. Find the probability that the student studies neither language. (5C) P(student studies neither language) 18 136 9 68 or 0 1335... ** Accept students answers from part (a)(i) if not oversimplified. Scale 5C (0,, 4, 5) Low partial credit: ( marks) Either #(E) or #(S) correct (answer must be shown in fraction format). High partial credit: (4 marks) 18 Finds, but fails to evaluate or 136 evaluates incorrectly. Correct answer, but no work shown. (iii) Two students are chosen at random from those surveyed who study at least one of these languages. Find the probability that both students study one language only. Give your answer correct to two decimal places. (5C) # students who study at least one language 33 + 54 + 31 118 P(1st student selected studies 1 language only) 33 + 31 118 64 3 // // 0 5437... 118 59 P(nd student selected studies 1 language only) 33 + 31 1 118 1 63 7 // // 0 538461... 117 13 P(both students selected study 1 language only) 64 63 118 117 4,03,016 // 13,806 6,903 // 0 9046... ** Accept students answers from part (a)(i) if not oversimplified. Scale 5C (0,, 4, 5) Low partial credit: ( marks) Some work of merit, e.g. any #(E) or #(S) correct (answer must be shown in fraction format). Finds one correct probability. Finds two correct probabilities, but adds values together instead of multiplying. High partial credit: (4 marks) 64 63 Finds (or consistent), but fails 118 117 to evaluate or evaluates incorrectly. Correct answer, but no work shown. 018.1 L.16/19_MS 39/76 Page 39 of 75 exams